Abstract

An overview of the recent developments in the field of cylindrical vector beams is provided. As one class of spatially variant polarization, cylindrical vector beams are the axially symmetric beam solution to the full vector electromagnetic wave equation. These beams can be generated via different active and passive methods. Techniques for manipulating these beams while maintaining the polarization symmetry have also been developed. Their special polarization symmetry gives rise to unique high-numerical-aperture focusing properties that find important applications in nanoscale optical imaging and manipulation. The prospects for cylindrical vector beams and their applications in other fields are also briefly discussed.

© 2009 Optical Society of America

1. Introduction

Polarization is one important property of light. This vector nature of light and its interactions with matter make many optical devices and optical system designs possible. Polarization propagation and interaction with materials have been extensively explored in optical inspection and metrology, display technologies, data storage, optical communications, materials sciences, and astronomy, as well as in biological studies. Most past research dealt with spatially homogeneous states of polarization (SOPs), such as linear, elliptical, and circular polarizations. For these cases, the SOP does not depend on the spatial location in the beam cross section. Recently there has been an increasing interest in light beams with spatially variant SOPs. Spatially arranging the SOP of a light beam, purposefully and carefully, is expected to lead to new effects and phenomena that can expand the functionality and enhance the capability of optical systems. One particular example is laser beams with cylindrical symmetry in polarization, the so-called cylindrical vector (CV) beams.

Because of their interesting properties and potential applications, there has been a rapid increase of the number of publications on CV beams. The purpose in the present paper is to provide a review of the recent developments. In Section 2, I introduce commonly used mathematical descriptions of these CV beams. Since 1972 [1, 2], various active and passive methods have been developed to generate CV beams. In Section 3, an overview of these methods is provided. Manipulation methods that can redirect, rotate, and modulate CV beams are summarized in Section 4. CV beams have attracted significant recent attention largely because of their unique properties under high-numerical-aperture (NA) focusing. In Section 5, an intuitive explanation of the focusing properties using electric dipole emission pattern is given, and the numerical computation formula is reviewed. Numerical calculations have shown that tighter focal spots can be obtained by using radial polarization, a subset of CV beams, owing to the existence of a strong and localized longitudinal field component [3]. Such an effect has been experimentally confirmed by several groups [4, 5, 6] and has found applications in high-resolution imaging, plasmonic focusing, and nanoparticle manipulation. The perspectives of CV beam applications in other fields such as laser machining, remote sensing, terahertz technology, and singular optics will be briefly mentioned toward the end of this paper.

2. Mathematical Description of Cylindrical Vector Beams

The CV beams are vector-beam solutions of Maxwell’s equations that obey axial symmetry in both amplitude and phase [7]. Modes with radial and azimuthal polarization are well known in waveguide theory [8]. However, their counterparts in free space are less familiar. For free space, the typical paraxial beamlike solutions with harmonic temporal dependence are obtained by solving the scalar Helmholtz equation:

(2+k2)E=0,
where k=2πλ is the wavenumber. For a beamlike paraxial solution in Cartesian coordinates, the general solution for the electric field takes the form
E(x,y,z,t)=u(x,y,z)exp[i(kzωt)].
By applying the slowly varying envelop approximation
2uz2k2u,2uz2kuz,
the Hermite–Gauss solution HGmn modes can be obtained by separation of variables in x and y. Mathematically, these solutions have the following form:
u(x,y,z)=E0Hm(2xw(z))Hn(2yw(z))w0w(z)exp[iφmn(z)]exp[ik2q(z)r2],
where Hm(x) denotes the Hermite polynomials that satisfy the differential equation
d2Hmdx22xdHmdx+2mHm=0
and E0 is a constant electric field amplitude, w(z) is the beam size, w0 is the beam size at the beam waist, z0=πw02λ is the Rayleigh range, q(z)=ziz0 is the complex beam parameter, and φmn(z)=(m+n+1)tan1(zz0) is the Gouy phase shift. For m=n=0, this solution reduces to the well-known fundamental Gaussian beam solution:
u(r,z)=E0w0w(z)exp[iφ(z)]exp[ik2q(z)r2],
where φ(z)=tan1(zz0) is the Gouy phase shift for the fundamental Gaussian beam. For beamlike paraxial solution in cylindrical coordinates, the solution takes the following general formula:
E(r,ϕ,z,t)=u(r,ϕ,z)exp[i(kzωt)].
Substituting Eq. (2.5) into the scalar Helmholtz equation (2.1) and applying the slowly varying envelop approximation leads to
1rr(rur)+1r22uϕ2+2ikuz=0.
From this equation, the Laguerre–Gauss solution LGpl modes can be obtained by using the separation of variables in r and ϕ:
u(r,ϕ,z)=E0(2rω)lLpl(2r2ω2)w0w(z)exp[iφpl(z)]exp[ik2q(z)r2]exp(ilϕ),
where Lpl(x) is the associated Laguerre polynomials that satisfy the differential equation
xd2Lpldx2(l+1x)dLpldx+pLpl=0
and φpl(z)=(2p+l+1)tan1(zz0) is the Gouy phase shift. For l=p=0, the solution also reduces to the fundamental Gaussian beam solution. For l0, the LG mode has a vortex phase term eilϕ. Another type of solution of Eq. (2.6) that obeys rotational symmetry (independent of the azimuthal angle ϕ) has also been found [9]. These solutions take the general form
u(r,z)=E0w0w(z)exp[iφ(z)]exp[ik2q(z)r2]J0(βr1+izz0)exp[β2z(2k)1+izz0],
where β is a constant scale parameter, φ(z) is the Gouy phase shift, z0 is the Rayleigh range, and J0(x) is the zeroth-order Bessel function of the first kind. This group of beamlike solution is the so-called scalar Bessel–Gauss beam solution. When β=0, the solution reduces to the fundamental Gaussian beam solution given in Eq. (2.4) as well.

The solutions derived above (Hermite–Gauss, Laguerre–Gauss, and Bessel–Gauss) are the paraxial beamlike solutions to the scalar Helmholtz equation (2.1) that correspond to spatially homogeneous polarization or scalar beams. For these beams, the electric field oscillation trajectory (i.e., SOP) does not depend on the location of observation points within the beam cross section. However, if we consider the full vector wave equation for the electric field [7],

××Ek2E=0,
then an axially symmetric beamlike vector solution with the electric field aligned in the azimuthal direction should have the form
E(r,z)=U(r,z)exp[i(kzωt)]eϕ,
where U(r,z) satisfies the following equation under the paraxial and slowly varying envelope approximation:
1rr(rUr)Ur2+2ikUz=0.
There is a clear difference in the second terms of Eqs. (2.6, 2.11). The solution that obeys azimuthal polarization symmetry has the trial solution
U(r,z)=E0J1(βr1+izz0)exp[iβ2z(2k)1+izz0]u(r,z),
where u(r,z) is the fundamental Gaussian solution given in Eq. (2.4) and J1(x) is the first-order Bessel function of the first kind. This solution corresponds to an azimuthally polarized vector Bessel–Gauss beam solution. Similarly, there should exist a transverse magnetic field solution
H(r,z)=H0J1(βr1+izz0)exp[iβ2z(2k)1+izz0]u(r,z)exp[i(kzωt)]hϕ,
where H0 is a constant magnetic field amplitude and hϕ is the unit vector in the azimuthal direction. For this azimuthal magnetic field solution, the corresponding electric field in the transverse plane is aligned in the radial direction. Hence Eq. (2.13) represents the radial polarization for the electric field. Clearly there should be a z component of the electric field as well. However, this component is weak and can be ignored under paraxial conditions.

For comparison, examples of the spatial distributions of the instantaneous electric field vector for several linearly polarized Hermite–Gauss and Laguerre–Gauss modes and the CV modes are illustrated together in Fig. 1. The SOPs of the modes shown in Figs. 1(a), 1(b), 1(c), 1(d), 1(e), 1(f) are considered spatially homogeneous, although the electric field may have an opposite instantaneous direction caused by the inhomogeneous phase distribution across the beam. The field illustrated in Fig. 1(g) has polarization aligned in the radial direction. This is called the radial polarization. Similarly, the polarization pattern shown in Fig. 1(h) is called the azimuthal polarization. The generalized CV beams shown in Fig. 1(i) are a linear superposition of these two. Due to the transverse field continuity, one of the features of these CV modes is the existence of a null of the transverse field.

In many applications, instead of the vector Bessel–Gauss solutions derived above, other simplified distributions have been used, especially for CV beams with large cross sections. For very small β, the vector Bessel–Gauss beam at the beam waist can be approximated as

E(r,z)=Arexp(r2w2)ei,i=r,ϕ.
The amplitude profile is exactly the LG01 mode [10] without the vortex phase term exp(iϕ). Using Eqs. (2.3, 2.7), it is easy to show that CV beams can also be expressed as the superposition of orthogonally polarized Hermite–Gauss HG01 and HG10 modes [11]:
Er=HG10ex+HG01ey,
Eϕ=HG01ex+HG10ey,
where Er and Eϕ denote radial and azimuthal polarization, respectively. This is illustrated schematically in Fig. 2. For some applications, particularly for those involving the generation of collimated CV beams through passive devices in applications requiring high-NA focusing, annular distributions with the center blocked by an opaque stop have also been frequently used. The field after these devices or in the pupil plane is [12]
E(r)=P(r)ei,i=r,ϕ,
where P(r) is the beam cross section or pupil function. For uniform annular illumination,
P(r)={1,r1<r<r20,0<r<r1}.

3. Generation of Cylindrical Vector Beams

Since 1972, many methods of generating CV beams have been reported, especially in the past decade or so. Depending on whether the generation methods involve amplifying media, these methods can be categorized as active or passive.

3.1. Active Generation Methods

Typically, active methods involve the use of laser intracavity devices that force the laser to oscillate in CV modes. Intracavity devices can be axial birefringent (intrinsic birefringent, form birefringent, or induced birefringent) component or axial dichroic component to provide mode discrimination against the fundamental mode. One of the earliest experiments utilized an intracavity axial birefringent component [1]. In this setup (Fig. 3), a calcite crystal is placed in a telescope setup with its crystal axis parallel to the optical axis of the cavity. Because of double refraction, the e polarization and o polarization experience slightly different magnifications. With a central stop aperture, one polarization is discriminated more because of higher loss. The cylindrical symmetry of the entire system ensures that the oscillation mode has cylindrical polarization symmetry. Since calcite is negatively birefringent, the azimuthal polarization was generated directly in this setup. Radial polarization was generated with optical active materials (quartz) to rotate the electric field by 90°. It is important to have the optical axis of the lasing medium aligned with the optical axis of the resonator if the laser medium is also anisotropic. Other papers extended these early reports [13, 14]. However, owing to the lack of practical applications of these special polarization modes, little attention was paid to this field until recently.

Driven by the potential applications of CV beams in imaging, machining, particle trapping, data storage, remote sensing, etc., this field has seen rapidly growth in the past decade or so. Active CV beam generation was revisited, and new methods were developed. For examples, CV beam generation with axial intracavity birefringence was reinvented and improved [15, 16, 17]. Besides axial birefringence, axial intracavity dichroism created with conical axicon [18] and Brewster angle reflectors [19, 20] were also utilized to provide polarization mode selection. One such configuration [20] is illustrated in Fig. 4.

The techniques mentioned above used bulk intracavity devices for creating axial birefringence or dichroism. The recent availability of microfabrication and nanofabrication tools enables the creation of diffractive phase plate or polarization selective end mirror devices [14, 21] for CV beam generation (Fig. 5). This type of device allows a much more compact laser design and can be exploited to generate high output power.

In addition to those methods that utilize either intracavity axial birefringence or dichroism to provide the necessary mode discrimination, CV beams can also be generated with intracavity interferometric methods using folded mirrors or prisms based on the linear superposition principle given by Eqs. (2.15, 2.16). A recently proposed intracavity Sagnac interferometer setup [22] is illustrated in Fig. 6. In this example, linearly polarized HG01 modes are created by placing a thin wire across the center of the cavity. A dove prism provides the necessary rotation to create the orthogonally polarized HG10 and HG01 modes. The Sagnac interferometer combines the two modes and creates the CV output.

3.2. Passive Generation Methods in Free Space

Passive methods have also been used to generate CV beams in free space. In general, these methods convert those more commonly known spatially homogeneous polarizations (typically linear or circular polarization) into spatially inhomogeneous CV polarizations. Consequently, devices with spatially variant polarization properties are normally required. For example, axial birefringence and dichroism have been applied to generate CV beam outside the laser cavity. Simple setups with a radial analyzer made either from birefringent materials [23] or from dichroic materials [24] can be used to generate the CV beams. A radial analyzer is a device that has its local polarization transmission axis aligned along either the radial or the azimuthal directions. In general, birefringent radial analyzers have better polarization purity than dichroic radial analyzers, while the setup for a dichroic radial analyzer is more compact. A circularly polarized collimated beam needs to be used as the input to the radial analyzer. The beam after the radial analyzer will be polarized either radially or azimuthally, depending on the type of radial analyzer used. However, one important factor that requires caution is the Berry’s phase [24]. For a circularly polarized input, the field can be expressed as

Ein=ex+jey=(cosϕersinϕeϕ)+j(sinϕer+cosϕeϕ)=ejϕ(er+jeϕ),
where ex and ey are the unit vectors in Cartesian coordinates and er and eϕ are the unit vectors in the polar coordinate system. After the beam passes through the radial analyzer, since the transmission axis of the radial analyzer is aligned along the radial direction, the SOP becomes
Eout=ejϕer.
This indicates that, although the electrical field is aligned along the radial direction, there is a spiral phase factor on top of it.

The geometric spiral phase has been confirmed by interferometric measurement (Fig. 7) [24]. To obtain a true CV beam, a spiral phase element (SPE) with the opposite helicity is necessary to compensate for the geometric phase (Fig. 8). A SPE can be fabricated with a variety of lithographic techniques, such as electron-beam lithography and gray-scale lithography, or can be generated by a liquid crystal (LC) spatial light modulator (SLM). An interesting simple tunable SPE using a deformed cracked glass plate was reported in [25]. Commercial SPE products are available now at several vendors [26]. This generation method has been successfully implemented by several groups to generate both continuous wave (CW) and ultrafast CV beams [6, 27, 28].

Spatially variant polarization rotation can also be utilized to produce CV beams. In this case, linearly polarized light is typically used as input and then locally rotated to the desired spatial polarization pattern. One such example is a device with twisted nematic LC sandwiched between linearly and circularly rubbed plates [29, 30]. Owing to the circular rubbing of the second plate, the twisted nematic LC molecules continuously rotate from the initial linear rubbing direction to the corresponding spatially distributed rubbing direction on the other plate. An incident beam that is linearly polarized perpendicular or parallel to the linear rubbing direction will follow the molecules’ rotation, creating a radial or azimuthal polarization on the exiting side. A π-step phase plate is necessary to correct a geometric phase similar to what we mentioned above. These types of devices are already available at several vendors [31].

Another very popular and powerful passive method uses a LC SLM. Despite its relatively high cost, a LC SLM offers the flexibility and capability to generate an almost arbitrary complex field distribution. One such example is shown in Fig. 9 [32]. In this setup, two LC SLMs were used. The first SLM provides pure phase modulation to the incoming beam to either correct certain aberrations in the system or to add the desired phase pattern to the beam. The combination of the λ4 plate and the second SLM essentially forms a polarization rotator [23, 33], where the amount of rotation is determined by the phase retardation of each pixel on the SLM. Properly designing the phase pattern on the second SLM allows the input linear polarization to be converted into any arbitrary polarization distribution, including CV beams.

Besides spatially variant polarization rotation, another class of methods use spatially variant retardation axis arrangements. For example, for a linear polarization input, λ2 plates with spatially variant fast axis directions can be used to convert the linear polarization into CV polarizations. This can be achieved by taping or gluing together several segmented λ2 plates with different discrete crystal angles (Fig. 10) [34]. Because of the discreteness, this type of device provides only rough spatial alignment of the polarization. A mode selector can be used to further clean up the polarization distribution pattern. One example is shown in Fig. 11 [4]. The polarization is initially roughly aligned by four segmented λ2 plates. Then a near-confocal Fabry–Perot interferometer is used to filter out the undesired mode and keep the radial polarization component. Recently, continuous rotation of linear polarization input using stress-induced space-variant plate [35] and photoaligned LC polymers [36] were reported, eliminating the need for a mode selector.

For circularly polarized input, a spatially variant λ4 retarder with axially symmetric local axis arrangement can be used to convert the input into cylindrical polarization. This type of device can be realized with spatial-variant subwavelength gratings (Fig. 12) [37]. The form birefringence of the subwavelength grating provides λ4 retardation, and its local orientation is continuously varied through lithographic patterning. However, owing to the requirement of a subwavelength period, extension into the visible and UV would be difficult. An interesting technique using a transparent electro-optic (EO) ceramic [Pb(Mg1/3Nb2/3)O3PbTiO3, PMN-PT] as a radial polarization retarder (Fig. 13, EO-RPR) was reported recently [38]. A voltage is applied across the electrodes to create λ4 retardation with a radially aligned retardation axis. This technique in principle could provide tunable CV generation from 500 to 7000nm, depending on the applied voltage. However, in both cases a geometrical spiral phase exists and needs to be compensated.

In addition to the methods using the spatially variant polarization properties described above, interferometric methods have also been used to create CV beams in free space. Methods using a Mach–Zehnder interferometer combined with a spiral phase plate [39] or a spiral phase created by a LC SLM [5] have been developed. A simple technique using π-phase step plates was reported in [40]. Recently a common path interferometer implemented with a LC SLM to generate CV beams and other more complex vector beams was also demonstrated (Fig. 14) [41].

3.3. Passive Generation Methods Using Optical Fiber

Generation of CV beams with few-mode fiber is another technique that deserves special attention. It is known that a multimode step-index optical fiber can support the TE01 and TM01 annular modes possessing cylindrical polarization symmetry, with the TE01 mode being azimuthally polarized and the TM01 mode being radially polarized (Fig. 15). Under the weakly guiding approximation, these modes have the same cutoff parameter that is lower than all the other modes except the HE11 fundamental mode.

In general, it is difficult to excite these modes in an optical fiber without exciting the fundamental mode. The presence of a strong fundamental mode would spoil the cylindrical polarization purity. CV mode excitation in fiber can be achieved with careful misalignment between a single-mode and a multimode fiber [42]. However, the conversion efficiency is fairly low for this method. The efficiency can be improved by preforming the incident polarization either in phase or polarization [27, 43]. A laboratory picture for CV mode excitation using an SPE is shown in Fig. 16. A collimated laser beam passes through a SPE. Then it is coupled into a fiber that is carefully chosen such that it supports only the fundamental mode and the second-higher-order modes. The optical fiber acts as a spatial filter and a polarization mode selector. The polarization symmetry is confirmed by inserting a linear polarizer between the fiber output end and the observation plane (shown in Fig. 16). Using optical fiber as mode selector, a tunable and narrow band CV beam laser source was demonstrated in [44].

4. Manipulation of Cylindrical Vector Beams

To make use of the CV beams in different applications, devices that can perform basic manipulations such as reflection, polarization rotation, and retardation are necessary. The key for these operations is to maintain the polarization symmetry. When CV beams are reflected and steered, polarization symmetry could be broken owing to the nonequal reflection coefficients for s and p polarizations. Even if the magnitudes of these reflection coefficients are close, the phase difference can still destroy the polarization symmetry. In principle, metallic mirrors should preserve the polarization symmetry better. However, many metallic mirrors have protective coatings that could give rise to different reflection coefficients for s and p polarizations. Combination of two identical beam splitters (picked up from the same coating run) with twisted orientation has been employed to maintain polarization symmetry while providing the steering function for CV beams [23]. Similar arrangements with two identically coated metallic mirrors can also be used to provide higher throughput.

Polarization rotation can be realized with an active material (such as quartz in [1]) or Faraday rotators. These types of rotator lack tunability in terms of the amount of rotation. And it is difficult to manufacture a Faraday rotator with the large clear aperture that is necessary for some applications. A polarization rotator using two cascaded λ2 plates has been designed and demonstrated [23]. If the angle between the fast axes of the two cascaded λ2 plates is Δφ, the Jones matrix of this combination can be shown to be

T=(cos(2Δφ)sin(2Δφ)sin(2Δφ)cos(2Δφ))=R(2Δφ),
which is a pure polarization rotation function independent of the incident polarization. The polarization rotation angle can be tuned by adjusting the angle between the fast axes of the two plates. This type of device can be used to rotate the polarization pattern of a certain CV beam into other desired generalized CV beam polarization patterns.

A nonmechanical polarization rotator using LC or EO retarders sandwiched between two orthogonally oriented λ4 plates (Fig. 17) has also been designed and demonstrated. The Jones matrix of these devices can be shown to be

T=R(π2)(100j)R(π2)R(π4)(100ejδ)R(π4)(100j)=jejδ2R(δ2),
where δ is the retardation of the LC or EO retarder. These devices can be made with very large clear apertures, and the amount of rotation can be adjusted and modulated rapidly.

For fiber optic devices, the torsion in the optical fiber induces a birefringence in the fiber. It can be shown that the uniformly twisted fiber behaves as a medium exhibiting rotatory power [27]. This may provide a convenient means of polarization manipulation. However, careful control is necessary due to the sensitivity. In addition, the fast EO radial polarization retarder [38] described above can be used to provide retardation between radial and azimuthal polarization components. Beams with cylindrical symmetrically distributed elliptical polarization states can be generated this way. This kind of polarization modulation may find applications such as high-resolution microellipsometry [23].

5. Focusing Properties and Applications

The rapid increase of interest in CV beams was driven largely by the unique focusing properties of these beams discovered recently. Particularly, it was found that radially polarized light can be focused into a tighter spot than those of spatially homogeneous polarization because of the creation of a strong and localized longitudinal component [3, 4]. In addition, the longitudinal component experiences an apodization effect that is different from the transverse component and is spatially separated from the transverse focal field. These effects enable 3D tailoring of the focus shape [12, 45, 46]. In this section, a dipole radiation pattern picture is given to explain the focusing advantages with radial polarization. Numerical methods based on Richards–Wolf theory will be summarized. Various applications of the focusing properties will be discussed.

5.1. Tighter Focusing with Radial Polarization Explained by Dipole Radiation

Let us consider a vertical electric dipole located at the focal point of a high-NA aplanatic objective lens (Fig. 18). The dipole oscillates along the optical axis of the lens. The well-known angular pattern of electric dipole radiation is illustrated with the local polarization indicated as well. The high-NA objective lens collects the dipole radiation into the top half-space and collimates the radiation. From the illustration, we can see that the polarization pattern at the lens pupil plane will be aligned along the radial directions. If we exactly reverse the optical path and start with the radial polarization pattern at the pupil plane, the corresponding focal field should recover the propagating components in the upper half-space. If another high-NA objective lens is also used in the lower half-space (4π setup), we should be able to recover the electric dipole field up to all the propagating components. From this argument, it can be seen qualitatively that the radial polarization should provide optimal focusing compared with other polarization distributions.

5.2. Numerical Calculation Methods

The focusing property of highly focused polarized beams can be numerically analyzed with the Richards–Wolf vectorial diffraction method [47, 48, 49]. The geometry of the problem is shown in Fig. 19. The illumination is a generalized CV beam, which assumes a planar wavefront over the pupil. The incident field can be written in the pupil plane cylindrical coordinates (ρ,φ,z) as

Ei(ρ,φ)=l0P(ρ)[cosφ0eρ+sinφ0eφ],
with l0 being the peak field amplitude at the pupil plane and P(ρ) being the axially symmetric pupil plane amplitude distribution normalized to l0, and the unit vectors are
eρ=cosφex+sinφey,
eφ=sinφex+cosφey.
An aplanatic lens produces a spherical wave converging to the focal point. The amplitude distribution over the pupil is mapped onto the spherical wavefront through the ray projection function g(θ) given by
ρf=g(θ),
where f is the focal length of the objective lens. With the ray projection function, the pupil apodization function P(θ) on the spherical wavefront can be found with the help of the power conservation requirement:
[l0P(ρ)]22πρdρ=[l0P(θ)]22πf2sinθdθ,
P(θ)=P(ρ)g(θ)g(θ)sinθ=P(fg(θ))g(θ)g(θ)sinθ.
For typical objective lenses that obey the sine condition, the ray projection function is given by
ρf=sinθ.
Thus the pupil plane apodization function becomes
P(θ)=P(fsinθ)cosθ.
The refraction of the objective lens also changes the polarization unit vectors. From Fig. 19, the polarization unit vectors after refraction can be found to be
er=cosθ(cosφex+sinφey)+sinθez,
eφ=eφ=sinφex+cosφey.
According to the Richards–Wolf method, the electric field near focus is given by the diffraction integral over the vector field on the spherical wavefront with radius equal to the objective lens focal length f:
E(r,ϕ,z)=ik2πΩa(θ,φ)eik(sr)dΩ=ik2π0θmaxdθ02πa(θ,φ)eik(sr)sinθdφ,
where θmax is the maximal angle determined by the NA of the objective lens, k is the wavenumber, and the field strength factor a(θ,φ) is given by
a(θ,φ)=l0fP(θ)[cosφ0er+sinφ0eφ].
For observation points in the vicinity of the focus, we have
sr=zcosθ+rsinθcos(φϕ).
Thus the field near the focal plane can be obtained as
E(r,ϕ,z)=ik2π0θmaxdθ02πl0fP(θ)[cosφ0er+sinφ0eφ]eik(zcosθ+rsinθcos(φϕ))sinθdφ=iAπ0θmaxdθ02πP(θ)[cosφ0er+sinφ0eφ]eik(zcosθ+rsinθcos(φϕ))sinθdφ=iAπ0θmaxdθ02πP(θ)[cosφ0(cosθcosφexcosθsinφeysinθez)+sinφ0(sinφexcosφey0ez)]eik(zcosθ+rsinθcos(φϕ))sinθdφ,
where the constant A is given by
A=πfl0λ.
This expression is still in Cartesian coordinates. The field components expressed in the focal plane cylindrical coordinates (r,ϕ,z) can be obtained by using the following transformations:
er=cosϕex+sinϕey,
eϕ=sinϕex+cosϕey.
After some straightforward mathematical manipulations, we can derive the field expression in cylindrical coordinates as
E(r,ϕ,z)=iAπ0θmaxdθ02πP(θ)[cosφ0(cosθcos(φϕ)er0eϕsinθez)+sinφ0(0ercos(φϕ)eϕ0ez)]eik(zcosθ+rsinθcos(φϕ))sinθdφ.
Notice that the radial component of the incident beam contributes to the radial and longitudinal field components near the focal plane, while the azimuthal component in the incident beam contributes only to the azimuthal field component near the focal plane.

The field expression can be further simplified with the following identity:

02πcos(nφ)eikrsinθcosφdφ=2πinJn(krsinθ),
where Jn(x) is the Bessel function of the first kind with order n.

Finally, the focal field of a generalized CV beam can be written as

E(r,ϕ,z)=Erer+Ezez+Eϕeϕ,
where er,ez,eϕ are the unit vectors along the radial, longitudinal, and azimuthal directions in the image space, respectively. Er, Ez, and Eϕ are the amplitudes of the three orthogonal components:
Er(r,ϕ,z)=2Acosφ00θmaxP(θ)sinθcosθJ1(krsinθ)eikzcosθdθ,
Ez(r,ϕ,z)=i2Acosφ00θmaxP(θ)sin2θJ0(krsinθ)eikzcosθdθ,
Eϕ(r,ϕ,z)=2Asinφ00θmaxP(θ)sinθJ1(krsinθ)eikzcosθdθ.
Besides the sine condition, other types of ray projection function are also used. For example, for an objective lens that obeys the Herschel condition
g(θ)=2sin(θ2),
P(θ)=P(fg(θ)).
For an objective lens that obeys the Lagrange condition (also known as the uniform projection condition)
g(θ)=θ,
P(θ)=P(fg(θ))θsinθ.
And for an objective lens that obeys the Helmholtz condition
g(θ)=tanθ,
P(θ)=P(fg(θ))(1cosθ)3.
However, these types of objective lens are not as common as objective lenses that obey the sine condition. Hence, for most of the examples in this paper, the pupil apodization function given by Eq. (5.8) corresponding to the sine condition will be used.

Note that all components given in Eqs. (5.21, 5.22, 5.23) are independent of ϕ, which means that the field maintains cylindrical symmetry. With these equations, one can numerically compute the intensity and amplitude distributions of different constituting components as well as the total field in the vicinity of focus. From the J0 Bessel function in the integral given by Eq. (5.22), one can see that a key feature of the focal field distribution is the existence of strong longitudinal component (z component). This can be understood from the axial symmetry of the setup (Fig. 18).

For comparison, the focal field of an x-polarized incident beam is found to be [50]

E(r,ϕ,z)=iA{[I0+cos(2ϕ)I2]ex+sin(2ϕ)I2ey2icosϕI1ez},
where
I0=0θmaxP(θ)sinθ(1+cosθ)J0(krsinθ)eikzcosθdθ,
I1=0θmaxP(θ)sin2θJ1(krsinθ)eikzcosθdθ,
I2=0θmaxP(θ)sinθ(1cosθ)J2(krsinθ)eikzcosθdθ.
The focal field is dominated by the transverse x-polarization component. The longitudinal component is zero on the optical axis owing to the J1 Bessel function in the integral given by Eq. (5.32).

One example of the field calculation is given in Fig. 20, which corresponds to CV polarization focused by an objective with NA=0.8 [12]. As one can see, for radially polarized incidence, the focal field consists of a strong and narrower longitudinal component and a radial component. The longitudinal component has its peak on the optical axis, while the radial component has a donut shape with zero amplitude on the optical axis. There is no azimuthal component for this situation. For azimuthally polarized incident, only a donut shape azimuthal component exists near the focal plane.

From Eq. (5.22), also notice that for radial polarization incidence the z component experiences an apodization function of sinθ arising from the polarization bending [Eq. (5.9)] compared with the apodization function (1+cosθ)2 for the dominant contribution I0 of the transverse field components given in Eq. (5.31). This apodization function for radial polarization focusing places more weight on the high-spatial-frequency components, consequently leading to a smaller spot size. It also indicates that in order to have prominent effects of this longitudinal component, a high-NA is required, and annular illumination is preferred.

5.3. Tighter Focusing and Applications in Imaging

Under very high-NA illumination, the z component could dominate the total focal field distribution and consequently determine the focal spot size. The existence of such a strong longitudinal focal component has been experimentally confirmed by direct measurements (Fig. 21) as well as indirect measurements [4, 5, 6]. A spot size as small as 0.161λ2 has been reported, compared with a 0.26λ2 spot created by linearly polarization focused under the same condition. Comprehensive studies have been done [51, 52, 53] to establish the advantages of using radial polarization over traditional spatially homogeneous polarization states, considering a variety of NA and apodization conditions. Besides the refractive objective lenses studied in most of the cases, focusing radial polarization with parabolic mirror has also been reported recently. This type of mirror can provide a NA of up to 1 in air, creating a very tight focus [54].

The unique vectorial focal distribution has apparent applications in high-resolution imaging, such as confocal microscopy, two-photon microscopy, second-harmonic generation microscopy [55], third-harmonic generation microscopy [56], and dark field imaging [57]. It is also reported that the smallest spherically symmetric spot can be obtained for 4π microscopy [58], which is expected from the dipole radiation pattern picture described in Subsection 5.1. The application of CV beams in microellipsometry has been suggested to improve spatial resolution [23]. Focusing radial polarization through a dielectric interface and a solid immersion lens to further reduce the spot size have also been studied [59]. The apodization effect shown in Eq. (5.22) leads to a longer depth of focus for the z component. Combined with a pupil plane binary phase mask, the depth of focus can be extended by two times compared with its linear polarization counterpart while maintaining a very small transverse focal spot [45].

A very interesting application of the strong longitudinal field has been developed to map the orientation of a single molecule [60]. The setup and the experimental results are summarized in Fig. 22. When excited by the longitudinal components of the focused radial polarization, depending on the molecule’s spatial orientation, the fluorescence emission pattern is different. For a molecule that is parallel to the longitudinal direction (θ=0° in Fig. 22), a symmetric emission pattern is expected. When the molecule tilts away from this direction, the emission pattern becomes distorted and asymmetric. Measuring the asymmetry allows the molecule’s orientation to be inferred. This experiment also confirmed the existence of the strong longitudinal field of the focused radial polarization.

5.4. Three-Dimensional Focus Engineering with Cylindrical Vector Beams

Observing the fact that the longitudinal and transversal components of the focal field are spatially separated (Fig. 20), if the illumination is a generalized CV beam with adjustable azimuthal angle φ0 from the radial direction [Fig. 1(i)], the relative strength of the transversal (radial plus azimuthal) and the longitudinal components can be adjusted continuously. For a fixed objective lens NA, at a suitable azimuthal angle φ0, a focal field with a transverse flattop profile can be obtained [12]. This focus shaping setup is illustrated in Fig. 23. A double λ2 plate polarization rotator converts the incident CV beam into the generalized CV polarization with the desired azimuthal angle φ0. A high-NA objective then focuses the generalized CV beam, and flattop focus can be obtained at the focal plane. One example of the created flattop profile is shown in Fig. 24. An objective with a NA of 0.8 was used, and the azimuthal angle to produce a flattop profile is found to be φ0=24°. This technique provides flattop focus for extremely high NA, which is very difficult to achieve with traditional beam shaping techniques using diffractive or refractive optical elements. In addition, traditional beam shaping techniques are typically designed with stringent requirements on the incident beam. If the beam does not meet the design conditions, the performance will degrade significantly. In that case, new elements would need to be designed and fabricated. The focus shaping technique using CV beams provides tunability and flexibility. Even if the incident beam conditions are not strictly satisfied, flattop focus still can be obtained by conveniently adjusting one of the half-wave plates shown in Fig. 23.

In the previous focus shaping setup, the polarization azimuthal angle of a generalized CV beam is utilized to achieve a flattop profile in the transverse plane. Introducing a pupil plane phase or amplitude mask provides additional degrees of freedom and enables extra focal field profile control in the longitudinal plane [46]. Such a 3D beam shaping technique using a pupil plane diffractive optical element (DOE) with several concentric zones is illustrated in Fig. 25. The adjustment of azimuthal angle of the CV beams controls the transverse profile, and adjustment of the concentric DOE permits longitudinal profile control. A binary DOE with three concentric rings with the transition points specified by the corresponding NA in the image space is used as an example (shown in Fig. 25). With appropriate combinations of these adjustments, a maximally homogenized field distribution in three dimensions can be generated. One such example is shown in Fig. 26. The parameters necessary to achieve this distribution are NA=0.8, NA1=0.35, NA2=0.55, and φ0=39.5°. For this particular example, an objective lens that obeys the Helmholtz condition is used in the calculation. As we can see, flattop profiles in both the longitudinal and transversal planes are obtained. The longitudinal intensity distribution for a linearly polarized incident beam focused by the same objective lens is also shown in Fig. 26(b) as a comparison. It can be seen that the depth of focus is expanded by nearly a factor of 3.

Utilizing the long depth of focus from the longitudinal component, more exotic focal field distributions such as an optical “bubble” can be achieved [46]. By adjusting the destructive interference from the middle ring, it is possible to further increase the destructive interference to carve into the focus and generate an optical bubble that has a total dark volume surrounded by high field distributions. Figure 27 illustrates one such example. The parameters for the binary DOE are found to be NA1=0.2 and NA2=0.65 with the objective lens NA=0.8. It has been reported that the smallest bubble can be created with CV beams [61], which could find applications in microscopy techniques such as stimulated emission depletion microscopy [62, 63].

Recently it was reported that an optical “needle” that consists of an almost pure longitudinal field can be generated by using a similar setup but with more concentric rings for the pupil plane DOE [64]. The proposed setup is illustrated in Fig. 28. A binary DOE with five concentric rings is used, with the transmission function defined as

T(θ)={1for0θ<θ1,θ2θ<θ3,θ4θ<sin1(NA)1forθ1θ<θ2,θ3θ<θ4},
with the parameters θi corresponding to angles in the image space for the transition edges of the binary DOE. A radially polarized light is used as the input. The NA of the objective lens is chosen to be 0.95. The focal field can be calculated as
Er(r,ϕ,z)=2Acosφ00θmaxP(θ)T(θ)sinθcosθJ1(krsinθ)eikzcosθdθ,
Ez(r,ϕ,z)=i2Acosφ00θmaxP(θ)T(θ)sin2θJ0(krsinθ)eikzcosθdθ.
In this example, the binary DOE performs a special polarization filter function that diffracts the radial field component away from focus center more than the longitudinal field, leaving the beam in the focal region substantially longitudinally polarized. The spatial distributions of the total field with its constituting radial and longitudinal components are shown on the right in Fig. 28 for the case when θ1=4.96°, θ2=21.79°, θ3=34.25°, and θ4=46.87°. It can be seen that the focal field is dominantly longitudinally polarized near the optical axis. An electric field vector map of the focal field shown at the bottom of Fig. 28 confirms the creation of such an optical needle field.

This work provides an example of controlling the vector field distribution near the focus to the desired states to a degree beyond the traditional optical field control on spatial intensity distribution. It has been reported that any desired 3D SOP at the focus can be obtained by using linearly polarization with a small set of low-order azimuthal spatial harmonics [65]. Full control of the optical field over the polarization in three dimensions may have tremendous impact on many applications, including single-molecule manipulation, imaging, and near-field optical imaging.

5.5. Phase Behavior Near the Focus

In addition to intensity and polarization distributions near the focus, the wavefront distribution of highly focused CV beams also displays interesting behavior. For example, it is found that the wavefront spacing is highly irregular [66] and is very different for wavefront spacing of highly focused linear or circular polarization. This difference can be attributed to the Gouy phase shift near the focus [67]. The Gouy phase shift is a well-known axial phase anomaly that a converging wave experiences through focus [68]. This phase shift is a general property for any waves that pass trough the focus. The existence of such a phase anomaly can be qualitatively understood with the uncertainty principle. The spatial confinement introduces a spread in the transverse momentum and hence an extra phase shift in the expectation value along the axial direction.

To quantitatively study the Gouy phase shift under paraxial as well as high-NA focusing situations, an exact formula using a tilted wave interpretation has been introduced [69]. As illustrated in Fig. 29, a tilted plane wave propagates at a direction that has an angle α with respect to the optical axis. Its effective phase retardation along the optical axis compared with an on-axis plane wave is given as

dφ(z)=(keffk0)dz=(k02k2k0)dz,
where k0 is the wave vector, k is the transverse component of the wave vector for the tilted wave, and keff is the effective wave vector along the optical axis for the tilted wave. For a converging wave with angular wave spectrum of F(kx,ky,z), the differential Gouy phase shift is given as
dφG(z)dz=++F(kx,ky,z)2(k02k2k0)dkxdky.
For most optical systems with rotational symmetry, the differential Gouy phase can be rewritten in the polar coordinate system as
dφG(z)dz=0kmax02πF(k,z)2(k02k2k0)kdkdϕ.
The overall Gouy phase is calculated by integrating Eq. (5.39) through the focus region:
φG=+dφG(z)dzdz=+0kmax02πF(k,z)2(k02k2k0)kdkdϕdz.
This is the general expression for the Gouy phase calculation using the tilted wave interpretation. In most cases, this exact formula does not yield an analytical expression. For paraxial (low-NA) situations, the following approximation can be used:
k02k2k0k22k0.
To study the Gouy phase in high-NA situations, we retain the expansion to its second order:
k02k2k0k0{112[(kk0)2]18[(kk0)2]2}k0=k22k0k48k03.
Considering the case illustrated in Fig. 30, a collimated uniform x-polarized beam illuminates the entrance pupil of an aplanatic objective lens that obeys the sine condition. The center part of the laser beam is blocked by a stop aperture in order to achieve an annular illumination. The electric field near the focal plane can be written as
E(r,ϕ,z)=iAπθ1θ2dθ02π{[cosθ+sin2ϕ(1cosθ)cosθexsinϕcosϕ(1cosθ)cosθeysinθcosϕcosθez]eik0cosθzei(kxx+kyy)sinθcosθdϕ},
where θ1 and θ2 are the angles corresponding to the annular illumination illustrated in Fig. 30. This equation is equivalent to the expressions given in Eqs. (5.30, 5.31, 5.32, 5.33). From this expression, the angular plane wave spectrum at the focal plane (z=0) can be found to be
F(k,z=0)=[cosθ+sin2ϕ(1cosθ)cosθ]N,
where N is a normalization constant that is given as
N=θ1θ2dθ02πcosθ+sin2ϕ(1cosθ)cosθ2k02sinθcosθdϕ=πk02[cos2θ1cos2θ2]+πk024[(1cosθ2)3(1cosθ1)3].
Consequently, combining Eqs. (5.44, 5.45) with Eqs. (5.39, 5.42), the differential Gouy phase shift at z=0 is given as
dφG(0)dz=k22k0k48k03=kx2+ky22k0kx4+2kx2ky2+ky48k03.
The denotes the average of the physical quantity weighted by the angular spectral density F2. Analytical expressions of these averaged quantities in Eq. (5.46) are polynomials of cosθ1 and cosθ2 that can be found in the Appendix of [69]. With this slope of the Gouy phase shift and an estimation of the effective focal field length, the total Gouy phase shift throughout the focus is given as
φG=dφG(0)dzΔz=[kx2+ky22k0+kx4+2kx2ky2+ky48k03]Δz,
with the effective focal field length Δz estimated by
Δz=2λcosθ1cosθ2λ2.
Thus, to the second-order approximation, one can analytically calculate the Gouy shift per unit length at the focal plane by using these polynomials of cosθ1 and cosθ2. Figure 31 illustrates one example of such a calculation with variable NA1 and fixed NA2=0.95. Analytical results calculated with Eq. (5.47) are plotted with the numerical results from the Richards–Wolf vectorial diffraction model for comparison, and very good agreement has been obtained. the discrepancy between the numerical calculation and the analytical prediction starts to increase for very high-NA values. In those cases, a higher-order approximation is necessary for more accurate predictions of the Gouy shifts. One important finding from Fig. 31 is that the Gouy phase shift can be continuously adjusted by changing the illumination annulus geometry. For example, from Fig. 31 we found that the Gouy phase shift can be adjusted to 2π if the inner size of the annulus is chosen to be NA1=0.5. The corresponding intensity as well as the Gouy phase shift distributions for this case are shown in Fig. 32, which confirms the analytical prediction of the Gouy phase shift.

The same notion can be applied to highly focused radial polarization. For radially polarized annular illumination, the longitudinal component is given by

Ez(r,Φ,z)=iAπθ1θ2dθ02πsinθcosθeik0zcosθei(kxx+kyy)sinθcosθdϕ.
From this expression, the angular plane wave spectrum for the longitudinal component from radially polarized illumination can be found as
F(k,z=0)=[sinθcosθ]N,
N=Aθ1θ2dθ02πsinθcosθ2k02sinθcosθdϕ=2πk02{13(cosθ13cosθ23)+(cosθ1cosθ2)}.
Apparently, for the longitudinal component, the angular plane wave spectrum of radially polarized illumination is different from the angular plane wave spectrum for the x component of the focal field due to x-polarized illumination given in Eq. (5.44). Consequently, the Gouy phase shift amount and distribution should be different as well. The differential Gouy phase shift at z=0 can be found to be
dφG(0)dz=0kmax02πF(kx,ky,0)2(k02k2k0)kdkdϕ=2πk03(sin4θ2sin4θ1)(4N)k0.
With the differential Gouy phase shift, we can examine the wavefront spacing irregularity near the focus. Using the definition of the Gouy phase shift given in Eq. (5.37), the effective wavelength for the z component is given by
keffΔzk0Δz=dφG(z)dzΔz,
λeff=λ1+dφG(z)dzλ2π.
The difference in Gouy phase behaviors for the focused radial polarization and linear polarization explains the different wavefront irregularity reported in [66]. One example of the wavefront spacing calculation for focused radially polarized light is shown in Fig. 33.

Just like the linear polarization case, the Gouy shift of radially polarization illumination can be adjusted with different illumination conditions (see Fig. 34) [67]. This finding may have important applications in nonlinear optical imaging such as third-harmonic generation or coherent anti-Stokes Raman scattering that are strongly influenced by the existence of the Gouy phase [70, 71]. For example, by adjusting the annular illumination geometry, it has been shown that third-harmonic generation from bulk materials may be detectable with better spatial resolution by using radially polarized illumination [72].

6. Plasmon Excitation and Focusing with Radial Polarization

Recently radial polarization was also discovered to be the ideal source for surface plasmon excitation with axially symmetric metal/dielectric structures. Surface plasmon resonance (SPR) is an electromagnetic excitation at the dielectric/metal interface and is due to the interaction of metals with the incident light [73]. The confinement, field enhancement effect, and short effective wavelength of the surface plasmon field make it very attractive for applications in sensing, imaging, and lithography. For example, it has been reported that much smaller patterns can be created by using plasmonic lithography [74, 75, 76]. As a wave phenomenon, surface plasmons can be focused by using appropriate excitation geometry and metallic/dielectric structures [77, 78]. In [77], plasmonic focusing with an annular ring structure under linear polarization excitation was demonstrated. The circular slit structure on metal film acts as plasmonic lenses, focusing the surface plasmon wave into a strongly confined spot.

Plasmon excitation has a strong dependence on excitation polarization. For example, in an attenuated total reflection configuration, only p polarization can excite SPR. This leads to an interesting application of radial polarization in plasmonic focusing. It is pointed out that optimal plasmonic focusing can be obtained with radial polarization for a rotationally symmetric setup [79, 80, 81]. When a radially polarized beam is launched into these dielectric/metal plasmonic lens structures, the entire beam is p polarized with respect to the dielectric/metal interface (Fig. 35), providing an efficient way to generate a highly focused surface plasmon through constructive interference and creating an enhanced local field. In contrast, if highly focused linear polarization is used to excite the surface plasmon, because of the destructive interference between counterpropagating surface plasmon waves, the focal field has a minimum at the geometric focus. Figure 36 illustrates an experimental setup for the generation and detection of the plasmon field near the silver interface, created by highly focused radial polarization [82]. Examples of a plasmonic focal field due to radial and linear polarizations are shown in the insets of Fig. 36 for comparison. An aplanatic oil immersion lens (NA=1.25) focuses the radially polarized beam onto a glass/silver interface. Immersion oil with a refractive index matched to the glass substrate (ns=1.516) is filled between the lens and the substrate. A 50nm silver film (ɛ=0.180.824i) is deposited onto the glass substrate. The optical excitation wavelength was chosen to be 532nm. The center part of the illumination corresponding to the incident angle below the critical angle θc=sin1(nmns)=41.27° is blocked by an annular photomask. The field right after the silver film can be calculated with the following integrals:

Er(r,ϕ,z)=2AθminθmaxP(θ)tp(θ)sinθcosθJ1(k1rsinθ)exp[iz(k22k12sin2θ)12]dθ,
Ez(r,ϕ,z)=i2AθminθmaxP(θ)tp(θ)sin2θJ0(k1rsinθ)exp[iz(k22k12sin2θ)12]dθ,
where θmin and θmax are the minimum and maximum incident angles on the glass/silver interface corresponding to the annular illumination, P(θ) is the pupil apodization function, tp(θ) is the transmission coefficient of p polarization at the incident angle of θ, Jm(x) is the mth-order Bessel function of the first kind, and k1 and k2 are the wavenumbers in the glass and air, respectively. Numerical results of the plasmonic field due to highly focused radial polarization excitation are shown in Fig. 37(a), 37(b), 37(c). From these plots, we can see that the plasmonic focal field generated this way is dominated by the z component.

A collection-mode near-field scanning optical microscope (NSOM) with a nanoaperture (with aperture size between 50 and 100nm) fiber probe was used to map out the plasmonic field distribution near the silver/air interface. For the metal-coated fiber NSOM probe with a nanoaperture, the detected signal is proportional to Ez2 owing to the symmetry of the HE11 mode propagating in the fiber core of the probe [11, 83]. The expected NSOM signal is calculated and shown in Fig. 37(d), which actually shows a donut pattern. This is confirmed by the experimental measurements shown in Fig. 38(a). Multiple concentric rings corresponding to surface plasmon wave propagation are observed. The plot is in logarithmic scale for better visualization of those outer rings. A comparison of the measured and calculated transverse profiles of the NSOM signal is shown in Fig. 38%(38%). Very good agreement between the experimental result and the simulated result was obtained. The main lobe and sidelobe locations of the experimental and calculated results match each other quite well. Because of the finite aperture size of the probe, a certain amount of radial component will also be detected even if the probe is placed at the center of the focus, leading to elevated NSOM signals at those minimum locations.

More interestingly, the focal field created this way is an evanescent Bessel beam because the sharp SPR resonance effectively acts as an axicon [81]. This axiconlike function can be seen from the very narrow dark ring corresponding to the SPR in Fig. 35. The excitation angle is measured to be 45.51° with a FWHM angular width of 1.28°. Only the portion of the incident beam corresponding to the dark ring is coupled into surface plasmon modes. Thus, the SPR excitation with highly focused radial polarization performs a rotationally symmetric angular filtering function for the transmitted field that mimics an axicon device, allowing the creation of an evanescent Bessel beam. This phenomenon is also confirmed by the experimental results. The normalized transverse profiles of intensity at different distances from the sample surface are plotted in Fig. 38(c). It can be seen that all curves almost overlap with each other, indicating the nonspreading property of the beam. Then the probe is moved to the peak of the innermost ring, and the signal is measured at a series of distances away from the surface. The evanescently decaying nature of the Bessel beam is clearly shown in Fig. 38(d), with a decay length of 143nm obtained through curve fitting.

These experimental results confirm that an evanescent Bessel beam is generated via surface plasmon excitation with a highly focused radially polarized beam. A spot size as small as 195nm (λ02.728) can be obtained with 532nm optical excitation. The combination of radial excitation symmetry and the SPR angular selectivity eliminates the need for a conical device and significantly simplifies the alignment procedure simultaneously. The evanescent Bessel beam could be used as virtual probes for near-field imaging and biosensing. In addition, the strong confinements along the longitudinal direction and the transversal direction may be useful for optical tweezing applications.

The optimal plasmonic focusing with radial polarization illumination further applies to other structures such as multiple rings [79] and conical probes [84, 85, 86]. An exemplary multiple-ring structure is shown in Fig. 39. Rings with 250nm width are etched into 200nm silver film deposited on quartz substrate. For a 632.8nm excitation wavelength, a spot with a FWHM of about 222nm (λ2.85) can be obtained. The peak intensity can be increased by adding more rings to the structure, but the spot size remains almost unchanged with the increasing number of rings.

A conical tip structure illustrated in Fig. 40 has also been investigated numerically. The entire glass tip is coated with 50nm silver film, and the tip radius is 5nm. The numerical results showed that with radial polarized illumination at 632.8nm, a FWHM of less than 10nm with an intensity enhancement higher than 105 can be obtained (Fig. 41) [85]. This type of apertureless probe is very useful for near-field optical microscopy, near-field Raman mapping, and metrology.

7. Applications in Optical Trapping

The focusing properties of CV beams have interesting applications in optical trapping. From Eqs. (5.21, 5.22, 5.23), it is noticed that the longitudinal and transversal components are π2 out of phase, which indicates that the z component does not carry average power. However, it still stores electrical energy. This property is very useful for 3D stable trapping of nanoparticles, especially metallic particles [87]. This advantage can be seen in the optical forces exerted on the particles. For very small particles, using Rayleigh scattering, the radiation forces exerted on the nanoparticle due to electric dipole interactions can be expressed as

Fgrad=Re(α)ɛ0E24,
Fscat,z=n1SzCscatc,
Fabs,z=n1SzCabsc,
where α is the polarizability of the metallic particles given by α=4πa3ɛ1(ɛ̂ɛ1)(ɛ̂+2ɛ1), with ɛ̂, ɛ1 being the dielectric constants of the metallic particle and the ambient medium, respectively; Sz is the time averaged axial component of the Poynting vector; Cscat and Cabs are the scattering and absorption cross sections. The gradient force Fgrad responsible for trapping particles in the focal volume is proportional to the gradient of the energy density. The scattering and absorption forces Fscat and Fabs that destabilize the trap are proportional to the Poynting vector. For metallic particles, the high scattering force causes difficulties for stable trapping in three dimensions.

However, if radial polarization is used, the tighter focusing provides stronger gradient force while the scattering force is zero due to the vanishing Sz along the optical axis [87]. These advantages can be illustrated by one realistic calculation. In this calculation, we choose λ=1.047μm, n1=ɛ1=1.33, and a simple annulus pupil apodization function

P(θ)={P0ifsin1(NA1)θsin1(NAn1)0otherwise},
where P0 is a constant amplitude factor and NA is the lens numerical aperture determined by the outer radius of the annulus and n1. The NA is chosen to be 0.95n1 in this study. NA1 corresponds to the inner radius of the annulus, which can be easily adjusted. The laser beam power is assumed to be 100mW. As NA1 is varied, the amplitude P0 is adjusted accordingly to maintain the power level. As one example, field strength distribution for NA1=0.6 is calculated and shown in Fig. 42. As a comparison, the transverse intensity distribution for linearly polarized light under the same focusing geometry is also shown in Fig. 42(a). From the results, it is evident that the E2 due to radially polarized illumination produces a smaller spot size and a higher peak value. Consequently, the trapping gradient force is greater for radially polarized illumination. Meanwhile, as shown in Figs. 42(b), 42(c), the time-averaged Poynting vector Sz is zero on the optical axis and remains negligible near the optical axis. This creates an ideal situation for 3D trapping of metallic nanoparticles. In addition to trapping of nanoparticles in the Rayleigh range, trapping of relative large particles by using CV beams has also been studied with ray optics, and advantages over traditional donut mode have been reported [88].

Combined with the focus shaping described in Section 5, an optical “chain” created with radial polarization that can be used for particle delivery has also been proposed [89]. Instead of a binary-only DOE pupil plane mask, a mask with both amplitude and phase modulation is used in this case. A three-zone pupil plane DOE is chosen as an example, shown in Fig. 43(b), where zone II is an opaque annulus that completely blocks the incident light. Zone I and zone III provide phase-only modulation. This pupil plane mask performs a function similar to the traditional Fresnel zone plate, creating multiple bright spots near the focal plane due to the interference between the focal field contributed from zone I (EI) and III (EIII). The field strength along the optical axis can be represented in the following form:

E(z)2=[EI(z)+EIII(z)][EI*(z)+EIII*(z)]=EI(z)2+EIII(z)2+2EI(z)2EIII(z)2cosδ=EI(z)2+EIII(z)2+2EI(z)2EIII(z)2cos(Δkzz+Δϕ),
where Δkz is the effective wave vector difference in the axial direction and Δϕ=ϕ3ϕ1 is the phase difference of the two zones introduced by the DOE. The effective wave vector difference is caused by the Gouy phase described in Subsection 5.5. It is the Gouy phase differences from regions I and III that gives rise to the periodicity of the field pattern and the multiple bright spots.

For radially polarized illumination, these bright spots are z polarized and connected to one another through the radial field components that are pushed away from the optical axis, leaving dark spots between those bright spots and forming an optical chain distribution [Fig. 43(c)]. For this example, the transition points of the DOE were chosen to be NA1=0.59, NA2=0.988, and NA=1 was assumed.

Such an optical chain can be used to trap multiple particles simultaneously. The period of the particle trapping site is given by Z=2πΔkz, which can be adjusted by the DOE design. Particles with index of refraction lower than the ambience can also be stably trapped and stacked in the dark region at a distance of Z2=πΔkz from the high-index particle. More interestingly, if the phase difference between the innermost and the outermost rings of the DOE, Δϕ=ϕ3ϕ1, is continuously adjusted (for example, with an SLM), according to Eq. (7.5), the location of the bright spots can be shifted continuously. This is illustrated in Fig. 44, obtained by inspecting the focal field pattern while changing Δϕ from 0 to 2π. Consequently, a desired number of particles can be trapped, accelerated, and then transported one by one in a prescribed way at a velocity v=[d(Δϕ)2πdt]Z, providing an optical conveyor belt function that can be used for controllable particle delivery.

Besides the trapping studies using radial polarization presented above, azimuthal polarization has also been explored for the trapping applications [90, 91]. For an azimuthal polarization, the electric field at the focal plane has a donut shape due to the polarization symmetry. Thus it is useful for the trapping of particles with a refractive index lower than the ambient or hollow particles [90]. More interestingly, its associated magnetic field at the pupil plane is aligned radially. Thus, the magnetic field distribution near the focal plane is similar to the electrical field distribution near the focal plane arising from radially polarized illumination. An example of the total electrical field distribution and magnetic field distribution for azimuthal polarization illumination is shown in Fig. 45. The magnetic field distribution is dominated by its longitudinal component. Such a unique focal field distribution could have applications in high-resolution imaging. For example, it has been suggested that the selection rules can be modified and the classical imaging resolution limit can be improved when various kind of nanostructures (such as quantum dots and quantum wires) are used with azimuthal polarization illumination [92].

It is also suggested that azimuthal polarization may be useful for magnetic particle trapping [91]. Trapping of Rayleigh magnetic particles may find applications in magnetorheological studies. Indirect trapping of magnetic particles has been used to study the micromechanical properties of dipolar chains and columns in a magnetorheological suspension [93]. Direct trapping of magnetic particles in this suspension would provide valuable understanding of the magnetorheological behaviors. According to the duality property of electromagnetic fields, one can obtain an expression for the magnetic counterpart by doing the following substitutions in the corresponding electric dipole force expression:

pmc,EcB,cBE,
where p is the electric dipole moment and m is the magnetic dipole moment. With these substitutions, the radiation force due to the magnetic dipole interaction for a Rayleigh magnetic particle can be written as
F=(αm2)B2+ωαm(E×B)c2,
where αm=αm+iαm is the magnetic polarizability of the particle. However, for nonmagnetic dielectric particles, this force typically is very small compared with the electric dipole force. In addition, the quasi-static approximation was used in the magnetic dipole force derivation, which means that the induced magnetic dipole should follow the electromagnetic fields in time. This requires the response time of the magnetic domain to be in the femtosecond range. For Rayleigh magnetic particles, this untrafast response may be possible because of their extremely small size, containing a monodomain or a few domains. Recent reports on surprisingly strong magnetic dipole scattering may be an indication of the validity of the quasi-static assumption [94]. In addition, with the recently rapid development in metamaterials, the magnetic response of these artificial materials at optical frequencies could be engineered. Research into the trapping of magnetic particles with azimuthal polarization may provide further understanding into the fundamental properties of these novel artificial materials.

8. Applications in Laser Machining

The capability of controlling the focus intensity profile and SOP in three dimensions has profound applications in laser machining. Laser machining is one of the fastest growing processes in industrial manufacturing and has been widely used in metal drilling, cutting and welding, etc. Compared with conventional tools, laser machining offers significant advantages in productivity, machining precision, part quality, material utilization, and flexibility. During the laser machining process, the energy of the focused laser beam is absorbed by the materials and is converted into heat in order to remove the materials. Laser beams that can be focused into a flattop spot will allow faster, high-quality laser cutting with better process uniformity and lower operating costs. More important, the laser machining efficiency of metals strongly depends on the polarization. It has been found that spatially homogeneous polarizations such as linear and circular polarizations have substantial disadvantages in laser machining applications. For example, it is reported that radial polarization is the optimal polarization formation in laser cutting applications [95]. In the case of cutting metals with a large aspect ratio of sheet thickness to width, the laser cutting efficiency for a radially polarized beam is shown to be 1.5–2 times higher than p-polarized and circularly polarized beams. This can be explained by the polarization dependence of absorption by the metal [96]. One example of such dependence is illustrated in Fig. 46. It is evident that p polarization is preferred to maximized the absorption and minimize the reflection loss. If linear polarization is used, the laser–metal interaction depends on the polarization orientation. The kerf of laser machining is directly related to how the linearly polarized beam is oriented with respect to the direction in which the cut is traveling. For circular polarization, the laser machining parameters are time averaged, making it optimized neither for minimum losses nor for maximum absorption. However, for radially polarized laser beams, the entire beam is p polarized with respect to the laser cutting kerf, maximizing the absorption efficiency and giving rise to an isotropic kerf (illustrated in Fig. 47).

To increase the laser cutting efficiency and velocity, sharper focusing of the incoming light is necessary for high-speed cutting. Generally, a focal spot with a smaller radius in the transversal direction will have a shorter depth of focus. The radiation intensity decreases quickly along the focal axis, giving rise to a small ratio of cutting depth to cutting width. In contrast, using CV beams with flattop focusing and a longer depth of focus, one can significantly increase the ratio of cutting depth to cutting width. Because of the threshold character of material removal, laser cutting works only when the densities of absorbed power exceed certain threshold values. It is possible to adjust the beam power to make sure that such threshold conditions are met by the power level in the main lobe only. Thus, the relative high sidelobes may not be a problem.

In contrast to laser cutting applications, the situation for laser drilling of very high-aspect ratio holes is quite different and more complicated [96]. In this case, the multiple reflections at the side wall, the waveguiding effect, and the absorption of radiation at the bottom of the hole all play important roles in determining the material removal efficiency. The optimal beam polarization strongly depends on the optical properties of the metal materials being processed. The effects of CV beams have been studied in laser drilling of different types of material. Depending on the materials being processed, it is found that either radial or azimuthal polarization could be more efficient compared with linear or circular polarization [88]. For example, radial polarization is discovered to be preferable for drilling in brass and copper, while azimuthal polarization is preferred for drilling of mild stainless steel. This difference is explained by the multiple reflection and guiding effect of the side wall of the holes (Figs. 48, 49). These differences were confirmed experimentally.

9. Summary and Prospects

Recent developments in the field of cylindrical vector (CV) polarization are reviewed in this paper. An overview of various generation and manipulation methods is given. Because of their polarization symmetry, CV beams were discovered to have many interesting properties when focused with high-NA objective lens. These focusing properties have a variety of applications in optical imaging and manipulation. Although the focusing properties and their corresponding applications are emphasized, the interests in CV beams are much broader. For example, second-order statistics of CV beams propagating through turbulent atmosphere has been investigated [97]. Further studies in this area may reveal whether using CV polarization can mitigate the adverse atmospheric effects and consequently have an effect on remote sensing and optical free-space communications [98]. Propagation of CV beams in nonlinear materials and their effects on modulation instability was studied [99]. Terahertz CV beams have been generated recently [100], and it was also found that azimuthally polarized surface plasmons along metal wires can be used as efficient terahertz waveguides [101]. Recent studies also linked the CV beams to the study of singular optics [102]. More research in this area may reveal the connections between orbital angular momentum and spin, which potentially could have an impact in quantum optics. It has been reported that the CV beams can be considered particular cases of a more general family of vectorial vortex beams, namely, the spirally polarized beams [103]. The nonparaxial propagation and focusing properties of the spirally polarized beams have been studied [104, 105]. The discussions of these very interesting topics are beyond the present scope. Nevertheless, it is safe to conclude that with simple and efficient generation methods becoming more and more accessible, this field will grow further, and the list of applications of CV beams will continue to expand rapidly.

Acknowledgments

I thank James R. Leger, Joseph W. Haus, John T. Fourkas, Yongping Li, Peter E. Powers, Hai Ming, Jianping Xie, Henning Heuer, and Yiqiong Zhao for collaborations and many enlightening discussions. Thanks also go to my graduate students, Weibin Chen, Shuangyang Yang, and Wen Cheng, for their contributions to my research work in the area of CV polarization.

Figures

 figure: Fig. 1

Fig. 1 Spatial distribution of instantaneous electric vector field for several conventional modes and CV modes: (a) x-polarized fundamental Gaussian mode; (b) x-polarized HG10 mode; (c) x-polarized HG01 mode; (d) y-polarized HG01 mode; (e) y-polarized HG01 mode; (f) x-polarized LG01 mode; (g) radially polarized mode; (h) azimuthally polarized mode; (i) generalized CV beams as a linear superposition of (g) and (h).

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 figure: Fig. 2

Fig. 2 Formation of radial and azimuthal polarizations using linear superposition of orthogonally polarized HG modes.

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 figure: Fig. 3

Fig. 3 Diagram of a ruby laser that generates CV beam output. The optical axis of the calcite crystal is parallel to the resonator axis. The combination of the calcite crystal, the telescope, and the aperture and stop provides the mode discrimination to force the laser operation in CV polarization mode. Figure adapted with permission from [1]. © 1972 American Institute of Physics.

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 figure: Fig. 4

Fig. 4 Nd:YAG laser that generates CV beams by using axial dichroism created by a conical Brewster prism (CBP). The structure of the conical Brewster prism is shown to the right. Figure adapted with permission from [20].

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 figure: Fig. 5

Fig. 5 Yb:YAG thin-disk laser that generates CV beam outputs using a circular multilayer polarizing grating end mirror. The output beam profile and polarization pattern are shown to the right. Figure adapted with permission from [21].

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 figure: Fig. 6

Fig. 6 Diagram of a laser cavity design with Sagnac interferometer as an end mirror to create CV beam output. BS, beam splitter. Figure reprinted with permission from [22].

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 figure: Fig. 7

Fig. 7 Mach–Zehnder interferometer setup to verify the spiral Berry’s phase given in Eq. (3.2). The Berry’s phase is shown in the lower right-hand corner.

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 figure: Fig. 8

Fig. 8 Generation of CV beam using radial analyzer and SPE. A circularly polarized input is used. The cascaded two λ2 plates rotate the polarization to the desired pattern.

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 figure: Fig. 9

Fig. 9 A polarization mode converter setup with two SLMs. The first SLM creates the desired phase pattern. The combination of the λ4 plate and the second reflection-type SLM converts the beam into the spatially variant polarization pattern. The second SLM can be programed to convert a linearly polarized beam into CV beam. Figure adapted with permission from [32].

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 figure: Fig. 10

Fig. 10 Segmented spatially variant λ2 plates that can convert linear polarization into CV polarization. Figure reprinted from [34] with permission from Elsevier.

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 figure: Fig. 11

Fig. 11 Experimental setup that uses a segmented λ2 plates polarization converter and a near-confocal Fabry–Perot interferometer (NCFPI) as a mode selector to generate CV beams. OD, optical diode; HWP, half-wave plate; PH, pinhole; TL1, TL2, telescope lenses; PC, polarization converter; FL, focusing lens; CL, collimating lens; M, four mirrors; MD, monitor diode; AS, aperture stop; MO, microscope objective; PD, photodiode. Figure reprinted with permission from [4] (http://prola.aps.org/abstract/PRL/v91/i23/e233901). © 2003 American Physical Society.

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 figure: Fig. 12

Fig. 12 (a) Geometry of spatially variant subwavelength grating. A scanning electron microscope image of the device is also shown. Experimentally measured spatial polarization patterns for (b) right-hand circular polarization and (c) left-hand circular polarization are illustrated. Figure reprinted with permission from [37].

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 figure: Fig. 13

Fig. 13 An experimental setup using an EO radial polarization retarder (EO-RPR) to generate CV beams. Figure adapted with permission from [38].

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 figure: Fig. 14

Fig. 14 A common-path interferometer setup implemented with a SLM to generate CV beams and other more complicated vector beams. Several CV beams with different polarization patterns are also shown to the right. Figure reprinted with permission from [41].

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 figure: Fig. 15

Fig. 15 Illustrations of the polarization patterns for LP01 and LP11 modes of optic fiber. It can be seen that the TE01 and TM01 modes from the second-higher-order mode group have azimuthal and radial polarization symmetry, respectively.

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 figure: Fig. 16

Fig. 16 Laboratory setup that generates CV beams by using few-mode optic fiber. The intensity patterns of radial polarization generated with this setup after passing through a linear polarizer are shown to the right.

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 figure: Fig. 17

Fig. 17 Diagram of a nonmechanical polarization rotator using LC or EO retarders sandwiched between two orthogonally oriented (λ4 plates. The Jones matrix of such a device is shown in the figure. It can be seen that such an arrangement provides polarization rotation that is independent of the incident polarization state.

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 figure: Fig. 18

Fig. 18 Illustration of the polarization pattern at the pupil plane for the radiation from a vertical electric dipole collected by a high-NA objective. If the optical path is reversed, a radial polarization input should recover the propagating components of the vertical dipole radiation and produce a very tight focus.

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 figure: Fig. 19

Fig. 19 Focusing of a CV beam. f is the focal length of the objective lens. Q(r,ϕ) is an observation point in the focal plane.

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 figure: Fig. 20

Fig. 20 Focal field calculation examples for CV polarization focused by a high-NA (NA=0.8) objective lens. (a)–(c) 2D plots of the longitudinal component, the radial component, and the total field strength in the rz plane for incident radial polarization. (d) Line scan across the focus (dashed lines). There is no azimuthal component for this situation. A strong and narrower longitudinal component is obtained. (e)–(f) Results for azimuthal polarization. Only azimuthal component exists for this case.

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 figure: Fig. 21

Fig. 21 Projections of a focal field experimentally measured by using a focal-plane knife-edge technique for a highly focused (a) radial and (b) azimuthal polarization. Actual focal intensity profiles for (c) radial and (d) azimuthal polarizations were obtained by using tomographic reconstruction. A focal spot as small as 0.16λ2 has been obtained for radial polarization with an annular aperture. Figures reprinted with permission from [4] (http://prola.aps.org/abstract/PRL/v91/i23/e233901). © 2003 American Physical Society.

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 figure: Fig. 22

Fig. 22 Molecule orientation imaging by using radial polarization. The output of an argon ion laser is converted into radial polarization through a mode converter. The radially polarized beam is focused by a high-NA lens (NA=1.4) on to a polymethyl methacrylate (PMMA) thin-film sample imbedded with fluorescent dye. The upper right-hand corner shows the predicted fluorescent emission rate patterns corresponding to different molecule orientations. Corresponding experimental results are shown in the lower right-hand corner. Figures reprinted with permission from [60] (http://prola.aps.org/abstract/PRL/v86/i23/p5251_1). © 2001 American Physical Society.

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 figure: Fig. 23

Fig. 23 Flattop focusing with generalized CV polarization. The double-λ2-plate polarization rotator converts the incident CV beam into generalized CV polarization with the desired pattern for flattop profile creation at the focal plane.

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 figure: Fig. 24

Fig. 24 Example of the flattop focusing generated with the setup shown in Fig. 23. Line scans of the field distribution across the focal plane are shown to the left. A 2D distribution in the rz plane in the vicinity of the focus is shown to the right.

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 figure: Fig. 25

Fig. 25 3D focus shaping using generalized CV polarization and pupil plane binary DOE. The pupil plane DOE consists of several concentric rings that give either 0 or π phase transmission for normal incidence. The transition points are specified with the corresponding NA in the image space.

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 figure: Fig. 26

Fig. 26 Example of 3D flattop focus generated with CV beam. (a) 3D plot of the E2 distribution in the vicinity of the focus. (b) Line scan of (a) along the z axis. A line scan for linear polarization is also shown for comparison. (c) Line scan in the focal plane.

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 figure: Fig. 27

Fig. 27 Optical bubble created by radial polarization with DOE pupil plane modulation.

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 figure: Fig. 28

Fig. 28 Generation of optical needle using radial polarization with pupil plane DOE modulation. The total field and the constituting radial and longitudinal components are shown in the column to the right. The electric vector distribution is shown in the plot at the left-hand bottom. It can be seen that the field near the optical axis is almost purely polarized in the longitudinal direction. Figure adapted and reprinted by permission from Macmillan Publishers Ltd, Nature Photonics [64], © 2008.

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 figure: Fig. 29

Fig. 29 Geometry for calculation of effective phase retardation of tilted wave.

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 figure: Fig. 30

Fig. 30 Setup for the Gouy phase shift calculation with uniform annular illumination. An x-polarized collimated beam uniformly illuminates the entrance pupil of an aplanatic objective lens. A circular blocking mask is placed at the entrance pupil of the lens to generate annular illumination. The illumination angles of the innermost and outermost rays are θ1 and θ2, respectively. Q(r,ϕ) is an observation point in the focal plane, and f is the focal length of the objective. The geometry for the focal field length calculation is also shown. ΔL is the optical path difference between the innermost ray and the outermost ray.

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 figure: Fig. 31

Fig. 31 Calculations of the Gouy phase shift under uniform annular illumination with NA2=sinθ2=0.95.

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 figure: Fig. 32

Fig. 32 Vectorial diffraction theory simulation of a tightly focused beam with a Gouy shift of 2π. Solid curve, axial intensity distribution; dotted curve, Gouy phase shift. A uniform annular illumination with NA1=sinθ1=0.5 and NA2=sinθ2=0.95 is used in the simulation. Only the positive side of the optical axis is shown.

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 figure: Fig. 33

Fig. 33 Calculated axial intensity distribution Iz (black solid curve), electric field component Ez (blue dotted curve) and the Gouy phase shift (red dashed curve) for uniform radial polarization illumination with NA=0.5 using the vectorial diffraction method. The space between the vertical dashed–dotted lines is wavefront spacing [67].

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 figure: Fig. 34

Fig. 34 Comparison of the differential Gouy phase shift at the focal plane (z=0) calculated by both vectorial diffraction simulation and the analytical expression. In this case, uniform illumination is used with variable sinθ2=NA and θ1 fixed at 0 [67].

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 figure: Fig. 35

Fig. 35 Diagram of SPR excitation with highly focused radial polarization. The entire beam is p polarized with respect to the silver/dielectric interface. A pupil plane image of the reflected beam is shown to the right. A full dark ring is observed that is due to the SPR excitation by radial polarization. The nonuniformity in the beam profile is caused by the difference in reflection coefficients for the s- and p-polarization components of a beam splitter that steers the reflected beam to the CCD camera.

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 figure: Fig. 36

Fig. 36 Experimental setup for the generation and detection of plasmonic focusing with highly focused radial polarization. PMT, photomultiplier tube; PM, photomask. A Veeco Aurora 3 NSOM with a nanoaperture fiber probe is used to detect the plasmonic field. The insets show the calculated plasmon focal fields at the metal surface for radial and linear polarization incidence, respectively. For linear polarization, because of the mismatch of polarization and the destructive interference of counterpropagating plasmon waves, the field at the focal plane splits into two lobes with much lower peak intensity.

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 figure: Fig. 37

Fig. 37 Calculated surface plasmon intensity distribution at silver/air interface with radially polarized illumination. (a) Total intensity E2; a homogeneous spot with a strongly enhanced field at the center is obtained. (b) Longitudinal component Ez2, which is much stronger than Er2 and dominates the total field distribution. (c) Radial component Er2, which has a donut shape. (d) Ez2 distribution, which is proportional to the NSOM signal detected by an apertured fiber probe.

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 figure: Fig. 38

Fig. 38 Experimental confirmation of evanescent Bessel beam creation via surface plasmon excitation. (a) Logarithmic plot of measured NSOM signal in the near field. (b) Comparison of the measured and calculated transverse profiles of the NSOM signal (Ez2) in the near field. (c) Measurements at different planes to reveal the nondiffracting nature of the evanescent Bessel beam. (d) Exponential decay of the evanescent Bessel beam [82].

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 figure: Fig. 39

Fig. 39 Plasmonic lens structure with multiple concentric rings illuminated by radial polarization. Surface plasmons generated by each of the rings propagate to the center and constructively interfere to produce a tight plasmonic focus (shown to the right).

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 figure: Fig. 40

Fig. 40 Conical glass tip fully coated with thin silver film. The entire radial polarization input is p polarized with respect to the glass/silver interface and creates surface plasmons that propagate to the tip apex.

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 figure: Fig. 41

Fig. 41 Logarithmic plots of (a) 2D and (b) 3D electric energy density distribution at the end of the probe tip under radial polarization illumination.

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 figure: Fig. 42

Fig. 42 (a) Calculated E2 distribution at the focal plane for a radially polarized beam focused by a high-NA objective. For comparison, the corresponding result from a linearly polarized beam incidence under the same focusing condition is also shown. (b) 2D plot of the time-averaged Poynting vector Sz in the rz plane for highly focused radial polarization. (c) Line scan of (b) at the focal plane.

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 figure: Fig. 43

Fig. 43 Generation of optical chain. (a) Incident radial polarization. (b) Pupil plane DOE that modulates both phase and amplitude of the incident radial polarization. (c) One example of optical chain created with the pupil plane DOE shown in (b). (d) Time-averaged Poynting vector Sz [89].

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 figure: Fig. 44

Fig. 44 Field distribution shifts as the phase difference Δφ between zone I and zone III varies from 0 to 2π [89].

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 figure: Fig. 45

Fig. 45 Magnetic field and electric field at the focal plane for a highly focused (NA=0.95) azimuthal polarization.

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 figure: Fig. 46

Fig. 46 Calculated absorption coefficients for s polarization and p polarization of mild steel [96].

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 figure: Fig. 47

Fig. 47 Expected laser machining cross section for radial and azimuthal polarizations [96].

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 figure: Fig. 48

Fig. 48 Calculated hole cross sections and corresponding absorbed laser fluence for mild steel with azimuthal and radial polarizations. The guiding effect is taken into consideration. From the simulations, it can be seen that azimuthal polarization is more energy efficient in this case. Figure reprinted from [96] with kind permission of Springer Science+Business Media.

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 figure: Fig. 49

Fig. 49 Calculated hole cross sections and corresponding absorbed laser fluence for brass with azimuthal and radial polarizations. The guiding effect is taken into consideration. From the simulations, it can be seen that radial polarization is more energy efficient in this case. Figure reprinted from [96] with kind permission of Springer Science+Business Media.

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1. D. Pohl, “Operation of a Ruby laser in the purely transverse electric mode TE01,” Appl. Phys. Lett. 20, 266–267 (1972). [CrossRef]  

2. Y. Mushiake, K. Matzumurra, and N. Nakajima, “Generation of radially polarized optical beam mode by laser oscillation,” Proc. IEEE 60, 1107–1109 (1972). [CrossRef]  

3. S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light into a tighter spot,” Opt. Commun. 179, 1–7 (2000). [CrossRef]  

4. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003). [CrossRef]   [PubMed]  

5. B. Jia, X. Gan, and M. Gu, “Direct measurement of a radially polarized focused evanescent field facilitated by a single LCD,” Opt. Express 13, 6821–6827 (2005). [CrossRef]   [PubMed]  

6. B. Hao and J. Leger, “Experimental measurement of longitudinal component in the vicinity of focused radially polarized beam,” Opt. Express 15, 3550–3556 (2007). [CrossRef]   [PubMed]  

7. D. G. Hall, “Vector-beam solutions of Maxwell’s wave equation,” Opt. Lett. 21, 9–11 (1996). [CrossRef]   [PubMed]  

8. For example, J. M. Senior, Optical Fiber Communications (Prentice Hall, 1992).

9. F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64, 491–495 (1987). [CrossRef]  

10. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley, 2007).

11. L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge U. Press, 2006). [CrossRef]  

12. Q. Zhan and J. R. Leger, “Focus shaping using cylindrical vector beams,” Opt. Express 10, 324–331 (2002). [CrossRef]   [PubMed]  

13. M. E. Marhic and E. Garmire, “Low-order TE0q operation of a CO2 laser for transmission through circular metallic waveguides,” Appl. Phys. Lett. 38, 743–745 (1981). [CrossRef]  

14. T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. Anderson, and M. J. Rooks, “Circularly symmetric operation of a concentric-circle-grating, surface-emitting, AlGaAsGaAs quantum-well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992). [CrossRef]  

15. K. Yonezawa, Y. Kozawa, and S. Sato, “Generation of a radially polarized laser beam by use of the birefringence of a c-cut Nd:YVO4 crystal,” Opt. Lett. 31, 2151–2153 (2006). [CrossRef]   [PubMed]  

16. G. Machavariani, Y. Lumer, I. Moshe, A. Meir, S. Jackel, and N. Davidson, “Birefringence-induced bifocusing for selection of radially or azimuthally polarized laser modes,” Appl. Opt. 46, 3304–3310 (2007). [CrossRef]   [PubMed]  

17. K. Yonezawa, Y. Kozawa, and S. Sato, “Compact laser with radial polarization using birefringent laser medium,” Jpn. J. Appl. Phys., Part 1 46, 5160–5163 (2007). [CrossRef]  

18. J. F. Bisson, J. Li, K. Ueda, and Y. Senatsky, “Radially polarized ring and arc beams of a neodymium laser with an intra-cavity axicon,” Opt. Express 14, 3304–3311 (2006). [CrossRef]   [PubMed]  

19. C.-C. Shih, “Radial polarization laser resonator,” U.S. patent 5,359,622 (Oct. 25, 1994).

20. Y. Kozawa and S. Sato, “Generation of a radially polarized laser beam by use of a conical Brewster prism,” Opt. Lett. 30, 3063–3065 (2005). [CrossRef]   [PubMed]  

21. M. A. Ahmed, A. Voss, M. M. Vogel, and T. Graf, “Multilayer polarizing grating mirror used for the generation of radial polarization in Yb:YAG thin-disk lasers,” Opt. Lett. 32, 3272–3274 (2007). [CrossRef]   [PubMed]  

22. V. G. Niziev, R. S. Chang, and A. V. Nesterov, “Generation of inhomogeneously polarized laser beams by use of a Sagnac interferometer,” Appl. Opt. 45, 8393–8399 (2006). [CrossRef]   [PubMed]  

23. Q. Zhan and J. R. Leger, “Microellipsometer with radial symmetry,” Appl. Opt. 41, 4630–4637 (2002). [CrossRef]   [PubMed]  

24. Q. Zhan and J. R. Leger, “Interferometric measurement of Berry’s phase in space-variant polarization manipulations,” Opt. Commun. 213, 241–245 (2002). [CrossRef]  

25. C. Rotschild, S. Zommer, S. Moed, O. Hershcovitz, and S. G. Lipson, “Adjustable spiral phase plate,” Appl. Opt. 43, 2397–2399 (2004). [CrossRef]   [PubMed]  

26. For example, the vortex phase plates from RPC Photonics (http://www.rpcphotonics.com/).

27. Q. Zhan, “Cylindrical polarization symmetry for nondestructive nano-characterization,” Proc. SPIE 5045, 85–92 (2003). [CrossRef]  

28. K. J. Moh, X. C. Yuan, J. Bu, D. K. Y. Low, and R. E. Burge, “Direct noninterference cylindrical vector beam generation applied in the femtosecond regime,” Appl. Phys. Lett. 89, 251114 (2006). [CrossRef]  

29. R. Yamaguchi, T. Nose, and S. Sato, “Liquid crystal polarizers with axially symmetrical properties,” Jpn. J. Appl. Phys. Part 1 28, 1730–1731 (1989). [CrossRef]  

30. M. Stalder and M. Schadt, “Linearly polarized light with axial symmetry generated by liquid-crystal polarization converters,” Opt. Lett. 21, 1948–1950 (1996). [CrossRef]   [PubMed]  

31. For example, the ZPol from Nanophoton (http://www.nanophoton.jp/) and the radial polarizer from ARCoptix (http://www.arcoptix.com/).

32. M. R. Beversluis, L. Novotny, and S. J. Stranick, “Programmable vector point-spread function engineering,” Opt. Express 14, 2650–2656 (2006). [CrossRef]   [PubMed]  

33. C. Ye, “Construction of an optical rotator using quarter-wave plates and an optical retarder,” Opt. Eng. 34, 3031–3035 (1995). [CrossRef]  

34. G. Machavariani, Y. Lumer, I. Moshe, A. Meir, and S. Jackel, “Spatially-variable retardation plate for efficient generation of radially- and azimuthally-polarized beams,” Opt. Commun. 281, 732–738 (2008). [CrossRef]  

35. A. K. Spilman and T. G. Brown, “Stress birefringent, space-variant wave plates for vortex illumination,” Appl. Opt. 46, 61–66 (2007). [CrossRef]  

36. S. C. McEldowney, D. M. Shemo, and R. A. Chipman, “Vortex retarders produced from photo-aligned liquid crystal polymers,” Opt. Express 16, 7295–7308 (2008). [CrossRef]   [PubMed]  

37. Z. Bomzon, G. Biener, V. Kleiner, and E. Hasman, “Radially and azimuthally polarized beams generated by space-variant dielectric subwavelength gratings,” Opt. Lett. 27, 285–287 (2002). [CrossRef]  

38. B. C. Lim, P. B. Phua, W. J. Lai, and M. H. Hong, “Fast switchable electro-optic radial polarization retarder,” Opt. Lett. 33, 950–952 (2008). [CrossRef]   [PubMed]  

39. S. C. Tidwell, D. H. Ford, and W. D. Kimura, “Generating radially polarized beams interferometrically,” Appl. Opt. 29, 2234–2239 (1990). [CrossRef]   [PubMed]  

40. N. Passilly, R. de Saint Denis, K. A-Ameur, F. Treussart, R. Hierle, and J. F. Roch, “Simple interferometric technique for generation of a radially polarized light beam,” J. Opt. Soc. Am. A 22, 984–991 (2005). [CrossRef]  

41. X. L. Wang, J. Ding, W. J. Ni, C. S. Guo, and H. T. Wang, “Generation of arbitrary vector beams with a spatial light modulator and a common path interferometric arrangement,” Opt. Lett. 32, 3549–3551 (2007). [CrossRef]   [PubMed]  

42. T. Grosjean, D. Courjon, and M. Spajer, “An all-fiber device for generating radially and other polarized light beams,” Opt. Commun. 203, 1–5 (2002). [CrossRef]  

43. G. Volpe and D. Petrov, “Generation of cylindrical vector beams with few-mode fibers excited by Laguerre–Gaussian beams,” Opt. Commun. 237, 89–95 (2004). [CrossRef]  

44. T. Hirayama, Y. Kozawa, T. Nakamura, and S. Sato, “Generation of a cylindrically symmetric, polarized laser beam with narrow linewidth and fine tunability,” Opt. Express 14, 12839–12845 (2006). [CrossRef]   [PubMed]  

45. C. Sun and C. Liu, “Ultrasmall focusing spot with a long depth of focus based on polarization and phase modulation,” Opt. Lett. 28, 99–101 (2003). [CrossRef]   [PubMed]  

46. W. Chen and Q. Zhan, “Three-dimensional focus shaping with cylindrical vector beams,” Opt. Commun. 265, 411–417 (2006). [CrossRef]  

47. E. Wolf, “Electromagnetic diffraction in optical systems I. An integral representation of the image field,” Proc. R. Soc. London Ser. A 253, 349–357 (1959). [CrossRef]  

48. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959). [CrossRef]  

49. K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical vector beams,” Opt. Express 7, 77–87 (2000). [CrossRef]   [PubMed]  

50. M. Gu, ed., Advanced Optical Imaging Theory, Vol. 75 of Springer Series in Optical Sciences (Springer-Verlag, 1999).

51. G. M. Lerman and U. Levy, “Effect of radial polarization and apodization on spot size under tight focusing conditions,” Opt. Express 16, 4567–4581 (2008). [CrossRef]   [PubMed]  

52. C. J. R. Sheppard and A. Choudhury, “Annular pupils, radial polarization, and superresolution,” Appl. Opt. 43, 4322–4327 (2004). [CrossRef]   [PubMed]  

53. S. F. Pereira and A. S. van de Nes, “Superresolution by means of polarisation, phase and amplitude pupil masks,” Opt. Commun. 234, 119–124 (2004). [CrossRef]  

54. J. Stadler, C. Stanciu, C. Stupperich, and A. J. Meixner, “Tighter focusing with a parabolic mirror,” Opt. Lett. 33, 681–683 (2008). [CrossRef]   [PubMed]  

55. D. P. Biss and T. G. Brown, “Polarization-vortex-driven second-harmonic generation,” Opt. Lett. 28, 923–925 (2003). [CrossRef]   [PubMed]  

56. S. Carrasco, B. E. A. Saleh, M. C. Teich, and J. T. Fourkas, “Second- and third-harmonic generation with vector Gaussian beams,” J. Opt. Soc. Am. B 23, 2134–2141 (2006). [CrossRef]  

57. D. P. Biss, K. S. Youngworth, and T. G. Brown, “Dark-field imaging with cylindrical-vector beams,” Appl. Opt. 45, 470–479 (2006). [CrossRef]   [PubMed]  

58. N. Bokor and N. Davidson, “Toward a spherical spot distribution with 4π focusing of radially polarized light,” Opt. Lett. 29, 1968–1970 (2004). [CrossRef]   [PubMed]  

59. L. E. Helseth, “Roles of polarization, phase and amplitude in solid immersion lens systems,” Opt. Commun. 191, 161–172 (2001). [CrossRef]  

60. L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001). [CrossRef]   [PubMed]  

61. N. Bokor and N. Davidson, “Generation of a hollow dark spherical spot by 4π focusing of a radially polarized Laguerre–Gaussian beam,” Opt. Lett. 31, 149–151 (2006). [CrossRef]   [PubMed]  

62. S. Hell and J. Wichmann, “Breaking the diffraction resolution limit by stimulated emission: stimulated-emission depletion fluorescence microscopy,” Opt. Lett. 19, 780–782 (1994). [CrossRef]   [PubMed]  

63. P. Török and P. Munro, “The use of Gauss–Laguerre vector beams in STED microscopy,” Opt. Express 12, 3605–3617 (2004). [CrossRef]  

64. H. F. Wang, L. P. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2, 501–505 (2008). [CrossRef]  

65. A. F. Abouraddy and K. C. Toussaint Jr., “Three-dimensional polarization control in microscopy,” Phys. Rev. Lett. 96, 153901 (2006). [CrossRef]   [PubMed]  

66. T. D. Visser and J. T. Foley, “On the wavefront spacing of focused, radially polarized beams,” J. Opt. Soc. Am. A 22, 2527–2531 (2005). [CrossRef]  

67. H. Chen, Q. Zhan, Y. L. Zhang, and Y. P. Li, “The Gouy phase shift of the highly focused radially polarized beam,” Phys. Lett. A 371, 259–261 (2007). [CrossRef]  

68. L. G. Gouy, “Sur une propriété nouvelle des ondes lumineuses,” C. R. Acad. Sci. Paris. 110, 1251–1253 (1890).

69. Q. Zhan, “Second-order tilted wave interpretation of the Gouy phase shift under high numerical aperture uniform illumination,” Opt. Commun. 242, 351–360 (2004). [CrossRef]  

70. J. Cheng and X. S. Xie, “Green’s function formulation for third-harmonic generation microscopy,” J. Opt. Soc. Am. B 19, 1604–1610 (2002). [CrossRef]  

71. J. Cheng, A. Volkmer, and X. S. Xie, “Theoretical and experimental characterization of coherent anti-Stokes Raman scattering microscopy,” J. Opt. Soc. Am. B 19, 1363–1375 (2002). [CrossRef]  

72. Y. Yang and Q. Zhan, “Third-harmonic generation microscopy with tightly focused radial polarization,” J. Opt. A 10, 125103 (2008). [CrossRef]  

73. H. Raether, Surface Plasmons on Smooth and Rough Surfaces and Gratings (Spinger-Verlag, 1988).

74. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998). [CrossRef]  

75. N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limit imaging with a silver superlens,” Science 308, 534–537 (2005). [CrossRef]   [PubMed]  

76. X. Luo and T. Ishihara, “Surface plasmon resonant interference nanolithography technique,” Appl. Phys. Lett. 84, 4780–4782 (2004). [CrossRef]  

77. Z. Liu, J. M. Steele, W. Srituravanich, Y. Pikus, C. Sun, and X. Zhang, “Focusing surface plasmons with a plasmonic lens,” Nano Lett. 5, 1726–1729 (2005). [CrossRef]   [PubMed]  

78. L. Yin, V. K. Vlasko-Vlasov, J. Pearson, J. M. Hiller, J. Hua, U. Welp, D. E. Brown, and C. W. Kimball, “Subwavelength focusing and guiding of surface plasmons,” Nano Lett. 5, 1399–1402 (2005). [CrossRef]   [PubMed]  

79. W. Chen and Q. Zhan, “Optimal plasmonic focusing with radial polarization,” Proc. SPIE 6450, 64500D (2007). [CrossRef]  

80. H. Kano, S. Mizuguchi, and S. Kawata, “Excitation of surface-plasmon polaritons by a focused laser beam,” J. Opt. Soc. Am. B 15, 1381–1386 (1998). [CrossRef]  

81. Q. Zhan, “Evanescent Bessel beam generation via surface plasmon resonance by radially polarized beam,” Opt. Lett. 31, 1726–1728 (2006). [CrossRef]   [PubMed]  

82. W. Chen and Q. Zhan, “Realization of evanescent Bessel beam via surface plasmon interference excited by radially polarized beam” Opt. Lett. (to be published).

83. A. Bouhelier, F. Ignatovich, A. Bruyant, C. Huang, G. Colas des Francs, J.-C. Weeber, A. Dereux, G. P. Wiederrecht, and L. Novotny, “Surface plasmon interference excited by tightly focused laser beams,” Opt. Lett. 32, 2535–2537 (2007). [CrossRef]   [PubMed]  

84. A. Bouhelier, J. Renger, M. R. Beversluis, and L. Novotny, “Plasmon-coupled tip-enhanced near-field optical microscopy,” J. Microsc. 210, 220–224 (2003). [CrossRef]   [PubMed]  

85. W. Chen and Q. Zhan, “Numerical study of an apertureless near field scanning optical microscope probe under radial polarization illumination,” Opt. Express 15, 4106–4111 (2007). [CrossRef]   [PubMed]  

86. W. Chen and Q. Zhan, “Field enhancement analysis of an apertureless near field scanning optical microscope probe with finite element method,” Chin. Opt. Lett. 5, 709–711 (2007).

87. Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12, 3377–3382 (2004). [CrossRef]   [PubMed]  

88. H. Kawauchi, K. Yonezawa, Y. Kozawa, and S. Sato, “Calculation of optical trapping forces on a dielectric sphere in the ray optics regime produced by a radially polarized laser beam,” Opt. Lett. 32, 1839–1841 (2007). [CrossRef]   [PubMed]  

89. Y. Q. Zhao, Q. Zhan, Y. L. Zhang, and Y. P. Li, “Creation of a three-dimensional optical chain for controllable particle delivery,” Opt. Lett. 30, 848–850 (2005). [CrossRef]   [PubMed]  

90. Q. Zhan, “Optical radiation forces on a dielectric sphere produced by highly focused cylindrical vector beams,” J. Opt. A, Pure Appl. Opt. 5, 229–232 (2003). [CrossRef]  

91. Q. Zhan, “Trapping nanoparticles with cylindrical polarization,” Proc. SPIE 5514, 275–282 (2004). [CrossRef]  

92. J. R. Zurita-Sanchez and L. Novotny, “Multipolar interband absorption in a semiconductor quantum dot. II. Magnetic dipole enhancement,” J. Opt. Soc. Am. B 19, 2722–2726 (2002). [CrossRef]  

93. E. M. Furst and A. P. Gast, “Micromechanics of dipolar chains using optical tweezers,” Phys. Rev. Lett. 82, 4130–4133 (1999). [CrossRef]  

94. Y. B. Du and P. Tong, “Light scattering properties of paramagnetic particles,” J. Chem. Phys. 107, 355–362 (1997). [CrossRef]  

95. V. G. Niziev and A. V. Nesterov, “Influence of beam polarization on laser cutting efficiency,” J. Phys. D 32, 1455–1461 (1999). [CrossRef]  

96. M. Meier, V. Romano, and T. Feurer, “Material processing with pulsed radially and azimuthally polarized laser radiation,” Appl. Phys. A 86, 329–334 (2007). [CrossRef]  

97. Y. Cai, Q. Lin, H. T. Eyyuboglu, and Y. Baykal, “Average irradiance and polarization properties of a radially or azimuthally polarized beam in a turbulent atmosphere,” Opt. Express 16, 7665–7673 (2008). [CrossRef]   [PubMed]  

98. W. Cheng, J. W. Haus, and Q. Zhan, “Propagation of scalar and vector vortex beams through turbulent atmosphere,” Proc. SPIE7200 (to be published).

99. J. W. Haus, Z. Mozumder, and Q. Zhan, “Azimuthal modulation instability for a cylindrically polarized wave in a nonlinear Kerr Medium,” Opt. Express 14, 4757–4764 (2006). [CrossRef]   [PubMed]  

100. G. Chang, C. J. Divin, C. H. Liu, S. L. Williamson, A. Galvanauskas, and T. B. Norris, “Generation of radially polarized terahertz pulses via velocity-mismatched optical rectification,” Opt. Lett. 32, 433–435 (2007). [CrossRef]   [PubMed]  

101. Q. Cao and J. Jahns, “Azimuthally polarized surface plasmons as effective terahertz waveguides,” Opt. Express 13, 511–518 (2005). [CrossRef]   [PubMed]  

102. Q. Zhan, “Properties of circularly polarization vortex beams,” Opt. Lett. 31, 867–869 (2006). [CrossRef]   [PubMed]  

103. F. Gori, “Polarization basis for vortex beams,” J. Opt. Soc. Am. A 18, 1612–1617 (2001). [CrossRef]  

104. R. Borghi and M. Santarsiero, “Nonparaxial propagation of spirally polarized optical beams,” J. Opt. Soc. Am. A 21, 2029–2037 (2004). [CrossRef]  

105. R. Borghi, M. Santarsiero, and M. A. Alonso, “Highly focused spirally polarized beams,” J. Opt. Soc. Am. A 22, 1420–1431 (2005). [CrossRef]  

References

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  1. D. Pohl, “Operation of a Ruby laser in the purely transverse electric mode TE01,” Appl. Phys. Lett. 20, 266–267 (1972).
    [Crossref]
  2. Y. Mushiake, K. Matzumurra, N. Nakajima, “Generation of radially polarized optical beam mode by laser oscillation,” Proc. IEEE 60, 1107–1109 (1972).
    [Crossref]
  3. S. Quabis, R. Dorn, M. Eberler, O. Glöckl, G. Leuchs, “Focusing light into a tighter spot,” Opt. Commun. 179, 1–7 (2000).
    [Crossref]
  4. R. Dorn, S. Quabis, G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
    [Crossref] [PubMed]
  5. B. Jia, X. Gan, M. Gu, “Direct measurement of a radially polarized focused evanescent field facilitated by a single LCD,” Opt. Express 13, 6821–6827 (2005).
    [Crossref] [PubMed]
  6. B. Hao, J. Leger, “Experimental measurement of longitudinal component in the vicinity of focused radially polarized beam,” Opt. Express 15, 3550–3556 (2007).
    [Crossref] [PubMed]
  7. D. G. Hall, “Vector-beam solutions of Maxwell’s wave equation,” Opt. Lett. 21, 9–11 (1996).
    [Crossref] [PubMed]
  8. For example, J. M. Senior, Optical Fiber Communications (Prentice Hall, 1992).
  9. F. Gori, G. Guattari, C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64, 491–495 (1987).
    [Crossref]
  10. B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley, 2007).
  11. L. Novotny, B. Hecht, Principles of Nano-Optics (Cambridge U. Press, 2006).
    [Crossref]
  12. Q. Zhan, J. R. Leger, “Focus shaping using cylindrical vector beams,” Opt. Express 10, 324–331 (2002).
    [Crossref] [PubMed]
  13. M. E. Marhic, E. Garmire, “Low-order TE0q operation of a CO2 laser for transmission through circular metallic waveguides,” Appl. Phys. Lett. 38, 743–745 (1981).
    [Crossref]
  14. T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. Anderson, M. J. Rooks, “Circularly symmetric operation of a concentric-circle-grating, surface-emitting, AlGaAs∕GaAs quantum-well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
    [Crossref]
  15. K. Yonezawa, Y. Kozawa, S. Sato, “Generation of a radially polarized laser beam by use of the birefringence of a c-cut Nd:YVO4 crystal,” Opt. Lett. 31, 2151–2153 (2006).
    [Crossref] [PubMed]
  16. G. Machavariani, Y. Lumer, I. Moshe, A. Meir, S. Jackel, N. Davidson, “Birefringence-induced bifocusing for selection of radially or azimuthally polarized laser modes,” Appl. Opt. 46, 3304–3310 (2007).
    [Crossref] [PubMed]
  17. K. Yonezawa, Y. Kozawa, S. Sato, “Compact laser with radial polarization using birefringent laser medium,” Jpn. J. Appl. Phys., Part 1 46, 5160–5163 (2007).
    [Crossref]
  18. J. F. Bisson, J. Li, K. Ueda, Y. Senatsky, “Radially polarized ring and arc beams of a neodymium laser with an intra-cavity axicon,” Opt. Express 14, 3304–3311 (2006).
    [Crossref] [PubMed]
  19. C.-C. Shih, “Radial polarization laser resonator,” U.S. patent 5,359,622 (Oct. 25, 1994).
  20. Y. Kozawa, S. Sato, “Generation of a radially polarized laser beam by use of a conical Brewster prism,” Opt. Lett. 30, 3063–3065 (2005).
    [Crossref] [PubMed]
  21. M. A. Ahmed, A. Voss, M. M. Vogel, T. Graf, “Multilayer polarizing grating mirror used for the generation of radial polarization in Yb:YAG thin-disk lasers,” Opt. Lett. 32, 3272–3274 (2007).
    [Crossref] [PubMed]
  22. V. G. Niziev, R. S. Chang, A. V. Nesterov, “Generation of inhomogeneously polarized laser beams by use of a Sagnac interferometer,” Appl. Opt. 45, 8393–8399 (2006).
    [Crossref] [PubMed]
  23. Q. Zhan, J. R. Leger, “Microellipsometer with radial symmetry,” Appl. Opt. 41, 4630–4637 (2002).
    [Crossref] [PubMed]
  24. Q. Zhan, J. R. Leger, “Interferometric measurement of Berry’s phase in space-variant polarization manipulations,” Opt. Commun. 213, 241–245 (2002).
    [Crossref]
  25. C. Rotschild, S. Zommer, S. Moed, O. Hershcovitz, S. G. Lipson, “Adjustable spiral phase plate,” Appl. Opt. 43, 2397–2399 (2004).
    [Crossref] [PubMed]
  26. For example, the vortex phase plates from RPC Photonics (http://www.rpcphotonics.com/).
  27. Q. Zhan, “Cylindrical polarization symmetry for nondestructive nano-characterization,” Proc. SPIE 5045, 85–92 (2003).
    [Crossref]
  28. K. J. Moh, X. C. Yuan, J. Bu, D. K. Y. Low, R. E. Burge, “Direct noninterference cylindrical vector beam generation applied in the femtosecond regime,” Appl. Phys. Lett. 89, 251114 (2006).
    [Crossref]
  29. R. Yamaguchi, T. Nose, S. Sato, “Liquid crystal polarizers with axially symmetrical properties,” Jpn. J. Appl. Phys. Part 1 28, 1730–1731 (1989).
    [Crossref]
  30. M. Stalder, M. Schadt, “Linearly polarized light with axial symmetry generated by liquid-crystal polarization converters,” Opt. Lett. 21, 1948–1950 (1996).
    [Crossref] [PubMed]
  31. For example, the ZPol from Nanophoton (http://www.nanophoton.jp/) and the radial polarizer from ARCoptix (http://www.arcoptix.com/).
  32. M. R. Beversluis, L. Novotny, S. J. Stranick, “Programmable vector point-spread function engineering,” Opt. Express 14, 2650–2656 (2006).
    [Crossref] [PubMed]
  33. C. Ye, “Construction of an optical rotator using quarter-wave plates and an optical retarder,” Opt. Eng. 34, 3031–3035 (1995).
    [Crossref]
  34. G. Machavariani, Y. Lumer, I. Moshe, A. Meir, S. Jackel, “Spatially-variable retardation plate for efficient generation of radially- and azimuthally-polarized beams,” Opt. Commun. 281, 732–738 (2008).
    [Crossref]
  35. A. K. Spilman, T. G. Brown, “Stress birefringent, space-variant wave plates for vortex illumination,” Appl. Opt. 46, 61–66 (2007).
    [Crossref]
  36. S. C. McEldowney, D. M. Shemo, R. A. Chipman, “Vortex retarders produced from photo-aligned liquid crystal polymers,” Opt. Express 16, 7295–7308 (2008).
    [Crossref] [PubMed]
  37. Z. Bomzon, G. Biener, V. Kleiner, E. Hasman, “Radially and azimuthally polarized beams generated by space-variant dielectric subwavelength gratings,” Opt. Lett. 27, 285–287 (2002).
    [Crossref]
  38. B. C. Lim, P. B. Phua, W. J. Lai, M. H. Hong, “Fast switchable electro-optic radial polarization retarder,” Opt. Lett. 33, 950–952 (2008).
    [Crossref] [PubMed]
  39. S. C. Tidwell, D. H. Ford, W. D. Kimura, “Generating radially polarized beams interferometrically,” Appl. Opt. 29, 2234–2239 (1990).
    [Crossref] [PubMed]
  40. N. Passilly, R. de Saint Denis, K. A-Ameur, F. Treussart, R. Hierle, J. F. Roch, “Simple interferometric technique for generation of a radially polarized light beam,” J. Opt. Soc. Am. A 22, 984–991 (2005).
    [Crossref]
  41. X. L. Wang, J. Ding, W. J. Ni, C. S. Guo, H. T. Wang, “Generation of arbitrary vector beams with a spatial light modulator and a common path interferometric arrangement,” Opt. Lett. 32, 3549–3551 (2007).
    [Crossref] [PubMed]
  42. T. Grosjean, D. Courjon, M. Spajer, “An all-fiber device for generating radially and other polarized light beams,” Opt. Commun. 203, 1–5 (2002).
    [Crossref]
  43. G. Volpe, D. Petrov, “Generation of cylindrical vector beams with few-mode fibers excited by Laguerre–Gaussian beams,” Opt. Commun. 237, 89–95 (2004).
    [Crossref]
  44. T. Hirayama, Y. Kozawa, T. Nakamura, S. Sato, “Generation of a cylindrically symmetric, polarized laser beam with narrow linewidth and fine tunability,” Opt. Express 14, 12839–12845 (2006).
    [Crossref] [PubMed]
  45. C. Sun, C. Liu, “Ultrasmall focusing spot with a long depth of focus based on polarization and phase modulation,” Opt. Lett. 28, 99–101 (2003).
    [Crossref] [PubMed]
  46. W. Chen, Q. Zhan, “Three-dimensional focus shaping with cylindrical vector beams,” Opt. Commun. 265, 411–417 (2006).
    [Crossref]
  47. E. Wolf, “Electromagnetic diffraction in optical systems I. An integral representation of the image field,” Proc. R. Soc. London Ser. A 253, 349–357 (1959).
    [Crossref]
  48. B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
    [Crossref]
  49. K. S. Youngworth, T. G. Brown, “Focusing of high numerical aperture cylindrical vector beams,” Opt. Express 7, 77–87 (2000).
    [Crossref] [PubMed]
  50. M. Gu, ed., Advanced Optical Imaging Theory, Vol. 75 of Springer Series in Optical Sciences (Springer-Verlag, 1999).
  51. G. M. Lerman, U. Levy, “Effect of radial polarization and apodization on spot size under tight focusing conditions,” Opt. Express 16, 4567–4581 (2008).
    [Crossref] [PubMed]
  52. C. J. R. Sheppard, A. Choudhury, “Annular pupils, radial polarization, and superresolution,” Appl. Opt. 43, 4322–4327 (2004).
    [Crossref] [PubMed]
  53. S. F. Pereira, A. S. van de Nes, “Superresolution by means of polarisation, phase and amplitude pupil masks,” Opt. Commun. 234, 119–124 (2004).
    [Crossref]
  54. J. Stadler, C. Stanciu, C. Stupperich, A. J. Meixner, “Tighter focusing with a parabolic mirror,” Opt. Lett. 33, 681–683 (2008).
    [Crossref] [PubMed]
  55. D. P. Biss, T. G. Brown, “Polarization-vortex-driven second-harmonic generation,” Opt. Lett. 28, 923–925 (2003).
    [Crossref] [PubMed]
  56. S. Carrasco, B. E. A. Saleh, M. C. Teich, J. T. Fourkas, “Second- and third-harmonic generation with vector Gaussian beams,” J. Opt. Soc. Am. B 23, 2134–2141 (2006).
    [Crossref]
  57. D. P. Biss, K. S. Youngworth, T. G. Brown, “Dark-field imaging with cylindrical-vector beams,” Appl. Opt. 45, 470–479 (2006).
    [Crossref] [PubMed]
  58. N. Bokor, N. Davidson, “Toward a spherical spot distribution with 4π focusing of radially polarized light,” Opt. Lett. 29, 1968–1970 (2004).
    [Crossref] [PubMed]
  59. L. E. Helseth, “Roles of polarization, phase and amplitude in solid immersion lens systems,” Opt. Commun. 191, 161–172 (2001).
    [Crossref]
  60. L. Novotny, M. R. Beversluis, K. S. Youngworth, T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
    [Crossref] [PubMed]
  61. N. Bokor, N. Davidson, “Generation of a hollow dark spherical spot by 4π focusing of a radially polarized Laguerre–Gaussian beam,” Opt. Lett. 31, 149–151 (2006).
    [Crossref] [PubMed]
  62. S. Hell, J. Wichmann, “Breaking the diffraction resolution limit by stimulated emission: stimulated-emission depletion fluorescence microscopy,” Opt. Lett. 19, 780–782 (1994).
    [Crossref] [PubMed]
  63. P. Török, P. Munro, “The use of Gauss–Laguerre vector beams in STED microscopy,” Opt. Express 12, 3605–3617 (2004).
    [Crossref]
  64. H. F. Wang, L. P. Shi, B. Lukyanchuk, C. Sheppard, C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2, 501–505 (2008).
    [Crossref]
  65. A. F. Abouraddy, K. C. Toussaint, “Three-dimensional polarization control in microscopy,” Phys. Rev. Lett. 96, 153901 (2006).
    [Crossref] [PubMed]
  66. T. D. Visser, J. T. Foley, “On the wavefront spacing of focused, radially polarized beams,” J. Opt. Soc. Am. A 22, 2527–2531 (2005).
    [Crossref]
  67. H. Chen, Q. Zhan, Y. L. Zhang, Y. P. Li, “The Gouy phase shift of the highly focused radially polarized beam,” Phys. Lett. A 371, 259–261 (2007).
    [Crossref]
  68. L. G. Gouy, “Sur une propriété nouvelle des ondes lumineuses,” C. R. Acad. Sci. Paris. 110, 1251–1253 (1890).
  69. Q. Zhan, “Second-order tilted wave interpretation of the Gouy phase shift under high numerical aperture uniform illumination,” Opt. Commun. 242, 351–360 (2004).
    [Crossref]
  70. J. Cheng, X. S. Xie, “Green’s function formulation for third-harmonic generation microscopy,” J. Opt. Soc. Am. B 19, 1604–1610 (2002).
    [Crossref]
  71. J. Cheng, A. Volkmer, X. S. Xie, “Theoretical and experimental characterization of coherent anti-Stokes Raman scattering microscopy,” J. Opt. Soc. Am. B 19, 1363–1375 (2002).
    [Crossref]
  72. Y. Yang, Q. Zhan, “Third-harmonic generation microscopy with tightly focused radial polarization,” J. Opt. A 10, 125103 (2008).
    [Crossref]
  73. H. Raether, Surface Plasmons on Smooth and Rough Surfaces and Gratings (Spinger-Verlag, 1988).
  74. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
    [Crossref]
  75. N. Fang, H. Lee, C. Sun, X. Zhang, “Sub-diffraction-limit imaging with a silver superlens,” Science 308, 534–537 (2005).
    [Crossref] [PubMed]
  76. X. Luo, T. Ishihara, “Surface plasmon resonant interference nanolithography technique,” Appl. Phys. Lett. 84, 4780–4782 (2004).
    [Crossref]
  77. Z. Liu, J. M. Steele, W. Srituravanich, Y. Pikus, C. Sun, X. Zhang, “Focusing surface plasmons with a plasmonic lens,” Nano Lett. 5, 1726–1729 (2005).
    [Crossref] [PubMed]
  78. L. Yin, V. K. Vlasko-Vlasov, J. Pearson, J. M. Hiller, J. Hua, U. Welp, D. E. Brown, C. W. Kimball, “Subwavelength focusing and guiding of surface plasmons,” Nano Lett. 5, 1399–1402 (2005).
    [Crossref] [PubMed]
  79. W. Chen, Q. Zhan, “Optimal plasmonic focusing with radial polarization,” Proc. SPIE 6450, 64500D (2007).
    [Crossref]
  80. H. Kano, S. Mizuguchi, S. Kawata, “Excitation of surface-plasmon polaritons by a focused laser beam,” J. Opt. Soc. Am. B 15, 1381–1386 (1998).
    [Crossref]
  81. Q. Zhan, “Evanescent Bessel beam generation via surface plasmon resonance by radially polarized beam,” Opt. Lett. 31, 1726–1728 (2006).
    [Crossref] [PubMed]
  82. W. Chen, Q. Zhan, “Realization of evanescent Bessel beam via surface plasmon interference excited by radially polarized beam” Opt. Lett. (to be published).
  83. A. Bouhelier, F. Ignatovich, A. Bruyant, C. Huang, G. Colas des Francs, J.-C. Weeber, A. Dereux, G. P. Wiederrecht, L. Novotny, “Surface plasmon interference excited by tightly focused laser beams,” Opt. Lett. 32, 2535–2537 (2007).
    [Crossref] [PubMed]
  84. A. Bouhelier, J. Renger, M. R. Beversluis, L. Novotny, “Plasmon-coupled tip-enhanced near-field optical microscopy,” J. Microsc. 210, 220–224 (2003).
    [Crossref] [PubMed]
  85. W. Chen, Q. Zhan, “Numerical study of an apertureless near field scanning optical microscope probe under radial polarization illumination,” Opt. Express 15, 4106–4111 (2007).
    [Crossref] [PubMed]
  86. W. Chen, Q. Zhan, “Field enhancement analysis of an apertureless near field scanning optical microscope probe with finite element method,” Chin. Opt. Lett. 5, 709–711 (2007).
  87. Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12, 3377–3382 (2004).
    [Crossref] [PubMed]
  88. H. Kawauchi, K. Yonezawa, Y. Kozawa, S. Sato, “Calculation of optical trapping forces on a dielectric sphere in the ray optics regime produced by a radially polarized laser beam,” Opt. Lett. 32, 1839–1841 (2007).
    [Crossref] [PubMed]
  89. Y. Q. Zhao, Q. Zhan, Y. L. Zhang, Y. P. Li, “Creation of a three-dimensional optical chain for controllable particle delivery,” Opt. Lett. 30, 848–850 (2005).
    [Crossref] [PubMed]
  90. Q. Zhan, “Optical radiation forces on a dielectric sphere produced by highly focused cylindrical vector beams,” J. Opt. A, Pure Appl. Opt. 5, 229–232 (2003).
    [Crossref]
  91. Q. Zhan, “Trapping nanoparticles with cylindrical polarization,” Proc. SPIE 5514, 275–282 (2004).
    [Crossref]
  92. J. R. Zurita-Sanchez, L. Novotny, “Multipolar interband absorption in a semiconductor quantum dot. II. Magnetic dipole enhancement,” J. Opt. Soc. Am. B 19, 2722–2726 (2002).
    [Crossref]
  93. E. M. Furst, A. P. Gast, “Micromechanics of dipolar chains using optical tweezers,” Phys. Rev. Lett. 82, 4130–4133 (1999).
    [Crossref]
  94. Y. B. Du, P. Tong, “Light scattering properties of paramagnetic particles,” J. Chem. Phys. 107, 355–362 (1997).
    [Crossref]
  95. V. G. Niziev, A. V. Nesterov, “Influence of beam polarization on laser cutting efficiency,” J. Phys. D 32, 1455–1461 (1999).
    [Crossref]
  96. M. Meier, V. Romano, T. Feurer, “Material processing with pulsed radially and azimuthally polarized laser radiation,” Appl. Phys. A 86, 329–334 (2007).
    [Crossref]
  97. Y. Cai, Q. Lin, H. T. Eyyuboglu, Y. Baykal, “Average irradiance and polarization properties of a radially or azimuthally polarized beam in a turbulent atmosphere,” Opt. Express 16, 7665–7673 (2008).
    [Crossref] [PubMed]
  98. W. Cheng, J. W. Haus, Q. Zhan, “Propagation of scalar and vector vortex beams through turbulent atmosphere,” Proc. SPIE7200 (to be published).
  99. J. W. Haus, Z. Mozumder, Q. Zhan, “Azimuthal modulation instability for a cylindrically polarized wave in a nonlinear Kerr Medium,” Opt. Express 14, 4757–4764 (2006).
    [Crossref] [PubMed]
  100. G. Chang, C. J. Divin, C. H. Liu, S. L. Williamson, A. Galvanauskas, T. B. Norris, “Generation of radially polarized terahertz pulses via velocity-mismatched optical rectification,” Opt. Lett. 32, 433–435 (2007).
    [Crossref] [PubMed]
  101. Q. Cao, J. Jahns, “Azimuthally polarized surface plasmons as effective terahertz waveguides,” Opt. Express 13, 511–518 (2005).
    [Crossref] [PubMed]
  102. Q. Zhan, “Properties of circularly polarization vortex beams,” Opt. Lett. 31, 867–869 (2006).
    [Crossref] [PubMed]
  103. F. Gori, “Polarization basis for vortex beams,” J. Opt. Soc. Am. A 18, 1612–1617 (2001).
    [Crossref]
  104. R. Borghi, M. Santarsiero, “Nonparaxial propagation of spirally polarized optical beams,” J. Opt. Soc. Am. A 21, 2029–2037 (2004).
    [Crossref]
  105. R. Borghi, M. Santarsiero, M. A. Alonso, “Highly focused spirally polarized beams,” J. Opt. Soc. Am. A 22, 1420–1431 (2005).
    [Crossref]

2008 (8)

2007 (14)

W. Chen, Q. Zhan, “Optimal plasmonic focusing with radial polarization,” Proc. SPIE 6450, 64500D (2007).
[Crossref]

A. Bouhelier, F. Ignatovich, A. Bruyant, C. Huang, G. Colas des Francs, J.-C. Weeber, A. Dereux, G. P. Wiederrecht, L. Novotny, “Surface plasmon interference excited by tightly focused laser beams,” Opt. Lett. 32, 2535–2537 (2007).
[Crossref] [PubMed]

W. Chen, Q. Zhan, “Numerical study of an apertureless near field scanning optical microscope probe under radial polarization illumination,” Opt. Express 15, 4106–4111 (2007).
[Crossref] [PubMed]

W. Chen, Q. Zhan, “Field enhancement analysis of an apertureless near field scanning optical microscope probe with finite element method,” Chin. Opt. Lett. 5, 709–711 (2007).

H. Kawauchi, K. Yonezawa, Y. Kozawa, S. Sato, “Calculation of optical trapping forces on a dielectric sphere in the ray optics regime produced by a radially polarized laser beam,” Opt. Lett. 32, 1839–1841 (2007).
[Crossref] [PubMed]

G. Chang, C. J. Divin, C. H. Liu, S. L. Williamson, A. Galvanauskas, T. B. Norris, “Generation of radially polarized terahertz pulses via velocity-mismatched optical rectification,” Opt. Lett. 32, 433–435 (2007).
[Crossref] [PubMed]

M. Meier, V. Romano, T. Feurer, “Material processing with pulsed radially and azimuthally polarized laser radiation,” Appl. Phys. A 86, 329–334 (2007).
[Crossref]

H. Chen, Q. Zhan, Y. L. Zhang, Y. P. Li, “The Gouy phase shift of the highly focused radially polarized beam,” Phys. Lett. A 371, 259–261 (2007).
[Crossref]

X. L. Wang, J. Ding, W. J. Ni, C. S. Guo, H. T. Wang, “Generation of arbitrary vector beams with a spatial light modulator and a common path interferometric arrangement,” Opt. Lett. 32, 3549–3551 (2007).
[Crossref] [PubMed]

A. K. Spilman, T. G. Brown, “Stress birefringent, space-variant wave plates for vortex illumination,” Appl. Opt. 46, 61–66 (2007).
[Crossref]

B. Hao, J. Leger, “Experimental measurement of longitudinal component in the vicinity of focused radially polarized beam,” Opt. Express 15, 3550–3556 (2007).
[Crossref] [PubMed]

G. Machavariani, Y. Lumer, I. Moshe, A. Meir, S. Jackel, N. Davidson, “Birefringence-induced bifocusing for selection of radially or azimuthally polarized laser modes,” Appl. Opt. 46, 3304–3310 (2007).
[Crossref] [PubMed]

K. Yonezawa, Y. Kozawa, S. Sato, “Compact laser with radial polarization using birefringent laser medium,” Jpn. J. Appl. Phys., Part 1 46, 5160–5163 (2007).
[Crossref]

M. A. Ahmed, A. Voss, M. M. Vogel, T. Graf, “Multilayer polarizing grating mirror used for the generation of radial polarization in Yb:YAG thin-disk lasers,” Opt. Lett. 32, 3272–3274 (2007).
[Crossref] [PubMed]

2006 (14)

V. G. Niziev, R. S. Chang, A. V. Nesterov, “Generation of inhomogeneously polarized laser beams by use of a Sagnac interferometer,” Appl. Opt. 45, 8393–8399 (2006).
[Crossref] [PubMed]

J. F. Bisson, J. Li, K. Ueda, Y. Senatsky, “Radially polarized ring and arc beams of a neodymium laser with an intra-cavity axicon,” Opt. Express 14, 3304–3311 (2006).
[Crossref] [PubMed]

M. R. Beversluis, L. Novotny, S. J. Stranick, “Programmable vector point-spread function engineering,” Opt. Express 14, 2650–2656 (2006).
[Crossref] [PubMed]

K. J. Moh, X. C. Yuan, J. Bu, D. K. Y. Low, R. E. Burge, “Direct noninterference cylindrical vector beam generation applied in the femtosecond regime,” Appl. Phys. Lett. 89, 251114 (2006).
[Crossref]

K. Yonezawa, Y. Kozawa, S. Sato, “Generation of a radially polarized laser beam by use of the birefringence of a c-cut Nd:YVO4 crystal,” Opt. Lett. 31, 2151–2153 (2006).
[Crossref] [PubMed]

T. Hirayama, Y. Kozawa, T. Nakamura, S. Sato, “Generation of a cylindrically symmetric, polarized laser beam with narrow linewidth and fine tunability,” Opt. Express 14, 12839–12845 (2006).
[Crossref] [PubMed]

W. Chen, Q. Zhan, “Three-dimensional focus shaping with cylindrical vector beams,” Opt. Commun. 265, 411–417 (2006).
[Crossref]

S. Carrasco, B. E. A. Saleh, M. C. Teich, J. T. Fourkas, “Second- and third-harmonic generation with vector Gaussian beams,” J. Opt. Soc. Am. B 23, 2134–2141 (2006).
[Crossref]

D. P. Biss, K. S. Youngworth, T. G. Brown, “Dark-field imaging with cylindrical-vector beams,” Appl. Opt. 45, 470–479 (2006).
[Crossref] [PubMed]

A. F. Abouraddy, K. C. Toussaint, “Three-dimensional polarization control in microscopy,” Phys. Rev. Lett. 96, 153901 (2006).
[Crossref] [PubMed]

N. Bokor, N. Davidson, “Generation of a hollow dark spherical spot by 4π focusing of a radially polarized Laguerre–Gaussian beam,” Opt. Lett. 31, 149–151 (2006).
[Crossref] [PubMed]

Q. Zhan, “Properties of circularly polarization vortex beams,” Opt. Lett. 31, 867–869 (2006).
[Crossref] [PubMed]

J. W. Haus, Z. Mozumder, Q. Zhan, “Azimuthal modulation instability for a cylindrically polarized wave in a nonlinear Kerr Medium,” Opt. Express 14, 4757–4764 (2006).
[Crossref] [PubMed]

Q. Zhan, “Evanescent Bessel beam generation via surface plasmon resonance by radially polarized beam,” Opt. Lett. 31, 1726–1728 (2006).
[Crossref] [PubMed]

2005 (10)

Y. Q. Zhao, Q. Zhan, Y. L. Zhang, Y. P. Li, “Creation of a three-dimensional optical chain for controllable particle delivery,” Opt. Lett. 30, 848–850 (2005).
[Crossref] [PubMed]

Q. Cao, J. Jahns, “Azimuthally polarized surface plasmons as effective terahertz waveguides,” Opt. Express 13, 511–518 (2005).
[Crossref] [PubMed]

R. Borghi, M. Santarsiero, M. A. Alonso, “Highly focused spirally polarized beams,” J. Opt. Soc. Am. A 22, 1420–1431 (2005).
[Crossref]

T. D. Visser, J. T. Foley, “On the wavefront spacing of focused, radially polarized beams,” J. Opt. Soc. Am. A 22, 2527–2531 (2005).
[Crossref]

N. Fang, H. Lee, C. Sun, X. Zhang, “Sub-diffraction-limit imaging with a silver superlens,” Science 308, 534–537 (2005).
[Crossref] [PubMed]

Z. Liu, J. M. Steele, W. Srituravanich, Y. Pikus, C. Sun, X. Zhang, “Focusing surface plasmons with a plasmonic lens,” Nano Lett. 5, 1726–1729 (2005).
[Crossref] [PubMed]

L. Yin, V. K. Vlasko-Vlasov, J. Pearson, J. M. Hiller, J. Hua, U. Welp, D. E. Brown, C. W. Kimball, “Subwavelength focusing and guiding of surface plasmons,” Nano Lett. 5, 1399–1402 (2005).
[Crossref] [PubMed]

N. Passilly, R. de Saint Denis, K. A-Ameur, F. Treussart, R. Hierle, J. F. Roch, “Simple interferometric technique for generation of a radially polarized light beam,” J. Opt. Soc. Am. A 22, 984–991 (2005).
[Crossref]

B. Jia, X. Gan, M. Gu, “Direct measurement of a radially polarized focused evanescent field facilitated by a single LCD,” Opt. Express 13, 6821–6827 (2005).
[Crossref] [PubMed]

Y. Kozawa, S. Sato, “Generation of a radially polarized laser beam by use of a conical Brewster prism,” Opt. Lett. 30, 3063–3065 (2005).
[Crossref] [PubMed]

2004 (11)

C. Rotschild, S. Zommer, S. Moed, O. Hershcovitz, S. G. Lipson, “Adjustable spiral phase plate,” Appl. Opt. 43, 2397–2399 (2004).
[Crossref] [PubMed]

G. Volpe, D. Petrov, “Generation of cylindrical vector beams with few-mode fibers excited by Laguerre–Gaussian beams,” Opt. Commun. 237, 89–95 (2004).
[Crossref]

N. Bokor, N. Davidson, “Toward a spherical spot distribution with 4π focusing of radially polarized light,” Opt. Lett. 29, 1968–1970 (2004).
[Crossref] [PubMed]

C. J. R. Sheppard, A. Choudhury, “Annular pupils, radial polarization, and superresolution,” Appl. Opt. 43, 4322–4327 (2004).
[Crossref] [PubMed]

S. F. Pereira, A. S. van de Nes, “Superresolution by means of polarisation, phase and amplitude pupil masks,” Opt. Commun. 234, 119–124 (2004).
[Crossref]

X. Luo, T. Ishihara, “Surface plasmon resonant interference nanolithography technique,” Appl. Phys. Lett. 84, 4780–4782 (2004).
[Crossref]

Q. Zhan, “Second-order tilted wave interpretation of the Gouy phase shift under high numerical aperture uniform illumination,” Opt. Commun. 242, 351–360 (2004).
[Crossref]

P. Török, P. Munro, “The use of Gauss–Laguerre vector beams in STED microscopy,” Opt. Express 12, 3605–3617 (2004).
[Crossref]

R. Borghi, M. Santarsiero, “Nonparaxial propagation of spirally polarized optical beams,” J. Opt. Soc. Am. A 21, 2029–2037 (2004).
[Crossref]

Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12, 3377–3382 (2004).
[Crossref] [PubMed]

Q. Zhan, “Trapping nanoparticles with cylindrical polarization,” Proc. SPIE 5514, 275–282 (2004).
[Crossref]

2003 (6)

Q. Zhan, “Optical radiation forces on a dielectric sphere produced by highly focused cylindrical vector beams,” J. Opt. A, Pure Appl. Opt. 5, 229–232 (2003).
[Crossref]

A. Bouhelier, J. Renger, M. R. Beversluis, L. Novotny, “Plasmon-coupled tip-enhanced near-field optical microscopy,” J. Microsc. 210, 220–224 (2003).
[Crossref] [PubMed]

D. P. Biss, T. G. Brown, “Polarization-vortex-driven second-harmonic generation,” Opt. Lett. 28, 923–925 (2003).
[Crossref] [PubMed]

C. Sun, C. Liu, “Ultrasmall focusing spot with a long depth of focus based on polarization and phase modulation,” Opt. Lett. 28, 99–101 (2003).
[Crossref] [PubMed]

Q. Zhan, “Cylindrical polarization symmetry for nondestructive nano-characterization,” Proc. SPIE 5045, 85–92 (2003).
[Crossref]

R. Dorn, S. Quabis, G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
[Crossref] [PubMed]

2002 (8)

2001 (3)

F. Gori, “Polarization basis for vortex beams,” J. Opt. Soc. Am. A 18, 1612–1617 (2001).
[Crossref]

L. E. Helseth, “Roles of polarization, phase and amplitude in solid immersion lens systems,” Opt. Commun. 191, 161–172 (2001).
[Crossref]

L. Novotny, M. R. Beversluis, K. S. Youngworth, T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[Crossref] [PubMed]

2000 (2)

K. S. Youngworth, T. G. Brown, “Focusing of high numerical aperture cylindrical vector beams,” Opt. Express 7, 77–87 (2000).
[Crossref] [PubMed]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, G. Leuchs, “Focusing light into a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[Crossref]

1999 (2)

E. M. Furst, A. P. Gast, “Micromechanics of dipolar chains using optical tweezers,” Phys. Rev. Lett. 82, 4130–4133 (1999).
[Crossref]

V. G. Niziev, A. V. Nesterov, “Influence of beam polarization on laser cutting efficiency,” J. Phys. D 32, 1455–1461 (1999).
[Crossref]

1998 (2)

H. Kano, S. Mizuguchi, S. Kawata, “Excitation of surface-plasmon polaritons by a focused laser beam,” J. Opt. Soc. Am. B 15, 1381–1386 (1998).
[Crossref]

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[Crossref]

1997 (1)

Y. B. Du, P. Tong, “Light scattering properties of paramagnetic particles,” J. Chem. Phys. 107, 355–362 (1997).
[Crossref]

1996 (2)

1995 (1)

C. Ye, “Construction of an optical rotator using quarter-wave plates and an optical retarder,” Opt. Eng. 34, 3031–3035 (1995).
[Crossref]

1994 (1)

1992 (1)

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. Anderson, M. J. Rooks, “Circularly symmetric operation of a concentric-circle-grating, surface-emitting, AlGaAs∕GaAs quantum-well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[Crossref]

1990 (1)

1989 (1)

R. Yamaguchi, T. Nose, S. Sato, “Liquid crystal polarizers with axially symmetrical properties,” Jpn. J. Appl. Phys. Part 1 28, 1730–1731 (1989).
[Crossref]

1987 (1)

F. Gori, G. Guattari, C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[Crossref]

1981 (1)

M. E. Marhic, E. Garmire, “Low-order TE0q operation of a CO2 laser for transmission through circular metallic waveguides,” Appl. Phys. Lett. 38, 743–745 (1981).
[Crossref]

1972 (2)

D. Pohl, “Operation of a Ruby laser in the purely transverse electric mode TE01,” Appl. Phys. Lett. 20, 266–267 (1972).
[Crossref]

Y. Mushiake, K. Matzumurra, N. Nakajima, “Generation of radially polarized optical beam mode by laser oscillation,” Proc. IEEE 60, 1107–1109 (1972).
[Crossref]

1959 (2)

E. Wolf, “Electromagnetic diffraction in optical systems I. An integral representation of the image field,” Proc. R. Soc. London Ser. A 253, 349–357 (1959).
[Crossref]

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[Crossref]

1890 (1)

L. G. Gouy, “Sur une propriété nouvelle des ondes lumineuses,” C. R. Acad. Sci. Paris. 110, 1251–1253 (1890).

A-Ameur, K.

N. Passilly, R. de Saint Denis, K. A-Ameur, F. Treussart, R. Hierle, J. F. Roch, “Simple interferometric technique for generation of a radially polarized light beam,” J. Opt. Soc. Am. A 22, 984–991 (2005).
[Crossref]

Abouraddy, A. F.

A. F. Abouraddy, K. C. Toussaint, “Three-dimensional polarization control in microscopy,” Phys. Rev. Lett. 96, 153901 (2006).
[Crossref] [PubMed]

Ahmed, M. A.

Alonso, M. A.

Anderson, E.

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. Anderson, M. J. Rooks, “Circularly symmetric operation of a concentric-circle-grating, surface-emitting, AlGaAs∕GaAs quantum-well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[Crossref]

Baykal, Y.

Beversluis, M. R.

M. R. Beversluis, L. Novotny, S. J. Stranick, “Programmable vector point-spread function engineering,” Opt. Express 14, 2650–2656 (2006).
[Crossref] [PubMed]

A. Bouhelier, J. Renger, M. R. Beversluis, L. Novotny, “Plasmon-coupled tip-enhanced near-field optical microscopy,” J. Microsc. 210, 220–224 (2003).
[Crossref] [PubMed]

L. Novotny, M. R. Beversluis, K. S. Youngworth, T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[Crossref] [PubMed]

Biener, G.

Biss, D. P.

Bisson, J. F.

J. F. Bisson, J. Li, K. Ueda, Y. Senatsky, “Radially polarized ring and arc beams of a neodymium laser with an intra-cavity axicon,” Opt. Express 14, 3304–3311 (2006).
[Crossref] [PubMed]

Bokor, N.

Bomzon, Z.

Borghi, R.

R. Borghi, M. Santarsiero, M. A. Alonso, “Highly focused spirally polarized beams,” J. Opt. Soc. Am. A 22, 1420–1431 (2005).
[Crossref]

R. Borghi, M. Santarsiero, “Nonparaxial propagation of spirally polarized optical beams,” J. Opt. Soc. Am. A 21, 2029–2037 (2004).
[Crossref]

Bouhelier, A.

Brown, D. E.

L. Yin, V. K. Vlasko-Vlasov, J. Pearson, J. M. Hiller, J. Hua, U. Welp, D. E. Brown, C. W. Kimball, “Subwavelength focusing and guiding of surface plasmons,” Nano Lett. 5, 1399–1402 (2005).
[Crossref] [PubMed]

Brown, T. G.

Bruyant, A.

Bu, J.

K. J. Moh, X. C. Yuan, J. Bu, D. K. Y. Low, R. E. Burge, “Direct noninterference cylindrical vector beam generation applied in the femtosecond regime,” Appl. Phys. Lett. 89, 251114 (2006).
[Crossref]

Burge, R. E.

K. J. Moh, X. C. Yuan, J. Bu, D. K. Y. Low, R. E. Burge, “Direct noninterference cylindrical vector beam generation applied in the femtosecond regime,” Appl. Phys. Lett. 89, 251114 (2006).
[Crossref]

Cai, Y.

Cao, Q.

Q. Cao, J. Jahns, “Azimuthally polarized surface plasmons as effective terahertz waveguides,” Opt. Express 13, 511–518 (2005).
[Crossref] [PubMed]

Carrasco, S.

S. Carrasco, B. E. A. Saleh, M. C. Teich, J. T. Fourkas, “Second- and third-harmonic generation with vector Gaussian beams,” J. Opt. Soc. Am. B 23, 2134–2141 (2006).
[Crossref]

Chang, G.

Chang, R. S.

Chen, H.

H. Chen, Q. Zhan, Y. L. Zhang, Y. P. Li, “The Gouy phase shift of the highly focused radially polarized beam,” Phys. Lett. A 371, 259–261 (2007).
[Crossref]

Chen, W.

W. Chen, Q. Zhan, “Optimal plasmonic focusing with radial polarization,” Proc. SPIE 6450, 64500D (2007).
[Crossref]

W. Chen, Q. Zhan, “Numerical study of an apertureless near field scanning optical microscope probe under radial polarization illumination,” Opt. Express 15, 4106–4111 (2007).
[Crossref] [PubMed]

W. Chen, Q. Zhan, “Field enhancement analysis of an apertureless near field scanning optical microscope probe with finite element method,” Chin. Opt. Lett. 5, 709–711 (2007).

W. Chen, Q. Zhan, “Three-dimensional focus shaping with cylindrical vector beams,” Opt. Commun. 265, 411–417 (2006).
[Crossref]

W. Chen, Q. Zhan, “Realization of evanescent Bessel beam via surface plasmon interference excited by radially polarized beam” Opt. Lett. (to be published).

Cheng, J.

J. Cheng, X. S. Xie, “Green’s function formulation for third-harmonic generation microscopy,” J. Opt. Soc. Am. B 19, 1604–1610 (2002).
[Crossref]

J. Cheng, A. Volkmer, X. S. Xie, “Theoretical and experimental characterization of coherent anti-Stokes Raman scattering microscopy,” J. Opt. Soc. Am. B 19, 1363–1375 (2002).
[Crossref]

Cheng, W.

W. Cheng, J. W. Haus, Q. Zhan, “Propagation of scalar and vector vortex beams through turbulent atmosphere,” Proc. SPIE7200 (to be published).

Chipman, R. A.

Chong, C. T.

H. F. Wang, L. P. Shi, B. Lukyanchuk, C. Sheppard, C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2, 501–505 (2008).
[Crossref]

Choudhury, A.

Colas des Francs, G.

Courjon, D.

T. Grosjean, D. Courjon, M. Spajer, “An all-fiber device for generating radially and other polarized light beams,” Opt. Commun. 203, 1–5 (2002).
[Crossref]

Davidson, N.

de Saint Denis, R.

N. Passilly, R. de Saint Denis, K. A-Ameur, F. Treussart, R. Hierle, J. F. Roch, “Simple interferometric technique for generation of a radially polarized light beam,” J. Opt. Soc. Am. A 22, 984–991 (2005).
[Crossref]

Dereux, A.

Ding, J.

Divin, C. J.

Dorn, R.

R. Dorn, S. Quabis, G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
[Crossref] [PubMed]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, G. Leuchs, “Focusing light into a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[Crossref]

Du, Y. B.

Y. B. Du, P. Tong, “Light scattering properties of paramagnetic particles,” J. Chem. Phys. 107, 355–362 (1997).
[Crossref]

Ebbesen, T. W.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[Crossref]

Eberler, M.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, G. Leuchs, “Focusing light into a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[Crossref]

Erdogan, T.

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. Anderson, M. J. Rooks, “Circularly symmetric operation of a concentric-circle-grating, surface-emitting, AlGaAs∕GaAs quantum-well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[Crossref]

Eyyuboglu, H. T.

Fang, N.

N. Fang, H. Lee, C. Sun, X. Zhang, “Sub-diffraction-limit imaging with a silver superlens,” Science 308, 534–537 (2005).
[Crossref] [PubMed]

Feurer, T.

M. Meier, V. Romano, T. Feurer, “Material processing with pulsed radially and azimuthally polarized laser radiation,” Appl. Phys. A 86, 329–334 (2007).
[Crossref]

Foley, J. T.

T. D. Visser, J. T. Foley, “On the wavefront spacing of focused, radially polarized beams,” J. Opt. Soc. Am. A 22, 2527–2531 (2005).
[Crossref]

Ford, D. H.

Fourkas, J. T.

S. Carrasco, B. E. A. Saleh, M. C. Teich, J. T. Fourkas, “Second- and third-harmonic generation with vector Gaussian beams,” J. Opt. Soc. Am. B 23, 2134–2141 (2006).
[Crossref]

Furst, E. M.

E. M. Furst, A. P. Gast, “Micromechanics of dipolar chains using optical tweezers,” Phys. Rev. Lett. 82, 4130–4133 (1999).
[Crossref]

Galvanauskas, A.

Gan, X.

Garmire, E.

M. E. Marhic, E. Garmire, “Low-order TE0q operation of a CO2 laser for transmission through circular metallic waveguides,” Appl. Phys. Lett. 38, 743–745 (1981).
[Crossref]

Gast, A. P.

E. M. Furst, A. P. Gast, “Micromechanics of dipolar chains using optical tweezers,” Phys. Rev. Lett. 82, 4130–4133 (1999).
[Crossref]

Ghaemi, H. F.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[Crossref]

Glöckl, O.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, G. Leuchs, “Focusing light into a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[Crossref]

Gori, F.

F. Gori, “Polarization basis for vortex beams,” J. Opt. Soc. Am. A 18, 1612–1617 (2001).
[Crossref]

F. Gori, G. Guattari, C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[Crossref]

Gouy, L. G.

L. G. Gouy, “Sur une propriété nouvelle des ondes lumineuses,” C. R. Acad. Sci. Paris. 110, 1251–1253 (1890).

Graf, T.

Grosjean, T.

T. Grosjean, D. Courjon, M. Spajer, “An all-fiber device for generating radially and other polarized light beams,” Opt. Commun. 203, 1–5 (2002).
[Crossref]

Gu, M.

Guattari, G.

F. Gori, G. Guattari, C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[Crossref]

Guo, C. S.

Hall, D. G.

D. G. Hall, “Vector-beam solutions of Maxwell’s wave equation,” Opt. Lett. 21, 9–11 (1996).
[Crossref] [PubMed]

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. Anderson, M. J. Rooks, “Circularly symmetric operation of a concentric-circle-grating, surface-emitting, AlGaAs∕GaAs quantum-well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[Crossref]

Hao, B.

Hasman, E.

Haus, J. W.

J. W. Haus, Z. Mozumder, Q. Zhan, “Azimuthal modulation instability for a cylindrically polarized wave in a nonlinear Kerr Medium,” Opt. Express 14, 4757–4764 (2006).
[Crossref] [PubMed]

W. Cheng, J. W. Haus, Q. Zhan, “Propagation of scalar and vector vortex beams through turbulent atmosphere,” Proc. SPIE7200 (to be published).

Hecht, B.

L. Novotny, B. Hecht, Principles of Nano-Optics (Cambridge U. Press, 2006).
[Crossref]

Hell, S.

Helseth, L. E.

L. E. Helseth, “Roles of polarization, phase and amplitude in solid immersion lens systems,” Opt. Commun. 191, 161–172 (2001).
[Crossref]

Hershcovitz, O.

Hierle, R.

N. Passilly, R. de Saint Denis, K. A-Ameur, F. Treussart, R. Hierle, J. F. Roch, “Simple interferometric technique for generation of a radially polarized light beam,” J. Opt. Soc. Am. A 22, 984–991 (2005).
[Crossref]

Hiller, J. M.

L. Yin, V. K. Vlasko-Vlasov, J. Pearson, J. M. Hiller, J. Hua, U. Welp, D. E. Brown, C. W. Kimball, “Subwavelength focusing and guiding of surface plasmons,” Nano Lett. 5, 1399–1402 (2005).
[Crossref] [PubMed]

Hirayama, T.

Hong, M. H.

Hua, J.

L. Yin, V. K. Vlasko-Vlasov, J. Pearson, J. M. Hiller, J. Hua, U. Welp, D. E. Brown, C. W. Kimball, “Subwavelength focusing and guiding of surface plasmons,” Nano Lett. 5, 1399–1402 (2005).
[Crossref] [PubMed]

Huang, C.

Ignatovich, F.

Ishihara, T.

X. Luo, T. Ishihara, “Surface plasmon resonant interference nanolithography technique,” Appl. Phys. Lett. 84, 4780–4782 (2004).
[Crossref]

Jackel, S.

G. Machavariani, Y. Lumer, I. Moshe, A. Meir, S. Jackel, “Spatially-variable retardation plate for efficient generation of radially- and azimuthally-polarized beams,” Opt. Commun. 281, 732–738 (2008).
[Crossref]

G. Machavariani, Y. Lumer, I. Moshe, A. Meir, S. Jackel, N. Davidson, “Birefringence-induced bifocusing for selection of radially or azimuthally polarized laser modes,” Appl. Opt. 46, 3304–3310 (2007).
[Crossref] [PubMed]

Jahns, J.

Q. Cao, J. Jahns, “Azimuthally polarized surface plasmons as effective terahertz waveguides,” Opt. Express 13, 511–518 (2005).
[Crossref] [PubMed]

Jia, B.

Kano, H.

Kawata, S.

Kawauchi, H.

Kimball, C. W.

L. Yin, V. K. Vlasko-Vlasov, J. Pearson, J. M. Hiller, J. Hua, U. Welp, D. E. Brown, C. W. Kimball, “Subwavelength focusing and guiding of surface plasmons,” Nano Lett. 5, 1399–1402 (2005).
[Crossref] [PubMed]

Kimura, W. D.

King, O.

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. Anderson, M. J. Rooks, “Circularly symmetric operation of a concentric-circle-grating, surface-emitting, AlGaAs∕GaAs quantum-well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[Crossref]

Kleiner, V.

Kozawa, Y.

Lai, W. J.

Lee, H.

N. Fang, H. Lee, C. Sun, X. Zhang, “Sub-diffraction-limit imaging with a silver superlens,” Science 308, 534–537 (2005).
[Crossref] [PubMed]

Leger, J.

Leger, J. R.

Lerman, G. M.

Leuchs, G.

R. Dorn, S. Quabis, G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
[Crossref] [PubMed]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, G. Leuchs, “Focusing light into a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[Crossref]

Levy, U.

Lezec, H. J.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[Crossref]

Li, J.

J. F. Bisson, J. Li, K. Ueda, Y. Senatsky, “Radially polarized ring and arc beams of a neodymium laser with an intra-cavity axicon,” Opt. Express 14, 3304–3311 (2006).
[Crossref] [PubMed]

Li, Y. P.

H. Chen, Q. Zhan, Y. L. Zhang, Y. P. Li, “The Gouy phase shift of the highly focused radially polarized beam,” Phys. Lett. A 371, 259–261 (2007).
[Crossref]

Y. Q. Zhao, Q. Zhan, Y. L. Zhang, Y. P. Li, “Creation of a three-dimensional optical chain for controllable particle delivery,” Opt. Lett. 30, 848–850 (2005).
[Crossref] [PubMed]

Lim, B. C.

Lin, Q.

Lipson, S. G.

Liu, C.

Liu, C. H.

Liu, Z.

Z. Liu, J. M. Steele, W. Srituravanich, Y. Pikus, C. Sun, X. Zhang, “Focusing surface plasmons with a plasmonic lens,” Nano Lett. 5, 1726–1729 (2005).
[Crossref] [PubMed]

Low, D. K. Y.

K. J. Moh, X. C. Yuan, J. Bu, D. K. Y. Low, R. E. Burge, “Direct noninterference cylindrical vector beam generation applied in the femtosecond regime,” Appl. Phys. Lett. 89, 251114 (2006).
[Crossref]

Lukyanchuk, B.

H. F. Wang, L. P. Shi, B. Lukyanchuk, C. Sheppard, C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2, 501–505 (2008).
[Crossref]

Lumer, Y.

G. Machavariani, Y. Lumer, I. Moshe, A. Meir, S. Jackel, “Spatially-variable retardation plate for efficient generation of radially- and azimuthally-polarized beams,” Opt. Commun. 281, 732–738 (2008).
[Crossref]

G. Machavariani, Y. Lumer, I. Moshe, A. Meir, S. Jackel, N. Davidson, “Birefringence-induced bifocusing for selection of radially or azimuthally polarized laser modes,” Appl. Opt. 46, 3304–3310 (2007).
[Crossref] [PubMed]

Luo, X.

X. Luo, T. Ishihara, “Surface plasmon resonant interference nanolithography technique,” Appl. Phys. Lett. 84, 4780–4782 (2004).
[Crossref]

Machavariani, G.

G. Machavariani, Y. Lumer, I. Moshe, A. Meir, S. Jackel, “Spatially-variable retardation plate for efficient generation of radially- and azimuthally-polarized beams,” Opt. Commun. 281, 732–738 (2008).
[Crossref]

G. Machavariani, Y. Lumer, I. Moshe, A. Meir, S. Jackel, N. Davidson, “Birefringence-induced bifocusing for selection of radially or azimuthally polarized laser modes,” Appl. Opt. 46, 3304–3310 (2007).
[Crossref] [PubMed]

Marhic, M. E.

M. E. Marhic, E. Garmire, “Low-order TE0q operation of a CO2 laser for transmission through circular metallic waveguides,” Appl. Phys. Lett. 38, 743–745 (1981).
[Crossref]

Matzumurra, K.

Y. Mushiake, K. Matzumurra, N. Nakajima, “Generation of radially polarized optical beam mode by laser oscillation,” Proc. IEEE 60, 1107–1109 (1972).
[Crossref]

McEldowney, S. C.

Meier, M.

M. Meier, V. Romano, T. Feurer, “Material processing with pulsed radially and azimuthally polarized laser radiation,” Appl. Phys. A 86, 329–334 (2007).
[Crossref]

Meir, A.

G. Machavariani, Y. Lumer, I. Moshe, A. Meir, S. Jackel, “Spatially-variable retardation plate for efficient generation of radially- and azimuthally-polarized beams,” Opt. Commun. 281, 732–738 (2008).
[Crossref]

G. Machavariani, Y. Lumer, I. Moshe, A. Meir, S. Jackel, N. Davidson, “Birefringence-induced bifocusing for selection of radially or azimuthally polarized laser modes,” Appl. Opt. 46, 3304–3310 (2007).
[Crossref] [PubMed]

Meixner, A. J.

Mizuguchi, S.

Moed, S.

Moh, K. J.

K. J. Moh, X. C. Yuan, J. Bu, D. K. Y. Low, R. E. Burge, “Direct noninterference cylindrical vector beam generation applied in the femtosecond regime,” Appl. Phys. Lett. 89, 251114 (2006).
[Crossref]

Moshe, I.

G. Machavariani, Y. Lumer, I. Moshe, A. Meir, S. Jackel, “Spatially-variable retardation plate for efficient generation of radially- and azimuthally-polarized beams,” Opt. Commun. 281, 732–738 (2008).
[Crossref]

G. Machavariani, Y. Lumer, I. Moshe, A. Meir, S. Jackel, N. Davidson, “Birefringence-induced bifocusing for selection of radially or azimuthally polarized laser modes,” Appl. Opt. 46, 3304–3310 (2007).
[Crossref] [PubMed]

Mozumder, Z.

Munro, P.

Mushiake, Y.

Y. Mushiake, K. Matzumurra, N. Nakajima, “Generation of radially polarized optical beam mode by laser oscillation,” Proc. IEEE 60, 1107–1109 (1972).
[Crossref]

Nakajima, N.

Y. Mushiake, K. Matzumurra, N. Nakajima, “Generation of radially polarized optical beam mode by laser oscillation,” Proc. IEEE 60, 1107–1109 (1972).
[Crossref]

Nakamura, T.

Nesterov, A. V.

Ni, W. J.

Niziev, V. G.

Norris, T. B.

Nose, T.

R. Yamaguchi, T. Nose, S. Sato, “Liquid crystal polarizers with axially symmetrical properties,” Jpn. J. Appl. Phys. Part 1 28, 1730–1731 (1989).
[Crossref]

Novotny, L.

Padovani, C.

F. Gori, G. Guattari, C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[Crossref]

Passilly, N.

N. Passilly, R. de Saint Denis, K. A-Ameur, F. Treussart, R. Hierle, J. F. Roch, “Simple interferometric technique for generation of a radially polarized light beam,” J. Opt. Soc. Am. A 22, 984–991 (2005).
[Crossref]

Pearson, J.

L. Yin, V. K. Vlasko-Vlasov, J. Pearson, J. M. Hiller, J. Hua, U. Welp, D. E. Brown, C. W. Kimball, “Subwavelength focusing and guiding of surface plasmons,” Nano Lett. 5, 1399–1402 (2005).
[Crossref] [PubMed]

Pereira, S. F.

S. F. Pereira, A. S. van de Nes, “Superresolution by means of polarisation, phase and amplitude pupil masks,” Opt. Commun. 234, 119–124 (2004).
[Crossref]

Petrov, D.

G. Volpe, D. Petrov, “Generation of cylindrical vector beams with few-mode fibers excited by Laguerre–Gaussian beams,” Opt. Commun. 237, 89–95 (2004).
[Crossref]

Phua, P. B.

Pikus, Y.

Z. Liu, J. M. Steele, W. Srituravanich, Y. Pikus, C. Sun, X. Zhang, “Focusing surface plasmons with a plasmonic lens,” Nano Lett. 5, 1726–1729 (2005).
[Crossref] [PubMed]

Pohl, D.

D. Pohl, “Operation of a Ruby laser in the purely transverse electric mode TE01,” Appl. Phys. Lett. 20, 266–267 (1972).
[Crossref]

Quabis, S.

R. Dorn, S. Quabis, G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
[Crossref] [PubMed]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, G. Leuchs, “Focusing light into a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[Crossref]

Raether, H.

H. Raether, Surface Plasmons on Smooth and Rough Surfaces and Gratings (Spinger-Verlag, 1988).

Renger, J.

A. Bouhelier, J. Renger, M. R. Beversluis, L. Novotny, “Plasmon-coupled tip-enhanced near-field optical microscopy,” J. Microsc. 210, 220–224 (2003).
[Crossref] [PubMed]

Richards, B.

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[Crossref]

Roch, J. F.

N. Passilly, R. de Saint Denis, K. A-Ameur, F. Treussart, R. Hierle, J. F. Roch, “Simple interferometric technique for generation of a radially polarized light beam,” J. Opt. Soc. Am. A 22, 984–991 (2005).
[Crossref]

Romano, V.

M. Meier, V. Romano, T. Feurer, “Material processing with pulsed radially and azimuthally polarized laser radiation,” Appl. Phys. A 86, 329–334 (2007).
[Crossref]

Rooks, M. J.

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. Anderson, M. J. Rooks, “Circularly symmetric operation of a concentric-circle-grating, surface-emitting, AlGaAs∕GaAs quantum-well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[Crossref]

Rotschild, C.

Saleh, B. E. A.

S. Carrasco, B. E. A. Saleh, M. C. Teich, J. T. Fourkas, “Second- and third-harmonic generation with vector Gaussian beams,” J. Opt. Soc. Am. B 23, 2134–2141 (2006).
[Crossref]

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley, 2007).

Santarsiero, M.

R. Borghi, M. Santarsiero, M. A. Alonso, “Highly focused spirally polarized beams,” J. Opt. Soc. Am. A 22, 1420–1431 (2005).
[Crossref]

R. Borghi, M. Santarsiero, “Nonparaxial propagation of spirally polarized optical beams,” J. Opt. Soc. Am. A 21, 2029–2037 (2004).
[Crossref]

Sato, S.

H. Kawauchi, K. Yonezawa, Y. Kozawa, S. Sato, “Calculation of optical trapping forces on a dielectric sphere in the ray optics regime produced by a radially polarized laser beam,” Opt. Lett. 32, 1839–1841 (2007).
[Crossref] [PubMed]

K. Yonezawa, Y. Kozawa, S. Sato, “Compact laser with radial polarization using birefringent laser medium,” Jpn. J. Appl. Phys., Part 1 46, 5160–5163 (2007).
[Crossref]

K. Yonezawa, Y. Kozawa, S. Sato, “Generation of a radially polarized laser beam by use of the birefringence of a c-cut Nd:YVO4 crystal,” Opt. Lett. 31, 2151–2153 (2006).
[Crossref] [PubMed]

T. Hirayama, Y. Kozawa, T. Nakamura, S. Sato, “Generation of a cylindrically symmetric, polarized laser beam with narrow linewidth and fine tunability,” Opt. Express 14, 12839–12845 (2006).
[Crossref] [PubMed]

Y. Kozawa, S. Sato, “Generation of a radially polarized laser beam by use of a conical Brewster prism,” Opt. Lett. 30, 3063–3065 (2005).
[Crossref] [PubMed]

R. Yamaguchi, T. Nose, S. Sato, “Liquid crystal polarizers with axially symmetrical properties,” Jpn. J. Appl. Phys. Part 1 28, 1730–1731 (1989).
[Crossref]

Schadt, M.

Senatsky, Y.

J. F. Bisson, J. Li, K. Ueda, Y. Senatsky, “Radially polarized ring and arc beams of a neodymium laser with an intra-cavity axicon,” Opt. Express 14, 3304–3311 (2006).
[Crossref] [PubMed]

Senior, J. M.

For example, J. M. Senior, Optical Fiber Communications (Prentice Hall, 1992).

Shemo, D. M.

Sheppard, C.

H. F. Wang, L. P. Shi, B. Lukyanchuk, C. Sheppard, C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2, 501–505 (2008).
[Crossref]

Sheppard, C. J. R.

Shi, L. P.

H. F. Wang, L. P. Shi, B. Lukyanchuk, C. Sheppard, C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2, 501–505 (2008).
[Crossref]

Shih, C.-C.

C.-C. Shih, “Radial polarization laser resonator,” U.S. patent 5,359,622 (Oct. 25, 1994).

Spajer, M.

T. Grosjean, D. Courjon, M. Spajer, “An all-fiber device for generating radially and other polarized light beams,” Opt. Commun. 203, 1–5 (2002).
[Crossref]

Spilman, A. K.

A. K. Spilman, T. G. Brown, “Stress birefringent, space-variant wave plates for vortex illumination,” Appl. Opt. 46, 61–66 (2007).
[Crossref]

Srituravanich, W.

Z. Liu, J. M. Steele, W. Srituravanich, Y. Pikus, C. Sun, X. Zhang, “Focusing surface plasmons with a plasmonic lens,” Nano Lett. 5, 1726–1729 (2005).
[Crossref] [PubMed]

Stadler, J.

Stalder, M.

Stanciu, C.

Steele, J. M.

Z. Liu, J. M. Steele, W. Srituravanich, Y. Pikus, C. Sun, X. Zhang, “Focusing surface plasmons with a plasmonic lens,” Nano Lett. 5, 1726–1729 (2005).
[Crossref] [PubMed]

Stranick, S. J.

Stupperich, C.

Sun, C.

Z. Liu, J. M. Steele, W. Srituravanich, Y. Pikus, C. Sun, X. Zhang, “Focusing surface plasmons with a plasmonic lens,” Nano Lett. 5, 1726–1729 (2005).
[Crossref] [PubMed]

N. Fang, H. Lee, C. Sun, X. Zhang, “Sub-diffraction-limit imaging with a silver superlens,” Science 308, 534–537 (2005).
[Crossref] [PubMed]

C. Sun, C. Liu, “Ultrasmall focusing spot with a long depth of focus based on polarization and phase modulation,” Opt. Lett. 28, 99–101 (2003).
[Crossref] [PubMed]

Teich, M. C.

S. Carrasco, B. E. A. Saleh, M. C. Teich, J. T. Fourkas, “Second- and third-harmonic generation with vector Gaussian beams,” J. Opt. Soc. Am. B 23, 2134–2141 (2006).
[Crossref]

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley, 2007).

Thio, T.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[Crossref]

Tidwell, S. C.

Tong, P.

Y. B. Du, P. Tong, “Light scattering properties of paramagnetic particles,” J. Chem. Phys. 107, 355–362 (1997).
[Crossref]

Török, P.

Toussaint, K. C.

A. F. Abouraddy, K. C. Toussaint, “Three-dimensional polarization control in microscopy,” Phys. Rev. Lett. 96, 153901 (2006).
[Crossref] [PubMed]

Treussart, F.

N. Passilly, R. de Saint Denis, K. A-Ameur, F. Treussart, R. Hierle, J. F. Roch, “Simple interferometric technique for generation of a radially polarized light beam,” J. Opt. Soc. Am. A 22, 984–991 (2005).
[Crossref]

Ueda, K.

J. F. Bisson, J. Li, K. Ueda, Y. Senatsky, “Radially polarized ring and arc beams of a neodymium laser with an intra-cavity axicon,” Opt. Express 14, 3304–3311 (2006).
[Crossref] [PubMed]

van de Nes, A. S.

S. F. Pereira, A. S. van de Nes, “Superresolution by means of polarisation, phase and amplitude pupil masks,” Opt. Commun. 234, 119–124 (2004).
[Crossref]

Visser, T. D.

T. D. Visser, J. T. Foley, “On the wavefront spacing of focused, radially polarized beams,” J. Opt. Soc. Am. A 22, 2527–2531 (2005).
[Crossref]

Vlasko-Vlasov, V. K.

L. Yin, V. K. Vlasko-Vlasov, J. Pearson, J. M. Hiller, J. Hua, U. Welp, D. E. Brown, C. W. Kimball, “Subwavelength focusing and guiding of surface plasmons,” Nano Lett. 5, 1399–1402 (2005).
[Crossref] [PubMed]

Vogel, M. M.

Volkmer, A.

Volpe, G.

G. Volpe, D. Petrov, “Generation of cylindrical vector beams with few-mode fibers excited by Laguerre–Gaussian beams,” Opt. Commun. 237, 89–95 (2004).
[Crossref]

Voss, A.

Wang, H. F.

H. F. Wang, L. P. Shi, B. Lukyanchuk, C. Sheppard, C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2, 501–505 (2008).
[Crossref]

Wang, H. T.

Wang, X. L.

Weeber, J.-C.

Welp, U.

L. Yin, V. K. Vlasko-Vlasov, J. Pearson, J. M. Hiller, J. Hua, U. Welp, D. E. Brown, C. W. Kimball, “Subwavelength focusing and guiding of surface plasmons,” Nano Lett. 5, 1399–1402 (2005).
[Crossref] [PubMed]

Wichmann, J.

Wicks, G. W.

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. Anderson, M. J. Rooks, “Circularly symmetric operation of a concentric-circle-grating, surface-emitting, AlGaAs∕GaAs quantum-well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[Crossref]

Wiederrecht, G. P.

Williamson, S. L.

Wolf, E.

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[Crossref]

E. Wolf, “Electromagnetic diffraction in optical systems I. An integral representation of the image field,” Proc. R. Soc. London Ser. A 253, 349–357 (1959).
[Crossref]

Wolff, P. A.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[Crossref]

Xie, X. S.

J. Cheng, A. Volkmer, X. S. Xie, “Theoretical and experimental characterization of coherent anti-Stokes Raman scattering microscopy,” J. Opt. Soc. Am. B 19, 1363–1375 (2002).
[Crossref]

J. Cheng, X. S. Xie, “Green’s function formulation for third-harmonic generation microscopy,” J. Opt. Soc. Am. B 19, 1604–1610 (2002).
[Crossref]

Yamaguchi, R.

R. Yamaguchi, T. Nose, S. Sato, “Liquid crystal polarizers with axially symmetrical properties,” Jpn. J. Appl. Phys. Part 1 28, 1730–1731 (1989).
[Crossref]

Yang, Y.

Y. Yang, Q. Zhan, “Third-harmonic generation microscopy with tightly focused radial polarization,” J. Opt. A 10, 125103 (2008).
[Crossref]

Ye, C.

C. Ye, “Construction of an optical rotator using quarter-wave plates and an optical retarder,” Opt. Eng. 34, 3031–3035 (1995).
[Crossref]

Yin, L.

L. Yin, V. K. Vlasko-Vlasov, J. Pearson, J. M. Hiller, J. Hua, U. Welp, D. E. Brown, C. W. Kimball, “Subwavelength focusing and guiding of surface plasmons,” Nano Lett. 5, 1399–1402 (2005).
[Crossref] [PubMed]

Yonezawa, K.

Youngworth, K. S.

Yuan, X. C.

K. J. Moh, X. C. Yuan, J. Bu, D. K. Y. Low, R. E. Burge, “Direct noninterference cylindrical vector beam generation applied in the femtosecond regime,” Appl. Phys. Lett. 89, 251114 (2006).
[Crossref]

Zhan, Q.

Y. Yang, Q. Zhan, “Third-harmonic generation microscopy with tightly focused radial polarization,” J. Opt. A 10, 125103 (2008).
[Crossref]

W. Chen, Q. Zhan, “Optimal plasmonic focusing with radial polarization,” Proc. SPIE 6450, 64500D (2007).
[Crossref]

H. Chen, Q. Zhan, Y. L. Zhang, Y. P. Li, “The Gouy phase shift of the highly focused radially polarized beam,” Phys. Lett. A 371, 259–261 (2007).
[Crossref]

W. Chen, Q. Zhan, “Field enhancement analysis of an apertureless near field scanning optical microscope probe with finite element method,” Chin. Opt. Lett. 5, 709–711 (2007).

W. Chen, Q. Zhan, “Numerical study of an apertureless near field scanning optical microscope probe under radial polarization illumination,” Opt. Express 15, 4106–4111 (2007).
[Crossref] [PubMed]

Q. Zhan, “Evanescent Bessel beam generation via surface plasmon resonance by radially polarized beam,” Opt. Lett. 31, 1726–1728 (2006).
[Crossref] [PubMed]

J. W. Haus, Z. Mozumder, Q. Zhan, “Azimuthal modulation instability for a cylindrically polarized wave in a nonlinear Kerr Medium,” Opt. Express 14, 4757–4764 (2006).
[Crossref] [PubMed]

Q. Zhan, “Properties of circularly polarization vortex beams,” Opt. Lett. 31, 867–869 (2006).
[Crossref] [PubMed]

W. Chen, Q. Zhan, “Three-dimensional focus shaping with cylindrical vector beams,” Opt. Commun. 265, 411–417 (2006).
[Crossref]

Y. Q. Zhao, Q. Zhan, Y. L. Zhang, Y. P. Li, “Creation of a three-dimensional optical chain for controllable particle delivery,” Opt. Lett. 30, 848–850 (2005).
[Crossref] [PubMed]

Q. Zhan, “Trapping nanoparticles with cylindrical polarization,” Proc. SPIE 5514, 275–282 (2004).
[Crossref]

Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12, 3377–3382 (2004).
[Crossref] [PubMed]

Q. Zhan, “Second-order tilted wave interpretation of the Gouy phase shift under high numerical aperture uniform illumination,” Opt. Commun. 242, 351–360 (2004).
[Crossref]

Q. Zhan, “Optical radiation forces on a dielectric sphere produced by highly focused cylindrical vector beams,” J. Opt. A, Pure Appl. Opt. 5, 229–232 (2003).
[Crossref]

Q. Zhan, “Cylindrical polarization symmetry for nondestructive nano-characterization,” Proc. SPIE 5045, 85–92 (2003).
[Crossref]

Q. Zhan, J. R. Leger, “Interferometric measurement of Berry’s phase in space-variant polarization manipulations,” Opt. Commun. 213, 241–245 (2002).
[Crossref]

Q. Zhan, J. R. Leger, “Microellipsometer with radial symmetry,” Appl. Opt. 41, 4630–4637 (2002).
[Crossref] [PubMed]

Q. Zhan, J. R. Leger, “Focus shaping using cylindrical vector beams,” Opt. Express 10, 324–331 (2002).
[Crossref] [PubMed]

W. Chen, Q. Zhan, “Realization of evanescent Bessel beam via surface plasmon interference excited by radially polarized beam” Opt. Lett. (to be published).

W. Cheng, J. W. Haus, Q. Zhan, “Propagation of scalar and vector vortex beams through turbulent atmosphere,” Proc. SPIE7200 (to be published).

Zhang, X.

N. Fang, H. Lee, C. Sun, X. Zhang, “Sub-diffraction-limit imaging with a silver superlens,” Science 308, 534–537 (2005).
[Crossref] [PubMed]

Z. Liu, J. M. Steele, W. Srituravanich, Y. Pikus, C. Sun, X. Zhang, “Focusing surface plasmons with a plasmonic lens,” Nano Lett. 5, 1726–1729 (2005).
[Crossref] [PubMed]

Zhang, Y. L.

H. Chen, Q. Zhan, Y. L. Zhang, Y. P. Li, “The Gouy phase shift of the highly focused radially polarized beam,” Phys. Lett. A 371, 259–261 (2007).
[Crossref]

Y. Q. Zhao, Q. Zhan, Y. L. Zhang, Y. P. Li, “Creation of a three-dimensional optical chain for controllable particle delivery,” Opt. Lett. 30, 848–850 (2005).
[Crossref] [PubMed]

Zhao, Y. Q.

Y. Q. Zhao, Q. Zhan, Y. L. Zhang, Y. P. Li, “Creation of a three-dimensional optical chain for controllable particle delivery,” Opt. Lett. 30, 848–850 (2005).
[Crossref] [PubMed]

Zommer, S.

Zurita-Sanchez, J. R.

Appl. Opt. (1)

A. K. Spilman, T. G. Brown, “Stress birefringent, space-variant wave plates for vortex illumination,” Appl. Opt. 46, 61–66 (2007).
[Crossref]

Appl. Phys. A (1)

M. Meier, V. Romano, T. Feurer, “Material processing with pulsed radially and azimuthally polarized laser radiation,” Appl. Phys. A 86, 329–334 (2007).
[Crossref]

Appl. Opt. (7)

Appl. Phys. Lett. (1)

X. Luo, T. Ishihara, “Surface plasmon resonant interference nanolithography technique,” Appl. Phys. Lett. 84, 4780–4782 (2004).
[Crossref]

Appl. Phys. Lett. (4)

M. E. Marhic, E. Garmire, “Low-order TE0q operation of a CO2 laser for transmission through circular metallic waveguides,” Appl. Phys. Lett. 38, 743–745 (1981).
[Crossref]

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. Anderson, M. J. Rooks, “Circularly symmetric operation of a concentric-circle-grating, surface-emitting, AlGaAs∕GaAs quantum-well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[Crossref]

D. Pohl, “Operation of a Ruby laser in the purely transverse electric mode TE01,” Appl. Phys. Lett. 20, 266–267 (1972).
[Crossref]

K. J. Moh, X. C. Yuan, J. Bu, D. K. Y. Low, R. E. Burge, “Direct noninterference cylindrical vector beam generation applied in the femtosecond regime,” Appl. Phys. Lett. 89, 251114 (2006).
[Crossref]

C. R. Acad. Sci. Paris. (1)

L. G. Gouy, “Sur une propriété nouvelle des ondes lumineuses,” C. R. Acad. Sci. Paris. 110, 1251–1253 (1890).

Chin. Opt. Lett. (1)

J. Opt. Soc. Am. A (1)

N. Passilly, R. de Saint Denis, K. A-Ameur, F. Treussart, R. Hierle, J. F. Roch, “Simple interferometric technique for generation of a radially polarized light beam,” J. Opt. Soc. Am. A 22, 984–991 (2005).
[Crossref]

J. Chem. Phys. (1)

Y. B. Du, P. Tong, “Light scattering properties of paramagnetic particles,” J. Chem. Phys. 107, 355–362 (1997).
[Crossref]

J. Microsc. (1)

A. Bouhelier, J. Renger, M. R. Beversluis, L. Novotny, “Plasmon-coupled tip-enhanced near-field optical microscopy,” J. Microsc. 210, 220–224 (2003).
[Crossref] [PubMed]

J. Opt. A (1)

Y. Yang, Q. Zhan, “Third-harmonic generation microscopy with tightly focused radial polarization,” J. Opt. A 10, 125103 (2008).
[Crossref]

J. Opt. Soc. Am. B (1)

J. Cheng, X. S. Xie, “Green’s function formulation for third-harmonic generation microscopy,” J. Opt. Soc. Am. B 19, 1604–1610 (2002).
[Crossref]

J. Opt. A, Pure Appl. Opt. (1)

Q. Zhan, “Optical radiation forces on a dielectric sphere produced by highly focused cylindrical vector beams,” J. Opt. A, Pure Appl. Opt. 5, 229–232 (2003).
[Crossref]

J. Opt. Soc. Am. A (2)

R. Borghi, M. Santarsiero, “Nonparaxial propagation of spirally polarized optical beams,” J. Opt. Soc. Am. A 21, 2029–2037 (2004).
[Crossref]

T. D. Visser, J. T. Foley, “On the wavefront spacing of focused, radially polarized beams,” J. Opt. Soc. Am. A 22, 2527–2531 (2005).
[Crossref]

J. Opt. Soc. Am. B (1)

S. Carrasco, B. E. A. Saleh, M. C. Teich, J. T. Fourkas, “Second- and third-harmonic generation with vector Gaussian beams,” J. Opt. Soc. Am. B 23, 2134–2141 (2006).
[Crossref]

J. Opt. Soc. Am. A (2)

J. Opt. Soc. Am. B (3)

J. Phys. D (1)

V. G. Niziev, A. V. Nesterov, “Influence of beam polarization on laser cutting efficiency,” J. Phys. D 32, 1455–1461 (1999).
[Crossref]

Jpn. J. Appl. Phys. Part 1 (1)

R. Yamaguchi, T. Nose, S. Sato, “Liquid crystal polarizers with axially symmetrical properties,” Jpn. J. Appl. Phys. Part 1 28, 1730–1731 (1989).
[Crossref]

Jpn. J. Appl. Phys., Part 1 (1)

K. Yonezawa, Y. Kozawa, S. Sato, “Compact laser with radial polarization using birefringent laser medium,” Jpn. J. Appl. Phys., Part 1 46, 5160–5163 (2007).
[Crossref]

Nano Lett. (2)

Z. Liu, J. M. Steele, W. Srituravanich, Y. Pikus, C. Sun, X. Zhang, “Focusing surface plasmons with a plasmonic lens,” Nano Lett. 5, 1726–1729 (2005).
[Crossref] [PubMed]

L. Yin, V. K. Vlasko-Vlasov, J. Pearson, J. M. Hiller, J. Hua, U. Welp, D. E. Brown, C. W. Kimball, “Subwavelength focusing and guiding of surface plasmons,” Nano Lett. 5, 1399–1402 (2005).
[Crossref] [PubMed]

Nat. Photonics (1)

H. F. Wang, L. P. Shi, B. Lukyanchuk, C. Sheppard, C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2, 501–505 (2008).
[Crossref]

Nature (1)

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[Crossref]

Opt. Commun. (3)

L. E. Helseth, “Roles of polarization, phase and amplitude in solid immersion lens systems,” Opt. Commun. 191, 161–172 (2001).
[Crossref]

T. Grosjean, D. Courjon, M. Spajer, “An all-fiber device for generating radially and other polarized light beams,” Opt. Commun. 203, 1–5 (2002).
[Crossref]

S. F. Pereira, A. S. van de Nes, “Superresolution by means of polarisation, phase and amplitude pupil masks,” Opt. Commun. 234, 119–124 (2004).
[Crossref]

Opt. Eng. (1)

C. Ye, “Construction of an optical rotator using quarter-wave plates and an optical retarder,” Opt. Eng. 34, 3031–3035 (1995).
[Crossref]

Opt. Express (2)

J. F. Bisson, J. Li, K. Ueda, Y. Senatsky, “Radially polarized ring and arc beams of a neodymium laser with an intra-cavity axicon,” Opt. Express 14, 3304–3311 (2006).
[Crossref] [PubMed]

Q. Cao, J. Jahns, “Azimuthally polarized surface plasmons as effective terahertz waveguides,” Opt. Express 13, 511–518 (2005).
[Crossref] [PubMed]

Opt. Lett. (2)

Y. Q. Zhao, Q. Zhan, Y. L. Zhang, Y. P. Li, “Creation of a three-dimensional optical chain for controllable particle delivery,” Opt. Lett. 30, 848–850 (2005).
[Crossref] [PubMed]

Y. Kozawa, S. Sato, “Generation of a radially polarized laser beam by use of a conical Brewster prism,” Opt. Lett. 30, 3063–3065 (2005).
[Crossref] [PubMed]

Opt. Commun. (7)

Q. Zhan, J. R. Leger, “Interferometric measurement of Berry’s phase in space-variant polarization manipulations,” Opt. Commun. 213, 241–245 (2002).
[Crossref]

F. Gori, G. Guattari, C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[Crossref]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, G. Leuchs, “Focusing light into a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[Crossref]

G. Machavariani, Y. Lumer, I. Moshe, A. Meir, S. Jackel, “Spatially-variable retardation plate for efficient generation of radially- and azimuthally-polarized beams,” Opt. Commun. 281, 732–738 (2008).
[Crossref]

G. Volpe, D. Petrov, “Generation of cylindrical vector beams with few-mode fibers excited by Laguerre–Gaussian beams,” Opt. Commun. 237, 89–95 (2004).
[Crossref]

W. Chen, Q. Zhan, “Three-dimensional focus shaping with cylindrical vector beams,” Opt. Commun. 265, 411–417 (2006).
[Crossref]

Q. Zhan, “Second-order tilted wave interpretation of the Gouy phase shift under high numerical aperture uniform illumination,” Opt. Commun. 242, 351–360 (2004).
[Crossref]

Opt. Express (13)

P. Török, P. Munro, “The use of Gauss–Laguerre vector beams in STED microscopy,” Opt. Express 12, 3605–3617 (2004).
[Crossref]

M. R. Beversluis, L. Novotny, S. J. Stranick, “Programmable vector point-spread function engineering,” Opt. Express 14, 2650–2656 (2006).
[Crossref] [PubMed]

G. M. Lerman, U. Levy, “Effect of radial polarization and apodization on spot size under tight focusing conditions,” Opt. Express 16, 4567–4581 (2008).
[Crossref] [PubMed]

K. S. Youngworth, T. G. Brown, “Focusing of high numerical aperture cylindrical vector beams,” Opt. Express 7, 77–87 (2000).
[Crossref] [PubMed]

T. Hirayama, Y. Kozawa, T. Nakamura, S. Sato, “Generation of a cylindrically symmetric, polarized laser beam with narrow linewidth and fine tunability,” Opt. Express 14, 12839–12845 (2006).
[Crossref] [PubMed]

B. Jia, X. Gan, M. Gu, “Direct measurement of a radially polarized focused evanescent field facilitated by a single LCD,” Opt. Express 13, 6821–6827 (2005).
[Crossref] [PubMed]

B. Hao, J. Leger, “Experimental measurement of longitudinal component in the vicinity of focused radially polarized beam,” Opt. Express 15, 3550–3556 (2007).
[Crossref] [PubMed]

Q. Zhan, J. R. Leger, “Focus shaping using cylindrical vector beams,” Opt. Express 10, 324–331 (2002).
[Crossref] [PubMed]

S. C. McEldowney, D. M. Shemo, R. A. Chipman, “Vortex retarders produced from photo-aligned liquid crystal polymers,” Opt. Express 16, 7295–7308 (2008).
[Crossref] [PubMed]

Y. Cai, Q. Lin, H. T. Eyyuboglu, Y. Baykal, “Average irradiance and polarization properties of a radially or azimuthally polarized beam in a turbulent atmosphere,” Opt. Express 16, 7665–7673 (2008).
[Crossref] [PubMed]

J. W. Haus, Z. Mozumder, Q. Zhan, “Azimuthal modulation instability for a cylindrically polarized wave in a nonlinear Kerr Medium,” Opt. Express 14, 4757–4764 (2006).
[Crossref] [PubMed]

W. Chen, Q. Zhan, “Numerical study of an apertureless near field scanning optical microscope probe under radial polarization illumination,” Opt. Express 15, 4106–4111 (2007).
[Crossref] [PubMed]

Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12, 3377–3382 (2004).
[Crossref] [PubMed]

Opt. Lett. (18)

H. Kawauchi, K. Yonezawa, Y. Kozawa, S. Sato, “Calculation of optical trapping forces on a dielectric sphere in the ray optics regime produced by a radially polarized laser beam,” Opt. Lett. 32, 1839–1841 (2007).
[Crossref] [PubMed]

Q. Zhan, “Evanescent Bessel beam generation via surface plasmon resonance by radially polarized beam,” Opt. Lett. 31, 1726–1728 (2006).
[Crossref] [PubMed]

G. Chang, C. J. Divin, C. H. Liu, S. L. Williamson, A. Galvanauskas, T. B. Norris, “Generation of radially polarized terahertz pulses via velocity-mismatched optical rectification,” Opt. Lett. 32, 433–435 (2007).
[Crossref] [PubMed]

Q. Zhan, “Properties of circularly polarization vortex beams,” Opt. Lett. 31, 867–869 (2006).
[Crossref] [PubMed]

A. Bouhelier, F. Ignatovich, A. Bruyant, C. Huang, G. Colas des Francs, J.-C. Weeber, A. Dereux, G. P. Wiederrecht, L. Novotny, “Surface plasmon interference excited by tightly focused laser beams,” Opt. Lett. 32, 2535–2537 (2007).
[Crossref] [PubMed]

Z. Bomzon, G. Biener, V. Kleiner, E. Hasman, “Radially and azimuthally polarized beams generated by space-variant dielectric subwavelength gratings,” Opt. Lett. 27, 285–287 (2002).
[Crossref]

B. C. Lim, P. B. Phua, W. J. Lai, M. H. Hong, “Fast switchable electro-optic radial polarization retarder,” Opt. Lett. 33, 950–952 (2008).
[Crossref] [PubMed]

M. Stalder, M. Schadt, “Linearly polarized light with axial symmetry generated by liquid-crystal polarization converters,” Opt. Lett. 21, 1948–1950 (1996).
[Crossref] [PubMed]

M. A. Ahmed, A. Voss, M. M. Vogel, T. Graf, “Multilayer polarizing grating mirror used for the generation of radial polarization in Yb:YAG thin-disk lasers,” Opt. Lett. 32, 3272–3274 (2007).
[Crossref] [PubMed]

D. G. Hall, “Vector-beam solutions of Maxwell’s wave equation,” Opt. Lett. 21, 9–11 (1996).
[Crossref] [PubMed]

K. Yonezawa, Y. Kozawa, S. Sato, “Generation of a radially polarized laser beam by use of the birefringence of a c-cut Nd:YVO4 crystal,” Opt. Lett. 31, 2151–2153 (2006).
[Crossref] [PubMed]

C. Sun, C. Liu, “Ultrasmall focusing spot with a long depth of focus based on polarization and phase modulation,” Opt. Lett. 28, 99–101 (2003).
[Crossref] [PubMed]

X. L. Wang, J. Ding, W. J. Ni, C. S. Guo, H. T. Wang, “Generation of arbitrary vector beams with a spatial light modulator and a common path interferometric arrangement,” Opt. Lett. 32, 3549–3551 (2007).
[Crossref] [PubMed]

J. Stadler, C. Stanciu, C. Stupperich, A. J. Meixner, “Tighter focusing with a parabolic mirror,” Opt. Lett. 33, 681–683 (2008).
[Crossref] [PubMed]

D. P. Biss, T. G. Brown, “Polarization-vortex-driven second-harmonic generation,” Opt. Lett. 28, 923–925 (2003).
[Crossref] [PubMed]

N. Bokor, N. Davidson, “Toward a spherical spot distribution with 4π focusing of radially polarized light,” Opt. Lett. 29, 1968–1970 (2004).
[Crossref] [PubMed]

N. Bokor, N. Davidson, “Generation of a hollow dark spherical spot by 4π focusing of a radially polarized Laguerre–Gaussian beam,” Opt. Lett. 31, 149–151 (2006).
[Crossref] [PubMed]

S. Hell, J. Wichmann, “Breaking the diffraction resolution limit by stimulated emission: stimulated-emission depletion fluorescence microscopy,” Opt. Lett. 19, 780–782 (1994).
[Crossref] [PubMed]

Phys. Lett. A (1)

H. Chen, Q. Zhan, Y. L. Zhang, Y. P. Li, “The Gouy phase shift of the highly focused radially polarized beam,” Phys. Lett. A 371, 259–261 (2007).
[Crossref]

Phys. Rev. Lett. (4)

A. F. Abouraddy, K. C. Toussaint, “Three-dimensional polarization control in microscopy,” Phys. Rev. Lett. 96, 153901 (2006).
[Crossref] [PubMed]

L. Novotny, M. R. Beversluis, K. S. Youngworth, T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[Crossref] [PubMed]

R. Dorn, S. Quabis, G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
[Crossref] [PubMed]

E. M. Furst, A. P. Gast, “Micromechanics of dipolar chains using optical tweezers,” Phys. Rev. Lett. 82, 4130–4133 (1999).
[Crossref]

Proc. IEEE (1)

Y. Mushiake, K. Matzumurra, N. Nakajima, “Generation of radially polarized optical beam mode by laser oscillation,” Proc. IEEE 60, 1107–1109 (1972).
[Crossref]

Proc. SPIE (1)

Q. Zhan, “Cylindrical polarization symmetry for nondestructive nano-characterization,” Proc. SPIE 5045, 85–92 (2003).
[Crossref]

Proc. R. Soc. London Ser. A (2)

E. Wolf, “Electromagnetic diffraction in optical systems I. An integral representation of the image field,” Proc. R. Soc. London Ser. A 253, 349–357 (1959).
[Crossref]

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[Crossref]

Proc. SPIE (2)

Q. Zhan, “Trapping nanoparticles with cylindrical polarization,” Proc. SPIE 5514, 275–282 (2004).
[Crossref]

W. Chen, Q. Zhan, “Optimal plasmonic focusing with radial polarization,” Proc. SPIE 6450, 64500D (2007).
[Crossref]

Science (1)

N. Fang, H. Lee, C. Sun, X. Zhang, “Sub-diffraction-limit imaging with a silver superlens,” Science 308, 534–537 (2005).
[Crossref] [PubMed]

Other (10)

W. Chen, Q. Zhan, “Realization of evanescent Bessel beam via surface plasmon interference excited by radially polarized beam” Opt. Lett. (to be published).

H. Raether, Surface Plasmons on Smooth and Rough Surfaces and Gratings (Spinger-Verlag, 1988).

W. Cheng, J. W. Haus, Q. Zhan, “Propagation of scalar and vector vortex beams through turbulent atmosphere,” Proc. SPIE7200 (to be published).

M. Gu, ed., Advanced Optical Imaging Theory, Vol. 75 of Springer Series in Optical Sciences (Springer-Verlag, 1999).

For example, the vortex phase plates from RPC Photonics (http://www.rpcphotonics.com/).

For example, the ZPol from Nanophoton (http://www.nanophoton.jp/) and the radial polarizer from ARCoptix (http://www.arcoptix.com/).

For example, J. M. Senior, Optical Fiber Communications (Prentice Hall, 1992).

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley, 2007).

L. Novotny, B. Hecht, Principles of Nano-Optics (Cambridge U. Press, 2006).
[Crossref]

C.-C. Shih, “Radial polarization laser resonator,” U.S. patent 5,359,622 (Oct. 25, 1994).

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Figures (49)

Fig. 1
Fig. 1 Spatial distribution of instantaneous electric vector field for several conventional modes and CV modes: (a) x-polarized fundamental Gaussian mode; (b) x-polarized HG 10 mode; (c) x-polarized HG 01 mode; (d) y-polarized HG 01 mode; (e) y-polarized HG 01 mode; (f) x-polarized LG 01 mode; (g) radially polarized mode; (h) azimuthally polarized mode; (i) generalized CV beams as a linear superposition of (g) and (h).
Fig. 2
Fig. 2 Formation of radial and azimuthal polarizations using linear superposition of orthogonally polarized HG modes.
Fig. 3
Fig. 3 Diagram of a ruby laser that generates CV beam output. The optical axis of the calcite crystal is parallel to the resonator axis. The combination of the calcite crystal, the telescope, and the aperture and stop provides the mode discrimination to force the laser operation in CV polarization mode. Figure adapted with permission from [1]. © 1972 American Institute of Physics.
Fig. 4
Fig. 4 Nd:YAG laser that generates CV beams by using axial dichroism created by a conical Brewster prism (CBP). The structure of the conical Brewster prism is shown to the right. Figure adapted with permission from [20].
Fig. 5
Fig. 5 Yb:YAG thin-disk laser that generates CV beam outputs using a circular multilayer polarizing grating end mirror. The output beam profile and polarization pattern are shown to the right. Figure adapted with permission from [21].
Fig. 6
Fig. 6 Diagram of a laser cavity design with Sagnac interferometer as an end mirror to create CV beam output. BS, beam splitter. Figure reprinted with permission from [22].
Fig. 7
Fig. 7 Mach–Zehnder interferometer setup to verify the spiral Berry’s phase given in Eq. (3.2). The Berry’s phase is shown in the lower right-hand corner.
Fig. 8
Fig. 8 Generation of CV beam using radial analyzer and SPE. A circularly polarized input is used. The cascaded two λ 2 plates rotate the polarization to the desired pattern.
Fig. 9
Fig. 9 A polarization mode converter setup with two SLMs. The first SLM creates the desired phase pattern. The combination of the λ 4 plate and the second reflection-type SLM converts the beam into the spatially variant polarization pattern. The second SLM can be programed to convert a linearly polarized beam into CV beam. Figure adapted with permission from [32].
Fig. 10
Fig. 10 Segmented spatially variant λ 2 plates that can convert linear polarization into CV polarization. Figure reprinted from [34] with permission from Elsevier.
Fig. 11
Fig. 11 Experimental setup that uses a segmented λ 2 plates polarization converter and a near-confocal Fabry–Perot interferometer (NCFPI) as a mode selector to generate CV beams. OD, optical diode; HWP, half-wave plate; PH, pinhole; TL1, TL2, telescope lenses; PC, polarization converter; FL, focusing lens; CL, collimating lens; M, four mirrors; MD, monitor diode; AS, aperture stop; MO, microscope objective; PD, photodiode. Figure reprinted with permission from [4] (http://prola.aps.org/abstract/PRL/v91/i23/e233901). © 2003 American Physical Society.
Fig. 12
Fig. 12 (a) Geometry of spatially variant subwavelength grating. A scanning electron microscope image of the device is also shown. Experimentally measured spatial polarization patterns for (b) right-hand circular polarization and (c) left-hand circular polarization are illustrated. Figure reprinted with permission from [37].
Fig. 13
Fig. 13 An experimental setup using an EO radial polarization retarder (EO-RPR) to generate CV beams. Figure adapted with permission from [38].
Fig. 14
Fig. 14 A common-path interferometer setup implemented with a SLM to generate CV beams and other more complicated vector beams. Several CV beams with different polarization patterns are also shown to the right. Figure reprinted with permission from [41].
Fig. 15
Fig. 15 Illustrations of the polarization patterns for LP 01 and LP 11 modes of optic fiber. It can be seen that the TE 01 and TM 01 modes from the second-higher-order mode group have azimuthal and radial polarization symmetry, respectively.
Fig. 16
Fig. 16 Laboratory setup that generates CV beams by using few-mode optic fiber. The intensity patterns of radial polarization generated with this setup after passing through a linear polarizer are shown to the right.
Fig. 17
Fig. 17 Diagram of a nonmechanical polarization rotator using LC or EO retarders sandwiched between two orthogonally oriented ( λ 4 plates. The Jones matrix of such a device is shown in the figure. It can be seen that such an arrangement provides polarization rotation that is independent of the incident polarization state.
Fig. 18
Fig. 18 Illustration of the polarization pattern at the pupil plane for the radiation from a vertical electric dipole collected by a high-NA objective. If the optical path is reversed, a radial polarization input should recover the propagating components of the vertical dipole radiation and produce a very tight focus.
Fig. 19
Fig. 19 Focusing of a CV beam. f is the focal length of the objective lens. Q ( r , ϕ ) is an observation point in the focal plane.
Fig. 20
Fig. 20 Focal field calculation examples for CV polarization focused by a high-NA ( NA = 0.8 ) objective lens. (a)–(c) 2D plots of the longitudinal component, the radial component, and the total field strength in the rz plane for incident radial polarization. (d) Line scan across the focus (dashed lines). There is no azimuthal component for this situation. A strong and narrower longitudinal component is obtained. (e)–(f) Results for azimuthal polarization. Only azimuthal component exists for this case.
Fig. 21
Fig. 21 Projections of a focal field experimentally measured by using a focal-plane knife-edge technique for a highly focused (a) radial and (b) azimuthal polarization. Actual focal intensity profiles for (c) radial and (d) azimuthal polarizations were obtained by using tomographic reconstruction. A focal spot as small as 0.16 λ 2 has been obtained for radial polarization with an annular aperture. Figures reprinted with permission from [4] (http://prola.aps.org/abstract/PRL/v91/i23/e233901). © 2003 American Physical Society.
Fig. 22
Fig. 22 Molecule orientation imaging by using radial polarization. The output of an argon ion laser is converted into radial polarization through a mode converter. The radially polarized beam is focused by a high-NA lens ( NA = 1.4 ) on to a polymethyl methacrylate (PMMA) thin-film sample imbedded with fluorescent dye. The upper right-hand corner shows the predicted fluorescent emission rate patterns corresponding to different molecule orientations. Corresponding experimental results are shown in the lower right-hand corner. Figures reprinted with permission from [60] (http://prola.aps.org/abstract/PRL/v86/i23/p5251_1). © 2001 American Physical Society.
Fig. 23
Fig. 23 Flattop focusing with generalized CV polarization. The double- λ 2 -plate polarization rotator converts the incident CV beam into generalized CV polarization with the desired pattern for flattop profile creation at the focal plane.
Fig. 24
Fig. 24 Example of the flattop focusing generated with the setup shown in Fig. 23. Line scans of the field distribution across the focal plane are shown to the left. A 2D distribution in the rz plane in the vicinity of the focus is shown to the right.
Fig. 25
Fig. 25 3D focus shaping using generalized CV polarization and pupil plane binary DOE. The pupil plane DOE consists of several concentric rings that give either 0 or π phase transmission for normal incidence. The transition points are specified with the corresponding NA in the image space.
Fig. 26
Fig. 26 Example of 3D flattop focus generated with CV beam. (a) 3D plot of the E 2 distribution in the vicinity of the focus. (b) Line scan of (a) along the z axis. A line scan for linear polarization is also shown for comparison. (c) Line scan in the focal plane.
Fig. 27
Fig. 27 Optical bubble created by radial polarization with DOE pupil plane modulation.
Fig. 28
Fig. 28 Generation of optical needle using radial polarization with pupil plane DOE modulation. The total field and the constituting radial and longitudinal components are shown in the column to the right. The electric vector distribution is shown in the plot at the left-hand bottom. It can be seen that the field near the optical axis is almost purely polarized in the longitudinal direction. Figure adapted and reprinted by permission from Macmillan Publishers Ltd, Nature Photonics [64], © 2008.
Fig. 29
Fig. 29 Geometry for calculation of effective phase retardation of tilted wave.
Fig. 30
Fig. 30 Setup for the Gouy phase shift calculation with uniform annular illumination. An x-polarized collimated beam uniformly illuminates the entrance pupil of an aplanatic objective lens. A circular blocking mask is placed at the entrance pupil of the lens to generate annular illumination. The illumination angles of the innermost and outermost rays are θ 1 and θ 2 , respectively. Q ( r , ϕ ) is an observation point in the focal plane, and f is the focal length of the objective. The geometry for the focal field length calculation is also shown. Δ L is the optical path difference between the innermost ray and the outermost ray.
Fig. 31
Fig. 31 Calculations of the Gouy phase shift under uniform annular illumination with NA 2 = sin θ 2 = 0.95 .
Fig. 32
Fig. 32 Vectorial diffraction theory simulation of a tightly focused beam with a Gouy shift of 2 π . Solid curve, axial intensity distribution; dotted curve, Gouy phase shift. A uniform annular illumination with NA 1 = sin θ 1 = 0.5 and NA 2 = sin θ 2 = 0.95 is used in the simulation. Only the positive side of the optical axis is shown.
Fig. 33
Fig. 33 Calculated axial intensity distribution I z (black solid curve), electric field component E z (blue dotted curve) and the Gouy phase shift (red dashed curve) for uniform radial polarization illumination with NA = 0.5 using the vectorial diffraction method. The space between the vertical dashed–dotted lines is wavefront spacing [67].
Fig. 34
Fig. 34 Comparison of the differential Gouy phase shift at the focal plane ( z = 0 ) calculated by both vectorial diffraction simulation and the analytical expression. In this case, uniform illumination is used with variable sin θ 2 = NA and θ 1 fixed at 0 [67].
Fig. 35
Fig. 35 Diagram of SPR excitation with highly focused radial polarization. The entire beam is p polarized with respect to the silver/dielectric interface. A pupil plane image of the reflected beam is shown to the right. A full dark ring is observed that is due to the SPR excitation by radial polarization. The nonuniformity in the beam profile is caused by the difference in reflection coefficients for the s- and p-polarization components of a beam splitter that steers the reflected beam to the CCD camera.
Fig. 36
Fig. 36 Experimental setup for the generation and detection of plasmonic focusing with highly focused radial polarization. PMT, photomultiplier tube; PM, photomask. A Veeco Aurora 3 NSOM with a nanoaperture fiber probe is used to detect the plasmonic field. The insets show the calculated plasmon focal fields at the metal surface for radial and linear polarization incidence, respectively. For linear polarization, because of the mismatch of polarization and the destructive interference of counterpropagating plasmon waves, the field at the focal plane splits into two lobes with much lower peak intensity.
Fig. 37
Fig. 37 Calculated surface plasmon intensity distribution at silver/air interface with radially polarized illumination. (a) Total intensity E 2 ; a homogeneous spot with a strongly enhanced field at the center is obtained. (b) Longitudinal component E z 2 , which is much stronger than E r 2 and dominates the total field distribution. (c) Radial component E r 2 , which has a donut shape. (d) E z 2 distribution, which is proportional to the NSOM signal detected by an apertured fiber probe.
Fig. 38
Fig. 38 Experimental confirmation of evanescent Bessel beam creation via surface plasmon excitation. (a) Logarithmic plot of measured NSOM signal in the near field. (b) Comparison of the measured and calculated transverse profiles of the NSOM signal ( E z 2 ) in the near field. (c) Measurements at different planes to reveal the nondiffracting nature of the evanescent Bessel beam. (d) Exponential decay of the evanescent Bessel beam [82].
Fig. 39
Fig. 39 Plasmonic lens structure with multiple concentric rings illuminated by radial polarization. Surface plasmons generated by each of the rings propagate to the center and constructively interfere to produce a tight plasmonic focus (shown to the right).
Fig. 40
Fig. 40 Conical glass tip fully coated with thin silver film. The entire radial polarization input is p polarized with respect to the glass/silver interface and creates surface plasmons that propagate to the tip apex.
Fig. 41
Fig. 41 Logarithmic plots of (a) 2D and (b) 3D electric energy density distribution at the end of the probe tip under radial polarization illumination.
Fig. 42
Fig. 42 (a) Calculated E 2 distribution at the focal plane for a radially polarized beam focused by a high-NA objective. For comparison, the corresponding result from a linearly polarized beam incidence under the same focusing condition is also shown. (b) 2D plot of the time-averaged Poynting vector S z in the r z plane for highly focused radial polarization. (c) Line scan of (b) at the focal plane.
Fig. 43
Fig. 43 Generation of optical chain. (a) Incident radial polarization. (b) Pupil plane DOE that modulates both phase and amplitude of the incident radial polarization. (c) One example of optical chain created with the pupil plane DOE shown in (b). (d) Time-averaged Poynting vector S z [89].
Fig. 44
Fig. 44 Field distribution shifts as the phase difference Δ φ between zone I and zone III varies from 0 to 2 π [89].
Fig. 45
Fig. 45 Magnetic field and electric field at the focal plane for a highly focused ( NA = 0.95 ) azimuthal polarization.
Fig. 46
Fig. 46 Calculated absorption coefficients for s polarization and p polarization of mild steel [96].
Fig. 47
Fig. 47 Expected laser machining cross section for radial and azimuthal polarizations [96].
Fig. 48
Fig. 48 Calculated hole cross sections and corresponding absorbed laser fluence for mild steel with azimuthal and radial polarizations. The guiding effect is taken into consideration. From the simulations, it can be seen that azimuthal polarization is more energy efficient in this case. Figure reprinted from [96] with kind permission of Springer Science+Business Media.
Fig. 49
Fig. 49 Calculated hole cross sections and corresponding absorbed laser fluence for brass with azimuthal and radial polarizations. The guiding effect is taken into consideration. From the simulations, it can be seen that radial polarization is more energy efficient in this case. Figure reprinted from [96] with kind permission of Springer Science+Business Media.

Equations (88)

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( 2 + k 2 ) E = 0 ,
E ( x , y , z , t ) = u ( x , y , z ) exp [ i ( k z ω t ) ] .
2 u z 2 k 2 u , 2 u z 2 k u z ,
u ( x , y , z ) = E 0 H m ( 2 x w ( z ) ) H n ( 2 y w ( z ) ) w 0 w ( z ) exp [ i φ m n ( z ) ] exp [ i k 2 q ( z ) r 2 ] ,
d 2 H m d x 2 2 x d H m d x + 2 m H m = 0
u ( r , z ) = E 0 w 0 w ( z ) exp [ i φ ( z ) ] exp [ i k 2 q ( z ) r 2 ] ,
E ( r , ϕ , z , t ) = u ( r , ϕ , z ) exp [ i ( k z ω t ) ] .
1 r r ( r u r ) + 1 r 2 2 u ϕ 2 + 2 i k u z = 0 .
u ( r , ϕ , z ) = E 0 ( 2 r ω ) l L p l ( 2 r 2 ω 2 ) w 0 w ( z ) exp [ i φ p l ( z ) ] exp [ i k 2 q ( z ) r 2 ] exp ( i l ϕ ) ,
x d 2 L p l d x 2 ( l + 1 x ) d L p l d x + p L p l = 0
u ( r , z ) = E 0 w 0 w ( z ) exp [ i φ ( z ) ] exp [ i k 2 q ( z ) r 2 ] J 0 ( β r 1 + i z z 0 ) exp [ β 2 z ( 2 k ) 1 + i z z 0 ] ,
× × E k 2 E = 0 ,
E ( r , z ) = U ( r , z ) exp [ i ( k z ω t ) ] e ϕ ,
1 r r ( r U r ) U r 2 + 2 i k U z = 0 .
U ( r , z ) = E 0 J 1 ( β r 1 + i z z 0 ) exp [ i β 2 z ( 2 k ) 1 + i z z 0 ] u ( r , z ) ,
H ( r , z ) = H 0 J 1 ( β r 1 + i z z 0 ) exp [ i β 2 z ( 2 k ) 1 + i z z 0 ] u ( r , z ) exp [ i ( k z ω t ) ] h ϕ ,
E ( r , z ) = A r exp ( r 2 w 2 ) e i , i = r , ϕ .
E r = H G 10 e x + H G 01 e y ,
E ϕ = H G 01 e x + H G 10 e y ,
E ( r ) = P ( r ) e i , i = r , ϕ ,
P ( r ) = { 1 , r 1 < r < r 2 0 , 0 < r < r 1 } .
E in = e x + j e y = ( cos ϕ e r sin ϕ e ϕ ) + j ( sin ϕ e r + cos ϕ e ϕ ) = e j ϕ ( e r + j e ϕ ) ,
E out = e j ϕ e r .
T = ( cos ( 2 Δ φ ) sin ( 2 Δ φ ) sin ( 2 Δ φ ) cos ( 2 Δ φ ) ) = R ( 2 Δ φ ) ,
T = R ( π 2 ) ( 1 0 0 j ) R ( π 2 ) R ( π 4 ) ( 1 0 0 e j δ ) R ( π 4 ) ( 1 0 0 j ) = j e j δ 2 R ( δ 2 ) ,
E i ( ρ , φ ) = l 0 P ( ρ ) [ cos φ 0 e ρ + sin φ 0 e φ ] ,
e ρ = cos φ e x + sin φ e y ,
e φ = sin φ e x + cos φ e y .
ρ f = g ( θ ) ,
[ l 0 P ( ρ ) ] 2 2 π ρ d ρ = [ l 0 P ( θ ) ] 2 2 π f 2 sin θ d θ ,
P ( θ ) = P ( ρ ) g ( θ ) g ( θ ) sin θ = P ( f g ( θ ) ) g ( θ ) g ( θ ) sin θ .
ρ f = sin θ .
P ( θ ) = P ( f sin θ ) cos θ .
e r = cos θ ( cos φ e x + sin φ e y ) + sin θ e z ,
e φ = e φ = sin φ e x + cos φ e y .
E ( r , ϕ , z ) = i k 2 π Ω a ( θ , φ ) e i k ( s r ) d Ω = i k 2 π 0 θ max d θ 0 2 π a ( θ , φ ) e i k ( s r ) sin θ d φ ,
a ( θ , φ ) = l 0 f P ( θ ) [ cos φ 0 e r + sin φ 0 e φ ] .
s r = z cos θ + r sin θ cos ( φ ϕ ) .
E ( r , ϕ , z ) = i k 2 π 0 θ max d θ 0 2 π l 0 f P ( θ ) [ cos φ 0 e r + sin φ 0 e φ ] e i k ( z cos θ + r sin θ cos ( φ ϕ ) ) sin θ d φ = i A π 0 θ max d θ 0 2 π P ( θ ) [ cos φ 0 e r + sin φ 0 e φ ] e i k ( z cos θ + r sin θ cos ( φ ϕ ) ) sin θ d φ = i A π 0 θ max d θ 0 2 π P ( θ ) [ cos φ 0 ( cos θ cos φ e x cos θ sin φ e y sin θ e z ) + sin φ 0 ( sin φ e x cos φ e y 0 e z ) ] e i k ( z cos θ + r sin θ cos ( φ ϕ ) ) sin θ d φ ,
A = π f l 0 λ .
e r = cos ϕ e x + sin ϕ e y ,
e ϕ = sin ϕ e x + cos ϕ e y .
E ( r , ϕ , z ) = i A π 0 θ max d θ 0 2 π P ( θ ) [ cos φ 0 ( cos θ cos ( φ ϕ ) e r 0 e ϕ sin θ e z ) + sin φ 0 ( 0 e r cos ( φ ϕ ) e ϕ 0 e z ) ] e i k ( z cos θ + r sin θ cos ( φ ϕ ) ) sin θ d φ .
0 2 π cos ( n φ ) e i k r sin θ cos φ d φ = 2 π i n J n ( k r sin θ ) ,
E ( r , ϕ , z ) = E r e r + E z e z + E ϕ e ϕ ,
E r ( r , ϕ , z ) = 2 A cos φ 0 0 θ max P ( θ ) sin θ cos θ J 1 ( k r sin θ ) e i k z cos θ d θ ,
E z ( r , ϕ , z ) = i 2 A cos φ 0 0 θ max P ( θ ) sin 2 θ J 0 ( k r sin θ ) e i k z cos θ d θ ,
E ϕ ( r , ϕ , z ) = 2 A sin φ 0 0 θ max P ( θ ) sin θ J 1 ( k r sin θ ) e i k z cos θ d θ .
g ( θ ) = 2 sin ( θ 2 ) ,
P ( θ ) = P ( f g ( θ ) ) .
g ( θ ) = θ ,
P ( θ ) = P ( f g ( θ ) ) θ sin θ .
g ( θ ) = tan θ ,
P ( θ ) = P ( f g ( θ ) ) ( 1 cos θ ) 3 .
E ( r , ϕ , z ) = i A { [ I 0 + cos ( 2 ϕ ) I 2 ] e x + sin ( 2 ϕ ) I 2 e y 2 i cos ϕ I 1 e z } ,
I 0 = 0 θ max P ( θ ) sin θ ( 1 + cos θ ) J 0 ( k r sin θ ) e i k z cos θ d θ ,
I 1 = 0 θ max P ( θ ) sin 2 θ J 1 ( k r sin θ ) e i k z cos θ d θ ,
I 2 = 0 θ max P ( θ ) sin θ ( 1 cos θ ) J 2 ( k r sin θ ) e i k z cos θ d θ .
T ( θ ) = { 1 for 0 θ < θ 1 , θ 2 θ < θ 3 , θ 4 θ < sin 1 ( NA ) 1 for θ 1 θ < θ 2 , θ 3 θ < θ 4 } ,
E r ( r , ϕ , z ) = 2 A cos φ 0 0 θ max P ( θ ) T ( θ ) sin θ cos θ J 1 ( k r sin θ ) e i k z cos θ d θ ,
E z ( r , ϕ , z ) = i 2 A cos φ 0 0 θ max P ( θ ) T ( θ ) sin 2 θ J 0 ( k r sin θ ) e i k z cos θ d θ .
d φ ( z ) = ( k eff k 0 ) d z = ( k 0 2 k 2 k 0 ) d z ,
d φ G ( z ) d z = + + F ( k x , k y , z ) 2 ( k 0 2 k 2 k 0 ) d k x d k y .
d φ G ( z ) d z = 0 k max 0 2 π F ( k , z ) 2 ( k 0 2 k 2 k 0 ) k d k d ϕ .
φ G = + d φ G ( z ) d z d z = + 0 k max 0 2 π F ( k , z ) 2 ( k 0 2 k 2 k 0 ) k d k d ϕ d z .
k 0 2 k 2 k 0 k 2 2 k 0 .
k 0 2 k 2 k 0 k 0 { 1 1 2 [ ( k k 0 ) 2 ] 1 8 [ ( k k 0 ) 2 ] 2 } k 0 = k 2 2 k 0 k 4 8 k 0 3 .
E ( r , ϕ , z ) = i A π θ 1 θ 2 d θ 0 2 π { [ cos θ + sin 2 ϕ ( 1 cos θ ) cos θ e x sin ϕ cos ϕ ( 1 cos θ ) cos θ e y sin θ cos ϕ cos θ e z ] e i k 0 cos θ z e i ( k x x + k y y ) sin θ cos θ d ϕ } ,
F ( k , z = 0 ) = [ cos θ + sin 2 ϕ ( 1 cos θ ) cos θ ] N ,
N = θ 1 θ 2 d θ 0 2 π cos θ + sin 2 ϕ ( 1 cos θ ) cos θ 2 k 0 2 sin θ cos θ d ϕ = π k 0 2 [ cos 2 θ 1 cos 2 θ 2 ] + π k 0 2 4 [ ( 1 cos θ 2 ) 3 ( 1 cos θ 1 ) 3 ] .
d φ G ( 0 ) d z = k 2 2 k 0 k 4 8 k 0 3 = k x 2 + k y 2 2 k 0 k x 4 + 2 k x 2 k y 2 + k y 4 8 k 0 3 .
φ G = d φ G ( 0 ) d z Δ z = [ k x 2 + k y 2 2 k 0 + k x 4 + 2 k x 2 k y 2 + k y 4 8 k 0 3 ] Δ z ,
Δ z = 2 λ cos θ 1 cos θ 2 λ 2 .
E z ( r , Φ , z ) = i A π θ 1 θ 2 d θ 0 2 π sin θ cos θ e i k 0 z cos θ e i ( k x x + k y y ) sin θ cos θ d ϕ .
F ( k , z = 0 ) = [ sin θ cos θ ] N ,
N = A θ 1 θ 2 d θ 0 2 π sin θ cos θ 2 k 0 2 sin θ cos θ d ϕ = 2 π k 0 2 { 1 3 ( cos θ 1 3 cos θ 2 3 ) + ( cos θ 1 cos θ 2 ) } .
d φ G ( 0 ) d z = 0 k max 0 2 π F ( k x , k y , 0 ) 2 ( k 0 2 k 2 k 0 ) k d k d ϕ = 2 π k 0 3 ( sin 4 θ 2 sin 4 θ 1 ) ( 4 N ) k 0 .
k eff Δ z k 0 Δ z = d φ G ( z ) d z Δ z ,
λ eff = λ 1 + d φ G ( z ) d z λ 2 π .
E r ( r , ϕ , z ) = 2 A θ min θ max P ( θ ) t p ( θ ) sin θ cos θ J 1 ( k 1 r sin θ ) exp [ i z ( k 2 2 k 1 2 sin 2 θ ) 1 2 ] d θ ,
E z ( r , ϕ , z ) = i 2 A θ min θ max P ( θ ) t p ( θ ) sin 2 θ J 0 ( k 1 r sin θ ) exp [ i z ( k 2 2 k 1 2 sin 2 θ ) 1 2 ] d θ ,
F grad = Re ( α ) ɛ 0 E 2 4 ,
F scat , z = n 1 S z C scat c ,
F abs , z = n 1 S z C abs c ,
P ( θ ) = { P 0 if sin 1 ( NA 1 ) θ sin 1 ( NA n 1 ) 0 otherwise } ,
E ( z ) 2 = [ E I ( z ) + E III ( z ) ] [ E I * ( z ) + E III * ( z ) ] = E I ( z ) 2 + E III ( z ) 2 + 2 E I ( z ) 2 E III ( z ) 2 cos δ = E I ( z ) 2 + E III ( z ) 2 + 2 E I ( z ) 2 E III ( z ) 2 cos ( Δ k z z + Δ ϕ ) ,
p m c , E c B , c B E ,
F = ( α m 2 ) B 2 + ω α m ( E × B ) c 2 ,

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