Abstract

We report selective excitations of higher-order Hermite-Gaussian and Ince-Gaussian (IG) modes in a laser-diode-pumped microchip solid-state laser and controlled generation of corresponding higher-order and multiple optical vortex beams of different shapes using an astigmatic mode converter (AMC). Simply changing the pump-beam diameter, shape, and lateral off-axis position of the tight pump beam focus on the laser crystal within a microchip semispherical cavity can produce the desired optical vortex beams in a well controlled manner. Pattern changes featuring different IG and HG modes obtained by rotating the AMC are also demonstrated. Numerical simulation shows that the vortex structure is changed by controlled off-axis laser diode pumping, which could lead toward precise optical manipulation of small particles.

© 2008 Optical Society of America

1. Introduction

Two transverse lasing modes, Hermite-Gaussian modes (HGMs) and Laguerre-Gaussian modes (LGMs), have been widely investigated using analytical, numerical, and experimental techniques. These two modes separately form two complete families of exact and orthogonal solutions of the paraxial wave equation (PWE) in rectangular and cylindrical coordinates. Researchers have recently proposed a third complete family of transverse modes, namely Ince-Gaussian modes (IGMs), as a third complete family of PWE solutions in elliptic coordinates [1,2]. They have observed IGMs in a stable resonator in a laser-diode-pumped (LD-pumped) solid-state laser by breaking the symmetry of the cavity under tight pump beam focusing conditions [3,4]. These attempts include introducing off-axis pumping with a cross hair inside the cavity [3] and adjusting the azimuthal symmetry of the short laser resonator [4,5]. We have verified off-axis pump-beam focusing conditions for generating desired IGMs by numerical simulations of the model of LD-pumped microchip solid-state lasers [6].

On the other hand, researchers have widely used optical vortex beams [7], which are optical beams that have phase singularities mixed with wavefront curvature, in the study of optical tweezers [8–12], trapping and guiding of cold atoms [13–15], rotational frequency shift [16,17], and entanglement states of photons [18]. Optical vortex beams have been reported to be created by passing HGM beams through spiral phase-plates [19], computer-generated holographic converters [20], or astigmatic mode converters (AMC) [21].

This paper demonstrates selective excitations of higher-order HGMs and IGMs in a microchip NdGdVO4 laser with off-axis LD pumping and controlled generation of several corresponding higher-order optical vortices by an AMC, featuring a novel type of non-circular optical vortex originating from higher-order IGMs.

2. Experimental results

2.1 Experimental setup

The experimental setup is shown in Fig. 1. The experiment was carried out using an LD-pumped 1-mm-thick 3 at.% Nd-doped a-plate Nd:GdVO4 crystal (refractive index n=2.0). A Nd:GdVO4 crystal was placed within a semispherical external cavity, where the sample was attached to a plane mirror M1 (99.8% reflective at 1064 nm and >95% transmissive at 808 nm), and a concave mirror M2 (99% reflective at 1064 nm, radius of curvature: 100 mm) was placed 10 mm away from the plane mirror. The two mirrors and the laser crystal were assembled into one body. An elliptical LD beam was transformed into a circular one and focused onto the Nd:GdVO4 crystal by using a microscope objective lens with a numerical aperture of 0.25 to obtain a tight focus giving a minimum spot size of approximately 75 µm at the crystal, while the lasing beam-waist spot size for the fundamental HG00 mode was 100 µm. The shape of the pump beam at the crystal was changed to non-circular by slightly changing the crystal along the lasing axis (z-axis) through an astigmatic aberration. The absorption coefficient for the LD wavelength of 808 nm was 74 cm-1. The resultant absorption length was as short as 135 µm. In all cases, linearly π-polarized emissions along the tetragonal c-axis were observed.

The lasing beam was delivered to an AMC through a mode-matching lens, as shown in Fig. 1, in which the separation of cylindrical lenses was precisely adjusted to L=21/2 f=70.71 mm (f=50 mm: focal length of cylindrical lenses). The optical design is summarized in Table 1. Both surfaces of each cylindrical lens were coated to be anti-reflective. The pair of cylindrical lenses was rotated by a step motor.

 

Fig. 1. Experimental setup.

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Tables Icon

Table 1. Design of optical elements and their optical distances.

2.2 Selective excitation of higher-order HGM operations and donut-like vortex beam generation

Any Laguerre-Gaussian (LG) mode propagating along the z-axis can be decomposed into a set of Hermite-Gaussian (HG) modes as follows [22].

ul,pLG(r,ϕ)=k=0NBkikuNk,kHG(x,y)

Here, u denotes the field function of eigenmodes, N=l+p, and Bk are real expansion coefficients that satisfy the normalization condition ∑Bk 2=1. In the case of p=0, each photon carries an orbital angular momentum of [23]. The factor ik, where i2=-1, corresponds to a π/2 relative phase difference between successive HG components.

On the other hand, any HG mode whose principal axes make an angle of 45° with the (x,y) axes (diagonal modes) can be decomposed into exactly the same constituents set with the same coefficients Bk but without the ik factor. For example, an HG1,0 mode, aligned at 45° to the (x,y) axes, can be expressed as two in-phase HG1,0 and HG0,1 modes aligned with the principal axes of the lenses. The decomposition of the diagonal HG2,0 and LG2,0 modes into u2,0, u1,1, u0,2 is shown in Fig. 2.

 

Fig. 2. Decomposition of the diagonal HG2,0 and LG2,0 modes.

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Therefore, in principle, any HG mode with indices m and n (denoted HGm,n), aligned at 45° to the principal axis of the lens, will then be converted into an LG mode with the same beam waist as a result of relative phase shifts introduced into the diagonal modes by an AMC [21]. For higher-order HGMs, the expansion and transformation is more complex, but the same principles apply. The separation of the cylindrical lenses L is made such that these orthogonal modes undergo different Gouy phase shifts. Any HG mode is transformed into a corresponding LG mode with l=m-n and p=min(m,n) after the lenses, where l is the azimuthal mode index and p describes the number of radial nodes in the field.

By shifting the position of tight beam focus laterally from the central axis of the laser cavity (i.e., off-axis pumping) and longitudinally to get a nearly ‘elliptic’ pump-beam focus due to aberration, we achieved a variety of higher-order HGm,0 mode oscillations. Mode number m was increased with increasing shift d, such that the pump-beam position was adjusted to the position of one of the brightest outermost spots, namely the ‘target spot’, depicted in Fig. 1.

When we increased d gradually, higher-order HGm,0 oscillations and their corresponding LGl,p modes (l=m, p=0) were easily obtained successively at a fixed pump power. Example results are shown in Fig. 3, where θ is the rotation angle of the AMC depicted in Fig. 1. The structural changes in beam patterns caused by the AMC rotation (i.e., azimuthal symmetry variation) for the HG4,0 mode is shown in Fig. 4 and the movie. Note that the same pattern repeated at every 90° rotation of the AMC. These patterns manifested themselves in optical vortices possessing angular momentum of (l: topological charge), whose diameter is proportional to l 1/2 [23]. The mode conversion efficiency was measured to be >95% on average in the entire pump-power region up to 400 mW from a comparison of the input HGM power and the corresponding LGM power.

 

Fig. 3. Successive generation of LGl,0 modes with increasing d. Pump power P=182 mW. Threshold pump power: 36 mW.

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Fig. 4. (1884 KB) Structural change of the HG4,0 mode with a rotation of the AMC. P=182 mW. [Media 1]

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Similar structural pattern changes for HGMs have been demonstrated using a variable-phase-shift mode converter consisting of two π/2 converters at ±45° to the beam and an image rotator R(ϕ) comprising two Dove prisms [24]. When the two parts of the image rotator are rotated with respect to each other through an angle ϕ/2, the image between the two π/2 converters is rotated through an angle ϕ. In this case, unlike in our present experiment, the variable-phase-shift mode converter acts as a 2ϕ converter and the pattern repeats at every 180° rotation of ϕ [24].

2.3 Forced IGM operations and non-circular vortex laser beam generation

The Ince-Gaussian (IG) modes propagating along the z-axis of an elliptic coordinate system of r=(ξ, η, z) with mode numbers p and m and ellipticity ε are given by [2]

IGep,m(r,ε)=C[w0w(z)]Cpm(iξ,ε)Cpm(η,ε)exp[r2w2(z)]
×expi[kz+{kr22R(z)}(p+1)ψz(z)],
IGop,m(r,ε)=S[w0w(z)]Spm(iξ,ε)Spm(η,ε)exp[r2w2(z)]
×expi[kz+{kr22R(z)}(p+1)ψz(z)],

where the elliptic coordinate is defined in a transverse z plane as x=f(z) cos ξ cos η, y=f(z) sin ξ sin η, and ξ∊[0, ∞], η ∊[0, 2π]. Here, f(z) is the semifocal separation of IGMs defined as the Gaussian beamwidth, i.e., f(z)=f 0 w(z)/w 0, where f 0 and w 0 are the semifocal separation and beamwidth at the z=0 plane, respectively. Here, w(z)=w 0 (1+z 2/z R 2)1/2 describes the beamwidth, zR=kw 0 2/2 is the Rayleigh length, and k is the wave number of the beam. C and S are normalization constants, the subscripts e and o refer to even and odd IGMs, and Cpm (., ε) and Sp m (., ε) are the even and odd Ince polynomials [2] of order p, degree m, and parameter ε, respectively. In Eqs. (1) and (2), r is the radial distance from the central axis of the cavity, R(z)=z+zR 2/z is the radius of curvature of the phase front, and Ψ z(z)=arctan(z/zR). The parameters of ellipticity ε, waist w 0, and semifocal separation f 0 are not independent, but related by ε=2f 0 2/w 0 2. The patterns of IGMs can be recognized by the rules that degree m corresponds to the number of hyperbolic nodal lines and (p-m)/2 is the number of elliptic nodal lines. Example analytic patterns of IGp,m modes are shown in Fig. 5.

 

Fig. 5. Some analytic patterns of the Ince-Gaussian modes.

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Let us examine the relationship between IGMs and HGMs. The transition from IGe,op,m to HGnx,ny occurs when ellipticity ε→∞, and their indices are related as follows [2]: For even IGMs, nx=m and ny=p-m, whereas for odd IGMs, nx=m-1 and ny=p-m-1.

Example results indicate that some IGop,1 and IGep,p modes with ε=4 approach their corresponding HGm,0 mode with ε=1000, as shown in Figs. 6(a)–6(c), respectively. Note that the shapes of the largest and most intense outside lobes of IG modes bent to become ‘half-elliptic’ or became more symmetric and circular than those of HGp,0 modes for IGop,1 and IGep,p modes, respectively [6], but their distances from the central axis are the same. Our idea for forcing single IG mode operations can be implemented by controlling the size and shape of the pump beam focus such that they match the outside lobes of the desired IG modes in a laser cavity, i.e., by performing controlled off-axis pumping [6].

 

Fig. 6. Approach from Ince-Gaussian to Hermite Gaussian modes with increasing ellipticity. (a) IGep,p modes and (b) IGep,0 and IGop,1. (Here, ellipticity ε : 4→103).

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For any plane z, the IG↔LG expansions are written as follows [2].

uσl,pLG(r,ϕ)=k=0NDkuσ2l+p,kIG(ξ,ε,η)
uσp,mIG(ξ,η,ε)=l,p=0NDl,puσl,pLG(r,ϕ)

Here, σ={e, o} is the parity and ∑D j 2=1. Once we know the IG↔LG relations, the IG↔HG formulas can be readily obtained by applying the already known LG↔HG expansions (1) in cascade with the IG↔LG expansions. Consequently, in principle, various mode patterns are expected to appear from the AMC with a change in the rotation angle, i.e., azimuthal symmetry. Below, we describe several examples.

When we introduced the appropriate asymmetry into the pump-beam focus profile by slightly tilting the pump-beam direction (oblique pumping) [6], we could achieve IGep,0 or IGop,1 mode oscillations. Delivering these IGMs into the AMC yielded the corresponding optical vortices. An example result is shown in Fig. 7, where the structural change of outgoing beam patterns from the AMC is demonstrated in the videos. Note that the asymmetric shapes of the vortices differ from the donut-like patterns formed from HGm,0 modes shown in Fig. 4. By controlling the pump focus to be tighter and more circular than that for selective excitation of HGm,0, we successfully obtained a variety of IGp,p mode oscillations by systematic off-axis pumping. A typical example demonstrating structural changes is shown in Fig. 8. In both cases, the intensity patterns are not circular, but have rectangular-like or squarish shapes and the intensity profiles of the vortices changed critically around θ=45°, as shown in Figs. 7 and 8. This point will be discussed again theoretically later. Mode conversions between IGo3,1 and HG3,0 as well as IGe3,3 and IGo3,1 modes took place when the AMC angle approached 0° and 90°. The pattern repeated at every 90° rotation of the AMC, similar to Fig. 4.

 

Fig. 7. (1839 KB) Structural pattern change with AMC rotation for the IGo3,1 mode. P=182 mW. [Media 2]

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Fig. 8. (2032 KB) Structural pattern change with AMC rotation for IGe3,3 mode. P=182 mW. [Media 3]

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As for HGm,n (n≠0) and IG modes with large p values, complicated beam pattern changes were observed by rotating the AMC, as shown in Figs. 9–10.

 

Fig. 9. (2000 KB) Structural pattern change with AMC rotation for HG3,1 mode. P=234 mW. [Media 4]

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Fig. 10. (1878 KB) Structural pattern change with AMC rotation for IGe4,4 mode. P=234 mW. [Media 5]

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3. Theoretical results

3.1 Simulation model

To obtain selective excitations of lasing modes, we simulated the initial stimulated field with a partially coherent field [25] in the space-frequency domain to avoid dependence between the initial field selection and the conversion field in a stable laser cavity using Endo’s method [26]. The stimulated initial field propagating back and forth in the resonator is mimicked by the Fresnel-Kirchhoff integration [27]; and the optical fields changed by laser mirrors and gain medium are introduced by modifying the optical field at each position under the off-axis pumping condition [6,28]. The loaded gain at each station is assumed to be homogeneously broadened. The gain medium was simulated as several gain sheets and the saturated gain at each gain sheet i is expressed as

gi(x,y)=gi0(x,y)(1+I~i+(x,y)+I~i(x,y)Is(x,y)),

where g0(x,y) is the loaded gain, gi0(x,y) is the small signal gain, and Is(x,y) is the saturation intensity. In this paper, we assume that Is≈1 kW/cm2, which is reasonable for general solid-state lasers [27]. The symbols Ĩ + i and Ĩ - i are the average right-going and left-going optical intensities defined as

I~i+(x,y)=(1α)i=0qαiIi+(qi),
I~i(x,y)=(1α)i=0qαiIi(qi).

Here, I + i (q) and I + i (q) denote the intensities of the qth iteration step and α is a summation over a period of the cavity’s photon decay time. For example, in this simulation, the reflectivities of the two mirrors were set to r1=100.0% and r2=99%. Thus, the parameter α is given by r1r2=0.99.

The amplification of the optical field E(x,y) passing through gain sheet i with thickness d is expressed as

Eiout(x,y)=Eiin(x,y)exp[12gi(x,y)d],

where the 1/2 is necessary because the gain is defined by the amplification of the optical intensity. With this simulation method, after a certain number of iterations, according to the boundary condition, the cavity will find the lasing mode distribution E(x, y) that satisfies

Eq+1(x,y)~Eq(x,y).

To model the performance of the mode converter we used a wave propagation algorithm, i.e., Fast Fourier transform method [29,30]. Outgoing fields from the AMC were numerically simulated.

3.2 Numerical results

Numerical simulations were carried out using relevant parameters for the laser cavity, mode-matching lens, and AMC that we used in the experiment. They are summarized in Table 1. Typical examples are shown in Figs. 11, 12, and 13, respectively. The numerical results well reproduced the experimental observations shown in Figs. 4, 7, and 8.

 

Fig. 11. Numerical result indicating a structural change in the HG4,0 mode intensity pattern with a rotation of AMC and the corresponding phase portrait of the donut-like multi-vortex at θ=45°.

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On the other hand, intensity profiles for IGM-originating vortices (Figs. 12 and 13) are not circular donut-like shapes unlike those for HGM-originating vortices (Fig. 11), but rectangular- or square-like shapes and they change critically around θ=45°, similar to the experimental results.

 

Fig. 12. Numerical result indicating a structural change in the IG° 3,1 mode intensity pattern with a rotation of AMC.

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Fig. 13. Numerical result indicating a structural change of the IGe3,3 mode intensity pattern with a rotation of AMC.

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Let us examine the structural change of such IGM-originating optical vortices with the rotation of the AMC around θ=45°. The computed interference fringes corresponding to intensity profiles at θ=35°, 45°, and 55° in Figs. 12 and 13 are shown in Fig. 14, where interference fringes were calculated by causing the vortex beam to interfere with a tilted plane wave, where the tilt angle relative to the propagation direction was 1°.

 

Fig. 14. Computed changes in phase portraits and interference fringes across θ=45°. (a) IGo3,1 mode and (b) IGe3,3 (right) mode. Points of phase singularity are indicated by red dots.

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Note that ‘in-line’ type vortices were formed along the longer elliptic axis and rotated by 90° across θ=45°, as indicated by red dots for both cases. The linear array of vortices (with high beam purity and power) may find applications for optical tweezers or for guiding condensates. Similar in-line vortices have been created in the form of helical IG modes with an argon laser operating at 514.5-nm wavelength by using complex amplitude and phase masks encoded onto a liquid-crystal display, in which an intensity pattern consists of an ‘elliptic’ ring [31].

The structure of optical vortices is expected to depend on the ellipticity ε even if IGMs possess the same mode numbers (p, m) and parity. The computed intensity profiles, phase portraits, and interference fringes for IG03,1 mode are shown in Fig. 15 together with those for HG3,0, where the LD pump-focus position was changed in the simulation. The intensity profile changed from a rectangular-like to circular donut-like shape as ε increased, and spatially separated singular points merged into the common central point accordingly. The numerical result suggests that one can control the property of optical vortices for optical manipulation of small particles by changing the ellipticity of IGMs with controlled off-axis LD pumping [6]. A systematic experimental study of this issue is in progress.

 

Fig. 15. Computed structural change in vortex beam created from IG03,1 modes with different values of ellipticity parameter ε and θ=45°. (a) Intensity patterns of IG03,1 modes, (b) intensity profiles of vortices, (c) phase portraits, and (d) interference fringe patterns. Results for HG3,0-originating vortices are shown in the bottom row for reference.

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4. Summary and discussion

We have produced circular donut-like, rectangular-like, and square-like vortices possessing multiple topological changes from a well-designed astigmatic mode converter (AMC) by using a microchip solid-state laser operating in higher-order Hermite and Ince-Gaussian modes with controlled laser-diode off-axis end pumping. Structural beam pattern changes with continuous rotation of the AMC, which feature conversions among three orthogonal modes (i.e., HGM, IGM, and LGM) leading to successful vortex formations, have been demonstrated experimentally and well reproduced by numerical simulation. It has been also verified by numerical simulation that the intensity profiles and spatial distribution of phase singularities of multi-vortex laser beams can be changed by the controlled laser-diode off-axis pumping and AMC rotation. The present control of optical vortices by adjusting the pump-focus lateral position and shape at the laser crystal with off-axis laser-diode pumping suggests the potential of IGMs for advanced manipulation of small particles using microchip solid-state lasers.

Finally, let us briefly address advantages and disadvantages of AMC approach for generating multiple vortex laser beams in comparison with the spatial light modulator (SLM) approach, which employs a holographic element [32] or an appropriate amplitude and phase mask encoded onto a liquid-crystal display [31].

In AMC approach, vortex laser beams are generated by a mode conversion of higher-order transverse lasing modes that are directly emitted from the microchip laser. Such beams are the superposition of eigenmodes of the paraxial wave equation: thus, they are also the solution of PWE. Therefore, the shape invariance of the resulting vortex beam is ensured against propagation [5] and the beam quality is not degraded in the far-field. While, in SLM approach, the performance of vortex generations (e.g., conversion efficiency and mode purity) from the fundamental transverse mode is limited by the precision of techniques for fabricating a sophisticated vortex lens. Other disadvantages of SLM include high loss, small acceptance angle and high cost [32]. In our AMC approach, high conversion efficiencies (i.e., more than 95 %) from HGMs or IGMs to various structures of vortices were achieved with the simplest, solid device configuration with low cost. The advantages of SLM approach might be the dynamic control of phase plates, i.e., flexibility [32]. In the liquid-crystal display type of SLM [31], for example, the electrical control of a phase plate can be utilized for dynamic control of vortices [31]. In AMC approach, on the other hand, such a dynamic control of phase shift is not attained in general. However, controlling spatial arrangements of phase singularities and shapes of vortex beams has been demonstrated to be possible based on controlled IGMs by simply controlling the pump position, shape and the “rotation” of AMC [Figs. 7–8 and Figs. 14–15] toward applications for optical tweezers or guiding condensates. A variety of mode conversions among three orthogonal HG, IG and LG orthogonal lasing eigenmodes and the potential of producing various laser-beam patterns by controlling the AMC angle have also been shown [Figs. 7–10 and Figs. 12–13]. In the conventional AMC approach, the AMC angle is fixed to 45 degree and such rotation-dependent non-trivial modal structural changes by the precise rotation control of AMC was not reported so far. It might require sophisticated control of phase plates or electrics in SLM approach for producing various laser-beam patterns and vortices.

Acknowledgment

This work was supported in part by a grant from the National Science Council of Taiwan, R.O.C., under contract no. NSC 96-2112-M-006 -019 -MY3.

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References

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  1. M. A. Bandres and J. C. Gutiérrez-Vega, "Ince-Gaussian beams," Opt. Lett. 29, 144 (2004).
    [CrossRef] [PubMed]
  2. M. A. Bandres and J. C. Gutiérrez-Vega, "Ince-Gaussian modes of the paraxial wave equation and stable resonators," J. Opt. Soc. Am. A 21, 873 (2004).
    [CrossRef]
  3. U. T. Schwarz, M. A. Bandres, and J. C. Gutiérrez-Vega, "Observation of Ince-Gaussian modes in stable resonators," Opt. Lett. 291870 (2004).
    [CrossRef] [PubMed]
  4. K. Otsuka, K. Nemoto, K. Kamikariya, and S.-C. Chu, "Linearly-polarized, single-frequency oscillations of laser-diode-pumped microchip ceramic Nd:YAG lasers with forced Ince-Gaussian mode operations," Jpn. J. Appl. Phys. 46, 5865 (2007).
    [CrossRef]
  5. T. Ohtomo, K. Kamikariya, K. Otsuka, and S.-C. Chu, "Single-frequency Ince-Gaussian mode operations of laser-diode-pumped microchip solid-state lasers," Opt. Express 15, 10705 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-17-10705.
    [CrossRef] [PubMed]
  6. S.-C. Chu and K. Otsuka, ""Numerical study for selective excitation of Ince-Gaussian modes in end-pumped solid-state lasers," Opt. Express 15, 16506 (2007).
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  8. N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, "Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner," Opt. Lett. 22, 52 (1997).
    [CrossRef] [PubMed]
  9. E. Santamato, A. Sasso, B. Piccirillo, and A. Vella, "Optical angular momentum transfer to transparent isotropic particles using laser beam carrying zero average angular momentum," Opt. Express 10, 871 (2002).
    [PubMed]
  10. K. T. Gahagan and G. A. Swartzlander, Jr., "Optical vortex trapping of particles," Opt. Lett. 21, 827 (1996).
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  11. L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, "Controlled rotation of optically trapped microscopic particles," Science 292, 912 (2001).
    [CrossRef] [PubMed]
  12. M. P. MacDonald, "Revolving interference pattern for the rotation of optically trapped particles," Opt. Commun. 201, 21 (2002).
    [CrossRef]
  13. Y. Song, D. Milam, and W. T. Hill, "Long, narrow all-light atom guide," Opt. Lett. 24, 1805 (1999).
    [CrossRef]
  14. X. Xu, K. Kim, W. Jhe, and N. Kwon, "Efficient optical guiding of trapped cold atoms by a hollow laser beam," Phy. Rev. A 63, 3401 (2001).
    [CrossRef]
  15. T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, "Novel optical trap of atoms with a doughnut beam," Phys. Rev. Lett. 78, 4713 (1997).
    [CrossRef]
  16. J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, and M. J. Padgett, "Rotational frequency shift of a light beam," Phys. Rev. Lett. 81, 4828 (1998).
    [CrossRef]
  17. J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, "Measurement of the rotational frequency shift imparted to a rotating light beam possessing orbital angular momentum," Phys. Rev. Lett. 80, 013601 (1998).
    [CrossRef]
  18. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, "Entanglement of the orbital angular momentum states of photons," Nature 412, 313 (2001).
    [CrossRef] [PubMed]
  19. M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, "Helical-wavefront laser beams produced with a spiral phase plate," Optics Commun. 112, 321 (1994).
    [CrossRef]
  20. N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, "Generation of optical phase singularities by computer-generated holograms," Opt. Lett. 17, 221 (1992).
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  21. M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, "Astigmatic laser mode converters and transfer of orbital angular momentum," Opt. Commun. 96, 123 (1993).
    [CrossRef]
  22. I. Kimel and L. R. Elias, "Relations between Hermite and Laguerre Gaussian modes," IEEE J. Quantum Electron. 29, 2562 (1993).
    [CrossRef]
  23. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes," Phys. Rev. A 45, 8185 (1992).
    [CrossRef] [PubMed]
  24. A. T. O’Neil and J. Courtial, "Mode transformations in terms of the constituent Hermite-Gaussian or Laguerre-Gaussian modes and the variable-phase mode converter," Opt. Commun. 181, 35 (2000).
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  31. J. B. Bentley, J. A. Davis, M. A. Bandres, and J. C. Gutiérrez-Vega, "Generation of helical Ince-Gaussian beams with a liquid-crystal display," Opt. Lett. 31, 649 (2006).
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2007

K. Otsuka, K. Nemoto, K. Kamikariya, and S.-C. Chu, "Linearly-polarized, single-frequency oscillations of laser-diode-pumped microchip ceramic Nd:YAG lasers with forced Ince-Gaussian mode operations," Jpn. J. Appl. Phys. 46, 5865 (2007).
[CrossRef]

S. -C. Chu and K. Otsuka, "Stable donut-like vortex beam generation from lasers with controlled Ince-Gaussian modes," Appl. Opt. 46, (2007). http://www.opticsinfobase.org/abstract.cfm?URI=ao-46-31-7709.
[CrossRef] [PubMed]

T. Ohtomo, K. Kamikariya, K. Otsuka, and S.-C. Chu, "Single-frequency Ince-Gaussian mode operations of laser-diode-pumped microchip solid-state lasers," Opt. Express 15, 10705 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-17-10705.
[CrossRef] [PubMed]

S.-C. Chu and K. Otsuka, ""Numerical study for selective excitation of Ince-Gaussian modes in end-pumped solid-state lasers," Opt. Express 15, 16506 (2007).
[CrossRef] [PubMed]

2006

2004

2002

2001

X. Xu, K. Kim, W. Jhe, and N. Kwon, "Efficient optical guiding of trapped cold atoms by a hollow laser beam," Phy. Rev. A 63, 3401 (2001).
[CrossRef]

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, "Controlled rotation of optically trapped microscopic particles," Science 292, 912 (2001).
[CrossRef] [PubMed]

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, "Entanglement of the orbital angular momentum states of photons," Nature 412, 313 (2001).
[CrossRef] [PubMed]

2000

A. T. O’Neil and J. Courtial, "Mode transformations in terms of the constituent Hermite-Gaussian or Laguerre-Gaussian modes and the variable-phase mode converter," Opt. Commun. 181, 35 (2000).
[CrossRef]

1999

1998

J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, and M. J. Padgett, "Rotational frequency shift of a light beam," Phys. Rev. Lett. 81, 4828 (1998).
[CrossRef]

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, "Measurement of the rotational frequency shift imparted to a rotating light beam possessing orbital angular momentum," Phys. Rev. Lett. 80, 013601 (1998).
[CrossRef]

1997

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, "Novel optical trap of atoms with a doughnut beam," Phys. Rev. Lett. 78, 4713 (1997).
[CrossRef]

N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, "Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner," Opt. Lett. 22, 52 (1997).
[CrossRef] [PubMed]

1996

1994

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, "Helical-wavefront laser beams produced with a spiral phase plate," Optics Commun. 112, 321 (1994).
[CrossRef]

1993

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, "Astigmatic laser mode converters and transfer of orbital angular momentum," Opt. Commun. 96, 123 (1993).
[CrossRef]

I. Kimel and L. R. Elias, "Relations between Hermite and Laguerre Gaussian modes," IEEE J. Quantum Electron. 29, 2562 (1993).
[CrossRef]

1992

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes," Phys. Rev. A 45, 8185 (1992).
[CrossRef] [PubMed]

N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, "Generation of optical phase singularities by computer-generated holograms," Opt. Lett. 17, 221 (1992).
[CrossRef] [PubMed]

1983

1975

Allen, L.

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, "Measurement of the rotational frequency shift imparted to a rotating light beam possessing orbital angular momentum," Phys. Rev. Lett. 80, 013601 (1998).
[CrossRef]

J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, and M. J. Padgett, "Rotational frequency shift of a light beam," Phys. Rev. Lett. 81, 4828 (1998).
[CrossRef]

N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, "Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner," Opt. Lett. 22, 52 (1997).
[CrossRef] [PubMed]

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, "Astigmatic laser mode converters and transfer of orbital angular momentum," Opt. Commun. 96, 123 (1993).
[CrossRef]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes," Phys. Rev. A 45, 8185 (1992).
[CrossRef] [PubMed]

Arlt, J.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, "Controlled rotation of optically trapped microscopic particles," Science 292, 912 (2001).
[CrossRef] [PubMed]

Bandres, M. A.

Beijersbergen, M. W.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, "Helical-wavefront laser beams produced with a spiral phase plate," Optics Commun. 112, 321 (1994).
[CrossRef]

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, "Astigmatic laser mode converters and transfer of orbital angular momentum," Opt. Commun. 96, 123 (1993).
[CrossRef]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes," Phys. Rev. A 45, 8185 (1992).
[CrossRef] [PubMed]

Bentley, J. B.

Bhowmik, A.

Bryant, P. E.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, "Controlled rotation of optically trapped microscopic particles," Science 292, 912 (2001).
[CrossRef] [PubMed]

Chu, S. -C.

S. -C. Chu and K. Otsuka, "Stable donut-like vortex beam generation from lasers with controlled Ince-Gaussian modes," Appl. Opt. 46, (2007). http://www.opticsinfobase.org/abstract.cfm?URI=ao-46-31-7709.
[CrossRef] [PubMed]

Chu, S.-C.

Coerwinkel, R. P. C.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, "Helical-wavefront laser beams produced with a spiral phase plate," Optics Commun. 112, 321 (1994).
[CrossRef]

Courtial, J.

A. T. O’Neil and J. Courtial, "Mode transformations in terms of the constituent Hermite-Gaussian or Laguerre-Gaussian modes and the variable-phase mode converter," Opt. Commun. 181, 35 (2000).
[CrossRef]

J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, and M. J. Padgett, "Rotational frequency shift of a light beam," Phys. Rev. Lett. 81, 4828 (1998).
[CrossRef]

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, "Measurement of the rotational frequency shift imparted to a rotating light beam possessing orbital angular momentum," Phys. Rev. Lett. 80, 013601 (1998).
[CrossRef]

Davis, J. A.

Dholakia, K.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, "Controlled rotation of optically trapped microscopic particles," Science 292, 912 (2001).
[CrossRef] [PubMed]

J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, and M. J. Padgett, "Rotational frequency shift of a light beam," Phys. Rev. Lett. 81, 4828 (1998).
[CrossRef]

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, "Measurement of the rotational frequency shift imparted to a rotating light beam possessing orbital angular momentum," Phys. Rev. Lett. 80, 013601 (1998).
[CrossRef]

N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, "Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner," Opt. Lett. 22, 52 (1997).
[CrossRef] [PubMed]

Elias, L. R.

I. Kimel and L. R. Elias, "Relations between Hermite and Laguerre Gaussian modes," IEEE J. Quantum Electron. 29, 2562 (1993).
[CrossRef]

Endo, M.

Fujioka, T.

Gahagan, K. T.

Gutiérrez-Vega, J. C.

Heckenberg, N. R.

Hill, W. T.

Hirano, T.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, "Novel optical trap of atoms with a doughnut beam," Phys. Rev. Lett. 78, 4713 (1997).
[CrossRef]

Jhe, W.

X. Xu, K. Kim, W. Jhe, and N. Kwon, "Efficient optical guiding of trapped cold atoms by a hollow laser beam," Phy. Rev. A 63, 3401 (2001).
[CrossRef]

Kamikariya, K.

K. Otsuka, K. Nemoto, K. Kamikariya, and S.-C. Chu, "Linearly-polarized, single-frequency oscillations of laser-diode-pumped microchip ceramic Nd:YAG lasers with forced Ince-Gaussian mode operations," Jpn. J. Appl. Phys. 46, 5865 (2007).
[CrossRef]

T. Ohtomo, K. Kamikariya, K. Otsuka, and S.-C. Chu, "Single-frequency Ince-Gaussian mode operations of laser-diode-pumped microchip solid-state lasers," Opt. Express 15, 10705 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-17-10705.
[CrossRef] [PubMed]

Kawakami, M.

Kim, K.

X. Xu, K. Kim, W. Jhe, and N. Kwon, "Efficient optical guiding of trapped cold atoms by a hollow laser beam," Phy. Rev. A 63, 3401 (2001).
[CrossRef]

Kimel, I.

I. Kimel and L. R. Elias, "Relations between Hermite and Laguerre Gaussian modes," IEEE J. Quantum Electron. 29, 2562 (1993).
[CrossRef]

Kristensen, M.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, "Helical-wavefront laser beams produced with a spiral phase plate," Optics Commun. 112, 321 (1994).
[CrossRef]

Kuga, T.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, "Novel optical trap of atoms with a doughnut beam," Phys. Rev. Lett. 78, 4713 (1997).
[CrossRef]

Kwon, N.

X. Xu, K. Kim, W. Jhe, and N. Kwon, "Efficient optical guiding of trapped cold atoms by a hollow laser beam," Phy. Rev. A 63, 3401 (2001).
[CrossRef]

MacDonald, M. P.

M. P. MacDonald, "Revolving interference pattern for the rotation of optically trapped particles," Opt. Commun. 201, 21 (2002).
[CrossRef]

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, "Controlled rotation of optically trapped microscopic particles," Science 292, 912 (2001).
[CrossRef] [PubMed]

Mair, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, "Entanglement of the orbital angular momentum states of photons," Nature 412, 313 (2001).
[CrossRef] [PubMed]

McDuff, R.

Milam, D.

Nanri, K.

Nemoto, K.

K. Otsuka, K. Nemoto, K. Kamikariya, and S.-C. Chu, "Linearly-polarized, single-frequency oscillations of laser-diode-pumped microchip ceramic Nd:YAG lasers with forced Ince-Gaussian mode operations," Jpn. J. Appl. Phys. 46, 5865 (2007).
[CrossRef]

O’Neil, A. T.

A. T. O’Neil and J. Courtial, "Mode transformations in terms of the constituent Hermite-Gaussian or Laguerre-Gaussian modes and the variable-phase mode converter," Opt. Commun. 181, 35 (2000).
[CrossRef]

Ohtomo, T.

Otsuka, K.

T. Ohtomo, K. Kamikariya, K. Otsuka, and S.-C. Chu, "Single-frequency Ince-Gaussian mode operations of laser-diode-pumped microchip solid-state lasers," Opt. Express 15, 10705 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-17-10705.
[CrossRef] [PubMed]

K. Otsuka, K. Nemoto, K. Kamikariya, and S.-C. Chu, "Linearly-polarized, single-frequency oscillations of laser-diode-pumped microchip ceramic Nd:YAG lasers with forced Ince-Gaussian mode operations," Jpn. J. Appl. Phys. 46, 5865 (2007).
[CrossRef]

S.-C. Chu and K. Otsuka, ""Numerical study for selective excitation of Ince-Gaussian modes in end-pumped solid-state lasers," Opt. Express 15, 16506 (2007).
[CrossRef] [PubMed]

S. -C. Chu and K. Otsuka, "Stable donut-like vortex beam generation from lasers with controlled Ince-Gaussian modes," Appl. Opt. 46, (2007). http://www.opticsinfobase.org/abstract.cfm?URI=ao-46-31-7709.
[CrossRef] [PubMed]

Padgett, M. J.

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, "Measurement of the rotational frequency shift imparted to a rotating light beam possessing orbital angular momentum," Phys. Rev. Lett. 80, 013601 (1998).
[CrossRef]

J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, and M. J. Padgett, "Rotational frequency shift of a light beam," Phys. Rev. Lett. 81, 4828 (1998).
[CrossRef]

N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, "Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner," Opt. Lett. 22, 52 (1997).
[CrossRef] [PubMed]

Paterson, L.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, "Controlled rotation of optically trapped microscopic particles," Science 292, 912 (2001).
[CrossRef] [PubMed]

Piccirillo, B.

Robertson, D. A.

J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, and M. J. Padgett, "Rotational frequency shift of a light beam," Phys. Rev. Lett. 81, 4828 (1998).
[CrossRef]

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, "Measurement of the rotational frequency shift imparted to a rotating light beam possessing orbital angular momentum," Phys. Rev. Lett. 80, 013601 (1998).
[CrossRef]

Santamato, E.

Sasada, H.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, "Novel optical trap of atoms with a doughnut beam," Phys. Rev. Lett. 78, 4713 (1997).
[CrossRef]

Sasso, A.

Schwarz, U. T.

Shimizu, Y.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, "Novel optical trap of atoms with a doughnut beam," Phys. Rev. Lett. 78, 4713 (1997).
[CrossRef]

Shiokawa, N.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, "Novel optical trap of atoms with a doughnut beam," Phys. Rev. Lett. 78, 4713 (1997).
[CrossRef]

Sibbett, W.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, "Controlled rotation of optically trapped microscopic particles," Science 292, 912 (2001).
[CrossRef] [PubMed]

Siegman, A. E.

Simpson, N. B.

Smith, C. P.

Song, Y.

Spreeuw, R. J. C.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes," Phys. Rev. A 45, 8185 (1992).
[CrossRef] [PubMed]

Swartzlander, G. A.

Sziklas, E. A.

Takeda, S.

Torii, Y.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, "Novel optical trap of atoms with a doughnut beam," Phys. Rev. Lett. 78, 4713 (1997).
[CrossRef]

van der Veen, H. E. L. O.

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, "Astigmatic laser mode converters and transfer of orbital angular momentum," Opt. Commun. 96, 123 (1993).
[CrossRef]

Vaziri, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, "Entanglement of the orbital angular momentum states of photons," Nature 412, 313 (2001).
[CrossRef] [PubMed]

Vella, A.

Weihs, G.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, "Entanglement of the orbital angular momentum states of photons," Nature 412, 313 (2001).
[CrossRef] [PubMed]

White, A. G.

Woerdman, J. P.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, "Helical-wavefront laser beams produced with a spiral phase plate," Optics Commun. 112, 321 (1994).
[CrossRef]

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, "Astigmatic laser mode converters and transfer of orbital angular momentum," Opt. Commun. 96, 123 (1993).
[CrossRef]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes," Phys. Rev. A 45, 8185 (1992).
[CrossRef] [PubMed]

Xu, X.

X. Xu, K. Kim, W. Jhe, and N. Kwon, "Efficient optical guiding of trapped cold atoms by a hollow laser beam," Phy. Rev. A 63, 3401 (2001).
[CrossRef]

Zeilinger, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, "Entanglement of the orbital angular momentum states of photons," Nature 412, 313 (2001).
[CrossRef] [PubMed]

Appl. Opt.

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Supplementary Material (5)

» Media 1: MOV (1883 KB)     
» Media 2: MOV (1838 KB)     
» Media 3: MOV (2032 KB)     
» Media 4: MOV (2058 KB)     
» Media 5: MOV (1877 KB)     

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Figures (15)

Fig. 1.
Fig. 1.

Experimental setup.

Fig. 2.
Fig. 2.

Decomposition of the diagonal HG2,0 and LG2,0 modes.

Fig. 3.
Fig. 3.

Successive generation of LGl,0 modes with increasing d. Pump power P=182 mW. Threshold pump power: 36 mW.

Fig. 4.
Fig. 4.

(1884 KB) Structural change of the HG4,0 mode with a rotation of the AMC. P=182 mW. [Media 1]

Fig. 5.
Fig. 5.

Some analytic patterns of the Ince-Gaussian modes.

Fig. 6.
Fig. 6.

Approach from Ince-Gaussian to Hermite Gaussian modes with increasing ellipticity. (a) IGe p,p modes and (b) IGe p,0 and IGo p,1 . (Here, ellipticity ε : 4→103).

Fig. 7.
Fig. 7.

(1839 KB) Structural pattern change with AMC rotation for the IGo 3,1 mode. P=182 mW. [Media 2]

Fig. 8.
Fig. 8.

(2032 KB) Structural pattern change with AMC rotation for IGe 3,3 mode. P=182 mW. [Media 3]

Fig. 9.
Fig. 9.

(2000 KB) Structural pattern change with AMC rotation for HG3,1 mode. P=234 mW. [Media 4]

Fig. 10.
Fig. 10.

(1878 KB) Structural pattern change with AMC rotation for IGe 4,4 mode. P=234 mW. [Media 5]

Fig. 11.
Fig. 11.

Numerical result indicating a structural change in the HG4,0 mode intensity pattern with a rotation of AMC and the corresponding phase portrait of the donut-like multi-vortex at θ=45°.

Fig. 12.
Fig. 12.

Numerical result indicating a structural change in the IG° 3,1 mode intensity pattern with a rotation of AMC.

Fig. 13.
Fig. 13.

Numerical result indicating a structural change of the IGe 3,3 mode intensity pattern with a rotation of AMC.

Fig. 14.
Fig. 14.

Computed changes in phase portraits and interference fringes across θ=45°. (a) IGo 3,1 mode and (b) IGe 3,3 (right) mode. Points of phase singularity are indicated by red dots.

Fig. 15.
Fig. 15.

Computed structural change in vortex beam created from IG0 3,1 modes with different values of ellipticity parameter ε and θ=45°. (a) Intensity patterns of IG0 3,1 modes, (b) intensity profiles of vortices, (c) phase portraits, and (d) interference fringe patterns. Results for HG3,0 -originating vortices are shown in the bottom row for reference.

Tables (1)

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Table 1. Design of optical elements and their optical distances.

Equations (12)

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u l , p LG ( r , ϕ ) = k = 0 N B k iku N k , k HG ( x , y )
IG e p , m ( r , ε ) = C [ w 0 w ( z ) ] C p m ( i ξ , ε ) C p m ( η , ε ) exp [ r 2 w 2 ( z ) ]
× exp i [ kz + { kr 2 2 R ( z ) } ( p + 1 ) ψ z ( z ) ] ,
IG o p , m ( r , ε ) = S [ w 0 w ( z ) ] S p m ( i ξ , ε ) S p m ( η , ε ) exp [ r 2 w 2 ( z ) ]
× exp i [ kz + { kr 2 2 R ( z ) } ( p + 1 ) ψ z ( z ) ] ,
u σ l , p LG ( r , ϕ ) = k = 0 N D k u σ 2 l + p , k IG ( ξ , ε , η )
u σ p , m IG ( ξ , η , ε ) = l , p = 0 N D l , p u σ l , p LG ( r , ϕ )
g i ( x , y ) = g i 0 ( x , y ) ( 1 + I ~ i + ( x , y ) + I ~ i ( x , y ) I s ( x , y ) ) ,
I ~ i + ( x , y ) = ( 1 α ) i = 0 q α i I i + ( q i ) ,
I ~ i ( x , y ) = ( 1 α ) i = 0 q α i I i ( q i ) .
E i out ( x , y ) = E i in ( x , y ) exp [ 1 2 g i ( x , y ) d ] ,
E q + 1 ( x , y ) ~ E q ( x , y ) .

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