## Abstract

We study focused fields which, for a given total power and a given numerical aperture, have maximum electric field amplitude in some direction in the focal point and are linearly polarized along this direction. It is shown that the optimum field is identical to the image of an electric dipole with unit magnification. In particular, the field which is the image of an electric dipole whose dipole vector is parallel to the optical axis, is identical to the field whose longitudinal component is maximum at the image point.

© 2011 OSA

## 1. Introduction

When light is focused by a lens with numerical aperture larger than 0.6, the rotation of polarization becomes important. The rotation of polarization of a diffraction limited lens of high numerical aperture is well described by the model of Ignatowsky [1, 2] and Wolf and Richards [3, 4]. When the field in the lens aperture is that of a linearly polarized plane wave, the dominant electric field component near the focal point is parallel to the direction of polarization of the incident plane wave. But, as the numerical aperture increases, the maximum value of the longitudinal component of the electric field in the focal plane becomes quite substantial, although it vanishes at the focal point itself.

In many applications such as optical recording, photolithography and microscopy, it is important to be able to shape the focused spot. A field in focus with maximum electric component in a certain direction is of interest for manipulating and probing single molecules and particles, and in materials processing [5–9]. The focused wavefront can be tailored by setting a proper amplitude, phase and polarization distributions in the pupil of the focusing lens [10,11]. Nowadays it is possible to realize almost any complex transmission function in the pupil plane, using for example liquid crystal-based devices [12–15].

The optimization of focused fields has been studied by [16–18] for rotational symmetric systems. In the first two papers it was noted that the focused linearly polarized plane wave in the theory of Ignatowsky and Richard and Wolf, is in the Debye approximation identical to the sum of fields that are truncated by the lens aperture and converge to orthogonal electric and magnetic dipoles, with a particular ratio of dipole strengths. A nice treatment is given in [19].

It was noted by several authors [16, 20–23] that when a radially polarized beam is focused, the distribution of the longitudinal component can be considerably narrower than the focused spot obtained by focusing a linearly polarized plane wave. In [24] the pupil field was determined which after focusing gives the maximum longitudinal component in the focal point for a given power incident on the lens. The optimum pupil field is radially polarized and its amplitude increases in a particular way with radial distance to the center of the pupil. Furthermore, the distribution in the focal plane of the optimized longitudinal component is narrower than the Airy spot. If a recording medium would be used which is sensitive for only the longitudinal component of the electric field, this optimized longitudinal component can yield higher resolution.

In [25] this analysis was generalized to optimum pupil fields of which the electric field amplitude in an arbitrary, not necessarily longitudinal, direction in the focal point is maximum. It was in particular found that the pupil field which gives maximum transverse amplitude in focus is similar but not identical to that of a linearly polarized plane wave.

An interesting feature of the optimum focused fields is that, although the electric field amplitude at the focal points is maximum in a required direction, the electric field is in general *not* linearly polarized in that direction but in another direction. Exceptions are the maximum longitudinal and maximum transverse components. In those cases the optimum fields are linearly polarized along the longitudinal and transverse directions, respectively.

For certain applications it is desirable to have maximum electric field amplitude in the focal point in a given direction under the additional constraint that the electric field is also linearly polarized in that direction. We shall therefore consider this optimization problem in this paper. A major result is that as the optimization direction is varied the set of solutions is identical to that of the problem without the constraint of parallel polarization studied in [25]. To be more precise: the electric field that has maximum amplitude in a given direction but which is not linearly polarized along that direction, is also the field that has maximum amplitude in the direction of its linear polarization.

As was already explained in [25], the optimum focused fields can be realized by focusing appropriate pupil fields. These pupil fields are linearly polarized with the direction of polarization and amplitude that are functions of the pupil coordinates. An alternative method to realize the optimum fields is by imaging an electric dipole by an optical system with unit magnification, e.g. two identical lenses. The image of an electric dipole was studied previously in [6, 17, 19] and, more recently, in [27]. We will show that when the imaging system has unit magnification, the image of an electric dipole with dipole vector **p** = (*p _{x}*,

*p*,

_{y}*p*) is, for the given power, identical to the electric field in image space that has maximum amplitude in the direction (

_{z}*p*,

_{x}*p*, –

_{y}*p*), where the

_{z}*z*-axis is the optical axis. It follows in particular that the image of a dipole that is parallel to the optical axis is the field for which the longitudinal component is maximum.

The paper is organized as follows. In Section 2 the optimization problem and the optimum fields that have been derived in [24] and [25] are summarized. The optimization problem is formulated in homogeneous space without considering the optical system by which the optimum fields are generated. The reason is that the results thus obtained are general and independent of any model used for the optical system. In Section 3 the results of this optimization are interpreted and the new optimization problem is studied where the electric field is required to be linearly polarized along the direction for which the amplitude of the electric field is maximum. It is then shown that when this direction is varied, the set of optimum fields is identical to the set of optimum fields that are the solutions of the first optimization problem. Finally in Section 4 two methods to realize the optimum fields are considered. The first method is by focusing an appropriate pupil field. We use here the vector diffraction theory of Ignatowksy and Wolf and Richards, which is based on the Debye approximation. The second method is by imaging an electric dipole with unit magnification.

## 2. The optimization problem

#### 2.1. Formulation of the optimization problem

We first define a few quantities and introduce some notations. As was stated in the Introduction, in the optimization problem we first leave the optical system out of consideration. The optimized fields are derived from the rigorous Maxwell equations and are therefore generally valid. We only assume that the field in image space has a certain numerical aperture NA ≤ *n*, where *n* is the refractive index in image space, and that the field has a prescribed power. Later we will discuss what fields in the entrance pupil of a lens with the chosen numerical aperture produce the optimum focused fields.

The plane wave vectors in image space are limited by the numerical aperture of the imaging system. Let (*x*,*y*,*z*) be a Cartesian coordinate system with the *z*-axis parallel to the optical axis. Using the plane wave expansion, the electric field in a point **r** in image space, which propagates in the direction of the positive *z*-axis, can be written as

**E**(

**r**) is the complex field,

**k**= (

*k*,

_{x}*k*,

_{y}*k*) the wave vector in the medium with refractive index

_{z}*n*,

*k*

_{0}= |

**k**|/

*n*= 2

*π*/

*λ*

_{0}with

*λ*

_{0}the wavelength in vacuum and

**A**(

*k*,

_{x}*k*) are the complex plane wave amplitudes. Since the field propagates in the positive

_{y}*z*-direction, we have ${k}_{z}=\sqrt{{k}_{0}^{2}{n}^{2}-{k}_{x}^{2}-{k}_{y}^{2}}>0$. The circle $\sqrt{{k}_{x}^{2}+{k}_{y}^{2}}\le {k}_{0}\text{NA}$ in the (

*k*,

_{x}*k*)-plane, which limits the integration area, will be denoted by Ω for brevity. Since the electric field is free of divergence, we have:

_{y}The time-averaged power flow is given by the integral over a plane *z* = constant of the *z*-component of the time averaged Poynting vector [25]:

**v̂**be a unit vector and consider the projection of the electric field at the origin

**r**=

**0**along this unit vector:

*ℰ*(

**0**,

*t*) ·

**v̂**is maximum for a given total power. Without restricting the generality we may assume that

**E**(

**0**) ·

**v**is real. In fact, if it is not real, applying a time shift can always make it real. Hence we will assume that

**A**(

*k*,

_{x}*k*) for which the component of the electric field along

_{y}**v̂**has maximum amplitude for a given total power

*P*

_{0}:

*ℋ*

_{0}is the space of plane wave amplitudes which have finite mean flow of power through a plane

*z*= constant and which satisfy Eq. (5). The problem is denoted as

*℘*

_{1}(

**v̂**) to indicate that we are optimizing in the direction of

**v̂**. It can be shown that optimization problem

*℘*

_{1}(

**v̂**) has a unique solution. The uniqueness is due to the imposed condition that

**E**(

**0**) ·

**v̂**is real. Without this condition the optimum solutions are unique except for a phase shift [25].

#### 2.2. Optimum plane wave amplitudes and field distributions

The solution of problem *℘*
_{1}(**v̂**) can be computed by applying the Lagrange multiplier rule for inequality constraints. We remark that since we optimize a linear functional on a sphere in a Hilbert space (the sphere is the set of all fields of finite given power), the solution can alternatively be derived by using the Cauchy-Schwarz inequality. In any case, the solution can be computed in closed form and is given by [25]:

*α*

_{max}is such that It follows from Eq. (6) that the electric field vector of the plane wave with wave vector

**k**is parallel to the projection of

**v̂**on the plane perpendicular to

**k**. The electric field in an arbitrary point

**r**in image space can be obtained by substitution of the optimum plane wave amplitudes into Eq. (1). By changing the integration variables to respective polar and azimuthal angles, 0 <

*α*<

*α*

_{max}and 0 <

*β*< 2

*π*, given by (Fig. 1) and using cylindrical coordinates

*ρ*,

*φ*,

*z*for the point of observation:

**r**=

*x*

**x̂**+

*y*

**ŷ**+

*z*

**ẑ**=

*ρ*

**+**

*$\widehat{\rho}$**z*

**ẑ**=

*ρ*cos

*φ*

**x̂**+

*ρ*sin

*φ*

**ŷ**+

*z*

**ẑ**, one finds for the optimum electric field [25]:

**x̂**,

**ŷ**,

**ẑ**}.

## 3. Interpretation of optimum fields and second optimization

#### 3.1. The electric field in focus

We consider the optimum electric field vector at the origin. Because
${\text{g}}_{l}^{\nu ,\mu}(\mathbf{0})=0$ when *l* ≥ 1 it follows from Eq. (13) that

**E**(

**0**) has real components, hence the electric field at

**r**=

**0**is linearly polarized. It is furthermore parallel to the plane through

**v̂**and the

*z*-axis. However, the direction of polarization is in general different from

**v̂**, i.e. from the direction along which the electric field amplitude is maximized. We shall write for convenience:

**û**be the unit vector of the direction of polarization, i.e.

We conclude that the direction of polarization at the origin is only parallel to the direction of optimization when *v _{z}* = 0 (transverse polarization) and when

*v*=

_{x}*v*= 0 (longitudinal polarization). An exception is the case that the numerical aperture NA =

_{y}*n*. Then

*α*

_{max}=

*π*/2, so that

*γ*

_{+}=

*γ*

_{−}= 4/3 and hence

**û**and

**v̂**are always identical.

Without restricting the generality we may assume that **v̂** lies in the (*x*,*z*)-plane, i.e. that *v _{y}* = 0. Then

**û**also is in the (

*x*,

*z*)-plane. Let ϑ

*and ϑ*

_{v}*be the angle that*

_{u}**v̂**and

**û**, respectively, make with the

*z*-axis. Then

*≥ ϑ*

_{u}*, i.e. the angle that*

_{v}**û**makes with the positive

*z*-axis is in general larger than that of

**v̂**. Furthermore, ϑ

*sweeps out all values between 0 and*

_{u}*π*/2 when ϑ

*varies between 0 and*

_{v}*π*/2. The difference between the two angles decreases when the numerical aperture is increased and the angles are the same in the limit NA =

*n*.

#### 3.2. Optimum linearly polarized electric field

Although the solutions of problem *℘*
_{1}(**v̂**) have at the origin maximum electric field amplitude in the direction of **v̂**, the electric field vector at the origin is, although linearly polarized, not parallel to **v̂**. In fact it is pointing in the direction of the unit vector **û**.

The question then arises: what are the optimum fields that have maximum amplitude in a given direction **v̂**
*and are in addition linearly polarized along*
**v̂**? The requirement that the electric field vector *ℰ*(**0**,*t*) = Re[**E**(**0**)exp(–*iωt*)] is parallel to **v̂** at all times *t* is equivalent to demanding that the vector **E**(**0**) is proportional to **v̂**, which in turn is equivalent to:

**v̂**=

**ẑ**these conditions become:

**E**(

**0**) ·

**x̂**= 0 and

**E**(

**0**) ·

**ŷ**= 0). Hence, the second optimization problem that we want to consider is:

*ℋ*

_{0}consists as before of all plane wave amplitudes

**A**(

*k*,

_{x}*k*) such that

_{y}**E**(

**0**) ·

**v̂**is real.

The solution of problem *℘*
_{2}(**v̂**) can be determined analogously to that of problem *℘*
_{1}(**v̂**) by using the Lagrange multiplier rule or alternatively using the Cauchy-Schwarz inequality. In the latter case the constraint that the electric field should be linearly polarized along the direction of **v̂**, should be incorporated into the Hilbert space of fields to which the Schwarz inequality is applied. Closed formulae are again obtained and the solution is again unique due to the fact that the phase is fixed by requiring that **E**(**0**) ·**v̂** is real. It is then found that when **v̂** is varied, more precisely, when ϑ* _{v}* is varied in the interval 0 ≤ ϑ

*≤*

_{v}*π*/2, the two sets of optimum solutions of the two optimization problems are

*identical*when the proper directions are chosen. For every

**v̂**there is a unit vector

**û**such that the optimum field that is the solution of

*℘*

_{1}(

**v̂**) is also the solution of

*℘*

_{2}(

**û**), and conversely. In fact, the relationship between the unit vectors

**v̂**and

**û**is given by Eq. (23) defined in the previous section.

This is easy to prove. Let
${\mathbf{\text{E}}}_{1}^{\widehat{\mathbf{\text{v}}}}$ and
${\mathbf{\text{E}}}_{2}^{\widehat{\mathbf{\text{u}}}}$ be the solution of problem *℘*
_{1}(**v̂**) and *℘*
_{2}(**û**), respectively, and let **v̂** and **û** be related by Eq. (23). Then we have to show that

**û**and are real vectors. For ${\mathbf{\text{E}}}_{1}^{\widehat{\mathbf{\text{v}}}}$ this follows from Eq. (15) while for ${\mathbf{\text{E}}}_{2}^{\widehat{\mathbf{\text{u}}}}$ it follows from the requirements for the field in optimization problem

*℘*

_{2}(

**û**).

It is clear that
${\mathbf{\text{E}}}_{1}^{\widehat{\mathbf{\text{v}}}}$ satisfies all constraints of problem *℘*
_{2}(**û**). Hence

**û**and since by Eq. (23)

**v̂**·

**û**> 0, it follows that

*℘*

_{1}(

**v̂**) and since ${\mathbf{\text{E}}}_{1}^{\widehat{\mathbf{\text{v}}}}$ is the solution, it follows that in Eq. (29) equality holds. Because the solution of problem

*℘*

_{1}(

**v̂**) is unique, we thus conclude that Eq. (27) is indeed true.

#### 3.3. Explicit expressions for the solution of problem ℘_{2}

With the results obtained above, explicit formulae for the optimum plane wave amplitudes that are solutions of optimization problem *℘*
_{2}(**û**) can be easily obtained. First we determine the unit vector **v̂** by inverting Eq. (23):

*℘*

_{2}(

**û**):

**r**becomes with Eq. (12):

*ℳ̿*(

*ρ*,

*φ*,

*z*) is still given by Eq. (13).

#### 3.4. Electric field strength in focus

Let
${\mathbf{\text{E}}}_{1}^{\widehat{\mathbf{\text{v}}}}$ be the solution of optimization problem *℘*
_{1}(**v̂**) for some unit vector **v̂** and let **û** be the unit vector in the direction of
${\mathbf{\text{E}}}_{1}^{\widehat{\mathbf{\text{v}}}}(\mathbf{0})$ as defined by Eq. (23). Without restricting the generality, we assume again that *v _{y}* =

*u*= 0. At the left-side of Fig. 3 the projections of ${\mathbf{\text{E}}}_{1}^{\widehat{\mathbf{\text{v}}}}(\mathbf{0})$ along the direction of optimization

_{y}**v̂**and along

**û**are compared as function of ϑ

*for several values of the numerical aperture and for total power*

_{v}*P*

_{0}= 1W. The right-side of Fig. 3 shows a comparison between the projections of the solutions of

*℘*

_{1}(

**v̂**) and

*℘*

_{2}(

**v̂**) along

**v̂**, i.e. ${\mathbf{\text{E}}}_{1}^{\widehat{\mathbf{\text{v}}}}(\mathbf{0})\xb7\widehat{\mathbf{\text{v}}}$ and ${\mathbf{\text{E}}}_{2}^{\widehat{\mathbf{\text{v}}}}(\mathbf{0})\xb7\widehat{\mathbf{\text{v}}}$ (again as functions of ϑ

*for several values of the numerical aperture and for*

_{v}*P*

_{0}= 1W). We see that for ϑ

*= 0*

_{v}*(longitudinal polarized optimum field) and ϑ*

^{o}*= 90*

_{v}*(transverse polarized optimum field), the projections are identical. As was pointed out before, this is also true for all angles 0 ≤ ϑ*

^{o}*≤*

_{v}*π*/2 when NA =

*n*.

## 4. Realization of the optimum fields

In this section we discuss two methods for realizing the optimum fields. In the first method an appropriate pupil distribution is focused by a lens of given numerical aperture. The optimum field is the field in image space of the lens with the focal point as point were the electric field is maximum. In the second method, an electric dipole with appropriate dipole moment is imaged by a lens system with magnification 1. The field component is then maximum at the image point of the dipole. It should be noted that both realizations of the image field are approximative because they are based on a model of the focusing and imaging of a high NA lens which are valid under certain, be it rather mild, assumptions [28].

#### 4.1. Pupil fields that when focused yield the optimum field distributions.

We consider a diffraction limited lens with (high) numerical aperture NA and a light beam which propagates parallel to the optical axis and is focused by this lens. Let {**x̂**,**ŷ**,**ẑ**} be a Cartesian basis in image space such that the optical axis is parallel to **ẑ** and the origin coincides with the focal point. Let {**x̂**
* _{p}*,

**ŷ**

*} be the coordinate system in the entrance pupil of the lens, with*

_{p}**x̂**

*and*

_{p}**ŷ**

*parallel to*

_{p}**x̂**and

**ŷ**, respectively. Finally, let

*ρ*and

_{p}*φ*be polar coordinates in the entrance pupil, i.e.

_{p}

*$\widehat{\mathbf{\rho}}$**,*

_{p}

*$\widehat{\mathbf{\phi}}$**,*

_{p}**ẑ**} is a positively orientated orthonormal basis. The electric field in a point (

*ρ*,

_{p}*φ*) of the entrance pupil of the lens is decomposed on this basis:

_{p}**ẑ**-component of the electric field is neglected because the incident beam is predominantly propagating parallel to the optical

*z*-axis. This system is illustrated by Fig. 4.

To determine the focused field we use the theory of Ignatowsky [1, 2] and Richards and Wolf [3,4]. This theory is based on (i) Debye’s approximation which expresses the plane wave amplitudes at the focal plane in terms of the tangential electric field in the entrance pupil, and (ii) the Abbe sine condition, which guarantees conservation of energy. Debye’s approximation is equivalent to a sharp cut-off of the plane wave spectrum as mentioned in [29] and is valid provided that [28, 30]:

where*f*is the focal distance and sin

*α*

_{max}= NA/

*n*. In the model, the pupil coordinates

*x*and

_{p}*y*are related to the wave vector components by (see Fig. 4):

_{p}**k̂**} is positively oriented. The vector amplitude

**A**(

*k*,

_{x}*k*) of the plane wave with wave vector

_{y}**k**can be written as:

*A*and

_{P}*A*, respectively [29, 31]:

_{S}*℘*

_{1}(

**v̂**), we obtain the entrance pupil field that after focusing by the lens yields this optimum field in the focal region:

*℘*

_{2}(

**û**) is:

It follows from the formulas that the optimum pupil fields are linearly polarized in all points of the pupil and that in all points they are in phase. Furthermore, for the case that *v _{x}* =

*v*=

_{y}*u*=

_{x}*u*= 0 (longitudinal case), the pupil field is radially polarized in all points of the pupil.

_{y}In Fig. 5 snapshots are shown of the optimum pupil distributions which after focusing give the solution of *℘*
_{1}(**v̂**) (left figure) and *℘*
_{2}(**v̂**) (right figure), for unit vector **v̂** with *v _{y}* = 0 and ϑ

*= 45°. The numerical aperture is NA/*

_{v}*n*= 0.75 in both cases. The direction of the arrows is that of the linear polarization and their length is proportional to the electric field amplitude. The optimum pupil field at the right of Fig. 5 is similar to a radially polarized pupil field but with center shifted in the negative

*x*-direction.

_{p}#### 4.2. Imaging of an electric dipole

Consider an imaging system consisting of two lenses as shown in Fig. 6. We choose a Cartesian coordinate system {**x̂**,**ŷ**,**ẑ**} in object space which is parallel to the coordinate system in image space as defined in Subsection 4.1 but the first focal point is at the origin of this coordinate system. Although the coordinate systems in object and image systems are denoted by the same symbols, no confusion will arise. Let there be an electric dipole with dipole vector **p** = (*p _{x}*,

*p*,

_{y}*p*) in the focal point of the first lens. The electric field in object space radiated by this dipole is [32]:

_{z}*G*(

**r**) is the free space Green’s function of the scalar Helmholtz equation: By Fourier transformation with respect to (

*x*,

*y*), the plane wave expansion of

**E**

*in a plane of constant*

_{d}*z*≥ 0 is:

*o*’ indicates a wave vector in object space. Let ${\mathbf{\text{A}}}^{\mathbf{\text{o}}}({k}_{x}^{o},\hspace{0.17em}{k}_{y}^{o})$ be the plane wave amplitudes in the source plane

*z*= 0. Then

*f*

_{2}. The pupil fields are again assumed to be related to the plane wave amplitudes by Ignatowsky’s model. After writing the plane wave amplitudes in the object plane on the basis {

**k̂**, ${\widehat{{\mathbf{\text{i}}}_{\mathbf{\text{P}}}}}^{\mathbf{\text{o}}}$, ${\widehat{{\mathbf{\text{i}}}_{\mathbf{\text{S}}}}}^{\mathbf{\text{o}}}$}, the electric field in the exit pupil of the left lens and the entrance pupil of the second lens are obtained by Eqs. (43) and (44). When the two lenses are identical, i.e.

^{o}*f*

_{1}=

*f*

_{2}, the (paraxial) magnification is equal to 1 and the plane wave amplitudes ${\mathbf{\text{A}}}^{\mathbf{\text{i}}}({k}_{x}^{i},{k}_{y}^{i})$ in the focal plane of the second lens become

*i*’ refers to image space and the wave vector components in image and object spaces are related by: ${k}_{x}^{i}=-{k}_{x}^{o}$, ${k}_{y}^{i}=-{k}_{y}^{o}$ and ${k}_{z}^{i}={k}_{z}^{o}$. These relationships and Eqs. (40) and (41) imply that in image space:

**p̃**= (

*p*,

_{x}*p*, −

_{y}*p*) and where we have omitted the image space superscript from the wave vector in image space. By comparing Eqs. (56) and (6), it follows that when the dipole vector has length:

_{z}**v̂**is defined by

*℘*

_{1}(

**v̂**) are identical (apart from a factor −

*i*). We conclude that when an electric dipole

**p**is imaged with unit magnification, the field in the image point has, for the given power, maximum field amplitude in the direction of the unit vector

**p̃**= (

*p*,

_{x}*p*, −

_{y}*p*)/|

_{z}**p̃**|. Because of the relationships between the solutions of optimization problems

*℘*

_{1}(

**v̂**) and

*℘*

_{2}(

**û**), the image of the electric dipole is also the field that has maximum amplitude and is linearly polarized in the direction of the unit vector

**û**given by

## 5. Conclusion

In a previous study focused fields were determined that have maximum electric field amplitude in a particular direction in the focal point. Although the optimum electric field is linear polarized in the focal point, the direction of polarization is in general not parallel to that for which the amplitude is optimum. We therefore studied the optimization problem in which it is required that the optimum field is linear polarized in the same direction as for which the amplitude is maximum. It was shown that when the direction is varied, the set of solutions of the two optimization problems are identical, i.e for every optimization direction in the unconstrained optimization problem there is a direction for the second optimization problem such that the solutions of the two problems are the same. The relationship between solutions of the two optimization problems for the same direction of optimization were derived. The optimum fields can be realized by focusing of appropriate pupil fields. Alternatively, the fields can be obtained by imaging an electric dipole with unit magnification. We demonstrated that the optimum field for a given direction is identical to the image of an electric dipole whose vector is the mirror image of the direction of optimization with the (*x*,*y*)-plane as mirror.

## References and links

**1. **V. S. Ignatowsky, “Diffraction by a lens having arbitrary opening,” Trans. Opt. Inst. Petrograd **I**, paper IV (1919), (in Russian).

**2. **V. S. Ignatowsky, “Diffraction by a parabolic mirror having arbitrary opening,” Trans. Opt. Inst. Petrograd , **I**, paper V (1920), (in Russian).

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

**4. **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]

**5. **X. S. Xie and R. C. Dunn, “Probing single molecule dynamics,” Science **265**, 361–364 (1994). [CrossRef] [PubMed]

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

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

**8. **L. E. Helseth, “Focusing of atoms with strongly confined light potentials,” Opt. Commun. **212**, 343–352 (2002). [CrossRef]

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

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

**11. **J. Wang, W. Chen, and Q. Zhan, “Engineering of high purity ultra-long optical needle field through reversing the electric dipole array radiation,” Opt. Express **18**, 21965–21972 (2010). [CrossRef] [PubMed]

**12. **N. Sanner, N. Huot, E. Audouard, C. Larat, J.-P. Huignard, and B. Loiseaux, “Programmable focal spot shaping of amplified femtosecond laser pulses,” Opt. Lett. **30**, 1479–1481 (2005). [CrossRef] [PubMed]

**13. **M. A. A. Neil, F. Massoumian, R. Juskaitis, and T. Wilson, “Method for the generation of arbitrary complex vector wave fronts,” Opt. Lett. **27**, 1929–1931 (2002). [CrossRef]

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

**15. **I. Iglesias and B. Vohnsen, “Polarization structuring for focal volume shaping in high-resolution microscopy,” Opt. Commun. **271**, 40–47 (2007). [CrossRef]

**16. **C. J. R. Sheppard and K. G. Larkin, “Optimal concentration of electromagnetic radiation,” J. Mod. Opt. **41**, 1495–1505 (1994). [CrossRef]

**17. **C. J. R. Sheppard and P. Török, “Electromagnetic field in the focal region of an electric dipole wave,” Optik **104**, 175–177 (1997).

**18. **A. J. E. M. Janssen, S. van Haver, J. J. M. Braat, and P. Dirksen, “Strehl ratio and optimum focus of high-numerical-aperture beams,” J. Eur. Opt. Soc. Rapid Publ. **2**, 07008 (2007). [CrossRef]

**19. **V. Dhayalan and J. J. Stamnes, “Focusing of mixed-dipole waves,” Pure Appl. Opt. **6**, 317–345 (1997). [CrossRef]

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

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

**22. **S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light–theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B **72**, 109–113 (2001).

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

**24. **H. P. Urbach and S. F. Pereira, “The field in focus with maximum longitudinal electric component,” Phys. Rev. Lett. **100**, 1233904 (2008). [CrossRef]

**25. **H. P. Urbach and S. F. Pereira, “Focused fields of given power with maximum electric field components,” Phys. Rev. A **79**, 013825 (2009). [CrossRef]

**26. **H. P. Urbach and S. F. Pereira, “Erratum: focused fields of given power with maximum electric field components,” Phys. Rev. A **81**, 059903 (2010). [CrossRef]

**27. **W. Chen and Q. Zhan, “Diffraction limited focusing with controllable arbitrary three-dimensional polarization,” J. Opt. **12**, 045707 (2010) [CrossRef]

**28. **E. Wolf and Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. **39**, 205–210 (1981). [CrossRef]

**29. **J. J. M. Braat, S. van Haver, A. J. E. M. Janssen, and P. Dirksen, “Assessment of optical systems by means of point-spread functions,” in “*Progress in Optics*,” Vol. 51, E. Wolf, ed. (Elsevier B.V., 2008), chap. 6, pp. 349–468. [CrossRef]

**30. **R. Aarts, J. J. M. Braat, P. Dirksen, S. van Haver, C. van Heesch, and A. Janssen, “Analytic expressions and approximations for the on-axis, aberration-free Rayleigh and Debye integral in the case of focusing fields on a circular aperture,” J. Eur. Opt. Soc. Rapid Publ. **3**, 08039 (2008). [CrossRef]

**31. **J. J. M. Braat, S. van Haver, A. J. E. M. Janssen, and S. F. Pereira, “Image formation in a multilayer using the extended Nijboer-Zernike theory,” J. Eur. Opt. Soc. Rapid Publ. **4**, 09048 (2009). [CrossRef]

**32. **L. Novotny and B. Hecht, “Principles of nano-optics,” Section 2.10.2, (Cambridge University Press, 2008).