4π Focusing of TM<sub>01</sub> beams under nonparaxial conditions

Alexandre April; Michel Piché

doi:10.1364/OE.18.022128

1. Introduction

Strongly focused optical beams are of considerable importance in many applications such as confocal microscopy, lithography, optical tweezers, and optical data storage. In particular, it was shown that smaller spot sizes can be achieved with radially polarized light instead of linearly polarized light [1,2], resulting in a growing interest for radially polarized beams such as the so-called TM₀₁ beam (transverse magnetic laser beam of lowest order). For instance, the focusing properties of TM₀₁ beams are exploited in high-resolution imaging [3] and in schemes of electron acceleration [4].

When a beam is tightly focused, the vector character of light becomes crucial to correctly describe such a nonparaxial beam. Furthermore, the expressions of its electromagnetic field have to satisfy Maxwell’s four equations beyond the paraxial regime. A theoretical approach by which strongly focused beams are accurately described, given the field distribution of the collimated input beam at the entrance pupil of the focusing system, was provided by Richards and Wolf [5]. Many authors considered the electromagnetic field of an optical beam focused by an aplanatic lens [6–8]; others investigated the fields at the focus of a parabolic mirror [9–12]. An alternative way to analyze tightly focused beams is based on the complex source-point method, in which a source is assumed to be located at an imaginary distance along the propagation axis of the optical beam [13,14]. A modified theory, in which a complex sink accompanies the complex source, avoids the occurrence of an axial discontinuity as well as a nonphysical annular singularity in the focal plane of the beam [15–17]. In this complex sink/source model, the nonparaxial beam is described as consisting in a superposition of an outgoing wave and an incoming wave of different amplitudes. The complex-source/sink solution provides an exact solution to the Maxwell's equations over all space. However a physical beam correctly described by such a solution requires a focusing system illuminated over a complete sphere. While a single lens or objective can only focus light within a 2π solid angle, a parabolic mirror of very large extent is capable of focusing light within nearly the 4π solid angle. The current work aims at recovering the complex-source/sink solution of a TM₀₁ beam using the Richards–Wolf vector field equation with a 4π focusing system, establishing a connection between the complex source/sink model and the Richards–Wolf theory.

It has been pointed out that a parabolic mirror shows distortion and severe coma when it is not illuminated exactly along its axis of revolution [10]. Thus, if the 4π focusing system is a parabolic mirror, the alignment of the optical axis of the beam incident on the mirror is critical. Furthermore, it remains relevant to describe the electromagnetic field of aberrated beams in order to understand the effect of each type of aberration on the fields focused by such a 4π system. In fact, it is not uncommon for even well-corrected optical systems to suffer from small amounts of aberrations. The link between the complex source/sink model and the Richards–Wolf theory allows to obtain analytical expressions for the electric and magnetic fields affected by aberrations.

The use of the vector magnetic potential can shorten and systematize the calculation of the electromagnetic field components. In fact, since the vector magnetic potential has only one nonzero component in the case of a TM beam [18], no more than one integral has to be solved in the context of the Richards–Wolf theory. Then, all six components of the electromagnetic field can be determined by straightforward derivatives of the vector magnetic potential. The vector magnetic potential method can advantageously be applied to efficiently compute the electromagnetic field of an aberrated TM beam. Specifically, the TM₀₁ beam affected by curvature of field, primary coma, astigmatism, or spherical aberration will be investigated.

The paper is divided as follows. In Section 2, we present the expressions for the field components of a strongly focused aberration-free TM₀₁ beam obtained, on the one hand, within the framework of the complex source/sink model and, on the other hand, using the Richards–Wolf vector field equation; the appropriate choice for the focusing system and for the far-field amplitude allows finding the complex-source/sink solution by means of the Richards–Wolf theory. In Section 3, we introduce the vector magnetic potential method to simplify the calculations of field components. In Section 4, we present a description of the primary aberrations and we determine the vector magnetic potentials of TM₀₁ beams affected by curvature of field, primary coma, astigmatism, or spherical aberration. In Section 5, we employ the vector magnetic potential method to find the electromagnetic field of an aberrated TM₀₁ beam.

2. Focusing of an aberration-free TM₀₁ beam

The complex source/sink model and the Richards–Wolf theory are both used to describe tightly focused optical beams. The former provides analytical solutions for the phasor of nonparaxial beams in a remarkably compact form, but it is useful ones only for some prescribed field distributions of the collimated input beam at the entrance pupil of the focusing system. The latter leads to an integral representation of the electromagnetic field, which applies to any input collimated beam, but several integrals have to be solved, numerically most often. First, we recall the fields of a TM₀₁ beam found with the complex source/sink model; second, we present the Richards–Wolf vector field integral applied to a radially polarized beam; finally, using the appropriate collimated input beam, the complex source/sink solution for the TM₀₁ beam is obtained using the Richards–Wolf theory.

2.1 The complex source/sink model

The complex source-point method is a helpful technique that may be exploited to convert a spherical wave into a nonparaxial Gaussian beam [13,14]. This approach consists in assuming the source of the wave to be positioned at an imaginary distance along the propagation axis, which is taken to be the z-axis. Mathematically, it means that the axial coordinate of the phasor of the wave is replaced by a complex quantity whose imaginary part is closely related to the beam divergence angle. However the complex-source wave has two shortcomings: an axial discontinuity as well as a circular singularity occur in the plane of the beam waist.

Both the discontinuity and the nonphysical singularity in the phasor of the nonparaxial beam can be removed by combining a sink to the source, leading to a complex-source/sink wave [15,17]. Thus, the complex source/sink model, as opposed to the complex point-source method, yields singularity-free phasors that describe nonparaxial, physically realizable beams. The complex-source/sink wave may be viewed as a superposition of an outgoing beam, produced by the source placed at a given imaginary distance along the z-axis, and an incoming beam, absorbed by the sink located at the same position, giving rise to a standing-wave component near the z = 0 plane. Hence, producing this superposition of two counter-propagating beams would require a focusing element that subtends a solid angle greater that 2π, such as a parabolic mirror of large extent.

The TM₀₁ beam is the lowest-order radially polarized beam. The electric and magnetic fields of a TM₀₁ beam propagating in free space along the z-axis, found in the context of the complex-source/sink model, may be written as [19,20]

E = E_{o} [j {\hat{a}}_{x} {\tilde{V}}_{0, 1}^{e} + j {\hat{a}}_{y} {\tilde{V}}_{0, 1}^{o} + {\hat{a}}_{z} {\tilde{U}}_{1, 0}^{e}],

H = j H_{o} [- {\hat{a}}_{x} {\tilde{U}}_{0, 1}^{o} + {\hat{a}}_{y} {\tilde{U}}_{0, 1}^{e}],

where

E_{o}

and

H_{o} = E_{o} / η_{0}

are constant amplitudes,

η_{0} = {(μ_{0} / ε_{0})}^{1 / 2}

is the intrinsic impedance of free space,

μ_{0}

and

ε_{0}

are the permeability and permittivity of free space, respectively. Here the time dependence

\exp (j ω t)

, where ω is the angular frequency of the beam, is omitted. The even functions

{\tilde{U}}_{p, m}^{e}

and

{\tilde{V}}_{p, m}^{e}

are defined in closed form by [21]:

{\tilde{U}}_{p, m}^{e} = (2 p)!! \sum_{s = 0}^{p} (\begin{matrix} p + m \\ s + m \end{matrix}) \frac{(4 s + 2 m + 1) (2 s - 1)!!}{(2 p + 2 s + 2 m + 1)!!} {\tilde{ψ}}_{2 s + m, m}^{e},

{\tilde{V}}_{p, m}^{e} = - j (2 p)!! \sum_{s = 0}^{p} (\begin{matrix} p + m \\ s + m \end{matrix}) \frac{(4 s + 2 m + 3) (2 s + 1)!!}{(2 p + 2 s + 2 m + 3)!!} {\tilde{ψ}}_{2 s + m + 1, m}^{e},

{\tilde{ψ}}_{n, m}^{e} = \exp (- k a) j_{n} (k \tilde{R}) P_{n}^{m} (\cos \tilde{θ}) \cos (m ϕ),

where k is the wave number,

\tilde{R} \equiv {[x^{2} + y^{2} + {(z + j a)}^{2}]}^{1 / 2}

,

\cos \tilde{θ} \equiv (z + j a) / \tilde{R}

, ϕ is the azimuthal angle,

j_{n} (k \tilde{R})

is the spherical Bessel function of order n, and

P_{n}^{m} (\cos \tilde{θ})

is the associated Legendre function [22]. The parameter a is a real constant that characterizes the divergence of the optical beam; the beam divergence angle increases as the value of a decreases. The arbitrary normalization constant in Ref [21]. has been chosen herein to be

K_{p, m} = \frac{1}{4} p! {(2 / k w_{o})}^{2 p + m + 2}

in order to simplify the expression of subsequent results. Odd functions

{\tilde{U}}_{p, m}^{o}

and

{\tilde{V}}_{p, m}^{o}

can be obtained from Eqs. (2a)–(2c) simply by replacing

\cos (m ϕ)

by

\sin (m ϕ)

. The functions given by Eqs. (2a) and (2b) are exact solutions of the Helmholtz equation in free space and both reduce to the phasors of the well-known elegant Laguerre–Gaussian beams in the paraxial limit, i.e. for ka >> 1 [21]. Furthermore, the electric and magnetic fields given by Eqs. (1a) and (1b) are rigorous solutions to Maxwell’s equations in free space [20].

The TM₀₁ beam is said to be radially polarized since its azimuthal electric field component is zero, i.e. $E_{ϕ} = 0$ , as it can be easily verified. Furthermore, the beam is transverse magnetic (TM), because its magnetic field does not have a longitudinal component, as opposed to its electric field. Also, it can be seen that the transverse components of the electromagnetic field of the TM₀₁ beam, that dominate in the paraxial regime, are proportional to elegant Laguerre–Gaussian modes of order (0,1).

2.2 The Richards–Wolf vector field equation

When a focusing system has a high numerical aperture, a vector diffraction theory is required to calculate the electric field of the tightly focused beam in the neighborhood of the focus. To investigate the electric field of a TM₀₁ beam focused by an optical system, we exploit the Richards and Wolf’s formulation of the vector diffraction theory of focusing systems, which is useful to analyze strongly focused electromagnetic beams [6,7,10,11]. For the sake of simplicity, only the electric field will be discussed here. Consider an incident collimated beam, whose electric field has prescribed spatial amplitude distribution and polarization state, at the entrance pupil of a given optical system of focal length f. The wave at the exit pupil of the system converges toward the focal point with a spherical wavefront. We employ cylindrical coordinates $(r, ϕ, z)$ near the focus, with the origin located at the focal point (Fig. 1 ). The electric field $E (r, ϕ, z)$ in the neighborhood of the focus is given as an integral over a specified vector field amplitude on a spherical aperture of focal radius f.

Fig. 1 The parabolic mirror of very short focal length f is an example of a 4π focusing system that can be used to generate tightly focused beams.

Download Full Size | PDF

With the help of the Richards–Wolf theory, we analyze a strongly focused radially polarized beam. The optical axis of the collimated beam coincides with the axis of revolution of the focusing system. For a radially polarized beam, the Richards–Wolf vector field equation becomes in an explicit form [6,7]:

[\begin{matrix} E_{x} \\ E_{y} \\ E_{z} \end{matrix}] = \frac{E_{o}}{4 π} \int_{0}^{2 π} \int_{0}^{α_{\max}} q (α) l_{0} (α) [\begin{matrix} \cos α \cos β \\ \cos α \sin β \\ \sin α \end{matrix}] \exp [- j k Γ (α, β)] sin α d α d β,

Γ (α, β) \equiv Φ (α, β) - r \sin α \cos (ϕ - β) + z \cos α,

where

E_{o}

is a normalization constant, k is the wave number of the illumination,

Φ (α, β)

is the aberration function [5], α and β are the polar and azimuthal angles, respectively, defining the orientation of the wave vector k, pointing toward the focus, of a typical plane wave component. The function

q (α)

is the apodization factor of the system, obtained from the energy conservation and geometric considerations. For instance, for an aplanatic lens, the apodization factor is

q (α) = \cos^{1 / 2} α

[5,8], for a parabolic mirror, it is

q (α) = 2 / (1 + \cos α)

[10,11], and for an optical system satisfying the Herschel condition, it is

q (α) = 1

[23,24]. The function

l_{0} (α)

is the amplitude distribution of the collimated input beam at the entrance pupil (assumed to be axially symmetric, i.e. β independent). In this section, we consider an aberration-free system, so that

Φ (α, β) = 0

.

Solving the vector field Eq. (3) with the appropriate choice of the field amplitude $l_{0} (α)$ yields the electric field components in the focal region. The integration is done over the polar angle that covers the entrance pupil of the optical system, i.e. $0 \leq β \leq 2 π$ and $0 \leq α \leq α_{\max}$ , where $α_{\max}$ is the aperture angle of the focusing system, which is usually less than $π / 2$ ; however, it can be as large as π, as it can occur for a parabolic mirror.

The integrations over β can be carried out using the following identity [7]:

\int_{0}^{2 π} \exp [j k r \sin α \cos (ϕ - β)] \cos (m β) d β = 2 π j^{m} J_{m} (k r \sin α) \cos (m ϕ),

where

J_{m} (\cdot)

is the Bessel function of the first kind of order m. A similar integral involving

\sin (m β)

in the integrand can be found by replacing the cosine functions by sine functions in Eq. (4). Using Eq. (4) in Eq. (3), one finds that the electric field near the focus is

[\begin{matrix} E_{x} \\ E_{y} \\ E_{z} \end{matrix}] = \frac{1}{2} E_{o} \int_{0}^{α_{\max}} q (α) l_{0} (α) \exp (- j k z \cos α) [\begin{matrix} j \cos α \cos ϕ J_{1} (k r \sin α) \\ j \cos α \sin ϕ J_{1} (k r \sin α) \\ \sin α J_{0} (k r \sin α) \end{matrix}] \sin α d α .

The cylindrical components of the electric field given by Eq. (5) are $E_{r} = E_{x} \cos ϕ + E_{y} \sin ϕ$ and $E_{ϕ} = - E_{x} \sin ϕ + E_{y} \cos ϕ \equiv 0$ ; it shows that the beam is ϕ independent and that it is radially polarized, because its electric field has no azimuthal component.

2.3 Connection between the complex-source/sink wave and the Richards–Wolf theory

Whilst the integrals defining the electric field components in Eq. (5) have to be solved numerically in general, analytical solutions can be expected in the case of the 4π focusing, i.e. when $α_{\max} = π$ . In fact, the nonparaxial TM₀₁ beam as given by Eq. (1) can be generate with the Richards–Wolf theory if one assumes a field distribution of the collimated input beam at the entrance pupil of the 4π focusing system of the form:

q (α) l_{0} (α) = \sin α \exp [- \frac{4 f^{2}}{W_{o}^{2}} \sin^{2} (\frac{1}{2} α)] = \sin α \exp [- \frac{2 f^{2}}{W_{o}^{2}} (1 - \cos α)],

where

W_{o}

is the spot size of the collimated incident beam and f is the focal length of the optical system. The field distribution given by Eq. (6) has a doughnut shape profile, as it is the case for a Laguerre–Gaussian beam of order (0,1) (Fig. 2 ). It is defined for all values of α (

0 \leq α \leq π

), i.e. it is valid for a complete sphere of incoming illumination. It should be noted that the Gaussian factor in Eq. (6) is of the form produced by focusing with an optical system satisfying the Herschel condition [23,24].

Fig. 2 The angular amplitude distribution given by Eq. (6) characterizes a doughnut shape beam in the far-field. The amplitude profile has a significant value on a broader range of α as $k a = 2 f^{2} / W_{o}^{2}$ decreases.

Download Full Size | PDF

For the angular amplitude distribution given by Eq. (6), the electric field in all space can be expressed in a closed form that corresponds to the complex-source/sink wave defined by Eq. (1a). Let us introduce the parameter $a \equiv 2 f^{2} / k W_{o}^{2}$ . Substituting Eq. (6) in Eq. (5), we obtain the following integrals to solve:

[\begin{matrix} E_{x} \\ E_{y} \\ E_{z} \end{matrix}] = \frac{1}{2} E_{o} \exp (- k a) \int_{0}^{π} \sin^{2} α \exp (- j k \tilde{z} \cos α) [\begin{matrix} j \cos α \cos ϕ J_{1} (k r \sin α) \\ j \cos α \sin ϕ J_{1} (k r \sin α) \\ \sin α J_{0} (k r \sin α) \end{matrix}] d α,

where

\tilde{z} \equiv z + j a

is a complex longitudinal coordinate. Equation (7) can be integrated analytically with the help of the integral representation of the functions

{\tilde{U}}_{p, m}^{e}

and

{\tilde{V}}_{p, m}^{e}

, given by

{\tilde{U}}_{p, m}^{e} = \frac{1}{2} \exp (- k a) \cos (m ϕ) \int_{0}^{π} \sin^{2 p + m + 1} α \exp (- j k \tilde{z} \cos α) J_{m} (k r \sin α) d α,

{\tilde{V}}_{p, m}^{e} = \frac{1}{2} \exp (- k a) \cos (m ϕ) \int_{0}^{π} \sin^{2 p + m + 1} α \cos α \exp (- j k \tilde{z} \cos α) J_{m} (k r \sin α) d α,

where p = 0 (or 1) and m = 1 (or 0) to connect with the transverse components (or the longitudinal component) in Eq. (7). Odd functions

{\tilde{U}}_{0, 1}^{o}

and

{\tilde{V}}_{0, 1}^{o}

can be obtained from Eqs. (8a) and (8b) by replacing

\cos ϕ

by

\sin ϕ

. Using Eqs. (8a) and (8b) to solve Eq. (7) leads exactly to the same expressions as Eqs. (1)a) and (1b).

A representative 4π focusing system requiring a single beam incident from one direction is a parabolic mirror whose entrance pupil is taken to infinity, in which case $α_{\max} = π$ [25]. However, the amplitude distribution $l_{0} (α)$ of a collimated TM₀₁ beam at the entrance pupil of a parabolic mirror is rigorously proportional to $2 \tan (\frac{1}{2} α) \exp [- 2 k a \tan^{2} (\frac{1}{2} α)]$ [23,25]. Although this expression is different from Eq. (6), both formulations become approximately equal in the paraxial limit, i.e. for ka >> 1. Even though the agreement is very close for large values of ka, we have to conclude that the use of the field distribution defined by Eq. (6) does not exactly lead to the electromagnetic field of a TM₀₁ beam focused by a parabolic mirror. Instead, Eqs. (1a) and (1b) can be thought of as the complete field of a TM₀₁ beam focused by a 4π system whose far-field amplitude $l_{0} (α)$ is modulated by a suitable filter in such a way that it is given by Eq. (6). In principle, any prescribed amplitude function $l_{0} (α)$ could be chosen by using a proper pupil filter.

3. The vector magnetic potential method

Electromagnetic field can efficiently be computed using the vector magnetic potential [18,20,26]. According to the Richards–Wolf theory, the vector field equation is solved for the electric field. This approach is rather complicated since the electric field of an optical beam has, in general, three nonzero components. The vector magnetic potential, in turn, is usually assumed to have only one component in all space. Consequently, the vector magnetic potential method allows solving Maxwell’s equations in a more efficient way: once the vector magnetic potential of the beam in determined after solving no more than one scalar field integral, the electric and magnetic fields are then deduced directly from the definition of the potential.

Expressions for the fields of a nonparaxial TM beam can be found using a vector magnetic potential A oriented along the propagation axis [18,20]. With the Lorenz gauge, the vector magnetic potential A satisfies the vector Helmholtz equation $\nabla^{2} A + k^{2} A = 0$ . Accordingly, the nonzero Cartesian component of the vector magnetic potential obeys a scalar Helmholtz equation, while the electric and magnetic fields still have to obey a vector Helmholtz equation, since they must necessarily have more than one component, as a result of Maxwell’s equations (except for the trivial case of a uniform plane wave). By definition, the magnetic field in free space is related to A by

H = \frac{1}{μ_{0}} \nabla \times A .

If A has only a z-component, it is seen from Eq. (9) that

H_{z} = 0

, giving rise to a TM beam, as expected. The electric field E is found from Maxwell’s equation

E = {(j ω ε_{0})}^{- 1} \nabla \times H

. Thus, the electric field can be expressed in term of the vector magnetic potential as

E = - \frac{j ω}{k^{2}} \nabla \times \nabla \times A .

We define the following vector magnetic potential, of the same form as the electric field defined by Eq. (3)a), that is an exact solution to the Hemlholtz equation:

A = \frac{1}{4 π} \int_{0}^{2 π} \int_{0}^{α_{\max}} A_{o} (α) \exp [- j k Γ (α, β)] \sin α d α d β,

where

Γ (α, β)

is still defined by Eq. (3b) and

A_{o} (α)

is the vector angular amplitude distribution of the potential. If

A_{o} (α) \equiv {\hat{a}}_{z} A_{o} (α)

where

E_{o} q (α) l_{0} (α) = - j ω A_{o} (α) \sin α

, it can be verified that the substitution of Eq. (11) in Eq. (10) is equivalent to E as given by Eq. (3a), since:

\nabla \times \nabla \times [{\hat{a}}_{z} \exp (- j k Γ)] = k^{2} \sin α ({\hat{a}}_{x} \cos α \cos β + {\hat{a}}_{y} \cos α \sin β + {\hat{a}}_{z} \sin α) \exp (- j k Γ),

The angular amplitude distribution of the vector magnetic potential of a TM₀₁ beam that enables us to recover the electric and magnetic fields given by Eqs. (1)a) and (1b) is

A_{o} (α) = a_{z} A_{o} \exp [- k a (1 - \cos α)],

where

A_{o}

is a constant amplitude. For an aberration-free 4π focusing system, we have

Φ (α, β) = 0

and

α_{\max} = π

. Substituting these values and Eq. (13) in Eq. (11) yields, after carrying out the integral over the azimuthal angle β with Eq. (4):

A = \frac{1}{2} {\hat{a}}_{z} A_{o} \exp (- k a) \int_{0}^{π} \exp (- j k \tilde{z} \cos α) J_{0} (k r \sin α) \sin α d α,

The remaining integral to be solved is the integral representation of

{\tilde{U}}_{0, 0}^{e}

[see Eq. (8a)], so that

A = {\hat{a}}_{z} A_{o} {\tilde{U}}_{0, 0}^{e}

. The magnetic and electric fields can be found using Eqs. (9) and (10), respectively. To evaluate the curls, one may use the following identities [20]:

\nabla \times ({\hat{a}}_{z} {\tilde{U}}_{0, 0}^{e}) = - k {\hat{a}}_{x} {\tilde{U}}_{0, 1}^{o} + k {\hat{a}}_{y} {\tilde{U}}_{0, 1}^{e},

\nabla \times \nabla \times ({\hat{a}}_{z} {\tilde{U}}_{0, 0}^{e}) = j k^{2} {\hat{a}}_{x} {\tilde{V}}_{0, 1}^{e} + j k^{2} {\hat{a}}_{y} {\tilde{V}}_{0, 1}^{o} + k^{2} {\hat{a}}_{z} {\tilde{U}}_{1, 0}^{e},

Using Eqs. (15a) and (15b) with the vector magnetic potential

A = {\hat{a}}_{z} A_{o} {\tilde{U}}_{0, 0}^{e}

leads to the same expressions for the electric field E and the magnetic field H given by Eqs. (1a) and (1b), provided that the amplitudes are related by

E_{o} = - j ω A_{o}

and

H_{o} = E_{o} / η_{0}

.

4. Vector magnetic potential of aberrated TM₀₁ beams

In the preceding sections, we have considered that the optical system was not affected by aberrations, i.e. the wavefront that converges toward the focal point of the focusing system is perfectly spherical. In some applications, such an assumption is not verified and it becomes relevant to take aberrations into account. For instance, focusing a beam whose optical axis is a slightly tilted with respect to the axis of revolution of a parabolic mirror results, among others, to a strong coma [10].

The aberrations are taken into account in Eq. (3b) with the function $Φ (α, β)$ , which represent the phase deviation relative to a spherical wavefront. In the following, we assume that the aberration function is of the form

Φ (α, β) = \sum_{n = 0}^{\infty} \sum_{m = 0}^{\infty} C_{n, m} \sin^{n} α \cos (m β),

where

C_{n, m}

are the aberration coefficients that give the strength of the aberration terms. It should be mentioned that the factor

\sin^{n} α

in Eq. (16) could be replaced by the associated Legendre function

P_{n}^{m} (\cos α)

, so that the aberration function would be expressed as a linear combination of tesseral harmonics, which are orthogonal over a complete spherical pupil [27]. Nevertheless, in this paper, we will work with the aberration function as given by Eq. (16). The primary aberrations considered in this paper are the distortion, the curvature of field, the primary coma, the astigmatism and the spherical aberration (Table 1 ) [27–31].

Table 1. Aberration function for each primary aberration.

View Table

4.1 Distortion

We consider first the case of distortion, for which the aberration function is $Φ (α, β) = C_{1, 1} \sin α \cos β$ . The argument of the exponential function [Eq. (3b)] of the vector magnetic potential becomes

\begin{matrix} Γ (α, β) = C_{1, 1} \sin α \cos β - r \sin α \cos (ϕ - β) + z \cos α \\ = - (r \cos ϕ - C_{1, 1}) \sin α \cos β - r \sin ϕ \sin α \sin β + z \cos α \\ = - r^{'} \sin α \cos (ϕ^{'} - β) + z \cos α, \end{matrix}

where

r^{'} \equiv {(r^{2} + C_{1, 1}^{2} - 2 C_{1, 1} r \cos ϕ)}^{1 / 2}

and

\tan ϕ^{'} \equiv r \sin ϕ / (r \cos ϕ - C_{1, 1})

. It is apparent that the last equality in Eq. (17) has the same form of a distortion-free function

Γ (α, β)

. In other words, Eq. (3b) with

Φ = 0

is formally identical to Eq. (17), provided that the substitutions

r \to r^{'}

and

ϕ \to ϕ^{'}

are made. Consequently, in presence of distortion, the vector magnetic potential is formally the same as the one in the absence of distortion. The quantities

r^{'}

and

ϕ^{'}

are the coordinates of a new reference system and they can be interpreted geometrically as follows. The length

r^{'}

is obtained by applying the cosine law in the triangle illustrated in Fig. 3 and the angle

ϕ^{'}

is then found by trigonometry. When the fields are affected by distortion, the observation point is located in the reference system

(r^{'}, ϕ^{'})

whose origin is translated by an amount

C_{1, 1}

with respect to the origin of the coordinate system

(r, ϕ)

. Thus, the shape of the field distribution at the focus remains unchanged by distortion, but the field distribution is shifted along the x-axis in the focal plane.

Fig. 3 The point P is located with the coordinates $(r, ϕ)$ or $(r^{'}, ϕ^{'})$ . The second reference system is translated horizontally with respect to the first by an amount $C_{1, 1}$ .

Download Full Size | PDF

4.2 Curvature of field and spherical aberration

We now analyze the case of the curvature of field and the spherical aberration, i.e. aberrations whose function is β independent. The aberration function $Φ (α) = C_{n, 0} \sin^{n} α$ (where n = 2 or n = 4) is substituted in Eq. (11). In order to evaluate the integral, one may use the following power series expansion of the exponential function:

\exp (- j k C_{n, 0} \sin^{n} α) = \sum_{s = 0}^{\infty} \frac{{(- j k C_{n, 0})}^{s}}{s!} \sin^{n s} α .

We describe here the special case of the TM₀₁ beam focused by a 4π system by using Eq. (13) and setting

α_{\max} = π

. The integral to be solve is

\begin{matrix} A = {\hat{a}}_{z} \frac{A_{o}}{4 π} \exp (- k a) \sum_{s = 0}^{\infty} \frac{{(- j k C_{n, 0})}^{s}}{s!} \\ \times \int_{0}^{2 π} \int_{0}^{π} \sin^{n s + 1} α \exp [j k r \sin α \cos (ϕ - β) - j k \tilde{z} \cos α] d α d β, \end{matrix}

where

\tilde{z} \equiv z + j a

. Carrying out the integration over β with the help of Eq. (4) yields

A = \frac{1}{2} {\hat{a}}_{z} A_{o} \exp (- k a) \sum_{s = 0}^{\infty} \frac{{(- j k C_{n, 0})}^{s}}{s!} \int_{0}^{π} \sin^{n s + 1} α \exp (- j k \tilde{z} \cos α) J_{0} (k r \sin α) d α .

This last integral can be evaluated analytically with the integral representation of the function

{\tilde{U}}_{p, m}^{e}

given by Eq. (8)a):

A = A_{o} \sum_{s = 0}^{\infty} \frac{{(- j k C_{n, 0})}^{s}}{s!} {\hat{a}}_{z} {\tilde{U}}_{n s / 2, 0}^{e} .

Note that the number ns/2 is an integer, because n is even in this context. Hence, the vector magnetic potential of an aberrated beam is expressed as a superposition of nonparaxial elegant Laguerre–Gaussian beams. As a consequence, since the function

{\tilde{U}}_{p, m}^{e}

is an exact solution of the Helmholtz equation in free space, the vector magnetic potential of the aberrated beam defined by Eq. (21) is also a rigorous solution to the Helmholtz equation.

The vector magnetic potential $A = {\hat{a}}_{z} A_{o} {\tilde{U}}_{p, m}^{e}$ generates what we call a ${TM}_{p, m + 1}$ beam; as a special case, the vector $A = {\hat{a}}_{z} A_{o} {\tilde{U}}_{0, 0}^{e}$ produces a TM₀₁ beam, as discussed in Section 3. Therefore, the beam generated by the vector magnetic potential defined by Eq. (21) for a given n (n = 2 for curvature of field and n = 4 for spherical aberration) consists in a superposition of the fundamental ${TM}_{01}$ beam and higher-order ${TM}_{n s / 2, 1}$ beams of decreasing amplitude as the radial number ns/2 increases.

4.3 Coma and astigmatism

We finally consider the case of primary coma and astigmatism, whose aberration function depends on the angle β. The aberration function $Φ (α, β) = C_{n, m} \sin^{n} α \cos (m β)$ is replaced in Eq. (11) and the following Bessel series expansion for the exponential function may be used:

\exp [- j ζ \cos (m β)] = J_{0} (ζ) + 2 \sum_{q = 1}^{\infty} {(- j)}^{q} J_{q} (ζ) \cos (q m β),

where

ζ \equiv k C_{n, m} \sin^{n} α

. Also, we have the power series expansion of the Bessel function of the first kind:

J_{q} (ζ) = \sum_{s = 0}^{\infty} \frac{{(- 1)}^{s} {(k C_{n, m} \sin^{n} α)}^{2 s + q}}{(2 s)!! (2 s + 2 q)!!} .

Once again, we consider the special case of the TM₀₁ beam focused by a 4π system by using Eq. (13) and

α_{\max} = π

, leading to

\begin{matrix} A = {\hat{a}}_{z} \frac{A_{o}}{4 π} \exp (- k a) \sum_{s = 0}^{\infty} \frac{{(- j k C_{n, m})}^{2 s}}{(2 s)!!} \int_{0}^{2 π} \int_{0}^{π} \sin α \exp (- j k \tilde{z} \cos α) [\frac{\sin^{2 n s} α}{(2 s)!!} \\ + 2 \sum_{q = 1}^{\infty} \frac{{(- j k C_{n, m})}^{q} \sin^{2 n s + n q} α}{(2 s + 2 q)!!} \cos (q m β)] \exp [j k r \sin α \cos (θ - β)] d α d β . \end{matrix}

Solving analytically these integrals with Eqs. (4) and (8a) gives

A = 2 A_{o} \sum_{s = 0}^{\infty} \sum_{q = 0}^{\infty} \frac{j^{m q} {(- j k C_{n, m})}^{2 s + q}}{(2 s)!! (2 s + 2 q)!! (1 + δ_{q, 0})} {\hat{a}}_{z} {\tilde{U}}_{n s + (n - m) q / 2, q m}^{e},

where

δ_{q, 0}

is the Kronecker delta which is one if q = 0 and zero otherwise. The beam generated by the vector magnetic potential defined by Eq. (25) for given n and m ({n,m} = {3,1} for primary coma and {n,m} = {2,2} for astigmatism) is made of the superposition of the fundamental

{TM}_{01}

beam and higher-order

{TM}_{p, q m + 1}

beams, where

p = n s + (n - m) q / 2

, of decreasing amplitude as the radial and/or the azimuthal numbers increase.

4. Electromagnetic field of an aberrated TM₀₁ beam

Once the vector magnetic potential is known for a given optical beam, the expressions for the electric and the magnetic fields can be found by straightforward analytical calculations. More precisely, the electromagnetic field of a TM₀₁ beam affected by curvature of field or spherical aberration are obtained by substituting Eq. (21) in Eqs. (9) and (10), whereas the fields of a TM₀₁ beam affected by primary coma or astigmatism are determined by substituting Eq. (25) in Eqs. (9) and (10). In each case, the amplitudes of the electromagnetic field and of the potential are related by $E_{o} = - j ω A_{o}$ and $H_{o} = E_{o} / η_{0}$ . In order to evaluate the curl of a vector of the form ${\hat{a}}_{z} {\tilde{U}}_{p, m}^{e}$ , the following identities are available [20]:

\nabla \times ({\hat{a}}_{z} {\tilde{U}}_{p, m}^{e}) = - \frac{1}{2} k {\hat{a}}_{x} ({\tilde{U}}_{p, m + 1}^{o} + {\tilde{U}}_{p + 1, m - 1}^{o}) + \frac{1}{2} k {\hat{a}}_{y} ({\tilde{U}}_{p, m + 1}^{e} - {\tilde{U}}_{p + 1, m - 1}^{e})

\nabla \times \nabla \times ({\hat{a}}_{z} {\tilde{U}}_{p, m}^{e}) = j \frac{1}{2} k^{2} {\hat{a}}_{x} ({\tilde{V}}_{p, m + 1}^{e} - {\tilde{V}}_{p + 1, m - 1}^{e}) + j \frac{1}{2} k^{2} {\hat{a}}_{y} ({\tilde{V}}_{p, m + 1}^{o} + {\tilde{V}}_{p + 1, m - 1}^{o}) + k^{2} {\hat{a}}_{z} {\tilde{U}}_{p + 1, m}^{e}

where

{\tilde{U}}_{p, m}^{e}

and

{\tilde{V}}_{p, m}^{e}

are defined by Eqs. (2a) and (2b), respectively. For m = 0, the following identities are useful:

{\tilde{U}}_{p + 1, - 1}^{e} = - {\tilde{U}}_{p, 1}^{e}

,

{\tilde{U}}_{p + 1, - 1}^{o} = {\tilde{U}}_{p, 1}^{o}

,

{\tilde{V}}_{p + 1, - 1}^{e} = - {\tilde{V}}_{p, 1}^{e}

, and

{\tilde{V}}_{p + 1, - 1}^{o} = {\tilde{V}}_{p, 1}^{o}

. We will not write out the explicit forms taken by the electric and magnetic fields. The fields of an aberrated TM₀₁ beam are analytical expressions that are exact solutions of Maxwell’s four equations. Note that these electric and magnetic fields reduce to Eqs. (1a) and (1b) when

C_{n, m} = 0

, as it must be.

The electric energy density of an optical beam is defined by $w_{e} \equiv \frac{1}{2} ε_{o} {| E |}^{2}$ . The electric energy density of an aberration-free TM₀₁ beam in the focal plane (z = 0) for ka = 1 (the numerical aperture approximately equals to 0.9) has a peak on the optical axis and exhibits rings of weak amplitude (Fig. 4a ). In fact, it is well-known that, in the paraxial limit (ka >> 1), the electric energy density on the axis is small compared to its maximum value, providing a beam with a dark center; however, as the beam is more tightly focused (as the value of ka decreases), the energy density associated to the longitudinal component of the electric field is enhanced and the dark center gradually disappears [19].

Fig. 4 Electric energy density distributions of a TM₀₁ beam for ka = 1 (a) without aberration, affected (b) by curvature of field with C _2,0 = 2λ, (c) by spherical aberration with C _4,0 = 3 λ, (d) by primary coma with C _3,1 = 2 λ, (e) and by astigmatism with C _2,2 = λ.

Download Full Size | PDF

The electric energy density profiles shown in Fig. 4 have the expected shape for each kind of primary aberration. The strongly focused TM₀₁ beam affected by curvature of field or spherical aberration shows more pronounced diffraction rings around the focal spot; the rotational symmetry of the fields near focus is not destroyed, as expected (Fig. 4b and 4c). The distribution of the beam affected by coma has a comet-like shape with unsymmetrical rings, exhibiting a comatic image flaring in the horizontal direction (Fig. 4d). Finally, the electric energy density distribution of a beam affected by astigmatism is not circular but has a definite cross-shaped aspect (Fig. 4e).

To conclude this section, we mention that the expressions for the aberrated electric and magnetic fields of the lower-order transverse electric beam, the TE₀₁ beam, can be easily found from those of an aberrated TM₀₁ beam by means of the duality transformation $E \to η_{0} H$ and $H \to - E / η_{0}$ .

5. Conclusion

The application of the Richards–Wolf vector field integral with a 4π system (such as a parabolic mirror of large extent) to the focusing of a TM₀₁ beam yields the complex sink/source solution, provided that the proper amplitude distribution of the collimated input beam at the entrance pupil of the focusing system is chosen. This establishes a connection between the Richards–Wolf theory and the complex sink/source model; these two approaches are commonly used to study strongly focused beams. The use of a 4π system to produce the focused beam gives a physical interpretation of the existence of the two counter-propagating beams in the complex-source/sink solution, especially in the case for which the beam is tightly focused.

The analytical expressions presented in this paper give a physical insight into the way that an aberrated TM₀₁ beam can be decomposed: it may be viewed as a superposition of the fundamental TM₀₁ beam and higher-order TM beams. The electromagnetic field of an optical beam affected by aberrations can be written as a linear combination of nonparaxial elegant Laguerre–Gaussian beams.

The complex-source/sink solution, which is expressed in a simple closed form, remains an accurate description of a nonparaxial beam, generated for instance by a parabolic mirror, and allows to investigate the behavior of strongly focused beams without having to deal with several integrals to solve.

Acknowledgements

The authors acknowledge support from Natural Sciences and Engineering Research Council of Canada (NSERC), Fonds québécois de recherche sur la nature et les technologies (FQRNT), Canadian Institute for Photonic Innovations (ICIP/CIPI), and the Centre d'optique, photonique et laser (COPL), Québec.

References and links

1. S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179(1-6), 1–7 (2000). [CrossRef]

2. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003). [CrossRef] [PubMed]

3. H. Dehez, M. Piché, and Y. De Koninck, “Enhanced resolution in two-photon imaging using a TM(₀₁) laser beam at a dielectric interface,” Opt. Lett. 34(23), 3601–3603 (2009). [CrossRef] [PubMed]

4. C. Varin, M. Piché, and M. A. Porras, “Acceleration of electrons from rest to GeV energies by ultrashort transverse magnetic laser pulses in free space,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(2), 026603 (2005). [CrossRef] [PubMed]

5. B. Richards and E. Wolf, “Electromagnetic Diffraction in Optical Systems. II. Structure of the Image Field in an Aplanatic System,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959). [CrossRef]

6. K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7(2), 77–87 (2000). [CrossRef] [PubMed]

7. 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).

8. L. Novotny, and B. Hecht, Principles of nano-optics, (Cambridge University Press, 2006, Chap. 3).

9. P. Varga and P. Török, “Focusing of electromagnetic waves by paraboloid mirrors. I. Theory,” J. Opt. Soc. Am. A 17(11), 2081–2089 (2000). [CrossRef]

10. M. A. Lieb and A. J. Meixner, “A high numerical aperture parabolic mirror as imaging device for confocal microscopy,” Opt. Express 8(7), 458–474 (2001). [CrossRef] [PubMed]

11. N. Davidson and N. Bokor, “High-numerical-aperture focusing of radially polarized doughnut beams with a parabolic mirror and a flat diffractive lens,” Opt. Lett. 29(12), 1318–1320 (2004). [CrossRef] [PubMed]

12. J. Stadler, C. Stanciu, C. Stupperich, and A. J. Meixner, “Tighter focusing with a parabolic mirror,” Opt. Lett. 33(7), 681–683 (2008). [CrossRef] [PubMed]

13. G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett. 7(23), 684–685 (1971). [CrossRef]

14. M. Couture and P.-A. Bélanger, “From Gaussian beam to complex-source-point spherical wave,” Phys. Rev. A 24(1), 355–359 (1981). [CrossRef]

15. C. J. R. Sheppard and S. Saghafi, “Beam modes beyond the paraxial approximation: A scalar treatment,” Phys. Rev. A 57(4), 2971–2979 (1998). [CrossRef]

16. C. J. R. Sheppard and S. Saghafi, “Electromagnetic Gaussian beams beyond the paraxial approximation,” J. Opt. Soc. Am. A 16(6), 1381–1386 (1999). [CrossRef]

17. Z. Ulanowski and I. K. Ludlow, “Scalar field of nonparaxial Gaussian beams,” Opt. Lett. 25(24), 1792–1794 (2000). [CrossRef]

18. L. W. Davis and G. Patsakos, “TM and TE electromagnetic beams in free space,” Opt. Lett. 6(1), 22–23 (1981). [CrossRef] [PubMed]

19. C. J. R. Sheppard and S. Saghafi, “Transverse-electric and transverse-magnetic beam modes beyond the paraxial approximation,” Opt. Lett. 24(22), 1543–1545 (1999). [CrossRef]

20. A. April, “Nonparaxial TM and TE beams in free space,” Opt. Lett. 33(14), 1563–1565 (2008). [CrossRef] [PubMed]

21. A. April, “Nonparaxial elegant Laguerre-Gaussian beams,” Opt. Lett. 33(12), 1392–1394 (2008). [CrossRef] [PubMed]

22. G. Arfken, Mathematical Methods for Physicists, 3rd ed., (Oxford, Ohio, Academic Press, Inc., 1985).

23. C. J. R. Sheppard and K. G. Larkin, “Optimal concentration of electromagnetic radiation,” J. Mod. Opt. 41(7), 1495–1505 (1994). [CrossRef]

24. N. Bokor and N. Davidson, “Toward a spherical spot distribution with 4π focusing of radially polarized light,” Opt. Lett. 29(17), 1968–1970 (2004). [CrossRef] [PubMed]

25. N. Bokor and N. Davidson, “4π Focusing with single paraboloid mirror,” Opt. Commun. 281(22), 5499–5503 (2008). [CrossRef]

26. L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19(3), 1177–1179 (1979). [CrossRef]

27. C. J. R. Sheppard, “Orthogonal aberration functions for high-aperture optical systems,” J. Opt. Soc. Am. A 21(5), 832–838 (2004). [CrossRef]

28. R. Kant, “An analytical method of vector diffraction for focusing optical systems with Seidel aberrations II: Astigmatism and coma,” J. Mod. Opt. 42(2), 299–320 (1995). [CrossRef]

29. C. J. R. Sheppard, “Vector diffraction in paraboloidal mirrors with Seidel aberrations: effects of small object displacements,” Opt. Commun. 138(4-6), 262–264 (1997). [CrossRef]

30. R. K. Singh, P. Senthilkumaran, and K. Singh, “Effect of primary coma on the focusing of a Laguerre–Gaussian beam by a high numerical aperture system; vectorial diffraction theory,” J. Opt. A, Pure Appl. Opt. 10(9pp), (2008). [CrossRef]

31. D. P. Biss and T. G. Brown, “Primary aberrations in focused radially polarized vortex beams,” Opt. Express 12(3), 384–393 (2004). [CrossRef] [PubMed]

	Values for n and m	Aberration function
Distortion	n = 1 and m = 1	$Φ (α, β) = C_{1, 1} \sin α \cos β$
Curvature of field	n = 2 and m = 0	$Φ (α) = C_{2, 0} \sin^{2} α$
Primary coma	n = 3 and m = 1	$Φ (α, β) = C_{3, 1} \sin^{3} α \cos β$
Astigmatism	n = 2 and m = 2	$Φ (α, β) = C_{2, 2} \sin^{2} α \cos (2 β)$
Spherical aberration	n = 4 and m = 0	$Φ (α) = C_{4, 0} \sin^{4} α$

4π Focusing of TM₀₁ beams under nonparaxial conditions

Abstract

1. Introduction

2. Focusing of an aberration-free TM₀₁ beam

2.1 The complex source/sink model

2.2 The Richards–Wolf vector field equation

2.3 Connection between the complex-source/sink wave and the Richards–Wolf theory

3. The vector magnetic potential method

4. Vector magnetic potential of aberrated TM₀₁ beams

4.1 Distortion

4.2 Curvature of field and spherical aberration

4.3 Coma and astigmatism

4. Electromagnetic field of an aberrated TM₀₁ beam

5. Conclusion

Acknowledgements

References and links

Cited By

Figures (4)

Tables (1)

Equations (33)

Optics Express

Abstract

1. Introduction

2. Focusing of an aberration-free TM01 beam

2.1 The complex source/sink model

2.2 The Richards–Wolf vector field equation

2.3 Connection between the complex-source/sink wave and the Richards–Wolf theory

3. The vector magnetic potential method

4. Vector magnetic potential of aberrated TM01 beams

4.1 Distortion

4.2 Curvature of field and spherical aberration

4.3 Coma and astigmatism

4. Electromagnetic field of an aberrated TM01 beam

5. Conclusion

Acknowledgements

References and links

Cited By

Figures (4)

Tables (1)

Equations (33)

Optics Express

2. Focusing of an aberration-free TM₀₁ beam

4. Vector magnetic potential of aberrated TM₀₁ beams

4. Electromagnetic field of an aberrated TM₀₁ beam