Focusing of an aberrated radially polarized beam by microscope objectives

. A rigorous method is presented to compute the vector diffraction pattern of microscope objectives considering on-axis aberrated and radially polarized wavefronts. For that, the diffraction pattern of the microscope objective is computed considering the polarization and the wavefront at the exit pupil of the optical system. These computations do not assume that at the exit pupil of the microscope objective the wave-front is spherical but aberrated. Examples of real microscope objectives ’ performance are presented.


Introduction
The first theories that described diffraction did not consider the vectorial nature of the electric field and magnetic field. 1 These theories are the Rayleigh-Sommerfeld and scalar Kirchhoff models. 2 Despite considering the electric field as a scalar quantity, these theories were available to describe the beam propagation on paraxial optical systems.A particular case of the scalar Kirchhoff model is the Fraunhofer approximation for far fields. 1,2The Fraunhofer approximation can compute the diffraction pattern of optical systems, the point-spread function, and the Airy disc. 3The Airy disc described the best-focused spot of light that a perfect optical system with a circular aperture could make, given that it is diffraction-limited. 1,2he Airy disc described by the Fraunhofer approximation is not the smallest possible spot.The spot size can be decreased if the vectorial nature of light is considered.Richards and Wolf obtained diffraction integral to describe the beam propagation considering the polarization of light. 4,5Using the Richards-Wolf integral with radially polarized light the Airy disc can be decreased.Therefore, the Richards-Wolf integral is used to describe the diffraction pattern beyond the limits imposed by the scalar theories.This treatment, beyond the scalar approximation, is necessary in several areas of science, including high-resolution microscopy, [6][7][8][9] optical trapping, 10 electron acceleration, 11 optical vortex knots, 12,13 beam shaping, etc. [14][15][16][17][18][19][20][21] Nevertheless, the Richards-Wolf integral only considers perfect spherical wavefronts 4,5 and real optical systems do not generate perfect spherical wavefronts at the exit pupil, but aberrated wavefronts.A new diffraction integral has been published in Refs.22 and 23 and this integral can compute the diffraction pattern of vectorial aberrated wavefronts with radial polarization.Here, the model presented in Ref. 23 is used to compute the diffraction pattern of several microscope objectives considering the polarization of the wavefront and its aberrations.The work in Ref. 23  differs from the models presented in Refs.15, 24, and 25 that the aberrations are taken directly from the wavefront using the Malus-Dupin theorem, 23 instead of considering the aberration function as an additional phase in over a spherical wavefront. 15,24,25This manuscript aims to present a rigorous method to compute the vector diffraction pattern of microscope objectives considering on-axis aberrated and radially polarized wavefronts.

Vectorial Diffraction Integral
Ref. 23 presents a new vector diffraction integral capable of computing the diffraction patterns for aspheric and freeform wavefronts, with any polarization.This paper does not consider an arbitrary polarization state since working with arbitrary polarization states is not straightforward, and it highly increases the complexity of the problem.The deduction of these integral starts from the angular spectrum representation of the optical fields.Like the Richards-Wolf integral, the integral presented in Ref. 23 is formulated by the assumption that the diffraction pattern is computed by the superposition of plane waves, and the contribution of evanescent waves is ignored.This integral is given as E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 1 ; 1 where Eðr; ϕ; zÞ is the electric field in cylindrical coordinates ðr; ϕ; zÞ, and x, ŷ, and ẑ, are the unit vectors in the x; y, and z directions.E 0 is a constant amplitude, qðx; yÞ is the apodization function, k is the wave number.The wavefront wðx; yÞ must be a continuous function of how independent variables are x and y. l 0 is the illumination function.∂ x w and ∂ y w are the partial derivatives of wðx; yÞ respect to are x and y.Pðx; yÞ is the exit pupil shape, a function that inside the pupil region is 1 and 0 otherwise.This region is delimited by x min and x max and y min and y max in the x and y directions, respectively.θ is the rotation angle along the plane perpendicular to k. Depending on the value of θ is the polarization state, if θ ¼ 0, we get a for a radially symmetric beam E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 2 ; 1 1 4 ; 2 6 9 Eðr; ϕ; zÞ ¼ E 0 2π In this manuscript, we are going to consider only an input on-axis field, thus the wavefront at the exit pupil is a spherical wavefront with some aberration.Without losing generality these aberrations can be expressed in terms of the Zernike polynomials, and since the field is on-axis and the system is radially symmetric the aberration is aspheric and can be expressed in Zernike terms which are radially symmetric, 26 for example E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 3 ; 1 1 7 ; 7 3 6 where ρ 2 ¼ x 2 þ y 2 , x ¼ ρ cosðβÞ, and y ¼ ρ sinðβÞ.Z 1 , Z 4 , Z 11 and Z 22 are the 1st, 4th, 11 th , and 22nd Zernike terms, and z 1 , z 4 , z 11 , and z 22 their respect coefficients.The method can be generalized to consider more terms.Then, the wavefront at the exit pupil is given as ; t e m p : i n t r a l i n k -; e 0 0 4 ; 1 1 7 ; 6 6 1 where R a is the radius of the reference sphere, W spherical is the reference sphere wavefront, and W zernike is the aberrated wavefront.The result of derivating Eq. ( 4) respect to x and y is in Eq. ( 5) ; t e m p : i n t r a l i n k -; e 0 0 5 ; 1 1 7 ; 5 6 8 where ; t e m p : i n t r a l i n k -; e 0 0 6 ; 1 1 7 ; 5 3 2 Notice that the partial derivative of wðx; yÞ for the on-axis field can always be expressed in the form of Eq. ( 5).There will be always a function ΞðρÞ that only depends on ρ due to the symmetry.The form of ΞðρÞ only depends on the number of terms of the Zernike polynomials implemented.If Z 1 , Z 4 , Z 11 , and Z 22 are the only terms considered, lastly the form of ΞðρÞ is given by Eq. ( 6).
If the input on-axis field is flat and radially polarized and replacing Eq. ( 5) in Eq. ( 2), we get E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 7 ; 1 1 7 ; 4 2 1 Eðr; ϕ; zÞ ¼ E 0 2π where Ξ 0 ðρÞ is the derivative of ΞðρÞ respect to ρ. Observe that the pupil function has been removed in Eq. ( 7) and the upper limit of the integral of ρ is one, these results come since the Zernike polynomials are defined in a circle of unity radius.The integral of β in Eq. ( 7) can be computed using well-known properties of Bessel functions 27 E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 8 ; 1 1 7 ; 2 7 7 Eðr; where J 0 and J 1 are the Bessel functions of the zeroth and first-order, respectively.Eq. ( 8) is the most important equation in the manuscript and it describes the diffraction pattern produced by an aspherical wavefront radially polarized at the exit pupil of an optical system.This integral is expressed in terms of ΞðρÞ, which is the common factor of the partial derivatives of the wavefront at the exit pupil.In this next section, we are going to use Eq. ( 8) to directly compute the diffraction pattern generated by microscope objectives and test their performance.Equation ( 8) is more compact and simple than Eq. ( 2), since it only depends in a single independent parameter.Computationally speaking this is a great advantage over Eq. ( 2). 23

Illustrative Examples of Diffraction Patterns in Microscopes Objectives
In this section are presented several illustrative examples of diffraction patterns in microscope objectives computed directly using Eq. ( 8).To use Eq. ( 8), it is needed to know the shape of the wavefront at the exit pupil of each microscope objective.Since the wavefront at the exit pupil can be expressed in Zernike polynomials, we express the wavefront at the exit pupil with Eq. ( 4).The coefficients of the Zernike polynomials of a field at the exit pupil of an optical system are computed with commercial optical software.The package used for the examples in this paper is Quadoa Optical CAD. 28In the following examples, we have set the apodization function q, and the illumination l 0 to be one and the wavelength to be 550 nm.
The first example is a microscope objective that is diffraction-limited according to raytracing and the scalar wave theory.This means, all the rays strike from a ray grid inside the Airy disc.Then, the exit pupil is almost a perfect spherical wavefront at the exit pupil, The numerical aperture of the microscope is NA = 0.749.The diffraction pattern in the z-axis and the r are plotted in Fig. 1.The lines that correspond to this example are in dashed orange.The black lines correspond to the classical formulation of Richards-Wolf. 4,5Observe that for the zaxis and in the r are plotted the orange and black lines overlap on every point.The result then is as predicted by Richards-Wolf. 4,5It is important to remark that the Richards-Wolf integral is only valid for stigmatic systems.Since the root mean square is much smaller than the Airy disc, the orange line almost overlaps the stigmatic case, the black line.The ray-tracing and stop diagrams are presented in Fig. 2. The grey circle surrounding the stop diagram is the Airy disc The dashed orange and red lines correspond to the results of the microscope objective in Tables 1  and 2, respectively.
Fig. 2 Microscope objective of Table 1.At left is the ray tracing and at right is the spot diagram.
described by the Fraunhofer approximation.The design parameters of this microscope are presented in Table 1.
The next example is another microscope objective that has a nearly diffraction-limited spot diagram, also with an NA = 0.749.The ray tracing and the spot diagram of this example can be seen in Fig. 3.The specifications are in Table 2.The profile of the diffraction pattern has been added in Fig. 1, under the red lines.The performance of this microscope objective is below the The results presented for the two microscopes are very illustrative since they show how two different systems with similar NA and can be considered diffraction-limited by geometrical optics and scalar diffraction theory they present different performances with a vectorial treatment.
There are other methods to compute diffraction patterns in similar circumstances.They are mostly based on the paradigm presented in Refs.15, 24, and 25.Reference 15 considers no aberrations, which he explicitly mentions in the mentioned paper before Eq. ( 5).Thus, its method is not considered directly the same problem that we consider in our manuscript.The difference between our method and Refs.24 and 25.The derivations presented in Refs.24 and 25 are too cumbersome in comparison with our method.Our method is straightforward.We only use eight equations, instead of 52 and 38 presented in Refs.24 and 25, respectively.Our method does not need additional so-called optical coordinates, like in Refs.24 and 25.Since our method comes from an expression that works for freeform wavefronts, Eq. ( 2).We do not need to use concepts like surfaces of revolution as our initial premise as can be seen in Eqs. ( 6) and (7) in Ref. 24 or Eq. ( 5) in Ref. 25.
References 24 and 25 are just limited to some Seidel coefficients.Our method works directly with an explicit wavefront function w.The computation of the method in Ref. 24 implies several approximations that we do not consider.To mention a few Eq. ( 27), the expression after Eq. ( 28) (not numbered in the paper), the expression after Eq. ( 35) (not numbered in the paper), and Eqs. ( 51) and (52).Reference 25 considers small aberrations, something we do not consider.It is explicitly mentioned before Eq. ( 20) in Ref. 25. Reference 25 considers several approximations that also we do not consider, such as their Eqs.( 20), ( 21), (22), and (43).Also, it is not clear and the author does not mention in Refs.24 and 25, that the propagation vector is normal to the wavefront, something that is a crucial step in our manuscript.
The approach outlined in this study diverges from the numerical strategy outlined in Ref. 29.In contrast, the method proposed by Wang in Ref. 29 relies on numerical approximations, such as the homeomorphic Fourier transform.The derivation of the closed-form equation Eq. ( 8) in our method is entirely analytical, devoid of any reliance on numerical approximations or iterative processes.In our manuscript, we can reduce our expressions to a single integral if and only if the wavefront is a surface of revolution and the polarization angle θ is constant.
It is important to remark that this analysis is not restricted to microscope objectives, but it can be applied in the radially symmetric optical system in general.Equation ( 8) can be used directly with the coefficients extracted from available tools.The natural step to follow in this research is to consider miss alignment and off-axial fields.This generalization considerably increases the complexity of the system and its non-linearity since the direction of the polarization in every point of the wavefront wðx; yÞ changes due to polarization aberrations. 30That problem is complex enough to be treated apart in a specific manuscript.

Conclusions
In this manuscript, we have derived an integral to compute the diffraction pattern of on-axis beams for radially symmetric systems, with radial polarization, Eq. ( 8).To validate Eq. ( 8), we tested it with a perfect spherical wavefront.The results are equal to the ones presented in the classical formulation of Richards-Wolf. 4,5Then, we compare two different microscope objectives and the results were as predicted by the theory presented in this work.

Disclosures
The author declares no conflicts of interest.

Fig. 1
Fig.1Normalized intensity profiles of several microscope objectives.On the left is the profile along z and on the right is a long r .The black line coincides with the diffraction pattern of a perfect spherical wavefront at the exit pupil computed with the classical formulation of Richards-Wolf.4,5The dashed orange and red lines correspond to the results of the microscope objective in Tables1 and 2, respectively.

Table 1
Specifications of the microscope of Fig. 2. Microscope objective of Table 2.At left is the ray tracing and at right is the spot diagram.otherexamples, as expected.It is diffraction-limited, but its geometrical spot size is bigger than the first example.Its full width at half maximum for the radial dimension is 1.2λ and 0.701λ for the on-axis component for the positive refraction index.The Zernike coefficients for this example are z 1 ¼ −3.1164 × 10 −04 , z 4 ¼ 3.2019 × 10 −04 , z 11 ¼ −1.1323 × 10 −05 , and z 22 ¼ −4.6723 × 10 −05 .

Table 2
Specifications of the microscope of Fig.3.