## Abstract

We study the effect of primary aberrations on the 3-D polarization of the electric field in a focused lowest order radially polarized beam. A full vector diffraction treatment of the focused beams is used. Attention is given to the effects of primary spherical, astigmatic, and comatic aberrations on the local polarization, Strehl ratio, and aberration induced degradation of the longitudinal field at focus.

©2004 Optical Society of America

## 1. Introduction

Inhomogeneously polarized light, such as it appears in radial or azimuthally-polarized beams, has seen increased interest in microscopy in recent years [1–5]. The focal fields from such beams have been investigated in several recent papers [2,6–8]. The point spread function due to radially and azimuthally polarized light in scanning microscopy has been studied [1,4]. Though much effort has been put forth to understand the imaging qualities of cylindrically symmetric vector beam the effect of wavefront aberrations present in the focused beams has not yet been studied.

In this paper, we study weakly aberrated high numerical aperture (NA) lenses such as might be used for linear confocal or nonlinear microscopy. It is not uncommon for even well-corrected objectives to suffer small amounts of spherochromatism or small amounts of coma or astigmatism due to slight misalignments and tilts of individual elements. Emphasis will therefore be placed on the effect of primary aberrations on local polarization at the focus of large NA lens systems.

## 2. Theory

Wavefront aberrations in linearly polarized beams focused with a large and small NA have been formulated and discussed in the literature [9–21]. For large focusing angles, these methods take into account, within the limits of the Debye approximation, the full vector diffraction formalism of a focused beam. We will take this formalism and apply it to the case of inhomogeneously polarized beams and, more specifically, to the case of lowest order radially polarized beams.

## 2.1. Focused radially polarized beams

Figure 1 describes the focusing geometry. A lens with focal length *f* is focusing a field at z -coordinate, *z*=0. The local field is situated at coordinates *x*,*y*, and *z*. The individual plane wave components can be described by azimuthal angle *φ* and polar angle *θ*, in which 0≤*φ*≤2*π* and 0≤*θ*≤*α*.

We use an angular spectrum representation of the focal fields for simplicity of calculation. This form is equivalent to the equations used by Youngworth and Brown, Helseth, and Biss and Brown to describe the fields near focus [6–8]. Our equation for the fields begin as,

$$\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\left[\begin{array}{c}\frac{{k}_{x}}{{k}_{\rho}}\frac{{k}_{z}}{k}\\ \frac{{k}_{y}}{{k}_{\rho}}\frac{{k}_{z}}{k}\\ -\frac{{k}_{\rho}}{k}\end{array}\right]{e}^{i\left({k}_{x}x+{k}_{y}y+{k}_{z}z\right)}d{k}_{x}d{k}_{y},$$

where *x*,*y*, and *z* are the spatial coordinates near the focus of the beam, *k* is the wavevector magnitude of the field, ${k}_{\mathit{max}}=\frac{kNA}{n}$, ${k}_{\rho}=\sqrt{{k}_{x}^{2}+{k}_{y}^{2}}$, and ${k}_{z}=\sqrt{1-{k}_{\rho}^{2}}$. Plane wave components *k _{x}* and

*k*can be related to the angles

_{y}*φ*and

*θ*in Fig. 1 through the relationship

*k*=

_{x}*k*cos(

*φ*) sin(

*θ*) and

*k*=

_{y}*k*sin(

*φ*) sin(

*θ*). Using the substitution for

*k*and

_{x}*k*we can retain the form of the focal fields used in Youngworth and Brown. The function

_{y}*l*(

_{o}*k*,

_{x}*k*) is the apodization function in the entrance pupil of the focusing system. We will take this apodization to be the radially polarized, paraxial solution to the vector wave equation as found in [22, 23]. If we consider the beam to be nearly collimated the resulting equation is,

_{y}where *r* is the normalized radial coordinate in the pupil, *β* is a constant, and *w*
_{o} is the Gaussian beam waist. In Greene and Hall *β* is related to the focusing angle of the paraxial beam, which allows us to relate *β* to *w*
_{o} [24]. After the switch into the angular spectrum coordinates and the substitution between *β* and *w*
_{o} is performed, the apodization function is,

## 2.2. Aberrations

With the focal fields known, we can now consider the effects of aberrations. We will not go into depth about determining the aberration function or the form it manifests itself in the equations of the focused field, but we will refer readers to the following articles [12, 14, 16, 18–20]. In a manner consistent with the aforementioned articles, aberrations in the entrance pupil of the system can be represented by a complex amplitude in the pupil,

where *k* is again the wave vector of the field and *W*(*ρ*,*φ*) is the phase deviation (aberration) relative to a spherical wavefront. We define the phase contribution of aberrations to the beam apodization as,

which is consistent with Kant [14, 16]. If we then insert Eq. (3) and (5) into Eq. (1) the final form of the focal fields becomes,

$$\times \mathrm{exp}(ik({w}_{040}{\left(\frac{{k}_{\rho}}{{k}_{max}}\right)}^{2}+{w}_{131}\frac{{k}_{x}}{{k}_{\rho}}{\left(\frac{{k}_{\rho}}{{k}_{max}}\right)}^{3}+{w}_{22}{\left(\frac{{k}_{x}}{{k}_{\rho}}\right)}^{2}{\left(\frac{{k}_{\rho}}{{k}_{max}}\right)}^{2}))$$

$$\times \mathrm{exp}\left(-{\beta}^{2}\frac{{k}_{\rho}^{2}}{{k}_{max}^{2}}\right){J}_{1}\left(2\beta \frac{{k}_{\rho}}{{k}_{max}}\right)\sqrt{\frac{{k}_{z}}{k}}\frac{1}{{k}_{z}}\left[\begin{array}{c}\frac{{k}_{x}}{\sqrt{{k}_{x}^{2}+{k}_{y}^{2}}}\frac{{k}_{z}}{k}\\ \frac{{k}_{y}}{\sqrt{{k}_{x}^{2}+{k}_{y}^{2}}}\frac{{k}_{z}}{k}\\ -\frac{\sqrt{{k}_{x}^{2}+{k}_{y}^{2}}}{k}\end{array}\right]$$

$$\times {e}^{i\left({k}_{x}x+{k}_{y}y+{k}_{z}z\right)}d{k}_{x}d{k}_{y}.$$

With the focal fields in this form, they can be easily integrated using standard 2-d fast fourier transform (FFT) algorithm. In the calculation, the pupil was sampled over a 512×512 grid, placed in a 2048×2048 FFT window, and each cartesian component calculated in the focal volume was according to Eq. 6.

## 3. Results and discussion

#### 3.1. Transverse and through-focus projections

Calculations of focused radially polarized beams with coma, astigmatism, and spherical aberration were performed. Both transverse and through-focus projections for the focused beams are shown in Fig. 2 through Fig. 4. The focusing system modeled has a *NA*=0.9 and is focused in air. In the figures axis dimensions are in wavelengths. We choose the beam to have *β*=1*e*-3. The cyan lines and ellipses overlaid on the figures depict the local state of polarization of the field. In each animation, the amount of the aberration varies from -2λ to 2λ. Figure 2(a), 3(a), and 4(a) are animations of the transverse projections at the geometrical focus (*z*=0) of the focused beams with aberrations. Figure 2(b), 3(b), and 4(b) are animations of through-focus projections of the focused beams with aberrations.

## 3.1.1. Spherical Aberration

Figure 2 shows two animations of focused radially polarized beams with varying amounts of spherical aberration. From these animations it can be seen that spherical aberration seems not to distort the focal spot for moderate amounts of aberration. As expected, spherical aberration does not destroy the rotational symmetry of the field near focus. At the geometrical focus, the transverse state of polarization remains unchanged. If we look at the through-focus projection, we see that at the focal spot the beam preserves both a longitudinal and radially polarized component.

## 3.1.2. Coma

Figure 3 shows two animations of focused radially polarized beams with varying amounts of coma. It is clearly seen that coma destroys any rotational symmetry of the spot. The distinct comet-like shape attributed coma can be seen in Fig. 3(a). If we focus our attention to the transverse projection the polarization of the spot can be partially described by considering a simple geometrical optics mapping of the polarization in the pupil to the focus. This mapping can explain the 45° polarization along the diagonal edge of the spot. Also, if one considers a simple ray based model of the through-focus field, we see that the polarization is mainly directed along the direction of the rays in the caustic region of the focus. This is seen especially for larger amounts of aberration.

## 3.1.3. Astigmatism

Figure 4 shows two animations of focused radially polarized beams with varying amounts of astigmatism. As with coma, astigmatism destroys the rotational symmetry of the focal spot. It is also seen that a phase shift between horizontal and vertically polarized components occurs, creating circularly polarized regions in the transverse planes of the focal fields. This phase shift can be partially explain by a ray-based examination of astigmatism. The tangential and sagittal ray fans have different foci when astigmatism is introduced in the system. In a focused radial beam the tangential ray fan is vertically polarized, the sagittal ray fan is horizontally polarized. The focusing of the horizontally and vertically polarized components to different foci introduces this phase shift which is seen in Fig. 4(a).

## 3.2. Strehl Ratio

We used the Strehl Ratio as a metric of the quality of the aberrated focused field for situations with varying amounts of spherical aberration and compared it to a similar focusing situation, but with a linearly polarized Gaussian mode illuminating the pupil. For the Strehl ratio we use,

where *S* is the Strehl ratio, *I*
_{o} is the maximum intensity of all polarization components of the unaberrated beam, ans *I _{ab}* is the maximum intensity of all the polarization components of the aberrated beam taken at the plane of the geometrical focus, which is consistent with the definition of Strehl intensity used in scalar theories [20]. For focused radially polarized beams, the total focal field is confined at focus in high NA systems much like a focused linearly polarized beam.

It is common in practice to compensate for a given amount of spherical aberration with a specified amount of defocus. We considered this case for varying amounts of spherical aberration and compared it to the case of a linearly polarized Gaussian beam illuminating the pupil of the focusing system. We also considered the case of a linearly polarized beam with an annular mask at the entrance pupil of the imaging system. The same field apodization (Eq. 3) as the radially polarized beam with the same value of *β* was used as the apodization function in this case.

Figure 5 shows the Strehl ratio for a few different cases. The solid lines are Strehl ratios for a linearly polarized Gaussian input beam with varying amounts of spherical and defocus aberration. The dotted lines are Strehl ratio plots for a linearly polarized annular input beam with varying amounts of spherical and defocus aberration. The dashed lines are Strehl ratio plots for a radially polarized input beam with varying amounts of spherical and defocus aberration. The blue lines correspond to a system with 0λ of spherical aberration. The green lines correspond to a system with 0.5λ of spherical aberration. The red lines correspond to a system with 1λ of spherical aberration. In the calculation, the linearly polarized annular beam has an apodization function equivalent to Eq. (3).

It is seen from the figure that the curves for the radially polarized case have a larger width of the peak, indicating a larger insensitivity to defocus aberration. When spherical aberration is added to the system, and then compensated for with defocus, it can clearly be seen that larger amounts of spherical aberration can be corrected for with the radially polarized case than the Gaussian linear case. The linearly polarized annulus shows that some of this insensitivity is due to the annular shape of the illumination, but the fact that the radially polarized beam has a larger insensitivity than the linear annulus also indicates the polarization of the field also has an effect.

Both Kant [14] and Sheppard [25] have noted that the wavefront error associated with defocus of the long conjugate (which Sheppard terms tube length error) differ from that due to defocus on the high NA side of the system. Figure 6 shows the Strehl ratio of focused radially and linearly polarized beams. In the figure the blue lines represent Strehl ratios of beams with long conjugate defocus coefficient *w*
_{020}=0λ, the green lines are Strehl ratios of beams with *w*
_{020}=0.5λ, and the red lines are Strehl ratios of beams with *w*
_{020}=1λ. The solid lines are Strehl ratios for focused radially polarized beams, the dashed lines are for focused linearly polarized beams. From this figure it can be seen that the focused radial beams has a larger insensitivity to tube length error and can be compensated to a better amount that focused linearly polarized beams.

## 3.3. Effect on longitudinal fields

In papers by Youngworth and Brown and Quabis *et al.*, imaging with just the longitudinal field of a focused radially polarized beam is considered [1,2,4,6]. To get a sense of the relative quality of longitudinal component, or z-polarized component, of the focused radial field a metric of the ratio of the maximum longitudinal intensity versus maximum radial intensity is introduced. This metric has a similar feel to that of the Strehl ratio, but is directed to the degradation of the longitudinal component.

Figure 7 shows the ratio of maximum longitudinal intensity versus maximum radial intensity for varying amounts of astigmatism, coma, and spherical aberration. The system considered is a 0.9*NA* air objective. We examine the field at the geometrical focus (*z*=0) and look at the transverse plane (*x*,*y*) intensity. The red line in Fig. 7 is the ratio for varying amounts of spherical aberration. The green line is the ratio for varying amounts of astigmatism. The blue line is the ratio for varying amounts of coma. The x-axis in the plot is the amount of the aberration in waves (λ). The y-axis is the ratio of maximum intensities, which is a unit-less quantity.

For small amounts of aberration (<0.25λ) coma degrades the longitudinal intensity more than astigmatism and spherical aberration. For aberration greater than 0.5λ astigmatism and spherical aberration lessen the longitudinal intensity more than coma, but at this amount of aberration the focal fields are already so distorted that they probably would be useless. It is easily seen from Fig. 7 that spherical aberration has the least effect on the longitudinal intensity compared to the radial intensity for aberration magnitudes less than 0.5λ.

It is important to emphasize that this work applies to primary aberrations such as might arise with spherechromatism (i.e., if an objective is used at a wavelength significantly different than the design wavelength) or misalignments. However, spherical aberration often occurs when focusing through a planar interface, Juskaitis has noted that in this situation, conventional expansions of wavefront aberrations do not converge rapidly [26]. In such a case, the calculation presented here could be combined with the treatment published by Biss and Brown [8].

## 4. Conclusion

This paper analyzed the effects of 3^{rd} order spherical aberration, coma, and astigmatism on strongly focused radially polarized beams. We found that the annular nature of the field and its polarization led to an insensitivity to spherical aberration compared to the same situation with linearly polarized light. We also found that the focused radially polarized beam with spherical aberration can be compensated by added defocus to a larger extend than with focused linearly polarized light.

Astigmatism and coma strongly distort the focal fields, and quickly degrade the longitudinal field intensity compared to the radial intensity. For systems that depend on the rotational symmetry of the focal fields, or the intensity of the longitudinal field, astigmatism and coma are of great concern.

For focusing systems that have a critical dependence on spherical aberration, such as lithographic systems or biological imaging in thick tissue, the effect of spherical aberration can be reduced with radially polarized beams, compared to linearly polarized Gaussian beams or linearly polarized annular illumination.

## Acknowledgments

We would like to acknowledge Alexis Lanning at the Institute of Optics for their helpful discussions. This work was conducted under a Frank Horton Fellowship from the Laboratory for Laser Energetics and a grant from the Semiconductor Research Corporation.

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