## Abstract

Polarization aberration (PA) is a serious issue that affects imaging quality for optical systems with high numerical aperture. Numerous studies have focused on the distribution rule of PA on the pupil, but the field remains poorly studied. We previously developed an orthonormal set of polynomials to reveal the pupil and field dependences of PA in rotationally symmetric optical systems. However, factors, such as intrinsic birefringence of cubic crystalline material in deep ultraviolet optics and tolerance, break the rotational symmetry of PA. In this paper, we extend the polynomials from rotationally symmetric to *M*-fold to describe the PA of *M*-fold optical systems. Two examples are presented to verify the polynomials.

© 2016 Optical Society of America

## 1. Introduction

In a previous paper [1], we developed an orthogonal set of polynomials, field-orientation Zernike polynomials (FOZP), to describe the polarization aberration (PA) of rotationally symmetric systems. FOZP can simultaneously reveal the pupil and field dependences of PA. However, when tolerances are considered, the rotationally symmetric property (RSP) can be broken. For precision optics in deep ultraviolet, cubic crystalline materials such as CaF_{2} are used. The combination of fused silica and cubic crystalline material can correct chromatic aberration in 193 nm lithography. Lithography with wavelength of 157 nm needs entirely crystalline refractive optics because fused silica is not sufficiently transparent at this wavelength [2]. However, intrinsic birefringence (IB) in cubic crystalline materials, which is not rotationally symmetric, breaks the RSP of retardance. In this paper, we extend FOZP from rotationally symmetric to *M*-fold to describe the PA of *M*-fold optical systems. An *M*-fold object means that rotation by an angle of *n·*(360°*/M*) (*n =* 1, 2 …, *M*) with respect to a particular point (in two dimensions) or axis (in three dimensions) does not change the object [3]. 0-fold means rotationally symmetric.

Xu [4] classified tolerances into one-dimensional (1D) tolerances, such as refractive index, radius, thickness, and element tilt; and two-dimensional (2D) tolerances such as material index inhomogeneity and surface figure errors. For simplicity, we only consider 1D tolerances in the current paper. We further classify 1D tolerances into two groups: the 0-fold tolerances such as refractive index, radius, and thickness, which do not change the rotational symmetry of an optical system; and 1-fold tolerances such as element tilt, decenter, and surface wedge, which change a rotationally symmetric system into 1-fold.

Burnett [2] indicated that the IB in cubic crystalline material exhibits interesting symmetries. Cubic crystalline materials with different crystal axis orientations exhibit different IB distributions. For example, the IBs of <100>-, <110>- and <111>-oriented CaF_{2} plates are 4-, 2- and 3-fold, respectively.

*M*-fold FOZP terms are derived in Section 2. In Section 3, two lenses of high numerical aperture (NA), namely, a microscope objective disturbed by 1D tolerances and a lithographic lens with a CaF_{2} plate, are used for simulation to verify the *M*-fold FOZP.

## 2. *M*-fold FOZP

In a previous paper [1], we derived four basic formulas (*f*_{0+}, *f*_{0−}, *f*_{1+}, and *f*_{1−}) to describe the PA of an optical system:

*h*,

*α*), and $O{Z}_{n,\epsilon}^{{}^{m}}$is the orientation Zernike polynomials (OZP) of pupil coordinates (

*ρ*,

*θ*):

It should be noted that (n−m) and (n'−m') are even. When describing PA, the eigenvalue and eigenvector of *f*_{0+} represent the diattenuation (or retardance) value and the direction of the bright (or fast) axis, respectively. The eigenvalue of *f*_{0+} is *R _{n}*

_{'}

^{m}^{'}(

*h*)

*R*(

_{n}^{m}*ρ*), which is radial and rotationally symmetric. Therefore, when the distribution of its eigenvector directions is

*M*-fold,

*f*

_{0+}is

*M*-fold. The corresponding eigenvector is

Note that E_{0+} is periodic in 180°, because a polarization element rotated through 180° operates identically to the one oriented at 0° [5]. To assess the *M*-fold symmetry, we rotate E_{0+} by an angle 2*π*/*M*_{0+} and require it to be the same with the original one obtained at the shifted angular coordinate position (*θ +* 2*π*/*M*_{0+}, *α +* 2*π*/*M*_{0+}), i.e.,

*M*

_{0+}

*=*−(

*m−m*'−2)/

*k*. The

*M*-fold symmetry of

*f*

_{0+}corresponds to the largest non-negative

*M*

_{0+}, therefore, we obtain

*M*

_{0+}

*=*|

*m−m*'−2|. Similarly, the

*M*-fold symmetries of

*f*

_{0−},

*f*

_{1+}, and

*f*

_{1−}are

*M*

_{0−}

*=*|

*m + m*'−2|,

*M*

_{1+}

*=*|

*m + m*'−2|, and

*M*

_{1−}

*=*|

*m−m*'−2|, respectively. Note that

*m*is an integer and

*m*' is a non-negative integer.

A series of *M*-fold polynomials is then obtained. Each term of the polynomials is a 2 × 2 matrix and a function of the pupil and field coordinates. The polynomials can be used to describe the PA of *M*-fold optical systems. The rotationally symmetric (i.e., 0-fold) terms of FOZP are given in [1], and the 1- to 4-fold terms of FOZP are shown in Tables 1 to 4, respectively, where different terms of FOZP are labeled as *FOZ _{Mj}*.

The product of *FOZ _{Mj}* and

*FOZ*follows Eq. (7):

_{Ni}*I*and

**denote the 2 × 2 unit matrix and 2 × 2 zero matrix, respectively, and**

*0**δ*

_{m}_{'0}is the Kronecker delta function. Equation (7) is defined as the orthogonality property of FOZP. Different terms of FOZP are orthogonal. Figure 1 shows the field maps of 1- to 4-fold FOZP terms. The field maps represent the field dependences of FOZP. When drawing field map, we view OZP terms as vectors, as shown in the fifth column of Fig. 1. The vectors multiplied by the coefficients of the corresponding OZP terms form a vector field (i.e., field map) across the field region. Note that the

*M*-fold property of field map is independent of the

*M*-fold property of the corresponding FOZP term.

## 3. Examples

In this section, a high-NA microscope objective disturbed by 1D tolerances and a high-NA lithographic lens with a CaF_{2} plate are used for simulation to verify the *M*-fold FOZP. The purpose of simulation is to assess what the distribution rule of PA on the field really is. Figure 2 shows the simulation flowchart. First, the tolerances or CaF_{2} plate turns the rotationally symmetric system into *M*-fold. Second, the polarization ray tracing function in the software Code V is used to obtain the Jones pupils for all the field points. Third, the Jones pupils are decomposed by single value decomposition (SVD), and the diattenuation and retardance are obtained. Fourth, the diattenuation and retardance are fitted by OZP. Finally, the OZP coefficients, which are functions of field coordinates, are further fitted by FZP.

#### 3.1 Disturbed microscope objective

An oil-immersion microscope objective of NA 1.28 [6] with anti-reflective (AR) coatings is used for simulation. Figure 3 shows the objective drawing and optical property of the AR coating. The objective is then randomly disturbed by 1D tolerances, as shown in Table 5, and becomes 1-fold. The simulation results are shown in Fig. 4.

Results show that different OZP terms and their coefficients can always be combined as FOZP terms, as marked with red rectangles in Fig. 4. There are 0-fold terms such as *F*_{5}*·OZ*_{1} *+ F*_{6}*·OZ _{−}*

_{1},

*F*

_{2}

*·OZ*

_{2}

*−F*

_{3}

*·OZ*

_{3}, and

*F*

_{5}

*·OZ*

_{4}

*+ F*

_{6}

*·OZ*

_{−}_{4}, 1-fold terms such as

*F*

_{2}

*·OZ*

_{1}

*+ F*

_{3}

*·OZ*

_{−}_{1},

*F*

_{3}

*·OZ*

_{1}

*−F*

_{2}

*·OZ*

_{−}_{1},

*F*

_{2}

*·OZ*

_{4}

*+ F*

_{3}

*·OZ*

_{−}_{4}, and

*F*

_{3}

*·OZ*

_{4}

*−F*

_{2}

*·OZ*

_{−}_{4}, and 2-fold terms such as

*OZ*

_{1},

*OZ*

_{−}_{1},

*OZ*

_{4}, and

*OZ*

_{−}_{4}.

The PA of the 1-fold lens is 1-fold, which is unchanged after being rotated by 360° along the optical axis. The 0-, 1- and 2-fold FOZP terms in the simulation results also remain unchanged after being rotated by 360°; thus, they can be used to describe the PA of 1-fold lens. In fact, all *M*-fold FOZP terms can be used to describe the 1-fold PA, but the dominant terms are low-order ones. Some opinions even hold that 1-fold symmetry is no symmetry, because there is no symmetry at all when one object has to undergo one full rotation to match itself [3].

#### 3.2 Lithographic lens with CaF_{2} plate

A water-immersion ArF lithographic lens of NA 1.35 [7] with AR coatings and high-reflective (HR) coatings is used for simulation. The lens material is SiO_{2} and the lens is rotationally symmetric. We then add a 10 mm thick CaF_{2} plate in the lens. The IB in CaF_{2} breaks the RSP of retardance. Figure 5 shows the lens drawing, optical properties of AR and HR coatings, and 2D distributions of IB of the CaF_{2} plate.

The retardances of the lithographic lens with the <100>-, <110>- and <111>-oriented CaF_{2} plates are 4-, 2- and 3-fold, respectively. The IB value of CaF_{2} is −3.4 × 10^{−7} at 193 nm wavelength [2,8]. The simulation is operated separately under four conditions: without CaF_{2}, with <100>-oriented CaF_{2}, with <110>-oriented CaF_{2}, and with <111>-oriented CaF_{2}. Figure 6 shows the simulation results of retardance piston *OZ*_{1} and *OZ*_{−1}, as well as retardance tilt *OZ*_{2} and *OZ*_{3}.

For retardance piston and tilt, the FOZP terms of the lithographic lens without CaF_{2} are *F*_{5}∙*OZ*_{1} + *F*_{6}∙*OZ*_{−1}, *F*_{12}∙*OZ*_{1} + *F*_{13}∙*OZ*_{−1}, *F*_{21}∙*OZ*_{1} + *F*_{22}∙*OZ*_{−1}, *F*_{2}∙*OZ*_{2}−*F*_{3}∙*OZ*_{3}, *F*_{7}∙*OZ*_{2}−*F*_{8}∙*OZ*_{3}, and *F*_{14}∙*OZ*_{2}−*F*_{15}∙*OZ*_{3}, which are 0-fold and in accordance with the RSP of the lens. Except for 0-fold terms, there are 4-fold terms (*F*_{5}∙*OZ*_{1}−*F*_{6}∙*OZ*_{−1} and *F*_{10}∙*OZ*_{2} + *F*_{11}∙*OZ*_{3}), 2-fold terms (*F*_{1}∙*OZ*_{1}, *F*_{4}∙*OZ*_{1} and *F*_{2}∙*OZ*_{2} + *F*_{3}∙*OZ*_{3}), and 3-fold terms (*F*_{3}∙*OZ*_{1} + *F*_{2}∙*OZ*_{-1} and *−F*_{6}∙*OZ*_{2} + *F*_{5}∙*OZ*_{3}) in the FOZP of lithographic lens with <100>-, <110>-, and <111>-oriented CaF_{2} plates, respectively, as shown in Fig. 6. The *M*-fold symmetries of the lithographic lens and the corresponding FOZP terms coincide, as concluded in Table 6.

## 4. Conclusion

We extend the FOZP proposed in a previous paper from rotationally symmetric to *M*-fold. The polynomials are classified by their *M*-fold symmetries and are orthonormal over a unit circle pupil and a unit circle field. FOZP can simultaneously reveal the pupil and field dependences of PA in *M*-fold optical systems. A microscope objective disturbed by 1D tolerances and a lithographic lens with a CaF_{2} plate are used for simulation to verify FOZP. Results show that 0-, 1- and 2-fold FOZP terms can fully describe the PA of disturbed microscope objective, which is 1-fold; the PA of the *M*-fold lithographic lens can be fully described by 0 and *M*-fold FOZP terms. The coincidence between the *M*-fold symmetries of optical systems and FOZP terms indicates the validity of the proposed polynomials.

## References and links

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