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

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References

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  1. X. Xu, W. Huang, and M. Xu, “Orthogonal polynomials describing polarization aberration for rotationally symmetric optical systems,” Opt. Express 23(21), 27911–27919 (2015).
    [Crossref] [PubMed]
  2. J. H. Burnett, Z. H. Levine, E. L. Shirley, and J. H. Bruning, “Symmetry of spatial-dispersion-induced birefringence and its implications for CaF2 ultraviolet optics,” J. Micro. Nanolithogr. MEMS MOEMS 1(3), 213–224 (2002).
    [Crossref]
  3. J. Ruoff and M. Totzeck, “Using orientation Zernike polynomials to predict the imaging performance of optical systems with birefringent and partly polarizing components,” Proc. SPIE 7652, 76521T (2010).
    [Crossref]
  4. W. Xu, W. Huang, C. Liu, and H. Shang, “Automatic clocking optimization for compensating two-dimensional tolerances,” Opt. Express 21(19), 22145–22152 (2013).
    [Crossref] [PubMed]
  5. J. Wolfe and R. Chipman, “Reducing symmetric polarization aberrations in a lens by annealing,” Opt. Express 12(15), 3443–3451 (2004).
    [Crossref] [PubMed]
  6. M. Laikin, Lens Design (CRC, 2007), Chap. 11.
  7. J. Dirk, “Projection exposure method, projection exposure system and projection objective,” United States Patent US9036129 (2015).
  8. J. H. Burnett, Z. H. Levine, and E. L. Shirley, “Intrinsic birefringence in calcium fluoride and barium fluoride,” Phys. Rev. B 64(24), 241102 (2001).
    [Crossref]

2015 (1)

2013 (1)

2010 (1)

J. Ruoff and M. Totzeck, “Using orientation Zernike polynomials to predict the imaging performance of optical systems with birefringent and partly polarizing components,” Proc. SPIE 7652, 76521T (2010).
[Crossref]

2004 (1)

2002 (1)

J. H. Burnett, Z. H. Levine, E. L. Shirley, and J. H. Bruning, “Symmetry of spatial-dispersion-induced birefringence and its implications for CaF2 ultraviolet optics,” J. Micro. Nanolithogr. MEMS MOEMS 1(3), 213–224 (2002).
[Crossref]

2001 (1)

J. H. Burnett, Z. H. Levine, and E. L. Shirley, “Intrinsic birefringence in calcium fluoride and barium fluoride,” Phys. Rev. B 64(24), 241102 (2001).
[Crossref]

Bruning, J. H.

J. H. Burnett, Z. H. Levine, E. L. Shirley, and J. H. Bruning, “Symmetry of spatial-dispersion-induced birefringence and its implications for CaF2 ultraviolet optics,” J. Micro. Nanolithogr. MEMS MOEMS 1(3), 213–224 (2002).
[Crossref]

Burnett, J. H.

J. H. Burnett, Z. H. Levine, E. L. Shirley, and J. H. Bruning, “Symmetry of spatial-dispersion-induced birefringence and its implications for CaF2 ultraviolet optics,” J. Micro. Nanolithogr. MEMS MOEMS 1(3), 213–224 (2002).
[Crossref]

J. H. Burnett, Z. H. Levine, and E. L. Shirley, “Intrinsic birefringence in calcium fluoride and barium fluoride,” Phys. Rev. B 64(24), 241102 (2001).
[Crossref]

Chipman, R.

Huang, W.

Levine, Z. H.

J. H. Burnett, Z. H. Levine, E. L. Shirley, and J. H. Bruning, “Symmetry of spatial-dispersion-induced birefringence and its implications for CaF2 ultraviolet optics,” J. Micro. Nanolithogr. MEMS MOEMS 1(3), 213–224 (2002).
[Crossref]

J. H. Burnett, Z. H. Levine, and E. L. Shirley, “Intrinsic birefringence in calcium fluoride and barium fluoride,” Phys. Rev. B 64(24), 241102 (2001).
[Crossref]

Liu, C.

Ruoff, J.

J. Ruoff and M. Totzeck, “Using orientation Zernike polynomials to predict the imaging performance of optical systems with birefringent and partly polarizing components,” Proc. SPIE 7652, 76521T (2010).
[Crossref]

Shang, H.

Shirley, E. L.

J. H. Burnett, Z. H. Levine, E. L. Shirley, and J. H. Bruning, “Symmetry of spatial-dispersion-induced birefringence and its implications for CaF2 ultraviolet optics,” J. Micro. Nanolithogr. MEMS MOEMS 1(3), 213–224 (2002).
[Crossref]

J. H. Burnett, Z. H. Levine, and E. L. Shirley, “Intrinsic birefringence in calcium fluoride and barium fluoride,” Phys. Rev. B 64(24), 241102 (2001).
[Crossref]

Totzeck, M.

J. Ruoff and M. Totzeck, “Using orientation Zernike polynomials to predict the imaging performance of optical systems with birefringent and partly polarizing components,” Proc. SPIE 7652, 76521T (2010).
[Crossref]

Wolfe, J.

Xu, M.

Xu, W.

Xu, X.

J. Micro. Nanolithogr. MEMS MOEMS (1)

J. H. Burnett, Z. H. Levine, E. L. Shirley, and J. H. Bruning, “Symmetry of spatial-dispersion-induced birefringence and its implications for CaF2 ultraviolet optics,” J. Micro. Nanolithogr. MEMS MOEMS 1(3), 213–224 (2002).
[Crossref]

Opt. Express (3)

Phys. Rev. B (1)

J. H. Burnett, Z. H. Levine, and E. L. Shirley, “Intrinsic birefringence in calcium fluoride and barium fluoride,” Phys. Rev. B 64(24), 241102 (2001).
[Crossref]

Proc. SPIE (1)

J. Ruoff and M. Totzeck, “Using orientation Zernike polynomials to predict the imaging performance of optical systems with birefringent and partly polarizing components,” Proc. SPIE 7652, 76521T (2010).
[Crossref]

Other (2)

M. Laikin, Lens Design (CRC, 2007), Chap. 11.

J. Dirk, “Projection exposure method, projection exposure system and projection objective,” United States Patent US9036129 (2015).

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Figures (6)

Fig. 1
Fig. 1 Samples of field maps of FOZP.
Fig. 2
Fig. 2 Simulation flowchart.
Fig. 3
Fig. 3 (a) A NA 1.28 microscope objective. (b) Transmittance and phase difference of s and p light of AR coating.
Fig. 4
Fig. 4 Simulation results of disturbed microscope objective. Top row: Diattenuation; Second row: Retardance.
Fig. 5
Fig. 5 (a) A NA 1.35 lithographic lens. (b) Transmittance and phase difference of s and p light of AR coating. (c) Reflectivity and phase difference of s and p light of HR coating. (d) IBs of CaF2 plate with crystal axis orientations <100> (top), <110> (middle) and <111> (bottom), where the color and short line represent the retardance value and fast axis direction, respectively.
Fig. 6
Fig. 6 Simulation results of (a) retardance piston OZ1 and OZ−1 and (b) retardance tilt OZ2 and OZ3. Top row: without CaF2; Second row: with <100>-oriented CaF2; Third row: with <110>-oriented CaF2; Bottom row: with <111>-oriented CaF2.

Tables (6)

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Table 1 1-fold terms of FOZP

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Table 2 2-fold terms of FOZP

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Table 3 3-fold terms of FOZP

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Table 4 4-fold terms of FOZP

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Table 5 Tolerance type and magnitude

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Table 6 M-fold symmetries of lithographic lens and FOZP terms

Equations (7)

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f 0+ = F n',0 m' O Z n,0 m + F n',1 m' O Z n,1 m , f 0 = F n',0 m' O Z n,0 m F n',1 m' O Z n,1 m , f 1+ = F n',1 m' O Z n,0 m + F n',0 m' O Z n,1 m , f 1 = F n',1 m' O Z n,0 m + F n',0 m' O Z n,1 m .
F n',0 m' (h,α)= R n' m' (h)cosm'α, F n',1 m' (h,α)= R n' m' (h)sinm'α.
O Z n,0 m = R n m (ρ)( cosmθsinmθ sinmθcosmθ ),O Z n,1 m = R n m (ρ)( sinmθcosmθ cosmθsinmθ ).
R n m (ρ)= s=0 (nm)/2 (1) s (ns)! s![(n+m)/2s]![(nm)/2s]! ρ n2s
Ε 0+ =( cos mθm'α 2 sin mθm'α 2 ).
mθm'α 2 + 2π M 0+ = m 2 (θ+ 2π M 0+ ) m' 2 (α+ 2π M 0+ )+kπ,kZ,
π 2 4 1 1+ δ m'0 0 2π 0 1 0 2π 0 1 (FO Z Ni FO Z Mj +FO Z Mj FO Z Ni )ρdρdθhdhdα ={ Iwheni=j,M=N; 0others.

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