Abstract

A description is given of a modeling technique that is used to explore three-dimensional image distributions formed by high numerical aperture (NA > 0.6) lenses in homogeneous, isotropic, linear, and source-free thin films. The approach is based on a plane-wave decomposition in the exit pupil. Factors that are due to polarization, aberration, object transmittance, propagation, and phase terms are associated with each plane-wave component. These are combined with a modified thin-film matrix technique in a derivation of the total field amplitude at each point in the film by a coherent vector sum over all plane waves. One then calculates the image distribution by squaring the electric-field amplitude. The model is used to show how asymmetries present in the polarized image change with the influence of a thin film. Extensions of the model to magneto-optic thin films are discussed.

© 1996 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968).
  2. F. H. Dill, A. R. Neureuther, J. A. Tuttle, E. J. Walker, “Modeling projection printing of positive photoresist,” IEEE Trans. Electron Devices ED-7, 456–464 (1975).
    [CrossRef]
  3. R. S. Herchel, “Partial coherence in projection printing,” in Developments in Semiconductor Microlithography III, D. R. Ciarlo, J. Dey, K. Hoeppner, R. L. Ruddell, eds., Proc. Soc. Photo-Opt. Instrum. Eng.135, 24–29 (1978).
    [CrossRef]
  4. B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
    [CrossRef]
  5. R. Kant, “A general numerical solution of vector diffraction for aplanatic systems,” Tech. Rep. TR02.1713.B (IBM, San Jose, Calif., 1991).
  6. T. D. Visser, S. H. Wiersma, “Spherical aberration and the electromagnetic field in high aperture systems,” J. Opt. Soc. Am. A 8, 1404–1410 (1991).
    [CrossRef]
  7. J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986).
  8. M. Mansuripur, “Distribution of light at and near the focus of high numerical aperture objectives,” J. Opt. Soc. Am A 3, 2086–2093 (1986).
    [CrossRef]
  9. H. Ling, S. W. Lee, “Focusing of electromagnetic waves through a dielectric interface,” J. Opt. Soc. Am. A 1, 965–973 (1984).
    [CrossRef]
  10. J. Gasper, G. C. Sherman, J. J. Stamnes, “Reflection and refraction of an arbitrary wave at a plane interface,” J. Opt. Soc. Am. 66, 955–961 (1976).
    [CrossRef]
  11. M. Yeung, “Modeling high numerical aperture optical lithography,” in Optical/Laser Microlithography, B. J. Lin, ed., Proc. Soc. Photo-Opt. Instrum. Eng.922, 149–167 (1988).
  12. M. Yeung, “Photolithography simulation on non-planar substrate,” in Optical/Laser Microlithography III, V. Pol, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1264, 309–321 (1990).
    [CrossRef]
  13. C. Yaun, A. Strojwas, “Modeling optical microscope images of integrated-circuit structures,” J. Opt. Soc. Am. A 8, 778–790 (1991).
    [CrossRef]
  14. D. G. Flagello, T. Milster, “Three-dimensional modeling of high numerical aperture imaging in thin films,” in Design, Modeling, and Control of Laser Beam Optics, Y. Kohanzadeh, G. N. Laurence, J. G. McCoy, H. Weichel, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1625, 246–261 (1992).
    [CrossRef]
  15. D. G. Flagello, A. E. Rosenbluth, C. Progler, J. Armitage, “Understanding high numerical aperture optical lithography,” Microcircuit Eng. 17, 105–108 (1991).
    [CrossRef]
  16. D. G. Flagello, A. E. Rosenbluth, “Lithographic tolerances based on vector diffraction theory,” J. Vac. Sci. Technol. 10, 2997–3003 (1992)
    [CrossRef]
  17. D. G. Flagello, A. E. Rosenbluth, “Vector diffraction analysis of phase-mask imaging in photoresist films,” in Optical/Laser Microlithography VI, J. D. Cuthbert, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1927, 395–412 (1993).
    [CrossRef]
  18. D. G. Flagello, “High numerical aperture imaging in homogeneous thin films,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1993).
  19. E. Wolf, “Electromagnetic diffraction in optical systems. I. An integral representation of the image field,” Proc. R. Soc. London Ser. A 253, 349–357 (1959).
    [CrossRef]
  20. H. H. Hopkins, “Image formation with coherent and partially coherent light,” Photogr. Sci. Eng. 21, 114–122 (1977).
  21. D. S. Goodman, “Stationary optical projectors,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1979).
  22. J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).
  23. R. A. Chipman, “Polarization aberrations,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1987).
  24. R. A. Chipman, “The mechanics of polarization ray tracing,” in Polarization Analysis and Measurement, R. A. Chipman, D. H. Goldstein, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1746, 62–75 (1989).
    [CrossRef]
  25. P. Debye, “Das Verhalten von Lichtwellen in der Nähe eines Brennpunktes oder einer Brennlinie” (“Behavior of light waves in the proximity of a focal point or focal line”), Ann. Phys. (Leipzig) 30, 755–776 (1909).
  26. R. K. Luneberg, Mathematical Theory of Optics (U. of California, Berkeley, Calif., 1944); reprint: (Brown U., Providence, R.I., 1964).
  27. E. Wolf, Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205–210 (1981).
    [CrossRef]
  28. H. A. Macleod, Thin Film Optical Filters (McGraw-Hill, New York, 1989).
  29. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).
  30. D. S. Goodman, “Topics in image formation,” short course presented at Optical Society of America Annual Meeting, Orlando, Fla., October 19, 1989.
  31. J. E. Harvey, “Fourier treatment of near-field scalar diffraction theory,” Am. J. Phys. 47, 974–980 (1979).
    [CrossRef]
  32. R. Hunt, “Magneto-optical scattering from thin solid films,” J. Appl. Phys. 38, 1652–1671 (1967).
    [CrossRef]

1992

D. G. Flagello, A. E. Rosenbluth, “Lithographic tolerances based on vector diffraction theory,” J. Vac. Sci. Technol. 10, 2997–3003 (1992)
[CrossRef]

1991

1986

M. Mansuripur, “Distribution of light at and near the focus of high numerical aperture objectives,” J. Opt. Soc. Am A 3, 2086–2093 (1986).
[CrossRef]

1984

1981

E. Wolf, Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205–210 (1981).
[CrossRef]

1979

J. E. Harvey, “Fourier treatment of near-field scalar diffraction theory,” Am. J. Phys. 47, 974–980 (1979).
[CrossRef]

1977

H. H. Hopkins, “Image formation with coherent and partially coherent light,” Photogr. Sci. Eng. 21, 114–122 (1977).

1976

1975

F. H. Dill, A. R. Neureuther, J. A. Tuttle, E. J. Walker, “Modeling projection printing of positive photoresist,” IEEE Trans. Electron Devices ED-7, 456–464 (1975).
[CrossRef]

1967

R. Hunt, “Magneto-optical scattering from thin solid films,” J. Appl. Phys. 38, 1652–1671 (1967).
[CrossRef]

1959

E. Wolf, “Electromagnetic diffraction in optical systems. I. An integral representation of the image field,” Proc. R. Soc. London Ser. A 253, 349–357 (1959).
[CrossRef]

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

1909

P. Debye, “Das Verhalten von Lichtwellen in der Nähe eines Brennpunktes oder einer Brennlinie” (“Behavior of light waves in the proximity of a focal point or focal line”), Ann. Phys. (Leipzig) 30, 755–776 (1909).

Armitage, J.

D. G. Flagello, A. E. Rosenbluth, C. Progler, J. Armitage, “Understanding high numerical aperture optical lithography,” Microcircuit Eng. 17, 105–108 (1991).
[CrossRef]

Chipman, R. A.

R. A. Chipman, “Polarization aberrations,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1987).

R. A. Chipman, “The mechanics of polarization ray tracing,” in Polarization Analysis and Measurement, R. A. Chipman, D. H. Goldstein, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1746, 62–75 (1989).
[CrossRef]

Debye, P.

P. Debye, “Das Verhalten von Lichtwellen in der Nähe eines Brennpunktes oder einer Brennlinie” (“Behavior of light waves in the proximity of a focal point or focal line”), Ann. Phys. (Leipzig) 30, 755–776 (1909).

Dill, F. H.

F. H. Dill, A. R. Neureuther, J. A. Tuttle, E. J. Walker, “Modeling projection printing of positive photoresist,” IEEE Trans. Electron Devices ED-7, 456–464 (1975).
[CrossRef]

Flagello, D. G.

D. G. Flagello, A. E. Rosenbluth, “Lithographic tolerances based on vector diffraction theory,” J. Vac. Sci. Technol. 10, 2997–3003 (1992)
[CrossRef]

D. G. Flagello, A. E. Rosenbluth, C. Progler, J. Armitage, “Understanding high numerical aperture optical lithography,” Microcircuit Eng. 17, 105–108 (1991).
[CrossRef]

D. G. Flagello, T. Milster, “Three-dimensional modeling of high numerical aperture imaging in thin films,” in Design, Modeling, and Control of Laser Beam Optics, Y. Kohanzadeh, G. N. Laurence, J. G. McCoy, H. Weichel, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1625, 246–261 (1992).
[CrossRef]

D. G. Flagello, A. E. Rosenbluth, “Vector diffraction analysis of phase-mask imaging in photoresist films,” in Optical/Laser Microlithography VI, J. D. Cuthbert, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1927, 395–412 (1993).
[CrossRef]

D. G. Flagello, “High numerical aperture imaging in homogeneous thin films,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1993).

Gaskill, J. D.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).

Gasper, J.

Goodman, D. S.

D. S. Goodman, “Stationary optical projectors,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1979).

D. S. Goodman, “Topics in image formation,” short course presented at Optical Society of America Annual Meeting, Orlando, Fla., October 19, 1989.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968).

Harvey, J. E.

J. E. Harvey, “Fourier treatment of near-field scalar diffraction theory,” Am. J. Phys. 47, 974–980 (1979).
[CrossRef]

Herchel, R. S.

R. S. Herchel, “Partial coherence in projection printing,” in Developments in Semiconductor Microlithography III, D. R. Ciarlo, J. Dey, K. Hoeppner, R. L. Ruddell, eds., Proc. Soc. Photo-Opt. Instrum. Eng.135, 24–29 (1978).
[CrossRef]

Hopkins, H. H.

H. H. Hopkins, “Image formation with coherent and partially coherent light,” Photogr. Sci. Eng. 21, 114–122 (1977).

Hunt, R.

R. Hunt, “Magneto-optical scattering from thin solid films,” J. Appl. Phys. 38, 1652–1671 (1967).
[CrossRef]

Kant, R.

R. Kant, “A general numerical solution of vector diffraction for aplanatic systems,” Tech. Rep. TR02.1713.B (IBM, San Jose, Calif., 1991).

Lee, S. W.

Li, Y.

E. Wolf, Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205–210 (1981).
[CrossRef]

Ling, H.

Luneberg, R. K.

R. K. Luneberg, Mathematical Theory of Optics (U. of California, Berkeley, Calif., 1944); reprint: (Brown U., Providence, R.I., 1964).

Macleod, H. A.

H. A. Macleod, Thin Film Optical Filters (McGraw-Hill, New York, 1989).

Mansuripur, M.

M. Mansuripur, “Distribution of light at and near the focus of high numerical aperture objectives,” J. Opt. Soc. Am A 3, 2086–2093 (1986).
[CrossRef]

Milster, T.

D. G. Flagello, T. Milster, “Three-dimensional modeling of high numerical aperture imaging in thin films,” in Design, Modeling, and Control of Laser Beam Optics, Y. Kohanzadeh, G. N. Laurence, J. G. McCoy, H. Weichel, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1625, 246–261 (1992).
[CrossRef]

Neureuther, A. R.

F. H. Dill, A. R. Neureuther, J. A. Tuttle, E. J. Walker, “Modeling projection printing of positive photoresist,” IEEE Trans. Electron Devices ED-7, 456–464 (1975).
[CrossRef]

Progler, C.

D. G. Flagello, A. E. Rosenbluth, C. Progler, J. Armitage, “Understanding high numerical aperture optical lithography,” Microcircuit Eng. 17, 105–108 (1991).
[CrossRef]

Richards, B.

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

Rosenbluth, A. E.

D. G. Flagello, A. E. Rosenbluth, “Lithographic tolerances based on vector diffraction theory,” J. Vac. Sci. Technol. 10, 2997–3003 (1992)
[CrossRef]

D. G. Flagello, A. E. Rosenbluth, C. Progler, J. Armitage, “Understanding high numerical aperture optical lithography,” Microcircuit Eng. 17, 105–108 (1991).
[CrossRef]

D. G. Flagello, A. E. Rosenbluth, “Vector diffraction analysis of phase-mask imaging in photoresist films,” in Optical/Laser Microlithography VI, J. D. Cuthbert, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1927, 395–412 (1993).
[CrossRef]

Sherman, G. C.

Stamnes, J. J.

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

Strojwas, A.

Tuttle, J. A.

F. H. Dill, A. R. Neureuther, J. A. Tuttle, E. J. Walker, “Modeling projection printing of positive photoresist,” IEEE Trans. Electron Devices ED-7, 456–464 (1975).
[CrossRef]

Visser, T. D.

Walker, E. J.

F. H. Dill, A. R. Neureuther, J. A. Tuttle, E. J. Walker, “Modeling projection printing of positive photoresist,” IEEE Trans. Electron Devices ED-7, 456–464 (1975).
[CrossRef]

Wiersma, S. H.

Wolf, E.

E. Wolf, Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205–210 (1981).
[CrossRef]

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

E. Wolf, “Electromagnetic diffraction in optical systems. I. An integral representation of the image field,” Proc. R. Soc. London Ser. A 253, 349–357 (1959).
[CrossRef]

Yaun, C.

Yeung, M.

M. Yeung, “Modeling high numerical aperture optical lithography,” in Optical/Laser Microlithography, B. J. Lin, ed., Proc. Soc. Photo-Opt. Instrum. Eng.922, 149–167 (1988).

M. Yeung, “Photolithography simulation on non-planar substrate,” in Optical/Laser Microlithography III, V. Pol, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1264, 309–321 (1990).
[CrossRef]

Am. J. Phys.

J. E. Harvey, “Fourier treatment of near-field scalar diffraction theory,” Am. J. Phys. 47, 974–980 (1979).
[CrossRef]

Ann. Phys. (Leipzig)

P. Debye, “Das Verhalten von Lichtwellen in der Nähe eines Brennpunktes oder einer Brennlinie” (“Behavior of light waves in the proximity of a focal point or focal line”), Ann. Phys. (Leipzig) 30, 755–776 (1909).

IEEE Trans. Electron Devices

F. H. Dill, A. R. Neureuther, J. A. Tuttle, E. J. Walker, “Modeling projection printing of positive photoresist,” IEEE Trans. Electron Devices ED-7, 456–464 (1975).
[CrossRef]

J. Appl. Phys.

R. Hunt, “Magneto-optical scattering from thin solid films,” J. Appl. Phys. 38, 1652–1671 (1967).
[CrossRef]

J. Opt. Soc. Am A

M. Mansuripur, “Distribution of light at and near the focus of high numerical aperture objectives,” J. Opt. Soc. Am A 3, 2086–2093 (1986).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Vac. Sci. Technol.

D. G. Flagello, A. E. Rosenbluth, “Lithographic tolerances based on vector diffraction theory,” J. Vac. Sci. Technol. 10, 2997–3003 (1992)
[CrossRef]

Microcircuit Eng.

D. G. Flagello, A. E. Rosenbluth, C. Progler, J. Armitage, “Understanding high numerical aperture optical lithography,” Microcircuit Eng. 17, 105–108 (1991).
[CrossRef]

Opt. Commun.

E. Wolf, Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205–210 (1981).
[CrossRef]

Photogr. Sci. Eng.

H. H. Hopkins, “Image formation with coherent and partially coherent light,” Photogr. Sci. Eng. 21, 114–122 (1977).

Proc. R. Soc. London Ser. A

E. Wolf, “Electromagnetic diffraction in optical systems. I. An integral representation of the image field,” Proc. R. Soc. London Ser. A 253, 349–357 (1959).
[CrossRef]

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

Other

R. Kant, “A general numerical solution of vector diffraction for aplanatic systems,” Tech. Rep. TR02.1713.B (IBM, San Jose, Calif., 1991).

R. S. Herchel, “Partial coherence in projection printing,” in Developments in Semiconductor Microlithography III, D. R. Ciarlo, J. Dey, K. Hoeppner, R. L. Ruddell, eds., Proc. Soc. Photo-Opt. Instrum. Eng.135, 24–29 (1978).
[CrossRef]

J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968).

D. G. Flagello, A. E. Rosenbluth, “Vector diffraction analysis of phase-mask imaging in photoresist films,” in Optical/Laser Microlithography VI, J. D. Cuthbert, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1927, 395–412 (1993).
[CrossRef]

D. G. Flagello, “High numerical aperture imaging in homogeneous thin films,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1993).

D. G. Flagello, T. Milster, “Three-dimensional modeling of high numerical aperture imaging in thin films,” in Design, Modeling, and Control of Laser Beam Optics, Y. Kohanzadeh, G. N. Laurence, J. G. McCoy, H. Weichel, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1625, 246–261 (1992).
[CrossRef]

M. Yeung, “Modeling high numerical aperture optical lithography,” in Optical/Laser Microlithography, B. J. Lin, ed., Proc. Soc. Photo-Opt. Instrum. Eng.922, 149–167 (1988).

M. Yeung, “Photolithography simulation on non-planar substrate,” in Optical/Laser Microlithography III, V. Pol, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1264, 309–321 (1990).
[CrossRef]

R. K. Luneberg, Mathematical Theory of Optics (U. of California, Berkeley, Calif., 1944); reprint: (Brown U., Providence, R.I., 1964).

D. S. Goodman, “Stationary optical projectors,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1979).

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).

R. A. Chipman, “Polarization aberrations,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1987).

R. A. Chipman, “The mechanics of polarization ray tracing,” in Polarization Analysis and Measurement, R. A. Chipman, D. H. Goldstein, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1746, 62–75 (1989).
[CrossRef]

H. A. Macleod, Thin Film Optical Filters (McGraw-Hill, New York, 1989).

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

D. S. Goodman, “Topics in image formation,” short course presented at Optical Society of America Annual Meeting, Orlando, Fla., October 19, 1989.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (18)

Fig. 1
Fig. 1

Köhler illuminated projection system focusing to a thin-film stack.

Fig. 2
Fig. 2

Propagation vector k.

Fig. 3
Fig. 3

Meridional plane through a high-NA optical system.

Fig. 4
Fig. 4

Diffraction geometry from object to entrance pupil.

Fig. 5
Fig. 5

Circular pupil defined in direction cosine space.

Fig. 6
Fig. 6

Diffraction geometry from exit pupil to image focus.

Fig. 7
Fig. 7

Mapping geometry between entrance and exit pupils.

Fig. 8
Fig. 8

Rotation of the polarization vector emerging from exit pupil.

Fig. 9
Fig. 9

Thin-film stack with incident plane waves and electric-field amplitudes.

Fig. 10
Fig. 10

PSF for NA′ = 0.95 given by values of normalized |E|2: (a) isoimage contours, (b) profiles along x and y.

Fig. 11
Fig. 11

Q distributions of PSF at interface I with NA′ = 0.95: total Q distribution on top surface with x, y, and z components.

Fig. 12
Fig. 12

Normalized PSF distribution in film: (a) meridional slices, (b) comparison of x and y polarizations at interface I and interface II.

Fig. 13
Fig. 13

Magnitudes of normalized system transfer function Ã1(α′, β′; z′ = 0) for x, y, and z components given in exit pupil coordinates for NA′ = 0.95. The magnitudes are normalized to the maximum x-component value.

Fig. 14
Fig. 14

(a) Tribar object with scaled image dimensions, (b) entrance pupil electric-field distribution normalized to unit magnitude.

Fig. 15
Fig. 15

Magnitude of Ã1(α′, β′; z′ = 0) components, with the object, scalar lens, polarization, and film terms at interface I for a tribar object and NA′ = 0.95.

Fig. 16
Fig. 16

Simulation of Q distribution of tribar object at interface I with NA′ = 0.95: Q distribution at top surface with x, y, and z components.

Fig. 17
Fig. 17

Q simulation of tribar image with NA′ = 0.95 and z0 = 0: (a) xz meridional plane, (b) profiles comparing x and y polarizations at interface I and interface II.

Fig. 18
Fig. 18

Q simulation of tribar image with NA′ = 0.95 and z0 = 45 μm: (a) xz meridional plane, (b) profiles comparing x and y polarizations at interface I and interface II.

Tables (3)

Tables Icon

Table 1 Reflection and Transmission Coefficients

Tables Icon

Table 2 Parameters for PSF Simulation

Tables Icon

Table 3 Parameters for Tribar Simulation

Equations (59)

Equations on this page are rendered with MathJax. Learn more.

U ( x , y , z ) = U 0 ( x , y , z ) exp ( i k · r ) = U 0 ( x , y , z ) exp [ - i 2 π N ( α x + β y + γ z ) ] ,
r = λ 0 ( x x + y y + z z ) , k = 2 π N λ 0 ( k x k x + k y k y + k z k z ) = 2 π N λ 0 [ ( cos ϕ sin θ ) x + ( sin ϕ sin θ ) y + ( cos θ ) z ] = 2 π N λ 0 ( α x + β y + γ z ) .
α 2 + β 2 + γ 2 = 1.
Q = · ( E × H ) = 1 2 σ E 2 = k 0 Y n κ E 2 ,
NA = sin θ max = α max 2 + β max 2 ,
O ˜ ( α , β ) = i γ exp ( - i 2 π r ) r - - O ( x , y ) × exp [ i 2 π ( α x + β y ) ] d x d y = i γ exp ( - i 2 π r ) r F { O ( x , y ) } ,
E ˜ ( α , β ) = O ˜ ( α , β ) T ( α , β ) ,
r E ˜ ( α , β ) = i γ exp ( - i 2 π r ) T ( α , β ) F { O ( x , y ) } ,
E ( x , y ; z ) = - i exp ( i 2 π r ) r S E ˜ ( α , β ) exp ( - i k · r ) d S = - i exp ( i 2 π r ) F - 1 { r E ˜ ( α , β ) exp ( - i 2 π γ z ) γ } ,
d S = r 2 d Ω = r 2 d α d β γ
m = h h ,
h N α = h N α ,             h N β = h N β ,
m = h h = α α = β β .
E ˜ ( α , β ) = 1 m 2 E ˜ ( m α , m β ) = 1 m 2 O ˜ ( m α , m β ) T ( m α , m β ) .
T ( α , β ) = T ( m α , m β ) ,
E ˜ ( α , β ) 2 d a = E ˜ ( α , β ) 2 d a ,
d a = r 2 d Ω = r 2 d α d β γ , d a = r 2 d Ω = r 2 d α d β γ .
r E ˜ ( α , β ) = r E ˜ ( α , β ) γ γ d α d β d α d β = r E ˜ ( α , β ) γ γ m .
E ( x , y ; z ) = - i exp ( i 2 π r ) × F - 1 { r E ˜ ( α , β ) exp ( - i 2 π γ z ) exp [ - i 2 π W ( α , β ) ] γ } = - i exp ( i 2 π r ) m F - 1 { r E ˜ ( m α , m β ) exp ( - i 2 π γ z ) × exp [ - i 2 π W ( α , β ) ] 1 γ γ } .
E ( x , y ; z ) = exp [ i 2 π ( r - r ) ] m × F - 1 { O ˜ ( m α , m β ) Ψ ˜ ( α , β ; z ) } ,
Ψ ˜ ( α , β ; z ) = T ( α , β ) exp ( - i 2 π γ z ) × exp [ - i 2 π W ( α , β ) ] γ γ .
O ˜ ( α , β ) = O ˜ ( α , β ) M i ,
M i = ( S x S y ) .
O ˜ ( α , β ) = ( O x 0 ) = O ˜ ( α , β ) ( 1 0 ) = O ˜ ( α , β ) M i .
O ˜ ( m α , m β ) = O ˜ ( m α , m β ) M i .
M ˜ P ( α , β ) = [ P x x S P y x S P x x P P y x P P x y S P y y S P x y P P y y P P x z P P y z P ] = [ β 2 1 - γ 2 - α β 1 - γ 2 γ α 2 1 - γ 2 α β γ 1 - γ 2 - α β 1 - γ 2 α 2 1 - γ 2 α β γ 1 - γ 2 γ β 2 1 - γ 2 - α - β ] .
M ˜ P = [ 0.5 - 0.5 0.35 0.35 - 0.5 0.5 0.35 0.35 - 0.5 - 0.5 ] .
A ˜ i I ( α , β ; z 0 ) = M ˜ P ( α , β ) U ˜ ( α , β ; z 0 ) = M ˜ P ( α , β ) O ˜ ( m α , m β ) Ψ ( α , β ; z 0 ) ,
A ˜ + ( α 1 , β 1 ) = A i II exp [ i 2 π N 1 γ 1 ( d 1 - z + z 0 ) ] × exp [ - i 2 π N 1 ( α 1 x + β 1 y ) ] ,
A ˜ - ( α 1 , β 1 ) = A r II exp [ - i 2 π N 1 γ 1 ( d 1 - z + z 0 ) ] × exp [ - i 2 π N 1 ( α 1 x + β 1 y ) ] ,
α = N j α j ,             β = N j β j .
γ j = [ 1 - sin 2 ( cos - 1 γ ) N j 2 ] 1 / 2 ,
Φ ( z ) = 2 π N 1 γ 1 ( d 1 - z + z 0 ) = 2 π ( d 1 - z + z 0 ) [ N 1 2 - sin 2 ( cos - 1 γ ) ] 1 / 2
A 1 ( α , β ; z ) = exp [ - i 2 π ( a x + β y ) ] × [ A i II exp i Φ + A r II exp ( - i Φ ) ] ,
M j = [ cos δ j i sin δ j / η j i η j sin δ j cos δ j ] ,
η j = { N j γ j S polarization N j / γ j P polarization , δ j = 2 π N j γ j d j .
( B C ) = ( j = 1 q M j ) ( 1 η m ) ,
( B s C s ) = ( j = 1 q M j ) ( 1 η m ) .
τ = 2 η B η + C ,             r = B η - C B η + C ,
τ II = 2 η 1 B s η 1 + C s ,             r II = B s η 1 - C s B s η 1 + C s ,
· E = k · E = 0.
τ z r II z = τ τ II N γ N 1 γ 1 ,             r II z = - r II .
A ˜ 1 ( α , β ; z ) = M ˜ F ( α , β ; z ) A ˜ i I ( α , β ; z 0 ) ,
M ˜ F ( α , β ; z ) = [ F S F P 0 0 0 0 0 F S F P 0 0 0 0 0 F z P ] ,
F n = ( τ τ II ) n [ exp ( i Φ ) + ( r II ) n exp ( - i Φ ) ] ,
F z P = N γ N 1 γ 1 ( τ τ II ) P [ exp ( i Φ ) - ( r II ) P exp ( - i Φ ) ] .
E 1 ( x , y ; z ) = c 0 F - 1 { A ˜ 1 ( α , β ; z ) } = c 0 F - 1 { M ˜ F ( α , β ; z ) A ˜ i I ( α , β ; z 0 ) } = c 0 F - 1 { M ˜ F ( α , β ; z ) M ˜ P ( α , β ) × O ˜ ( α , β ) Ψ ˜ ( α , β ; z 0 ) } .
E 1 ( x , y ; z ) = c 0 M F ( x , y ; z ) M P ( x , y ) O ( x m , y m ) Ψ ( x , y ; z 0 ) .
Q ( x , y ; z ) = k 0 Y n 1 κ 1 E 1 ( x , y ; z ) 2 .
M SP O = [ sin ϕ - cos ϕ cos ϕ sin ϕ ] ( O x O y ) .
cos ϕ = α 1 - γ 2 ,             sin ϕ = β 1 - γ 2 ,
M SP = [ β 1 - γ 2 - α 1 - γ 2 α 1 - γ 2 β 1 - γ 2 ] .
A ˜ + ( α , β ) = M F + ( α , β ; z ) M SP ( α , β ) O ( α , β ) × Ψ ( α , β ; z 0 ) ,
M F + ( α , β ; z ) = exp ( i Φ ) [ cos ψ sin ψ - sin ψ cos ψ ] × [ ( τ τ II ) S 0 0 ( τ τ II ) P ] ,
= [ - i q 0 i q 0 0 0 ] .
A ˜ - ( α , β ) = M F - ( α , β ; z ) M SP ( α , β ) O ( α , β ) × ψ ( α , β ; z 0 ) ,
M F - ( α , β ; z ) = exp ( - i Φ ) [ cos ψ - sin ψ sin ψ cos ψ ] × [ ( τ τ II ) S r IIS 0 0 ( τ r II τ II ) P r IIP ] .
M x y z ± = [ β 1 - γ 2 α γ 1 - γ 2 - α 1 - γ 2 β γ 1 - γ 2 0 ± N γ N 1 γ 1 1 - γ 2 ] ,
E 1 ( x , y ; z ) = c 0 F - 1 { M x y z + ( α , β ) A ˜ + ( α 1 , β 1 ) + M x y z - ( α , β ) A ˜ - ( α 1 , β 1 ) } .

Metrics