Abstract

Theoretical modeling of a slit-scan-type aerial image measurement sensor used for optical lithography is presented. Slit transmission properties are fully represented by the slit transfer function in terms of incident and scattering angles of light, which is then incorporated into the scheme of a partially coherent imaging formula to obtain an expression for image profiles measured by slit scanning. As an exemplary case, we analyze the influence of a 100nm width slit used in an ArF lithography system. To understand the mechanism of image profile changes by slit transmission, we focus on frequency transfer characteristics of sinusoidal patterns.

© 2011 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. H. J. Levinson, “Optical pattern formation,” in Principles of Lithography, 2nd ed. (SPIE, 2005), pp. 9–54.
  2. Y. Wei and R. L. Brainard, Advanced Processes for 193nmImmersion Lithography (SPIE, 2009).
    [PubMed]
  3. W. N. Partlo, C. H. Fields, and W. G. Oldham, “Direct aerial image measurement as a method of testing high numerical aperture microlithographic lenses,” J. Vac. Sci. Technol. B 11, 2686–2691 (1993).
    [CrossRef]
  4. C. H. Fields, W. G. Oldham, A. K. Ray-Chaudhuri, K. D. Krenz, and R. H. Stulen, “Direct aerial image measurements to evaluate the performance of an extreme ultraviolet projection lithography system,” J. Vac. Sci. Technol. B 14, 4000–4003 (1996).
    [CrossRef]
  5. T. Hagiwara, M. Hamatani, N. Kondo, K. Suzuki, H. Nishinaga, J. Inoue, K. Kaneko, and Higashibata, “Self-calibration of wafer scanners using an aerial image sensor,” Proc. SPIE 4691, 871–881 (2002).
    [CrossRef]
  6. R. R. Kunz, D. D. Rathman, S. J. Spector, and M. Yeung, “Monolithic detector array comprised of >1000 aerial image sensing elements,” Proc. SPIE 5040, 1441–1455 (2003).
    [CrossRef]
  7. J. Xue, K. Moen, and C. J. Spanos, “Integrated aerial image sensor: design, modeling, and assembly,” J. Vac. Sci. Technol. B 24, 3088–3093 (2006).
    [CrossRef]
  8. A. George and T. D. Milster, “Characteristics of a scanning nano-slit image sensor for line-and-space patterns,” Appl. Opt. 49, 3821–3830 (2010).
    [CrossRef] [PubMed]
  9. A. K. Pfau, R. Hsu, and W. G. Oldham, “A two-dimensional high-resolution stepper image monitor,” Proc. SPIE 1674, 182–192 (1992).
    [CrossRef]
  10. S. Hunsche, M. J. Gassner, J. A. Schefske, E. R. Kenb, and A. Acheta, “Characterization and applications of an in-scanner aerial image detection system,” Proc. SPIE 6152, 61522U(2006).
    [CrossRef]
  11. C. H. Wei, P. H. Tsao, W. Fann, P.-K. Wei, J. O. Tegenfeldt, and R. H. Austin, “Polarization dependence of light intensity distribution near a nanometric aluminum slit,” J. Opt. Soc. Am. B 21, 1005–1012 (2004).
    [CrossRef]
  12. Y. Xie, A. R. Zakharian, J. V. Moloney, and M. Mansuripur, “Transmission of light through slit aperture in metallic films,” Opt. Express 12, 6106–6121 (2004).
    [CrossRef] [PubMed]
  13. Y. Xie, A. R. Zakharian, J. V. Moloney, and M. Mansuripur, “Optical transmission at oblique incidence through a periodic array of sub-wavelength slits in a metallic host,” Opt. Express 14, 10220–10227 (2006).
    [CrossRef] [PubMed]
  14. M. Born and E. Wolf, Principles of Optics, 7th ed. (Pergamon, 1999), Chap. 10.
  15. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).
  16. J. W. Goodman, “The angular spectrum of plane waves,” in Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996), pp. 55–61.
  17. V. M. Domnenko, “Computer simulation of a partially coherent image based on the vector theory of diffraction,” Proc. SPIE 3780, 168–179 (1999).
    [CrossRef]
  18. A. K.-K. Wong, Optical Imaging in Projection Microlithography (SPIE, 2005).
    [CrossRef]
  19. T. D. Milster, J. S. Jo, and K. Hirota, “Roles of propagation and evanescent waves in solid immersion lens systems,” Appl. Opt. 38, 5046–5057 (1999).
    [CrossRef]
  20. M. D. Levenson, N. S. Viswanathan, and R. A. Simpson, “Improving resolution in photolithography with a phase-shifting mask,” IEEE Trans. Electron Devices 29, 1828–1836(1982).
    [CrossRef]
  21. A. K.-K. Wong, “Alternating phase-shifting mask,” in Resolution Enhancement Techniques in Optical Lithography (SPIE, 2001), pp. 117–138.
    [CrossRef]

2010 (1)

2006 (3)

S. Hunsche, M. J. Gassner, J. A. Schefske, E. R. Kenb, and A. Acheta, “Characterization and applications of an in-scanner aerial image detection system,” Proc. SPIE 6152, 61522U(2006).
[CrossRef]

Y. Xie, A. R. Zakharian, J. V. Moloney, and M. Mansuripur, “Optical transmission at oblique incidence through a periodic array of sub-wavelength slits in a metallic host,” Opt. Express 14, 10220–10227 (2006).
[CrossRef] [PubMed]

J. Xue, K. Moen, and C. J. Spanos, “Integrated aerial image sensor: design, modeling, and assembly,” J. Vac. Sci. Technol. B 24, 3088–3093 (2006).
[CrossRef]

2004 (2)

2003 (1)

R. R. Kunz, D. D. Rathman, S. J. Spector, and M. Yeung, “Monolithic detector array comprised of >1000 aerial image sensing elements,” Proc. SPIE 5040, 1441–1455 (2003).
[CrossRef]

2002 (1)

T. Hagiwara, M. Hamatani, N. Kondo, K. Suzuki, H. Nishinaga, J. Inoue, K. Kaneko, and Higashibata, “Self-calibration of wafer scanners using an aerial image sensor,” Proc. SPIE 4691, 871–881 (2002).
[CrossRef]

1999 (2)

V. M. Domnenko, “Computer simulation of a partially coherent image based on the vector theory of diffraction,” Proc. SPIE 3780, 168–179 (1999).
[CrossRef]

T. D. Milster, J. S. Jo, and K. Hirota, “Roles of propagation and evanescent waves in solid immersion lens systems,” Appl. Opt. 38, 5046–5057 (1999).
[CrossRef]

1996 (1)

C. H. Fields, W. G. Oldham, A. K. Ray-Chaudhuri, K. D. Krenz, and R. H. Stulen, “Direct aerial image measurements to evaluate the performance of an extreme ultraviolet projection lithography system,” J. Vac. Sci. Technol. B 14, 4000–4003 (1996).
[CrossRef]

1993 (1)

W. N. Partlo, C. H. Fields, and W. G. Oldham, “Direct aerial image measurement as a method of testing high numerical aperture microlithographic lenses,” J. Vac. Sci. Technol. B 11, 2686–2691 (1993).
[CrossRef]

1992 (1)

A. K. Pfau, R. Hsu, and W. G. Oldham, “A two-dimensional high-resolution stepper image monitor,” Proc. SPIE 1674, 182–192 (1992).
[CrossRef]

1982 (1)

M. D. Levenson, N. S. Viswanathan, and R. A. Simpson, “Improving resolution in photolithography with a phase-shifting mask,” IEEE Trans. Electron Devices 29, 1828–1836(1982).
[CrossRef]

Acheta, A.

S. Hunsche, M. J. Gassner, J. A. Schefske, E. R. Kenb, and A. Acheta, “Characterization and applications of an in-scanner aerial image detection system,” Proc. SPIE 6152, 61522U(2006).
[CrossRef]

Austin, R. H.

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Pergamon, 1999), Chap. 10.

Brainard, R. L.

Y. Wei and R. L. Brainard, Advanced Processes for 193nmImmersion Lithography (SPIE, 2009).
[PubMed]

Domnenko, V. M.

V. M. Domnenko, “Computer simulation of a partially coherent image based on the vector theory of diffraction,” Proc. SPIE 3780, 168–179 (1999).
[CrossRef]

Fann, W.

Fields, C. H.

C. H. Fields, W. G. Oldham, A. K. Ray-Chaudhuri, K. D. Krenz, and R. H. Stulen, “Direct aerial image measurements to evaluate the performance of an extreme ultraviolet projection lithography system,” J. Vac. Sci. Technol. B 14, 4000–4003 (1996).
[CrossRef]

W. N. Partlo, C. H. Fields, and W. G. Oldham, “Direct aerial image measurement as a method of testing high numerical aperture microlithographic lenses,” J. Vac. Sci. Technol. B 11, 2686–2691 (1993).
[CrossRef]

Gassner, M. J.

S. Hunsche, M. J. Gassner, J. A. Schefske, E. R. Kenb, and A. Acheta, “Characterization and applications of an in-scanner aerial image detection system,” Proc. SPIE 6152, 61522U(2006).
[CrossRef]

George, A.

Goodman, J. W.

J. W. Goodman, “The angular spectrum of plane waves,” in Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996), pp. 55–61.

Hagiwara, T.

T. Hagiwara, M. Hamatani, N. Kondo, K. Suzuki, H. Nishinaga, J. Inoue, K. Kaneko, and Higashibata, “Self-calibration of wafer scanners using an aerial image sensor,” Proc. SPIE 4691, 871–881 (2002).
[CrossRef]

Hagness, S. C.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).

Hamatani, M.

T. Hagiwara, M. Hamatani, N. Kondo, K. Suzuki, H. Nishinaga, J. Inoue, K. Kaneko, and Higashibata, “Self-calibration of wafer scanners using an aerial image sensor,” Proc. SPIE 4691, 871–881 (2002).
[CrossRef]

Higashibata,

T. Hagiwara, M. Hamatani, N. Kondo, K. Suzuki, H. Nishinaga, J. Inoue, K. Kaneko, and Higashibata, “Self-calibration of wafer scanners using an aerial image sensor,” Proc. SPIE 4691, 871–881 (2002).
[CrossRef]

Hirota, K.

Hsu, R.

A. K. Pfau, R. Hsu, and W. G. Oldham, “A two-dimensional high-resolution stepper image monitor,” Proc. SPIE 1674, 182–192 (1992).
[CrossRef]

Hunsche, S.

S. Hunsche, M. J. Gassner, J. A. Schefske, E. R. Kenb, and A. Acheta, “Characterization and applications of an in-scanner aerial image detection system,” Proc. SPIE 6152, 61522U(2006).
[CrossRef]

Inoue, J.

T. Hagiwara, M. Hamatani, N. Kondo, K. Suzuki, H. Nishinaga, J. Inoue, K. Kaneko, and Higashibata, “Self-calibration of wafer scanners using an aerial image sensor,” Proc. SPIE 4691, 871–881 (2002).
[CrossRef]

Jo, J. S.

Kaneko, K.

T. Hagiwara, M. Hamatani, N. Kondo, K. Suzuki, H. Nishinaga, J. Inoue, K. Kaneko, and Higashibata, “Self-calibration of wafer scanners using an aerial image sensor,” Proc. SPIE 4691, 871–881 (2002).
[CrossRef]

Kenb, E. R.

S. Hunsche, M. J. Gassner, J. A. Schefske, E. R. Kenb, and A. Acheta, “Characterization and applications of an in-scanner aerial image detection system,” Proc. SPIE 6152, 61522U(2006).
[CrossRef]

Kondo, N.

T. Hagiwara, M. Hamatani, N. Kondo, K. Suzuki, H. Nishinaga, J. Inoue, K. Kaneko, and Higashibata, “Self-calibration of wafer scanners using an aerial image sensor,” Proc. SPIE 4691, 871–881 (2002).
[CrossRef]

Krenz, K. D.

C. H. Fields, W. G. Oldham, A. K. Ray-Chaudhuri, K. D. Krenz, and R. H. Stulen, “Direct aerial image measurements to evaluate the performance of an extreme ultraviolet projection lithography system,” J. Vac. Sci. Technol. B 14, 4000–4003 (1996).
[CrossRef]

Kunz, R. R.

R. R. Kunz, D. D. Rathman, S. J. Spector, and M. Yeung, “Monolithic detector array comprised of >1000 aerial image sensing elements,” Proc. SPIE 5040, 1441–1455 (2003).
[CrossRef]

Levenson, M. D.

M. D. Levenson, N. S. Viswanathan, and R. A. Simpson, “Improving resolution in photolithography with a phase-shifting mask,” IEEE Trans. Electron Devices 29, 1828–1836(1982).
[CrossRef]

Levinson, H. J.

H. J. Levinson, “Optical pattern formation,” in Principles of Lithography, 2nd ed. (SPIE, 2005), pp. 9–54.

Mansuripur, M.

Milster, T. D.

Moen, K.

J. Xue, K. Moen, and C. J. Spanos, “Integrated aerial image sensor: design, modeling, and assembly,” J. Vac. Sci. Technol. B 24, 3088–3093 (2006).
[CrossRef]

Moloney, J. V.

Nishinaga, H.

T. Hagiwara, M. Hamatani, N. Kondo, K. Suzuki, H. Nishinaga, J. Inoue, K. Kaneko, and Higashibata, “Self-calibration of wafer scanners using an aerial image sensor,” Proc. SPIE 4691, 871–881 (2002).
[CrossRef]

Oldham, W. G.

C. H. Fields, W. G. Oldham, A. K. Ray-Chaudhuri, K. D. Krenz, and R. H. Stulen, “Direct aerial image measurements to evaluate the performance of an extreme ultraviolet projection lithography system,” J. Vac. Sci. Technol. B 14, 4000–4003 (1996).
[CrossRef]

W. N. Partlo, C. H. Fields, and W. G. Oldham, “Direct aerial image measurement as a method of testing high numerical aperture microlithographic lenses,” J. Vac. Sci. Technol. B 11, 2686–2691 (1993).
[CrossRef]

A. K. Pfau, R. Hsu, and W. G. Oldham, “A two-dimensional high-resolution stepper image monitor,” Proc. SPIE 1674, 182–192 (1992).
[CrossRef]

Partlo, W. N.

W. N. Partlo, C. H. Fields, and W. G. Oldham, “Direct aerial image measurement as a method of testing high numerical aperture microlithographic lenses,” J. Vac. Sci. Technol. B 11, 2686–2691 (1993).
[CrossRef]

Pfau, A. K.

A. K. Pfau, R. Hsu, and W. G. Oldham, “A two-dimensional high-resolution stepper image monitor,” Proc. SPIE 1674, 182–192 (1992).
[CrossRef]

Rathman, D. D.

R. R. Kunz, D. D. Rathman, S. J. Spector, and M. Yeung, “Monolithic detector array comprised of >1000 aerial image sensing elements,” Proc. SPIE 5040, 1441–1455 (2003).
[CrossRef]

Ray-Chaudhuri, A. K.

C. H. Fields, W. G. Oldham, A. K. Ray-Chaudhuri, K. D. Krenz, and R. H. Stulen, “Direct aerial image measurements to evaluate the performance of an extreme ultraviolet projection lithography system,” J. Vac. Sci. Technol. B 14, 4000–4003 (1996).
[CrossRef]

Schefske, J. A.

S. Hunsche, M. J. Gassner, J. A. Schefske, E. R. Kenb, and A. Acheta, “Characterization and applications of an in-scanner aerial image detection system,” Proc. SPIE 6152, 61522U(2006).
[CrossRef]

Simpson, R. A.

M. D. Levenson, N. S. Viswanathan, and R. A. Simpson, “Improving resolution in photolithography with a phase-shifting mask,” IEEE Trans. Electron Devices 29, 1828–1836(1982).
[CrossRef]

Spanos, C. J.

J. Xue, K. Moen, and C. J. Spanos, “Integrated aerial image sensor: design, modeling, and assembly,” J. Vac. Sci. Technol. B 24, 3088–3093 (2006).
[CrossRef]

Spector, S. J.

R. R. Kunz, D. D. Rathman, S. J. Spector, and M. Yeung, “Monolithic detector array comprised of >1000 aerial image sensing elements,” Proc. SPIE 5040, 1441–1455 (2003).
[CrossRef]

Stulen, R. H.

C. H. Fields, W. G. Oldham, A. K. Ray-Chaudhuri, K. D. Krenz, and R. H. Stulen, “Direct aerial image measurements to evaluate the performance of an extreme ultraviolet projection lithography system,” J. Vac. Sci. Technol. B 14, 4000–4003 (1996).
[CrossRef]

Suzuki, K.

T. Hagiwara, M. Hamatani, N. Kondo, K. Suzuki, H. Nishinaga, J. Inoue, K. Kaneko, and Higashibata, “Self-calibration of wafer scanners using an aerial image sensor,” Proc. SPIE 4691, 871–881 (2002).
[CrossRef]

Taflove, A.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).

Tegenfeldt, J. O.

Tsao, P. H.

Viswanathan, N. S.

M. D. Levenson, N. S. Viswanathan, and R. A. Simpson, “Improving resolution in photolithography with a phase-shifting mask,” IEEE Trans. Electron Devices 29, 1828–1836(1982).
[CrossRef]

Wei, C. H.

Wei, P.-K.

Wei, Y.

Y. Wei and R. L. Brainard, Advanced Processes for 193nmImmersion Lithography (SPIE, 2009).
[PubMed]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Pergamon, 1999), Chap. 10.

Wong, A. K.-K.

A. K.-K. Wong, Optical Imaging in Projection Microlithography (SPIE, 2005).
[CrossRef]

A. K.-K. Wong, “Alternating phase-shifting mask,” in Resolution Enhancement Techniques in Optical Lithography (SPIE, 2001), pp. 117–138.
[CrossRef]

Xie, Y.

Xue, J.

J. Xue, K. Moen, and C. J. Spanos, “Integrated aerial image sensor: design, modeling, and assembly,” J. Vac. Sci. Technol. B 24, 3088–3093 (2006).
[CrossRef]

Yeung, M.

R. R. Kunz, D. D. Rathman, S. J. Spector, and M. Yeung, “Monolithic detector array comprised of >1000 aerial image sensing elements,” Proc. SPIE 5040, 1441–1455 (2003).
[CrossRef]

Zakharian, A. R.

Appl. Opt. (2)

IEEE Trans. Electron Devices (1)

M. D. Levenson, N. S. Viswanathan, and R. A. Simpson, “Improving resolution in photolithography with a phase-shifting mask,” IEEE Trans. Electron Devices 29, 1828–1836(1982).
[CrossRef]

J. Opt. Soc. Am. B (1)

J. Vac. Sci. Technol. B (3)

W. N. Partlo, C. H. Fields, and W. G. Oldham, “Direct aerial image measurement as a method of testing high numerical aperture microlithographic lenses,” J. Vac. Sci. Technol. B 11, 2686–2691 (1993).
[CrossRef]

C. H. Fields, W. G. Oldham, A. K. Ray-Chaudhuri, K. D. Krenz, and R. H. Stulen, “Direct aerial image measurements to evaluate the performance of an extreme ultraviolet projection lithography system,” J. Vac. Sci. Technol. B 14, 4000–4003 (1996).
[CrossRef]

J. Xue, K. Moen, and C. J. Spanos, “Integrated aerial image sensor: design, modeling, and assembly,” J. Vac. Sci. Technol. B 24, 3088–3093 (2006).
[CrossRef]

Opt. Express (2)

Proc. SPIE (5)

V. M. Domnenko, “Computer simulation of a partially coherent image based on the vector theory of diffraction,” Proc. SPIE 3780, 168–179 (1999).
[CrossRef]

T. Hagiwara, M. Hamatani, N. Kondo, K. Suzuki, H. Nishinaga, J. Inoue, K. Kaneko, and Higashibata, “Self-calibration of wafer scanners using an aerial image sensor,” Proc. SPIE 4691, 871–881 (2002).
[CrossRef]

R. R. Kunz, D. D. Rathman, S. J. Spector, and M. Yeung, “Monolithic detector array comprised of >1000 aerial image sensing elements,” Proc. SPIE 5040, 1441–1455 (2003).
[CrossRef]

A. K. Pfau, R. Hsu, and W. G. Oldham, “A two-dimensional high-resolution stepper image monitor,” Proc. SPIE 1674, 182–192 (1992).
[CrossRef]

S. Hunsche, M. J. Gassner, J. A. Schefske, E. R. Kenb, and A. Acheta, “Characterization and applications of an in-scanner aerial image detection system,” Proc. SPIE 6152, 61522U(2006).
[CrossRef]

Other (7)

H. J. Levinson, “Optical pattern formation,” in Principles of Lithography, 2nd ed. (SPIE, 2005), pp. 9–54.

Y. Wei and R. L. Brainard, Advanced Processes for 193nmImmersion Lithography (SPIE, 2009).
[PubMed]

A. K.-K. Wong, Optical Imaging in Projection Microlithography (SPIE, 2005).
[CrossRef]

A. K.-K. Wong, “Alternating phase-shifting mask,” in Resolution Enhancement Techniques in Optical Lithography (SPIE, 2001), pp. 117–138.
[CrossRef]

M. Born and E. Wolf, Principles of Optics, 7th ed. (Pergamon, 1999), Chap. 10.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).

J. W. Goodman, “The angular spectrum of plane waves,” in Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996), pp. 55–61.

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 (13)

Fig. 1
Fig. 1

Cross-sectional view of an aerial image measurement sensor using a scanning slit. The slit and the aerial image are both assumed to be extended in the x direction, and the aerial image that is positionally fixed is scanned by moving the slit in the y direction to detect the change of light intensity transmitted through the slit.

Fig. 2
Fig. 2

Change of image contrast C R ^ ( η ) with respect to η calculated by a convolution model. The variable d is normalized by λ, and η is in the unit of 1 / λ . For example, d = 0.5 corresponds to a slit of 96.5 [nm] width ( λ = 193 nm ), and η = 2.0 gives a pattern frequency of 1 / 96.5 [ 1 / nm ].

Fig. 3
Fig. 3

Two-dimensional model of an optical lithography system, combined with an aerial image measurement sensor. The slit is placed on the image plane to measure the profile of an aerial image by scanning. TE and TM polarization vectors are presented associated with one optical path connecting the object and the image planes.

Fig. 4
Fig. 4

Schematic model of a beam transmission through a slit of sub-λ width. An incident plane wave specified by the normalized wave vector ( α , 1 α 2 ) is converted to quasi-cylindrical wavefronts, which are decomposed into plane waves, each represented by a wave vector ( β , 1 β 2 ) .

Fig. 5
Fig. 5

Slit structure used for the FDTD simulations. M1 is water ( n 1 = 1.440 , k 1 = 0 ), M2 is fused silica ( n 2 = 1.560 , k 2 = 0 ), and M3 is tantalum ( n 3 = 1.358 , k 3 = 1.734 ). TE and TM incident beams with incident angles α ^ ( 70 ° α ^ 70 ° ) are assumed.

Fig. 6
Fig. 6

FDTD simulation results for α = 0 ( α ^ = 0 ° ) presented as the amplitude and phase distributions on the y z plane. The contour threshold levels are the same in G x a ( 0 ; y , z ) , G y a ( 0 ; y , z ) , and G z a ( 0 ; y , z ) , to enable their direct comparison in magnitude. The phase distributions show the behavior of wavefronts created by slit transmission.

Fig. 7
Fig. 7

FDTD simulation results for α = 0.5 ( α ^ = 30 ° ), similarly presented as Fig. 6. The transmitted beam is unsymmetrical with respect to the z axis.

Fig. 8
Fig. 8

Amplitudes of Ψ x ( α ; β ) and Ψ y ( α ; β ) obtained for α = 0.0 and 0.5. The narrow oscillating peaks originate from field components that directly transmit the M3 layer shown in Fig. 5. The unit of the vertical axis is arbitrary, and relative comparison is possible between Ψ x a ( α ; β ) and Ψ y a ( α ; β ) .

Fig. 9
Fig. 9

Amplitude and phase distributions of Ψ x ( α ; β ) and Ψ y ( α ; β ) expressed by polynomial expansions in terms of α and β whose ranges are given by Eqs. (26, 27), respectively. The amplitudes are normalized by the value of Ψ x a ( 0 ; 0 ) , whereas the phases are given in radian. The narrow oscillating peaks seen in Fig. 8 have been removed in the polynomial expansions.

Fig. 10
Fig. 10

Slit transfer function Φ TE ( α ; β ) presented as a function of α for the cases of β = 0.0 , 0.3, 0.6, and 0.9. The amplitudes are normalized by the value of Φ TE a ( 0 ; 0 ) .

Fig. 11
Fig. 11

Slit transfer function Φ TM ( α ; β ) presented for β = 0.0 , 0.3, 0.6, and 0.9. The amplitudes are normalized by Φ TE a ( 0 ; 0 ) to allow quantitative comparison between Figs. 10, 11.

Fig. 12
Fig. 12

Change of image contrasts with respect to η, obtained for symmetrically configured interfering beams. The graph for “Convolution” is obtained by Eq. (4) assuming d = 100 / 193 .

Fig. 13
Fig. 13

Change of image contrasts with respect to η, obtained for asymmetrically configured interfering beams. The graph for “Convolution” is obtained by Eq. (4) assuming d = 100 / 193 .

Equations (47)

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

R ^ ( y ) = y d / 2 y + d / 2 I ^ ( y ) d y = I ^ ( y ) W ( y ) ,
I ^ ( y ) = 1 + cos ( 2 π η y ) ,
C I ^ ( η ) = 1 ,
C R ^ ( η ) = sin ( π η d ) π η d ,
I TE ( y ) = u L ( u ) | α max α max F ( α u ) exp ( i 2 π n 1 α y ) d α | 2 d u ,
α max 0.94 .
{ Ω y ( α ) = 1 α 2 Ω z ( α ) = α ,
I TM ( y ) = ξ = y , z u L ( u ) | α max α max Ω ξ ( α ) F ( α u ) exp ( i 2 π n 1 α y ) d α | 2 d u ,
E ( α ; u ; y 0 ) = L ( u ) F ( α u ) exp ( i 2 π n 1 α y 0 ) .
T Λ ( α ; β ; u ; y 0 ) = E ( α ; u ; y 0 ) Φ Λ ( α ; β ) exp ( i 2 π n 2 β y 0 ) ,
R Λ ( y ) = | β max β max u α max α max T Λ ( α ; β ; u ; y ) d α d u d β | 2 = β max β max u L ( u ) | α max α max Φ Λ ( α ; β ) F ( α u ) exp ( i 2 π n 1 α y ) d α | 2 d u d β ,
G ξ ( α ; y , z ) = G ξ a ( α ; y , z ) exp [ i G ξ p ( α ; y , z ) ] ,
{ G x ( α ; y , z ) = G x ( α ; y , z ) G y ( α ; y , z ) = G y ( α ; y , z ) G z ( α ; y , z ) = G z ( α ; y , z )
Ψ ξ ( α ; β ) b / 2 b / 2 G ξ ( α ; y , z ) exp ( i 2 π n 2 β y ) d y ,
G ξ ( α ; y , z ) β max β max Ψ ξ ( α ; β ) exp ( i 2 π n 2 β y ) d β ,
Ψ z ( α ; β ) = β 1 β 2 Ψ y ( α ; β ) ,
{ Φ TE ( α ; β ) = Ψ x ( α ; β ) Φ TM ( α ; β ) = Ψ y ( α ; β ) / 1 β 2 ,
Ψ ξ ( α ; β ) = Ψ ξ a ( α ; β ) exp [ i Ψ ξ p ( α ; β ) ]
Ψ ξ p ( α ; β ) Ψ ξ p ( α ; β ) Ψ ξ p ( α ; 0 ) ,
Ψ ξ p ( α ; β ) Ψ ξ p ( α ; β ) Ψ ξ p ( 0 ; β ) = Ψ ξ p ( α ; β ) Ψ ξ p ( α ; 0 ) Ψ ξ p ( 0 ; β ) + Ψ ξ p ( 0 ; 0 ) ,
{ Ψ x ( α ; β ) = Ψ x ( α ; β ) Ψ y ( α ; β ) = Ψ y ( α ; β )
Ψ ¯ ξ a ( α ; β ) = c ξ , 0 , 0 + m = 0 2 c ξ , 2 , m α 2 m β m + m = 0 4 c ξ , 4 , m α 4 m β m + = q = 0 q max m = 0 2 q c ξ , 2 q , m α 2 q m β m ,
Ψ ¯ ξ p ( α ; β ) = m = 0 2 d ξ , 2 , m α 2 m β m + m = 0 4 d ξ , 4 , m α 4 m β m + = q = 1 q max m = 0 2 q d ξ , 2 q , m α 2 q m β m ,
W ξ a α , β [ Ψ ¯ ξ a ( α ; β ) Ψ ξ a ( α ; β ) ] 2 ,
W ξ p α , β Ψ ¯ ξ p ( α ; β ) Ψ ξ p ( α ; β ) 2 ,
0.94 α 0.94 ,
0.9 β 0.9 ,
L ( u ) = δ ( u u ) ,
F ( α u ) = 2 2 δ ( α u η 2 ) + 2 2 δ ( α u + η 2 ) ,
I TE ( y ) = 1 2 | exp { i 2 π n 1 ( u + η 2 ) y } + exp { i 2 π n 1 ( u η 2 ) y } | 2 ,
I TM ( y ) = 1 2 ξ = y , z | Ω ξ ( u + η 2 ) exp { i 2 π n 1 ( u + η 2 ) y } + Ω ξ ( u η 2 ) exp { i 2 π n 1 ( u η 2 ) y } | 2 ,
R Λ ( y ) = 1 2 β max β max | Φ Λ ( u + η 2 ; β ) exp { i 2 π n 1 ( u + η 2 ) y } + Φ Λ ( u η 2 ; β ) exp { i 2 π n 1 ( u η 2 ) y } | 2 d β ,
α max u η 2 < u + η 2 α max ,
0 η 1.88 ,
I TE ( y ) = 1 + cos ( 2 π n 1 η y ) ,
I TM ( y ) = 1 + ( 1 η 2 2 ) cos ( 2 π n 1 η y ) ,
R Λ ( y ) = 1 2 β max β max | Φ Λ ( η 2 ; β ) exp ( i π n 1 η y ) + Φ Λ ( η 2 ; β ) exp ( i π n 1 η y ) | 2 d β ,
C I , TE ( η ) = 1 ,
C I , TM ( η ) = 1 η 2 / 2 ,
C R , Λ ( η ) = β max β max { Φ Λ ( η ; β ) Φ Λ * ( η ; β ) + Φ Λ * ( η ; β ) Φ Λ ( η ; β ) } d β β max β max { | Φ Λ ( η ; β ) | 2 + | Φ Λ ( η ; β ) | 2 } d β ,
0 η 0.94 .
I TE ( y ) = 1 + cos ( 2 π n 1 η y ) ,
I TM ( y ) = 1 + 1 η 2 cos ( 2 π n 1 η y ) ,
R Λ ( y ) = 1 2 β max β max | Φ Λ ( η ; β ) exp ( i 2 π n 1 η y ) + Φ Λ ( 0 ; β ) | 2 d β ,
C I , TE ( η ) = 1 ,
C I , TM ( η ) = 1 η 2 ,
C R , Λ ( η ) = β max β max { Φ Λ ( η ; β ) Φ Λ * ( 0 ; β ) + Φ Λ * ( η ; β ) Φ Λ ( 0 ; β ) } d β β max β max { | Φ Λ ( η ; β ) | 2 + | Φ Λ ( 0 ; β ) | 2 } d β ,

Metrics