Abstract

This paper focuses on scatterometry problems arising in lithography production of periodic gratings. Namely, the paper introduces a theoretical and numerical-modeling-oriented approach to scatterometry problems and discusses its capabilities. The approach allows for reliable detection of deviations in gratings’ critical dimensions (CDs) during the manufacturing process. The core of the approach is the one-to-one correspondence between the electromagnetic (EM) characteristics and the geometric/material properties of gratings. The approach is based on highly accurate solutions of initial boundary-value problems describing EM waves’ interaction on periodic gratings. The advantage of the approach is the ability to perform simultaneously and interactively both in frequency and time domains under conditions of possible resonant scattering of EM waves by infinite or finite gratings. This allows a detection of CDs for a wide range of gratings, and, thus is beneficial for the applied scatterometry.

© 2013 Optical Society of America

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References

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  1. A. Kato and F. Scholze, “Effect of line roughness on the diffraction intensities in angular resolved scatterometry,” Appl. Opt. 49, 6102–6110 (2010).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  4. Y.-S. Ku, S.-C. Wang, D.-M. Shyu, and N. Smith, “Scatterometry-based metrology with feature region signatures matching,” Opt. Express 14, 8482–8491 (2006).
    [CrossRef]
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    [CrossRef]
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  7. Y. K. Sirenko, S. Strom, and N. Yashina, Modeling and Analysis of Transient Processes in Open Resonant Structures. New Methods and Techniques (Springer, 2007).
  8. M. Neviere and E. Popov, Light Propagation in Periodic Media: Differential Theory and Design (Dekker, 2003).
  9. R. Petit, ed., Electromagnetic Theory of Gratings (Springer, 1980).
  10. V. P. Shestopalov, L. N. Litvinenko, S. A. Masalov, and V. G. Sologub, Wave Diffraction by Gratings (Kharkiv University, 1973) (in Russian).
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  12. V. P. Shestopalov and Y. K. Sirenko, Dynamic Theory of Gratings (Naukova Dumka, 1989) (in Russian).
  13. A. V. Tishchenko, “Numerical demonstration of the validity of the Rayleigh hypothesis,” Opt. Express 17, 17102–17117 (2009).
    [CrossRef]
  14. K. Yasuura and Y. Okuno, “Numerical analysis of diffraction from a grating by the mode-matching method with a smoothing procedure,” J. Opt. Soc. Am. 72, 847–852 (1982).
    [CrossRef]
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    [CrossRef]
  16. A. G. Kyurkchan, A. I. Sukov, and A. I. Kleev, “Singularities of wave fields and numerical methods of solving the boundary-value problems for Helmholtz equation,” in Generalized Multipole Techniques for Electromagnetic and Light Scattering, T. Wriedt, ed. (Elsevier, 1999), pp. 81–109.
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    [CrossRef]
  18. G. Bao, “Finite element approximation of time harmonic waves in periodic structures,” SIAM J. Numer. Anal. 32, 1155–1169 (1995).
    [CrossRef]
  19. K. Y. Sirenko, Y. K. Sirenko, and N. P. Yashina, “Modeling and analysis of transients in periodic gratings. I. Fully absorbing boundaries for 2-D open problems,” J. Opt. Soc. Am. A 27, 532–543 (2010).
    [CrossRef]
  20. K. Sirenko, V. Pazynin, Y. K. Sirenko, and H. Bagci, “An FFT-accelerated FDTD scheme with exact absorbing conditions for characterizing axially symmetric resonant structures,” Prog. Electromagn. Res. 111, 331–364 (2011).
    [CrossRef]
  21. K. Sirenko, M. Liu, and H. Bagci, “Incorporation of exact boundary conditions into a discontinuous Galerkin finite element method for accurately solving 2D time-dependent Maxwell equations,” IEEE Trans. Antennas Propag. 61, 472–477 (2013).
    [CrossRef]
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    [CrossRef]
  23. I. Gereige, S. Robert, G. Granet, D. Jamon, and J. J. Rousseau, “Rapid control of submicrometer periodic structures by a neural inversion from ellipsometric measurement,” Opt. Commun. 278, 270–273 (2007).
    [CrossRef]
  24. Y. Sirenko, L. Velychko, and O. Shafalyuk, “Time-domain analysis of open resonators. Analytical grounds,” Prog. Electromagn. Res. 61, 1–26 (2006).
    [CrossRef]
  25. L. G. Velychko and Y. K. Sirenko, “Controlled changes in spectra of open quasi-optical resonators,” Prog. Electromagn. Res. B 16, 85–105 (2009).
    [CrossRef]
  26. K. Y. Sirenko, Y. K. Sirenko, and N. P. Yashina, “Modeling and analysis of transients in periodic gratings. II. Resonant wave scattering,” J. Opt. Soc. Am. A 27, 544–552 (2010).
    [CrossRef]
  27. P. Melezhik, A. Poyedinchuk, N. Yashina, and G. Granet, “Periodic boundary of metamaterial: eigen regimes and resonant radiation,” Pure Appl. Opt. 9, S403–S409 (2007).
    [CrossRef]

2013 (1)

K. Sirenko, M. Liu, and H. Bagci, “Incorporation of exact boundary conditions into a discontinuous Galerkin finite element method for accurately solving 2D time-dependent Maxwell equations,” IEEE Trans. Antennas Propag. 61, 472–477 (2013).
[CrossRef]

2011 (1)

K. Sirenko, V. Pazynin, Y. K. Sirenko, and H. Bagci, “An FFT-accelerated FDTD scheme with exact absorbing conditions for characterizing axially symmetric resonant structures,” Prog. Electromagn. Res. 111, 331–364 (2011).
[CrossRef]

2010 (4)

2009 (2)

A. V. Tishchenko, “Numerical demonstration of the validity of the Rayleigh hypothesis,” Opt. Express 17, 17102–17117 (2009).
[CrossRef]

L. G. Velychko and Y. K. Sirenko, “Controlled changes in spectra of open quasi-optical resonators,” Prog. Electromagn. Res. B 16, 85–105 (2009).
[CrossRef]

2008 (2)

2007 (2)

I. Gereige, S. Robert, G. Granet, D. Jamon, and J. J. Rousseau, “Rapid control of submicrometer periodic structures by a neural inversion from ellipsometric measurement,” Opt. Commun. 278, 270–273 (2007).
[CrossRef]

P. Melezhik, A. Poyedinchuk, N. Yashina, and G. Granet, “Periodic boundary of metamaterial: eigen regimes and resonant radiation,” Pure Appl. Opt. 9, S403–S409 (2007).
[CrossRef]

2006 (2)

Y. Sirenko, L. Velychko, and O. Shafalyuk, “Time-domain analysis of open resonators. Analytical grounds,” Prog. Electromagn. Res. 61, 1–26 (2006).
[CrossRef]

Y.-S. Ku, S.-C. Wang, D.-M. Shyu, and N. Smith, “Scatterometry-based metrology with feature region signatures matching,” Opt. Express 14, 8482–8491 (2006).
[CrossRef]

1998 (1)

1995 (1)

G. Bao, “Finite element approximation of time harmonic waves in periodic structures,” SIAM J. Numer. Anal. 32, 1155–1169 (1995).
[CrossRef]

1982 (1)

1980 (2)

T. Miyamoto, “Numerical analysis of a rib optical waveguide with trapezoidal cross section,” Opt. Commun. 34, 35–38(1980).
[CrossRef]

J. Chandezon, D. Maystre, and G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. 11, 235–241 (1980).
[CrossRef]

Badran, F.

Bagci, H.

K. Sirenko, M. Liu, and H. Bagci, “Incorporation of exact boundary conditions into a discontinuous Galerkin finite element method for accurately solving 2D time-dependent Maxwell equations,” IEEE Trans. Antennas Propag. 61, 472–477 (2013).
[CrossRef]

K. Sirenko, V. Pazynin, Y. K. Sirenko, and H. Bagci, “An FFT-accelerated FDTD scheme with exact absorbing conditions for characterizing axially symmetric resonant structures,” Prog. Electromagn. Res. 111, 331–364 (2011).
[CrossRef]

Bao, G.

G. Bao, “Finite element approximation of time harmonic waves in periodic structures,” SIAM J. Numer. Anal. 32, 1155–1169 (1995).
[CrossRef]

Chandezon, J.

J. Chandezon, D. Maystre, and G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. 11, 235–241 (1980).
[CrossRef]

Gereige, I.

I. Gereige, S. Robert, S. Thiria, F. Badran, G. Granet, and J. J. Rousseau, “Recognition of diffraction-grating profile using a neural network classifier in optical scatterometry,” J. Opt. Soc. Am. A 25, 1661–1667 (2008).
[CrossRef]

I. Gereige, S. Robert, G. Granet, D. Jamon, and J. J. Rousseau, “Rapid control of submicrometer periodic structures by a neural inversion from ellipsometric measurement,” Opt. Commun. 278, 270–273 (2007).
[CrossRef]

Granet, G.

I. Gereige, S. Robert, S. Thiria, F. Badran, G. Granet, and J. J. Rousseau, “Recognition of diffraction-grating profile using a neural network classifier in optical scatterometry,” J. Opt. Soc. Am. A 25, 1661–1667 (2008).
[CrossRef]

I. Gereige, S. Robert, G. Granet, D. Jamon, and J. J. Rousseau, “Rapid control of submicrometer periodic structures by a neural inversion from ellipsometric measurement,” Opt. Commun. 278, 270–273 (2007).
[CrossRef]

P. Melezhik, A. Poyedinchuk, N. Yashina, and G. Granet, “Periodic boundary of metamaterial: eigen regimes and resonant radiation,” Pure Appl. Opt. 9, S403–S409 (2007).
[CrossRef]

Jamon, D.

I. Gereige, S. Robert, G. Granet, D. Jamon, and J. J. Rousseau, “Rapid control of submicrometer periodic structures by a neural inversion from ellipsometric measurement,” Opt. Commun. 278, 270–273 (2007).
[CrossRef]

Kallioniemi, I.

Kato, A.

Kirilenko, A. A.

V. P. Shestopalov, A. A. Kirilenko, S. A. Masalov, and Y. K. Sirenko, Resonance Wave Scattering, Vol. 1. Diffraction Gratings (Naukova Dumka, 1986) (in Russian).

Kleev, A. I.

A. G. Kyurkchan, A. I. Sukov, and A. I. Kleev, “Singularities of wave fields and numerical methods of solving the boundary-value problems for Helmholtz equation,” in Generalized Multipole Techniques for Electromagnetic and Light Scattering, T. Wriedt, ed. (Elsevier, 1999), pp. 81–109.

Ku, Y.-S.

Kyurkchan, A. G.

A. G. Kyurkchan, A. I. Sukov, and A. I. Kleev, “Singularities of wave fields and numerical methods of solving the boundary-value problems for Helmholtz equation,” in Generalized Multipole Techniques for Electromagnetic and Light Scattering, T. Wriedt, ed. (Elsevier, 1999), pp. 81–109.

Li, L.

Litvinenko, L. N.

V. P. Shestopalov, L. N. Litvinenko, S. A. Masalov, and V. G. Sologub, Wave Diffraction by Gratings (Kharkiv University, 1973) (in Russian).

Liu, M.

K. Sirenko, M. Liu, and H. Bagci, “Incorporation of exact boundary conditions into a discontinuous Galerkin finite element method for accurately solving 2D time-dependent Maxwell equations,” IEEE Trans. Antennas Propag. 61, 472–477 (2013).
[CrossRef]

Masalov, S. A.

V. P. Shestopalov, L. N. Litvinenko, S. A. Masalov, and V. G. Sologub, Wave Diffraction by Gratings (Kharkiv University, 1973) (in Russian).

V. P. Shestopalov, A. A. Kirilenko, S. A. Masalov, and Y. K. Sirenko, Resonance Wave Scattering, Vol. 1. Diffraction Gratings (Naukova Dumka, 1986) (in Russian).

Maystre, D.

J. Chandezon, D. Maystre, and G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. 11, 235–241 (1980).
[CrossRef]

Melezhik, P.

P. Melezhik, A. Poyedinchuk, N. Yashina, and G. Granet, “Periodic boundary of metamaterial: eigen regimes and resonant radiation,” Pure Appl. Opt. 9, S403–S409 (2007).
[CrossRef]

Miyamoto, T.

T. Miyamoto, “Numerical analysis of a rib optical waveguide with trapezoidal cross section,” Opt. Commun. 34, 35–38(1980).
[CrossRef]

Neviere, M.

M. Neviere and E. Popov, Light Propagation in Periodic Media: Differential Theory and Design (Dekker, 2003).

Oja, E.

Okuno, Y.

Pazynin, V.

K. Sirenko, V. Pazynin, Y. K. Sirenko, and H. Bagci, “An FFT-accelerated FDTD scheme with exact absorbing conditions for characterizing axially symmetric resonant structures,” Prog. Electromagn. Res. 111, 331–364 (2011).
[CrossRef]

Popov, E.

M. Neviere and E. Popov, Light Propagation in Periodic Media: Differential Theory and Design (Dekker, 2003).

Poyedinchuk, A.

P. Melezhik, A. Poyedinchuk, N. Yashina, and G. Granet, “Periodic boundary of metamaterial: eigen regimes and resonant radiation,” Pure Appl. Opt. 9, S403–S409 (2007).
[CrossRef]

Raoult, G.

J. Chandezon, D. Maystre, and G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. 11, 235–241 (1980).
[CrossRef]

Robert, S.

I. Gereige, S. Robert, S. Thiria, F. Badran, G. Granet, and J. J. Rousseau, “Recognition of diffraction-grating profile using a neural network classifier in optical scatterometry,” J. Opt. Soc. Am. A 25, 1661–1667 (2008).
[CrossRef]

I. Gereige, S. Robert, G. Granet, D. Jamon, and J. J. Rousseau, “Rapid control of submicrometer periodic structures by a neural inversion from ellipsometric measurement,” Opt. Commun. 278, 270–273 (2007).
[CrossRef]

Rousseau, J. J.

I. Gereige, S. Robert, S. Thiria, F. Badran, G. Granet, and J. J. Rousseau, “Recognition of diffraction-grating profile using a neural network classifier in optical scatterometry,” J. Opt. Soc. Am. A 25, 1661–1667 (2008).
[CrossRef]

I. Gereige, S. Robert, G. Granet, D. Jamon, and J. J. Rousseau, “Rapid control of submicrometer periodic structures by a neural inversion from ellipsometric measurement,” Opt. Commun. 278, 270–273 (2007).
[CrossRef]

Saarinen, J.

Scholze, F.

Shafalyuk, O.

Y. Sirenko, L. Velychko, and O. Shafalyuk, “Time-domain analysis of open resonators. Analytical grounds,” Prog. Electromagn. Res. 61, 1–26 (2006).
[CrossRef]

Shestopalov, V. P.

V. P. Shestopalov, L. N. Litvinenko, S. A. Masalov, and V. G. Sologub, Wave Diffraction by Gratings (Kharkiv University, 1973) (in Russian).

V. P. Shestopalov, A. A. Kirilenko, S. A. Masalov, and Y. K. Sirenko, Resonance Wave Scattering, Vol. 1. Diffraction Gratings (Naukova Dumka, 1986) (in Russian).

V. P. Shestopalov and Y. K. Sirenko, Dynamic Theory of Gratings (Naukova Dumka, 1989) (in Russian).

Shyu, D.-M.

Sirenko, K.

K. Sirenko, M. Liu, and H. Bagci, “Incorporation of exact boundary conditions into a discontinuous Galerkin finite element method for accurately solving 2D time-dependent Maxwell equations,” IEEE Trans. Antennas Propag. 61, 472–477 (2013).
[CrossRef]

K. Sirenko, V. Pazynin, Y. K. Sirenko, and H. Bagci, “An FFT-accelerated FDTD scheme with exact absorbing conditions for characterizing axially symmetric resonant structures,” Prog. Electromagn. Res. 111, 331–364 (2011).
[CrossRef]

Sirenko, K. Y.

Sirenko, Y.

Y. Sirenko, L. Velychko, and O. Shafalyuk, “Time-domain analysis of open resonators. Analytical grounds,” Prog. Electromagn. Res. 61, 1–26 (2006).
[CrossRef]

Sirenko, Y. K.

K. Sirenko, V. Pazynin, Y. K. Sirenko, and H. Bagci, “An FFT-accelerated FDTD scheme with exact absorbing conditions for characterizing axially symmetric resonant structures,” Prog. Electromagn. Res. 111, 331–364 (2011).
[CrossRef]

K. Y. Sirenko, Y. K. Sirenko, and N. P. Yashina, “Modeling and analysis of transients in periodic gratings. I. Fully absorbing boundaries for 2-D open problems,” J. Opt. Soc. Am. A 27, 532–543 (2010).
[CrossRef]

K. Y. Sirenko, Y. K. Sirenko, and N. P. Yashina, “Modeling and analysis of transients in periodic gratings. II. Resonant wave scattering,” J. Opt. Soc. Am. A 27, 544–552 (2010).
[CrossRef]

L. G. Velychko and Y. K. Sirenko, “Controlled changes in spectra of open quasi-optical resonators,” Prog. Electromagn. Res. B 16, 85–105 (2009).
[CrossRef]

Y. K. Sirenko, S. Strom, and N. Yashina, Modeling and Analysis of Transient Processes in Open Resonant Structures. New Methods and Techniques (Springer, 2007).

V. P. Shestopalov and Y. K. Sirenko, Dynamic Theory of Gratings (Naukova Dumka, 1989) (in Russian).

V. P. Shestopalov, A. A. Kirilenko, S. A. Masalov, and Y. K. Sirenko, Resonance Wave Scattering, Vol. 1. Diffraction Gratings (Naukova Dumka, 1986) (in Russian).

Smith, N.

Sologub, V. G.

V. P. Shestopalov, L. N. Litvinenko, S. A. Masalov, and V. G. Sologub, Wave Diffraction by Gratings (Kharkiv University, 1973) (in Russian).

Strom, S.

Y. K. Sirenko, S. Strom, and N. Yashina, Modeling and Analysis of Transient Processes in Open Resonant Structures. New Methods and Techniques (Springer, 2007).

Sukov, A. I.

A. G. Kyurkchan, A. I. Sukov, and A. I. Kleev, “Singularities of wave fields and numerical methods of solving the boundary-value problems for Helmholtz equation,” in Generalized Multipole Techniques for Electromagnetic and Light Scattering, T. Wriedt, ed. (Elsevier, 1999), pp. 81–109.

Thiria, S.

Tishchenko, A. V.

Velychko, L.

Y. Sirenko, L. Velychko, and O. Shafalyuk, “Time-domain analysis of open resonators. Analytical grounds,” Prog. Electromagn. Res. 61, 1–26 (2006).
[CrossRef]

Velychko, L. G.

L. G. Velychko and Y. K. Sirenko, “Controlled changes in spectra of open quasi-optical resonators,” Prog. Electromagn. Res. B 16, 85–105 (2009).
[CrossRef]

Wang, S.-C.

Wei, S.

Yashina, N.

P. Melezhik, A. Poyedinchuk, N. Yashina, and G. Granet, “Periodic boundary of metamaterial: eigen regimes and resonant radiation,” Pure Appl. Opt. 9, S403–S409 (2007).
[CrossRef]

Y. K. Sirenko, S. Strom, and N. Yashina, Modeling and Analysis of Transient Processes in Open Resonant Structures. New Methods and Techniques (Springer, 2007).

Yashina, N. P.

Yasuura, K.

Appl. Opt. (4)

IEEE Trans. Antennas Propag. (1)

K. Sirenko, M. Liu, and H. Bagci, “Incorporation of exact boundary conditions into a discontinuous Galerkin finite element method for accurately solving 2D time-dependent Maxwell equations,” IEEE Trans. Antennas Propag. 61, 472–477 (2013).
[CrossRef]

J. Opt. (1)

J. Chandezon, D. Maystre, and G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. 11, 235–241 (1980).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

Opt. Commun. (2)

I. Gereige, S. Robert, G. Granet, D. Jamon, and J. J. Rousseau, “Rapid control of submicrometer periodic structures by a neural inversion from ellipsometric measurement,” Opt. Commun. 278, 270–273 (2007).
[CrossRef]

T. Miyamoto, “Numerical analysis of a rib optical waveguide with trapezoidal cross section,” Opt. Commun. 34, 35–38(1980).
[CrossRef]

Opt. Express (2)

Prog. Electromagn. Res. (2)

K. Sirenko, V. Pazynin, Y. K. Sirenko, and H. Bagci, “An FFT-accelerated FDTD scheme with exact absorbing conditions for characterizing axially symmetric resonant structures,” Prog. Electromagn. Res. 111, 331–364 (2011).
[CrossRef]

Y. Sirenko, L. Velychko, and O. Shafalyuk, “Time-domain analysis of open resonators. Analytical grounds,” Prog. Electromagn. Res. 61, 1–26 (2006).
[CrossRef]

Prog. Electromagn. Res. B (1)

L. G. Velychko and Y. K. Sirenko, “Controlled changes in spectra of open quasi-optical resonators,” Prog. Electromagn. Res. B 16, 85–105 (2009).
[CrossRef]

Pure Appl. Opt. (1)

P. Melezhik, A. Poyedinchuk, N. Yashina, and G. Granet, “Periodic boundary of metamaterial: eigen regimes and resonant radiation,” Pure Appl. Opt. 9, S403–S409 (2007).
[CrossRef]

SIAM J. Numer. Anal. (1)

G. Bao, “Finite element approximation of time harmonic waves in periodic structures,” SIAM J. Numer. Anal. 32, 1155–1169 (1995).
[CrossRef]

Other (8)

A. G. Kyurkchan, A. I. Sukov, and A. I. Kleev, “Singularities of wave fields and numerical methods of solving the boundary-value problems for Helmholtz equation,” in Generalized Multipole Techniques for Electromagnetic and Light Scattering, T. Wriedt, ed. (Elsevier, 1999), pp. 81–109.

Y. K. Sirenko and S. Strom, eds., Modern Theory of Gratings. Resonant Scattering: Analysis Techniques and Phenomena (Springer, 2010).

Y. K. Sirenko, S. Strom, and N. Yashina, Modeling and Analysis of Transient Processes in Open Resonant Structures. New Methods and Techniques (Springer, 2007).

M. Neviere and E. Popov, Light Propagation in Periodic Media: Differential Theory and Design (Dekker, 2003).

R. Petit, ed., Electromagnetic Theory of Gratings (Springer, 1980).

V. P. Shestopalov, L. N. Litvinenko, S. A. Masalov, and V. G. Sologub, Wave Diffraction by Gratings (Kharkiv University, 1973) (in Russian).

V. P. Shestopalov, A. A. Kirilenko, S. A. Masalov, and Y. K. Sirenko, Resonance Wave Scattering, Vol. 1. Diffraction Gratings (Naukova Dumka, 1986) (in Russian).

V. P. Shestopalov and Y. K. Sirenko, Dynamic Theory of Gratings (Naukova Dumka, 1989) (in Russian).

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Figures (11)

Fig. 1.
Fig. 1.

Geometry of an infinite grating.

Fig. 2.
Fig. 2.

Geometry of a compact periodic structure.

Fig. 3.
Fig. 3.

(a) Geometry of transparent gratings. (b) Efficiency of reflection, Wn0R(κ). (c) Efficiency of transition, Wn0T(κ). The gray arrows indicate the direction of “shifts” of resonance peaks when f is increasing.

Fig. 4.
Fig. 4.

Same characteristics as in Fig. 3.

Fig. 5.
Fig. 5.

Photoresist grating on a PEC substrate. Autocollimation reflection of the minus first harmonic.

Fig. 6.
Fig. 6.

Efficiency of reflection, Wn0R(κ), of the minus second (n=2), the minus first (n=1), the zeroth (n=0), and the plus first (n=1) harmonics.

Fig. 7.
Fig. 7.

(a) Geometry of the grating. (b) Efficiency of the zeroth-order reflection, W00R(κ).

Fig. 8.
Fig. 8.

Inverse of radiation efficiency, W11(κ)=1η(κ).

Fig. 9.
Fig. 9.

(a) Field pattern U(g,t), t=465—regime of eigenoscillations. (b) Field U(g,t) in the point g=g1; the source is turned off at the moment marked by the vertical dashed line. (c) Inverse of radiation efficiency. (d) Radiation directive pattern.

Fig. 10.
Fig. 10.

Radiation pattern, D(ϕ,κ,), from Fig. 9(d) computed at frequencies (a) κ0.39, (b) κ0.404, and (c) κ0.41.

Fig. 11.
Fig. 11.

Efficiency of the zeroth-order reflection, Wn0R(κ).

Equations (7)

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

[εt2σt+y2+z2]U(g,t)=0;t>0U(g,0)=0,tU(g,t)|t=0=0;U(l,z,t)=e2πiΦU(0,z,t),yU(l,z,t)=e2πiΦyU(0,z,t);D±[U±s(g,t)]|gL±=0.
U±s(y,L±,t)=n=μn(y){0tJ0([tτ]Φn)[0lzU±s(y˜,L±,τ)μn*(y˜)dy˜]dτ};0yl,t0.
[y2+z2+ε˜k2]U˜(g,k)=0;U˜(l,z,k)=e2πiΦU˜(0,z,k),yU˜(l,z,k)=e2πiΦyU˜(0,z,k);U˜(g,k)={U˜pinc(g,k)+n=RnpAAexp[i(Φny+Γn(zL+))];zL+n=TnpBAexp[i(ΦnyΓn(zL))];zL.
RnpAA(k)=u˜n+(z,k)v˜p(z,k)|z=L+andTnpBA(k)=u˜n(z,k)|z=Lv˜p(z,k)|z=L+.
[εt2σt+y2+z2]U(g,t)=0;t>0U(g,t)|t=0=0,tU(g,t)|t=0=0;D1[Us(g,t)]|gL1=0andD[U(g,t)]|gL=0;t0.
Us(g1,t)=U(g1,t)Uinc(g1,t);g1Q1,Us(g1,t)=n=1un1(z1,t)μn1(y1),Uinc(g1,t)=n=1vn1(z1,t)μn1(y1).
D(ϕ,k,M)=|U˜(M,ϕ,k)|2maxϕ1<ϕ<ϕ2|U˜(M,ϕ,k)|2;K1kK2,

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