Abstract

A theory is presented for the application of Hill’s matrix method to the calculation of the reflection and transmission spectra of multitone holographic interference filters in which the permittivity is modulated by a sum of repeating functions of arbitrary period. Such filters are important because they may have two or more independent reflection bands. Guidelines are presented for accurately truncating the Hill matrix, and numerical methods are described for finding the exponential coefficient and the coefficients of the Floquet-Bloch waves within the filter. The latter calculation is performed by use of a computational technique known as inverse iteration. The Hill matrix for such problems is sparse, and thus, even though the matrix can be quite large, it may be efficiently stored and processed by a desktop computer. It is shown that the results of using Hill’s matrix method are in close agreement with numerical calculations based on thin-film decomposition, a transfer-matrix technique. An important result of this research is the demonstration that Hill’s matrix method may, in principle, be used to analyze any multiperiodic problem, so long as the periods are known to finite precision.

© 2004 Optical Society of America

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References

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  1. O. Wiener, “Stehende Lichtwellen und die Schwingungrichtung polarisirten Lichtes,” Ann. Phys. Chem. 40, 203–243 (1890).
  2. G. Lippmann, “Sur la théorie de la photographie des coulars simples et compusées par la méthode interférentielle,” J. Phys. (Paris) 3, 97–107 (1894).
  3. P. Connes, “Silver salts and standing waves: the history of interference colour photography,” J. Opt. (Paris) 18, 147–166 (1987).
    [CrossRef]
  4. T. W. Stone, B. J. Thompson, eds., Selected Papers on Holographic and Diffractive Lenses and Mirrors, Vol. MS 34 of SPIE Milestones Series (SPIE Optical Engineering Press, Bellingham, Wash., 1991).
  5. J. B. Adolph, R. W. Bertram, K. L. Yan, P. Zhou, R. de Leon, “Design and fabrication of multi-line inhomogeneous rejection filters,” in Inhomogeneous and Quasi-Inhomogeneous Optical Coatings, J. A. Dobrowolski, P. G. Verly, eds., Proc. SPIE2046, 141–146 (1993).
    [CrossRef]
  6. K. W. Steijn, “Multicolor holographic recording in DuPont holographic recording film: determination of exposure conditions for color balance,” in Holographic Materials II, T. J. Trout, ed., Proc. SPIE2688, 123–134 (1996).
    [CrossRef]
  7. D. W. Diehl, N. George, “Holographic interference filters for infrared communications,” Appl. Opt. 42, 1203–1210 (2003).
    [CrossRef] [PubMed]
  8. R. Jacobsson, “Light reflection from films of continuously varying refractive index,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1966), Vol. 5, Chap. 5, pp. 247–286.
    [CrossRef]
  9. R. Alferness, “Analysis of optical propagation in thick holographic gratings,” Appl. Phys. 7, 29–33 (1975).
    [CrossRef]
  10. R. Alferness, “Analysis of propagation at the second-order Bragg angle of a thick holographic grating,” J. Opt. Soc. Am. 66, 353–362 (1976).
    [CrossRef]
  11. T. W. Stone, N. George, “Wavelength performance of holographic optical elements,” Appl. Opt. 24, 3797–3810 (1985).
    [CrossRef] [PubMed]
  12. T. W. Stone, “Holographic optical elements,” Ph.D. dissertation (University of Rochester, Rochester, N.Y., 1986).
  13. C. V. Raman, N. S. N. Nath, “The diffraction of waves by high frequency sound waves. I, II,” Proc. Ind. Acad. Sci. 2, 406–420 (1935).
  14. C. V. Raman, N. S. N. Nath, “The diffraction of waves by high frequency sound waves. III, IV,” Proc. Ind. Acad. Sci. 3, 75–84, 119–125 (1936).
  15. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
  16. M. Chang, N. George, “Holographic dielectric grating: theory and practice,” Appl. Opt. 9, 713–719 (1970).
    [CrossRef] [PubMed]
  17. M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 71, 811–818 (1981).
    [CrossRef]
  18. M. G. Moharam, T. K. Gaylord, “Planar dielectric grating diffraction theory,” Appl. Phys. B 28, 1–14 (1982).
    [CrossRef]
  19. P. St. J. Russell, “Optics of Floquet-Block waves in dielectric gratings,” Appl. Phys. B 39, 231–246 (1986).
    [CrossRef]
  20. M. G. Moharam, T. K. Gaylord, “Chain-matrix analysis of arbitrary-thickness dielectric reflection gratings,” J. Opt. Soc. Am. 72, 187–190 (1982).
    [CrossRef]
  21. Z. Zylberberg, E. Marom, “Rigorous coupled-wave analysis of pure reflection gratings,” J. Opt. Soc. Am. 73, 392–398 (1983).
    [CrossRef]
  22. M. G. Moharam, T. K. Gaylord, “Comments on analyses of reflection gratings,” J. Opt. Soc. Am. 73, 399–401 (1983).
    [CrossRef]
  23. G. W. Hill, “On the part of the motion of the lunar perigee which is a function of the mean motions of the Sun and Moon,” Acta Math. 8, 1–36 (1886).
    [CrossRef]
  24. Lord Rayleigh, “On the maintenance of vibrations by forces of double frequency, and on the propagation of waves through a medium endowed with a periodic structure,” Phil. Mag. J. Sci. 24, 145–159 (1887).
    [CrossRef]
  25. E. T. Whittaker, G. N. Watson, A Course of Modern Analysis, 4th ed. (Cambridge U. Press, Cambridge, 1927), Chap. 19, pp. 412–417.
  26. W. Magnus, S. Winkler, “Hill’s equation,” in Interscience Tracts in Pure and Applied Mathematics, L. Bers, R. Courant, J. J. Stoker, eds. (Wiley Interscience, New York, 1966), Vol. 20, Chap. 1, pp. 3–10.
  27. S. N. Biswas, K. Datta, R. P. Saxena, P. K. Srivastava, V. S. Varma, “The Hill determinant: an application to the anharmonic oscillator,” Phys. Rev. D 4, 3617–3620 (1971).
    [CrossRef]
  28. C. Elachi, “Waves in active and passive periodic structures: a review,” Proc. IEEE 64, 1666–1698 (1976).
    [CrossRef]
  29. D. L. Jaggard, C. Elachi, “Floquet and coupled-wave analysis of higher-order Bragg coupling in a periodic medium,” J. Opt. Soc. Am. 66, 674–682 (1976).
    [CrossRef]
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    [CrossRef]
  31. J. H. Wilkinson, “The calculation of eigenvectors of codiagonal matrices,” Comput. J. 1, 90–96 (1958).
    [CrossRef]
  32. J. H. Wilkinson, “Inverse iteration in theory and practice,” in Symposium Mathematica (Instituto Nazionale de Alta Matematica, Rome, 1972), Vol. X, pp. 361–379.
  33. I. C. F. Ipsen, “Computing an eigenvector with inverse iteration,” Soc. Indust. Appl. Math. Rev. 39, 254–291 (1997).
  34. H. A. Macleod, Thin-Film Optical Filters, 2nd ed. (McGraw-Hill, New York, 1989).
  35. P. Sharlandjiev, T. Mateeva, “Normal incidence holographic mirrors by the characteristic matrix method. Numerical examples,” J. Opt. (Paris) 16, 185–189 (1985).
    [CrossRef]

2003 (1)

1997 (1)

I. C. F. Ipsen, “Computing an eigenvector with inverse iteration,” Soc. Indust. Appl. Math. Rev. 39, 254–291 (1997).

1990 (1)

1987 (1)

P. Connes, “Silver salts and standing waves: the history of interference colour photography,” J. Opt. (Paris) 18, 147–166 (1987).
[CrossRef]

1986 (1)

P. St. J. Russell, “Optics of Floquet-Block waves in dielectric gratings,” Appl. Phys. B 39, 231–246 (1986).
[CrossRef]

1985 (2)

P. Sharlandjiev, T. Mateeva, “Normal incidence holographic mirrors by the characteristic matrix method. Numerical examples,” J. Opt. (Paris) 16, 185–189 (1985).
[CrossRef]

T. W. Stone, N. George, “Wavelength performance of holographic optical elements,” Appl. Opt. 24, 3797–3810 (1985).
[CrossRef] [PubMed]

1983 (2)

1982 (2)

M. G. Moharam, T. K. Gaylord, “Chain-matrix analysis of arbitrary-thickness dielectric reflection gratings,” J. Opt. Soc. Am. 72, 187–190 (1982).
[CrossRef]

M. G. Moharam, T. K. Gaylord, “Planar dielectric grating diffraction theory,” Appl. Phys. B 28, 1–14 (1982).
[CrossRef]

1981 (1)

1976 (3)

1975 (1)

R. Alferness, “Analysis of optical propagation in thick holographic gratings,” Appl. Phys. 7, 29–33 (1975).
[CrossRef]

1971 (1)

S. N. Biswas, K. Datta, R. P. Saxena, P. K. Srivastava, V. S. Varma, “The Hill determinant: an application to the anharmonic oscillator,” Phys. Rev. D 4, 3617–3620 (1971).
[CrossRef]

1970 (1)

1969 (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

1958 (1)

J. H. Wilkinson, “The calculation of eigenvectors of codiagonal matrices,” Comput. J. 1, 90–96 (1958).
[CrossRef]

1936 (1)

C. V. Raman, N. S. N. Nath, “The diffraction of waves by high frequency sound waves. III, IV,” Proc. Ind. Acad. Sci. 3, 75–84, 119–125 (1936).

1935 (1)

C. V. Raman, N. S. N. Nath, “The diffraction of waves by high frequency sound waves. I, II,” Proc. Ind. Acad. Sci. 2, 406–420 (1935).

1894 (1)

G. Lippmann, “Sur la théorie de la photographie des coulars simples et compusées par la méthode interférentielle,” J. Phys. (Paris) 3, 97–107 (1894).

1890 (1)

O. Wiener, “Stehende Lichtwellen und die Schwingungrichtung polarisirten Lichtes,” Ann. Phys. Chem. 40, 203–243 (1890).

1887 (1)

Lord Rayleigh, “On the maintenance of vibrations by forces of double frequency, and on the propagation of waves through a medium endowed with a periodic structure,” Phil. Mag. J. Sci. 24, 145–159 (1887).
[CrossRef]

1886 (1)

G. W. Hill, “On the part of the motion of the lunar perigee which is a function of the mean motions of the Sun and Moon,” Acta Math. 8, 1–36 (1886).
[CrossRef]

Adolph, J. B.

J. B. Adolph, R. W. Bertram, K. L. Yan, P. Zhou, R. de Leon, “Design and fabrication of multi-line inhomogeneous rejection filters,” in Inhomogeneous and Quasi-Inhomogeneous Optical Coatings, J. A. Dobrowolski, P. G. Verly, eds., Proc. SPIE2046, 141–146 (1993).
[CrossRef]

Alferness, R.

R. Alferness, “Analysis of propagation at the second-order Bragg angle of a thick holographic grating,” J. Opt. Soc. Am. 66, 353–362 (1976).
[CrossRef]

R. Alferness, “Analysis of optical propagation in thick holographic gratings,” Appl. Phys. 7, 29–33 (1975).
[CrossRef]

Bertram, R. W.

J. B. Adolph, R. W. Bertram, K. L. Yan, P. Zhou, R. de Leon, “Design and fabrication of multi-line inhomogeneous rejection filters,” in Inhomogeneous and Quasi-Inhomogeneous Optical Coatings, J. A. Dobrowolski, P. G. Verly, eds., Proc. SPIE2046, 141–146 (1993).
[CrossRef]

Biswas, S. N.

S. N. Biswas, K. Datta, R. P. Saxena, P. K. Srivastava, V. S. Varma, “The Hill determinant: an application to the anharmonic oscillator,” Phys. Rev. D 4, 3617–3620 (1971).
[CrossRef]

Chang, M.

Connes, P.

P. Connes, “Silver salts and standing waves: the history of interference colour photography,” J. Opt. (Paris) 18, 147–166 (1987).
[CrossRef]

Datta, K.

S. N. Biswas, K. Datta, R. P. Saxena, P. K. Srivastava, V. S. Varma, “The Hill determinant: an application to the anharmonic oscillator,” Phys. Rev. D 4, 3617–3620 (1971).
[CrossRef]

de Leon, R.

J. B. Adolph, R. W. Bertram, K. L. Yan, P. Zhou, R. de Leon, “Design and fabrication of multi-line inhomogeneous rejection filters,” in Inhomogeneous and Quasi-Inhomogeneous Optical Coatings, J. A. Dobrowolski, P. G. Verly, eds., Proc. SPIE2046, 141–146 (1993).
[CrossRef]

Diehl, D. W.

Elachi, C.

Gaylord, T. K.

George, N.

Hill, G. W.

G. W. Hill, “On the part of the motion of the lunar perigee which is a function of the mean motions of the Sun and Moon,” Acta Math. 8, 1–36 (1886).
[CrossRef]

Ipsen, I. C. F.

I. C. F. Ipsen, “Computing an eigenvector with inverse iteration,” Soc. Indust. Appl. Math. Rev. 39, 254–291 (1997).

Jacobsson, R.

R. Jacobsson, “Light reflection from films of continuously varying refractive index,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1966), Vol. 5, Chap. 5, pp. 247–286.
[CrossRef]

Jaggard, D. L.

Kogelnik, H.

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

Lippmann, G.

G. Lippmann, “Sur la théorie de la photographie des coulars simples et compusées par la méthode interférentielle,” J. Phys. (Paris) 3, 97–107 (1894).

Macleod, H. A.

H. A. Macleod, Thin-Film Optical Filters, 2nd ed. (McGraw-Hill, New York, 1989).

Magnus, W.

W. Magnus, S. Winkler, “Hill’s equation,” in Interscience Tracts in Pure and Applied Mathematics, L. Bers, R. Courant, J. J. Stoker, eds. (Wiley Interscience, New York, 1966), Vol. 20, Chap. 1, pp. 3–10.

Marom, E.

Mateeva, T.

P. Sharlandjiev, T. Mateeva, “Normal incidence holographic mirrors by the characteristic matrix method. Numerical examples,” J. Opt. (Paris) 16, 185–189 (1985).
[CrossRef]

Moharam, M. G.

Nath, N. S. N.

C. V. Raman, N. S. N. Nath, “The diffraction of waves by high frequency sound waves. III, IV,” Proc. Ind. Acad. Sci. 3, 75–84, 119–125 (1936).

C. V. Raman, N. S. N. Nath, “The diffraction of waves by high frequency sound waves. I, II,” Proc. Ind. Acad. Sci. 2, 406–420 (1935).

Ning, X.

Raman, C. V.

C. V. Raman, N. S. N. Nath, “The diffraction of waves by high frequency sound waves. III, IV,” Proc. Ind. Acad. Sci. 3, 75–84, 119–125 (1936).

C. V. Raman, N. S. N. Nath, “The diffraction of waves by high frequency sound waves. I, II,” Proc. Ind. Acad. Sci. 2, 406–420 (1935).

Rayleigh, Lord

Lord Rayleigh, “On the maintenance of vibrations by forces of double frequency, and on the propagation of waves through a medium endowed with a periodic structure,” Phil. Mag. J. Sci. 24, 145–159 (1887).
[CrossRef]

Russell, P. St. J.

P. St. J. Russell, “Optics of Floquet-Block waves in dielectric gratings,” Appl. Phys. B 39, 231–246 (1986).
[CrossRef]

Saxena, R. P.

S. N. Biswas, K. Datta, R. P. Saxena, P. K. Srivastava, V. S. Varma, “The Hill determinant: an application to the anharmonic oscillator,” Phys. Rev. D 4, 3617–3620 (1971).
[CrossRef]

Sharlandjiev, P.

P. Sharlandjiev, T. Mateeva, “Normal incidence holographic mirrors by the characteristic matrix method. Numerical examples,” J. Opt. (Paris) 16, 185–189 (1985).
[CrossRef]

Srivastava, P. K.

S. N. Biswas, K. Datta, R. P. Saxena, P. K. Srivastava, V. S. Varma, “The Hill determinant: an application to the anharmonic oscillator,” Phys. Rev. D 4, 3617–3620 (1971).
[CrossRef]

Steijn, K. W.

K. W. Steijn, “Multicolor holographic recording in DuPont holographic recording film: determination of exposure conditions for color balance,” in Holographic Materials II, T. J. Trout, ed., Proc. SPIE2688, 123–134 (1996).
[CrossRef]

Stone, T. W.

T. W. Stone, N. George, “Wavelength performance of holographic optical elements,” Appl. Opt. 24, 3797–3810 (1985).
[CrossRef] [PubMed]

T. W. Stone, “Holographic optical elements,” Ph.D. dissertation (University of Rochester, Rochester, N.Y., 1986).

Varma, V. S.

S. N. Biswas, K. Datta, R. P. Saxena, P. K. Srivastava, V. S. Varma, “The Hill determinant: an application to the anharmonic oscillator,” Phys. Rev. D 4, 3617–3620 (1971).
[CrossRef]

Watson, G. N.

E. T. Whittaker, G. N. Watson, A Course of Modern Analysis, 4th ed. (Cambridge U. Press, Cambridge, 1927), Chap. 19, pp. 412–417.

Whittaker, E. T.

E. T. Whittaker, G. N. Watson, A Course of Modern Analysis, 4th ed. (Cambridge U. Press, Cambridge, 1927), Chap. 19, pp. 412–417.

Wiener, O.

O. Wiener, “Stehende Lichtwellen und die Schwingungrichtung polarisirten Lichtes,” Ann. Phys. Chem. 40, 203–243 (1890).

Wilkinson, J. H.

J. H. Wilkinson, “The calculation of eigenvectors of codiagonal matrices,” Comput. J. 1, 90–96 (1958).
[CrossRef]

J. H. Wilkinson, “Inverse iteration in theory and practice,” in Symposium Mathematica (Instituto Nazionale de Alta Matematica, Rome, 1972), Vol. X, pp. 361–379.

Winkler, S.

W. Magnus, S. Winkler, “Hill’s equation,” in Interscience Tracts in Pure and Applied Mathematics, L. Bers, R. Courant, J. J. Stoker, eds. (Wiley Interscience, New York, 1966), Vol. 20, Chap. 1, pp. 3–10.

Yan, K. L.

J. B. Adolph, R. W. Bertram, K. L. Yan, P. Zhou, R. de Leon, “Design and fabrication of multi-line inhomogeneous rejection filters,” in Inhomogeneous and Quasi-Inhomogeneous Optical Coatings, J. A. Dobrowolski, P. G. Verly, eds., Proc. SPIE2046, 141–146 (1993).
[CrossRef]

Zhou, P.

J. B. Adolph, R. W. Bertram, K. L. Yan, P. Zhou, R. de Leon, “Design and fabrication of multi-line inhomogeneous rejection filters,” in Inhomogeneous and Quasi-Inhomogeneous Optical Coatings, J. A. Dobrowolski, P. G. Verly, eds., Proc. SPIE2046, 141–146 (1993).
[CrossRef]

Zylberberg, Z.

Acta Math. (1)

G. W. Hill, “On the part of the motion of the lunar perigee which is a function of the mean motions of the Sun and Moon,” Acta Math. 8, 1–36 (1886).
[CrossRef]

Ann. Phys. Chem. (1)

O. Wiener, “Stehende Lichtwellen und die Schwingungrichtung polarisirten Lichtes,” Ann. Phys. Chem. 40, 203–243 (1890).

Appl. Opt. (3)

Appl. Phys. (1)

R. Alferness, “Analysis of optical propagation in thick holographic gratings,” Appl. Phys. 7, 29–33 (1975).
[CrossRef]

Appl. Phys. B (2)

M. G. Moharam, T. K. Gaylord, “Planar dielectric grating diffraction theory,” Appl. Phys. B 28, 1–14 (1982).
[CrossRef]

P. St. J. Russell, “Optics of Floquet-Block waves in dielectric gratings,” Appl. Phys. B 39, 231–246 (1986).
[CrossRef]

Bell Syst. Tech. J. (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

Comput. J. (1)

J. H. Wilkinson, “The calculation of eigenvectors of codiagonal matrices,” Comput. J. 1, 90–96 (1958).
[CrossRef]

J. Opt. (Paris) (2)

P. Sharlandjiev, T. Mateeva, “Normal incidence holographic mirrors by the characteristic matrix method. Numerical examples,” J. Opt. (Paris) 16, 185–189 (1985).
[CrossRef]

P. Connes, “Silver salts and standing waves: the history of interference colour photography,” J. Opt. (Paris) 18, 147–166 (1987).
[CrossRef]

J. Opt. Soc. Am. (6)

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

J. Phys. (Paris) (1)

G. Lippmann, “Sur la théorie de la photographie des coulars simples et compusées par la méthode interférentielle,” J. Phys. (Paris) 3, 97–107 (1894).

Phil. Mag. J. Sci. (1)

Lord Rayleigh, “On the maintenance of vibrations by forces of double frequency, and on the propagation of waves through a medium endowed with a periodic structure,” Phil. Mag. J. Sci. 24, 145–159 (1887).
[CrossRef]

Phys. Rev. D (1)

S. N. Biswas, K. Datta, R. P. Saxena, P. K. Srivastava, V. S. Varma, “The Hill determinant: an application to the anharmonic oscillator,” Phys. Rev. D 4, 3617–3620 (1971).
[CrossRef]

Proc. IEEE (1)

C. Elachi, “Waves in active and passive periodic structures: a review,” Proc. IEEE 64, 1666–1698 (1976).
[CrossRef]

Proc. Ind. Acad. Sci. (2)

C. V. Raman, N. S. N. Nath, “The diffraction of waves by high frequency sound waves. I, II,” Proc. Ind. Acad. Sci. 2, 406–420 (1935).

C. V. Raman, N. S. N. Nath, “The diffraction of waves by high frequency sound waves. III, IV,” Proc. Ind. Acad. Sci. 3, 75–84, 119–125 (1936).

Soc. Indust. Appl. Math. Rev. (1)

I. C. F. Ipsen, “Computing an eigenvector with inverse iteration,” Soc. Indust. Appl. Math. Rev. 39, 254–291 (1997).

Other (9)

H. A. Macleod, Thin-Film Optical Filters, 2nd ed. (McGraw-Hill, New York, 1989).

J. H. Wilkinson, “Inverse iteration in theory and practice,” in Symposium Mathematica (Instituto Nazionale de Alta Matematica, Rome, 1972), Vol. X, pp. 361–379.

T. W. Stone, B. J. Thompson, eds., Selected Papers on Holographic and Diffractive Lenses and Mirrors, Vol. MS 34 of SPIE Milestones Series (SPIE Optical Engineering Press, Bellingham, Wash., 1991).

J. B. Adolph, R. W. Bertram, K. L. Yan, P. Zhou, R. de Leon, “Design and fabrication of multi-line inhomogeneous rejection filters,” in Inhomogeneous and Quasi-Inhomogeneous Optical Coatings, J. A. Dobrowolski, P. G. Verly, eds., Proc. SPIE2046, 141–146 (1993).
[CrossRef]

K. W. Steijn, “Multicolor holographic recording in DuPont holographic recording film: determination of exposure conditions for color balance,” in Holographic Materials II, T. J. Trout, ed., Proc. SPIE2688, 123–134 (1996).
[CrossRef]

T. W. Stone, “Holographic optical elements,” Ph.D. dissertation (University of Rochester, Rochester, N.Y., 1986).

E. T. Whittaker, G. N. Watson, A Course of Modern Analysis, 4th ed. (Cambridge U. Press, Cambridge, 1927), Chap. 19, pp. 412–417.

W. Magnus, S. Winkler, “Hill’s equation,” in Interscience Tracts in Pure and Applied Mathematics, L. Bers, R. Courant, J. J. Stoker, eds. (Wiley Interscience, New York, 1966), Vol. 20, Chap. 1, pp. 3–10.

R. Jacobsson, “Light reflection from films of continuously varying refractive index,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1966), Vol. 5, Chap. 5, pp. 247–286.
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Index of refraction versus hologram depth for one period of the example problem.

Fig. 2
Fig. 2

Value of the Hill determinant, Δ(iμ), as a function of matrix size. It can be seen that, as matrix size increases, the values of μ found by solution of Eq. (12) become better approximations of the root of the equation Δ(iμ) = 0. The dashed line shows that a matrix of 2497 × 2497 is sufficient to ensure that the Hill determinant is within 10-6 of 0. The test wavelength is 350 nm.

Fig. 3
Fig. 3

Re(μ) versus free-space wavelength. The regions where Re(μ) > 0 correspond to the reflection bands of the hologram.

Fig. 4
Fig. 4

Im(μ) versus free-space wavelength. The flat regions of the curve correspond to the wavelengths for which Re(μ) is nonzero.

Fig. 5
Fig. 5

Reflection spectrum of a three-tone holographic interference filter, as calculated by Hill’s matrix technique.

Fig. 6
Fig. 6

Reflection spectrum of a three-tone holographic interference filter, as calculated by thin-film decomposition.

Fig. 7
Fig. 7

Difference between the reflection spectra as calculated by Hill’s matrix method and by thin-film decomposition. The result of subtracting the thin-film-decomposition spectrum from Hill’s matrix method spectrum is shown.

Equations (20)

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

z=l=1N flz,
2z2 Exz+k020 zExz=0,
2πΛ22ξ2 Exξ+k020 ξExξ=0.
2ξ2 Exξ+m=- θm expimξExξ=0,
Exξ=expμξϕξ=expμξl=- bl expilξ,
expμξl=-μ+il2bl expilξ+m=- θm expimξbl expilξ=0.
l=-μ+il2bl expilξ+m=- θm expimξbl expilξ=0.
l=-expilξμ+il2bl+m=- θmbl-m=0.
μ+il2bl+m=- θmbl-m=0
iμ-l2l2-θ0 bl-m=- θmbl-ml2-θ0=0
iμ+22-θ022-θ0-θ122-θ0-θ222-θ0-θ322-θ0-θ422-θ0-θ-112-θ0iμ+12-θ012-θ0-θ112-θ0-θ212-θ0-θ312-θ0-θ-2-θ0-θ-1-θ0iμ2-θ0-θ0-θ1-θ0-θ2-θ0-θ-312-θ0-θ-212-θ0-θ-112-θ0iμ-12-θ012-θ0-θ112-θ0-θ-422-θ0-θ-322-θ0-θ-222-θ0-θ-122-θ0iμ-22-θ022-θ0b-2b-1b0b1b2=0.
Δiμ=Δ0-sin2πiμsin2πθ0=0,
Exz=A expikz+B exp-ikz,
μ=ReiπarcsinΔ0sinπθ0+iθ0
M2m+1iμb-mb-1b0b1bm=0.
A-λˆIvk=skvk-1, k1,
Exz=1exp-ik0ncz+ρ exp+ik0nczz0exp-μ 2πΛ za-l=-mm bl exp-il 2πΛ z+ exp+μ 2πΛ za+l=-mm bl exp+il 2πΛ z0zLτ exp-ik0nsz-LzL,
1+ρ=a-l=-mm bl+a+l=-mm bl,τ=exp-μ 2πΛ La-l=-mm bl×exp-il 2πΛ L+ exp+μ 2πΛ La+×l=-mm bl exp+il 2πΛ L,ik0nc-1+ρ=-a-2πΛl=-mm blμ+il+a+2πΛl=-mm blμ+il,-ik0nsτ=-a- exp-μ 2πΛ L2πΛl=-mm blμ+ilexp-il 2πΛ L+ a+ exp+μ 2πΛ L2πΛ×l=-mm blμ+ilexp+il 2πΛ L.
z=a+j=13 j cos2πΛj z
θm=ΛLCMk02π210am=03/2m=±202/2m=±281/2m=±350all other m.

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