OSA's Digital Library

Applied Optics

Applied Optics

APPLICATIONS-CENTERED RESEARCH IN OPTICS

  • Vol. 43, Iss. 1 — Jan. 1, 2004
  • pp: 88–96

Analysis of multitone holographic interference filters by use of a sparse Hill matrix method

Damon W. Diehl and Nicholas George  »View Author Affiliations


Applied Optics, Vol. 43, Issue 1, pp. 88-96 (2004)
http://dx.doi.org/10.1364/AO.43.000088


View Full Text Article

Enhanced HTML    Acrobat PDF (138 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

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

OCIS Codes
(000.3860) General : Mathematical methods in physics
(090.2890) Holography : Holographic optical elements
(090.4220) Holography : Multiplex holography
(260.2110) Physical optics : Electromagnetic optics
(310.6860) Thin films : Thin films, optical properties
(350.2460) Other areas of optics : Filters, interference

History
Original Manuscript: June 6, 2003
Revised Manuscript: September 2, 2003
Published: January 1, 2004

Citation
Damon W. Diehl and Nicholas George, "Analysis of multitone holographic interference filters by use of a sparse Hill matrix method," Appl. Opt. 43, 88-96 (2004)
http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-43-1-88


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  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]
  30. X. Ning, “Analysis of multiplexed-reflection holographic gratings,” J. Opt. Soc. Am. A 7, 1436–1440 (1990). [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]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.


« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited