Abstract

We present an efficient and accurate method for synthesis of optical thin-film structures. The method is based on a differential inverse-scattering algorithm and considers therefore both phase and amplitude reflectance data. We apply the algorithm to the synthesis of filters with arbitrary index layers and two-material filters consisting of only high- and low-index layers. The layered structure is approximated by a stack of discrete reflectors with equal distance between all reflectors. This mirror stack is in turn determined from the desired, complex reflection spectrum by a layer-peeling inverse-scattering algorithm. The complexity of the design algorithm is approximately the same as that of the forward problem of computing the spectrum from a known structure.

© 2001 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. A. Dobrowolski, “Numerical methods for optical thin films,” Opt. Photon. News (June1997), pp. 24–33.
  2. N. Matuschek, F. X. Kärtner, U. Keller, “Analytical design of double-chirped mirrors with custom-tailored dispersion characteristics,” IEEE J. Quantum Electron. 35, 129–137 (1999).
    [CrossRef]
  3. S. R. A. Dods, Z. Zhang, M. Ogura, “Highly dispersive mirror in Ta2O5/SiO2 for femtosecond lasers designed by inverse spectral theory,” Appl. Opt. 38, 4711–4719 (1999).
    [CrossRef]
  4. See, for example, commercial thin-film design software on the Internet: http://www.qis.net/~bnichols/opticsnotes/thincad.htm .
  5. J. A. Dobrowolski, D. Lowe, “Optical thin film synthesis program based on the use of Fourier transforms,” Appl. Opt. 17, 3039–3050 (1978).
    [CrossRef] [PubMed]
  6. J. A. Dobrowolski, S. H. C. Piotrowski, “Refractive index as a variable in the numerical design of optical thin-film systems,” Appl. Opt. 21, 1502–1511 (1997).
    [CrossRef]
  7. A. M. Bruckstein, T. Kailath, “Inverse scattering for discrete transmission-line models,” SIAM (Soc. Ind. Appl. Math.) Rev. 29, 359–389 (1987).
  8. A. M. Bruckstein, B. C. Levy, T. Kailath, “Differential methods in inverse scattering,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 45, 312–335 (1995).
    [CrossRef]
  9. H. A. Macleod, Thin-Film Optical Filters (Adam Hilger, Bristol, UK, 1985).
  10. J. G. Proakis, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1996).
  11. R. Feced, M. N. Zervas, M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform fibre Bragg gratings,” J. Quantum Electron. 35, 1105–1115 (1999).
    [CrossRef]

1999 (3)

N. Matuschek, F. X. Kärtner, U. Keller, “Analytical design of double-chirped mirrors with custom-tailored dispersion characteristics,” IEEE J. Quantum Electron. 35, 129–137 (1999).
[CrossRef]

S. R. A. Dods, Z. Zhang, M. Ogura, “Highly dispersive mirror in Ta2O5/SiO2 for femtosecond lasers designed by inverse spectral theory,” Appl. Opt. 38, 4711–4719 (1999).
[CrossRef]

R. Feced, M. N. Zervas, M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform fibre Bragg gratings,” J. Quantum Electron. 35, 1105–1115 (1999).
[CrossRef]

1997 (2)

1995 (1)

A. M. Bruckstein, B. C. Levy, T. Kailath, “Differential methods in inverse scattering,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 45, 312–335 (1995).
[CrossRef]

1987 (1)

A. M. Bruckstein, T. Kailath, “Inverse scattering for discrete transmission-line models,” SIAM (Soc. Ind. Appl. Math.) Rev. 29, 359–389 (1987).

1978 (1)

Bruckstein, A. M.

A. M. Bruckstein, B. C. Levy, T. Kailath, “Differential methods in inverse scattering,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 45, 312–335 (1995).
[CrossRef]

A. M. Bruckstein, T. Kailath, “Inverse scattering for discrete transmission-line models,” SIAM (Soc. Ind. Appl. Math.) Rev. 29, 359–389 (1987).

Dobrowolski, J. A.

Dods, S. R. A.

Feced, R.

R. Feced, M. N. Zervas, M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform fibre Bragg gratings,” J. Quantum Electron. 35, 1105–1115 (1999).
[CrossRef]

Kailath, T.

A. M. Bruckstein, B. C. Levy, T. Kailath, “Differential methods in inverse scattering,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 45, 312–335 (1995).
[CrossRef]

A. M. Bruckstein, T. Kailath, “Inverse scattering for discrete transmission-line models,” SIAM (Soc. Ind. Appl. Math.) Rev. 29, 359–389 (1987).

Kärtner, F. X.

N. Matuschek, F. X. Kärtner, U. Keller, “Analytical design of double-chirped mirrors with custom-tailored dispersion characteristics,” IEEE J. Quantum Electron. 35, 129–137 (1999).
[CrossRef]

Keller, U.

N. Matuschek, F. X. Kärtner, U. Keller, “Analytical design of double-chirped mirrors with custom-tailored dispersion characteristics,” IEEE J. Quantum Electron. 35, 129–137 (1999).
[CrossRef]

Levy, B. C.

A. M. Bruckstein, B. C. Levy, T. Kailath, “Differential methods in inverse scattering,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 45, 312–335 (1995).
[CrossRef]

Lowe, D.

Macleod, H. A.

H. A. Macleod, Thin-Film Optical Filters (Adam Hilger, Bristol, UK, 1985).

Matuschek, N.

N. Matuschek, F. X. Kärtner, U. Keller, “Analytical design of double-chirped mirrors with custom-tailored dispersion characteristics,” IEEE J. Quantum Electron. 35, 129–137 (1999).
[CrossRef]

Muriel, M. A.

R. Feced, M. N. Zervas, M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform fibre Bragg gratings,” J. Quantum Electron. 35, 1105–1115 (1999).
[CrossRef]

Ogura, M.

Piotrowski, S. H. C.

Proakis, J. G.

J. G. Proakis, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1996).

Zervas, M. N.

R. Feced, M. N. Zervas, M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform fibre Bragg gratings,” J. Quantum Electron. 35, 1105–1115 (1999).
[CrossRef]

Zhang, Z.

Appl. Opt. (3)

IEEE J. Quantum Electron. (1)

N. Matuschek, F. X. Kärtner, U. Keller, “Analytical design of double-chirped mirrors with custom-tailored dispersion characteristics,” IEEE J. Quantum Electron. 35, 129–137 (1999).
[CrossRef]

J. Quantum Electron. (1)

R. Feced, M. N. Zervas, M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform fibre Bragg gratings,” J. Quantum Electron. 35, 1105–1115 (1999).
[CrossRef]

Opt. Photon. News (1)

J. A. Dobrowolski, “Numerical methods for optical thin films,” Opt. Photon. News (June1997), pp. 24–33.

SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. (1)

A. M. Bruckstein, B. C. Levy, T. Kailath, “Differential methods in inverse scattering,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 45, 312–335 (1995).
[CrossRef]

SIAM (Soc. Ind. Appl. Math.) Rev. (1)

A. M. Bruckstein, T. Kailath, “Inverse scattering for discrete transmission-line models,” SIAM (Soc. Ind. Appl. Math.) Rev. 29, 359–389 (1987).

Other (3)

H. A. Macleod, Thin-Film Optical Filters (Adam Hilger, Bristol, UK, 1985).

J. G. Proakis, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1996).

See, for example, commercial thin-film design software on the Internet: http://www.qis.net/~bnichols/opticsnotes/thincad.htm .

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 (10)

Fig. 1
Fig. 1

Stack of discrete reflectors. The distance between the reflectors is d, and the complex fields before the jth section are A j and B j .

Fig. 2
Fig. 2

Refractive-index profile of the thin-film filter. The defined Bragg period, or unit cell, is the region between the dotted vertical lines. The discrete reflectors ρ j are indicated below the profile.

Fig. 3
Fig. 3

Refractive-index profile of the inhomogeneous layer nondispersive bandpass filter.

Fig. 4
Fig. 4

Layer thicknesses of the two-material nondispersive bandpass filter. Dots, thicknesses of the low-index layers (d l,j ); dotted curve, five times the thicknesses of the high-index layers (d h,j ).

Fig. 5
Fig. 5

Reflectivity of the nondispersive bandpass filter. Dashed curve, spectrum of the inhomogeneous layer filter; solid curve, spectrum of the two-material filter. The spectra are shown in both linear and logarithmic scales.

Fig. 6
Fig. 6

Group-delay dispersion for the nondispersive bandpass filters. Dashed curve, dispersion spectrum of the inhomogeneous-layer filter; solid curve, dispersion of the two-material filter.

Fig. 7
Fig. 7

Refractive-index profile of the inhomogeneous layer dispersive bandpass filter.

Fig. 8
Fig. 8

Layer thicknesses of the two-material dispersive bandpass filter. Dots, thicknesses of the low-index layers (d l,j ); dotted curve, five times the thicknesses of the high-index layers (d h,j ).

Fig. 9
Fig. 9

Reflectivity of the dispersive bandpass filter. Dashed curve, spectrum of the inhomogeneous layer filter; solid curve, spectrum of the two-material filter. The spectra are given in both linear and logarithmic scales.

Fig. 10
Fig. 10

Group-delay dispersion of the dispersive bandpass filters. Dashed curve, dispersion spectrum of the inhomogeneous layer filter; solid curve, dispersion of the two-material filter.

Equations (13)

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

A1kB1k=1rk.
Td=expikd00exp-ikd
Tρ,j=1-|ρj|2-1/21-ρj*-ρj1
ρ1=dπperiodB1kA1kdk  or ρ1=1Mm=1MB1A1m,
B2kA2k=exp-i2kdB1k/A1k-ρ11-ρ1*B1k/A1k.
qj=-1dρj*|ρj|arctanh|ρj|.
|ρj|<nh-nl/nh+nl.
rj=2ir sin φh,jexp-iφh,j-r2 exp+iφh,j,
|ρj|=|rj|k=kB=2r sin φh,j1+r4-2r2 cos2φh,j1/2k=kB,
φr,j=arg rj=arctantanφh,j1+r21-r2,
φj-φj-1+2kd=φr,j-φr,j-1+2φh,j-1+2φl,j-1
|ρj|=2r sin φh,j,
2φl,j=φj+1-φj+m×2π-φh,j-φh,j+1.

Metrics