Abstract

When unity reflectance is approached, the Fourier-transform method of calculating the reflectance spectrum of an optical grating modulated by a slowly varying envelope becomes unacceptably inaccurate. The modified Fourier transform method of Bovard [Appl. Opt. 29, 24 (1990)] can achieve complete accuracy for quarter-wave gratings. We report herein the extension of Bovard’s method to non-quarter-wave gratings. We demonstrate the accurate deployment of our simplified modified Fourier-transform method to apodized linear gratings and optically apodized nonlinear gratings.

© 2002 Optical Society of America

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References

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  1. B. G. Bovard, “Rugate filter design: the modified Fourier transform technique,” Appl. Opt. 29, 24–30 (1990).
    [CrossRef] [PubMed]
  2. E. Delano, “Fourier synthesis of multilayer filters,” J. Opt. Soc. Am. 57, 1529–1533 (1967).
    [CrossRef]
  3. B. G. Bovard, “Rugate filter theory: an overview,” Appl. Opt. 32, 5427–5442 (1993).
    [CrossRef] [PubMed]
  4. B. G. Bovard, “Fourier transform technique applied to quarterwave optical coatings,” Appl. Opt. 27, 3062–3063 (1988).
    [CrossRef] [PubMed]
  5. P. G. Verly, J. A. Dobrowolski, “Iterative correction process for optical thin film synthesis with the Fourier transform method,” Appl. Opt. 29, 3672–3684 (1990).
    [CrossRef] [PubMed]
  6. J. A. Dobrowolski, “Optical thin film synthesis program based on the use of Fourier transforms,” Appl. Opt. 17, 3039–3050 (1978).
    [CrossRef] [PubMed]
  7. P. G. Verly, “Fourier transform technique with frequency filtering for optical thin-film design,” Appl. Opt. 34, 688–694 (1995).
    [CrossRef] [PubMed]
  8. H. Takata, M. Yamada, Y. Yamane, M. Ahmed, “A Bessel function-based design method of a periodic multireflection optical filters,” Electron. Commun. Jpn. Part 2 Electron. 81, 19–29 (1998).
    [CrossRef]
  9. P. G. Verly, J. A. Dobrowolski, W. J. Wild, R. L. Burton, “Synthesis of high rejection filters with the Fourier transform method,” Appl. Opt. 28, 2864–2875 (1989).
    [CrossRef] [PubMed]
  10. P. G. Verly, J. A. Dobrowolski, R. R. Willey, “Fourier-transform method for the design of wideband antireflection coatings,” Appl. Opt. 31, 3836–3846 (1992).
    [CrossRef] [PubMed]
  11. G. P. Agrawal, Fiber-Optic Communication Systems, 2nd ed. (Wiley, New York, 1997).
  12. P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).
  13. L. Brzozowski, E. H. Sargent, “Optical signal processing using nonlinear distributed feedback structures,” IEEE J. Quantum Electron. 36, 550–555 (2000).
    [CrossRef]
  14. L. Brzozowski, E. H. Sargent, “Nonlinear distributed feedback structures as passive optical limiter,” J. Opt. Soc. Am. B 17, 1360–1365 (2000).
    [CrossRef]

2000

L. Brzozowski, E. H. Sargent, “Optical signal processing using nonlinear distributed feedback structures,” IEEE J. Quantum Electron. 36, 550–555 (2000).
[CrossRef]

L. Brzozowski, E. H. Sargent, “Nonlinear distributed feedback structures as passive optical limiter,” J. Opt. Soc. Am. B 17, 1360–1365 (2000).
[CrossRef]

1998

H. Takata, M. Yamada, Y. Yamane, M. Ahmed, “A Bessel function-based design method of a periodic multireflection optical filters,” Electron. Commun. Jpn. Part 2 Electron. 81, 19–29 (1998).
[CrossRef]

1995

1993

1992

1990

1989

1988

1978

1967

Agrawal, G. P.

G. P. Agrawal, Fiber-Optic Communication Systems, 2nd ed. (Wiley, New York, 1997).

Ahmed, M.

H. Takata, M. Yamada, Y. Yamane, M. Ahmed, “A Bessel function-based design method of a periodic multireflection optical filters,” Electron. Commun. Jpn. Part 2 Electron. 81, 19–29 (1998).
[CrossRef]

Bovard, B. G.

Brzozowski, L.

L. Brzozowski, E. H. Sargent, “Nonlinear distributed feedback structures as passive optical limiter,” J. Opt. Soc. Am. B 17, 1360–1365 (2000).
[CrossRef]

L. Brzozowski, E. H. Sargent, “Optical signal processing using nonlinear distributed feedback structures,” IEEE J. Quantum Electron. 36, 550–555 (2000).
[CrossRef]

Burton, R. L.

Delano, E.

Dobrowolski, J. A.

Sargent, E. H.

L. Brzozowski, E. H. Sargent, “Optical signal processing using nonlinear distributed feedback structures,” IEEE J. Quantum Electron. 36, 550–555 (2000).
[CrossRef]

L. Brzozowski, E. H. Sargent, “Nonlinear distributed feedback structures as passive optical limiter,” J. Opt. Soc. Am. B 17, 1360–1365 (2000).
[CrossRef]

Takata, H.

H. Takata, M. Yamada, Y. Yamane, M. Ahmed, “A Bessel function-based design method of a periodic multireflection optical filters,” Electron. Commun. Jpn. Part 2 Electron. 81, 19–29 (1998).
[CrossRef]

Verly, P. G.

Wild, W. J.

Willey, R. R.

Yamada, M.

H. Takata, M. Yamada, Y. Yamane, M. Ahmed, “A Bessel function-based design method of a periodic multireflection optical filters,” Electron. Commun. Jpn. Part 2 Electron. 81, 19–29 (1998).
[CrossRef]

Yamane, Y.

H. Takata, M. Yamada, Y. Yamane, M. Ahmed, “A Bessel function-based design method of a periodic multireflection optical filters,” Electron. Commun. Jpn. Part 2 Electron. 81, 19–29 (1998).
[CrossRef]

Yeh, P.

P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).

Appl. Opt.

Electron. Commun. Jpn. Part 2 Electron.

H. Takata, M. Yamada, Y. Yamane, M. Ahmed, “A Bessel function-based design method of a periodic multireflection optical filters,” Electron. Commun. Jpn. Part 2 Electron. 81, 19–29 (1998).
[CrossRef]

IEEE J. Quantum Electron.

L. Brzozowski, E. H. Sargent, “Optical signal processing using nonlinear distributed feedback structures,” IEEE J. Quantum Electron. 36, 550–555 (2000).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. B

Other

G. P. Agrawal, Fiber-Optic Communication Systems, 2nd ed. (Wiley, New York, 1997).

P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).

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Figures (7)

Fig. 1
Fig. 1

Periodic stack consisting of N periods of alternating layers H with high index, n h and L with low index, n l . The incident and substrate media are assumed to have refractive index Hn h . x is the centered double optical thickness. The special case a = b describes the quarter-wave stack.

Fig. 2
Fig. 2

Comparison of the reflectance of a non-quarter-wave stack as computed by the TMM and the FT methods. The layers have thicknesses a = 250 nm and b = 200 nm, with N = 50, n h = 1.5, and n l = 1.4.

Fig. 3
Fig. 3

Comparison of the reflectance of a non-quarter-wave stack as computed by the TMM and the MFT methods. The layers have thicknesses b = 200 nm and a = 250 nm, with N = 50, n h = 1.5, and n l = 1.4.

Fig. 4
Fig. 4

Comparison of the reflectance of a non-quarter-wave stack as computed by the TMM and the MFT methods. The layers have thicknesses b = 100 nm and a = 250 nm, with N = 50, n h = 1.5, and n l = 1.4.

Fig. 5
Fig. 5

Comparison of apodized-grating reflectance calculated by use of the simplified MFT, FT, and recursion (exact) methods. (a) Hanning window, (b) Hamming window, (c) Kaiser (β = 5), window, and (d) Gaussian window.

Fig. 6
Fig. 6

(a) Intensity profile I(z) inside the nonlinear structure with n 0 = 1.45, n nl = 0.075, and I out = 0.05 (top); and (b) N = 100 periods and the corresponding refractive-index profile.

Fig. 7
Fig. 7

Reflectance of the nonlinear grating of Fig. 6 and of a linear grating (quarter-wave) that has n 0 = 1.45 and index difference Δ = 0.15 as calculated with the MFT method.

Equations (18)

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rx=r0m=-pp-1δx-4m+34σ0-δx-4m+14σ0,
QνexpiΦν=-+rxexp-2iπvxdx,
v=2σ0πcosh-1cosh r0 sinπσ2σ0,
|Qν|=-+rxexp-2iπνxdx.
|Qν|=r0m=-pp-1exp-2iπ 4m+3v4σ0-exp-2iπ 4m+1v4σ0.
R=exp2|Q|-1exp2|Q|+12.
R=|Q|21+|Q|2.
rx=r0m=-pp-1δx-2a+b+2ma+b-δx-b+2ma+b.
a+b=λ1/2,
a1=b1=λ1/4.
|Qν|=r0m=-pp-1exp-2iπν2a+b+2ma+b-exp-2iπνb+2ma+b.
nx=Cx+n¯,
Napodx=ExCx+n¯.
rapodx=½ln Napodx=½ExCx+ExCxExCx+n¯.since ExCx n¯½ExCxn¯+ExCxn¯,slowly varying envelope Ex½ExCxn¯.
n=n0+nnlI,
Ix=2I1x-Iout,
I1x=1+cos4IoutnnlL-xΛn02 cos4IoutnnlL-xΛn0 Iout
Icontrol=I10=121cos4Iouta+1 Iout,

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