Abstract

The symmetry characteristics in various optical waveguide filters that employ the Bragg-scattering mechanism are generalized in terms of scattering coefficients and the coupling potential. A formalism known as the Zakharov–Shabat system of coupled-mode equations is employed, along with the associate Gel’fand–Levitan–Marchenko inverse-scattering formalism. The resulting symmetry characteristics between the coupling potential and the filter spectral response resemble the Fourier relationship. It is found, however, that complications exist because there are two scattering coefficients involved. Common variations of perturbation are then associated with special symmetry characteristics in the two scattering coefficients. The misinterpretation of physical inversion in the literature is also corrected.

© 1994 Optical Society of America

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  1. S. E. Miller, “Some theory and applications of periodically coupled waves,” Bell Syst. Tech. J. 48, 2189–2219 (1969).
  2. D. C. Flanders, H. Kogelnik, R. V. Schmidt, C. V. Shank, “Grating filters for thin-film optical waveguides,” Appl. Phys. Lett. 24, 194–196 (1974).
    [CrossRef]
  3. S. Valette, P. Gidon, J. P. Jadot, “New integrated multiplexer-demultiplexer realised on silicon substrate,” in Proceedings of the 4th European Conference on Integrated Optics (Scottish Electronics Technical Group, Glasgow, Scotland, 1987), pp. 145–147.
  4. C. Dragone, C. A. Edwards, R. C. Kistler, “Integrated optics N× Nmultiplexer on silicon,”IEEE Photon. Tech. Lett. 3, 896–899 (1991).
    [CrossRef]
  5. L. N. Binh, J. Livingstone, “A wide-band acoustooptic TE-TM mode converter using doubly-confined structure,” IEEE J. Quantum Electron. QE-16, 964–971 (1980).
    [CrossRef]
  6. R. C. Alferness, L. L. Buhl, “Low loss, wavelength tunable, waveguide electro-optic polarization controller for λ = 1.32 μm,” Appl. Phys. Lett. 47, 1137–1139 (1985).
    [CrossRef]
  7. M. Matsuhara, K. O. Hill, A. Watanabe, “Optical waveguide filters; synthesis,”J. Opt. Soc. Am. 65, 804–809 (1975).
    [CrossRef]
  8. H. Kogelnik, “Filter response of nonuniform almost-periodic structures,” Bell Syst. Tech. J. 55, 109–126 (1976).The third line from the bottom of p. 112 reads, “For a structure with symmetrical κ(z) and symmetric ϕ(z), we have ϕ′= ϕ′(–z),” but this replacement does not mean that we have physically reversed the filter. This situation corresponds to the case of Theorem 5 in the present paper.
  9. G.-H. Song, S.-Y. Shin, “Inverse-scattering problem for the coupled-wave equations when the reflection coefficient is a rational function,” Proc. IEEE 71, 266–268 (1983).
    [CrossRef]
  10. W. Warzanskyj, F. Heismann, R. C. Alferness, “Polarization-independent electro-optically tunable narrowband wavelength filter,” Appl. Phys. Lett. 53, 13–15 (1988).
    [CrossRef]
  11. C. M. Ragdale, D. Reid, D. J. Robbins, J. Buus, I. Bennion, “Narrowband fiber grating filters,” IEEEJ. Sel. Areas Commun. 8, 1146–1150 (1990).
    [CrossRef]
  12. F. Laurell, G. Arvidsson, “Frequency doubling in Ti:MgO:LiNbO3channel waveguides,” J. Opt. Soc. Am. B 5, 292–299 (1988).
    [CrossRef]
  13. D. A. Smith, A. d’Alessandro, J. E. Baran, H. Herrmann, “Source of sidelobe asymmetry in integrated acousto-optic filters,” Appl. Phys. Lett. 63, 814–816 (1993).
    [CrossRef]
  14. W. R. Trutna, D. W. Dolfi, C. A. Flory, “Anomalous sidelobes and birefringence apodization in acousto-optic tunable filters,” Opt. Lett. 18, 28–30 (1993).
    [CrossRef] [PubMed]
  15. G. H. Song, “Synthesis of corrugated waveguide filters by the Gel’fand–Levitan–Marchenko inverse-scattering method,” M.S. dissertation (Department of Electrical Engineering, Korea Advanced Institute of Science and Technology, Seoul, Korea, 1982).
  16. G. H. Song, S.-Y. Shin, “Design of corrugated waveguide filters by the Gel’fand–Levitan–Marchenko inverse-scattering method,” J. Opt. Soc. Am. A 2, 1905–1915 (1985).Below Eq. (A1) in that paper the phrase “a skew-Hermitian” should be corrected to “not a Hermitian.”
    [CrossRef]
  17. G. P. Bava, G. Ghione, “Inverse scattering for optical couplers. Exact solution of Marchenko equations,”J. Math. Phys. 25, 1900–1904 (1984).
    [CrossRef]
  18. A. W. Snyder, “Coupled-mode theory for optical fibers,”J. Opt. Soc. Am. 62, 1267–1277 (1972).
    [CrossRef]
  19. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974).In that book βμ is defined to be positive when it is real, whereas it is signed depending on the propagation direction in the present paper.
  20. V. E. Zakharov, A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).
  21. H. Kogelnik, “Theory of dielectric waveguides,” Chap. 2 of Integrated Optics, T. Tamir, ed. (Springer, New York, 1975).
    [CrossRef]
  22. R. E. Collin, Foundations for Microwave Engineering (McGraw-Hill, Tokyo, 1965), pp. 181–182.
  23. G. R. Fowles, Introduction to Modern Optics, 2nd ed. (Holt, Rinehart and Winston, New York, 1975).
  24. The integration has been replaced by integration over the real axis, since small coupling potentials [∫−∞∞|q(z)|dz<0.904] (Ref. 25) do not give any discrete eigenvalues and thus give no zeros of a(ζ) for Im ζ> 0, even for the non-Hermitian case of −q*.
  25. M. J. Ablowitz, D. J. Kaup, A. C. Newell, H. Segur, “The inverse-scattering transform—Fourier analysis for nonlinear problems,” Stud. Appl. Math. LIII, 249–315 (1974).Errata on p. 273: Eqs. (4.42c) and (4.42d) should have −1/2 in place of 1/2. Note that, ϕ¯, b, b¯, and L in that reference are defined to be equivalent to, −ϕ¯, b¯, b, and −L, respectively, in the present paper.
  26. A. Kar-Roy, C. S. Tsai, “Integrated acousto-optic tunable filters using bidirectional surface acoustic waves,” in Topical Meeting on Integrated Photonics Research, Vol. 10 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), paper ITuB4.
  27. H. A Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, N.J., 1984), p. 257 and 243–247.
  28. H. A. Haus, “Narrow-band optical channel-dropping filter,” J. Lightwave Technol. 10, 57–62 (1992).
    [CrossRef]
  29. Y. Yamamoto, C. S. Tsai, K. Esteghamat, “Guided-wave acoustooptic tunable filters using simple coupling weighting technique,” in Proceedings of the IEEE 1990 Ultrasonics Symposium (Institute of Electrical and Electronics Engineers, New York, 1990), Vol. 2, pp. 605–608.
  30. D. A. Smith, J. J. Johnson, “Surface-acoustic-wave directional coupler for apodization of integrated acousto-optic filters,”IEEE Trans. Ultrason. Ferroelectron. Freq. Control 40, 22–25 (1993).
    [CrossRef]
  31. H. Herrmann, S. Schmid, “Integrated acousto-optical mode-converters with weighted coupling using surface acoustic wave directional couplers,” Electron. Lett. 28, 979–980 (1992).
    [CrossRef]
  32. A. Kar-Roy, C. S. Tsai, “Low-sidelobe integrated acoustooptic tunable filter using focused surface acoustic waves,” in Topical Meeting on Integrated Photonics Research, Vol. 10 of the 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), paper ME2, pp. 90–91.
  33. H. Nishimoto, N. Shimosaka, T. Hosoi, Y. Yamamoto, “Low sidelobe acousto-optic tunable wavelength filter using weighted SAW slot waveguide,” presented at the1992 Institute of Electronics, Information, and Communication Engineers Spring Conference, March 24–27, 1992, Chiba Prefecture, Japan.
  34. G. H. Song, “Simulation of directional coupling of anisotropic surface-acoustic waves in acousto-optic filters,” Internal memorandum (Bellcore, Red Bank, N.J., 1994).

1993 (3)

D. A. Smith, A. d’Alessandro, J. E. Baran, H. Herrmann, “Source of sidelobe asymmetry in integrated acousto-optic filters,” Appl. Phys. Lett. 63, 814–816 (1993).
[CrossRef]

W. R. Trutna, D. W. Dolfi, C. A. Flory, “Anomalous sidelobes and birefringence apodization in acousto-optic tunable filters,” Opt. Lett. 18, 28–30 (1993).
[CrossRef] [PubMed]

D. A. Smith, J. J. Johnson, “Surface-acoustic-wave directional coupler for apodization of integrated acousto-optic filters,”IEEE Trans. Ultrason. Ferroelectron. Freq. Control 40, 22–25 (1993).
[CrossRef]

1992 (2)

H. Herrmann, S. Schmid, “Integrated acousto-optical mode-converters with weighted coupling using surface acoustic wave directional couplers,” Electron. Lett. 28, 979–980 (1992).
[CrossRef]

H. A. Haus, “Narrow-band optical channel-dropping filter,” J. Lightwave Technol. 10, 57–62 (1992).
[CrossRef]

1991 (1)

C. Dragone, C. A. Edwards, R. C. Kistler, “Integrated optics N× Nmultiplexer on silicon,”IEEE Photon. Tech. Lett. 3, 896–899 (1991).
[CrossRef]

1990 (1)

C. M. Ragdale, D. Reid, D. J. Robbins, J. Buus, I. Bennion, “Narrowband fiber grating filters,” IEEEJ. Sel. Areas Commun. 8, 1146–1150 (1990).
[CrossRef]

1988 (2)

F. Laurell, G. Arvidsson, “Frequency doubling in Ti:MgO:LiNbO3channel waveguides,” J. Opt. Soc. Am. B 5, 292–299 (1988).
[CrossRef]

W. Warzanskyj, F. Heismann, R. C. Alferness, “Polarization-independent electro-optically tunable narrowband wavelength filter,” Appl. Phys. Lett. 53, 13–15 (1988).
[CrossRef]

1985 (2)

1984 (1)

G. P. Bava, G. Ghione, “Inverse scattering for optical couplers. Exact solution of Marchenko equations,”J. Math. Phys. 25, 1900–1904 (1984).
[CrossRef]

1983 (1)

G.-H. Song, S.-Y. Shin, “Inverse-scattering problem for the coupled-wave equations when the reflection coefficient is a rational function,” Proc. IEEE 71, 266–268 (1983).
[CrossRef]

1980 (1)

L. N. Binh, J. Livingstone, “A wide-band acoustooptic TE-TM mode converter using doubly-confined structure,” IEEE J. Quantum Electron. QE-16, 964–971 (1980).
[CrossRef]

1976 (1)

H. Kogelnik, “Filter response of nonuniform almost-periodic structures,” Bell Syst. Tech. J. 55, 109–126 (1976).The third line from the bottom of p. 112 reads, “For a structure with symmetrical κ(z) and symmetric ϕ(z), we have ϕ′= ϕ′(–z),” but this replacement does not mean that we have physically reversed the filter. This situation corresponds to the case of Theorem 5 in the present paper.

1975 (1)

1974 (2)

D. C. Flanders, H. Kogelnik, R. V. Schmidt, C. V. Shank, “Grating filters for thin-film optical waveguides,” Appl. Phys. Lett. 24, 194–196 (1974).
[CrossRef]

M. J. Ablowitz, D. J. Kaup, A. C. Newell, H. Segur, “The inverse-scattering transform—Fourier analysis for nonlinear problems,” Stud. Appl. Math. LIII, 249–315 (1974).Errata on p. 273: Eqs. (4.42c) and (4.42d) should have −1/2 in place of 1/2. Note that, ϕ¯, b, b¯, and L in that reference are defined to be equivalent to, −ϕ¯, b¯, b, and −L, respectively, in the present paper.

1972 (2)

A. W. Snyder, “Coupled-mode theory for optical fibers,”J. Opt. Soc. Am. 62, 1267–1277 (1972).
[CrossRef]

V. E. Zakharov, A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

1969 (1)

S. E. Miller, “Some theory and applications of periodically coupled waves,” Bell Syst. Tech. J. 48, 2189–2219 (1969).

Ablowitz, M. J.

M. J. Ablowitz, D. J. Kaup, A. C. Newell, H. Segur, “The inverse-scattering transform—Fourier analysis for nonlinear problems,” Stud. Appl. Math. LIII, 249–315 (1974).Errata on p. 273: Eqs. (4.42c) and (4.42d) should have −1/2 in place of 1/2. Note that, ϕ¯, b, b¯, and L in that reference are defined to be equivalent to, −ϕ¯, b¯, b, and −L, respectively, in the present paper.

Alferness, R. C.

W. Warzanskyj, F. Heismann, R. C. Alferness, “Polarization-independent electro-optically tunable narrowband wavelength filter,” Appl. Phys. Lett. 53, 13–15 (1988).
[CrossRef]

R. C. Alferness, L. L. Buhl, “Low loss, wavelength tunable, waveguide electro-optic polarization controller for λ = 1.32 μm,” Appl. Phys. Lett. 47, 1137–1139 (1985).
[CrossRef]

Arvidsson, G.

Baran, J. E.

D. A. Smith, A. d’Alessandro, J. E. Baran, H. Herrmann, “Source of sidelobe asymmetry in integrated acousto-optic filters,” Appl. Phys. Lett. 63, 814–816 (1993).
[CrossRef]

Bava, G. P.

G. P. Bava, G. Ghione, “Inverse scattering for optical couplers. Exact solution of Marchenko equations,”J. Math. Phys. 25, 1900–1904 (1984).
[CrossRef]

Bennion, I.

C. M. Ragdale, D. Reid, D. J. Robbins, J. Buus, I. Bennion, “Narrowband fiber grating filters,” IEEEJ. Sel. Areas Commun. 8, 1146–1150 (1990).
[CrossRef]

Binh, L. N.

L. N. Binh, J. Livingstone, “A wide-band acoustooptic TE-TM mode converter using doubly-confined structure,” IEEE J. Quantum Electron. QE-16, 964–971 (1980).
[CrossRef]

Buhl, L. L.

R. C. Alferness, L. L. Buhl, “Low loss, wavelength tunable, waveguide electro-optic polarization controller for λ = 1.32 μm,” Appl. Phys. Lett. 47, 1137–1139 (1985).
[CrossRef]

Buus, J.

C. M. Ragdale, D. Reid, D. J. Robbins, J. Buus, I. Bennion, “Narrowband fiber grating filters,” IEEEJ. Sel. Areas Commun. 8, 1146–1150 (1990).
[CrossRef]

Collin, R. E.

R. E. Collin, Foundations for Microwave Engineering (McGraw-Hill, Tokyo, 1965), pp. 181–182.

d’Alessandro, A.

D. A. Smith, A. d’Alessandro, J. E. Baran, H. Herrmann, “Source of sidelobe asymmetry in integrated acousto-optic filters,” Appl. Phys. Lett. 63, 814–816 (1993).
[CrossRef]

Dolfi, D. W.

Dragone, C.

C. Dragone, C. A. Edwards, R. C. Kistler, “Integrated optics N× Nmultiplexer on silicon,”IEEE Photon. Tech. Lett. 3, 896–899 (1991).
[CrossRef]

Edwards, C. A.

C. Dragone, C. A. Edwards, R. C. Kistler, “Integrated optics N× Nmultiplexer on silicon,”IEEE Photon. Tech. Lett. 3, 896–899 (1991).
[CrossRef]

Esteghamat, K.

Y. Yamamoto, C. S. Tsai, K. Esteghamat, “Guided-wave acoustooptic tunable filters using simple coupling weighting technique,” in Proceedings of the IEEE 1990 Ultrasonics Symposium (Institute of Electrical and Electronics Engineers, New York, 1990), Vol. 2, pp. 605–608.

Flanders, D. C.

D. C. Flanders, H. Kogelnik, R. V. Schmidt, C. V. Shank, “Grating filters for thin-film optical waveguides,” Appl. Phys. Lett. 24, 194–196 (1974).
[CrossRef]

Flory, C. A.

Fowles, G. R.

G. R. Fowles, Introduction to Modern Optics, 2nd ed. (Holt, Rinehart and Winston, New York, 1975).

Ghione, G.

G. P. Bava, G. Ghione, “Inverse scattering for optical couplers. Exact solution of Marchenko equations,”J. Math. Phys. 25, 1900–1904 (1984).
[CrossRef]

Gidon, P.

S. Valette, P. Gidon, J. P. Jadot, “New integrated multiplexer-demultiplexer realised on silicon substrate,” in Proceedings of the 4th European Conference on Integrated Optics (Scottish Electronics Technical Group, Glasgow, Scotland, 1987), pp. 145–147.

Haus, H. A

H. A Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, N.J., 1984), p. 257 and 243–247.

Haus, H. A.

H. A. Haus, “Narrow-band optical channel-dropping filter,” J. Lightwave Technol. 10, 57–62 (1992).
[CrossRef]

Heismann, F.

W. Warzanskyj, F. Heismann, R. C. Alferness, “Polarization-independent electro-optically tunable narrowband wavelength filter,” Appl. Phys. Lett. 53, 13–15 (1988).
[CrossRef]

Herrmann, H.

D. A. Smith, A. d’Alessandro, J. E. Baran, H. Herrmann, “Source of sidelobe asymmetry in integrated acousto-optic filters,” Appl. Phys. Lett. 63, 814–816 (1993).
[CrossRef]

H. Herrmann, S. Schmid, “Integrated acousto-optical mode-converters with weighted coupling using surface acoustic wave directional couplers,” Electron. Lett. 28, 979–980 (1992).
[CrossRef]

Hill, K. O.

Hosoi, T.

H. Nishimoto, N. Shimosaka, T. Hosoi, Y. Yamamoto, “Low sidelobe acousto-optic tunable wavelength filter using weighted SAW slot waveguide,” presented at the1992 Institute of Electronics, Information, and Communication Engineers Spring Conference, March 24–27, 1992, Chiba Prefecture, Japan.

Jadot, J. P.

S. Valette, P. Gidon, J. P. Jadot, “New integrated multiplexer-demultiplexer realised on silicon substrate,” in Proceedings of the 4th European Conference on Integrated Optics (Scottish Electronics Technical Group, Glasgow, Scotland, 1987), pp. 145–147.

Johnson, J. J.

D. A. Smith, J. J. Johnson, “Surface-acoustic-wave directional coupler for apodization of integrated acousto-optic filters,”IEEE Trans. Ultrason. Ferroelectron. Freq. Control 40, 22–25 (1993).
[CrossRef]

Kar-Roy, A.

A. Kar-Roy, C. S. Tsai, “Integrated acousto-optic tunable filters using bidirectional surface acoustic waves,” in Topical Meeting on Integrated Photonics Research, Vol. 10 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), paper ITuB4.

A. Kar-Roy, C. S. Tsai, “Low-sidelobe integrated acoustooptic tunable filter using focused surface acoustic waves,” in Topical Meeting on Integrated Photonics Research, Vol. 10 of the 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), paper ME2, pp. 90–91.

Kaup, D. J.

M. J. Ablowitz, D. J. Kaup, A. C. Newell, H. Segur, “The inverse-scattering transform—Fourier analysis for nonlinear problems,” Stud. Appl. Math. LIII, 249–315 (1974).Errata on p. 273: Eqs. (4.42c) and (4.42d) should have −1/2 in place of 1/2. Note that, ϕ¯, b, b¯, and L in that reference are defined to be equivalent to, −ϕ¯, b¯, b, and −L, respectively, in the present paper.

Kistler, R. C.

C. Dragone, C. A. Edwards, R. C. Kistler, “Integrated optics N× Nmultiplexer on silicon,”IEEE Photon. Tech. Lett. 3, 896–899 (1991).
[CrossRef]

Kogelnik, H.

H. Kogelnik, “Filter response of nonuniform almost-periodic structures,” Bell Syst. Tech. J. 55, 109–126 (1976).The third line from the bottom of p. 112 reads, “For a structure with symmetrical κ(z) and symmetric ϕ(z), we have ϕ′= ϕ′(–z),” but this replacement does not mean that we have physically reversed the filter. This situation corresponds to the case of Theorem 5 in the present paper.

D. C. Flanders, H. Kogelnik, R. V. Schmidt, C. V. Shank, “Grating filters for thin-film optical waveguides,” Appl. Phys. Lett. 24, 194–196 (1974).
[CrossRef]

H. Kogelnik, “Theory of dielectric waveguides,” Chap. 2 of Integrated Optics, T. Tamir, ed. (Springer, New York, 1975).
[CrossRef]

Laurell, F.

Livingstone, J.

L. N. Binh, J. Livingstone, “A wide-band acoustooptic TE-TM mode converter using doubly-confined structure,” IEEE J. Quantum Electron. QE-16, 964–971 (1980).
[CrossRef]

Marcuse, D.

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974).In that book βμ is defined to be positive when it is real, whereas it is signed depending on the propagation direction in the present paper.

Matsuhara, M.

Miller, S. E.

S. E. Miller, “Some theory and applications of periodically coupled waves,” Bell Syst. Tech. J. 48, 2189–2219 (1969).

Newell, A. C.

M. J. Ablowitz, D. J. Kaup, A. C. Newell, H. Segur, “The inverse-scattering transform—Fourier analysis for nonlinear problems,” Stud. Appl. Math. LIII, 249–315 (1974).Errata on p. 273: Eqs. (4.42c) and (4.42d) should have −1/2 in place of 1/2. Note that, ϕ¯, b, b¯, and L in that reference are defined to be equivalent to, −ϕ¯, b¯, b, and −L, respectively, in the present paper.

Nishimoto, H.

H. Nishimoto, N. Shimosaka, T. Hosoi, Y. Yamamoto, “Low sidelobe acousto-optic tunable wavelength filter using weighted SAW slot waveguide,” presented at the1992 Institute of Electronics, Information, and Communication Engineers Spring Conference, March 24–27, 1992, Chiba Prefecture, Japan.

Ragdale, C. M.

C. M. Ragdale, D. Reid, D. J. Robbins, J. Buus, I. Bennion, “Narrowband fiber grating filters,” IEEEJ. Sel. Areas Commun. 8, 1146–1150 (1990).
[CrossRef]

Reid, D.

C. M. Ragdale, D. Reid, D. J. Robbins, J. Buus, I. Bennion, “Narrowband fiber grating filters,” IEEEJ. Sel. Areas Commun. 8, 1146–1150 (1990).
[CrossRef]

Robbins, D. J.

C. M. Ragdale, D. Reid, D. J. Robbins, J. Buus, I. Bennion, “Narrowband fiber grating filters,” IEEEJ. Sel. Areas Commun. 8, 1146–1150 (1990).
[CrossRef]

Schmid, S.

H. Herrmann, S. Schmid, “Integrated acousto-optical mode-converters with weighted coupling using surface acoustic wave directional couplers,” Electron. Lett. 28, 979–980 (1992).
[CrossRef]

Schmidt, R. V.

D. C. Flanders, H. Kogelnik, R. V. Schmidt, C. V. Shank, “Grating filters for thin-film optical waveguides,” Appl. Phys. Lett. 24, 194–196 (1974).
[CrossRef]

Segur, H.

M. J. Ablowitz, D. J. Kaup, A. C. Newell, H. Segur, “The inverse-scattering transform—Fourier analysis for nonlinear problems,” Stud. Appl. Math. LIII, 249–315 (1974).Errata on p. 273: Eqs. (4.42c) and (4.42d) should have −1/2 in place of 1/2. Note that, ϕ¯, b, b¯, and L in that reference are defined to be equivalent to, −ϕ¯, b¯, b, and −L, respectively, in the present paper.

Shabat, A. B.

V. E. Zakharov, A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

Shank, C. V.

D. C. Flanders, H. Kogelnik, R. V. Schmidt, C. V. Shank, “Grating filters for thin-film optical waveguides,” Appl. Phys. Lett. 24, 194–196 (1974).
[CrossRef]

Shimosaka, N.

H. Nishimoto, N. Shimosaka, T. Hosoi, Y. Yamamoto, “Low sidelobe acousto-optic tunable wavelength filter using weighted SAW slot waveguide,” presented at the1992 Institute of Electronics, Information, and Communication Engineers Spring Conference, March 24–27, 1992, Chiba Prefecture, Japan.

Shin, S.-Y.

Smith, D. A.

D. A. Smith, A. d’Alessandro, J. E. Baran, H. Herrmann, “Source of sidelobe asymmetry in integrated acousto-optic filters,” Appl. Phys. Lett. 63, 814–816 (1993).
[CrossRef]

D. A. Smith, J. J. Johnson, “Surface-acoustic-wave directional coupler for apodization of integrated acousto-optic filters,”IEEE Trans. Ultrason. Ferroelectron. Freq. Control 40, 22–25 (1993).
[CrossRef]

Snyder, A. W.

Song, G. H.

G. H. Song, S.-Y. Shin, “Design of corrugated waveguide filters by the Gel’fand–Levitan–Marchenko inverse-scattering method,” J. Opt. Soc. Am. A 2, 1905–1915 (1985).Below Eq. (A1) in that paper the phrase “a skew-Hermitian” should be corrected to “not a Hermitian.”
[CrossRef]

G. H. Song, “Synthesis of corrugated waveguide filters by the Gel’fand–Levitan–Marchenko inverse-scattering method,” M.S. dissertation (Department of Electrical Engineering, Korea Advanced Institute of Science and Technology, Seoul, Korea, 1982).

G. H. Song, “Simulation of directional coupling of anisotropic surface-acoustic waves in acousto-optic filters,” Internal memorandum (Bellcore, Red Bank, N.J., 1994).

Song, G.-H.

G.-H. Song, S.-Y. Shin, “Inverse-scattering problem for the coupled-wave equations when the reflection coefficient is a rational function,” Proc. IEEE 71, 266–268 (1983).
[CrossRef]

Trutna, W. R.

Tsai, C. S.

Y. Yamamoto, C. S. Tsai, K. Esteghamat, “Guided-wave acoustooptic tunable filters using simple coupling weighting technique,” in Proceedings of the IEEE 1990 Ultrasonics Symposium (Institute of Electrical and Electronics Engineers, New York, 1990), Vol. 2, pp. 605–608.

A. Kar-Roy, C. S. Tsai, “Integrated acousto-optic tunable filters using bidirectional surface acoustic waves,” in Topical Meeting on Integrated Photonics Research, Vol. 10 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), paper ITuB4.

A. Kar-Roy, C. S. Tsai, “Low-sidelobe integrated acoustooptic tunable filter using focused surface acoustic waves,” in Topical Meeting on Integrated Photonics Research, Vol. 10 of the 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), paper ME2, pp. 90–91.

Valette, S.

S. Valette, P. Gidon, J. P. Jadot, “New integrated multiplexer-demultiplexer realised on silicon substrate,” in Proceedings of the 4th European Conference on Integrated Optics (Scottish Electronics Technical Group, Glasgow, Scotland, 1987), pp. 145–147.

Warzanskyj, W.

W. Warzanskyj, F. Heismann, R. C. Alferness, “Polarization-independent electro-optically tunable narrowband wavelength filter,” Appl. Phys. Lett. 53, 13–15 (1988).
[CrossRef]

Watanabe, A.

Yamamoto, Y.

H. Nishimoto, N. Shimosaka, T. Hosoi, Y. Yamamoto, “Low sidelobe acousto-optic tunable wavelength filter using weighted SAW slot waveguide,” presented at the1992 Institute of Electronics, Information, and Communication Engineers Spring Conference, March 24–27, 1992, Chiba Prefecture, Japan.

Y. Yamamoto, C. S. Tsai, K. Esteghamat, “Guided-wave acoustooptic tunable filters using simple coupling weighting technique,” in Proceedings of the IEEE 1990 Ultrasonics Symposium (Institute of Electrical and Electronics Engineers, New York, 1990), Vol. 2, pp. 605–608.

Zakharov, V. E.

V. E. Zakharov, A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

Appl. Phys. Lett. (1)

D. A. Smith, A. d’Alessandro, J. E. Baran, H. Herrmann, “Source of sidelobe asymmetry in integrated acousto-optic filters,” Appl. Phys. Lett. 63, 814–816 (1993).
[CrossRef]

Appl. Phys. Lett. (3)

W. Warzanskyj, F. Heismann, R. C. Alferness, “Polarization-independent electro-optically tunable narrowband wavelength filter,” Appl. Phys. Lett. 53, 13–15 (1988).
[CrossRef]

D. C. Flanders, H. Kogelnik, R. V. Schmidt, C. V. Shank, “Grating filters for thin-film optical waveguides,” Appl. Phys. Lett. 24, 194–196 (1974).
[CrossRef]

R. C. Alferness, L. L. Buhl, “Low loss, wavelength tunable, waveguide electro-optic polarization controller for λ = 1.32 μm,” Appl. Phys. Lett. 47, 1137–1139 (1985).
[CrossRef]

Bell Syst. Tech. J. (2)

H. Kogelnik, “Filter response of nonuniform almost-periodic structures,” Bell Syst. Tech. J. 55, 109–126 (1976).The third line from the bottom of p. 112 reads, “For a structure with symmetrical κ(z) and symmetric ϕ(z), we have ϕ′= ϕ′(–z),” but this replacement does not mean that we have physically reversed the filter. This situation corresponds to the case of Theorem 5 in the present paper.

S. E. Miller, “Some theory and applications of periodically coupled waves,” Bell Syst. Tech. J. 48, 2189–2219 (1969).

Electron. Lett. (1)

H. Herrmann, S. Schmid, “Integrated acousto-optical mode-converters with weighted coupling using surface acoustic wave directional couplers,” Electron. Lett. 28, 979–980 (1992).
[CrossRef]

IEEE J. Quantum Electron. (1)

L. N. Binh, J. Livingstone, “A wide-band acoustooptic TE-TM mode converter using doubly-confined structure,” IEEE J. Quantum Electron. QE-16, 964–971 (1980).
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IEEE Photon. Tech. Lett. (1)

C. Dragone, C. A. Edwards, R. C. Kistler, “Integrated optics N× Nmultiplexer on silicon,”IEEE Photon. Tech. Lett. 3, 896–899 (1991).
[CrossRef]

IEEE Trans. Ultrason. Ferroelectron. Freq. Control (1)

D. A. Smith, J. J. Johnson, “Surface-acoustic-wave directional coupler for apodization of integrated acousto-optic filters,”IEEE Trans. Ultrason. Ferroelectron. Freq. Control 40, 22–25 (1993).
[CrossRef]

IEEEJ. Sel. Areas Commun. (1)

C. M. Ragdale, D. Reid, D. J. Robbins, J. Buus, I. Bennion, “Narrowband fiber grating filters,” IEEEJ. Sel. Areas Commun. 8, 1146–1150 (1990).
[CrossRef]

J. Lightwave Technol. (1)

H. A. Haus, “Narrow-band optical channel-dropping filter,” J. Lightwave Technol. 10, 57–62 (1992).
[CrossRef]

J. Math. Phys. (1)

G. P. Bava, G. Ghione, “Inverse scattering for optical couplers. Exact solution of Marchenko equations,”J. Math. Phys. 25, 1900–1904 (1984).
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J. Opt. Soc. Am. A (1)

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

Opt. Lett. (1)

Proc. IEEE (1)

G.-H. Song, S.-Y. Shin, “Inverse-scattering problem for the coupled-wave equations when the reflection coefficient is a rational function,” Proc. IEEE 71, 266–268 (1983).
[CrossRef]

Sov. Phys. JETP (1)

V. E. Zakharov, A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

Stud. Appl. Math. LIII (1)

M. J. Ablowitz, D. J. Kaup, A. C. Newell, H. Segur, “The inverse-scattering transform—Fourier analysis for nonlinear problems,” Stud. Appl. Math. LIII, 249–315 (1974).Errata on p. 273: Eqs. (4.42c) and (4.42d) should have −1/2 in place of 1/2. Note that, ϕ¯, b, b¯, and L in that reference are defined to be equivalent to, −ϕ¯, b¯, b, and −L, respectively, in the present paper.

Other (13)

A. Kar-Roy, C. S. Tsai, “Integrated acousto-optic tunable filters using bidirectional surface acoustic waves,” in Topical Meeting on Integrated Photonics Research, Vol. 10 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), paper ITuB4.

H. A Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, N.J., 1984), p. 257 and 243–247.

Y. Yamamoto, C. S. Tsai, K. Esteghamat, “Guided-wave acoustooptic tunable filters using simple coupling weighting technique,” in Proceedings of the IEEE 1990 Ultrasonics Symposium (Institute of Electrical and Electronics Engineers, New York, 1990), Vol. 2, pp. 605–608.

H. Kogelnik, “Theory of dielectric waveguides,” Chap. 2 of Integrated Optics, T. Tamir, ed. (Springer, New York, 1975).
[CrossRef]

R. E. Collin, Foundations for Microwave Engineering (McGraw-Hill, Tokyo, 1965), pp. 181–182.

G. R. Fowles, Introduction to Modern Optics, 2nd ed. (Holt, Rinehart and Winston, New York, 1975).

The integration has been replaced by integration over the real axis, since small coupling potentials [∫−∞∞|q(z)|dz<0.904] (Ref. 25) do not give any discrete eigenvalues and thus give no zeros of a(ζ) for Im ζ> 0, even for the non-Hermitian case of −q*.

A. Kar-Roy, C. S. Tsai, “Low-sidelobe integrated acoustooptic tunable filter using focused surface acoustic waves,” in Topical Meeting on Integrated Photonics Research, Vol. 10 of the 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), paper ME2, pp. 90–91.

H. Nishimoto, N. Shimosaka, T. Hosoi, Y. Yamamoto, “Low sidelobe acousto-optic tunable wavelength filter using weighted SAW slot waveguide,” presented at the1992 Institute of Electronics, Information, and Communication Engineers Spring Conference, March 24–27, 1992, Chiba Prefecture, Japan.

G. H. Song, “Simulation of directional coupling of anisotropic surface-acoustic waves in acousto-optic filters,” Internal memorandum (Bellcore, Red Bank, N.J., 1994).

S. Valette, P. Gidon, J. P. Jadot, “New integrated multiplexer-demultiplexer realised on silicon substrate,” in Proceedings of the 4th European Conference on Integrated Optics (Scottish Electronics Technical Group, Glasgow, Scotland, 1987), pp. 145–147.

G. H. Song, “Synthesis of corrugated waveguide filters by the Gel’fand–Levitan–Marchenko inverse-scattering method,” M.S. dissertation (Department of Electrical Engineering, Korea Advanced Institute of Science and Technology, Seoul, Korea, 1982).

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974).In that book βμ is defined to be positive when it is real, whereas it is signed depending on the propagation direction in the present paper.

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Figures (6)

Fig. 1
Fig. 1

Two dispersion curves β1(ω)and β2(ω) and the determination of ξ(ω): (a) contradirectional case and (b) codirectional case.

Fig. 2
Fig. 2

Translation of q(z) along z by z1 illustrates how the incoming and reflected waves of ψ(z,τ) change in the two-component scattering system.

Fig. 3
Fig. 3

Complex coupling potential in the case of a linearly chirped corrugated-waveguide filter.

Fig. 4
Fig. 4

Coupling potential in the case of a grating-assistedcoupler filter when the two waveguides are overly close in the middle. The Δβ(z, ω0) increases in the middle because of the close proximity of the two guides, whereas 2π/Λ is constant throughout.

Fig. 5
Fig. 5

Coupling potential in the case of an acousto-optic tunable filter on LiNbO3, where Ti is indiffused with a profile. The doping density of Ti is depicted by the gray scale. The Δβ(z, ω0) increases in the middle because of less diffused Ti in the middle, whereas the surface acoustic wave (SAW) propagation constant βl is still constant throughout. The phase θ(z) is similar to that of Fig. 4.

Fig. 6
Fig. 6

Schematic diagram of phase mismatch in the case of the apodized acousto-optic interaction profile of Ref. 29.

Equations (47)

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K μ ν = i ω | β μ | 4 s μ e ̂ μ * · ( 0 ) × ( e ̂ ν , t β μ + 1 0 e ̂ ν , z β μ * ) d 2 ρ ,
[ υ 1 ( z , ξ ) υ 2 ( z , ξ ) ] [ V 1 ( z , ω ) exp [ i z ξ ( z , ω ) d z ] V 2 ( z , ω ) exp [ i z ξ ( z , ω ) d z ] ] ,
2 ξ ( z , ω ) Δ β ( z , ω ) Δ β ( z , ω 0 ) ,
Δ β ( z , ω ) β 2 ( z , ω ) β 1 ( z , ω ) ,
d υ 1 ( z , ξ ) / d z + i ξ υ 1 ( z , ξ ) = q ( z ) υ 2 ( z , ξ ) ,
d υ 2 ( z , ξ ) / d z i ξ υ 2 ( z , ξ ) = ± q * ( z ) υ 1 ( z , ξ ) .
i κ 12 U ( z , t ) = K 12 ( z , t ) + R 12 ( z ) ,
U ( z , t ) 2 u ( z ) cos [ z Δ β ( z , ω 0 ) d z + θ ( z ) σ ω l t ] .
q ( z , t ) i κ 12 u ( z ) exp [ i θ ( z ) + i σ ω ι t ] .
q ( z ) = ± 2 L 2 * ( z , z ) b ( ξ ) a ( ξ ) exp ( i 2 ξ z ) d ξ π .
( d / d z + i ξ ) υ 1 * ( z , ξ ) = q ( z ) υ 2 * ( z , ξ ) ,
( d / d z i ξ ) υ 2 * ( z , ξ ) = ± q ( z ) υ 1 * ( z , ξ ) .
[ b ( ξ ) exp ( i ξ z ) a ( ξ ) exp ( i ξ z ) ] ψ ( z , ξ ) [ 0 exp ( i ξ z ) ] ,
[ b * ( ξ ) exp ( i ξ z ) a * ( ξ ) exp ( i ξ z ) ] ψ * ( z , ξ ) [ 0 exp ( i ξ z ) ]
( d / d z + i ξ ) υ 1 ( z , ξ ) = q * ( z ) υ 2 ( z , ξ ) ,
( d / d z i ξ ) υ 2 ( z , ξ ) = ± q ( z ) υ 1 ( z , ξ ) ,
( d / d z + i ξ ) [ υ 2 ( z , ξ ) ] = q ( z ) υ 1 ( z , ξ ) ,
( d / d z i ξ ) υ 1 ( z , ξ ) = ± q * ( z ) [ υ 2 ( z , ξ ) ] .
[ b ¯ ( ξ ) exp ( i ξ z ) a ( ξ ) exp ( i ξ z ) ] [ ϕ 2 ( z , ξ ) ϕ 1 ( z , ξ ) ] [ 0 exp ( i ξ z ) ]
[ ± L 2 * ( z , τ ) L 1 * ( z , τ ) ] [ F ( z τ ) 0 ] z [ L 1 ( z , y ) L 2 ( z , y ) ] × F ( y τ ) d y = 0 , τ < z ,
[ L 1 * ( z , τ ) ± [ L 2 * ( z , τ ) ] ] + [ 0 F ( z + τ ) ] + z [ L 2 ( z , y ) L 1 ( z , y ) ] F ( y + τ ) d y = 0 , z < τ .
( d / d z + i ξ ) υ 1 ( z , ξ ) = q ( z ) υ 2 ( z , ξ ) ,
( d / d z i ξ ) υ 2 ( z , ξ ) = ± q * ( z ) υ 1 ( z , ξ ) .
[ b ( ξ ) exp ( i ξ z ) ā ( ξ ) exp ( i ξ z ) ] ϕ ¯ ( z , ξ ) [ 0 exp ( i ξ z ) ]
Ψ ( z , τ ) ψ ( ζ , ξ ) exp ( i ξ τ ) d ξ 2 π .
[ d / d z + i ( ξ + η ) ] υ 1 ( z , ξ + η ) = q ( z ) υ 2 ( z , ξ + η ) ,
[ d / d z i ( ξ + η ) ] υ 2 ( z , ξ + η ) = ± q * ( z ) υ 1 ( z , ξ + η ) .
β 2 ( ω 0 ) β 1 ( ω 0 ) β 2 ( ω 0 ) β 1 ( ω 0 ) + 2 η
ω 0 ω 0 + 2 η / [ d β 2 / d ω d β 1 / d ω ] ω 0 .
d θ ( z ) / d z 2 π / Λ ( z ) Δ β ( z , ω 0 ) .
U ( z , t ) = 2 u ( z ) cos [ z β l ( z , ω l ) d z ω l t ] ,
q ( z , t ) = { i κ 12 u ( z ) exp [ i θ ( z ) i ω l t ] L < z < 0 i κ 12 u ( z ) exp [ i θ ( z ) + i ω l t ] 0 < z < L 0 otherwise .
[ i ( d / d z ) i q ( z ) i r ( z ) i ( d / d z ) ] [ υ 1 ( z ) υ 2 ( z ) ] = ζ [ υ 1 ( z ) υ 2 ( z ) ]
[ exp ( i ζ z ) 0 ] ϕ ( z , ζ ) , [ 0 exp ( i ζ z ) ] ϕ ¯ ( z , ζ )
ψ ¯ ( z , ζ ) [ exp ( i ζ z ) 0 ] , ψ ( z , ζ ) [ 0 exp ( i ζ z ) ]
ψ ( z , ζ ) = a ( ζ ) ϕ ¯ ( z , ζ ) + b ( ζ ) ϕ ( z , ζ ) ,
ψ ¯ ( z , ζ ) = ā ( ζ ) ϕ ( z , ζ ) + b ¯ ( ζ ) ϕ ¯ ( z , ζ ) .
ā ( ζ ) a ( ζ ) b ¯ ( ζ ) b ( ζ ) = 1 .
ϕ ( z , ζ ) = a ( ζ ) ψ ¯ ( z , ζ ) b ¯ ( ζ ) ψ ( z , ζ ) ,
ϕ ¯ ( z , ζ ) = ā ( ζ ) ψ ( z , ζ ) b ( ζ ) ψ ¯ ( z , ζ ) .
[ ± L 2 * ( z , τ ) L 1 * ( z , τ ) ] + [ G ( z + τ ) 0 ] + z [ L 1 ( z , y ) L 2 ( z , y ) ] G ( y + τ ) d y = 0 , τ < z ,
[ K 2 * ( z , τ ) ± K 1 * ( z , τ ) ] + [ 0 G ( z + τ ) ] + z [ K 1 ( z , y ) K 2 ( z , y ) ] F ( y + τ ) d y = 0 , z < τ ,
q ( z ) = ± 2 L 2 * ( z , z ) = 2 K 1 ( z , z ) .
V ( z 1 , ω ) = T ( z 1 , z 2 ; ξ ) V ( z 2 , ω ) , z 1 < z 2 .
T ( z 1 , z 2 ; ξ ) = [ ā ( ξ ) b ( ξ ) b ¯ ( ξ ) a ( ξ ) ]
a ( ξ ) = { i ξ Δ z sinc [ ( ξ 2 | q 0 | 2 ) 1 / 2 Δ z ] + cos [ ( ξ 2 | q 0 | 2 ) 1 / 2 Δ z ] } exp ( i ξ Δ z ) ,
b ( ξ ) = q 0 Δ z sinc [ ( ξ 2 | q 0 | 2 ) 1 / 2 Δ z ] exp ( i ξ 2 z 0 + i Ω t ) ,

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