Abstract

This paper describes the adaptation of Rouard’s method, a familiar computational technique used in thin-film coating design, to the analysis of waveguide diffraction gratings. The approach can be used to generate the spectral and the angular characteristics of arbitrary straight-line gratings (periodic or nonperiodic), is easy to implement on a computer, and provides an intuitively appealing picture of the operation of such gratings. The application of the thin-film method is extended to periodic gratings as well as to gratings having either a linear or a quadratic chirp in period. Specific examples are used to compare the results of the thin-film computational method with those predicted by previous authors.

© 1985 Optical Society of America

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References

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  1. R. W. Gruhlke, D. G. Hall, “Comparison of two approaches to the waveguide scattering problem: TM polarization,” Appl. Opt. 23, 127 (1984).
    [CrossRef] [PubMed]
  2. H. Kogelnik, in Integrated Optics, T. Tamir, ed., Vol. 7 of Topics in Applied Physics (Springer-Verlag, New York, 1975), pp. 66–79.
    [CrossRef]
  3. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974), pp. 95–126, 132–145.
  4. W. Streifer, D. R. Scifres, R. D. Burnham, “Coupling coefficients for DFB single- and double-heterostructure diode lasers,” IEEE J. Quantum Electron. QE-11, 867 (1975).
    [CrossRef]
  5. W. Streifer, “TM-mode coupling coefficients in guided-wave DFB lasers,” IEEE J. Quantum Electron. QE-12, 74 (1976).
    [CrossRef]
  6. T. Tsai, H. Tuan, “Reflection and scattering by a single groove in integrated optics,” IEEE J. Quantum Electron. QE-10, 326 (1974).
    [CrossRef]
  7. Y. Yamamoto, T. Kamiya, H. Yanai, “Improved coupled mode analysis of corrugated waveguides and lasers,” IEEE J. Quantum Electron. QE-14, 245 (1978).
    [CrossRef]
  8. A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. QE-9, 919 (1973).
    [CrossRef]
  9. J. Marcou, N. Gremillet, G. Thomin, “Polarization conversion by Bragg deflection in isotropic planar integrated optics waveguides; part I: theoretical study,” Opt. Commun. 32, 63 (1980).
    [CrossRef]
  10. N. Gremillet, G. Thomin, J. Marcou, “Polarization conversion by Bragg deflection in isotropic planar integrated optics waveguides; part II: experimental study,” Opt. Commun. 32, 69 (1980).
    [CrossRef]
  11. S. R. Seshadri, “TE–TE coupling at oblique incidence in a periodic dielectric waveguide,” Appl. Phys. 25, 211 (1981).
    [CrossRef]
  12. G. I. Stegeman, D. Sarid, J. J. Burke, D. G. Hall, “Scattering of guided waves by surface periodic gratings for arbitrary angles of incidence: perturbation theory and implications to normal-mode analysis,” J. Opt. Soc. Am. 71, 1497 (1981).
    [CrossRef]
  13. J. VanRoey, P. E. Lagasse, “Coupled wave analysis of obliquely incident waves in thin film gratings,” Appl. Opt. 20, 423 (1981).
    [CrossRef]
  14. K. Wagatsuma, H. Sakaki, S. Saito, “Mode conversion and optical filtering of obliquely incident waves in corrugated waveguide filters,” IEEE J. Quantum Electron. QE-15, 632 (1979).
    [CrossRef]
  15. K. Wagatsuma, K. Yokoyama, H. Sakaki, S. Saito, “Mode couplings in corrugated-waveguide optical demultiplexers,” in Proceedings of Fifth European Conference on Optical Communication (Philips Research Laboratories, Eindhoven, The Netherlands, 1979) pp. 15.3/1–4.
  16. T. Fukuzawa, M. Nakamura, “Mode coupling in thin-film chirped gratings,” Opt. Lett. 4, 343 (1979).
    [CrossRef] [PubMed]
  17. C. S. Hong, J. B. Shellan, A. C. Livanos, A. Yariv, A. Katzir, “Broad-band grating filters for thin-film optical waveguides,” Appl. Phys. Lett. 31, 276 (1977).
    [CrossRef]
  18. A. C. Livanos, A. Katzir, A. Yariv, C. S. Hong, “Chirped-grating demultiplexers in dielectric waveguides,” Appl. Phys. Lett. 30, 519 (1977).
    [CrossRef]
  19. J. B. Shellan, C. S. Hong, A. Yariv, “Theory of chirped gratings in broad band filters,” Opt. Commun. 23, 398 (1977).
    [CrossRef]
  20. H. Kogelnik, “Filter response of nonuniform almost-periodic structures,” Bell Syst. Tech. J. 55, 109 (1976).
  21. D. C. Flanders, H. Kogelnik, R. V. Schmidt, C. V. Shank, “Grating filters for thin-film optical waveguides,” Appl. Phys. Lett. 24, 194 (1974).
    [CrossRef]
  22. A. W. Crook, “The reflection and transmission of light by any system of parallel isotropic films,” J. Opt. Soc. Am. 38, 954 (1948).
    [CrossRef] [PubMed]
  23. O. S. Heavens, Optical Properties of Thin Solid Films (Butter-worth, London, 1955), pp. 63–66.
  24. M. P. Rouard, “Etudes des propriétés optiques des lames métalliques très minces,” Ann. Phys. II 7, 20 (1937).
  25. A. Vasicek, “The reflection of light from glass with double and multiple films,” J. Opt. Soc. Am. 37, 623 (1947).
    [CrossRef] [PubMed]
  26. W. L. Wilcock, “On a paper by Vasicek concerning reflection from multilayer films,” J. Opt. Soc. Am. 39, 889 (1949).
    [CrossRef]
  27. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1965), Sec. 1.6.
  28. Typically, the coupled-mode analysis of a Bragg-matched periodic grating of length L yields the grating amplitude reflectivity, r = i tanh(KL). In such treatments, the surface corrugation of the grating is described byx=h+Δhcos(2πΛz),0≤z≤L.In this paper, we have specified the grating corrugation in the slightly different form [see Eq. (4)]:x=h+Δhsin(2πΛz),0≤z≤L.When this form is used in the coupled-mode equations, the resulting grating amplitude reflectivity is given byr=−tanh(KL).For the case of a detuned periodic grating, the amplitude reflectivity is written in the more general formr=−Ksinh(αL)αcosh(αL)−iδsinh(αL).
  29. We consider only TE–TE coupling in the examples presented in this paper and use the TE–TE coupling coefficient derived by Wagatsuma et al.using a coupled-mode analysis.14,15 It should be noted that there is little agreement among the coupling coefficients presented in the literature, 2–5,8,12–15 a point that has been discussed by several authors.1,4,12 Consequently, a judicious choice of coupling coefficient is required in order to obtain the most accurate predictions of the grating-response characteristics.
  30. M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (U.S. Department of Commerce, National Bureau of Standards, Washington, D.C., 1972), Eq. (4.5.26).
  31. When propagating in the +z direction, the grating corrugation is described byx=h+Δhsin(2πΛz),0≤z≤Λ,resulting in the amplitude reflectivity r = −tanh(KΛ). When propagating in the −z direction, the grating appears reversed and is described byx=h−Δsin(2πΛz),0≤z≤Λ.The change in sign of the sinusoidal corrugation results in a corresponding change in the grating-period reflectivity to r = +tanh(KΛ).Since a third-order reflection consists of two reflections of the forward-propagating mode and one reflection of the backward-propagating mode, the sign of a third-order reflection is positive.
  32. The method of Hong et al.17 must be modified to predict the width of the grating response as well as the amplitude. We have done this by ascribing a reflectivity of zero percent for spectral or angular components having propagation constants β not satisfyingβL≤β≤β0,whereβL=πcos(θ)Λ(z=L)andβ0=πcos(θ)Λ(z=0).
  33. The effective-length approximation16,18 takes a slightly different form in the two references cited. The formulation presented by Fukuzawa and Nakamura16 is used throughout this paper.

1984

1981

1980

J. Marcou, N. Gremillet, G. Thomin, “Polarization conversion by Bragg deflection in isotropic planar integrated optics waveguides; part I: theoretical study,” Opt. Commun. 32, 63 (1980).
[CrossRef]

N. Gremillet, G. Thomin, J. Marcou, “Polarization conversion by Bragg deflection in isotropic planar integrated optics waveguides; part II: experimental study,” Opt. Commun. 32, 69 (1980).
[CrossRef]

1979

K. Wagatsuma, H. Sakaki, S. Saito, “Mode conversion and optical filtering of obliquely incident waves in corrugated waveguide filters,” IEEE J. Quantum Electron. QE-15, 632 (1979).
[CrossRef]

T. Fukuzawa, M. Nakamura, “Mode coupling in thin-film chirped gratings,” Opt. Lett. 4, 343 (1979).
[CrossRef] [PubMed]

1978

Y. Yamamoto, T. Kamiya, H. Yanai, “Improved coupled mode analysis of corrugated waveguides and lasers,” IEEE J. Quantum Electron. QE-14, 245 (1978).
[CrossRef]

1977

C. S. Hong, J. B. Shellan, A. C. Livanos, A. Yariv, A. Katzir, “Broad-band grating filters for thin-film optical waveguides,” Appl. Phys. Lett. 31, 276 (1977).
[CrossRef]

A. C. Livanos, A. Katzir, A. Yariv, C. S. Hong, “Chirped-grating demultiplexers in dielectric waveguides,” Appl. Phys. Lett. 30, 519 (1977).
[CrossRef]

J. B. Shellan, C. S. Hong, A. Yariv, “Theory of chirped gratings in broad band filters,” Opt. Commun. 23, 398 (1977).
[CrossRef]

1976

H. Kogelnik, “Filter response of nonuniform almost-periodic structures,” Bell Syst. Tech. J. 55, 109 (1976).

W. Streifer, “TM-mode coupling coefficients in guided-wave DFB lasers,” IEEE J. Quantum Electron. QE-12, 74 (1976).
[CrossRef]

1975

W. Streifer, D. R. Scifres, R. D. Burnham, “Coupling coefficients for DFB single- and double-heterostructure diode lasers,” IEEE J. Quantum Electron. QE-11, 867 (1975).
[CrossRef]

1974

T. Tsai, H. Tuan, “Reflection and scattering by a single groove in integrated optics,” IEEE J. Quantum Electron. QE-10, 326 (1974).
[CrossRef]

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

1973

A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. QE-9, 919 (1973).
[CrossRef]

1949

1948

1947

1937

M. P. Rouard, “Etudes des propriétés optiques des lames métalliques très minces,” Ann. Phys. II 7, 20 (1937).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1965), Sec. 1.6.

Burke, J. J.

Burnham, R. D.

W. Streifer, D. R. Scifres, R. D. Burnham, “Coupling coefficients for DFB single- and double-heterostructure diode lasers,” IEEE J. Quantum Electron. QE-11, 867 (1975).
[CrossRef]

Crook, A. W.

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 (1974).
[CrossRef]

Fukuzawa, T.

Gremillet, N.

J. Marcou, N. Gremillet, G. Thomin, “Polarization conversion by Bragg deflection in isotropic planar integrated optics waveguides; part I: theoretical study,” Opt. Commun. 32, 63 (1980).
[CrossRef]

N. Gremillet, G. Thomin, J. Marcou, “Polarization conversion by Bragg deflection in isotropic planar integrated optics waveguides; part II: experimental study,” Opt. Commun. 32, 69 (1980).
[CrossRef]

Gruhlke, R. W.

Hall, D. G.

Heavens, O. S.

O. S. Heavens, Optical Properties of Thin Solid Films (Butter-worth, London, 1955), pp. 63–66.

Hong, C. S.

A. C. Livanos, A. Katzir, A. Yariv, C. S. Hong, “Chirped-grating demultiplexers in dielectric waveguides,” Appl. Phys. Lett. 30, 519 (1977).
[CrossRef]

J. B. Shellan, C. S. Hong, A. Yariv, “Theory of chirped gratings in broad band filters,” Opt. Commun. 23, 398 (1977).
[CrossRef]

C. S. Hong, J. B. Shellan, A. C. Livanos, A. Yariv, A. Katzir, “Broad-band grating filters for thin-film optical waveguides,” Appl. Phys. Lett. 31, 276 (1977).
[CrossRef]

Kamiya, T.

Y. Yamamoto, T. Kamiya, H. Yanai, “Improved coupled mode analysis of corrugated waveguides and lasers,” IEEE J. Quantum Electron. QE-14, 245 (1978).
[CrossRef]

Katzir, A.

A. C. Livanos, A. Katzir, A. Yariv, C. S. Hong, “Chirped-grating demultiplexers in dielectric waveguides,” Appl. Phys. Lett. 30, 519 (1977).
[CrossRef]

C. S. Hong, J. B. Shellan, A. C. Livanos, A. Yariv, A. Katzir, “Broad-band grating filters for thin-film optical waveguides,” Appl. Phys. Lett. 31, 276 (1977).
[CrossRef]

Kogelnik, H.

H. Kogelnik, “Filter response of nonuniform almost-periodic structures,” Bell Syst. Tech. J. 55, 109 (1976).

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

H. Kogelnik, in Integrated Optics, T. Tamir, ed., Vol. 7 of Topics in Applied Physics (Springer-Verlag, New York, 1975), pp. 66–79.
[CrossRef]

Lagasse, P. E.

Livanos, A. C.

A. C. Livanos, A. Katzir, A. Yariv, C. S. Hong, “Chirped-grating demultiplexers in dielectric waveguides,” Appl. Phys. Lett. 30, 519 (1977).
[CrossRef]

C. S. Hong, J. B. Shellan, A. C. Livanos, A. Yariv, A. Katzir, “Broad-band grating filters for thin-film optical waveguides,” Appl. Phys. Lett. 31, 276 (1977).
[CrossRef]

Marcou, J.

N. Gremillet, G. Thomin, J. Marcou, “Polarization conversion by Bragg deflection in isotropic planar integrated optics waveguides; part II: experimental study,” Opt. Commun. 32, 69 (1980).
[CrossRef]

J. Marcou, N. Gremillet, G. Thomin, “Polarization conversion by Bragg deflection in isotropic planar integrated optics waveguides; part I: theoretical study,” Opt. Commun. 32, 63 (1980).
[CrossRef]

Marcuse, D.

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974), pp. 95–126, 132–145.

Nakamura, M.

Rouard, M. P.

M. P. Rouard, “Etudes des propriétés optiques des lames métalliques très minces,” Ann. Phys. II 7, 20 (1937).

Saito, S.

K. Wagatsuma, H. Sakaki, S. Saito, “Mode conversion and optical filtering of obliquely incident waves in corrugated waveguide filters,” IEEE J. Quantum Electron. QE-15, 632 (1979).
[CrossRef]

K. Wagatsuma, K. Yokoyama, H. Sakaki, S. Saito, “Mode couplings in corrugated-waveguide optical demultiplexers,” in Proceedings of Fifth European Conference on Optical Communication (Philips Research Laboratories, Eindhoven, The Netherlands, 1979) pp. 15.3/1–4.

Sakaki, H.

K. Wagatsuma, H. Sakaki, S. Saito, “Mode conversion and optical filtering of obliquely incident waves in corrugated waveguide filters,” IEEE J. Quantum Electron. QE-15, 632 (1979).
[CrossRef]

K. Wagatsuma, K. Yokoyama, H. Sakaki, S. Saito, “Mode couplings in corrugated-waveguide optical demultiplexers,” in Proceedings of Fifth European Conference on Optical Communication (Philips Research Laboratories, Eindhoven, The Netherlands, 1979) pp. 15.3/1–4.

Sarid, D.

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 (1974).
[CrossRef]

Scifres, D. R.

W. Streifer, D. R. Scifres, R. D. Burnham, “Coupling coefficients for DFB single- and double-heterostructure diode lasers,” IEEE J. Quantum Electron. QE-11, 867 (1975).
[CrossRef]

Seshadri, S. R.

S. R. Seshadri, “TE–TE coupling at oblique incidence in a periodic dielectric waveguide,” Appl. Phys. 25, 211 (1981).
[CrossRef]

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 (1974).
[CrossRef]

Shellan, J. B.

J. B. Shellan, C. S. Hong, A. Yariv, “Theory of chirped gratings in broad band filters,” Opt. Commun. 23, 398 (1977).
[CrossRef]

C. S. Hong, J. B. Shellan, A. C. Livanos, A. Yariv, A. Katzir, “Broad-band grating filters for thin-film optical waveguides,” Appl. Phys. Lett. 31, 276 (1977).
[CrossRef]

Stegeman, G. I.

Streifer, W.

W. Streifer, “TM-mode coupling coefficients in guided-wave DFB lasers,” IEEE J. Quantum Electron. QE-12, 74 (1976).
[CrossRef]

W. Streifer, D. R. Scifres, R. D. Burnham, “Coupling coefficients for DFB single- and double-heterostructure diode lasers,” IEEE J. Quantum Electron. QE-11, 867 (1975).
[CrossRef]

Thomin, G.

N. Gremillet, G. Thomin, J. Marcou, “Polarization conversion by Bragg deflection in isotropic planar integrated optics waveguides; part II: experimental study,” Opt. Commun. 32, 69 (1980).
[CrossRef]

J. Marcou, N. Gremillet, G. Thomin, “Polarization conversion by Bragg deflection in isotropic planar integrated optics waveguides; part I: theoretical study,” Opt. Commun. 32, 63 (1980).
[CrossRef]

Tsai, T.

T. Tsai, H. Tuan, “Reflection and scattering by a single groove in integrated optics,” IEEE J. Quantum Electron. QE-10, 326 (1974).
[CrossRef]

Tuan, H.

T. Tsai, H. Tuan, “Reflection and scattering by a single groove in integrated optics,” IEEE J. Quantum Electron. QE-10, 326 (1974).
[CrossRef]

VanRoey, J.

Vasicek, A.

Wagatsuma, K.

K. Wagatsuma, H. Sakaki, S. Saito, “Mode conversion and optical filtering of obliquely incident waves in corrugated waveguide filters,” IEEE J. Quantum Electron. QE-15, 632 (1979).
[CrossRef]

K. Wagatsuma, K. Yokoyama, H. Sakaki, S. Saito, “Mode couplings in corrugated-waveguide optical demultiplexers,” in Proceedings of Fifth European Conference on Optical Communication (Philips Research Laboratories, Eindhoven, The Netherlands, 1979) pp. 15.3/1–4.

Wilcock, W. L.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1965), Sec. 1.6.

Yamamoto, Y.

Y. Yamamoto, T. Kamiya, H. Yanai, “Improved coupled mode analysis of corrugated waveguides and lasers,” IEEE J. Quantum Electron. QE-14, 245 (1978).
[CrossRef]

Yanai, H.

Y. Yamamoto, T. Kamiya, H. Yanai, “Improved coupled mode analysis of corrugated waveguides and lasers,” IEEE J. Quantum Electron. QE-14, 245 (1978).
[CrossRef]

Yariv, A.

J. B. Shellan, C. S. Hong, A. Yariv, “Theory of chirped gratings in broad band filters,” Opt. Commun. 23, 398 (1977).
[CrossRef]

A. C. Livanos, A. Katzir, A. Yariv, C. S. Hong, “Chirped-grating demultiplexers in dielectric waveguides,” Appl. Phys. Lett. 30, 519 (1977).
[CrossRef]

C. S. Hong, J. B. Shellan, A. C. Livanos, A. Yariv, A. Katzir, “Broad-band grating filters for thin-film optical waveguides,” Appl. Phys. Lett. 31, 276 (1977).
[CrossRef]

A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. QE-9, 919 (1973).
[CrossRef]

Yokoyama, K.

K. Wagatsuma, K. Yokoyama, H. Sakaki, S. Saito, “Mode couplings in corrugated-waveguide optical demultiplexers,” in Proceedings of Fifth European Conference on Optical Communication (Philips Research Laboratories, Eindhoven, The Netherlands, 1979) pp. 15.3/1–4.

Ann. Phys. II

M. P. Rouard, “Etudes des propriétés optiques des lames métalliques très minces,” Ann. Phys. II 7, 20 (1937).

Appl. Opt.

Appl. Phys.

S. R. Seshadri, “TE–TE coupling at oblique incidence in a periodic dielectric waveguide,” Appl. Phys. 25, 211 (1981).
[CrossRef]

Appl. Phys. Lett.

C. S. Hong, J. B. Shellan, A. C. Livanos, A. Yariv, A. Katzir, “Broad-band grating filters for thin-film optical waveguides,” Appl. Phys. Lett. 31, 276 (1977).
[CrossRef]

A. C. Livanos, A. Katzir, A. Yariv, C. S. Hong, “Chirped-grating demultiplexers in dielectric waveguides,” Appl. Phys. Lett. 30, 519 (1977).
[CrossRef]

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

Bell Syst. Tech. J.

H. Kogelnik, “Filter response of nonuniform almost-periodic structures,” Bell Syst. Tech. J. 55, 109 (1976).

IEEE J. Quantum Electron.

K. Wagatsuma, H. Sakaki, S. Saito, “Mode conversion and optical filtering of obliquely incident waves in corrugated waveguide filters,” IEEE J. Quantum Electron. QE-15, 632 (1979).
[CrossRef]

W. Streifer, D. R. Scifres, R. D. Burnham, “Coupling coefficients for DFB single- and double-heterostructure diode lasers,” IEEE J. Quantum Electron. QE-11, 867 (1975).
[CrossRef]

W. Streifer, “TM-mode coupling coefficients in guided-wave DFB lasers,” IEEE J. Quantum Electron. QE-12, 74 (1976).
[CrossRef]

T. Tsai, H. Tuan, “Reflection and scattering by a single groove in integrated optics,” IEEE J. Quantum Electron. QE-10, 326 (1974).
[CrossRef]

Y. Yamamoto, T. Kamiya, H. Yanai, “Improved coupled mode analysis of corrugated waveguides and lasers,” IEEE J. Quantum Electron. QE-14, 245 (1978).
[CrossRef]

A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. QE-9, 919 (1973).
[CrossRef]

J. Opt. Soc. Am.

Opt. Commun.

J. B. Shellan, C. S. Hong, A. Yariv, “Theory of chirped gratings in broad band filters,” Opt. Commun. 23, 398 (1977).
[CrossRef]

J. Marcou, N. Gremillet, G. Thomin, “Polarization conversion by Bragg deflection in isotropic planar integrated optics waveguides; part I: theoretical study,” Opt. Commun. 32, 63 (1980).
[CrossRef]

N. Gremillet, G. Thomin, J. Marcou, “Polarization conversion by Bragg deflection in isotropic planar integrated optics waveguides; part II: experimental study,” Opt. Commun. 32, 69 (1980).
[CrossRef]

Opt. Lett.

Other

O. S. Heavens, Optical Properties of Thin Solid Films (Butter-worth, London, 1955), pp. 63–66.

K. Wagatsuma, K. Yokoyama, H. Sakaki, S. Saito, “Mode couplings in corrugated-waveguide optical demultiplexers,” in Proceedings of Fifth European Conference on Optical Communication (Philips Research Laboratories, Eindhoven, The Netherlands, 1979) pp. 15.3/1–4.

H. Kogelnik, in Integrated Optics, T. Tamir, ed., Vol. 7 of Topics in Applied Physics (Springer-Verlag, New York, 1975), pp. 66–79.
[CrossRef]

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974), pp. 95–126, 132–145.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1965), Sec. 1.6.

Typically, the coupled-mode analysis of a Bragg-matched periodic grating of length L yields the grating amplitude reflectivity, r = i tanh(KL). In such treatments, the surface corrugation of the grating is described byx=h+Δhcos(2πΛz),0≤z≤L.In this paper, we have specified the grating corrugation in the slightly different form [see Eq. (4)]:x=h+Δhsin(2πΛz),0≤z≤L.When this form is used in the coupled-mode equations, the resulting grating amplitude reflectivity is given byr=−tanh(KL).For the case of a detuned periodic grating, the amplitude reflectivity is written in the more general formr=−Ksinh(αL)αcosh(αL)−iδsinh(αL).

We consider only TE–TE coupling in the examples presented in this paper and use the TE–TE coupling coefficient derived by Wagatsuma et al.using a coupled-mode analysis.14,15 It should be noted that there is little agreement among the coupling coefficients presented in the literature, 2–5,8,12–15 a point that has been discussed by several authors.1,4,12 Consequently, a judicious choice of coupling coefficient is required in order to obtain the most accurate predictions of the grating-response characteristics.

M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (U.S. Department of Commerce, National Bureau of Standards, Washington, D.C., 1972), Eq. (4.5.26).

When propagating in the +z direction, the grating corrugation is described byx=h+Δhsin(2πΛz),0≤z≤Λ,resulting in the amplitude reflectivity r = −tanh(KΛ). When propagating in the −z direction, the grating appears reversed and is described byx=h−Δsin(2πΛz),0≤z≤Λ.The change in sign of the sinusoidal corrugation results in a corresponding change in the grating-period reflectivity to r = +tanh(KΛ).Since a third-order reflection consists of two reflections of the forward-propagating mode and one reflection of the backward-propagating mode, the sign of a third-order reflection is positive.

The method of Hong et al.17 must be modified to predict the width of the grating response as well as the amplitude. We have done this by ascribing a reflectivity of zero percent for spectral or angular components having propagation constants β not satisfyingβL≤β≤β0,whereβL=πcos(θ)Λ(z=L)andβ0=πcos(θ)Λ(z=0).

The effective-length approximation16,18 takes a slightly different form in the two references cited. The formulation presented by Fukuzawa and Nakamura16 is used throughout this paper.

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Figures (9)

Fig. 1
Fig. 1

Two-layer thin-film stack illustrating the application of Rouard’s method, (a) Superstate–layer 1–layer 2–substrate stack with interfaces 0,1, and 2 having reflectivities r0, r1, and r2, respectively. The thicknesses of layers 1 and 2 are given by d1 and d2. The complex reflectivity of layer 2 is given by ρ2. (b) Two-layer stack with the second layer replaced by an interface having the same complex reflectivity. The total reflectivity of the stack is given by ρ.

Fig. 2
Fig. 2

Diagram of the slab waveguide with surface corrugation. The parameters θi, θr, h, Δh, L, and Λ denote incident angle, reflected angle, unperturbed film thickness, corrugation depth, grating length, and grating period, respectively.

Fig. 3
Fig. 3

Side and top views of a periodic waveguide grating illustrating the thin-film–waveguide-grating analogy. Each period of the grating depicted in the side view is represented by an effective interface, as is shown by the dashed lines in the top view. The effective interfaces have reflectivities r0r5.

Fig. 4
Fig. 4

Side and top views of a three-period periodic waveguide grating of length L = 3Λ, illustrating the application of Rouard’s method. As is defined in Fig. 1, the reflectivities of the effective interfaces are given by r0, r1, and r2. The complex reflectivity of the second effective layer is defined by ρ2; that of the entire grating is given by ρ.

Fig. 5
Fig. 5

(a) Spectral response of the periodic grating defined in the text. The peak reflectivity of approximately 83% is centered at λ = 830 nm. (b) Angular response of the same grating centered on θ = 60 deg. In both (a) and (b), the dotted curves were obtained by using the coupled-mode theory. The solid curves were generated by Rouard’s method.

Fig. 6
Fig. 6

Spectral response of the periodic grating defined in the text generated by using first-order (plot A) and third-order (plot B) multiple-reflection calculations as well as by coupled-mode theory (plot C).

Fig. 7
Fig. 7

Spectral-response curves generated by using Rouard’s method (plot A), the approximation of Hong et al.17 (plot B), and the effective-length calculation of Fukuzawa and Nakamura16 (plot C). (a) Chirp parameter, F/2π = 1.0; grating parameters, Λ(0) = 0.5523 μm, Λ(L) = 0.5594 μm, L = 85.29 μm; (b) chirp parameter, F/2π = 3.0; grating parameters, Λ(0) = 0.5453 μm, Λ(L) = 0.5668 μm, L = 85.29 μm; (c) frequency-response curves generated by Rouard’s method for the chirp rates F/2π = 0,1.0, and 3.0. The grating parameters for chirps 1.0 and 3.0 are as given (b) and (c), respectively.

Fig. 8
Fig. 8

Spectral-response curves generated by using Rouard’s method (plot A), the approximation of Hong et al.17 (plot B), and the effective-length calculation of Fukuzawa and Nakamura16 (plot C). (a) Chirp parameter, F/2π = 5.6; grating parameters, Λ(0) = 0.5489 μm, Λ (L) = 0.5628 μm, L = 250 μm; (b) chirp parameter, F/2π = 16.9; grating parameters, Λ(0) = 0.5350 μm, Λ(L) = 0.5768 μm, L = 250 μm; (e) chirp parameter, F/2π = 28.3; grating parameters, Λ(0) = 0.5211 μm, Λ(L) = 0.5907 μm, L = 250 μm.

Fig. 9
Fig. 9

Spectral-response curves generated by using Rouard’s method (plot A), the approximation of Hong et al.11 (plot B), and the effective-length calculation of Fukuzawa and Nakamura16 (plot C). (a) Chirp parameter, F/2π = 7.9; grating parameters, Λ(0) = 0.5489 μm, Λ(L) = 0.5628 μm, L = 350 μm; (b) chirp parameter, F/2π = 11.2; grating parameters, Λ(0) = 0.5489 μm, Λ(L) = 0.5628 μm, L = 500 μm.

Equations (25)

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ρ 2 = r 1 + r 2 exp ( 2 i Δ 2 ) 1 + r 1 r 2 exp ( 2 i Δ 2 ) ,
Δ 2 = 2 π λ n 2 d 2 cos ( θ 2 ) .
ρ = r 0 + ρ 2 exp ( 2 i Δ 1 ) 1 + r 0 ρ 2 exp ( 2 i Δ 1 ) ,
x = h + Δ h sin ( 2 π Λ z ) , 0 z L ,
r n = tanh ( K Λ n ) ,
2 Δ n = d n [ β i cos ( θ i ) + β r cos ( θ r ) ] ,
r = tanh ( K L ) ,
2 δ = β i cos ( θ i ) + β r cos ( θ r ) 2 m π Λ , m = 1 , 2 , 3 ,
ρ 2 = r 1 + r 2 exp ( 2 i Δ 2 ) 1 + r 1 r 2 exp ( 2 i Δ 2 ) = r 1 + r 2 1 + r 1 r 2 ,
arctanh ( ρ 2 ) = arctanh ( r 1 ) + arctanh ( r 2 ) .
ρ = r 0 + ρ 2 exp ( 2 i Δ 1 ) 1 + r 0 ρ 2 exp ( 2 i Δ 1 ) = r 0 + ρ 2 1 + r 0 ρ 2 ,
arctanh ( ρ ) = arctanh ( r 0 ) + arctanh ( r 1 ) + arctanh ( r 2 )
ρ = tanh ( K L ) ,
r = K sinh ( α L ) α cosh ( α L ) i δ sinh ( α L ) ,
x = h + Δ h sin [ 2 π Λ ( z ) ( z L / 2 ) ] ,
2 π Λ ( z ) = 2 π Λ ( L / 2 ) + F L 2 ( z L / 2 ) ,
x=h+Δhcos(2πΛz),0zL.
x=h+Δhsin(2πΛz),0zL.
r=tanh(KL).
r=Ksinh(αL)αcosh(αL)iδsinh(αL).
x=h+Δhsin(2πΛz),0zΛ,
x=hΔsin(2πΛz),0zΛ.
βLββ0,
βL=πcos(θ)Λ(z=L)
β0=πcos(θ)Λ(z=0).

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