Abstract

Recently we introduced a new computational tool for the analysis of waveguide diffraction gratings based on Rouard’s method, a recursive technique used in thin-film coating design. In this paper we compare the reflectivities predicted by Rouard’s method with those found by using the ideal mode expansion of coupled-mode theory for both periodic and aperiodic gratings. We find empirically that the two methods are in excellent agreement, with typical differences in reflectivity of less than 1%.

© 1987 Optical Society of America

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  1. Throughout this paper we compare the results of Rouard’s method with those of an ideal mode expansion of coupled-mode theory.2–4 The coupled-mode-theory results are derived by using a synchronous approximation resulting in the general coupled equations given by Eqs. (5a) and (5b) in the text.5
  2. A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. QE-9, 919–933 (1973).
    [CrossRef]
  3. H. Kogelnik, in Integrated Optics, Vol. 7 of Topics in Applied Physics, T. Tamir, ed. (Springer-Verlag, New York, 1975), pp. 66–79.
    [CrossRef]
  4. D. Marcuse, Theory of Dielecteic Optical Waveguides (Academic, New York, 1974), pp. 95–126 and 132–145.
  5. H. Kogelnik, “Filter response of nonuniform almost-periodic structures,” Bell Syst. Tech. J. 55, 109–126 (1976).
  6. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
  7. T. Fukuzawa, M. Nakamura, “Mode coupling in thin-film chirped gratings,” Opt. Lett. 4, 343–345 (1979).
    [CrossRef] [PubMed]
  8. 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, 276–278 (1977).
    [CrossRef]
  9. J. B. Shellan, C. S. Hong, A. Yariv, “Theory of chirped gratings in broad band filters,” Opt. Commun. 23, 398–400 (1977).
    [CrossRef]
  10. L. A. Weller-Brophy, D. G. Hall, “Analysis of waveguide gratings: application of Rouard’s method,” J. Opt. Soc. Am. A 2, 863–871 (1985).
    [CrossRef]
  11. M. P. Rouard, “Etudes des propriétés optiques des lames métalliques trés minces,” Ann. Phys. (Paris) Ser. II 7, 291–384 (1937).
  12. IMSL Library User’s Manual, Ed. 9.2 (IMSL, Houston, Tex., 1984), Chap. D, pp. DVERK-1–DVERK-9.
  13. We will consider only TE–TE coupling in the examples presented in this paper and will use the TE–TE coupling coefficient derived by Wagatsuma using a coupled-mode analysis.14,15 It should be noted that there is little agreement among the coupling coefficients presented in the literature, a point that has been discussed by several authors.16–19 Consequently, a judicious choice of coupling coefficient is required in order to obtain the most accurate predictions of the grating response characteristics.
  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, 632–637 (1979).
    [CrossRef]
  15. K. Wagatsuma, K. Yokoyama, H. Sakaki, S. Saito, “Mode couplings in corrugated-waveguide optical demultiplexers,” presented at the Fifth European Conference on Optical Communication, Amsterdam, September 1979.
  16. R. W. Gruhlke, D. G. Hall, “Comparison of two approaches to the waveguide scattering problem: TM polarization,” Appl. Opt. 23, 127–133 (1984).
    [CrossRef] [PubMed]
  17. W. Streifer, D. R. Scifres, R. D. Burnham, “Coupling coefficients for DFB single- and double-heterostructure diode lasers,” IEEE J. Quantum Electron. QE-11, 867–873 (1975).
    [CrossRef]
  18. 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, 1497–1507 (1981).
    [CrossRef]
  19. L. A. Weller-Brophy, D. G. Hall, “Waveguide diffraction gratings in integrated optics,” in Integrated Optical Circuit Engineering II, S. Sriram, ed., Proc. Soc. Photo-Opt. Instrum. Eng.578, 173–177 (1985).
    [CrossRef]

1985 (1)

1984 (1)

1981 (1)

1979 (2)

T. Fukuzawa, M. Nakamura, “Mode coupling in thin-film chirped gratings,” Opt. Lett. 4, 343–345 (1979).
[CrossRef] [PubMed]

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, 632–637 (1979).
[CrossRef]

1977 (2)

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, 276–278 (1977).
[CrossRef]

J. B. Shellan, C. S. Hong, A. Yariv, “Theory of chirped gratings in broad band filters,” Opt. Commun. 23, 398–400 (1977).
[CrossRef]

1976 (1)

H. Kogelnik, “Filter response of nonuniform almost-periodic structures,” Bell Syst. Tech. J. 55, 109–126 (1976).

1975 (1)

W. Streifer, D. R. Scifres, R. D. Burnham, “Coupling coefficients for DFB single- and double-heterostructure diode lasers,” IEEE J. Quantum Electron. QE-11, 867–873 (1975).
[CrossRef]

1973 (1)

A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. QE-9, 919–933 (1973).
[CrossRef]

1969 (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

1937 (1)

M. P. Rouard, “Etudes des propriétés optiques des lames métalliques trés minces,” Ann. Phys. (Paris) Ser. II 7, 291–384 (1937).

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, 867–873 (1975).
[CrossRef]

Fukuzawa, T.

Gruhlke, R. W.

Hall, D. G.

Hong, C. S.

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, 276–278 (1977).
[CrossRef]

J. B. Shellan, C. S. Hong, A. Yariv, “Theory of chirped gratings in broad band filters,” Opt. Commun. 23, 398–400 (1977).
[CrossRef]

Katzir, A.

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, 276–278 (1977).
[CrossRef]

Kogelnik, H.

H. Kogelnik, “Filter response of nonuniform almost-periodic structures,” Bell Syst. Tech. J. 55, 109–126 (1976).

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

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

Livanos, A. C.

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, 276–278 (1977).
[CrossRef]

Marcuse, D.

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

Nakamura, M.

Rouard, M. P.

M. P. Rouard, “Etudes des propriétés optiques des lames métalliques trés minces,” Ann. Phys. (Paris) Ser. II 7, 291–384 (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, 632–637 (1979).
[CrossRef]

K. Wagatsuma, K. Yokoyama, H. Sakaki, S. Saito, “Mode couplings in corrugated-waveguide optical demultiplexers,” presented at the Fifth European Conference on Optical Communication, Amsterdam, September 1979.

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, 632–637 (1979).
[CrossRef]

K. Wagatsuma, K. Yokoyama, H. Sakaki, S. Saito, “Mode couplings in corrugated-waveguide optical demultiplexers,” presented at the Fifth European Conference on Optical Communication, Amsterdam, September 1979.

Sarid, D.

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, 867–873 (1975).
[CrossRef]

Shellan, J. B.

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, 276–278 (1977).
[CrossRef]

J. B. Shellan, C. S. Hong, A. Yariv, “Theory of chirped gratings in broad band filters,” Opt. Commun. 23, 398–400 (1977).
[CrossRef]

Stegeman, G. I.

Streifer, W.

W. Streifer, D. R. Scifres, R. D. Burnham, “Coupling coefficients for DFB single- and double-heterostructure diode lasers,” IEEE J. Quantum Electron. QE-11, 867–873 (1975).
[CrossRef]

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, 632–637 (1979).
[CrossRef]

K. Wagatsuma, K. Yokoyama, H. Sakaki, S. Saito, “Mode couplings in corrugated-waveguide optical demultiplexers,” presented at the Fifth European Conference on Optical Communication, Amsterdam, September 1979.

Weller-Brophy, L. A.

L. A. Weller-Brophy, D. G. Hall, “Analysis of waveguide gratings: application of Rouard’s method,” J. Opt. Soc. Am. A 2, 863–871 (1985).
[CrossRef]

L. A. Weller-Brophy, D. G. Hall, “Waveguide diffraction gratings in integrated optics,” in Integrated Optical Circuit Engineering II, S. Sriram, ed., Proc. Soc. Photo-Opt. Instrum. Eng.578, 173–177 (1985).
[CrossRef]

Yariv, A.

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, 276–278 (1977).
[CrossRef]

J. B. Shellan, C. S. Hong, A. Yariv, “Theory of chirped gratings in broad band filters,” Opt. Commun. 23, 398–400 (1977).
[CrossRef]

A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. QE-9, 919–933 (1973).
[CrossRef]

Yokoyama, K.

K. Wagatsuma, K. Yokoyama, H. Sakaki, S. Saito, “Mode couplings in corrugated-waveguide optical demultiplexers,” presented at the Fifth European Conference on Optical Communication, Amsterdam, September 1979.

Ann. Phys. (Paris) Ser. II (1)

M. P. Rouard, “Etudes des propriétés optiques des lames métalliques trés minces,” Ann. Phys. (Paris) Ser. II 7, 291–384 (1937).

Appl. Opt. (1)

Appl. Phys. Lett. (1)

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, 276–278 (1977).
[CrossRef]

Bell Syst. Tech. J. (2)

H. Kogelnik, “Filter response of nonuniform almost-periodic structures,” Bell Syst. Tech. J. 55, 109–126 (1976).

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

IEEE J. Quantum Electron. (3)

A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. QE-9, 919–933 (1973).
[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, 867–873 (1975).
[CrossRef]

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, 632–637 (1979).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Opt. Commun. (1)

J. B. Shellan, C. S. Hong, A. Yariv, “Theory of chirped gratings in broad band filters,” Opt. Commun. 23, 398–400 (1977).
[CrossRef]

Opt. Lett. (1)

Other (7)

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

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

Throughout this paper we compare the results of Rouard’s method with those of an ideal mode expansion of coupled-mode theory.2–4 The coupled-mode-theory results are derived by using a synchronous approximation resulting in the general coupled equations given by Eqs. (5a) and (5b) in the text.5

L. A. Weller-Brophy, D. G. Hall, “Waveguide diffraction gratings in integrated optics,” in Integrated Optical Circuit Engineering II, S. Sriram, ed., Proc. Soc. Photo-Opt. Instrum. Eng.578, 173–177 (1985).
[CrossRef]

K. Wagatsuma, K. Yokoyama, H. Sakaki, S. Saito, “Mode couplings in corrugated-waveguide optical demultiplexers,” presented at the Fifth European Conference on Optical Communication, Amsterdam, September 1979.

IMSL Library User’s Manual, Ed. 9.2 (IMSL, Houston, Tex., 1984), Chap. D, pp. DVERK-1–DVERK-9.

We will consider only TE–TE coupling in the examples presented in this paper and will use the TE–TE coupling coefficient derived by Wagatsuma using a coupled-mode analysis.14,15 It should be noted that there is little agreement among the coupling coefficients presented in the literature, a point that has been discussed by several authors.16–19 Consequently, a judicious choice of coupling coefficient is required in order to obtain the most accurate predictions of the grating response characteristics.

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Figures (5)

Fig. 1
Fig. 1

Diagram of the slab waveguide with surface corrugation grating. 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. 2
Fig. 2

Reflectivity of the periodic gratings described in the text. The grating reflectivity is plotted as a function of normalized detuning, δL/π. In each of the plots the results of the coupled-mode analysis and of Rouard’s method are indistinguishable. The grating lengths vary in each plot, with (a) L = 135 μm, (b) L = 270 μm,(c) L = 814 μm.

Fig. 3
Fig. 3

(a)–(c) Illustrate the difference ΔR = RRMRCMT for the three gratings whose reflectivities are shown in Fig. 2. Note the change in vertical scale from (a) to (b).

Fig. 4
Fig. 4

Reflectivity of the linearly chirped gratings described in the text. In each of the plots, the results of the coupled-mode analysis and of Rouard’s method are indistinguishable. While each grating analyzed has the length L = 1000 μm, the chirp parameters are different. (a) Chirp parameter F/2π = 0.1; grating parameters Λ(0) = 0.87223 μm, Λ(L) = 0.87207 μm. (b) Chirp parameter F/2π = 5; grating parameters Λ(0) = 0.87596 μm, Λ(L) = 0.86836 μm. (c) Chirp parameter F/2π = 10; grating parameters Λ(0) = 0.87981 μm, Λ(L) = 0.86461 μm.

Fig. 5
Fig. 5

(a)–(c) Illustrate the difference ΔR = RRMRCMT for the three aperiodic gratings whose reflectivities are shown in Fig. 4. Note the change in vertical scale from (a) to (b).

Equations (8)

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x = h + Δ h sin ( 2 π Λ z ) ,             0 z L ,
r = - κ sinh ( a L ) a cosh ( a L ) - i δ sinh ( a L ) ,
2 δ = β i cos ( θ i ) + β r cos ( θ r ) - 2 m π Λ ,             m = 1 , 2 , 3 , ,
Δ R max ( κ Λ ) 2 π [ 1 + ( κ L π ) 4 ] ,
R + i δ R = - i κ S e - i ϕ ( z )
S - i δ S = i κ R e i ϕ ( z ) ,
ϕ ( z ) = F ( z L ) 2 ,
F = 2 π L { Λ ( L / 2 ) - Λ ( L ) [ Λ ( L / 2 ) ] 2 }

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