Abstract

A study of second- and higher-order waveguide grating filters is considerably complicated by the fact that the coupling between the guided waves and the radiated waves has to be taken into account. In this paper, we review the theoretical calculations of the spectral response of these grating filters that have been reported by several authors and present a new approach based on the use of the Green’s functions of the waveguide structure. Excellent agreement is found between the results of our method of calculation and the results of other, more rigorous but much more complicated, methods. The Green’s function method is thus a reasonably simple but fairly accurate method for calculating the spectral response of grating filters.

© 1979 Optical Society of America

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  1. A. Basu, J. M. Ballantyne, Appl. Opt. 18, 0000 (1979).
  2. A. A. Zlenko, V. A. Kiselev, A. M. Prokhorov, A. A. Spikhalskii, V. A. Sychugov, Sov. J. Quantum Electron. 5, 1325 (1976).
    [CrossRef]
  3. K. Handa, S. T. Peng, T. Tamir, Appl. Phys. 5, 325 (1975).
    [CrossRef]
  4. T. Tamir, S. T. Peng, Appl. Phys. 14, 235 (1977).
    [CrossRef]
  5. W. Streifer, D. Scifres, R. D. Burnham, IEEE J. Quantum Electron QE-12, 422 (1976).
    [CrossRef]
  6. W. Streifer, R. D. Burnham, D. R. Scifres, IEEE J. Quantum Electron. QE-12, 494 (1976).
    [CrossRef]
  7. W. Streifer, D. R. Scifres, R. D. Burnham, IEEE J. Quantum Electron. QE-13, 134 (1977).
    [CrossRef]
  8. W. H. Lee, W. Streifer, J. Opt. Soc. Am. 68, 1701 (1978).
    [CrossRef]
  9. Y. Yamamoto, T. Kamiya, H. Yanai, IEEE J. Quantum Electron. QE-14, 245 (1978).
    [CrossRef]
  10. Y. Yamamoto, T. Kamiya, H. Yanai, IEEE J. Quantum. Electron. QE-14, 620 (1978).
    [CrossRef]
  11. W. Streifer, D. R. Scifres, R. D. Burnham, IEEE J. Quantum Electron. QE-11, 867 (1975).
    [CrossRef]
  12. C. C. Ghizoni, “Wave Coupling in Periodic Optical Waveguides,” Ph.D. dissertation, Cornell University, Ithaca, New York (1976).
  13. C. C. Ghizoni, B. U. Chen, C. L. Tang, IEEE J. Quantum Electron. QE-12, 69 (1976).
    [CrossRef]
  14. C. C. Ghizoni, J. M. Ballantyne, C. L. Tang, IEEE J. Quantum Electron. QE-13, 843 (1977).
    [CrossRef]
  15. A. Basu, “A Study of Grating Filters in Thin-Film Optical Waveguides,” Ph.D. dissertation, Cornell University, Ithaca, New York (1979).

1979 (1)

A. Basu, J. M. Ballantyne, Appl. Opt. 18, 0000 (1979).

1978 (3)

Y. Yamamoto, T. Kamiya, H. Yanai, IEEE J. Quantum Electron. QE-14, 245 (1978).
[CrossRef]

Y. Yamamoto, T. Kamiya, H. Yanai, IEEE J. Quantum. Electron. QE-14, 620 (1978).
[CrossRef]

W. H. Lee, W. Streifer, J. Opt. Soc. Am. 68, 1701 (1978).
[CrossRef]

1977 (3)

C. C. Ghizoni, J. M. Ballantyne, C. L. Tang, IEEE J. Quantum Electron. QE-13, 843 (1977).
[CrossRef]

W. Streifer, D. R. Scifres, R. D. Burnham, IEEE J. Quantum Electron. QE-13, 134 (1977).
[CrossRef]

T. Tamir, S. T. Peng, Appl. Phys. 14, 235 (1977).
[CrossRef]

1976 (4)

W. Streifer, D. Scifres, R. D. Burnham, IEEE J. Quantum Electron QE-12, 422 (1976).
[CrossRef]

W. Streifer, R. D. Burnham, D. R. Scifres, IEEE J. Quantum Electron. QE-12, 494 (1976).
[CrossRef]

A. A. Zlenko, V. A. Kiselev, A. M. Prokhorov, A. A. Spikhalskii, V. A. Sychugov, Sov. J. Quantum Electron. 5, 1325 (1976).
[CrossRef]

C. C. Ghizoni, B. U. Chen, C. L. Tang, IEEE J. Quantum Electron. QE-12, 69 (1976).
[CrossRef]

1975 (2)

W. Streifer, D. R. Scifres, R. D. Burnham, IEEE J. Quantum Electron. QE-11, 867 (1975).
[CrossRef]

K. Handa, S. T. Peng, T. Tamir, Appl. Phys. 5, 325 (1975).
[CrossRef]

Ballantyne, J. M.

A. Basu, J. M. Ballantyne, Appl. Opt. 18, 0000 (1979).

C. C. Ghizoni, J. M. Ballantyne, C. L. Tang, IEEE J. Quantum Electron. QE-13, 843 (1977).
[CrossRef]

Basu, A.

A. Basu, J. M. Ballantyne, Appl. Opt. 18, 0000 (1979).

A. Basu, “A Study of Grating Filters in Thin-Film Optical Waveguides,” Ph.D. dissertation, Cornell University, Ithaca, New York (1979).

Burnham, R. D.

W. Streifer, D. R. Scifres, R. D. Burnham, IEEE J. Quantum Electron. QE-13, 134 (1977).
[CrossRef]

W. Streifer, D. Scifres, R. D. Burnham, IEEE J. Quantum Electron QE-12, 422 (1976).
[CrossRef]

W. Streifer, R. D. Burnham, D. R. Scifres, IEEE J. Quantum Electron. QE-12, 494 (1976).
[CrossRef]

W. Streifer, D. R. Scifres, R. D. Burnham, IEEE J. Quantum Electron. QE-11, 867 (1975).
[CrossRef]

Chen, B. U.

C. C. Ghizoni, B. U. Chen, C. L. Tang, IEEE J. Quantum Electron. QE-12, 69 (1976).
[CrossRef]

Ghizoni, C. C.

C. C. Ghizoni, J. M. Ballantyne, C. L. Tang, IEEE J. Quantum Electron. QE-13, 843 (1977).
[CrossRef]

C. C. Ghizoni, B. U. Chen, C. L. Tang, IEEE J. Quantum Electron. QE-12, 69 (1976).
[CrossRef]

C. C. Ghizoni, “Wave Coupling in Periodic Optical Waveguides,” Ph.D. dissertation, Cornell University, Ithaca, New York (1976).

Handa, K.

K. Handa, S. T. Peng, T. Tamir, Appl. Phys. 5, 325 (1975).
[CrossRef]

Kamiya, T.

Y. Yamamoto, T. Kamiya, H. Yanai, IEEE J. Quantum Electron. QE-14, 245 (1978).
[CrossRef]

Y. Yamamoto, T. Kamiya, H. Yanai, IEEE J. Quantum. Electron. QE-14, 620 (1978).
[CrossRef]

Kiselev, V. A.

A. A. Zlenko, V. A. Kiselev, A. M. Prokhorov, A. A. Spikhalskii, V. A. Sychugov, Sov. J. Quantum Electron. 5, 1325 (1976).
[CrossRef]

Lee, W. H.

Peng, S. T.

T. Tamir, S. T. Peng, Appl. Phys. 14, 235 (1977).
[CrossRef]

K. Handa, S. T. Peng, T. Tamir, Appl. Phys. 5, 325 (1975).
[CrossRef]

Prokhorov, A. M.

A. A. Zlenko, V. A. Kiselev, A. M. Prokhorov, A. A. Spikhalskii, V. A. Sychugov, Sov. J. Quantum Electron. 5, 1325 (1976).
[CrossRef]

Scifres, D.

W. Streifer, D. Scifres, R. D. Burnham, IEEE J. Quantum Electron QE-12, 422 (1976).
[CrossRef]

Scifres, D. R.

W. Streifer, D. R. Scifres, R. D. Burnham, IEEE J. Quantum Electron. QE-13, 134 (1977).
[CrossRef]

W. Streifer, R. D. Burnham, D. R. Scifres, IEEE J. Quantum Electron. QE-12, 494 (1976).
[CrossRef]

W. Streifer, D. R. Scifres, R. D. Burnham, IEEE J. Quantum Electron. QE-11, 867 (1975).
[CrossRef]

Spikhalskii, A. A.

A. A. Zlenko, V. A. Kiselev, A. M. Prokhorov, A. A. Spikhalskii, V. A. Sychugov, Sov. J. Quantum Electron. 5, 1325 (1976).
[CrossRef]

Streifer, W.

W. H. Lee, W. Streifer, J. Opt. Soc. Am. 68, 1701 (1978).
[CrossRef]

W. Streifer, D. R. Scifres, R. D. Burnham, IEEE J. Quantum Electron. QE-13, 134 (1977).
[CrossRef]

W. Streifer, R. D. Burnham, D. R. Scifres, IEEE J. Quantum Electron. QE-12, 494 (1976).
[CrossRef]

W. Streifer, D. Scifres, R. D. Burnham, IEEE J. Quantum Electron QE-12, 422 (1976).
[CrossRef]

W. Streifer, D. R. Scifres, R. D. Burnham, IEEE J. Quantum Electron. QE-11, 867 (1975).
[CrossRef]

Sychugov, V. A.

A. A. Zlenko, V. A. Kiselev, A. M. Prokhorov, A. A. Spikhalskii, V. A. Sychugov, Sov. J. Quantum Electron. 5, 1325 (1976).
[CrossRef]

Tamir, T.

T. Tamir, S. T. Peng, Appl. Phys. 14, 235 (1977).
[CrossRef]

K. Handa, S. T. Peng, T. Tamir, Appl. Phys. 5, 325 (1975).
[CrossRef]

Tang, C. L.

C. C. Ghizoni, J. M. Ballantyne, C. L. Tang, IEEE J. Quantum Electron. QE-13, 843 (1977).
[CrossRef]

C. C. Ghizoni, B. U. Chen, C. L. Tang, IEEE J. Quantum Electron. QE-12, 69 (1976).
[CrossRef]

Yamamoto, Y.

Y. Yamamoto, T. Kamiya, H. Yanai, IEEE J. Quantum Electron. QE-14, 245 (1978).
[CrossRef]

Y. Yamamoto, T. Kamiya, H. Yanai, IEEE J. Quantum. Electron. QE-14, 620 (1978).
[CrossRef]

Yanai, H.

Y. Yamamoto, T. Kamiya, H. Yanai, IEEE J. Quantum. Electron. QE-14, 620 (1978).
[CrossRef]

Y. Yamamoto, T. Kamiya, H. Yanai, IEEE J. Quantum Electron. QE-14, 245 (1978).
[CrossRef]

Zlenko, A. A.

A. A. Zlenko, V. A. Kiselev, A. M. Prokhorov, A. A. Spikhalskii, V. A. Sychugov, Sov. J. Quantum Electron. 5, 1325 (1976).
[CrossRef]

Appl. Opt. (1)

A. Basu, J. M. Ballantyne, Appl. Opt. 18, 0000 (1979).

Appl. Phys. (2)

K. Handa, S. T. Peng, T. Tamir, Appl. Phys. 5, 325 (1975).
[CrossRef]

T. Tamir, S. T. Peng, Appl. Phys. 14, 235 (1977).
[CrossRef]

IEEE J. Quantum Electron (1)

W. Streifer, D. Scifres, R. D. Burnham, IEEE J. Quantum Electron QE-12, 422 (1976).
[CrossRef]

IEEE J. Quantum Electron. (6)

W. Streifer, R. D. Burnham, D. R. Scifres, IEEE J. Quantum Electron. QE-12, 494 (1976).
[CrossRef]

W. Streifer, D. R. Scifres, R. D. Burnham, IEEE J. Quantum Electron. QE-13, 134 (1977).
[CrossRef]

Y. Yamamoto, T. Kamiya, H. Yanai, IEEE J. Quantum Electron. QE-14, 245 (1978).
[CrossRef]

W. Streifer, D. R. Scifres, R. D. Burnham, IEEE J. Quantum Electron. QE-11, 867 (1975).
[CrossRef]

C. C. Ghizoni, B. U. Chen, C. L. Tang, IEEE J. Quantum Electron. QE-12, 69 (1976).
[CrossRef]

C. C. Ghizoni, J. M. Ballantyne, C. L. Tang, IEEE J. Quantum Electron. QE-13, 843 (1977).
[CrossRef]

IEEE J. Quantum. Electron. (1)

Y. Yamamoto, T. Kamiya, H. Yanai, IEEE J. Quantum. Electron. QE-14, 620 (1978).
[CrossRef]

J. Opt. Soc. Am. (1)

Sov. J. Quantum Electron. (1)

A. A. Zlenko, V. A. Kiselev, A. M. Prokhorov, A. A. Spikhalskii, V. A. Sychugov, Sov. J. Quantum Electron. 5, 1325 (1976).
[CrossRef]

Other (2)

C. C. Ghizoni, “Wave Coupling in Periodic Optical Waveguides,” Ph.D. dissertation, Cornell University, Ithaca, New York (1976).

A. Basu, “A Study of Grating Filters in Thin-Film Optical Waveguides,” Ph.D. dissertation, Cornell University, Ithaca, New York (1979).

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Figures (4)

Fig. 1
Fig. 1

Geometry for Green’s function analysis of second- or higher-order grating filters.

Fig. 2
Fig. 2

Variation of effective index of corrugated waveguide with the groove depth, calculated by two different methods.

Fig. 3
Fig. 3

Reflection, transmission, and radiation loss coefficients of a second-order grating filter, plotted against the wavelength deviation from the Bragg wavelength λ B . Dotted curve, method of Zlenko et al.2; Solid curve, Green’s function method. Grating parameters: n1 = 1.0, n2 = 1.56, n3 = 1.515; Λ = 3918 Å, w = 5000 Å; h = 500 Å, λ B = 5965 Å, L = 1.25 mm. Groove profile: sinusoidal.

Fig. 4
Fig. 4

Reflection, transmission, and radiation loss coefficients of a second-order grating filter, plotted against the wavelength deviation from the Bragg wavelength λ B . Dotted curve, method of Streifer et al.7; solid curve, Green’s function method. Grating parameters: n1 = 1.0, n2 = 1.56, n3 = 1.515; w = 5000 Å, h = 500 Å; Λ = 3918 Å, λ B = 5965 Å, L = 1.25 mm. Grove profile: triangular.

Equations (35)

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Λ = N λ 0 / ( 2 n eff ) ,
( 2 + k 0 2 n i 2 ) E y = - k 0 2 Δ n 2 ( x , z ) E y
Δ n 2 ( x , z ) = Δ n 2 l D l ( x ) exp ( i G l z ) ,
E y = E H + d x d z Δ n 2 ( x , z ) G ( x , z x , z ) × E y ( x , z ) .
G = m G m + + m G m - + G r ,
G = G m + + G m - + G r .
E y = E m + + E m - + E r + E H .
E m + = G ^ m + ( E m + + E m - + E r + E H ) E m - = G ^ m - ( E m + + E m - + E r + E H ) E r = G ^ r ( E m + + E m - + E r + E H ) } ,
G ^ m + Ψ ( x , z ) = k 0 2 d x d z Δ n 2 ( x , z ) G m ± ( x , z x , z ) × Ψ ( x , z ) G ^ r Ψ ( x , z ) = k 0 2 d x d z Δ n 2 ( x , z ) G r ( x , z x , z ) × Ψ ( x , z ) }
E m + = G ^ m + ( E m - + E r ) E m - = G ^ m - ( E m + + E r + E H ) E r = G ^ r ( E m + + E m - + E H ) }
G ^ A E m + = G ^ B E H ,
G ^ A = 1 - G ^ m + G ^ m - - G ^ m + ( 1 + G ^ m - ) 1 1 - G ^ r G ^ m - G ^ r ( 1 + G ^ m - ) G ^ B = G ^ m + [ G ^ m - + ( 1 + G ^ m - ) 1 1 - G ^ r G ^ m - G ^ r ( 1 + G ^ m - ) ] , }
G ^ C E m - = G ^ D E H ,
G ^ C = 1 - G ^ m - G ^ m + - G ^ m - ( 1 + G ^ m + ) 1 1 - G ^ r G ^ m + G ^ r ( 1 + G ^ m + ) G ^ D = G ^ m - ( 1 + G ^ m + ) 1 1 - G ^ r G ^ m + G ^ r }
G ^ A = 1 - G ^ m + + G ^ m - - G ^ m + ( 1 + G ^ m - ) G ^ r ( G ^ E + G ^ F ) , G ^ B = G ^ m + [ G ^ m - + ( 1 + G ^ m - ) G ^ r ( G ^ E + G ^ F ) ] ,
G ^ E = 1 + G ^ m - G ^ r + G ^ m - G ^ r G ^ m - G ^ r + , G ^ F = G ^ E G ^ m - .
E H = A m [ cos ( p 2 m x ) + ( p 3 m / p 2 m ) sin ( p 2 m x ) ] exp ( i k m z ) ,
p 1 m = k m 2 - n 1 2 k 0 2 p 2 m = n 2 2 k 0 2 - k m 2 p 3 m = k m 2 - n 3 2 k 0 2 } .
E m + = [ A 1 exp ( σ 1 z ) + A 2 exp ( σ 2 z ) ] E H ,
F ( p 1 , p 2 , p 3 , q 1 , q 2 , q 3 , l ) = d x D l ( x ) { p 2 cos [ p 2 ( x - w ) ] - p 1 sin [ p 2 ( x - w ) ] } × [ cos ( q 2 x ) - i ( q 3 / q 2 ) sin ( q 2 x ) ] .
K m = k 0 2 Δ n 2 F 0 N sin ( p 2 m w ) k m w ( p 1 m + p 3 m ) ,
F 0 N = F ( p 1 m , p 2 m , p 3 m , i p 1 m , p 2 m , i p 3 m , N ) , and w = w + 1 / p 1 m + 1 / p 3 m .
γ r = k 0 4 ( Δ n 2 ) 2 sin ( p 2 m w ) k m w ( p 1 m + p 3 m ) l [ F 1 l F 2 l / Δ ( k l ) ] ,
0 k m - G l k 0 n 1 , k 0 n 3 .
ϕ m = k 0 4 ( Δ n 2 ) 2 sin ( p 2 m w ) k m w ( p 1 m + p 3 m ) l [ F 1 l F 2 l / Δ ( k l ) ] ,
Δ m = 2 k m - G n
σ 1 = ( - i Δ m + S ) / 2 σ 2 = ( - i Δ m - S ) / 2 S = [ 4 ( K m - ϕ m ) 2 - ( Δ m - 2 γ r ) 2 ] 1 / 2 A 1 = - [ S + i ( Δ m - 2 γ r ) ] exp ( - S L / 2 ) / D A 2 = - [ S + i ( Δ m - 2 γ r ) ] exp ( - S L / 2 ) / D D = - S cosh ( S L 2 ) + i ( Δ m - 2 γ r ) sinh ( S L 2 ) }
E + = E m + + E H = { - S cosh [ S 2 ( L - z ) ] + i ( Δ m - 2 γ r ) sinh [ S 2 ( L - z ) ] } × exp ( - i Δ m z / 2 ) E H / D ,
G ^ C = 1 - G ^ m - G ^ m + - G ^ m - ( 1 + G ^ m + ) G ^ r ( 1 + G ^ I ) ,
G ^ I = ( G ^ m + + G ^ r + G ^ m + G ^ r G ^ m + G ^ r + ) + ( G ^ m + + G ^ m + G ^ r G ^ m + + G ^ m + G ^ r G ^ m + G ^ r G ^ m + + ) , G ^ D = G ^ m - [ 1 + ( 1 + G ^ m + ) G ^ r G ^ J ] ,
G ^ J = 1 + G ^ m + G ^ r + G ^ m + G ^ r G m + G ^ r + .
E m - = E H - [ B 1 exp ( θ 1 z ) + B 2 exp ( θ 2 z ) ] ,
θ 1 = ( i Δ m + S ) / 2 θ 2 = ( i Δ m - S ) / 2 B 1 = - ( K m - ϕ m ) exp ( - S L / 2 ) / i D B 2 = ( K m - ϕ m ) exp ( S L / 2 ) / i D }
E m - = - 2 i ( K m - ϕ m ) sinh [ S 2 ( L - z ) ] exp ( i Δ m z / 2 ) E H - / D .
C refl = | E m - ( z = 0 ) E H | 2 = 4 K m - ϕ m 2 | sinh ( S L 2 ) | 2 D 2 C trans = | E + ( z = L ) E H | 2 = S 2 D 2 }

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