Abstract

We derive the coupled equations for the ideal modes of an optical waveguide, using a Green function technique, and use these to determine the TM–TM coupling coefficient for a periodic waveguide diffraction grating. The results are consistent with experimental observations, in marked contrast to the results of more conventional versions of the ideal mode expansion. Further, our approach can deal with more general corrugated structures than can easily be done with other methods.

© 1990 Optical Society of America

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References

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  1. A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. QE-9, 919–933 (1973).
    [Crossref]
  2. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974).
  3. H. Kogelnik, “Theory of dielectric waveguides,” in Integrated Optics, 2nd ed., T. Tamir, ed. (Springer, New York, 1982), p. 15.
  4. W. Streifer, D. R. Scifres, R. D. Burnham, “TM-mode coupling coefficients in guided-wave DFB lasers,” IEEE J. Quantum Electron. QE-12, 74–78 (1976).
    [Crossref]
  5. 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]
  6. J. van Roey, P. E. Lagasse, “Coupled wave analysis of obliquely incident waves in corrugated waveguide filters,” Appl. Opt. 20, 423–429 (1981).
    [Crossref] [PubMed]
  7. L. A. Weller-Brophy, D. G. Hall, “Local normal mode analysis of guided mode interactions with waveguide gratings,” IEEE J. Lightwave Technol. 6, 1069–1082 (1988).
    [Crossref]
  8. 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 field theory and implications to normal mode analysis,” J. Opt. Soc. Am. 71, 1497–1507 (1981).
    [Crossref]
  9. M. T. Wlodarczyk, S. R. Seshadri “Analysis of grating couplers in planar waveguides for waves at oblique incidence,” J. Opt. Soc. Am. A 2, 171–185 (1985).
    [Crossref]
  10. H. Kogelnik in Guided-Wave Optoelectronics, T. Tamir, ed. (Springer-Verlag, Berlin, 1988), p. 85.
  11. L. A. Weller-Brophy, D. G. Hall, “Measured TM–TM coupling in waveguide gratings,” Opt. Lett. 12, 756–758 (1987).
    [Crossref] [PubMed]
  12. G. A. Reider, M. Huemer, A. J. Schmidt, “Surface second harmonic generation spectroscopy without interference of substrate contributions,” Opt. Commun. 68, 149–152 (1988).
    [Crossref]
  13. J. E. Sipe, G. I. Stegeman, “Comparison of normal mode and total field analysis techniques in planar integrated optics,” J. Opt. Soc. Am. 69, 1676–1683 (1979).
    [Crossref]
  14. G. W. Ford, W. H. Weber, “Electromagnetic effects on a molecule at a metal surface,” Surf. Sci. 109, 451–481 (1981).
    [Crossref]
  15. W. Lukosz, “Light emission by multiple sources in thin layers,” J. Opt. Soc. Am. 71, 744–754 (1981).
    [Crossref]
  16. J. E. Sipe, “New Green-function formalism for surface optics,” J. Opt. Soc. Am. B 4, 481–489 (1987).
    [Crossref]
  17. This conclusion is not valid if w is too small or if |k| ≈ ν. However, a polarization with period 2π/ν can never couple two counterpropagating guided modes.

1988 (2)

L. A. Weller-Brophy, D. G. Hall, “Local normal mode analysis of guided mode interactions with waveguide gratings,” IEEE J. Lightwave Technol. 6, 1069–1082 (1988).
[Crossref]

G. A. Reider, M. Huemer, A. J. Schmidt, “Surface second harmonic generation spectroscopy without interference of substrate contributions,” Opt. Commun. 68, 149–152 (1988).
[Crossref]

1987 (2)

1985 (1)

1981 (4)

1979 (2)

J. E. Sipe, G. I. Stegeman, “Comparison of normal mode and total field analysis techniques in planar integrated optics,” J. Opt. Soc. Am. 69, 1676–1683 (1979).
[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]

1976 (1)

W. Streifer, D. R. Scifres, R. D. Burnham, “TM-mode coupling coefficients in guided-wave DFB lasers,” IEEE J. Quantum Electron. QE-12, 74–78 (1976).
[Crossref]

1973 (1)

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

Burke, J. J.

Burnham, R. D.

W. Streifer, D. R. Scifres, R. D. Burnham, “TM-mode coupling coefficients in guided-wave DFB lasers,” IEEE J. Quantum Electron. QE-12, 74–78 (1976).
[Crossref]

Ford, G. W.

G. W. Ford, W. H. Weber, “Electromagnetic effects on a molecule at a metal surface,” Surf. Sci. 109, 451–481 (1981).
[Crossref]

Hall, D. G.

Huemer, M.

G. A. Reider, M. Huemer, A. J. Schmidt, “Surface second harmonic generation spectroscopy without interference of substrate contributions,” Opt. Commun. 68, 149–152 (1988).
[Crossref]

Kogelnik, H.

H. Kogelnik in Guided-Wave Optoelectronics, T. Tamir, ed. (Springer-Verlag, Berlin, 1988), p. 85.

H. Kogelnik, “Theory of dielectric waveguides,” in Integrated Optics, 2nd ed., T. Tamir, ed. (Springer, New York, 1982), p. 15.

Lagasse, P. E.

Lukosz, W.

Marcuse, D.

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974).

Reider, G. A.

G. A. Reider, M. Huemer, A. J. Schmidt, “Surface second harmonic generation spectroscopy without interference of substrate contributions,” Opt. Commun. 68, 149–152 (1988).
[Crossref]

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]

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]

Sarid, D.

Schmidt, A. J.

G. A. Reider, M. Huemer, A. J. Schmidt, “Surface second harmonic generation spectroscopy without interference of substrate contributions,” Opt. Commun. 68, 149–152 (1988).
[Crossref]

Scifres, D. R.

W. Streifer, D. R. Scifres, R. D. Burnham, “TM-mode coupling coefficients in guided-wave DFB lasers,” IEEE J. Quantum Electron. QE-12, 74–78 (1976).
[Crossref]

Seshadri, S. R.

Sipe, J. E.

Stegeman, G. I.

Streifer, W.

W. Streifer, D. R. Scifres, R. D. Burnham, “TM-mode coupling coefficients in guided-wave DFB lasers,” IEEE J. Quantum Electron. QE-12, 74–78 (1976).
[Crossref]

van Roey, J.

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]

Weber, W. H.

G. W. Ford, W. H. Weber, “Electromagnetic effects on a molecule at a metal surface,” Surf. Sci. 109, 451–481 (1981).
[Crossref]

Weller-Brophy, L. A.

L. A. Weller-Brophy, D. G. Hall, “Local normal mode analysis of guided mode interactions with waveguide gratings,” IEEE J. Lightwave Technol. 6, 1069–1082 (1988).
[Crossref]

L. A. Weller-Brophy, D. G. Hall, “Measured TM–TM coupling in waveguide gratings,” Opt. Lett. 12, 756–758 (1987).
[Crossref] [PubMed]

Wlodarczyk, M. T.

Yariv, A.

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

Appl. Opt. (1)

IEEE J. Lightwave Technol. (1)

L. A. Weller-Brophy, D. G. Hall, “Local normal mode analysis of guided mode interactions with waveguide gratings,” IEEE J. Lightwave Technol. 6, 1069–1082 (1988).
[Crossref]

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, “TM-mode coupling coefficients in guided-wave DFB lasers,” IEEE J. Quantum Electron. QE-12, 74–78 (1976).
[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. (3)

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

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

Opt. Commun. (1)

G. A. Reider, M. Huemer, A. J. Schmidt, “Surface second harmonic generation spectroscopy without interference of substrate contributions,” Opt. Commun. 68, 149–152 (1988).
[Crossref]

Opt. Lett. (1)

Surf. Sci. (1)

G. W. Ford, W. H. Weber, “Electromagnetic effects on a molecule at a metal surface,” Surf. Sci. 109, 451–481 (1981).
[Crossref]

Other (4)

This conclusion is not valid if w is too small or if |k| ≈ ν. However, a polarization with period 2π/ν can never couple two counterpropagating guided modes.

H. Kogelnik in Guided-Wave Optoelectronics, T. Tamir, ed. (Springer-Verlag, Berlin, 1988), p. 85.

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974).

H. Kogelnik, “Theory of dielectric waveguides,” in Integrated Optics, 2nd ed., T. Tamir, ed. (Springer, New York, 1982), p. 15.

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Figures (3)

Fig. 1
Fig. 1

Schematic of the ideal waveguide. The media are interchangeably denoted either by their refractive index (as in this figure) or by their relative dielectric constant.

Fig. 2
Fig. 2

Schematic of the grating structure that we consider. The figure is slightly misleading since the thickness of the deposited material is much less than suggested. The grating rulings are perpendicular to the z axis.

Fig. 3
Fig. 3

Top view of the reflection of a TM waveguide mode and its associated electric fields off a diffraction grating at an oblique angle of incidence. The directions of the field components are in accordance with Eqs. (2.4). Note that for this scattering process the angle of incidence equals the angle of reflection. This will not be so for a scattering process involving two different modes.

Equations (78)

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f ( r , t ) = f ( r ) e i ω t + c . c . ,
× E ( r ) i ω μ 0 H ( r ) = 0 ,
× H ( r ) + i ω ( x ) E ( r ) = i ω P ( r ) ,
F ( r ) = [ E l ( r ) H l ( r ) E t ( r ) H t ( r ) ] .
F e ( r ) = a ± e [ 0 ± h l e ( x ) ± e t e ( x ) h t e ( x ) ] exp ( ± i β e z ) , F m ( r ) = a ± m [ ± e l m ( x ) 0 e t m ( x ) ± h t m ( x ) ] exp ( ± i β m z ) ,
e t e ( x ) = e t e ( x ) ŷ , h t e ( x ) = h t e ( x ) x ̂ , e t m ( x ) = e t m ( x ) x ̂ , h t m ( x ) = h t m ( x ) ŷ .
· d x e t i ( x ) × [ h t j ( x ) ] = 2 S δ i j ,
P z ( x , z ) = P z ( x ) δ ( z z ) ,
P ( x , z ) = d z P z ( x , z ) ,
E ( r ) = ϑ ( z z ) E + ( x , z ) + ϑ ( z z ) E ( x , z ) + ( x ) δ ( z z ) , H ( r ) = ϑ ( z z ) H + ( x , z ) + ϑ ( z z ) H ( x , z ) ,
( x ) = p l ( x ) / ( x )
× [ E + ( x , z ) E ( x , z ) ] = ŷ d d x [ p l ( x ) ( x ) ] , × [ H + ( x , z ) H ( x , z ) ] = i ω p t ( x ) .
a ± e = i ω S exp ( i β e z ) d x e t e ( x ) · p t ( x ) .
a ± m = i ω S exp ( i β m z ) d x [ ± e l m ( x ) p l ( x ) + e t m ( x ) · p t ( x ) ] .
e ± e ( x ) = e t e ( x ) , e ± m ( x ) = e t m ( x ) ± e l m ( x ) .
a ± n = i ω S exp ( i β n z ) d x e t n ( x ) · p ( x ) ,
E ( x , z ) = P ( x , z ) · ( x ) + i ω S n d x d z × { exp [ i β n ( z z ) ] ϑ ( z z ) e + n ( x ) e + n ( x ) + exp [ i β n ( z z ) ] ϑ ( z z ) × e n ( x ) e n ( x ) } P ( x , z ) ,
P ( x , z ) = P ( x , k z ) exp ( i k z z ) ,
E ( x , z ) = E ( x , k z ) exp ( i k z z ) ,
E ( x , k z ) = d x G ( x , x ; k z ) · P ( x , k z )
G ( x , x , k z ) = δ ( x x ) ( x ) + ω S n [ e + n ( x ) e + n ( x ) k z β n e n ( x ) e n ( x ) k z + β n ] .
f ± m ( x , k ̂ ) = x ̂ e t m ( x ) ± k ̂ e l m ( x ) ,
e + n ( x ) e + n ( x ) k z β n e n ( x ) e n ( x ) k z + β n = f + n ( x , k ̂ ) f + n ( x , k ̂ ) k β n f n ( x , k ̂ ) f n ( x , k ̂ ) k + β n
f ± e ( x , k ̂ ) = ± ( k ̂ × x ̂ ) e e t ( x ) ,
P ( x , R ) = P ( x , k ) exp ( i k · R ) ,
E ( x , R ) = E ( x , k ) exp ( i k · R ) ,
E ( x , k ) = d x G ( x , x ; k ) · P ( x , k ) ,
G ( x , x ; k ) = k ̂ k ̂ δ ( x x ) ( x ) + ω S n [ f + n ( x , k ) f + n ( x , k ̂ ) k β n f n ( x , k ̂ ) f n ( x , k ̂ ) k + β n ]
P ( x , R ) = d k ( 2 π ) 2 P ( x , k ) exp ( i k · R ) ,
E ( x , R ) = d k ( 2 π ) 2 E ( x , k ) exp ( i k · R ) ,
k 0 = β n ( ω 0 ) ,
[ k β n ( ω ) ] E ( x , k ; ω ) = k ̂ k ̂ ( x ; ω ) [ k β n ( ω ) ] P ( x , k ; ω ) + ω S n d x [ k β n ( ω ) k β n ( ω ) ] f + n ( x , k ̂ ; ω ) f + n ( x , k ̂ ; ω ) · P ( x , k ; ω ) + ω S n d x [ k β n ( ω ) k + β n ( ω ) ] × f n ( x , k ̂ ; ω ) f n ( x , k ̂ ; ω ) · P ( x , k ; ω ) .
[ k β n ( ω ) ] E ( x , k ; ω ) = ω 0 S d x f + n ( x , k 0 ; ω 0 ) f + n ( x , k ̂ 0 ; ω 0 ) · P ( x , k ; ω ) .
P ( x , k ; t ) = d ω 2 π P ( x , k ; ω ) exp ( i ω t ) ,
E ( x , k ; t ) = d ω 2 π E ( x , k ; ω ) exp ( i ω t ) ,
d k ( 2 π ) 2 d ω 2 π [ k β n ( ω ) ] E ( x , k ; ω ) exp ( i k · R ) exp ( i ω t ) = ω 0 S d x f + n ( x , k ̂ 0 ; ω 0 ) f + n ( x , k ̂ 0 ; ω 0 ) · P ( x , R ; t ) .
P ( x , R ; t ) = P 0 ( x , R ; t ) exp ( i k 0 · R ) exp ( i ω 0 t ) ,
d k ( 2 π ) 2 d ω 2 π [ k β n ( ω ) ] E ( x , k ; ω ) exp [ i ( k k 0 ) · R ] × exp [ i ( ω ω 0 ) t ] = ω 0 S d x f + n ( x , k ̂ 0 ; ω 0 ) × f + n ( x , k ̂ 0 ; ω 0 ) · P 0 ( x , R ; t ) .
( k k 0 ) [ β n ( ω ) β n ( ω 0 ) ] ,
( k k 0 ) 1 c [ c β n ( ω ) ω ] ω = ω 0 ( ω ω 0 ) ,
t i ( ω ω 0 ) ,
η n 0 c ( ω ω 0 ) i η n 0 c t ,
( k k 0 ) i ( k ̂ 0 · i | k ̂ 0 × | 2 2 k 0 ) ,
ŷ y + z
i O d k ( 2 π ) 2 d ω 2 π E ( x , k ; ω ) exp [ i ( k k 0 ) · R ] × exp [ i ( ω ω 0 ) t ] = ω 0 S f + n ( x , k ̂ 0 ; ω 0 ) d x f + n ( x , k ̂ 0 ; ω 0 ) · P 0 ( x , R ; t ) ,
O k ̂ 0 · i | k ̂ 0 × | 2 2 k 0 + η n 0 c t .
E ( x , R ; t ) = n ( k 0 , R ; t ) f + n ( x , k ̂ 0 ; ω 0 ) × exp ( i k 0 · R ) exp ( i ω 0 t ) + c . c . ,
O n ( k 0 , R ; t ) = i ω 0 S d x f + n ( x , k ̂ 0 ; ω 0 ) · P ( x , R ; t ) .
E ( x , R ; t ) = q j q ( k j , R ; t ) f + q ( x , k ̂ j ; ω j ) × exp ( i k j · R ) exp ( i ω j t ) + c . c . ,
P ( x , R ; t ) = q j σ ( x , R ) · q ( k j , R ; t ) f + q ( x , k ̂ j ; ω j ) × exp ( i k j · R ) exp ( i ω j t ) + c . c . ,
σ ( x , R ) = { 0 ( b c ) ( c b x ̂ x ̂ + ŷ ŷ + ) if = 0 b 0 otherwise .
O n ( k 0 , R ; t ) = i q j C n , 0 q , j ( R ) exp [ i ( k 0 k j ) · R ] × exp [ i ( ω 0 ω j ) t ] q ( k j , R ; t ) ,
C n , 0 q , j ( R ) = ω 0 S d x f + n ( x , k ̂ 0 ; ω 0 ) · σ ( x , R ) · f + q ( x , k ̂ j ; ω j ) .
f + n ( x , k ̂ 0 ; ω 0 ) = x ̂ e t m ( x ) + e l m ( x ) , f + q ( x , k ̂ j ; ω j ) = x ̂ e t m ( x ) e l m ( x ) .
rect ( z / d ) = { 1 ( 4 n 1 ) / 4 z / d < ( 4 n + 1 ) / 4 0 ( 4 n + 1 ) / 4 z / d < ( 4 n + 3 ) / 4 ( n = 0 , ± 1 , ± 2 , .
C ( z ) = 0 ( b c ) rect ( z d ) ω 0 S h h + δ h d x ( x ̂ e t m + z e l m ) · ( c b x ̂ x ̂ + ŷ ŷ + ) · ( x ̂ e t m e l m ) ,
2 π λ n f 2 N TM 2 N TM δ h h effTM ( 1 N TM 2 n c 2 ) 1 q c n b 2 n c 2 n f 2 n c 2 rect ( z / d ) ,
2 π λ n f 2 N TM 2 N TM δ h h effTM N TM 2 n f 2 1 q c n b 2 n c 2 n f 2 n c 2 rect ( z / d ) ,
q c = N TM 2 n c 2 + N TM 2 n f 2 1.
C = 2 π λ n f 2 N TM 2 N TM δ h h effTM [ ( 1 N TM 2 n c 2 ) cos ( 2 θ ) + N TM 2 n f 2 ] × 1 q c n b 2 n c 2 n f 2 n c 2 rect ( z d ) .
C = κ [ exp ( 2 π i z / d ) + exp ( 2 π i z / d ) ]
κ TM TM = π λ n f 2 N TM 2 N TM Δ h h effTM × [ ( 1 N TM 2 n c ) cos ( 2 θ ) + N TM 2 n f 2 ] 1 q c ,
κ TE TE = π λ n f 2 N TE 2 N TE Δ h η effTE cos ( 2 θ ) ,
κ TE TM = κ TM TM = i π λ Δ h ( h e ffTE ) 1 / 2 ( h e ffTM ) 1 / 2 ( 1 q c ) 1 / 2 × ( ( N TM 2 n c 2 ) 1 / 2 n c ) ( n f 2 N TE 2 N TE ) 1 / 2 × ( n f 2 N TE 2 N TM ) 1 / 2 sin ( θ TM + θ TE ) ,
P new = 0 ( b 1 ) E b new ( x ̂ x ̂ + ŷ ŷ + ) ,
( b x ̂ x ̂ + ŷ ŷ + ) E b new = ( f x ̂ x ̂ + ŷ ŷ + ) E f new ,
E f new E f old .
E c old = ( f x ̂ x ̂ + ŷ ŷ + ) E f old ,
P new 0 ( b 1 ) ( 1 b x ̂ x ̂ + ŷ ŷ + ) E c old .
P s 0 ( b 1 ) ( 1 b x ̂ x ̂ ̂ + ŷ ŷ + ) E c old ,
G ( x , k ) = 1 0 [ i ω 2 2 w c 2 ( ŝ ŝ + p ̂ + p ̂ + ϑ ( x ) e iwx + i ω 2 2 w c 2 ( ŝ ŝ ̂ + p ̂ p ̂ ) ϑ ( x ) e iwx x ̂ x ̂ c δ ( x ) ] ,
P ind = 0 ( c 1 ) E = c 1 c P s x ̂ x ̂ .
Δ P = P s ( 1 c x ̂ x ̂ + ŷ ŷ + ) .
( c x ̂ x ̂ + ŷ ŷ + ) E c old = ( f x ̂ x ̂ + ŷ ŷ + ) E f old ,
P new 0 ( b 1 ) ( c b x ̂ x ̂ + ŷ ŷ + ) E c old .
P old = 0 ( c 1 ) E c old ,
Δ P P new P old 0 ( b c ) ( 1 b x ̂ x ̂ + ŷ ŷ + ) E c new .
P s = 0 ( b c ) ( c b x ̂ x ̂ + ŷ ŷ + ) E c old .

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