Abstract

Normal mode and total field analysis techniques for calculating the fields generated by a polarization source are discussed. The amplitudes of the normal modes are evaluated using a Green’s-function approach and the results are compared to a previous treatment by Yariv. The generated fields are also calculated by solving the polarization driven wave equation subject to the usual electromagnetic boundary conditions at all of the pertinent interfaces. It is shown that these two approaches yield equivalent results for a simple case, and their relative merits for solving problems in different regions of a waveguide are discussed.

© 1979 Optical Society of America

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References

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  1. R. V. Schmidt, “Acoustooptic Interactions Between Guided Optical Waves and Acoustic Surfaces Waves,” IEEE Trans. Sonics Ultrason. SU-23, 22–30 (1976).
    [Crossref]
  2. E. M. Conwell, “Theory of Second-Harmonic Generation in Optical Waveguides,” IEEE J. Quantum Electron. QE-10, 608–612 (1974).
    [Crossref]
  3. P. K. Tien, “Integrated Optics and New Wave Phenomena in Optical Waveguides,” Rev. Mod. Phys. 49, 361–420 (1977).
    [Crossref]
  4. D. Marcuse, Theory of Dielectric Waveguides (Academic, New York1974).
  5. A. Yariv, “Coupled Mode Theory for Guided Wave Optics,” IEEE J. Quantum Electron. 9, 919–933 (1973).
    [Crossref]
  6. A. Yariv, Quantum Electronics, 2nd edition (Wiley, New York, 1975), Chap. 19.
  7. R. Normandin, V. C. Y. So, N. Rowell, and G. I. Stegeman, “Scattering of guided optical beams by surface acoustic waves in thin films,” J. Opt. Soc. Am. 69, 1153–1165 (1979).
    [Crossref]
  8. V. C. Y. So, R. Normandin, and G. I. Stegeman, “Field Analysis of Harmonic Generation in Thin Film Integrated Optics,” J. Opt. Soc. Am. 69, 1166–1171 (1979).
    [Crossref]
  9. M. Born and E. Wolfe, Principles of Optics (Macmillan, New York1964).
  10. Eq. (25) is derived in Yariv5,6 from the equation∇2E(r,t)=μ∊(z)∂2E(r,t)∂t2+μ∂2∂t2P(r,t), which follows from the Maxwell equations only if ∇·E(r,t) = 0. Such an assumption is, as shown above, not necessary to derive Eq. (25).
  11. K. W. Loh, W. S. C. Chang, W. R. Smith, and T. Grudkowski, “Bragg coupling efficiency for guided acoustooptic interaction in GaAs,” Appl. Opt. 15, 156–166 (1976).
    [Crossref] [PubMed]

1979 (2)

1977 (1)

P. K. Tien, “Integrated Optics and New Wave Phenomena in Optical Waveguides,” Rev. Mod. Phys. 49, 361–420 (1977).
[Crossref]

1976 (2)

R. V. Schmidt, “Acoustooptic Interactions Between Guided Optical Waves and Acoustic Surfaces Waves,” IEEE Trans. Sonics Ultrason. SU-23, 22–30 (1976).
[Crossref]

K. W. Loh, W. S. C. Chang, W. R. Smith, and T. Grudkowski, “Bragg coupling efficiency for guided acoustooptic interaction in GaAs,” Appl. Opt. 15, 156–166 (1976).
[Crossref] [PubMed]

1974 (1)

E. M. Conwell, “Theory of Second-Harmonic Generation in Optical Waveguides,” IEEE J. Quantum Electron. QE-10, 608–612 (1974).
[Crossref]

1973 (1)

A. Yariv, “Coupled Mode Theory for Guided Wave Optics,” IEEE J. Quantum Electron. 9, 919–933 (1973).
[Crossref]

Born, M.

M. Born and E. Wolfe, Principles of Optics (Macmillan, New York1964).

Chang, W. S. C.

Conwell, E. M.

E. M. Conwell, “Theory of Second-Harmonic Generation in Optical Waveguides,” IEEE J. Quantum Electron. QE-10, 608–612 (1974).
[Crossref]

Grudkowski, T.

Loh, K. W.

Marcuse, D.

D. Marcuse, Theory of Dielectric Waveguides (Academic, New York1974).

Normandin, R.

Rowell, N.

Schmidt, R. V.

R. V. Schmidt, “Acoustooptic Interactions Between Guided Optical Waves and Acoustic Surfaces Waves,” IEEE Trans. Sonics Ultrason. SU-23, 22–30 (1976).
[Crossref]

Smith, W. R.

So, V. C. Y.

Stegeman, G. I.

Tien, P. K.

P. K. Tien, “Integrated Optics and New Wave Phenomena in Optical Waveguides,” Rev. Mod. Phys. 49, 361–420 (1977).
[Crossref]

Wolfe, E.

M. Born and E. Wolfe, Principles of Optics (Macmillan, New York1964).

Yariv, A.

A. Yariv, “Coupled Mode Theory for Guided Wave Optics,” IEEE J. Quantum Electron. 9, 919–933 (1973).
[Crossref]

A. Yariv, Quantum Electronics, 2nd edition (Wiley, New York, 1975), Chap. 19.

Appl. Opt. (1)

IEEE J. Quantum Electron. (2)

E. M. Conwell, “Theory of Second-Harmonic Generation in Optical Waveguides,” IEEE J. Quantum Electron. QE-10, 608–612 (1974).
[Crossref]

A. Yariv, “Coupled Mode Theory for Guided Wave Optics,” IEEE J. Quantum Electron. 9, 919–933 (1973).
[Crossref]

IEEE Trans. Sonics Ultrason. (1)

R. V. Schmidt, “Acoustooptic Interactions Between Guided Optical Waves and Acoustic Surfaces Waves,” IEEE Trans. Sonics Ultrason. SU-23, 22–30 (1976).
[Crossref]

J. Opt. Soc. Am. (2)

Rev. Mod. Phys. (1)

P. K. Tien, “Integrated Optics and New Wave Phenomena in Optical Waveguides,” Rev. Mod. Phys. 49, 361–420 (1977).
[Crossref]

Other (4)

D. Marcuse, Theory of Dielectric Waveguides (Academic, New York1974).

A. Yariv, Quantum Electronics, 2nd edition (Wiley, New York, 1975), Chap. 19.

M. Born and E. Wolfe, Principles of Optics (Macmillan, New York1964).

Eq. (25) is derived in Yariv5,6 from the equation∇2E(r,t)=μ∊(z)∂2E(r,t)∂t2+μ∂2∂t2P(r,t), which follows from the Maxwell equations only if ∇·E(r,t) = 0. Such an assumption is, as shown above, not necessary to derive Eq. (25).

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Figures (1)

FIG. 1
FIG. 1

Thin film waveguide with a polarization source confined to the film between x = 0 and x = L. The a ± m ( x ) are growing normal mode amplitudes obtained from normal mode analysis and the a ± 0 m and a ± L m are the mode amplitudes generated at the boundaries in the total field approach.

Equations (76)

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f ( r , t ) = 1 2 [ f ( r ) e i ω t + c . c . ] = Re [ f ( r ) e i ω t ] ,
× E ( r ) + i ω μ H ( r ) = 0 , × H ( r ) i ω ( z ) E ( r ) = i ω P ( r ) ,
· ( z ) E ( r ) = · P ( r ) , · μ H ( r ) = 0 ,
F = [ E x ( x , z ) H x ( x , z ) E t ( x , z ) H t ( x , z ) ] ,
P ( x , z ) = p ( z ) δ ( x x ) .
F ± = [ E x ± ( x , z ) H x ± ( x , z ) E t ± ( x , z ) H t ± ( x , z ) ] ,
× E ± + i ω μ H ± = 0 , × H ± i ω ( z ) E ± = 0 ,
E ( r ) = θ ( x x ) E + ( x , z ) + θ ( x x ) E ( x , z ) + e ( z ) x ˆ δ ( x x ) H ( r ) = θ ( x x ) H + ( x , z ) + θ ( x x ) H ( x , z ) ,
e ( z ) = p x ( z ) / ( z ) ,
x ˆ × [ E + ( x , z ) E ( x , z ) ] = y ˆ d d z [ p x ( z ) / ( z ) ] , x ˆ × [ H + ( x , z ) H ( x , z ) ] = i ω p t ( z ) .
F ± m = [ 0 H x m ( z ) E t m ( z ) ± H t m ( z ) ] e i β m x ,
E y m ( z ) = C m e q m z = C m { cos ( h m z ) + ( q m / h m ) sin ( h m z ) } = C m { cos ( h m d ) + ( q m / h m ) sin ( h m d ) } e p m ( z d ) . air film substrate
( z ) = 1 = 2 = 3 air film substrate
tan h m d = ( q m + p m ) / h m ( 1 p m q m h m 2 ) .
1 2 + ( E t m × H t n ) · x ˆ d z = ( 1 W / m 2 ) δ m n ,
C m = 2 h [ β m ω μ ( d + 1 q m + 1 p m ) ( h m 2 + q m 2 ) ] 1 / 2 ( 1 W / m 2 ) 1 / 2 .
1 2 + ( E t m × H t 0 ) · x ˆ d z = 1 2 + ( E t 0 × H t m ) · x ˆ d z = 0 ,
F ± = F ± 0 + m a ± m F ± m ,
a ± m = ( i ω / 4 ) e ± i β n ι x + P ( z ) · E t m ( z ) d z .
P ( x , z ) = p ( x , z ) δ ( x x ) d x ,
a + m ( x ) = x e i β m x s m ( x ) d x , a m ( x ) = x e i β m x s m ( x ) d x ,
s m ( x ) = i ω 4 + P ( x , z ) · E m ( z ) d z .
a ± m = 0 L e ± i β m x s m ( x ) d x .
× ( × E ( r ) ) = ω 2 μ ( z ) E ( r ) + ω 2 μ P ( r ) .
( 2 x 2 + 2 z 2 ) E v ( x , z ) = ω 2 μ ( z ) E y ( x , z ) ω 2 μ P y ( x , z ) ,
E y ( x , z ) = m a + m ( x ) e i β m x E y m ( z ) + m a m ( x ) e i β m x E y m ( z ) + E ¯ y ( x , z ) ,
e i β m x d c + m ( x ) d x e i β m x d c m ( x ) d x = s m ( x ) ,
c ± m ( x ) = a ± m ( x ) + i 2 β m d a ± m ( x ) d x .
c ± m ( x ) a ± m ( x ) ,
e i β m x d a + m ( x ) d x e i β m x d a m ( x ) d x s m ( x ) .
e i β m x d a + m ( x ) d x s m ( x ) ,
a + m ( x ) x e i β m x s m ( x ) d x .
e i β m x d a + m ( x ) d x = e i β m x d a m ( x ) d x = s m ( x ) .
s m ( x ) = 0 x < 0 x > L , s m ( x ) = e i β m x 0 x L .
a + m ( x ) = x 0 < x < L ,
c + m ( x ) = x + i 2 β m 0 < x < L ,
a m ( x ) = e 2 i β m x 1 2 i β m ,
F ± m = [ E x m ( z ) 0 ± E t m ( z ) H t m ( z ) ] e i β m x ,
H y m ( z ) = ( C m h m / q ¯ ) e q m z = C m { ( h m / q ¯ m ) cos ( h m z ) + sin ( h m z ) } = C m { ( h m / q ¯ m ) cos ( h m d ) + sin ( h m d ) } e p m ( z d ) air film substrate
tan h m d = ( p ¯ m + q ¯ m ) / h m ( 1 p ¯ m q ¯ m h m 2 ) ,
C m = 2 q ¯ m [ β m ω ( d 2 + h m 2 + q m 2 h m 2 + q ¯ m 2 1 q m 1 + h m 2 + p m 2 h m 2 + p ¯ m 2 1 q m 3 ) ] 1 / 2 ( 1 W / m 2 ) 1 / 2 .
b + m ( x ) = x e i β m x s + m ( x ) d x , b m ( x ) = x e i β m x s m ( x ) d x ,
s ± m ( x ) = i ω 4 + [ P x ( x , z ) x ˆ ± P t ( x , z ) ] · E m ( z ) d z .
b ± m ( x ) = 0 L e ± i β m x s ± m ( x ) d x .
2 π ( n 2 / c 2 ) π ¨ = P / 0 n 2 ,
E p = × ( × π ) P 0 n 2 , D p = 0 n 2 E p + P , H p = n 2 μ c 2 × π ˙ , B p = μ H p ,
P ( r ) = P ( z ) e i β x ,
[ E y a , s , E x a , s , H y a , s , H x a , s ] e i β x ,
E ( x , z ) = y ˆ C m exp ( i β m x + q m z ) air = y ˆ C m E m f exp ( i β m x ) × [ exp ( i h m z ) + exp ( i h m z + i ϕ m ) ] film = y ˆ C m E m s exp [ i β m x p m ( z d ) ] , susbstrate
E ( x , z ) = y ˆ C m D ( 1 ) exp ( i β x + q z ) , air = y ˆ C m E m f exp ( i β x ) × [ D ( 2 ) exp ( i h z ) + D ( 3 ) exp ( i h z + i ϕ m ) ] , film = y ˆ C m E m s D ( 4 ) exp [ i β x p ( z d ) ] , susbstrate
y ˆ C m D exp ( i β x + q z ) { 1 exp [ Δ β ( i x q ˜ z ) ] } / Δ β , air y ˆ C m E m f D exp ( i β x ) [ exp ( i h z ) { 1 exp [ i Δ β ( x + h ˜ z ) ] } / Δ β + exp ( i h z + i ϕ m ) { 1 exp [ i Δ β ( x h ˜ z ) ] } / Δ β ] , film y ˆ C m E m s D exp [ i β x p ( z d ) ] { 1 exp [ Δ β ( i x + p ˜ z ) ] } / Δ β ,
D 1 = ( 1 + i q / h ) ( ω μ H x s i p E y s ) + e i h d ( 1 i p / h ) ( ω μ H x a + i q E y a ) , D 2 = ( 1 + i q / h ) ( ω μ H x s i p E y s ) + e i h d ( 1 i p / h ) ( ω μ Δ H x a + i q Δ E y a ) ,
a + 0 m = 1 2 Z m [ Δ E y ( 0 + , z ) + ω μ β m Δ H z ( 0 + , z ) ] E y m * ( z ) d z , a 0 m = 1 2 Z m [ Δ E y ( 0 + , z ) ω μ β m Δ H z ( 0 + , z ) ] E y m * ( z ) d z ,
a + L m = e i β m L 2 Z m [ Δ E y ( L , z ) + ω μ β m Δ H z ( L , z ) ] E y m * ( z ) d z , a L m = e i β m L 2 Z m [ Δ E y ( L , z ) ω μ β m Δ H z ( L , z ) ] E y m * ( z ) d z ,
Z m = E y m ( z ) E y m * ( z ) d z
a + m = a + 0 m + a + L m ,
a m = a 0 m + a L m .
y ˆ C m D exp ( i β x + q z ) { i x + β z / q } , air y ˆ C m E m f D exp ( i β x ) { i x [ exp ( i h z ) + exp ( i h z + i ϕ ) ] β z [ exp ( i h z ) exp ( i h z + i ϕ ) ] / h } , film y ˆ C m E m f D exp [ i β x p ( z d ) ] { i x β ( z d ) / p } , substrate
D 2 = β ( h + i q ) ( h 2 + q 2 ) 1 / 2 ( h 2 + p 2 ) 1 / 2 i h 3 ( d + p 1 + q 1 ) .
x M = Max ( β / q 2 , β d / h , β / p 2 ) .
a + m = i D L ( x > L ) a m = ( D / 2 β ) { e 2 i β L 1 } ( x < 0 ) .
H = y ˆ C m H m a exp ( i β m x + q m z ) , air = y ˆ C m H m f exp ( i β m x ) { exp ( i h m x ) + exp ( i h m z + i ϕ m ) } , film = y ˆ C m H m s exp [ i β m x p m ( z d ) ] substrate
D 1 = ( 1 + i q ¯ / h ¯ ) ( i p ¯ H y s ω 1 E x s ) e i h z ( 1 i p ¯ / h ¯ ) ( i q ¯ H y a ω 2 E x a ) , D 2 = ( 1 + i q ¯ / h ¯ ) ( i p ¯ H y s ω 1 E x s ) e i h z ( 1 i p ¯ / h ¯ ) ( i q ¯ Δ H y a ω 2 E x a ) ,
b + 0 m = 1 2 Y m [ Δ M y ( 0 + , z ) ω β m Δ D z ( 0 + , z ) ] H m * ( z ) d z , b 0 m = 1 2 Y m [ Δ M y ( 0 + , z ) + ω β m Δ D z ( 0 + , z ) ] H m * ( z ) d z ,
b + L m = e i β m L 2 Y m [ Δ H y ( L , z ) ω β m Δ D z ( L , z ) ] H m * ( z ) d z , b L m = e i β m L 2 Y m [ Δ H y ( L , z ) + ω β m Δ D z ( L , z ) ] H m * ( z ) d z ,
Y m = H m ( z ) H m * ( z ) d z .
b + m = b + 0 m + b + L m ,
b m = b 0 m + b L m ,
D 2 = β ( h + i q ¯ ) ( h 2 + p ¯ 2 ) 1 / 2 ( h 2 + q ¯ 2 ) 1 / 2 h 2 n f 2 q ¯ i × { h 2 + q 2 h 2 + q ¯ 2 1 n a 2 q + h 2 + p 2 h 2 + p ¯ 2 1 n s 2 p + d n f 2 } ,
b + m = i D L ( x > L ) , b m = ( D / 2 β ) { e 2 i β L 1 } ( 1 / 2 Y m ) { e 2 i β L 1 } P z ( z ) H y * m ( z ) d z ,
P ( r ) = ( x ˆ P x + y ˆ P y + z ˆ P z ) e i β x γ z ,
D = ω 2 μ h P y ( γ 2 + h 2 ) ( h 2 + q 2 ) ( h 2 + p 2 ) 1 / 2 β C m × { ( h 2 + q 2 ) ( p γ ) e γ d + ( h 2 + p 2 ) ( q + γ ) d + q 1 + p 1 } ,
D = ω h q ¯ { ( h 2 + q ¯ 2 ) 1 / 2 [ β P z ( p ¯ γ ) + i P x ( h 2 + γ p ¯ ) ] e γ d ( h 2 + p ¯ 2 ) 1 / 2 [ β P z ( q ¯ + γ ) + i P x ( h 2 γ q ¯ ) ] } β ( γ 2 + h 2 ) n f 2 ( h 2 + p ¯ 2 ) 1 / 2 ( h 2 + q ¯ 2 ) C m { h 2 + q 2 h 2 + q ¯ 2 1 q n a 2 + h 2 + p 2 h 2 + p ¯ 2 1 n p s 2 + d n f 2 } .
a + m = i D L exp ( i Δ β L / 2 ) sin ( Δ β L / 2 ) ( Δ β L / 2 ) ,
y ˆ C m D exp ( i β x + q z ) { 1 exp ( i Δ β x ) Δ β } , air y ˆ C m E m f D exp ( i β x ) { 1 exp ( i Δ β x ) Δ β } × { exp ( i h z ) + exp ( i h z + i ϕ m ) } , film y ˆ C m E m s D exp [ i β x p ( z d ) ] { 1 exp ( i Δ β x ) Δ β } , substrate
2E(r,t)=μ(z)2E(r,t)t2+μ2t2P(r,t),