Abstract

For TM-polarized waves, a mode power measure is applied to characterize nonlinear thin-film optical waveguides in an approach analogous to that we recently proposed for TE-polarized waves. For design conditions in which all the guided waves are induced by the nonlinearity of the film, we study how the power level threshold needed for wave propagation differs between the TE and the TM modes of polarization. Since our description is based on universal parameters, our results are applicable to different geometries of waveguides through simple scaling rules.

© 1992 Optical Society of America

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References

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  1. Th. Peschel, P. Dannberg, U. Langbein, F. Lederer, “Investigation of optical tunneling through nonlinear films,” J. Opt. Soc. Am. B 5, 29–36 (1988).
    [CrossRef]
  2. P. Li, Kam Wa, P. N. Robson, J. S. Roberts, M. A. Pate, J. P. R. David, “All-optical switching between modes of a GaAs/GaAlAs multiple quantum well waveguide,” Appl. Phys. Lett. 52, 2013–2014 (1988).
    [CrossRef]
  3. M. Fontaine, “Scaling rules for nonlinear thin film optical waveguides,” Appl. Opt. 29, 3891–3899 (1990). We would like to mention to the reader that expressions (29) and (30) of Ref. 3 used to generate the normalized power curves P/P0 in the TE-polarization mode were obtained by defining P as the power per unit of length, and not as the mean power per unit of length. Consequently, the factor 1/2 in definition (28) of P in Ref. 3 must be dropped.
    [CrossRef] [PubMed]
  4. S. Chelkowski, J. Chrostowski, “Scaling rules for slab waveguides with nonlinear substrate,” Appl. Opt. 26, 3681–3686 (1987).
    [CrossRef] [PubMed]
  5. A. Boardman, P. Egan, “Optically nonlinear waves in thin films,” IEEE J. Quantum Electron. QE-22, 319–324 (1986).
    [CrossRef]
  6. K. Hayata, M. Nagai, M. Koshiba, “Finite-element formalism for nonlinear slab-guides waves,” IEEE Trans. Microwave Theory Tech. 36, 1207–1215 (1988).
    [CrossRef]
  7. M. Fontaine, “Universal dispersion and power curves for transverse magnetic waves propagating in slab waveguides with a nonlinear self-focusing substrate,” J. Appl. Phys. 69, 2826–2834 (1991).
    [CrossRef]
  8. H. Kogelnik, V. Ramaswamy, “Scaling rules for thin-film optical waveguides,” Appl. Opt. 13, 1857–1862 (1974).
    [CrossRef] [PubMed]

1991 (1)

M. Fontaine, “Universal dispersion and power curves for transverse magnetic waves propagating in slab waveguides with a nonlinear self-focusing substrate,” J. Appl. Phys. 69, 2826–2834 (1991).
[CrossRef]

1990 (1)

1988 (3)

K. Hayata, M. Nagai, M. Koshiba, “Finite-element formalism for nonlinear slab-guides waves,” IEEE Trans. Microwave Theory Tech. 36, 1207–1215 (1988).
[CrossRef]

Th. Peschel, P. Dannberg, U. Langbein, F. Lederer, “Investigation of optical tunneling through nonlinear films,” J. Opt. Soc. Am. B 5, 29–36 (1988).
[CrossRef]

P. Li, Kam Wa, P. N. Robson, J. S. Roberts, M. A. Pate, J. P. R. David, “All-optical switching between modes of a GaAs/GaAlAs multiple quantum well waveguide,” Appl. Phys. Lett. 52, 2013–2014 (1988).
[CrossRef]

1987 (1)

1986 (1)

A. Boardman, P. Egan, “Optically nonlinear waves in thin films,” IEEE J. Quantum Electron. QE-22, 319–324 (1986).
[CrossRef]

1974 (1)

Boardman, A.

A. Boardman, P. Egan, “Optically nonlinear waves in thin films,” IEEE J. Quantum Electron. QE-22, 319–324 (1986).
[CrossRef]

Chelkowski, S.

Chrostowski, J.

Dannberg, P.

David, J. P. R.

P. Li, Kam Wa, P. N. Robson, J. S. Roberts, M. A. Pate, J. P. R. David, “All-optical switching between modes of a GaAs/GaAlAs multiple quantum well waveguide,” Appl. Phys. Lett. 52, 2013–2014 (1988).
[CrossRef]

Egan, P.

A. Boardman, P. Egan, “Optically nonlinear waves in thin films,” IEEE J. Quantum Electron. QE-22, 319–324 (1986).
[CrossRef]

Fontaine, M.

Hayata, K.

K. Hayata, M. Nagai, M. Koshiba, “Finite-element formalism for nonlinear slab-guides waves,” IEEE Trans. Microwave Theory Tech. 36, 1207–1215 (1988).
[CrossRef]

Kogelnik, H.

Koshiba, M.

K. Hayata, M. Nagai, M. Koshiba, “Finite-element formalism for nonlinear slab-guides waves,” IEEE Trans. Microwave Theory Tech. 36, 1207–1215 (1988).
[CrossRef]

Langbein, U.

Lederer, F.

Li, P.

P. Li, Kam Wa, P. N. Robson, J. S. Roberts, M. A. Pate, J. P. R. David, “All-optical switching between modes of a GaAs/GaAlAs multiple quantum well waveguide,” Appl. Phys. Lett. 52, 2013–2014 (1988).
[CrossRef]

Nagai, M.

K. Hayata, M. Nagai, M. Koshiba, “Finite-element formalism for nonlinear slab-guides waves,” IEEE Trans. Microwave Theory Tech. 36, 1207–1215 (1988).
[CrossRef]

Pate, M. A.

P. Li, Kam Wa, P. N. Robson, J. S. Roberts, M. A. Pate, J. P. R. David, “All-optical switching between modes of a GaAs/GaAlAs multiple quantum well waveguide,” Appl. Phys. Lett. 52, 2013–2014 (1988).
[CrossRef]

Peschel, Th.

Ramaswamy, V.

Roberts, J. S.

P. Li, Kam Wa, P. N. Robson, J. S. Roberts, M. A. Pate, J. P. R. David, “All-optical switching between modes of a GaAs/GaAlAs multiple quantum well waveguide,” Appl. Phys. Lett. 52, 2013–2014 (1988).
[CrossRef]

Robson, P. N.

P. Li, Kam Wa, P. N. Robson, J. S. Roberts, M. A. Pate, J. P. R. David, “All-optical switching between modes of a GaAs/GaAlAs multiple quantum well waveguide,” Appl. Phys. Lett. 52, 2013–2014 (1988).
[CrossRef]

Wa, Kam

P. Li, Kam Wa, P. N. Robson, J. S. Roberts, M. A. Pate, J. P. R. David, “All-optical switching between modes of a GaAs/GaAlAs multiple quantum well waveguide,” Appl. Phys. Lett. 52, 2013–2014 (1988).
[CrossRef]

Appl. Opt. (3)

Appl. Phys. Lett. (1)

P. Li, Kam Wa, P. N. Robson, J. S. Roberts, M. A. Pate, J. P. R. David, “All-optical switching between modes of a GaAs/GaAlAs multiple quantum well waveguide,” Appl. Phys. Lett. 52, 2013–2014 (1988).
[CrossRef]

IEEE J. Quantum Electron. (1)

A. Boardman, P. Egan, “Optically nonlinear waves in thin films,” IEEE J. Quantum Electron. QE-22, 319–324 (1986).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

K. Hayata, M. Nagai, M. Koshiba, “Finite-element formalism for nonlinear slab-guides waves,” IEEE Trans. Microwave Theory Tech. 36, 1207–1215 (1988).
[CrossRef]

J. Appl. Phys. (1)

M. Fontaine, “Universal dispersion and power curves for transverse magnetic waves propagating in slab waveguides with a nonlinear self-focusing substrate,” J. Appl. Phys. 69, 2826–2834 (1991).
[CrossRef]

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

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Figures (8)

Fig. 1
Fig. 1

Nonlinear slab waveguide configuration.

Fig. 2
Fig. 2

Universal dispersion curves for a symmetrical (a = 0) thin-film optical waveguide for TM0 and TM1 modes. The curves are labeled with different values of b1 = b2. The solid curves correspond to configurations where γ = 1.001, and the dotted curves correspond to configurations where γ = 1.01.

Fig. 3
Fig. 3

Relation between parameter b1 (equal to b2) and the normalized power flow P/P0 in the TM0 mode. The solid curves correspond to configurations where γ = 1.001, and the dotted curves correspond to configurations where γ = 1.01. The asymmetry coefficient is 0.

Fig. 4
Fig. 4

Relation between parameter b1 (equal to b2) and the normalized power flow P/P0 in the TM1 mode. The solid curves correspond to configurations where γ = 1.001, and the dotted curves correspond to configurations where γ = 1.01. The asymmetry coefficient is 0.

Fig. 5
Fig. 5

Relation between the effective index and the normalized power flow P/P0 in the TM1 mode for V = 3. The asymmetry coefficient is 0.

Fig. 6
Fig. 6

Relation between the effective index and the normalized power flow P/P0 in the TM1 mode for V = 4. The asymmetry coefficient is 0.

Fig. 7
Fig. 7

Relation between the effective index and the normalized power flow P/P0 in the TM1 mode for V = 5. The asymmetry coefficient is 0.

Fig. 8
Fig. 8

Relation between the effective index and the normalized power flow P/P0 in the TM1 mode for V = 5 for an asymmetrical waveguide.

Equations (40)

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V = 2 π λ d ( n f 2 n s 2 ) 1 / 2 = k 0 d ( n f 2 n s 2 ) 1 / 2 ,
a = ( n f n c ) 4 ( n s 2 n c 2 n f 2 n s 2 ) ,
γ = n s / n f .
b = 1 γ 2 q s ( N 2 n s 2 n f 2 n s 2 ) , N = β / k 0 , q s = N 2 n f 2 + N 2 n s 2 1 ,
q s = ( n s / n f ) 2 1 b + b ( n s / n f ) 4 = γ 2 1 b + b γ 4 .
V = 2 π λ d ( n s 2 n f 2 ) 1 / 2 = k 0 d ( n s 2 n f 2 ) 1 / 2 ,
a = ( n f n c ) 4 ( n s 2 n c 2 n s 2 n f 2 ) ,
γ = n s / n f ( γ > 1 ) ,
b = 1 γ 2 q s ( N 2 n s 2 n s 2 n s 2 ) , N = β / k 0 .
q s = N 2 n f 2 + N 2 n s 2 1 = n s / n f ) 2 1 + b b ( n s / n f ) 4 = γ 2 1 + b b γ 4 .
ω 0 y E y = β H x ,
ω 0 z E z = d H x d y ,
β E y ωμ 0 H x = d E z d y .
h x ( i ) ( y ) = { H x ( i ) ( y ) H 0 ω 0 d [ n ( i ) ] 2 } ,
f z ( i ) ( y ) = [ E z ( i ) ( y ) H 0 ] .
f z ( 1 ) ( y ) = A exp [ ( V 2 b γ 4 1 + b b γ 4 ) 1 / 2 y ] ,
h x ( 1 ) ( y ) = A ( 1 + b b γ 4 ) 1 / 2 V b 1 / 2 γ 2 exp [ ( V 2 b γ 4 1 + b b γ 4 ) 1 / 2 y ] ,
f z ( 3 ) ( y ) = K exp { V [ ( 1 Y 2 γ 2 1 ) + X ] 1 / 2 y } ,
h x ( 3 ) ( y ) = K V [ ( 1 Y 2 γ 2 1 ) + X ] 1 / 2 × exp { V [ ( 1 Y 2 γ 2 1 ) + X ] 1 / 2 y } ,
X ( b , γ ) = 1 + b 1 + b b γ 4 ,
Y 2 ( a , γ ) = [ 1 + 4 a γ 2 ( γ 2 1 ) ] 1 / 2 1 2 a ( γ 2 1 ) .
A = ( γ 2 1 ) ( b 1 + b b γ 4 ) 1 / 2 .
b 1 = ( 1 2 α 1 H 0 2 n f 2 ) = ( 1 2 c 0 n f 2 n ¯ 2 , 1 H 0 2 n f 2 ) = ( 1 2 Z 0 n ¯ 2 , 1 H o 2 ) ,
b 2 = ( 1 2 α 2 H 0 2 n f 2 ) = ( 1 2 c 0 n f 2 n ¯ 2 , 2 H 0 2 n f 2 ) = ( 1 2 Z 0 n ¯ 2 , 2 H 0 2 ) .
y ( 2 ) = n f 2 { 1 + 2 b 1 [ E y ( 2 ) H 0 ] 2 + 2 b 2 [ E z ( 2 ) H 0 ] 2 } ,
z ( 2 ) = n f 2 { 1 + 2 b 2 [ E y ( 2 ) H 0 ] + 2 b 1 [ E z ( 2 ) H 0 ] 2 } .
d h x ( 2 ) d y = [ z ( 2 ) n f 2 ] f z ( 2 ) ,
d f z ( 2 ) d y = V 2 { 1 [ y ( 2 ) / n f 2 ] ( b γ 4 1 + b b γ 4 + γ 2 γ 2 1 ) 1 γ 2 1 } h x ( 2 ) ,
[ E y ( 2 ) H 0 ] = s 3 [ t 2 + ( s 3 27 + t 2 4 ) 1 / 2 ] 1 / 3 + [ t 2 + ( s 3 27 + t 2 4 ) 1 / 2 ] 1 / 3 ,
s = 1 2 b 1 { 1 + 2 b 2 [ f z ( 2 ) ] 2 } ,
t = 1 2 b 1 [ b γ 4 1 + b b γ 4 + γ 2 ( γ 2 1 ) ] 1 / 2 V h x ( 2 ) .
[ z ( 2 ) n f 2 ] = [ z ( 2 ) n f 2 ] [ b , γ , V , b 1 , b 2 , f z ( 2 ) , h x ( 2 ) ] , [ y ( 2 ) n f 2 ] = [ y ( 2 ) n f 2 ] [ b , γ , V , b 1 , b 2 , f z ( 2 ) , h x ( 2 ) ]
f z ( 2 ) ( y = 0 ) = f z ( 1 ) ( y = 0 ) = A = ( γ 2 1 ) ( b 1 + b b γ 4 ) 1 / 2 ,
h x ( 2 ) ( y = 0 ) = γ 2 h x ( 1 ) ( y = 0 ) = ( γ 2 1 ) V .
P = E y ( y ) H x * ( y ) d y = N c 0 | H x ( y ) | 2 y d y .
P 0 = N d ( n s 2 n f 2 ) | n ¯ 2 , 1 | n f 2 = N ( n s 2 n f 2 ) 1 / 2 V ( γ 2 1 ) k 0 | n ¯ 2 , 1 | .
P P 0 = b 1 [ F 1 ( b , γ , V ) + F 2 ( b , γ , V , b 1 , b 2 ) + F 3 ( a , b , γ , V ) ] ,
F 1 ( b , γ , V ) = ( 1 + b b γ 4 ) 1 / 2 V γ 4 b 1 / 2 ,
F 2 ( b , γ , V , b 1 , b 2 ) = 2 × 0 1 [ h x ( 2 ) ( y ) h x ( 2 ) ( 0 ) ] 2 1 ( y ( 2 ) / n f 2 ) d y ,
F 3 ( a , b , γ , V ) = K 2 [ Y 2 ( a , γ ) V ( γ 2 1 ) 2 ] ( 1 Y 2 γ 2 1 + X ) 3 / 2 × { exp [ 2 V ( 1 Y 2 γ 2 1 + X ) 1 / 2 ] } .

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