Abstract

The wave equation of a planar waveguide with Fermi refractive-index profile is treated theoretically. An analytical solution has been obtained in terms of hypergeometric functions. It has been shown that the dispersion equation can be expressed as an infinite series of functions of modal index which converges rapidly. Highly accurate results can be obtained by taking the zero-order approximation only.

© 1985 Optical Society of America

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References

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  1. J. L. Jacket, “Variation in Waveguides Fabricated by Immersion of LiNbO3 in AgNO3 and TlNO3: The Role of Hydrogen,” Appl. Phys. Lett. 14, 508 (1982).
    [CrossRef]
  2. J. L. Jackel, “High-Δn Optical Waveguides in LiNB3: Thalium-Lithium Ion Exchange,” Appl Phys. Lett. 37, 739 (1980).
    [CrossRef]
  3. Y. X. Chen et al., “Characterization of LiNbO3 Waveguides Exchanged in TlNO3 Solution,” Appl. Phys. Lett. 40, 10 (1982).
    [CrossRef]
  4. J. L. Jackel et al., “Proton Exchange for High-Index Waveguides in LiNbO3,” Appl. Phys. Lett. 41, 607 (1982).
    [CrossRef]
  5. E. M. Conwell, “Modes in Optical Waveguides Formed by Diffusion,” Appl. Phys. Lett. 23, 327 (1973).
    [CrossRef]
  6. D. Marcuse, “TE Modes of Graded-Index Slab Waveguides,” IEEE J. Quantum Electron. QE-9, 1000 (1973).
    [CrossRef]
  7. D. Marcuse, “Losses and Impulse Response in Parabolic Index Fibers With Square Cross Section,” Bell Syst. Tech. J. 53, 195 (1974).
  8. I. Savationova, E. Nadiakov, “Modes in Diffused Optical Waveguides (Parabolic and Gaussian Modes),” Appl. Phys. 8(3), 245 (1975).
    [CrossRef]
  9. P. K. Tien et al., “Optical Waveguide Modes in Single-Crystalline LiNbO3-LiTaO3 Solid-Solution Films,” Appl. Phys. Lett. 24, 503 (1974).
    [CrossRef]
  10. H. C. Dong, J. Q. Fan, “Fabrication of Glass Optical Waveguides by Tl2SO4 Ion Exchange,” in Digest of Second National Conference on Integrated Optics (Optical Society of China, 1983), p. 72.
  11. See, for example, Z. S. Wang, D. R. Gwo, Introduction to Special Functions (Science Publishing House, Beijing, 1985), Chap. 4, p. 178, in Chinese; A. Erdelyi, Higher Transcendental Functions (McGraw-Hill, New York, 1953), Vol. 1.

1982 (3)

Y. X. Chen et al., “Characterization of LiNbO3 Waveguides Exchanged in TlNO3 Solution,” Appl. Phys. Lett. 40, 10 (1982).
[CrossRef]

J. L. Jackel et al., “Proton Exchange for High-Index Waveguides in LiNbO3,” Appl. Phys. Lett. 41, 607 (1982).
[CrossRef]

J. L. Jacket, “Variation in Waveguides Fabricated by Immersion of LiNbO3 in AgNO3 and TlNO3: The Role of Hydrogen,” Appl. Phys. Lett. 14, 508 (1982).
[CrossRef]

1980 (1)

J. L. Jackel, “High-Δn Optical Waveguides in LiNB3: Thalium-Lithium Ion Exchange,” Appl Phys. Lett. 37, 739 (1980).
[CrossRef]

1975 (1)

I. Savationova, E. Nadiakov, “Modes in Diffused Optical Waveguides (Parabolic and Gaussian Modes),” Appl. Phys. 8(3), 245 (1975).
[CrossRef]

1974 (2)

P. K. Tien et al., “Optical Waveguide Modes in Single-Crystalline LiNbO3-LiTaO3 Solid-Solution Films,” Appl. Phys. Lett. 24, 503 (1974).
[CrossRef]

D. Marcuse, “Losses and Impulse Response in Parabolic Index Fibers With Square Cross Section,” Bell Syst. Tech. J. 53, 195 (1974).

1973 (2)

E. M. Conwell, “Modes in Optical Waveguides Formed by Diffusion,” Appl. Phys. Lett. 23, 327 (1973).
[CrossRef]

D. Marcuse, “TE Modes of Graded-Index Slab Waveguides,” IEEE J. Quantum Electron. QE-9, 1000 (1973).
[CrossRef]

Chen, Y. X.

Y. X. Chen et al., “Characterization of LiNbO3 Waveguides Exchanged in TlNO3 Solution,” Appl. Phys. Lett. 40, 10 (1982).
[CrossRef]

Conwell, E. M.

E. M. Conwell, “Modes in Optical Waveguides Formed by Diffusion,” Appl. Phys. Lett. 23, 327 (1973).
[CrossRef]

Dong, H. C.

H. C. Dong, J. Q. Fan, “Fabrication of Glass Optical Waveguides by Tl2SO4 Ion Exchange,” in Digest of Second National Conference on Integrated Optics (Optical Society of China, 1983), p. 72.

Fan, J. Q.

H. C. Dong, J. Q. Fan, “Fabrication of Glass Optical Waveguides by Tl2SO4 Ion Exchange,” in Digest of Second National Conference on Integrated Optics (Optical Society of China, 1983), p. 72.

Gwo, D. R.

See, for example, Z. S. Wang, D. R. Gwo, Introduction to Special Functions (Science Publishing House, Beijing, 1985), Chap. 4, p. 178, in Chinese; A. Erdelyi, Higher Transcendental Functions (McGraw-Hill, New York, 1953), Vol. 1.

Jackel, J. L.

J. L. Jackel et al., “Proton Exchange for High-Index Waveguides in LiNbO3,” Appl. Phys. Lett. 41, 607 (1982).
[CrossRef]

J. L. Jackel, “High-Δn Optical Waveguides in LiNB3: Thalium-Lithium Ion Exchange,” Appl Phys. Lett. 37, 739 (1980).
[CrossRef]

Jacket, J. L.

J. L. Jacket, “Variation in Waveguides Fabricated by Immersion of LiNbO3 in AgNO3 and TlNO3: The Role of Hydrogen,” Appl. Phys. Lett. 14, 508 (1982).
[CrossRef]

Marcuse, D.

D. Marcuse, “Losses and Impulse Response in Parabolic Index Fibers With Square Cross Section,” Bell Syst. Tech. J. 53, 195 (1974).

D. Marcuse, “TE Modes of Graded-Index Slab Waveguides,” IEEE J. Quantum Electron. QE-9, 1000 (1973).
[CrossRef]

Nadiakov, E.

I. Savationova, E. Nadiakov, “Modes in Diffused Optical Waveguides (Parabolic and Gaussian Modes),” Appl. Phys. 8(3), 245 (1975).
[CrossRef]

Savationova, I.

I. Savationova, E. Nadiakov, “Modes in Diffused Optical Waveguides (Parabolic and Gaussian Modes),” Appl. Phys. 8(3), 245 (1975).
[CrossRef]

Tien, P. K.

P. K. Tien et al., “Optical Waveguide Modes in Single-Crystalline LiNbO3-LiTaO3 Solid-Solution Films,” Appl. Phys. Lett. 24, 503 (1974).
[CrossRef]

Wang, Z. S.

See, for example, Z. S. Wang, D. R. Gwo, Introduction to Special Functions (Science Publishing House, Beijing, 1985), Chap. 4, p. 178, in Chinese; A. Erdelyi, Higher Transcendental Functions (McGraw-Hill, New York, 1953), Vol. 1.

Appl Phys. Lett. (1)

J. L. Jackel, “High-Δn Optical Waveguides in LiNB3: Thalium-Lithium Ion Exchange,” Appl Phys. Lett. 37, 739 (1980).
[CrossRef]

Appl. Phys. (1)

I. Savationova, E. Nadiakov, “Modes in Diffused Optical Waveguides (Parabolic and Gaussian Modes),” Appl. Phys. 8(3), 245 (1975).
[CrossRef]

Appl. Phys. Lett. (5)

P. K. Tien et al., “Optical Waveguide Modes in Single-Crystalline LiNbO3-LiTaO3 Solid-Solution Films,” Appl. Phys. Lett. 24, 503 (1974).
[CrossRef]

Y. X. Chen et al., “Characterization of LiNbO3 Waveguides Exchanged in TlNO3 Solution,” Appl. Phys. Lett. 40, 10 (1982).
[CrossRef]

J. L. Jackel et al., “Proton Exchange for High-Index Waveguides in LiNbO3,” Appl. Phys. Lett. 41, 607 (1982).
[CrossRef]

E. M. Conwell, “Modes in Optical Waveguides Formed by Diffusion,” Appl. Phys. Lett. 23, 327 (1973).
[CrossRef]

J. L. Jacket, “Variation in Waveguides Fabricated by Immersion of LiNbO3 in AgNO3 and TlNO3: The Role of Hydrogen,” Appl. Phys. Lett. 14, 508 (1982).
[CrossRef]

Bell Syst. Tech. J. (1)

D. Marcuse, “Losses and Impulse Response in Parabolic Index Fibers With Square Cross Section,” Bell Syst. Tech. J. 53, 195 (1974).

IEEE J. Quantum Electron. (1)

D. Marcuse, “TE Modes of Graded-Index Slab Waveguides,” IEEE J. Quantum Electron. QE-9, 1000 (1973).
[CrossRef]

Other (2)

H. C. Dong, J. Q. Fan, “Fabrication of Glass Optical Waveguides by Tl2SO4 Ion Exchange,” in Digest of Second National Conference on Integrated Optics (Optical Society of China, 1983), p. 72.

See, for example, Z. S. Wang, D. R. Gwo, Introduction to Special Functions (Science Publishing House, Beijing, 1985), Chap. 4, p. 178, in Chinese; A. Erdelyi, Higher Transcendental Functions (McGraw-Hill, New York, 1953), Vol. 1.

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Figures (2)

Fig. 1
Fig. 1

(a) Planar waveguide; (b) a Fermi index distribution curve in (a).

Fig. 2
Fig. 2

Schematic field distribution in three regions of the waveguide with a Fermi index profile.

Tables (1)

Tables Icon

Table I Calculated and Experimentally Observed10 Modal Index Nm

Equations (40)

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n 2 ( x ) = { n b 2 + n 2 f ( x ) ( x 0 ) n 0 2 ( x < 0 ) ,
f ( x ) = [ 1 exp ( d a ) + exp ( x d a ) ] 1 .
E , H J ( x ) exp i ( ω t k z z ) ,
d 2 J ( x ) d x 2 k 2 [ N 2 n 2 ( x ) ] J ( x ) = 0 ,
d 2 J ( x ) d 2 x 1 n 2 ( x ) d n 2 ( x ) d x d J ( x ) d x k 2 [ N 2 n 2 ( x ) ] J ( x ) = 0 ,
H z ( x ) = i ω μ E y x and H x ( x ) = K z ω μ E y .
η = a 2 k 2 ( N 2 n b 2 ) ι = exp ( d a ) / [ exp ( d a ) 1 ] , τ 0 2 = a 2 k 2 ι Δ n 2 ,
ζ ( x ) = exp ( x d a ) / ι , x = d a ln ( ι ζ ) .
[ 1 ζ ( x ) ] d 2 J ( x ) d x 2 1 a 2 { [ 1 ζ ( x ) ] η 2 + τ 0 2 ζ ( x ) } J ( x ) = 0 .
J ( x ) = D ( ζ ) = ζ η u ( ζ ) ,
ζ ( 1 ζ ) d 2 u ( ζ ) d ζ 2 + ( 1 ζ ) ( 1 + 2 η ) d u ( ζ ) d ζ τ 0 2 u ( ζ ) = 0 .
ζ ( 1 ζ ) d 2 F d ζ 2 + [ γ ( α + β + 1 ) ζ ] d F d ζ α β F = 0
u ( ζ ) = C 0 F ( α , β , γ , ζ ) + C 1 F ( α γ + 1 , β γ + 1 , 2 γ , ζ ) ζ 1 γ , | ζ | < 1 ,
γ = 1 + 2 η , β = η i τ , α = η + i τ , τ 2 = τ 0 2 η 2 .
F ( α , β , γ , ζ ) = n = 0 ( α ) n ( β ) n n ! ( γ ) n ζ n , ( | ζ | < 1 ) ,
( λ ) 0 = 1 , ( λ ) n = λ ( λ + 1 ) ( λ + n 1 ) = Γ ( λ + n ) Γ ( λ ) ,
D ( ζ ) = C 0 ζ η F ( α , β , γ , ζ ) + C 1 ζ η F ( α β + 1 , β γ + 1 , 2 γ , ζ ) .
D ( ζ ) = C 0 ζ η F ( α , β , γ , ζ ) ( 1 ζ 0 ) .
D ( ζ ) = C 0 ζ η ( 1 ζ ) α F ( α , γ β , γ , ζ ζ 1 ) ( ζ 0 ) .
D ( ζ ) = C 0 [ ( 1 ) η C ( ζ ) i τ F ( α , α γ + 1 , α β + 1 , ζ 1 ) + c . c . ] ( < ζ < 1 ) ,
C = Γ ( γ ) Γ ( β α ) Γ ( γ α ) Γ ( β ) .
E z = i ω H y x , E x = K z ω H y ,
H y ( x ) = n ( x ) w ( x ) .
a λ / ( 2 2 π n b ) λ = wavelength .
J ( x ) = A exp ( h x / a ) x < 0 ,
A = C 0 [ C ( 1 ) η ( ζ 0 ) i τ F c ( ζ 0 1 ) + c . c . ] ,
h a A = ( 1 ) η C 0 a { C ( ζ 0 ) i τ [ ( i τ α ) F c ( ζ 0 1 ) + α F c ( α + 1 , ζ 0 1 ) ] + c . c . } ,
d F d ζ = α β γ F ( α + 1 , β + 1 , γ + 1 ) , γ F β ζ F ( β + 1 , γ + 1 ) γ F ( α 1 ) = 0 .
| 1 C ( ζ 0 ) i τ F c ( ζ 0 1 ) + c . c . h C ( ζ 0 ) i τ [ F c ( ζ 0 1 ) ( i τ α ) + α F c ( α + 1 , ζ 0 1 ) ] + c . c . | = 0 .
Γ ( β α ) Γ ( γ α ) Γ ( β ) ( ζ 0 ) i τ n = 0 ( h i τ n ) × ( α ) n ( β c ) n n ! ( γ c ) n ζ 0 n + c . c . = 0 ,
Γ ( β α ) Γ ( γ α ) Γ ( β ) ( ζ 0 ) i τ ( h i τ ) + c . c . = 0 .
arg ( Γ ( β α ) Γ ( γ α ) Γ ( β ) ) δ τ tan 1 ( τ h ) + ( 2 m + 1 ) π 2 = 0 ,
1 Γ ( z ) = z exp ( γ 0 z ) p = 1 ( 1 + z p ) exp ( z p ) ,
arg [ Γ ( β α ) Γ ( γ α ) Γ ( β ) ] = π 2 tan 1 ( τ η ) + p = 1 [ tan 1 ( 2 τ p ) 2 tan 1 ( τ p + η ) ] ,
δ τ + tan 1 ( τ h ) + tan 1 ( τ η ) p = 1 [ tan 1 ( 2 τ p ) 2 tan 1 ( τ p + η ) ] = ( m + 1 ) π m = 0 , 1 , 2 , .
( τ h ) 1 = [ ( 1 + 4 τ 2 ) ζ 0 + τ 0 2 ( 2 h 3 ) ] τ / [ ( 1 + 4 τ 2 ) ζ 0 τ 0 2 ( 2 τ 2 + h 1 ) ] .
Γ ( β α ) Γ ( γ α ) Γ ( β ) ( ζ 0 ) i τ n = 0 [ n s 2 n 0 2 h + Δ n 2 2 n s 2 exp ( d a ) i τ n ] × ( α ) n ( β c ) n n ! ( γ c ) n ζ 0 n + c . c .
δ τ + tan 1 ( τ η ) + tan 1 ( n 0 2 τ n a 2 h ) p = 1 [ tan 1 ( 2 τ p ) 2 tan 1 ( τ p + η ) ] = ( m + 1 ) π m = 0 , 1 , 2 , .
δ τ 0 + tan 1 ( τ 0 a k n b 2 n 0 2 ) p = 1 [ tan 1 ( 2 τ 0 p ) 2 tan 1 ( τ 0 p ) ] = 3 2 π .
M = int { 8 τ 0 π + 1 π tan 1 ( τ 0 a k n b 2 n 0 2 ) 1 π p = 1 tan 1 ( 2 τ 0 p ) 2 tan 1 ( τ 0 p ) 1 2 } ,

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