Abstract

Analytic expressions for the field and propagation constants of the mode along a uniformly curved three-layered slab waveguide are presented in terms of the power series of the curvature. To compare the accuracy of approximate solutions, numerical analysis is carried out. New bending loss formulas are also derived which consider the field deformation and the change of the propagation constant. It is shown that they are very accurate over a wide range of normalized frequency.

© 1981 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. E. A. J. Marcatili, Bell Syst. Tech. J. 48, 2103 (1969).
  2. D. Marcuse, Bell Syst. Tech. J. 50, 2551 (1971).
  3. L. Lewin, IEEE Trans. Microwave Theory Tech. MTT-22, 718 (1974).
    [CrossRef]
  4. J. A. Arnaud, Bell Syst. Tech. J. 53, 1379 (1974).
  5. E. F. Kuester, D. C. Chang, IEEE J. Quantum Electron. QE-11, 903 (1975).
    [CrossRef]
  6. S. Kawakami, M. Miyagi, S. Nishida, Appl. Opt. 14, 2588 (1975);Appl. Opt.15, 1681 (1976).
    [CrossRef] [PubMed]
  7. Y. Takuma, S. Kawakami, N. Nishida, Trans. IECE Jpn. 60-C, 706 (1976).
  8. M. Miyagi, Appl. Opt. 20, 1221 (1981).
    [CrossRef] [PubMed]
  9. W. Magnus, F. Oberhettinger, R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics (Springer, New York, 1966), p. 141.

1981 (1)

1976 (1)

Y. Takuma, S. Kawakami, N. Nishida, Trans. IECE Jpn. 60-C, 706 (1976).

1975 (2)

1974 (2)

L. Lewin, IEEE Trans. Microwave Theory Tech. MTT-22, 718 (1974).
[CrossRef]

J. A. Arnaud, Bell Syst. Tech. J. 53, 1379 (1974).

1971 (1)

D. Marcuse, Bell Syst. Tech. J. 50, 2551 (1971).

1969 (1)

E. A. J. Marcatili, Bell Syst. Tech. J. 48, 2103 (1969).

Arnaud, J. A.

J. A. Arnaud, Bell Syst. Tech. J. 53, 1379 (1974).

Chang, D. C.

E. F. Kuester, D. C. Chang, IEEE J. Quantum Electron. QE-11, 903 (1975).
[CrossRef]

Kawakami, S.

Y. Takuma, S. Kawakami, N. Nishida, Trans. IECE Jpn. 60-C, 706 (1976).

S. Kawakami, M. Miyagi, S. Nishida, Appl. Opt. 14, 2588 (1975);Appl. Opt.15, 1681 (1976).
[CrossRef] [PubMed]

Kuester, E. F.

E. F. Kuester, D. C. Chang, IEEE J. Quantum Electron. QE-11, 903 (1975).
[CrossRef]

Lewin, L.

L. Lewin, IEEE Trans. Microwave Theory Tech. MTT-22, 718 (1974).
[CrossRef]

Magnus, W.

W. Magnus, F. Oberhettinger, R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics (Springer, New York, 1966), p. 141.

Marcatili, E. A. J.

E. A. J. Marcatili, Bell Syst. Tech. J. 48, 2103 (1969).

Marcuse, D.

D. Marcuse, Bell Syst. Tech. J. 50, 2551 (1971).

Miyagi, M.

Nishida, N.

Y. Takuma, S. Kawakami, N. Nishida, Trans. IECE Jpn. 60-C, 706 (1976).

Nishida, S.

Oberhettinger, F.

W. Magnus, F. Oberhettinger, R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics (Springer, New York, 1966), p. 141.

Soni, R. P.

W. Magnus, F. Oberhettinger, R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics (Springer, New York, 1966), p. 141.

Takuma, Y.

Y. Takuma, S. Kawakami, N. Nishida, Trans. IECE Jpn. 60-C, 706 (1976).

Appl. Opt. (2)

Bell Syst. Tech. J. (3)

E. A. J. Marcatili, Bell Syst. Tech. J. 48, 2103 (1969).

D. Marcuse, Bell Syst. Tech. J. 50, 2551 (1971).

J. A. Arnaud, Bell Syst. Tech. J. 53, 1379 (1974).

IEEE J. Quantum Electron. (1)

E. F. Kuester, D. C. Chang, IEEE J. Quantum Electron. QE-11, 903 (1975).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

L. Lewin, IEEE Trans. Microwave Theory Tech. MTT-22, 718 (1974).
[CrossRef]

Trans. IECE Jpn. (1)

Y. Takuma, S. Kawakami, N. Nishida, Trans. IECE Jpn. 60-C, 706 (1976).

Other (1)

W. Magnus, F. Oberhettinger, R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics (Springer, New York, 1966), p. 141.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1
Fig. 1

(a) Refractive-index profile of a three-layered dielectric slab waveguide and (b) the cylindrical coordinate system (r,θ,y) and the local rectangular coordinate system (x,y,z) in a bent slab waveguide.

Fig. 2
Fig. 2

Field distribution of the TE0 mode in the bent symmetric slab waveguide at (a) υ = 2.5 and (b) υ = 4 for ℛ = 60. The solid lines represent numerical results. The dashed and dotted lines represent those predicted by Eq. (22) and by the WKB method, respectively.

Fig. 3
Fig. 3

β/k0)/Δn of the TE0 mode in the symmetric waveguide for ℛ = 60 as a function of υ. The solid line represents the numerical result. The dashed and dotted lines represent those predicted by using Eq. (25) where F is replaced by F [Eq. (27)] and by Fw [Eq. (28)], respectively. The long dashed line represents that predicted by using F [Eq. (26)]. The dot–dash line represents that in the edge guidance [Eq. (C7)].

Fig. 4
Fig. 4

(Δβ/k0)/Δn of the TE0 mode in various asymmetric waveguides obtained by the numerical analysis for ℛ = 60 as a function of υ.

Fig. 5
Fig. 5

(Δβ/k0)/Δn of the TE0 mode in the asymmetric waveguide for ℛ = 60 and A = 0.6. The solid line represents the numerical result. The dotted and dashed lines represent those predicted by using Gm [Eq. (30)] and G [Eq. (31)], respectively. The dot–dash line represents that in the edge guidance [Eq. (C7)]. The long dashed line represents that evaluated only by δβ1.

Fig. 6
Fig. 6

β/k0)/Δn of the TE0 mode in the asymmetric waveguide for ℛ = 60 and A = 1.25. Curve symbols are the same as those in Fig. 5.

Fig. 7
Fig. 7

Normalized bending loss of the TE0 mode in the symmetric (a) and asymmetric [(b) A = 0.6, (c) A = 0.8] waveguide for ℛ = 60 as a function of υ. The solid lines represent the numerical results. The dashed lines represent those predicted by Eq. (40). The dotted and dot–dash lines represent those in the edge guidance predicted by Eqs. (41) and (43), respectively. Results given in Refs. 1, 2, and 6 are also shown.

Equations (83)

Equations on this page are rendered with MathJax. Learn more.

d 2 E y d r 2 + 1 r d E y d r + [ k 0 2 n 2 ( r ) ν 2 r 2 ] E y = 0 ,
r = R + x ,
E y = ( 1 + x R ) 1 / 2 E ( x ) exp ( j β z ) ,
d 2 E d x 2 + [ k 0 2 n 2 ( x ) β 2 1 / ( 4 R 2 ) ( 1 + x / R ) 2 ] E = 0 .
E ( x ) = E 0 ( x ) + 1 R f ( x ) + 1 R 2 g ( x ) + ,
β = β 0 + 1 R δ β 1 + 1 R 2 δ β 2 + β 0 + Δ β ,
d 2 E 0 d x 2 + [ k 0 2 n 2 ( x ) β 0 2 ] E 0 = 0 ,
d 2 f d x 2 + [ k 0 2 n 2 ( x ) β 0 2 ] f = 2 β 0 ( δ β 1 β 0 x ) E 0 ,
d 2 g d x 2 + [ k 0 2 n 2 ( x ) β 0 2 ] g = [ 2 β 0 δ β 2 + ( δ β 1 ) 2 1 4 4 β 0 δ β 1 x + 3 β 0 2 x 2 ] E 0 + 2 β 0 ( δ β 1 β 0 x ) f .
2 β 0 ( δ β 1 β 0 x ) E 0 2 = d d x ( E 0 d f d x f d E 0 d x ) .
δ β 1 = β 0 x E 0 2 d x E 0 2 d x ,
[ 2 β 0 δ β 2 + ( δ β 1 ) 2 1 4 4 β 0 δ β 1 x + 3 β 0 2 x 2 ] E 0 2 + 2 β 0 ( δ β 1 β 0 x ) E 0 f = d d x ( E 0 d g d x g d E 0 d x ) .
[ 2 β 0 δ β 2 + ( δ β 1 ) 2 1 4 4 β 0 δ β 1 x + 3 β 0 2 x 2 ] E 0 2 d x = 2 β 0 2 ( x δ β 1 β 0 ) E 0 f d x .
δ β 2 = β 0 ( x δ β 1 β 0 ) E 0 f d x E 0 2 d x .
E 0 ( x ) = { cos ( u + φ ) exp [ w ̂ T ( x T ) ] ; x > T , cos ( u T x + φ ) ; | x | T , cos ( u φ ) exp [ w T ( x + T ) ] ; x < T ,
tan ( 2 u ) = u ( w + w ̂ ) u 2 w w ̂ ,
u 2 + w 2 = 2 ( 1 b ) ( n 0 k 0 T ) 2 υ 2 ,
u 2 + w ̂ 2 = 2 ( 1 a ) ( n 0 k 0 T ) 2 υ ̂ 2 .
φ = u tan 1 ( w u ) = u + tan 1 ( w ̂ u ) .
δ β 1 = β 0 T 2 ( 1 w 1 w ̂ ) ,
f ( x ) = { β 0 2 T 3 2 u 2 cos ( u + φ ) [ ( 1 + 1 w ) ( 1 u 2 w ̂ ) + u 2 w w ̂ x T + u 2 w ̂ ( x T ) 2 ] exp [ w ̂ T ( x T ) ] ; x > T , β 0 2 T 3 2 u 2 { u [ ( 1 + 1 w ) ( 1 + 1 w ̂ ) ( 1 w 1 w ̂ ) x T ( x T ) 2 ] sin ( u T x + φ ) x T cos ( u T x + φ ) } ; | x | T , β 0 2 T 3 2 u 2 cos ( u φ ) [ ( 1 + 1 w ̂ ) ( 1 u 2 w ) + u 2 w w ̂ x T u 2 w ( x T ) 2 ] exp [ w T ( x + T ) ] ; x < T ,
E y = ( 1 + x R ) 1 / 2 [ E 0 + 1 R f ( x ) + 1 R 2 g ( x ) + ] exp ( j β z ) [ E 0 + 1 R f ( x ) ] exp ( j β z ) .
= 2 n 0 k 0 [ 2 ( 1 b ) ] 3 / 2 R ,
A = ( 1 a ) ( 1 b ) .
δ β 2 = β 0 3 T 4 6 u 2 1 1 + 1 w F ,
F = ( w υ ) 2 [ 1 15 4 u 2 + 3 w ( 1 5 4 u 2 ) + u 2 w 2 ( 1 3 4 u 2 ) 3 2 w 3 3 u 2 w 4 3 u 4 w 5 ( 1 + 1 2 u 2 ) 6 u 4 w 6 15 u 4 4 w 7 ] .
F = ( w υ ) 2 [ 1 15 4 u 2 + 3 w ( 1 5 4 u 2 ) ] .
F w = 1 + 3 w ( 1 u 2 w 2 ) .
δ β 2 = β 0 3 T 4 6 u 2 1 1 + 1 2 ( 1 w + 1 w ̂ ) G m ,
G m = 1 15 4 u 2 + 3 2 ( 1 5 4 u 2 ) ( 1 w + 1 w ̂ ) .
G = ( w w ̂ υ υ ̂ ) [ 1 15 4 u 2 + 3 2 ( 1 5 4 u 2 ) ( 1 w + 1 w ̂ ) ] .
Δ n = n 0 T R .
2 α = P r P g .
2 ( 1 b ) α R = w ̂ e 4 υ 2 E y 2 ( T ) T E y 2 ( x ) d x exp [ 2 n 0 k 0 R 3 ( w ̂ e a n 0 k 0 T ) 3 ] ,
w ̂ e = [ β 2 ( 1 2 T R ) ( a n 0 k 0 ) 2 ] 1 / 2 T .
E y ( T ) = u υ ̂ [ 1 + υ 3 u 2 ( 1 + 1 w ) ] ,
E y 2 ( x ) d x = T [ 1 + 1 2 ( 1 w + 1 w ̂ ) + υ 3 u 2 ( 1 w 1 w ̂ ) ] .
S = 2 n 0 k 0 R 3 { 2 [ β ( 1 T R ) a n 0 k 0 1 ] } 3 / 2 ,
S = w ̂ 3 3 υ 3 { 1 2 υ 3 w ̂ 2 ( 2 + 1 w 1 w ̂ ) 4 3 w υ 5 u 2 w ̂ υ ̂ 2 × [ 1 15 4 u 2 + 3 2 ( 1 5 4 u 2 ) ( 1 w + 1 w ̂ ) ] / [ 1 + 1 2 ( 1 w + 1 w ̂ ) ] } 3 / 2 .
2 ( 1 b ) α R = u 2 w w ̂ 4 υ 2 υ ̂ 2 1 + 2 υ 3 u 2 ( 1 + 1 w ) w + 1 2 ( 1 + w w ̂ ) + υ 3 u 2 ( 1 w w ̂ ) exp ( S ) .
2 ( 1 b ) α R = 1 A exp ( ̂ 3 { 1 [ 6 ( n + 3 4 ) π ̂ ] 2 / 3 + 4 ̂ } 3 / 2 ) ,
w ̂ e = { 1 [ 6 ( n + 3 4 ) π ̂ ] 2 / 3 } 1 / 2 υ ̂ .
2 ( 1 b ) α R = 1 A { 1 [ 6 ( n + 3 4 ) π ̂ ] 2 / 3 } 1 / 2 × exp ( ̂ 3 { 1 [ 6 ( n + 3 4 ) π ̂ ] 2 / 3 + 4 ̂ } 3 / 2 ) .
d 2 ψ d x 2 + 2 β 0 2 x R ψ + [ n 2 ( x ) k 0 2 β 2 ] ψ = 0
E y = ψ r 1 / 2 ,
r = R + x ,
ν = β R .
ψ = { C 1 α t 1 / 2 exp ( T x α t d x ) ; x < T , β t 1 / 2 sin ( T x β t d x + ϕ ) ; | x | T , C 2 α t 1 / 2 exp ( T x α t d x ) ; x > T ,
α t = [ β 2 ( a n 0 k 0 ) 2 2 β 0 2 x / R ] 1 / 2 ,
β t = [ ( n 0 k 0 ) 2 β 2 + 2 β 0 2 x / R ] 1 / 2 .
1 1 ( u 2 Δ 2 + Δ 1 x ) 1 / 2 d x + tan 1 ( u 2 Δ 1 Δ 2 w 2 + Δ 1 + Δ 2 ) 1 / 2 + tan 1 ( u 2 + Δ 1 Δ 2 w 2 Δ 1 + Δ 2 ) 1 / 2 = ( n + 1 ) π
Δ 1 = 2 β 0 2 T 3 R ,
Δ 2 = 2 β 0 T 2 Δ β .
Δ β = β 0 3 T 4 R 2 [ 1 6 u 2 + 1 2 w ( 1 u 2 1 w 2 ) ] / ( 1 + 1 w ) .
E y = Ψ / r ,
Ψ = { M exp [ R 1 2 T r ( ν 2 r 2 b 2 n 0 2 k 0 2 ) 1 / 2 d r ] ; r < R 1 2 T , [ 3 ( n 0 k 0 ) 3 ν ] 1 / 6 τ 1 / 3 [ C Z 1 / 3 ( τ ) + D Z 1 / 3 ( τ ) ] ; R 1 2 T < r < R 1 , N exp [ R 1 r ( ν 2 r 2 a 2 n 0 2 k 0 2 ) 1 / 2 d r ] ; r > R 1 ,
τ = 2 3 [ 2 ( n 0 k 0 ) 3 ν ] 1 / 2 | r ν n 0 k 0 | 3 / 2 ,
Z ± 1 / 3 ( τ ) = { J ± 1 / 3 ( τ ) ; r ν / n 0 k 0 , I ± 1 / 3 ( τ ) ; r ν / n 0 k 0 ,
X = ( ν R 1 n 0 k 0 ) T / 2 ( 1 b ) ,
| a 11 a 12 a 21 a 22 | = 0 ,
a 11 = Z 1 / 3 ( τ 1 ) ( 2 X 8 υ 2 / 1 υ + 2 X + 8 υ 2 / 1 ) 1 / 2 Z 2 / 3 ( τ 1 ) ,
a 12 = Z 1 / 3 ( τ 1 ) + ( 2 X 8 υ 2 / 1 υ + 2 X + 8 υ 2 / 1 ) 1 / 2 Z 2 / 3 ( τ 1 ) ,
a 21 = J 1 / 3 ( τ 11 ) + ( 2 X A υ + 2 X ) 1 / 2 J 2 / 3 ( τ 11 ) ,
a 22 = J 1 / 3 ( τ 11 ) ( 2 X A υ + 2 X ) 1 / 2 J 2 / 3 ( τ 11 ) ,
τ 1 = 1 6 ( 2 X 8 υ 2 / 1 ) 3 / 2 / υ 3 / 2 ,
τ 11 = 1 6 ( 2 X ) 3 / 2 / υ 3 / 2 ,
Z ± 2 / 3 ( τ 1 ) = { J ± 2 / 3 ( τ 1 ) ; R 1 2 T ν / n 0 k 0 , I ± 2 / 3 ( τ 1 ) ; R 1 2 T ν / n 0 k 0 .
Δ β / k 0 Δ n = 1 + 2 υ 2 ( X + u 2 2 υ )
2 ( 1 b ) α R = ( A υ + 2 X ) 1 / 2 1 24 υ 3 / 2 ( 2 X ) ( J 1 / 3 P J 1 / 3 ) 2 [ 6 2 / 3 1 1 / 3 Q + E / ( 12 υ A ) ] × exp [ 1 3 ( A υ + 2 X ) 3 / 2 1 υ 3 / 2 ] ,
1 = 2 n 0 k 0 [ 2 ( 1 b ) ] 3 / 2 R 1 ,
P = a 11 / a 12 = a 21 / a 22 , Q = 3 2 [ τ 11 4 / 3 ( J 1 / 3 2 + J 2 / 3 2 ) τ 1 4 / 3 ( Z 1 / 3 2 ± Z 2 / 3 2 ) ]
+ 3 2 P 2 [ τ 11 4 / 3 ( J 1 / 3 2 + J 2 / 3 2 ) τ 1 4 / 3 ( Z 1 / 3 2 ± Z 2 / 3 2 ) ] 3 P [ τ 11 4 / 3 ( J 1 / 3 J 1 / 3 J 2 / 3 J 2 / 3 ) τ 1 4 / 3 ( ± Z 1 / 3 Z 1 / 3 Z 2 / 3 Z 2 / 3 ) ] ,
E = ( 2 X 8 υ 2 / 1 ) ( Z 1 / 3 P Z 1 / 3 ) 2 + ( 2 X ) ( J 1 / 3 P J 1 / 3 ) 2 .
E y = ( 2 π ν ) 1 / 2 [ 2 ( n 0 k 0 r / ν 1 ) ] 1 / 4 × sin { ν 3 [ 2 ( n 0 k 0 r / ν 1 ) ] 3 / 2 + π 4 } exp ( j ν θ ) ,
d E y d r / E y r = R 1 0 [ β e 2 ( a n 0 k 0 ) 2 ] 1 / 2 n 0 k 0 [ 2 ( 1 a ) ] 1 / 2 .
η = ( n 0 k 0 R 1 3 ) 1 / 3 [ 2 ( 1 ν n 0 k 0 R 1 ) ] 1 / 2
η 1 tan ( η 3 + π 4 ) = ( 6 ̂ 1 ) 1 / 3 ,
̂ 1 = 2 n 0 k 0 [ 2 ( 1 a ) ] 3 / 2 R 1 ̂ .
η = [ ( n + 3 4 ) π ] 1 / 3 1 3 ( 6 ̂ 1 ) 1 / 3 1 [ ( n + 3 4 ) π ] 1 / 3 ,
( Δ β / k 0 ) Δ n = 1 + u 2 4 υ 3 4 υ { ( 6 ̂ 1 ) 1 / 3 [ ( n + 3 4 ) π ] 1 / 3 1 3 ( 6 ̂ 1 ) 2 / 3 1 A [ ( n + 3 4 ) π ] 1 / 3 } .
E y 2 ( R 1 ) T 0 E y 2 ( r ) d r = 4 υ A 1 ,
w ̂ e = [ β e 2 ( a n 0 k 0 ) 2 ] 1 / 2 T n 0 k 0 [ 2 ( 1 a ) ] 1 / 2 T [ 1 η 2 ( 6 ̂ 1 ) 2 / 3 ] 1 / 2 A υ { 1 [ 6 ( n + 3 4 ) π ̂ 1 ] 2 / 3 } 1 / 2 .
2 ( 1 b ) α R = 1 A { 1 [ 6 ( n + 3 4 ) π ̂ ] 2 / 3 } 1 / 2 × exp ( ̂ 3 { 1 [ 6 ( n + 3 4 ) π ̂ ] 2 / 3 + 4 ̂ } 3 / 2 ) ,

Metrics