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Asymptotic eigenvalues of vector waves in an inhomogeneous circular waveguide
Volume 34, number 1
OPTICS COMMUNICATIONS
July 1980
ASYMPTOTIC EIGENVALUES OF VECTOR WAVES IN AN INHOMOGENEOUS CIRCULAR WAVEGUIDE Masahiro HASHIMOTO Department of Applied Electronic Engineering, Osaka Electro-Communication University, Neyagawa, Osaka 5 72, Japan Received 1 1 March 1980
Based on the vector wave equations of an inhomogeneous circular waveguide, asymptotic eigenvalues of the system of the coupled second-order differential equations are given, of which the first-order eigenvalues coincide with the well-known WKB solutions. A parabolic-index optical fiber is examined to obtain the propagation constants. These asymptotic results are compared with the perturbational solutions and both are shown to be in agreement.
Recently, the author has succeeded [1] in obtaining asymptotic eigenvalues of cylindrical scalar waves in a radially inhomogeneous medium by using WentzelDunham's complex-contour-integral quantum condition [2]. The usefulness of this technique appears to cover a wide range of eigenvalue problems related to acoustic, electromagnetic and quantum mechanical fields. In this Communications, we present the results of application of the technique to electromagnetic vector waves in an optical waveguide. We also give a comparison with other solutions in a specific case. Suppose that q~ and xI, are the solutions of the coupled differential equations of the second order, t
(2)
where the primes denote differentiation with respect to a variable r, e is a small parameter, m is an integer (--- 0,1,2 . . . . ), X is an eigenvalue, f i s an analytic evenfunction of r satisfying )'(0) = 0, and h 1, h2, g l , g2, q are smooth functions of r. The mathematical problem of solving eqs. (1) asymptotically can be treated by means of the method presented in in ref. [1 ], expanding X and ~ into )t = )t (1) + eZx (2) + e4X (3) + ...
(3a)
= ~(0) + e~(l) + e2~(2) + . . .
(3b)
t
q¢' + (1/r - 2Po/Po)~
+[p2[e 2
PO = PO (r) =- X/X - f(r),
where
m2r 2 +(h2
P'0 +gz)p0-
hl-gl
}
e a = +- lira ~(r)/rb(r) (4) r--*0 and assuming that the values of X(1) are the WKB solutions determined from
~
t
_ 2m (1 + q)P0 xp, r
(la)
?lP0 0
t
q/' + (1/r - 2Po/Po)q~ +[p2
m2
r2
+ (h 2
(I+m-ll+l) 2
--dr=Tr n+
P00
'
(5)
(n = 0,1,2 .... ),
t
-g2)
e
P0
where r 1 is the turning point at which X = f(r 1 ) and thus depends on the value of X (r 1 = f--1 ()k)). The secondorder solutions can be summarized in simple form as follows
-h 1 + g l } ~
t
P0 d~, =2mr (1 + q ) p 0
(lb)
43
Volume 34, number 1
x {2) = }I
OPTICS COMMUNICATIONS
I 32 / '
e = 1/k(O),
(=f,)2 dr
July 1980 q(r) = [/3/k(r) -1] x=xi1)+cz?,l:l+.. .,
h2(r ) = k'(r)/k(r),
a ~X
/
l [(m 2 + 3/4)/r + h, - g 2 c o t h ~ {0)] .f' -
dr
PO
hl
0
-
o
rt
PO
(6) X=X(l)~
where T=/1 0
' ' ) + ' [1/r + ½h2(r)] hz(r ) + ¼g~(r), hl(r )-- ~h2(r 10)
i , gl(r) = ~g2(r) + ~~ [ l / r + h v (_r ) ] g ~ (,r ) -
gl c°th~(°)
. . . . .
g2(r) = y'(r)/y(r).
dr~
(7)
By virtue of eqs. (10) the value of ~l 1) determined from eq. (9)vanishes exactly. The longitudinal components of electric and magnetic waves propagating with exp( ~/3z),t?z and H e , can be represented I3,41 by k"z = q,(r)/[k(r)y(ri[ l.'2
( I la)
H z = -jq-,(r)/[kfr)/ytr)] 1/2
( 1 Ib }
PO ?,=?,(~)"
and
In case of a parabolic-index dielectric optical fiber [41
-/'?o0
gI d r l I ~ 7 ~ / a (1 + q ) j c ' d r l - 1 } i
g2(0) = + oo
(8a)
[ q'(rt)
for m = O,
x
l'~
~ ]g2(rl
(8b)
d
0
rPo
~-A2r2,
(13a)
h2(r)=g2(r)~---ar
A2r 3.
(13b)
q ( r ) ~-
_½(?,~l) + (741)/2)2)
XI1) = ~'A. ( 4 n + 2 j + m k(O) .
(13c)
Substitution of eqs. (13) into eqs. (5)-(8) leads to
(9)
"(°)=¥sinh-l{m
_ lJ+2),
/+~X(1-)),
(14)
(15t
x=x (~)
For electromagnetic vector waves of a radially inhomogeneous waveguide in which the medium is described by the local wavenmnber k(r) = k(O)[1 - f i r ) ] 1/2 and the local wave admittance y(r) and the propagation constant/3 along the waveguide axis z is defined by/3 = k(0)(1 - ~.)1/2, the remaining parameter and functions can be written in terms of k(r) and y(r).
44
A
+ ½(1 - ~(l~/2)Ar2 + ~A2r 4.
)1
I¢i' (! +q).r[ rl-'
where A is a small constant, the functions h I , h 2 , ,¢'1 ' g2 and q may be approximated hl(r)=gl(r)m
0 -"po-
t ~2(1) _ n 1 + q(rl) ~2 gl(rl ) - g2(rl ) 5 r l f , ( r 1)
+ ~ i +q(rl ~
(12)
fir) = A r 2
fZ(0) ~'+-sinh-1 51/7 [_3~.~ --P0 d/*
~(1) =0, X(2) = 3A~(1)(1 -- cothg2(°)),
(I6) (17a)
Volume 34, number 1
OPTICS COMMUNICATIONS
and X(2) = { ~ _A~(1)
for m = 0.
(18a) (18b)
The upper and lower signs in eqs. (14)-(17) correspond to HEm,n+ 1 and EHm,n+ 1 modes [3], respectively, and eqs. (18a) and (18b) indicate the eigenvalues of TE0,n+ 1 and TM0,n+ 1 modes [3], respectively. To check the validity of these solutions we quote the perturbational solutions due to Yamada and Inabe [5]. Although they presented only the first-order perturbational solutions in ref. [5], eqs. (17) and (18) have been verified by means of the second-order perturbation involving lengthy calculations. For HE (EH) modes the values of ~2(0) become negative (positive) and this indicates that the longitudinal electric (magnetic) fields of these hybrid modes dominate the longitudinal magnetic (electric) fields.
July 1980
This conclusion seems to be true for general classes of refractive-index profile because the same conclusion can be drawn for hybrid modes of a step-index optical fiber [6]. In addition, the family of hybrid modes of a radially inhomogeneous waveguide has been seen to have a strong resemblance to that of a step-index waveguide. The mathematical detail for derivation of eqs. ( 6 ) (9) and further applications will be reported elsewhere.
References
[1] [2] [3] [4]
M. Hashimoto, Optics Comm. 32 (1980) 383. See references in ref. [1]. M. Hashimoto, Int. J. Electron, 46 (1979) 125. M. Hashimoto, S. Nemoto and T. Makimoto, IEEE Trans. MicrowaveTheory Tech. MTT-25 (1977) 11. [5] R. Yamada and Y. Inabe, J. Opt. Soc. Am. 64 (1974) 964. [6] H.G. Unger, Planar optical waveguidesand fibers (Clarendon Press, Oxford, 1977) p. 372.
45
OPTICS COMMUNICATIONS
July 1980
ASYMPTOTIC EIGENVALUES OF VECTOR WAVES IN AN INHOMOGENEOUS CIRCULAR WAVEGUIDE Masahiro HASHIMOTO Department of Applied Electronic Engineering, Osaka Electro-Communication University, Neyagawa, Osaka 5 72, Japan Received 1 1 March 1980
Based on the vector wave equations of an inhomogeneous circular waveguide, asymptotic eigenvalues of the system of the coupled second-order differential equations are given, of which the first-order eigenvalues coincide with the well-known WKB solutions. A parabolic-index optical fiber is examined to obtain the propagation constants. These asymptotic results are compared with the perturbational solutions and both are shown to be in agreement.
Recently, the author has succeeded [1] in obtaining asymptotic eigenvalues of cylindrical scalar waves in a radially inhomogeneous medium by using WentzelDunham's complex-contour-integral quantum condition [2]. The usefulness of this technique appears to cover a wide range of eigenvalue problems related to acoustic, electromagnetic and quantum mechanical fields. In this Communications, we present the results of application of the technique to electromagnetic vector waves in an optical waveguide. We also give a comparison with other solutions in a specific case. Suppose that q~ and xI, are the solutions of the coupled differential equations of the second order, t
(2)
where the primes denote differentiation with respect to a variable r, e is a small parameter, m is an integer (--- 0,1,2 . . . . ), X is an eigenvalue, f i s an analytic evenfunction of r satisfying )'(0) = 0, and h 1, h2, g l , g2, q are smooth functions of r. The mathematical problem of solving eqs. (1) asymptotically can be treated by means of the method presented in in ref. [1 ], expanding X and ~ into )t = )t (1) + eZx (2) + e4X (3) + ...
(3a)
= ~(0) + e~(l) + e2~(2) + . . .
(3b)
t
q¢' + (1/r - 2Po/Po)~
+[p2[e 2
PO = PO (r) =- X/X - f(r),
where
m2r 2 +(h2
P'0 +gz)p0-
hl-gl
}
e a = +- lira ~(r)/rb(r) (4) r--*0 and assuming that the values of X(1) are the WKB solutions determined from
~
t
_ 2m (1 + q)P0 xp, r
(la)
?lP0 0
t
q/' + (1/r - 2Po/Po)q~ +[p2
m2
r2
+ (h 2
(I+m-ll+l) 2
--dr=Tr n+
P00
'
(5)
(n = 0,1,2 .... ),
t
-g2)
e
P0
where r 1 is the turning point at which X = f(r 1 ) and thus depends on the value of X (r 1 = f--1 ()k)). The secondorder solutions can be summarized in simple form as follows
-h 1 + g l } ~
t
P0 d~, =2mr (1 + q ) p 0
(lb)
43
Volume 34, number 1
x {2) = }I
OPTICS COMMUNICATIONS
I 32 / '
e = 1/k(O),
(=f,)2 dr
July 1980 q(r) = [/3/k(r) -1] x=xi1)+cz?,l:l+.. .,
h2(r ) = k'(r)/k(r),
a ~X
/
l [(m 2 + 3/4)/r + h, - g 2 c o t h ~ {0)] .f' -
dr
PO
hl
0
-
o
rt
PO
(6) X=X(l)~
where T=/1 0
' ' ) + ' [1/r + ½h2(r)] hz(r ) + ¼g~(r), hl(r )-- ~h2(r 10)
i , gl(r) = ~g2(r) + ~~ [ l / r + h v (_r ) ] g ~ (,r ) -
gl c°th~(°)
. . . . .
g2(r) = y'(r)/y(r).
dr~
(7)
By virtue of eqs. (10) the value of ~l 1) determined from eq. (9)vanishes exactly. The longitudinal components of electric and magnetic waves propagating with exp( ~/3z),t?z and H e , can be represented I3,41 by k"z = q,(r)/[k(r)y(ri[ l.'2
( I la)
H z = -jq-,(r)/[kfr)/ytr)] 1/2
( 1 Ib }
PO ?,=?,(~)"
and
In case of a parabolic-index dielectric optical fiber [41
-/'?o0
gI d r l I ~ 7 ~ / a (1 + q ) j c ' d r l - 1 } i
g2(0) = + oo
(8a)
[ q'(rt)
for m = O,
x
l'~
~ ]g2(rl
(8b)
d
0
rPo
~-A2r2,
(13a)
h2(r)=g2(r)~---ar
A2r 3.
(13b)
q ( r ) ~-
_½(?,~l) + (741)/2)2)
XI1) = ~'A. ( 4 n + 2 j + m k(O) .
(13c)
Substitution of eqs. (13) into eqs. (5)-(8) leads to
(9)
"(°)=¥sinh-l{m
_ lJ+2),
/+~X(1-)),
(14)
(15t
x=x (~)
For electromagnetic vector waves of a radially inhomogeneous waveguide in which the medium is described by the local wavenmnber k(r) = k(O)[1 - f i r ) ] 1/2 and the local wave admittance y(r) and the propagation constant/3 along the waveguide axis z is defined by/3 = k(0)(1 - ~.)1/2, the remaining parameter and functions can be written in terms of k(r) and y(r).
44
A
+ ½(1 - ~(l~/2)Ar2 + ~A2r 4.
)1
I¢i' (! +q).r[ rl-'
where A is a small constant, the functions h I , h 2 , ,¢'1 ' g2 and q may be approximated hl(r)=gl(r)m
0 -"po-
t ~2(1) _ n 1 + q(rl) ~2 gl(rl ) - g2(rl ) 5 r l f , ( r 1)
+ ~ i +q(rl ~
(12)
fir) = A r 2
fZ(0) ~'+-sinh-1 51/7 [_3~.~ --P0 d/*
~(1) =0, X(2) = 3A~(1)(1 -- cothg2(°)),
(I6) (17a)
Volume 34, number 1
OPTICS COMMUNICATIONS
and X(2) = { ~ _A~(1)
for m = 0.
(18a) (18b)
The upper and lower signs in eqs. (14)-(17) correspond to HEm,n+ 1 and EHm,n+ 1 modes [3], respectively, and eqs. (18a) and (18b) indicate the eigenvalues of TE0,n+ 1 and TM0,n+ 1 modes [3], respectively. To check the validity of these solutions we quote the perturbational solutions due to Yamada and Inabe [5]. Although they presented only the first-order perturbational solutions in ref. [5], eqs. (17) and (18) have been verified by means of the second-order perturbation involving lengthy calculations. For HE (EH) modes the values of ~2(0) become negative (positive) and this indicates that the longitudinal electric (magnetic) fields of these hybrid modes dominate the longitudinal magnetic (electric) fields.
July 1980
This conclusion seems to be true for general classes of refractive-index profile because the same conclusion can be drawn for hybrid modes of a step-index optical fiber [6]. In addition, the family of hybrid modes of a radially inhomogeneous waveguide has been seen to have a strong resemblance to that of a step-index waveguide. The mathematical detail for derivation of eqs. ( 6 ) (9) and further applications will be reported elsewhere.
References
[1] [2] [3] [4]
M. Hashimoto, Optics Comm. 32 (1980) 383. See references in ref. [1]. M. Hashimoto, Int. J. Electron, 46 (1979) 125. M. Hashimoto, S. Nemoto and T. Makimoto, IEEE Trans. MicrowaveTheory Tech. MTT-25 (1977) 11. [5] R. Yamada and Y. Inabe, J. Opt. Soc. Am. 64 (1974) 964. [6] H.G. Unger, Planar optical waveguidesand fibers (Clarendon Press, Oxford, 1977) p. 372.
45