Volume 77, number 1
OPTICS COMMUNICATIONS
1 June 1990
P O L A R I Z A T I O N C O U P L I N G IN S I N G L E - M O D E S I N G L E - P O L A R I Z A T I O N OPTICAL FIBERS: P O W E R C O U P L I N G Chao-Xiang SHI and You-Ju MAO Radio Engineering Department, Chongqing Institute of Posts and Telecommunication, Chongqing, Sichuan, P.R. China
Received 23 March 1989; revised manuscript received 30 September 1989
A set of coupled powerequations derived from that of amplitude equations for single-modesingle-polarization (SMSP) optical fibers has been presented. Our results show that the power coupling coefficientsare related not only to the effect of the birefringence of SMSP fibers, but also to the leaky mode loss.
1. Introduction Polarization maintaining optical fibers (PM) are essential for coherent optical fiber communication systems and fiber sensor systems. Many types of polarization-maintaining fibers have been proposed and fabricated [ 1,2 ]. However, the polarization maintaining ability of PM fibers is influenced by the random polarization mode coupling. The crosstalk of PM fibers (two polarizations) degrades with fiber length due to the random mode coupling [ 3 ]. Recently, single-mode single-polarization optical fibers have been developed to overcome the degradation of the polarization crosstalk. The crosstalk of SMSP fibers becomes almost a constant - 30 dB, and is independent of the fiber length beyond 200 m [4]. Snyder made a prediction in 1983 that fibers composed of highly birefringent materials have a single polarization property because of the leaky mode effect [ 5,6 ]. A recent paper also indicated that the tilting of the anisotropic axis in SMSP fibers can amplify their single polarization property [ 7 ]. There have been two other papers discussing the anisotropic fiber with finite cladding [ 8,9 ], especially, a further research for finite cladding anisotropic fibers by considering the influence of their lossy outer jacket has been completed [ 10]. In addition, single-mode, single-polarization fibers have been realized by using the bending effect of the practical bow-tie fibers [ 11,12 ]. An analysis of the polarization coupling in single-mode, single-polarization optical fibers has been given, and a set of amplitude coupled equations also has been derived [ 13 ]. This paper is also about the polarization coupling concerned with the coupled power equations.
2. Theory According to Snyder's theories, there are two fundamental polarization modes in a SMSP fiber: a guided mode and a leaky mode. When an optical fiber suffers external perturbations, polarization coupling will be included, there is a continuous exchange of the energy between the guided and leaky modes. The guided mode couples a part of its energy to the leaky mode, the leaky mode can also give back a small part of its energy to the guided mode, in addition, it leaks some energy to the cladding. Therefore, the energy coupled from the guided mode to the leaky mode. This coupling procedure can be described well by a set of coupled power equations. We assume that f ( x ) is a stationary random process of disturbances to the fiber that can be described by an autocorrelation with a finite correlation length D. The average o f f ( z ) vanishes i.e. ( f ( z ) ) = 0, or if z - z ' >> D, 0030-4018/90/$03.50 © Elsevier Science Publishers B.V. (North-Holland)
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I Z'f ( z ) d z = 0. Besides, we can make use of the fact that, due to the finite correlation length of the disturbances function f ( z ) , the field amplitude C,,,(z') and f ( z ) are uncorrelated based on z - z ' >> D, expressed as
(C,,(z') C*(z') f(z))=(C;,,(z') C ; , ( z ' ) ) (.f(z))
.
We will then derive the coupled power equation for SMSP fibers from the amplitude coupled equations. The amplitude coupled equations are following [ 13 ], dCl/dz=Kll
(-'1 q-Kj2(~2 exp (ifl~, -ifl~ +c~) z,
( la )
dC2/dz=K21 Cj exp ( i f l , - i f l v - o ~ ) z + K 2 2 C 2 .
(ib)
In order to be in accord with the previous paper, we still let the x-polarization mode be a guided mode, and y-polarization mode be a leaky mode, as expressed by the following equations at a fiber length z [ 13 ], E~I = C I ( z ) e,1 e x p ( - i f l v - o e ) z,
Et2 =C2(z) e,zexp(-ifi~z) .
The average power carried by each mode at the fiber length z is (here we omit the normalization coefficient p for the same of brevity) PI = ( C l ( z ) C'~(z)) e x p ( - 2 ~ z ) ,
P2=(C2(z) C * ( z ) )
.
In above calculation, the integral over the squared magnitude of the leaky mode diverges. However, in practical fibers, if we consider the influence of the lossy outer jacket, the cladding of the fiber is finitely extended [ 10 ], and the oscillatory fields of the leaky mode exist only in the finite cladding region, thus, we only need to substitute the integral section by the practical fiber cross section to calculate the above integral. Next, we take the z derivation of the average power of y-polarization mode
dPt/dz= -2oeP~ + ( ( d G / d z ) C ' f ) e x p ( - 2c~z)+c.c.
(2)
By using eq. ( 1 ), we have
(dC,/dz)C•) =/~11 (C'fC, f ( z ) ) + f 1 2 ( C T C z f ( z ) ) exp(iflv-iflx +o~) : .
(3)
The notation c.c. indicates that the complex conjugate of terms already appearing in the equation is to bc added. According to the point of weak coupling assumption and ideas of perturbation theory, we can assume that the field amplitudes C,,(z) ( m = 1, 2) change very slightly over a distance that is large compared to the correlation length D. So, the perturbed solution eq. ( 1 ) is C~ (z) = C, ( z ' ) + R ~ C~ ( z ' ) j f ( x ) d.x"~/~12 C2 ( z ' ) f ( x ) exp(ifl,,- iflx + o~) x d x , 2' :'
(4a)
.f(x) exp(ifi~ -iris,-c~) x dx.
(4b)
(-'2(Z) : (:2 (Z') "4-J~22 C2 ( Z ' ) j r ( X ) :
C I(z') :
In the following derivation, we use the proximate relation J§,f(z) d x = 0 ( z - z ' >> D), and the eq. (4),
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( C,(z) C'~(z) f(z) ) = ( C,(z' ) CT(z' ) f(z) ) + (CT(z')
C2(z' )) Kl2 i (f(z)f(x)) exp(ifly--ipx + a ) x d x z'
+ ( C,(z' ) C~(z' ) ) KTz i (f(z) f(x) ) e x p ( i f l x - i f l y + a ) x d x z'
+ ( Cz(z') C'~(z') ) /£Tz/(zi i (f(z) f(x) ) e x p ( i f l x - i f l y + a ) x d x z'
× i (f(z)f(x)) e x p ( i f l y - i f l x + a ) x d x . z'
We can then simplify the above equations by using previous proximation assumptions. (i) The first term on the right hand side of above equation vanishes
(Cl(Z') CT(z')f(z))=(C,(z') CT(z')) ( f ( z ) ) = 0 . (ii) We expected that the phase of the coupled field amplitude is sufficiently random, so that we have
(C*(z)C,(z)) = (ICm(Z) 12)6,,,,, and the second and third terms on the right hand side of above equations vanish. (iii) The product of the fourth right hand side term is proportional to/(T2/(2~, in the spirit of perturbation theory and the weak coupling assumption, this fourth term can be neglected. So, the devotion of the first term in the right hand side of eq. ( 3 ) can be neglected based on the above discussions. We also use the same method to operate and obtain,
KIz(CT(z ) C2(z)f(z)) exp(i/~y-ipx+a) z C,(z' ) ) exp(i/~y-i/~x+a) z i (f(x) f(z) ) exp(i//x-i/~v-a) x d x
=/(12/(2~ (CT(z')
z'
+ IK, z 13(C~(z') C2(z') ) exp(i/~y-i/~x + a ) z i (f(x)f(z) ~ exp(i/~-i/~v + a ) x dx.
(5)
z'
Next, we introduce the autocorrelation function R (u) = R ( - u) = ( f ( z ) f ( z - u ) ). Because the autocorrelation function contributes over a range only on the order of the correlation length D, we can change the lower integration limit from z to -o0, and obtain exp(i/~y-i~x+a)
z i (f(z)f(x)) e x p ( i / ~ x - i / ~ y - a ) x d x = i R(u)exp(i~v-i//x + a ) u du. z'
In deriving the above equation we use a relation we can obtain z
exp(i/~-i/~x+a) zJ z'
(6)
0
z-x=u and R(u)--R(-u). By using the same method oo
(f(z)f(x)) exp(i/~x-i/~y+a) xdx=exp(2az) j R ( u ) e x p ( i / / y - i / / x - a ) udu.
(7)
0
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Considering the field amplitude changes very slowly over the distance from a section z' to z, and almost being constant, we have
C~(z' ) Ca (z') ) exp( - 2o~z) = (CT(z) Ca ( z ) ) exp( - 2c~z) =Pr , ( C ~ ( z ' ) C 2 ( z ' ) ) = (C'~(7_.) C 2 ( z ) ) = P 2 Finally, one of the power coupled equations can be derived by using eqs. (2), (3), (5), (6), and eq. (7)
dP1/dz=-2o~Pj +hlaP1 +hl2P2, where
i i
R(u) exp(ifly-iflx+a) udu+c.c. ,
(8a)
0
hi2 = r/~,212
R(u) e x p ( i f l y - i f l x - a )
u
d u + c.c.
8b)
0
The other equation can be derived by using the same method as above, so we have a set of power coupled equations as follows
dP~/dz=-2otPa +haaP~ +ha2P2,
dP2/dz=h2aP1 +h22P2,
where oo
h2a = I/~2~ 12 j R(u) exp(ifl~ - i f l y + a ) u d u + c . c . ,
(9a)
O
h22 =/£~2/(2a i R(u) exp(ifl.~ - i f l y - c O u du+c.c.
(9b)
0
We find that power coupling coefficients are Laplace transforms of the autocorrelation function. If Laplace transforms exist, the autocorrelation function must be sectionally continuous on every finite interval in the range u > 0, and satisfies
R(u)
m>0.
The Laplace transforms are convergent for all a < m (this will be satisfied in most real cases). If the above conditions are not true, the analysis to the polarization coupling using the power coupled equations are invalid, and we can only solve the random amplitude coupling equations rigorously with a few assumptions [ 14 ]. This kind of analysis will be given in another paper. Coming back to eqs. (8) and (9), we find that the power coupling coefficients apparently depend on the autocorrelation function of the disturbancesf(x). We can a s s u m e f ( x ) to be a stationary process characterized by an exponential autocorrelation function,
R(u)=ag exp(- ]ul/D) , where ao2 is the variance of the disturbance, D is the correlation length. From eqs. (8) and (9) we obtain
h,2 = IR~2 120"02(l/D-t-oL)/(fly--flx) 2+ (0/+ 1/O) 2, hzl = [K21 120"02(1/O-oL)/(flv-)~,:) 2+ (ot- 1/O) 2. 16
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Volume 77, number 1
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0.95
'o
/
0.90
~w
m
O. 8 5
e-
©
0.80 J
0.75 0
O. 7 0 ~.
0.65
O
0.60
1'0 Leaky
{o
io*
Node Loss
e~
(kin -I)
Fig. 1. Power coupling coefficients against the leaky mode loss.
In fig. 1 we show coupling coefficients hi2, h2, against the leaky mode loss a. When plotting, we assume the correlation length D = 0.02 m, I/~ 21 a 2 = [/~2t I ao2 = 0.36, the birefringence B = 1 × 10 - 4 (fl~_ fly= 2riB~R), and the parameter of the wavelength is 2 = 1.3 ~m. The plot in fig. 1 indicates that the power coupling coefficient h~2 increases with increasing a but h2~ decreases, which means that the guided mode couples more energy to the leaky mode, and this case is just inverse for the coupling procedure from leaky mode back to the guided mode. In fig. 2 and fig. 3 the coupling coefficients are reported against the birefringence B as well as the correlation
1.6 #
i0-~.
O
v
1.2
10 "G~ c-
~
0.8
10"7"
Q
c)
o
¢..)
1 0-8.
0.4
O
¢,:'
O U
10- 9
0 i
i
'a '4
Birefringence
;
'7 "8 9 B
011
( Y, 10"4)
Fig. 2. Variations of power coupling coefficients with birefringence B, for D = 10 m, ~ = 50 k m - 1.
0.0
hz' .0
!
3,0
-*
w
5.0
Correlation
7.0
length
'
I
9.0
D
'
11.0
(m)
Fig. 3. Power coupling coefficients against the correlation length
D, for B = l X I0 -4, and t~= I0 kin-t.
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length D. We conclude from the above curves that the polarization coupling in SMSP fibers is an unequal procedure, which means that the energy carried by each polarization mode exchanges unequally. The coupling coefficient h~2 is always greater than that of h2~, this tendency will be strengthened by increasing the leaky mode loss.
3. Conclusion We have derived the power coupled equations from the amplitude coupled equations for SMSP fibers, in our equations, the power coupling coefficients are decided not only by the birefringence of the SMSP fiber, but also by the leaky mode loss. The coefficient hi2 is not equal to h2~, which indicates that the energy carried by each polarization mode exchanges unequally while the polarization coupling occurs. The energy coupled from the guided to the leaky mode is always greater than that from the leaky mode back to the guided mode, so the guided mode has an additional loss through the polarization mode coupling. While light propagates in the SMSP fiber, the polarization crosstalk can also be calculated by using the coupled power equations.
Acknowledgements The authors are very grateful to Dr. M.S. D e m o k a n for his encouragement. Thanks are also given to the F o u n d a t i o n of Applied Science of Sichuan for their support.
References [ l ] T. Hosaka, K. Okamoto, T. Miya, Y. Sasoki and Edahiro, Electron. Lett. 17 ( 1981 ) 530. [2 ] T. Katsuyama, H. Matsumura and T. Suganuma, Electron. Left. 17 ( 1981 ) 473. [ 3 ] Y. Sasaki, T. Hosaka and J. Noda, Elec. Lett. 20 (1984) 784. [4] T. Hosaka, Y. Sasaki and K. Okamoto, Electron. Len. 21 (1985) 1023. [ 5 ] A.W. Snyder and F. Ruhl, Electron. Lett. 19 (1983) 687. [ 6 ] A.W. Snyder and F. Ruhl, IEEE J. Lightwave Technology,LT2 (1984) 284. [7] A.W. Snyder and A. Ankiewicz,J. Opt. Soc. Am. A 76 (1986) 856. [8 ] A.W. Snyder and A. Ankiewicz,Electron. Lett. 21 (1986) 1105. [9] Shi Chao-Xiang,Optics Comm. 70 (1989) 384. [ 10] Shi Chao-Xiang, J. Opt. Soc. Am. A 6 (1989) 550. [ 11 ] M.P. Verham, D.N. Payne, R.D. Brich and E.J. Tarbox, Electron. Lett. 19 (1983) 246. [ 12] M.P. Vernham, D,N. Payne, R.D. Brich and E.J. Tarbox, Electron. Lett. 19 (1983) 679. [ 13 ] Chi Chao-Xiang, Optics Lett. 13 (1988) 1120. [ 14] FengTian, Yi-Zun Wu and Pei-Da Yei, J. LightwaveTechnology 5 ( 1987 ) I 165.
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