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Influence of Raman scattering on the cross phase modulation in optical fibers

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Optics Communications 94 ( 1992) 477-484
OPTICS
North-Holland
COMMUNICATIONS
Full length article
Interaction of two optical waves in a nonlinear medium resulting in Bragg diffraction
Shiva Kumar, G.V. Anand and A. Selvarajan
Department of Electrical Communication Engineering Indian Institute of Science Bangalore-560012 India
Received 25 September 199 1; revised manuscript received 11 June 1992 The interaction of two optical waves in a Kerr medium is considered. One of the waves is initially unmodulated and the other wave is amplitude modulated. Due to the Kerr effect, the amplitude modulated wave produces a refractive index grating and the first wave gets diffracted because of this grating. It is shown that there are two Bragg angles for a single modulating frequency at which the incident wave can be diffracted. The Bragg angle for the second order diffracted wave is approximately twice that for the first order wave and the interaction length required for complete power transfer into the second order diffracted wave is more than that for the first order wave. Some numerical calculations are made to find the interaction length for the first and second order diffracted waves and the optical power requirements.
1. Introduction The acousto-optic Bragg scattering in the Kerr medium has been studied in the past
[
l-41. In this case, the incident beam and the scattered beam form a travelling interference pattern, inducing a volume index grating. It is shown that the diffraction effi- ciency increases as a function of optical intensity for downshifted scattering at low acoustic power. In this paper, we have considered the Bragg scattering in the absence of an acoustic field. We consider the interaction of two optical waves propagating almost perpendicular to each other (fig.
1 ,
in a Kerr medium. The first wave is initially un- modulated and the other wave is amplitude modu- lated. In the present analysis, the amplitudes of the carrier and the sidebands of the second wave are as- sumed to be much larger than the amplitude of the first wave. In addition, the ratio of the amplitudes of the carrier and the sidebands is so selected that these amplitudes remain unchanged during the propaga- tion. Due to the Kerr effect, the second wave pro- duces a refractive index grating and the first wave gets diffracted because of this grating. The present interaction is analogous to the Bragg diffraction by acoustic waves of two different frequencies. How- ever, in the present interaction the grating moves at
z t
I
err medium
X
2
Fig. 1. Two optical waves propagating almost perpendicular to each other interacting in a Kerr medium.
the group velocity of light and therefore, the Bragg angle is very small compared to the case of Bragg dif- fraction by acoustic waves. Also, there can be two Bragg angles for a single modulating frequency, at which the first wave can be diffracted. 2. Theory Consider the wave equation,
0030-4018/92/ 05.00 0 1992 Elsevier Science Publishers B.V. All rights reserved,
477
Volume 94 number 5 OPTICS COMMUNICATIONS 1 December 1992
1)
with
E=EI +E,, (2)
where
El
and E2 are the electric fields of the optical waves 1 and 2, respectively, and
P,_
and
PNL
are re- spectively the linear and nonlinear components of the polarization. We can write
E,= [l?,(r,t)
exp(io,t)+c.c.], j=l, 2, (3) where j is the polarization unit vector and
Ej(r, t)
is a slowly varying function of time. With the as- sumption of instantaneous nonlinear response, the polarization components can be written as
PL(r, t) =c,X(‘)E(r, t)
,
(4)
P&r, t)=~,~(~)iE(r, t)E(r, t)E(r, t) =fP{h(w) w(iwt)+PNL(w2) ew(bt) +PNL(2w1 -01~)
exp[i(2w, -oz)t] +PNL(2w2-wl) exp[i(2w,-w,)t] +...+ c.c.},
5)
where PNL(W,)=e r(IEl
12+21~212)G > 6) PNL ~2)=wefdl~2 12+21~1 I’@2 9
(7) xeff=3x’3’/4.
8)
The induced nonlinear polarization has terms oscil- lating at new frequencies 20, - o2 and 20~ - wl. The phase matching condition is required to build up the new frequency components significantly, a condition not satisfied in practice
[ 5 1.
Hence, these frequency components are neglected in the ensuing analysis. The first term in eq. (6) represents the effect of self-phase modulation (SPM )
_
The second term rep- resents the effect of phase modulation of
El
by E2 and is responsible for cross-phase modulation (XPM). The two terms is eq. (7) can be similarly interpreted. Before entering the nonlinear medium, let the first wave be unmodulated plitude modulated, so E,(r, t)= (r)
,
and the second wave be am- that 478 (9)
E2(r, t)=B,(r)+B,(r)
exp(iL2t)
+B_,(r)
exp( -iQt) ,
10)
and Q<w,,~. Substituting eqs. (9) and (10) in eq. (6), we get +2(B,BT+B_,B?j) exp( -i V) +2( B*, +BIB;) exp(iQt) +2B_,B; exp( -i2Qt) +2B,B*, exp(i2Qt)]A,.
11)
The medium is assumed to be dispersionless and lossless. Without any loss of generality, we can as- sume propagation in the x-z plane so that the field has no y-dependence. So, eq. (1) becomes On substituting eq. ( 11) into eq. (5) and finally into eq. ( 12)) we see that the right hand side of eq. ( 12 ) has terms proportional to exp
[i
(w, + n-Q) t ] for n = 0, 1,2. Therefore, because of the periodic refractive in- dex grating, diffracted waves of frequencies w1 + Q, wi + 2Q and o, - 252 are produced. For an arbitrary angle of incidence of the first wave, it is difficult to derive expressions for the power in each diffracted wave, analytically. However, for those angles of in- cidence which are close to the Bragg angle corre- sponding to the first order diffracted wave, we can neglect the diffracted waves of frequencies o1 k 2Q and thus obtain the expressions for the power in the first order diffracted wave. In our analysis, we have assumed that the second wave is propagating in the positive z direction and the first wave is incident al- most at right angles to it as shown in fig. 1. Also, we have assumed the amplitudes of the carrier and the sidebands of the second wave to be much larger than the amplitude of the first wave so that the cross phase modulation of the second wave by the first wave can be neglected. In this case, the electric field of the first wave in the nonlinear medium can be written as
Volume 94, number 5 OPTICS COMMUNICATIONS 1 December 1992
El =tj{a(x, z) exp[i(o,t-o,x-Az)] +a+(~, z)
exp[i(ol+t-a,+x-p,+z)l +a_(x, z)
exp[i(w,_t-cr,_x-jI,_z)]
c.c.} ,
(13) where &+P:=k:) (14) a:? +P:t =k:, , (15) w,, =o, +.Q, (16) w,_ =w, -52 (17) and
kI
and
k, f
are the wavenumbers in the medium for the incident and 2 1 order diffracted waves, re- spectively. oi+ are the angular frequencies for the diffracted waves. The electric field of the second wave can be writ- ten as E2 = @{b,(z) expCi(WZf-PZz)
1 +h (z) ewG[ (w +Q)t- (A +WzlJ
+b_,(z)exp{i[(w,-Q)t-(~z-K)z]}+c.c.},
18)
where
19)
is the velocity of light in the medium. .Q and
K
de- note respectively the deviations in the frequency and the wavenumber of the sidebands from the carrier, and
b.
and
b,
, are the amplitudes of the carrier and the sidebands, respectively. We now impose the following constraints on the spatially varying amplitudes appearing in eqs. ( 13 ) and ( 18 ). The amplitudes of the carrier and the side- bands of the second wave are assumed to be large compared to the amplitude of the first wave, so that the magnitude of the second degree terms in a or ai is negligibly small compared to the magnitude of the second degree terms in
b.
or
b+ ,.
Secondly the am- plitudes appearing in eq. ( 13) are assumed to be slowly varying functions of the spatial variables x and z. Hence, their second order spatial derivatives are negligibly small
[
6
1.
Finally, since the z-component of the propagation vector of the first wave is very small, the z-dependence of a and a? can be ignored altogether. Hence, if we substitute eqs. ( 13) and ( 18) into eq. ( 12) and compare the coefficients of exp(io,t) and exp(iw,+t), we get (-2i) [cx, da/d-x] exp[ -i(c.uix+/Iiz)
1
=-k&,{b ruexp[-i(cr,x+j3,z)]
+2[b,b:+b_,b;;]u+ x ew{-i[aI+X+ (A+
-WI)
+2[bobY, +b,b:]u_ x
exp{-i[a,_x+ (P1-
+K)zl}}
and
20)
(-2i)(c_u*da./dx) exp[-i(criix+B1+z)]
=-k&ff{b&z+
exp[-i(a,+x+A~z)l
+2[bob:, +b+lb:lu x em{-i[a,x+ (PI
*K)zlII,
21)
where
b eff=
~~~I~o12+~,12+~-,12~, 22)
and ko=w1JLZ, (23) is the free space wavenumber of the first wave. The ratio of the amplitudes of the carrier and the sidebands can be so selected that the amplitudes re- main unchanged during the propagation. Specifi- cally, when
Ib,(0)12=lb_,(0)12=(2/3)lbo(0)12,
it is known
[
7,8] that no exchange of energy be- tween the carrier and the side bands takes place. In other words, the amplitudes of the carrier and the sidebands are independent of z. In this case, we can write b,(z)
=bo
, (24)
bl(z)=-b_,(z)=b;/2i, (25)
and
E2=f {[bo+b;
sin(Bt-Kz)] X exp[i(c02t-j?2z>]+~.~.},
26)
where
b.
and b; are real constants and related through the equation 479
Volume 94, number 5 OPTICS COMMUNICATIONS 1 December 1992
6 =
J8/3
b.
(27) Substituting eqs. (24) and (25) in eqs. (20) and (21). we get (-2i)(cr,da/dx) exp[-i(a,xfp,z)] =
-ki {b a
exp[ -i(crrx+j3,z)]
+ib,b,a+
exp(-i[a,+x+(Pr+ -K)z]}
-ib,b,a_
exp{-i[a,_x+(/?r_ +K)z]}} (28) and (-2i)(cr r*da+ldx) exp[-i(aIix+PIkz)l =-kkyew{b&z. exp[ -i(aIkx+P1kz)] Tib&aexp{-i[a,x+(p,fK)z]}}. (29) Since a and a, are functions of x only, z-depen- dent factors appearing in eqs. (28 ) and (29 ) must cancel out. Therefore, we have PI+ =PI +K, (30) PI- =P, -K, (31) which are nothing but the Bragg conditions
[
9
1.
3.
Particle picture
The first wave can be thought of as made up of a stream of particles with momentum Ak, and energy fiwr [6]. This wave can be considered to be inter- acting with a particle having momentum
fiK
and en- ergy ML?. The diffraction of the first wave by the ap- proaching second wave can be described as a series of collisions, each of which involves an interaction between an incident photon at frequency w, and a particle at frequency Q and a simultaneous creation of a new photon at frequency w, +. From the law of conservation of momentum, the total momentum fi(k, +
K)
of the colliding particles must be equal to the momentum Ak, + of the scattered photon, i.e., kr, =k,
K.
The conservation of energy yields w,+ =w, +C?, which is in agreement with eq. ( 16). (32) (33) 480
Fig. 2. Momentum conservation relation for an optical beam that is diffracted by an approaching amplitude modulated optical wave.
To show that the conservation of momentum con- dition is equivalent to the Bragg condition, consider fig. 2. Since .Q;2<< , and w, + z w,, it follows that
k , z k,
and that the magnitude of the two optical wave vectors can be taken as
k,.
From fig. 2 we see that
k, cos(O)=k,+ cos(O+),
or equivalently, aI =a1+
>
and
34)
(35)
k,
sin(B)+k,+ sin(o+)=K. (36) Since /3,=-k, sin(@) and &+=k,+ sin(8+), eq. ( 36 ) becomes P,+ =P, +K> (37) which is the same as eq. (30). Since
k, = k,
+ , from eq. (34) we get 19 , and eq. (36) reduces to, 0=sin-‘(K/2k,), (38) which gives the Bragg angle. 4.
Coupled analysis of the first order diffracted wave
When the angle of incidence of the first wave is such that cur ZCX~+, we can neglect the diffracted wave of frequency w, -Q, since the difference CY, LX, _ is too large. Hence, neglecting a_ in eqs. (28) and (29 ),
we
get the following equations. dZ/dx=rc+d+ exp(iAa+x)-iiyrZ, (39)
Volume 94, number 5 OPTICS COMMUNICATIONS 1 December 1992
dii+/dx=-K+Ciexp(-iAa+x)-iyE+, where (40) 8= (a,/20,/hJ)“
)
(41) d+ = (~,+/2~1~o)“2a+
,
(42)
K+=Ib~b;k~X~ff(ala,+)- *,
(43) y= t (b:&~err/~,)
>
(44) Aa, =a, -a,‘,,
.
(45) The amplitudes of the incident and the diffracted waves are multiplied by suitable factors in eqs. (4 1) and (42) so that
1 i 1 2
and
15, *
represent the pow- ers in the incident and the diffracted waves, respec- tively. The parameter y is a some kind of propaga- tion constant which is proportional to the power in the second wave, and
K,
is the coupling coefficient which determines how much power is coupled into the diffracted wave. When a wave of unit power is incident at x= 0, we have the initial conditions G(O)=l, a+(o)=o. (46) The solution of the coupled amplitude equations ( 39) and (40) is given by (appendix A) rT(x)= [cos6x-i(Aa+/26)sin6x] Xexp[MAa+/2-y)
1 ,
G+(x)= (K+/d)sin(Gx) (47) x
ew[-Ny+Aa+/2 1 ,
(48) where 6=
[
(Aa+)2/4+K:]“2. (49) Expressions for power in the incident and the dif- fracted waves are P(x)= In(x) I’=cos*(~x)+ (A(-U+/28)2sin2(6x)
, 50) P+ x)=
l~?+(x)I*=(K+/8)*sin*(fJx).
51)
As can be seen from eqs. (50) and (5 1
),
there is a periodic exchange of energy between the incident and diffracted waves with a period 7~16, and the period decreases as A(Y+ is increased. From eqs. (47 ) and (48 ), we see that y introduces only a phase shift to the incident and diffracted waves and does not effect the coupling of power into the diffracted wave. When the angle of incidence is equal to the Bragg angle, we have Aa+=O and
d=lc+,
and eqs. (50) and (5 1) reduce to P(X)=
Ia 12(X)=COS2(K+,X),
(52) P+(x)= la”, 12(x)=sin2(K+,x). (53) In this case, complete power transfer from the in- cident to the diffracted wave is possible, and the in- teraction length for complete power transfer is given by>
L, =nn/ 2ic+),
n=l,3,5 .
.
(54) The interaction length for complete power transfer is inversely proportional to the coupling coefficient
K+,
which in turn is proportional to the product of the amplitudes of the carrier and the sidebands. When the angle of incidence deviates from the Bragg angle, (Y, will be different from cr,+ and we have Acr, ~0. In this case, total power transfer from the incident wave to the diffracted wave is not possible. When the angle of incidence of the first wave is such that czr =a,_, we can neglect (I+ in eqs. (28) and (29 ), and derive an expression for power in the - 1 order diffracted wave, similar to that for the + 1 order wave. 5.
nalysis of the second order diffracted waves
In the previous section, we considered the inter- action between the incident wave and the first order diffracted waves and we neglected the second order diffracted waves. This is justified for those angles of incidence which are close to the Bragg angle corre- sponding to the frequency Q. Since the induced re- fractive index change is proportional to the square of the modulus of the electric field, the moving re- fractive index will have a frequency component 2Q. Therefore, power transfer into the second order dif- fracted waves cannot be neglected if the angle of in- cidence of the first wave is close to the Bragg angle corresponding to the frequency 252. In this case, we can, however, neglect the first order diffracted waves and the electric field of the first wave can be written as 481

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