Gain for Seeded Brillouin Scattering
ref: R.W. Boyd Nonlinear Optics (3rd ed) Chap 9
Define the backwards-propagating Stokes gain as G I S (0) / I S (L) in the seeded case. In
this paper, the gain is calculated for Brillouin amplification in a fiber with negligible scattering
loss ( = 0 ).
The coupled intensity equations in the absence of loss are
dI L
dz
= g B I L IS
(1)
for the laser as a function of length z along the fiber ranging from 0 to L, and
dIS
dz
= g B I L IS
(2)
for the Stokes beam. By equating the left-hand sides of Eqs. (1) and (2), and integrating, one
obtains
I L = IS + C
(3)
where C is some constant intensity difference. This result is an expression of conservation of
energy: any amplification in the Brillouin power propagating backwards must come at the
expense of the laser power propagating forwards. Substitute Eq. (3) into (1) to get
IS ( z)
z
dIS 1
1 =
g
C I I + C B dz S
S
(0)
0
IS
IS (z)
IS (0)
=
IS (z) + C
IS (0) + C
exp(g BCz) .
(4)
Replacing C by I L (0) IS (0) , Eq. (4) rearranges into
IS (z) =
R(1 R)
I (0)
exp (1 R)g B I L (0)z R L
(5)
where R IS (0) / I L (0) is the Brillouin reflectivity. Evaluating Eq. (5) at z = L and dividing
both sides by IS (0) = R I L (0) leads to
IS (0)
IS (L)
=
exp (1 R)g B I L (0)L R
1 R
.
(6)
In the limit as R 0 , this result gives rise to the familiar small-signal gain,
G = exp g B I L (0)L .
(7)
However, unlike for unseeded Brillouin scattering, the reflectivity does not have an upper limit
of unity because it starts with a value of IS (L) at the back end of the fiber. So rather than
expressing equations in terms of the reflectivity, it is more useful to introduce the Brillouin
transfer efficiency T IS (0) IS (L) / I L (0) , which varies from 0 to 1 because it equals the
fraction of the input laser power that is converted to newly created Stokes power. Subtracting
1
unity from both sides of Eq. (6) gives
T exp (1 T S)g 1
=
1 T S
S
(8)
where to reduce clutter I have defined g g B I L (0)L and introduced the normalized Stokes seed
intensity S IS (L) / I L (0) so that R = T + S . If the seed is weak, i.e., S 1, then T R . In
that case it makes sense to consider the limit of Eq. (6) as R 1 . Applying l’Hopital’s rule then
gives rise to an analogous result to Eq. (7),
G = 1 + g B I L (0)L
(9)
for the large-signal gain, where the factor of unity on the right-hand side can be neglected,
because the following graph shows that g B I L (0)L 1 for large T and small S. This result
implies that the gain saturates, IS (0) I L (0) , as makes sense because the laser is being
completely converted into Brillouin power. (Note in contrast from the same graph that small T
does not guarantee that g 1 for small S.)
Assuming that the seed S has a given value under experimental control, Eq. (8) implicitly
gives T (which specifies the Stokes output) as a function of g (which is proportional to the laser
input). However, we can only explicitly solve for the inverse function,
T
g = (1 T S)1 ln 1 + (1 T S) .
S
(10)
Equation (10) is plotted below for several different values of S. For the weakest two seeds, the
threshold where the Stokes output begins to rise occurs at the familiar value of g 21 .
2
If we define threshold more rigorously as the value of g when T equals 1%, then we can use
Eq. (10) to semi-logarithmically plot the threshold as a function of the seed strength. Notice from
the graph that gth asymptotically approaches zero as S exceeds unity.
By estimating the intensity of the seed noise at the back end of the fiber due to thermal
fluctuations, this graph gives a theoretical approximation of the threshold value. In particular, in
the spontaneous limit, T 0 so that S = R0 , where the spontaneous reflectivity R0 is estimated
by Eq. (11) in my JQE 48:1542 paper.
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