i
vf
vz
+
1
2k
v
2
vx
2
+
v
2
vy
2
!
f + k
0
ðn n
0
Þf = 0 (Equation 2)
Ifweassumethefieldintheformf = jðx; yÞe
imz
, the equa tion can be fur ther simplified to
1
2k
v
2
vx
2
+
v
2
vy
2
!
j + ½k
0
ðn n
0
Þmj = 0 (Equation 3)
where m is related to propagation constant b by the equation b = k + m.
Figures 2A–2E show the modal fields j in the cross secti on of a single-mode fiber when different amounts
of gain are applied. A conventional single-mode optical fiber ( n
I,1
= n
I,2
= constant) only supports t he
LP01 mode (Agrawal, 2007; Snyder and Love, 1 983) . As shown in Figure 2A, this mode has an intensity
distribution that is similar to that of a Gau ssia n beam. Figure 2B shows the mode intensity profile with
relatively low gain in the left section. In this case, the fiber still possesses a G aussian-like distributed
mode int ensity, just like its conventional counterparts. If the gain keeps increasing, t he mode intensity
starts showing asymmetry with more energy residing in the doped section, as displayed in Figure 2C.
As the gain coefficient g increases and exceeds a critical value, the fundamental mode will be confined
in the left section (Figure 2D). Also, one can find that another new mode appears in the undoped section
(Figure 2E).
The physics behind this phenomeno n can be e xplaine d by noting the im aginary parts of the refractive index
imposing on j. Both real and imaginary refractive indice s can create optical p ote ntia ls confining light fi eld.
When the amount of gain i s small, lig ht can be f ree ly exchanged bet ween core secti ons; hence the mode
field distribution in the core is just like conventional fibers. With the increasing gain gradually creating
stronger imaginary potentials, th e energy flow between two sections begins to be reduced and light field
starts to be trapped in t he tw o pot entials. As the imaginary pot ential strength increases beyond the
threshold, the weaker ene rgy flow is no longer abl e to sustain the int egri ty of the field, and, as suc h, the
fundamental mode becomes two isolated modes in each site.
To give a broader insight into the modes supported by the fibers with asymmetric gain, we further present
an analysis in the complex
e
V
2
b space, where mathematical transformations are taken to make the
results independent of particular fiber parameters. These tr ansformation s will not change th e system’s
action. If we d efine a contrast factor Dn
I
= ðn
I;1
n
I;2
Þ=2, ga uge tra ns form atio n can be e stab li shed by
f = f
0
exp½k
0
ðn
I;1
Dn
I
Þz or f
0
exp
k
0
Dn
I
+ n
I;2
z
. The real and imaginary parts of the complex-valued
index are expressed in the dimensionless forms ( Siegman, 2003, 2007).
Figure 2. The Fiber Modes wi th Asymm etric Gain
(A–E) Typical behavior of modal field when gain coefficient is (A) zero, (B) low, (C) medium, and (D and E) high.
(F) Real and (G) imaginary parts of the propagating mo des in the complex
e
V
2
b space. The transformation
b
jbj
logð1 + jbjÞ is used to better visualize data. For
the projections shown in the
e
V
2
plane, no localized states exist in the gray region; orange region, only fundamental modes exist; pink region, fundamental
modes and higher-order modes coexist; blue region, each fundamental mode bifurcates into two separate modes; the region beyond blue, bifurcated
higher-order modes.
See also Figure S1.
iScience 22, 423–429, December 20, 2019 425