Beam shaping in spatially modulated broad-area
semiconductor amplifiers
R. Herrero,
1,
* M. Botey,
2
M. Radziunas,
3
and K. Staliunas
1,4
1
Departament de Física i Enginyeria Nuclear, Universitat Politècnica de Catalunya, Colom 11, Terrassa 08222, Spain
2
Departament de Física i Enginyeria Nuclear, Universitat Politècnica de Catalunya, Urgell 187, Barcelona 08036, Spain
3
Weierstrass Institute, Mohrenstrasse 39, Berlin 10117, Germany
4
Institució Catalana de Recerca i Estudis Avançats (ICREA), Pg. Lluis Companys, 23, Barcelona 08010, Spain
*Corresponding author: ramon.herrero@upc.edu
Received October 16, 2012; revised November 28, 2012; accepted November 28, 2012;
posted November 30, 2012 (Doc. ID 177939); published December 14, 2012
We propose and analyze a beam-shaping mechanism that in broad-area semiconductor amplifiers occurs due to
spatial pump modulation on a micrometer scale. The study, performed under realistic parameters and conditions,
predicts a spatial (angular) filtering of the radiation, which leads to a substantial improvement of the spatial quality
of the beam during amplification. Quantitative analysis of spatial filtering performance is presented based on
numerical integration of the paraxial propagation model and on analytical estimations. © 2012 Optical Society
of America
OCIS codes: 160.6000, 230.4480, 250.5980, 160.4236.
Broad-area semiconductor (BAS) lasers (also called
edge-emitting lasers) are technologically relevant light
sources. The main advantage of such lasers is their high
conversion efficiency, as the planar configuration en-
ables efficient access of the pump to the whole volume
of the active amplifying medium. BAS lasers, however,
have a serious disadvantage as the spatial and temporal
quality of the emitted beam is relatively low [
1]. If no spe-
cial mechanisms are incorporated in the design, such as
different schemes of optical injection [
2,3] or optical
feedback [
4,5], among others, the emission exhibits
spatiotemporal fluctuations and is of a broad and noisy
optical and angular spectrum. The poor spatial quality of
the emitted beam is principally due to the absence of a
natural angular selection mechanism in the large-
aspect-ratio cavity of such devices. In addition, the
Bespalov and Talanov [
6] modulation instability in
strongly nonlinear regimes leads to filamentation and
deteriorates the quality of the emission.
In the absence of cavity mirrors, such planar semicon-
ductor structures can act as light amplifiers. In the
present work, we study the influence of periodic micro-
structuring of BAS amplifiers on the spatial quality of the
amplified beam. We consider a two-dimensional modula-
tion of the gain function, which can be achieved using a
periodic grid of electrodes for electrically pumped semi-
conductors, as illustrated in Fig.
1. The main result pre-
sented in this Letter is that the spatial qua lity of the
amplified beam can be substantially improved by peri-
odic modulation of the spatial pump profile on a scale
of several wavelengths.
Previous studies show that a periodic gain/loss (GL)
modulation on the wavelength scale can lead to particu-
lar beam propagation effects, such as self-collimation,
spatial (angular) filtering, or beam focalizatio n [
7,8]. In
semiconductor media, due to the linewidth enhancement
factor (Henry factor) α
H
a periodic spatial pump distribu-
tion causes a combined gain and refraction index mod-
ulation (GIM). We note that a specific and different
GIM case is being intensively studied in systems with bro-
ken parity-time symmetry [
9,10]. Here we show that the
angular spectrum of the radiation through a GIM ampli-
fier becomes narrower while being amplified. For suffi-
ciently long propagation distances (of the order of
millimeters), the normalized beam quality factor M
2
[11] can reduce down to unity, indicating that the BAS
amplifier output becomes perfectly Gaussian even for
strongly random input beam profiles.
A typical BAS amplifier is 1–3 mm long and 0.3–0.5 mm
wide, and the electromagnetic field is well confined in the
vertical waveguiding plane. The gain coefficient in in-
verted (pumped) semiconductor media can reach values
of 10
4
m
−1
[12]. Note that patterning the semiconductor
through a fishnet electrode, as here considered, is tech-
nologically feasible. Moreover, other approaches provid-
ing periodic gain, such as periodic microstructuring of
the amplifying media, are possible. Given that the exact
form of the pump modulation function is not critical, we
assume that the pump profile is harmonically modulated.
Furthermore, carrier diffusion usually smoothens spatial
distributions, making them harmonic despite the steplike
pump profile of electrodes.
In this Letter we consider a simplified model,
∂A
∂z
i
2
∂
2
A
∂x
2
px; z − 1
1 jAj
2
1 − iα
H
− iα
H
− γ
A; (1)
which governs the evolution of the optical beam injected
at z 0. The complex amplitude of the electric field,
Fig. 1. (Color online) Planar semiconductor amplifier struc-
ture with fishnet electrodes. The pump profile is periodically
modulated in space with transverse and longitudinal periods
d
⊥
and d
∥
. The incident beam of low spatial quality is amplified
while its spatial structure is progressively improved. A part of
the radiation is, however, lost in sideband components.
December 15, 2012 / Vol. 37, No. 24 / OPTICS LETTERS 5253
0146-9592/12/245253-03$15.00/0 © 2012 Optical Society of America
Ax; z, evolves in paraxial approximation experiencing
diffraction, nonlinearities (due to the gain and refractive
index dependence on the carrier density), and linear
losses, γ. The spatial coordinates are normalized to re-
duced wavelength
¯
λ λ∕2π, where λ is the central wave-
length of the injected beam inside the semiconductor.
Equation (
1) is derived from the traveling-wave model
[
13–15] neglecting gain dispersion, assuming only linear
spontaneous recombination, and adiabatically eliminat-
ing the carrier density.
An important ingredient in Eq. (
1) is the spatially
modulated gain px; zp
0
4m · cosq
⊥
x cosq
∥
z,
which is proportional to the pump profile smoothed by
carrier diffusion with a factor 1 Dq
2
⊥
q
2
∥
−1
. The
geometry is defined by the normalized longitudinal,
q
∥
λ∕d
∥
, and transverse, q
⊥
λ∕d
⊥
, components of
the lattice wave-vectors. They define the adimensional
geometry factor Q 2q
∥
∕q
2
⊥
2d
2
⊥
∕λd
∥
.
We study the amplification within the linear regime of
Eq. (
1), which is suitable for relatively weak fields with
no sensible gain depletion.
For analytical estimations we apply a harmonic expan-
sion of field in terms of the periodicities of the pump
modulation,
Ax; ze
ik
⊥
x
⌊a
0
za
−1
ze
−iq
⊥
x−iq
∥
z
a
1
ze
iq
⊥
x−iq
∥
z
⌋ (2)
which, inserted into Eq. (
1), gives
da
0
∕dz
−i
2
k
2
⊥
g
a
0
1− iα
H
ma
1
a
−1
; (3a)
da
1
∕dz
−i
2
k
⊥
q
⊥
2
iq
∥
g
a
1
1 − iα
H
ma
0
. (3b)
Note that the constant part of the complex gain g
p
0
1 − iα
H
− 1 − γ implies an exponential growth/decay
of the total field and can be neglected when studying
the spatial effects of a propagating beam.
We perform a standard analysis of Eq. (
3), calculating
the propagation eigenmodes (analogs of Bloch modes),
which grow or decay with the complex propagation
wavenumber as expik
z
zexpik
z;Re
− k
z;Im
z. The
analysis is analogou s to that performed in [7] for a pure
GL modulation and in [
16] for photonic crystals. The re-
sults are summarized in Fig. 2. We consider a wavelength
(in vacuum) of 1 μm, a semiconductor with effective re-
fractive index n 3, a gain coefficient 10
4
m
−1
that cor-
responds to m 10
−4
, and a transverse period d
⊥
4 μm. The longitudinal period is d
∥
96 μ matQ 1.
It is important to note that the imaginary part of the
eigenvectors, k
∥;Im
, is negative at least for one of the
branches of the Bloch modes, indicating an angular range
where the corresponding eigenmodes are amplified. The
most efficient spatial filtering regime is characterized by
a well-defined gain peak. Hence, modes propagating at
small angles to the optical axis (z axis) are amplified
while modes at larger angles decay or are amplified less;
that is, the amplified radiation is spatially filtered. This
occurs for Q ≈ 1 and corresponds to a resonance between
all three interacting harmonics [Figs.
2(b), 2(e), and 2(i)].
The branch with positive (negative) gain corresponds to
a Bloch mode with intensity maxima located at gain
(loss) areas of the GIM media. The third branch is a re-
latively homogeneous Bloch mode showing no substan-
tial gain.
At resonance, Q ≈ 1, the dispersion curves of the three
Bloch modes have a simple analytic form:
k
∥;0
−k
2
⊥
2
;k
∥;1
−k
2
⊥
2
2m
2
i α
H
2
k
2
⊥
q
2
⊥
q
; (4)
which allows estimating the half-width of the gain peak
responsible for the spatial filtering:
Δk
⊥
m
1.5 6α
2
H
q
∕q
⊥
. (5)
Considering previously used parameters, the half-
width corresponds to 0.5 deg inside the semiconductor
media and a divergence about 1.5 deg in free space.
The dispersion curves k
∥;Re
of Fig. 2 show concave
shapes for the maximally amplified modes, indicating po-
sitive diffraction and divergent propagation behind the
BAS. The situation is different for purely GL modulated
materials, where convex and concave curvatures for the
amplified modes are possible [
7].
The numerical integration of the resonant case, Q 1,
shows a diffusive broadening inside the BAS with
approximately a square-root dependence of the beam
width on the propagated distance that evidences the
spatial filtering effect. Intensity losses to sidebands are
Fig. 2. (Color online) Spatial dispersion curves showing real
and imaginary parts of k
∥
, for (a), (d) Q 0.8, (b), (e) Q 1,
and (c), (f) Q 1.6 as obtained from Eq. (
3). (g) Transverse
profile of the noisy injected beam determined by a Gaussian
(width 13 μm, peak intensity I
0
). (h)–(j) Intensities of the pro-
pagating beam according to Eq. (
1) for the three above consid-
ered values of Q. Thick vertical lines indicate interfaces
between the 1.5 mm long amplifier (left) and transparent homo-
geneous media (right). Curves at the right side of panels show
transverse field intensity distributions at the right limit of the
integration domain.
5254 OPTICS LETTERS / Vol. 37, No. 24 / December 15, 2012
always compensated by the beam amplification for
Q 1; intensity increases by a factor of 80 in the specific
case shown in Fig.
2(i). Filtering decreases sufficien tly
far away from resonance. For Q<1, spatial filtering is
interfered with by another Bloch mode with positive gain
[Fig.
2(d)], resulting in a three-peaked distribution in the
far-field (or angular) domain and a modulated structure
in the near field [Fig.
2(h)]. For Q<1, a double-peak an-
gular gain profile develops [Fig.
2(f)], which increases the
half-width of the spatial filtering angle; also the amplified
beam becomes distorted [Fig.
2(j)]. An efficient spatial
filtering is obtained for jQ − 1j ≤ jα
H
jm∕q
∥
, or equiva-
lently jq
∥
− q
2
⊥
∕2j ≤ jα
H
jm, as estimated from the series
expansions of propagation eigenvalues in the limit
jQ − 1j ≪ 1.
We analyze the spatial filtering effect, considering am-
plification of a noisy Gaussian beam [Fig.
2(h)]. A noisy
beam is prepared by randomizing the phases in the spa-
tial domain but keeping the width of the beam limited in
the angular domain. Physically, this corresponds to pro-
pagation through a set of scatters with a Gaussian distri-
bution of sizes and mean width smaller than the input
beam width.
The quantitative characteristics of the filtering are
summarized in Fig.
3. Behind the amplifier, the sidebands
generated by the GIM rapidly separate from the central
beam [Figs.
2(i) and 2(j)]. We calculate the width, W,of
the central part of the beam and its divergence, θ.We
numerically propagate the beam backward to reach
the minimum value of the product Wθ, that is, to the focal
plane (note that beams diverge behind the modulated
BAS). At the focal plane, we calculate the beam quality
factor, M
2
W θπ ∕λ, which has a minimum value of 1 for
a perfect Gaussian beam.
Figure
3(a) depicts the beam quality factor for two dif-
ferent amplifier lengths and initial noise levels M
2
0
3.25
and M
2
0
5.07. M
2
reaches minimal values of 1.005
and 1.014 at Q ≈ 1.1 for the long amplifier and 1.11 and
1.6 at Q ≈ 1.4 for the less efficient short amplifier. The
shift of the efficient filtering range to higher Q values
for shorter amplifiers is related to an interplay between
a broad angular gain profile and a strong beam-shape dis-
tortion induced for Q values larger than unity [Figs.
2(e)
and
2(f)]. Figure 3(b) shows the beam quality factor
depending on the amplifier length for different Q values.
For longer structures, M
2
decreases and rapidly
approaches unity. The lowest M
2
value is obtained for
Q 1.1 in accordance with the analytical estimate:
jQ − 1j ≤ jα
H
jm∕q
∥
0.1.
To conclude, we show that a spatial gain modulation in
BAS amplifiers can lead to substantial improvement of
the spatial structure of the amplified beam. The study
is performed under realistic semiconductor parameters
and technically realizable modulation periods. In parti-
cular, we show that for the proposed spatial filtering
mechanism, for a single pass through a GIM BAS am-
plifier, a length on the order of 1 mm is sufficient for
a substantial improvement of the beam quality. Beyond
what is here presented , this new technique could be im-
plemented to improve the spatial quality of emission of
BAS lasers.
We acknowledge financial support by the Spanish
Ministerio de Educación y Ciencia, European FEDER
(project FIS2011-29734-C02-01), and Generalitat de
Catalunya (2009 SGR 1168). The work of M. R. was
supported by German Research Foundation Research
Center Matheon.
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Fig. 3. (Color online) (a) Dependence of the beam quality
factor on geometry factor Q for short (L 0.380 mm ≈ 4d
∥
)
(orange) and long (L 1.52 mm ≈ 16d
∥
) amplifier (green) for
random beams with initial M
2
0
3.25 (solid) and M
2
0
5.07
(dashed). (b) Dependence of the beam quality factor on the am-
plifier length for M
2
0
5.07 and Q 0.8 (black), 0.9 (blue), 1.0
(red), 1.1 (purple), 1.2 (orange), and 1.3 (green).
December 15, 2012 / Vol. 37, No. 24 / OPTICS LETTERS 5255
