The variable stripe-length method revisited: Improved analysis
C. Lange, M. Schwalm, S. Chatterjee, W. W. Rühle, N. C. Gerhardt, S. R. Johnson, J.-B. Wang, and Y.-H. Zhang
Citation: Applied Physics Letters 91, 191107 (2007); doi: 10.1063/1.2802049
View online: http://dx.doi.org/10.1063/1.2802049
View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/91/19?ver=pdfcov
Published by the AIP Publishing
Articles you may be interested in
Carrier-depletion in the stripe-length method: Consequences for gain measurement
J. Appl. Phys. 108, 103119 (2010); 10.1063/1.3504222
Spectroscopic method of strain analysis in semiconductor quantum-well devices
J. Appl. Phys. 96, 4056 (2004); 10.1063/1.1791754
Evaluating the continuous-wave performance of AlGaInP-based red (667 nm) vertical-cavity surface-emitting
lasers using low-temperature and high-pressure techniques
Appl. Phys. Lett. 78, 865 (2001); 10.1063/1.1342049
6.1 W continuous wave front-facet power from Al-free active-region (λ=805 nm) diode lasers
Appl. Phys. Lett. 72, 4 (1998); 10.1063/1.120628
III-V interband 5.2 μm laser operating at 185 K
Appl. Phys. Lett. 71, 3764 (1997); 10.1063/1.120499

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
209.147.144.20 On: Sat, 07 Feb 2015 23:19:59

APPLIED PHYSICS LETTERS 91, 191107 共2007兲

The variable stripe-length method revisited: Improved analysis
C. Lange,a兲 M. Schwalm, S. Chatterjee, and W. W. Rühle
Materials Sciences Center, Faculty of Physics and Materials Sciences Center, Philipps-Universität Marburg,
Renthof 5, D-35032 Marburg, Germany

N. C. Gerhardt
AG Optoelektronische Bauelemente und Werkstoffe, Ruhr-Universität Bochum, D-44780 Bochum, Germany

S. R. Johnson, J.-B. Wang, and Y.-H. Zhang
Center for Solid State Electronics Research and Department of Electrical Engineering, Arizona State
University, Tempe, Arizona 85287-6206, USA

共Received 24 August 2007; accepted 4 October 2007; published online 6 November 2007兲
The variable stripe length method described by Shaklee and Leheny, 关Appl. Phys. Lett. 18, 475
共1971兲兴 is a straightforward way to determine the steady-state gain spectrum of laser material. Here,
common sources of error are identified and several new, robust ways of calculating the gain from the
data are presented. The advantages of these methods are underlined by applying them to data
obtained from a Ga共AsSb兲 / GaAs/ 共AlGa兲As heterostructure. © 2007 American Institute of Physics.
关DOI: 10.1063/1.2802049兴
Several ways have been used to measure material gain,
e.g., the Hakki-Paoli method,1 time-resolved,2,3 and cw4
transmission gain spectroscopy. A relatively straightforward
technique to determine the steady-state gain spectrum for an
optically active medium is the variable stripe length method,
originally introduced in Ref. 5. Here, the emission from a
homogeneously illuminated stripe of varying length is collected out of a cleaved edge as a function of the stripe length.
Usually, the gain value is then computed using the 具l / 2l典
method6 or by fitting an exponential function to the data.7 In
this letter, we present several significantly improved, robust
ways of calculating the gain from typical data sets.
In the experimental setup, 200 ps pulses at 527 nm and
10 kHz repetition rate were used for a quasicontinuous excitation. The beam is focused to a homogeneous stripe on the
sample of 16 ␮m width, passing its cleaved edge on one
side. A slit aperture after the lens controls the length of the
stripe on the sample. Amplified spontaneous emission 共ASE兲
along the pump channel to the sample edge is analyzed using
a spectrometer and a liquid nitrogen cooled InGaAs linear
array detector without discriminating between polarizations.
The width of the stripe should be small in comparison to the
typical length that is used to calculate the gain value in order
to reduce the leakage of ASE to the side of the pump channel. In our case, this length is in the order of 300 ␮m, about
20 times larger than the width. At both edges of the stripe,
the step in refractive index reflects a certain portion of the
ASE which then round-trips the pumped region and causes
an additional carrier depletion and premature gain saturation.
This was avoided by a 5° angle between the pump stripe and
sample edge normal.8,9 Details of the sample may be found
in Ref. 10.
The experimental data are analyzed as follows. Assuming that from every unit volume along the stripe, spontaneous emission emerges and is amplified along the stripe, the
following differential is found for the resulting intensity
I共l兲:11 ⳵lI共l兲 = 关⌫gm共l兲 − ␣兴I共l兲 + A共l兲. Here, the material gain
gm is modified with the confinement factor ⌫ and the propagation loss coefficient ␣. Spontaneous emission is reprea兲

Electronic mail: christoph.lange@physik.uni-marburg.de

sented by A共l兲. For a homogeneous sample, the solution is
I共l兲 =

A0
关exp共gmodl兲 − 1兴,
gmod

共1兲

with the modal gain gmod = ⌫gm − ␣, the length l of the stripe,
and a scaling factor A0. The latter factor is depending on the
Einstein coefficient for spontaneous emission, pump intensity, and, most indefinite, geometrical form factors. If desired, the photoluminescence can also extracted from these
data.9 The 具l / 2l典 method uses the intensity values at a certain
stripe length l and at 2l to calculate the gain value. One
obtains gmod = 1 / l ln关I共2l兲 / I共l兲 − 1兴. The derivation of this result can be found in Ref. 12 or easily retraced along the
derivation of the 具l / xl典 method below. One of the drawbacks
of the 具l / 2l典 method is that it requires two data points eventually too far apart from each other, which will be discussed
below. This often leads to gain saturation,11,13 and the gain
value is underestimated. On the other hand, the advantage of
this method is the existence of an analytical solution to the
equation. We will show in the following that a numerical
solution using two data points at the positions l and xl for
x ⬎ 1 does not require mentionable computational effort but
eliminates the saturation issue.
We start with the intensity values at these two stripe
lengths and introduce z = exp共gl兲, obtaining for their ratio rx,
rx =

I共xl兲 exp共gxl兲 − 1 zx − 1
=
⬅
.
I共l兲
exp共gl兲 − 1
z−1

共2兲

At this point, the analytical solution for the 具l / 2l典 method is
found for x = 2. Actually, another analytical solution can be
found introducing a new variable y 2 = z,
g=

冋

2
r1.5 − 1
ln
+
2
l

冑

册

共r1.5 − 1兲2
+ 共r1.5 − 1兲 ,
4

共3兲

where the unphysical result with negative values of y has
been dropped. This result is obtained solving the resulting
third order polynomial in y with the trivial zero y = 1. For
arbitrary x, Eq. 共2兲 has to be solved numerically for z. Introducing f共z兲 = 共zx − 1兲 − rx共z − 1兲, the problem is relocated to

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
0003-6951/2007/91共19兲/191107/3/$23.00
91, 191107-1
© 2007 American Institute of Physics
209.147.144.20 On: Sat, 07 Feb 2015 23:19:59

191107-2

Appl. Phys. Lett. 91, 191107 共2007兲

Lange et al.

finding the zero of f共z兲 for z ⬎ 0, which can be done using
the midpoint method.14
The function has the trivial zero z = 1, which corresponds
to gl = 0. The first and second derivative of f共z兲 show that the
function has its minimum at zmin = 共rx / x兲1/x−1, and has to have
another zero if zmin ⫽ 1, which is the nontrivial solution to the
equation. In particular, for zmin ⬍ 1, the zero has to be
searched for in 关0 , zmin兴. The result is again z = 1 for zmin = 1.
The zero is located in 关zmin , ⬁兴 for zmin ⬎ 1. A practical upper
searching boundary is z⬁ = exp共lmaxgmax兲, where lmax is the
maximum length of the stripe and gmax is the maximum gain
value expected. In our calculation, z⬁ = 10250 was used. Despite the huge search interval, the calculation time for a spectrally resolved measurement 共512 wavelengths times 300 intervals for the length of the stripe兲 was only 4 s on a standard
3 GHz PENTIUM 4 office personal computer.
An alternative to using the integrated emission over the
whole stripe is taking into account the differential values.
Considering the first and second derivatives of Eq. 共1兲, the
gain g can be extracted,
g=

⳵2l I共l兲
.
⳵lI共l兲

共4兲

gation, the resulting noise in the gain value is expressed as
␴gain = ␥␴syn. Both methods show very similar, negligible dependencies for ␥ on detector noise. Regarding laser noise,
the 具l / xl典 method turns out to be less sensitive, since ␥
quickly drops below 10 as l becomes greater than 0.2 mm.
For the 具dl / xdl典 method, ␥ is more than two orders of magnitude larger and is merely constant for all values of g and l.
The 具l / xl典 method is especially suitable to analyze experimental data in the absorptive regime, as ␥ decreases for
lower gain values. These numerical findings coincide with
those from the evaluation of experimental data, as shown
below.
Next, optical sources of error are discussed. Diffraction
effects cause spatially inhomogeneously illuminated stripe
edges for small stripe lengths and thereby produce artificial
modulations in the emission spectrum.11,15 Diffraction itself
can only be reduced, e.g., by placing the sample as close as
possible behind the apertures that constrain the stripe. Therefore, a method insensitive to diffraction is presented, referred
to as the 具difflog典 method. A standard solving scheme for the
differential version of Eq. 共1兲 and separation of the integral
at a midpoint x0 yields

再冕
冕

x0

This equation, further referred to as the 具d / d典 method, requires the data at exactly one position l0 while the equation
itself does not explicitly include the experimental value for
l0. Therefore, it is not susceptible to errors in determining the
stripe offset. However, determining the second derivative
from experimental data is rather sensitive to noise. In our
calculation, a local polynomial fit of sixth order was used.
Here, each data point around l is assigned a weight according
to a Gaussian distribution, which has been proven to show
the best results.
Another method to determine the gain value from the
derivative uses the data at two different positions l and xl,
similarly to the 具l / xl典 method discussed above. Since the
derivative removes the additive constant in Eq. 共1兲, the following analytical expression is found: g = 1 / 关l共x
− 1兲兴ln关⳵lI共xl兲 / ⳵lI共l兲兴. Due to the implicit dependence on l,
this analysis is sensitive to errors in determining the absolute
stripe length. In contrast to the previous one, however, this
method is less susceptible to noise because only the first
derivative is used. This method will be referred to as the
具dl / xdl典 method from now on. In order to investigate the
effect of noise on the stability of the methods, we perform an
error estimation. Here, synthetic gain data 共i.e., numerical
ASE data generated according to Eq. 共1兲 for gmod and A0
such that the resulting data is comparable to the experiment兲
are used. The data are scaled by Gaussian-distributed noise
centered at unity to account for the variations of the excitation power. Furthermore, a constant noise floor representing
detector noise is added,
2

I共l兲 =

冋冕
冋冕

A共x⬘兲exp −

0

A共x⬘兲exp −

x0

冋 冉冕

⫻ exp

x⬘

0

x0

0

册
册 冎

g共x⬙兲dx⬙ dx⬘

0

l

+

x⬘

g共x⬙兲dx⬙ dx⬘

冊 冉冕

l

g共x⬘兲dx⬘ exp

g共x⬘兲dx⬘

x0

冊册

.

共6兲

Here, g共x兲 and A共x兲 are the gain and the spontaneous emission form factor, both depending on the position in the
sample. Dropping these dependencies for large x0 yields
Ix0共l兲 =

A0
兵exp关g共l − x0兲兴 − 1其 + Ii共x0兲exp关g共l − x0兲兴. 共7兲
g

The first term is the amplified spontaneous emission over the
homogeneous part of the stripe from x0 to l 关Eq. 共1兲兴. The
contribution Ii共x0兲 accounts for the ASE collected from 0 to
x0. It is emitted in the region of inhomogeneous pumping
conditions and passes the homogeneous region before being
detected. The latter is the reason for the amplification factor
exp关g共l − x0兲兴. Simple math yields
ln关⳵lI共l兲兴 = 兵ln关A0 + gIi共x0兲兴 − gx0其 + gl,

共8兲

where the term in braces is a constant, so that g can be
calculated by determining the slope of this function. This
method does not require the experimental setup to be perfectly free of diffraction effects, as long as the stripe is homogeneous along the region behind x0. Similar to Eq. 共4兲, it
is insensitive to errors in determining the absolute length of
the stripe.
共5兲
Inoise共l兲 = Isyn共l兲 ⫻ 关1 + nlaser共l兲兴 + ndetector共l兲.
Now, these techniques are applied to experimental data
The noise is scaled such that its amplitude is about 5% of the
acquired as described above. A three-dimensional plot of the
signal amplitude at the typical optimum stripe length for this
raw data can be seen in the inset of Fig. 1. The pump intensample of 0.25 mm. The dependence of the noise terms nlaser
sity on the sample during excitation was 1.3 W / cm2. As
and ndetector on l is stochastic. Derivatives by l are treated as
mentioned in the introduction, a proper calculation of the
the difference of random variables scaled with 1 / l. The stangain value is delicately dependent on the right choice of the
dard deviation increases by a factor of 2 / ⌬l and is dependent
stripe length interval. If a too low starting value is chosen,
on the experimental resolution. Both Eqs. 共2兲 and 共4兲 are
edge effects due to an improperly cleaved sample and a
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
applied to the synthetic data and, by means of error propagreater emission angle will cause deviations from Eq. 共1兲.
209.147.144.20 On: Sat, 07 Feb 2015 23:19:59

191107-3

Appl. Phys. Lett. 91, 191107 共2007兲

Lange et al.

FIG. 1. 共Color online兲 Gain value as a function of the interval of stripe
length considered for the 具l / xl典 method 共solid: x = 1.2; dashed: x = 1.4兲. The
inset shows the raw experimental data.

Noise from the detector and shot noise from the sample
emission play a considerable role as the signal is still small.
The maximum stripe length on the other hand must not exceed the saturation length.11
To illustrate this issue, Fig. 1 shows the gain calculated
with the 具l / xl典 method for x = 1.4 and x = 1.2 as a function of
l. The datapoints for each gain value g共l兲 are thus taken at l
and 1.2l or 1.4l, respectively. For ␭ = 1340 nm and ␭
= 1303 nm, the gain value is the maximum of the curve, located at l ⬇ 0.25 mm and l ⬇ 0.30 mm, respectively. It is important to see that the gain maximum is located at different
stripe lengths for different wavelengths. This demonstrates
that if the spectra are calculated for a fixed stripe length l,
they only avoid saturation for a part of the spectral range.
Note that for x = 1.2, there is a plateau where the gain value is
constant for a certain range of stripe lengths. Here, the true
gain value is not decreased, neither by noise in the low emission regime nor by saturation effects. For both values of
separation x, gain values are very similar, where the higher
gain value is more accurate since it suffers less from saturation effects. Despite the smaller distance of the data points
for x = 1.2, the noise level is still very low, which demonstrates the utility of this method. For the absorptive regime at
␭ = 1008 nm, the gain value is best estimated by the average
over a range of large stripe lengths which is indicated by the
dashed box. Here, no saturation effects are expected, and
therefore values for large l are most reliable. Figure 2 共left兲
shows the derived gain spectra for a selection of the methods. Results obtained with the conventional 具l / 2l典 method
may be misleading, as illustrated by the black line, where a
nonoptimal stripelength was intentionally chosen. Although
the spectrum looks convincingly smooth and consistent, it is
actually wrong. For optimum l, gain values are much higher.
The 具l / 1.2l典 method clearly shows that even the optimum
choice of l for the 具l / 2l典 method does not yield proper results
due to saturation effects. Lower values of x down to x

= 1.02 show that the gain value converges for
x → 1.0 without introducing a critically high noise level, as
can be seen in Fig. 2 共right兲. Here, a series of gain spectra for
different values of x are shown to illustrate the convergence.
The conventional 具l / 2l典 method turns out to underestimate
the proper gain values by about 20%. With the 具dl / xdl典
method and the 具d2 / d典 method 共not shown, similar to
具dl / xdl典兲, the analysis was only possible on a limited spectral
range within the gain region due to noise effects, as predicted
by the estimation above. Similarly, the 具difflog典 method does
not cover a broad spectral range, but delivers results comparable with the 具l / 1.2l典 method. For this particular material
system, the advantages of the latter three methods do not
countervail the drawback of their sensitivity to noise. For the
absorptive regime below 1000 nm where saturation is not an
issue, both 具l / xl典 and 具l / 2l典 are suitable and deliver exactly
the same result. The overall gain bandwith of this material
system of roughly 300 nm is very large. From the measurement, a bandgap of about 0.88 eV was determined.
In summary, four methods for calculating the gain for a
variable stripe-length experiment were presented. Their sensitivity to noise was discussed theoretically and verified experimentally. A model insensitive to inhomogeneities at the
stripe edges was presented. The standard 具l / 2l典 method has
been extended with a numerical technique in order to drastically improve the insensitivity of this class of methods
against saturation effects, requiring data over a range as little
as only 10% apart in stripe length. A GaAsSb-based sample
has been investigated with these techniques, showing that the
sample is suitable for laser emission over a broad spectral
range of approximately 300 nm in width.
The authors thank Jürgen Vollmer and Martin Hofmann
for helpful discussions and gratefully acknowledge financial
support by the Deutsche Forschungsgemeinschaft and the
Optodynamics Research Centre.

B. W. Hakki and T. L. Paoli, J. Appl. Phys. 44, 4413 共1973兲.
C. Lange, S. Chatterjee, C. Schlichenmaier, A. Thränhardt, S. W. Koch,
W. W. Rühle, J. Hader, J. V. Moloney, G. Khitrova, and H. M. Gibbs,
Appl. Phys. Lett. 90, 251102 共2007兲.
3
H. Giessen, U. Woggon, B. Fluegel, G. Mohs, Y. Z. Hu, S. W. Koch, and
N. Peyghambarian, Opt. Lett. 21, 1043 共1996兲.
4
C. Ellmers, A. Girndt, M. Hofmann, A. Knorr, W. W. Rühle, F. Jahnke, S.
W. Koch, C. Hanke, L. Korte, and C. Hoyler, Appl. Phys. Lett. 72, 1647
共1998兲.
5
K. L. Shaklee and L. F. Leheny, Appl. Phys. Lett. 18, 475 共1971兲.
6
J. M. Hvam, J. Appl. Phys. 49, 3124 共1978兲.
7
S. Borck, S. Chatterjee, B. Kunert, K. Volz, W. Stolz, J. Heber, W. W.
Rühle, N. C. Gerhardt, and M. R. Hofmann, Appl. Phys. Lett. 89, 031102
共2006兲.
8
G. M. Lewis, P. M. Smowton, J. D. Thomson, H. D. Summers, and P.
Blood, Appl. Phys. Lett. 80, 1 共2002兲.
9
G. M. Lewis, P. M. Smowton, P. Blood, G. Jones, and S. Bland,
Appl. Phys. Lett. 80, 3488 共2002兲.
10
G. Blume, T. J. C. Hosea, and S. J. Sweeney, Phys. Status Solidi A 202,
1244 共2005兲.
11
L. Dal Negro, P. Bettotti, M. Cazzanelli, D. Pacifici, and L. Pavesi,
Opt. Commun. 229, 337 共2004兲.
12
Claus F. Klingshirn, Semiconductor Optics, 2nd ed. 共Springer, Berlin,
2007兲, Vol. 1, pp. 701–702.
13
H. Kalt, M. Umlauff, M. Kraushaar, M. Scholl, and J. Söllner, J. Cryst.
Growth 184/185, 627 共1998兲.
14
J. D. Faires, and R. L. Burden, Numerische Methoden 1st ed. 共Spektrum
Akademischer Verlag, Heidelberg, Berlin Oxford, 1994兲, 1, 30.
FIG. 2. 共Color online兲 Left: gain spectra for the different methods. Right:
15
Eugene Hecht, Optik 3rd ed. 共Oldenbourg Wissenschaftsverlag, München,
gain spectra computed using the 具l / xl典 method with different spacings from
2001兲, 1, 743.
x = 3.0 down to x = 1.02 and corresponding positions of the maximum gain.
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
1
2

209.147.144.20 On: Sat, 07 Feb 2015 23:19:59