IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL.52, NO.7, June 2016
1
Small-Signal Analysis of Ultra-High-Speed
Multi-Mode VCSELs
Wissam Hamad, Stefan Wanckel and Werner Hofmann, Member, IEEE

Abstract—In this paper we show that the dynamic performance of multi-mode vertical-cavity surface-emitting lasers
(VCSELs) can be modeled by single-mode rate equations
developed for edge-emitters as long as the lasing modes share a
common carrier reservoir. However, this assumption does not
hold for ultra-high performing VCSEL devices. Due to the high
photon densities inside these optimized VCSELs, the common
carrier reservoir splits up as a result of the spatial hole burning
(SHB) effect. This is caused by the high intensity of the multiple
transverse modes. In this case, a small-signal modulation
response with a different shape is expected. We derive an easy-toapply fitting function which allows the extraction of consistently
expanded figures of merit. This novel function works for all
VCSELs, particularly including devices with carrier reservoir
splitting. Further, we use this new model to perform a detailed
analysis of our latest VCSEL generation with a modulation
bandwidth of up to 32.7 GHz.
Index Terms—Direct Modulation, High-Speed, Multi-Mode,
Small-Signal-Analysis,
Small-Signal-Modulation
Response,
Transfer Function, VCSEL.
D
I. INTRODUCTION
irectly modulated vertical-cavity surface-emitting lasers
(VCSELs) are playing an important role in today’s
standardized serial data rates and in the emerging technology
of optical interconnect abolishing copper lines. Consequently,
the demand for higher bandwidth optical interconnects has
become urgent. Interconnects based on silicon photonics either
lack the cost-effective multiple-wavelength laser source [1] or
the matched wave-guide if using orbital angular momentums
for multiplexing [2]-[3]. Directly modulated VCSELs,
however, match existing photo-detector and waveguide
technologies, can be scaled up meeting future demands [4],
and can therefore be readily applied as directly modulated
light sources in optical interconnects. In order to reach a datarate of 100 Gb/s, the VCSEL bandwidth has to increase by
30 to 40 % towards the 40 GHz level [5]-[6].
Intensive research to meet these requirements yielded highspeed VCSELs with direct modulation bandwidths up to
20 GHz [7]-[8] covering several wavebands. Recently, further
Manuscript received March 2, 2016, revised April 27, 2016, accepted May
25, 2016, published June 2016.
W. Hamad, S. Wanckel and W. Hofman are with the Institut für
Festkörperphysik and Zentrum für Nanophotonik, Technische Universität
Berlin, EW 5-2, 10623 Berlin, Germany
(e-mail: werner.hofmann@tu-berlin.de).
According to IEEE regulations, authors and/or their employers are allowed
to post the accepted version of IEEE-copyrighted papers on their own
personal servers. ©IEEE 2016. 10.1109/JQE.2016.2574540.
progress has been made boosting the modulation performance
via 24 GHz [9]-[5] towards 28 GHz [10].
High-speed short-distance optical interconnects rely on
multi-mode light sources transmitting via multi-mode fibers to
achieve better bit error rate (BER) performance. As the latest
research and experiments have shown [11]-[12], multi-mode
fiber interconnection systems utilizing multi-mode VCSELs
provide better immunity to intensity and modal noise
compared to the single-mode VCSELs. This enhanced
immunity can be explained by the fact that multi-mode
VCSELs exhibit low coherent characteristics, minimizing the
coherence of the reflected signal and reducing both relative
intensity noise and BER.
It has been experimentally observed [13]-[14] that the
spectral emission characteristics of high-speed, oxide-confined
VCSELs depend mainly on the aperture size, confining the
modulation current, and the geometry of the injection
electrodes. In general, these optoelectronic devices tend to
operate at a single longitudinal mode when driven at low
modulation currents near their lasing threshold. This is
especially the case in small aperture VCSELs with diameters
below 3 µm where the typical emission wavelength is of the
same order as the confining oxide aperture. However, devices
having apertures ranging from approximately 3 to 10 µm, the
emission spectra shows an increase in the number of excited
higher order transverse modes at higher injection currents. On
one hand, this multi-mode behavior is mainly caused by the
raise in the modal gain of these higher-order modes, and on
the other hand, the impact of the self-focusing effects grows,
leading also to a multi-mode operation. The increase in selffocusing effects can be directly attributed to the increase in
spatial hole burning (SHB) of the carrier concentrations and
thermal lensing [15].
Thus, in order to establish highly performing multi-mode
VCSELs in multi-mode fiber links at ultra-high-speeds,
sufficient knowledge of the intrinsic laser dynamics is of great
importance. Moreover, a deep understanding of the smallsignal modulation response of these multi-mode VCSELs
becomes indispensable for a reliable system design, modelling
and characterization. While large-signal transmission experiments show the feasibility of a proposed system, the final
performance only depends on the VCSEL device directly. This
is especially true, if this VCSEL device turns out to be the
bottleneck. Better large-signal performance can always be
achieved through the optimization of the system by matching
drivers, emitters, waveguides and receivers. Consequently,
2
large-signal experiments cannot reliably produce figures of
merit for the isolated high-speed VCSEL. Small-signal
experiments, on the other hand, can be calibrated accurately to
isolate the VCSEL performance. The difficulty lies here in the
proper modelling of the measured response to acquire reliable
figures of merit to judge the device performance.
Despite the intensive research conducted to understand the
underlying physics behind the multi-mode behavior in oxide
VCSELs and their impact on the intrinsic laser dynamics,
many ambiguities still exist concerning the nature of the multipeak phenomenon occurring in the small-signal modulation
response of VCSELs [16]-[17]. These multiple local maxima
which appear in the transfer function signal of the multi-mode
VCSELs, deviate substantially from the standard single-mode
model normally applied to characterize these multimode
devices. Moreover, the adapted single-mode fit model, which
is based on a single-mode rate equation analysis, can
reproduce only one resonance peak.
In addition to the historic assumption that the modulation
response of “highly” index guided VCSELs show typical
single mode transfer function characteristic [17]-[18],
anomalies in the modulation response data are experimentally
observed and reported by several research groups [13], [19],
[7], [14], and [20]-[21]. Several explanations are presented for
these maxima and minima ranging from optical reflections
[19] to self-induced pulsation [22] and mode partition noise
[18]. Some of the more promising approaches [23]-[24] are
based on the interaction of the different lasing modes through
their carrier reservoirs. This is due to the onset of strong
modal competition behavior. The modes that are generated in
the optical cavity of the VCSEL compete for the carrier
density in the active region.
Investigating the majority of published VCSEL modulation
data, we found strong deviations between the experimental
data and the utilized modelling, especially for certain driving
condition of the VCSEL devices. The small-signal modulation
response of an index guided multi-mode VCSEL can typically
be divided into three different driving-current regions. In the
low-current region, which is characterized by the onset of
strong modal competition, we frequently observe an
interference-like pattern in the frequency response. This effect
is caused by the large difference in the power of the different
modes. In the intermediate-current region, the modulation
frequency response is mainly dominated by the relaxation
oscillation frequency. Surprisingly, we see the shape of this
transfer function resembling a single mode laser, even though
the device is clearly operating in the multi-mode regime. The
most interesting region for ultra-high-speed operation,
however, is the so called high current region.
Besides the well understood mechanisms which control the
strength and the form of relaxation oscillation frequency (e.g.
carrier diffusion, nonlinear gain suppression and carrier
transport effects), the contribution of co-dominant higher
order modes is still under discussion. Despite of the longitudinal single-mode emission and the fact that small aperture
VCSELs support mainly the fundamental mode, higher order
transverse modes can be detected in the emission spectra at
high injection currents. In this case the transverse modes have
IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL.52, NO.7, June 2016
low spatial overlap. This causes different carrier reservoirs to
establish each serving its spatially separated mode.
Even though the experiments deviated from the single-mode
case, only theorists [25] considered advanced modeling. Up to
now, experimentalists are modelling their multi-mode VCSEL
devices exclusively with single-mode rate equations, even
inconsistencies occur frequently, and some measured data can
only be fitted with large error. In this paper we investigate
whether in some cases this common practice might still be
justified. Furthermore, in order to resolve the discrepancies
mentioned above, we derive an advanced fitting model and
show under which conditions it is obligatory to be used.
Finally, we analyze our latest generation of ultra-high speed
VCSEL devices with this novel model.
Deriving the multi-mode transfer functions, the model
based on the standard single-mode rate equations was
expanded to consider the coupling of two (or more) different
modes through their lateral spatial distribution and overlap in
the carrier reservoirs. If the modal competition for the carrier
reservoir is not included explicitly in the interaction matrix as
coupling factors, it is not possible to obtain more than one
resonance peak in the modulation response fit model. Thus,
solving the resulting interaction matrix gives rise to a singlemode transfer function even if two or more modes are
included. However, if the spatial distribution of the modal
fields in the carrier reservoir is taken into consideration, two
or more resonance peaks can be fitted. This approach led to
the development of our multi-mode transfer function. As this
model was consistently expanded from the commonly used
single-mode case, we also could easily expand well-known
figures of merit. Our latest generation of VCSELs serve as
sample-devices for detailed investigation.
II. SMALL-SIGNAL TRANSFER-FUNCTION MODELS
We will begin our discussion of the different models by
going through the most fundamental model, the single-mode
case, where one mode interacts with its carrier reservoir in the
active region. This case is important to get familiar with the
different derivation steps and the matrix representations of the
linearized rate equations. The later derivation of the different
multi-mode cases is going to be based on a generalized matrix
presented at the end of the single-mode discussion.
A. Single-Mode Model
As already known, the lasing dynamics of a single mode
laser can be, to a certain extent, fully described by a model
consisting of a reservoir containing the electron carriers and a
photon reservoir [26]-[27]. This model is depicted in Fig. 1.
Based on particle conservation, the rates of change in both,
carrier- and photon densities can be expressed as
dN
 J inj - J th - R stim  S
dt
dS
 S (  R stim -  g ( i   m ))   J sp
dt
(1)
(2)
where N is the carrier density and S is the photon density in
HAMAD et al.: SMALL-SIGNAL ANALYSIS OF ULTRA-HIGH-SPEED VCSELs
3
 SS =  v g g  v g ( i   m )   v g a p S  0
(8)
where g represents the optical material gain under all
operation conditions and gth at threshold. Further, a and ap are
the differential gain and the negative gain derivative after
photon density, respectively. The assumptions made in (5-8)
are approximations which can be made for a VCSELs
operating above threshold and neglecting gain compression.
The above system of linearized rate equations in (3-4) can be
represented in the following matrix form
 j   NN
 
SN

Fig. 1. Single Mode Reservoir Model. This model was used for deriving the
rate equations in this section. A single drive current Jinj pumps a single carrier
reservoir. Electron-hole pairs from this reservoir N supply a single photon
reservoir S. The arrows symbolize the interaction in between and the supply
of the reservoirs. This diagram is also valid for particle densities and their
differential changes.
the active region and the optical cavity, respectively. Г is the
confinement factor. The first equation describes the total
carrier density change. With the injected carrier density Jinj,
the carrier recombination due to spontaneous emission or
losses Jth, and the stimulated net-emission coefficient Rstim
representing the stimulated emission rate.
The second equation states the net increase in the photon
density being equal to the effective spontaneous generation
rate Jsp multiplied by the confinement factor Г plus the photons
generated by the stimulated emission which are not coupled
out or being absorbed. Further, vg is the group velocity, αi and
αm represent the internal- and mirror losses, respectively.
In order to study the dynamic behavior, or the so called
modulation response of the driven laser, the derived rate
equations have to be analyzed with the time derivatives
included. As there is no analytical solution of the full rate
equations, small-signal analysis have to be carried out. Thus
linearizing and rewriting the system of equations in (1-2)
yields
j dN  dJ inj   NN dN   NS dS
(3)
j dS   SN dN   SS dS
(4)
with the following simplified rate coefficients:
 NN =

J  v g aS  v g aS
 N th
 NS = v g g  v g a p S  v g g th
 SN = 

J   v g aS   v g aS
 N sp
(9)
 M x = i
Solving this matrix yields
x  M 1  i 
dJ inj  j   SS

det M   SN
  dN 


  dS 
(10)
where the det M can be already expressed in terms of figures
of merit as:
det M   R2  j   2
(11)
Introducing the relaxation oscillation frequency ωR and
damping factor γ in terms of the rate coefficients gives
 R2 =  NN  SS   NS  SN
and
 =  NN   SS
(12)
Taking the approximations made for a VCSELs operating
above threshold into consideration and neglecting gain
compression (12) can be further simplified to
 R2   NS  SN
and
   NN
(13)
Rewriting (10) in terms of the small-signal response dS to
the perturbation in the carrier density dJinj, and later replacing
the carrier density by the laser power P leads to the formation
of the theoretical transfer function H(ω). This transfer function
describes the small-signal modulation response of the laser to
a driving sinusoidal current I and is defined as
(5)
H ( ) =
(6)
 NS   dN   dJ inj 

=
j   SS   dS   0 

dP h
=
 d v g g th  SN
dI
e
detM
h
 R2

d  2
e
 R  j   2
(7)
where ηd is the differential quantum efficiency.
(14)
IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL.52, NO.7, June 2016
4
B. Multi-Mode Model
The matrix representation (9) of the single mode rate
equations can be expanded to include various effects. These
expanded models can be used to analyze the laser dynamics
and derive the modulation transfer function in almost all
operation and configuration cases.
N N
 j   N1N1 ...
1 n





 
... j   N N
N n N1
n n

 
...
S N
S1N1
1 n






 S N
...
S N
m 1
m n

Fig. 2. Shared Reservoir Multi-Mode Model. A single drive current Jinj
pumps a single carrier reservoir. Electron-hole pairs from this reservoir N
supply several photon reservoirs Sm. Our calculation shows that this case falls
back to a single-mode-like transfer function.
As we are primarily interested in the relative modulation
response, the constant front in (14) is scaled to one. Thus, any
constant which can be factorized is set to one. This step is
carried out without losing the generality of the modulation fit
function and can be later compensated by inserting a variable
offset value. In order to initiate the modulation fit function by
zero attenuation response, this fit function is further multiplied
by the inverse of H(0) which represents the modulation
response at zero. In this paper we omit this last technical fit
step for a better visual representation of the derived equations.
Thus the modulation response is given in proportion form.
The relative modulation response function is used to
characterize the performance of VCSEL devices. Extracting
the different parameters from the fitted scattering parameter
S21 data, yields the well-known figures of merit. Thus, until
now the measured data are fitted to a relative single-mode
function proportional
H () 
1
 2R  j    2
(15)
This intrinsic modulation response resembles a second-order
filter function with two complex conjugated poles. However,
for modeling the total transfer function of the VCSELs, this
intrinsic response is multiplied by a third pole which models
external parasitics. This gives the well-known 3-Pole transfer
function in (16). An elaborate discussion on the modelling of
the parasitics as a single-pole low-pass filter is given in
chapter III of this work.
H ( ) = H ( ) Intrinsic  H ( ) Extrinsic

1

1
 R2  j   2 1  j ( /  P )
(16)
N S

N
n S1
j   S S
 M x = i
11

S
mS1
N S

  dN1   dJ inj1 



   

 


...
N S
n m
  dN n  =  dJ inj(n  k ) 
  dS   0 
...
S S

1 m
 1  


   


  dS   0 

... j  S S   m  
m m 
...
11
1 m
(17)
Thus, the developed multi-mode interaction Matrix
presented above in (17), can model the interaction and
coupling of the different lasing modes among each other, and
with their underlying carrier reservoirs. More importantly, this
model also supports the formation of multiple carrier
reservoirs, one for each lasing mode distinctly. Furthermore,
the interaction between these different carrier reservoirs
among one another and themselves is supported.
The different matrix entries in (17) represent the rate
coefficients of the linearized multi-mode rate equations
analogous to the case of the single-mode matrix
representation. However, expanding the number of carrierand photon reservoirs to n and m reservoirs, respectively,
induces the introduction of a second subscript character to the
already existing ones. This step is essential in order to be able
to distinguish the impact resulting from the different reservoirs
on themselves and on one another. For example, μN1S2 defines
the impact on the first carrier reservoir N1 caused by the
changes in the second photon reservoir S2. In obvious cases,
this second subscript is omitted as in the single-mode, singlecarrier reservoir case discussed previously.
It is important to mention at this stage that the rate
coefficients (entries) of the multi-mode matrix for some of the
upcoming multi-mode rate equation analysis are still
consistent with the single mode coefficients. However, particle
conservation for carrier- and photon reservoir densities should
be taken into consideration.
Finally, this general interaction matrix can be sized and
adapted to model the different dynamics in small-signal
analysis resulting from different considerations and device
configurations. In the following, coupling and interaction
effects are analyzed by tailoring the interaction matrix to
satisfy these boundary conditions and considerations with the
final aim to derive an analytical transfer function model. This
analytical function is utilized to fit the experimental S21 data
allowing us to extract reliable information about the dynamic
properties of the real-world device. Furthermore, this
analytical model can be used to categorize the different
operation regions as discussed in the introduction.
HAMAD et al.: SMALL-SIGNAL ANALYSIS OF ULTRA-HIGH-SPEED VCSELs
As already mentioned, higher order transverse modes
evolve in real-world VCSEL devices under certain operation
currents or geometrical layouts. Thus, in order to include the
impact of the different interacting lasing modes on the transfer
function, we start our investigation by the simplest multi-mode
case. Here, we assume one carrier density reservoir N serving
as a carrier source for the two different lasing mode densities
S1 and S2 as depicted in Fig. 2. Consequently, the interaction
matrix becomes a 3 x 3 matrix
 j    NN

  S N
1

  S N
2

 NS
j   S S
1 1
S
2 S1

  dN 
   dS1  =
S S

1 2
 

j    S S   dS 2 
2 2 
 NS
1
2
 dJ inj 
 0 


 0 


 m11 1 
 dN 
 dJ inj 
 dS  = M  1   0  = dJ   m  1 
21 
inj 
 1


1 
 dS 
 0 
 m 31
 2




and are therefore set to zero. Rewriting (19) in terms of the
small-signal response dP to the perturbation in the carrier
density dJinj, and taking into consideration that the total output
power is now the sum over all mode powers (P = PS1+PS2),
results in the following intrinsic modulation response:
H ( ) = c  ( dS1  dS 2 )
 c 
H ( ) 




1
 NS   S N   NS   S

 j NN   2
(23)
1
  j     2
2
R'
with
 R2 '   NS1   S1N   NS 2   S 2 N
and
   NN
(24)
The analysis above for one carrier- and two photon
reservoirs can be generalized as long as all lasing modes share
the same carrier reservoir. Thus, expanding the interaction
matrix to model m different photon density reservoirs, and
calculating the intrinsic modulation response results in

(25)
m m 11
m Modes
(19)
and can be also expressed as
(20)
and
1
(21)
  S N  ( j   S S )
2
1 1
det M
As in the single mode case μS1S1 and μS2S2 can be neglected
1
m 31

 NS1   S1 N   NS 2   S 2 N  j   NN   2
N
1
1
2
2

2
R '
H () 
 m  NS  S
m
mN

2
 R ''
matrix M-1 with
1
  S N  ( j   S S )
1
2 2
det M
 S1 N   S 2 N
where cη is a constant. Again here we are just interested in the
relative small-signal frequency response. Rewriting and
comparing this derived intrinsic transfer function to that of the
single mode case, reveals in a very interesting finding. This
model, consisting of one carrier reservoir and two lasing
modes, yields also a second-order filter function with two
complex conjugated poles as a modulation response. Thus, the
intrinsic modulation response can be expressed as
H ( ) 
1
represent the first column entries of the inverse
m111 to m31
1
m 21

(22)
1
1
 c  ( m 21
 m31
)
(18)
To avoid modelling of any mode beating effects which are
not supported by our VCSEL geometrical configuration, the
two rate coefficients μS1S2 and μS2S1 are set to zero. Thus, any
interaction between these two lasing modes through their
photon reservoirs is inhibited and just the interaction of both
modes with their common carrier reservoir is permitted.
Before proceeding with the discussion, it is important to
point out that the impact of the interaction of the two modes
among each other through their photon reservoirs, results in
many interesting phenomena in the modulation response.
Many mechanisms are employed to realize this mode coupling
ranging from injection-locking techniques [28] to schemes that
are based on coupled cavities and modulator integration [29].
These so called optical feed-back mechanisms are a result of
the photon–photon resonance between two transverse
oscillating modes under strong slow-light feedback.
In this paper, the focus is exclusively on the modulation
response effects resulting from the interaction between the
different carrier reservoirs, each belonging to a different
resonating mode, among one another and with their own- and
other photon density reservoirs.
Going back to the interaction matrix (18), and solve
for x = M -1i , gives the following system of equations:
5

1
2
 j 
NN  


(26)
1
 2R ''  j      2
which also represents a two complex conjugated pole transfer
function just as a single-mode laser would have. Thus, singlemode rat equations can be used to model these cases.
6
Fig. 3. Multiple Reservoir Multi-Mode Model. In this case the modes confine
their independent carrier reservoirs. This behavior corresponds to spatial hole
burning of multiple transverse mode ensembles of a VCSEL. A single drive
current Jinj also splits up proportional to a power ratio defined by the two
transverse modes. This model can explain the dynamic behavior of real-world
high-speed VCSEL devices well.
Until now the standard applied technique to characterize the
performance of both, single- and multi-mode oxide confined
VCSELs, is fitting the measured S21 data with a two complex
conjugated pole transfer function. This approach originated
from experimental observations which, in many cases, show a
very similar profile to that of the single mode.
Equation (26) above, confirms the methodology to extract
important device parameters of multi-mode VCSELs from the
measured response by single-mode fitting. This behavior in
modulation response becomes obvious when taking a closer
look at the crucial parameters, damping factors and relaxation
resonance frequencies, in the derived transfer functions.
The damping factor γ, as defined in (5) for the single mode
case, remains unchanged throughout the derivation of the
modulation responses for the two- and multi-mode cases.
However, the photon density S in the two- or more mode
cases, is redefined to be the sum over all photon densities.
Despite this fact, the value of the rate coefficient μNN stays
almost the same in all cases except for some negligible factors
added for each further lasing mode.
Until now, the developed models and the derived transfer
functions can simulate a modulation response with only one
maximum. These models represent a classical singleresonance frequency response. However, under certain
operation conditions, a S21 measured curve which is rich in
maxima and minima is observed. In this case the deviation
from the classical single mode model becomes significant. The
assumptions matching this case are presented in Fig. 3. Here,
the extraction of figures of merits using the classical fit yields
unreliable performance parameters. In order to develop a more
reliable transfer function which can model the S21 data of highspeed VCSELs and give a deeper understanding of the laser
mode dynamics of multi-mode oxide VCSELs, the interaction
matrix is expanded to include more carrier reservoirs. In the
following we are going to derive a transfer function from an
interaction matrix which includes two photon reservoir
densities S1 and S2 along with two carrier reservoir densities
N1 and N2. The interaction of these reservoirs is ensured
IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL.52, NO.7, June 2016
through the sixteen different matrix entries as can be seen in
(27). Fig. 3 shows a schematic representation of these
interactions which are based on the governing rate equations.
By implementing the approximations and boundary
conditions discussed in the single mode cases, the matrix
entries in general matrix (17) are then reduced to fourteen
having twelve different interaction coefficients.
In order to further simplify the interaction matrix, only the
cross interactions between carrier and photon density
reservoirs which are shown in Fig. 3 are modelled in (27).
Thus, the interaction of the carrier reservoirs with the photon
reservoirs, μS1N2 and μS2N1, are set to zero in agreement with
the established device physics.
Again, photon-photon interaction is not supported by our
device structure, and therefore μS1S2 and μS2S1 are omitted as
well. These assumptions lead to the following system matrix:
 j   N1N1

  N2 N1

   S1N1

0

 N1N2
j   N
 N1S1
2 N2
N
2 S1
0
j
S
0
2 N2
 N1S2 
  dN1   dJ inj1 
 N S   dN   dJ 
2 2 
2
 inj 2 

0   dS1

j   dS 2




=
(27)
 0 


 0 
This matrix representation of the linearized rat equations is
used for the derivation of the multi-mode transfer function.
This matrix has ten different interaction coefficients of which
six are already introduced in the single mode model. Two new
entries represent mainly the interaction or communication
between the two different carrier reservoir densities N1 and N2,
and can expressed by
 N 1 N 2  v g aS 2  V 2 / V12   N 2 N 2  V 2 / V12
(28)
 N 2 N 1  v g aS1  V1 / V12   N 1 N 1  V1 / V12
(29)
and the following interaction coefficients are needed to model
cross reabsorption:
 N 1 S 2  v g g th (mod e 2 )  V 2 / V12   N 2 S 2  V 2 / V12
(30)
 N 2 S 1  v g g th (mod e1)  V1 / V12   N 1 S 1  V1 / V12
(31)
In the derivation of (28) and (29), diffusion terms are
neglected. On one hand, diffusion effects are already
considered as first-order effects in the definition of the steadystate quantities, S and N. They are also included in the photondensity-ratio q which is going to be represented later. On the
other hand, this approximation does not affect the shape of the
modulation response for high-speed VCSELs [17]-[25].
We define V1 / V12 = s1 and V2 / V12 = s2 representing the
spatial dependency of the two interacting carrier reservoirs
with respect to the two different mode volumes [30].
Furthermore, Vi has a volume dimension and represents the
effective volume of the ith mode. The degree of spatial overlap
between the two lasing modes, mode 1 and mode 2, is given
by s = s1 s2 = V1V2 / (V12)2. If this factor s is small, which means
the overlap between the two modes is weak, the two modes
can barely interact through their carrier reservoirs. This leads
HAMAD et al.: SMALL-SIGNAL ANALYSIS OF ULTRA-HIGH-SPEED VCSELs
to the so called decoupled model where the total transfer
function of this system is given by the sum of the individual
single mode function of the different modes. The transfer
function belonging to this case is represented at the end of this
section.
For a VCSEL with an oxide aperture diameter ranging from 3
to 10 µm, the typical two lasing super-modes have the shape
of LP01 and LP11 modes. In a circular oxide aperture, the
LP01 has its main optical intensity in the center of this
aperture, whereas the LP11 mode power is mainly in the outer
region. Typical values of s1 and s2 for this type of modes are
0.67 and 0.94 for the two super-modes LP01 and LP11,
respectively [30].
The last two parameters which are still to be defined in the
above matrix representation in (27) are the driving current
densities dJinj1 and dJinj2. As shown in Fig. 3, the VCSEL is
uniformly injected with the carrier density Jinj. Before
reaching the carrier reservoirs, this injected carrier density Jinj
is split among the two reservoir in proportion to the photon
densities of the respective modes. We define p1 and p2 as two
pumping currents which are proportional to the respective
current densities and the supplied photon reservoir densities.
Thus, p1 ~ dJinj1 ~ S1 and p2 ~ dJinj2 ~ S2. As we are just
interested in the mode power ratio, we introduce the parameter
q. This q represents the photon-density-ratio or the modepower-ratio between the two modes and is defined as q = (Г2
vg2 a2 S2)/ (Г1 vg1 a1 S1) ≈ S2 / S1. The lower case numbers 1
and 2 distinguish between the two set of parameters belonging
to the two different lasing modes. These parameters are
discussed in the single mode case.
Finally, solving the matrix in (27) and summing over all
mode photon densities, the modulation response can be
expressed as
7
Even though (33) appears to be bulky, it consists of four well
know figures of merit, two damping factors along with two
relaxation oscillation frequencies. Moreover, if the
geometrical configurations of the device are known, s1 and s2
can be calculated or extracted and set as constants.
For the decoupled case, where the degree of spatial overlap
is negligible, s1 and s2 can be set to zero. Thus the total
modulation response in (33) reduces to a power-weighted sum
of the individual single mode responses
H ( ) 
q
 R2 2  j 2   2

1/ q
(34)
 R21  j 1   2
Finally, if one of the mode powers dominates over the other
mode powers, the modulation response collapses again to the
single-mode model. For example, if the power of mode one
becomes very dominant compared to the second mode, S1 >>
S2, the modulation response reduces consistently to the singlemode case
H ( ) 
1
(35)
  j 1   2
2
R1
III. VCSEL DEVICE STRUCTURE AND PERFORMANCE
The device architecture is an optimized version of our very
successful high-speed, temperature-stable 980 nm VCSEL
design [5]. It shares the very short half-lambda cavity and a
binary bottom-mirror with 32 pairs. Doping levels were further
optimized minimizing internal loss. Like the previous design
two oxide apertures and highly conducting current-spreading
materials are used for low RC-parasitic. For higher differential
gain we used as active region a 5x InGaAs multiple quantum
well structure with strain compensated GaAsP
H ( ) = c  ( dS1  dS 2 )
-pair
barriers.
Instead
of
the
22
Al12Ga88As/Al90Ga10As top-mirror we utilize an
improved 18 -pair GaAs/Al90Ga10As mirror with
1
1
p m 1  p 2 m32
p m 1  p 2 m 42
lower photon lifetime, better confinement and better
 c  ( 1 31
 1 41
)
det M
det M
heat extraction properties. The structure was grown
by IQE Europe ensuring the commercialization
1
1
1
1
potential. Like the previous structure the VCSEL
p1 ( m31
 m 41
 m 42
)  p 2 ( m32
)
 c 
was processed in our two-mesa design with
det M
Cyclotene™ passivation utilizing inductively
coupled plasma (ICP) dry etching with a coplanar
2
2
2
2
q ( R1  j ( 1   2 s 2 (1 / q ))   )  (1 / q )( R 2  j ( 2   1 s1q )   )
ground-signal-ground (GSG) pad layout.

2
2
2
2
2
( R1  j 1   )( R 2  j 2   )   1 2 s1 s 2
The VCSEL device performance was
(32)
characterized by a calibrated 40-GHz vector
with cκ being a constant. Again, the lower case numbers 1 and network analyzer (HP8722C). To suppress large-signal
2 distinguish the relaxation oscillation frequencies ωRi and artefacts,
the
modulation
amplitude
was
damping factor γi of each lasing mode and are defined as set as low as -15 dBm. Calibration was done until the wafer Ri2   NiSi  SiNi
and
 i   NiNi .
plane of a Cascade GSG150 probe with the matched
If the cross interactions between the carrier- and photon calibration substrate. Optical coupling was done with a
density reservoirs are modeled as discussed above, then (32) cleaved multimode-fiber of 1 m length. To minimize distortion
of the signal by optical feedback, the coupling was reduced to
can be expanded to
a reasonable level of 0 dBm in the fiber. Remaining small
H ( ) 
q ((1  s 2 ) R2 1  j ( 1   2 s 2 (1 / q ))   2 )  (1 / q )((1  s1 ) R2 2  j ( 2   1 s1 q )   2 )
( R2 1  j 1   2 )( R2 2  j 2   2 )   1 2 s1 s 2 2  j ( s1 s 2 ( R21 2   R2  1 ))  s1 s 2 R2 2 R21
2
(33)
8
IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL.52, NO.7, June 2016
Fig. 4. L-I-V Characteristics of the oxide-confined 980-nm VCSEL at room
temperature. The characterized VCSEL has an aperture diameter of ~7 µm.
reflections caused by the straight FC/PC connectors could be
extracted from the whole dataset allowing the systematic error
to be de-embedded with no influence on the measured
bandwidth. The high-speed photo-detector was calibrated by a
femtosecond laser-pulse allowing its transmission behavior to
be subtracted from the measurement yielding correct data even
beyond 30 GHz.
L-I-V characteristics of the device are presented in Fig. 4.
The 980-nm VCSEL features an aperture for current
confinement of about 7 µm in diameter with a threshold
current below 0.8 mA, and a differential slope of ~ 0.64 W/A
at 25°C (equaling a differential quantum efficiency of 50 %).
The maximum optical output power is above 7 mW, and the
differential series resistance is 75 Ω.
An elaborate small-signal analysis was carried out. The
original data of the modulation responses at the indicated bias
level are calibrated and plotted in Fig. 5. These measured S21
data are later fitted to various rate-equation based models and
several figure of merit could be extracted as depicted in
Fig. 6-8. The maximum 3 dB bandwidth of the device
including chip-parasitics was found to exceed 32.7 GHz at
14 mA. The VCSEL reached thermal-rollover at 17 mA. For
better visibility, the corresponding S21 curve at this current is
not depicted. It is worth to mention that the small response
offset appearing in Fig. 5 at the first two S21 data sets is not
artificially introduced to the plot. It is probably caused by low
power levels at low driving currents.
In order to demonstrate the potential of the multi-mode
model presented in this work (33), the S21 data in Fig. 5 is
analyzed. For comparison, we used the established singlemode model (16) as well, and showing its pitfalls. In analogous to (16), (33) is further multiplied by a parasitic pole in
order to model the device parasitics. It is important to mention, that the frequency response corresponding to the electrical parasitic, can be well approximated by a single-pole lowpass filter. This simple first order approximation is consistent
with an elaborate analysis by equivalent circuit modelling of
the S11 signal measured simultaneously with our S21 data as
presented in Fig. 6. The method used here requires a high level
of understanding of the device layout to reduce the degrees of
freedom created by introducing several equivalent circuit
elements. Even though this method delivers slightly higher
Fig. 5. Calibrated small-signal modulation response of a 980-nm multi-mode
and oxide-confined VCSEL with an aperture diameter of ~7 µm. The curves
depict the relative response data S21 for various driving currents at room
temperature. The modulation current is increased gradually up to 14 mA
where thermal rollover was reached. At this current the measured 3-dB
bandwidth was found to approach 33 GHz.
Fig. 6. Parasitic response of the same device, measured at 14 mA bias point.
The solid line represents the single-pole approximation as in (16) with a
parasitic cut-off frequency of fPMM. The scatters represent the extrinsic
response extracted from device impedance measurement and fitting with the
given equivalent circuit. The measurement data, the fit and the parasitic
equivalent circuit with extracted device values are inset.
precision in theory, the remaining uncertainties might well
cancel out this advantage by larger error-bars. Consequently
we suggest using the single parasitic pole method if the device
under investigation features a low-parasitic chip layout similar
to the devices presented here. The parasitic small-signal
response of such a device is dominated by an electrical
equivalent-circuit including a series resistance Rm of both
Bragg mirrors as well as an current crowding resistance Ra
parallel to a capacitance Ca caused by the aperture. This
simple electrical equivalent-circuit sufficiently models the
parasitic behavior well as long as the capacitance of the
bonding pads Cp is kept small pushing their effects to be in the
range of 50 GHz. Therefore, the parasitic response of this kind
of devices can be well modeled by a simple single-pole lowpass filter [27] in the given frequency range.
HAMAD et al.: SMALL-SIGNAL ANALYSIS OF ULTRA-HIGH-SPEED VCSELs
Fig. 7. Three different small-signal modulation response data at various
driving currents 2.5 mA, 5.0 mA and 14 mA are presented. These S21 curves
are selected from the calibrated data in Fig. 5. For clarity, the selected curves
are shifted along the y-axes. The dashed lines are fits to the data and they are
based on the single mode model (SM). The solid line fits are based on the
advanced multi-mode multi-reservoir model (MM). The position of the
extracted relaxation oscillation frequencies fR, fR1 and fR2 along with the
parasitic cutoff frequencies fP from both models are also depicted. Two
straight lines show the propagation direction of the parasitic cutoff
frequencies as the current increases for the two models.
9
(a)
(b)
Fig. 8. Extracted figures of merit from the small-signal modulation response
data using the single-mode model. Plotting the resonance frequency fR and
the bandwidth f-3dB versus the square-root of the driving current above
threshold allow us to estimate the MCEF and D-factor (a). In (b), the Kfactor is obtained by plotting the gradient of the damping-rate γ versus the
squared resonance frequency. Above a certain driving current, the parameter
extraction reliability of K and D of the single-mode model is questioned.
This region is shaded.
The figures of merit extracted using (33) can then be
directly compared to that of (16) with the detailed definitions
of the single-mode model derived from (12) as follows:
   NN   SS  v g aS   v g a p S 

J th  J sp

 N 
S


 o
 R2


 1  a a
p
 v g aS  v g ( i   m ) 
0
v g ( i   m )

(36)
 K 4 π2
 f K  0
2
R
defining the K-factor K, a fitting parameter and damping offset
γ0 in laser parameters as defined in (5) – (8). For the multimode case, we can define multiple K-factors Ki for each
independent mode ensemble using the same definition. The
relaxation oscillation frequency depends on the biasing
conditions. Therefore, we can define the D-factor as follows:
f R  D I  I th
where
D
1
2π
v g i a
eV res
(37)
with Vres as the volume of the optical resonator [27]. Again,
for the multi-mode case, we can define multiple D-factors Di
for each independent mode ensemble using the same
definition. A similar interrelationship is also observed for the
laser bandwidth f3dB. Here, we speak from a modulation
current efficiency factor MCEF with
f 3 dB  MCEF I  I th
(38)
(a)
(b)
Fig. 9. Extracted figures of merit from the small-signal modulation response
data of Fig. 5 using the advanced multi-mode model (33). Plotting the
resonance frequencies fR1 and fR2 versus the square-root of the driving current
above the threshold of each mode, allow us to estimate the D-factors D1 and
D2 (a). In (b), the K-factors K1 and K2 are obtained by plotting the gradients
of the damping-rates γ1 and γ2 versus the squared resonance frequencies of
the corresponding modes.
A comparison between the extracted figures of merit from
both models is depicted in Fig. 7-9 and the values of these
parameters are stated in Table I.
The first interesting contrasts between both models is
graphically illustrated in Fig. 7 through the plotting and fitting
of three different modulation response data sets at various
driving currents; 2.5 mA, 5.0 mA, and 14 mA. These S21
curves are selected data sets taken from Fig. 5. For clarity, the
selected curves are shifted along the y-axes. The dashed lines
represent the fits which are based on the single mode model
(SM), whereas, the solid lines show fits based on the advanced
10
multi-mode multi-carrier reservoir model (MM). The position
of the extracted relaxation oscillation frequencies fR, fR1, and
fR2 along with the parasitic cut-off frequencies fP from both
models are also depicted. The tendency in the propagation
direction of the parasitic cutoff frequencies, which varies as
the current increases, is also indicated by straight line.
Based on this new evaluation, we can make several
findings. Firstly, the novel fits match all curves much better.
In particular, the fitting of the 2.5 mA-data, where no one
would doubt the validity of the single-mode fit at the first
place, are well fitted. For the two other curves, it is much more
obvious that the old model, represented by the dashed lines,
can neither explain the form of the measured data, nor the
physics behind. The figure from 0 to 10 GHz of the VCSEL
biased at 5 mA cannot be replicated at all by the single-mode
fit. The fit attempts to follow the front notch and the resonance
peak. Even though unrealistic device parasitics are assumed,
the first 10 GHz are still way off. Other fitting strategies led to
even worse or even less consistent results. At 14 mA, the
single-mode fit only coincides with the measured data from
20 GHz onwards. To achieve this level of matching in both
single-mode fits, the parasitic pole had to be allowed to range
around 14 GHz for the small drive current and dropping from
there towards 6 GHz, whilst the differential resistance of the
devices drops with current as can be observed in the declining
slope in Fig. 4. This is contra intuitive and unphysical,
however, common practice. This might be the reason why the
validity of small-signal experiments is questioned from time to
time in the community. The fitting according to (33) does not
have this problem. The parasitic cutoff frequency starts from
22 GHz raising to 27 GHz with lower differential resistance.
These values of electrical parasitics coincide with those
extracted from the reflected S11 measurements. Li Hui et al.
also observed similar tendency in the parasitic cutoff
frequency when extracting data from the reflected S11
originating from a similar VCSEL architecture [31].
Even though (33), with its 5 fitting parameters, could give
the impression of being able to fit arbitrary transfer-functions,
this is not the case. Basically, this equation can be seen as the
sum of two well-defined single-mode transfer functions with a
coupling term determined by the device geometry. However,
effective fitting strategies are helpful for the extraction of
reliable parameters.
All the measured data presented in Fig. 5 was fit
accordingly, and the results are plotted in Fig. 8 and Fig. 9 for
extraction of figures of merit like K- and D-factors. Whilst the
modulation current efficiency MCEF is derived from the 3-dB
bandwidth and is therefore independent of curve fitting, the
extraction of K- and D-factors relies on correct fitting. The
quality of the fits can no longer be verified in these values.
Therefore, we shaded the areas of fitting with high error and
and unphysical parasitics in Fig. 8, cancelling out most of the
plot. While the D-factor seems to be still usable as it is derived
from data with better fits, the K-factor can no longer be trusted
by using the simple model for evaluation.
The multi-mode fit using (33), does not only give better
fits and physical parasitics, but also more information about
the VCSEL device. We can extract two D-factors, one for
each of the two dominant modes forming their individual
IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL.52, NO.7, June 2016
carrier reservoirs. Interestingly, these two super-modes have
different dynamics. This can explain experimental findings in
large-signal experiments, where the achievable bit-rates and
error-flooring strongly depends on coupling strategies of
multi-mode devices. In Fig. 9(b) we show two K-factors being
extracted corresponding to different damping characteristics
for the different modes competing with each other for material
gain. From the fits we found the mode-power-ratio q to be
around 0.25 … 2 raising with the bias current. We also see no
damping offset for the second mode ensemble, which is
expected, as the VCSEL is already lasing while these higherorder modes evolve. The differences in K- and D-factor could
be interpreted by assuming the differential gain to be six times
higher for the first mode ensemble which is also suffering
from twice as high gain compression. More detailed studies
are planned for the future.
The device architecture optimization towards short photon
life time results in a moderate, but still sufficiently high
modulation current efficiency factor and D-factor of
12.5 GHz/mA0.5 and 9.1 GHz/mA0.5, respectively. The higher
MCEF can be explained by the very small K-factor as low as
0.12 ns, making the VCSEL no longer damping limited. The
second mode ensemble seems to suffer from slightly lower
dynamics. By filtering out the slower mode, e.g. by the
implementation of high-contrast-gratings (HCG), large-signal
modulation characteristics are expected to be improved. Even
though the device design is clearly optimized for heat
extraction, the bandwidth is still limited by thermal effects as
can be seen in Fig. 9(a) with the performance still saturating
for higher currents.
All the three different current regions, as discussed in the
introduction, can be accurately modelled by our multi-mode
theory (33).
TABLE I
EXTRACTED FIGURE OF MERIT FROM BOTH MODELS
Fit Model
K1 [ns]
K2 [ns]
D1 [GHz/√mA]
D2 [GHz/√mA]
SM-Model
0,07
-
8,8
-
MM-Model
0,12
0,36
9,1
3,5
The extracted VCSELs performance parameters the K-factors and Dfactors from both models, are depicted for a direct comparison. The first row
shows the data which are extracted using the single-mode model. The
proposed multi-mode model was applied to extract the data which are
presented in the second row.
IV. CONCLUSION
In this paper, we have consistently expanded the established
single-mode laser rate equations towards a multi-mode highspeed VCSEL rate-equation model. We could show that the
commonly used practice of modelling multi-mode VCSELs by
single-mode rate-equations is justified as long as the carrier
reservoirs do not split up. For the highest performing
VCSELs, however, this assumption does not hold up, and our
more comprehensive model has to be used. Even though this
model includes many effects such as spatial hole burning or
HAMAD et al.: SMALL-SIGNAL ANALYSIS OF ULTRA-HIGH-SPEED VCSELs
cross carrier reabsorption, only as little as five free fitting
parameters are needed. This allows reliable small-signal
analysis and the extraction of figures of merit by means of
curve-fitting. Therefore, we believe this method should be the
technique of choice in judging the dynamic performance of
VCSEL devices.
Further, we used this model to investigate our latest
generation of ultra-high-speed VCSEL devices with
modulation bandwidths of up to 32.7 GHz and a modulation
current efficiency of 12.5 GHz/mA0.5. The K-factor for the
dominant set of modes was found to be 0.12 ns, whilst the
K‐factor for the higher order transverse modes was 0.36 ns
indicating a higher damping of these modes, however, without
a damping offset as these modes evolve competing with other
lasing modes rather than spontaneous emission below
threshold. The D-factor of the dominant modes was as high as
9.1 GHz/mA0.5 while the higher-order transverse modes being
pumped at high drive currents show lower efficiency of
3.5 GHz/mA0.5 only. Modelling this VCSEL device with the
former theory would have required to set the parasitic
response to unphysical low values. This would cause the
intrinsic damping to be underestimated yielding unphysical
values for the K-factor as well.
Analyzing this VCSEL with the newly proposed model not
only gives much better fits, but also lets us derive very
physical figures of merit allowing us to understand our device
in detail. This gives us the knowledge for further optimization
which is the basis of the next device generation.
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12
Wissam Hamad has studied B.Eng. and M.Sc. in Engineering
Physics at the University of Oldenburg,
Germany. After his graduation in 2012
with the specialization “Microrobotics
and Micro Systems Engineering” he
worked as a researcher in the division of
Prof. S. Fatikow in the field of handling
and characterization of nanomaterials and
nanoobjects. In 2014 he joined
Prof. Hofmann´s research group at TU
Berlin were his research focus is the high-speed
nanostructured VCSELs.
Stefan Wanckel is a graduate student in Physics at the
Technical University of Berlin (TU Berlin), Germany. In 2015
he started his master-program in Engineering Physics at the
same place. In 2015 he joined Prof. Hofmann’s group focusing
on the modelling of VCSEL-dynamics.
Werner H. E. Hofmann (M’06)
received his Dipl-Ing (M.S.) degree in
electrical engineering and information
technology in 2003 and the Dr.-Ing.
(Ph.D.) degree in 2009, both from the
Technical University of Munich,
Germany. From 2003 to 2008, he was
with the the group of Prof. Amann at the
Walter Schottky Institute, where he was
engaged in research on long-wavelength vertical-cavity
surface-emitting lasers (VCSELs). Subsequently, he joined
Prof. Chang-Hasnain´s Group, University of California,
Berkeley (UCB), where he worked on the incorporation of
high-contrast gratings into VCSEL devices. In 2010, he joined
the Technical University of Berlin (TU Berlin), Germany, as
principal scientist in the group of Prof. Bimberg. Since 2013
Prof. Hofmann is leading his own group focusing on ultrahigh-speed nanophotonic devices. Prof. Hofmann is also the
Chief Technical Officer at the Centre of Nanophotonics at TU
Berlin. Since 2014 he is also with the Xiamen University,
P. R. China as guest professor and highly qualified foreign
expert.
He received the E.ON Future Award in 2009 and the SPIE
Green Photonics Award in 2012 for his research on highspeed, energy efficient VCSELs. Prof. Hofmann authored over
100 papers cited over 500 times and has presented over 10
invited talks and tutorials on international conferences. He is a
member of the Association of the German Engineers (VDI),
the German Association of University Professors (DHV) and
the IEEE Photonics Society. His main research interests are
high-speed surface emitters and their advance by nanophotonics.
IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL.52, NO.7, June 2016