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Keysight Technologies
DesignCon 2014
Modeling, Extraction and Verification of
VCSEL Model for Optical IBIS AMI
White Paper
Abstract
A technique of modeling and extraction of VCSEL devices for IBIS-AMI has been
proposed. With the measured L-I and V-I curves of the VCSEL, a behavior model is
extracted from rate-equation model that includes thermal effects. Here two suitable
curve fitting algorithms are used. This model exhibits the observed performance in
both time and frequency domains. It has been verified on several mainstream VCSEL
devices where a consistent fitting between the measured data and abstracted one is
achieved. And the results prove that the modeling, extraction and verification process
can abstract a real VCSEL device accurately. The generation of an IBIS-AMI model
facilitates system designs that include VCSEL devices. Also a certain VCSEL can
be published into a dynamic link library. This technique is intended to help VCSEL
vendors build and publish behavior models with the measured device characteristics.
By following the IBIS-AMI standard, VCSEL users can simulate their designs more
conveniently.
Authors'
Zhaokai Yuan, Keysight Technologies, Inc.
M. V. Ramana Murty, Avago Technologies, Inc.
Sanjeev Gupta, Avago Technologies, Inc.
Amolak Badesha, Avago Technologies, Inc.
Authors Biographies
Zhaokai Yuan is at Keysight Technologies EEsof division as a R&D system engineer
of the SystemVue team. He mainly focuses on Wireless Communication libraries and
High Speed Digital libraries developing.
Ramana Murty joined the Fiber Optics III-V Division at Avago Technologies in 2007.
He led the development of 850 nm 10G VCSELs for high performance computing and
100 GbE applications. His current interests include the development of VCSELs and
p-i-n photodetectors for 25G applications.
Sanjeev Gupta is currently employed by Avago Technology Fiber Optic division as Sr.
R&D Manager and is leading a team of Signal Integrity, EMI and Layout engineers.
From 1994 to 2011 he was employed by Keysight Technologies and held various
application engineering positions. He has co-authored numerous papers and was
recipient of DesignCon best paper award consecutively for three years (2008, 2009
and 2010).
Amolak Badesha is Program Director at Avago Technologies, driving strategic
initiatives championed by executive management at Fiber Optics Division. Previously,
Amolak built and lead the SI/EMI team at Avago's Fiber Optic Division. Amolak also
spent 7 years at Keysight's EEsof Division, specializing in high-speed design. While
at Keysight, Amolak made key contributions to innovative products like Automated
AMI model generation and Optical AMI modeling.
2
I Introduction
The transceiver market has been growing rapidly over the past few years due to the
increasing demand of large-scale data communications. An optical link is an effective
solution because of high bandwidth and low power consumption.
Figure 1 Block Diagram of an Optical Link
As illustrated in Figure 1, an optical link consists of Laser Driver, Vertical cavity
surface emitting laser (VCSEL), Photo-detector, trans-impedance amplifier (TIA),
Limiting Amplifier, optical fiber and clock-data-recovery (CDR). The electrical signal is
received from the host board through pluggable or embedded optical module. The
received signal goes through signal conditioning to improve signal quality in terms of its
amplitude and timing before it is converted to an optical signal. Electrical receiver on the
Optical transmitter can utilize various equalization schemes such as Continuous Time
Equalizer or De-emphasis before the signal jitter performance is improved using clock
and data recovery circuit. Laser driver following receives the signal from CDR output
and convert it into a current waveform which drives VCSEL diode. For short range
communication, VCSEL provides many advantages such as low power requirements and
low cost over other types of electro-optics devices such as DFBs. The output signal from
the VCSEL is coupled to a multimode optical fiber using precision optical techniques. On
the receiver end, High speed optical signal is received by the p-i-n photo-detector which
converts incoming light to a current waveform. The TIA following the p-i-n diode
transforms the current waveform into an output voltage waveform. The signal passes
through limiting amplifier stages before this data is re-timed and equalized to the host
receiver.
In this data link, VCSEL is a key device due to its unique electro-optic
characteristics [1] [2] [3]and it is important to model the VCSEL to simulate the
performance of an integrated data communication system. Here, a first principles model
[4] of a VCSEL is not required. An effective theory in the form of rate equations captures
the behavior of the VCSEL [5] [6] [7] [8].
The paper is organized as follows. A rate equation model for the VCSEL
including thermal effects is described in section II and the method extracting the model
3
parameters is described in section III. The VCSEL model is applied to three devices from
literature in section IV followed by a conclusion.
II VCSEL Modeling and Parameter Extraction
Rate Equation Based Thermal VCSEL Model
The VCSEL device characteristics can be described by a set of coupled of non-linear rate
equations. While the rate equations are traditionally expressed in terms of carrier and
photon densities, here we use the carrier N and photon S numbers. Electrons and holes
(here represented by N) are injected into the active region of the VCSEL by an applied
voltage across the junction, and lost through stimulated emission and other (spontaneous
emission and non-radiative) paths. The non-radiative and spontaneous emission paths are
captured through a carrier lifetime n in the rate equation. Photons are generated by
stimulated and spontaneous emission, and lost through the partially transparent mirrors
and other (absorption, scattering) optical losses. The total optical loss is captured through
a photon lifetime p. Stimulated emission depends on the gain in the active region and is a
function of carrier density and a (weak) function of photon density. The rate equations
can be descripted as Equation 1 and Equation 2.
dN i ( I I off (T )) N Go ( N N o )S
Equation 1
dt q n 1 S
dS S N Go ( N N o ) S
3
Equation 2
dt p n 1 S
where i is the carrier injection efficiency, G0 is the gain coefficient, N0 is the carrier
transparency number, is the spontaneous emission factor, is the gain compression
factor, and q is the electron charge. The drive current is I and Ioff(T) is a temperature
dependent offset current described below. Certain simplifications have been made in the
description of the VCSEL and are duly noted. Most VCSELs used in practical
applications are multi-mode with significant photon populations in several transverse
modes. They are more accurately represented by a rate equation for the photon number S 1,
S2, ... in each transverse mode. Here, a single photon number S = S1 + S2 + ... is used to
capture the behavior of the device. The active region gain has been approximated as a
linear function of N. Finally, carrier transport effects have been ignored [7].
Light emitted P0 from the VCSEL is proportional to the photon density
Po kS Equation 3
and can also be expressed in terms of the drive current
Po ( I I tho I off (T )) Equation 4
4
Here k is a scale-factor that includes the output coupling efficiency through the
distributed Bragg mirror in the VCSEL, the photon energy and the photon lifetime [7].
The parameter represents the slope efficiency, and Ith0 is the threshold current at a
reference temperature. Light emitted by a VCSEL is a function of temperature and this is
incorporated through the Ioff(T) term. An empirical expression is used for the offset
current
M
I off ak T k Equation 5
k 0
A polynomial of order M = 4 is found to provide an adequate description. It is
noted that the temperature variation of both slope efficiency and threshold current of a
VCSEL is captured by the Ioff(T) term.
The junction temperature T can be related to the ambient temperature T 0 and the input
electrical power by
dT
T To ( IV Po ) Rth th Equation 6
dt
where V is the applied voltage and Rth is the device thermal resistance. The last term with
the time constant th represents a reactive load. It vanishes under dc conditions. Also,
junction temperature does not change significantly during high data rate (> 1 Gb/s)
modulation. With increasing drive current, the junction temperature T increases causing
Ioff(T) term to increase. Eventually, at a certain drive current I = Ioff(T) and the light
emitted does not increase any further. This is the rollover current and a further increase in
drive current causes P0 to decrease.
Calculation of the junction temperature requires knowledge of the applied voltage and
this is achieved by modeling the IV characteristic of the VCSEL. Three different
functions between V and I are proposed in [5], Equation 7, Equation 8 and Equation 9.
I
V c0 Rs c1 ln(1 ) Equation 7
c2
P Q
V b p I p cqT q Equation 8
p 0 q 0
P
V bp I p Equation 9
p 0
Equation 7 describes a resistance in series with a diode. Equation 8 treats the
voltage as a polynomial function of current and temperature. Equation 9 is a reduced
version of Equation 8 where the impact of temperature has been ignored.
The VCSEL can be modeled and simulated using the equations described above. The
non-linear nature of the equations requires the development of a curve fitting algorithm
for extraction of the parameters.
5
Curve Fitting Algorithms
Two curve fitting algorithms have been used to fit the experimentally measured
device characteristics. The first is the linear least squares method, and the second is the
non-linear least squares method for equations that involve products of quantities.
Linear Least Squares Curve Fitting
The linear least squares curve fitting algorithm is used to extract the coefficients in
polynomial equations such as the relationship between the offset current and the
temperature, and between laser voltage and current. Consider a polynomial function f
describing the relationship between x and y
L
y f ( x) ak x k Equation 10
k 0
The goal is to extract the optimum values of coefficients ak from a data set
( x1 , y1 ), ( x2 , y2 ), ( xn , yn ) . The optimum values are obtained by minimizing the sum of
squares [5]:
n
[ yi f ( xi )]2 Equation 11
i 1
The result is a system of L simultaneous linear equations in the coefficients {ak }
that can be solved for the coefficients by standard linear algebra technique, e.g. LU
decomposition of the coefficient matrix. Another approach is to use an orthonormal basis
instead of the original basis. The orthonormal basis can be got by the Schmidt
orthogonalization process, where an iteration process is taken instead which is more
easily implemented on computer.
P0 ( x) 1,
P ( x) ( x 1 ) P0 ( x)
1
Equation 12
Pk 1 ( x) ( x k 1 ) P0 ( x) k Pk 1 ( x)
(k 0,1,..., n 1)
Where the Pk (x) is a polynomial with the order of k, and
m m
x P i k
2
( xi )
( xPk , Pk ) P k
2
( xi )
( Pk , Pk )
k 1 i 0
m
, k i 0
m
.
P P
2 ( Pk , Pk ) 2 ( Pk 1 , Pk 1 )
k ( xi ) k 1 ( xi )
i 0 i 0
Non-Linear Least Squares Curve Fitting
The non-linear least squares curve fitting algorithm is used to extract parameter values
from the more complex equations, such as the rate equations and the stationary LI model
equations. Both the two kinds of equations cannot be solved directly and contain multiple
6
parameters. So a total different thought is taken compared to the Linear LS algorithm.
The final solution is achieved by multiple trying. One or more measured data is used as a
target item. For each loop, one possible set of parameter values are preset and then the
target item is calculated with the equations. The differences are collected together and
multiplied with the according weight separately. And this is the error of current set of
parameter values. During the multiple try, the minimal error can be got and the
corresponding set of parameter values is taken as the final solution. In this algorithm, the
chosen of possible values are very important. A reasonable range of the being extracted
variable should be a precondition. Within the rage, a random value generation method is
taken that is how the certain set of value is chosen out. So the issue that how many values
are selected exists in this method. If the number is too small, the best value cannot be
picked out correctly. And if more details are considered, the calculation will cost too
much time. To resolve this issue, an iteration process is brought in. The range is reduced
gradually along with the increasing of iteration number.
III VCSEL Parameter Extraction
The procedure for extracting the various coefficients from measured VCSEL
characteristics is described in this subsection. Commonly measured characteristics of a
VCSEL include the LI, VI and small-signal modulation response S21. Measurement at
multiple temperatures provides a more complete data set for generating the VCSEL
model.
Coefficients in the LI Curve
A typical LI curve shows a threshold current for lasing, a linear increase in output power
P0 for small currents above threshold current, and a rollover point beyond which P0
decreases with increasing current. Equations 4