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IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 37, NO. 5, MAY 2001
Dynamical Simulation of Quantum-Well Structures
Marco Wiedenhaus, Andreas Ahland, Dirk Schulz, and Edgar Voges
Abstract—For the design and development of optical semiconductor devices based on quantum-well structures, the investigation
of saturation phenomena is necessary for high optical power operation. By applying stationary physical models, nonlinear effects
cannot be described adequately; hence, transient models are important for an accurate analysis. By utilizing transient models, saturation phenomena, signal delays, and distortions can be investigated. For the analysis of integrated optoelectronic devices, such as
lasers and modulators, transient transport or density matrix equations for carriers and photons and the Poisson equation have to be
solved self-consistently.
A transient model which is useful for the investigation of a wide
range of optoelectronic applications is presented. Quantum optical
phenomena are included by applying the interband density matrix formalism in real-space representation, where the Coulomb
singularity is treated exactly in the limits of the discretization. As
we focus on electroabsorption modulators, a drift-diffusion model
adequately approximates the transport properties. Here, quantum
effects are considered by a quantum correction, the Bohm potential. The model is applied to investigate transport effects in InPbased waveguide electroabsorption modulators including strained
lattices.
Index Terms—Dynamical solutions, excitons, modulators,
quantum wells, semiconductor device modeling.
I. INTRODUCTION
O
PTOELECTRONIC devices are based on both the electrical properties of a semiconductor and the optical characteristics. It is important, therefore, to couple the simulations
of these two systems to describe important optoelectronic effects like the Franz–Keldysh effect (FKE), the quantum-confined Stark effect (QCSE), and the Wannier–Stark effect (WSE).
As these devices are operated at high frequencies, special attention is directed toward the simulation of dynamical processes
like saturation effects. This is the basis for the determination
of, e.g., cutoff frequencies and phase shifts. Simulation tools allowing an adequate coupling of electrical, optical, and transient
properties of applicable structures have not been presented up
to now. The dynamical simulation of transport phenomena in
optoelectronic components demands a treatment that considers
computational effort, stability of the numerical algorithms and
realistic results. We present an approach in this paper that meets
these requirements.
For the first time, to our knowledge, dynamical simulations
are performed, which describe a whole optoelectronic system.
In particular, the optical system and the coupling of optics and
electronics are a new feature of transient device simulations. It
becomes possible, therefore, to realize novel time-varying sim-
Manuscript received October 3, 2000; revised February 1, 2001. This work
was supported by the Deutsche Forschungsgemeinschaft (DFG).
The authors are with the Universität Dortmund, D-44221 Dortmund, Germany.
Publisher Item Identifier S 0018-9197(01)03489-3.
ulations of electroabsorption modulators, as described in this
paper.
Electrical properties are obtained from the solution of the
fundamental semiconductor equations. These are the Poisson
equation and an adequate transport model. Classical transport
can be described by a drift diffusion model (DDM) [1] or a
hydrodynamic model (HDM) [2], which accounts for temperature dependence. For electroabsorption modulators (EAMs) the
DDM is adequate. A HDM should be used for the simulation
of lasers. A quantum correction given by Bohm [3] specifies
the quantum mechanical transport in the semiconductor. These
fundamental equations can be implemented in a coupled scheme
[1]. This leads to the introduction of an artificial time derivative into Poisson’s equation, which actually does not depend on
time [4], [5]. Several approaches solved a decoupled scheme by
a stable explicit algorithm, including a numerical damping term
[6]–[8]. But this resulted in rapid damping or unphysical oscillations. In this paper, the hydrodynamic model and the quantum
correction are implemented. An implicit decoupled algorithm is
preferred here as it provides wider time-steps.
Optical properties, on the other hand, are determined by the
interband density matrix equation describing the excitonic effect [9]. The singularity of the Coulomb interaction term in this
equation makes a decisive contribution to the optical spectrum.
It can be treated exactly in the limits of the discretization [10],
which is realized here for the first time. The density matrix formalism allows one to account for nanostructures, which, due to
the QCSE, yield a higher bandwidth and an improved modulation depth. In contrast to [11], where a “leap-frog” algorithm
is used to handle the time dependence, we prefer an implicit
scheme allowing wider time-steps due to the stability of implicit
methods.
As outlined above, the coupling of the optical and electrical
systems is essential for an analysis of optoelectronic effects.
The interband density matrix equation includes the impact of
slanted band edges and, thus, describes an influence of the electrical properties on the optical system. Saturation terms in this
equation cause an optical generation of carriers regarded in the
recombination-generation-rate of the continuity equations. The
influence of the optical on the electrical system is thereby as
well described. Coupling is therefore an important feature of
our calculations.
We implemented different algorithms to investigate derivatives in time. This enables a flexible choice concerning stability
and oscillations. Explicit algorithms provide a simple calculation of each time-step [1]. Its main drawback is the restrictive
stability condition implying very short steps. Hence, we perform implicit algorithms, which allow much longer time-steps.
Results are computed faster this way although a system of linear
equations has to be solved.
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First, we introduce the fundamental semiconductor equations,
which describe the electrical properties (Section II). Briefly, the
interband density matrix equation is presented in Section III. Sections IV and V show the implementation of the equations. In our
opinion, these should be presented in detail, as important features of device simulations are outlined. For a more superficial
reading, these sections may be skipped. Finally, examples are
given in Section VI to demonstrate the usefulness of dynamical
solutions in the simulation of modulators.
The argument
reads
(10)
and is determined by the Fermi energy
and the potential
energy .
contains the intrinsic Fermi energy
and the
, the
quasi-Fermi potential , includes the band energy
self-consistent potential calculated in Poisson’s equation and
the quantum potential in the form given by Bohm [3]
II. SEMICONDUCTOR EQUATIONS
(11)
A system containing Poisson’s equation (1), the continuity
equations (2) and (3) and carrier transport equations of the DDM
with an additional quantum correction (4) and (5) is solved in a
decoupled manner to determine the electrical properties of the
structure [1]
(12)
(1)
(2)
(3)
(4)
(5)
These equations contain the permittivity , the self-consistent
(electrons) and
potential , the carrier concentrations
(holes) and the concentrations of singly ionized accepters
and donors
. Further, the difference of recombination- and
generation-rate , the carrier current densities
and
for
electrons ( ) and holes ( ), respectively, and the mobilities
and
are used, as well as the quasi-Fermi potentials
and
. is the elementary charge.
In addition to these fundamental semiconductor equations,
carrier densities are calculated for the Fermi distribution and
parabolic bands from
(6)
(7)
is the effective density of states for parabolic bands
The optical properties of a semiconductor are described by
the interband density matrix equation in the real-space representation linearized in the electric field , i.e., saturation terms
for carrier densities comparable to the saturation density are neglected [9]. These neglected terms depend on the carrier densities. They account for carrier density based effects such as the
Burnstein–Moss effect described in Section VI. Beyond this,
the source terms can be used to describe the optical generation
of carriers [see (21)]. The effective mass approximation and an
approach corresponding to the slowly varying envelope (SVE)
yield
function
(13)
The solution is the “pair wave function” describing a coherent
wave function due to the well defined phase relation of the linearization on the right-hand side (13). Its envelope function
leads to the observable , the macroscopic polarization by (16).
An incoming optical wave of photon energy
causes this polarization damped by a phenomenological relaxation term according to the inverse relaxation time. This accounts for a dephasing of the transition amplitude under influence of random
is set to 7 meV for transitions implying heavy
local fields.
holes and to 8 meV for light holes.
represents the undisturbed Hamiltonian
(14)
(8)
where
effective mass;
temperature;
Boltzmann constant;
reduced Planck constant
In general, the Fermi integral of order
III. OPTICAL PROPERTIES
.
is defined as
(9)
It cannot be solved in closed form and is treated by applying
asymptotic approximations [1] or, for arguments close to zero,
by a polynomial expression that yields an error below 0.5% [12].
where the second term, the Coulomb term, includes the absolute
distance between electron and hole. The potential energy is
;
and
are the potentreated as
[10].
tials at the respective positions of electron and hole
Consequently, the main mechanisms in electroabsorption modulators, the effect of electroabsorption (the FKE), the QCSE,
and the WSE, can be described. The FKE leads to a flattened
absorption edge, which can be shifted by the QCSE in quantum
confined structures. Thus, in a quantum well (QW) it is possible
to control the absorption of the material by an external voltage
and, therefore, to modulate an incoming wave. A distinction between the radial and -directions (growth and confinement direction) becomes necessary to describe this effect. Beyond that,
the density matrix formalism accounts for free states as well
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IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 37, NO. 5, MAY 2001
as for bound and quasibound states. Accordingly, it is possible
to describe field dependent effects like, as mentioned above,
QCSE, WSE, and FKE. The real-space representation in relative coordinates simplifies the quantum mechanical calculation
because the termination of the calculation domain by applying
a perfectly matched layer (PML) [13] becomes possible.
As already mentioned, the source term on the right-hand side
of (13) is linear in the electric field
(15)
weighted with the delta funcis the dipole matrix element
. It is assumed to be of rotational symmetry. The polartion
ization is given by
(16)
and the complex linear interband susceptibility
is defined as
(17)
which determines the optical absorption coefficient and the refractive index.
As we merely consider -excitons, a radial symmetry is assumed. Higher excitations can be neglected [14] having only an
indirect impact on the optical absorption.
Since the complex susceptibility is calculated, there is no
need for using the Kramers–Kronig relation. From this, we derive an effective dielectric constant [10]
Fig. 1. Flowchart of the nonlinear calculation of the electrical and the optical
properties.
IV. IMPLEMENTATION OF THE SEMICONDUCTOR EQUATIONS
Poisson’s equation (1) can be efficiently solved by an
algorithm introduced by Gummel et al. [15]–[17]. There are no
time derivatives in this equation. Solving a decoupled system
of equations there is no need for introducing an artificial time
derivative as described in [1], [4], [5]. At each time-step,
Poisson’s equation is solved using the carrier density values of
the last time-step. The calculated potential in turn is used for
the subsequent solution of the continuity equations (2) and (3).
A general time-dependent problem
(22)
is discretized according to
(18)
(23)
(integration over the center-of-mass coordinate ). Assuming
the absorption and the effective refractive index
are
including a parameter (
), where
is the time-step
size. Our programs are implemented using this parameter, which
for an explicit algorithm,
can be chosen to be
for a Crank–Nicolson scheme, or
for a fully implicit
algorithm.
Selberherr [1] remarks that for a coupled system, the best
results are obtained by using a fully implicit scheme. For the
decoupled system, we compared this fully implicit method to a
Crank–Nicolson scheme. Since the latter unfortunately tended
to oscillations [18], the fully implicit scheme also gives best
results for the decoupled system.
, no restrictive stability condition has to be reFor
garded. Hence, the step size can be chosen much larger than for
an explicit algorithm. Its choice may be aligned to the feature
that the main variations of the electrical properties take place
within less than 1 ps. Though no stability problem occurs, numerical dispersion must be taken into account because step sizes
on the order of 0.1 ps are not suitable for description of the given
problem.
The semiconductor equations (1)–(5) are expressed by the not
). But two other sets of varimandatory set of variables (
) based on the
ables are feasible as well, e.g., the set (
quasi-Fermi potentials. The latter leads to nonlinear continuity
equations and, therefore, is not advantageous as the solution of
the decoupled scheme converges quite slowly. These variables
and
(19)
(20)
The vacuum wave number
is given by the wavelength
of the incident light.
The electrical and the optical system can be treated separately
near the local equilibrium. The density matrix equation even
holds for processes out of equilibrium, whereas the continuity
equations (2) and (3) have to be modified.
The saturation terms in the source terms of the density matrix equation neglected in (13) lead to an optical generation of
carriers. This generation can be accounted for in the continuity
equations (2) and (3)—a coupling of the electrical (Section II)
and the optical system just mentioned can be realized this way.
Consequently, an additive generation-recombination rate
can be derived
(21)
leading to a nonlinear calculation, as depicted in Fig. 1.
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WIEDENHAUS et al.: DYNAMICAL SIMULATION OF QW STRUCTURES
687
should be considered in a coupled solution because they are of
nearly the same order of magnitude and provide an improved
behavior concerning convergence.
), with and defined by
We prefer the set (
(24)
(25)
Though these variables reveal an enormous range of 32 orders
of magnitude for a difference of voltage from 1 to 1 V at a temperature of 300 K [1], this set supplied stable results in our case
because no noteworthy differences between two succeeding results occurred in an iterative method.
The spatial derivatives are implemented by a simple finite
differences approach.
V. IMPLEMENTATION OF THE DENSITY MATRIX EQUATION
Different from the implementation of the semiconductor
equations, a polynomial finite element scheme is used in
the spatial domain as mentioned in [10]. As the Coulomb
in the second term on the right-hand side
singularity for
of (14) is constitutive for the solution of the density matrix
equation, it is of interest to include this contribution as exactly
as possible. Our finite-elements approach allows an exact
treatment of this Coulomb singularity within the limits of the
discretization since the integral over this singularity is finite.
In particular, the application of spherical coordinates yields a
simple solution of this integral because angular dependence
can be neglected. In bulk material and ideal 2-D collocations, a
volume element containing emerges from the integral trans-dependence of the Coulomb
formation. This cancels the
term. The second derivative in the first term on the right-hand
side of the Hamiltonian is subject to an integration by parts as
the resulting first derivative is applicable to the finite elements
scheme. A nonequidistant discretization near the Coulomb
singularity can be avoided in contrast to [19], as well as it is
not necessary to calculate the ground state for the effective
potential as described in [11].
In order to reduce computational effort, it is advisable to
shorten the calculation domain. Reflections at the boundaries
are eliminated by a PML absorber [13], a nonphysical absorber
which is matched independent of the angle of incidence. Here,
we assume a small Coulomb term and a plane wave in cartesian
coordinates.
Again, we implemented different algorithms to treat the time
dependence. The “leap-frog” scheme uses central differences
for the time derivatives. Thus, the problem (22) mentioned
above, which is comparable to beam propagation methods [20],
[21], can be rewritten as
(26)
leading to a directly solvable scheme in the general form
(27)
Fig. 2. Structure of a GaInAsP QW system.
[A matrix in front of the sought
emerging from the finite
elements method can be constricted to a lumped stiffness matrix. This diagonal matrix arises by taking the sum of each row.
It can easily be inverted, which again leads to a directly solvable
scheme (27).] This explicit algorithm underlies very restricted
stability conditions. We found step widths in the order of 0.1 fs, a
really restrictive condition, which does not allow efficient computations. An implicit algorithm, on the other hand, leads to
tedious calculations counterbalanced by a possible choice of
wider time-steps.
In opposition to the solution of the semiconductor equations,
a fully implicit scheme should not be applied here, since energy conservation is not guaranteed this way. Wave functions
are preferably solved by a Crank–Nicolson scheme, which does
not tend to oscillations as much as in the electrical system. Expressed in terms of (23), the parameter has to be chosen to
. If the impact of oscillations is not acceptable, can be
set to a higher value, e.g., 0.7 or 0.8. The main aspect to be considered, is that dephasing is sufficiently sampled. The step size
should, therefore, be about an order below the relaxation time,
which is typically of order 500 fs in our calculations. Compared
to the stability condition for the “leap-frog” method, this enables
wider time-steps up to about an order of magnitude below the
relaxation time.
Equation (13) becomes, discretized in time, the following:
(28)
VI. EXAMPLES
First, we show the results for a simple GaInAsP system. The
QW structure has a special behavior because of a field-induced
absorption edge merging (see [14] for details). Its structure is
shown in Fig. 2.
The structure consists of the following material. Barriers I and
V are of lattice-matched Ga In As P , layer II of compressively strained InAs P and layer IV of tensile strained
Ga In As. A well layer of InAs P (layer III) provides
an increased overlap of the electron and hole wavefunctions; but
there is no bound state within this narrow layer.
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IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 37, NO. 5, MAY 2001
Fig. 3. Potential energies inside the QW for a voltage step of 0.5 V. Calculation
1), without optics.
Fig. 5. Band edges at two different times. At 0.75 ps, the outer region of the
device near the contact is affected, while the inner parts are nearly unaffected.
Fig. 4. Potential energies inside the QW for a voltage step of 0.5 V. Calculation
2), nonlinear with optical system.
Fig. 6. Actuated voltage and imaginary part of the optical susceptibility for an
incoming wave of 0.675 eV.
Here, a voltage step of 0.5 V is actuated to show how the material follows the external voltage and approaches the stationary
state again. Two calculations were performed:
1) a linear calculation, where the optical system has no influence on the electrical properties;
2) a nonlinear simulation, where the optical system affects
the potential energies.
Figs. 3 and 4 exhibit the evolution of the potential energies in
the inner QW for an incoming wave of 0.81 eV (1530 nm) and
V/m.
From the initial state, the thermodynamic equilibrium, it takes
around 3–4 ps in calculation 1) to reach the stationary state.
Calculation 2) results in a much slower behavior of the device,
which can be explained by a generation of carriers as described
in (21). The transport of these carriers is the reason for the retarded reaction.
Fig. 5 shows the temporal behavior of the inner parts of the
structure. First of all, the actuated voltage affects the outer region of the device near the active contact. At 0.75 ps, no remarkable change can be seen in the QW region. A retarded reaction
of the active region is the consequence.
The presented method achieves stable results for a discretization of 1 fs. The electrical system is calculated by a fully implicit scheme, while we chose
for the optical system.
No further attention is devoted to a detailed description of the
absorption edge merging and strained material parameters, for
these are given in [14].
For an applied sinusoidal voltage of 40 GHz, the result of the
optical calculations is shown in Fig. 6. The voltage alternates
with an amplitude of 0.5 V around 0.5 V, the energy of the wave
is set to 0.675 eV and the electric field to 300 V/m. A short delay
can be seen due to the evolution of the material.
Again, the outer parts are affected first. After around 1 ps, the
active region responds to the external voltage, which leads to the
delay of the device resulting in a phase shift of the modulated
signal. Here, the electrical and the optical system are coupled.
It is interesting to see how this algorithm copes with more
complicated structures controlled by a sinusoidal voltage. We
give a demonstration of a wave-guide electroabsorption modulator that accounts for the QCSE in a SPQW structure. This
property, in the present case, results in a blue-shift of the absorption edge [22], [23].
Normally, the Stark effect results in a red-shift of the edge, but
in structures with an asymmetrically stepped-potential, a blueshift can be obtained for small electric fields directed from the
narrow to the wide bandgap. The SPQW has a small QW at one
side of the main QW, as shown in Fig. 7.
In principle, a tilting of the bands leads to a slightly raised
electron resonance level followed by a blue-shift, as mentioned.
But a further increase of the field yields the Stark effect, a redshift again. In this case, a barrier is included in the QW separating the small trap from the rest of the well (Fig. 8), which
may lead to a blue-shift of more than 13 meV. All information
concerning these structures can be obtained from [23]–[25].
Fig. 8 shows the layered structure of the modulator applied.
The periodically arranged active region is depicted in detail for
the shaded region. The doping of the n- and p-contact regions is
cm .
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Fig. 7. Principle band diagram of a stepped-potential QW.
Fig. 9. Optical susceptibility for an incoming wave of 0.81 eV, stationary
calculations.
Fig. 8. Structure of the applied blue-shift modulator: band energies and Fermi
level. The shaded region is depicted in detail.
Fig. 10. Actuated voltage and imaginary part of the optical susceptibility for
an incoming wave of 0.81 eV, transient calculations.
The calculations presented are done with a discretization in
space of 1 nm and a discretization in time of 1 fs. In the beginning, the left contact is biased by 1 V while the right contact
is fixed at 0 V. Thus, we first calculate the stationary state. A sinusoidal voltage with an amplitude of 1 V and a frequency of
40 GHz is actuated in addition to the stationary bias. The material is exposed to a wave of energy 0.81 eV and an electric
field of 300 V/m.
The electrical system is again coupled to the optical system.
Saturation effects have to be considered [23]. A small
bandgap and a small density of states cause a fully occupied
band near the band edge, the so-called Burnstein–Moss effect,which appears in terms of a screened Coulomb term.
This is another reason for a blue-shift since the generation of
electron-hole pairs needs higher energies. As a counterpart to
the Burnstein–Moss effect, a reduction of the bandgap has to be
regarded. Concerning high carrier concentrations, this follows
from exchange terms in the interband density matrix equations.
For small carrier densities, the two effects compensate one
another. The reduction of the Coulomb potential, even a
rejection for higher concentrations, results in a decrease of the
absorption.
The time derivatives of the electrical system are handled by
a fully implicit algorithm, whereas the optical properties are
treated by an implicit scheme with the parameter
in
(28). It is chosen this way because a parameter
tended
to oscillations. A compromise had to be found that accounts for
), as well as avoids oscillations
energy conservation (
).
(
Fig. 9 shows the optical susceptibility depending on the actuated voltage in the stationary case. This graph is quite useful
for comparison.
Fig. 11. Actuated voltage (dashed line), band energies and Fermi levels in one
of the QWs, transient solutions.
Starting from the stationary case at 1 V the dynamical calculations are done. Its result is shown in Fig. 10.
The graph reveals that the device again has a retarded reaction. It takes about 0.5 ps before the inner parts of the device are
subject to a significant variation of the band energies and the
Fermi levels. The band energies and the Fermi levels inside the
left QW are depicted in Fig. 11.
Variations of the actuated voltage have an immediate influence on the outer parts of the device. Gradually, this effects the
inner parts of the structure and, therefore, causes a delayed variation of the optical behavior in the active region leading to the
phase shift.
The calculations for such a complicated structure are a question of days, and therefore, only the first few picoseconds are
shown. Fortunately, it is possible to choose a simpler, but more
improper treatment, e.g., a noncoupled simulation. The simula-
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IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 37, NO. 5, MAY 2001
tions can be adjusted to individual requirements. It should, at
any rate, be mentioned that transient calculations demand an increased effort.
The last example shows that the implemented algorithms provide a stable calculation for even a quite complex device. This
is a remarkable confirmation of our approach.
VII. CONCLUSION
We have demonstrated an efficient method to apply dynamical solutions of the optical and electrical properties on EAMs,
which account for computational time and numerical stability.
The response of a blue-shift waveguide EAM on field-induced
effects was calculated concerning delay and nonlinearities. The
methods presented allow the investigation of transient effects of
QW semiconductor devices at high frequencies, which are constitutive for the study of cutoff frequencies, time constants and
phase shifts.
Implicit schemes were preferred to explicit algorithms in
order to treat time dependence in our calculations. This is
due to wider time-steps, which are not feasible in an explicit
method because of stability considerations. In particular, the
transient solutions of the density matrix approach are a new
feature of simulating QW structures providing such a wide
field of applications.
The tools developed are a successful approach to close the
gap between existing investigations of the physical basics and
an applicable and fast simulation of optoelectronic devices.
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Marco Wiedenhaus was born in Lippstadt, Germany, in 1974. He received the
Dipl.-Ing. degree in electrical engineering from the Universität Dortmund, Dortmund, Germany, in 2000.
In March 2000, he joined the Department for High Frequency, Universität
Dortmund, and is currently working in the field of modeling and simulation of
semiconductor devices and optical communication systems.
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Andreas Ahland was born in Münster, Germany, in 1971. He received the
Dipl.-Ing. degree in electrical engineering from the Universität Dortmund, Dortmund, Germany, in 1996.
In April 1996 he joined the Department for High Frequency, Universität Dortmund, and is currently working in the field of modeling and simulation of semiconductor devices and optical communication systems.
Dirk Schulz was born in Düsseldorf, Germany, in 1965. He received the
Dipl.-Ing. degree in electrical engineering from the RWTH Aachen, Aachen,
Germany, and the Dr.-Ing. degree from the Universität Dortmund, Dortmund,
Germany, in 1990 and 1994, respectively, and is now pursuing his venia
legendi.
Since August 1990, he has been a Research Assistant at the Institute for High
Frequency Techniques, Universität Dortmund, where he worked on beam propagation techniques for guided wave devices in integrated optics and optoelectronics. His current interests are in optical communication systems, modeling,
and simulation techniques for integrated optical, optoelectronic, and high-frequency devices, and high-frequency circuits.
Edgar Voges was born in Braunschweig, Germany, in 1941. He received
the Dipl.-Phys. and Dr.-Ing. degrees from the Technische Universität Braunschweig, Braunschweig, Germany, where he was engaged in investigations on
microwave semiconductor devices and acoustic surface waves.
From 1974 to 1977, he was a Professor with the Department of Electrical Engineering, Universität Dortmund, Dortmund, Germany. From 1977 to 1982, he
was a Professor of Communication Engineering at the Fern Universität, Hagen,
Germany. Since 1982, he has again been with the Department of Electrical Engineering, Universität Dortmund, as Head of the Institute for High Frequency
Techniques. His current interests are in optical communication, integrated optics, and high-frequency integrated circuits.