1. 1244 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 57, NO. 5, MAY 2009
Embedded Grating n+-i-n+
Traveling-Wave Photodetectors
Guido Torrese, Cailin Wei, Matthew J. Frank, Isabelle Huynen, Senior Member, IEEE,
André Vander Vorst, Life Fellow, IEEE, and Patrice Mégret
Abstract—This paper describes the design of embedded-grating expense of a reduced photocurrent since half of the photogener-
n+ -i-n+ traveling-wave photodetectors (TWPDs). In the ideal case ated current is dissipated in the output termination impedance
of perfect electrically matched photodetectors, the TWPD band- [1]–[3]. In order to support traveling electrical waves, particular
width is determined by the velocity mismatch between the group
velocity of the optical beam and the phase velocity of the elec- care must be paid in designing the TWPD electrodes, while the
trical photogenerated signal, as well as by the attenuation of the RF external circuitry must insure the transmission line matching
signal propagating along the electrodes. Since the group velocity is [6]. While TWPDs with matched input ends do not suffer from
determined once the layer’s thickness is selected, which, in turn, the ripple resulting when connecting the detector to the external
is optimized for coupling the light into the photodetector, only an electrical circuitry, the bandwidth of the input end mismatched
accurate electrode design allows for the velocity matching. In this
paper, we propose a new structure that utilizes a grating integrated
photodiodes is almost insensitive to the mismatching between
within a single-mode waveguide in order to tailor the optical group optical group velocity and electrical phase velocity. In this
velocity and match it to the phase velocity of the photogenerated paper, we consider photodetectors perfectly matched at both
signal. input and output ends. In this case, the device frequency re-
Index Terms—Bandwidth, grating, photodiode, semiconductor sponse will depend mostly on the mismatch between optical
device, traveling-wave photodetector (TWPD). group velocity and electrical phase velocity. When terminating
both input and output ends of the detector by the characteristic
impedance of the line, no electrical wave is reflected back at
I. INTRODUCTION the TWPD ends, and the detector can be made long enough
VER THE last several years, interest in high-speed to guarantee 100% internal quantum-efficiency. Moreover,
O traveling-wave photodetectors has increased [1]–[4].
Ultra-high-bandwidth traveling-wave photodetectors (TWPDs)
long photodetectors allow, at least in principle, power levels
approaching the optical damage limit. As discussed in [4],
that overcome the traditional limits of vertically illuminated when considering high optical power incident on the active
photodiodes (VPDs) have been successfully demonstrated [5]. region, in order to avoid damage the light should be absorbed
Although TWPD distributed structures can overcome the RC along the entire structure since the active region is quite small
bandwidth limitation of both VPDs and waveguide photode- and cladding materials are fully transparent. Meanwhile, the
tectors (WGPDs) by matching the optical group velocity to absorption length (i.e., with being the absorption coef-
the phase velocity of the electrical signal [6], the design of ficient) has to be kept short to avoid bandwidth degradation [1].
the intrinsic region is limited by the geometry of the trans- In this paper, we show how, by integrating a grating within
mission line [3]. By terminating the input and output ends of a TWPD, it is possible to tune the speed of the optical wave in
the photodetector with loads matching the line characteristic order to match the electrical phase velocity, while achieving a
impedance, electrical backward propagating waves can be theoretically infinite bandwidth even for small absorption coef-
completely absorbed. It should be noted, however, that the ficients. This paper is organized as follows. In Section II, we de-
bandwidth improvement achieved by minimizing the phase lag scribe the design of the embedded-grating traveling-wave pho-
between forward and backward traveling waves is gained at the todetector (EG-TWPD). Section III deals with the calculation
of the phase velocity of the RF signal and the group velocity
Manuscript received December 05, 2008; revised February 10, 2009. First
of the optical wave. In Section IV, we discuss how a grating
published April 17, 2009; current version published May 06, 2009. etched in the waveguide core allows the optical group velocity
G. Torrese was with the Electromagnetism and Telecommunication Depart- to be tuned as desired. Finally, Section V summarizes the main
ment, Faculté Polytechnique de Mons, B-7000 Mons, Belgium. He is now with
SEE Telecom, BE-1420 Braine-l’Alleud, Belgium. results of this paper.
C. Wei is with the Lumera Corporation, Bothell, WA 98011 USA (e-mail:
cwei@lumera.com).
M. J. Frank is with the RSoft Design Group Inc., Ossining, NY 10562 USA II. TWPD DESIGN
(e-mail: matthew@rsoftdesign.com).
I. Huynen and A. Vander Vorst are with the Microwave Laboratory, Univer-
The general perspective of a TWPD consists of an elec-
sité catholique de Louvain (UCL), B-1348 Louvain-la-Neuve, Belgium (e-mail: trical coplanar transmission line coupled to an optical wave-
isabelle.huynen@uclouvain.be; andre.vandervorst@uclouvain.be). guide. The latter is formed by sandwiching a low doped
P. Mégret is with the Electromagnetism and Telecommunication De- N–In Alx Ga As x absorbing layer among transparent di-
partment, Faculté Polytechnique de Mons, B-7000 Mons, Belgium (e-mail:
patrice.megret@fmps.ac.be). electric cladding layers. By varying the aluminum composition,
Digital Object Identifier 10.1109/TMTT.2009.2017364 it is possible to adjust the absorption coefficient . In order
0018-9480/$25.00 © 2009 IEEE
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2. TORRESE et al.: EG-TWPDs 1245
TABLE I III. DEVICE OPERATION
MATERIAL GROWTH STRUCTURE
As previously discussed, the largest bandwidth in a TWPD
terminated at both input and output ends by the transmission line
characteristic impedance is achieved by matching the phase ve-
locity of the RF signal to the group velocity of the optical wave.
Usually the phase velocity is adjusted by varying the dimensions
of the electrodes. At high frequency, the electrodes give rise to a
finite signal delay, and the electrical traveling wave is distorted
due to the dispersive characteristic of the line. Depending on
the electrode geometry, semiconductor resistivity, and operating
frequency, electrodes can support a slow-wave mode propaga-
tion or a propagation mode dominated by the skin effect [9].
Due to the high conductivity of the N layers, the electric en-
ergy density is mainly confined into the intrinsic region, which
behaves as a lossless dielectric. On the other hand, the mag-
netic field can penetrate into the structure. When thin electrodes
are used, the skin effect can be neglected and the electrodes of
TWPDs support a slow-wave propagation mode [10]. Although
a full wave approach is required for an accurate characteriza-
tion of the propagation mechanism [11], for our purpose, the
quasi-TEM approach developed in [12] is satisfactory.
According to [12], the admittance per unit length (F/m) is
(1)
where is the electron effective mass, is the electron charge,
is the momentum relaxation time, is the semiconductor
permittivity, is the donor concentration, is the photode-
tector width, is the intrinsic region thickness, and is the an-
gular frequency. Equation (1) takes into account the finite con-
ductivity of the intrinsic region through the finite nonzero donor
Fig. 1. Optical field amplitude contour plot for the fundamental quasi-TE mode concentration. For perfect intrinsic materials, the admittance per
of the n -i-n TWPD discussed in this paper. The effective refractive index at unit length reduces to a simple capacitance. The impedance per
= 1 33
: m is n = 3 180413 + 0 0008415
: j : .
unit length m given by
(2)
to increase the saturation current, the photoabsorption should
be distributed along the waveguide by reducing the modal accounts for the electrode thickness through the series resistivity
absorption coefficient . Although it is possible to distribute , and the magnetic energy density storage through the induc-
the photoabsorption by embedding a very thin absorption layer tance . When increasing the frequency, the field
in laterally tapered transparent intrinsic layers [7], the com- only penetrates a small distance into the structure so that the se-
plexity of the design increases, and particular care is required ries resistance increases with the RF frequency. However, the
when designing the electrodes in order to avoid a high series skin effect becomes important only for very thick electrodes.
resistance. Details of the material structure and geometrical In the remainder of this study, we assume thin electrodes, and
dimensions of our design are given in Table I. therefore, consider a constant series resistance. The complex
The optical mode has been calculated with the commercial propagation constant and characteristic impedance are
software GratingMod, RSoft Design Group Inc., Ossining, NY, then given by [1]
by solving the 2-D semivectorial wave equations [8], while the
fraction of the mode power contained inside the core has been
calculated by direct integration of the amplitude squared of the and (3)
field. In order to minimize the carrier transit time, the intrinsic
region is quite small, which, in turn, implies a low external For thin electrodes, the series resistance is independent of
quantum efficiency. The waveguide is monomodal, and allows the operating frequency. Consequently, as the RF frequency
the propagation of the fundamental quasi-TE mode at the oper- increases, the ohmic losses become negligible and the charac-
ation wavelength of 1.33 m. The amplitude field profile of the teristic impedance will be almost independent of frequency,
optical mode, corresponding to a complex effective refractive provided the ideal dielectric approximation holds. When
index , is shown in Fig. 1. writing the electrical propagation constant as ,
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3. 1246 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 57, NO. 5, MAY 2009
optical wave propagating inside the waveguide, the absorption
coefficient of the core layer cannot be very high. A small ab-
sorption length not only will reduce the maximum temperature
rise that the TWPD can tolerate, but will also decrease the ef-
fective spatial extent of the grating as the light is absorbed over
a very short distance. Due to the periodic nature of the index
variation, forward and backward propagating waves couple to-
gether at wavelengths close to the Bragg wavelength. Conse-
quently, the grating acts as a dispersive material to the incident
optical signal [13]. By optimizing the grating parameters, the
group velocity of the optical traveling wave can be adjusted. In
this study, we considered a uniform grating in which the refrac-
tive index along the -length varies periodically. By assuming
that the transverse electric and magnetic field components for
the and modes of the unperturbed waveguide satisfy
the orthogonality relations, the field components for the pertur-
bated waveguide can be written as the superposition of orthog-
Fig. 2. 3-dB bandwidth for a TWPD electrically matched at both input and
output ends versus the velocity mismatch v =v for three different values of
onal modes as
the series resistance. Geometrical parameters are w =2
m, d =04
: m, and
L =1 mm. Electrical and optical parameters are V = 10
V, = 1 33
: m, (6)
n = 3 180413 + 0 0008415
: j : , and P = 0 001
: nW.
(7)
where is the attenuation coefficient and is the phase
constant, the electrical phase velocity can be written as where is the propagation constant of the mode, while
and are the amplitude of the forward and backward
(4) propagating components, respectively. According to coupled-
mode theory (CMT) [13], the following set of coupled ordinary
On the other hand, the group velocity of the light is differential equations:
(5)
with defined as the group index. The group velocity is fixed (8)
when the dimensions and the material composition of the wave-
guide are chosen, so that in traditional TWPDs, the electrical
phase velocity has to be matched to the optical group velocity.
As previously explained, when the photodetector is electrically
matched at both input and output ends, the velocity matching is
critical in order to achieve a large bandwidth. The photodetector (9)
bandwidth versus the velocity mismatch is shown in Fig. 2. In
TWPDs with highly conductive electrodes m, describes the field distribution inside the structure, where
as the phase velocity approaches the group velocity, the band- and are the coupling coefficients between the and
width increases dramatically, and a theoretically infinite 3-dB modes for the transverse and longitudinal directions, respec-
frequency response can be obtained when the velocity matching tively. Equations (8) and (9) are rigorous for any linear media
condition is satisfied. However, perfect velocity matching in the independent of their lossy or absorbing nature. The coupled or-
slow-wave structure is difficult to achieve in practice. For a typ- dinary differential equation can now be written in the form
ical photodetector width of a few micrometers, the phase ve-
locity of the microwave signal is slower than the group velocity (10)
of the light. To speed up the electrical signal, thick electrodes are
required. However, at high frequencies, the thickness of larger where is the matrix of the coupling coefficients. By applying
electrodes becomes comparable to the skin depth. Conductor the appropriate boundary conditions and solving the coupled
losses can no longer be neglected and the increased series resis- mode equations by assuming that the matrix is independent
tance ( m and m) reduces the of over an interval , the forward and backward propagating
photodetector bandwidth. fields at the two ends of the grating are simply connected by the
An alternative way to achieve the velocity matching con- transfer matrix as
sists of reducing the speed of the optical beam by embedding a
grating within the core of the single-mode waveguide. In order
(11)
to be able to achieve a large saturation current and slow down the
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4. TORRESE et al.: EG-TWPDs 1247
where is the grating length. Once is found, the left to right
reflection and transmission coefficient are
(12)
(13)
where are the elements of the matrix. Once the trans-
mission coefficient is known, the transmission spectrum
and the transmission phase shift accumulated
by the light traveling along the structure can be calculated. By
taking the derivative of the transmission phase with respect to
the angular frequency , the group delay in transmission is
(14)
The latter is related to the group velocity by
Fig. 3. Group velocity normalized by the speed of light in vacuum versus the
normalized frequency f =f : solid line shows results obtained with GratingMod
(15) software, dashed line corresponds to simulations using chain matrix formalism
[see (18) and (19)], while the dashed–dotted line shows normalized speed of
Since the group delay is not constant over the bandwidth of in- light in vacuum.
terest, the optical wave exhibits a dispersive behavior.
IV. RESULTS field distribution has been calculated by using the commercial
The modal analysis of the optical waveguide discussed in software GratingMod, RSoft Design Group Inc. We designed a
Section II and the computation of the optical field traveling uniform grating of length mm having peri-
along the -direction have been carried out by assuming a uni- odic sections of respective refractive indices
form refractive index distribution along the propagation direc- and .
tion. When a grating is etched into the waveguide core, for- The grating period is m, while the length of
ward and backward traveling modes couple together. Providing each and section is equal to a quarter wavelength at the
a complex propagation constant, the coupling coefficients operating frequency THz corresponding to the
and are Bragg wavelength m: and
. Once the forward and backward fields are
(16) known, the wave transmission coefficient is calculated by using
(13), which, in turn, allows us to determine the optical group
velocity through (14) and (15). The group velocity normalized
and
with respect to the speed of light in vacuum versus the normal-
ized frequency is shown via a solid line in Fig. 3. For com-
parison, we also calculated the group velocity using the chain
matrix formalism, applied to each type of section ( or ),
(17) shown in (18) at the bottom of this page. Coefficients of the
where is the index profile of the unperturbed wave- chain matrix include effective refractive indices pro-
guide, is the refractive index of the grating, and vided by GratingMod as and
is the transverse cross section. The solution of (8) and (9) re- , wave impedance
quires the specification of the mode amplitude at the input of , and the length of each or section.
the waveguide. For our calculation, we assume the quasi-TE The cascade of sections of type and sections of type
mode amplitude shown in Fig. 1 is guided within the waveguide is represented as the product of their chain matrices
core. Of course, the amplitude decays exponentially along the
propagation axis because of the material absorption. The electric (19)
(18)
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5. 1248 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 57, NO. 5, MAY 2009
V. CONCLUSIONS
In this paper, we have proposed a new TWPD able to handle
high optical power while maintaining a ultra-wide bandwidth.
The optical group velocity has been tailored by using a grating
embedded in the core of the waveguide structure. When the pho-
todiode is terminated at both the input and output end by the line
characteristic impedance, and the velocity matching is ensured
by an optimized grating design, the photodetector bandwidth
is only limited by the electrodes resistivity. For thin electrodes
1 m , the RF signal is characterized by a slow wave prop-
agation and the velocity matching results from slowing the op-
tical group velocity.
ACKNOWLEDGMENT
The authors extend their acknowledgment and appreciation
Fig. 4. Phase velocity of the electrical signal (dashed line) and group velocity to the Research Science Foundation (FRS-FNRS), Brussels,
of the optical beam (solid line) normalized by the speed of light in vacuum Belgium.
versus the RF frequency. The series resistance is R = 10000
=m. Insets
show enlarged views of lower and upper frequency ranges.
REFERENCES
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Authorized licensed use limited to: Isabelle Huynen. Downloaded on May 21, 2009 at 09:36 from IEEE Xplore. Restrictions apply.

6. TORRESE et al.: EG-TWPDs 1249
Guido Torrese received the Electronic Engineering Isabelle Huynen (S’90–A’95–M’96–SM’06) re-
degree from the Universitá degli Studi di Genova, ceived the Electrical Engineer degree and Ph.D.
Genova, Italy, in 1997, and the Ph.D. degree in degree in applied sciences from the Université
applied sciences from the Université catholique de catholique de Louvain (UCL), Louvain-la-Neuve,
Louvain, Louvain-la-Neuve, Belgium, in 2002. Belgium, in 1989 and 1994, respectively.
From October 2002 to December 2004, he was In 1989, she joined the Microwave Laboratory,
with the Centre for Research in Photonics, School of UCL, where she is currently a Senior Research
Information Technology and Engineering, University Associate with the Research Science Foundation
of Ottawa, where he was involved with theoretical (FRS-FNRS), Brussels, Belgium, and a Part-Time
and experimental investigation of pulse dynamics Professor. She has authored or coauthored one book
in photonic crystals, design of tunable lasers, and and over 200 publications in journals and conference
integrated photonics devices. In 2005, he joined the Electromagnetism and proceedings. She holds one patent. She has particular interest in the develop-
Telecommunication Department, Faculté Polytechnique de Mons, Mons, ment of devices based on nanoscaled materials and topologies for applications
Belgium, where he developed metrology for characterization of WDM systems. at microwave, millimeter-wave, and optical wavelengths.
He is currently a Research and Developoment Manager with SEE Telecom,
Braine-l’Alleud, Belgium, a leading developer of innovative solutions and
technologies for communications providers around the world.
André Vander Vorst (M’64–SM’68–F’86–LF’01)
received the Electrical and Mechanical Engineer
degrees and Ph.D. degree in applied sciences
Cailin Wei received the Masters degree from the Xian Institute of Optics and from the Université catholique de Louvain (UCL),
Precision Mechanics, Xían, Shaanxi, China, in 1988, and the Ph.D. degree from Louvain-la-Neuve,Belgium, in 1958 and 1965,
Ghent University, Gent, Belgium in 1997. respectively, and the M.Sc. degree in electrical
Since 1997, he been with major telecom firms in Ottawa, ON, Canada, as engineering from the Massachusetts Institute of
a Research and Development Scientist, where he has designed and developed Technology (MIT), Cambridge, in 1965.
various photonic components and systems. He is currently with the Lumera Cor- In 1966, as an Assistant Professor, he founded
poration, Bothell, WA. the Microwave Laboratory, UCL, where he became
Emeritus Professor in 2001. He has authored or
coauthored three textbooks and a variety of scientific and technical papers in
international journals and proceedings.
Matthew J. Frank received the Bachelors degree Dr. Vander Vorst is active in the IEEE Region 8, as well as in the European Mi-
in electrical engineering from Cornell University, crowave Association. He was the recipient of the 1994 IEEE Microwave Theory
Ithaca, NY, in 1998, and the Masters degree in and Techniques Society (IEEE MTT-S) Meritorious Service Award.
electrical engineering from Columbia University,
New York, NY, in 1999. While with Columbia
Univeristy, he was focused on merging standard
electronic devices with state-of-the-art optical sys- Patrice Mégret received the Electrical Engineering
tems for telecommunications. While with Cornell degree and Ph.D. degree in applied science from the
University, he was involved with optical systems Faculté Polytechnique de Mons, Mons, Belgium, in
for the detection and analysis of the combustion of 1987 and 1993, respectively.
hazardous materials in incinerators. He is currently Head of the Electromagnetism and
In 2000, he joined the RSoft Design Group, Ossining, NY, where he is cur- Telecommunication Department, Faculté Polytech-
rently a Senior Application Engineer. He has been instrumental in the develop- nique de Mons, and possesses 20 year of experience
ment and the support of worldwide-leading design software for photonic com- in the photonic field. He has authored or coauthored
ponents. He also possesses extensive application experience in areas including over 280 publications in journals and conference
photonic integrated circuits, fiber devices, and photonic-bandgap crystals. proceedings. He has supervised 11 Ph.D. theses.
His main research interests are in the metrology
of components and telecom systems with emphasis on optical transmission
quality.
Dr. Mégret was president of the IEEE LEOS Benelux Chapter (2003–2005).
He is a member of the Optical Society of America (OSA). He is an associate
member of URSI.
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