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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 43, NO. 9, SEPTEMBER 1995 2181
Introduction to Special Issue on Microwave
and Millimeter Wave Photonics
T HE RAPID evolution of photonics and microwave and allowed for this Special Issue. Therefore, a few papers that
millimeter wave electronics and their related technologies could not be included here will appear in a subsequent regular
is resulting in new device developments and novel system con issue of the TRANSACTIONS.
figurations. The interface between microwaves and millimeter The Special Issue is divided into six sections. The first two
waves on the one hand and lightwaves on the other is an area of groups of papers deal with optical modulators and transmitters
growing interest with a broad range of emerging applications. and the optical generation of microwaves and millimeter
This Special Joint Issue of the IEEE TRANSACTIONS ON waves. The next two groups are concerned with optical de
MICROWAVE THEORY AND TECHNIQUES and the JOURNAL tectors and receivers. The fifth group contains articles on
OF LIGHTWAVE TECHNOLOGY is devoted to Microwave and photonic signal processing and its application in microwave
Millimeter Wave Photonics. systems. The final group covers optical-microwave interaction
The vitality of the microwave-photonics research area is in devices and circuits.
evidenced by the nearly 70 papers that were considered It is clear that the field of microwave and millimeter wave
for this special issue. Papers were received from all five photonics represents a rapidly developing area of research
continents-from universities as well as industrial and gov with inspiring new results. We hope that this present Special
ernmental laboratories. The transnational aspect of this effort Issue will provide a useful picture of the current state of the
is reflected by the fact that the five guest editors represent technology and serve as a stimulus for further advances.
Australia, Asia, Europe, and both North and South America.
It should be noted that manuscripts submitted by any of the
guest editors were handled independently by two of the other
guest editors. PETER R. HERCZFELD
The quality of the papers was generally excellent and the H!ROYO OGAWA
editors are grateful to reviewers for their thorough and timely ALVARO AUGUSTO A. DE SALLES
response. We selected 38 papers for publication in the Special ALWYN SEEDS
Issue from those submitted. In a few cases, it was not possible RODNEY S. TuCKER
to complete the review/revision process in the tight time-frame Guest Editors
Peter R. Herczfeld (S'66-M'67-SM'89-F'91), born in Budapest, Hungary, in 1936 and now
a U.S. citizen, received the B.S. degree in physics from Colorado State University in 1961, the
M.S. degree in physics in 1963, and the Ph.D. degree in electrical engineering in 1967, both
from the University of Minnesota.
Since 1967, he has been on the faculty of Drexel University, where he is a Professor of
Electrical and Computer Engineering. He has published over 300 papers in solid-state electronics,
microwaves, photonics, solar energy, and biomedical engineering. He is the Director of the Center
for Microwave-Lightwave Engineering at Drexel, a Center of Excellence that conducts research
in microwaves and photonics. He has served as project director for more than 70 projects.
Dr. Herczfeld, a member of APS, SPIE, and the ISEC, is a recipient of several research and
publication awards, including the Microwave Prize ( 1986 and 1994).
2182 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 43, NO. 9, SEPTEMBER 1995
Hiroyo Ogawa (M'84) received the B.S., M.S., and Dr.Eng. degrees in electrical engineering
from Hokkaido University, Sapporo, in 1974, 1976, and 1983, respectively.
He joined the Yokosuka Electrical Communication Laboratories, Nippon Telegraph and
Telephone Public Corporation, Yokosuka, in 1976. He has been engaged in research
on microwave and millimeter-wave integrated circuits, monolithic integrated circuits, and
development of subscriber radio systems. From 1985 to 1986, he was a Postdoctoral
Research Associate at the University of Texas at Austin, on leave from NTT. From 1987
to 1988, he was engaged in design of the subscriber radio equipment at the Network
System Development Center of NTI. From 1990- 1992, he was engaged in the research of
optical/microwave monolithic integrated circuits and microwave and millimeter-wave fiber
optic links for personal communication systems at ATR Optical and Radio Communication
Research Laboratories. Since 1993, he has been researching microwave and millimeter-wave
photonics for communication satellites at NTI Wireless Systems Laboratories.
Dr. Ogawa is a member of the Institute of Electronics, Information and Communication Engineers (IEICE) of Japan.
Alvaro Augusto A. de Salles was born in Bage, RGS, Brazil, on March 6, 1946. He received
the B.Sc. degree in electrical engineering from the Federal University of Rio Grande do Sul
(UFRGS), Porto Alegre, Brazil, in 1968, the M.Sc. degree in electrical engineering from the
Catholic University of Rio de Janeiro (PUC/RJ), Brazil, in 1971, and the Ph.D. degree in
electrical engineering from University College London, England, in 1982.
From 1970-1978 he was an Assistant Professor at the Catholic University Center for Research
and Development in Telecommunications (CETUC), in Rio de Janeiro, where his major interest
...-..... was microstrip passive devices, including circulators and filters. From 1978- 1982 he was at
University College London working on solid-state phased array radars design and development
and on optical control of GaAs MESFET oscillators and amplifiers. From 1982-1990 he was
at CETUC, performing research and development on microwave and optical communication
semiconductor devices and components. From 1991-1994 he was a Visiting Professor at the
Federal University of Rio Grande do Sul (UFRGS) in Porto Alegre, RGS, Brazil, where he
is now Professor. He was also an Associate Professor at PUC/RJ. His area of research interest is optical interactions with
semiconductor devices, including HEMT's and HBT's, for microwave and optical communication applications. He has authored
more than 50 papers in brazilian and international periodics and conferences.
Dr. de Salles was Chairman of the 1987 SBMO (Brazilian Microwave and Optoelectronics Society) International Microwave
Symposium and is a founding member of SBMO and of the Brazilian Telecommunication Society (SBT).
Alwyn Seeds (M'81-SM'92) received the B.Sc. degree in electronics in 1976 and the Ph.D.
degree in electronic engineering in 1980, both from the University of London.
From 1980-1983 he was a Staff Member at ·Lincoln Laboratory, Massachusetts Institute of
Technology, where he worked on monolithic millimetre-wave integrated circuits for use in
phased-array radar. He was appointed Lecturer in Telecommunications at Queen Mary College,
University of London, in 1983. In 1986 he moved to University College London, where he
is currently BNR Professor of Opto-electronics and leader of the Microwave Opto-electronics
Group. He is author of over 100 papers on microwave and opto-electronic devices and their
systems applications and presenter of the video "Microwave Opto-electronics" in the IEEE
Emerging Technologies series. His current research interests include microwave bandwidth
tunable lasers, optical control of microwave devices, mode-locked lasers, optical phase-lock
loops, optical frequency synthesis, dense WDM networks, optical soliton transmission and the
application of optical techniques to microwave systems.
IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 43, NO. 9, SEPTEMBER 1995 2183
Rodney S. Tucker (S'72-M'75-SM'85-F'90) was born in Melbourne, Australia, in 1948.
He received the B.E. and Ph.D. degrees from the University of Melbourne, Australia, in 1969
and 1975, respectively.
From 1973-1975 he was a Lecturer in Electrical Engineering at the University of
Melbourne. During 1975 and 1976 he was a Harkness Fellow with the Department of
Electrical Engineering and Computer Sciences, University of California, Berkeley, and from
1976-1977 he was a Harkness Fellow with the School of Electrical Engineering, Cornell
University, New York. From 1977 to 1978 he was with Plessey Research (Caswell) Ltd., UK,
and from 1978-1983 he was with the Department of Electrical Engineering at the University
of Queensland, Birsbane, Australia. During the period from 1973-1983 he worked on high
speed electronic and optoelectronic devices, the synthesis of ultra-wideband amplifiers, and
modelling of high-speed semiconductor lasers. From 1984-1990 he was with AT&T Bell
Laboratories, Crawford Hill Laboratory, Holmdel, NJ, where he worked in the area of high
speed optoelectronics and lightwave cmnmunications. He is presently with the Department of Electrical and Electronic
Engineering at the University of Melbourne, where he is a Professor of Electrical Engineering, Director of the Photonics
Research Laboratory, and a Director and Deputy Chief Executive officer of the Australian Photonics Cooperative Research
Center. His research interests are in the areas of high-speed semiconductor lasers and photonic networks and systems.
Dr. Tucker has served on the Technical Program Committee of a number of international conferences. From 1989-1990 he
was Editor of the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES. He is a member of the Optical Society
of America, a Fellow of the Institution of Engineers, Australia, and a Fellow of the Australian Academy of Technological
Sciences and Engineering.
2184 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 43, NO. 9, SEPTEMBER 1995
Distortion in Linearized Electrooptic Modulators
William B. Bridges, Fellow, IEEE, and James H. Schaffner, Member, IEEE
Abstract-Intermodulation and harmonic distortion are cal
OPTICAL MACH-ZEHNDER Ps
culated for a simple fiber-optic link with a representative set WAVEGUIDE INTERFEROMETERS
of link parameters and a variety of electrooptic modulators: tp 1M OPTICAL
simple Mach-Zehnder, linearized dual and triple Mach-Zehnder, 0 •
simple directional coupler (two operating points), and linearized ERECEIVER
directional coupler with one and two de electrodes. The resulting
dynamic ranges, gains, and noise figures are compared for these OPTICAL FIBER ~ 12 a P1 + P2
modulators. A new definition of dynamic range is proposed to 0
accommodate the more complicated variation of intermodula
tion with input power exhibited by linearized modulators. The
effects of noise bandwidth, preamplifier distortion, and errors in
modulator operating conditions are described.
OPTICAL DIRECTIONAL COUPLER
I. INTRODUCTION Fig. I. Dual-parallel modulator configured with equal length electrodes and
one input optical signal. This particular approach requires two photodiodes at
E LECTROOPTIC modulators, both discrete interference the optical receiver. An alternative approach would use two lasers and then
types such as the Mach-Zehnder modulator and dis combine the optical signals at the modulators' outputs into one detector.
tributed interference types such as the directional-coupler
modulator, have inherently nonlinear transfer curves. As a
(e.g., a slope 3 line on the dBout versus dBin graph for third
consequence, they may limit the dynamic range of the photonic
order intermodulation), and the photonic link dynamic range
link in which they are embedded by generating harmonics and
no longer depends on the noise level in a simple way; a clearer
intermodulation products. Various modulator configurations
definition of "dynamic range" is really required. Finally, the
have been proposed and demonstrated . in the last several
improved modulator dynamic range can easily be eroded by
years [I ]-[8] to address this problem and increase the link
the nonlinear behavior of the electronic amplifiers required by
dynamic range. All of these schemes depend on generating
the photonic link to realize reasonable gain and noise Fig. [9).
two or more modulation samples with different ratios of signal
This paper uses a simple photonic link model to find the
to distortion and then combining the samples so that the
gain, noise figure, harmonics, intermodulation, and dynamic
distortions cancel (to some order) while the signals do not
range for a number of the modulator schemes listed above,
cancel. Jn some cases it is easy to identify where the two
and it uses the model to optimize the modulator parameters.
modulations occur and where the combinations take place, as
The sensitivity of representative Mach-Zehnder modulator
in the dual Mach-Zehnder schemes [l], [2], [6]; in others it
(MZM) and directional coupler modulator (DCM) schemes
is not so obvious, such as the directional-coupler modulator
to modulator and link parameters are calculated and com
and its variations [3)-[5].
pared. A refined definition of "dynamic range" is proposed
The various linearized modulator schemes predict, and in
to eliminate possible ambiguities resulting from the definition
some cases have demonstrated [l], [4)-[7], significant reduc
based on simple slopes. Finally, the results of adding electronic
tion in harmonics and intermodulation products, which should
amplifiers to the photonic link are calculated.
lead to the realization of photonic links with higher dynamic
ranges. However, in all cases, the cancellation turns out to
be critically dependent upon the modulator device parameters, II. DUAL MACH-ZEHNDER MODULATORS
so that these parameters will likely have to be controlled by The Mach-Zehnder modulator i.s a simple two-channel
active means, especially if the distortion cancellation is to be interference device, resulting in a sine-squared dependence of
maintained over a large operating bandwidth. In addition, the light output on drive voltage. The modulator is biased to the
dependence of the harmonic or intermodulation product on most linear portion of the transfer curve, which for a perfect
the signal drive level is no longer a simple constant exponent modulator also assures no even-harmonic generation.
However, the nonlinearity of the transfer curve is respon
Manuscript received January 9, 1995; revised May 5, 1995. This work was
supported in part by Contract no. F30602-9 l-C-O I 04 to Hughes Research sible for the generation of all odd-harmonics and all possible
Laboratories from the US Air Force, Rome Laboratories (N. P. Bernstein intermodulation products. The dual MZM scheme uses two
technical monitor) and by the ARPA Technology Reinvestment Project on
MZM's, driven at different RF levels and fed with different op
Analog Optoelectronic Modules, Agreement No. MDA972-94-3-0016.
W. B. Bridges is with the California Institute of Technology, Pasadena, CA tical powers, as illustrated in Fig. 1. The RF and optical power
91125 USA. splitting ratios are chosen so that the modulator receiving the
J. H. Schaffner is with Hughes Research Laboratories, Malibu, CA 90265
larger optical power receives the smaller RF drive power. This
USA.
IEEE Log Number 9413707. modulator may be thought of as the "main" modulator, with
0018-9480/95$04.00 © 1995 TEEE
BRJDGES AND SCHAFFNER: DISTORTION IN LINEARJZED ELECTROOPTIC MODULATORS 2185
some distortion created by the finite RF drive power. The other
0
modulator receives only a little optical power, but is driven
relatively much harder, thus yielding a much more distorted "CUBIC" SPLIT
signal. The two optical outputs are combined incoherently, for E' -40
example, by combining the electrical outputs of two separate !l:l!.
detectors as shown in Fig. 1.1 If the bias points of the two .JJJ
iU -80
modulators are chosen so that the modulations are out of phase, .J
and the ratios of both optical and RF powers are properly z.<J
chosen, then the sum of the two distortions (Pr M) can exactly ~ -120
cancel, while the signals (Ps) do not completely cancel. This I:>
~
exact cancellation can only occur for a specific drive level, :::>
0 -160
with distortion reappearing at both lower and higher drive NOISE FLOOR
levels.
There are various strategies to determine the optimum ratio -200
of optical and RF power splits to maximize the dynamic range. -160 -120 -80 -40 0 40
One strategy, first proposed and demonstrated by Johnson INPUT SIGNAL LEVEL (dBm)
and Rousell [I 0], was arrived at by expanding the distorted Fig. 2. Output RF signal power and third-order intermodulation power as a
function of the input signal power for a fiber-optic link, with the parameters
output signal of each modulator in a Fourier series including
in Table I. The dual-parallel modulator is arranged for the "optimum" split
the signal, odd harmonics, and intermodulation products. The so that the small-signal cubic intermodulation terms cancel, leaving a residual
coefficients in this well-known series are the products of Bessel intermodulation at 2w 1-w2 that varies as the fifth power of t~e input signal
level.
functions. If the input signal consists of equal amplitudes at
two frequencies w1 and w2, then the coefficient giving the
intermodulation at frequency 2w1-w2 contains the product TABLE I
of Bessel functions J (B)h(B), where the argument () is FIBER-OPTIC LINK COMMON PARAMETERS
1
proportional to the RF drive voltage. Johnson and Rousell then Laser Power Pi. 0.1 w
approximated this product with the first terms in the power
series expansions of J (B) and J (B), so that the coefficient is
1 2 Laser Noise RIN -165 dB
proportional to the RF voltage cubed. To cancel this coefficient
in the summed output of two modulators, they found that the
optical power split ratio should be the inverse cube of the Total Optical~ iv -10.0 dB
RF drive voltage split ratio. In their particular experiment, the
RF voltage split was fixed at 1 : 3, so that the optical power Modulator Sensitivity Vs orV11: 10 v
split was set to 27 : 1.2 Although this particular condition
cancels the cubic term 'in the Bessel function expansion, there Modulator Impedance RM 50 n
remain 5th, 7th, gth, · · ·power terms in the RF modulation.
Thus, the intermodulation at 2w1-w2 is not exactly canceled, Detector Responsivity TID 0.7 A/V{
but exhibits a roughly 5th power dependence on Pin· This is
illustrated in Fig. 2, which shows the intermodulation in a dual
MZM with the inverse cubic relation prescribed by Johnson Detector Load Ro 50 n
and Rousell. (The method of calculation and link parameters
used are discussed in detail in the link model section, and in Noise Bandwidth BW Hz
the Appendix.) The resulting dynamic range is 126.2 dB for
this particular link, which has its component parameters given Combination 11.ivTtD 7 mA
in Table I. An RF voltage split of 2.62 rather than 3 was used
as discussed later.
Alternatively, the intermodulation distortion may be exactly than 5, while the ultimate slope to the left of the auxiliary
canceled using a slightly different optical or RF splitting ratio, maximum is 3. Note that it is now possible for the IMD
but only for a single power level, as illustrated by the null in curve to have three intersections with the noise level line.
Fig. 3. Slight adjustments of the splits move the exact position We must specify which intersection to use to define "dynamic
of the zero. The slope just to the right of the zero is steeper range." There will be no ambiguity if we define the spurious
free dynamic range as that distance in dB from the signal
1A ilcrna1ely, a 90° polarization could be added to one output if a single to the intermodulation level where the intermodulation level
detector is desired or the two modulators could be driven by two independent
equals the noise level at the smallest input level. With this
lasers with the receiver, comprised of a single detector.
2 Johnson and Rousell's "dual MZM" was actually a single MZM on x definition, we see that the dynamic range will now depend
cut LiNbO:i with the light polarized before entering the modulator such that discontinuously on the noise level. The maximum dynamic
27 times as much optical power was in the TM polarization as in the TE range occurs when the auxiliary maximum to the left of the
polarization. A single set of electrodes modulate both optical polarizations,
minimum is just below the noise level, and the dynamic range
but the TE state is three times as sensitive to the drive voltage, as fixed by
the electrooptic properties of lithium niobate. will drop discontinuously when that maximum increases above
2186 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 43, NO. 9, SEPTEMBER 1995
"MAXIMUM DYNAMIC RANGE"
SPLIT
-40
'E
m
:!:!.
..J
>w -80
w
..J
..J
<z(
£! -120
(/)
:~::>
c..
i~5
-160
NOISE FLOOR
1 2
v,,;v v,,;vrt
5 or
-200
Fig. 4. Transfer curves of simple directional coupler and Mach-Zehnder
-160 -120 -80 -40 0 40 modulators from zero voltage to twice the switching voltage applied to the
INPUT SIGNAL LEVEL (dBm) electrodes.
Fig. 3. Same modulator as Fig. 2 but the splitting ratio is adjusled for
maximum dynamic range, which results in complete cancellation of the
large-signal 2w 1 -w2 intermodulation term at one particular signal level. the transfer voltage (Vs), and is analogous to the half-wave
voltage of the MZM. Fig. 4 shows the theoretical modulation
transfer functions for a directional coupler modulator (DCM);
the noise level. The maximum dynamic range of this link is
there are two complementary transfer functions YsR(V) and
now 129.7 dB, compared to 126.2 dB for the "cubic" condition
Yss(V) since the DCM has two output channels for an input
in Fig. 2. One important consequence of the more complicated
behavior of the IMD and harmonics is that we must now treat into one arm. The MZM transfer curve YM z (V) with a half
wave voltage V7r equal to the DCM transfer voltage Vs is also
the whole photonic link rather than analyze just the modulator
shown for comparison. The two modulator transfer curves are
to determine the dynamic range, since the dynamic range
very much alike from zero up to one switching voltage, but
depends on the relationship of the noise level to the kinks and
beyond that they depart; the MZM is periodic in 2Vr., while
bends in the harmonic and IMD curves. The best adjustment
increasing t:,,(3 further spoils the transfer from one arm back to
of the modulator parameters will depend on the actual values
the other. The mathematical form of the DCM transfer function
of the other link parameters.
[12] is
There is an additional degree of freedom in the true dual
MZM. The condition discussed by Johnson and Rousell spec
ifies the ratio of optical split in terms of the RF split to cancel
the cubic contribution to the intermodulation. But the RF split
ratio can be specified independently if a true dual MZM is (I)
used as in Fig. I instead of the two polarization states of a
single modulator, where the equivalent voltage ratio is fixed
at 3. The true optimum in the voltage ratio is about 2.62, but
The transfer voltage Vs is defined by
only one dB in dynamic range is sacrificed in the example
given in Fig. 2 if the ratio is 1.8 or 4.8. However, as shown
Vs 4r;,2g2,\2
later, the dynamic range is very rapidly degraded if the voltage (2)
3 7r2(2n~r2
and optical power are not near the inverse cube relation.
where l is the length of the coupling region and "' is the
III. LINEARIZED DIRECTIONAL COUPLER MODULATORS = =
coupling constant. When V 0 and d 7r /2, the signal is
Integrated-optic directional couplers made on electrooptic transferred completely from one guide to the other. The other
substrates can also be used as optical modulators [ l I]. If variables in (2) are n the optical index of refraction for the
0
the guides are physically identical, then complete transfer of guide, r the relevant electrooptic coefficient, g the electrode
the optical input from guide I to guide 2 is possible in one gap spacing, ( the overlap integral between the optical and
coupling length, which is determined by the optical waveguide electrical fields, and ,\ the free space optical wavelength. Vs
dimensions and refractive indices of the guide and substrate. is usually determined experimentally. Unfortunately, a Fourier
Modulating electrodes are applied to the two waveguide chan series for the output from a modulator with this transfer
nels so that the propagation constants of the guides are changed function is not available in closed from. One must use a power
incrementally in opposite directions when a voltage is applied. series expansion, as in [3], or input the transfer function with
The differential change in the propagation constants, t:,,(3, a two-tone time variation and find the Fourier components
depends upon the electrode configuration and the electrooptic numerically-as in [4] and the present work.
coefficient of the modulator material. By applying sufficient The intermodulation distortion produced by a simple DCM
voltage, the optical signal may be transferred from guide 2 is usually very much like that of an MZM driven to produce
back to guide I. The voltage required to do this is termed the same modulation percentage, as pointed out by Halemane
BRIDGES AND SCHAFFNER: DISTORTION IN LINEARIZED ELECTROOPTIC MODULATORS 2187
·.1~ \:Z:J
INPUT
0.2J = v : 3
MODULATOR SECTION TWO PASSIVE SECTIONS
LENGTH eMOO RADIANS LENGTHS eA, 99 RADIANS
Fig. 5. Linearized directional coupler modulator with a modulator section
followed by two biased passive sections. The angle B is shorthand for ,.,/.
and Korotky [12). However, there are subtle differences. For
example, biasing to the zero second-harmonic point does not ···~ ~
eliminate higher-order even harmonics. More interesting, a
zero in the third derivative curve, which is primarily responsi
ble for both third harmonic and 2w1 -w2 IM D, occurs where Vp
the signal is not zero, at about 0.7954 Vs. This is unlike the
~ FJ=:;:J
MZM, where zeros in all odd derivatives occur at the same
value of Vs /2. We shall return to this point later. Vs 0.6
Attempts to linearize the transfer function given in (I) by
adding elements to a basic DCM have been made by several ..7 J JS J
workers [3)-[5]. Farwell et al. [4] have analyzed and built the
configuration illustrated in Fig. 5, a directional coupler that
has three sets of electrodes. The first set is used to apply the
modulation signal plus a de bias voltage. The second and third
(passive) electrodes have only de bias voltages applied. The
two "extra" degrees of freedom introduced by these sections · · · l J:S:J
are used to linearize the modulation transfer function.
Before treating the modulator with three electrodes, it is
instructive to look at a simpler modulator, namely a DCM with
only one extra set of bias electrodes as described by Lam and
Tangonan [3). The reader may think of this as the modulator -2 0 +2
= = = =
of Fig. 5= wi th VA Va =Vp and ()p ()A+ Ba[BA V~5
,,,1A, Ba da and thus ()p ,,,(zA +la)]. We can illustrate
Fig. 6. Evolution of the transfer function of a directional coupler modulator
the development of a "more linear" region by plotting the with a passive bias section as the normalized voltage V p /Vs is increased
transmission Yss versus the voltage on the first section with from 0 to 0.8. Note the "linearized" region on the 0. 7 curve.
the normalized voltage on the second section Vp /Vs as a
parameter. The result is shown in Fig. 6 for the particular very little change occurs above that voltage. Jn the limit of
case where both the modulator section and the biased sections very high voltage applied to the second section, 6.(3 becomes
are electrically 7r /2 radians long: that is, () M = () p = 7r /2. so large that there is little coupling between the two guides,
The figures give the modulation transfer curves for - 2 < and the second section effectively becomes two independent
VM /Vs < 2, or a range of four transfer voltages. Thus, with guides (with equal and opposite phase shifts that still depend
zero voltage applied to all sections the optical input on branch on the applied voltage).
1 is completely transferred to branch 2 in () M and then back to It is interesting to look at the shape of the derivatives of
branch 1 in ()A +Ba. If VM /Vs = 1 is applied to the modulator the modulation transfer function as the bias on the second
=
section with Vp/Vs 0, the transfer is complete from branch section is varied. Fig. 7 repeats the transfer function from
1 to branch 2. With Vp /Vs = 0, we would bias the modulator 0 < Vi\!l /Vs < 1 and adds the first three derivatives with
section to VM /Vs = 0.4394 to obtain the minimum second Vp/Vs = 0. The first derivative produces most of the signal,
harmonic output. We note that with Vp/Vs = 0.7 applied the second derivative produces most of the second harmonic,
to the second section, the region about the modulator bias and the third derivative produces most of the third harmonic
point VM /Vs ::::: 0.5 begins to look much more linear. As the and the 2w1-w2 intermodulation (and a very small amount of
voltage is increased further, Vp /Vs = 0.8, this added linearity signal), etc. Clearly, biasing for a zero in the second derivative
disappears, and at Vp/Vs = 1, the transfer curve is identical to will nearly maximize the third derivative, an undesirable
Vp /Vs = 0, but it is inverted. Further increase in the voltage situation. What we really wish to do to is make the second and
applied to the second section continues to change the shape of third derivatives simultaneously zero, and this can be realized
the transfer curves but never yields such an improvement in if Vp /Vs is changed to 0.73193; the resulting transfer function
linearity over Vp/Vs ::::: 0. At Vp/Vs = JS, the modulation and its derivatives are shown in Fig. 8. This condition is near
transfer curve is exactly the same as that at zero voltage, and the "0.7" curve in Fig. 6. By making the second derivative
2188 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 43, NO. 9, SEPTEMBER 1995
y:~
YJ ~·I
J
Y' _: 1~ :~1 J
Y'
1
r'_::t:?~~:: ~ ~L
/ !
r·_:1
I
<: L 7 4
Y"'1:~/j
-10
0 0 1
Vrl'/s Vrl'/s
Fig. 7. The transfer curve and its first three derivatives for a directional cou Fig. 8. Same modulator as Fig. 7, but biased to. Vp /Vs 0.73193 to
pler modulator of electrical length() M = r. /2 followed by an identical passive simultaneously zero the second and third derivatives.
= =
section of length () p 7r /2, with normalized bias voltage Vp /Vs 0.0.
The proper bias for minimum second harmonic, V,\l /Vs = 0.4394 is
shown by the arrow; the star indicates a 'possible bias that would make the·
intermodulation distortion zero, but would result in a large second harmonic.
just touch the to zero line at its maximum, we make both
second and third derivatives zero simultaneously, assuring that
the second harmonic, third harmonic, and 2w1-w2 outputs are
nearly minimized. There will be small remainders at these
frequencies produced by the nonzero higher derivatives, which
may be canceled by a slight adjustment of the second bias
voltage away from 0.73193 Vs at a single value of modulation
drive voltage, just as in the dual MZM previously discussed.
We can apply this same strategy to the three section modu
lator shown in Fig. 5 in order to find optimum values of VA
and V3. Fig. 9 shows the transfer function and its first three
derivatives for the particular case that (}MOD = 7r /2, fJ A =
BB = 7r/4, VA/Vs = 0.73805 and V8 /Vs = 0.77002. For
these values (found by trial and error), second, third, and fourth
=
derivatives are all zero at a modulator bias of VM /Vs 0.509.
Thus, the fourth harmonic will be greatly reduced, the second
harmonic will be reduced somewhat from the case of the two
section modulator, and the third harmonic and the 2w -w
1 2
intermodulation will be of the same order.
It is tempting to speculate that adding further biased sections
Fig. 9. Transfer function and first three derivatives for the directional coupler
will add still more degrees of freedom that could be used modulator of length () M = 7r /2 followed by two passive sections of lengths
to set additional derivatives to zero and improve the 2w -w ()A = 7r/4,()s = 7r/4 as shown in Fig. 5. The biases VA and Vs shown
1 2
were found by trial and error to the maximum dynamic range. The optimum
intermodulation. In a study by Sheehy [ 19] it appears that the modulator bias is VM /V · = 0.509.
fifth derivative may be set to zero, not by adding an additional
section, but by moving the second biased section to precede
IV. LINK MODEL
the modulator, and adding phase-shifting lengths between the
modulator section and the biased sections. Sheehy also shows We now introduce a model for a complete optical link illus
that adding further biased electrodes or phase shift sections to trated in Fig. LO, containing a laser source with power PL [W],
the DCM can do no better than this. and a relative intensity noise RIN [dB/Hz]. The laser feeds a
