2184
IEEb
TKANSACTIONS
ON
MICKOWAVb
rHEORY
AND
TECHNIQUES,
VOL
33.
NO
9.
SEPTEMBER
1995
Distortion in Linearized Electrooptic Modulators
William
B.
Bridges,
Fellow,
IEEE,
and
James
H.
Schaffner,
Member,
IEEE
Abstruct-
Intermodulation
and
harmonic distortion
are
cal-
culated
for
a simple fiber-optic
link
with
a
representative
set
of
link
parameters and a
variety
of
electrooptic modulators:
simple
Mach-Zehnder,
linearized dual
and
triple
Mach-Zehnder,
simple directional coupler
(two
operating
points),
and linearized
directional coupler with one and two
dc
electrodes.
The
resulting
dynamic ranges, gains, and noise figures
are
compared
for
these
modulators.
A
new
definition
of
dynamic range is proposed
to
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.
I.
INTRODUCTION
LECTROOPTIC
modulators, both discrete interference
E
types such as the
Mach-Zehnder
modulator and
dis-
tributed
interference types such
as
the directional-coupler
modulator, have inherently nonlinear transfer curves.
As
a
consequence,
they may
limit the
dynamic
range
of
the photonic
link
in
which
they
are embedded
by
generating harmonics and
intermodulation products.
Various
modulator configurations
have
been
proposed and demonstrated in the last several
years
[1]-[8]
to address this problem and increase the
link
dynamic
range.
All
of
these schemes depend on generating
two
or
more
modulation samples
with
different ratios
of
signal
to distortion and
then
combining the samples
so
that
the
distortions
cancel
(to some order)
while the signals
do
not
cancel.
In
some cases
it is
easy
to
identify
where the
two
modulations occur and where the combinations take place, as
in
the dual
Mach-Zehnder
schemes
[l],
121,
161;
in
others it
is
not
so
obvious, such
as
the directional-coupler modulator
and its variations
[31-[51.
The
various linearized modulator schemes predict, and
in
some
cases have demonstrated
[I],
[4]-[7],
significant reduc-
tion in
harmonics and intermodulation products, which should
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,
so
that
these parameters
will likely
have to
be
controlled
by
active
means,
especially
if the distortion cancellation
is
to
be
maintained over a
large
operating bandwidth.
In
addition, the
dependence
of
the harmonic
or
intermodulation product on
the signal drive
level
is
no
longer a simple constant exponent
Manuscript received
January
9. 1995;
revised May
5, 1995.
This
work
was
supported
in
part
by
Contract
no.
F30602-9
I
-C-0
104
to Hughes
Research
Laboratories
from
the
IJS
Air
Force,
Rome
Laboratories
(N.
P.
Bemstein
technical
monitor)
and
hy
the
ARPA
Technology
Reinvestment
Project on
Analog Optoelectronic
Modules,
Agreemcnt
No.
MDA972-94-3-0016.
W.
6.
Bridges
is
with the
California
Institute of Technology, Pasadena,
CA
91
12.5
USA.
J.
H.
Schaffnor
is
with
Hughes
Research Laboratories,
Malihu, CA
90265
USA.
IEEE
Log
Number
9413707.
ps
OPTICAL
MACH-ZEHNDER
WAVEGUIDE INTERFEROMETERS
A
OPTICAL DIRECTIONAL
COUPLER
Fig.
I.
Dual-parallel modulator configured with
equal
length
electrodes
and
one
input optical
signal. This
particular approach requires
two photodiodes
at
the
optical receiver.
An
alternative
approach
would
use
two
lasers and then
combine the
optical signals
at
the modulators’
outputs into
one
detector.
(e.g.,
a slope
3
line on the
dBOut
versus
dB,,
graph
for
third-
order intermodulation), and the photonic link dynamic range
no
longer depends
on
the noise level
in
a simple
way; a clearer
definition
of
“dynamic range”
is really
required. Finally, the
improved modulator dynamic range can easily
be
eroded
by
the nonlinear behavior
of
the electronic amplifiers required
by
the photonic
link
to realize reasonable gain and noise Fig.
[9].
This
paper uses
a simple photonic link
model
to
find
the
gain,
noise
figure, harmonics, intermodulation, and
dynamic
range
for
a number
of
the modulator schemes
listed
above,
and
it uses the model to optimize the modulator parameters.
The
sensitivity
of
representative
Mach-Zehnder
modulator
(MZM)
and directional coupler modulator
(DCM)
schemes
to modulator and
link
parameters are calculated and
com-
pared.
A
refined
definition
of
“dynamic range”
is
proposed
to eliminate possible ambiguities
resulting
from the definition
based
on
simple
slopes. Finally, the results
of
adding electronic
amplifiers
to
the photonic
link
are
calculated.
11.
DUAL
MACH-ZEHNDER
MODULATORS
The
Mach-Zehnder
modulator
is
a simple two-channel
interference device, resulting
in
a sine-squared dependence
of
light output on drive voltage.
The
modulator
is
biased
to
the
most linear
portion
of
the transfer
curve,
which for a perfect
modulator
also
assures no even-harmonic generation.
However, the
nonlinearity
of
the
transfer
curve is
respon-
sible for the generation
of all
odd-harmonics and
all
possible
intermodulation products.
The
dual
MZM
scheme uses
two
MZM’s,
driven at different
RF
levels and
fed
with different op-
tical
powers,
as
illustrated in Fig.
I.
The
RF
and optical power
splitting ratios are chosen
so
that
the modulator receiving the
larger optical power receives the smaller
RF
drive power.
This
modulator
may be
thought
of
as
the “main” modulator, with
OOl8-9480/95$04.(x)
0
1995
IEEE
BRIDGES
AND
SCHAFFNEK.
DISTORTION
IN
LINEARIZED
FLF.(’TROOPTIC
MODULATORS
2185
some distortion created
by
the
finite
RF
drive
power.
The
other
modulator receives only
a
little optical power,
but is
driven
relatively much harder, thus yielding
a
much
more distorted
signal.
The
two
optical
outputs are combined incoherently,
for
example,
by
combining the electrical outputs
of
two separate
detectors
as
shown
in
Fig.
1.’
If
the
bias
points
of
the two
modulators are chosen
so
that the modulations are
out
of
phase,
and the ratios
of
both
optical
and
RF
powers are properly
chosen,
then
the sum
of
the
two
distortions
(If.41
j can exactly
cancel, while the signals
(Ps)
do
not
completely cancel. This
exact cancellation can only occur for a
specific
drive level,
with distortion reappearing
at
both lower
and
higher drive
levels.
There
are various strategies to determine the optimum
ratio
of
optical
and
RF
power
splits to maximize the dynamic
range.
One
strategy,
first
proposed and demonstrated
by
Johnson
and
Rousell
[lo],
was
arrived at
by
expanding the distorted
output signal
of
each modulator
in
a
Fourier series including
the signal, odd harmonics,
and
intermodulation products.
The
coefficients
in
this well-known series are the products
of
Bessel
functions.
If
the input signal consists
of
equal amplitudes
at
two frequencies
w1
and
Q,
then
the coefficient giving the
intermodulation
a1
frequency
2wl-wz
contains the product
of
Bessel functions
J1(H)J2(H),
where the argument
H
is
proportional
to
the RF
drive voltage.
Johnson
and
Rousell
then
approximated
this
product
with
the
first
terms
in
the
power
series expansions
of
Jl(0)
and
J2(0),
so
that the
coefficient is
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
RF
drive voltage split ratio.
In
their particular experiment, the
RF
voltage
split was
fixed
at
1
:
3,
so
that
the optical power
split was
set to
27
:
1.’
Although
this
particular condition
cancels the cubic term
in
the Bessel function expansion, there
remain
5th3
7th,
911‘,
. .
.power
terms
in
the
RF
modulation.
Thus, the intermodulation
at
~W~--LJ~
is not
exactly canceled,
but exhibits
a
roughly
StiL
power dependence
on
Pi7>.
This is
illustrated
in
Fig.
2,
which
shows the intermodulation
in
a dual
MZM
with
the inverse cubic
relation
prescribed
by
Johnson
and Rousell.
(The
method
of
calculation and
link
parameters
used
are discussed
in
detail
in
the
link model
section, and
in
the Appendix.)
The
resulting dynamic range
is
126.2
dB
for
this
particular
link. which has
its component parameters
given
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
canceled
using
a
slightly
different optical
or
RF
splitting ratio,
but
only
for
a
single power level,
as
illustrated
by
the null
in
Fig.
3.
Slight adjustments
of
the
splits
move the
exact
position
of
the
zero.
The
slope
just
to
the right
of
the zero
is
steeper
’
Alternately,
a
YO“
polarization could be added
to
one
output
if a single
detector
is desired
or
the two
modulators
could
be
driven
by
two
independent
lasers with
the
receiver, comprised
of
a single
detector.
’Johnson and
Rourell’s
“dual MZM” was actually
a single
MZM
on
x-
cut
LiNb0:j
with the light polarized before
entering the
modulator such that
27
times
as
much opiical power
was
in
the
TM
polari7ation
as
in
the
TE
polarization.
A
single
set
of
electrodes
modulate both optical polarizations.
but the
TE
state is thrcc
times
as
sensitive to the
drive
voltage,
as
fixed
by
the
clcctruoptic
propcr-tics
of
lithium
niobate.
0
I
I
I
I
‘CUBIC’
SPLIT
E-
40
A
Y
5
>
-80
8
-120
c
0
-160
-200
y
I
I
I
I
-160
-120
-BO
-40
0
40
INPUT
SIGNAL LEVEL (dBm)
Fig.
2.
Output
RF
+pal
power and third-order intermodulation power
as
a
function
of
the input 3ignal power
for
a fiber-optic link, with
the
parameters
in Table
1.
The
dual-parallel modulator
is
arranged
for the
“optimum” split
so
that the rmall-signal
cubic
intermodulation terms
cancel, leaving a
residual
intermodul:ition
at
L1-d~
that varies
as
the fifth
power
of
the
input signal
level.
TABLE
I
FIBER-OPTIC
LINK
COMMON
PARAMETERS
Laser
Power
PL
0.1
W
Laser
Noise
m
-165
dB
Total
Optical
Loss
r,
-10.0
dB
Modulatorhpedance
RM
50
n
DetectorRespansivity
slD
0.7
AIW
DekctorLoad
RD
50
n
Noise
Bandwidth
BW
1
HZ
than
5,
while the
ultimate
slope
to
the
left
of
the auxiliary
maximum is
3.
Note that it is now possible
for
the
IMD
curve to have
three
intersections with the
noise
level line.
We
must
specify
which
intersection to use
to
define “dynamic
range.”
There
will
be no ambiguity
if we define the spurious-
free dynamic range as that distance
in
dB
from
the signal
to
the intermodulation
level
where the intermodulation level
equals
the
noise
level
NI
the
smallest
input
level.
With
this
definition,
we
see that the dynamic range
wit1
now
depend
discontinuously on the
noise level.
The
maximum
dynamic
range occurs
when
the auxiliary maximum to the left
of
the
minimum is
just
below
the
noise level,
and
the
dynamic range
will
drop
discontinuously
when
that maximum increases
above
2186
IEEE
TRANSACTIONS
ON
MICROWAVE
THEORY
AND
TECHNIQUES,
VOL.
43,
NO.
9,
SEITEMBEK
1995
0
I
I
I
I
“MAXIMUM DYNAMIC RANGE’
-40
I
E
s
-I
w
-I
z
g
-80
E
.120
5
2
-160
-200
-160
-120
-80
-40
0
40
INPUT
SIGNAL
LEVEL
(dBm)
Fig.
3.
Same
modulator
as
Fig.
2
but
the splitting ratio
is
adjusted
for
maximum dynamic range, which results
in
complete cancellation
of
the
large-signal
261
“‘2
interinodulation term at
one
particular signal level.
the noise level.
The
maximum
dynamic
range
of
this
link
is
now
129.7
dB,
compared to
126.2
dB for
the “cubic” condition
in
Fig.
2.
One important consequence
of
the
more complicated
behavior
of
the
IMD
and
harmonics
is
that
we must now treat
the whole photonic
link
rather than
analyze just
the modulator
to
detennine
the dynamic range,
since
the dynamic
range
depends
on
the relationship
of
the noise
level
to the
kinks
and
bends
in
the harmonic and
IMD curves.
The
best
adjustment
of
the modulator parameters
will
depend
on
the
actual
values
of
the other
link
parameters.
There
is an
additional degree
of
freedom
in
the
true
dual
MZM.
The
condition discussed
by
Johnson
and
Rouse11
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
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
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
later, the
dynamic
range is very
rapidly degraded
if
the voltage
and
optical power are
not
near the inverse cube relation.
111.
LINEARIZED
DIRECTIONAL COUPLER
MODULATORS
Integrated-optic directional couplers
made
on electrooptic
substrates can also be
used
as optical modulators
[I
I].
If
the
guides are
physically
identical, then complete transfer
of
the optical input from guide
1
to
guide
2
is
possible
in
one
coupling length,
which
is dctermined
by
the optical waveguide
dimensions
and
refractive indices
of
the guide
and
substrate.
Modulating electrodes are applied
to
the two waveguide
chan-
nels
so
that the
propagation
constants
of
the guides are changed
incrementally
in
opposite directions when a voltage
is applied.
The
differential change
in
the propagation constants,
A@,
depends
upon
the electrode configuration
and the
electrooptic
coefficient
of‘
the
modulator material.
By
applying
sufficient
voltage, the optical signal may be transferred from guide
2
back
to
guide
1.
The
voltage
required to
do
this
is
termed
1
.o
z
0
u)
2
I
0.5
u)
z
+
d
0
0
1
2
VflS
or
V&,
Fig.
4.
Transfer curves
of
simple directional coupler and Mach-Zehnder
modulators from
zero
voltage to twice the switching voltage applied to the
electrodes.
the
transjer
voltage
(V?),
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);
there are two complementary transfer functions
YsR(V)
and
Yss(1’)
since the DCM has two output channels
for
an
input
into
one arm.
The MZM
transfer curve
Yh{z(V)
with a
half
wave
voltage
V,
equal
to the DCM transfer voltage
V,
is also
shown for comparison.
The
two modulator transfer curves are
very much
alike
from
zero
up
to one switching voltage, but
beyond that they
depart; the
MZM
is periodic
in
2V,,
while
increasing
A/j
further spoils the transfer from
one
arm
back to
the other.
The
mathematical form
of
the
DCM
transfer function
[I21
is
The
transfer voltage
Vs
is defined
by
where
1
is the length
of
the
coupling region and
K
is the
coupling constant. When
=
0
and
61
=
~/2,
the signal
is
transferred completely
from
one guide to the other.
The
other
variables
in
(2)
are
7b0
the optical index
of
refraction
for
the
guide,
r
the
relevant electrooptic coefficient,
g
the electrode
gap
spacing.
C
the overlap integral between the optical
and
electrical
fields,
and
X
the free space optical wavelength.
Vs
is usually
determined experimentally. Unfortunately, a Fourier
seriej
for
the
output from a modulator
with this
transfer
function
ic not
available
in
closed from.
One
must
use a power
series expansion, as
in
[3],
or
input
the transfer function
with
a two-tone time
variation
and
find
the Fourier components
numerically-as
in
[4]
and the present work.
The
intermodulation distortion produced
by
a simple
DCM
is
usually very much
like
that
of
an
MZM
driven
to
produce
the same
rnodulation
percentage,
as
pointed
out
by
Halemane
BRIDGES
AND
SC‘HAFFNER.
DISTORTION
IN LINEARIZED
ELECTROOPTIC
MODULATORS
OUTPUT
2
--
MODULATOR
SECTION
TWO
PASSIVE SECTIONS
LENGTH
e,,,
RADIANS
LENGTHS
e*.
eB
RADIANS
Fig.
5.
followed
by
two
biased passive sections.
The
angle
0
is
shorthand
for
~l.
Linearized directional
coupler
modulator
with
a
modulator section
and
Korotky
1121.
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-u2
IMD,
occurs where
the signal
is
not
zero, at about
0.7954
Vs.
This
is
unlike the
MZM. where zeros
in
all
odd
derivatives
occur
at
the
same
value
of
1~>/2.
We
shall return
to
this
point
later.
Attempts to linearize the transfer function given
in
(I)
by
adding elements to a
basic
DCM
have
been
made
by
several
workers
[3]-[5].
Farwell
et
a/.
[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
dc
bias
voltage.
The
second and
third
(passive) electrodes have only
dc
bias voltages applied.
The
two “extra” degrees
of
freedom introduced
by
these sections
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
of Fig.
5
with
v~
=
V,
E
I/r
and
HP
d.4,
OB
z
td~
and thus
fjp
G
PG(~A
+
l~)].
We
can
illustrate
the development
of
a
“more linear” region
by
plotting the
transmission
YSS
versus
the voltage
on
the first
section
with
the normalized
voltage on
the
second
section
Vp/Vs
as
a
parameter.
The
result
is
shown
in
Fig.
6
for
the particular
case where
both
the modulator section
and.
the
biased
sections
are electrically
T/Z
radians
long: that
is,
Bitf
=
6’p
=
n/2.
The
figures
give
the modulation transfer curves
for
-2
<
Vl~~/~~
<
2)
or
a
range
of
four
transfer voltages.
Thus,
with
zero voltage applied
to
all
sections the optical input on
branch
1
is completely transferred to
branch
2
in
6’~
and
then
back
to
branch
1
in
0-4
+OB.
If
VI\I/V~
=
1
is applied
to
the modulator
section with
Vp/l/:s
=
0.
the transfer
is
complete from branch
1
to branch
2.
With
c;P/I/s
=
0,
we
would bias
the modulator
section
to
~~,~/~c;
=
O.zL394
to
obtain the minimum second
harmonic output.
We
note that with
Vp/Ci
=
0.7
applied
to
the second section, the region about the modulator bias
point
C:lf/V,
!z
0.5
begins
to
look
much more linear.
As
the
voltage
is
increased further,
Vp/&
=
0.8,
this added linearity
disappears. and at
Vp/&
=
1,
the transfer curve
is
identical
to
Vp/CS
=
0.
but
it
is
inverted. Further increase
in
the voltage
applied to the second section continues
to
change
the
shape
of
the transfer curves but never yields
such
an
improvement
in
linearity
over
Vp/Vs
z
0.
At
Vp/Vs
=
&.
the
modulation
transfer
curve
is
exactly the same as that at
zero
voltage, and
0.4
+
HD[tl.A
0
0
2187
V-TTJOI
0.4
]
v,
0‘
I
1
0’
I
0.8
’
0
-2
0
+2
“MNS
Fig,
6.
Evolution
of
the
transfer function
of
a directional
coupler
modulator
with
a passive
bias
section
as
the
normalized voltage
Vp/l.s
is
increased
from
0
to
0.8.
Note
the
“linearized” region on the
0.7
curve.
very
little
change
occurs above that voltage.
In
the limit
of
very high voltage applied to the second section,
AB
becomes
so
large that
there
is
little coupling
between
the two
guides,
and
the
second
section effectively becomes
two
independent
guides
(with
equal
and
opposite
phase
shifts that still depend
on
the applied voltage).
It
is
interesting to
look
at the shape
of
the
derivatives
of
the modulation transfer function
as
the
bias
on the second
section
is
varied. Fig.
7
repeats the transfer function
from
0
<
V;$[/Vs
<
1
and adds the
first
three derivatives with
Vp/V,
=
0.
The
first
derivative produces most
of
the signal,
the second derivative produces most
of
the second harmonic,
and the third derivative produces most
of
the
third
harmonic
and the
Ewl-t4
intermodulation (and
a
very
small amount
of
signal). etc. Clearly, biasing
for
a zero
in
the second
derivative
will nearly
maximize the third derivative,
an
undesirable
situation. What
we really wish
to
do
to
is make
the second and
third
derivatives
simultanecrusly
zero,
and
this
can
be realized
if
Vp/1’5
is changed to
0.73193;
the resulting transfer function
and its derivatives are shown
in Fig.
8.
This condition
is near
the
“0.7”
curve
in
Fig.
6.
By
making the second derivative