Bandwidth of Linearized Electrooptic Modulators

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JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 16, NO. 8, AUGUST 1998

Uri V. Cummings and William B. Bridges,

Life Fellow, IEEE, Fellow, OSA

Abstract—

Many schemes have been proposed to make high

dynamic range analog radio frequency (RF) photonic links by

linearizing the transfer function of the link’s modulator. This

paper studies the degrading effects of finite transit time and

optical and electrical velocity dispersion on such linearization

schemes. It further demonstrates that much of the lost dynamic

range in some modulators may be regained by segmenting and

rephasing the RF transmission line.

Index Terms—

Bandwidth, electrooptic, linearized, modulators,

photonic-link.

I. I

NTRODUCTION

LECTROOPTIC intensity modulators have inherently

nonlinear transfer functions which may limit the dy-

namic range of the photonic link through the production

of harmonic and intermodulation distortion. Many schemes

have been proposed to reduce the distortion byproducts of

these modulators by linearizing their transfer functions; for

example see the review paper by Bridges and Schaffner [1],

and the references therein. The proposed applications for the

resulting high dynamic range links include antenna remoting,

photonic-coupled phased-array antennas, and cable television

transmission.

Linearizing a modulator is a challenge. All of the proposed

linearization schemes involve the cancellation of selected

distortion terms, and this cancellation depends critically on

modulator device parameters. Electrical biases very likely

will require active control, and in some modulators, radio

frequency (RF) or optical levels will require more accuracy

than is realizable with current fabrication techniques.

For applications higher than 1 GHz, traveling wave elec-

trode structures are mandatory to overcome the limitations

resulting from interelectrode capacitance and finite transit time.

A further difficulty with some popular electrooptic materials,

such as lithium niobate, is that the electrical and optical waves

travel at different velocities over the finite interaction length of

the device, a result of material dispersion. This property limits

the modulation-index

voltage product at high frequencies.

Given the critical dependence of the linearization scheme

on modulator parameters, it is a fair question to ask, “How

will velocity mismatch effect the linearization results?” This

paper addresses that question for several popular modulator

types. The summary result is that good velocity matching

is essential to successfully linearize some but not all of the

Manuscript received October 24, 1997; revised April 21, 1998. This work

was supported in part by the United States Air Force Rome Laboratories under

Contract F30602-C-96-0020.

The authors are with the California Institute of Technology, Pasadena, CA

91125 USA.

Publisher Item Identifier S 0733-8724(98)05646-1.

TABLE I

ARAMETERS OF A

ANONICAL

PTICAL

INK

modulators. The details differ significantly from one modulator

type to another. This paper treats the frequency dependence

of six modulator configurations: a standard Mach–Zehnder

modulator (MZM), a dual parallel Mach–Zehnder modula-

tor (DPMZM) and a dual series Mach–Zehnder modulator

(DSMZM), a simple directional coupler modulator (DCM) at

two different bias points, and a directional coupler modulator

linearized with two additional dc biased directional couplers

in series optically (DCM2P). The results of linearizing these

modulators (except for the DSMZM)

without

regard for transit

time were reported in [1]. Now, we report the comparisons

including transit time and velocity mismatch.

II. L

INK

ODEL

This paper assumes the same simplified photonic link and

parameters as [1], but now adds the parameters for the finite

frequency calculations. They are the effective index mismatch

, the modulator length

, and the frequency

. The link

consists of a laser, an electrooptic modulator, a length of fiber,

and a detector. The model excludes electronic preamplification

and postamplification. Table I shows all of the parameters

associated with this link. These parameters are assumed to

have no frequency dependence in the model. The parameters

for the active length and the velocity mismatch are typical

for LiNbO

modulators using simple parallel strip electrodes

with no velocity matching. That is,

and

, and thus

. Velocity matching will

result in a lower value of

As in [2], the results are calculated numerically, since

no closed-form solution exists for the transfer function of

some modulator types. A program, written in C, calculates

the frequency-dependent gain and dynamic range. A two-tone

electrical test signal, with frequencies

and

drives the

modulator. The Fourier transform of the output is evaluated to

0733–8724/98$10.00



1998 IEEE

CUMMINGS AND BRIDGES: BANDWIDTH OF LINEARIZED ELECTROOPTIC MODULATORS

1483

find the gain, the harmonic content, and the intermodulation

content of the link.

Let

be the electrical signal power after the detector,

given the modulator RF drive power

. Let

be the

Fourier transform of

The gain is

Gain

(1)

The small signal gain is obtained by evaluating (1) at suffi-

ciently small

that the log-log plot of

is linear

with slope one. In practice, a good value of

for this

calculation is the geometric mean of

(the power that drives

the modulator voltage to about

) and the precision of double

precision floating point numbers, or about

dBm.

The spur free dynamic range, DR

, is the power interval

that spans the input power level at which the signal is just

distinguishable from the noise and the input power level at

which the strongest distortion term becomes distinguishable

from the noise. The calculation of DR

(2)

(3)

(4)

is the maximum of the relevant distortion terms

minus the noise level in dB. Of the roots of

is the root that occurs at the lowest power level. DR

the difference between

and the input power level at which

the signal intersects the noise floor. Since the log-log plot

of the signal has slope one, this interval is equivalent to

the signal power minus the noise power at the RF drive

power at which the distortion power equals the noise level

which is (4). Fig. 1 describes the dynamic range calculation

for a simple Mach–Zehnder modulator. While dynamic range

generally refers to all harmonics and intermodulation products,

in practice, there are two dominant distortion terms, the second

harmonic,

, and the first intermodulation product,

. The linearized modulators in this paper are

separated into two categories, each with a different definition

of dynamic range. Equation (2) applies to broadband, or

supe-

roctave

modulators. That is,

is the maximum of the

second harmonic and the intermodulation product.

is defined differently in narrow band or

suboctave

modulators.

In this second category,

equals only the third order

intermodulation product. In ordinary modulators, the log–log

plots of the distortion terms are linear intersecting the noise

floor only once. In linearized modulators, the distortion terms

are nulled at discrete power levels. They may cross the noise

level more than once. It is then necessary to find all of the

roots of the distortion minus the noise,

, and take the root

representing the lowest RF drive power in the definition of

dynamic range.

Fig. 1. The signal and third-order intermodulation product of a simple

Mach–Zehnder modulator. Since there is effectively no second-harmonic

distortion,

equals the intermodulation product.

crosses the noise

floor only once at an input power level of

dBm. Since the signal has

slope one, the dynamic range may be found from the difference between the

signal and

either horizontally or vertically in the plot, giving the familiar

dynamic range triangle.

A. Computation of Modulators with Velocity Dispersion

The frequency dependent output of any modulator with

a known dc transfer function is calculable. Farwell gives a

detailed computational method for this in [2]. The mathematics

are straightforward. Let

and

be the input

and output complex amplitudes of the optical wave in a

modulator. Let

be the transfer function, where

is the normalized modulator drive voltage and

is the active

length of the modulator. If the optical and electrical waves

travel at the same velocity, or if the operating frequency is

so low that

is effectively constant over

, then the output

is given by

(5)

Even if there is significant velocity dispersion while the two

waves travel the distance

, over a short enough section of the

guide

, the change in the complex optical amplitudes may

be described with the dc transfer function. This is the basis for

the frequency dependent calculation. Let

be the coordinate

along the optical waveguide, and let

be time. Then

(6)

The optical and electrical signals are now functions of two

variables,

and

. In (6) the elapsed time equals the incre-

mental length divided by the electrical velocity

Let there be

equally spaced increments of

and

equally spaced increments of

, and let

and

The modulator is divided into

sections over which the

optical and electrical fields are approximately constant. The

finite product of the resulting

unitary dc transfer matrices

gives the overall transfer function from

Let

, and

, and note

that

for

. The approximate modulator

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JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 16, NO. 8, AUGUST 1998

transfer function is

(7)

The function

representing the two tone test is

(8)

(9)

The parameter

depends on the operating frequency

, the

difference between the optical and electrical indices,

, the

active length

, and the velocity of light

. It is important to

note that the calculation results depend solely on

and not on

, and

independently. Thus the results of different

lengths or relative wave velocities at different frequencies

will be the same if

is the same. The curves shown below

are universal in the sense that they apply to more than the

“worst case” velocity mismatch, which is

for

lithium niobate modulators. Any change in

leads

to a rescaling of the frequency axis for the gain and dynamic

range plots.

Equation (7) is general and is the basis for the frequency

dependent computations in the model. However, when the

active region of a modulator consists of only simple phase

shifts, as it does for the Mach–Zehnder modulator, a further

simplification may be made. The transfer function is just

a diagonal matrix of exponentials. Instead of multiplying

exponentials, their arguments are summed. This is equivalent

to integrating the location variable out of the voltage function.

That is

(10)

(11)

The approximation introduced in (11) comes from the substi-

tution of a summation for an integral, the relationship between

and

shown in (10) is exact. It may seem odd to use

an approximation for a function for which a trivial analytic

solution exists [integral of (8)]. However, this is done to

mirror the calculation technique for directional couplers and

to support the modeling of modulator voltage functions which

may not have an analytic integral representation.

In the C-program mentioned above, the temporal incre-

ments,

are restricted to be powers of two, so that a radix-2

fast Fourier transform (FFT) algorithm may be used for the

spectral analysis.

Fig. 2 shows the convergence of the gain of

a Mach–Zehnder modulator as a function of spatial increments

at 5, 10, 20, and 40 GHz. The error is normalized, and

It is customary to use the FFT algorithm which is

(

log(

))

instead

of the DFT algorithm which is

(

2

)

. However, it should be noted that the

algorithm to compute the modulator output is

(

2

)

, so the time spent in

the FFT algorithm is inconsequential.

Fig. 2. The convergence of the calculation of the gain of a Mach–Zehnder

modulator (MZM), with an increasing number of modulator sections. The

error is the magnitude of the calculated gain (not in dB) minus the analytical

value normalized by the dc analytical value for the gain. The curves of the

log-log plot are for 5, 10, 20, and 40 GHz.

thus the

-axis value “one” corresponds to a 3 dB error and

“0.1” corresponds to a 0.4 dB error in the calculated gain.

The curves in Fig. 2 are linear until the error is six-to-seven

orders of magnitude below the dc value of the gain (since

the plot is log-log, constant slopes do not indicate geometric

convergence). The convergence saturates at a normalized error

of about 10

because the RF drive power for the small

signal gain calculation was arbitrarily chosen to be

100

dBm. Whether in the calculation for gain or dynamic range,

the numbers have components that differ by 6–7 orders of

magnitude. Since these components occupy the same mantissa,

there is a loss in accuracy not recovered by the floating

decimal point. Double precision numbers must be used to

attain a satisfactory accuracy. It is interesting to note that the

calculated points form a horizontal line across the curves in

the linear regime. This indicates that doubling

(by doubling

the frequency for instance) exactly requires a doubling of the

modulator sections to achieve the same error. Efficient code

allows the calculation hundreds of frequency points with a

128-point FFT and a comparable number of spatial increments

in seconds on a contemporary desk-top machine (120 MHz

Pentium processor).

III. M

ACH

–Z

EHNDER AND

IRECTIONAL

OUPLER

ODULATORS

The

most

common

electrooptic

modulator

the

Mach–Zehnder interferometer (MZM). It has a sine-squared

transfer function, and the gain is a sinc function of the

frequency-length-index product,

. When biased at the

half-wave voltage,

, it attains its maximum

linearity and dynamic range. All even-order harmonics are

identically zero. The intermodulation distortion product

solely determines the dynamic range, even in a super-octave

system. The dynamic range is independent of frequency;

the signal decays with frequency, but the intermodulation

product decays identically. Thus, the range of RF drive

powers (in dB), over which their are no spurs above the noise,

shifts with a change in frequency, but it does not expand or

CUMMINGS AND BRIDGES: BANDWIDTH OF LINEARIZED ELECTROOPTIC MODULATORS

1485

Fig. 3. Gain and dynamic range of a standard Mach–Zehnder modulator.

The pole and zero in the dynamic range are the frequencies at which the

signal and intermodulation products go to zero, respectively.

contract. Given its analytical simplicity and widespread use,

the Mach–Zehnder is used first to evaluate the accuracy of the

numerical calculation, and then it is used for a comparison

to the linearized modulators.

Fig. 3 shows the calculations for gain and dynamic range

of a simple MZM. The gain has the form

with zeros at multiples of 16.2 GHz (where

for the

canonical parameters from Table I), and a low frequency link

gain of

25.5 dB (also appropriate for the link parameters).

The dynamic range is flat except for a null and singularity

near the gain null. This is a simple numerical artifact, resulting

from the finite frequency difference between the two tones in

the driving function. The signal,

, and the intermodulation

product,

, are at slightly different frequencies, and

hence, they null at different frequencies. The dynamic range

goes to zero at the signal null and it goes to infinity at the

distortion null.

Fig. 4 shows the analogous calculations for a directional

coupler modulator. The low frequency gain is

24.8 dB, 0.7

dB better than Mach–Zehnder.

The first null of the gain of

the directional coupler occurs at 26 GHz,

, compared

to 16 GHz for the MZM. The first lobe of the gain curve

does not correspond to the sinc function of the Mach–Zehnder.

However, subsequent nulls are periodic with a 16 GHz period,

resulting from the underlying

of the directional coupler.

The frequency at which the gain has fallen by 3 dB is 40%

higher than that of the Mach–Zehnder modulator with the same

index-length product.

The dynamic range compares unfavorably to that of the

Mach–Zehnder. At low frequency it is similar to that of the

MZM, and it is approximately flat with frequency. However

there is a kink in the curve at 1.8 GHz (left vertical arrow),

after which the dynamic range decays rapidly with frequency.

Unlike the Mach–Zehnder, where all even-order derivatives of

the transfer function are identically zero when the modulator

The comparison of the gain bandwidth product between the MZM and

the DCM assumes that

V

=

V

; that is, the modulators have the same

normalization voltages. While these voltages should be similar in the same

manufacturing process, it is hard to directly compare them since the electrode

geometries and crystal orientations of the two modulator types may differ.

Fig. 4. Gain and dynamic range of a directional coupler modulator biased

at the 0.43

V

point. The two arrors indicate that the dynamic range changes

from intermodulation-limited to second-harmonic limited or vice versa. If the

second harmonic is ignored (suboctave operation), the dynamic range follows

the dashed curve.

is biased at

, in the directional coupler only the sec-

ond derivative is zero at its optimal bias,

. Distortion

migrates from nonzero fourth-, sixth-, eighth-, etc., order

derivatives to the frequency at which the second harmonic

occurs. It does so rapidly with increasing frequency, and it

equals the third order intermodulation product at 1.8 GHz.

At higher frequencies, distortion from these nonzero even

derivatives limits the dynamic range. The intermodulation-

limited dynamic range (as if this were a suboctave modulator)

is shown with a dotted line to demonstrate the intermingling of

the second harmonic and third order intermodulation product.

The bias voltage that nulls the total second harmonic, which

arises from all even order derivatives, is thus a function of

frequency in the DCM, but it is independent of frequency in

the MZM. At any finite frequency, there is a bias value which

nulls the total second harmonic, but the null will hold only

over a narrow bandwidth. While the gain compares favorably

to the MZM, the dynamic range makes the directional coupler

inferior to the Mach–Zehnder for superoctave, high-frequency

applications.

IV. B

ROAD

-B

AND

INEARIZED

ODULATORS

Dynamic range values are calculated for two linearized

broad-band, or superoctave, modulators: the dual parallel

Mach–Zehnder (DPMZM) and the directional coupler with

two passive sections (DCM2P) which are described in [4]

and [5], respectively. The DPMZM has two identical, single

Mach–Zehnder modulators in parallel optically and electrically

but with unequal levels of optical and RF power driving the

two modulators. Both modulators are biased at the

point,

but with opposite slopes, so the modulators are 180 degrees

out of phase. Most of the optical power and a small fraction

of the RF power drive one modulator. A small amount of

optical power and the majority of the RF power drive the

other modulator, creating relatively larger distortion products

than in the first modulator. The two signals are combined

incoherently in the photodetector. The RF and optical splits

are adjusted so that the distortion terms cancel exactly, but the

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Fig. 5. Dynamic range comparison of two linearized modulators: a dual par-

allel Mach–Zehnder modulator (DPMZM) and a directional coupler modulator

with two passive sections (DCM2P). As in Fig. 3 for the simple MZM the

dynamic range of the DPMZM is independent of frequency. However, the

dynamic range of the DCM2P drops precipitously with frequency.

signals do not. Since these two paths are in parallel, the effects

of velocity mismatch apply equally to each Mach–Zehnder.

Thus, the distortion terms still precisely cancel regardless of

frequency, and this is the result shown in Fig. 5. While the

DPMZM is robust to velocity mismatch, in practice it is hard

to make broadband. A precise RF split must be maintained

over the desired frequency band. If it varies, the dynamic range

will decrease at all but the narrow frequency at which the RF

split is optimized.

The linearized directional coupler with two passive sections

can be adjusted to provide high dynamic range with both inter-

modulation and second harmonic reduction in the absence of

velocity mismatch. However, it suffers severely from velocity

mismatch as shown in Figs. 5 and 6. By 80 MHz it is no

better than the ordinary Mach–Zehnder and by 8 GHz it is

18 dB worse than the ordinary MZM. Fig. 6 shows the first

50 MHz of the DCM2P dynamic range in more detail. Unlike

the DPMZM, in the DCM2P the mismatch between the RF

drive and the modulated signal upsets the critical distortion

cancellation conditions.

It has previously been reported [1] that the distortion can-

cellation condition is critically sensitive to the modulator

parameters, particularly bias voltage. The voltage on the bias

electrodes must be maintained to a very high accuracy. The

accuracy required depends on the operating bandwidth, also

explained in [1]. This requires active bias stabilization. Given

the critical bias conditions and the fact that the distortion

cancellation sections are in series, unlike the DPMZM, the

rapid degradation with dynamic range is reasonable. The

original experiments on this modulator were performed at

audio frequencies where these effects would not be noticed [5].

Subsequently, measurements at 1 and 2 GHz [6] were single-

frequency measurements, with the bias values reoptimized for

the operating frequency; no bandwidth measurements around

1 and 2 GHz were made.

V. S

UBOCTAVE

INEARIZED

ODULATORS

For suboctave applications, the second harmonic may be

ignored; the third-order intermodulation product alone de-

Fig. 6. Expanded plot of the dynamic range of the DCM2P shown in Fig. 5.

Fig. 7. Dynamic range comparison of two suboctave linearized modulators:

a dual series Mach–Zehnder modulator (DSMZM) and a directional coupler

modulator biased at 0.79

V

s

(SDCM).

termines the dynamic range. Two suboctave modulators are

analyzed in this section.

1) The dual series Mach–Zehnder modulator (DSMZM), as

described in [7], which has two MZM’s in series opti-

cally, the same bias voltage on each pair of electrodes,

and a single RF electrode covering both modulators.

2) A suboctave directional coupler modulator (SDCM), i.e.,

a directional coupler biased at

istead of

as described in [1]. Unlike the MZM, in the directional

coupler, the intermodulation product nulls at a different

voltage (

) than the signal (

). Thus no extra

electrode sections (as in the DCM2P) are needed to make

a suboctave directional coupler.

Fig. 7 shows the dynamic range as a function of frequency

for the DSMZM and SDCM compared to the standard MZM

reference (horizontal dotted line). Both of these modulators

suffer from the effects of velocity mismatch and transit time.

However, unlike the DCM2P, the DSMZM shows an advan-

There are other cascaded Mach–Zehnders proposed in the literature. In

some there are different bias voltages on the two electrodes. In [8] there is

a time delay between the first and second modulator so that the RF drive

and the modulated signal are rephased at the second Mach–Zehnder. While

this version may be more common in the literature, for the purposes of

a fair comparison, this modulation scheme is addressed in the section on

periodic rephasing below. Additionally, in [9] a mixed directional coupler and

Mach–Zehnder scheme purports to minimize the second and third harmonic.

CUMMINGS AND BRIDGES: BANDWIDTH OF LINEARIZED ELECTROOPTIC MODULATORS

1487

Fig. 8. Reoptmization of the suboctave linearized directional coupler mod-

ulator (SDCM). Slightly adjusting the bias voltage around 0.79

V

s

fully

recovers the low-frequency dynamic range optimum, but only over a narrow

bandwidth.

Fig. 9. Reoptimization of the dual series Mach–Zehnder modulator

(DSM-ZM). Adjusting the bias voltage fully recovers the low-frequency

dynamic range optimum, but only over a narrow bandwidth.

tage up to 4 GHz over the standard Mach–Zehnder, and the

SDCM is better still. The SDCM is consistently about 5 dB

better than the DSMZM. At the SDCM bias (0.79

), the dc

optical output is quite small, thus reducing the shot noise at

the detector, which is the dominant term in the total noise (as

described in [1]).

The bias voltage of the SDCM controls the frequency of the

dynamic range optimum. Small adjustments in the bias around

the dc value of

allow the full recovery of the low-

frequency dynamic range optimum at any center frequency.

Unfortunately, this results in a relatively narrow operating

bandwidth around the reoptimization frequency, as shown in

Figs. 8 and 9. The critical control of bias voltage will doubtless

require the use of pilot tones to minimize the harmonic or

intermodulation products. However, the results of Figs. 8 and

9 indicate that these pilot tones will have to be within the band

of interest, not at low frequencies.

VI. T

FFECTS OF

OISE

Noise effects linearized modulators somewhat differently

than standard modulators. In a standard MZM, the noise

bandwidth reduces the dynamic range by (BW)

2/3

. In Fig. 1 the

signal is a line with slope one, and the third-order intermodu-

Fig. 10. Dynamic range comparison with 1 MHz noise bandwidth of five

modulator configuration: the superoctave dual parallel Mach–Zehnder mod-

ulator (DPMZM), the superoctave directional coupler modulator linearized

by two passive couplers in series (DCM2P), the suboctave dual series Mach

Zehnder (DSMZM), the suboctave linearized directional coupler (SDCM), and

the standard directional coupler modulator (DCM).

lation product is a line with slope three. The “noise floor” is a

third, horizontal line which forms a triangle with the signal and

the intermodulation line. The length of the base of this triangle

is the dynamic range (in dB). The vertical position of the noise

line is proportional to the

(BW), so from simple geometry,

it is clear that the dynamic range goes as (BW)

2/3

. However, in

a linearized modulator the third-order intermodulation product

has been nulled. The dominant intermodulation

term is at

, and it grows as the fifth power of the RF

drive. Thus the noise bandwidth reduces the dynamic range by

(More complicated linearization schemes can result

in even steeper slopes for intermodulation, as discussed in [1]).

The excess dynamic range,

, of a linearized modulator over

a ordinary modulator is

(BW)

(12)

where BW is a the bandwidth in Hertz, and

is a constant

in dB equal to the difference between the dynamic range

of a linearized modulator and a standard modulator with a

1-Hz noise bandwidth. This equation is only valid in the

approximation that the signal and intermodulation curves are

straight lines. For a comparison of the suboctave DCM versus

the standard MZM,

equals 26 dB. For a 1 MHz noise

bandwidth,

dB. The dynamic range is often given

for a 1-Hz noise bandwidth, and this simple scaling rule is

applied to find the dynamic range of a system with a realistic

bandwidth.

Apart from the different scaling rules, the noise bandwidth

has an additional effect on linearized modulators. The dynamic

range becomes less sensitive to device variations, either bias

voltages or fabrication parameters, as the noise bandwidth

increases. This makes it easier to maintain the distortion

cancellation than is implied by some of the previous figures.

Fig. 10 shows the dynamic range for the four modulators

discussed above, but now with a 1-MHz noise bandwidth.

The DPMZM and the MZM reference lines are flat as in

the 1-Hz bandwidth case. The simple DCM is flat out to the

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Fig. 11.

Dynamic range versus frequency for the standard directional coupler

modulator (DCM) with multiple electrode segments. The arrows show the

breaks between the intermodulation and the second-harmonic limitation of

the dynamic range.

frequency where the second harmonic exceeds the intermod-

ulation distortion, as in Fig. 3. However, the crossover now

occurs at 3.8 GHz instead of 1.8 GHz. The DSMZM, SDCM,

and DCM2P dynamic ranges, which all roll off with frequency,

now do so at a slower rate than in the 1-Hz bandwidth case.

While the DCM2P still rolls off too quickly to be a useful

modulator, the suboctave modulators are starting to show

reasonable frequency performance. A similar reduction in peak

dynamic range, with a broadening of the bandwidth over which

it occurs, is also obtained in the bias reoptimized curves of

Figs. 8 and 9.

VII. B

ANDWIDTH

ECOVERY

HROUGH

ERIODIC

EPHASING

One method of overcoming the degradation in dynamic

range due to velocity mismatch is to break the transmission

line into a number of segments and rephase the signal at the

beginning of each segment. This is velocity-matching “on the

average.” The technique has been used successfully in a num-

ber of forms, for example, [10]–[12]. The program written for

this study is easily modified to make calculations for such pe-

riodically rephased modulators, since the modulator is already

broken up into a cascade of matrices. Thus, the modulator is

incrementally velocity mismatched for a few matrices and is

then rephased for the next section, and so on. Fig. 11 shows the

results of such a calculation for the simple DCM link with the

parameters used above, but with the modulator’s transmission

line having 1, 2, 3, and 4 segments. The 1-segment curve

repeats the result in Fig. 4 (no rephasing) for reference. With

only two segments (one rephasing), the bandwidth over which

the dynamic range is flat improves vastly, and using four

segments gives an essentially flat dynamic range. Of course,

it would still be preferable to use a standard MZM for broad-

band links since it does not require rephasing.

A similar dramatic improvement is obtained in the SDCM,

biased at the 0.79

point, as shown in Fig. 12. The curve

from Fig. 7 is shown for reference along with curves for two-,

three-, and four-electrode segments. The curve for one segment

initially shows a deep roll-off in dynamic range and then

a more gradual roll-off, with only 5 dB of dynamic range

improvement over the standard MZM remaining at 8 GHz.

Fig. 12. Dynamic range versus frequency for the suboctave linearized direc-

tional coupler (SDCM) with multiple electrode segments.

Fig. 13. Dynamic range versus frequency for the the suboctave dual series

Mach–Zehnder (DSMZM) with multiple electrode segments. Note that two

(and any even number) segment give frequency independent dynamic range.

With just two segments, the roll-off is made gradual over the

whole range. With four segments, there is 18 dB of dynamic

range improvement remaining at 8 GHz. This figure assumes

a 1-Hz noise bandwidth. When a 1-MHz noise bandwidth is

used, the roll-off is more gradual. For instance, the dynamic

range of the two-electrode segment SDCM is better than the

two-electrode DSMZM (shown in Fig. 13) up to 5.9 GHz.

A very interesting result of the application of rephasing is

seen in the dual series Mach–Zehnder modulator. This modu-

lator has been shown to give very high dynamic range values

based on the intermodulation distortion, although it has, like

the SDCM, a very large second harmonic content, restricting

it to suboctave applications as stated in [8]. The decrease

in dynamic range with frequency is shown in Fig. 13 in the

curve labeled “1 segment,” meaning one set of traveling wave

electrodes spanning two Mach–Zehnder modulators (with an

admittedly unrealistic zero distance assumed between the two

modulators). This is simply a repeat of the curve in Fig. 7.

The surprise is that with only one rephasing (two segments)

the modulator exhibits frequency-independent performance.

Breaking the electrodes into three segments yields an improve-

ment over one segment, but not as good as two or any even

number of segments, all of which yield perfect performance.

This behavior results from the symmetry of the modulators

and their bias points.

CUMMINGS AND BRIDGES: BANDWIDTH OF LINEARIZED ELECTROOPTIC MODULATORS

1489

Fig. 14.

Dynamic range versus frequency for the superoctave linearized

directional coupler (DCM2P) with one to eight electrode segments. While

rephasing yields improvements, the results remain inferior to other modulators.

Fig. 15.

The gain of a standard Mach–Zehnder with multiple electrode

segments.

The results for the DCM2P superoctave modulator remain

discouraging, even with rephasing. Fig. 14 shows the dynamic

range from one-to-eight rephased electrode segments. This plot

spans 2 GHz only, which is enough to capture the interesting

range. Even with eight electrodes, the frequency at which

the dynamic range improvement over the simple MZM has

fallen 10 dB is only 1.3 GHz. While this plot assumes an

unrealistically low 1 Hz noise bandwidth, it also assumes

perfect bias control. Given the modulators sensitivity to its

three bias voltages, it is unlikely that this modulator is suitable

for microwave applications with reasonable bandwidth.

Segmenting the RF electrode does have a low-frequency

gain penalty. While flattening the gain versus bandwidth curve,

it also lowers the absolute gain at low frequencies. The reduc-

tion occurs from the splitting of the RF power. It is split

ways to feed the

electrodes, reducing the voltage by

each electrode segment. However, if the rephasing is achieved

by a series intermittent interaction method such as described

in [12], then there is no power splitting and subsequently no

penalty. But this scheme involves long, curved, trans-

mission lines and thus may have unacceptable losses. Fig. 15

shows the calculated gain of a simple Mach–Zehnder with

one-, two-, three-, and four-electrode segments. Above some

cross-over frequency each curve has a better absolute gain then

the curve with one fewer electrode segments. Clearly, true

velocity matching, by somehow making

is preferable

if possible, since there is no

penalty.

We wish to remind the reader that all the results obtained

in this study may be applied to modulators with any degree

of velocity matching by rescaling the frequency axis by the

change in

, the frequency-length-index product.

CKNOWLEDGMENT

The authors would like to thank E. Ackerman, G. E. Betts,

and C. H. Cox III of Lincoln Laboratories, Cambridge, MA,

N. P. Bernstein of the Air Force Rome Laboratory, and J. H.

Schaffner of Hughes Research Laboratories. The authors also

wish to acknowledge the support and enthusiasm of the late

B. Hendrickson of Rome Laboratories and ARPA.

EFERENCES

[1] W. B. Bridges and J. H. Schaffner, “Distortion in linearized electrooptic

modulators,”

IEEE Trans. Microwave Theory and Tech.

, vol. 43, pp.

2184–2197, Sept. 1995.

[2] M. L. Farwell and W. S. C. Chang, “Simulating the response of coupled

channel and interferometric modulator designs,”

J. Lightwave Technol.

vol. 13, Oct. 1995.

[3] B. Pucel, L. Riviere, and J. L. Bihan, “New model for the active

directional coupler,”

J. Lightwave Technol.

, vol. 14, pp. 1501–1506,

June 1996.

[4] S. K. Korotky and R. M. Ridder, “Dual parallel modulation schemes

for low-distortion analog optical transmission,”

IEEE J. Select. Areas

Commun.

, vol. 8, pp. 1377–1381, Sept. 1990.

[5] M. L. Farwell, Z. Q. Lin, E. Wooten, and W. S. C. Chang, “An

electrooptic intensity modulator with improved linearity,”

IEEE Photon.

Technol. Lett.

, vol. 3, pp. 792–795, Sept. 1991.

[6] M. L. Farwell, W. S. C. Chang, and J. H. Schaffner, “A directional

coupler modulator with improved linearity (operating up to 1 GHz),”

Leos Summer Topical Meet. Dig. Broadband Analog and Digital

Optoelectron.

, 1992, vol. WA3, pp. 8–9.

[7] G. E. Betts and F. J. O’Donnell, “Microwave analog optical links using

suboctave linearized modulators,”

IEEE Photon. Technol. Lett.

, vol. 8,

pp. 1273–1275, Sept. 1996.

[8] G. E. Betts, “Linearized modulator for suboctave-bandpass optical

analog links,”

IEEE Trans. Microwave Theory Tech.

, vol. 42, pp.

2642–2649, Dec. 1994.

[9] H. Skeie and R. V. Johnson, “Linearization of electrooptic modulators

by a cascade coupling of phase modulating electrodes,” in

Proc. SPIE

1991, vol. 1583, pp. 153–164.

[10] W. B. Bridges, F. T. Sheehy, and J. H. Schaffner, “Wave-coupled elec-

trooptic modulators for microwave and millimeter-wave modulation,”

IEEE Photon. Technol. Lett.

, vol. 3, pp. 133–135, Feb. 1991.

[11] R. C. Alferness, S. K. Korotky, and E. A. J. Marcatili, “Velocity-

matching techniques for integrated optic traveling wave switch mod-

ulators,”

IEEE J. Quantum. Electron.

, vol. QE-20, pp. 301–309, Mar.

1984.

[12] J. H. Schaffner, “Analysis of a millimeter wave integrated electro-optic

modulator with a periodic electrode,” in

Proc. SPIE OE-LASE Conf.

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, Los Angeles, CA, Jan. 16–17, 1990, pp. 101–110.

Uri V. Cummings

was born in Vallejo, CA, on

November 9, 1971. He received the B.A. degree

in english from Wesleyan University, Middletown,

CT, and the B.S. degree from the California Institute

of Technology (Caltech), Pasadena, simultaneously

in 1994. He received the M.S. degree in electrical

engineering from the Caltech in 1995. Currently,

he is working toward the Ph.D. degree in electrical

engineering at Caltech.

His research interests include the study of lin-

earized electrooptic modulators and the design and

experimental verification of antenna-arrayed electrooptic modulators at 100

GHz.

1490

JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 16, NO. 8, AUGUST 1998

William B. Bridges

(S’53–M’61–F’70–LF’98) was

born in lnglewood, CA, in 1934. He received the

B.S., M.S., and Ph.D. degrees in electrical engineer-

ing from the University of California, Berkeley, in

1956, 1957, and 1962, respectively. His graduate

research dealt with noise in microwave tubes and

electron-stream instabilities (which later became the

basis of the Vircator.)

From 1957 to 1959, he was an Associate Pro-

fessor in the Department of Electrical Engineering

at the University of California, Berkeley, teaching

courses in communication and circuits. Summer jobs at RCA and Varian

provided stimulating experience with microwave radar systems, ammonia

beam masers, and the early development of the ion vacuum pump. In 1960,

he joined the Hughes Research Laboratories, Malibu, CA, as a Member

of the Technical Staff, and from 1968 to 1977, he was a Senior Scientist

with a brief tour as Manager of the Laser Department in 1969–1970. His

research at Hughes Research Laboratories involved gas lasers of all types

and their application to optical communication, radar, and imaging systems.

He is the discoverer of laser oscillation in noble gas ions and spent several

years on the engineering development of practical high-power visible and

ultraviolet ion lasers for military applications. He joined the faculty of the

California Institute of Technology, Pasadena, CA, in 1977 as Professor of

Electrical Engineering and Applied Physics, and serving as Executive Officer

for Electrical Engineering from 1979 to 1981. In 1983, he was appointed

Carl F. Braun Professor of Engineering and conducted research in optical

and millimeter-wave devices and their applications. Current studies include

the millimeter-wave modulation of light and high-fidelity analog microwave

photonic links. He is coauthor (with C. K. Birdsall) of

Electron Dynamics of

Diode Regions

(New York: Academic, 1966.)

Dr. Bridges is a member of Eta Kappa Nu, Tau Beta Pi, Phi Beta

Kappa, and Sigma Xi, receiving Honorable Mention from Eta Kappa Nu

as an “Outstanding Young Electrical Engineer” in 1966. He received the

Distinguished Teaching Award in 1980 and 1982 from the Associated Students

of Caltech, the Arthur L. Schawlow Medal from the Laser Institute of America

in 1986, and the IEEE LEOS Quantum Electronics Award in 1988. He is a

member of the National Academy of Engineering and the National Academy

of Sciences, and a Fellow of the Optical Society of America (OSA) and the

Laser Institute of America. He was a Sherman Fairchild Distinguished Scholar

at Caltech, Pasadena, CA, in 1974–1975, and a Visiting Professor at Chalmers

Technical University, G

oteborg, Sweden, in 1989. He has served on various

committees of both IEEE and OSA and was formerly Associate Editor of

the IEEE J

OURNAL OF

UANTUM

LECTRONICS

and the

Journal of the Optical

Society of America

. He was the President of the Optical Society of America

in 1988, a member of the United States Air Force Scientific Advisory Board

1985–1989, and a member of the Board of Directors of Uniphase Corporation

1986–1998.