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2573

Nonlinear phase shifts induced by semiconductor
optical amplifiers with control
pulses at repetition frequencies
in the 40–160-GHz range for use inultrahigh-speed all-optical signal processing
Yoshiyasu Ueno, Shigeru Nakamura, and Kazuhito Tajima
Networking Laboratories, NEC Corporation, 34 Miyukigaoka, Tsukuba, Ibaraki 305-8501, Japan
Received February 27, 2002; revised manusript received June 5, 2002
In a semiconductor optical amplifier (SOA) with copropagating optical pump pulses, the application of a nonlinear phase shift to optical signals provides the driving force for all-optical interferometric switching. We
study, both analytically and experimentally, the dependencies of the nonlinear phase shift on the driving frequency (42–168 GHz) and on the SOA parameters. We have found that the nonlinear phase shift (⌬⌽ NL)
decreases with the driving frequency but that this decrease is only linear, i.e., ⌬⌽ NL ⬀ f ⫺1 . We have also
found that the nonlinear phase shift in the SOA linearly increases with the injection current (I op), i.e.,⌬⌽ NL⬀I op , even in this ultrahigh-frequency range. © 2002 Optical Society of America
OCIS codes: 060.0060, 120.5060, 190.5970, 190.7110, 320.7080, 320.7160.

1. INTRODUCTION
Ultrafast all-optical signal processors whose bandwidths
possibly approach optical frequencies (e.g., 200 THz in optical communication) have fascinated optics researchers
for more than three decades, i.e., since the birth of nonlinear optics.1 The most basic unit of such a processor
would be what is referred to as an all-optical switch.
Thus far, however, the trade-off between the speed ofswitching and the power required for all-optical control,
both of which are determined by the nonlinear opticalproperties applied to the switch, are perceived to have
prevented the large-scale application of all-opticalswitches. When an all-optical switch is based on a nonlinear phenomenon that appears at some distance fromits one-photon resonance, the switch requires a relatively
strong optical control pulse or an extremely long interaction length; the latter approach is used in all-opticalswitches that are made from silica fibers. The typicalcontrol-pulse energy (peak power) for an all-optical
AlGaAs waveguide switch operating in a band well below
its one-photon resonance (slightly below its two-photon
resonance) was 100 pJ (with a peak power of 200 W).2
Such strong pulses are readily generated in laboratories
but are not suitable for the optical communications industry. A switch may also be based on a nonlinearity in thevicinity of one-photon resonance of such a semiconductor
waveguide, in which case the power required for alloptically controlling the switch is several orders of magnitude smaller. In this case it was generally assumed forsome time that switching is necessarily slowed down by
the relaxation time that is needed for the resonantly excited electrons in the material to return to the groundstate.
Researchers began to challenge that assumption in the0740-3224/2002/112573-17$15.00

late 1980s and the early 1990s, as they found that the relaxation time does not necessarily limit the speed ofswitching. The limitation does not apply when theswitch consists of an optical nonlinear material embedded
in an interferometer in a certain way; the symmetricMach–Zehnder (SMZ) all-optical switch3,4 and the terahertz optical asymmetric demultiplexer5 are two examples. In these structures, as we explain later in thetext, the interference effectively masks the relaxation of
the nonlinear material. The interference is between two
split optical components, where one of the two components is given a finite delay time, ⌬t. This interference
effectively forms a rectangularlike window for switchingby the all-optical switch, and the rise and fall times of thewindow are determined by the width of the control pulse
rather than by the relaxation time. The window’s widthis determined by the delay time, ⌬t. Both an ultrafast
response and a nearly rectangular shape for the switchingwindow are important in obtaining timing-jitter-tolerant
signal processing for an optical communications system.
In fact, a record response time (the sum of rise and falltimes) of 200 fs has been reported for a SMZ switch.6 In
this device, each 130-fs-long control pulse in an 82-MHztrain was absorbed by InGaAsP waveguides inside theswitch. The control pulses generated electron–hole pairs(carriers), i.e., induced nonlinear changes in the refractive
index because of the band-filling effect, imparted nonlinear phase shifts to the copropagating input signal andswitched the signal after the interference while masking
the relaxation time. The carrier relaxation time of thesemiconductor material (100–200 ps, 3 orders of magnitude longer than the measured response time) did not
limit the speed of the switch’s response. The control
pulse required for the 200-fs switching had 3 pJ of energy
and a peak power level of 23 W. Near-rectangular© 2002 Optical Society of America

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switching windows have been reported.4,7 The control
pulses for a SMZ switch that contains semiconductor optical amplifiers (SOAs) as the nonlinear elements require
even lower levels of power. This is so because a smaller
number of control-pulse photons is enough to generate a
specific change in the number of carriers, which in turncreates a specific nonlinear phase shift (e.g., a shift of ␲).
Response times of 0.67–1.0 ps have been recorded forsuch devices, with control pulses in the range 82 MHz–10
GHz.8,9 The near-rectangular quality of the switchingwindows in these devices has also been confirmed.9 Taking advantage of these fast response times and switchingwindows, the present authors and some of their colleagues have also demonstrated error-free 16:1 demultiplexing from 168 to 10.5 Gbits/s.10,11 Only 50–80 fJ ofenergy was required in the control pulse (the peak power
level of which was 25–40 mW). Note that the frequencyof repetition for switching in the demultiplexer was 10.5
GHz (not 168 GHz).
For flexible processing and logical gating of ultrafast
optical-time-division multiplexing (OTDM) signals, thefrequency of repetition for switching (repetition frequency) is as important as the response time. For example, all-optical wavelength conversion,12,13 reamplifying, reshaping, and retiming regeneration,14 and asynchronous packet gating15 have recently been attracting
attention as devices that will play key roles in the building of efficient and flexible OTDM packet networks. In
each of these signal-processing functions the switch must
operate at the repetition frequency of the OTDM signal(e.g., 160 GHz) and must be driven by digitally coded
pulses. The repetition frequencies in early experiments
were in the range 80 MHz–10 GHz. Several teams of scientists have reported operation of the switches at 40
GHz.12,16–19 Research has recently entered the 80–160
GHz range; examples include logic gating at 100 GHz,20
3R regeneration of random signals at 84 GHz,21–23 andwavelength conversion of random signals at 168
GHz21,24,25 (in all of the research cited in these publications the all-optical switches used were of the SMZ type).
However, no reports to date of which we are aware haveincluded an analysis or investigation of the dynamics ofthe nonlinear phase shift in SOAs driven at ultrahigh frequencies. Specifically, although the nonlinear phaseshift is known to decrease with the driving frequency, nobody has reported studying the steepness of this decrease.
Durhuus et al.26 and Joergensen et al.27 have analyzed
the SOA’s nonlinear response to sinusoidally modulated
input lights as a function of repetition frequency. The
approach of each team was similar to the small-signal
analysis that is conventionally used to analyze the directmodulation of semiconductor lasers. The driving force
for SMZ-type switching is, however, the impulselike nonlinear phase shift (jump) that is induced by short control
pulses. Being able to obtain a phase shift of ␲ is particularly important not only because this is necessary to ensure perfectly constructive interference in forming theswitching window3 but also because the sinusoidal transfer function of the interferometer inside the switch suppresses noise in the signal intensity at a phase shift of
␲.28
Despite its importance, most previous highrepetition-rate experiments (with a few exceptions16,29)

Ueno et al.

have not included measurement of the magnitude of thenonlinear phase shift. Part of the reason for this is thedifficulty of measuring the magnitude of a nonlinear
phase shift that coexists with strong gain modulation.
We report here our experimental study of the dynamics
of the nonlinear phase shift applied by a SOA operating atrepetition frequencies in the 42–168-GHz range. First,
in Section 2 we introduce several all-optical semiconductor switches of the SMZ type. In Section 3 we introduce atheoretical background to the nonlinear dynamics and aform of numerical simulation that we recently developed.
After describing our verification of the correspondence between the results of simulation and experiment in the 10–
40-GHz range, we discuss the use of the simulation to investigate the dependencies of the nonlinear phase shift on
many important SOA parameters. We also describe a
new method for measuring the nonlinear phase shift.
This method was specifically designed for SOA nonlinearity. In Section 4 we show our measurements of nonlinear
phase shift in the 42–168-GHz range. We found that thenonlinear phase shift was straightforwardly dependent
on the repetition frequency and the injection current, andthis result was in line with our predictions. In Section 5
we introduce examples of results for 168-GHz regularswitching (i.e., switching driven by periodic control
pulses) and determine the characteristics of the nonlinear
phase shift under such switching conditions. Although
all the results and designs presented in Sections 3–5 arelimited to those in the regular switching regime, we believe that these results provide a basis for the design ofrandom switching in real applications such as 3R andwavelength conversion, which is still a subject of our research. Section 6 is a brief discussion, on the basis of experimental results, of switching driven by digitally coded
control pulses. We also introduce an example of 168Gbit/s pseudorandom wavelength conversion and determine the nonlinear phase shift under the conditions of theconversion. In Section 7 we draw conclusions from ourresults.

2. SYMMETRIC-MACH–ZEHNDER-TYPE
ALL-OPTICAL SWITCHES
Three all-optical switches, each a variant of the (SMZ)
switch, are depicted schematically in Fig. 1. Two of theseswitches are depicted, each with a different signal configuration, in Fig. 2. The types and their applications arealso tabulated in Table 1. The switches in Fig. 1 are being
driven (controlled) by digitally coded pulses, whereas
those in Fig. 2 are being driven by divided-clock pulses.
The switch in Figs. 1(a) and 2(a) has the original SMZ
structure.3 The switch in Figs. 1(b) and 2(b) is referred
to as a polarization-discriminating SMZ (PD-SMZ), which
is a modification of the SMZ structure.30 The switch inFig. 1(c), which is a simplification of the PD-SMZ structure, is referred to as a delayed-interference signalwavelength converter (DISC).11 The switching windowsthat are formed in all the switches in Figs. 1 and 2 can benearly rectangular and have ultrafast rise and fall times.
The formation of these windows is based on the SMZ
switching mechanism.

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Vol. 19, No. 11 / November 2002 / J. Opt. Soc. Am. B

Fig. 1. SMZ-type all-optical semiconductor switches driven by digitally coded pulses:
type 3R regenerator, (c) a DISC-type wavelength converter. pol, polarizer.

Fig. 2. SMZ-type all-optical semiconductor switches driven by regular clock pulses:
demultiplexer. pol, polarizer.

2575(a) a SMZ-type 3R regenerator, (b) a PD-SMZ-(a) a SMZ-type demultiplexer, (b) a PD-SMZ-type

Table 1. Three Types of SMZ Switch and Their Applications in OTDM Communications Systems
Control Scheme of
the Switch

SMZ or PD-SMZ Switcha

DISCb

Data pulses control the switch (OTDM rate)

3R regeneration,
asynchronous gating, logic gates (XOR, etc.)

Wavelength conversion,
polarization conversion,
2R regeneration

Clock pulses control the switch (OTDM rate)
Divided-clock pulses control the switch(electronic-time-division multiplexing rate)

Soliton modulation
Demultiplexingab

Pulses are gated.
cw light is gated.

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Ueno et al.

form such as that of the demultiplexer in Fig. 2 (the irregular form of switching, as required in 3R regeneration,is discussed in Section 6). When a control pulse enters a
SOA, it stimulates the recombination of carriers as a reaction to stimulated amplification. The carrier density
decreases sharply on each arrival of a pulse, and it recovers slowly until the next pulse arrives [Fig. 3(a)]. The
fall time is approximately equal to the width of the control pulse. The change in the carrier density induceschange in both the refractive index [Fig. 3(b)] and thegain [Fig. 3(c)].
As a result of the nonlinear change in refractive index
as shown in Fig. 3(b), the SMZ switch [see, e.g., Fig. 2(a)]forms a rectangularlike switch window for the signal
pulse, as is schematically depicted in Fig. 4. The signal
pulse is split into two components at the input of the interferometer inside the SMZ switch. The optical phase ofeach component is nonlinearly shifted by the nonlinearchange in refractive index of the respective SOA, as isshown in Fig. 4(a). This shifting of phase is slightly laterfor one of the components [dashed curve in Fig. 4(a)] thanfor the other [solid curve in Fig. 4(a)]. This delay time ⌬t

Fig. 3. Schematic view of the nonlinear phenomena inside a
SOA: (a) the change in the carrier density that is induced by the
optical control pulses, which causes (b) a nonlinear change in refractive index and (c) a change in the gain. The dashed curvesin (a) indicate the waveforms of the input pulses.

The configurations shown in Figs. 1(a) and 1(b) implement 3R regeneration. The digitally coded data pulsesthat drive the switch arrive in an irregular (random) but
nearly clock-synchronous train. Each pulse opens anear-rectangular switching window and thus gates theprecisely timed clock pulses. Consequently, each 1 bit in
the train of data pulses is replaced by a new, clean, andprecisely synchronized, that is, reshaped and retimed,
pulse. The configuration in Fig. 1(c) implements wavelength conversion.
Switches in the configurations shown in Fig. 2 work as
demultiplexers (or as soliton modulators), with a train of
divided-frequency clock pulses (or signal-frequency clock
pulses) driving the switch at regular intervals. Note thatthe structure of the demultiplexer in Fig. 2(a) is exactly
the same as that of the 3R regenerator in Fig. 1(a); thetwo devices differ only in terms of the inputs to which therespective pulses are applied. In the same manner, only
the input configuration differentiates the demultiplexer
in Fig. 2(b) from the 3R regenerator in Fig. 1(b).
Figure 3 is a schematic depiction of nonlinear phenomena as they take place over time in the operation of a
SOA. In the remainder of this section, for the sake ofsimplicity, switching is assumed to be regular, i.e., of a

Fig. 4. Mechanism in the SMZ-type all-optical switches: (a)Each of the two split signal components inside the interferometerobtains a nonlinear phase shift. (b) The phase difference between the two components and (c) the switching window.

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2577

Fig. 5 shows the switch window’s shape. The dashedcurve in Fig. 5 follows the trace of the intensity of thecross-gain modulated continuous-wave (CW) light at the
SOA output inside the DISC, which indicates a carrier relaxation time of 200 ps. Figure 5 shows that the cancellation depicted in Fig. 4 makes it possible to form aswitching window that is much shorter than the carrierrelaxation time in a SMZ-type switch.

Fig. 5. Comparison of the DISC output (solid curve) and the
SOA output (dashed curve) as observed with a streak camera,
showing that the SMZ-type switch can form a switching window
whose width is much shorter than the carrier lifetime.

is produced by the difference between the times of arrivalof the control pulses at the respective SOAs [Fig. 2(a)].
The two split signal components collide and interferewith each other at the interferometer’s output. The resultant complex amplitude of the signal pulse at the interference output is determined as
E OUT共 t 兲 ⫽ 1 / 2 兵 冑G 共 t 兲 exp关 i⌽ 共 t 兲 ⫹ ⌬⌽ b 兴 E IN共 t 兲⫹ 关 G 共 t ⫺ ⌬t 兲兴 1/2 exp关 i⌽ 共 t ⫺ ⌬t 兲兴 E IN共 t 兲 其

3. THEORY OF THE DYNAMICS OF THENONLINEAR PHASE SHIFT IN A
SEMICONDUCTOR OPTICAL AMPLIFIER
A. Rate Equation and Other Basic Equations
for Numerical Simulation
Our approach to building a set of basic equations for use
in numerically simulating the nonlinear phase shift andthe all-optical switching at ultrahigh repetition frequencies is described in the following passages. First, we assume for simplicity that the changes in the refractive index and the material gain are proportional to the excess
carrier density.16 Because of this assumption, both thenonlinear phase shift and the SOA chip gain are determined by the average excess carrier density n c (t) over thelength of the SOA’s active region, L. n c (t) is defined as⫽ 1 / 2 兵 冑G 共 t 兲 exp关 i⌽ 共 t 兲 ⫹ ⌬⌽ b 兴

n c共 t 兲 ⬅

⫹ 关 G 共 t ⫺ ⌬t 兲兴 1/2 exp关 i⌽ 共 t ⫺ ⌬t 兲兴 其 E IN共 t 兲⬅ T W 共 t 兲 E IN共 t 兲 .

(1)

The phase bias (⌬⌽ b ) between the two components is adjusted such that the phase difference before t 0 and aftert 0 ⫹ ⌬t equals ␲. Consequently, the switch’s transmissivity 兩 T w (t) 兩 2 , that is, the switch window, may be approximated as a sinusoidal function of the phase difference [Fig. 4(b)] between the two components:

1

冕

L

n c 共 z, t 兲 dz,(3)n c 共 z, t 兲 ⬅ n 共 z, t 兲 ⫺ n tr .

(4)

L

z⫽0

where

Here n c is the excess carrier density, n is the carrier density, and n tr is the transparency carrier density 关 G(n tr)⫽ 0 兴 . With the average excess carrier density n c (t), thenonlinear phase shift [Fig. 4(a)] given to an optical component (either a pulse or a continuous wave) copropagating through the SOA is expressed as兩 T W 共 t 兲 兩 2 ⫽ 兩 1/2兵冑G 共 t 兲 exp关 i⌽ 共 t 兲 ⫹ ⌬⌽ b 兴⌽ 共 t 兲 ⫽ k 0 dn r /dn c 关 n c 0 ⫺ n c 共 t 兲兴 ⌫L,⫹ 关 G 共 t ⫺ ⌬t 兲兴 1/2 exp关 i⌽ 共 t ⫺ ⌬t 兲兴 其 兩 2⫽ 关 G 共 t 兲 G 共 t ⫺ ⌬t 兲兴 1/2⫻ cos2⌽ 共 t 兲 ⫺ ⌽ 共 t ⫺ ⌬t 兲 ⫹ ⌬⌽ b2

⫹ 1/4兵 冑G 共 t 兲 ⫺ 关 G 共 t ⫺ ⌬t 兲兴 1/2其 2 .

(2)Thus, in the switching window shown in Fig. 4(c), there isno carrier-relaxation-induced effect before t 0 and after t 0⫹ ⌬t. The relaxation-induced effects in the two split
components have canceled each other out. The rise andfall times of the rectangularlike switching window are determined by the width of the control pulse. The width of
the switching window is determined by delay time ⌬t.
Figure 5 shows an example of the cancellation of therelatively slow relaxation-induced effects. We used
1560-nm CW light and a streak camera to examine theformation of a switching window in a DISC by 0.7-ps
1530-nm control pulses [Fig. 1(c)]. The lower curve in(5)where k 0 is the wave number in vacuum, n r is the refractive index, and ⌫ is the optical confinement factor of these
optical components within the cross section of the active
layer in the SOA. The equilibrium excess carrier density,n c 0 , is defined as the excess carrier density at which the
SOA does not receive any input light. The gain of the
SOA chip [Fig. 3(c)] is expressed as
G 共 t 兲 ⬅ exp关 dg/dn c n c 共 t 兲 ⌫L 兴 .

(6)Here g is the material gain. As a reference, unsaturated
gain G 0 is expressed as
G 0 ⬅ exp共 dg/dn c n c 0 ⌫L 兲 .

(7)

Note that we no longer have to consider the distribution of carrier density along the propagation axis of the
SOA because of the proportionality assumption at the be-2578

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Fig. 6. Measured (solid curves) and calculated (dashed curves) SPM spectra of the amplified 10.5-GHz 2.0-ps control pulses at the SOA
output. The input pulse energies were (a) 340, (b) 34, and (c) 3.4 fJ.

ginning of this section. The dynamics of both the nonlinear phase shift and the chip gain as a function of time aregoverned by a rate equation for the average excess carrier
density, which is assumed to be

Table 2. Independent SOA Parameters Used for
Numerical Calculation
Parameter
G0ddt

n c共 t 兲 ⫽

I op
qV
⫻

⫺

n c共 t 兲

␶c

⫺

1
V

兩 E control共 t 兲 兩 2 ⫹ 兩 E signal兩 2ប␻⌬⌽ max,(8)where I op is the injected current, q is the elementarycharge, ␶ c is the carrier lifetime, and V is the volume of
the current-injected active layer. The first, the second,
and the third terms on the right-hand side of Eq. (8) represent, respectively, the injection of carriers, the relaxation of carriers, and the stimulated recombination thatis induced by input control pulses 关 E control(t) 兴 and input
signals 关 E signal(t) 兴 . Note that the basic equations (5)–(8)are written in their simplest forms, on the assumptionthat there are no higher-order gain, higher-order changes
in refractive index, or higher-order relaxation terms withrespect to the carrier density.
It should be noted that the effects of carrier heating
and spectral hole burning are not taken into account inbasic equations (5)–(8), either. These effects play significant roles inside SOAs when the width of the input pulses
to them is shortened to less than specific values.31–33
Nevertheless, in our simulation throughout this study weintentionally neglected these effects because the pulse
widths with which we were working were slightly greaterthan the values given above and also because we havebeen interested in how well one can reproduce our experimental results numerically without taking these effects
into account. In what follows, we show that several setsof our experimental results have successfully been reproduced with this simplified model.
Finally, we take width and shape of the input pulses
into account by assuming sech-type pulse shapes:

E control共 t 兲 ⫽兺Emcontrol sech冉m ⫽ 1, 2 ,..., ⬁.

t ⫺ Tm
Tw

冊

,
(9)

Value

Unsaturated gain
(at I op ⫽ 150 mA)
Gain saturation energy(at I op ⫽ 150 mA)
Phase shift at complete
depletion, ⌬⌽(⌬n c ⫽ n c 0 )Carrier lifetime (at I op ⫽ 200 mA)
Injection current

P sat

兵 G 关 n c 共 t 兲兴 ⫺ 1 其

Description

␶c
I op

28 dB
180 fJ⫹3.0␲
60 ps
150–250 mA

Table 3. Derived SOA Parameters
Parameter␣n c0dg/dn cdn r /dn c

Description

Value␣ parameter
Equilibrium excess carrier
density
Differential gain
Nonlinear change in refractive
index

⫹2.9
4.8 ⫻ 1016 cm⫺3
4.1 ⫻ 10⫺15⫺1.2 ⫻ 10⫺19 cm3

B. Comparison of Measured and Calculated
Semiconductor Optical Amplifier Output Spectra inOperation at 10–40 GHz
We initially verified our numerical simulation by comparing the calculated pulse gains by using Eqs. (6)–(9) andthe calculated pulse gains by using a well-known form
from the conventional Franz–Nodvik analysis.34 After
this verification, we further verified our numerical simulation by comparing calculated and measured results inthe following way:
Figure 6 shows measured (solid curves) and calculated(dashed curves) spectra of the amplified 2.0-ps 1550-nm
control pulses at the SOA output. The repetition frequency was 10.496 GHz (a 128th-harmonic of the repetition frequency, 82 MHz, of our synchronous streak camera). As the input pulse energy was increased, we
observed strongly self-phase-modulated (SPM) spectra.
The injection current to the SOA was set to 150 mA. A
polarization-insensitive high-gain bulk-InGaAsP-activelayer SOA was used throughout the research reported

Ueno et al.

here. The SOA parameters treated in this paper are
listed in Table 2. The unsaturated gain, the gain saturation energy, and the carrier lifetime were directly measured in separate experiments, in which either lowrepetition rate 82-MHz pulses or a CW light were used.
We only used fitting to obtain one of the parameters in
Table 2 to calculate the SPM spectra in Fig. 6, that is, thephase shift at complete depletion:

Vol. 19, No. 11 / November 2002 / J. Opt. Soc. Am. B

2579⌬⌽ max ⬅ ⌽ 共 n c ⫽ n c 0 兲 ⫺ ⌽ 共 n c ⫽ 0 兲⫽ k 0 dn r /dn c n c 0 ⌫L.

(10)With a ⌬⌽ max value of ⫹3.0␲, our numerical simulation
accurately reproduced the measured SPM spectra. Our
simulation revealed that the nonlinear phase shifts involved in the respective spectra were 2.0 ␲ [Fig. 6(a)],1.4␲ [Fig. 6(b)], and 0.75␲ [Fig. 6(c)].

Fig. 7. Measured [(a)–(c)] and calculated [(d)–(f)] XPM spectra of the 1546.5-nm CW light at the SOA output. The input control-pulse
energies were (a), (d) 100 fJ, (b), (e) 10 fJ; (c), (f) 0.3 fJ.

Fig. 8. Measured (solid curves) and calculated (dashed curves) autocorrelator traces (a), (c) and spectra (b), (d) of 42-GHz 7.0-ps
wavelength-converted pulses at the output of the DISC: (a), (b) with the phase bias (⌬⌽ b ) set to ␲ and (c), (d) with ⌬⌽ b optimized to1.10 ␲. The dotted curves in (b) and (d) are the calculated spectrum of a transform-limited 4-ps pulse, which is given as a reference.

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Fig. 9. Calculated nonlinear phase shift (a), pulse gain (b), and peak carrier density (c). The repetition frequencies of the input pulses
are 10.5 (crosses), 42 (triangles), and 168 (circles) GHz.

The other fundamental SOA parameters were derived
on the basis of the experimentally obtained parameters(Table 2) and are listed in Table 3. The ␣ parameter isderived from Eqs. (7) and (10) as␣ ⬅ ⫺2k 0⫽ ⫺2

dn r /dn c
dg/dn c⌬⌽ maxln G 0.

(11)

The equilibrium excess carrier density n c 0 was obtained
from this equation:
Fig. 10. Dependences of the calculated nonlinear phase shifts of
the injection current to the SOA. The repetition frequencies are
10.5 (crosses), 42 (triangles), and 168 (circles) GHz.

n c0 ⫽

1 P sat ln G 0
V

ប␻

.

(12)

The differential gain and the nonlinear change in refractive index were derived from Eqs. (7) and (10), respectively, with the value of n c 0 .
Here, note that the value of n c 0 is listed in Table 3
rather than in Table 2. We do not explicitly need thevalue of n c 0 for our simulation because of assumptions ofa linear change in refractive index [Eq. (5)] and linear
gain [Eq. (6)] that are bases of our model.
Figure 7 shows measured and calculated spectra of the1546.5-nm CW light at the SOA output. Here the phaseof the CW light was being cross-phase modulated by astream of 2.0-ps 1560-nm control pulses at 42 GHz. Eachcross-phase-modulation (XPM) spectrum was split into
discrete components with a spacing of 42 GHz (0.34 nm).
Note that this repetition frequency is above the SOA’s cutoff frequency 关 (60 ps) ⫺1 ⫽ 17 GHz兴 . Recovery of the refractive index should therefore be almost linear over timeat this frequency, as is shown in Fig. 3(b).16 The calculated XPM spectra match the measured spectra acrossthe range from low to high amounts of input pulse energy.
Our simulation revealed that the nonlinear phase shifts⌬⌽ NL [see Fig. 4(a)] induced by the 42-GHz control pulses
were 0.89␲ [Fig. 7(a)], 0.58 ␲ [Fig. 7(b)], and 0.37␲ [Fig.
7(c)].

Fig. 11. Dependences of the calculated nonlinear phase shift (a)and pulse gain (b) on the differential gain. The unsaturated
gain is assumed to be either 7000 (solid curves) or 700 (dashedcurves).

C. Comparison of Measured and Calculated AllOptically Switched Outputs at 40 GHz
Figure 8 shows measured and calculated autocorrelation
traces and spectra of wavelength-converted pulses at theoutput of the DISC.16 In this experiment the inputs to
the DISC were a 42-GHz stream of 5-ps 1548-nm pulses

Ueno et al.

and 1560.2-nm CW light; 7-ps switching windows were
all-optically formed in the DISC. The CW light wasgated through these windows to generate the 7-ps
1560-nm pulses at the output, which are shown in Fig. 8.
When the interferometer’s phase bias ⌬⌽ b was set to exactly ␲, however, the extinction ratio of the output pulses
was very poor [Fig. 8(a)]. In this case, the output pulse’s
spectrum lacked its central component [Fig. 8(b)].
Here, note that the phase bias between the two interferometer components should be adjusted not to ␲ but toan optimal value that is slightly greater than ␲, such thatthe two interference components outside the switchingwindow interfere destructively with each other [i.e.,phase difference ⌬⌽(t) ⫽ ⫹1.0␲ at t ⬍ t 0 or t ⬎ t 0⫹ ⌬t in Fig. 4(b)]. In fact, we obtained good extinction
for the stream of output pulses [Fig. 8(c)] by optimizing⌬⌽ b to 1.10␲.16 Moreover, the spectrum for these pulses
has a smooth envelope [Fig. 8(d)]. All these results forthe output of the all-optical switch (DISC) were reproduced surprisingly well by our simulation, which was
based on Eqs. (1)–(8) and the parameters in Table 2.
There was also a good match between calculated andmeasured transmittance of the DISC (the ratio of thepeak power level of the output pulses to the power of the
input cw).

D. Designing Semiconductor Optical Amplifiers for the
Nonlinear Phase Shift Induced by UltrahighRepetition Pulses
1. Dependencies of the Nonlinear Phase Shift on the
Repetition Frequency and the Injection Current
The nonlinear phase shift imparted to the optical pulses
by the SOA (Figs. 6 and 7) decreases with the repetition
frequency. As a step toward the processing of signals at
much higher repetition frequencies (in the 160–640-GHz
range) we studied the dependencies of the nonlinear
phase shift on the conditions of operation (the repetition
frequency and the injection current), the parameters ofthe material (the nonlinear change in refractive index,
the differential gain, the ␣ parameter, and the carrier lifetime), and the structural parameters (the dimensions of
the active layer and the optical confinement factor).
Figure 9(a) shows the dependence of the calculated
nonlinear phase shift on the repetition frequency. The
gain for the input pulses and the peak carrier density are
shown for reference in Figs. 9(b) and 9(c), respectively.
The peak carrier density in Fig. 9(c) represents the relative carrier density just before the arrival of each control
pulse [Fig. 3(a)]. When the control pulses have little energy, the increase in the nonlinear phase shift with thepulse energy is linear. As the pulse energy is increased
to increase the nonlinear phase shift, the phase shift
starts to enter saturation because of gain compression[Fig. 9(b)]. Although the gain falls rapidly with increasing amounts of pulse energy, the nonlinear phase shift
does not decrease but converges to a maximum value, behaving in a manner that is similar to that of the extracted
pulse energy in the standard pulse-amplification
regime.34 This behavior is the result of the direct relation of nonlinear phase shift ⌬⌽ NL to the extracted pulse
energy, E out ⫺ E in :Vol. 19, No. 11 / November 2002 / J. Opt. Soc. Am. B

2581⌬⌽ NL ⫽ k 0 dn r /dn c ⌬n c ⌫L
⫽ k 0 dn r /dn c

1 E out ⫺ E in
ប␻

V

⌫L.

(13)

When the repetition frequency is 10.5 GHz, which islower than the carrier’s cutoff frequency ( ␶ c ⫺1
⫽ 16.7 GHz), the depletion of the carrier density by one
pulse has almost recovered by the time the next pulse arrives, as is indicated by the peak carrier density’s being
relatively close to that at the lowest level of the input
pulse energy [Fig. 9(c)]. At this frequency, saturation of
the nonlinear phase shift behaves in a manner similar tothat of pulse amplification.
When the repetition frequency is greater than the cutoff frequency, as is the case at 42–168 GHz, the carrier
density no longer recovers before the arrival of each input
pulse. The values of gain fall below those at 10.5 GHz[Fig. 9(b)]. As a result, there is less nonlinear phase shift
than at 10.5 GHz [Fig. 9(a)]. We believe that these are
typical conditions for SOA inside an all-optical switchthat is being operated at an ultrahigh repetition frequency. In our simulation of these highly saturated conditions we found that the nonlinear phase shift is inversely proportional to the repetition frequency, which isindicated to some extent by Fig. 9(a). More precisely, themaximum nonlinear phase shift induced by a train of sufficiently strong input pulses is inversely proportional to
the repetition frequency as long as the repetition frequency ( f REP) is sufficiently greater than the carrier’s
cutoff frequency and sufficiently lower than the inverse of
the pulse width (⌬t pulse), i.e.,1␶c⬍ f REF ⬍1⌬t pulse.

(14)

Figure 10 shows the dependencies of the calculated
nonlinear phase shifts on the injection current to the
SOA. As indicated in the figure, we have also found thatthe nonlinear phase shift under a strong saturation SOA
condition increases proportionally to the injection current(the dashed guidelines are for reference and indicate perfect proportionality). On the basis of Figs. 9 and 10 weconclude that, although the nonlinear phase shift at ultrahigh repetition frequency decreases with frequency, it
is possible to restore the shift to ␲ by increasing the injection current.
These theoretical proportionalities to both the repetition frequency and the injection current will be helpful inboth the design and the characterization of SOAs for use
in all-optical switching. The reason for these proportionalities is expressed by the rate equation, Eq. (8). Whenthe repetition frequency is much larger than the carrier’s
cutoff frequency, the second term on the right-hand-sideof Eq. (8) is negligible in comparison with the other two
terms, that is, those that represent the carrier injection
and the stimulated carrier recombination by the input
pulses. These proportionalities are compared in Section
4 with their equivalents obtained by measurement.

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J. Opt. Soc. Am. B / Vol. 19, No. 11 / November 2002

2. Dependencies of the Nonlinear Phase Shift on the
Material and Structural Parameters of the
Semiconductor Optical Amplifier
Figure 11(a) indicates the dependence of the calculated
nonlinear phase shift on the unsaturated gain (i.e., its dependence on the differential gain, dg/dn c ). The corresponding pulse gains are shown for reference in Fig. 11(b).
The values with the unsaturated gain of 700 (dashedcurves) in Figs. 11(a) and 11(b) show, for reference, thesame phase shifts and pulse gains at 168 GHz as were already shown in Figs. 9(a) and 9(b), respectively. The
solid curves show the phase shifts and pulse gains when
an increase by a factor of 10 in the SOA’s unsaturated
gain is assumed. As is indicated in the figures, the nonlinear phase shift increases under the unsaturated SOA
conditions but does not increase at all under the highly
saturated conditions in which we are most interested
here.
The smallness of the dependence on the unsaturated
gain (equivalently, on the differential gain) leads us to thenegative conclusion that the ␣ parameter [Eq. (11)], which
has been widely used as the standard measure of nonlinearity within SOAs, is not a good measure of the nonlinear phase shift at ultrahigh repetition frequencies. For
example, the ␣ parameter decreases when the differential
gain of a SOA increases, but the saturation level of thenonlinear phase shift remains constant, according to theresults shown in Fig. 11(a).
The dependencies of the nonlinear phase shift on theother material parameters, i.e., the nonlinear change inthe refractive index and the carrier lifetime, are clear.
The increase in the nonlinear phase shift should be proportional to the nonlinear change in the refractive index,⌬n r /⌬n c , according to Eq. (5). Note that carrier lifetime
␶ c does not appear in Eq. (5) because it does not affect thenonlinear phase shift as long as the repetition frequency
is sufficiently larger than the cutoff frequency of the carriers ( ␶ c ⫺1 ).
We also consider the dependencies of the nonlinear
phase shift on the SOA’s structural parameters. First,
the nonlinear phase shift in Eq. (5) appears to increase
with the SOA length, L. We have theoretically concluded, however, that the nonlinear phase shift is independent of L when the injection current (not the injection
current density) is kept constant. This is so because theincrease in the nonlinear phase shift with L is compensated for by the decrease with the carrier density that iscaused by the decrease in injection current density. In
contrast, the nonlinear phase shift increases with L when
the injection current density (not the injection current) is
kept constant.
Here we must recognize that the dependence on L mentioned above is a guideline that indicates the theoreticallimit in the representation that we are using. Changing
L while keeping the injection current constant will affectcertain other conditions inside the SOA, such as its temperature and the quantum efficiency of carrier injection.
We must take these factors into account separately fromthe theoretical predictions.
We used a similar approach to study the theoretical dependencies of the nonlinear phase shift on the otherstructural parameters. We concluded that the nonlinear

Ueno et al.

phase shift is proportional to the optical confinement factor (⌫, the ratio of the optical field confined to the active
layer) and is inversely proportional to both the thickness(d a ) and the width (W) of the active layer. We summarize these dependencies as a single proportionality:⌬⌽ NL ⬀ ⌫

I op

(15)

d aW

or
⌬⌽ NL ⬀⌫d aW

I op .

(16)

Relation (15) is suitable for intuitive understanding andindicates that the nonlinear phase shift increases with ⌫and also with the cross-sectional injection current density.
Relation (16), however, is suitable for use in optimizing
the shape of the SOA’s active layer in the following way:
Both ⌫/d a and ⌫/W have maxima, which are functions ofd a and W, respectively. This is a well-known phenomenon that is used in the design of semiconductor lasers.
So we conclude from Relation (16) that there are optimum
d a and W for the active layer that maximize the value of⌫/(d a W). The optimum d a is equal to the optimum W,
which means that the optimum shape of the cross sectionof the active layer is square. Such an active layer maximizes the contribution of the injection current to the nonlinear phase shift.
All the dependencies of the nonlinear phase shift that
we have discussed in this subsection are summarized inTable 4.
E. New Method for Measuring the Nonlinear Phase
Shift
In general, the XPM spectrum is broadened as the nonlinear phase shift is increased (Fig. 7). The XPM specTable 4. Calculated Dependences of the
Nonlinear Phase Shift ⌬⌽ NL Induced by SOAs
with Input Pulses at Relatively High Repetition
Frequencies f REP

Property
Operating conditionfR
I op
Material parameterdn r /dn c

Description

Dependency
of Nonlinear
Phase Shift,⌬⌽ NL

Repetition frequency
Injection current⌬⌽ NL ⬀ 1/f REP⌬⌽ NL ⬀ I op

Nonlinear change inrefractive index
Differential gain
Alpha parameter
Carrier lifetime⌬⌽ NL ⬀ dn r /dn c

dg/dn c␣␶cStructural parameter⌫
Optical confinement
factorL
Active-layer lengthdaWIndependent
Independent
Independent⌬⌽ NL ⬀ ⌫Independent(for a fixed I op)
Active-layer thickness ⌬⌽ NL ⬀ 1/d a
Active-layer width⌬⌽ NL ⬀ 1/W

Ueno et al.

Vol. 19, No. 11 / November 2002 / J. Opt. Soc. Am. B

2583

ratio of the first blue spectrum component to the central
spectrum component, 兩 a 1 兩 2 / 兩 a 0 兩 2 , which were obtainedwith control pulses at repetition frequencies of 42 GHz(triangles) and 168 GHz (circles). We varied the amountof energy per pulse to change the nonlinear phase shift.
Note that the correlation between the two values is almost independent of the repetition frequency.
In a similar way, we have numerically demonstrated
that the correlation between ⌬⌽ NL and 兩 a 1 兩 2 / 兩 a 0 兩 2 is independent not only of the repetition frequency [under theinequality condition given above, Eq. (14)] but also of theinjection current, the material parameters, and the structural parameters. The correlation between ⌬⌽ NL and兩 a 1 兩 2 / 兩 a 0 兩 2 as calculated with various parameter values
stayed close to the following function obtained by fitting:⌬⌽ NL ⫽ 0.938共 兩 a 1 兩 2 / 兩 a 0 兩 2 兲 0.372.

Fig. 12. Correlation of the calculated intensity ratios between
components of the XPM spectrum with the calculated nonlinear
phase shift induced by pulses at 42 GHz (triangles) and 168 GHz(circles). The dashed line in (a) is a line of fit to the set of data
points. The dashed curve in (b) was calculated with Eq. (22) of
the text.

(17)

This function produces the dashed curve shown in Fig.
12(a). Inasmuch as the correlation between ⌬⌽ NL and兩 a 1 兩 2 / 兩 a 0 兩 2 is independent of all the parameters that determine the phase shift, we propose using the XPM spectrum and Eq. (17) as a way of measuring the nonlinear
phase shift.
Our analytical derivation of the above correlation follows. In the derivation we assumed that (a) the pulse
width is negligibly short, (b) the gain modulation is negligibly small, and (c) the carrier’s cutoff frequency is sufficiently low. Under these assumptions, the modulationof the nonlinear phase by the train of control pulses is ultimately simplified to a saw-toothed shape [Fig. 13(a)]that has only two parameters, ⌬⌽ NL and ⌬T (⬅1/f REP).
The nonlinear phase is expressed in terms of these twoparameters in the following way:
E 共 t 兲 ⬅ exp共 ⫺iht 兲⫽ a 0 ⫹ a 1 exp共 ⫺i ␻ t 兲 ⫹ a 2 exp共 ⫺2i ␻ t 兲⫹ a 3 exp共 ⫺3i ␻ t 兲 ⫹ ...
⫹ a ⫺1 exp共 i ␻ t 兲 ⫹ a ⫺2 exp共 2i ␻ t 兲 ⫹ a 3 exp共 3i ␻ t 兲⫹ ...,(18)whereh ⬅ ⌬⌽ NL /⌬T,(19)

␻ ⬅ 2 ␲ /⌬T ⫽ 2 ␲ f REP .

(20)

The intensity of each Fourier-transformed XPM component is derived as
Fig. 13. Analytical derivation of the XPM spectra: (a) a simplified model of the nonlinear phase shifts, (b) 42-GHzmodulated XPM spectrum, analytically derived on the assumptions of the simplified phase-shift model and negligible gain
modulation.

trum is composed of discrete components at a spacing of
the repetition frequency. In carrying out the numericalsimulation we noticed a useful correlation between thetwo phenomena. Figure 12(a) shows an example of thecorrelation between nonlinear phase shift ⌬⌽ NL and the

an ⫽

⫽

1
⌬T

冕

⫹⌬T/2

⫺⌬T/2

exp共 ⫺iht 兲 exp共 ni ␻ t 兲 dt

2
i 共 h ⫺ n ␻ 兲 ⌬Tsin

共 h ⫺ n ␻ 兲 ⌬T22i2n ␲ ⫺ ⌬⌽ NL.
⫽⫺
sin
2n ␲ ⫺ ⌬⌽ NL
2(21)

Thus the ratio between the XPM components is expressed
as

2584

兩 a 1兩 2
兩 a 0兩

⫽2

J. Opt. Soc. Am. B / Vol. 19, No. 11 / November 2002

冉

⌬⌽ NL
2 ␲ ⫺ ⌬⌽ NL

冊

2

sin2

2 ␲ ⫺ ⌬⌽ NL
2

冒

sin2

Ueno et al.

g⌬⌽ NL
2.

(22)

The XPM spectrum at a ⌬⌽ NL of 0.5␲ as analytically calculated by use of Eq. (21) is shown in Fig. 13(b). It provides a fairly good reproduction of the measured XPM
spectrum of Fig. 7(b), for which the ⌬⌽ NL was estimated
The correlation between ⌬⌽ NL andto be 0.58␲.
兩 a 1 兩 2 / 兩 a 0 兩 2 that is derived from Eq. (22) is shown by thedashed curve in Fig. 12(b). The numerically calculated
data on the correlation (circles and triangles) are again
surprisingly close to the analytically derived curve for thecorrelation. Note that
(1) The asymmetric XPM spectrum was analyticallyrelated to the saw-toothlike profile of the nonlinear phaseshift,(2) The gain modulation has very little effect on the
shape of the XPM spectrum, and(3) This method for measuring the nonlinear phaseshift will be applicable to other materials that have relatively long relaxation times as well as to semiconductormaterials.

4. MEASURED NONLINEAR PHASE SHIFTSAT REPETITION FREQUENCIES IN THE
40–160-GHz RANGE
We applied the new method of measurement describedabove to the experimental study of the nonlinear phaseshift. We applied 1.5-ps pulses at repetition frequencies
of 42–168 GHz and a 1547-nm CW probe light to the SOA,

Fig. 15. Model of the nonlinear phase shift induced by a SOA by
input pulses at a high repetition rate.

observed XPM spectra at wavelengths near that of theCW, and used Eq. (17) to obtain the corresponding nonlinear phase shifts, as are summarized in Fig. 14.
Figure 14(a) shows the measured nonlinear phaseshifts induced by trains of pulses at 42 (crosses), 84 (triangles), and 168 (circles) GHz, as functions of the input
pulse energy. The injection current to the SOA was set to
250 mA. The saturation of the nonlinear phase shiftseen in Fig. 14(a) is similar to that for the calculated
phase shift in Fig. 9(a). The nonlinear phase shift atsaturation was approximately inversely proportional to
the repetition frequency.
Figure 14(b) shows the dependence of the nonlinear
phase shift on the injection current. The input pulse energy was set to 100 fJ. The dashed guidelines are for reference and indicate proportionalities. As the figure
shows, the nonlinear phase shift was approximately proportional to the injection current. Thus the theoretically
predicted proportional dependencies on the repetition frequency and the injection current,⌬⌽ NL ⬀

I op
f REP

,

(23)

had been verified.
On the basis of these experimental results we have proposed an intuitive model for the nonlinear phase shift induced in a SOA by input pulses at high repetition frequencies. When a SOA receives a train of input pulses at
a high repetition rate, as depicted in Fig. 15, each input
pulse almost completely depletes the carrier number
within the SOA. As a result, the number of carriers thatare recombined on the arrival of each pulse is approximately equal to the number of carriers injected between
the times of arrival for the previous pulse and for thepresent pulse. The nonlinear phase shift that is produced by the arrival of each pulse is proportional to thenumber of recombined carriers. Thus the nonlinear
phase shift is inversely proportional to the repetition frequency and proportional to the injection current.

5. REGULAR SWITCHING AT 160 GHz
Fig. 14. Nonlinear phase shifts as measured from respective
XPM spectrum ratios at the SOA’s output, induced by 1.5-ps
pulses at 42 (crosses), 84 (triangles), and 168 (circles) GHz.

We performed experiments with switching at 168 GHz todemonstrate the behavior of the SMZ-type all-opticalswitch,35 for which the nonlinear phase shift mentioned
above is applied. Figure 16 shows the setup for our ex-Ueno et al.

periments with switching. The high-repetition-rate trainof control pulses was generated with an actively modelocked fiber-ring laser (Pritel, Inc., Naperville, Ill.), fiber
multiplexers, and Er-doped fiber amplifiers (EDFAs).
The mode-locked fiber laser generates a 10.496-GHz trainof 1565-nm pulses. Four stages of Mach–Zehnderinterferometer-type multiplexers precisely multiplex therepetition frequency from 10.496 to 168 GHz. The pulse
width of the generated 168-GHz pulses was measured
with an autocorrelator to be 2.0 ps. The maximum pulse
energy coupled to the SOA was 100 fJ at 168 GHz. The
1547-nm CW light was generated with an external-cavity
semiconductor laser. The CW light and the 168-GHz control pulses were combined in a 50:50 coupler for input tothe SOA via a pigtailed input fiber. The coupling coefficient was 50%.
The all-optical switch used in the experiment was a
DISC, i.e., the switch schematically depicted in Fig. 1(c).
We used a calcite crystal, a quarter-wave plate, aBabinet–Soleil phase retarder, and polarizers to build theVol. 19, No. 11 / November 2002 / J. Opt. Soc. Am. B

2585

Mach–Zehnder interferometer (MZI) inside the DISC.
The calcite crystal is birefringent to one of the two orthogonally polarized components and thus gives a delay
time of 2.0 ps to the other component. An EDFA was inserted between the SOA and the Mach–Zehnder interferometer.
Figure 17 shows typical DISC outputs at 168 GHz as
observed with a streak camera [Fig. 17(a)], an autocorrelator [Fig. 17(b)], and a spectrum analyzer [Fig. 17(c)].
Each input pulse had 100 fJ of energy and an average
power level of 16.8 mW, and the level of input CW power
was 10 ␮W. A large extinction ratio of the output pulses(at least 12, but measurement was limited by the streak
camera’s time resolution) was obtained, as shown in Fig.
17(a). The output pulse width was measured with the
autocorrelator to be 2.0 ps, which was a good match withthe calcite’s delay time. This result indicates that the ultrafast response mechanism of the SMZ switch is stillvalid in switching at 168 GHz. The output spectrum’s
envelope approximately matched the calculated spectrum

Fig. 16. Experimental setup for all-optical switching at 168 GHz. The DISC switch structure was used.
larizers; R, phase retarder; Q, quarter-wave plate; ML, mode locked; Mod, modulator.

Mux, multiplexer; P’s po-

Fig. 17. Observed 168-GHz switch output waveforms: (a) streak-camera image, (b) autocorrelator trace, and (c) XPM spectrum. The
nonlinear phase shift involved here was measured as 0.33␲ on the basis of the XPM spectrum (d) observed at the SOA output.

2586

J. Opt. Soc. Am. B / Vol. 19, No. 11 / November 2002

Fig. 18. Comparison of typical output waveforms when theswitch is driven at 168 GHz by digitally coded control pulses with(a) weak and (b) strong CW input.

Fig. 19. Typical eye diagram after 168-GHz random switching(wavelength conversion). The eye diagram was measured after
demultiplexing of the 168-Gbit/s wavelength-converted output to10.5-Gbit/s signals because of the limited bandwidth of our sampling scope.

for a transform-limited 1.6-ps pulse [dashed curve in Fig.
17(c)], which suggests that the output pulse contains little
chirping.17
We measured the nonlinear phase shift under the condition of this 168-GHz switching. The XPM component
ratio observed at the SOA’s output indicated [i.e., the result in Fig. 17(d) and Eq. (17) indicated] that the nonlinear phase shift in this switching was 0.33␲.

6. RANDOM SWITCHING (WAVELENGTHCONVERSION) AT 160 GHz DRIVENBY DIGITALLY CODED SIGNAL PULSES
Using the same DISC, we demonstrated wavelength conversion [i.e., the DISC was controlled by a stream of digitally coded pulses at 168 GHz (Refs. 21, 24, and 25)] andmeasured the nonlinear phase shift involved under theconditions of operation. To generate the pseudorandom

Ueno et al.

control pulses we used a 10-GHz LiNbO3 modulator
driven by a 12.5-GHz pulse-pattern generator (Anritsu)
between the synchronized fiber-ring laser and the multiplexers of the arrangement in Fig. 16. As has been wellestablished, relatively strong CW input is required for operating the SMZ-type all-optical switch in such a randomswitching regime.12,36 Figure 18 shows a comparison of
typical output waveforms with weak [Fig. 18(a)] andstrong [Fig. 18(b)] CW input. Here, a 16-bit-long patternof control pulses temporarily replaced the pseudorandom
pulses, which allowed us to synchronize the waveforms
with our streak camera. The strong CW input in our experiment has effectively reduced the relaxation time of
the SOA and consequently suppressed the signal-patterninduced effects, as shown in Fig. 18(b). This output behavior was confirmed by numerical simulation with Eqs.
(1)–(8).
For the pseudorandom operation of wavelength conversion at 168 GHz, the levels of input pulse energy and CW
power were optimized to 60 fJ and 24 mW, respectively.
The pseudorandom pattern had a length of 2 7 – 1 and amark ratio of 50%. Figures 19 and 20 show our results
for the 168-GHz wavelength conversion. Figure 19
shows a typical output eye diagram as measured with asampling scope. Our sampling scope’s bandwidth was 30
GHz, so the output eye was measured after the 168-GHz
output was demultiplexed by a SMZ demultiplexer10,11 to
a 10.496-GHz signal. Under these operating conditions,error-free wavelength conversion (i.e., wavelength conversion with error rates below 10⫺10) had successfully been
demonstrated.25 Figure 20 shows the observed pulse
waveforms and spectra under the wavelength-conversion
conditions. The extinction ratio of the output pulses was
12 [Fig. 20(c)]. The output pulse width as measured froman autocorrelation trace [Fig. 20(d)] was 2.0 ps. The output pulse width was consistent with the width of theswitching window, which is in principle determined by thedelay time in the calcite crystal. This indicates that theultrafast-response mechanism of the SMZ switches isvalid for both regular and random switching at 168 GHz.
The nonlinear phase shift involved in the pseudorandom wavelength conversion was measured as 0.2-0.3␲.
In the measurement of the nonlinear phase shift, 168GHz continuous pulses, instead of the 168-Gbits/s pseudorandom pulses, had been applied to the DISC to generate the XPM spectrum with discrete spectral components
at the SOA output, as is shown in Fig. 20(e). As had been
expected, the strong CW input decreased the nonlinear
phase shift in random switching from its 0.33␲ value inregular switching.
Thus the performance in 168-GHz wavelength conversion was limited by the nonlinear phase shift induced by
our SOA, which was fabricated according to a conventional design for use with high-gain high-saturationpower applications. We believe that it will be possible toenhance the nonlinear phase shift by applying the new
SOA design that was discussed in Subsection 3.D.

7. CONCLUSIONS
We have studied the nonlinear phase shifts induced by asemiconductor optical amplifier at driving frequencies in
the range from 42 to 168 GHz. In the study we used

Ueno et al.

Vol. 19, No. 11 / November 2002 / J. Opt. Soc. Am. B

2587

Fig. 20. Observed pulse waveforms and spectra under the wavelength-conversion conditions: For the input pulses, (a) a streak-camera
image and (b) an autocorrelation trace, and for the pulses at the DISC output, (c) a streak-camera image, (d) an autocorrelation trace,and (e) an XPM spectrum. The nonlinear phase shift involved in this operation was measured as 0.2–0.3␲ on the basis of the XPM
spectrum observed at the SOA output.

theory to deduce proportional dependencies of the nonlinear phase shift on the repetition frequency (⌬⌽ NL⬀ f REP⫺1 ) and on the injection current (⌬⌽ NL ⬀ I op), bothof which had been believed to be more complicated because of the interplay between strong gain modulationand the nonlinear change in refractive index. We experimentally verified both proportionalities, which are bothvery simple because the SOA carriers at such ultrahighrepetition frequencies are strongly depleted. The two dependencies indicate, in a straightforward manner, that increasing the injection current is a promising way to compensate for the decrease of the nonlinear phase shift at
ultrahigh repetition frequencies.
We have also theoretically determined all the other basic design factors for SOAs, i.e., the dependencies of thenonlinear phase shift on the material and structural parameters of the SOAs. This includes the unexpected conclusion that the ␣ parameter is not necessarily an appropriate measure. Although most of the results anddesigns presented in this paper were limited to those in
the regular switching regime, we believe that most of thedesigns (such as the dependence of the nonlinear phaseshift on the injection current and the optimization of thecross section of the SOA active layer) will be applicable toreal applications in the random-switching regime such as
wavelength conversion and 3R regeneration.
The results presented in this paper will be helpful inthe design of SOAs as ultrafast nonlinear elements insymmetric Mach–Zehnder switches and also in the development of new nonlinear optical materials and waveguidestructures for use in such ultrahigh-speed signal processors operating in the 160–640 GHz range and to push the
speed limit for such devices into the region above 1 THz.

ACKNOWLEDGMENTS
We gratefully acknowledge valuable technical discussions
on semiconductor optical amplifiers with Takemasa
Tamanuki, Hiroshi Hatakeyama, and Tatsuya Sasaki.
We thank Jun’ichi Sasaki and Akio Furukawa for collaboration on the integration of the SMZ demultiplexer for use
in the wavelength-conversion experiment. Part of thisresearch was performed under the management of the
Femtosecond Technology Research Association, supportedby the New Energy and Industrial Technology Development Organization.

2588

J. Opt. Soc. Am. B / Vol. 19, No. 11 / November 2002

Present address, Department of Electronic Engineering, University of Electro-Communications, 1-5-1 Chofugaoka, Chofu, Tokyo 182-8585, Japan; e-mail address,
y.ueno@osa.org.

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