Asymmetric optical loop mirror:analysis of an all-optical switch

Michael G. Kane, Ivan Glesk, Jason P. Sokoloff, and Paul R. Prucnal

We present an analysis of the optical loop mirror in which a nonlinear optical element is asymmetricallyplaced in the loop. This analysis provides a general framework for the operation of a recently inventedultrafast all-optical switch known as the terahertz optical asymmetric demultiplexer. We show that aloop with small asymmetry, such as that used in the terahertz optical asymmetric demultiplexer, permitslow-power ultrafast all-optical sampling and demultiplexing to be performed with a relatively slow opticalnonlinearity. The size of the loop is completely irrelevant to switch operation as long as the requireddegree of asymmetry is accommodated. This is therefore the first low-power ultrafast all-optical switchthat can be integrated on a single substrate.

Key words: Optical loop mirror, semiconductor optical amplifier, optical switching, optical demultiplex-ing.

Introduction

Various all-optical switches have been proposed anddemonstrated. All of these switches use an opticalnonlinearity of some type, since all-optical switchingrequires a nonlinear function to be performed on oneor more optical fields. The optical nonlinearities areproduced by electronic excitations of a material,usually a semiconductor or a glass. Nonlinearitiestend to be relatively strong and slow when thefrequency of an optical field resonates with the fre-quency of a long-lived excitation, permitting slowswitching with small switching energy. An exampleof such a nonlinearity is modification of the refractiveindex in a semiconductor through the photogenera-tion of electron-hole pairs. On the other hand,excitations produced when the frequency of an opticalfield is detuned from the frequencies of excitation orwhen the excitations are short lived lead to relativelyweak, fast nonlinearities. These permit fast switch-ing but require large switching energy. An example

M. G. Kane is with Siemens Corporate Research, Inc., 755College Road East, Princeton, New Jersey 08540; he is also withthe Department of Electrical Engineering, Princeton University,Princeton, New Jersey 08544. I. Glesk, J. P. Sokoloff, and P. R.Pruenal are with the Center for Photonics and OptoelectronicMaterials, Department of Electrical Engineering, Princeton Univer-sity, Princeton, New Jersey 08544.

Received 16 June 1993; revised manuscript received 22 April1994.

0003-6935/93/296833-10$06.00/0.e 1994 Optical Society of America.

of this type of excitation is the optical Kerr effect inoptical fiber, arising from nonresonant excitation ofbound electrons in the silica glass.

The fast response time of the Kerr nonlinearity insilica fiber has prompted the development of severaltypes of all-fiber switches, capable of switching pulsesthat are a picosecond long or less. Because theoptical Kerr effect in silica fiber is weak, these devicesrequire large switching energies or long lengths offiber to extend the interaction distance, or both.14

In order to reduce the size and the power require-ments of an all-optical switch, researchers have inves-tigated other materials with stronger nonlinearities.5-7

In particular, switches have been constructed byincorporation of a nonlinear element (NLE) in a loopmirror, permitting the switch to take advantage ofthe loop's highly stable zero-background output.8' 2

However, in most switches of this type the basicspeed-power trade-off of the nonlinear element re-mains. For example, Eiselt demonstrated a switchwith a semiconductor optical amplifier in a fiber loop;the amplifier's strong, slow nonlinearity permits theswitch to operate with a switching energy of only 0.4pJ, but the time resolution of the switch is limited bythe amplifier's 400-ps gain recovery time.'2

In contrast, a terahertz optical asymmetric demul-tiplexer (TOAD) has been demonstrated that directlyaddresses the problem created by the speed-powertrade-off by permitting a relatively slow, sensitivenonlinearity to perform ultrafast switching.13 Byplacing a NLE in a short loop mirror with slightasymmetry, one can reduce the time resolution to far

10 October 1994 / Vol. 33, No. 29 / APPLIED OPTICS 6833

less than the recovery time of the optical nonlinearity.Using semiconductor optical amplifiers with a gainrecovery time of 1 ns, researchers have used thisarchitecture to perform optical time-division demulti-plexing at 9 Gbit/s (Ref. 14) and 50 Gbit/s.15

Recently demultiplexing was demonstrated at 250Gbit/s with a bit-error rate of less than 10-9 and aswitching energy of 800 fj.1 6

Our purpose in this paper is to provide a generalanalytical framework for the operation of the TOADswitch and to place it within the broader context ofother switches that use a nonlinear optical element ina loop mirror. After the analysis, we provide experi-mental data to illustrate the main results and de-scribe some applications.

Analysis of Switch Operation

General Formulation

A linear optical loop mirror consists of a 2 x 2 3-dBcoupler with two ports joined. Its properties andapplications are described in Refs. 17-20. Here weconsider a loop mirror in which a NLE is placedasymmetrically in the loop (Fig. 1). The degree ofasymmetry is represented by the distance Ax betweenthe NLE and the midpoint of the loop. The controlis exercised by applying an optical control pulse to theNLE, altering its absorption and/or refractive index,and switching some portion of the input signal to theoutput for the duration of the sampling windowcreated by the control pulse. The optical inputenters the loop through port 1 of the coupler, and theswitched optical output exits from port 2.

AX

M6- Control

Input OutputFig. 1. Optical loop mirror with a nonlinear element (NLE) placedasymmetrically in the loop at a distance Ax from the loop's

midpoint M. The optical control modifies the optical properties ofthe NLE, causing the input signal to be switched to the output.

In the absence of a control signal the switchoperates as a linear loop mirror. The input signalenters port 1 of the coupler with an intensity I, thensplits into two fields of intensity 1/2 at ports 3 and 4,one propagating clockwise (CW) in the loop and theother counterclockwise (CCW). (CW does not standfor continuous wave in what follows.) As it crossesfrom one side of the coupler to the other, the CCWfield undergoes a rr/2 phase delay relative to the CWfield. Then both fields travel the same distance inthe loop. To reach the output at port 2, the CCWfield must cross the coupler again. As a result, at theoutput there is a superposition of two fields havingequal amplitudes but a r phase difference. Theresulting destructive interference prevents opticalpower from exiting from the output, provided thatthe relative polarization of the two fields does notchange as they traverse the loop. Therefore all ofthe optical power entering the input must be reflected.(In actual use an optical isolator can be placed at theinput to absorb the reflected power.) The loop canthus be regarded as a white-light interferometer inwhich the two arms are distinguished by the twodirections of propagation.

If a constant control signal is applied to the NLE,its optical characteristics will be modified. However,the two counterpropagating modes will interfere inthe same way as in the absence of the control signalbecause the counterpropagating fields still experienceidentical propagation characteristics. But if a time-varying control signal is applied to the NLE, theoptical properties of the NLE become time dependent.As a result, the destructive interference that leads tofield cancellation at the output is disturbed. If aportion of the light traveling in the loop experiencesan absorption or an index difference relative to itscounterpropagating complement, light will emergefrom the output of the coupler.

Now we analyze the time-dependent operation ofthe loop in more detail. In our analysis we makeseveral assumptions:

(1) Optical couplers are ideal. Commerciallyavailable couplers approximate this ideal adequately.

(2) The input signal's intensity is sufficiently smallthat it does not modify the optical properties of theNLE, and neither the input nor the control signalinduces nonlinearities in the other parts of the loop.

(3) The relative polarization of the counterpropa-gating CW and CCW fields is maintained as theytraverse the loop.

(4) No reflections occur at the interface betweenthe NLE and the loop medium. This assumption iseasily approximated in practice by use of antireflec-tion coatings.

(5) The input signal can be distinguished from thecontrol signal. One realizes this in practice by keep-ing the control signal out of the loop or by usingsuitable wavelength or polarization filters. Simi-larly we ignore noise that may be generated by the

6834 APPLIED OPTICS / Vol. 33, No. 29 / 10 October 1994

NLE, since it can largely be blocked with suitableoutput filters.

The total field at port 2 of the loop is the sum of twocomponents, one from the CW field and one from theCCW field. The output intensity is

cnI0 .t(t) = 8r I Ecw(t) + Eccw(t) 12, (1)

where n is the index of the medium and Ecw ccw)(t)represents the complex amplitude of the CW CCW)field as it exits from port 2. For a monochromaticcontinuous-wave optical input of frequency o, theamplitudes at port 2 are

Ecw(t) = 21rlin) exp[-iwt + i-lcw(t) - ycw(t)], (2)

Eccw(t) = -( cn )exp[-iw)t + iccw(t) - yccw(t)],

(3)

where Iin is the input intensity incident upon port 1and 'bcw(ccw)(t) and ycw(ccw)(t) represent the time-dependent phase shift and attenuation that the CW(CCW) field emerging from port 2 at time t experi-enced as it traversed the NLE. The phase shift andattenuation can depend on the frequency o. Theexpression for Eccw(t) has a negative sign to accountfor the two w/2 phase shifts that the counterclock-wise field experiences crossing the coupler twice.

The attenuation and phase responses of the NLE tothe control signal can be complicated functions oftime. These responses represent changes in absorp-tion and refractive index, and they depend on thecontrol signal and the properties of the NLE. Thesedetails are not of interest here, although we do makesome observations about certain types of responsecharacteristics later in the paper.

Using exponential notation would be burdensomefor our purposes, so we represent attenuation with-out it, defining the NLE's transmission A(t) for theclockwise and counterclockwise fields as

Acw(ccw)(t) exp[-ycw(ccw)(t)]. (4)

We define the sampling function S(t) as the time-dependent output intensity I)ut(t) divided by the con-tinuous-wave input intensity Iin. Then, combiningEqs. (1)-(4), we get

S(t) = /4{A2w(t) + ACw(t) - 2AcW(t)Accw(t)

x cos[Lw(t) - ccw(t)]1 (5)

This is the general expression describing the time-dependent sampling of the loop's optical input by itsoutput.

Often an NLE is used in a manner in which thetime variation of either the transmission or the phaseshift is dominant. In these cases, we can simplify

Eq. (5), either by approximating cw(t) and PCcw(t) asa constant phase shift +0 or by approximating ACw(t)and Accw(t) as a constant transmission AO. We referto these as amplitude-modulated and phase-modu-lated operation, respectively.

Zero-Length Approximation

We now employ an approximation that we call thezero-length approximation. The condition for valid-ity of this approximation is as follows. We define theloop's time asymmetry At - Ax/V1 P, where v100 isthe speed of light in the loop medium, and similarlywe define the transit time of the NLE asTtransit L/VNLE, where L is the length of the NLEand VNLE is the speed of light in the NLE. Thezero-length approximation applies if Ttransit << At.This approximation leads to a simpler analysis, butlater in the paper we remove it.

In the zero-length approximation we thereforeignore the finite length of the NLE and treat it as anasymmetrically placed point element. Then the am-plitude (and phase) changes experienced by the twofields are identical except for a time shift, and we canintroduce new transmission and phase-shift variableswithout the CW and CCW subscripts:

Acw(t - At) = Accw(t + At) - A(t),

cw(t - At) = kccw(t + At) - (t).

(6)

(7)

Substituting Eqs. (6) and (7) into Eq. (5), we get

S(t) = ¼/,{A2(t + At) + A2(t - At) - 2A(t + At)A(t - At)

x cos[4(t + At) - (t - At)]I. (8)

Loop with Large AsymmetryIt is useful to analyze two special loop configurations.The first is a loop with large asymmetry. Represen-tatives of this type of switch can be found in theresearch described in Refs. 9-12. To define thisconfiguration, we consider the control signal to be apulse of some shape. The NLE responds to thispulse. Before the control pulse appears, the trans-mission A(t) and the phase shift P(t) of the NLE havetheir steady-state values Ao and j0, respectively, andafter some finite time Tcontroi, they return to thesevalues. We define a zero-background transmissionA(t) and phase shift +(t) such that

A(t) A(t) - Ao,

+(t A () - k).

(9)

(10)

Thus these two new variables are zero outside someperiod of duration Tcontrol.

A loop with large asymmetry is one in which theinequality 2At > Tcontrol is satisfied. Continuing touse the zero-length approximation for now, wecan see that the period of time that A(t + At) and4(t + At) are nonzero does not overlap the period oftime that A(t - At) and 4(t - At) are nonzero. (Weassume that successive control pulses are separated

10 October 1994 / Vol. 33, No. 29 / APPLIED OPTICS 6835

by more than 2At + Tontroi)tions to Eq. (8), we get

Applying these condi-

S(t) = A2t+A)+ A0 A(t + At)sin 2t A)

+A2(t - At) + \* -A~i2 4(t - At)+ 4 0 t-tiisn 2 (11)

The first two terms on the right-hand side areidentical to the second two terms, apart from a timeshift 2At. Thus each control pulse causes twononoverlapping sampling windows to be opened be-tween input and output. The two windows areidentical but separated in time by 2At. The durationof each window is Tontroi, and therefore switchingcannot be performed faster than the recovery time ofthe NLE. As a result, in this configuration the basicspeed-power trade-off of the NLE remains.

Loop with Small AsymmetryNow we consider the second special loop configura-tion, representing the configuration used in the TOADswitch.13 In some ways it is the opposite of the loopwith large asymmetryjust described. Now the loop'stime asymmetry is much shorter than the response ofthe NLE to the control pulse; thus the inequality2At << Tontroi is satisfied. In general this yieldsrather complicated results unless we specify that theresponse of the NLE's optical properties to the con-trol pulse consists of a fast transition with a short risetime, followed by a period of slow relaxation with along fall time, with Trise < At << Tl. Thereforewhen we call this a loop with small asymmetry, it iswith the understanding that the small asymmetry iscombined with occasional rapid transitions in thetransmission and/or the phase shift.

To analyze this configuration, we apply the approxi-mations T

rise = 0 and Tfall = oo. Thus at some time tothe transmission A(t) steps from a value ofAo to Al, orthe phase shift cb(t) steps from a value of 4o to 1,, orboth. After the step, variations in A(t) and +(t) areneglected until the next step. By inspection of Eq.(8) we can determine the response of the samplingfunction to the step. Before t = to - At and aftert = to + At, the sampling function S(t) is approxi-mately zero. The first two terms in the braces areapproximately canceled by the third term because ofthe slow variation in the transmission and the phaseshift. But for times to - At < t < to + At thecancelation does not occur, and the sampling functionhas a constant level:

1/4[A + A' - 2AoA cos(4o -

SWt) = for to - At < t < to + At . (12)

0 otherwise

Thus the loop with small asymmetry permits arectangular sampling window to be created from asingle rapid transition, with the width of the sam-pling window determined by the time asymmetry of

the loop. This explains why the TOAD switch canachieve a time resolution far below the recovery timeof the optical nonlinearity. Furthermore, the size ofthe loop is completely irrelevant to switch operationas long as there is enough room for the requiredasymmetry.

The only restriction on operation is that anothersampling window cannot be created until a timegreater than Tfal has elapsed, permitting the nonlinear-ity to recover. But modeling the transmission andthe phase shift as a step function involves twoapproximations whose validity must be addressed.

First, in reality, T rise > 0. As a result, the risingand falling transitions of the rectangular pulse de-scribed by Eq. (12) are not instantaneous, but requirea time Trise. This transition time can be quite small,even for slow nonlinearities such as those associatedwith interband transitions in semiconductors. Thereason is as follows: When we speak of an opticalnonlinearity in a material as fast or slow, we arecharacterizing the time required for the optical prop-erties to return to their normal, linear values afterthe incident optical power is removed. The timerequired for the nonlinearity to turn on can be muchshorter than this, since in the presence of an incidentoptical field the electronic excitations can be inducedrapidly. As a first approximation, the time requiredfor an excited state to be generated is shorter than itsdecay time by a factor equal to the average photonoccupation number of all the field modes capable ofinducing the excitation. The large occupation num-bers achievable with laser excitation permit slownonlinearities to turn on rapidly. For example, gainnonlinearities associated with interband transitionsin semiconductor optical amplifiers can exhibit subpi-cosecond turn-on times.21 22

Second, rfi, is not infinite. As a result, the top ofthe rectangular pulse described by Eq. (12) has adescending slope. In addition, after the pulse hasended, a small residual intensity remains at theoutput for a period Tfall. However, it is straightfor-ward to show that this error is proportional to(At/Trfla) 2 and can be reduced to small values.

Effect of Finite Nonlinear-Element Length

Up to this point, we have considered the asymmetricloop mirror with the approximation that the NLE haszero length. This is usually a good approximation ina loop with large asymmetry, since the loop's asymme-try is generally much larger than the length of theNLE. But in a loop with small asymmetry, such asthe TOAD switch, this approximation leads to anincorrect conclusion. It suggests that the minimumduration of the sampling window is determined ulti-mately by the rise time of the nonlinearity, since inprinciple the NLE can be placed arbitrarily close tothe center of the loop. In this subsection we showthat the finite length L of the NLE leads to anadditional limitation: a nonlinear element can pro-duce a sampling window with a total duration noshorter than twice the NLE's transit time, with a full

6836 APPLIED OPTICS / Vol. 33, No. 29 / 10 October 1994

width at half-maximum (FWHM) approximately equalto the transit time. If a typical semiconductingmaterial is used as the nonlinear element, this impliesthat, with a 100-rim-long NLE, the FWHM of thesampling window is limited by the finite-length effectto -1 ps.

If the NLE has finite length, we must be morespecific about the definition of the time asymmetryAt. We define it as the propagation delay from themidpoint of the loop to the nearest edge of the NLE.The midpoint of the loop is defined as the point atwhich half the time required to traverse the loop haselapsed, including the NLE's transit time.

Equation (5) expresses the sampling function interms of the transmission functions Acw(t) andAccw(t) and the phase shifts cw(t) and)ccw(t). With the zero-length approximation the

two transmission functions are time-shifted replicasof each other, as are the phase shifts [Eqs. (6) and (7)].In general this is not true for a finite-length NLE.Instead Acw(t) can have a different shape from Accw(t), as can 4cw(t) from ccw(t).

First, we analyze amplitude-modulated operation.We must work in terms of the attenuation y(t) insteadof the transmission A(t); these are related throughEq. (4). The attenuation ycw(t) represents the at-tenuation that the CW field emerging from port 2 ofthe coupler at time t experienced as it traversed theNLE; the attenuation is similar for the CCW field.To calculate the attenuation, we represent the NLE'sattenuation per unit length at a point x in the NLEand a time t as at(x, t), where the origin x = 0 is at theend of NLE nearest to the loop's midpoint. Weperform an integral through the NLE of the attenua-tion per unit length in the frame of reference of theCW or the CCW fields, which propagate at a speedVNLE. The two moving frames of reference aretreated by the introduction of an additional timeintegral and a delta function that moves with theappropriate frame of reference:

ycw(t) = f dr dx a(x, T)

X 8(T - t + T1/2 - At - X/vNLE), (13)A AL

yccw(t) = dT dx a(x, T)

X 8(T - t + T1/2 + At + X/VNLE). (14)

Here we have introduced a fixed time delay T1/ 2,representing the propagation delay from the loop'smidpoint to port 2 of the main coupler.

In order to make the effects of finite NLE lengtheasier to visualize, we apply some simplifying assump-tions to Eqs. (13) and (14). The first assumption isthat the NLE does not deviate strongly from transpar-ency. Consider the fact that the optical control pulsepropagates through the NLE with a fixed speed.Changes in the NLE attenuation a(x, t) are induced

by the control pulse. If the NLE is nearly transpar-ent, the control pulse does not change much as itpropagates through the NLE, and the NLE attenua-tion (x, t) propagates through the NLE withoutchanging shape, much as a pulse traveling at thesame speed as the control pulse. In general thistransparency assumption leads to results that arecorrect to first order, and it makes it easier tounderstand the finite-length effect.

Various methods of introducing the control pulseinto the NLE are possible. For example, if it isinjected normal to the loop, as suggested by Fig. 1, theentire length of the NLE experiences the effect of thecontrol pulse simultaneously, and ax(x, t) = a(t). Buthere we assume a configuration in which the controlpulse is injected into the loop through an intraloopcoupler so that it propagates in the loop in a clockwisedirection (Fig. 2 ). [If the intraloop coupler does notdiscriminate between the control and the input sig-nals, using, for example, wavelength or polarizationdiscrimination, it will permit some of the input signalto leave the loop. This loss reduces the samplingfunction S(t) by a constant factor.] We assume thatthe control pulse propagates through the NLE at thesame speed as the CW and the CCW input fields.This assumption is usually satisfactory, although itmay not be if the control and input signals havedifferent polarization states or are widely separatedin wavelength. With these assumptions the attenua-tion in the NLE takes the form of a traveling wave,

4(X, t) = a(t - X/VNLE).

AX

InputFig. 2. Optical loop mirror withoptical control is introduced intocoupler.

(15)

L

Outputa finite-length NLE. Thethe loop with an intraloop.

10 October 1994 / Vol. 33, No. 29 / APPLIED OPTICS 6837

Substituting Eq. (15) into Eqs. (13) and (14), we get

,ycw(t) = La(t - T1/2 + At), (16)

AL

yCcw(t) = f dx a(t - T1/2- At - 2X/VNLE)- (17)

Because the control pulse and the clockwise inputfield propagate in the same direction and at the samespeed through the NLE, the effect of the control pulseappears to be stationary in the frame of the clockwiseinput field. This is why the copropagation integralreduces to the simple form of Eq. (16), and the finitelength of the NLE has no effect on the shape of'yCW(t). But the counterpropagation integral in Eq.(17) is more complicated because the control pulseand the counterclockwise input field move past eachother in the NLE, and the NLE's finite length altersthe shape of yccw(t).

An analysis of phase-modulated operation followsthe same lines. The phase shifts through the NLE(cw(t) and ccw(t) can be calculated from the space-and time-dependent phase shift per unit length,which is the NLE's propagation wave numberk(x, t) = wn(x, t)/c, where n(x, t) is the space- andtime-dependent refractive index in the NLE. How-ever, in this case, simple propagation integrals suchas those in Eqs. (13) and (14) cannot be writtenbecause the CW and the CCW input fields can nolonger be treated as sampling functions traveling at afixed speed VNLE. Instead, the speed of each sam-pling function changes as it propagates through theNLE because of the changing index. In spite of theadded complexity the qualitative effect of finite NLElength is the same for phase-modulated operation asit is for amplitude-modulated operation.

Let us consider a specific example of the finite-length effect for amplitude-modulated operation.In this example the control pulse induces a steplikechange in the attenuation of the NLE from ao to al.Therefore

a(x, t) = a(t - X/VNLE){to (t x/VNLE < 0, 0 < x < L)

a1 (t X/VNLE > 0, 0 < x < L)

It is straightforward to evaluate Eqs. (16) and (17) forthis simple case, then to substitute ycw(t) and yccw(t)into Eqs. (4) and (5). In Fig. 3(a) we plot S(t) for thecase in which the NLE lies entirely to one side of themidpoint. Arbitrary time units are used, withTtransit = 1 and At = 1.5. The finite length of theNLE adds a tail of total length 2Ttransit to the trailingedge of the 2At-wide rectangular sampling window.Although the detailed shape of the tail depends on theexact form of a(x, t), in general the effect is the samefor all NLE's with the same transit time. By chang-ing the sign of At and VNLE, we can show that, if theNLE lies on the other side of the midpoint, the extraregion is added to the leading edge of the rectangular

(e) <

CD

12 34 5 67 8Time

Fig. 3. Finite-length effect for different positions of the NLE.The sampling function S(t) is plotted for the NLE position shown inthe corner of each plot. (a), (b) The NLE is to the side of themidpoint; (c), (d) one edge of the NLE is at the midpoint; (e) theNLE is exactly centered in the loop.

6838 APPLIED OPTICS / Vol. 33, No. 29 / 10 October 1994

sampling window, as in Fig. 3(b). If the NLE ismoved closer to the midpoint of the loop, At can bereduced to zero as one edge of the NLE reaches themidpoint of the loop. Then all that remains is theextra region with a total length of 2 Ttransit and aFWHM of approximately Ttransit, as in Figs. 3(c) and(d). Centering the NLE even more closely in the loopdoes not reduce the duration of the sampling window.However, it does provide more symmetry, as shown inFig. 3(e), in which the NLE is exactly centered in theloop. The improved symmetry of the sampling win-dow is obtained at the expense of a lower peak value;compare Fig. 3(e) with Figs. 3(a)-3(d), which areplotted on the same vertical scale. In spite of thefact that the NLE is perfectly centered in the loop, itis nevertheless the asymmetry of the loop that causesthe sampling window to exist. The asymmetry isexpressed by the fact that the intraloop couplerappears on one side of the NLE rather than the other,as shown in Fig. 2.

If the length of the NLE is reduced, Ttransit can bereduced and a shorter sampling window can beobtained. However, the height of the sampling func-tion is then reduced. Since the required opticaloutput power of the switch is determined by subse-quent processing stages, the power of the controlpulse must be increased to restore the loss introducedby shortening the NLE. As a result, shortening theNLE does not permit any fundamental improvementin the gain-bandwidth product achievable from agiven optical nonlinearity.

In the foregoing analysis we have disregarded theeffect of the NLE's group-velocity dispersion. Thisis a legitimate approximation because the finite-length effect that we have described limits the short-

:3

1-

M

a)C-

. . . . , -. . . . , . . . . , . . . . , . . . . , . . . . , . . . . , . . . . , . . . . , . . . a

* (a) 4

, , , , ,,,,,,. . - . , . . . . , . . . . , . . . . , . . . . , . , . . , . . . . , . . . . , . . . . , . . . Inr

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b 11 ,. I I .

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- , ,,,,,,,,.,,,,,,j

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est possible sampling window to approximatelyTtransit = Ln/c, where n is the refractive index, but thetemporal broadening that is due to group-velocitydispersion limits the shortest sampling window to(LXOAX/c)d2n/dX2, where Xo and AX are the wave-length and the spectral width of the control signal,respectively. The first quantity is much larger thanthe second, and the finite-length effect thereforeimposes a much stronger limitation than the group-velocity dispersion effect.

Experimental Results and Discussion

In the case of a loop with large asymmetry ouranalysis shows that each control pulse causes twononoverlapping sampling windows to be opened be-tween input and output, identical but separated intime by 2At. To illustrate this result, we con-structed a fiber loop mirror with the effect of the NLEsimulated by a LiNbO3 electro-optic phase modulator,which is placed in the loop with a time asymmetryAt = 360 ns. (In order to accommodate this muchasymmetry, the loop is 75 m long.) Using thiselectrically controlled element, we can simulate thebehavior of a phase-modulated loop mirror for anarbitrary time-varying phase shift +(t) by using awaveform generator to produce the electrical input tothe phase modulator. In this experiment the electri-cal input is a 400-ns long ramp with its amplitudeadjusted for 180° peak-to-peak phase modulation,shown in Fig. 4(a). Since 2At > Tcontrol, the condi-tion for large asymmetry is satisfied. Although thegenerator and the phase modulator have relativelylow bandwidth, we can simulate ultrafast operationby time scaling the time asymmetry At, the phaseshift ¢(t), and the sampling function S(t) by the same

1.2 1.6 2.0

Time (s)Fig. 4. Experimental data obtained with an electro-optic phase modulator used to simulate the effect of the nonlinear element. (a) The400-ns ramp applied to the phase modulator. The output intensity is shown for two different positions of the phase modulator relative tothe loop's midpoint. They correspond to a time asymmetry of (b) 360 ns (large asymmetry loop) and (c) 22 ns (small asymmetry loop).

10 October 1994 / Vol. 33, No. 29 / APPLIED OPTICS 6839

factor. A continous-wave optical signal from a1320-nm semiconductor laser is introduced into theloop's input so that the output intensity representsthe sampling function S(t). Figure 4(b) shows thattwo identical pulses emerge from the loop, separatedby 2At. By simplifying Eq. (11) for the case ofphase-modulated operation, we obtain an expressionfor the predicted sampling function:

S(t) = A [sin2 ) (t + At) + sin2 (t - At)] (19)

The phase shift induced by the phase modulatordepends linearly on voltage, and therefore +(t) is alinear function of time during the control pulse.Thus the intensity of each output pulse has a sin2(t)time dependence.

In the case of a loop with small asymmetry, ouranalysis shows that a short rectangular samplingwindow can be created from a single rapid transition,with the width of the window determined by the timeasymmetry of the loop. To illustrate this result, wereduce the time asymmetry At from 360 ns to 22 ns.The LiNbO3 electro-optic phase modulator and thecontrol pulse are the same as used for Fig. 4(b), but,because of the reduced asymmetry, the loop is nowonly a few meters long. The conditions for smallasymmetry are satisfied: the rise time for the con-trol pulse transitions is 10 ns; thus 'Trise < At << Tfall-

Figure 4(c) shows the resulting sampling functionS(t). It consists of a rectangular pulse 2At wide,generated by each control transition, in accordancewith Eq. (12). After the pulse, a residual outputintensity remains for the duration of the controlpulse, caused by the finite fall time of the phase shift(t).

Now we cite our earlier results using a TOADswitch13 to demonstrate the feasibility of all-opticalswitching in a loop with small asymmetry and toillustrate the finite-length effect. The configurationis the same as that of Fig. 3(a). The NLE is a500-pm InGaAsP polarization-insensitive traveling-wave semiconductor optical amplifier with a transittime Ttransit of 5 ps. Optical pulses with a 2-psFWHM are generated with a pulse-compressedNd:YLF laser, and they are split into fixed-delay,20-fJ input pulses and variable-delay, 600-fJ controlpulses. The input pulses and the control pulses areorthogonally polarized before they are introducedinto the loop, and the control pulses are blocked at theoutput with a polarizer. The length of the loop inthese experiments is 2 m, but this length is irrel-evant to the speed of the switch, which is controlledby the asymmetry of the NLE in the loop, whichranges from 10 cm to 1 mm. The dominant opticalnonlinearity arises from the gain saturation inducedby the control pulse. The rise time T

rise of thetransmission modification is 1 pS,

2 2 but its fall timeTfall and total duration Tcontroj are 800 ps. Thispump-probe experiment actually measures the convo-lution of S(t) with the input intensity Iin(t). As a

result, the measured sampling function has slowertransitions than the true sampling function S(t), andthis effect is noticeable when the sampling functionhas a duration of the order of the 2-ps input pulseduration.

Figure 5(a) shows the measured sampling functionfor a time asymmetry At of 345 ps. Since At isneither much smaller nor much larger than the800-ps recovery time of the NLE, this loop does notsatisfy the conditions for large or small asymmetry.However, the zero-length approximation is valid,since Ttrasit << A t. Equation (8) describes the sam-pling function under these conditions. Because theNLE is predominantly transmission modulated, thephase shift +)(t) can be treated as a constant, and Eq.(8) reduces to

S(t) = /4[A(t + At) - A(t - At)]2 . (20)

When the first term in brackets turns on, it yields theinitial transient. Then the first term diminishes,and, after a time 2At has elapsed, the second termturns on, creating a notch in the sampling function.Following this, both terms diminish with time, andthe sampling function slowly returns to zero.

In Fig. 5(b) the time asymmetry At is reduced to 65ps, and therefore Trise < A t << Tfall, and this is a loopwith small asymmetry. The zero-length approxima-

-.

800800 ~ 8 (a)

400

200

0-700 . I .600 5 600 _ ,8litlt (b)

400- 300-

O 200 -100

0 I800 _ __ __ __ __ __ ___ (c)

80060060400 - 400

200200 0_____________

0 ~ 30 40 50 60 70 50

0 200 400 600 800

Time (ps)Fig. 5. Results of an all-optical switching experiment obtainedwith a semiconductor optical amplifier used as a NLE. The

pump-probe experiment yields a profile of the sampling windowopened by the control pulse. The figure shows three differentpositions of tlie optical amplifier, corresponding to a time asymme-try of (a) 345 ps, (b) 65 ps, and (c) 4 ps.

6840 APPLIED OPTICS / Vol. 33, No. 29 / 10 October 1994

tion is still valid, since Ttr~a,,it << At. The samplingfunction is negligible outside a time window of dura-tion 2At, and within the window it has a relativelyconstant level, given by Eq. (12). The rapid turn onof the gain saturation mechanism causes both thefast turn on and the fast turn off of the samplingfunction.

In Fig. 5(c) the time asymmetry At is reduced to 4ps. This is also a small asymmetry loop, but in thiscase the zero-length approximation does not holdvery well because Ttransit = At. As a result, thesampling function shown in the inset detail is asym-metrical. Its leading edge has a rise time of a fewpicoseconds, partly because the measurement con-volves the sampling function with the 2-ps long inputpulse. But there is an extra region lasting 10 psattached to the trailing edge. The extra region arisesfrom the finite-length effect, which adds an additionalregion 2Ttransit long to the trailing edge of the sam-pling function, as noted in Fig. 3(a). If subpicosec-ond control pulses were used and the NLE werecentered more closely in the loop, the total duration ofthe sampling function would reach the minimumpermitted by this NLE, which is 2Ttransit = 10 ps, withaFWHMof -5 ps.

Applications of the Small-Asymmetry Loop Mirror

Our analysis has shown that, with sufficiently smallasymmetry, a loop mirror with a nonlinear element inthe loop can perform ultrafast switching. The timeresolution of the switch is determined by the asymme-try of the NLE within the loop, not by the relaxationtime of the nonlinearity. However, the relaxationtime of the nonlinearity does limit the repetition rateat which switching can be performed. Therefore theapplications for the switch are those that demand of ahigh sampling bandwidth but at a repetition rate thatis much smaller than the bandwidth.

Two such applications are time-division opticaldemultiplexing and time sampling of repetitive opti-cal signals. The usefulness of the small-asymmetryloop mirror for time-division demultiplexing has beendemonstrated experimentally.'4' 6 For demultiplex-ing and sampling applications the duration of thesampling window must be short enough to sample themodulation bandwidth of the optical input effectively.However, these applications are not as demanding asgeneral purpose ultrafast all-optical switching be-cause they give the switch a long time to recoverbetween sampling events. In the case of samplingrepetitive signals the sampling can be performed at alow repetition rate, as is done frequently in electronicsampling. Similarly in the case of optical demulti-plexing the switch does not need to route anotheroptical pulse to the output until the next data frame,providing a recovery time as long as the data frameitself. If the small-asymmetry loop mirror is used inapplications such as these, size and cost constraintsdemand that it be integrated on a substrate. Fortu-nately the size of the loop is completely irrelevant toswitch operation as long as there is enough room in

the loop for the required degree of asymmetry. As aresult, the switch can easily be integrated, perhaps aspart of a more complex optoelectronic integratedcircuit.

Conclusions

In this paper we have described a general analyticalframework for an optical loop mirror in which anonlinear optical element is placed in the loop. Wehave shown that, when a NLE is placed in an opticalloop mirror, two special types of loop exist, thesmall-asymmetry loop and the large-asymmetry loop.In a loop with small asymmetry a strong, slownonlinearity can perform ultrafast all-optical switch-ing in demultiplexing and sampling applications.Our analysis provides an analytical framework forthe operation of the TOAD switch.

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