A number of optical switching and logic devices havebeen proposed and demonstrated. Some of the morepopular devices include the bulk1 and fiber2 optical
err gates, the nonlinear bistable Fabry–Perot eta-on,3 the nonlinear directional coupler,4 the nonlinear
Mach–Zehnder interferometer,5 and the relatedfiber-based nonlinear optical loop mirror.6 Recentstudies have concentrated on derivatives of these de-vices in fiber, such as the terahertz optical asymmet-ric demultiplexor7 and the ultrafast nonlinearnterferometer8; however, momentum is currently
pushing toward the future use of integrated devices9
with greater device stability and much lower latency.A different class of optical logic device, which is
based on optical solitons and fulfills many of the re-quirements for a general logic gate,10 has recentlybeen studied in temporal11 and spatial12 forms. Theuse of solitons or solitary waves in optical logic iscritical in that solitons beat the dispersion or thediffraction limit or both over distances that are muchlonger than the characteristic linear lengths. In this
S. Blair [email protected]! is with the Department of Electricalngineering, University of Utah, Salt Lake City, Utah 84112-9206.. Wagner is with the Optoelectronic Computing Systems Center,epartment of Electrical and Computer Engineering, University ofolorado, Boulder, Boulder, Colorado 80309-0425.Received 14 January 1999; revised manuscript received 2 Au-
paper, the main focus is on the use of lateral spatialconfinement over distances larger than the linear dif-fraction distance, which allows for spatially resolv-able deflection of a spatial soliton to be induced by aless than resolvable angular change. Before dis-cussing the spatial soliton angular deflection logicgate in Section 2, we review the analogous temporalsoliton dragging logic gates.
The soliton trapping13 and dragging11 gates arebased on the temporal trapping mechanism.14,15
Two nonlinear waves of different polarization, wave-length, or both propagate down a fiber at differentgroup velocities. If the waves initially overlap intime, in linear propagation one wave will reach theend of the fiber before the other. In nonlinear prop-agation, however, the waves can trap each otherthrough nonlinear cross-phase modulation such thatthey both propagate at a mean group velocity. Inthis case, each wave must experience a frequencyshift to propagate at the common, weighted-mean,group velocity. The two nonlinear waves do notneed to be the same for trapping to occur,15 and thefrequency shift experienced by the smaller wave isgreater than that experienced by the larger wave.
Figure 1 shows the generic three-terminal gate ge-ometry. In this geometry the temporal trapping anddragging gates perform an inversion operation inwhich the pump is passed in the absence of the signal,or data, pulse and blocked by a spectral filter or timegate in the presence of the signal pulse. Note thatthe signal is always blocked at the output, whichresults in true three-terminal operation with input–output isolation. Inasmuch as the pump pulse may
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be passed on to later gates ~such as in the implemen-tation of a multi-input NOR gate11!, it is convenient forthe pulse to be a temporal soliton to maintain itsshape after propagation through many tens to hun-dreds of meters of fiber, thereby ensuring logic levelrestoration. The Islam trapping–dragging gatesuse pulses of the same wavelength and orthogonalpolarization and are directly cascadable if a waveplate is used to rotate the output pump polarizationto the correct state for an input signal or if the defi-nitions of pump and signal polarization are alter-nated. Note that one can construct similar gates byusing pulses of different colors,16 based on the anal-gous trapping mechanism,17 but this gate is cascad-
able only through the use of nonlinear frequencyshifters or by alternation of the definitions of pumpand signal color at every gate at the expense of re-quiring two pump pulse sources with a constant dif-ference frequency. In either implementation oforthogonal polarizations or different colors, the gateoperation is independent of the relative phase be-tween the pump and the signal ~when nonlinear four-wave mixing effects are neglected!.
The Islam soliton trapping gate relies on a resolv-able spectral shift of the pump soliton,13 so an un-shifted pump will pass through a spectral bandpassfilter but a shifted pump ~in the presence of the sig-nal, which is blocked by an analyzer at the output!will not. This operation results in amplitude keyedlogic, which is compatible with common high-speedoptical detectors for eventual conversion into the elec-tronic domain. For high-contrast operation to occur,the fiber birefringence must be sufficiently large toproduce the necessary difference in group velocitybetween the pulse propagating down the slow axisand the pulse propagating down the fast axis, which,when compensated for by trapping, results in a spec-trally resolved shift. The pump and signal pulsesmust also be of nearly the same amplitude so thespectral shift will not be weighted preferentially to-ward the signal. As a result, this gate cannot pro-vide significant large-signal gain. A final note isthat, because complete trapping can occur within a
Fig. 1. Temporal soliton trapping–dragging gate. Soliton pumppulses of different wavelength, polarization, or both. Cross-inducgroup velocity. This frequency shift must be resolvable for the traptiming shift after propagation through a dispersive fiber, delay linewavelength-division multiplexer; PBS, polarizing beam splitter.
750 APPLIED OPTICS y Vol. 38, No. 32 y 10 November 1999
few soliton periods, the gate length need be only a fewtens of meters.
The temporal soliton dragging logic gate utilizesthe fiber dispersive delay line as a lever arm to allowa weak signal soliton to drag a strong pump in time.18
With this gate, even a small spectral shift of thepump ~i.e., less than a p phase shift induced by a
eak signal pulse! can result in a large time delaywing to nonzero group-delay dispersion. Becausehe pump is a temporal soliton that does not broadenith propagation distance, a resolvable temporal
hift can be achieved by choice of the appropriateength delay line, according to the expression
dvk00L . tFWHM, (1)
where the frequency shift dv is a function of the pumpand signal pulses and the fiber birefringence, k00 ishe group-delay dispersion coefficient, L is the fiber
length, and tFWHM 5 1.7627t is the intensity fullwidth at half-maximum ~FWHM! for a solitonech~tyt! amplitude profile. Therefore, for fixed bi-efringence, there is a trade-off between gain ~whichs approximately the ratio of pump to signal pulsenergies and determines the amount of spectral shiftv! and gate length L. If the pump were a linear
dispersive wave, broadening would occur at the samerate as the temporal separation ~when higher-orderdispersion is neglected!, and, in general, temporalresolvability can be achieved only by at least a re-solvable spectral shift. For soliton pulses that prop-agate as a unit at a single group velocity, however,temporal resolvability can be achieved without a re-solvable spectral shift.
Temporal dragging logic gates must be time-shiftkeyed, which are directly cascadable in the opticaldomain ~but may require large temporal guard bandso achieve practical three-terminal operation in theresence of timing jitter! for a clocked system but are
not compatible with high-speed electronic detection.Detection can be achieved by use of a trapping gate inthe last stage18 or of an ultrafast optical gating mech-anism, such as a fiber Kerr gate.19 Note that, in
es ~always present! initially overlap in time with data ~or signal!irp causes the pump and the signal to copropagate with the samegate. The change in frequency is manifest as an arbitrarily large
or the dragging gate, this timing shift must be resolvable. WDM,
pulsed chping. F
S
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contrast to other fiber switching gates, which sufferthe problems imposed by birefringent walk-off,19 thetemporal soliton trapping–dragging gates use bire-fringent walk-off ~in combination with nonlinearcross-phase modulation! to induce the switching op-eration. The only constraint is that the walk-offlength be greater than approximately one soliton pe-riod, z0 5 pt2y2uk00u, because the bulk of the interac-tion occurs within the distance z0.20
In the rest of this paper we consider analogous logicgates based on the angular deflection of a spatialsoliton pump wave by a weaker signal wave. Thisspatial geometry also results in a restoring three-terminal logic gate with gain and has the advantagesof being integratable onto active or passive substratesby use of a wide variety of nonlinear materials, hav-ing an additional degree of freedom for spatial par-allelism, and having much reduced latency comparedwith its fiber-based, temporal counterparts. In Sec-tion 2 we present the basic inverting angular deflec-tion geometry and discuss the specific spatialinteractions between orthogonally polarized solitonsthat produce this behavior. In Section 3 we considerin detail the spatial collision and dragging interac-tions and calculate a contrast metric that we use todetermine valid operating regions within the space ofinteraction parameters. Logic gate transfer func-tions are introduced in Section 4 and are calculatedfor specific examples of collision and dragging gatesthat demonstrate small-signal gain and large-signalgain with wide noise margins. Finally, in Section 5we summarize the results of this paper.
2. Spatial Soliton Angular Deflection Logic Gates
A one-dimensional ~1-D! spatial soliton is a station-ary solution to the ~1 1 1! 2 D spatial nonlinear
chrodinger ~NLS! equation
2ik0
]A]z
1]2A]x2 1 2k0
2 n2
n0uAu2 A 5 0. (2)
The fundamental soliton solution to Eq. ~1! is
A~x, z! 51
k0 w0În0
n2sechS x
w0Dexp~izy2k0 w0
2!, (3)
here k0 is the material propagation constant, w0 isthe soliton width parameter ~such that the intensityspatial FWHM is 1.7627 w0!, n0 is the material re-fractive index, and n2 is the nonlinear Kerr index.The peak intensity and power of the fundamentalsoliton are given by
I0 5n0
k02w0
2n2I , (4)
P0 52n0 D
k02w0 n2
I , (5)
respectively, where n2I is the Kerr index in intensity
units and D is the effective thickness of the confininglab waveguide.
1
In analogy to temporal soliton trapping and drag-ing gates, spatial soliton trapping21 and dragging
gates12 are one instantiation of a more-general classf angular deflection logic gates studied in this paper.he first spatial trapping gate demonstrated experi-entally21 used spatial solitons of the same polariza-
tion ~i.e., phase dependent! but propagating atifferent angles that were not necessarily resolvable.ecause of cross focusing, which is the spatial anal-gy to cross-phase modulation, the solitons are mu-ually attracted and, under favorable conditions,orm a trapped pair. The analogous temporal trap-ing interaction between two temporal solitons of dif-erent colors ~and hence of different group velocities!as been studied analytically17 and experimentally.16
The direct spatial analogy to the Islam temporal trap-ping13 and dragging11 gates is the interaction be-tween spatial solitons in the orthogonaleigenpolarization states of a uniaxial crystal.22 Inhe case of an off-axis cut, the soliton in the ordinaryolarization will propagate straight and the soliton inhe extraordinary polarization will experience powerow walk-off away from the optic axis. These soli-ons will mutually attract and form a bound pairnder the appropriate conditions of crystal cut ~which
determines the walk-off angle! and soliton power ra-tio. Because the nonlinear cross focusing is betweenthe intensity profiles, nearly the same behavior re-sults when two initially overlapping solitons withtilted angles are launched in isotropic media, as hasalso been analyzed theoretically12,23 and is furtherstudied in this paper. This spatial interaction wasrecently demonstrated experimentally24 but in a pa-rameter regime that did not allow for gain to be ob-served. The temporal analog to this interaction istemporal solitons of different colors and orthogonalpolarizations.
One benefit of spatial soliton logic gates comparedwith temporal trapping and dragging gates is thatcascadable amplitude keyed logic is straightforwardto implement while providing gain. In the spatialinteractions one can use an aperture to discriminatethe output. For temporal dragging, though, imple-menting amplitude keyed logic requires an ultrafasttime gating mechanism instead of the more naturaltime-shift keyed logic.18 A spectral filter is all thatis necessary to implement amplitude keyed logic fortemporal trapping, but the leverage of the dispersivedelay line is lost, and gain cannot be provided.Therefore, only the spatial interaction geometrieshave the simultaneous advantages of simple outputstate determination and the leverage of nondiffract-ing propagation, which allows for large-signal gain.Additional advantages are spatial parallelism, whichresults from only one dimension of linear confine-ment by a slab waveguide, the flexibility to use highspace–bandwidth-product optical interconnects and awide variety of highly nonlinear media, compatibilitywith integrated optics technology, and latency. Theuse of spatial soliton logic gates may allow for phys-ical implementations with centimeter gate lengths,resulting in a factor-of-1000 reduction in latency com-
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pared with that of their temporal counterparts.Much shorter temporal solitons can be used to reducethe length of the fiber-based gates, however, buthigher-order temporal effects such as Raman scatter-ing can degrade performance.11
A. Angular Deflection Geometry
Before discussing the angular deflection logic gate,we first define parameters that describe the interac-tion geometry. The parameter k is defined as thepatial frequency separation between two tiltedech¼ profiles divided by the sum of their spectralntensity half-widths at half-maxima ~HWHM! and is
referred to as the normalized interaction angle.Therefore, for mutually incoherent but otherwiseidentical beams, k 5 1 corresponds to the Sparrowresolvability condition25 such that, in the far field oflinear propagation, the beam intensity profiles willoverlap at the half-power points.
Assuming that one spatial soliton propagates downthe optical axis ~i.e., the z axis!, the following Fouriertransform relation holds26:
sechS xw1
D7 pw1
2sechSpw1 kx
2 D . (6)
The spectral intensity FWHM Dkx is given by thecondition
pw1Dkxuw1
25 1.7627f Dkxuw1
51.1222
w1. (7)
The other beam propagates at an angle u to the op-tical axis and, by the Fourier shift theorem,
sechS xw2
Dexp~idkx x!7pw2
2sechFpw2
2~kx 2 dkx!G ,
(8)
where dkx 5 k0 sin u. Now, k is evaluated:
k ;2dkx
Dkxuw11 Dkxuw2
5k0 w1 w2 sin u
0.5611~w1 1 w2!
511.20w1 w2 sin u
l~w1 1 w2!. (9)
The condition k 5 1 means physically that the beamsropagate with a relative angle given by the sum ofheir respective diffraction angles. Another usefulelation gives the transverse spatial frequency offsetn terms of the normalized interaction angle k:
dkx 50.5611~w1 1 w2!k
w1 w2. (10)
For the 1-D spatial soliton interactions studied in thispaper, w1 5 w0 is the signal soliton width and w2 [wp 5 w0yr is the pump soliton width, where r is thenitial power ratio between the pump and the signal.
752 APPLIED OPTICS y Vol. 38, No. 32 y 10 November 1999
Figure 2 illustrates the physical, linear interpreta-tions of the definition of k in both real and Fourierspace. In physical space, the k 5 1 condition indi-cates that the rate of separation between two beamsis the same as the rate of diffraction. Thus the pro-portion of spatial overlap is constant with distance.In Fourier space, k , 1 indicates large spatial fre-quency overlap between two beams ~i.e., not angu-arly resolvable or spatially resolvable in the far field!nd k . 1 denotes small overlap ~i.e., angularly re-
solvable!.The length of a logic gate is measured in terms of
the confocal distance, which is twice the Rayleighrange, or the distance over which the intensityFWHM increases by a factor of =2 in linear propa-gation. Here, for sech¼-shaped beams, the confocaldistance is Z0 5 p2w0
2yl 5 pk0w02y2 for a signal
soliton of width parameter w0. In the case of aGaussian beam the confocal distance is k0w0
2. Theconfocal distance for a sech¼ profile was determinednumerically27 through linear nonparaxial beam prop-agation.
The generalized spatial soliton angular deflectiongeometry is illustrated in Fig. 3, which shows thebasic logic gate for the spatial trapping–dragging in-teraction. A pump soliton ~left-hand side! propa-gates the length of the gate ~which is assumed to bein a slab waveguide geometry! and passes through aspatial aperture at the output, providing a standard-ized high-output state of the device. Because thespatial soliton does not diffract, the size of the aper-ture can be the same as the size of the pump beam at
Fig. 2. Illustration of the normalized interaction angle k as givenby the spatial frequency separation between two beams divided bythe sum of the individual power spectral HWHM. Left, the k 5 1condition in linear real-space propagation where the beam on theleft is propagating down the optical axis and the beam on the rightis propagating with normalized angle k 5 1. The heavy linesindicate the spatial FWHM of the respective beams, and the con-tours are spaced at 3-dB intervals relative to the initial peakintensity. Right, the spatial frequency power spectra of twobeams for different values of k. When k 5 1, the beams are at theangular resolvability condition ~i.e., the somewhat arbitrary con-dition that the spatial frequency spectra overlap at their half-power points!. The normalized distance Z0 5 p2w0
2yl.
idltlthspatttp
nt
the input, independently of the actual gate length~when absorption is neglected!.
One performs the switching operation by disturb-ng the propagation of the pump beam such that itoes not exit the spatial aperture, thus providing theow-output state of the device. One can accomplishhis by inducing a change in propagation angle thateads to a spatially resolved shift at the output. Ifhe beam propagates linearly ~as shown on the right-and side of Fig. 3!, one can produce a spatially re-olved shift only by inducing a change in theropagation angle ~with a change in phase across theperture of the beam of at least p! that is greater thanwice the linear diffraction angle. In nonlinear soli-on propagation, however, an induced angle changehat is less than linearly resolvable ~i.e., less than a phase change across the spatial aperture! results in a
differential spatial shift that can be integrated overnondiffracting propagation such that a spatially re-solvable shift occurs at the output. Thus, as shownin the figure, the gate length must be at least theminimum resolvable dragging distance, which de-pends on the spatial width of the soliton beam andthe amount of angular change.
The angular change is produced through the non-linear interaction between the pump and another,orthogonally polarized, beam, called the signal,which initially overlaps the pump and propagates ata nonzero relative angle, as illustrated for the drag-ging interaction. The signal beam must be strongenough to induce a nonlinear index change that is felt
Fig. 3. Logic gate geometry based on the light-induced deflectionof an optical pump beam away from a spatial aperture placed at theoutput. At one extreme the pump can propagate nonlinearly as aspatial soliton ~left!; at the other extreme the pump can propagatelinearly and diffract ~right!. Deflection is induced by the cross-
onlinear interaction with a signal beam that is tilted with respecto the pump. The dashed contours represent the deflected pump.
1
by the pump through cross focusing. If the pumppropagates linearly, large-signal gain is not possiblebecause the signal beam is necessarily stronger. Inthis case there is no mutual nonlinear interaction,and the nominal effect of the signal is to create anonlinear prism that deflects the pump in the direc-tion of the signal. The deflection angle depends onthe relative propagation angle and intensity of thesignal, with the maximum deflection occurring whenthe pump is completely guided, or trapped, by thelarger signal. Therefore the relative angle u must beat least the diffraction angle of the pump ~sin u $0.5611yk0wp! because, in the case of complete trap-ping, the deflected pump will be nonlinearly guidedby the signal, which may be a nondiffracting spatialsoliton.
If the pump is a spatial soliton, then large-signalgain is possible, but the interaction is more compli-cated because of mutual nonlinear coupling. Now,during propagation, each beam affects the otherthrough cross focusing. If large-signal gain is to berealized, then the pump must exert a greater attrac-tive force on the signal than the signal on the pump.Nevertheless, as was mentioned above, only a smallangular deviation of the pump is needed. As in thecase with the linear pump, two interaction situationscan occur: Each soliton simply deflects the other orthe two solitons form a bound, possibly orbiting, pairpropagating at the weighted-mean angle.12 Theformer is typically referred to as dragging,11 and thelatter is referred to as trapping.28 Soliton trappingis more likely to occur when the relative propagationangle is small and the solitons are nearly the samesize, whereas deflection, or dragging, occurs for largeangles, large gain, or both. Note also that the opti-mal interaction may occur not in the regimes of puretrapping or dragging but instead in the highly inelas-tic regime where part of the signal, the shadow,29
remains bound to the pump while the rest propagatesat a much larger angle as an unbound linearly dif-fracting wave. Therefore this particular interactionwill be generically referred to as dragging because inall cases the signal transfers transverse momentumto the pump. Other interactions can also producethis angular deflection, as discussed in Subsection2.B below.
These gates have two inputs and only one outputbecause the signal is blocked by the aperture and apolarizer. As a result, these gates are three-terminal devices with input–output isolation. Onlythe undeflected pump is passed on to subsequentgates, and it is important that the pump propagatestably over the length of the gate with little change inits physical parameters. This stable propagation al-lows not only for restoration of the logic level but alsofor restoration of timing, position, polarization, color,and shape, which are crucial to cascaded operation30
and can allow for more complex logic functionality.Because the pump and the signal have orthogonalpolarizations, there is no linear interference, and theinteraction is nominally phase insensitive. In addi-tion, nonlinear interference can be eliminated exactly
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by the use of orthogonal circular polarizations in iso-tropic media. However, for linear polarizations~which must be used in the waveguide geometry!, thiss not strictly true because of the presence of phase-ependent vectorial four-wave mixing terms in theonlinear polarization. Here it is assumed thathese terms can be neglected because of waveguideirefringence, which causes each polarization eigen-ode to propagate with a different phase velocity.his assumption is valid when the interaction length
s much greater than the birefringence beat length.Figure 4 shows a typical inverting input–output
elation for an angular deflection logic gate. Theost important feature to note is the region of small-
ignal gain at the input threshold level near 0.3, sur-ounded by saturated levels. The transfer functionhows that the angular deflection gate has the sameperational characteristics as an n–metal-oxide semi-onductor inverter.31 Here, the role of the electric
power supply is provided by the pump wave and therole of the gate voltage is played by the signal wave.The presence of small-signal gain at the thresholdlevel allows the output to be driven low with sharpswitching characteristics. The threshold level is thepoint on the curve at which the input and outputlevels are the same. For an input level beyondthreshold, the output of the device is switched into avalid low state; therefore the threshold level is theminimum input signal level required for switchingthe output state of the device and determines the~unstable! operating point on the transfer curve
here large-signal gain is unity. When small-signalain is greater than unity at the threshold level, thehe input switching level can be chosen such that
754 APPLIED OPTICS y Vol. 38, No. 32 y 10 November 1999
arge-signal gain greater than unity is obtained.he transfer curve is terminated by saturated levels,howing that overswitching is not possible. As a re-ult, for operation well into the on and off regimes,mall variations in the input-signal level are attenu-ted and do not affect proper gate operation, thusroviding large noise margin. These transfer func-ions are considered more formally in Section 4 below.
B. Examples of Angular Deflection Gates
We must define some metric to evaluate the potentialfor a particular soliton angular deflection interactionto form the basis for a useful logic gate. A thresholdcontrast metric t is defined here as the power of theinput ~fundamental! signal soliton divided by thepower of the deviated pump that exits the aperture.This metric depends on pump power, signal power,gate length, aperture size, and interaction angle andis used initially to provide a comparison among thedifferent interactions and to locate the optimal inter-action parameters, such as the choice of normalizedinteraction angle k for a given gate length, powerratio, and aperture size. With this measure, thresh-old contrast greater than unity indicates that suffi-cient deviation has occurred that the power exitingthe aperture is less than the input-signal power; con-trast much greater than unity ~i.e., 10 or greater! isan indication that a restoring logic gate can be im-plemented for the given interaction parameters, asdiscussed in Section 4 below. The choice of referenceinput-signal level, which determines the unity con-trast condition, is somewhat arbitrary. The twomain options are the power of the fundamental signalsoliton and the fraction of that energy that is requiredfor reaching some acceptable low output level of thedevice. For simplicity, the fundamental signal soli-ton power is used, such that t 5 1 indicates that thegate is operating at the switching threshold point.
The phase sensitivity of the interaction between soli-tons in the same state of polarization depends on theamount of spectral overlap. When there is very littleoverlap, such as is the case for a large relative propa-gation angle ~k .. 1!, even in regions of complete spa-tial overlap the interaction will be phase insensitivebecause the solitons pass through each other withlarge relative transverse velocity. This also meansthat the interaction is weak, though, and consequentlyineffective for a logic gate. Therefore, for phase-sensitive interactions to perform high-contrast logicoperations with gain, the solitons must have signifi-cant spatial frequency overlap. The result is that theinteraction necessarily depends on the relative phasebetween the two solitons, such that the contrast of thegate also depends on the relative phase.
In a computing or switching system consisting of alarge number of gates with arbitrary interconnectionsand feedback paths, a fixed relative phase may be dif-ficult if not impossible to maintain; therefore it is es-sential to reduce or completely eliminate phasedependence. This can be accomplished by use of or-thogonally polarized solitons such that, in a nonlinearKerr material, the soliton interaction will be based
Fig. 4. Transfer function for the r 5 1 dragging gate of length10Z0 with normalized angle k 5 0.8 and separation rs 5 0. Ab-orption is included, with sopt 5 0.2627, and the threshold contrasts 131. This gate provides large-signal gain G 5 1.4 with NML 5
0.13 and NMH 5 0.07. The high noise margin represents 14%eviation about IH, and the filled diamonds denote the operating
points and threshold. The transfer function characteristics aredefined in Subsection 4.A below.
Tg
only on phase-insensitive nonlinear cross focusing~typically referred to as cross-phase modulation fortemporal solitons! between their intensity profiles.Phase dependence is completely eliminated in the caseof pseudo-1-D soliton interactions in bulk32 or in aperfect fiber through the use of orthogonal circularpolarizations. In the case of orthogonal linear polar-izations, an additional phase-dependent nonlinearterm ~the vector four-wave mixing term! enters thevector NLS ~vNLS! equation. In this situation, phasedependence can be reduced by use of material orwaveguide birefringence to mismatch the phase-dependent terms in the induced nonlinear materialpolarization when the interaction length is greaterthan the birefringence beat length. The latter is themethod used here because the 1-D spatial solitons areconfined in one dimension by a slab waveguide thatadmits of orthogonal linearly polarized TE and TMsolutions, each propagating with a different phase ve-locity. This difference is due to the form birefringenceof the waveguide and can be tailored by proper choiceof the waveguide parameters.
Neglecting the phase-dependent nonlinear ~i.e., thevector four-wave mixing! terms, then one can de-scribe the interaction between orthogonally polarizedsolitons in isotropic media by the vNLS equationswritten here in spatial form12:
2ik0
]Ax
]z1
]2Ax
]x2 1 2k02 n2
n0~uAxu2 1 2DuAyu2!Ax 5 0,
(11a)
2ik0
]Ay
]z1
]2Ay
]x2 1 2k02 n2
n0~uAyu2 1 2DuAxu2!Ay 5 0,
(11b)
where 2D represents the strength of the nonlinearcoupling between the polarizations. These equa-tions are the spatial analogs of those used for tempo-ral solitons in fiber.33,34 Even though largebirefringence is necessary for strong phase mismatchof the vector four-wave mixing terms, birefringenceterms can be neglected as well, because the resultingevolution equations are coupled only through the op-tical intensity. In isotropic Kerr media, D 5 1y3,which is the value used here. For this choice of D,the coupled equations are not integrable @in fact, theyare integrable only when 2D 5 1 ~Ref. 35!#, and ap-proximate analytical or numerical techniques mustbe used. The result is that some interactions areinelastic,36 which means that the solitons do not nec-essarily retain their initial character after collisionand that generation of dispersive radiation can re-sult. As will be shown in what follows, this behaviorcan be advantageous for optical switching. A conse-quence of the inelastic nature is that the solitons canbe pulled apart by the interaction, resulting inshadow beams ~or pulses29 in the case of temporalinteractions! copropagating with the stronger solitoninvolved. Note that, when 2D 5 1, the coupled equa-tions are known as the Manakov equations, whichare integrable,35 resulting in interactions with behav-
1
ior similar to that between same-polarized solitonsbut without the phase dependence. The case 2D 5 1corresponds not exactly to any material symmetryclass but rather to a coincidental combination of thethird-order nonlinear susceptibility components ofcubic AlGaAs at sub-half bandgap.37
The point was argued that the strength of an in-teraction between same polarized solitons depends onthe amount of spectral overlap between the solitonsand results in phase dependence owing to scalar lin-ear interference. In the case of interaction betweensolitons of orthogonal polarization, though, there isno linear interference, and the interaction dependsonly on the nonlinear coupling between the intensityprofiles. It is expected that these interactions willalso be stronger for k , 1 because this implies thatthe transverse velocities are small, such that eachsoliton remains bound by the potential well38 of theother soliton. This potential well is created by crossfocusing and gives rise to an escape velocity17,39 thatis the minimum relative velocity necessary for onesoliton to escape the potential well of the other.
Some deflection interactions are now examined forthe phase-insensitive case of orthogonally polarizedsolitons. Here the initial electric field amplitudedistributions are taken to be of the form
Ax~x, z 5 0! 5r
k0 w0În0
n2sechSx 2 zy2
w0yr Dexp~2idkx xy2!,
(12a)
Ay~x, z 5 0! 51
k0 w0În0
n2sechSx 1 zy2
w0Dexp~idkx xy2!,
(12b)
where Ax~x, z 5 0! is the initial field envelope of thepump soliton, Ay~x, z 5 0! is the envelope of the signalsoliton, w0 is the width parameter of the signal, w0yris the width of the pump, z 5 rsw0~1 1 1yr!y2 is theinitial spatial separation defined such that the sepa-ration in terms of the sum of the spatial HWHM ofthe two beams is constant with r, and dkx is the initialrelative transverse wave number as given by Eq. ~10!.
he initial power of the fundamental pump soliton isiven by Pp 5 rP0, and the confocal distance of the
pump is Zp 5 p2wp2yl 5 Z0yr2. Note that, even if
the solitons are otherwise identical ~i.e., r 5 1!, theyare distinguishable by polarization state in the ab-sence of the vectorial four-wave mixing term, and anysignal leakage through the aperture can be blockedby a linear polarization analyzer. The signal andthe pump can also be brought into coincidence withk , 1 through the use of a polarizing beam splitterwithout incurring any input coupling loss, which isnot the case for copolarized solitons that are initiallyoverlapping.
1. Soliton CollisionCollision between optical ~scalar! NLS solitons is con-servative in the sense that the solitons pass througheach other without change except for a small spatial ortemporal shift.40 Theoretical studies of multisoliton
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collisions have been performed for the purposes ofuantifying timing jitter in wavelength-division mul-iplexing soliton systems. Experimental demonstra-ions in optics have been carried out for temporaloliton collisions in fiber43,44 and dark spatial soliton
collisions in self-defocusing liquid45 and solid46 media.he last-named experiments prompted a study of us-
ng the dark soliton collisions for general-purpose log-c.47 The scalar collision interaction between bright
spatial solitons has also been suggested as a mecha-nism for a photonic switch.48
The scalar spatial collision interaction occurs whentwo copolarized spatial solitons, propagating at dif-ferent angles, symmetrically collide within a nonlin-ear medium, producing a spatial shift in thepropagation of each soliton with no permanentchange in propagation angle. After the collision,when the solitons are well separated, the contrast isindependent of gate length because the initial prop-agation angles are restored and the spatial shift as-sumes a fixed value. Therefore, shorter gate lengthsresult from greater angles of collision. One problemwith the collision interaction, however, is that thespatial shift decreases with increasing angle of colli-sion, meaning that long gate lengths are required forhigh-contrast operation by use of small collision an-gles. Small collision angles additionally result inhighly phase-sensitive operation.
For soliton collision in the integrable case of thecoupled vNLS equations with orthogonal polariza-tions and 2D 5 1, the solitons pass through eachother,35 simply producing a small spatial shift as inthe copolarized interaction, with no change in angle.But in nonintegrable cases ~such as when 2D Þ 1!,
ore-interesting behavior occurs, because the anglesan change after collision.49 A theoretical study of
the collision between two orthogonally polarized tem-poral solitons with 2D 5 2y3 showed that reflection,annihilation, and fusion could result after the colli-sion.36 That study was concerned with the deleteri-ous effects that the inelastic collision had on opticalcommunications and switching. With the type ofthree-terminal logic gate studied here, though, whathappens to the pump soliton does not matter, as longas the soliton does not pass through the aperture inthe presence of the signal. Therefore these inelasticinteractions are potentially useful for three-terminaloptical switching.
One result of the previous study was the derivationof a resonance condition that determines when theinteraction is highly inelastic such that the solitonsform a bound pair ~i.e., trapping! after collision.36
This resonance condition depends on the group-velocity difference ~relative angle for the present pur-poses! and the relative amplitude of the two solitons.A later theoretical study50 determined the permanentfrequency ~angle! shifts versus fiber birefringence af-ter inelastic collision between equal-amplitude tem-poral ~spatial! solitons and noted that theserequency ~angle! shifts become temporal ~spatial!hifts as a result of nonzero group-delay dispersionparaxial diffraction!. The same group of research-
756 APPLIED OPTICS y Vol. 38, No. 32 y 10 November 1999
rs numerically investigated the use of the inelasticollision between orthogonally polarized spatial soli-ons as switching gates.51
Figures 5 and 6 show some examples of these in-elastic collision interactions obtained through nu-merical simulation ~described below! by use ofrthogonally polarized spatial solitons with 2D 5 2y3.n each case the initial separation is rs 5 5.29. Col-
lision between equal-amplitude r 5 1 solitons isshown in Fig. 5. The top contour plots show ~elastic!collision at large angles, which behaves similarly tothe ~phase-sensitive! scalar collision. In the case ofunequal amplitudes r 5 3 as shown at the top of Fig., part of the weaker signal soliton breaks away afterollision with the pump, resulting in an inelastic in-
Fig. 5. Collision interaction between orthogonally polarized spa-tial solitons with equal amplitudes r 5 1 and initial separation 3.0
WHM of the signal width, or rs 5 5.29. Left, with D 5 0, forhich there is no nonlinear interaction; right, interaction withD 5 2y3. The top interaction has a gate length 5.0Z0 and
achieves a threshold contrast of 1.1, and the bottom fusion inter-action has a gate length of 15Z0 and achieves a threshold contrastof 23. Each soliton propagates at an angle given by the tangent ofthe initial separation divided by the gate length, and the value ofk 5 5.599w0 sin uyl in each plot corresponds to the total angle ofpropagation u between the solitons.
u~tu
1lb
teraction. In either case, though, gates based onlarge-angle collision have low contrast.
Small-angle collision, on the other hand, can pro-vide for high-contrast logic gates, as shown at thebottom of the two figures. Small-angle collision istermed slow collision in the temporal soliton litera-ture.36 In this case, nearly equal-amplitude solitons~Fig. 5! will fuse into a bound orbiting pair, whereas
nequal-amplitude solitons will be mutually reflectedFig. 6!. This reflection interaction is useful for op-ical switching because it occurs between solitons ofnequal amplitude ~thus providing gain! and allows
for as much as twice the transfer of transverse mo-mentum between the interacting solitons. As shownin Fig. 6, these small-angle collisions can provide thebasis for high-contrast angular deflection logic gates
Fig. 6. Collision interaction between orthogonally polarized spa-tial solitons with unequal amplitudes r 5 3 and initial separation.5 FWHM~1 1 1yr!, or rs 5 5.29. The top interaction has a gateength 5.0Z0 and achieves a threshold contrast of 9.6, and theottom deflection interaction has a gate length of 15Z0 and
achieves a threshold contrast of .1000. Each soliton propagatesat an angle given by the tangent of the initial separation divided bythe gate length, and the value of k 5 11.20w0 sin uyl~r 1 1!corresponds to the total angle of propagation u between the soli-tons. The output aperture width is now given by 3.5w0yr.
1
with gain, and are studied below in Sections 3 and 4for this purpose.
When r . 1, the inelastic collision interaction tendsto produce behavior that is analogous to the slingshoteffect in orbital dynamics.39 In this situation thesignal soliton enters the potential well of the pump~which is created through nonlinear cross-phase mod-ulation and is a three-dimensional function of thesoliton widths and separation! at a shallow angle.The signal is not trapped in the potential well andinstead makes a partial orbit and emerges with op-positely directed transverse velocity. The physicalmanifestation of this dynamic is illustrated at thebottom of Fig. 6. It is also possible that the signalundergoes multiple orbits before escaping,39 but onlythe simplest case is considered here. For a purelyelastic interaction, twice the initial transverse mo-mentum ~derated by the cross-phase modulation co-efficient! of the signal will be transferred to the pump,resulting in a large permanent angle change. In thecase of inelastic interaction studied here, part of thesignal may remain bound within the potential well ofthe pump, and some part may even pass through thepotential well completely without orbiting, as shownat the top of Fig. 6, resulting in less than twice thetransverse momentum of the signal’s being trans-ferred to the pump. For fixed initial signal trans-verse momentum, a larger fraction of the signalremains bound to the pump with increasing r.
2. Soliton Attraction–RepulsionThe first experimental study of soliton interactions inoptical fiber demonstrated the scalar attraction–repulsion interaction between temporal solitons ofthe same frequency.52 Experimental demonstra-tions of spatial soliton attraction and repulsion havealso been reported in liquid CS2 cells53 and in glassslab waveguides.54 An all-optical switch based onthe spatial interaction has also been proposed andanalyzed.55
In this copolarized spatial interaction, two spatialsolitons launched along parallel paths separated by afew beam widths attract, repel, or exhibit more-complicated behavior, depending on the relativephase. This interaction is strongly phase sensitivebecause the angular spectra of the interacting soli-tons overlap completely, which is the limiting case ofsmall-angle collision. Greater initial spatial overlapbetween the two solitons leads to a shorter interac-tion distance, but coherent beam combining can re-sult in random transmission when the sources are notphase locked to within a fraction of a radian, as theymust be in order that the nonlinear interaction berepeatable. The use of a beam splitter to combinethe two beams results in a 3-dB loss in the nonover-lapping regions because the solitons initially propa-gate with the same angle and are of the samepolarization. This 3-dB loss is unacceptable for op-tical logic because it ultimately reduces the gate fan-out by a factor of 2, with the loss taken not at theoutput of a gate but at the inputs to subsequent gates.
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tons can overlap at the input without loss and repre-sent the limiting case of inelastic small-anglecollision; however, it is not clear that the phase-insensitive vector attraction–repulsion interactionhas been studied in the literature as the basis for alogic gate. As in orthogonally polarized collision,solitons with nearly the same amplitudes will attract,and with sufficiently different amplitudes they willreflect or repel. In addition, the same type of inelas-tic behavior occurs in this case such that the solitonsdo not necessarily exit the interaction intact.
Examples of orthogonally polarized attraction andrepulsion interactions are shown in Fig. 7. Attrac-tion between equal-amplitude solitons is shown atthe top; repulsion between unequal-amplitude soli-tons is shown at the bottom. The repulsion exampleshown is an inelastic interaction in that part of thesignal breaks off and becomes bound to the pump.Repulsion becomes stronger as the initial separation
Fig. 7. Attraction–repulsion interaction between parallel-propagating orthogonally polarized spatial solitons with a gatelength of 20Z0. Top, attraction between equal-amplitude solitons
ith initial separation of 3.0 FWHM, or rs 5 5.29, achieving athreshold contrast of 9.8. Bottom, repulsion between solitons,where the pump power is r 5 3 times the signal and the initialseparation is 1.0 FWHM~1 1 1yr!, or rs 5 3.53, achieving a thresh-old contrast of 69.
758 APPLIED OPTICS y Vol. 38, No. 32 y 10 November 1999
is decreased from the well-separated condition, butthen a point is reached where the interaction weak-ens. One can understand this intuitively by notingthat, when the solitons completely overlap at the in-put, symmetry dictates that no angular or spatialdeviation of the center of mass is possible, and thethreshold contrast drops to approximately 1yr.Therefore there must be an optimal separation thatlies between the well-separated and overlapping con-ditions. One can break the symmetry at completeoverlap by allowing one or both of the solitons topropagate initially at an angle, which is the collisioninteraction with complete initial overlap. This in-teraction is known as spatial soliton dragging and isexamined next.
3. Soliton DraggingIn copolarized spatial collision, there is no net anglechange after the interaction, but a permanent spatialshift. During the collision, though, there is a tem-porary change in angle, such that the change is larg-est at the point when the solitons completely overlapand returns to zero when the solitons are well sepa-rated. The integral of this temporary angle changeresults in a small, fixed spatial shift. When the sym-metry of this interaction is broken by absorption–amplification41,56 or by initial overlap,17,57 some anglechange remains after the solitons separate. This ef-fect is detrimental to wavelength-multiplexed opticalcommunications58 ~in which a permanent colorchange can result! but is useful for optical switch-ng16,21 because the permanent angle change can be
integrated over long distances to provide large spatialshift, as discussed in Subsection 2.A.
The copolarized spatial soliton trapping interac-tion21,59 breaks the symmetry of the collision interac-tion by launching two solitons that are initiallyoverlapping in space and propagating at different an-gles. In this asymmetric configuration there is nointeraction before the point of complete overlap, soany angle change that occurs after overlap is not apriori compensated for by an equal and opposite an-gle change before overlap. This interaction is simi-lar to the orthogonally polarized spatial draggingthat is a focus of this paper but is highly phase de-pendent. During propagation, the solitons mutuallyattract, and, when the initial relative propagationangle is not too great, form a bound state. The netresult is a change in propagation angle such that thecontrast depends on the gate length, a property of thegeneral angular deflection gate. When the overlap-ping components of the two spatial frequency spectraare not in phase, more-complicated behavior can re-sult. As the relative phase is changed, interferencefringes move through the overlapping intensity re-gion such that, for k 5 1, a p relative phase shiftplaces a single dark fringe at the center of the overlapregion. Changing the relative phase can result innontrapping behavior, and the final position of thepump at the output can be moved. This effect hasbeen called soliton steering16 and is detrimental for
rdepawlr
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l
optical switching when phase cannot be precisely con-trolled.
Similar trapping behavior occurs in the phase-insensitive case of solitons of orthogonal polarizationas well, as was discussed in detail in Subsection 2.Aand described by the vNLS equations. Figure 8shows small- ~top! and large- ~bottom! angle spatialdragging between orthogonally polarized spatial soli-tons. Note that, in contrast to the situation for co-polarized trapping, now the solitons can beoverlapped at the input at small angles ~less thanesolvable! without power loss owing to constant ra-iance. Small-angle dragging k , 1 ~for nearlyqual-amplitude solitons! typically results in com-lete trapping such that the interacting solitons formbound orbiting pair ~as shown! and propagate at theeighted mean angle. When k . 1 ~bottom!, there is
ess spatial frequency overlap ~resulting in greaterelative transverse velocity! and the signal and the
Fig. 8. Dragging interaction between orthogonally polarized spa-tial solitons of unequal amplitudes ~r 5 3!. The initial total prop-agation angles between the solitons are given by k, and the gatelength is 5Z0 in each case. Each soliton acquires an angle shiftthat is due to mutual trapping, so the final displaced positionsdepend on gate length. The threshold contrasts of these gates are.1000.
1
ump break free from the respective potential wellsreated by their mutual interaction. However, theignal still imparts some of its oppositely directedomentum to the pump, which induces an angular
hange, such that the pump is literally dragged off tohe side and misses the spatial aperture. As shownn the figure, the signal does not completely escapehe potential well of the pump, and part of it remainsound to the pump. This part is known as a daugh-er, or shadow, beam, the formation of which leads toignal leakage through the aperture and can be re-oved by a polarization analyzer at the output.his phase-insensitive, initially overlapping interac-ion is referred to in general as spatial dragging, and,s shown by both cases illustrated in the figure, veryarge threshold contrast can be obtained.
3. Soliton Collision and Dragging Logic Gates
In this section we evaluate in detail logic gates basedon the phase-insensitive, inelastic interaction be-tween orthogonally polarized spatial solitons. Inparticular, the collision and dragging interactions areexamined because they belong to the class of angulardeflection gate that was discussed in Subsection 2.Aand are shown to possess desirable properties. Notethat the attraction and repulsion interactions arespecial cases of the more-general collision interaction~at zero angle! and will not be treated separately.
In addition, the effects of material absorption areconsidered in this section and in Section 4. The ma-terial example used for the studies of this paper isfused silica, mainly because of its large transparencyregion in the visible and the near infrared. Another,related, study focused on lead-doped silicate glass,23
which has smaller figures of merit than fused silica,and discussed other material candidates with largernonresonant third-order nonlinearity. Fused silicaat lf 5 1.55 mm ~Ref. 60! has the following properties:inear index n0 5 1.444, nonlinear Kerr index n2
I 53.3 3 10216 cm2yW and two-photon absorption coef-ficient b2
I 5 5.5 3 10215 cmyW, and a linear absorp-tion coefficient a0 5 0.1 cm21 owing to scattering inthe slab waveguide geometry is assumed.
A. Effects of Material Absorption
The main considerations with absorption are the rateof broadening of the pump, which is detrimental toangular deflection logic gates, and the amount of un-deflected pump power that remains at the end of thegate, which is the power that must be used when oneis calculating the large-signal gain of the gate asdiscussed in Section 4 below. In the presence of lin-ear and two-photon absorption, the following expres-sions describe the evolution of the spatial widthparameter and 1-D power of the pump soliton23:
wp2~d! 5
4K3a0 k0
~e2s 2 1! 1 ~w0yr!2e2s, (13)
Pp2~d! 5
r2P02
k03~n2
I!2Kr2P02~e2s 2 1!y3a0 n0
2 1 e2s , (14)
0 November 1999 y Vol. 38, No. 32 y APPLIED OPTICS 6759
a
i
w
ow
ti
g
mc
~tt
T
6
where dZ0 is the total gate length, a0 is the linearbsorption coefficient, s 5 a0dZ0 is the linear absorp-
tion parameter that represents the number of absorp-tion lengths, K 5 b2y2kfn2 is the normalized two-photon absorption coefficient, and kf 5 2pylf is thefree-space propagation constant. The first expres-sion for the pump width at the output determines thesize of the spatial aperture, which is always set to 3.5wp~d!, allowing 94.2% of the pump power to pass.
Linear amplification can be achieved in a glasswaveguide by rare-earth doping and external opticalpumping such that a balance between gain and two-photon absorption is obtained for which the pump canpropagate without changing. This is advantageousin angular deflection geometry because there is nobroadening and the gate throughput is unity ~theaperture is neglected!. This balance is achievedwhen
a0 5 24Kr2
3k0 w02 . (15)
It is expected that, in this situation, interactions willbehave much as for the case without absorption; how-ever, the signal will amplify and narrow during prop-agation when r . 1. In addition, increased gain willalso result in pump broadening and amplification,which could provide further benefit for the angulardeflection logic gates. These possibilities are pre-sented here simply as an interesting aside; we intendto present a further study elsewhere.
In the remainder of this section we focus instead onangular deflection soliton interactions in absorbingmedia. Before the general case is considered inmore detail, however, the two specific cases of linearand two-photon absorption in isolation are discussed.
1. Linear Absorption OnlyIn the absence of two-photon absorption ~TPA!, theevolution of the pump soliton width and power re-duces to
wp~d! 5 ~w0yr!es, (16)
Pp~d! 5 rP0 e2s, (17)
respectively. These expressions clearly illustratethe detrimental effects of absorption on soliton angu-lar deflection logic gates: exponential broadeningand exponential attenuation of the pump with dis-tance. Note that the rate of broadening and atten-uation of the signal is the same as for the pump.
Broadening leads to the minimum deflection angleof the pump that is necessary to produce spatial re-solvability at the output. As a function of propaga-tion distance, the minimum deflection angle is givenby
tan umin 51.7627wp~d!
dZ05
1.7627pr Îa0l
sdes. (18)
The value of s for which umin ~as a function of r and d!s minimized is s 5 0.5, which fixes the physical gate
760 APPLIED OPTICS y Vol. 38, No. 32 y 10 November 1999
length dZ0 5 0.5ya0. In addition, umin decreasesith increasing normalized gate length d and initial
ratio r because w02 } Z0 } 1yd and wp
2 } 1yr2d; hencea smaller deflection angle is necessary because wpdecreases and the spatial broadening factor is inde-pendent of r ~and independent of d for fixed s!. As isdiscussed next, the product r=d cannot be arbitrarilylarge, so there is a limit to how small umin can bemade.
The initial signal and pump soliton widths can bewritten as
w0 5 ^al, wp 5 ^alyr, (19)
respectively, where the linear absorption figure ofmerit ^a is defined as23
^a 5 S sp2a0 dlD
1y2
(20)
and is determined for a given physical gate lengthdZ0 and absorption parameter s. Using the condi-tion on the signal width parameter given by Eqs. ~19!ensures that attenuation over the length of the gate isgiven by e2s. The restriction that ^a . r is placed inrder that the initial pump soliton width parameterp be greater than approximately a material wave-
length. In the nonparaxial regime r ; ^a, addi-ional linear and nonlinear terms would need to bencluded in the evolution equations.61 For the logic
gate to possess large-signal gain, the undeflectedpump power at the output must be greater than thesignal switching power at the input, or re2s ; 1 orreater. Assuming that rmax 5 ^a, the optimal
value of s that maximizes pump throughput is sopt 50.5 ~so rmax 5 ^a . 1.65 is desirable!. The maxi-
um pump throughput can be further increased byhoice of small d, which allows for larger values of r.
The linear absorption figure of merit given by Eq.20! establishes an upper bound on r. Inasmuch ashe minimum deviation angle depends inversely onhe product r=d, it is useful to consider the maxi-
mum value of this quantity:
rmaxÎd 5 S sp2a0l
D1y2
, (21)
which depends only on s and material parameters.he minimum deflection angle can now be written as
tan umin 51.7627a0l
ses. (22)
Under this condition of maximum ratio, the mini-mum value of umin occurs when sopt 5 1.0. Whenthis value is used for s, the maximum pump through-put will drop by a factor of 2.33 from the optimalvalue when s 5 0.5. Alternatively, when s 5 0.5 isused for maximum throughput, umin will increase bya factor of 1.21 from its optimal value.
This simple analysis shows that there are two op-timal choices for s: one for which the pump through-put is maximized ~sopt 5 0.5! and one for which the
Fm
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ha
o0r
required deflection angle is minimized ~sopt 5 1.0!.or all studies in this paper, s is chosen to allow foraximum throughput.
. Two-Photon Absorption Onlyn the absence of linear absorption, the evolution ofhe pump width and power reduces to
wp~d! 5w0
r~1 1 4pdKr2y3!1y2, (23)
Pp~d! 5rP0
~1 1 4pdKr2y3!1y2 , (24)
respectively, so the pump broadens by the same fac-tor by which it attenuates. Because TPA provides alimiting mechanism that permits transmission ofsome maximum intensity ~or power in the case of abeam! only a given distance, it is useful to look atthese expressions as the initial ratio becomes large.Allowing r 3 ` then yields the expressions
wp~d! 5 w0y^TPA, (25)
Pp~d! 5 ^TPAP0, (26)
where the TPA figure of merit is defined as23
^TPA 5 ~3y4pdK!1y2, (27)
which does not depend on any physical distance.Expressions ~25! and ~26! indicate that it is desirablethat ^TPA be large to minimize spatial broadeningand maximize throughput.
Again, the minimum deviation angle of the pumpcan be written as
tan umin 51.7627wp~d!
dZ05
1.7627w0
^TPAdZ0. (28)
For some fixed value of Z0 ~which is arbitrary in theabsence of linear absorption!, larger normalized gatelength d leads to a longer physical gate length as wellut to a smaller minimum deflection angle as tan umin
} 1y=d. However, this benefit for the angular de-flection geometry comes at the cost of reduced pumpthroughput, because ^TPA } 1y=d. The constrainton this scaling is that ^TPA . 1 for unity or greateroutput-pump to input-signal ratio, which places theupper bound d , 3y4pK.
3. Linear and Two-Photon AbsorptionIn terms of the material figures of merit, the pumpsoliton width parameter and power can be rewrittenas
wp~d! 5l@r2^a
2~e2s 2 1! 1 2s~w0^TPAyl!2e2s#1y2
Î2sr^TPA
, (29)
Pp~d! 5Î2sr^TPAP0
@r2~e2s 2 1! 1 2s^TPA2 e2s#1y2 . (30)
1
As before, Eq. ~29! is used to calculate the minimumdeflection angle, which in general occurs for a differ-ent value of s than for linear absorption alone. Aslong as the rate of broadening owing to absorptionremains much less than would be produced by lineardiffraction, the leverage of the angular deflection ge-ometry can be utilized to increase the spatial resolv-ability at the output. The linear diffraction angle isgiven by
sin udiff 51.7627lr
p2w0, (31)
which is defined as the propagation angle of the spec-tral component at the intensity HWHM @cf. Eq. ~7!#and is valid in the paraxial far field of propagation.The necessary deflection angle is then ulin 5 2udiff,
hich is the minimum angle for linear resolvability.Using the material parameters of fused silica, we
ave plotted the spatial soliton minimum deflectionngle and the linear resolvability angle versus s for
two normalized gate lengths d [ $5, 10% and initialpower ratio r 5 3. Figure 9 shows that, even in thepresence of absorption, the minimum soliton deflec-tion angle is approximately 2 orders of magnitudesmaller than the linear resolvability angle. Theminimum soliton deflection angle occurs near s 5 0.5for both gate lengths. The linear resolvability angledecreases with increasing s because of increasingwidth, but the soliton deflection angle increases ~for slarger than the optimal! owing to increasing absorp-tion. Note that linear absorption does not changethe rate of broadening in linear propagation, and non-linear absorption ~which would affect the apparentrate of broadening because of greater attenuation atthe center than in the wings! cannot occur in linearpropagation. The soliton width, however, is affectedby both forms of absorption.
The maximum initial ratio is limited to r 5 ^a
owing to the restrictions placed by linear absorption.
Fig. 9. Linear ~dashed curves! and spatial soliton ~solid curves!minimum deflection angle as a function of linear absorption pa-rameter for normalized gate lengths d 5 5 ~thin curves! and d 5 10~heavy curves!. The initial power ratio r 5 3 and the materialparameters of fused silica at lf 5 1.55 mm are used. The minimaf the soliton deflection angle curves occur at s 5 0.505 and s 5.510 for d 5 5 and d 5 10, respectively. The angle scale is inadians.
0 November 1999 y Vol. 38, No. 32 y APPLIED OPTICS 6761
0m
r
pr
o
tastmfistbcflmamtcwtc
ast
m
6
For a given set of material parameters, the pumpthroughput factor &~d! 5 Pp~d!yP0 can be maximizedby the value of s that satisfies
e22sopt
1 2 2sopt5 S3pa0l
2K1 1D , (32)
which depends only on material parameters. Noticethat 0 # sopt , 0.5 for K . 0, and sopt 5 0.5 when K 5
because of linear absorption alone. For a givenaterial, the value of sopt fixes the total gate length
dZ0. Then the desired normalized gate length d de-termines the signal soliton width parameter w0 andallows ^a to be calculated, which sets the upperbound on initial ratio r. In general, smaller valuesof d lead to larger w0 and smaller input power and toless attenuation of the pump by TPA ~for fixed r!.
Figure 10 plots Eq. ~30! for the pump throughputfactor & versus s and parameterized by two normal-ized propagation distances, d [ $5, 10%. For bothdevice lengths the optimal value occurs when s 50.2627, as given by Eq. ~32!, providing a maximumoutput-pump to input-signal ratio of &~5! 5 10.5when r 5 ^a~0.2627.5! 5 22.3 and of &~10! 5 7.42when r 5 ^a~0.2627, 10! 5 15.7.
Using this optimal value for s, in Fig. 11 we haveplotted the soliton minimum deflection angle versusinitial ratio r. For each value of normalized gatelength, the initial ratio is limited to the value of^a~d!. As discussed above, the minimum deflectionangle decreases with normalized gate length, but Fig.11 also shows that the deflection angle decreases withr as well. In the linear absorption–dominated re-gime of small r, the broadening factor of the pump isapproximately given by esopt, which is constant withespect to r. Inasmuch as the total gate length re-
mains constant and the initial pump width wp } 1yr,the minimum deflection angle must scale as 1yr.For larger r, for which TPA becomes important,
Fig. 10. Plot of the pump throughput factor & given by Eq. ~30!versus the linear absorption parameter s parameterized by nor-malized gate lengths d 5 5 and d 5 10. The material parametersfor fused silica are used, and the optimum gain upper bound occursin both cases at sopt 5 0.2627, as given by Eq. ~32!. For theseparameters, &~0.2627, 5! 5 10.5 and &~0.2627, 10! 5 7.42.
762 APPLIED OPTICS y Vol. 38, No. 32 y 10 November 1999
broadening is enhanced compared with the rate thatis due to linear absorption alone, such that umin ap-
roaches an asymptotic minimum value obtained for3 `.The effects of material absorption on pump
throughput and minimum deflection angle have ram-ifications for the expected performance of a solitonangular deflection logic gate. The threshold con-trast of a gate is determined by the normalized ~to theutput aperture width! spatial shift and normalized
~to the input signal power! output power of the pumpsoliton. For increasing normalized gate length dand constant ratio r, Fig. 11 shows that the minimumdeflection angle of the pump decreases. Therefore itis expected that, for constant pump deflection angle,the normalized spatial shift will increase, which willresult in higher threshold contrast. In addition, thesignal transverse momentum increases as =d ~owingo the scaling of k!, potentially leading to greaterngular deflection as well. The greater pump ab-orption for larger d ~owing to TPA! will inflate thehreshold contrast, but the reduced pump throughputay decrease the obtainable large-signal gain. Forxed d, Fig. 11 shows that the same argumentshould hold true for increasing r. Here the signal’sransverse momentum increases as ~1 1 r! ~againecause of the scaling of k!, but the pump mass in-reases as r, with the expectation that angular de-ection remains constant, but the decreasinginimum deflection angle ~as 1yr in the linear
bsorption–dominated regime! suggests that nor-alized spatial shift and contrast also increase in
his case. However, the pump throughput factor in-reases with r, so a constant normalized spatial shiftill result in decreasing contrast. The net result of
hese effects is that the threshold contrast shouldhange weakly with r.
Next, the operation of the spatial soliton collisionnd dragging gates is studied in the presence of ab-orption. The phase-insensitive interaction be-ween orthogonally polarized solitons is modeled by
Fig. 11. Soliton minimum deflection angle for the optimized valuesopt 5 0.2627 versus initial power ratio r parameterized by nor-
alized gate lengths d 5 5 and d 5 10. The material parametersfor fused silica are used. The angle scale is in radians.
2
wnt
l
rt
aet
t
~ipdm
w
aoc
the vNLS equations with the inclusion of absorp-tion23:
2ik0
]Ax
]z1
]2Ax
]x2 1 2k02 n2
n0~1 1 iK!~uAxu2
1 2DuAyu2!Ax 1 ik0a0 Ax 5 0, (33a)
ik0
]Ay
]z1
]2Ay
]x2 1 2k02 n2
n0~1 1 iK!~uAyu2
1 2DuAxu2!Ay 1 ik0a0 Ay 5 0, (33b)
here any material or form birefringence has beeneglected along with the phase-dependent term inhe nonlinear polarization. Again, 2D 5 2y3 for an
isotropic material and orthogonal linear polariza-tions away from one- and two-photon resonances,which is consistent with the requirement for smallabsorption. For each value of gate length d, the fun-damental signal soliton width w0 is chosen such thatthe amount of linear absorption over the length ofthe gate is given by e2sopt, with the value sopt 50.2627 5 a0dZ0. This initial signal width is calcu-ated from the expression
w0 5 S soptl
p2a0 dD1y2
5 ^al (34)
and sets the maximum initial power ratio to ^a forthat gate length. Then, from Eq. ~30!, the maximumpump throughput is given by &~sopt, d!.
The simulations are performed by the well-knownsymmetric split-step Fourier method.62 The evolu-tion of the pump, for example, over a small longitu-dinal step Dz, can be written as
Ax~x, Dz! 5 exp~iDzQLy2!exp~iDzQIH!
3 exp~iDzQLy2!Ax~x, 0!, (35)
which is nominally second-order accurate in Dz.This separation into linear and inhomogeneous stepsis the main characteristic of the split-step method.Using Eq. 33~a!, we can write the linear and inhomo-geneous split-step operators as
QL 51
2k0
]2
]x2 1 ia0
23 Fk0~k0 1 ia0! 1
]2
]x2G1y2
2 k0,
QL 5 kz 5 @k0~k0 1 ia0! 2 kx2#1y2 2 k0, (36a)
QIH 5 kf n2~1 1 iK!~uAxu2 1 2DuAyu2!. (36b)
Here the expressions within the square roots are theexact ~assuming weak absorption! operator and Fou-ier phase representations of linear spatial diffrac-ion. A similar result ~without absorption! was
obtained previously63 with an alternative, but equiv-lent, form of the linear nonparaxial operator. Lin-ar propagation is performed in the more naturalransverse Fourier space with the relation
Ax~kx, Dzy2! 5 exp~iQLDzy2!Ax~kx, 0! (37)
1
for the initial half-step, for example. For correct-ness with off-axis plane-wave components the linearabsorption term must be applied in the Fourier do-main. TPA must be applied in the real-space do-main along with nonlinear refraction. Note that thenumerical scheme contains higher-order linear non-paraxial effects that are not included in the vNLSequations, but in the regime of the simulations in thispaper these additional effects are negligible.
B. Evaluation of the Soliton Collision Logic Gate
In this section we calculate the threshold contrastmetric for the collision interaction for a range of gatelengths d, normalized interaction angles k, and ini-ial power ratios r. Using the material parameters
of fused silica, we determine the operating regionswithin which a restoring logic gate with large-signalgain is expected.
It is particularly useful to obtain an optimal valuefor the normalized interaction angle k for use in therest of the paper. The optimal angle varies for agiven choice of interaction parameters but will bedetermined here for the initial ratio r 5 3. To thisend, in Fig. 12 we have plotted threshold contrastcontours versus normalized gate length and normal-ized angle. As expected for an angular deflectionlogic gate, the contrast increases with normalizedgate length. For all values of gate length, the nor-malized interaction angle k 5 0.33 provides nearlyoptimal threshold contrast. The minimum normal-ized gate length for contrast, t $ 10, is approximatelyd 5 6.2. The final power ratio for this gate length is&~0.2627, 6.2! 5 2.27, which is the expected large-signal gain. Note that, for a gate length d 5 10, theexpected large-signal gain is &~0.2627, 10! 5 2.24and is slightly smaller because of TPA!. For highernitial ratios ~or for materials with larger TPA!, theresence of nonlinear absorption will amplify thisifference, in which case it may become important toinimize the value of d at the expense of threshold
contrast.
Fig. 12. Threshold contrast contours for the spatial collision in-teraction for r 5 3, with the material parameters of fused silica
ith sopt 5 0.2627. The contours are spaced in decades, and theheavy contours represent the minimum desired contrast of 101 and
contrast of 104. Dashed line, the value of k that serves as a goodperating point for all normalized gate lengths. In the case of theollision interaction, this value is 0.33.
0 November 1999 y Vol. 38, No. 32 y APPLIED OPTICS 6763
sl
wursttptts
Il
r
aoaiaataaWab
1
tl
saF
adndsdtph
6
With the optimal interaction angle k 5 0.33, Fig. 13hows threshold contrast versus normalized gateength and initial ratio r. The leftmost heavy con-
tour represents the minimum desired contrast of 10;therefore the region enclosed within this contour~which extends toward the upper right-hand corner!meets or exceeds this criterion, indicating that alarge-signal gain of approximately &~r, d! ~denoted bythe dashed curves! should be obtainable. The con-tour levels are separated by powers of 10, so largethreshold contrasts are obtained for longer gatelengths and larger values of r. In these regions ofhigh contrast, for example within the other heavycontour, which represents t $ 104, it is expected thatgate operation will be well within the saturated lowoutput level and that large-signal gains greater than&~r, d! can be obtained.
Note that, for the parameters of fused silica, theminimum value of ^a~sopt, d! is 15.7 when d 5 10,
hich is larger than the highest initial ratio of 10sed here and in the following sections. The maineason why the maximum value is not used is thatufficiently high values of &, for which the interac-ions produce t $ 10, can be obtained. In fact, forhe longer gate lengths, greater values for r do notroduce significantly greater values for &, which ishe result of the rapid attenuation at high peak in-ensities that is due to TPA. Another practical con-ideration is that, for larger r approaching ^a~sopt, d!,
more time-consuming fully nonparaxial simulationswould need to be performed as wp 3 l.
The minimum gate length is a strong function ofthe initial power ratio r for the collision interaction.t is clear that, when r # 2.0, the minimum gateength increases rapidly with decreasing r, to the
point that the gate length is greater than 10Z0 when5 1. Figure 5 shows that the r 5 1 collision inter-
Fig. 13. Threshold contrast contours for the collision interaction,with the parameters of fused silica and the value sopt 5 0.2627 forll normalized gate lengths. The contour levels are spaced inecades, and the heavy curves show t 5 101 and t 5 104. Theormalized interaction angle is fixed at k 5 0.33. Absorptionecreases the ratio of the undeflected pump at the output to theignal at the input. This final ratio &~r, d! is indicated by theashed curves, which do not include the 5.8% power loss that is dueo clipping by the output aperture. For small values of r, two-hoton absorption is weak, and the dashed contours are nearlyorizontal.
764 APPLIED OPTICS y Vol. 38, No. 32 y 10 November 1999
ction at small k results in the formation of a boundrbiting pair, with little induced change in the prop-gation angle of the pump. This behavior manifeststself in Fig. 13, which illustrates that the contrast isweak function of gate length when r 5 1. It shouldlso be noted that the unity contrast contour extendso zero gate length; this is due to material absorptionnd to the choice of aperture size, the latter of whichllows 94.2% of the undeviated pump power to pass.hen r 5 1, TPA can be neglected, and the undevi-
ted pump output power ~after the aperture! is giveny 0.942P0e2sopt 5 0.724P0, which sets the lower
bound on the threshold contrast to t $ 1y0.724 5.38.As expected for an angular deflection logic gate, the
hreshold contrast increases with normalized gateength d for the collision interaction. However, Fig.
13 also shows that contrast increases with r, which isnot expected for an angular deflection gate, for whichthe threshold contrast should remain approximatelyconstant with r. It is straightforward to explain thisphenomenon by considering the scaling of the colli-sion interaction parameters with r for a fixed value ofd. The normalized spatial separation is fixed at rs 55, meaning that the separation in physical units de-creases as ~1 1 1yr!. When the separation is com-bined with the increase in real propagation angle ~ortransverse momentum! of the signal that is due to thescaling of k, the actual distance to interaction de-creases, causing an effective increase in the gatelength, which leads to additional integrated spatialshift. Thus the decrease in effective interaction dis-tance is the main contributing factor in the observedincrease in contrast with r.
C. Evaluation of the Soliton Dragging Logic Gate
In this subsection we calculate the threshold contrastmetric for the spatial dragging interaction over thesame range of gate lengths and initial power ratios aswere used for the collision interaction. The differ-ences for the dragging interaction are that largervalues of k ~;0.8 versus ;0.3! produce optimal re-ults and that the minimum normalized gate lengthsre shorter. These observations are illustrated inig. 14, which plots threshold contrast versus k and
normalized gate length for r 5 3. The minimumnormalized gate length is approximate d 5 2.4, witha final output ratio of 2.29, versus a required lengthof approximately d 5 6.2 for the collision gate. Thisfigure also illustrates that the dragging interaction istolerant of a much wider variation of angle, such thatconsistent gate operation is expected without criticaltolerancing.
Figure 15 plots threshold contrast versus initialpower ratio and normalized gate length for the drag-ging interaction, with the optimized angle k 5 0.8obtained from Fig. 14. As before, the leftmost heavycontour represents the minimum desired contrastof 10, and large threshold contrast is obtained forlonger gate lengths, as expected. The minimumnormalized gate length for the dragging interactionis relatively constant with initial power ratio and
tamoedilb
fih
ltpess
siiip
wt
tgm
atiri
approaches d 5 2.0 for large r. As was discussedabove, this property provides an indication that thedeflection angle of the pump remains approximatelyconstant, resulting in a normalized ~to the aperturewidth! spatial shift that increases with r, which com-pensates for the increased throughput factor of thepump.
In the case of complete trapping, as illustrated atthe top of Fig. 8, the induced angle change of thepump must in fact remain constant with r in orderhat both solitons propagate at the weighted meanngle and conserve transverse momentum. Theaximum contrast is obtained when spatial dragging
ccurs, however, that is, when all or part of the signalscapes the potential well of the pump and literallyrags the pump to the side in the process, also induc-ng a permanent angle change that is not necessarilyinearly resolvable. This situation is shown at theottom of Fig. 8. Part of the signal, known as the
Fig. 14. Threshold contrast contours for the spatial dragging in-teraction for r 5 3, with the parameters of fused silica with sopt 50.2627. The contours are spaced in decades, and the heavy con-tours represent the minimum desired contrast of 101 and a con-trast of 104. Dashed line, the value of k that serves as a goodoperating point for all normalized gate lengths. In the case of thedragging interaction, this value is 0.8.
Fig. 15. Threshold contrast contours for the dragging interaction,with the parameters of fused silica and the value sopt 5 0.2627 forll gate lengths. The contour levels are spaced in decades, andhe heavy contours show t 5 101 and t 5 104. The normalizednteraction angle is fixed at k 5 0.8. Absorption decreases theatio of the undeflected pump at the output to the signal at thenput. This final ratio &~s, d! is indicated by the dashed curves,
which do not include the 5.8% power loss that is due to clipping bythe output aperture.
1
soliton shadow, may also remain bound to the pump,resulting in an orbiting pair. This orbiting behavioris in part responsible for the numerous disjoint con-tour regions in Fig. 15, but the major contributingfactor is discussed in Subsection 3.D. Because thenormalized interaction angle k is held constant, theinitial transverse momentum of the signal increaseswith respect to r. Inasmuch as the solitons initiallyoverlap in space, the effective interaction distance~i.e., the distance over which the induced anglechange becomes fixed! for a given value of r is fixed to
rst order, which does not result in strongly en-anced contrast as with the collision interaction.
D. Comparison of Logic Gates
Here the collision and dragging interactions are com-pared in terms of their properties for the implemen-tation of logic gates. Particular attention is paid tothe minimum normalized gate length for a givenvalue of r and the maximum value of r at a given gateength, under the condition that threshold contrast
$ 10. Large values of r are needed to maximizeump throughput, but optimized interaction param-ters for small ratios can result in lower power dis-ipation owing to TPA and in Section 4 below arehown to provide large-signal gain.The dragging interaction generally produces
horter normalized gate lengths than the collisionnteraction. The main reason is that in the collisionnteraction the solitons are not overlapping at thenput. As a result, the actual interaction occurs ap-roximately 1–2Z0 inside the material. One could
reduce this extra distance by decreasing the initialseparation at the input such that there is some, butnot complete, overlap. However, the distinction be-tween the collision and dragging interactions be-comes blurred, such that they may be no longerconsidered separate, and reveals the tolerance to~asymmetric! spatial misalignments that these de-flection interactions possess. The same argumentsshow that absorption does not influence dragging sostrongly because the interaction occurs at the begin-ning of the nonlinear material as a result of completeoverlap, before significant absorption occurs. Colli-sion, on the other hand, takes place within the ma-terial such that absorption has already attenuatedthe pump and the signal, causing their widths toincrease. This increase in width decreases the dis-tance to interaction somewhat, but the interactionwill subsequently require a longer physical distancefor high contrast to be achieved after the collisionbecause of the scaling of the confocal distance ~to
hich the gate length is defined! with increasing soli-on width.
Another important difference between the interac-ions is that the operating regimes of the draggingate are much less sensitive to variations in the nor-alized angle k than those of the collision gate. As
a result, the collision gate will require more-precisetolerancing in any real implementation. The largeacceptance angle of the dragging gate also suggestsanother advantage compared with the collision gate
0 November 1999 y Vol. 38, No. 32 y APPLIED OPTICS 6765
ps
esiina
i1iwcwtNcpf
ndkhi
6
in that multiple signal inputs can be used to imple-ment a multi-input NOR gate in a single stage. Inthis implementation the signals must be greater thanlinearly resolved ~i.e., differences in k between thesignals greater than 1! to avoid phase-sensitive be-havior. Figure 14 shows that this should indeed bepossible and that high contrast can be obtained forgate lengths of ;3.0Z0 or longer when either signal is
resent when k1 5 0.6 between the pump and firstignal and k2 5 1.2 between the pump and second
signal, so Dk 5 0.6. In terms of the normalized anglebetween the two signals, Dk9 5 ~r 1 1!Dky2 5 1.2when r 5 3, which is more than angularly resolvable.Of course, the case when both signals are presentneeds to be investigated to guarantee robust opera-tion, the study of which is beyond the scope of thispaper.
Figure 16 shows numerical simulations of example
Fig. 16. Spatial soliton collision ~top! and dragging ~bottom! logicgates in the presence of linear absorption and TPA. Here the gatelengths are 10Z0 with sopt 5 0.2627 and initial ratio r 5 3. The
ormalized initial separations are rs 5 5 for collision and rs 5 0 forragging, and the normalized interaction angles are k 5 0.33 and5 0.8, respectively. The pump soliton is represented by the
eavier curves, and the intensity contours are spaced in 3-dBntervals. The threshold contrasts of these gate are 3.1 3 104 and
7.4 3 103. Note that the dominant diffracting wave in the drag-ging interaction collides with and reflects off the absorbing bound-ary conditions of the simulation, but this does not affect gateoperation.
766 APPLIED OPTICS y Vol. 38, No. 32 y 10 November 1999
interactions for the collision ~top! and dragging ~bot-tom! gates. In each case, r 5 3 and the gate lengthis 10Z0. The plots clearly show that large deflectionof the pump soliton occurs in both interactions, lead-ing to very high-contrast operation such that thelarge-signal gain of each gate is expected to be at leastthe output ratio &~10! 5 2.25 ~multiplied by the ap-rture throughput factor 0.942!. The interactionshown are inelastic, with significant signal breakupn each case. In addition, the plots show that thentense pump is more strongly absorbed at the begin-ing of the interaction than at the end under thection of TPA.In general, the r . 1 collision interaction is highly
nelastic for the signal. In the case shown here ~Fig.6, top!, at the point of collision the signal breaks upnto transmitting and reflecting diffracting waves,ith spatial broadening of the reflected signal wave
learly shown. As a result of the weak transmittedave, less than twice the transverse momentum of
he signal is transferred to the pump on reflection.ote that in some cases ~not that shown in Fig. 16 for
ollision!, part of the signal will remain bound to theump, as discussed above, and result in the mutualocusing ~termed here cross focusing! of both the
bound pump and the signal. The values of interac-tion parameters r and k determine the precise behav-ior. As the reflected signal wave propagates in thedirection opposite the deflected pump, some power inthe signal polarization may escape through the aper-ture. Blocking this leakage requires an analyzer atthe output, which is assumed to be present for allsimulations. Even in the inelastic collision interac-tion, the pump tends to return to a soliton shape wellafter the collision. This means that almost no dif-fracting radiation is generated from the pump, whichremains nearly exponentially localized. There issome diffracting pump radiation generated by theinteraction of Fig. 16, however, which is weaklybound to the reflected signal, but the amplitude ofthis radiation is more than 5 orders of magnitudesmaller than the main peak.
A simulation of the spatial dragging gate is shownat the bottom of Fig. 16. For this simulation theinteraction is highly inelastic for the signal, whichbreaks into two components: the shadow, which re-mains bound to the pump, and a freely propagating,diffracting component. Because of this breakup, notall the initial transverse momentum of the signal isshared in the bound state, but sufficient momentumis transferred to the bound pair that a large spatialshift of the pump does occur, leading to high thresh-old contrast. Because the signal always propagatesto the same side of the aperture as the pump, noanalyzer is required at the output to block leakage ofthe signal polarization. This interaction also illus-trates other vectorial effects besides cross focusing.Here, cross TPA between the pump and the orthog-onally polarized signal shadow results in strongerattenuation of the signal compared than that whichwould occur with the signal alone and in slightlymore attenuation of the pump than occurs in the
t
o~tl
srp
f
collision interaction, for which there is nominal trap-ping. In addition, initial cross focusing appears tocompensate for the rapid initial decrease in pumpintensity that is due to TPA.
Exponential localization of the pump after interac-tion gives rise to threshold contrast that increasesexponentially for more than spatially resolvable de-flection. This factor explains the regular contoursfor the collision gate in Figs. 12 and 13 and the factthat the maximum threshold contrast is more thanan order of magnitude higher than for the dragginggate. The dragging interaction is inelastic for thepump soliton as well as for the signal. The result isthat, during the interaction, diffracting radiation isgenerated from the pump, leading to exponential lo-calization of the main radiation with well-definedsidelobes. These sidelobes are typically 3–4 ordersof magnitude smaller in intensity than the mainpump, but, in the large-contrast regime t . 103 forlarge deflection, they form the major contribution tothe power that flows through the aperture. This ef-fect is seen in the contours of Figs. 14 and 15. Thedifference between the maximum obtainable contrastfor the collision and dragging gates is not a majorfactor because the large-signal gain calculated in Sec-tion 4 does not change noticeably between t 5 103 and
5 104, for example.
4. Soliton Logic Gate Transfer Functions
In Section 3 it was demonstrated that the inelasticcollision and dragging interactions can produce suf-ficient angular deflection of the pump soliton to pro-duce large threshold contrast over large regions inthe parameter space spanned by normalized interac-tion angle, initial power ratio, and normalized gatelength. However, the properties of logic gates aretypically determined based on the plot of gate outputversus gate input, called the gate transfer function.In this section the calculation of the transfer functionis addressed, and small-signal gain, large-signalgain, and noise margins, which are useful for char-acterizing the performance of these soliton logic gates~and of logic gates in general!, are defined. If theoperation of all gates in the system were identical, thelarge-signal gain ~within the constraints of accept-able noise margin! would give the fan-out of the gate.This is the situation assumed here, with furtherstudy of cascaded operation ~which one must considerto properly obtain the fanout factor! presented else-where.64 From this section, the large-signal gainand the switching power serve as a basis for compar-ison among gates based on collision and dragginginteractions.
A. Definition of Transfer Function
Large threshold contrast provides an indication thatsmall-signal gain @and therefore large-signal gainwhen &~d! . 1# can be obtained. For a particularinitial power ratio r, the gate transfer function isobtained as the plot of output power of the gate ver-sus the signal’s input power, as illustrated schemat-ically in Fig. 17, where the input signal amplitude is
1
varied but constant width is maintained such thatthe signal is a fundamental soliton at only one par-ticular amplitude, or power. At that particularpower ~to which the input and output levels are nor-malized!, the threshold contrast t is calculated.Therefore, for unity input level ~given by P0, thepower of the fundamental signal soliton!, the normal-ized output level is given by 1yt, and the transfercurve must pass through point ~1, 1yt!. When t . 1,nly a fraction of the original signal soliton powerinput level less than unity! is required for reachinghe threshold switching level, which is defined as theevel at which the input and the output are the same.
Because little deviation of the pump occurs for amall input level, the transfer curve is flat, or satu-ated, at the high output state. Therefore the out-ut remains saturated near the output level POH
until an input level of PIL is reached. Likewise, athigh input levels beyond the point at which spatiallyresolvable deviation of the pump occurs, further in-crease of the input leads to little change in the output.Here POL is the output level at the knee of this sat-urated state and PIH is the associated minimum in-put level. As a result, the transfer curve must alsopass through points ~PIL, POH! and ~PIH, POL!. Thethin solid curve in Fig. 17 indicates the canonical, orstraight-line,31 representation of the transfer func-tion that intersects the three known points, whereasthe heavy solid curve is representative of a moretypical function. In terms of small-signal gain, thestraight-line representation is the worst-case trans-fer function; any other transfer function must possessa region of larger small-signal gain.
Fig. 17. Generic inverting logic gate transfer function. Thethreshold contrast t is calculated at the unity input level ~normal-ized by the fundamental signal soliton power!, and 1yt gives thecorresponding output level. Thin solid curve, straight-line trans-fer function; heavy solid curve, a more-realistic transfer function,which necessarily has greater small-signal gain. These transferfunctions must pass through points ~0, POH!, ~PIL, POH!, ~1, 1yt!,and ~PIH, POL!. Dashed curves represent the inverted transferunctions and are used to locate the stable operating points.
0 November 1999 y Vol. 38, No. 32 y APPLIED OPTICS 6767
sTdltt
w
v
6
For small-signal gain, the slope of the canonicaltransfer function, defined by
m 5POH 2 POL
PIL 2 PIH, (38)
must satisfy the relation m , 21. When m satisfiesthis criterion, an inverting, restoring logic gateshould result for the given choice of parameters. Interms of the threshold contrast, a bound on the slopeis given by
m #POH 2 1yt
0 2 1,
1t
2 &~d!. (39)
This bound takes into account only the pumpthroughput factor and threshold contrast and is eas-ily determined from the parameter plots of Subsec-tions 3.B and 3.C. and shows that the restriction on tis relaxed considerably when &~d! is large.
The large-signal gain of the gate is defined as theoutput level of the valid high device state divided bythe input level of the valid low device state, or G 5OHyIH, where OH and IH are obtained from the stableoperating points. For the present purposes, large-signal gain is approximated by G 5 &~d!yPIH and canbe estimated with the knowledge of &~d! and t.When 1yt # POL, then it is clear that PIH # 1. Inthis situation, an approximate lower bound to thelarge-signal gain is given by &~d!. It is found in latersimulations that typically POL ; 0.1, so t . 10 pro-vides an indication that a large-signal gain of at least&~d! ~derated by the throughput factor of the outputaperture! can be achieved. This does not mean thata large-signal gain cannot be achieved when t , 10,because a large-signal gain of at least unity is ob-tained as long as PIH , &~d!, which could occur forsmall t. In addition, by relation ~39!, for t $ 10 amall-signal gain is obtained when &~d! . 1.1.herefore the condition t . 10 provides a strong in-ication that a gate will function with small- andarge-signal gain, but it should be emphasized that, 10 does not guarantee that a gate will not possess
hese properties.The regions of valid high and low output levels,ith the corresponding valid low ~I # PIL! and high
~I $ PIH! input levels, as illustrated in Fig. 17, aredefined as the regions of the transfer curve for whichthe slope is less than unity, such that any variationabout the stable operating points of the system~which must lie within these valid regions! is atten-uated. Therefore, as long as the noise marginsabout the operating points are greater than the sta-tistical variation in the input levels, the gates willfunction properly in a complex system. If no addi-tional sources of noise were present in the system,because the local slope in the region near the stableinput levels were less than unity, the variance in thefluctuation about the input levels would asymptoti-cally collapse to zero as the digital signal propagatedthrough a chain of gates. The input levels that cor-respond to the operating points of the device are de-
768 APPLIED OPTICS y Vol. 38, No. 32 y 10 November 1999
noted IL and IH and locate the two self-consistent,stable points of the transfer function that would beobtained in a ring oscillator configuration. Thethreshold level is also a self-consistent point of thetransfer function, but it is not a stable operating pointbecause the local slope, or small-signal gain, isgreater than unity.
The noise margins of a particular device are tradi-tionally defined for an electronic gate as31
NML $ PIL 2 POL (40a)
NMH $ POH 2 PIH, (40b)
where powers are used here as the relevant gateinput and output quantities ~for optics! instead ofoltages and NML and NMH are, respectively, the
high- and low-noise margins. These relations pro-vide lower bounds on the noise margins, which aremore precisely defined for the present studies by useof the stable operating points. The probability-density function that represents the distribution ofinput-signal values about the stable input levels ILand IH ~which includes effects such as the laser’srelative intensity noise, scattering, and variation indevice parameters! must fit within the noise marginsof the gate to within a number of standard deviationss.
For a particular value of fan-out, the noise marginsare recalculated. The self-consistent operatingpoints must shift ~assuming that fan-out equalslarge-signal gain!, such that the input levels are IL 5OLyG and IH 5 OHyG, where G is the desired large-signal gain. The single-sided noise margins arethen defined more precisely:
NML 5 PIL 2 OLyG 5 PIL 2 IL, (41a)
NMH 5 OHyG 2 PIH 5 IH 2 PIH. (41b)
Notice that there is a trade-off between large noisemargin and large fan-out, such that the maximumfan-out of the gate is obtained when the noise mar-gins reach the minimum acceptable width in terms ofs. Fan-out can be increased for high-contrast elec-tronic logic gates through the use of level shifters, forwhich there is no direct analog in optics. This is thereason why the noise margins are defined as abovefor optical logic, but it should be noted that adding abias beam to the gate signal input would alter thesedefinitions. The statistical distributions about ILand IH are not considered in this paper. Instead,based on the obtained transfer functions, the allow-able noise margins are calculated, and it is arguedthat sufficient noise margin exists for the chosenvalue of fan-out to ensure proper operation in a more-complex system of gates.
B. Collision Gate Transfer Functions
Figure 13 shows that the collision gate can providevery large threshold contrast when r . 1. Thereforeit is expected that these gates will function with bothsmall- and large-signal gain. The specific case of r 5
1
shPlaITttf
gssisfilst
gle
d
s
gN~
o
3 was examined in more detail in Fig. 12, from whichthe value k 5 0.33 was obtained. Even including theeffects of absorption, the threshold contrast for ther 5 3 collision gate exceeds 10 for gate lengths of;6.2Z0 or longer with &~d! $ 2.24. It is thereforeexpected that, in this range of gate lengths, restoringlogic gates with large-signal gains of at least 2 can beobtained.
Figure 18 shows the transfer functions for the r 53 collision gate of lengths 5Z0 and 10Z0, includingabsorption with sopt 5 0.2627. The threshold level ofthe d 5 5 gate is 0.97, which is expected inasmuch as
Fig. 18. Transfer functions for the r 5 3 collision gate of lengths5Z0 ~top! and 10Z0 ~bottom! with normalized angle k 5 0.33 andeparation rs 5 5. Absorption is included using sopt 5 0.2627 for
each gate length. The threshold contrasts are t 5 1.2 for the 5Z0
gate length and t 5 4.7 3 104 for the 10Z0 gate length. Theseates provide large-signal gains of G 5 1.6 with NML 5 0.60 andMH 5 0.11 ~top! and G 5 2.8 with NML 5 0.18 and NMH 5 0.07
bottom!. The high noise margins represent 8.2% and 9.7% devi-ation about IH, respectively, and the filled diamonds denote theperating points and threshold.
1
the threshold contrast is t 5 1.2. The rule of thumbused in obtaining the minimum level of t 5 10 wasthat POL ' 0.1, and POL 5 0.14 here, but, given that
yt 5 0.83 .. POL, it is clear that the high input levelPIH . 1. But it was mentioned above that large-ignal gain is still obtained when PIH , &~5!, as it isere, with the chosen value G 5 1.6. Given thatIL 5 0.63, relation ~39! suggests that t $ 0.56 for at
east unity small-signal gain. Because t 5 1.2, anpproximate lower bound is given by 1.2y0.56 5 2.1.n fact, the small-signal gain of this gate is 4.8.herefore this gate provides one example of a situa-ion in which small- and large-signal gains are ob-ained when t , 10. As the output saturates rapidlyor input levels less than PIL and greater than PIH,
this gate exhibits large noise margins.The threshold contrast t 5 4.7 3 104 for the d 5 10
ate indicates that point ~1, 1yt! lies well within theaturated region of valid low states. Thereforemall-signal gain is guaranteed @because the normal-zed high output 0.942 &~10! 5 2.11 . 1# given amooth transfer function ~the small-signal gain is 5.7or this gate!, and a large-signal gain of at least 2.11s expected. For G 5 2.8, the stable operating pointsie well within the region of valid high and low devicetates ~i.e., the regions where small-signal gain is lesshan unity!, leading to noise margins of NML 5 0.18
and NMH 5 0.07, which should be sufficient for sus-tained, cascaded operation. Note that this providesonly an indication of the cascadability of these de-vices, because even though the output power of thedevice ~which is divided by the fan-out and serves asthe signal input to the next gate! may correspond toa particular input power, the shape of the new inputbeam is not exactly the same as the original input-signal soliton shape as a result of clipping at theoutput aperture. This effect is minimized in the ab-sence of the signal such that the pump is undeviated,where symmetric clipping of the wings resulting in asmall power loss occurs, but the overall effect is toalter the shape of the transfer characteristic. As aresult, true logic-level restoration is not obtained forthe first gate ~because the shape is not restored!, butit is obtained in cascaded operation, for which theclipped output is the same input for every gate.
C. Dragging Gate Transfer Functions
The comparisons of Subsection 3.D showed that thedragging gate generally has a shorter length than thecollision gate for given values of r and t, which sug-ests that the dragging gates can achieve greaterarge-signal gain at a fixed gate length. These prop-rties are verified here.Figure 4 shows the transfer functions for an r 5 1
ragging gate of length 10Z0 including the effects ofabsorption with sopt 5 0.2627 at 10Z0. In this case,t ' 100, which indicates that the threshold level is,1, and POL ' 0.1, which satisfies the rule of thumbemployed to place the condition t . 10. The r 5 1dragging gate provides large-signal gain of G 5 1.4,with NML 5 0.13 and NMH 5 0.07. Of course, if
0 November 1999 y Vol. 38, No. 32 y APPLIED OPTICS 6769
rl6l
casml
m
FS
t
a
t
6
necessary, one can increase the noise margins by re-ducing the large-signal gain.
Even though large-signal gain ~which should leadto fan-out in a cascaded system! can be obtained forthe r 5 1 dragging gates, it is desirable to obtaingreater values for increased fan-out. For generalpurpose logic, a minimum fan-out of 2 is necessary.Because of material absorption, this minimum maynot be achievable for the r 5 1 dragging gate, forwhich the maximum large-signal gain is 1.4 for d 510 within the constraint of a reasonably high noisemargin of ;10% variation about IH. Therefore wehave investigated the r 5 3 dragging gates to provideincreased large-signal gain.
The transfer functions for the r 5 3 dragging gatesare shown in Fig. 19, for gate lengths of 5Z0 and 10Z0,each with sopt 5 0.2627. With the same device pa-ameters, the dragging gates provide much greaterarge-signal gains, G 5 3.5 for the d 5 5 gate and G 5.0 for the d 5 10 gate, than the corresponding col-ision gates, with similar high noise margins. Be-
Fig. 19. Transfer functions for the r 5 3 dragging gate of lengths5Z0 ~top! and 10Z0 ~bottom! with normalized angle k 5 0.8 andseparation rs 5 0. Absorption is included using sopt 5 0.2627 foreach gate length. The threshold contrasts are t 5 1.8 3 103 forhe 5Z0 gate length and t 5 7.4 3 103 for the 10Z0 gate length.
The gates provide large-signal gains of G 5 3.5 with NML 5 0.05nd NMH 5 0.07 ~top! and G 5 6.0 with NML 5 0.02 and NMH 5
0.05 ~bottom!. The high noise margins represent 12% and 15%deviation about IH, respectively, and the filled diamonds denotehe operating points and threshold.
770 APPLIED OPTICS y Vol. 38, No. 32 y 10 November 1999
ause the dragging interaction works so well, thellowable noise margin about the low input is muchmaller than for the collision gates, which possessuch broader saturated regions at the low input
evel. Here, PIL 5 0.06 for d 5 5 and PIL 5 0.02 ford 5 10, which result in low noise margins of NML 50.05 and NML 5 0.02 for IL 5 0.01 and IL 5 0.0,respectively, and show that the dragging gates intrin-sically provide higher gain with smaller NML ~be-cause PIL is so small! and comparable high noise
argin NMH ~on a percentage basis!.
5. Discussion and Conclusions
In this paper a novel spatial soliton angular deflec-tion geometry has been considered that forms thebasis for three-terminal, restoring optical logic gateswith large-signal gain. Because of the nondiffract-ing nature of spatial solitons, a resolvable spatialshift at the gate output can be produced by an angu-lar deflection that is not resolvable and can be real-ized by a weak nonlinear phase shift induced across astrong pump soliton by a smaller-signal wave. Ex-amples of soliton interactions that produce angulardeflection were discussed, with two instances of theseinteractions—soliton collision and dragging—studiedin further detail.
The effects of linear and nonlinear material absorp-tion were also considered. Absorption is detrimen-tal to angular deflection logic gates because ofsimultaneous attenuation and broadening of thepump soliton such that a minimum deflection angle~which remains much smaller than the linear resolv-ability angle! is needed to produce spatial resolvabil-ity. However, simulations through the deviceparameter space showed that the collision and drag-ging interactions produce a large enough angular de-flection to achieve spatial resolvability even with theincreased initial ratio needed to compensate for ab-sorption and maintain a large pump throughput fac-tor.
Finally, based on the calculation of the gate trans-fer function, specific examples of the spatial solitoncollision and dragging gates were studied and shownto possess the basic requirements of a restoring logicgate. These simulations suggested that large-signalgain ~sufficient for fan-out of two or greater! withwide noise margins could be obtained, with the ex-pectation that fully cascadable operation with therealization of logically complete multi-input NOR
gates is also possible, which is the subject of furtherinvestigation.
It is useful at this point to consider physical pa-rameters of the r 5 3 angular deflection logic gates.
or the logic gate transfer function simulations inection 4, sopt 5 0.2627, and it was assumed that a0 5
0.1 cm21 for fused silica waveguides. All physicalgate lengths are obtained from the condition a0z 50.2627, or z 5 2.63 cm, but the soliton widths andpowers are determined by the value of the normalizedgate length d.
For the d 5 5 normalized gate lengths, w0 5 23.9mm ~FWHM 42.2 mm!, and wp 5 w0yr 5 7.97 mm
2
i
4p
s;erellA
plications of new low-power nonlinear effects in InSb,” Appl.
1
1
1
1
1
1
1
~FWHM 14.0 mm! for the initial ratio r 5 3. Withthe nonlinear index of fused silica taken as n2
I 53.3 3 10216 cm2yW, the peak intensities of the fun-damental solitons are I0 5 2.23 3 1011 and Ip 5.01 3 1012 Wycm2. Assuming a 4-mm-thick
waveguide, the corresponding fundamental solitonpowers are P0 5 4.27 3 105 and Pp 5 12.8 3 105 W,neither of which represents the actual switchingpower. The switching power is determined from theproduct of the signal power and the high input levelIH. For the d 5 5 collision gate, the switching powers IHP0 5 1.34P0 5 5.7 3 105 W. The d 5 5 drag-
ging gate has a much lower switching threshold, withswitching power 0.61P0 5 2.6 3 105 W, more than afactor of 2 smaller, with corresponding large-signalgain more than a factor of 2 greater.
The physical length of the d 5 10 gates is z 5 2.63cm as well, with w0 5 16.9 mm ~FWHM 29.8 mm! andwp 5 5.63 mm ~FWHM 9.93 mm!. The correspondingpeak soliton intensities are a factor of 2 greater thanfor the d 5 5 gate, with the powers greater by a factorof =2, or P0 5 6.05 3 105 and Pp 5 18.1 3 105 W.The switching power for the collision gate is 0.75P0 5.6 3 105 W, and, for the dragging gate, the switchingower is 0.35P0 5 2.1 3 105 W. Again, the switch-
ing power of the dragging gate is less than half thatof the collision gate, so the large-signal gain is morethan twice, but it is also interesting to note that theswitching power of the d 5 10 dragging gate is lessthan that of the d 5 5 dragging gate, providing aquantitative indication that larger values of d can bebeneficial because of the smaller required deflectionangle even though nonlinear absorption is enhanced.
In closing, spatial soliton angular deflection logicgates possess the necessary properties of their elec-tronic counterparts to suggest their use in special-purpose digital switching and logic applications.Modern pulsed laser systems readily produce thepeak powers required ~;1 MW! for exciting the pumpoliton for these gates. A 1-ps pulse would contain1 mJ of energy to yield this power, with switching
nergies of ;100 nJ. Further reduction in pulse du-ations can lead to lower energies, but spatiotemporalffects65 need to be carefully considered when theinear dispersion length is of the order of the gateength. More highly nonlinear materials, such aslGaAs at sub-half-bandgap,66 can also be utilized,
which could potentially lower the energy require-ments by an additional 3 orders of magnitude. Theuse of semiconductor optical amplifiers may providefurther orders-of-magnitude reduction, but the gainrecovery time will ultimately limit the bit rate.67
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