472 J. Opt. Soc. Am. B/Vol. 5, No. 2/February 1988 Caglioti et al.
Limitations to all-optical switching using nonlinear couplersin the presence of
linear and nonlinear absorption and saturation
E. Caglioti, S. Trillo, and S. Wabnitz
Fondazione Ugo Bordoni, Viale Europa 190, 00144 Rome, Italy
G. I. Stegeman
Optical Sciences Center, University of Arizona, Tucson, Arizona 85721
Received August 14, 1987; accepted October 22, 1987The basic limitations to light-controlled switching and logic gating when using nonlinear couplers fabricated frommaterials whose nonlinear response departs from ideal Kerr-like behavior are discussed. A critical value (10 dB)of total linear loss along a one-beat-length-long coupler exists: for greater absorptions the achievable power-dependent sharp switching at the output of a coupler is degraded greatly. Our analysis of spatial instabilities in thecoupled-mode equations with linear-loss terms yields an analytical estimate of this critical absorption as well as ofthe switching efficiency. When paired with the exact results obtainable for a saturable nonlinearity withoutabsorption, our description also may provide useful qualitative insight (in terms of trade-offs between switchingpower, efficiency, and throughput) when correct modeling of resonantly enhanced nonlinearities (e.g., a two-levelabsorber) requires numerical solutions of the coupled-mode equations.
1. INTRODUCTION
Materials supporting nonlinear guided waves have becomeof increased interest because of their potential use in devicesfor ultrafast all-optical switching, logic gating, and modula-tion of light beams.' The operational characteristics ofmany proposed all-optical signal-processing devices arebased on the presence of an intensity-dependent contribu-tion to the refractive index (optical Kerr effect), originatingfrom the third-order term in the nonlinear polarization ex-pansion. Ideally, Kerr-law nonlinearity predicts nonlinearindex changes whose absolute value increases linearly andindefinitely with a growing optical intensity. Transparentmaterials exhibiting a purely Kerr-like fast nonlinear re-sponse, such as silicate glasses, typically possess such a weaknonlinearity that power consumption and dimension consid-erations restrict their potential application to complex opti-cal logic processing. Much larger nonlinearities are found inintegrated waveguides fabricated from semiconductors(multiple quantum wells)2 or semiconductor-doped glass-es,3'4 and switching powers of the order of 1 mW or 1 W,respectively, are possible when using guides a few millime-ters long. Only recently has it been appreciated, however,that switching using devices based on resonantly enhancednonlinearities exhibits a rather strict wavelength selectivitythat is imposed by the presence of absorption losses andsaturation. 5-9
As a specific example, we will be considering in this paperthe operation of the nonlinear directional coupler (NLDC),first proposed by Jensen.' 0 He gave exact solutions for thecoherent exchange of power between two codirectionalmodes traveling in a guiding structure constituted by twoweakly coupled parallel waveguides in the presence of an
ideal, absorption-free Kerr-law material nonlinearity. (Forthis paper we will refer to this case as the ideal NLDC.11,12)Subsequently, a number of analyses and novel physical ef-fects (such as spatial instabilities) based on the same modelhave been discussed, showing the potential of a NLDC for avariety of all-optical signal-processing operations.3 -19
Experiments using silica-glass-fiber nonlinear couplershave demonstrated power-dependent switching at the cou-pler output ports by using hundreds of watts of peak powerpulses.20 -2 2 Substantial reduced switching power (down tocw values of 1 mW) has been reported for a GaAs-GaAlAsmulti-quantum-well coupler.2 However, in that experimenta relatively large absorption was measured, and the observ-able switching was incomplete. These features gave con-vincing evidence that a more accurate description requiresimprovements on the ideal NLDC theory. To this end,limitations on NLDC power-dependent switching character-istics as imposed by the presence of linear absorption andsaturation of the nonlinear index have been studied first bynumerically solving the coupled-mode equations.7 The re-sults concerning the effects of the saturation were later con-firmed by using an analytically solvable model, which alsoallowed us to point out that nonzero saturation forces a one-beat-length-long coupler either to switch its output field at adouble number of discrete powers or not to switch at all (forvalues of the maximum index change below a certain criticalvalue).8 Our numerical study of the switching characteris-tics of a half-beat-length-long coupler, on the basis of thecomplex susceptibility of a two-level saturable nonlinearity,has demonstrated that the interplay of nonlinear absorptionand saturation may lead to a relatively intricate switchingbehavior, which may impose a careful choice of the opera-tional detuning from resonance.9
Vol. 5, No. 2/February 1988/J. Opt. Soc. Am. B 473
Our aim in this paper is to show that a basic qualitativeunderstanding of the combined effects of nonlinear loss andsaturation can be gained by separately investigating simpli-fied models that may involve, for example, either linear
absorption or saturations only. Sharp power-dependentswitching behavior in a nonlinear coupler is related inti-mately to the spatial instability property of steady-statesolutions, originating from a rather subtle balancing of lin-ear coupling and nonlinear mismatch. Thus we attack theproblem by considering a phase-space representation of theevolution of the mode amplitudes along the coupler. Ourapproach shows that the main consequence of linear (con-stant) absorption for a one-beat-length-long coupler is theexistence of a critical limit to the total allowable power lossalong the device. For slightly larger losses, the switchingbecomes considerably broader and incomplete. On the oth-er hand, for absorptions below this critical value, sharpswitching characteristics are preserved: the net effect is anoverall shift toward larger input powers of the nonlineartransmission curves, and the allowable power transfer is stillsignificant. Furthermore, our analysis will yield an analyti-cal estimate for the switching power.
Along similar lines, our phase-space treatment will be ableto furnish a critical value for the maximum nonlinear refrac-tive-index change below which power-dependent saturationper se (i.e., neglecting losses) would prevent efficient switch-ing.
The second part of the paper presents a comparison of theabove predictions with the numerical results obtained whensimulating a resonant nonlinearity by taking the complexsusceptibility of a two-level medium. This modeling neces-sarily includes both power-dependent saturation of the(real) nonlinear index changes and saturable absorption.Our detailed description, in substantial qualitative agree-ment with the exact analysis, shows how different experi-mentally relevant parameters, such as throughput, switch-ing fraction, and switching power, may indeed critically de-pend on the maximum index change and detuning fromresonance, which in turn fixes the power dependence ofsaturation and absorption.
2. NONLINEAR COUPLED-MODE EQUATIONS
To exploit the power-orthogonality condition between guid-ed modes, coupled-mode analyses of a composite waveguid-ing structure would require using the array modes as a cor-rect expansion basis for the field.23 In the limiting case ofidentical and weakly linearly coupled waveguides, however,a nonlinear coherent coupler can be investigated by consid-ering the usual nonlinear coupled-mode equations on thebasis of the eigenmodes of the two individual (isolated)waveguides.19 The coupled-mode equations for the com-plex field amplitudes E1 ,2 of the modes of two identicalwaveguides 1 and 2 can be extended to include both a power-dependent wave vector koAn and absorption a (possiblypower dependent as well), in the form 7
where K is the linear coupling constant and ko = w/c is thevacuum wave vector. Equations (1) imply that both the
waveguide effective index change An and attenuation coeffi-cient a may depend on the local power in their respectivewaveguide only. Furthermore, we normalize the complexfield amplitudes in Eqs. (1) so that their squared modulus isequal to the power traveling in the respective mode.
In Sections 3-6 we will assign Eqs. (1) to three differentsituations of increasing complexity. This action will permitus to single out the different origins of the limitations tononlinear power transfer along a nonlinear coupler. First,we will be dealing with a constant absorption a coupled to anideal Kerr-like nonlinearity. Second, we will examine brief-ly the case of a saturable two-level nonlinearity in the limit-ing situation of very large detunings (i.e., negligible losses).Then we will consider fully the two-level nonlinearity, tak-ing into account both saturation of the nonlinear index andnonlinear absorption.
3. LINEAR LOSSES
In this section we consider the effect of linear absorption onthe transmission characteristics of a NLDC when assumingan ideal Kerr-law power dependence for the wave vector [i.e.,kohndEl 2) = k1AnoIEI2 - RIEI2]. Equations (1) simplify to
-jdE,/dz' = -7rE + LbRIEI2El + ja'LbE,,
-jdE 2 /dz' rEl + LbRIE22E2 + ja'LbE 2 ,
(2a)
(2b)
where a' is the field-amplitude absorption coefficient (expo-nential loss per unit length), R is a mode-field overlap inte-gral,"°10 and z' is the propagation length in units of the linearbeat length Lb /K.
The coupler equations in their general form [Eqs. (1)] arerepresentable as a nonlinear dynamic system of three inde-pendent real variables, since power transfer between theguides is regulated in a self-consistent manner by the evolu-tion of the sum and the difference of the modal powers andtheir relative phase. In the present case, however, becausethe absorption rate a' is constant with distance, Eqs. (2) canbe reduced to a particularly simple z-dependent one-degree-of-freedom Hamiltonian system. This is done by introduc-ing a z-dependent total optical power traveling along thecoupler. Then we define the new variables k = 02- 01 and y
= (P2 - P/(P1 + P2 ), with E, 2 = (P,,2)12 exp(1,2), and fornotational convenience we set as the normalized length =--27rz'. (, y), which represent phase and normalized powerdifference along the coupler, are conjugate variables whoseevolution is governed by
where p0 [Pl(z = 0) + P2(z = 0)]/P, = Po/P, is the inputpower normalized to the critical power of an ideal loss-freecoupler P, - 4K/R and a0 = a'Lb/27r is the normalized absorp-tion coefficient. At any distance z the power left in onewaveguide when all the power is initially in the same guide is
P2(1)(z)/P2(1)(z = 0) = exp(-47raoz/Lb)1l i y [k(z)]1/2, (4)
where the upper (lower) sign holds for waveguide 2 (1).The loss-free coupler as described by the Hamiltonian
system [Eqs. (3a) and (3b)] [where p(r) = constant = p] is
Caglioti et al.
474 J. Opt. Soc. Am. B/Vol. 5, No. 2/February 1988
1.0
zV)Y)
U)Z
0.8
0.6
0.4
0.2
0.00 1 2 3
INPUT POWER p0
Fig. 1. Straight-through fraction (transmission) of the outputpower versus the incident power for different values of the normal-ized total linear loss ao and L = Lb-
exactly solvable in terms of Jacobian elliptic functions.10-13
Therefore linear absorption plays the role of a parametricperturbation to the integrable Hamiltonian.
With the nonlinear switching application in mind, we areinterested in examining the possibility of switching on (off)the straight-through transmittance (defined as the fractionof the output power in the channel that is excited at theinput) by slightly changing the input power Po. In Fig. 1 thecurve with a = 0 shows the transmittance of the ideal (i.e.,loss-free) coupler of length L = Lb. As the input power israised across a critical value (p = Po/P = 1), a sharp switch-ing of the transmission from 0 (say, off) to 1 (say, on) occurs.This effect is described nicely on the basis of a stabilityanalysis of the coupler equations [Eqs. (3)],11,15 whose mainresults are outlined briefly now.
Rapid switching is brought about by doubling the spatialfrequency of the power exchange between the channels.This event occurs precisely when the input power crosses PC(p = 1). Frequency doubling is in turn originated by a lossof stability (as p = 1/2) of the even coupler eigenmode.Consider the specific case of input excitation with p = 1 ofthe mode field of an individual waveguide. After one cou-pling distance 1 = Lb/2, the field approaches the spatiallyunstable state of the coupler and eventually stays there forany coupler length. This situation, however, is unstable;even the smallest variation to the input excitation condi-tions yields periodic evolutions for the field. The corre-sponding period doubles as Po is changed from a value justabove P to a slightly smaller value just below P.
This effect can be described pictorially by representingthe solution of Eqs. (3) as a trajectory on the phase space (y,q) or on the Poincar6 sphere" [the Hamiltonian coordinates(y, ) are expressible as functions of the coordinates on thePoincar6 sphere (S1, S2, S3) through the relationships y = s,and = tan'1(s3 / 2 )]. For Po = P the representative pointof the field evolves along a separatrix orbit, which dividesdifferent regions of periodic motion. In Fig. 2(a) we show theunperturbed (ao = 0) evolutions in the phase plane. In thephase plane (y, 0), the condition y = 1, whatever the phasevalue of , corresponds to the same physical state of thecoupler, namely, all the power in one of the two guides. Forgraphic convenience, we have shown in Figs. 2 and 3 theevolution in the plane [y, (1 - y 2
)1/ 2 sin ]- (s1, S3). In Fig.2(a) trajectories 1, 3, and 2 refer to input powers p = p < 1,p = 1, and p = P2 > 1, respectively. Trajectory 1 is outsidethe separatrix (henceforth, oscillation) and corresponds to
Caglioti et al.
the deep minimum in the a 0 = 0 transmission curve of Fig. 1.Trajectory 2 is inside the separatrix (rotation) and corre-sponds to the first transmission maximum obtained for theslightly larger power P2. When guide 2 is excited at theinput, trajectories 1-3 have common origin in (y, )z=o = (1,0), and, as can be seen, they follow neighboring paths untilafter a certain distance ' the unstable eigenmode (y, 0) = 0,0) [point B in Fig. 2(a)] is approached closely. Successively,trajectories 1 and 2 diverge very rapidly, and for a coupler oflength Lb all the input power ends up in the same guide 2 forP = P2 > I [point A in Fig. 2(a)] or in guide 1 for p = Pi < 1
1.0
-0 0.5.
'N- 0.0N
-0.5
-1.0
1.0
r 0.5
'EN
A_- 0.0N
I -0.5
-1.0
1 -0.5
c
0 0.5 1y
(a)
1 .. . . ... . . , . . _ _
2
B- A-
I . .l . - . . ..ll l l l l -1 -0.5 0 0.5 1
Y(b)
Fig. 2. (a) Evolution of the unperturbed system on the phase plane[y, (1 y 2)1/2 sin ]: 1, 2, and 3 correspond to input powers p < 1p > 1, and p = 1, respectively. Points A and C represent all thepower in guides 2 and 1, respectively, and point B is the unstablecoupler eigenmode. (b) Perturbed trajectories on the phase planefor the coupler with losses (ao = 0.1): 1 and 2 correspond to inputpowers p = 1.18 and P2 = 1.3 for the minimum and the maximum,respectively, in the transmission. The points are the same as in (a).
1.0
0.5u)
'N3
A_- 0.0cQ
1 -0.5
-1.0-1 -0.5 0 0.5
1yFig. 3. Schematic of separatrix crossing on the phase plane [y, (1 -y 2 )1/
2 sin 4k].
_ . .. I I I IBA
- N1 N- \ I-' I I
.1 II I..
...
l l l
I
1
Vol. 5, No. 2/February 1988/J. Opt. Soc. Am. B 475
(point C). In the following discussion we show how the
presence of losses modifies the switching characteristics of anonlinear coupler.
Let us consider the perturbed system. A Hamiltoniansystem with a z-dependent parameter can be described inthe framework of the adiabatic approximation 24 whenever
the parameter itself slowly varies with respect to the periodsof the unperturbed orbits. In the present case [Eqs. (3)], theadiabatic approximation is valid only for values of ao, satis-fying the condition Idp/dd1 = 2 aop << Q, where Q is a charac-
teristic fundamental spatial frequency of the unperturbedmotion. Unfortunately, the adiabatic invariance fails todescribe correctly the motion for trajectories in the vicinityof the separatrix, 2 5 since the frequency Q approaches zero in
this case.11,15"16 Now, switching in the unperturbed systemoccurs precisely when the power is varied in such a way that
the resulting (y, 0) trajectories cross the separatrix. As a
conclusion, adiabatic invariance cannot be used to investi-gate the switching characteristics of the coupler in the pres-ence of losses. Thus we need to follow a different approach,involving an analytical approximation to near-separatrixevolutions that allows us to predict and estimate a criticalvalue for the linear absorption.
The separatrix is defined as the particular orbit that goesthrough the unstable eigenmode [henceforth referred to asseparatrix node; see point B in Fig. 2(a)] and whose energy H- H, = 1 (a separatrix exists only when p ' 1/2). For
oscillations (rotations) one has H < 1 (> 1).Because H is conserved in the unperturbed case, no sepa-
ratrix-crossing trajectories exist [see Fig. 2(a)]. As we have
seen, this is why switching between the channels may beinduced by a slight change in input power, and trajectorieswith well-separated behaviors result. As ao 5d 0, H (as well
as the input power) is no longer conserved so that an individ-ual trajectory may cross the virtual separatrix at some dis-tance along the coupler (see Fig. 3). [The virtual separatrixat position v in the propagation direction is the separatrix ofthe unperturbed trajectories obtained for p = p(v).] Thisseparatrix-crossing phenomenon 2 5 is the very origin of thespoiling of power-induced switching in a lossy nonlinearcoupler. In fact, consider any two input powers yieldingtrajectories located on opposite sides of the virtual separa-trix at r _ 0. One understands immediately that, if one ofthe two trajectories crosses the virtual separatrix before theunstable node is approached, then for any successive dis-tance the trajectories will remain close to each other, whichimplies that switching is inhibited. In the following discus-sion we will support this description with quantitative argu-ments.
Examples of numerically calculated transmissivities areshown in Fig. 1 for different losses ao. The first effect of
increasing absorption above zero is to move the input switch-ing power to higher values. Also, the fractional switch (i.e.,the depth of the minimum) is reduced. As the critical valueac, - 0.2 is reached, the minimum and maximum merge, andthe transmission notch becomes much broader and uselessfor switching purposes.
We characterize position and sharpness of the off-onswitching as functions of a( by evaluating the input powersPt and P2 defined above (their value corresponds to the firstminimum and successive maximum of transmissions as inFig. 1, respectively). In Fig. 2(b) we show, for ao = 0.1, the
perturbed trajectories corresponding to the two input pow-ers Pt and P2-
In the unperturbed case, pi and P2 are obtainable from theexact solutions10-'3 by imposing the condition that the pow-er-dependent beat length [distance of first return to point Afor orbit 2 in Fig. 2(a)] or half-beat length [distance betweenpoints A and C for orbit 1 in Fig. 2(a)], respectively, be just
equal to the coupler length.As soon as ao 5d 0, we may still define the switching power
as ps 8 Pi [another choice could be ps = (PI + p2)!2]. As ithappens in the unperturbed case for weak absorptions, thetwo input powers Pi and P2 yield trajectories that, after adistance A', closely approach the separatrix node. Corre-spondingly, their energy is H(D') _ 1. When the mode isexcited in, say, channel 2, the initial energy is
Ho = H(O = Oy = 1; = 0) =Po = P(0). (5)
Furthermore, the energy (H) rate of change associated with agiven trajectory is
dH/d =-2aop(r)y'G)- (6)
Our goal is to estimate at what length (c') the two trajec-tories will reach their closest approach to the separatrixnode. Now, the trajectories slow down considerably when
approaching the node. Therefore, given the input powerP1(P2), it is reasonable to approximate the distance ' withthe quarter-period (half-period) A pertaining to the virtualoscillation (rotation) fixed by p' = p(t') and H' = H('%which reads as (see Appendix A and Ref. 26)
A 1 ln 8(2p'- 1)2 (p'> '/2).22t- 1)/2 p'2 H' - 1
(7)
For the perturbed orbit, H' and p' are obtainable from theinitial condition H( = 0) = Ho = Po = p(0) through the
relationships
p' = Po exp(-2a0 oA),
H' = Ho - AH = Ho(1 - 2ao).
(8)
(9)
Equation (8) is immediately obtainable from Eq. (3c), andAH is a kind of first-order approximation obtained by inte-grating Eq. (6) along the above-defined virtual orbit andretaining only terms up to first order in ao.
Imposing 2A 2D' = L, one finally finds that Pt and P2 are
defined implicitly as the two roots of Eq. (7). Figure 4 shows
0.6
Nl0.4
0.2
0.0
0 0.1 0.2
ABSORPTION a0
Fig. 4. Input power variation of the minimum (APi = p, - 1) ormaximum (AP2 = P2 - 1) in the transmission as a function of thenormalized total linear loss ao. The analytical and numerical calcu-lations are shown as solid and dashed lines, respectively.
Caglioti et al.
476 J. Opt. Soc. Am. B/Vol. 5, No. 2/February 1988
1.0
0
C-)
C/)
0.8
0.6
0.4
0.2
0.00 0.1 0.2
ABSORPTION o
Fig. 5. Maximum and minimum off-on switched fraction of theoutput power versus absorption ao for a one-beat-length-long cou-pler.
the resulting AP1,2 P1,2 - Ps(ao = 0) - P1,2 - 1 whencomputed from Eq. (7). The comparison with correspond-ing numerical data [obtained from transmissions as in Fig. 1]demonstrates subtantially good quantitative agreement upto ao0 < 0.1. Both curves also show that the maximum andthe minimum eventually merge (at the critical ao), thusindicating that the off-on switching feature has disappearedfrom the transmission.
As we have briefly mentioned in this section, completedeterioration of the switching occurs when, owing to separa-trix crossing, both trajectories corresponding to pi and P2happen to be located on the same side of the separatrixbefore distance '. Let us see if either of the two trajectoriesever had a chance to cross the virtual separatrix somewherealong the way. To accomplish this means that at some > 0the condition H(v) = H = 1 is fulfilled. Given that dH/dA <0, the trajectory with initial power pi (i.e., located outsidethe virtual separatrix) will not be able to cross the separatrixat any propagation distance . Conversely, the evolutionoriginating with P2 might eventually cross after a certaindistance, or even collapse toward the separatrix node. Be-cause the condition dp/dd > dH/dfl holds [see Eq. (5)],then p' < H' in Eqs. (8) and (9). Now, the critical absorptionao = 0.18 [as obtained from Eqs. (7)-(9)] can also be inter-preted in terms of separatrix crossing as follows. When a =ao,, owing to propagation from 0 to ', the input power po =P1 - P2 decreases to p' = 1/2; correspondingly, H decreasesfrom p0 to unity. In fact, the condition p' = 1/2 means thatthe virtual separatrix at ' is just vanishing.
For a 0 larger than a few per cent, switching efficiencydeteriorates as well. Note that the efficiency of the switch-ing includes both the throughput T defined as the ratio oftotal output power to input power, and the off-on switchedfraction, which is the maximal fraction of the output powertransferred from one channel to the other, as the criticalpower is crossed.
The throughput for constant losses is expressible immedi-ately from Eq. (4) as
T(z; e) = exp(-4raoz/Lb). (10)
In Fig. 5 we show the maximum (upper branch) and mini-mum (lower branch) fractions of the total output power,which is left in the excited guide after distance Lb, obtained(at input powers P2 and pl, respectively) from transmissionsas in Fig. 1 with different ao. As can be seen for smallabsorptions (ao < 0.02), the only effect of losses is a net shift
(proportional to 2o) toward higher input powers of thetransmittance, and the switched fraction remains nearlyunity. As the absorption is further increased, however, theoff-on fractional switching drops quite rapidly and eventu-ally vanishes at the critical value [the critical loss corre-sponds to quite a high absorption, T(Lb; aoc) 10 dB].Although the above-defined switched fraction [which is giv-en, for a certain a 0, by the difference between the branches ofFig. 5] vanishes for a = aoc, for ao > ac a considerablefraction of the total output power can still be switched be-tween the two coupler channels (see the curve labeled a =0.21 in Fig. 1). One could also define another fractionalswitching, with reference to switching from the on state atlow powers to the off state for input powers approaching pi.In this section, however, we were concerned mainly withdetermining the effect of losses on the sharp off-on switch-ing caused by power-induced separatrix crossing. Con-versely, in situations when saturation of the nonlinear indexchanges also is taken into account, it may be of interest toconsider the deterioration of the on-off fractional switchingas well (see Section 6).
4. TWO-LEVEL SATURABLE NONLINEARITY
The susceptibility of a two-level absorber can be written, in aconvenient form for the present analysis, as27
2nOvO 6ko (1 + 2 )(1 + I/Isat)
where 6 is the normalized line detuning from resonance, no isthe background refractive index, vo is the peak absorption(amplitude) coefficient, and Isat is (1 + 62) times the line-center saturation intensity. For I/Isat << 1, the absolutevalue of the dispersive part of the nonlinearity has a maxi-mum for = 1. Therefore the nonlinear index can be en-hanced with respect to the off-resonance case by operatingat small detunings. However, as we will see in the followingsections, this type of operation results in a large increase ofthe absorption, which may deteriorate the device perfor-mances. Equations (1) can be expressed in terms of thepower-dependent change in the waveguide effective indexand absorption as9
An(dE12 ) = -A#,at/(1 + 1E12 /Psat)
a = pl 1V0 /[(l + 62)(1 + IE12/ps)],
27rAn = Xa,
(12a)
(12b)
(12c)
where An and a are evaluated at the detuning and Pnl is thefraction of the power guided in the nonlinear material foreach mode2 8 (the saturation value of the effective waveguideindex M/3sat is Pnl times the saturation value of the bulkmaterial index Ansat). The usual saturation intensity for abulk medium can be shown to be simply Isat = PcAsatlc/(PnlA~eff), where Aeff is the effective waveguide transversearea and 1 = Lb/2 is the coupling length of the coupler. Psatitself is dependent on the detuning.
We were not able to solve analytically the coupled-modeequations in their general form [Eqs. (1) and (12)]. There-fore we computed the transmission characteristics of a one-beat-length-long coupler by numerical integration. Thecrucial parameters for the present case of a two-level nonlin-earity are detuning 6 and normalized maximum index
Caglioti et al.
Vol. 5, No. 2/February 1988/J. Opt. Soc. Am. B 477
change w AatLb/2X [sat also depends on the detuningthrough the term 6(1 + 62)-1]. In fact, once 6 and Afsat arespecified, both An and a are fixed at a given power level.
In Section 5 we will only examine how the switching isaffected by the power-dependent saturation characteristicsof the dispersive part of the nonlinearity, i.e., we will consid-er cases when the absorption is negligible. In successivesections we will deal with the general situation, namely,when both the saturation of the nonlinear index and theabsorption are fully taken into account.
5. NONLINEAR INDEX SATURATION WITHNEGLIGIBLE ABSORPTION
Saturation of the nonlinear index change may affect thebehavior of the nonlinear coupler in a substantial way.7 8 Itwas recently shown (with reference to an exponential modelof the saturation) that one can solve analytically for thepower transfer along the coupler by expressing the coupled-mode equations in terms of a one-degree-of-freedom Hamil-tonian in analogy with Eq. (3).8 In general, the resultingsystem then can be solved, at least through a quadrature. 8
The main feature of a saturable nonlinear index is theexistence of a critical value of the saturation, say, w, whichdivides two different switching regimes. For w > w,, switch-ing in a coupler of length Lb occurs at two distinct inputpowers, as shown, for example, in the transmission with w =1.7 in Fig. 6 (6 = 50). As one permits the saturation parame-ter to increase indefinitely, the lower switching power ap-proaches, in practice, the critical power P, of the ideal cou-pler; also, the second larger switching power grows monoton-ically to much higher values. For saturation values lower
8 =50
C0.i
,n.EU)C
than the critical value, only one switching power occurs, butthe transmission minimum becomes much broader and fea-tureless (see the curve in Fig. 6 where w = 1). This behavioris a typical consequence of the saturation of the nonlinearindex, no matter which particular model of the saturationone assumes. 8 9
The real and imaginary parts of the susceptibility of a two-level absorber are proportional to 6(1 + 62)-l and (1 + 6
2)-
1,respectively. Therefore for large detunings (6 > 100) ab-sorption can be neglected, and the coupled-mode equationsbecome amenable to analytical solution. Under the aboveassumptions, the saturation characteristics of a two-levelmedium lead to the Hamiltonian
where p = Po/P, is the conserved normalized power. Here,the power P, is defined as the critical power of the idealcoupler, which has the same nonlinear index of the two-levelsystem for PO/Psat << 1. Following this definition, the rela-tion Pc = Psatw holds. The critical power so defined is anincreasing function of detuning [P, is proportional to (1 +62)2/6].
Integrating the Hamilton equation dy/d = -H/q[where sin(o) is expressed as a function of the HamiltonianHo = H(y = +1, q = 0)], we obtain the quadrature
-(27rz/Lb) = + J dy ((1 - y2 ) -Ho + (4W2/p)
X ln[(1 + p12w)2 - (py/2w)2 ]2) 1 /2, (14a)
Ho = -(4w 2/p)ln[(1 + p1w)]. (14b)
Following a procedure analogous to the one given in Ref. 8(a detailed description of this procedure is given in Appen-dix B), the stability analysis yields for the switching power Ps
= P/Pc the following relationship:
4w2. (1 + p,/2w) 2
p5 (1 + p5 /w)(15)
PS/PCFig. 6. Same as in Fig. 1 for a two-level system with 6 = 50 anddifferent values of the saturation parameter w - Asatlc/X. On theabscissa we have indicated the normalized input power that givesrise to switching as Ps = Ps/P.
15
10
0al 5
00 1 2 3 4 5
SATURATION W
Fig. 7. Switching power ps = Ps/Pc versus the saturation parameterw, for large detuning 6 > 100.
Equation (15) defines implicitly the critical value w, = 1.68.Furthermore, for w > w, the two switching powers are thetwo implicit solutions of Eq. (15). Their values are shown inFig. 7, illustrating the above-discussed behavior.
6. SWITCHING IN THE PRESENCE OFSATURATION AND NONLINEAR ABSORPTION
In this section we examine the transmission characteristicsof a nonlinear coupler for an arbitrary detuning 6, i.e., whenthe nonlinear absorption and the index saturation of thetwo-level absorber are simultaneously taken into consider-ation in Eqs. (1) and (12). In the present case, most of theswitching limitations are predictable in terms of the samephenomenologies associated with the two previous simpli-fied examples, namely, either a pure Kerr-law nonlinearitywith constant absorption or a saturable loss-free nonlinear-ity.
We investigate in particular how switching power, frac-tional switching, and throughput depend on the normalizedsaturation value of the nonlinear index w and the detuning.
Caglioti et al.
478 J. Opt. Soc. Am. B/Vol. 5, No. 2/February 1988
15.
8=20-8:50
....... 8= 200
7.5 -
5.0 -
2.5 -
I I I I I l I1.0 1.5 2.0 2.5
A 3sat )c/XoFig. 8. Switching power normalized to the critical power as a func-tion of saturation w /3flstlC/X and for different detunings 6.
Our study will show that although the index saturation in-duces the switching power to follow basically the same be-havior as in the off-resonance case, different characteristics(such as the throughput) may exhibit unexpected features.This fact is due to the intrinsic power dependence of theabsorption coefficient.
A typical straight-through transmission versus incidentpower (normalized by the critical power of the ideal coupler)is shown in Fig. 6 for 6 = 50 and with different values of thesaturation parameter w = Afsatl,/X-
The switching power P, normalized to the critical power P,is plotted against the saturation parameter w in Fig. 8. Aswas shown in Section 5 for large resonance detunings, for anarbitrary 6 a critical value for the minimal saturation indexchange still exists, below which the two-level coupler switch-es twice with respect to an ideal coupler (see a correspondingtransmittance in Fig. 6 for w = 1.7). For large values of w thelower branches tend to a constant value asymptotically, thatis, slightly larger for smaller detunings. This behavior oc-curs because once w is fixed, a lower w results in largerabsorption, which in turn causes a higher switching power.More important is the fact that the critical power dependson the detunings, as discussed in Section 5, so that the actualinput power P required to switch the device is larger forhigher detunings.
Moreover, as can be inferred from Fig 8, the critical satu-ration w does not depend significantly on the detuning, sothat an almost universal key limitation for material parame-ters is that the maximal change of the effective index satisfythe inequality
Mmsa > 17 /1, (16)Conversely, as shown in Fig. 9, the throughput T (ratio of
total output power to input power) is sensitive to detuning.The absorption coefficient is proportional to 6(1 + 62)-1, and
Fig. 9. Throughput as a function of the normalized saturationparameter w for various 6.
1.0 ...0.8 -
0°-0.6 -/
V.b~C)
C
U~~q -- 8 20U) -__8-50
........... A= 200
I..t
0.2 _-
/
1.0 1.5 2.0 2.5AGsat c/Xo
Fig. 10. Fraction of the on-off switched output power (F) as afunction of saturation parameter w and different detunings 6.
Caglioti et al.
Vol. 5, No. 2/February 1988/J. Opt. Soc. Am. B 479
8=10
C0
.EC,,C0
Fig. 11. As in Fig. 6 with 3 = 10.
2.5
8=10
\I
Apsat c/XoFig. 12. As in Fig. 8 with 6 = 10.
working close to resonance strongly deteriorates the efficien-cy since T decreases dramatically. The upper branches ofthe curves shown in Fig. 9 for different values of 3 correspondto the minimum in the transmission at large switching pow-ers (see the curve for w = 1.7 in Fig. 6). The lower branchesrefer to the lower-power switching. The throughput in-creases for the upper branches since for increasing satura-tion w the transmission minimum moves to higher powers,and the saturation tends to bleach out the absorption. Inany case a throughput of greater than 90% at the low-powertransmission minimum would require that 6 > 100.
The on-off switching fraction F,, which measures thedepth of the transmission minimum, is shown in Fig. 10 as afunction of the saturation parameter w and for the samedetunings as in Fig. 9. For w < wc the switching fractiondrops rapidly to negligibly small values. For w > wc, F8
maintains itself to values larger than 95% for large detunings(3 > 100), whereas it may decrease considerably for smalldetunings (6 < 100). For low detunings (see the curve for 6 -
20 in Fig. 10) for w > w, F8 remains almost unity for thetransmission minimum obtained at larger switching powerbut decreases as a function of w for the lower-power mini-mum. We interpreted this reduction in the switching frac-tion as being due to an increase of the peak absorptioncoefficient. In fact, once the detuning 6 is fixed, an increas-
1.0 _
8=10
0.8 _
0.6 -Is-c
0
0.4
0.2
1.0 1.5 2.0 2.5A/3sat Ic/Xo
Fig. 13. As in Fig. 9 with 3 = 10.
1.0
0.8 -
0~~~~~~~~~~~~~~~~~~
°0.6/ '
C /
*0.4 /
Fig. 14. As in Fig. 10 with 6 = 10.
Caglioti et al.
O?
ff.
480 J. Opt. Soc. Am. B/Vol. 5, No. 2/February 1988
ing w also implies, for the two-level system, a growing peakabsorption coefficient [see Eq. (12c)].
Once the material parameters are chosen so that Eq. (16)is satisfied with good confidence, high switching efficiencyrequires that the optical wavelength be tuned so that islarge enough.
Finally, in Fig. 11 we show the transmittance in a near-resonance case of 6 = 10. Figure 12 shows the correspondingnormalized switching power ps versus the saturation param-eter w. The behavior is quite similar to the off-resonancecase shown in Figs. 5 and 6, with an almost identical value forthe critical saturation w,. However, as is shown in Fig. 11 forw = 2, the first transmission minimum exhibits a double-notch structure. (A second, broader minimum occurs at alarger power, which, however, for w = 2 does not appear inthe scale of the figure.) Analogous behavior was observedpreviously in the case of purely linear absorptive losses (seeFig. 1). The origin of this behavior was ascribed to separa-trix-crossing phenomenon, with associated degradation ofthe off-on switch, which occurs as the critical loss is ap-proached [ is proportional to (1 + 62)- 1]. For low absorp-tions, a sharp notch corresponding to the transmission zerois followed by a second notch, which is much smaller, thatinitiates the rippled structure displayed in Fig. 1. As dis-cussed in Section 3, progressively larger absorptions leadeventually to merging of the minimum and maximum trans-mission points corresponding to Pi and P2, respectively (seealso Fig. 4). In this limit the first transmission notch hasdisappeared, and only the second notch is left as a broadminimum. The connection to the present situation is givenagain by the fact that a larger value of the saturation param-eter w also implies a larger peak absorption. Also, becausethe absorption decreases at higher powers, the depth of thehigher-power notch is not reduced so rapidly as for the firstone.
The switching power, throughput, and switching fractionrelative to the higher-power notch are shown in Figs. 12-14as dashed lines. In particular, in Fig. 12 we indicated by asolid line (lower branch) and a dashed line the on-off switch-ing fractions of the lower- and higher-power notches, respec-tively; the solid-line (upper branch) pertains to the mainminimum of the transmission occurring for higher powers(see Fig. 11). For each of the two notches the switchingfraction behaves, as a function of w, in an opposite way. Infact, as w grows larger than w,, the first notch, which used tobe (i.e., for w < w) the deepest minimum, tends to disap-pear. Meanwhile, the second notch replaces it as a trans-mission minimum.
Furthermore, owing to absorption saturation with power,the throughput of the higher-power notch is slightly higherthan that of the first one.
7. CONCLUSIONS
In conclusion, we have investigated the interplay of variousmechanisms that may set fundamental limitations to all-optical signal processing by using nonlinear couplers withresonantly enhanced nonlinearities. First, we demonstrat-ed that linear absorption alone may substantially degradethe sharp input-power-induced switching occurring in a one-beat-length-long nonlinear coupler. The relative simplicity
of this case allowed us to derive an analytical estimate for thecritical total loss along Lb, in good agreement with the nu-merical findings.
When considering resonant nonlinearities one may distin-guish between two different regimes (we discussed the modelof a two-level absorber, but the results below are extendablequalitatively to other mechanisms as well). At large detun-ing from resonance, the saturation of the nonlinear indexchange is the dominant process, because absorption effectsare negligible. In this limit, the coupled-mode equations aresolvable exactly. Our solution led to the prediction of acritical value for the maximum index change. When largerchanges are allowed, then a coupler of length Lb may exhibittwice as many separated switching powers with respect tothe ideal NLDC case. For smaller values of the maximalindex change, power-dependent transmission characteris-tics are broad, and the switching deteriorates completely.
Finally, we demonstrated that when resonant operation isdesired to enhance the absolute value of the nonlinear indexchange, absorption and saturation can degrade the switchingcharacteristics of the coupler severely. A certain detuningfrom resonance is imposed because the absorption otherwisemay be so large that drastic degradation of modulationdepth results in both the transmission and throughput.High detunings, however, increase the actual switching pow-er of a device. In any case, a detailed knowledge of linearand nonlinear dispersion characteristics of the material aswell as a rather strict choice of the source wavelength arenecessary.
APPENDIX A
In this appendix we derive the approximate expression [Eq.(7)] for the spatial period A associated with near-separatrixorbits. The Hamiltonian describing the nonlinear couplingprocess [see Eqs. (3)] in the presence of absorption is
H = H(0, y; ) = (1 - y2)"I2 cos 0 + p()y 2. (Al)
Thus the evolution of the normalized difference between thepower traveling in the two guides, y = (P2 - PD/(P + P2 ), isgoverned by
dy/dr = -H/a = (1 - y2)1/2 sin A. (A2)
As discussed in Section 3, we are interested in estimating atwhat distance (say, ') the representative point of the fieldreaches its closest approach to the unstable coupler eigen-mode. To this end, it is reasonable to approximate theactual trajectory with one pertaining to a virtual loss-free[p(r) = constant] coupler orbit, corresponding to fixed val-ues of power p = p(D') and energy Ho = H(%'). The corre-sponding period is given by
A0 = f dy(dA/dy) = dy[(1 - y2
)1/2sin 0]-1
(A3)
where the integration is taken over the whole trajectory andthe fourth-degree polynomial
M(y) = -p 2y4 + (2pE - 1)y2 + 1 - H02
Caglioti et al.
= I dy[M(y)]-112,
Vol. 5, No. 2/February 1988/J. Opt. Soc. Am. B 481
has been obtained by expressing sin(0) as a function of y andorbit parameters Ho and p. M(y) may admit either fourdistinct real roots, say, -a, -b, b, a, for Ho > 1 [see curve 2 inFig. 2(a)] or two real and two purely imaginary conjugatedroots, say, -a, a, jc, -jc, for Ho < 1 [see curve 3 in Fig. 2(a)].For Ho = 1 (corresponding to the separatrix trajectory) M(y)has two coincident roots in y = 0. For Ho > 1 (rotations) thespatial period is29
AO(HO, p) = 2[p(a + b)]-'K(k), (A4)
where k = (a - b)/(a + b), K(k) is the complete ellipticintegral of the first kind [a and b being the real positive rootsof M(y) corresponding to the two extremal y coordinates ofthe orbit in the phase plane]. For Ho < 1 (oscillations), weobtain2 9
Ao(HO, p) = 2p-1A-K(k), (A5)
with k = (1 + c2/a 2 )-1/2, A 2 = (a2 + C2).The spatial period of the power exchange diverges for
initial conditions corresponding to evolution along the sep-aratrix. In fact for Ho = 1 (p > 1/2), k = 1, and K(k) tends toinfinity. The spatial period A0 can be approximated for Ho
1 by means of the asymptotic expression 30
K(k) (1/2)ln[16/(1 - k2)], (A6)
which is valid for k 1. Finally, we obtain
Ao = (2p - )-112 ln[8(2p -1) 2/p 2(HO - 1)]
A0 = 2(2p - 1)-1/2 ln[8(2p - 1)2/p21HO - 11]
if Ho > 1,
(A7)
if Ho < 1.
(In Section 3 we defined A = Ao/4 if Ho < 1 or A = Ao/2 if Ho> 1.)
APPENDIX B
The stability analysis of the steady solutions of the saturablenonlinear coherent coupler coupled-mode equations is car-ried out in a standard manner by studying the behavior ofthe Hamiltonian around the corresponding fixed points.The Hamiltonian that describes the power transfer in adirectional coupler with saturable nonlinear index changesis
H(y, k) = (1 y 2 )1/2 COS k - (4W2 /p)
X ln[(1 + p/2w)2 - (py/2w) 2 ]. (B1)
The condition for the existence of the separatrix in thepresence of saturation of the nonlinear index becomes
p > /2(l + p/2w)2. (B2)
For w , this condition reduces to p > 1/2. Inequality(B2) is immediately obtained by assuming that the equilibri-um point is a point of flex for the Hamiltonian.
When only one channel of the coupler is excited at theinput, one can calculate the switching power p5 [given in Eq.(15)] by imposing a more stringent constraint (switchingpower p5 corresponds to the critical power defined in Section3). For input power po = p 8 the field of the coupler evolvesasymptotically toward the unstable 3-dB eigenstate, at
which equisplitting of power between the two channels isreached. In the phase plane, the system evolves along aseparatrix toward the unstable point (y, 0) = (0, 0). More-over, excitation of an individual guide requires that theseparatrix also contain the point (y, k) = (+1, 0). ThereforeEq. (15) is obtained easily by solving for the energy associat-ed with the separatrix trajectory:
H(y = O, = O) = H(y = 1, = ). (B3)
ACKNOWLEDGMENTS
We are grateful to G. De Biase and F. Gori for makingavailable to us the computing facilities of the Dipartimentodi Fisica, University of Rome. This research was sponsoredby the Fondazione Ugo Bordoni, by the National ScienceFoundation (ECS-8501249), by the U.S. Army Research Of-fice (DAAG29-85-K-0173), and by the National SecurityAdministration.
S. Trillo is also with Dipartimento di Fisica, Universita diRoma, La Sapienza, Rome, Italy.
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E. Caglioti
E. Caglioti received the Laurea degree inphysics in 1986 from the University ofRoma, La Sapienza. In 1986 he went tothe Fondazione Ugo Bordoni, Rome, ona two-year research grant. Since 1987he has also been a research associate atInstituto Nazionale di Fisica Nucleare.His current research interests concernnonlinear propagation in optical guides,with particular interest in soliton propa-gation in fibers and chaotic behavior oflight.
S. Trillo received the Laurea degree inelectrical engineering in 1982 from theUniversity of Roma, La Sapienza. Hereceived the Ph.D. degree in appliedelectromagnetism from the same univer-sity in 1986. He was a research associateat the Laboratori Nazionali di Frascati,Istituto Nazionale di Fisica Nucleare,from 1982 to 1983. Since 1986 he hasbeen with the research staff at the Fon-dazione Ugo Bordoni in Rome. His cur-rent research interests include propaga-tion in optical fibers, nonlinear optics in
bulk and guidinig structures, phase conjugation, and free-electronlasers.
S. Wabnitz
S. Wabnitz received the Laurea degreein electronics engineering in 1982 fromthe University of Rome, La Sapienza.Subsequently he spent one year at theCalifornia Institute of Technology, Pas-adena, where he obtained the M.S. de-gree in electrical engineering in 1983.Since 1983 he has been with the Fonda-zione Ugo Bordoni. In 1987 he obtainedthe Ph.D. degree in applied electromag-
,I, _ netism from the University of Rome.He has been working on the theory ofsuperresolution and inverse scattering
applied to electromagnetic imaging and on mode-coupling effects insingle-mode fibers. His current research activity involves nonlin-ear beam propagation in optical waveguides, applications to all-optical switching devices, optical bistability, instabilities and chaos,and optical phase conjugation.
