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arXiv:1008.2408v1

[quant-ph]14Aug2010

Interaction-Free All-Optical Switching via Quantum-Zeno Effect

Yu-Ping Huang, Joseph B. Altepeter, and Prem KumarCenter for Photonic Communication and Computing, EECS Department

Northwestern University, 2145 Sheridan Road, Evanston, IL 60208-3118(Dated: August 17, 2010)

We propose a novel interaction-free scheme for all-optical switching which does not rely on thephysical coupling between signal and control waves. The interaction-free nature of the scheme

allows it to overcome the fundamental photon-loss limit imposed by the signal-pump coupling. Thesame phenomenon protects photonic-signal states from decoherence, making devices based on thisscheme suitable for quantum applications. Focusing on (2) waveguides, we provide device designsfor traveling-wave and Fabry-Perot switches. In both designs, the performance is optimal when thesignal switching is induced by coherent dynamical evolution. In contrast, when the switching isinduced by a rapid dissipation channel, it is less efficient.

PACS numbers: 42.65.Wi, 42.50.Nn, 42.79.Ta, 03.67.-a

I. INTRODUCTION

All-optical networks hold the promise of fast trans-mission speed and ultra-high data capacity owing to the

non-interacting nature and large bandwidth of photonicsignals. A necessary resource for such networks is all-optical switching, in which the state of a signal wave isswitched in the presence of a control wave (pump). Thisis routinely achieved using direct coupling between thetwo waves in nonlinear materials, such as semiconduc-tors [1], (3) fibers [25], and cascaded (2) waveguides[610]. A fundamental limit in these devices is due to thephoton loss resulting from signal-pump coupling. Whenextended to the quantum domain, this photon loss ad-ditionally causes the quantum states of photonic signalsto collapse (i.e., decohere). To overcome this difficulty,we propose interaction-free switching that eliminates

direct signal-pump coupling.The concept of interaction-free phenomenon wasfirst proposed as measurement without touching [11,12] and has recently been extended to quantum logicgates [1316] and counterfactual quantum computation[17]. In our application to all-optical switching, we as-sume the signal wave is initially in state, say, |0. Ifun-disrupted, it will coherently evolve into an orthogo-nal output state |1, for example, after passing througha nonlinear waveguide. To induce switching, we applya pump wave which couples |1 to an ancillary state|2. This coupling provides a source of disruption to the|0 |1 evolution, resulting in the signal being frozenin state

|0. Thus the signal output will be switched be-

tween |0 and |1 depending on the presence or absence ofthe pump. Note that the signal and pump do not directlyinteract during operation of the device, as (ideally) thesignal remains in |0 when the pump is on. Therefore, inthe ideal case, any photon loss from signal-pump couplingis eliminated. Moreover, the quantum states of both thesignal and the pump are protected from decoherence.

Each of the above schemes exploits the quantum Zenoeffect [18, 19], whereby unitary evolution is suppressedvia quantum decoherence (i.e., quantum measurement)

[20]. Physically, the decoherence can be induced via ei-ther incoherent and irreversible coupling to many un-known quantum modes (e.g., spontaneous emission into alarge number of electromagnetic modes [21]), or coherent

and reversible coupling to a known quantum mode (e.g.,polarization decoherence via a polarization-dependenttime delay [22]). In the context of quantum measure-ment, these two processes are mathematically identical.In the context of interaction-free switching, however, wewill show that the incoherent quantum Zeno (IQZ) ef-fect and the coherent quantum Zeno (CQZ) effect canlead to very different switching performances.

An IQZ-based interaction-free switch was recently pro-posed by Jacobs and Franson (hereafter referred to as theJF switch), which is composed of a microring embed-ded in an atomic vapor [23]. We have recently designedanother type of CQZ-based, interaction-free switch which

is based on the coherent dynamical effect of Autler-Townes splitting [24]. The latter switch is composed of a(2) microdisk coupled to two fibers (or waveguides). Athird kind of switch has been demonstrated which re-lies on the absorption or scattering of pump photonsto change the optical properties of resonators [2531].Although implemented without direct signal-pump cou-pling, switches of this kind are not lossless and involvesignificant pump dissipation. They are thus not consid-ered to be interaction-free.

In our switch design, both the |0 and |1 states areassumed to be lossless (a necessary condition for high-fidelity switching). The ancillary state |2, on the otherhand, can be lossy. Depending on its lifetime, theinteraction-free disruption will be induced by either thecoherent or incoherent quantum Zeno effects. First, if|2 is short lived, then the |1 |2 coupling effectivelyopens a dissipation channel for |1. The inhibition of the|0 |1 transition is therefore a consequence of the IQZeffect. In the second case, that of a long-lived |2 sate,there is no dissipation involved. Instead, the |0 |1transition is suppressed by an Autler-Townesbased CQZeffect [32], similar to the electromagnetic-induced trans-parency [33]. The former IQZ effect corresponds to broad-

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ening of the |1 state, whereas the latter CQZ effect cor-responds to shifting of the same state. The two processesare equally effective for suppressing the |0 |1 transi-tion, but lead to very different pump-power requirementsfor similar switching setups. To see why, we consider thetoy model described above with Rabi frequencies ofs and p for the |0 |1 and |1 |2 transitions,respectively. The system evolution time t is set to be

/s, such that the signal evolves to |1 with pump off.The |2 state is assumed to decay at rate . Note thata necessary condition is |p| |s|. In the IQZ regimewith p, one finds via adiabatic elimination thatthe output power fraction of the signal in |1 with pumpon is t|s|2/|p|2. In contrast, in the CQZ regime with p, the power fraction in |1 is |s|2/|p|2. Ast |p|t , the fraction of the signal evolving into|1, even with the pump on, is much higher in the IQZregime than in the CQZ regime. This strongly suggeststhat the CQZ effect is more efficient in suppressing the|0 |1 transition, which will lead to better switchingperformance in the CQZ regime. Physically, this is be-cause in the presence of a strong dissipation for

|2, the

|1 |2 coupling is adversely suppressed by the IQZeffect. As a result, for a given pump, the |0 |1 tran-sition is in fact less disrupted in the IQZ regime.

In order to perform a systematic study of theinteraction-free switching, in this paper we consider sys-tems composed of(2) waveguides. Two designs are pro-vided. The first is a traveling-wave design composed of asingle-pass waveguide. With no pump, the signal under-goes a coherent second-harmonic generation (SHG) pro-cess in the waveguide, resulting in frequency doubling or-phase shifting. In this design, the fundamental (signalinput) and the second-harmonic (SH) wave correspondto

|0

and|1, respectively, in the above prototypical

model. To induce switching, a pump wave suppressesthe SHG by coupling the SH state to an ancillary state(|2). This coupling can be either sum-frequency gener-ation (SFG) or difference-frequency generation (DFG),but for concreteness, in this traveling-wave design we fo-cus on DFG. Using both the IQZ and CQZ effects, weare able to construct frequency, polarization, and spatial-mode switches. To achieve similar performances, a muchstronger pump is required in the IQZ regime than in theCQZ regime. All switches turn out to work well forcontinuous-wave inputs. For the realistic case of non-flat pulses, however, only the frequency switch is poten-tially practical. This is because both the polarization

and spatial-mode switches are subject to significant pulsedistortion, which fundamentally arises from the nonlin-ear nature of the SHG process. Such distortion wouldseverely limit the cascadability of this type of switches,making them not very useful for potential practical ap-plications.

To overcome pulse distortion, our second design uti-lizes a Fabry-Perot cavity, specifically, a (2) waveguidecoated with reflective layers on the two end faces. Thecavity is designed to be resonant for both the signal and

pump waves. In the absence of the pump, the signalwave, say, applied from the left, is resonantly coupledwith the cavity mode. Eventually, it exits from the rightend after a time delay. To induce switching, we apply apump wave to the cavity, also through the left-end layer.In the cavity, the signal and pump waves undergo SFG(or DFG). The SFG dynamics shifts the cavity out ofresonance, resulting in the signal being reflected from

the cavity. Note here that the SFG only potentiallyhappens, as ideally the signal would never enter the cav-ity. Moreover, by applying the pump pulse slightly aheadof the signal pulse, the pump will pass through the cav-ity unaffected (as there is ideally no signal field in thecavity). Compared to the traveling-wave design, in thiscase the quasi-linear cavity-coupling process would over-come pulse distortion, provided that the cavity passbandis wider than the Fourier spectrum of the pulses. Again,in this design, better switching performance is obtainedin the CQZ regime than in the IQZ regime. We notethat, in the IQZ regime, our cavity design is physicallyequivalent to the JF switch [23]. Both are implementedvia avoided two-photon absorption. Physically, the JFdevice is composed of a microring evanescently coupledto an atomic vapor. In contrast, our device utilizes an allsolid-state design. In the CQZ regime, the present Fabry-Perot switch can be mapped onto our recently proposedmicrodisk switch [24]. The goal of this paper is to system-atically study the IQZ and CQZ effects in interaction-freeswitching by varying the system dissipation. We presentthe traveling-wave and Fabry-Perot designs in sections IIand III, respectively, before presenting a brief conclusionin section IV.

II. TRAVELING-WAVE SWITCH

In this section we present the traveling-wave design forinteraction-free all-optical switching. In section II A wedescribe the schematic setups for three types of devices.In section IIB, we analytically solve the system dynamicsin the CQZ and IQZ regimes. Using these results, insection II C we analyze the switching performance forboth continuous-wave and pulsed inputs.

A. Setup

The schematic setups for our frequency, polarization,and spatial-mode switching devices are shown in Figs. 1(a), (b), and (c), respectively. The frequency switchtransforms a signal wave originally at angular frequencys into an identical signal at angular frequency s or2s, depending on the presence or absence of the controlwave. To achieve this, the waveguide is designed to bephase-matching (PM) or quasi-phase-matching (QPM)for SHG of the signal wave. Phase-matching can beachieved in some birefringent crystals by angle tuning[34] or temperature tuning [35], whereas quasi-phase-

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WG-IIWG-II WG-IIWG-II

WG-IIWG-II WG-IIWG-II

(c)

|H+i|V H-iV |H+iV |H+i|V

|V |V(b)

WG-IWG-I

Zs

(a)

Zs

Zs

Zs

Signalwave Pumpwave 50/50Beamsplitter

WG-IWG-I

WG-IWG-I PM/QPMwaveguide WG-IIWG-II PNMwaveguide

pq

q

pq

q

FIG. 1: (Color online) Schematic setups for frequency switch-ing (a), polarization switching (b), and spatial-mode switch-ing (c).

matching is more flexibly realized in periodically-poledmaterials whose crystal-axis orientation is periodicallyinverted along the wave-propagation direction [36, 37].In the absence of the control wave, the signal wave un-dergoes a complete SHG cycle as it travels through thewaveguide, where its power is monotonically transferredto the harmonic wave (at angular frequency 2s). Inthe presence of the control wave the SHG process is sup-

pressed, resulting in the majority of the signal power re-maining at angular frequency s. In effect, the presenceof the control wave switches the output signal frequencyfrom 2s to s.

Analogously, the polarization switch (see Fig. 1(b))performs a polarization rotation (while leaving the sig-nal frequency unchanged) that is conditional on the pres-ence or absence of the pump wave. This type of switchemploys phase-mismatched (PNM) waveguides to avoidmonotonous transfer of power from the signal to thesecond-harmonic wave. Using a uniaxial crystal, thewaveguide is designed such that one of the two orthogonalsignal polarizations experiences a phase shift during theSHG process. This can be achieved, for example, by us-ing a periodically-poled (PP) uniaxial waveguide in whichthe ordinary-polarized light is close to QPM, whereas theextraordinary-polarized light is far from phase matching.As above, the presence of the control wave suppresses theSHG process and the phase shift associated with it.As an example, consider a waveguide designed such thatthe polarization state |V gains an SHG-induced phaseshift, while the orthogonal polarization state |H experi-ences no SHG and hence no phase shift. This waveguidewill rotate an input signal whose polarization state is

|+ 12

(|H + i|V) to an output signal with polariza-tion state | 1

2(|H i|V). When the pump wave is

applied, the SHG process is suppressed, leaving the inputsignals polarization state |+ unchanged. To implementthis switch in isotropic crystals, the waveguide must bedesigned such that both polarizations experience an iden-tical -phase shift in the absence of the pump beam. Aninput signal

|+

will transform to the state

|+

, i.e., un-

changed except for a global phase shift. The switchingoperation is induced by a vertically-polarized pump ofappropriate intensity which suppresses the SHG processfor vertically-polarized inputs, thus eliminating the SHG-induced phase shift. This type of operation is feasiblein type-I (2) crystals because the diagonal nonlinear co-efficient d33 is usually much larger than the off-diagonalones. For a control wave of a particular intensity, onecan strongly disturb the |V-polarized harmonic wave(up-converted from the |V-polarized signal wave), whileleaving the |H-polarized harmonic wave nearly undis-turbed. (An analogous design is also possible for biaxialcrystals.)

By applying the same type of -phase suppression toan input signal in a waveguide inside one arm of an in-terferometer, we can realize an analogous spatial-modeswitching device. The schematic setup in Fig. 1(c) showsa Mach-Zehnder interferometer containing a waveguidein the lower arm. This waveguide is designed such thatthe SHG process induces a phase shift at the signalwavelength. In the absence of a control wave, the rel-ative phase between the two arms of the interferometercan be adjusted such that a signal wave entering the qinput port of the interferometer will exit from the qoutput port of the interferometer. In the presence of thecontrol wave, the SHG process is suppressed along with

the associated phase shift. As a result, the signal wavewill exit from the p output port of the interferometer.This setup can be modified to incorporate two symmetricwaveguides with opposite-sign phase-mismatching coeffi-cients, one in each arm of the interferometer, such thatin the absence of the control wavephase shifts of /2and /2 will be applied to the signal wave in each arm.

Fundamentally, the switches described here rely onsecond-order nonlinearities to enable the optical control.More specifically, they rely on the suppression of cer-tain second-order nonlinear optical processes in the pres-ence of a pump wave. There are two ways to achievethis: sum-frequency generation (SFG) and difference-frequency generation (DFG). When using SFG, the har-monic wave (with angular frequency h = 2s) is cou-pled to the pump wave (with angular frequency p) andup-converted into a sum-frequency (SF) wave of angularfrequency = h + p. When using DFG, the har-monic and pump waves generate a difference-frequency(DF) wave of angular frequency d = h p (or equiv-alently, d = ph), as shown in Fig. 2. In both cases,the harmonic level (following atomic-physics terminol-ogy) will be disturbed and, provided that the couplingis sufficiently strong, the SHG process will be suppressed

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FIG. 2: (Color online) The level scheme for DFG-basedswitching, showing the cases for pump on (a) and pump off(b).

due to either the IQZ or the CQZ effect. If the signalwave is in a single-mode of the waveguide, the SF wavewill be well guided; such may not be the case for the DFwave, however. For example, when using a 1565-nm sig-nal wave and a 1535-nm pump wave, the resulting DFwave will be in the terahertz (THz) regime. For typicalwaveguides, such as annealed proton-exchanged waveg-uides in PPLN with a small circumference (

100 m),

such a THz wave will be strongly diffracted, leading toenergy dissipation from the system. This provides a nat-ural dissipation channel for the IQZ effect via the purelygeometrical effect of diffraction. (If desired, this diffrac-tion could be overcome by applying reflective coatingsto the waveguide side surfaces.) In contrast, for the SFwave the potential dissipation must rely solely on mate-rial absorption. For concreteness, in the following we willfocus on the DFG scheme as shown in Fig. 2. Extendingto the SFG case is straightforward.

In all of the optical switches described above, the pumpwave does not directly interact with the signal wave. Thisis because the signals second-harmonic levelthe level

coupled to the pump wavenever gets populated whenunder the suppressive influence of the pump. Again, thisinteraction-free feature eliminates photon loss and pro-tects against decoherence.

B. Wave Dynamics

To model the system dynamics, we denote the positive-frequency electric fields for the signal, harmonic, pump,and DF waves as Es, Eh, Ep, and Ed, respectively. Weassume all waves travel in the z direction and are lin-early x-polarized. Note whereas only the type-I dynam-ics are considered here, generalization to the type-II caseis straightforward.

To proceed, we define the following dimensionless vari-ables,

Aj = i

0V nj2j

Ejeijzijt, (1)

with j = s,h,p,d. Here, V is the quantization volume,0 is the permittivity of free space, kj is the wave-vectormagnitude, and nj is the refractive index. Defined as

such, the quantity |Aj |2 gives the average number of pho-tons inside the volume V. Further, we introduce the ef-fective Rabi-frequencies in the space domain as

1 = 4deff

3s

n2snh0c2V

, (2)

2 = 2deff 2hpd

npndnh0c2V , (3)

where deff is the effective second-order nonlinear coeffi-cient.

In this paper, we assume a matched group velocity vgfor all the waves and neglect any group-velocity disper-sion. For switching, we consider a phase-matched DFGprocess. These two conditions lead to optimal switchingperformance. Furthermore, for simplicity we assume allwaves to be lossless except for the DF wave. The coupled-wave equations in the moving frame z z t/vg are

As

z = 1eikz

AhAs, (4)Ahz

=12

eikzA2s + 2ApAd, (5)

Apz

= 2AdAh, (6)Adz

= 2ApAh Ad. (7)

Here, k = 2ks kh is the phase-mismatch per unitlength and 2 is the decay rate of the DF wave.

1. Case I: Pump off

In the absence of the pump wave, i.e., Ap(0) = 0,the equations of motion are reduced to the well-knownSHG equations [36, 38]. For our purposes, there arethree dynamical regimes of interest: a) Perfect phase-matching with k = 0; b) small PNM with k 1|As|(k = 0); and c) large PNM with k 1|As|. Defin-ing the field amplitude and phase as Aj = jeij , fork = 0, the SHG dynamics have an analytical solution[38]:

s(z) = 0s sech

0s1z2 , (8)

h(z) =0s

2tanh

0s1z

2

, (9)

s(z) = s(0), (10)

h(z) = 2s(0). (11)

where 0s s(0). Thus the system undergoes amonotonous up-conversion process, transferring all powerfrom s to 2s. An example of this PM dynamics isshown in Fig. 3.

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5

0

0.2

0.4

0.6

0.8

1

As

0 1 2 3 4 50

0.2

0.4

0.6

0.8

z (mm)

Ap

pump off

pump on

pump off

pump on

(b)

(a)

FIG. 3: (Color online) Wave dynamics in a PM waveguidewith the pump off and on, for (a) the signal wave and (b)the harmonic wave. Parameters are chosen as follows: eff =10 mm1, 1 = 1 mm

1, As(0) = 1, k = 0 mm1 and

= 0 mm1.

For a PNM waveguide with k = 0, a general analyt-ical solution was also derived [36], where the amplitudesare given by

s(z) = 0s

1 1

2JacobiSN2

0s1z

2,

1

4

, (12)

h(z) = 0s

12

JacobiSN

0s1z2

,1

4

, (13)with

=

2 k

41 0s+

1 +

k2

821 2s(0)

. (14)

The solutions (12) and (13) are oscillatory functions witha period

z0 =2

2

0s1EllipticK

1

4

. (15)

In the small PNM regime with k < 10s, the signaland pump waves undergo a complete power-conversionprocess, whereas in the opposite regime of k > 10s,

the majority of power is trapped in the signal wave. Inthe limit k 10s, we have

s(z) = 0s

21(0s)3

k2sin2

kz

2

, (16)

h(z) =(0s(0))

21k

sin

kz

2

. (17)In this case, only a small fraction (

22s(0)2

1

k2 1) of thesignal power oscillates between the fundamental and SH

0

0.2

0.4

0.6

0.8

1

s

0 5 10 15 200

1

2

3

z (mm)

s

pump off

pump on

pump off

pump on

(b)

(a)

FIG. 4: (Color online) Wave dynamics for small PNM withthe pump off and on. In (a) we plot the amplitude and in(b) the phase of the signal wave. Parameters are: eff =10 mm1, 1 = 1 mm

1, As(0) = 1, k = 0.01 mm1 and

= 0.

levels. As an example, in Figs. 4(a) and 5(a) we plotthe amplitude dynamics of the fundamental wave for thecases of small and large PNM, respectively.

The phase dynamics, on the other hand, yield a com-plicated and nonintuitive expression. Nonetheless, fork 0.0110s, the periods for the fundamental waveto restore its power while gaining a phase shift of /2and are approximately z0 and 2z0, respectively. In thelarge PNM limit (k 10s), the phase shift increaseslinearly with z, giving

s(z) =21

2s(0)

2kz. (18)

This result also shows a linear dependence of the phaseshift on the intensity of the signal wave. As we will showlater, this intensity-dependence constitutes a major ob-stacle to applying the present traveling-wave switchingscheme to time-varying pulses. The phase dynamics inthe small and large PNM regimes are shown in Figs. 4(b)and 5(b), respectively.

2. Case II: Pump on

In the presence of the pump wave, the system dynam-ics will be disturbed. For switching purposes, the dy-namical regime of interest is where the SHG process isstrongly suppressed by either the IQZ or the CQZ ef-fect. In either case, a strong pump wave will be requiredsuch that 2Ap 1As, as we will show later. For suc-cessful operation, the harmonic wave must remain un-pumped while the pump wave remains undepleted (theinteraction-free regime). Under the undepleted-pump

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In effect, only a fraction of the input power (22

1|As(0)|22

eff

1) undergoes frequency conversion, while most remains inthe fundamental wave. We note that during this dynam-ics no dissipation of any kind is involved; rather, the SHGis suppressed by the coherent CQZ effect. In principle,all the input power can be restored to the fundamentalwave (the desired signal output), thus implementing alossless switch. In practice, however, it may be notice-

ably difficult to achieve this, as the frequency-conversionoscillation has a very short period ( 1/eff).

In Figs. 3, 4, and 5 we compare the wave dynamics withthe pump off and on in the regimes of PM (Fig. 3), smallphase-mismatching (Fig. 4), and large PNM (Fig. 5), re-spectively. In all the dynamical regimes, the amplitudeand phase dynamics are strongly suppressed by the CQZeffect.

Finally, we note that by replacing eff with k, theleading-order term in the Taylor series of Eq. (31) is thesame as Eq. (16). This result clearly shows that the effectof strong DFG coupling is to shift the SHG off resonance(i.e., off phase-matching).

In the first dynamical regime considered above, theCQZ effect dominates ( eff). In contrast, in thesecond, opposite, dynamical regime of interest with eff, the wave dynamics are dominated by the IQZ effect.To analytically solve for the dynamics in this regime, weadiabatically eliminate the DF wave in Eqs. (19)(21),obtaining

Asz

= 1eikzAhAs, (32)Ahz

12

eikzA2s effAh (33)

with the effective decay rate for the SH level given by

eff =2eff

. (34)

Here, the effect of coupling to a lossy DF wave is to opena decay channel for the SH level. Interestingly, the effec-tive decay rate eff is inversely proportional to the DFloss rate . This is a consequence of the IQZ-suppressioneffect on the DFG dynamics (not SHG). More specifi-cally, in the presence of a strong dissipation for the DFwave, the DFG dynamics are suppressed by the IQZ ef-

fect, leading to Ad 2

eff

2 Ah Ah. Thus the DF waveremains nearly unpumped. As a result, the effective dis-sipation rate for the SH level is Ad/Ah, giving Eq. (34).

In the IQZ regime, a necessary condition for successfulswitching is eff 1s in the case of PM SHG or small-PNM SHG, or eff 212s/k in the case of large-PNMSHG. Given that, the wave dynamics can be solved byadiabatically eliminating the SH wave and we obtain

As = As(0)

1 +

212s(0)

eff + ikz

1/2(35)

As(0) exp

21

2s(0)

22effz

exp

i

212s(0)

2kSIQZz

0 4 8 12 16 200

0.2

0.4

0.6

0.8

1

z (mm)

s

=100

=10

=1

=0

pump off

FIG. 6: (Color online) Amplitude dynamics of the signal wavewith the pump off (solid line) and pump on (dashed lines).The chosen parameters are: eff = 10 mm

1, 1 = 1 mm1,

As(0) = 1, k = 0.01 mm1. The DF dissipation rates

marked in the figure are in units of mm1.

with a phase suppression coefficient

SIQZ = k22

4eff. (36)

Here, both the amplitude loss and the phase shift of thesignal wave increase with . Thus a better switching per-formance is achieved with a weaker loss of the DF wave;in fact, the switch is optimal for = 0 correspondingto the CQZ regime. To show this, in Fig. 6 we plot theSHG dynamics for various values of . As shown, theSHG dynamics is most efficiently suppressed with = 0,i.e., in the CQZ regime. As increases, the suppression

effect becomes weaker. Eventually, for , the SHGdynamics approach the pump-off result.Of course, the fact that the SHG-suppression is weaker

with a large does not imply that the IQZ effect is inap-plicable to all-optical switching. Our conclusion is thatfor a given pump intensity, the SHG-suppression effect,and thus the switching efficiency, is highest when the sys-tem is operated in the CQZ regime. In other words, toachieve a certain switching performance, a much weakerpump would be required when the switch is operated inthe CQZ regime than in the IQZ regime. In terms of pho-ton loss, a switching operation in the CQZ regime yields afraction of 2212s(0)/2eff compared to 212s(0)z /2effin the IQZ regime. For the same pump intensity, the lat-ter is a factor of z 1 greater than the former. (Here,z 1 is from 10s and 10sz > 1.) In termsof the SHG suppression, which determines the switchingcontrast, for the same pump intensity we obtain

SCQZSIQZ =

2eff2

1, (37)

showing a much stronger suppression effect in the CQZregime.

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C. Switching Performance

In the previous section, we have solved for the wavedynamics of light propagation in a (2) waveguide, whereanalytical results are obtained for the three dynamicalregimes of interests. In this section we use these results tocharacterize the switching performance. We focus on theCQZ regime with = 0, as it leads to the strongest SHG-suppression effect. We consider nearly degenerate signaland pump waves and use the approximation nd np.

We consider super-Gaussian input signal and pumpwaves in the form

s,p(t) exp

t2m

22m

, (38)

where m = 0, 1, 2 . . . is the order-number for the super-Gaussian waves. m = 0 corresponds to a flat continuouswave (CW), m = 1 corresponds to a Gaussian pulse,and m > 1 corresponds to super-Gaussian pulses with

increasingly flatter top. In the following, we assume thepulse shapes to be the same for the signal and pumpwaves.

To characterize the switching performance, two impor-tant metrics are normally used: the photon loss duringswitching and the switching contrast. The switching lossis the fraction of energy missing from the two outputs.In all of our switches, the mean energy loss for CW op-eration is

L = IsIp

, (39)

with an upper-bound of 2 IsIp . Here, Is (Ip) is the input

intensity of the signal (pump) wave. The universalityof L over all dynamical regimes has been discussed insection II B. For pulsed operation, where the signal andthe pump have similar shapes, L matches the CW re-sult (39). Thus, for all switches, a pump wave of powerover 100 times greater than the signal wave is requiredto achieve below 1% loss. In practice, this would funda-mentally limit the present switch from being applied ina fan-in/fan-out all-optical network. This difficulty canbe overcome with a Fabry-Perot design, which we willpresent later.

The switching contrast, on the other hand, is defined asthe power ratio with pump on and off in a certain outputchannel. As there are two outputs in our switches, we de-fine the flipped (unflipped) contrast as the power ra-tio in the flipped (unflipped) output state. In Figs. 1 (a),(b), and (c), the flipped state respectively corresponds tothe frequency state 2s, the polarization state |H i|V,and the output state in the q port. In each case, theunflipped state then corresponds to the complementaryoutput.

1. Frequency switching

For CW operation, the flipped-state contrast is

FSCW =2

Ltanh2

0s1L

2

, (40)

which increases monotonically with the waveguide length

L. This behavior arises from the fact that the SHGis a one-way process (assuming no spontaneous down-conversion). It approaches 2/L when L 0s1. Theunflipped contrast, on the other hand, is given by

UFCW = (1 L)cosh2

0s1L2

, (41)

which monotonically approaches infinity as L andall the power is transferred to the SH level.

In the case of pulsed operation with the pump off,the pulse center up-converts more quickly than the pulsetails, due to the nonlinear nature of the SHG process. Af-ter the switching, the SH pulse-width is reduced by half

compared to the signal. With the pump on, the SHG-dynamic is equally suppressed across the whole pulsearea. Therefore, the switching contrasts with pump offor on are about the same as for CW inputs.

2. Polarization and Spatial-Mode Switching

The polarization and spatial-mode switches operate ona similar principle; in fact, they yield an identical math-ematical formulism. Therefore, here we only analyze thespatial-mode switch, noting that all results can be ex-actly applied to the polarization-mode switch.

In the case of CW inputs and small PNM (with k 1

0s), we set L = 2z0 to necessarily acquire a phase

shift as the signal passes through the waveguide. Wenote that z0 is insensitive to k, but is proportional to1/0s; thus this switch can only function appropriately forsignal waves of a certain known intensity. Therefore, itcan not be applied to non-flat pulses. The flipped-statecontrast in this case is given by

FSCW =16k2

2414s(0)L

2S2CQZ>

I2pI2s

. (42)

This results in a high switching efficiency. The unflipped-

state contrast, on the other hand, is infinity in the idealcase, as no wave exits from the p output port with thepump off.

In the case of large PNM with k 212s(0), thewaveguide length is set to be

L =2k

212s(0)

2z0, (43)

to achieve a phase shift for the signal wave. Thus fora given pump, a much longer waveguide is needed in the

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0

0.2

0.4

0.6

0.8

1

SignalAmplitude

0.6 0.4 0.2 0 0.2 0.4 0.60

1

2

3

t (ns)

SignalPhase

input, m=1

output, m=1

input, m=2

output, m=2

output, m=1

output, m=2

FIG. 7: (Color online) Amplitude and phase profiles of theoutput signal wave passing through a waveguide in the caseof small PNM and pump off, shown for Gaussian (m=1) andsuper-Gaussian (m=2) pulses. Here = 0.2 ns and all otherrelevant parameters are the same as in Fig. 4.

large PNM regime than that in the small PNM regime.The flipped-state contrast in this case is given by

FSCW =4

2S2CQZ 4

4eff

2k41, (44)

which shows a moderate switching efficiency. For exam-ple, for Ip = 100Is and a typical k = 101

0s, we have

FSCW = 40. The unflipped-state switching contrast hereis infinity, for the same reason as in the small PNM case.

When non-flat pulses are used, the system behavioris similar to the CW case with the pump on. This is

because the SHG-suppression effect is determined by theratio of the signal and pump intensities, and not by theirabsolute values. Thus the flipped-state contrast is aboutthe same as in the CW case. On the other hand, withthe pump offdue to the nonlinear nature of SHGtheoutput signal pulse will be significantly distorted. Thisgives rise to a low unflipped-state contrast.

First, for small PNM, the pulse distortion arises fromthe linear dependence of the SHG period on the signalamplitude s. This is a fundamental difficulty inherentto SHG; to overcome this pulse distortion one would re-quire nearly-flat-top super-Gaussian pulses. In Fig. 7 weshow the amplitude and phase profiles of the output sig-nal in the fundamental level with the pump off. For aGaussian pulse, the amplitude is very distorted and thephase is highly inhomogeneous. By using m = 2 super-Gaussian pulses, the distortion is somewhat mitigated.A direct consequence of the pulse distortion is a lowunflipped-state switching contrast. For the parametersused in Fig. 7, we numerically find the contrast to be 10for m = 1; and the contrast near-linearly increases to 90for m = 10.

For large PNM, the signal pulse largely maintains itsshape with the pump off, while its phase evolves inhomo-

0.4 0.2 0 0.2 0.4 0.60

0.2

0.4

0.6

0.8

1

t (ns)

SignalAmplitude

input, m=1

output, m=1

input, m=2

output, m=2

FIG. 8: (Color online) Amplitude profile of the signal waveat the output p ort I for Gaussian (m=1) and super-Gaussianinputs (m=2) with pump off in the case of large PNM, com-pared to the input. Relevant parameters are the same as inFig. 5.

geneously. For an mth-order Gaussian input, the result-ing phase shift is

s = exp

t

2m

2m

. (45)

This arises from the linear proportionality of the phase tothe intensity, as seen in Eq. (18). As a result, the outputpulse in the p channel in Fig. 1(c) is

Aps exp

i exp

t

2m

2m

1

exp

t

2m

22m

. (46)

Here, the first exponential term gives rise to pulse dis-tortion. As an example, we plot the signal output inFig. 8 for Gaussian (m=1) and super-Gaussian (m=2)

input pulses. Comparatively, the distortion is signifi-cantly mitigated for super-Gaussian pulses. Lastly, forthe parameters used in Fig. 5, we numerically find theunflipped-state contrast to be 8 for m = 1, and near-linearly increasing to 54 for m = 10.

III. WAVEGUIDE FABRY-PEROT DESIGN

The fundamental difficulty with the traveling-wave de-sign is pulse distortion. To overcome this difficulty, inthis section we present a Fabry-Perot design. The or-ganization is as follows: In section IIIA, we describethe switch setup and present the dynamical model. In

section IIIB, we use semi-static analysis to characterizethe switching performance. Finally, in section III C, wepresent numerical simulations for pulsed inputs.

A. The Model

The schematic setup of our Fabry-Perot switch isshown in Fig. 9. It is composed of a (2) waveguide de-signed to support PM or QPM SFG for the signal and

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FIG. 9: (Color online) Schematic setup of the noninteracting

all-optical switch configured using a (2) waveguide. Here (a)and (b) illustrate the ideal light paths with the pump off andon, respectively.

pump waves. The two waveguide ends are coated withhighly-reflective layers. The waveguide length is chosensuch that both the signal and pump waves are in cavityresonance. With the pump off, as shown in Fig. 9(a), the

signal wave resonantly couples to the cavity, say, fromthe left, which then eventually exits from the right. Forswitching, we apply a pump pulse to change the cavityresonance condition via SFG, as shown in Fig. 9(b). As aresult the signal will be reflected (which is the switchingoperation). For optimal performance, the pump pulse isapplied slightly ahead of the signal pulse to allow a suffi-cient pump field to built up in the cavity by the time thesignal pulse arrives. This is necessary to reflect the frontof the signal pulse. By excluding the signal wave from thecavity, the pump pulse is able to pass through the cavityunaffected. Note that while Fig. 9 shows a planar Fabry-Perot cavity, in practice a spherical cavity will be more

robust against mirror divergence or misalignment. Ofcourse, one can also replace the reflective coatings withtwo mirrors. It is straightforward to extend our analysisto these types of setups.

For simplicity, in the following we consider identical,lossless coatings on the two end faces. The coating reflec-tivity is R (thus the transmissivity is T = 1 R), whichis assumed to be the same for both the signal and pumpwaves. For the SF field, it is taken as R and in generalR = R. For further simplicity, we neglect the waveguideloss for both the signal and pump waves. This approx-imation is valid given that the single-round-trip loss isexpected to be much smaller than the coating transmis-sivity. The loss rate for the SF wave, which determines

whether the switch is in the IQZ or CQZ regime, is de-noted by 2.

To model the SFG process, we define the following di-mensionless variables in a similar fashion as in Eq. ( 1):

Aj = i

0V nj2j

Ej , (47)

where Ej (j = s,p,f) is the slowly-varying electric fieldof the signal, pump, and SF waves, respectively. We also

introduce an effective Rabi-frequency similar to Eqs. (2)and (3),

= 2deff

2spf

nsnpnf0c2V. (48)

At phase matching, the SFG dynamics in the cavityare governed by

Asz

= ApAf, (49)Apz

= AsAf, (50)Afz

= AsAp Af. (51)The above equations of motion have analytical solutionsunder the undepleted-pump approximation, in whichAp(z) = A

0pe

ip(z). In the IQZ regime, correspondingto |Ap|, we have

As(z) exp2effz/As(0), (52)

where we have introduced eff = A0p. In the opposite

CQZ regime, we have

As(z) = As(0) cos(effz) Af(0)eip sin(effz), (53)Af(z) = Af(0) cos(effz) + As(0)e

ip sin(effz). (54)

B. Quasi-Static Analysis

We now use quasi-static analysis to characterize theswitching performance. With the pump off, the signalwave is resonant with the cavity, as shown in Fig. 9(a).Assuming a /2 phase shift on the reflected light by thecoating, a straightforward self-consistent analysis reveals

ATs =AIsT eiksL

1 + Re2iksL, (55)

ARs = i

RAIs

1 +

T

R + e2iksL

, (56)

where kj = njj/c, and ATs and ARs are the transmit-

ted and reflected signal fields, respectively. The cavitytransmittance and reflectance spectra are then given by

Toff(ks) =(1 R)2

|1 + Rei2ksL|2 , (57)

Roff(ks) = 1 (1 R)2

|1 + Rei2ksL

|2

. (58)

An example of a typical cavity spectrum with the pumpoff is shown in Fig. 10.

When the pump is on, the cavity spectrum is alteredby the SFG process. In the IQZ regime with eff,we find

ATs =AIsT e

iksLe2

effL/

1 + Re2iksLe22effL/, (59)

ARs = i

RAIs

1 +

T

R + e2iksLe22effL/

, (60)

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which lead to the following transmission and reflectionspectra in the IQZ regime:

TIQZ(ks) =(1 R)2e22effL/

|1 + Rei2ksLe22effL/ |2 , (61)

RIQZ(ks) = R2

1 + e2iksLe22

effL/

R + e2iksLe22

effL/

2

. (62)

On the signal resonance with ksL = (n + 1/2) (n is aninteger), they become

T0IQZ =(1 R)2e22effL/|1 Re22effL/|2 , (63)

R0IQZ = R2

1 e22effL/R e22effL/

2. (64)

Equation (64) shows that the cavity reflectance R0IQZ de-creases with , indicating that the switching is in factless efficient in the presence of a strong loss for the SF

wave.In the CQZ regime with eff, a similar quasi-static

analysis can be performed. The cavity transmittance andreflectance at the signal resonance are approximately

T0CQZ =T2(2 + R22)1 R2 iR[(iR + R) + R]2 , (65)

R0CQZ =(R + R)2(

RR + )21 R2 iR[(iR + R) + R]2 , (66)

with = cos(effL) and = sin(effL).

To show the switching performance, in Fig. 10 we plotthe cavity spectra in three cases: pump off, pump on inthe IQZ regime, and pump on in the CQZ regime. Thesystem parameters are chosen to be L = 1 mm, R = 0.99,R = 0.90, and eff = 0.5 mm1. In the IQZ and CQZregimes, the dissipation of the SF wave is taken to be =5 mm1 and = 0, respectively. As shown in Fig. 10(a),the transmission window is significantly altered in bothregimes. The cavity transmittance is lowered to < 1% ( 81% (> 98%) [see Fig. 10(b)].Here the fact that the transmittance and reflectance donot sum to 1 is indicative of signal losses. Comparatively,for similar parameters, the switch is more efficient in theCQZ regime than in the IQZ regime.

To further compare the IQZ and CQZ effects, in Fig. 11we plot the cavity transmittance and reflectance at thesignal resonance as a function of . Unlike in Fig. 10where we applied a static self-consistent analysis, herewe obtain the results via an asymptotical analysis usinga dynamical model. As shown in Fig. 11(a), the cav-ity transmittance increases slowly with , indicating thatboth the IQZ and CQZ mechanisms are effective in mak-ing the cavity opaque. In contrast, the reflectance drops

0 . 0 0 0

0 . 0 0 3

0 . 0 0 6

0 . 0 0 9

- 0 . 1 0 - 0 . 0 5 0 . 0 0 0 . 0 5 0 . 1 0

0 . 8 0

0 . 8 5

0 . 9 0

0 . 9 5

1 . 0 0

P u m p O f f

I Q Z

C Q Z

T

r

a

n

s

m

s

s

o

n

S

p

e

c

t

r

u

m

P u m p O f f

I Q Z

C Q Z

R

e

f

e

c

t

o

n

S

p

e

c

t

r

u

m

( n m )

FIG. 10: (Color online) Cavity transmission (a) and reflection(b) spectra, for the cases of pump off (solid), pump on inthe IQZ regime (dash-dotted), and pump on in CQZ regime(dashed). The various parameters are chosen to b e: L = 1mm, R = 0.99, R = 0.90, and eff = 0.5 mm

1. The IQZand CQZ regimes correspond to = 5 mm1 and = 0,respectively.

dramatically from 99% to 80% when increases from 0to 6, as seen in Fig. 11(b). This behavior shows that thecavity is highly reflective only in the CQZ regime. In

the IQZ regime, on the other hand, a significant portionof the signal is lost owing to the presence of the SF-dissipation channel. It is, therefore, optimal to operatethe switch in the CQZ regime.

We now give a physical explanation as to why all-optical switching is more efficient in the CQZ regime thanin the IQZ regime. It is well known that light transmis-sion through a cavity relies on the destructive interfer-ence of the reflected light and the light leaking from thecavity through the input coupler (the left-end coating inour design). Cavity reflectivity is then achieved by elim-inating this destructive interference. There are two waysto accomplish this: (a) by diminishing the amplitude ofthe leaking light or (b) by modulating its phase. TheJF proposal is based on the principle in (a), where thesignal field in the resonator is eliminated via two-photonabsorption [23]. In this case, the cavity resonance is de-stroyed. The switches in Ref. [2531], for example, arebased on the principle in (b), where the cavity resonanceis shifted by modifying the refractive index of the intra-cavity material. Comparatively, the latter design allows ahigher reflectance and weaker photon loss for similar pa-rameters. This is because unlike in the JF switch, here aportion of the light entering the cavity is recycled as it

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0 . 0 0 6

0 . 0 0 8

0 . 0 1 0

0 . 0 1 2

0 1 2 3 4 5 6

0 . 8 0

0 . 8 5

0 . 9 0

0 . 9 5

1 . 0 0

C

a

v

t

y

T

r

a

n

s

m

t

t

a

n

c

e

C Q Z r e g i m e

C Q Z r e g i m e

I Q Z r e g i m e

I Q Z r e g i m e

C

a

v

t

y

R

e

f

e

c

t

a

n

c

e

( m m

FIG. 11: Cavity transmittance (a) and reflectance (b) plot-ted as a function of at the signal resonance. The systemparameters are the same as in Fig. 10.

leaks backwards from the cavity and eventually recom-bines with the reflected beam.

In our design, both the amplitude and phase modu-lation effects exist. In the IQZ regime, the dominantprocesses are a combination of the two and subsequentloss of the signal and the pump fields from the cavity.In this case, the switch operates on amplitude modula-tion, or more precisely, amplitude dissipation. This is

exactly analogous to the mechanism in the JF switch, asboth are based on two-photon absorption. The ultimatecavity reflectance is R, which is achieved when the two-photon absorption is 100%. In the CQZ regime, on theother hand, the cavity fields undergo a coherent dynam-ics. Depending on the value of R (the reflectivity of thecoatings for the SF light), the switch can be operated oneither principle. In the first case, for R 1, the SF fieldis mostly lost through the coatings, resulting in a strongloss of the signal field from the cavity after each cavitycycle. The switch is then effectively a IQZ-based switch,working on amplitude modulation. Note, however, thatin this case the depletion of the signal field in single passthrough the cavity is efficient, as it is not suppressed bythe Zeno effect. In contrast, in the JF switch (as wellas in our switch in the IQZ regime), the two-photon ab-sorption is Zeno-suppressed by the fast decay of theiratomic level 3 (of the SF wave in our design) [23]. In thesecond case, for R 1, the scattered SF light is mostlystored in the cavity. The signal field then undergoes a co-herent, oscillatory dynamical evolution, incurring a /2phase shift. As a result, the destructive interference atthe input coupler is destroyed, leading to reflectance fromthe cavity. The ultimate reflectance is (1 + R)/2, which

0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0

0 . 8 7

0 . 9 0

0 . 9 3

0 . 9 6

0 . 9 9

0 . 9 0 0 . 9 5 1 . 0 0

0 . 9 8 8

0 . 9 9 2

C

a

v

t

y

R

e

f

e

c

t

a

n

c

e

T h e f u n d a m e n t a l l i m i t

o f I Q Z - b a s e d s w i t c h e s

FIG. 12: (Color online) Solid line: Cavity reflectance as afunction ofR, where = 0 and all other parameters are thesame as in Fig. 11. Dashed line: Fundamental limit for anIQZ-based switch. For the JF switch and the present Fabry-Perot switch in the (effective) IQZ regime, this limit is ob-tained when the (effective) two-photon absorption is 100%.

is higher than in the IQZ regime above by an amountof (1 R)/2. This improvement can be very useful forcavity-based all-optical switching, especially when shortpulses are used [40]. Working on phase-modulation, thisswitch is similar to the designs in Refs. [2531]. How-ever, there is an important difference. In the cited de-signs, the pump photons are destroyed, for example, tocreate free carriers, leading to physical energy dissipa-tion. This feature also prevents their potential operationin the quantum domain due to an increased number of

noise channels that are coupled in. Our scheme, however,does not involve dissipation of any kind. In fact, it is thepotential for the SFG process that changes the cavity fromtransmissive to reflective. In this sense, our switch is veryclean and, therefore, suitable for quantum applications(as we recently showed for an equivalent microdisk model[24]).

To graphically present these arguments, in Fig. 12 weplot the cavity reflectance as a function of R. As shown,the cavity reflectance increases monotonically with R.At R = 0, for which the cavity is effectively in the IQZregime, the reflectance is only 86.3%. [Note that by in-creasing the pump power or the cavity length, this re-flectance can eventually approach the limit of 99%(= R)for an IQZ-based switch.] At R = 1, the reflectanceincreases to 99.5%; a result which agrees with our pre-diction above. From the inset in Fig. 12, one sees that forR > 0.93 the reflectance exceeds the fundamental limit(99%) of IQZ-based switches, e.g., the JF switch and ourswitch in the (effective) IQZ-regime. We note that thecrossing will occur at a smaller R as one increases thepump power.

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C. Pulse Dynamics

In the last subsection we characterized the Fabry-Perotswitch using a continuous-wave model. In this section weexamine the switching performance for pulsed operation,focusing on the optimal CQZ regime with R = 1. Wenote first that for any cavity-based switch, the spectralwidth of the cavity must be much wider than the band-width of the incident pulse in order to avoid distortion ofthe switched output pulse [40]. In our design, the spec-tral width of the cavity is

cavity =vc(1 R)

L, (67)

where vc is the speed of light in cavity. For typical vc =1011 mm/s and L = 1 mm, one would then require R