High-efficiency degenerate four-wave mixing in triply resonant nanobeam cavities

transcript

High-efficiency degenerate four wave-mixing in triply resonant nanobeam cavities

Zin Lin1,∗ Thomas Alcorn2, Marko Loncar1, Steven G. Johnson2, and Alejandro W. Rodriguez31School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138

2Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139 and3Department of Electrical Engineering, Princeton University, Princeton, NJ, 08544

(Dated: December 16, 2013)

We demonstrate high-efficiency, degenerate four-wave mixing in triply resonant Kerr (χ(3) ) photonic crystal(PhC) nanobeam cavities. Using a combination of temporal coupled mode theory and nonlinear finite-differencetime-domain (FDTD) simulations, we study the nonlinear dynamics of resonant four-wave mixing processes anddemonstrate the possibility of observing high-efficiency limit cycles and steady-state conversion correspondingto≈ 100% depletion of the pump light at low powers, even including effects due to losses, self- and cross-phasemodulation, and imperfect frequency matching. Assuming operation in the telecom range, we predict close toperfect quantum efficiencies at reasonably low∼ 50mW input powers in silicon micrometer-scale cavities.

PACS numbers: Valid PACS appear here

I. INTRODUCTION

Optical nonlinearities play an important role in numer-ous photonic applications, including frequency conversionand modulation [1–7], light amplification and lasing [1, 8–10], beam focusing [1, 11], phase conjugation [1, 12], sig-nal processing [13, 14], and optical isolation [15, 16]. Re-cent developments in nanofabrication are enabling fabricationof nanophotonic structures, e.g. waveguides and cavities,thatconfine light over long times and small volumes [17–21], min-imizing the power requirements of nonlinear devices [22, 23]and paving the way for novel on-chip applications based onall-optical nonlinear effects [18, 24–33]. In addition to greatlyenhancing light–matter interactions, the use of cavities canalso lead to qualitatively rich dynamical phenomena, includ-ing multistability and limit cycles [34–40]. In this paper,weexplore realistic microcavity designs that enable highly effi-cient degenerate four-wave mixing (DFWM) beyond the un-depleted pump regime. In particular, we extend the resultsof our previous work [41], which focused on the theoreticaldescription of DFWM in triply resonant systems via the tem-poral coupled-mode theory (TCMT) framework, to accountfor various realistic and important effects, including linearlosses, self- and cross-phase modulation, and frequency mis-match. Specifically, we consider the nonlinear process de-picted in Fig. 1, in which incident light at two nearby fre-quencies, a pumpω0 and signalωm = ω0 − ∆ω photon, isup-converted into output light at another nearby frequency, anidler ωp = ω0 +∆ω photon, inside a triply resonant photoniccrystal nanobeam cavity (depicted schematically in Fig. 8).We demonstrate that 100% conversion efficiency (completedepletion of the pump power) can be achieved at a criticalpower and that detrimental effects associated with self- andcross-phase modulation can be overcome by appropriate tun-ing of the cavity resonances. Surprisingly, we find that criticalsolutions associated with maximal frequency conversion areultra-sensitive to frequency mismatch (deviations from per-

∗Electronic address: zlin@seas.harvard.edu

fect frequency matching resulting from fabrication imperfec-tions), but that there exist other robust, dynamical states(e.g.“depleted” states and limit cycles) that, when properly ex-cited, can result in high conversion efficiencies at reasonablepump powers. We demonstrate realistic designs based on PhCnanobeam cavities that yield 100% conversion efficiencies at∼ 50mW pump powers and over broad bandwidths (modallifetimesQ ∼ 1000s). Although our cavity designs and powerrequirements are obtained using the TCMT framework, wevalidate these predictions by checking them against rigorous,nonlinear FDTD simulations.

Although chip-scale nonlinear frequency conversion hasbeen a topic of interest for decades [33], most theoretical andexperimental works have been primarily focused on large-etalon and singly resonant systems exhibiting either largefootprints and small bandwidths [25, 26, 42, 43], or low con-version efficiencies (the undepleted pump regime) [22, 44–46]. These include studies ofχ(2) processes such as sec-ond harmonic generation [26, 47–49], sum and differencefrequency generation [50], and optical parametric amplifica-tion [27, 28, 51], as well asχ(3) processes such as third har-monic generation [47, 52], four-wave mixing [53–55] and op-tical parametric oscillators [22, 56–58]. Studies that go be-yond the undepleted regime and/or employ resonant cavitiesreveal complex nonlinear dynamics in addition to high effi-ciency conversion [23, 37–39, 41, 59, 60], but have primarilyfocused on ring resonator geometries due to their simplicityand high degree of tunability [60]. Significant efforts are un-derway to explore similar functionality in wavelength-scalephotonic components (e.g. photonic crystal cavities) [49,50],although high-efficiency conversion has yet to be experimen-tally demonstrated. Photonic crystal nanobeam cavities notonly offer a high degree of tunability, but also mitigates thewell-known volume and bandwidth tradeoffs associated withring resonators [61], yielding minimal device footprint andon-chip integrability [62, 63], in addition to high qualityfac-tors [21, 64–67].

In what follows, we investigate the conditions and designcriteria needed to achieve high efficiency DFWM in realis-tic nanobeam cavities. Our paper is divided into two pri-mary sections. In Sec. II, we revisit the TCMT framework

sp₋Qsp

FIG. 1: Schematic diagram of a degenerate four-wave mixing pro-cess in which a pump photon at frequencyω0 and a signal photonat frequencyωm = ω0 − ∆ω are converted into an idler photon atωp = ω0 + ∆ω and an additional signal photon atωm, inside of atriply resonantχ(3) nonlinear cavity. The cavity supports three res-onant modes with frequenciesωck, lifetimesQk, and modal ampli-tudesak, which are coupled to a waveguide supporting propagatingmodes at the incident/output frequenciesωk, with coupling lifetimesQsk. The incident and output powers associated with thekth modeare given by|sk+|2 and|sk−|2.

introduced in Ref. 41, and extend it to include new effectsarising from cavity losses (Sec. II A), self- and cross-phasemodulation (Sec. II B), and frequency mismatch (Sec. II C). InSec. III, we consider specific designs, starting with a simple2d design (Sec. III B) and concluding with a more realistic 3ddesign suitable for experimental realization (Sec. III C).Thepredictions of our TCMT are checked and validated in the 2dcase against exact nonlinear FDTD simulations.

II. TEMPORAL COUPLED-MODE THEORY

In order to obtain accurate predictions for realistic designs,we extend the TCMT predictions of [41] to include impor-tant effects associated with the presence of losses, self- andcross-phase modulation, and imperfect frequency-matching.We consider the DFWM process depicted in Fig. 1, in whichincident light from some input/output channel (e.g. a waveg-uide) at frequenciesω0 andωm is converted to output light ata different frequencyωp = 2ω0 − ωm inside a triply-resonantχ(3) cavity. The fundamental assumption of TCMT (accu-rate for weak nonlinearities) is that any such system, regard-less of geometry, can be accurately described by a few setof geometry-specific parameters [41]. These include, the fre-quenciesωck and corresponding lifetimesτk (or quality fac-torsQk = ωckτk/2) of the cavity modes, as well as nonlin-ear coupling coefficients,αkk′ andβk, determined by over-lap integrals between the cavity modes (and often derivedfrom perturbation theory [23]). Note that in general, the to-tal decay rate (1/τk) of the modes consist of decay into theinput/output channel (1/τsk), as well as external (e.g. ab-sorption or radiation) losses with decay rate1/τek, so that1/τk = 1/τsk + 1/τek. Lettingak denote the time-dependentcomplex amplitude of thekth cavity mode (normalized so that|ak|2 is the electromagnetic energy stored in this mode), andletting sk± denote the time-dependent amplitude of the inci-

dent (+) and outgoing (−) light (normalized so that|sk±|2 isthe power at the incident/output frequencyωk), it follows thatthe field amplitudes are determined by the following set ofcoupled ordinary differential equations [23]:

= iωc0(

1− α00|a0|2 − α0m|am|2 − α0p|ap|2)

− a0τ0

− iωc0β0a∗

0amap +

τs0s0+, (1)

dt= iωcm

1− αm0|a0|2 − αmm|am|2 − αmp|ap|2)

− am

τm− iωcmβma

τsmsm+, (2)

dt= iωcp

1− αp0|a0|2 − αpm|am|2 − αpp|ap|2)

− ap

τp− iωcpβpa

m, (3)

s0− =

τs0a0 − s0+, sm− =

τsmam − sm+,

sp− =

τspap, (4)

where the nonlinear coupling coefficients [41],

αkk =1

d3x ǫ0χ(3)

2|Ek · E∗

k|2 + |Ek · Ek|

d3x ǫ|Ek|2)2 (5)

αkk′ =1

d3x ǫ0χ(3)

|Ek ·E∗

k′ |2 + |Ek · Ek′ |2 + |Ek|2|Ek′ |2

d3x ǫ|Ek|2) (∫

d3x ǫ|Ek′ |2)

αkk′ = αk′k (7)

β0 =1

d3xǫ0χ(3) [(E∗

0 ·E∗

0)(Em ·Ep) + 2(E∗

0 · Em)(E∗

0 · Ep)](∫

d3x ǫ|E0|2) (∫

d3x ǫ|Em|2)1/2 (∫

d3x ǫ|Ep|2)1/2

(8)βm = βp = β∗

0/2 (9)

express the strength of the nonlinearity for a given mode, withtheα terms describing SPM and XPM effects and theβ termscharacterizing energy transfer between the modes. (Techni-cally speaking, this qualitative distinction betweenα andβ isonly true in the limit of small losses [23]).

A. Losses

Eqs. 1–4 can be solved to study the steady-state conversionefficiency of the system [η = |sp−|2/(|s0+|2 + |sm+|2)] inresponse to incident light at the resonant cavity frequencies(ωk = ωck), as was done in Ref. 41 in the ideal case of perfectfrequency-matching (ωcp = 2ωc0−ωcm), no losses (τk 6= τsk),and no self- or cross-phase modulation (α = 0). In this idealcase, one can obtain analytical expressions for the maximumefficiencyηmax and critical powers,P crit

0 = |scrit0+|2 andP crit

|scritm+|2, at which 100% depletion of the total input power is

attained [41]. Performing a similar calculation, but this time

including the possibility of losses, we find:

P crit0 =

τs0|β0|√τmτpωmωp(10)

ηmax =τp

2− τs0

2ω0. (11)

With respect to the lossless case, the presence of losses merelydecreases the maximum achievable efficiency by a factor ofτp/τsp(2− τs0/τ0) while increasing the critical powerP crit

0 bya factor of

τsmτsp/τmτp. As in the case of no losses, 100%depletion is only possible in the limit asPm → 0, from whichit follows that the maximum efficiency is independent ofτm.As noted in [41], the existence of a limiting efficiency (Eq. 11)can also be predicted from the Manley–Rowe relations gov-erning energy transfer in nonlinear systems [68] as can thelimiting conditionPm → 0. While theoretically this suggeststhat one should always employ as small aPm as possible, aswe show below, practical considerations make it desirable towork at a small but finite (non-negligible)Pm.

B. Self- and cross-phase modulation

Unlike losses, the presence of self- and cross-phase mod-ulation dramatically alters the frequency-conversion process.Specifically, a finiteα leads to a power-dependent shift in theeffective cavity frequenciesωNL

ck = ωck(1 − ∑

j αkj |Aj |2)that spoils both the frequency-matching condition as wellas the coupling of the incident light to the correspondingcavity modes. One approach to overcome this difficulty isto choose/design the linear cavity frequencies to have fre-quencyωck slightly detuned from the incident frequenciesωk,such that at the critical powers, the effective cavity frequen-cies align with the incident frequencies and satisfy the fre-quency matching condition [41]. Specifically, assuming inci-dent light atω0 andωm, it follows by inspection of Eqs. 1–4that preshifting the linear cavity resonances away from thein-cident frequencies according to the transformation,

ωcritc0 =

1− α00|acrit0 |2 − α0m|acrit

m |2 − α0p|acritp |2 (12)

ωcritcm =

1− αm0|acrit0 |2 − αmm|acrit

m |2 − αmp|acritp |2 (13)

ωcritcp =

2ω0 − ωm

1− αp0|acrit0 |2 − αpm|acrit

m |2 − αpp|acritp |2 , (14)

yields the same steady-state critical solution obtained for α =0, whereacrit

k denote the critical, steady-state cavity fields.An alternative approach to excite the critical solution above

in the presence of self- and cross-phase modulation is to de-tune the incident frequencies away fromωc0 andωcm, keepingthe two cavity frequencies unchanged, while pre-shiftingωcp

to enforce frequency matching. Specifically, by inspectionofEqs. 12–14, it follows that choosing input-light frequencies

ωcrit0 = ωc0

1− α00|acrit0 |2 − α0m|acrit

m |2 − α0p|acritp |2

ωcritm = ωcm

1− αm0|acrit0 |2 − αmm|acrit

m |2 − αmp|acritp |2

and tuningωcp such that

ωcritcp =

2ωc0(1−∑

α0k|acritk |2)− ωcm(1 −

αmk|acritk |2)

1−∑

αpk|acritk |2 ,

(17)yields the same steady-state critical solution above. Thisap-proach is advantageous in that the requirement that all threecavity frequencies be simultaneously and independently tuned(post-fabrication) is removed in favor of tuning a single cavitymode. Given a scheme to tune the frequencies of the cavitymodes that achieves perfect frequency matching at the criticalpower, what remains is to analyze the stability and excitabil-ity of the new critical solution, which can be performed usinga staightforward linear stability analysis of the coupled modeequations [38]. Before addressing these questions, however,is important to address a more serious concern.

C. Frequency mismatch

Regardless of tuning mechanism, in practice one can neverfully satisfy perfect frequency matching (even when self- andcross-phase modulation can be neglected) due to fabricationimperfections. In general, one would expect the finite band-width to mean that there is some tolerance∼ 1/Qp on anyfrequency mismatch∆ω = 2ωc0−ωcm−ωcp . ωcp/Qcp [60].However, here we find that instabilities and strong modifica-tions of the cavity lineshapes arising from the particular na-ture of this nonlinear process lead to extreme, sub-bandwidthsensitivity to frequency deviations that must be carefullyex-amined if one is to achieve high-efficiency operation.

To illustrate the effects of frequency mismatch, we firstconsider an ideal, lossless system with zero self- and cross-phase modulation (α = 0) and with incident light at fre-quenciesω0 = ωc0 andωm = ωcm, and powersP crit

0 andPm, respectively. With the exception ofα, the coupling co-efficients and cavity parameters correspond to those of the2d design described in Sec. III B. Figure 2 (top) shows thesteady-state conversion efficiencyη (solid lines) as a func-tion of the frequency mismatch∆cp = ωcp − ωcrit

cp awayfrom perfect frequency-matching, for multiple values ofPm ={0.001, 0.01, 0.1}P crit

0 , with blue/red solid lines denoting sta-ble/unstable steady-state fixed points. As shown, solutionscome in pairs of stable/unstable fixed points, with the sta-ble solution approaching the maximum-efficiencyηmax criti-cal solution asPm → 0. Moreover, one observes that as∆cp

increases for finitePm, the stable and unstable fixed points ap-proach and annihilate one other, with limit cycles appearingin their stead (an example of what is known as a “saddle-nodehomoclinic bifurcation” [69]). The mismatch at which this bi-furcation occurs is proportional toPm, so that, asPm → 0,the regime over which there exist high-efficiency steady statesreduces to a singlefixed pointoccurring at∆cp = 0. Beyondthis bifurcation point, the system enters a limit-cycle regime(shaded regions) characterized by periodic modulations oftheoutput signal in time [37, 38, 70]. Interestingly, we find that

0 2 4-2 6

stable

unstable τm

stable

unstable

FIG. 2: (Top:) Steady-state conversion efficiencyη (normalizedby the maximum achievable efficiencyηmax) as a function of fre-quency mismatch∆cp = ωcp − ωcrit

cp (in units of ωcritcp /2Qp), for

the cavity system depicted in Fig. 4, but in the absence of self-and cross-phase modulation (α = 0). Incident frequencies arechosen to beω0 = ωcrit

0 and ωm = ωcritm , with corresponding

powersP0 = P crit0 and Pm, where we consider multiplePm =

{0.1, 0.01, 0.001}P crit0 . Note that sinceα = 0, critical frequencies

are independent of incident powers, so thatωcrit0 = ωc0, ωcrit

m = ωcm,andωcrit

cp = 2ωc0 − ωcm. Blue/red solid lines denote stable/unstablefixed points, whereas shaded areas indicate regimes lackingfixedpoint solutions and exhibiting limit-cycle behavior, shown only forPm = {0.01, 0.001}P crit

0 , with smaller amplitudes corresponding tosmallerPm. Dashed lines denote the average efficiency of the limitcycles η, whereas the top/bottom of the shaded regions denote themaximum/minimum efficiency per period. The inset shows the ef-ficiency as a function of time for a typical limit cycle, obtained at∆cp ≈ 3ωcrit

cp /2Qp. (Bottom:) η and η for the same system above,but in the presence of self- and cross-phase modulation (α 6= 0), andonly for Pm = 0.01P crit

0 . Note that additional stable and unstablefixed points arise due to the non-zeroα, and that limit-cycle behav-iors arise only for∆cp > 0. Inset shows the temporal shape of theincident power needed to excite the desired limit cycles, correspond-ing to a Gaussian pulse superimposed over CW inputs.

the average efficiency of the limit cycles (dashed lines),

η = limT→∞

dt η(t), (18)

remains large and. ηmax even as∆cp is several fractionalbandwidths. The inset of Fig. 2 (top) shows the efficiency ofthis system as a function of time (in units of the lifetimeτ0)for large mismatch∆cp = 3ωcrit

cp /2Qp. As expected, the mod-ulation amplitude and period of the limit cycles depend onthe input power and mismatch, and in particular we find thatthe amplitude goes to zero and the period diverges∼ 1/∆cp

as∆cp → 0. This behavior is observed across a wide rangeof Pm, with largerPm leading to lowerη and larger ampli-tudes. For small enough mismatch, the modulation frequencyenters the THz regime, in which case standard rectificationsprocedures [71] can be applied to extract the useful THz os-cillations [4, 5, 72–75].

Frequency mismatch leads to similar effects for finiteα,including homoclinic bifurcations and corresponding high-efficiency limit cycles that persist even for exceedingly largefrequency mismatch. One important difference, however, isthat the redshift associated with self- and cross-phase modu-lation creates a strongly asymmetrical lineshape that preventshigh-efficiency operation for∆cp < 0. Figure 2 (bottom)shows the stable/unstable fixed points (solid lines) and limitcycles (dashed regions) as a function of∆cp for the same sys-tem of Fig. 2 (top) but with finiteα, for multiple values ofPm = {0.001, 0.01}P crit

0 . As before, the coupling coefficientsand cavity parameters correspond to those of the 2d design de-scribed in Sec. III B. Here, in contrast to theα = 0 case, thecritical incident frequenciesωcrit

0 andωcritm are chosen accord-

ing to Eqs. 15–16 in order to counter the effects of self- andcross-phase modulation, and are therefore generally differentfrom ωc0 andωcm. Aside from the asymmetrical lineshape,one important difference from theα = 0 case is the presenceof additional stable/unstable low-efficiency solutions. Multi-stability complicates matters since, depending on the initialconditions, the system can fall into different stable solutionsand in particular, simply turning on the source at the criticalinput power may result in an undesirable low-efficiency solu-tion. One well-known technique that allows such a system tolock into the desired high-efficiency solutions is to superim-pose a gradual exponential turn-on of the pump with a Gaus-sian pulse of larger amplitude [37]. We found that a singleGaussian pulse with a peak power of4P crit

0 and a temporalwidth ∼ τm, depicted in the right inset of Fig. 2 (bottom), issufficient to excite high-efficiency limit cycles in the regime∆cp > 0.

Despite their high efficiencies (even for large∆cp & 1),the limit-cycle solutions above leave something to be desired.Depending on the application, it may be desirable to operateathigh-efficiency fixed points. One way to achieve this for non-zero frequency mismatch is to abandon the critical solutionand instead choose incidence parameters that exploit self-andcross-phase modulation in order to enforce perfect frequency

-2 0 2

frequency detuning cp (units of ωcp / 2 Qp )crit

FIG. 3: Steady-state conversion efficiency (normalized by the max-imum achievable efficiencyηmax) and required incident powersP p

andPmm (normalized by the critical powerP crit

0 ) corresponding to de-pleted steady states of the system of Fig. 4, as a function of frequencymismatch∆cp = ωcp − ωcrit

cp (in units ofωcritcp /2Qp). As described in

Sec. III B 2, depleted states yield 100% depletion ofP0, and are ex-cited by appropriate combinations of incident frequenciesωk = ωdep

and powersPk = P depk . Blue/red lines denote stable/unstable solu-

tions, with solid and dashed lines, and circles, denotingη, P depm , and

P dep0 , respectively.

matching and 100% depletion of the pump as follows,

ωNLcp + ωNL

cm = 2ωNLc0 , (19)

s0− = 0 (20)

Specifically, enforcing Eqs. 19–20 by solving Eqs. 1–4 forωdep0 , ωdep

m , P dep0 , andP dep

m , we obtain adepletedsteady-statesolutionadep

k that, in contrast to the critical solutionacritk , yields

a steady-state efficiency that corresponds to 100% depletionof the pump regardless of frequency mismatch. Note that weare not explicitly maximizing the conversion efficiency butrather enforcing complete conversion of pump energy in thepresence of frequency mismatch, at the expense of a non-negligible inputP dep

m . Figure 3 shows the depleted steady-state efficiencyηdep (solid line) and corresponding incidentpowers (solid circles and dashed line) as a function of∆cp,for the same system of Fig. 2 (bottom). We find that for mostparameters of interest, depleted efficiencies and powers areuniquely determined by Eqs. 19–20. As expected, the op-timal efficiency occurs at∆cp = 0 and corresponds to thecritical solution, so thatP dep

0 = P crit0 , P dep

m = P critm = 0,

andηdep = ηmax. For finite∆cp 6= 0, the optimal efficien-cies are lower due to the finiteP dep

m , but there exist a broadrange of∆cp over which one obtains relatively high efficien-cies∼ ηmax. Power requirementsP dep

0 andP depm follow dif-

ferent trends depending on the sign of∆cp. Away from zerodetuning,P dep

m can only increase whereasP dep0 decreases for

∆cp < 0 and increases for∆cp > 0. In the latter case, the totalinput power exceedsP crit

0 leading to the observed instability ofthe fixed-point solutions.

Finally, we point out that limit cycles and depleted steadystates reside in roughly complementary regimes. Although nostable high-efficiency fixed points can be found in the∆cp >0 regime, it is nevertheless possible to excite high-efficiencylimit cycles. Conversely, although no such limit cycles existfor ∆cp < 0, it is possible in that case to excite high-efficiencydepleted steady states.

III. NANOBEAM DESIGNS

In this section, we consider concrete and realistic cavity de-signs in 2d and 3d, and check the predictions of our TCMTby performing exact nonlinear FDTD simulations in 2d. Ourdesigns are based on a particular class of PhC nanobeam struc-tures, depicted schematically in Figs. 4 and 8, where a cavityis formed by the introduction of a defect in a lattice of air holesin dielectric, and coupled to an adjacent waveguide formed bythe removal of holes on one side of the defect. We restrictour analysis to dielectric materials with high nonlinearities atnear- and mid-infrared wavelengths [1], and in particular fo-cus on undoped silicon, whose refractive indexn ≈ 3.4 andKerr susceptibilityχ(3) ∼ 10−18 m2/V 2 [76].

A. General design considerations

Before delving into the details of any particular design, wefirst describe the basic considerations required to achievethedesired high efficiency characteristics. To begin with, we re-quire three modes satisfying the frequency-matching condi-tion to within some desired bandwidth (determined by thesmallest of the mode bandwidths). We begin with the lin-ear cavity design, in which case we seek modes that approx-imately satisfyωcm + ωcp = 2ωc0. The final cavity design,incorporating self- and cross-phase modulation, is then ob-tained by additional tuning of the mode frequencies as de-scribed above. Second, we seek modes that have large non-linear overlapβ, determined by Eq. 8. (Ideally, one wouldalso optimize the cavity design to reduceα/β, but such anapproach falls beyond the scope of this work.) Note that theoverlap integralβ replaces the standard “quasiphase match-ing” requirement in favor of constraints imposed by the sym-metries of the cavity [1]. In our case, the presence of reflectionsymmetries means that the modes can be classified as eithereven or odd and also as “TE-like” (E · z ≈ 0) or “TM-like”(H · z ≈ 0) [77], and hence only certain combinations ofmodes will yield non-zero overlap. It follows from Eq. 8 thatany combination of even/odd modes will yield non-zero over-lap so long asEm andEp have the same parity, and as long asall three modes have similar polarizations: modes with differ-ent polarization will cause the term∼ (E∗

0 ·Em) (E∗0 · Ep) in

Eq. 8 to vanish. Third, in order to minimize radiation losses,we seek modes whose radiation lifetimes are much greaterthan their total lifetimes, as determined by any desired op-erational bandwidth. In what follows, we assume operationalbandwidths withQ ∼ 103. Finally, we require that our systemsupport a single input/output port for light to couple in/out of

the cavity, with coupling lifetimesQsk ≪ Qrk in order tohave negligible radiation losses.

B. 2d design

In what follows, we consider two different 2d cavities withdifferent mode frequencies but similar lifetimes and couplingcoefficients. (Note that by 2d we mean that electromagneticfields are taken to be uniform in thez direction.) The two cav-ities follow the same backbone design shown in Fig. 4 whichsupports three TE-polarized modes (H · z = 0) with radiativelifetimesQrad

0 = 6×104, Qradm = 6×104, andQrad

p = 3×103,and total lifetimesQ0 = 1200, Qm = 1100, andQp = 700,respectively. The nonlinear coupling coefficients are calcu-lated from the linear modal profiles (shown on the inset ofFig. 4) via Eqs. 8–7, and are given by:

β = (23.69 + 5.84i) × 10−5

ǫ0a2h

α00 = 4.935 × 10−4

ǫ0a2h

, αmm = 5.096 × 10−4

ǫ0a2h

αpp = 4.593 × 10−4

ǫ0a2h

, α0m = 6.540 × 10−4

ǫ0a2h

α0p = 5.704 × 10−4

ǫ0a2h

, αmp = 5.616 × 10−4

ǫ0a2h

where the additional factor ofh allows comparison to the re-alistic 3d structure below and accounts for finite nanobeamthickness (again, assuming uniform fields in thez direction).

Compared to the optimalβmax = 34n4wd

, correspond-

ing to modes with uniform fields inside and zero fields outsidethe cavity, we find thatβ = 5.5 × 10−3βmax is significantlysmaller due to the fact that these TE modes are largely con-centrated in air. In the 3d design section below, we choosemodes with peaks in the dielectric regions, which leads tomuch largerβ ≈ 0.4βmax.

In order to arrive at this 2d design, we explored a widerange of defect parameters, with the defect formed by mod-ifying the radii of a finite set of holes in an otherwise periodiclattice of air holes of perioda and radiusR = 0.36a in adielectric nanobeam of widthw = 1.2a and index of refrac-tionn = 3.4. The defect was parametrized via an exponentialadiabatic taper of the air-hole radiir, in accordance with the

formular(x) = R

1− 34e

−4⌊x⌋2

, where the parameterd is

an “effective cavity length”. Such an adiabatic taper is chosento reduce radiation/scattering losses at the interfaces ofthecavity [78]. The removal of holes on one side of the defectcreates a waveguide, with corresponding cavity–waveguidecoupling lifetimesQsk determined by the number of holesremoved [79–81]. To illustrate the dependence of the modeproperties on the cavity parameters, Fig. 5(top) shows the evo-lution of the cavity-mode frequencies as a function ofd, withblue/red dots denoting even/odd modes and with larger dotsdenoting longer modal lifetimes. As expected, the volumesof the modes decrease with decreasingd, leading to largerβ

cavity center

FIG. 4: Schematic of two-dimensional, triply-resonant cavity designinvolving a PhC nanobeam of refractive indexn = 3.4, widthw =1.2a and adiabatically varying hole radii (see text). The effectivecavity lengthd = 6.6a and the radius of the central holeR0 arechosen so as to fine-tune the relative frequency spacing and lifetimesof the modes. Also shown are theEy electric field components of thethree modes relevant to DFWM. The cavity is coupled to a waveguideformed by the removal of holes to the right of the defect.

(smaller critical powers) but causing the frequency gap be-tween the modes and radiation losses to increase. We find thatthe desired modal parameters for FWM lie at some intermedi-ated ≈ 6.6a. In order to tune the relative frequencies betweenthe modes, an additional tuning parameter is required. Specif-ically, it follows from perturbation theory [61] that changingthe central hole radiusR0 allows control of the even-modefrequencies while leaving odd-mode frequencies unchanged.Figure 5(bottom) shows the evolution of the cavity-mode fre-quencies as a function ofR0, for a fixed d = 6.6a. Asdescribed below, the particular choice ofR0 will depend onwhether one seeks to operate with high-efficiency limit cyclesversus high-efficiency steady-state solutions.

1. Limit cycles

In this section, we consider a design supporting high-efficiency limit cycles. ChoosingR0 = 0.149a, we ob-tain critical parametersωcrit

0 = 0.2319(

, ωcritm =

0.2121(

, ωcritcp = 0.2530

, and P crit0 =

10−3(2πcǫ0ahχ(3) ), corresponding to frequency mismatch∆cp ≈

3ωcritcp /2Qp and critical efficiencyηmax = 0.51. Choosing

a small but finitePm = 0.01P crit0 , it follows from Fig. 2

(dashed line) that the system will support limit cycles withaverage efficienciesη ≈ 0.65ηmax. To excite these solutions,we employed the priming technique described in Sec. II C.Figure 6 showsη as a function ofP0, for incident frequen-ciesωk = ωcrit

k determined by Eqs. 15–16, as computed byour TCMT (gray line) and by exact, nonlinear FDTD simula-tions (solid circles). The two show excellent agreement. For0.7 < P0/P

crit0 < 3, we observe limit cycles with relatively

high η , in accordance with the TCMT predictions, whereasoutside of this regime, we find that the system invariably fallsinto low-efficiency fixed points. The periodic modulation of

0.26 8 10 124

0.1 0.15 0.2 0.30.250.2

FIG. 5: Mode frequencies (in units of2πc/a) as a function of effec-tive cavity lengthd (top), for fixed center-hole radiusR0 = 0.9a, andas a function ofR0 (bottom), for fixedd = 6.6a. Red/blue circlesindicate symmetric/anti-symmetric mode profiles, where the size ofthe circle is proportional to the modal lifetime (quality factor) of thecorresponding mode. The shaded area indicates the parameter regionexplored in the Sec. B1, B2.

the limit cycles means that instead of a single peak, the spec-trum of the output signal consists of a set of equally spacedpeaks surroundingωp. The top and bottom insets of Fig. 6show the corresponding frequency spectra of the TCMT andFDTD output signals aroundωp, for a particular choice ofP0 ≈ P crit

0 (red circle), showing agreement both in the relativemagnitude and spacing≈ 2.5× 10−3

of the peaks.

2. Depleted steady states

In this section, we consider a design supporting high-efficiency, depleted steady states. ChoosingR0 = 0.143a,one obtains critical parametersωcrit

0 = 0.2320(

,ωcrit

m = 0.2118(

, ωcritcp = 0.2532

, andP crit0 =

10−3(2πcǫ0ahχ(3) ), corresponding to frequency mismatch∆cp ≈

0.252 0.2530.251

FDTDωp

TCMT(simulation)

1 2 �0

0.252 0.2530.251

critincident power

FIG. 6: Average conversion efficiencyη (normalized by the max-imum achievable efficiencyηmax) of limit cycles as a function ofpowerP0 (normalized byP crit

0 ) at the critical frequenciesωcrit0 and

ωcritm and a fixedPm = 0.01P crit

0 . The modal parameters are obtainedfrom the 2d cavity of Fig. 4, with chosenR0 = 0.149a leadingto a detuning∆cp ≈ 3ωcrit

cp /2Qp corresponding to the dashed linein Fig. 2 (bottom). Solid circles and gray lines denote results as com-puted by FDTD and TCMT. Insets show the spectra of the outputlight for a givenP0 (red circle), and for both FDTD and TCMT.

−0.6ωcritcp /2Qp and critical efficiencyηmax = 0.51. Choos-

ing incident frequenciesωdep0 = 0.2320

, ωdepm =

0.2119(

, and incident powersP dep0 ≈ 0.7P crit

P depm ≈ 0.04P crit

0 , it follows from Fig. 3 (dashed line) thatthe system supports stable, depleted steady states with effi-ciencies≈ 0.95ηmax. Figure 7 shows the efficiency of thesystem as a functionP0, with all other incident parametersfixed to the depleted-solution values above, where blue/redlines denote stable/unstable solutions. As before, we employthe priming technique of Sec. II C in order to excite the de-sired high efficiency solutions and obtain excellent agreementbetween our TCMT (gray line) and FDTD simulations (solidcircles). Exciting the high-efficiency solutions by steady-stateinput “primed” with a Gaussian pulse is convenient in FDTDbecause it leads to relatively short simulations, but is prob-lematic forP0 > 0.8P crit

0 , where the system becomes verysensitive to the priming parameters, and it became impracticalin for us to find the optimal FDTD source conditions in Fig. 7.In realistic experimental situations, however, one can useadifferent technique to excite the high-efficiency solutionin away that is very robust to errors, based on adiabatic tuning ofthe pump power [37].

C. 3d design

We now consider a 3d design, depicted in Fig. 8, as afeasible candidate for experimental realization. The cav-ity supports three TE00 modes (Ez = 0 atz = 0) of fre-

0.4 0.6 0.8 10

(simulation)

stable

unstable

frequency ω

ωm ωp

critincident power

FIG. 7: Conversion efficiencyη (normalized by the maximumachievable efficiencyηmax) of depleted states as a function of powerP0 (normalized byP crit

0 ), at incident frequenciesωdep0 andωdep

m , anda fixed powerP dep

m ≈ 0.2P crit0 . The modal parameters are obtained

from the 2d cavity of Fig. 4, withR0 = 0.143a leading to a detun-ing∆cp ≈ −0.6ωcrit

cp /2Qp corresponding to the dashed line in Fig. 3.Ey component of the steady-state electric field inside the cavity isshown as an inset (left-bottom). Solid circles and gray lines denoteFDTD and TCMT, while blue/red lines denote stable/unstablesteadystates. Inset (left-top) shows the spectral profile (in arbitrary units)of the system, showing full depletion of the pump (blue) and corre-spondingly high conversion of the signal/idler frequencies (red). ForP0 & 0.8P crit

0 , the system becomes ultra sensitive to the priming pa-rameters, in which case high-efficiency solutions can only be excitedby adiabatic tuning of the pump power (see text).

quenciesωc0 = 0.2848(

, ωcm = 0.2801(

andωcp = 0.2895

, radiative lifetimesQrad0 = 106, Qrad

3× 104, Qradp = 2× 104. As before, the total lifetimes can be

adjusted by removing air holes to the right or left of the defect,which would allow coupling to the resulting in-plane waveg-uides. (Alternatively, one might consider an out-of-planecou-pling mechanism in which a fiber carrying incident light atbothω0 and/orωm is brought in close proximity to the cav-ity [79, 82].) In what follows, we do not consider any oneparticular coupling channel and focus instead on the isolatedcavity design. Nonlinear coupling coefficients are calculatedfrom the linear modal profiles (shown on the inset of Fig. 8)via Eqs. 9–7, and are given by:

β = 2× 10−4

α00 = 8.1× 10−4

, αmm = 4.6× 10−4

αpp = 11.5 × 10−4

, α0m = 6.2× 10−4

α0p = 12.7 × 10−4

, αmp = 5.5× 10−4

FIG. 8: Schematic of three dimensional, triply resonant cavity de-sign involving a PhC nanobeam of refractive indexn = 3.4, widthw = a, and heighth = 0.51a, and linearly tapered air holes,as described in the text. The central cavity lengthL ≈ 0.4a andnumber of taper segments are chosen so as to fine-tune the rela-tive frequency spacing and lifetimes of the modes. Also shownare theEy electric-field components of three TE-like modes withfundamental TE00 transverse profiles, and with frequenciesωc0 =0.2848

, ωcm = 0.2801(

andωcp = 0.2895(

. Ra-diation lifetimes are found to beQrad

0 = 106, Qradm = 3 × 104, and

Qradp = 2× 104.

Here, in contrast to the 2d design of Sec. III B, we chose modeswhose amplitudes are concentrated in dielectric regions, and there-fore find appreciably largerβ ≈ 0.4βmax .

In order to arrive at the above 3d design, we explored a cavityparametrization similar to the one described in [83]. Specifically,we employed a suspended nanobeam of widthw = a, thicknessh = 0.51a, and refractive indexn = 3.4. The beam is schemati-cally divided into a set of2N lattice segments, each having lengthai, i ∈ {±1, ...,±N} and corresponding air-hole radiiRi = 0.3ai,wherea1 (a−1) is the length of the lattice segment immediately tothe right (left) of the beam’s center. The cavity defect is induced viaa linear taper ofai over a chosen set of2N segments, according tothe formula:

ai = a

fa +(1− fa)

(N − 1)(|i| − 1)

, |i| ≤ N

= a, |i| > N.

In order to arrive at our particular design, we chosefa = 0.85, N =21, N = 9 and varied the central cavity lengthL to obtainthe desired TE00 modes. Assuming total modal lifetimesQ0 =8500, Qm = 3000, and Qp = 3000 and using these designparameters, we obtain critical parametersωcrit

0 = 0.2843(

,ωcrit

m = 0.2798(

, ωcritcp = 0.2895

, andP crit0 = 5 ×

10−5(

2πcǫ0a2

, corresponding to frequency mismatch∆cp ≈

−0.07ωcritcp /(2Qp) and ηmax = 0.42. Note that because the ra-

diative losses in this system are non-negligible, the maximum effi-ciency of this system is≈ 82% of the optimal achievable efficiencyωp/(2ω0) ≈ 0.51. At these small∆cp, we find that depletion ofthe pump is readily achieved through the critical parameters asso-ciated with perfect frequency matching. However, as illustrated inSec. III B 1, it is indeed possible to choose a design that leads tohighly efficient limit cycles or other dynamical behaviors.

We now express the power requirements of this particular designusing real units instead of the dimensionless units of2πcε0a

2/χ(3)

we have employed thus far. Choosing to operate at telecom wave-lengthsλc0 ≡ 2πc/ωc0 = 1.5µm, with correspondingn ≈ 3.4 and

χ(3) = 2.8 × 10−18 m2

V 2 [1], we find thata = 0.2848 × 1500 =

427nm andP crit0 ≈ 50mW . Although our analysis above incor-

porates effects arising from linear losses (e.g. due to material ab-sorption or radiation), it neglects important and detrimental sourcesof nonlinear losses in the telecom range, including two-photon andfree carrier absorption [84, 85]. Techniques that mitigatethe lat-ter exist, e.g. reverse biasing [86], but in their absence itmay besafer to operate in the spectral region below the half-bandgap of sil-icon [76]. One possibility is to operate atλc0 = 2.2µm, in whichcaseχ(3) ≈ 1.5 × 10−18 m2

V 2 [76], leading toa = 627nm andP crit0 ≈ 200mW . For a more detailed analysis of nonlinear ab-

sorption in triply resonant systems, the reader is referredto Ref. 87.While that work does not consider the effects of nonlinear disper-sion, self- and cross-phase modulation, or frequency mismatch, itdoes provide upper bounds on the maximum efficiency in the pres-ence of two-photon and free-carrier absorption.

IV. CONCLUDING REMARKS

In conclusion, using a combination of TCMT and FDTD simula-tions, we have demonstrated the possibility of achieving highly effi-

cient DFWM at low input powers (∼ 50mW ) and large bandwidths(Q ∼ 1000) in a realistic and chip-scale (µm) nanophotonic platformconsisting of a triply resonant silicon nanobeam cavity. Our the-oretical analysis includes detrimental effects stemming from linearlosses, self- and cross-phase modulation, and mismatch of the cavitymode frequencies (e.g. arising from fabrication imperfections), andis checked against the predictions of a full nonlinear Maxwell FDTDsimulation. Although power requirements in the tens of mW s are notoften encountered in conventional chip-scale silicon nanophotonics,they are comparable if not smaller than those employed in conven-tional centimeter-scale DFWM schemes [86, 88, 89]. Our proof-of-concept design demonstrates that full cavity-based DFWM not onlyreduces device dimensions down toµm scales, but also allows de-pletion of the pump with efficiencies close to unity. However, weemphasize that there is considerable room for additional design opti-mization. In particular, we find that increasing the radiative lifetimesof the signal and converted modes (currently almost two orders ofmagnitudes lower than the pump) can significantly lower the powerrequirements of the system.

Acknowledgements:We acknowledge support from the MIT Un-dergraduate Research Opportunities Program and the U.S. Army Re-search Office through the Institute for Soldier Nanotechnology undercontract W911NF-13-D-0001.

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High-efficiency degenerate four-wave mixing in triply resonant nanobeam cavities

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