Generation of Rabi frequency radiation using exciton-polaritons

Fabio Barachati,1, ∗ Simone De Liberato,2 and Stephane Kena-Cohen1, †

1Department of Engineering Physics, Ecole Polytechnique de Montreal, Montreal H3C 3A7, QC, Canada2School of Physics and Astronomy, University of Southampton, Southampton SO17 1BJ, United Kingdom

(Dated: October 6, 2018)

We study the use of exciton-polaritons in semiconductor microcavities to generate radiation span-ning the infrared to terahertz regions of the spectrum by exploiting transitions between upper andlower polariton branches. The process, which is analogous to difference-frequency generation (DFG),relies on the use of semiconductors with a nonvanishing second-order susceptibility. For an organicmicrocavity composed of a nonlinear optical polymer, we predict a DFG irradiance enhancementof 2.8 · 102, as compared to a bare nonlinear polymer film, when triple resonance with the funda-mental cavity mode is satisfied. In the case of an inorganic microcavity composed of (111) GaAs,an enhancement of 8.8 · 103 is found, as compared to a bare GaAs slab. Both structures show highwavelength tunability and relaxed design constraints due to the high modal overlap of polaritonmodes.

PACS numbers: 42.65.-k,42.65.Ky,71.36.+c,78.20.Bh

I. INTRODUCTION

Half-light, half-matter quasiparticles called polari-tons arise in systems where the light-matter interactionstrength is so strong that it exceeds the damping dueto each bare constituent. In semiconductor microcavi-ties, polaritons have attracted significant attention due totheir ability to exhibit strong resonant nonlinearities andto condense into their energetic ground state at relativelylow densities. Such polaritons result from the mixingbetween an exciton transition (EX) and a Fabry-Perotcavity photon (EC). They exhibit a peculiar dispersion,which is shown in Fig. 1. Around the degeneracy pointof both bare constituents, the lower and upper polari-ton (LP and UP) branches anticross and their minimumenergetic separation is called the vacuum Rabi splitting(hΩR). It can range from a few meV in inorganic semi-conductors to ∼ 1 eV in organic ones1–5. Radiative tran-sitions from the upper to the lower polariton branch cantherefore provide a simple route towards tunable infrared(IR) and terahertz (THz) generation.

Such transitions can be understood as resulting froma strongly coupled χ(2) nonlinear interaction in whichtwo photons, dressed by the resonant interaction withexcitons, interact emitting a third photon. As a conse-quence of the usual χ(2) selection rule, such polariton-polariton transitions are forbidden in centrosymmetricsystems. To overcome this issue several solutions havebeen proposed, including the use of asymmetric quan-tum wells,6,7 the mixing of polariton and exciton stateswith different parity,8,9 and the use of transitions otherthan UP to LP.10,11

Here, we study the use of non-centrosymmetric semi-conductors, possessing an intrinsic second-order suscepti-bility χ(2), to allow for the generation of Rabi-frequencyradiation. The irradiance of the resulting UP to LPtransitions, which are analogous to classical difference-frequency generation (DFG), have been calculated usinga semiclassical model, yielding DFG irradiance enhance-

ħω1

ħω2

ħΩR

UP

LP

EX

EC

E

k//

x

θi

θ3

E31 N. . . . . . . .

z

E1,2

ω1

ω2

(2)

y

FIG. 1. Dispersion relation of exciton-polaritons as a func-tion of in-plane wavevector. The interaction between an ex-citon transition (EX) and a Fabry-Perot cavity mode (EC),both represented by dashed lines, leads to the appearance oflower and upper polariton (LP and UP) branches (solid blue).A radiative transition at the Rabi energy (hΩR) occurs be-tween two incident pumps at frequencies ω1 and ω2 throughdifference-frequency generation in a second-order nonlinearsemiconductor (χ(2) 6= 0). Inset: microcavity showing the

two pump beams ( ~E1,2), incident at angle θi, and the Rabi

radiation ( ~E3), reflected at angle θ3. The solid blue lines in

the χ(2) layer illustrate the high modal overlap of polaritonfields.

ments up to almost four orders of magnitude comparedto the ones due to the bare χ(2) nonlinearity. These en-hancements can also be related to those expected forparametric fluorescence. Finally, we highlight the useof a triply-resonant scheme to obtain polariton opticalparametric oscillation (OPO).

Semiconductor microcavities are advantageous for non-linear optical mixing due to their ability to spatially and

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temporally confine the interacting fields. For small in-teraction lengths, the efficiency of the nonlinear processdoes not depend on phase-matching, but instead on max-imizing the field overlap.12 To overcome mode orthogo-nality, while simultaneously satisfying the symmetry re-quirements of the χ(2) tensor, a number of strategies havebeen proposed such as mode coupling between crossedbeam photonic crystal cavities with independently tun-able resonances13–15 and the use of single cavities sup-porting both TE and TM modes.16,17 Exciton-polaritonsprovide a simple solution to this problem because theyarise from coupling to a single cavity mode and thus nat-urally display good modal overlap. Many of the fascinat-ing effects observed in strongly-coupled semiconductormicrocavities exploit this property, but these have beenprincipally limited to the resonant χ(3) nonlinearity in-herited from the exciton.

Note that in this paper we define the vacuum Rabifrequency as being equal to the resonant splitting dueto light-matter coupling. Although this definition iscommonly used in the study of quantum light-matterinteractions,18 it differs from that often employed in thefield of microcavity polaritons,19 where the vacuum Rabifrequency is defined as being equal to half of the resonantsplitting.

This paper is organized as follows. Section II reviewsthe nonlinear transfer matrix scheme used to calculatefrequency mixing in the small-signal regime. In Sec. III,we calculate the enhancement in irradiance at the Rabifrequency over a bare nonlinear slab for organic and inor-ganic microcavities and in Sec. IV we discuss the resultsand highlight some of the peculiarities of both materialsets. Conclusions are presented in Sec. V.

II. THEORY

To calculate the propagation of the incident pumpfields and the difference-frequency contribution due tononlinear layers, we use the nonlinear transfer matrixmethod introduced by Bethune.20 This method is ap-plicable to structures with an arbitrary number of par-allel nonlinear layers,21 but is restricted to the unde-pleted pump approximation, where the three fields areessentially independent. First, we propagate the incidentpump fields using the standard transfer matrix method.Within each nonlinear layer, these behave as source termsin the inhomogeneous wave equation. Then, we solve forthe particular solution and determine the correspondingsource field vectors. Finally, we use the boundary con-ditions and propagate the free fields using the transfermatrix method to obtain the total field in each layer.

A. Propagation of the pump fields

We begin by calculating the field distribution of thetwo incident pumps as shown in Fig. 1 by using the

standard transfer matrix method.22–24 To simplify thediscussion, we consider the pumps to be TE (y) polarized.In our notation, the electric field in each layer i is givenby sum of two counter-propagating plane waves

E±i (z, x, t) = ReE±i exp[i(±kizz + kxx− ωt)]

, (1)

where the kiz and kx components of the ~ki wavevectorsatisfy the relationship k2iz + k2x = n2i (ω)ω2/c2, with nithe refractive index of layer i. The forward and backwardcomplex amplitudes of the electric field are represented

in vector form as Ei =[E+i E−i

]T.

For a given incident field E1, the field in layer i iscalculated by Ei = TiE1, where Ti is the partial transfermatrix

Ti = Mi(i−1)φi−1 · · ·M21. (2)

The interface matrixMij , that relates fields in adjacentinterfaces i and j, and the propagation matrix φi, thatrelates fields on opposite sides of layer i with thicknessdi, are given by

Mij =1

2kiz

[kiz + kjz kiz − kjzkiz − kjz kiz + kjz

](3)

and

φi =

[exp(ikizdi) 0

0 exp(−ikizdi)

]. (4)

B. Inclusion of nonlinear polarizations

To obtain the difference-frequency contribution withina nonlinear layer, we must solve the inhomogeneous waveequation for the electric field

∇2E − µε∂2E∂t2

= µ∂2PNL

∂t2, (5)

where the source term

PNL(z, x, t) = ε0χ(2)E2(z, x, t) (6)

is the second-order nonlinear polarization, µ is the mag-netic permeability and ε the permittivity. By using apolarization term of the same form as Eq. (1), Eq. (5)can be written in the frequency domain as[−(kNL)2 + ω2

NLn2(ωNL)µ0ε0

]E = −ω2

NLµ0PNL, (7)

with wavevector kNL, µ(ωNL) = µ0 and ε(ωNL) =n2(ωNL)ε0. The nonlinear polarization thus generatesa bound source field at the same frequency given by

Es =PNL

(kNL)2

ω2NLµ0

− n2(ωNL)ε0. (8)

3

If we consider the presence of two pump fields E1(ω1)and E2(ω2), with ω1 > ω2, the E2(z, x, t) term in Eq. (6)can be written as

E2(z, x, t) = ReE+

1 exp[i(k1zz + k1xx− ω1t

)]+E−1 exp

[i(−k1zz + k1xx− ω1t

)]+E+

2 exp[i(k2zz + k2xx− ω2t

)]+E−2 exp

[i(−k2zz + k2xx− ω2t

)]2.

(9)

Expanding E2(z, x, t) leads to terms related to fre-quency doubling (ωNL = 2ω1 or 2ω2) and rectifica-tion (ωNL = 0), sum-frequency (ωNL = ω1 + ω2) anddifference-frequency generation (ωNL = ω1 − ω2). Theterms contributing to the latter (≡ ω3) are given by

P3(z, x, t) = ε0χ(2) Re

(E+

1 E+∗2 exp

[i(k1z − k2z

)z]

+E+1 E−∗2 exp

[i(k1z + k2z

)z]

+E−1 E+∗2 exp

[−i

(k1z + k2z

)z]

+E−1 E−∗2 exp

[−i

(k1z − k2z

)z])

× exp(i[(k1x − k2x

)x− ω3t

]).

(10)

Co-propagating waves (±,±) generate terms with per-pendicular wavevector k3−z = k1z − k2z , whereas counter-propagating waves (±,∓) generate terms with k3+z =k1z + k2z . Their contributions can be handled separatelywhen pump depletion is ignored, so we divide the polar-ization term into two components

P3− = ε0χ(2)

[E+

1 E+∗2

E−1 E−∗2

](11a)

P3+ = ε0χ(2)

[E+

1 E−∗2

E−1 E+∗2

], (11b)

with their source fields given by Eq. (8) and the per-pendicular component of kNL taking the values of k3−zor k3+z , respectively.

In addition to the bound fields, there are also free fieldswith frequency ω3 that are solutions to the homogeneouswave equation. The free field in a nonlinear layer j isobtained from the bound field amplitudes Ejs and theboundary conditions at the interfaces. By imposing con-tinuity of the total tangential electric and magnetic fieldsacross interfaces i–j and j–k, an effective free field sourcevector can be defined as

Sj =(φ−1j Mjsφjs −Mjs

)Ejs, (12)

where the source matrices with the subscript s, Mjs andφjs, are identical to the ones given by Eqs. (3) and (4),

with kiz and kjz taking the values of k3jz and k3±jz , re-spectively.

The total nonlinear field is then given by the sum ofindependent source field vectors Sj propagated using the

transfer matrix method reviewed in Sec. II A. In par-ticular, for the case where only layer j is nonlinear, weobtain [

E3T

0

]= MN(N−1) · · ·M21

[0E3R

]+MN(N−1) · · ·M(j+1)jSj

= TN

[0E3R

]+

[R+j

R−j

],

(13)

with

Rj = TNTj−1Sj . (14)

Therefore, the reflected and transmitted componentsof the E3 field can be calculated by

E3R = −R−jT22

(15a)

E3T = R+j −

T12T22

R−j . (15b)

The angle dependence of the reflected difference-frequency field can be expressed as

|k3| sin θ±3 = |k1| sin θ1 ± |k2| sin θ2, (16)

where the ± sign must match the wavevector compo-nent k3±z when both pumps are incident on the same sideof the normal.25 Because the first layer is taken to be airwith n(ω) = 1, if we consider both pumps to be incidentwith the same angle θ1 = θ2 = θi, we obtain for the casesof k3−z and k3+z

sin θ−3 =ω1 sin θi − ω2 sin θi

ω1 − ω2= sin θi (17a)

sin θ+3 =

(ω1 + ω2

ω1 − ω2

)sin θi. (17b)

Equation (17a) shows that the DFG component dueto co-propagating waves exits the structure at the sameangle as the incident pumps, resembling the law of reflec-tion. Conversely, according to Eq. (17b), the componentdue to counter-propagating pump waves is very sensitiveto any angle mismatch between the pumps and easilybecomes evanescent for small DFG frequencies.

III. RESULTS

A. Organic polymer cavity

In this section, we investigate the use of organic mi-crocavities for Rabi frequency generation. Due to thelarge binding energy of Frenkel excitons, organic micro-cavities can readily reach the strong coupling regime atroom temperature and have shown Rabi splittings ofup to 1 eV.3,4 Demonstrations of optical nonlinearities

4

have been more limited than in their inorganic counter-parts, but a variety of resonant26,27 and non-resonantnonlinearities28–30 have nevertheless been observed inthese systems.

Although most organic materials possess a negligiblesecond-order susceptibility, a number of poled nonlinearoptical (NLO) chromophores have been shown to exhibithigh electro-optic coefficients that exceed those of con-ventional nonlinear crystals such as LiNbO3 by over anorder of magnitude.31,32 In addition, the metallic elec-trodes needed for polling can also be used as mirrors,providing high mode confinement and a means for elec-trical injection.

We will consider a thin NLO polymer film enclosed by apair of metallic (Ag) mirrors of thicknesses 10 nm (front)and 100 nm (back). The model polymer is taken to pos-sess a dielectric constant described by a single Lorentzoscillator

ε(ω) = εB +fω0

2

ω02 − ω2 − iΓω

, (18)

where εB is the background dielectric constant, f is theoscillator strength, ω0 is the frequency of the optical tran-sition and Γ its full width at half maximum (FWHM).The parameters are chosen to be εB = 4.62, f = 0.91,hω0 = 1.55 eV and hΓ = 0.12 eV. Experimental valuesare used for the refractive index of Ag.33 For simplic-ity, we ignore the dispersive nature of the second-ordernonlinear susceptibility and take χ(2) = 300 pm/V. Inprinciple, the Lorentz model could readily be extendedto account for the dispersive resonant behavior.31

Figure 2 shows the linear reflectance, calculated at nor-mal incidence, as a function of polymer film thickness.The reflectance for film thicknesses below 200 nm showsonly the fundamental cavity mode (M1), which is splitinto UP and LP branches. For these branches, the Rabienergy falls below the LP branch, where there are no fur-ther modes available for difference-frequency generation.

By increasing the thickness of the film, low-ordermodes shift to lower energies and provide a pathwayfor the DFG radiation to escape. For example, at 300nm, a triple-resonance condition occurs where the Rabisplitting of the M2 cavity mode matches the M1 en-ergy (EUP − ELP = hΩR = EM1 = 0.68 eV). Asecond resonance occurs between M3 and LP becauseEM3 − ELP = ELP = 1.25 eV, but with reduced modaloverlap.

The enhancement in DFG irradiance from the micro-cavity, as compared to a bare nonlinear slab, is shown inFig. 3 as a function of the pump energies. The two peakscorrespond to the triple-resonance conditions mentionedabove, where the left peak corresponds to an enhance-ment of 2.8 · 102 at the Rabi energy (λ3 = 1.82 µm) andthe right peak to an enhancement of 3.3 · 102 at the LPenergy (λLP = 996 nm).

The inset shows the normalized electric field profiles ofthe relevant modes, which highlight the good modal over-lap of the two pump fields in the strong-coupling regime.

1.0

0.8

0.6

0.4

0.2

0.0

Polymer thickness (nm)

Pum

p en

ergy

(eV

)

0.5

1.0

1.5

2.0

2.5

0 100 200 300 400 500

UP

M1

Pump1

M3

LP

ΩRPump2

ћω0

M2

FIG. 2. Reflectance as a function of pump energy and thick-ness of the polymer film. Front and back Ag mirrors havethicknesses of 10 nm and 100 nm, respectively. Dielectric pa-rameters: εB = 4.62, f = 0.91, hω0 = 1.55 eV, hΓ = 0.12 eVand χ(2) = 300 pm/V. Dashed horizontal line indicates theexciton energy. At the thickness of 300 nm, indicated by avertical dashed line, the M1 cavity mode is resonant with thedifference-frequency generation of pumps 1 and 2 such thatEUP − ELP = hΩR = EM1.

The small thickness of the front metallic mirror lowersthe mutual orthogonality of different modes and accountsfor the lack of symmetry of the fields with respect to thecenter of the film. This loss of orthogonality allows theoverlap integral between M3 and LP to be non-zero andthe enhanced DFG extraction due to the triple-resonancecondition leads to the appearance of the second peak athω3 = 1.25 eV in Fig. 3.

Additionally, oblique incidence of the pump beams canbe used to tune the DFG energy. As indicated by Eq.(17b), the k3+z component of the DFG signal rapidly be-comes evanescent and therefore we shall consider only thek3−z component. Figure 4 shows the dependence of DFGenergy and irradiance on the angle of incidence whenθ1 = θ2 = θi. In the lower panel, as the interactingmodes move to higher energies, the triple-resonance con-dition at the Rabi (EUP −ELP ) energy is maintained forincidence angles up to 79o. The maximum irradiance isobtained at 57o for hωNL = 0.72 eV (λNL = 1.72 µm).This enhancement is reduced by 3 dB at hω3 = 0.74 eV(λ3 = 1.68 µm) for 79o. The upper panel shows that thepeak at hω3 = 1.25 eV falls out of the triple resonancecondition faster with a 3 dB roll-off at 40o.

B. (111) GaAs cavity

The vast majority of resonant nonlinearities observedin inorganic semiconductor microcavities are due to aχ(3) nonlinearity inherited from the exciton.34 In the typ-

5

ћω1 (eV)

ћω2

(eV

)

1.0

1.1

1.2

1.3

1.4

1.8 2.0 2.2 2.4 2.6 2.81.6

0.68 eV 1.25 eVћω3 ћω3

350

280

210

140

0

70

UP

M1LP

M3

FIG. 3. DFG irradiance enhancement of the poled NLOpolymer model structure with respect to a bare film of equalthickness. Due to the thickness of the second mirror, onlyreflected fields are considered. The tilted dashed lines corre-spond to pairs of pump energies that generate the same DFGenergy and that match the M1 (left, hω3 = hωM1 = 0.68 eV)and LP (right, hω3 = hωLP = 1.25 eV) energies in the triple-resonance condition. Inset: normalized electric field profilesof the relevant modes, illustrating the excellent modal overlapof the LP and UP branches.

ical χ(3) four-wave mixing process, two pump (p) polari-tons interact to produce signal (s) and idler (i) compo-nents such that their wave-vectors satisfy 2kp = ks + ki.Second-order susceptibilities tend to be much larger thantheir χ(3) counterparts, but conventionally used (001)-microcavities only allow for nonlinear optical mixing be-tween three orthogonally polarized field components.

A number of commonly used inorganic semiconductorsare known to be non-centrosymmetric and to possess highsecond-order susceptibility tensor elements. Examplesinclude III-V semiconductors, such as gallium arsenide(GaAs) and gallium phosphide (GaP), and II-VI semi-conductors, such as cadmium sulfide (CdS) and cadmiumselenide (CdSe).31,35 To allow for the nonlinear opticalmixing of co-polarized waves to occur, we will consider(111) GaAs as the microcavity material,36,37 in contrastto the typical (001)-oriented material.

We consider a λ/2 (111) bulk GaAs microcavity sand-wiched between 20 (25) pairs of AlAs/Al0.2Ga0.8As dis-tributed Bragg reflectors (DBRs) on top (bottom). Thestructure is followed by a bulk GaAs substrate with thesame dielectric constant as the cavity material, mod-eled by Eq. (18) with experimental values εB = 12.53,f = 1.325 · 10−3, hω0 = 1.515 eV and hΓ = 0.1 meV.38

Experimental values are also used for the refractive indexof AlxGa1−xAs.39 The nonlinear susceptibility was keptthe same as for the NLO polymer (χ(2) = 300 pm/V)to allow for a direct comparison of the irradiances. Theabsolute value chosen has no effect on the enhancementfactor. In practice, the largest contribution to the back-

Angle of incidence (deg) 0 10 20 30 40 50 60 70 80

0.60

0.65

0.70

0.75

0.80

1.2

1.3

1.4

1.5

ћω3

(eV

)

50 meV

100 meV

EUP-ELP

EM3-ELP

LP

M1

0

2.05

4.10

6.15

8.20

10.3

FIG. 4. Angle dependence of DFG energy and irradiance(kW/m2) for TE polarized pumps incident on the structurewith NLO polymer and Ag mirrors when θ1 = θ2 = θi. Onlywaves with k3−z = k1z − k2z are considered. Lower and upperpanels show the DFG at the the Rabi and LP energies, re-spectively. Solid black lines illustrate the energies of the M1(bottom) and LP (top) modes where DFG radiation can beextracted in triple-resonance. Dashed black lines illustrate atypical linewidth of 100 meV for the LP branch and 50 meVfor the M1 mode. Solid white lines indicate the angle depen-dence of the DFG energy. For the upper panel, as the whiteline moves out of resonance with the black LP line, the DFGpeak is suppressed. For the lower one, a slight increase is ob-served around 57o and corresponds to an enhancement of thetriple-resonance condition, after which the irradiance rolls off.

ground χ(2) in GaAs is due to interband transitions andfor simplicity we ignore the resonant contribution to χ(2).

The enhancement in DFG irradiance as compared toa bare GaAs slab of equal thickness is shown in Fig. 5.Due to the much smaller oscillator strength in GaAs,as compared to the NLO polymer, the Rabi splitting ofhω3 = 5.52 meV falls in the THz range (ν3 = 1.33 THz)with an enhancement of 8.8 · 103.

Figure 6 shows the angle dependence of the DFG en-ergy and irradiance when θ1 = θ2 = θi. The dashed blackline in the upper panel traces the DFG energy, where alogarithmic scale for the irradiance was used due to itsrapid decrease with angle of incidence. The lower panelshows a segment of the same data on a linear scale. Tun-ability down to 3 dB can be obtained up to hω3 = 7.21meV (ν3 = 1.74 THz) at 17o.

IV. DISCUSSION

In Sec. III we showed that the use of polaritonic modesfor Rabi frequency generation can lead to irradiance en-hancements of almost four orders or magnitude with re-spect to bare nonlinear slabs. Quantitative estimatescan be obtained by considering equal pump irradiances

6

ћω2

(eV

)

1.5128

1.5125

1.5123

1.5120

1.5118

ћω1 (eV) 1.5173 1.5175 1.5178 1.5180 1.5183

5.52 meVћω3

9000

7500

6000

4500

3000

1500

0

1.5115

1.5130

1.5170 1.5185

UP

DFG

LP

FIG. 5. DFG enhancement of a λ/2 (111) GaAs cavitystructure with respect to a bare slab. GaAs parameters: εB =12.53, f = 1.325 · 10−3, hω0 = 1.515 eV and hΓ = 0.1 meV.38

The same value of χ(2) = 300 pm/V was used as for the NLOpolymer. Due to the presence of the substrate, only reflectedfields are considered. The tilted dashed line corresponds topairs of pump energies that generate the same DFG energy.Inset: normalized electric field profiles inside the GaAs layerillustrating the excellent modal overlap of the LP and UPbranches.

I1 = I2 = 10 GW/m2. Figure 7 shows the maximumDFG irradiances for the two structures and the referenceslabs. For the NLO film with Ag mirrors, the calcu-lated peak DFG irradiances are IDFG = 7.69 kW/m2 athω3 = 0.68 eV and IDFG = 4.05 kW/m2 at hω3 = 1.25eV. For the λ/2 (111) GaAs microcavity with DBRs, wefind IDFG = 45 W/m2 at hω3 = 5.52 meV.

For the organic microcavity, Fig. 7 shows that theirradiance due to DFG at the Rabi energy exceeds theone at the LP energy, as expected due to the higher modeoverlap. In Fig. 3, however, a higher enhancement wasfound at the LP energy. This apparent contradictionarises from normalizing each point by the correspondingDFG irradiances of the bare polymer slab.

There is also a substantial difference in cavity field en-hancement for both material sets. Metal losses in thepolymer cavity prevent a significant enhancement of theUP and LP electric fields with |Epeak/Ein| = 1.2, whereEpeak and Ein are the peak and incident fields, respec-tively. In contrast, for the GaAs microcavity an enhance-ment of 15 is obtained. Despite this field enhancement,the irradiance shown in Fig. 7 is 170 times lower at theRabi energy for the inorganic microcavity than for theorganic one. This is a consequence of the ω2

NL factor inthe source field given by Eq. (8), making DFG at smallerenergies increasingly difficult.

Finally, we can use Fig. 7 to evaluate the tunability ofthe structures at normal incidence. For the first struc-ture, the FWHM of the hω3 = 0.68 eV DFG peak is0.045 eV, indicating that the same structure can be used

ћω3

(meV

)

Angle of incidence (deg) 0 5 10 15 20 25 30

3

6

9

12

15

18 50

40

30

20

10

0

0 10 20 30 40 50 60 70 80

10

20

30

40

50

60 100

10-1

10-4

10-6

10-9

EUP-ELP

FIG. 6. Angle dependence of DFG energy and irradiance(W/m2) for TE polarized pumps incident on the λ/2 (111)GaAs structure with DBR mirrors when θ1 = θ2 = θi. Onlywaves with k3−z = k1z − k2z are considered. The upper panelshows the angle dependence of DFG irradiance in logarithmicscale, with the dashed black line tracing the DFG energy. Thelower panel shows a smaller angular range of the same datain linear scale where a fast decrease of DFG irradiance can beobserved as the angle of incidence increases.

for DFG generation from 1.76 µm to 1.88 µm by adjust-ment of the pumps only. For the GaAs structure, theFWHM of the hω3 = 5.52 meV DFG peak is 0.12 meV,indicating a tunability from ν3 = 1.32 THz to ν3 = 1.35THz.

We should note that although in our calculation twopumps were used, similar enhancements are anticipatedfor (spontaneous) parametric fluorescence (I2 = 0). Inaddition, the triply-resonant scheme introduced for theorganic microcavity where the signal is resonant has fur-ther consequences. First, coupled-mode theory analy-sis of triply-resonant systems has shown the existence ofcritical input powers to maximize nonlinear conversionefficiency.13,40 These are found to be inversely propor-tional to the product of the Q-factors. Lower Q-factorsare thus advantageous for high power applications. Sec-ond, the scheme is also well-suited for realizing a moreconventional χ(2) polariton OPO. In this case, the oscil-lation threshold can be shown to depend inversely on theproduct of Q-factors.

Since in general, any χ(2) medium will also have a non-zero χ(3), these structures will display a change in refrac-tive index proportional to the square of the applied elec-tric field, an effect known as self/cross-phase modulation.The power dependance of the refractive index can leadto rich dynamics such as multistability and limit-cyclesolutions.41,42

7

Irrad

ianc

e (W

/m2 )

ћω3 (eV)

10000

1000

100

10

0.1

0.01

0.001

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.84.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0×10-3

Ag

Slab

Slab

DBR

FIG. 7. Comparison of the calculated DFG irradiances forthe two structures studied. Solid blue (dash-dot red) linerepresents the NLO polymer (GaAs) cavity with Ag (DBR)mirrors and dotted lines directly below represent the corre-sponding bare slabs. Top blue (bottom red) energy scale re-lates to the NLO polymer (GaAs) cavity. The curves havebeen extracted from the maps shown in Fig. 3 and Fig. 5 bypicking out the maximum values among all pairs of pump en-ergies that generate the same DFG energy. Pump irradiancesare I1 = I2 = 10 GW/m2.

V. CONCLUSION

We studied the potential for generating Rabi-frequencyradiation in microcavities possessing a non-vanishingsecond-order susceptibility. Using a semiclassical modelbased on nonlinear transfer matrices in the undepletedpump regime, we calculated the Rabi splitting and theDFG irradiance enhancement for an organic microcavity,composed of a poled nonlinear optical polymer, and foran inorganic one, composed of GaAs. In the first case,we obtained a Rabi splitting of hω3 = 0.68 eV (λ3 = 1.82µm) and an enhancement of two orders of magnitude,as compared to a bare polymer film. In the second case,we found a Rabi splitting of hω3 = 5.52 meV (ν3 = 1.33THz) and an enhancement of almost four orders ofmagnitude, as compared to a bare GaAs slab. Theseresults show the potential of the use of polaritonic modesfor IR and THz generation. Both model structures dis-play a high degree of frequency tunability by changingthe wavelength and angle of incidence of the incomingpump beams. Similar enhancements are anticipated forparametric fluorescence and the triply-resonant schemeintroduced for the optical microcavity can be exploitedto realize monolithic χ(2) OPOs.

ACKNOWLEDGMENTS

FB and SKC acknowledge support from the NaturalSciences and Engineering Research Council of Canada.SDL acknowledges support from the Engineering andPhysical Sciences Research Council (EPSRC), researchgrant EP/L020335/1. SDL is Royal Society ResearchFellow.

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