pattern formation without diffraction matching in optical parametric oscillators with a metamaterial

$: Pattern formation without diffraction matching in optical parametric oscillators with a metamaterial$
Pattern formation without diffractionmatching in optical parametricoscillators with a metamaterial

Philippe Tassin,1 Guy Van der Sande,1 Irina Veretennicoff,1Pascal Kockaert,2 Mustapha Tlidi3

1Department of Applied Physics and Photonics; Vrije Universiteit Brussel (VUB);Pleinlaan 2; B-1050 Brussel; Belgium

2OPERA-photonics, CP 194/5; Universite Libre de Bruxelles (ULB);50, Av. F. D. Roosevelt; B-1050 Bruxelles; Belgium

3Optique non lineaire theorique, CP 231; Universite Libre de Bruxelles (ULB);Campus Plaine; B-1050 Bruxelles; Belgium

[email protected]

Abstract: We consider a degenerate optical parametric oscillator con-taining a left-handed material. We show that the inclusion of a left-handedmaterial layer allows for controlling the strength and sign of the diffractioncoefficient at either the pump or the signal frequency. Subsequently,we demonstrate the existence of stable dissipative structures withoutdiffraction matching, i.e., without the usual relationship between thediffraction coefficients of the signal and pump fields. Finally, we investigatethe size scaling of these light structures with decreasing diffraction strength.

© 2009 Optical Society of America

OCIS codes: (190.4420) Nonlinear optics, transverse effects in; (160.3918) Metamaterials.

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#109071 - $15.00 USD Received 23 Mar 2009; revised 14 May 2009; accepted 16 May 2009; published 21 May 2009

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1. Introduction

Frequency conversion by means of quadratic media in degenerate optical parametric oscilla-tors (DOPO) is a fundamental technique for the generation of tunable coherent radiation [1, 2].When used in broad area devices, the coupling between diffraction and nonlinearity can inducemodulational instability of the homogeneous output beam, leading to the formation of dissipa-tive periodic structures [3]. Besides spatially periodic structures, DOPOs also support localizedstructures. In the study of these structures, three different regimes may be distinguished de-pending on the magnitude of kM, the most unstable wavenumber at the onset of modulationalinstability. (i) If kM is finite, modulational instability appears subcritically and there exists a



pinning domain where localized structures are stable [4,5]. (ii) If kM is small (long-wavelengthregime), the modulational instability occurs close to the limit point associated with a domain ofbistability. The long-wavelength pattern formation process is altered and leads to the formationof localized structures [6]. (iii) If kM vanishes, the homogeneous steady states are modulation-ally stable. In this case, the stabilization of localized structures requires bistability between twohomogeneous steady states; the resulting structures are often called phase solitons or domainwalls [7–12]. The latter two types of spatial confinement of light [cases (ii) and (iii)] are gen-erated from noise and the late-time kinetics of their formation obeys a power law [6, 12]. Anoverview of this subject can be found in Refs. [13–17].

In a different subfield of optics, scientists have recently established metamaterials with neg-ative permittivity and permeability, often called left-handed materials (LHM) [18, 19]. Suchmaterials have been first fabricated at microwave frequencies [20, 21], but are now also real-ized in the optical domain [22, 23]. Metamaterials are shown to exhibit novel electromagneticphenomena [18, 24] and can be used in nonlinear optical devices [25–29]. In particular, theformation of dissipative structures in a Kerr resonator containing a LHM has been studied inRefs. [30–32], where it was shown that the addition of a layer of LHM strongly alters thespatiotemporal dynamics and provides the ability of diffraction management.

In this paper, we apply the technique of diffraction management to degenerate optical para-metric oscillators containing a LHM layer in their cavity. We show that this device can operatewithout diffraction matching, i.e., without the usual relationship between the diffraction coef-ficients of signal and pump fields. The technique allows to control the magnitude and the signof the diffraction coefficient of the signal and pump fields. This can be achieved by tuning thethickness of the left-handed layer and of the quadratic crystal. Finally, we present a study of thesize scaling of the emerging diffractive patterns.

2. Mean field model, diffraction management and steady-state solutions

We consider a ring cavity containing two material layers and driven by a coherent input beamat angular frequency ω (see Fig. 1). The first layer contains a material with a quadratic nonlin-earity, coupling the pump wave (ω) to the signal wave (ω/2). Both waves are phase matched inthis layer to ensure high conversion efficiency. The second layer consists of a LHM. We assumethat the LHM has a linear optical response, so that there is no need for phase matching in thislayer; the nonlinearities in the system are only due to the presence of the χ(2) medium. In thecurrently existing metamaterials, the left-handedness is limited to a very small frequency banddue to the resonant coupling with electromagnetic radiation. Therefore, only the case whereone of both optical fields experiences a negative index of refraction should be considered. Inthe remainder of this paper, we will take the index of refraction of the signal field negative(ns < 0) and the index of the pump field positive (np > 0).

χ(2)LHM

phase

matching

no phase

matching

pump ω:

signal ω/2:

input output

Fig. 1. Schematic setup of the degenerate optical parametric oscillator with a LHM.

Here, we consider type I parametric oscillation, where polarization effects are unimportant.



The propagation of light in the quadratic crystal can be described by the reduced Maxwell’sequations. Using the technique from Ref. [30], we have derived the following propagationequations for the pump and signal amplitudes As and Ap, which are valid in both layers (withχ(2) = 0 in the LHM, since we assume that this layer is linear):

∂As

∂ z+

ns

c∂As

∂ t=

iωχ(2)

cηsApA∗s + i

c2ωns

∇2⊥As, (1)

∂Ap

∂ z+

np

c∂Ap

∂ t= i

iωχ(2)

2cηpA2

s + ic

ωnp∇2⊥Ap. (2)

We thus find that the propagation equations keep the same form in LHMs, but that the differencebetween the index of refraction ns,p and the characteristic impedance ηs,p in such media mustbe carefully taken into account. Since ns < 0, Eqs. (1)-(2) show that the diffraction of the signalbeam acts with a negative sign in the LHM.

The signal wave propagates first through the quadratic crystal with positive index and thenthrough the left-handed layer with negative index. This means that the diffraction in both layerswill counteract and partly compensate. This property is reflected in the mean-field model thatwe have derived in earlier work from Eqs. (1)-(2) with appropriate boundary conditions at themirrors [32]:

∂As

∂ t=−(1+ i∆s)As +ApA∗s + iDs∇2

⊥As, (3)

∂Ap

∂ t= E− (

1+ i∆p)

Ap−A2s + iDp∇2

⊥Ap, (4)

where ∇2⊥ is the transverse Laplacian. ∆s,p are the normalized detunings between the wave and

the cavity mode, and the diffraction strengths are given by a weighted average over the layers:

Ds =cF2πω

(lQC

nQC+

lLHM

nLHM(ω/2)

), Dp =

cF4πω

(lQC

nQC+

lLHM

nLHM(ω)

), (5)

where F is the finesse of the cavity, lQC and lLHM are the lengths of the layers, and nQC andnLHM the indices of refraction. Eqs. (3)-(4) are obtained under the so-called mean-field approx-imations, which require (1) that reflections at the surfaces between the layers can be neglected;(2) that the dissipative Fresnel number is large; and (3) that the roundtrip length of the cavityis shorter than the diffraction, dispersion and nonlinearity space scales. Note that there is nophase matching in the LHM and, as a result, that there is no longer a fixed relationship betweenthe two diffraction coefficients, unlike traditional DOPOs where Dp = 2Ds. Finally, we want tomention that losses in the LHM can be accounted for in the mean-field model. In Ref. [32], itis shown that such losses naturally add to the cavity losses, and eventually result in a rescalingof the finesse of the cavity. By using a thin layer of LHM, the material losses will neverthelessbe negligible to the cavity losses.

The homogeneous steady-state solutions of Eqs. (3)-(4) are (i) the nonlasing solution As = 0,Ap = E/(1+ i∆p) and (ii) the lasing solutions

As =±eiφ

√−1+∆s∆p +

√|E|2− (

∆s +∆p)2

, (6)

Ap = eiθ√

1+∆2s , φ = − 1

2 atan(∆s + ∆p)/(E2− (∆s + ∆p)), and eiθ = (1 + i∆s)ei2φ /√

1+∆2s .

The lasing solution emerges from the nonlasing solution in a pitchfork bifurcation at the lasingthreshold ET = (1 + ∆2

p)1/2(1 + ∆2

s )1/2. Note that the left-handed layer in the ring cavity does

not modify the homogeneous steady states or the lasing threshold. However, as we will see inthe following section, their stability is strongly affected by the presence of the left-handed layer.



3. Linear stability analysis and size scaling

A linear stability analysis of the homogeneous steady states (i) and (ii) with respect to finitewavelength perturbations of the form exp(ikkk · rrr⊥−λ t), which are compatible with large Fres-nel number systems, shows that the modulational (Turing) instability occurs when one of theeigenvalues of the linear operator vanishes. This happens when

4 |As|2[|As|2 +Qs(1−Qp)+Qp

]+Qp(Qs−

∣∣Ap∣∣2) = 0, (7)

where Qs,p = Qs,p(k) = 1 +(∆s,p + Ds,pk2)2. Eq. (7) shows that the nonlasing solution (i) be-comes unstable with respect to the modulational instability in the range EM < E < ET, withEM = (1+∆2

p)1/2. At the modulational instability point (E = EM), the wavenumber is given byDsk2

M =−∆s. The unstable wavelength at the modulational instability threshold is therefore

ΛM = 2π√−Ds/∆s. (8)

Negative diffraction of the signal field thus allows for dissipative structures in cavities withpositive detuning ∆s > 0. This is surprising since the pattern formation instability arises onlyfor negative signal detuning in the absence of the LHM. To study analytically the stability ofthe lasing state (ii), let us assume that both signal and pump field are in perfect resonance, i.e.,∆s = ∆p. Since we do not have to satisfy the phase matching condition, let us suppose thatthe diffraction coefficients of both fields are the same, i.e., Ds = Dp. Under these assumptions,

the critical wavelength is k2c = −∆s/Ds−

√(2 |AsM|2−1−∆2

s )/(2(4 |AsM|2−1))/Ds and the

threshold associated with the modulation instability, |AsM|2, satisfies the following cubic equa-tion 64 |AsM|6 + 48 |AsM|4− 16(1 + ∆2

s ) |AsM|2 + (1 + ∆2s )2 = 0. This equation determines the

critical field amplitude EM at which the modulational instability takes place.An important motivation for adding a LHM to nonlinear optical devices is the size scal-

ing of the emerging dissipative structures made possible by the altered diffraction. It has beenshown that dissipative structures can be scaled down beyond the diffraction limit by reducingthe diffraction coefficient [30], even though other effects can impose a new size limit [33, 34].However, it is not straightforward that dissipative structures in an OPO would have similar scal-ing properties, since one cannot reduce the diffraction coefficients of the signal and the pumpboth at the same time. Indeed, most metamaterials are only left-handed in a narrow frequencyband. Therefore, we have studied numerically how the dissipative structures in the DOPO scalein size when the signal diffraction coefficient Ds is decreased (Dp is kept constant). We willtreat structures in two regimes: stripes below and phase solitons above the lasing threshold.

(a)

0.6 0.80.04

0.06

0.08

0.10

0.12

0.14

Λn

um

(a

.u.)

(b)

1.0 1.2 1.4

|D |1/2

s

Fig. 2. Dissipative structures due to modulational instability below threshold. Signaldiffraction strength is decreased from left to right: Ds = −2.0, Ds = −1.0, Ds = −0.5,and Ds =−0.3. Maxima are plain white and the mesh integration is 128×128.



For the stripes below the threshold, we have taken the parameters E = 2.5, ∆s = ∆p = 2.0,and Dp = 1.5. When Ds is decreased (in magnitude) from −2.0 to −0.3, we indeed observethat the wavelength of the stripes becomes smaller [see Fig. 2(a)]. This figure is obtained fromnumerical simulations of the mean field model [Eqs. (3)-(4)]. The initial condition consists of asmall amplitude noise added to the homogeneous steady state, and the boundary conditions areperiodic in both transverse directions. We have calculated the period of the stripes by countingthe number of stripes in the simulation area and we have plotted the result in Fig. 2(b). From thisfigure, one can observe that the scaling nicely follows the square root law predicted by Eq. (8).Further simulations have also shown that the pump diffraction Dp has almost no influenceon the size of the structures. This behavior can be understood from a perturbation analysisaround the onset of modulational instability, which was presented in Ref. [32]. The pump fieldis homogeneous up to first order, implying a vanishing pump diffraction term in Eq. (4).

(a)

Λn

um

(a

.u.)

(b)

0.6 0.8 1.0 1.2 1.4 1.6 1.8

14

12

10

8

6

4

2

|D |1/2

s

Fig. 3. Phase domains and solitons above the lasing threshold for decreasing signal diffrac-tion strength (from left to right: Ds = −3.0, Ds = −1.5, Ds = −1.0, Ds = −0.8, andDs =−0.5). Maxima are plain white and the mesh integration is 128×128.

A similar analysis was performed for phase solitons above the lasing threshold, with parame-ters E = 6.5, ∆s =−∆p = 2.0, and Dp = 1.5. Numerical simulations show labyrinthic structures[Fig. 3(a)]. By decreasing the signal diffraction strength, we observe the formation of domainsin the form of phase solitons as shown in Fig. 3(a). Measuring the spatial size of the domains,we find again a size scaling proportional to the square root of the signal diffraction coefficient[Fig. 3(b)]. That the size scaling is valid without reducing the pump diffraction coefficient canhere be explained by the phase indetermination of the two homogeneous lasing solutions; thephase solitons are made up of these two solutions. But since the pump field is actually the samefor both lasing solutions, the pump diffraction term in Eq. (4) does not play a major role. Thisexplains that the size can be scaled by adjusting only the signal diffraction coefficient.

4. Conclusion

We have studied a nonlinear resonator containing a LHM and a χ(2) crystal. The LHM providesthe ability to manage the diffraction of the signal field. We show that stable dissipative structureswithout diffraction matching (Dp 6= 2Ds) can exist for positive signal and pump detunings. Inthe absence of the LHM, these structures do not exist. The fact that our device can operatewithout diffraction matching also allows for the reduction of the wavelength of the emergingdissipative structures. Finally, we show that the size of the structures follows a square root lawwith respect to the signal diffraction coefficient.

Acknowledgments

The authors acknowledge financial suppport by the Belgian Science Policy Office under grantno. IAP-VI/10 (Photonics@be). P. T. is an Aspirant and G. V. is a Postdoctoral Researcher ofthe FWO-Vlaanderen. M. T. is a Research Associate of the FRS-FNRS.



pattern formation without diffraction matching in optical parametric oscillators with a metamaterial

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