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B148 J. Opt. Soc. Am. B/Vol. 26, No. 12 /December 2009 Kockaert et al.

Beyond the zero-diffraction regime in opticalcavities with a left-handed material

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

1OPERA-Photonics CP194/5, Université Libre de Bruxelles (U.L.B.), 50 Avenue F. D. Roosevelt,B-1050 Bruxelles, Belgium

2Department of Applied Physics and Photonics, Vrije Universiteit Brussel (V.U.B.), Pleinlaan 2,B-1050 Bruxelles, Belgium

3Optique Non linéaire Théorique, CP 231, Université Libre de Bruxelles (U.L.B.), Campus Plaine,B-1050 Bruxelles, Belgium

*Corresponding author: Pascal.Kockaert@ulb.ac.be

Received August 6, 2009; accepted September 27, 2009;posted October 22, 2009 (Doc. ID 115291); published November 30, 2009

The combination of right-handed and left-handed materials offers the possibility to design devices in which themean diffraction is zero. Such systems are encountered, for example, in nonlinear optical cavities, where a truezero-diffraction regime could lead to the formation of patterns with arbitrarily small sizes. In practice, theminimal size is limited by nonlocal terms in the equation of propagation. We study the nonlocal properties oflight propagation in a nonlinear optical cavity containing a right-handed and a left-handed material. We obtaina model for the propagation, including two sources of nonlocality: the spatial dispersion of the materials in thecavity, and the higher-order terms of the mean field approximation. We apply these results to a particular caseand derive an expression for the parameter fixing the minimal size of the patterns. © 2009 Optical Society ofAmerica

OCIS codes: 190.6135, 050.1940, 160.3918, 140.4780, 160.4330.

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2MCd

. INTRODUCTIONince the first realization of a left-handed metamaterial

n the radio-frequency domain [1,2], the work by Veselago3] on the optical properties of materials with negativeermittivity and permeability has attracted a lot of atten-ion and triggered substantial research activities. Mostfforts have concentrated on the fabrication of these ma-erials and the characterization of their linear properties4–10]. Left-handed materials allow phase compensationn optical structures combining left-handed and right-anded materials; this has led to, e.g., imaging systemsith subwavelength resolution [11], photonic devices go-

ng beyond the diffraction limit [12–14], and exotic appli-ations including invisibility cloaks and perfect opticaloncentrators [15–17].

A few authors have recently pointed out that left-anded metamaterials can have a dramatic impact on theonlinear propagation of light [18–22]. Systems involvingcombination of metamaterials and nonlinear propaga-

ion could be used to improve telecommunication systemsy lowering power thresholds [23] and increasing the dataensity [24] in optical memories.In particular, it was shown recently that the combina-

ion of nonlinear and left-handed materials in an opticalavity allows the existence of a zero-diffraction regime25]. Such a low-diffraction regime leads to the formationf sub-diffraction-limited dissipative structures [26,27].lthough it is predictable that some physical limitationust forbid data to be stored in infinitely small spatial

ells, as the writing in such cells would result in infiniteower densities, the model that was used to predict the

0740-3224/09/12B148-8/$15.00 © 2

ero-diffraction regime does not provide information onhese physical limitations. Nonlinear diffraction was al-eady suggested by Boardman et al. as a possible limita-ion [28,29].

In this paper, we derive two other possible limitations.e consider the limitation that will arise from the nonlo-

al response of the material and that from the inherentonlocal response of the optical feedback in the resonator.aking some assumptions on the symmetry properties of

he metamaterial, we derive a model of the optical reso-ator that includes higher-order diffraction and diffusionerms.

In Section 2, we briefly review how metamaterials withpatial dispersion can be modeled from a macroscopicoint of view, following the three-field approach of Agra-ovich [30,31]. In Section 3, we study the propagtion in aonlinear left-handed metamaterial in the framework ofhe slowly-varying-envelope approximation (SVEA). Aimplified model is derived for materials exhibiting inver-ion and rotation symmetry around the propagation di-ection. The nonlocal terms coming from the mean-fieldpproximation are introduced in Section 4 and studied inhe low-power regime, where it is shown that the ex-ended Lugiato–Lefever model, including a bi-Laplacianerm, is a generalized version of the model already stud-ed in [26].

. GENERAL RESPONSE OF AN OPTICALATERIAL

lassical textbooks describe the response of dielectric me-ia with two linear and local parameters: the electric and

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Kockaert et al. Vol. 26, No. 12 /December 2009 /J. Opt. Soc. Am. B B149

agnetic susceptibilities [32,33]. By considering the re-ulting model in the optical domain, the conclusion ariseshat the magnetic susceptibility must be equal to that in aacuum [34]. This last point is questionable when differ-nt laboratories are manufacturing materials with a mag-etic response in the optical domain [35].A partial answer to this apparent contradiction comes

rom the fact that the concepts of the displacement fieldnd the magnetic field are valid in materials whereharges are either free or bound to an atom. By movingith respect to the atom, the electron cloud will induce lo-

ally a dipole that accounts for the difference between thelectric field and the displacement field. The response ofhe material can therefore be assumed to be local, and theescription of classical textbooks is valid.The nonlocal nature of the material response is also

alled “spatial dispersion.” Its description requires a moreeneral model than the E, D, B, H approach [36]. The in-bility of the EDBH model to describe some anisotropicedia was already pointed out a few years ago [30] and

as been repeated since then [37]. Very recently, newathematical tools and methods to describe the macro-

copic properties of metamaterials have been proposed38,39]. These methods are in agreement with the sugges-ion of [36] that a single susceptibility would better de-cribe the electromagnetic properties of electromagneticedia.Therefore, in this paper, we consider that the material

esponse is given solely by the relation between the polar-zation vector P and the electric field E:

P�E� = �0��1��

t,r

E + PNL�E�, �1�

here the convolution product � applies to the time t andhe spatial coordinate r, ��1� accounts for the linear com-onent, and the nonlinear response PNL�E� will be de-ailed below. Because spatial dispersion is introduced in, we can set the magnetic response of the medium M0, or equivalently B=�0H [30].

. NONLOCAL EFFECTS IN THEROPAGATION EQUATIONhe Maxwell’s equations for the macroscopic fields E, B,nd P in a medium without applied currents can be recastnto the generalized wave equation

curl curl E +1

c02

�2

�t2 ���1�� E� + �0

�2

�t2PNL = 0 �2�

r, equivalently,

�E −1

c02��1�

�

�2E

�t2 = grad�div E� + �0

�2PNL

�t2 , �3�

ith � the three-dimensional Laplacian operator.In the case of a local response in a centrosymmetric me-

ium, the SVEA can be applied to transform Eq. (3) into aonlinear Schrödinger equation (NLSE) with a cubic non-

inearity of the form �E�2E [40]. As our model is more gen-ral than the one contained in the cubic NLSE, we can ex-

ect that the SVEA will lead to a propagation equationith additional terms that should disappear when spatialispersion is suppressed.

. Complex Equationhe wave equation (3) is valid for real quantities. Intro-ucing a carrier wave,

E�t,r� = a�t,r�cos��0t − k0r + ��t,r��, �4�

ith a and � real functions, and the parameters �0 and0 to be fixed later, we rewrite the left-hand side of Eq. (3)

n the form

L =1

2��ae+i��0t−k0r+��� −

1

2

��1�

c02 �

�2

�t2 �ae+i��0t−k0r+��� + c.c.

=1

2LWE + c.c., �5�

ith c.c. denoting the complex conjugate, and the obviouselation L=Re�LWE�. A last definition of A=aei� allows toewrite the expression as

LWE = ��Aei��0t−k0r�� −��1�

c02 �

�2

�t2 �Aei��0t−k0r��. �6�

t this stage, the parameters of the carrier wave are notxed. The particular case of k0=�0=0 corresponds to EA. This shows clearly that the introduction of an enve-

ope does not restrain the possible expressions for E. Theinear properties of the material sustaining the propaga-ion are completely described by ��1��t ,r�.

. Equation for the Envelopentil now, we introduced no hypothesis about the varia-

ions of A�t ,r�, and we will pursue our calculation as fars possible without introducing further assumptions. Inhe last steps of our computation, we will assume that theariations of the envelope are slow with respect to theariations in the exponential factor:

��Aei��0t−k0r�� = ei�0t��Ae−ik0r�, �7�

��Ae−ik0r� = ��e−ik0r�A + 2��e−ik0r � �A + e−ik0r�A

= ��e−ik0r�A − 2ie−ik0r�k0 � �A + e−ik0r�A

= �− k02A − 2i�k0 � �A + �A�e−ik0r. �8�

imilarly, we find

�2

�t2 �Aei�0t� = �− �02A + 2i�0

�A

�t+

�2A

�t2 �ei�0t. �9�

In order to simplify the expressions of the convolutionroduct appearing in Eq. (6), we introduce

D�0=

1

c02�− �0

2 + 2i�0

�

�t+

�2

�t2� , �10�

nd we denote the integration over space and time withrackets,

Iw

w

CNt

wsncu

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−tmf�=fi

o

wt

f

I

T

t

B150 J. Opt. Soc. Am. B/Vol. 26, No. 12 /December 2009 Kockaert et al.

¯ =�� �E�

−�

�

. . . dt�d3r�. �11�

n this notation, the convolution product of Eq. (6) can beritten as

��1�� ��D�0

A�ei��0t−k0r��

= ��1��t�,r���D�0A�ei��0�t−t��−k0�r−r���

= ei��0t−k0r� � ��1��t�,r��e−i��0t�−k0r���D�0A�

= ei��0t−k0r����1��t,r�e−i��0t−k0r�� � �D�0A�. �12�

This leads to the introduction of

�0�1��t,r� = ��1��t,r�e−i��0t−k0r�, �13�

hich allows us to rewrite Eq. (6) in the form

ei�k0r−�0t��LWE� = �− k02A − 2i�k0 � �A + �A� − �0

�1�� �D�0

A�.

�14�

. Weak Temporal and Spatial Dispersiononlocal and dispersive effects are contained in the last

erm of Eq. (14):

�0�1�

� �D�0A� = �0

�1��t,r� � ��−�0

2

c02 + 2i

�0

c02

�

�t+

1

c02

�2

�t2��t,r

A

= �−�0

2

c02 + 2i

�0

c02

�

�t+

1

c02

�2

�t2���0�1�

� A�

= D�0��0

�1�� A�. �15�

The convolution product can conveniently be written as

�0�1�

� A = �0�1��t�,r��A�t − t�,r − r��. �16�

To go further, we must assume that the medium iseakly nonlocal and dispersive. This means that the re-

ponse of the material will not last for a very long time,or extend for a very broad volume. The integration in theonvolution product can therefore be limited to small val-es of t� and r�.As a first illustration, we consider the limiting case

here nonlocality and dispersion disappear. In this par-icular case, the response of the medium is ��1��t ,r��t ,r�. As the convolution by a function is the identityperation, it is sufficient to know A�t ,r� in order to com-ute the convolution integral in �t ,r�.In a more general case, the expression of A�t− t� ,r

r�� will be expanded in a Taylor series. For our purpose,his series can be limited to the second order. It wouldake no sense to compare t� and r�, as their units are dif-

erent. Therefore, we replace the time coordinate t with=c0t. To keep the notation simple, we introduce �0�0 /c0, which will play the role of a frequency, and we de-ne U�� ,r�=A�t ,r�, and X0�� ,r�=�0

�1��t ,r�. Therefore

¯ =1

c0�� �

E�

−�

�

. . . d��d3r�, �17�

D�0= − �0

2 + 2i�0

�

��+

�2

��2 = �i�0 +�

���2

. �18�

The first-order Taylor expansion of Eq. (16) is

�0�1�

� A = X0���,r��U�� − ��,r − r��

= X0���,r��U��,r� − ��X0���,r��� �

���

��,r�

U�− �r�X0���,r��� � ���,r��U + O�2�. �19�

Before going further, we simplify the zeroth- and first-rder terms as follows:

X0���,r��U��,r� = X0U��,r�, �20�

��X0���,r���U

�� � = ��X0�U

��, �21�

�r�X0���,r�� � �U = x�X0�U

�x+ y�X0

�U

�y+ z�X0

�U

�z,

�22�

here we made the assumption that the susceptibility ofhe material is scalar, which is quite a strong assumption.

The particular form of X0 allows us to compute the dif-erent moments quite easily. We have

��X0 = ��Xe−i��0��−k0r�� = �i�

��0�Xe−i��0��−k0r���

= i�

��0X0 = ic0

�

��0�0

�1�. �23�

n a similar way, we find

r�X0 = − i�k,0X0 = − i�k,0�0�1�. �24�

he mean value X0 is straightforward to compute:

X0 =1

c0F�X�0 =

X̃0

c0= F���1��0 = �̃0

�1� = n2��0,k0� − 1.

�25�

Now, we develop the convolution product in Eq. (16) upo the second order:

X0���,r��U�� − ��,r − r��

= X0U − ��X0�U

��− �r�X0 � �U +

1

2��2X0

�2U

��2

+1

2x�2X0

�2U

�x2 +1

2y�2X0

�2U

�y2 +1

2z�2X0

�2U

�z2

+ x�y�X0�2U

�x�y+ x�z�X0

�2U

�x�z+ y�z�X0

�2U

�y�z

+ ��x�X0�2U

���x+ ��y�X0

�2U

���y+ ��z�X0

�2U

���z+ O�3�.

�26�

Taeaobtt

DF(tteaaplekgh

aa

w

Ftt

seE

f

or

fi

ETneiasoD

FsNcw

Kockaert et al. Vol. 26, No. 12 /December 2009 /J. Opt. Soc. Am. B B151

his expression contains information about the temporalnd spatial dispersive properties of the material. Thevaluation of these terms would require more informationbout the medium sustaining the propagation. In the casef a metamaterial, the methods presented in [37,38] coulde applied in order to calculate the macroscopic quanti-ies, on the basis of the microscopic subwavelength struc-ure.

. Symmetriesor symmetry reasons, some coefficients appearing in Eq.

26) might be related, or zero. If the material is supposedo be isotropic, then ��1��t ,r� must be invariant for any ro-ation, as well as for inversion. If the material admits, forxample, cubic symmetry, ��1��t ,r� should be invariant forny 90° rotation around a principal axis, for inversion,nd therefore, for reflection with respect to a principallane. By rewriting the electric field as an envelope modu-ated by a carrier wave, we have selected a particular ori-ntation, namely, k0. If the material is not isotropic, but0 corresponds to some principal direction in the propa-ation medium, then some symmetry properties can stillelp to simplify Eq. (26).Below, we assume that k0 is aligned in the z direction

nd that ��1� admits inversion and reflection symmetriesround the axis of the orthogonal basis �x ,y ,z�. Therefore,

k0 = k01z, �27�

��1���,x,y,z� = ��1��t,− x,y,z� = ��1��t,x,− y,z� = ��1��t,x,y,− z�,

�28�

�0�1��t,x,y,z� = ��1��t,x,y,z�e−i��0t−k0z� = �0

�1��t,− x,y,z�

= �0�1��t,x,− y,z� = �0

�1��t,x,y,− z�e2ik0z, �29�

hich implies similar relations for X0; hence

r�X0 = z�X01z, �30�

x�y�X0 = − x�y�X0 = 0, �31�

x���X0 = − x���X0 = 0. �32�

inally, we rewrite the convolution product of Eq. (26) inhe particular case where the properties along the x andhe y directions are the same, so that x�2X0= y�2X0, and

X0���,r��U�� − ��,r − r��

= X0U − ��X0�U

��+

1

2��2X0

�2U

��2 − z�X0�U

�z

+1

2z�2X0

�2U

�z2 +1

2x�2X0��U + O�3�. �33�

The last step before explicitly obtaining the left-handide of wave equation (6) is to apply the differential op-rator of Eq. (18) to the simplified convolution product ofq. (33).The result contains 18 terms that we order into 4 dif-

erent categories: an algebraic term A, a term containing

nly time derivatives T, a term containing only spatial de-ivatives S, and a mixed term M:

D�0�0

�1�� A = A + T + S + M. �34�

Using Eqs. (23)–(25), and defining 0=�0n��0 ,k0�, wend

A = − 02A, �35�

T = i� 0

2

��0

�A

�t+

1

2

�2 02

��02

�2A

�t2 −i

c02

�

��0��0

�n2

��0� �3A

�t3

−1

2c02

�2n2

��02

�4A

�t4 , �36�

S = − i� 0

2

�kz

�A

�z+

1

2

�2 02

�kz2

�2A

�z2 +1

2

�2 02

�kx2 ��A,

M = − i1

�0

�2 02

�kx2 ��

�A

�t−

1

2�02

�2 02

�kx2 ��

�2A

�t2 −2

�0

� 02

�kz

�2A

�t�z

+ i1

�02

� 02

�kz

�3A

�t2�z− i

1

�0

�2 02

�kz2

�3A

�t�z2 −1

2�02

�2 02

�kz2

�4A

�t2�z2 .

�37�

. Slowly Variable Envelopehe Taylor expansion in Eq. (26) was valid for a weaklyonlocal medium. We should notice that this expansion isqually valid if we assume slow variations of the envelopen space and time, together with important nonlocalitynd dispersion. But, if we make this last assumption, theecond term in Eq. (18) is one order smaller than the firstne. With this in mind, the second-order expansion of

�0�0

�1�� A reduces to

A = − 02A, �38�

T = i� 0

2

��0

�A

�t+

1

2

�2 02

��02

�2A

�t2 , �39�

S = − i� 0

2

�kz

�A

�z+

1

2

�2 02

�kz2

�2A

�z2 +1

2

�2 02

�kx2 ��A, �40�

M = −2

�0

� 02

�kz

�2A

�t�z. �41�

urther simplification is obtained if we limit these expan-ions to the first order. In this case, we find the classicalLSE. To get more insight into the nonlocal effects, we fo-

us on the spatial second-order term. Therefore, we re-rite Eq. (6) as

Ii

�

FFsofi

npt

wfIdTmm

pSt

4MTcdh

Bmtr

ptit

dta

we

wf

tl

atwri

w

t

Fap

B152 J. Opt. Soc. Am. B/Vol. 26, No. 12 /December 2009 Kockaert et al.

�LWE�e−i��0t−k0z� + T + M

= � 02 − k0

2�A − 2ik0�1 − 0

k0

� 0

�kz� �A

�z+ �1 −

�2 02

�kz2� �2A

�z2

+ �1 −�2 0

2

�kx2���A. �42�

t is now time to fix the value of k0. To avoid quickly vary-ng phase factors, the obvious choice is k0= 0.

Furthermore, the classical SVEA implies ��A /�z���2A /�z2�, so that the only remaining spatial terms are

�LWE�e−i��0t−k0z� + T + M

= − 2ik0�1 − 0

k0

� 0

�kz� �A

�z+ �1 −

�2 02

�kx2���A. �43�

. Commentsrom the result obtained in Eq. (43), it is seen that in thetationary case, nonlocal terms do not change the naturef the propagation equation, but slightly modify the coef-cients.This is probably the reason why nonlocal effects can be

eglected to a certain extent, i.e., as far as the spatial dis-ersion presents sufficient symmetries, with respect tohe direction of propagation.

Now we turn back to our initial aim in this paper,hich is to determine the physical limitations that will

orbid writing information in infinitely small spatial cells.t appears that for small values of the spatialispersion—for which a second-order expansion in theaylor series fairly reflects the nonlocal behavior of theedium—the nature of the propagation equation is notodified by the nonlocality.If all the above mentioned symmetries hold, and the in-

ut beam is continuous (T=0, M=0), in the frame of theVEA, a new term in the propagation equation appears athe fourth order in the Taylor expansion:

S4 = −1

4!

�4 22

�kx4 ��

2 A. �44�

. NONLOCAL EFFECTS IN THEEAN-FIELD MODEL

he zero-diffraction regime can be obtained in differentonfigurations [25,41]. We consider here the case of two-imensional zero diffraction in a cavity filled with a left-anded and a right-handed material, as studied in [25].In Fig. 1, we present the geometry under consideration.

y adjusting the relative thickness (lL and lR) of the twoaterials (LHM and RHM), it is possible to cancel diffrac-

ion at each round trip, which leads to the zero-diffractionegime in two dimensions.

The mean-field model is explicitly obtained in the ap-endix of [25]. In particular, the propagation in each ofhe two materials is described with the help of a general-zed NLSE, where care is taken to avoid the confusion be-ween the refractive index and the impedance of the me-

ium. In the mean-field approximation, the variation ofhe field over one round trip in the cavity is very small. Asresult, the propagation equation of the form

�A

��= i�D + N�A, �45�

here D is a linear differential operator and N a nonlin-ar operator, i.e.,

D =1

2k0��

2 − 2

2

�2

�t2 , �46�

N =3�0

2c�0�0

�3���A�2 + ��B�2�A, �47�

here � is the nonlinear coupling coefficient between theorward �A� and the backward �B� waves.

In [25], this equation is integrated with the assumptionhat �=O�1�, so that a first-order integration of Eq. (45)eads to

A�� + �� = A��� + �i��D + N��� A + O�2�

= e�i��D + N��� A + O�2� = e�i�D�� e�i�N�� A + O�2�

= e�i�N��e�i�D�� A + O�2�. �48�

Here, we are concerned with the nonlocality that willpppear in the model. A first physical understanding ofhe nonlocal terms is obtained in the continuous regime,here the differential term D contains only spatial de-

ivatives. The fourth-order term S4 from Eq. (44) is takennto account so that

D =1

2k0��1 −

1

2

�2 22

�kx2��� +

1

4!

�4 22

�kx4 ��

2 � �1

2k0�1 + ������,

�49�

ith

� =1

4!

�4 22

�kx4 . �50�

Expanding the propagation operator over one roundrip up to the second order, we get

RHM LHM

�lR 0 lL

Ein

z

ig. 1. (Color online) Cavity containing right-handed (RHM)nd left-handed (LHM) materials. The cavity is driven by the in-ut field Ei.

AWlwa

Eo

id

wsttc

fp

at

Ltto

fiiati

ittw

BTwipdsLcLb

5Inmstt

chtafL

qtd

dp

Kockaert et al. Vol. 26, No. 12 /December 2009 /J. Opt. Soc. Am. B B153

A�� + �� = ei��D + N��A = 1 + i�D + N�� +i2

2�D + N��

2 + O�3�

= 1 + i�D + N�� −1

2D2�2 −

1

2N2�2

−1

2�DN + ND���2 + O�3�.

. Low-Power Regimee require that our model be valid at low power. In this

imit, the diffraction term can be considered of first order,hile the nonlinear term is of second order, i.e., D=O�1�nd N=O�2�. This assumption leads to

A�� + �� = 1 + i�D + N�� −1

2D2�2 + O�3�.

Together with the additional considerations of [25], andq. (49), this result provides the nonlocal generalizationf the Lugiato–Lefever equation:

�A

���= Ein − �1 + i��A + i��

2 A + �i� −1

22���

4 A + i��A�2A,

�51�

n which we assumed a continuous pump beam and intro-uce the reduced quantities as in the appendix of [25]:

F = 2����

1 − ���, �52�

�� =�2�

FT�, �53�

Ein =F2�

Ein, �54�

� = −F2�

�, �55�

=F4�

� lL

kL+

lR

kR� , �56�

� =F4�

� lL�L

kL+

lR�R

kR� , �57�

� =F2�

�lL�L + lR�R�, �58�

here F denotes de finesse of the cavity, �� is the dimen-ionless time scale, Ein is the reduced driving field, � ishe detuning, is the effective diffraction coefficient, � ishe effective coefficient of spatial dispersion, and � ac-ounts for the effective Kerr-type nonlinearity.

It is important to notice that the nonlocal term resultsrom the contribution of two quantities with very differenthysical meanings: the spatial dispersion in the medium,

nd the second-order term in the mean-field approxima-ion of the intracavity diffraction.

In particular, we observe that the real part of the bi-aplacian term will asymptotically evolve to zero when

he diffraction is reduced, while the imaginary contribu-ion will dominate at this time and prevent the formationf infinitely small features, as shown in [26].

The zero-diffraction regime is reached when , as de-ned in Eq. (56), equals zero. Because kL�0, this regime

s obtained by tuning the relative thickness of the left-nd the right-handed media, i.e., lL and lR. If the struc-ure of the right-handed medium is of atomic size, then its safe to set �R=0, so that

� = −F4�

lL

�kL��L. �59�

The sign of this parameter is of paramount importancen order to define the minimal size of the nonlinear pat-erns that can form in the cavity [26]. More precisely, inhe vicinity of =0, the parameter defining the dynamicsas shown to be

�

=

�L

1 − �kLlR

kRlL� . �60�

. Local Mediumhe mean field equation (51) takes a particular formhen the spatial dispersion of the material is negligible,

.e., when �=0. In this case, it is seen that both the La-lacian and the bi-Laplacian terms disappear in the zero-iffraction regime, where =0. However, a simple analy-is shows that close to this zero-diffraction regime, the bi-aplacian term stabilizes the system. In particular, asould be expected from a physical point of view, the bi-aplacian diffusion term induces a shrinking of the insta-ility domains.

. CONCLUSIONSn this paper, we obtained a clear picture of the origin ofonlocal effects in cavities containing left-handedetamaterials. We studied separately the spatial disper-

ion in the material itself and the higher-order diffractionerms coming from the mean-field approximation leadingo a slow-time description of the cavity dynamics.

The mean-field model was applied in the particularase of a cavity filled with both a left-handed and a right-anded material. Some assumptions on the symmetries ofhe materials together with the slowly-variable envelopepproximation allowed for a simple description, in theorm of a Lugiato–Lefever equation with an additional bi-aplacian term with complex coefficient.The model that we obtain provides an answer to the

uestion of the physical limitations on the pattern forma-ion beyond the first-order approximation of a zero-iffraction regime.As the detailed derivation of our model was provided,

ifferent cases of practical interest could be studied by ap-lying the right assumptions on the symmetries of the

mw

pos

sdfi

APttgSoVFTA

R

1

1

1

1

1

1

1

1

1

1

2

2

2

2

2

2

2

2

2

2

3

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B154 J. Opt. Soc. Am. B/Vol. 26, No. 12 /December 2009 Kockaert et al.

aterials, or by computing their effective permittivityith the use of recent homogenization techniques [38,39].In the example that we studied, we calculated the ex-

ression of the parameter � /, appearing in the limit casef [26], where the authors predict the minimal size of dis-ipative structures in such cavities.

The model derived here above could be studied in aimilar way in other limiting cases in order to reveal theifferent possible dynamics in such cavities and also tond the best design parameters for a given application.

CKNOWLEDGMENTS. Kockaert thanks Gregory Kozyreff for introducing himo some homogenization techniques. This work was par-ially funded by the Belgian Science Policy Office, underrant IAP6-10, “Photonics@be.” P. Tassin and G. Van derande are, respectively, a PhD and a Postdoctoral Fellowf the Fonds voor Wetenschappelijk Onderzoek-laanderen (FWO, Belgium). M. Tlidi is a fellow of theonds de la Recherche Scientifique (FRS, Belgium). P.assin acknowledges financial support from the Belgianmerican Educational Foundation.

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