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Kockaert et al.
Beyond the zero-diffraction regime in optical cavities with a left-handed material Pascal Kockaert,1,* Philippe Tassin,2 Irina Veretennicoff,2 Guy Van der Sande,2 and Mustapha Tlidi3 1
OPERA-Photonics CP194/5, Université Libre de Bruxelles (U.L.B.), 50 Avenue F. D. Roosevelt, B-1050 Bruxelles, Belgium 2 Department of Applied Physics and Photonics, Vrije Universiteit Brussel (V.U.B.), Pleinlaan 2, B-1050 Bruxelles, Belgium 3 Optique Non linéaire Théorique, CP 231, Université Libre de Bruxelles (U.L.B.), Campus Plaine, B-1050 Bruxelles, Belgium *Corresponding author:
[email protected] 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 the mean diffraction is zero. Such systems are encountered, for example, in nonlinear optical cavities, where a true zero-diffraction regime could lead to the formation of patterns with arbitrarily small sizes. In practice, the minimal size is limited by nonlocal terms in the equation of propagation. We study the nonlocal properties of light propagation in a nonlinear optical cavity containing a right-handed and a left-handed material. We obtain a model for the propagation, including two sources of nonlocality: the spatial dispersion of the materials in the cavity, and the higher-order terms of the mean field approximation. We apply these results to a particular case and derive an expression for the parameter fixing the minimal size of the patterns. © 2009 Optical Society of America OCIS codes: 190.6135, 050.1940, 160.3918, 140.4780, 160.4330.
1. INTRODUCTION Since the first realization of a left-handed metamaterial in the radio-frequency domain [1,2], the work by Veselago [3] on the optical properties of materials with negative permittivity and permeability has attracted a lot of attention and triggered substantial research activities. Most efforts have concentrated on the fabrication of these materials and the characterization of their linear properties [4–10]. Left-handed materials allow phase compensation in optical structures combining left-handed and righthanded materials; this has led to, e.g., imaging systems with subwavelength resolution [11], photonic devices going beyond the diffraction limit [12–14], and exotic applications including invisibility cloaks and perfect optical concentrators [15–17]. A few authors have recently pointed out that lefthanded metamaterials can have a dramatic impact on the nonlinear propagation of light [18–22]. Systems involving a combination of metamaterials and nonlinear propagation could be used to improve telecommunication systems by lowering power thresholds [23] and increasing the data density [24] in optical memories. In particular, it was shown recently that the combination of nonlinear and left-handed materials in an optical cavity allows the existence of a zero-diffraction regime [25]. Such a low-diffraction regime leads to the formation of sub-diffraction-limited dissipative structures [26,27]. Although it is predictable that some physical limitation must forbid data to be stored in infinitely small spatial cells, as the writing in such cells would result in infinite power densities, the model that was used to predict the 0740-3224/09/12B148-8/$15.00
zero-diffraction regime does not provide information on these physical limitations. Nonlinear diffraction was already suggested by Boardman et al. as a possible limitation [28,29]. In this paper, we derive two other possible limitations. We consider the limitation that will arise from the nonlocal response of the material and that from the inherent nonlocal response of the optical feedback in the resonator. Making some assumptions on the symmetry properties of the metamaterial, we derive a model of the optical resonator that includes higher-order diffraction and diffusion terms. In Section 2, we briefly review how metamaterials with spatial dispersion can be modeled from a macroscopic point of view, following the three-field approach of Agranovich [30,31]. In Section 3, we study the propagtion in a nonlinear left-handed metamaterial in the framework of the slowly-varying-envelope approximation (SVEA). A simplified model is derived for materials exhibiting inversion and rotation symmetry around the propagation direction. The nonlocal terms coming from the mean-field approximation are introduced in Section 4 and studied in the low-power regime, where it is shown that the extended Lugiato–Lefever model, including a bi-Laplacian term, is a generalized version of the model already studied in [26].
2. GENERAL RESPONSE OF AN OPTICAL MATERIAL Classical textbooks describe the response of dielectric media with two linear and local parameters: the electric and © 2009 Optical Society of America
Kockaert et al.
Vol. 26, No. 12 / December 2009 / J. Opt. Soc. Am. B
magnetic susceptibilities [32,33]. By considering the resulting model in the optical domain, the conclusion arises that the magnetic susceptibility must be equal to that in a vacuum [34]. This last point is questionable when different laboratories are manufacturing materials with a magnetic response in the optical domain [35]. A partial answer to this apparent contradiction comes from the fact that the concepts of the displacement field and the magnetic field are valid in materials where charges are either free or bound to an atom. By moving with respect to the atom, the electron cloud will induce locally a dipole that accounts for the difference between the electric field and the displacement field. The response of the material can therefore be assumed to be local, and the description of classical textbooks is valid. The nonlocal nature of the material response is also called “spatial dispersion.” Its description requires a more general model than the E, D, B, H approach [36]. The inability of the EDBH model to describe some anisotropic media was already pointed out a few years ago [30] and has been repeated since then [37]. Very recently, new mathematical tools and methods to describe the macroscopic properties of metamaterials have been proposed [38,39]. These methods are in agreement with the suggestion of [36] that a single susceptibility would better describe the electromagnetic properties of electromagnetic media. Therefore, in this paper, we consider that the material response is given solely by the relation between the polarization vector P and the electric field E: t,r
P关E兴 = ⑀0共1兲 丢 E + PNL关E兴,
共1兲
where the convolution product 丢 applies to the time t and the spatial coordinate r, 共1兲 accounts for the linear component, and the nonlinear response PNL关E兴 will be detailed below. Because spatial dispersion is introduced in P, we can set the magnetic response of the medium M ⬅ 0, or equivalently B = 0H [30].
pect that the SVEA will lead to a propagation equation with additional terms that should disappear when spatial dispersion is suppressed. A. Complex Equation The wave equation (3) is valid for real quantities. Introducing a carrier wave, E共t,r兲 = a共t,r兲cos共0t − k0r + 共t,r兲兲,
1
L = ⌬关ae 2
1 2
2
c02 t
t
共共1兲 丢 E兲 + 0 2
P =0 2 NL
⌬E −
c02
E
共1兲 丢
t2
丢
t2
关ae+i共0t−k0r+兲兴 + c.c.
with c.c. denoting the complex conjugate, and the obvious relation L = Re关LWE兴. A last definition of A = aei allows to rewrite the expression as LWE = ⌬关Ae
i共0t−k0r兲
兴−
共1兲 c02
丢
2 t2
关Aei共0t−k0r兲兴.
PNL t2
B. Equation for the Envelope Until now, we introduced no hypothesis about the variations of A共t , r兲, and we will pursue our calculation as far as possible without introducing further assumptions. In the last steps of our computation, we will assume that the variations of the envelope are slow with respect to the variations in the exponential factor:
共3兲
with ⌬ the three-dimensional Laplacian operator. In the case of a local response in a centrosymmetric medium, the SVEA can be applied to transform Eq. (3) into a nonlinear Schrödinger equation (NLSE) with a cubic nonlinearity of the form 兩E兩2E [40]. As our model is more general than the one contained in the cubic NLSE, we can ex-
共7兲
⌬关Ae−ik0r兴 = ⌬共e−ik0r兲A + 2共ⵜe−ik0r ⵜ 兲A + e−ik0r⌬A = ⌬共e−ik0r兲A − 2ie−ik0r共k0 ⵜ 兲A + e−ik0r⌬A = 关− k02A − 2i共k0 ⵜ 兲A + ⌬A兴e−ik0r .
t ,
共6兲
At this stage, the parameters of the carrier wave are not fixed. The particular case of k0 = 0 = 0 corresponds to E = A. This shows clearly that the introduction of an envelope does not restrain the possible expressions for E. The linear properties of the material sustaining the propagation are completely described by 共1兲共t , r兲.
Similarly, we find
2
= grad共div E兲 + 0
2 c02
2
共5兲
2 2
兴−
1 共1兲
1 = LWE + c.c., 2
共2兲
or, equivalently, 1
+i共0t−k0r+兲
⌬关Aei共0t−k0r兲兴 = ei0t⌬关Ae−ik0r兴,
The Maxwell’s equations for the macroscopic fields E, B, and P in a medium without applied currents can be recast into the generalized wave equation
共4兲
with a and real functions, and the parameters 0 and k0 to be fixed later, we rewrite the left-hand side of Eq. (3) in the form
3. NONLOCAL EFFECTS IN THE PROPAGATION EQUATION
curl curl E +
B149
冉
关Aei0t兴 = − 02A + 2i0 2
A t
+
2A t2
冊
e i0t .
共8兲
共9兲
In order to simplify the expressions of the convolution product appearing in Eq. (6), we introduce D 0 =
1 c02
冉
− 02 + 2i0
t
+
2 t2
冊
,
共10兲
and we denote the integration over space and time with brackets,
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具¯典=
冕冕冕 冕 E
Kockaert et al.
⬁
. . . dt⬘d3r⬘ .
D0 = − 02 + 2i0
共11兲
−⬁
In this notation, the convolution product of Eq. (6) can be written as
= 具 共t⬘,r⬘兲共D0A兲e
i关0共t−t⬘兲−k0共r−r⬘兲兴
= 具X0共⬘,r⬘兲U共,r兲典 −
典
= ei共0t−k0r兲共共1兲共t,r兲e−i共0t−k0r兲兲 丢 共D0A兲.
= 共t,r兲e
共12兲
冓
共13兲
,
共14兲 C. Weak Temporal and Spatial Dispersion Nonlocal and dispersive effects are contained in the last term of Eq. (14): 共D0A兲 =
冉
= − =
02 c02
冏冉
+ 2i
D0共0共1兲 丢
−
0 c02 t
02
+ 2i
c02 +
1
0 c02
2
c02 t2
冊
t
+
1 2 c02 t2
共0共1兲 丢
冊冏
A
0共1兲 丢 A = 具0共1兲共t⬘,r⬘兲A共t − t⬘,r − r⬘兲典.
c0
冕冕冕 冕 E
. . . d ⬘d 3r ⬘ ,
U x
⬘X0共⬘,r⬘兲
冏冏
U 共,r兲
冔 共19兲
U
+ 具y⬘X0典
U y
共21兲
,
+ 具z⬘X0典
U z
,
where we made the assumption that the susceptibility of the material is scalar, which is quite a strong assumption. The particular form of X0 allows us to compute the different moments quite easily. We have
=i
冓冉 冊 i
0
具X0典 = ic0
0
0
Xe−i共0⬘−k0r⬘兲
冔
具0共1兲典.
共23兲
In a similar way, we find 具r⬘X0典 = − iⵜk,0具X0典 = − iⵜk,0具0共1兲典.
共16兲
共24兲
The mean value 具X0典 is straightforward to compute: 具X0典 =
1 c0
F关X兴0 =
˜ X 0 c0
= F关共1兲兴0 = ˜0共1兲 = n2共0,k0兲 − 1. 共25兲
Now, we develop the convolution product in Eq. (16) up to the second order: 具X0共⬘,r⬘兲U共 − ⬘,r − r⬘兲典 = 具X0典U − 具⬘X0典
U
1 2U − 共具r⬘X0典 ⵜ 兲U + 具⬘2X0典 2 2
1 2U 1 2U 1 2U + 具x⬘2X0典 2 + 具y⬘2X0典 2 + 具z⬘2X0典 2 2 x 2 y 2 z + 具x⬘y⬘X0典 + 具 ⬘x ⬘X 0典
共17兲
= 具 ⬘X 0典
共20兲
共22兲
A兲
⬁
−⬁
冓
冔
具⬘X0典 = 具⬘Xe−i共0⬘−k0r⬘兲典 =
To go further, we must assume that the medium is weakly nonlocal and dispersive. This means that the response of the material will not last for a very long time, nor extend for a very broad volume. The integration in the convolution product can therefore be limited to small values of t⬘ and r⬘. As a first illustration, we consider the limiting case where nonlocality and dispersion disappear. In this particular case, the response of the medium is 共1兲共t , r兲 ⬀ ␦共t , r兲. As the convolution by a ␦ function is the identity operation, it is sufficient to know A共t , r兲 in order to compute the convolution integral in 共t , r兲. In a more general case, the expression of A共t − t⬘ , r − r⬘兲 will be expanded in a Taylor series. For our purpose, this series can be limited to the second order. It would make no sense to compare t⬘ and r⬘, as their units are different. 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 define U共 , r兲 = A共t , r兲, and X0共 , r兲 = 0共1兲共t , r兲. Therefore 1
U
具共r⬘X0共⬘,r⬘兲 ⵜ 兲U典 = 具x⬘X0典
共15兲
A兲.
⬘X0共⬘,r⬘兲
t,r
The convolution product can conveniently be written as
具¯典=
共18兲
.
具X0共⬘,r⬘兲U共,r兲典 = 具X0典U共,r兲,
−i共0t−k0r兲
ei共k0r−0t兲共LWE兲 = 关− k02A − 2i共k0 ⵜ 兲A + ⌬A兴 − 0共1兲 丢 共D0A兲.
0共1兲共t,r兲 丢
冊
2
Before going further, we simplify the zeroth- and firstorder terms as follows:
which allows us to rewrite Eq. (6) in the form
0共1兲 丢
冉
− 具共r⬘X0共⬘,r⬘兲兩 ⵜ 兩共,r兲兲U典 + O共2兲.
This leads to the introduction of 共1兲
2
= i0 +
0共1兲 丢 A = 具X0共⬘,r⬘兲U共 − ⬘,r − r⬘兲典
= ei共0t−k0r兲 ⫻ 具共1兲共t⬘,r⬘兲e−i共0t⬘−k0r⬘兲共D0A兲典
0共1兲共t,r兲
2
+
The first-order Taylor expansion of Eq. (16) is
共1兲 丢 关共D0A兲ei共0t−k0r兲兴 共1兲
2U xy 2U x
+ 具x⬘z⬘X0典 + 具 ⬘y ⬘X 0典
2U xz 2U
y
+ 具y⬘z⬘X0典 + 具 ⬘z ⬘X 0典
2U yz
2U z
+ O共3兲. 共26兲
Kockaert et al.
Vol. 26, No. 12 / December 2009 / J. Opt. Soc. Am. B
This expression contains information about the temporal and spatial dispersive properties of the material. The evaluation of these terms would require more information about the medium sustaining the propagation. In the case of a metamaterial, the methods presented in [37,38] could be applied in order to calculate the macroscopic quantities, on the basis of the microscopic subwavelength structure. D. Symmetries For symmetry reasons, some coefficients appearing in Eq. (26) might be related, or zero. If the material is supposed to be isotropic, then 共1兲共t , r兲 must be invariant for any rotation, as well as for inversion. If the material admits, for example, cubic symmetry, 共1兲共t , r兲 should be invariant for any 90° rotation around a principal axis, for inversion, and therefore, for reflection with respect to a principal plane. By rewriting the electric field as an envelope modulated by a carrier wave, we have selected a particular orientation, namely, k0. If the material is not isotropic, but k0 corresponds to some principal direction in the propagation medium, then some symmetry properties can still help to simplify Eq. (26). Below, we assume that k0 is aligned in the z direction and that 共1兲 admits inversion and reflection symmetries around the axis of the orthogonal basis 共x , y , z兲. Therefore, 共27兲
k 0 = k 01 z ,
B151
only time derivatives T, a term containing only spatial derivatives S, and a mixed term M: D00共1兲 丢 A = A + T + S + M.
共34兲
Using Eqs. (23)–(25), and defining 0 = 0n共0 , k0兲, we find A = − 02A,
T=i
02 A 0 t
−
2 02 t2
2c02 02 t4
02 A kz z
A
0 kx
t
⌬ 2 ⬜
02
A
i c02 0
冉 冊 0
n 2 3A
0
t3 共36兲
1 202 2A 2 k z2 z 2
−
3
−i
+
1 202 202 kx
202
1
02 kz t2z
−
,
+
1 202
1 +i
1 202 2A
1 2n 2 4A
S=−i
M=−i
+
共35兲
⌬ 2 ⬜
A
1 202 2 k x2
2A t2 −
2 02 2A
0 kz tz
1 202 4A
3
0 k z2 t z 2
−
⌬⬜A,
202 kz2 t2z2
. 共37兲
共1兲共,x,y,z兲 = 共1兲共t,− x,y,z兲 = 共1兲共t,x,− y,z兲 = 共1兲共t,x,y,− z兲, 共28兲
0共1兲共t,x,y,z兲 = 共1兲共t,x,y,z兲e−i共0t−k0z兲 = 0共1兲共t,− x,y,z兲 = 0共1兲共t,x,− y,z兲 = 0共1兲共t,x,y,− z兲e2ik0z ,
共29兲
which implies similar relations for X0; hence 具r⬘X0典 = 具z⬘X0典1z ,
共30兲
具x⬘y⬘X0典 = 具− x⬘y⬘X0典 = 0,
共31兲
具x⬘⬘X0典 = 具− x⬘⬘X0典 = 0.
E. Slowly Variable Envelope The Taylor expansion in Eq. (26) was valid for a weakly nonlocal medium. We should notice that this expansion is equally valid if we assume slow variations of the envelope in space and time, together with important nonlocality and dispersion. But, if we make this last assumption, the second term in Eq. (18) is one order smaller than the first one. With this in mind, the second-order expansion of D00共1兲 丢 A reduces to A = − 02A,
共32兲
Finally, we rewrite the convolution product of Eq. (26) in the particular case where the properties along the x and the y directions are the same, so that 具x⬘2X0典 = 具y⬘2X0典, and
T=i
具X0共⬘,r⬘兲U共 − ⬘,r − r⬘兲典 = 具X0典U − 具⬘X0典
U
1
2U
S=−i
U
+ 具⬘2X0典 2 − 具z⬘X0典 2 z
1 2U 1 + 具z⬘2X0典 2 + 具x⬘2X0典⌬⬜U + O共3兲. 2 z 2
共33兲
The last step before explicitly obtaining the left-hand side of wave equation (6) is to apply the differential operator of Eq. (18) to the simplified convolution product of Eq. (33). The result contains 18 terms that we order into 4 different categories: an algebraic term A, a term containing
02 A 0 t
02 A kz z
+
+
1 202 2A 2 02 t2
1 202 2A 2 k z2 z 2
M=−
共38兲
+
2 02 2A
0 kz tz
1 202 2 k x2
.
共39兲
,
⌬⬜A,
共40兲
共41兲
Further simplification is obtained if we limit these expansions to the first order. In this case, we find the classical NLSE. To get more insight into the nonlocal effects, we focus on the spatial second-order term. Therefore, we rewrite Eq. (6) as
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J. Opt. Soc. Am. B / Vol. 26, No. 12 / December 2009
共LWE兲e−i共0t−k0z兲 + T + M
冉
= 共02 − k02兲A − 2ik0 1 −
冉
+ 1−
202 kx
2
冊
冊 冉
0 0 A k0 kz
z
+ 1−
冊
202 2A
⌬⬜A.
k z2
z2
共LWE兲e−i共0t−k0z兲 + T + M
冉
Ein
共42兲
It is now time to fix the value of k0. To avoid quickly varying phase factors, the obvious choice is k0 = 0. Furthermore, the classical SVEA implies 兩A / z兩 Ⰶ 兩2A / z2兩, so that the only remaining spatial terms are
= − 2ik0 1 −
Kockaert et al.
冊 冉
0 0 A k0 kz
z
+ 1−
202 k x2
冊
⌬⬜A. 共43兲
RHM
lR
0
S4 = −
1 422 4! kx
4
2 ⌬⬜ A.
dium. In the mean-field approximation, the variation of the field over one round trip in the cavity is very small. As a result, the propagation equation of the form
The zero-diffraction regime can be obtained in different configurations [25,41]. We consider here the case of twodimensional zero diffraction in a cavity filled with a lefthanded and a right-handed material, as studied in [25]. In Fig. 1, we present the geometry under consideration. By adjusting the relative thickness (lL and lR) of the two materials (LHM and RHM), it is possible to cancel diffraction at each round trip, which leads to the zero-diffraction regime in two dimensions. The mean-field model is explicitly obtained in the appendix of [25]. In particular, the propagation in each of the two materials is described with the help of a generalized NLSE, where care is taken to avoid the confusion between the refractive index and the impedance of the me-
= i关D + N兴A,
共45兲
where D is a linear differential operator and N a nonlinear operator, i.e., D=
N=
1 2k0
30 2c0
2 ⵜ⬜ −
2 2 2 t2
共46兲
,
0共3兲共兩A兩2 + 兩B兩2兲A,
共47兲
where is the nonlinear coupling coefficient between the forward 共A兲 and the backward 共B兲 waves. In [25], this equation is integrated with the assumption that ␦ = O共1兲, so that a first-order integration of Eq. (45) leads to A共 + ␦兲 = A共兲 + 兩i␦关D + N兴兩 A + O共2兲 = e兩i␦关D + N兴兩 A + O共2兲 = e兩i␦D兩 e兩i␦N兩 A + O共2兲 = e兩i␦N兩e兩i␦D兩 A + O共2兲.
共44兲
4. NONLOCAL EFFECTS IN THE MEAN-FIELD MODEL
z
lL
Fig. 1. (Color online) Cavity containing right-handed (RHM) and left-handed (LHM) materials. The cavity is driven by the input field Ei.
A F. Comments From the result obtained in Eq. (43), it is seen that in the stationary case, nonlocal terms do not change the nature of the propagation equation, but slightly modify the coefficients. This is probably the reason why nonlocal effects can be neglected to a certain extent, i.e., as far as the spatial dispersion presents sufficient symmetries, with respect to the direction of propagation. Now we turn back to our initial aim in this paper, which is to determine the physical limitations that will forbid writing information in infinitely small spatial cells. It appears that for small values of the spatial dispersion—for which a second-order expansion in the Taylor series fairly reflects the nonlocal behavior of the medium—the nature of the propagation equation is not modified by the nonlocality. If all the above mentioned symmetries hold, and the input beam is continuous (T = 0, M = 0), in the frame of the SVEA, a new term in the propagation equation appears at the fourth order in the Taylor expansion:
LHM
共48兲
Here, we are concerned with the nonlocality that will apppear in the model. A first physical understanding of the nonlocal terms is obtained in the continuous regime, where the differential term D contains only spatial derivatives. The fourth-order term S4 from Eq. (44) is taken into account so that D=
1 2k0
冋冉
1−
1 222 2 kx
2
冊
⌬⬜ +
1 422 4! kx
4
册
2 ⌬⬜ ⬇
1 2k0
共1 + ⌬⬜兲⌬⬜ , 共49兲
with
=
1 422 4! kx4
.
共50兲
Expanding the propagation operator over one round trip up to the second order, we get
Kockaert et al.
Vol. 26, No. 12 / December 2009 / J. Opt. Soc. Am. B
A共 + ␦兲 = ei␦关D + N兴A = 1 + i关D + N兴 +
i2 2
关D + N兴2 + O共3兲
1 1 = 1 + i关D + N兴 − D2␦2 − N2␦2 2 2 1 − 关DN + ND兴␦2 + O共3兲. 2 A. Low-Power Regime We require that our model be valid at low power. In this limit, the diffraction term can be considered of first order, while the nonlinear term is of second order, i.e., D = O共1兲 and N = O共2兲. This assumption leads to 1 A共 + ␦兲 = 1 + i关D + N兴 − D2␦2 + O共3兲. 2 Together with the additional considerations of [25], and Eq. (49), this result provides the nonlocal generalization of the Lugiato–Lefever equation:
冊
冉
A
1 2 4 A + i⌫兩A兩2A, = Ein − 共1 + i⌬兲A + i␦ⵜ⬜ A + i − ␦2 ⵜ⬜ ⬘ 2
and the second-order term in the mean-field approximation of the intracavity diffraction. In particular, we observe that the real part of the biLaplacian term will asymptotically evolve to zero when the diffraction is reduced, while the imaginary contribution will dominate at this time and prevent the formation of infinitely small features, as shown in [26]. The zero-diffraction regime is reached when ␦, as defined in Eq. (56), equals zero. Because kL ⬍ 0, this regime is obtained by tuning the relative thickness of the leftand the right-handed media, i.e., lL and lR. If the structure of the right-handed medium is of atomic size, then it is safe to set R = 0, so that
=−
F = 2
⬘ =
Ein =
␦=
=
␦
4
冉
F 2
kL
冏 冏
.
共60兲
k Rl L
共55兲
5. CONCLUSIONS
2
l L L
1−
k Ll R
共54兲
F
lL
L
=
Ein ,
FT
4 kL
F
⌫=
冉
F
共59兲
共53兲
冑2
⌬=−
L .
,
1 − 兩兩
2
4 兩kL兩
B. Local Medium The mean field equation (51) takes a particular form when the spatial dispersion of the material is negligible, i.e., when = 0. In this case, it is seen that both the Laplacian and the bi-Laplacian terms disappear in the zerodiffraction regime, where ␦ = 0. However, a simple analysis shows that close to this zero-diffraction regime, the biLaplacian term stabilizes the system. In particular, as could be expected from a physical point of view, the biLaplacian diffusion term induces a shrinking of the instability domains.
兩兩
F
F lL
The sign of this parameter is of paramount importance in order to define the minimal size of the nonlinear patterns that can form in the cavity [26]. More precisely, in the vicinity of ␦ = 0, the parameter defining the dynamics was shown to be
共51兲 in which we assumed a continuous pump beam and introduce the reduced quantities as in the appendix of [25]:
B153
共52兲
,
,
+
+
lR kR
冊
l R R kR
共56兲
,
冊
共lL␥L + lR␥R兲,
,
共57兲
共58兲
where F denotes de finesse of the cavity, ⬘ is the dimensionless time scale, Ein is the reduced driving field, ⌬ is the detuning, ␦ is the effective diffraction coefficient, is the effective coefficient of spatial dispersion, and ⌫ accounts for the effective Kerr-type nonlinearity. It is important to notice that the nonlocal term results from the contribution of two quantities with very different physical meanings: the spatial dispersion in the medium,
In this paper, we obtained a clear picture of the origin of nonlocal effects in cavities containing left-handed metamaterials. We studied separately the spatial dispersion in the material itself and the higher-order diffraction terms coming from the mean-field approximation leading to a slow-time description of the cavity dynamics. The mean-field model was applied in the particular case of a cavity filled with both a left-handed and a righthanded material. Some assumptions on the symmetries of the materials together with the slowly-variable envelope approximation allowed for a simple description, in the form of a Lugiato–Lefever equation with an additional biLaplacian term with complex coefficient. The model that we obtain provides an answer to the question of the physical limitations on the pattern formation beyond the first-order approximation of a zerodiffraction regime. As the detailed derivation of our model was provided, different cases of practical interest could be studied by applying the right assumptions on the symmetries of the
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J. Opt. Soc. Am. B / Vol. 26, No. 12 / December 2009
materials, or by computing their effective permittivity with the use of recent homogenization techniques [38,39]. In the example that we studied, we calculated the expression of the parameter / ␦, appearing in the limit case of [26], where the authors predict the minimal size of dissipative structures in such cavities. The model derived here above could be studied in a similar way in other limiting cases in order to reveal the different possible dynamics in such cavities and also to find the best design parameters for a given application.
Kockaert et al. 12.
13.
14. 15. 16. 17.
ACKNOWLEDGMENTS P. Kockaert thanks Gregory Kozyreff for introducing him to some homogenization techniques. This work was partially funded by the Belgian Science Policy Office, under grant IAP6-10, “Photonics@be.” P. Tassin and G. Van der Sande are, respectively, a PhD and a Postdoctoral Fellow of the Fonds voor Wetenschappelijk OnderzoekVlaanderen (FWO, Belgium). M. Tlidi is a fellow of the Fonds de la Recherche Scientifique (FRS, Belgium). P. Tassin acknowledges financial support from the Belgian American Educational Foundation.
18. 19.
20. 21.
REFERENCES 1.
2. 3.
4.
5.
6. 7. 8.
9.
10.
11.
D. R. Smith, W. J. Padilla, D. C. Vier, D. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000). R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001). V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ⑀ and ,” Sov. Phys. Usp. 10, 509–514 (1968). R. A. Shelby, D. R. Smith, S. C. Nemat-Nasser, and S. Schultz, “Microwave transmission through a twodimensional, isotropic, left-handed metamaterial,” Appl. Phys. Lett. 78, 489–491 (2001). C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah, and M. Tanielian, “Experimental verification and simulation of negative index of refraction using Snell’s law,” Phys. Rev. Lett. 90, 107401 (2003). A. A. Houck, J. B. Brock, and I. L. Chuang, “Experimental observations of a left-handed material that obeys Snell’s law,” Phys. Rev. Lett. 90, 137401 (2003). D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 6, 788–792 (2004). K. Aydin, M. Guven, L. Kafesaki, C. Zhang, M. Soukoulis, and M. Ozbay, “Experimental observation of true lefthanded transmission peaks in metamaterials,” Opt. Lett. 29, 2623–2625 (2004). C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. 95, 203901 (2005). T. F. Gundogdu, I. Tsiapa, A. Kostopoulos, G. Konstantinidis, N. Katsarakis, R. S. Penciu, M. Kafesaki, E. N. Economou, T. Koschny, and C. M. Soukoulis, “Experimental demonstration of negative magnetic permeability in the far-infrared frequency regime,” Appl. Phys. Lett. 89, 084103 (2006). J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
22. 23.
24. 25.
26.
27.
28.
29.
30. 31. 32. 33. 34.
N. Engheta, “An idea for thin subwavelength cavity resonators using metamaterials with negative permittivity and permeability,” IEEE Antennas Wireless Propag. Lett. 1, 10–13 (2002). A. Alu, N. Engheta, A. Erentok, and R. W. Ziolkowski, “Single-negative, double-negative and low index metamaterials and their electromagnetic applications,” IEEE Trans. Antennas Propag. 49, 23–36 (2007). P. Tassin, X. Sahyoun, and I. Veretennicoff, “Miniaturization of photonic waveguides by the use of left-handed materials,” Appl. Phys. Lett. 92, 203111 (2008). U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006). J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006). M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations,” Photonics Nanostruct. Fundam. Appl. 6, 87–95 (2008). A. A. Zharov, N. A. Zharova, I. V. Shadrivov, and Y. S. Kivshar, “Subwavelength imaging with opaque nonlinear left-handed lenses,” Appl. Phys. Lett. 87, 091104 (2005). N. Lazarides and G. P. Tsironis, “Coupled nonlinear Schrödinger field equations for electromagnetic wave propagation in nonlinear left-handed materials,” Phys. Rev. E 71, 036614 (2005). G. D’Aguanno, N. Mattiucci, M. Scalora, and M. J. Bloemer, “Bright and dark gap solitons in a negative index Fabry–Perot etalon,” Phys. Rev. Lett. 93, 213902 (2004). V. M. Agranovich, Y. R. Shen, R. H. Baughman, and A. A. Zakhidov, “Linear and nonlinear wave propagation in negative refraction metamaterials,” Phys. Rev. B 69, 165112 (2004). I. V. Shadrivov, A. A. Zharov, and Y. S. Kivshar, “Secondharmonic generation in nonlinear left-handed metamaterials,” J. Opt. Soc. Am. B 23, 529–534 (2006). P. Tassin, L. Gelens, J. Danckaert, I. Veretennicoff, G. Van der Sande, P. Kockaert, and M. Tlidi, “Dissipative structures in left-handed material cavity optics,” Chaos 17, 037116 (2007). K. Staliunas, “Midband dissipative spatial solitons,” Phys. Rev. Lett. 91, 053901 (2003). P. Kockaert, P. Tassin, G. V. der Sande, I. Veretennicoff, and M. Tlidi, “Negative diffraction pattern dynamics in nonlinear cavities with left-handed materials,” Phys. Rev. A 74, 033822 (2006). L. Gelens, G. Van der Sande, P. Tassin, M. Tlidi, P. Kockaert, D. Gomila, I. Veretennicoff, and J. Danckaert, “Impact of nonlocal interactions in dissipative systems: towards minimal-sized localized structures,” Phys. Rev. A 75, 063812 (2007). L. Gelens, D. Gomila, G. Van der Sande, J. Danckaert, P. Colet, and M. A. Matías, “Dynamical instabilities of dissipative solitons in nonlinear optical cavities with nonlocal materials,” Phys. Rev. A 77, 033841 (2008). A. D. Boardman, K. Marinov, D. I. Pushkarov, and A. Shivarova, “Influence of nonlinearly induced diffraction on spatial solitary waves,” Opt. Quantum Electron. 32, 49–62 (2000). A. D. Boardman, P. Egan, R. C. Mitchell-Thomas, Y. G. Rapoport, and N. J. King, “Weakly and strongly nonlinear waves in negative phase metamaterials,” Proc. SPIE 7029, 70291F (2008). V. M. Agranovich and V. L. Ginzburg, “Crystal optics with spatial dispersion” in Progress in Optics, E. Wolf, ed. (North-Holland, 1971), Vol. IX, Chap. 6, pp. 235–280. S. Tretyakov, Analytical Modeling in Applied Electromagnetics (Artech House, 2003). M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge Univ. Press, 1999). J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, 1925). L. Landau and E. Lifchitz, Électrodynamique des Milieux Continus (Mir, 1969).
Kockaert et al. 35.
36.
37. 38.
G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, “Negative-index metamaterial at 780 nm wavelength,” Opt. Lett. 32, 53–55 (2007). K. Cho, “A single susceptibility scheme of macroscopic Maxwell equations: beyond the ‘E, D, B, H’ approach,” J. Phys. Condens. Matter 20, 175202 (2008). A. P. Vinogradov and A. V. Aivazyan, “Scaling theory for homogenization of the Maxwell equations,” Phys. Rev. E 60, 987–993 (1999). D. Sjöberg, “Homogenization of dispersive material
Vol. 26, No. 12 / December 2009 / J. Opt. Soc. Am. B
39. 40. 41.
B155
parameters for Maxwell’s equations using a singular value decomposition,” Multiscale Model. Simul. 4, 760–789 (2005). D. Sjöberg, “A modified Drude-Born-Fedorov model for isotropic chiral media, obtained by finite scale homogenization,” J. Phys. D 41, 155412 (2008). R. W. Boyd, Nonlinear Optics (Academic, 1992). K. Staliunas, C. Serrat, R. Herrero, C. Cojocaru, and J. Trull, “Subdiffractive light pulses in photonic crystals,” Phys. Rev. E 74, 016605 (2006).
