G Van Der Sande

I Veretennicoff

M Tlidi

P Kockaert

P Tassin

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Left-Handed Materials 
in Nonlinear Fabry-Perot Resonators 
P. Tassin1, G. Van der Sande1,2, I. Veretennicoff1, M. Tlidi2, P. Kockaert3 
1Department of Applied Physics and Photonics, Vrije Universiteit Brussel,  
Pleinlaan 2, B-1050 Brussel, Belgium 
2Optique non linéaire théorique, Université libre de Bruxelles, CP 231,  
Campus Plaine, B-1050 Bruxelles, Belgium 
3Service d’optique et acoustique, Université libre de Bruxelles, CP 194/5,  
50, Av. F. D. Roosevelt, B-1050 Bruxelles, Belgium 
We have studied a double layered Fabry-Perot resonator filled with both a traditional 
nonlinear material and a nonlinear left-handed material. First, we derived a set of 
equations for the propagation of light through a nonlinear left-handed material. These 
equations were subsequently used for developing a mean-field model for the nonlinear 
resonator. We show that the introduction of left-handed materials has a number of 
interesting implications, of which the most important is the elimination of diffraction. It 
is also possible to design a resonator in a negative diffraction regime, for which 
stability is enhanced with respect to the classical resonator. 
Introduction 
Materials with simultaneously negative electric permittivity and negative magnetic 
permeability have been first studied theoretically by Veselago [1]. Because the electric 
field, the magnetic field and the wavevector of a plane monochromatic wave form a 
left-handed system of reference in these materials, he termed them “left-handed”. 
Veselago showed that left-handed materials exhibit a number of remarkable properties 
unlike those of any known materials, such as reversed Doppler effect, reversed 
Cerenkov radiation, radiation tension instead of radiation pressure and negative 
refraction. Although no natural materials with left-handedness have been identified, 
artificial materials – or metamaterials – with negative electromagnetic parameters have 
recently been designed and experimentally demonstrated  in the microwave range [2]. 
These metamaterials derive their electromagnetic properties from their (sub-
wavelength) structure, rather than from their bulk properties.  
Linear devices filled with left-handed materials have received a lot of attention. Most 
noteworthy, it has been predicted that a parallel-sided plate of left-handed material is an 
imaging system with sub-wavelength resolution properties [3]. This sub-wavelength 
resolution can be explained by the amplification of evanescent wave components that 
occurs in the plate. The nonlinear properties of left-handed materials, on the other hand, 
are still largely unexplored. One of the first contributions was by Zharov et al., who 
analysed the nonlinear behaviour of the microwave metamaterial constructed by 
Shelby [4]. Agranovich et al. showed that well-known nonlinear effects like second 
harmonic generation, wave mixing and Raman scattering are more or less identical in 
left- and right-handed materials [5]. Feise et al. observed bistability in the transmission 
through a number of alternating slabs of two nonlinear materials with positive and 
negative refractive index [6]. Although the publications mentioned above provide a first 
step in the study of nonlinear optics in left-handed materials, a lot of work still needs to 
be done. 
Proceedings Symposium IEEE/LEOS Benelux Chapter, 2005, Mons
39
The Double Layered Resonator 
In this contribution, we will present our study of a new device containing left-handed 
materials: a double layered Fabry-Perot resonator, of which one layer is filled with a 
traditional nonlinear material and the other layer with a nonlinear left-handed material 
(see Figure 1). The Kerr type nonlinear optical materials are assumed to be 
homogeneous, cubic, centrosymmetric and impedance matched. The sides of the 
resonator are coated with dielectric mirrors to enhance the finesse of the cavity.  
 
Figure 1: The device under test: a double layered Fabry-Perot resonator. 
Propagation of Light in a Nonlinear Left-Handed Material 
Light coupled into the resonator will first propagate through the right-handed layer and 
then through the left-handed layer. A rigorous model of our device should therefore 
include a description of both interactions. Propagation through a traditional Kerr type 
nonlinear material can be properly treated with the nonlinear Schrödinger equation. 
However, propagation through a nonlinear left-handed material has not been considered 
before. It is not obvious at all that the nonlinear Schrödinger equation is still valid, 
because it is commonly derived for nonmagnetic materials with µ(ω) = µ0. In [7], we 
have presented a generalisation for magnetic media. In the first step, we have derived 
from Maxwell’s equations the wave equation governing the evolution of the electric 
field of light in a nonlinear dispersive magnetic dielectric: 
 
2 2 (3)
02 2 2
1rot rot ( ) ( ) ( , )d d ( ) d .r r rt r tc
µ ξ ε ξ τ τ τ ξ µ µ ξ ξ
ξ ξ
∞ ∞ ∞
−∞ −∞ −∞
∂ ∂− − − − = −
∂ ∂∫ ∫ ∫
PE E  (1) 
Subsequently, we assumed the nonlinear term in (1) to be small and applied the slowly 
varying envelope approximation. A perturbation analysis then yields 
 ( )
2 2 2
2 20
02 2 2
0
3'' 2 ,
2 2 2
A c ik Ai A i A B A
n x y t c
ω χ
ξ ω η
 ∂ ∂ ∂ ∂= + − + + ∂ ∂ ∂ ∂ 
 (2) 
 ( )
2 2 2
2 20
02 2 2
0
3'' 2 ,
' 2 2 2
B c ik Bi B i B A B
n x y t c
ω χ
ξ ω η
 ∂ ∂ ∂ ∂= − + + − + ∂ ∂ ∂ ∂ 
 (3) 
where A and B are the envelopes of forward and backward waves, respectively. Please 
note the difference between Eqs. (2)-(3) and the traditional nonlinear Schrödinger 
equations for right-handed media. The coefficient of the nonlinear term of the former 
contains the characteristic impedance η of the medium instead of the index of refraction 
n written traditionally. Nevertheless, the coefficient of the diffraction term remains 
inversely proportional to the index of refraction, which implies that the diffraction term 
becomes negative in a left-handed material. 
Left-Handed Materials in Nonlinear Fabry-Perot Resonators
40
The Mean-Field Model 
The propagation model, which consists of the nonlinear Schrödinger equations 
completed with appropriate boundary conditions, is a rigorous model, because it 
describes in every point the interaction of light with the materials and the structure of 
the nonlinear Fabry-Perot resonator. However, it is also too complex for a detailed 
analytical study. Therefore we have considered an approximation to the propagation 
model for cavities near to resonance [8]. This means that (1) the cavity has a high 
finesse – i.e. that it has a high photon lifetime – and (2) that the optical path length of 
light in the cavity is approximately an integer multiple of the wavelength of the laser 
light. In this case, we find the following equation relating the input field Σ and the 
output field A of the double layered resonator: 
 ( ) 221 ,A i A i A i A A
t
α ⊥
∂ = Σ − + ∆ + ∇ +
∂
 (4) 
where ∆ is the detuning from perfect resonance and α the diffraction coefficient. Eq. (4) 
is called the mean field equation. Since the same equation also describes traditional 
nonlinear Fabry-Perot resonators [9], it looks like our resonator exhibits the same 
behaviour as a traditional resonator. Nevertheless, there is an important difference. Let 
us look to the equation for the diffraction coefficient α: 
 0 .
2 R L
l L l
n n
λα
πσ
 −= + 
 
 (5) 
The first term of the right-hand side of Eq. (5) is positive, while the second term is 
negative (remember that nL is negative. Both terms will thus cancel each other to some 
extent. In this way, it is possible to engineer the strength of the diffraction by adjusting 
the lengths l and L – l of the layers. This should be compared with classical resonators, 
where the diffraction coefficient can only be positive. 
Diffraction Compensation and Negative Diffraction 
If the ratio of the thicknesses of the layers equals the ratio of the indices of refraction, 
the diffraction coefficient vanishes. This is the regime of diffraction compensation. It is 
straightforward that this regime is very advantageous for a large number of applications 
of the nonlinear Fabry-Perot resonator, such as high finesse resonators, optical 
transistors and optical memory elements. With the diffraction compensated resonator, it 
is indeed possible to eliminate diffraction losses, which leads naturally to resonators 
with higher quality factors. Elimination of diffraction losses is even more important for 
nonlinear Fabry-Perot resonators, where a too small interaction length currently limits 
their use. A higher quality factor will here increase the photon life time of the cavity, so 
that the nonlinear phase change is accumulated over more roundtrips. 
 
Figure 2: A Gaussian beam in the double layered resonator will first be widened due to diffraction in the 
right-handed layer, but will then again be focused due to the negative diffraction in the left-handed layer. 
Overall, it looks as if there is no widening and, hence, as if there is no diffraction. 
Proceedings Symposium IEEE/LEOS Benelux Chapter, 2005, Mons
41
If the negative term in Eq. (5) is larger than the positive term, the overall diffraction 
coefficient α becomes negative. The resonator is said to be in the regime of negative 
diffraction. We have studied the dynamics of this new regime and showed that the 
resonator’s input-output characteristic has the same form as for positive diffraction (see 
Figure 3). The difference here lies in the stability. For the classical resonator with 
positive diffraction, the upper branch of the curves is unstable, whereas the upper 
branch is stabilised for a resonator in the negative diffraction regime. This stabilisation 
is again advantageous for applications, such as switches or optical memories, where the 
resonator is switched from one to another branch. 
 
0 1 2 3
0
0.5
1
1.5
2
∆² < 3
|Σ|2
|A
|2
0 5
0
1
2
3
4
∆² > 4
|Σ|2   0 1 2 3
0
0.5
1
1.5
2
∆² < 3
|Σ|2
0 10 20
0
1
2
3
4
5
6
∆² > 4
|Σ|2  
Figure 3: Input-output characteristics of the double layered resonator. (a) Positive diffraction. 
(b) Negative diffraction. Stable states are denoted by a solid line, unstable states by a dashed line. 
Acknowledgements 
P.T. and G.V. are Research Assistants of the Research Foundation – Flanders (FWO –  
Vlaanderen). M.T. is a Research Associate of the Fonds national de la recherche 
scientifique (FNRS, Belgium). This work was supported by the Belgian Science Policy 
Office under grant IAP V/18 (Belgian Photon Network) and by the Research Council 
(OZR) of the Vrije Universiteit Brussel. 
References 
[1] V. Veselago, “The Electrodynamics of Substances with Simultaneously Negative Values of ε and µ”, 
Sov. Phys. Usp., vol. 10, pp. 509-514, 1968.  
[2] R. Shelby, D. Smith and S. Schultz, “Experimental Verification of a Negative Index of Refraction”, 
Science, vol. 292, p. 77, 2001. 
[3] J. Pendry, “Negative Refraction Makes a Perfect Lens”, Phys. Rev. Lett., vol. 85, pp. 3966-3969, 
2000. 
[4] A. Zharov, I. Shadrivov and Y. Kivshar, “Nonlinear Properties of Left-Handed Metamaterials”, 
Phys. Rev. Lett., vol. 91, 037401, 2003. 
[5] V. Agranovich, Y. Shen, R. Baughman and A. Zakhidov, “Linear and nonlinear propagation in 
negative refraction metamaterials”, Phys. Rev. B, vol. 69, 165112, 2004. 
[6] M. Feise, I. Shadrivov and Y. Kivshar, “Tunable transmission and bistability in left-handed band-gap 
structures”, Appl. Phys. Lett., vol. 85, pp. 1451-1453, 2004. 
[7] P. Tassin, G. Van der Sande, I. Veretennicoff, M. Tlidi and P. Kockaert, “Analytical model for the 
optical propagation in a nonlinear left-handed material”, Proc. SPIE, vol. 5955, 59550X, 2005. 
[8] P. Kockaert, P. Tassin, G. Van der Sande, I. Veretennicoff and M. Tlidi, “Spatial dissipative structures 
in a Fabry-Perot cavity with a negative refractive index”, Proc. SPIE, vol. 5955, 595510, 2005. 
[9] L. A. Lugiato and R. Lefever, “Spatial Dissipative Structures in Passive Optical Systems”, Phys. 
Rev. Lett., vol. 58, pp. 2209-2211, 1987. 
(a) α > 0 (b) α < 0 
Left-Handed Materials in Nonlinear Fabry-Perot Resonators
42


Left-Handed Materials in Nonlinear Fabry-Perot Resonators

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Left-Handed Materials in Nonlinear Fabry-Perot Resonators

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