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(1)World Academy of Science, Engineering and Technology 61 2010. Numerical Simulations of Negative-Index Nanocomposites and Backward-Wave Photonic Microdevices Alexander K. Popov and Sergey A. Myslivets. Abstract—Optical and nonlinear-optical properties of negativeindex nanocomposite as well as the feasibility of a design of novel photonic microdevices and all-optical data processing chips are numerically simulated. Keywords—Numerical simulation, nano materials and nanostructures, negative-index metamaterials, nanophotonics, nano and micro devices, all-optical switches, optical frequency narrow-band filters, optical parametric amplifiers and cavityless oscillators.. O. I. I NTRODUCTION. PTICAL negative-index (NI) metamaterials (NIMs) form a novel class of artificial electromagnetic media that promises revolutionary breakthroughs in photonics [1]. Nanostructured metamaterials are expected to play a key role in the development of all-optical data processing chips. Negative refraction does not exists in natural media. Its engineering has become possible only recently with the advent of nanotechnologies and can be implemented to develop a wide variety of devices with enhanced and uncommon functions. Unlike ordinary positive-index (PI) materials (PIMs), the energy flow and wave vector (phase velocity) are counterdirected in NIMs, which determines their extraordinary linear and nonlinear optical (NLO) properties. The majority of NIMs realized to date consist of metal-dielectric nanostructures that have highly controllable magnetic and dielectric responses. Significant progress has been achieved recently in the design of bulk, multilayered, negative-index, plasmonic slabs [2]–[5]. The problem, however, is that these structures introduce strong losses inherent to metals that are difficult to avoid, especially in the visible range of frequencies. Irrespective of their origin, losses constitute a major hurdle to the practical realization of the unique optical applications of these structures. Therefore, developing efficient loss-compensating techniques is of a paramount importance. So far, the most common approach to compensating losses in NIMs is associated with the possibility to embed amplifying centers in the host matrix [1]. The amplification is supposed to be provided through a population inversion between the energy levels of the embedded centers. Extraordinary properties of three- and four-wave mixing processes in NIMs and their application for compensating Alexander K. Popov is with the Department of Physics and Astronomy, University of Wisconsin-Stevens Point, Stevens Point, WI 54481 U.S.A. email: apopov@uwsp.edu Sergey A. Myslivets is with the Institute of Physics of the Siberian Division of Russian Academy of Sciences and Siberian Federal University, Krasnoyarsk, Russian Federation. e-mail: sam@iph.krasn.ru Manuscript received Month Date, 2009. losses in NIMs were explored in [6]–[8]. Herein, the numerical simulations are implemented of optical and nonlinear-optical properties of doped negative-index metamaterials as well as of NLO propagation processes of coupled ordinary and backward electromagnetic waves in such artificial electromagnetic media in order to draw from the numerical results of the simulation a better knowledge of background processes and their physical understanding and numerically demonstrate the possibility to implement such processes for a design a novel class of photonic microdevices. Two options are explored. One is threewave (TW) optical parametric amplification (OPA), which implements nonlinearities attributed to the building blocks of the NIMs [9]. The other option, is four-wave mixing (FWM) OPA [7], which suggests independent engineering of a χ(3) nonlinearity through embedded, resonant, NLO centers. In the vicinity of the resonances, χ(3) is exceptionally strong. In addition, optical properties of the composite can be tailored by means of quantum control in such a case. It is shown that the outcomes are strongly dependent on the relaxation properties of the coupled optical transitions and on the specific coupling schemes. Comparative analysis of several promising schemes is given. II. BACKWARD WAVES AND PARAMETRIC INTERACTION IN A NIM: MASTER EQUATIONS A. Poynting and wave-vectors in a lossless NIM Consider a traveling electromagnetic wave, E(r, t) = (1/2)E0 exp[i(k · r − ωt)] + c.c., H(r, t) = (1/2)H0 exp[i(k · r − ωt)] + c.c. From the equations ∇×E=−. 1 ∂D 1 ∂B , B = μH, ∇ × H = , D = E c ∂t c ∂t. one finds that k×E=. √ ω ω √ μH, k × H = − E, E = − μH. c c. (1). Equations (1) show that the vector triplet E, H, k forms a right-handed system for an ordinary medium with i > 0 and μi > 0. Simultaneously negative i and μi result in a lefthanded triplet and negative refractive index √ n = − μ, k 2 = n2 (ω/c)2 .. 107.

(2) World Academy of Science, Engineering and Technology 61 2010. All indices of , μ and n are assumed real numbers. The direction of the wave-vector k with respect to the energy flow (Poynting vector) [10]–[12] depends on the signs of  and μ: S(r, t) =. c2 k 2 c c2 k 2 [E × H] = H = E . 4π 4πω 4πωμ. (2). At i < 0 and μi < 0, S and k become contradirected, which is in contrast with the electrodynamics of ordinary media and opens opportunities for many revolutionary breakthroughs in photonics. B. Coupling geometry and coherent energy transfer from the ordinary control fields to the backward signal in a NIM. k ,S. k4, S4. k ,S. 2. k3, S3. k ,S. k3, S3. k2, S2. k2, S2. 4. 2. 4. 3. 3. k1. k1. k1. S1(BW). S1(BW). S1(BW). L. 0. (a). L. 0. L:. 0. (b). (c). m. m g. ω. 1. ω. g ω2. 3. ω. ω. 2. 4. n. ω. ω1. 3. ω. 4. n. l. C. Equations for coupled contrapropagating backward and ordinary waves. l. (d). two waves enter the slab, strong control field at ω3 and weak signal at ω1 , which then generate a difference-frequency idler at ω2 = ω3 − ω1 . The idler contributes back to the signal through the similar TWM process, ω1 = ω3 − ω2 , and thus provides OPA of the signal. The signal is assumed negativeindex, n(ω1 ) < 0, and therefore backward wave (BW). This means that the energy flow S1 is antiparallel to k1 [Fig. 1 (a)], which contrasts with the early proposals [13], [14] and their recent realization [15], [16] of BWOPO in PI materials. The idler and the control field are the ordinary waves with parallel k2,3 and S2,3 along the z axis. Consequently, the control beam enters the slab at z = 0, whereas the signal at z = L. In the second, FWM case, Fig. 1(b) and (c), the slab is illuminated by two PI control (pump) waves at ω3 and ω4 . In both cases, all wave-vectors are co-directed along the the z-axis. Due to the parametric interaction, the control and signal fields generate a difference-frequency idler at ω2 = ω4 + ω3 − ω1 (FWM), which is also assumed to be a PI wave (n2 > 0). The idler contributes back into the wave at ω1 through the same type of the parametric interaction and thus enables OPA at ω1 by converting the energy of the control fields into the signal. Thus, all of the coupled waves have their wave-vectors co-directed along z, whereas the energy flow of the signal wave, S1 , is counter-directed to the energy flows of all the other waves, which are codirected with their wave-vectors. Such coupling schemes are in contrast both with the conventional phasematching scheme for OPA in ordinary materials, where all energy-flows and phase velocities are co-directed, as well as with TWM backward-wave mirrorless OPO [13]–[16], where both the energy flow and wave-vector of one of the waves are opposite to all others.. (e). Fig. 1. Coupling geometry and alternative schemes of four-wave mixing in the embedded resonant nonlinear-optical centers. [(a)] Coupling geometry. S1 – negative-index signal, S3 – positive-index control field, S2 – positive-index idler. [(b), (c)] Coupling geometry for four-wave mixing of the backward and ordinary electromagnetic waves. S1 , k1 and ω1 are energy flux, wavevector and frequency for the backward-wave signal; S2 , k2 and ω2 – for the ordinary idler; S3,4 , k3,4 and ω3,4 – for the ordinary control fields. [(d), (e)] Corresponding alternative schemes of quantum controlled four-wave mixing in the embedded resonant nonlinear-optical centers with different ratio of the signal and the idler absorption rates and nonlinear susceptibilities. [(d)] Shortestwavelength negative-phase signal, where, depending on the partial relaxation rates, parametric amplification may be assisted by the idler’s populationinversion or Raman-type amplification. [(e)] Longer-wavelength negativephase signal, where depending on the partial relaxation rates, parametric amplification may be assisted by the signal incoherent amplification attributed to population-inversion or Raman-type gain.. Two options are considered, a NIM slab with a tree-wave (TWM) mixing optical nonlinearity χ(2) attributed to its building blocks [Fig. 1(a)] and the option with four-wave mixing (FWM) optical nonlinearity χ(3) attributed to the embedded nonlinear optical centers [Fig. 1(b) and (c)]. In the first case, three coupled optical electromagnetic waves with wave vectors k1,2,3 co-directed along the z axis propagate through a slab of thickness L with quadratic, TWM, nonlinearity χ(2) . Only. First, it will be shown that magnetic and electric TWM and FWM processes can be treated identically. Two alternative types of nonlinearities will be considered– electric, D = E + 4πPN L , B = μH; and magnetic, B = μH + 4πMN L , D = E. Nonlinear polarization and magnetization are sought in the form  · r − ωt)] + c.c., PN L (r, t) = (1/2)P0 N L (r) exp[i(k NL NL  · r − ωt)] + c.c. M (r, t) = (1/2)M0 (r) exp[i(k Accounting for Eqs. (1), one can derive ∇ × ∇ × E = −(μ/c2 )∂ 2 D/∂t2 , −ΔE = μ(ω 2 /c2 )[E + 4πPN L ], ∇ × ∇ × H = −(/c2 )∂ 2 B/∂t2 , −ΔH = (ω 2 /c2 )[μH + 4πMN L ]. For the medium with the electric nonlinearity, the equation for the slowly varying amplitude E0 of the wave with the wavevector along the z-axis takes the form: dE0 /dz = iμ(2πω 2 /kc2 )P0N L exp[i( k − k)z]. For the magnetic nonlinearity, the equation is. 108. dH0 /dz = i(2πω 2 /kc2 )M0N L exp[i( k − k)z]..

(3) World Academy of Science, Engineering and Technology 61 2010. The equations are symmetric and can be converted from one to the other by replacing μ ←→ . For the electric quadratic nonlinearity, (2). P1N L. =. χe1 E3 E2∗ exp{i[(k3 − k2 )z − ω1 t]},. P2N L. =. χe2 E3 E1∗ exp{i[(k3 − k1 )z − ω2 t]},. (2). where ω2 = ω3 − ω1 and kj = |nj |ωj /c > 0. Then the equations for the slowly-varying amplitudes of the signal and idler in the lossy medium can be given in the form dE1 /dz dE2 /dz. = =. iσe1 E2∗ exp[iΔkz] + (α1 /2)E1 ,. (3). exp[iΔkz] − (α2 /2)E2 .. (4). iσe2 E1∗. (2) (kj /j )2πχej E3 ,. Δk = k3 − k2 − k1 , and αj Here, σej = are the absorption indices. The depletion of the control (pump) wave E3 due to the NLO energy conversion is neglected here. Hence, the equation for the control field with the account for absorption takes a standard form and can be solved independently. For the magnetic type of quadratic nonlinearity, (2). M1N L. =. χm1 H3 H2∗ exp{i[(k3 − k2 )z − ω1 t]},. M2N L. =. χm2 H3 H1∗ exp{i[(k3 − k1 )z − ω2 t]},. D. Density matrix equations and local optical parameters of the medium. (2). the equations for the slowly-varying amplitudes are: dH1 /dz dH2 /dz. = iσm1 H2∗ exp[iΔkz] + (α1 /2)H1 , = iσm2 H1∗ exp[iΔkz] − (α2 /2)H2 .. (5) (6). (2). Here, σmj = (kj /μj )2πχmj H3 , H3 = const, and the other notations remain the same. For the electric-type FWM, the equations for the slowlyvarying amplitudes are similar: dE1 /dz dE2 /dz. = iγ1 E2∗ exp[iΔkz] + (α1 /2)E1 , = iγ2 E1∗ exp[iΔkz] − (α2 /2)E2 .. (7) (8). (3). Here, γj = (kj /j )2πχj E3 E4 and Δk = k3 + k4 − k1 − k2 . It is convenient to introduce effective amplitudes, ae,m,j , and nonlinear coupling parameters, ge,m,j , which for the electric and magnetic types of quadratic nonlinearity are defined as   (2) aej = |j /kj |Ej , gej = |k1 k2 /1 2 |2πχej E3 ,   (2) amj = |μj /kj |Hj , gmj = |k1 k2 /μ1 μ2 |2πχmj H3 , and for FWM as   (3) aj = |j /kj |Ej , gj = |k1 k2 /1 2 |2πχj E3 E4 . The quantities |aj |2 are proportional to the photon numbers in the energy fluxes. Equations for amplitudes aj are identical for all of the types of nonlinearities studied here: da1 /dz da2 /dz da3 /dz da4 /dz. = −g1 a∗2 exp(iΔkz) + (α1 /2)a1 , = g2 a∗1 exp(iΔkz) − (α2 /2)a2 , = −(α3 /2)a3 ,. The following three fundamental differences in equations (3)-(8) should be noted as compared with their counterpart in ordinary, PI materials. First, the signs of σ1 and γ1 are opposite to those of σ2 and γ2 because 1 < 0 and μ1 < 0. Second, the opposite sign appears with α1 because the energy flow S1 is against the z-axis. Third, the boundary conditions for the signal are defined at the opposite side of the sample as compared to the idler because the energy flows S1 and S2 are counter-directed. Consequently, sign with the right side of Eq. (9) is opposite to that in Eq. (10). As will be discussed below, it leads to dramatic changes in the solutions to the equations and in general behavior of the NLO system. It is convenient to introduce energy distribution for the backward 2 wave, T1 (z) = |a1 (z)/a1L | , and for the PI idler, η2 (z) = ∗ 2 |a2 (z)/a1L | , across the slab. Then the transmission factor for the backward-wave signal at z = 0, T10 , and the output 2 idler at z = L, η2L , are given by T10 = |a1 (0)/a1L | , η2L = 2 |a2 (L)/a1L | . At a1L = 0, a2 (z = 0) = a20 , the slab serves as an NLO mirror with a reflectivity r10 (output conversion 2 efficiency) at ω1 given by the equation r10 = |a1 (0)/a∗20 | .. (9). A significant difference between the resonant and nearresonant processes attributed to the embedded centers and offresonant NLO processes associated with the nonlinearity of the host metamaterial is that all local resonance optical parameters become intensity-dependent, and hence their spectral properties may experience a radical change near resonance. In particular, the NLO susceptibilities and, therefore, the parameters γ1 and γ2 become complex and differ from each other in the vicinity of the resonances. Hence, the factor g 2 may become negative or complex. This indicates an additional phase shift between the NLO polarization and the generated wave that causes further radical changes in the nonlinear propagation features, which can be tailored. With account for the embedded centers, the macroscopic parameters in Eqs. (9) and (10) are convenient for calculations with the density matrix technique, which allows one to account for various relaxation and incoherent excitation processes. For the case of Fig. 1(d), the power-dependent susceptibility χ1 responsible for absorption and refraction at ω1 can be found as P (ω1 ) = χ1 E1 , P (ω1 ) = N ρlm dml + c.c., where P is the polarization of the medium oscillating with the frequency ω1 , N is the number density of molecules, dml is the electric dipole moment of the transition, and ρlm is the density matrix element. Other polarizations are determined in the same way. The density matrix equations for a mixture of pure quantum mechanical ensembles in the interaction representation can be written in a general form as Lnn ρnn = qn − i[V, ρ]nn + γmn ρmm , Llm ρlm = L1 ρ1 = −i[V, ρ]lm , Lij = d/dt + Γij ,. (10). = −(α4 /2)a4 .. Vlm = Glm · exp{i[Ω1 t − kz]},. Here, last two equations account for absorption of the control fields.. Glm = −E1 · dlm /2,. where Ω1 = ω1 , −ωml is the frequency detuning from the corresponding resonance; Γmn - homogeneous half-widths. 109.

(4) World Academy of Science, Engineering and Technology 61 2010. of the corresponding transition (in  the collisionless regime Γmn = (Γm + Γn )/2); Γn = j γnj - inverse lifetimes rate of relaxation from level m to n; and of levels; γ  mn qn = j wnj rj - rate of incoherent excitation to state n from underlying levels. The equations for the other elements are written in the same way. It is necessary to distinguish the open and closed energylevel configurations. In the open case (where the lowest level is not the ground state), the rate of incoherent excitation to various levels by an external source essentially does not depend on the rate of induced transitions between the considered levels. In the closed case (where the lowest level is the ground state), the excitation rate to different levels and velocities depends on the value and velocity distribution at other levels, which are dependent on the intensity of the driving fields. For open configurations, qi are primarily determined by the population of the ground state and do not depend on the driving fields. Inhomogeneous broadening of the transitions can be accounted for in the final formulas by substituting Ωi for Ωi = Ωi −δi , δi is resonance shift for individual center embedded in the host material, and then by averaging over the shifts. Equations for the case of Fig. 1(e) are obtain by replacing indices 1 ↔ 2. In a steady-state regime, the solution of a set of densitymatrix equations can be cast in the following form: ρii = ri , ρlg = r3 · exp(iΩ3 t), ρnm = r4 · exp(iΩ4 t), ρng = r2 · exp(iΩ2 t) + r˜2 · exp[i(Ω3 + Ω4 − Ω1 )t], ρlm = r1 · exp(iΩ1 t) + r˜1 · exp[i(Ω3 − Ω2 + Ω4 )t], ρln = r32 · exp[i(Ω3 − Ω2 )t] + r41 · exp[i(Ω1 − Ω4 )t]. The density matrix amplitudes ri determine the absorption/gain and refraction indexes, and r˜i determine the fourwave mixing driving nonlinear polarizations. Then the problem reduces to the set of algebraic equations ∗ ∗ + ir32 G3 , P2 r2 = iG2 Δr2 − iG4 r42 ∗ ∗ d2 r˜2 = −iG4 r13 + ir14 G3 , P3 r3 = i [G1 Δr1 − G1 r13 + r14 G4 ] , d1 r˜1 = −iG3 r42 + ir32 G4 P13 r13 = −iG∗3 r1 + ir3∗ G1 , P14 r14 = −iG1 r4∗ + ir1 G∗4 ,. (11). P42 r42 = −iG∗2 r4 + ir2∗ G4 , P32 r32 = −iG3 r2∗ + ir3 G∗2 , P3 r1 = iG3 Δr3 , P4 r4 = iG4 Δr4 ,. Γm rm = −2 Re{iG∗4 r4 } + qm , Γn rn = −2 Re{iG∗4 r4 } + γgn rg + γmn rm + qn , Γg rg = −2 Re{iG∗3 r3 } + qg , (12). Γl rl = −2 Re{iG∗3 r3 } + γgl rg + γml rm + ql ,. where G1 = −E1 dml /2, G2 = −E2 dgn /2, G3 = −E3 dgl /2, G4 = −E4 dmn /2, P1 = Γml + iΩ1 , P2 = Γng + iΩ2 , P3 = Γgl + iΩ3 , P4 = Γmn + iΩ4 , P32 = Γln + i(Ω3 −Ω2 ), P14 = Γln +i(Ω1 −Ω4 ), P42 = Γgm +i(Ω4 −Ω2 ), P13 = Γgm + i(Ω1 − Ω3 ), d2 = Γng + i(Ω3 + Ω4 − Ω1 ), d1 = Γlm + i(Ω3 − Ω2 + Ω4 ), Ω1 = ω1 − ωlm , Ω3 = ω3 −. ωlg , Ω2 = ω2 − ωgn , Ω4 = ω4 − ωmn , Δr1 = rl − rm , Δr2 = rn − rg , Δr3 = rl − rg , Δr4 = rn − rm . For a closed scheme, Eq. (12) must be replaced by rl = 1 − rn − rg − rm . The solution of these equations as applied to the problem under consideration is given in Appendix B. III. L ASER - INDUCED TRANSPARENCY, AMPLIFICATION AND GENERATION OF THE BACKWARD WAVE. The fundamental difference between the spatial distribution of the signal in ordinary and NI slabs is explicitly seen at αj = Δk = 0. Then, equation (A5) reduces to T10 = 1/[cos(gL)]2 .. (13). Equation (13) shows that the output signal and idler experience a sequence of geometrical resonances at gL → (2j + 1)π/2, (j = 0, 1, 2, ...), as functions of the slab thickness L and of the intensity of the control field (factor g). Such behavior is in drastic contrast with that in an ordinary medium, where the signal would grow exponentially as T1 ∝ exp(2gL). The resonances indicate that strong absorption of the lefthanded wave and of the idler can be turned into transparency, amplification and even into cavity-free self-oscillation when the denominator tends to zero. The reflection factor, r10 , and conversion factor η2L experience a similar resonance increase. Self-oscillations would provide for the generation of entangled counter-propagating left-handed, ω1 , and righthanded, ω2 , photons without a cavity. A similar behavior is characteristic for distributed-feedback lasers and is equivalent to a great extension of the NLO coupling length. It is known that even weak amplification per unit length may lead to lasing provided that the corresponding frequency coincides with high-quality cavity or feedback resonances. Numerical simulations described below in subsection III-A show that absorption and phase mismatch Δk = k3 − k2 − k1 may essentially change the properties of the spatial distribution and the output values of the signal. However, basic cardinal difference between the field distributions and transmission properties of an ordinary and NIM slab of the same optical thickness is explicitly seen at the numerical examples shown in Fig. 2. Figure 2(a) displays ”geometrical” resonances where amplification may exceed oscillation threshold, which provides mirrorless OPO. It contrasts with the exponential dependence depicted in Fig. 2(b) which is computed for a positive index slab. Figures 2(c)-(f) display the differences in the corresponding spatial distributions. A. Effect of absorption and phase mismatch on the laserinduced transparency resonances The fact that the waves decay towards opposite directions causes a specific strong dependence of the entire propagation process and, consequently, of the transmission properties of the slab on the ratio of the decay rates. A typical plasmonic NIM slab absorbs about 90% of light at the frequencies which are in the NI frequency-range. Such absorption corresponds to α1 L ≈ 2.3. As outlined above, the transparency exhibits. 110.

(5) World Academy of Science, Engineering and Technology 61 2010. 5. T1. α L=−3, α L=5, Δk=0. α2L=−3, α1L=5, Δk=0. 2. 10. 5. 1. T 8. 4. T1 α2L=3, α1L=2.3, Δk=0. 5. 1. 4. 4. 2. η. 3. 3. 3. 6. 2. 2. 2. 4. 1. 1. 1. 2. 0 0. 0 0. 0 0. 5. gL. 10. 15. 10. 1 gL 2. 3. 5 T1. (b). α2L=−3, α1L=5,. α L=−3, α L=5,. gL=1.815, Δk=0. gL=1.815, Δk=0. 10. 1. η2. 0 0. 0 0. 10 8 6. T1. 4. η2/10. T1. 10. 0.5 z/L 1 (e). ΔkL=π ΔkL=π/2 ΔkL=π/4 ΔkL=π/8. 0 0. 15. 5. gL. 10. 15. (d). α L=2.3, α L=4, Δk=0 1. 2. 5 T1. 4. 3. gL=1.98, Δk=0. 2. 2. 1. 1. T1. 6. η. 0 0. 5. 2. gL. 10. 0 0. 15. (a). 4. 5 T1. 0.5 z/L 1 (f). α L=2.3, α L=4, α =0 1. 2. 3. 4. 3. 8. 0 0. 10. α2L=−3, α1L=5,. 2. 2 0 0. 5. 0.5 z/L 1 (d). gL=1.98, Δk=0. 15. Fig. 3. Effect of inhomogeneity of the control field across the slab caused by its absorption on the transmission resonances at different phase mismatch.. 2. α2L=−3, α1L=5,. gL. 10. α1L=2.3, α2L=3, α3L=2.1. (c). 4. (c). 12. 5. gL. 4. 1. 6. 0.5 z/L 1. 5 T1. 2. 6. 0 0. α1L=2.3, α2L=3, α3L=2.1, Δk=0. 2 1. 2. 5. (b). 3. T1. η2/10. 0 0. 15. 3. 8. 4. 10. 4. 8. T1. gL. (a). (a) 2. 5. T1 α2L=3, α1L=2.3, Δk=π. ΔkL=π ΔkL=π/2 ΔkL=π/4 ΔkL=π/8. 5. gL. 10. 15. (b). α1L=2.3, α2L=4, α3L=2.1, Δk=0. 5 T1. 4. α1L=2.3, α2L=4, α3L=2.1 ΔkL=π ΔkL=π/2 ΔkL=π/4 ΔkL=π/8. 4. 3. 3. 2. 2. 1. 1. Fig. 2. [(a)] Transmitted negative-index signal, T1 (z = 0), and generated idler, η2 (z = L). [(b)] T1 (z = L) and generated idler, η2 (z = L) for positive-index slab of the same optical thickness α1 L and α2 L. α3 L = 0. [(c)-(f)] The difference in the distribution of the fields across the negativeindex, [(c) and (e)], and positive-index, [(d) and (f)], slabs. Panels (c) and (d) correspond to the left and (e) and (f) to the right slopes of the first peak in panel (a). Here, the materials are absorptive at the frequency of the signal and amplifying for the idler, Δk = 0.. Fig. 4. Changes in transparency with increase of the idler absorption. At given ratio of the absorption indices, transparency does not fall below 100% in the transmission minimums.. an extraordinary resonance behavior as a function of the intensity of the control field and the NIM slab thickness, which occurs due to the backwardness of the light waves in NIMs. Basically, such resonances are narrow and the sample. remains opaque anywhere beyond the resonance intensity of the control field. If nonlinear susceptibility varies within the negative-index frequency domain, this translates into relatively narrow-band filtering. Alternatively, the slab would become transparent within the broad range of the slab thickness and. 0 0. 5. gL. (c). 111. 10. 15. 0 0. 5. gL. 10. 15. (d).

(6) World Academy of Science, Engineering and Technology 61 2010. α1L=2.3, α2L=1, Δk=0. 5 T1. T1. 3. 4. α3L=1 α L=2. 3. 4. 2. 2. 1. 1 5. 10. gL. 0 0. 15. 5. (a). 4. 3. 2. 2. 1. 1 10. gL. 15. 15. ΔkL=π ΔkL=π/2 ΔkL=π/4 ΔkL=π/8. 4. 3. 5. 10. α1L=4, α2L=4, α3=0. 5 T1. 0 0. gL. (b). α1L=4, α2L=4, Δk=0. 5 T1. is transparent for the control field, which is often not the case in the real world. In order to demonstrate the major effects of absorption of all three fields and of the phase mismatch, the model is adopted here where the dependence of the local optical and NLO parameters on the intensity of the control field can be neglected and the parameter g is real. Such a model is relevant to, e.g., off-resonant quadratic and cubic nonlinearities attributed to the structural elements of metaldielectric nanocomposites [9]. The results will be used in Section V for optimization of transparency achievable through embedded resonant FWM centers with power-dependent optical parameters.. ΔkL=π Δk=π/2 Δk=π/4 Δk=π/8. 3. 3. 0 0. α1L=2.3, α2L=1, α3L=2.1. 5. α L=0. 0 0. 5. (c). gL. 10. 15. (d). Fig. 5. Transmission of the negative-index slab at α2 L  α1 L. Minimum transmission does not exceed 100%.. 5 T1. α L=5, α L=−3, α =2.1, Δk=0 1. 2. gL=9.51, α L=5, α L=−3, α =2.1, Δk=0 1. 1. 3. 4. 0.8. 3. 0.6. 2. 0.4. 1. 0.2. 0 0. T1. 5. 10. gL. 15. 2. 0 0. 0.5. 1. 2. z/L. The distribution of the signal and the idler inside the slab would also dramatically change with the ratio of the depletion rates (Fig. 6). Unless optimized, the signal maximum inside the slab may appear much greater than its output value at z = 0. The spatial distributions of the signal and the idler also experience a strong dependence on phase mismatch. Such dependencies are in strong contrast with their counterparts in PI materials and are determined by the backwardness of the coupled waves that is inherent to NIMs.. 1. (b). α L=2.3, α L=0.1, α =2.1, Δk=0. 1. 3. gL=9.51, α L=2.3, α L=0.1, α =2.1, Δk=0 1. 2. 3. T1. 4. 0.8. 3. 0.6. 2. 0.4. 1. 0.2. 0 0. 3. η. (a) 5. 2. T1. 5. gL. (c). 10. 15. 0 0. η. 2. 0.5. z/L. Figures 3-5 present numerical simulations of the dependence of transparency on ratio of the absorption indices at the frequencies of the coupled waves and on phase mismatch. Figures 3-4 show the feasibility of achieving robust transparency and amplification in a NIM slab at the signal frequency through a wide range of the control field intensities by the appropriate adjustment of the absorption indices α2 ≥ α1 . It is seen that the transmissions does not drop below 1 at α2 > α1 . Figure 5 proves that larger absorption for the idler is advantageous for robust transmission of the signal, which is counterintuitive. The increase of the idler’s absorption is followed by the relatively small shift of the resonances to larger magnitudes of gL. Oscillation amplitudes grow sharply near the resonances, which indicates cavity-less generation. Phase mismatch causes the decrease of maximums of the first resonances. Inhomogeneity of the control field due to its absorption causes the decrease of next maximums and their insignificant shift to the larger intensities.. 1. (d). Fig. 6. Transmission resonances and distribution of the signal and the idler inside the negative-index slab in the vicinity of second transmission minimum.. the control field intensity if the transmission in all of the minimums is about or more than 1. The results of the numerical simulations presented below (see. also [17]) show that such robust transparency can be achieved thorough the appropriate adjustment of the absorption indices at the frequencies of the coupled fields. The model used in [17] assumed that the slab. Qualitative explanation of the described dependencies revealed through numerical simulations is as follows. Besides the factor g, the local NLO energy conversion rate for each of the waves is proportional to the amplitude of another coupled wave and depends on the phase mismatch Δk. Hence, the fact that the waves decay in opposite directions causes a specific, strong dependence of the entire propagation process and, consequently, of the transmission properties of the slab on the ratio of their decay rates. Since the idler and the control field are absorbed toward the back facet of the slab and the signal experiences absorption in the opposite direction, the maximum of the signal for the given parameters is located somewhere inside the slab. A change in the slab optical thickness or in the intensity of the control fields leads to significant changes in the distributions of the signal and idler along the slab. Hence, the simulations suggest a general procedure of optimization and control of the output signal and the slab transparency without a change in its composition and structure.. 112.

(7) World Academy of Science, Engineering and Technology 61 2010. IV. N ONLINEAR - OPTICAL NEGATIVE - INDEX MIRROR. 5 4 3. r. 4. 10. 4. 2. α1L=2.3, α2L=3, α3L=2.1 ΔkL=π ΔkL=π/4 ΔkL=π/8. r1. 2. 2. 1. 1. 1. 1. 0 0. 0 0. 0 0. (a) 5. 5. 10. 15. 0 0. 5. gL. 10. 15. (d). Fig. 8. Effect of of absorption on reflectivity and transmittance of the NLO mirror at different phase mismatch.. r. 1. T. 4. T2. 3. 3. 2. 2. 1. 1. 0 0. 0 0. 5. 10. gL. 15. 5. 2. 10. gL. 15. (d). α L=2.3, α L=4, α =0, Δk=0 1. 10 r1 8. 2. (c). 4. gL. ΔkL=π/4. (c). α1L=2.3, α2L=2.3, α3L=2.1, Δk=0. 1. 5. 5. 15. (b). α1L=2.3, α2L=2.3, α3=0, Δk=0 r. 4. 10. gL. ΔkL=π. ΔkL=π/8. 2. 5. α1L=2.3, α2L=4, α3L=2.1. 3 2. 15. 15. 3 2. 10. 10. ΔkL=π/2. 3. gL. gL. 4. 4. T. 5 r1. 3. 5. 5. (b). 1. T. 0 0. 15. (a). r. 1. 3. 1 gL. ΔkL=π ΔkL=π/4 ΔkL=π/8. 4. 1 5. α1L=2.3, α2L=2.3, α3L=2.1. r1. 2. 0 0. α1L=2.3, α2L=1, α3L=2.1, Δk=0. 5. 5. 2. 5 α1L=2.3, α2L=1, α3=0, Δk=0. ΔkL=π ΔkL=π/2 ΔkL=π/4 ΔkL=π/8. r1. Alternatively, at a1L = 0, a2 (z = 0) = a20 , where only positive-index signal at ω2 and control field at ω3 enter a slab at z = 0 and generated difference-frequency electromagnetic wave at ω1 = ω3 − ω2 falls into negativeindex frequency domain, the slab serves as a generator of a backward wave at ω1 , i.e., as a NLO mirror. Basically, reflected wave has differen frequency and reflectivity may significantly exceed 100%. Results of numerical simulations of such process and properties of the mirror are presented below. Figure 7 displays reflectivity resonances which 5. α1L=2.3, α2L=1, α3L=2.1. 3. 5. 1. T2. T. 2. 2. 1. 1. 0 0. 0 0. 5. gL. 10. (e). 15. 3. gL=2 gL=2.5 gL=3. 4. 4. 2. 2. 3. 0.5. ΔkL/π. 1. 10 r1 8. gL=2 gL=2.5 gL=3. 15. 1. 2. 3. gL=2 gL=2.5 gL=3. 0.5. ΔkL/π. 1. 1.5 r. α1L=2.3, α2L=4, α3L=2.1 gL=2 gL=2.5 gL=3. 1. 1. 4 10. α L=2.3, α L=1, α L=2.1. (b). α1L=2.3, α2L=4, α3=0. 6. gL. 0 0. (a). 2. 5. 10 r1 8 6. 1. 3. 2. 6. 0 0. 2. r. 3. 1. α L=2.3, α L=4, α L=2.1, Δk=0. r1. 4. α L=2.3, α L=1, α =0. 0.5. 2. (f). 0 0. Fig. 7. Effect of absorption on reflectivity and transmittance of the NLO mirror. Here, reflectivity can be switched from zero to the magnitudes exceeding 100%.. may exceed self-oscillation threshold. Transmission minimums depend on the ratio of absorption rates, whereas reflectivity minimums remain robust, which is in strict contrast with the process investigated in preceding Section IV. Alternatively, phase mismatch causes decrease of the reflectivity maximums and increase the minimums (Fig. 8). Reflectivity becomes relatively robust against phase mismatch with increase of intensity of the control field. It drops dramatically in the range. 0.5. (c). ΔkL/π. 1. 0 0. 0.5. ΔkL/π. 1. (d). Fig. 9. Effect of phase mismatch on reflectivity and transmittance of the NLO mirror at different intensities of the control field.. of small phase mismatch and then remains relatively robust witin the range of greater phase mismatch (Fig. 9 and 10). The outlined properties of NLO mirror are determined by interplay of several processes which have strong effect on NLO coupling of contrapropagating waves as indicated in Section IV and, consequently, on their distribution inside the slab [Figs. 11. 113.

(8) World Academy of Science, Engineering and Technology 61 2010. α L=2.3, α L=1, α L=0 1. 2. α L=2.3, α L=1, α L=2.1. 3. 1. r 5. 1. 5. 0 −1. 3. gL=3, α1L=2.3, α2L=4, α3L=2.1, Δk=0. 1.5. 1.5. r. gL=3, α1L=2.3, α2L=1, α3L=2.1, ΔkL=π/2. r. 1. 1. 1. T2. T2. 1. 1. 0.5. 0.5. 0 −1. ΔkL/π. 0. 10 5 1 0. gL. ΔkL/π. 0. 10 5 1 0. (a) α L=2.3, α L=4, α L=0 1. 2. gL. 0 0. (b) 1. 2. 5. 1. 0 0. 1. 1. 2. 3. 1.5. r1. gL=8, α L=2.3, α L=1, α L=2.1, ΔkL=π/2 1. 2. 3. r1. T. T. 2. 0 −1 ΔkL/π. 0 −1 0. 10 5 1 0. (c). gL. ΔkL/π. 0. 1. (b). gL=10, α L=2.3, α L=4, α L=2.1, Δk=0. 1.5. z/L. 0.5. (a). 3. r. 1. z/L. 0.5. α L=2.3, α L=4, α L=2.1. 3. r 5. 2. r. 2. 1. 1. 0.5. 0.5. 10 5 1 0. gL. (d) 0 0. Fig. 10. Reflectivity vs. intensity of the control field and phase mismatch for different absorption indices for the coupled waves.. 1. 0 0. 2. 3. 1.5. r1. gL=10, α L=2.3, α L=4, α L=2.1, ΔkL=π 1. 2. 3. r1. T. T. 2. 2. 1. 1. 0.5. 0.5. 0 0. 0.5. (e). 1. (d). gL=10, α L=2.3, α L=4, α L=2.1, ΔkL=π/8 1. z/L. 0.5. (c) 1.5. and 12]. Ultimately, the simulations show possibility to tailor and to switch reflectivity of such a mirror in the wide range by changing intensity of the control field. Only rough estimations can be made regarding χ(2) attributed to metal-dielectric nanostructures. Assuming χ(2) ∼ 10−6 ESU (∼ 103 pm/V), which is on the order of that for CdGeAs2 crystals, and a control field of I ∼ 100 kW focused on a spot of D ∼ 50μm in diameter, one can estimate that the typical required value of the parameter gL ∼ 1 can be achieved for a slab thickness in the microscopic range of L ∼ 1μm, which is comparable with that of the multilayer NIM samples fabricated to date [2], [4].. z/L. 0.5. z/L. 1. 0 0. 0.5. z/L. 1. (f). Fig. 11. Intensity distribution for the ordinary and backward waves inside the slab for different absorption indices, phase mismatch and intensity of the control field.. V. E MBEDDED N ONLINEARITY The above described features allow to propose and to optimize the feasibility of independently engineering the NI and the resonantly enhanced higher-order (χ(3) ) NLO response of a composite metamaterial with embedded NLO centers (ions or molecules) [Fig. 1(d)-(e)]. The sample is illuminated by two control fields, E3 and E4 , so that the amplification of the NI signal, E1 , and the generation of the counter-propagating PI idler, E2 , occur due to the FWM process ω1 + ω2 = ω3 + ω4 . The transmission factor for the signal, T1 (z = 0) can be computed as described in Subsection II-C. Due to resonant or near-resonant coupling, all local parameters here become strongly dependent on the intensity of the control fields and can be tailored by the means of quantum control. The schemes Fig. 2, (b), (d), and Fig. 2, (c), (e), provide for different relations between local linear and NLO parameters at ω1 and ω2 and for their dependencies on the control fields. Linear and NLO local parameters attributed to the embedded centers are calculated by the density-matrix method as described in Appendix B. The strength of the control fields is represented by the coupling Rabi frequencies G3 = E3 dlg /2 and. G4 = E4 dnm /2, where dij are electrodipole transition matrix elements. The quantities α10 and α20 denote the value of fully resonant absorption introduced by the embedded centers at frequencies of the corresponding transitions with all driving fields turned off. The absorption, attributed to the host slab at the frequencies of the signal and the idler, are taken αh1 L=2.3, and αh2 L=2.1 for both schemes Fig. 2, (d) and (e). For scheme Fig. 1(e), the electrical linear and nonlinear polarizations, Eq. (3), are calculated as L exp(ik1 z) P1 (z, t) = (1/2){P01 NL +P01 exp[i(k3 + k4 − k2 )z]} exp(−iω1 t) + c.c. = N (ρng dgn + ρgn dng ); L P2 (z, t) = (1/2){P02 exp(ik2 z) NL +P02 exp[i(k3 + k4 − k1 )z]} exp(−iω2 t) + c.c. = N (ρml dlm + ρlm dml ).. Here, ρij are the density matrix elements, and dij are the transition dipole elements. For scheme Fig. 1(d), they are calculated in the similar way. Effective linear, χ1,2 , and NLO,. 114.

(9) World Academy of Science, Engineering and Technology 61 2010. gL=3, α L=2.3, α L=1, α =0 1. 2. 2. gL=3, α L=2.3, α L=1, α L=2.1. 3. 1. 2. 3. r. r1. 15. 5 6 T1 10 4 104. 1. 10. 3. 5. 1. ΔkL/π 0. 2. 1.5 r 1 1. 2. 0.5 1. ΔkL/π 0. 0.5 1 0. z/L. 4. T. 0. 100. 2. 200. 1. 0 60. 0 −0.4 −0.2. 100 α10L. 80. (a). z/L. 0.5 1 −1. 0 ΔkL/π. Fig. 12. Dependence of distribution of generated backward wave inside the slab on phase mismatch.. (3). χ1,2 , susceptibilities dependent on the intensities of the driving control fields E3 and E2 are defined as χ2 E2 ,. ys. (b). G =0.29GHz, G =2.27GHz, 3 4 Ω =Ω =Ω =0 2 3 4 6. T1. 10. =. 0.2. A similar dependence for the alternative option Fig. 2, (c) and (e), is shown in Figs. 14, (a) and (b). The plots also prove the possibilities of compensating strong. 1.5. χ1 E1 ,. 0. 1. (d). =. 4 3. 0. 10. 1. 0.5 z/L. 0 0. NL P01 NL P02. G3=0.361GHz, G4=2.773GHz, Ω =Ω =0, α L=98.65 3 4 10. Fig. 13. Transmission of the negative-index signal in the vicinity of the higher frequency transition [Fig. 2, (d)], tailored by the fully resonant control fields. [(a)] Transmission of the signal vs. optical thickness of the slab, α10 L. ω1 = ωml . [(b)] Transmission of the signal at α10 L=98.65 vs. signal resonance detuning. ys = (ω1 − ωml )/Γln , ω2 = ω3 + ω4 − ω1 . Coupling Rabi frequencies for the control fields are: G3 =0.361 GHz, G4 =2.773 GHz.. (c). L P01 L P02. 5. 1. gL=3, α1L=2.3, α2L=4, α3L=2.1. 3. 4. 0 −1. −1 0. (b). gL=3, α L=2.3, α L=4, α =0, 1. 1. 0. (a) r1. 3. 10. 2. 0 1 ΔkL/π. 0.5 z/L. −1 0. 1. 2. 1 0 1. G3=0.361GHz, G4=2.773GHz, Ω =Ω =Ω =0. = =. 1. 4. 20. 0.8. 2. 10. 0.6. 0. 10. 0.5. 3. 1. 4. 10. (3) χ1 E3 E4 E2∗ ; (3) χ2 E3 E4 E1∗ .. G =0.29GHz, G =2.27GHz, 3 4 Ω =Ω =0, α L=60. 0. 100. 200. 0.4 0.2. The linear susceptibilities determine the intensity-dependent contributions to absorption and to the refractive indices of the composite attributed to the embedded centers, while the NLO susceptibilities determine the FWM. Here, ω1 +ω2 = ω3 +ω4 , and kj = |nj |ωj /c > 0. A. Fully resonant control fields The results of numerical simulations for two examples of fully resonant control fields, Ω3 = ω3 − ωgl = 0 and Ω4 = ω4 − ωmn = 0, are presented in Fig. 3 and 4. Relaxation properties of the model are taken as follows: energy level relaxation rates Γn = 20, Γg = Γm = 120; partial transition probabilities γgn = 50, γmn = 90, (all in 106 s−1 ); homogeneous transition half-widths Γlg = 1, Γlm = 1.9, Γng = 1.5, Γnm = 1.8 (all in 1011 s−1 ); Γgm = 5, Γln = 0.5 (all in 109 s−1 ); λ1 = 756 nm and λ2 = 480 nm. Figures 13, (a) and (b), depict the dependence of the transmission of the signal on the thickness of the doped slab and on its resonance frequency for coupling schemes Fig. 2, (b) and (d). The plots displays the appearance of transmission and amplification that may exceed the oscillation threshold. For the given fields and relaxation parameters, the ratios of the power-dependent energy level populations occur rl ≈ 0.42, rn ≈ 0.2, rg ≈ 0.19, rm ≈ 0.19 and, hence, no population-inversion or Raman-type gain is involved in the coupling. Figure 13 proves the feasibility of compensating losses, producing narrow-band transparency, amplification, and mirror-less generation.. 0 0. 20. 40 α20L 60. (a). 0 −10. 0. ys. 10. (b). Fig. 14. Resonant coupling in the scheme Fig. 1 (e). [(a)] Transmission of the signal vs. optical thickness of the slab, α20 L. ω1 = ωgn . [(b)]: Transmission of the signal at α20 L=60 vs. signal resonance detuning. ys = (ω1 − ωgn )/Γln , ω2 = ω3 + ω4 − ω1 . Coupling Rabi frequencies for the control fields are: G3 =0.29 GHz, G4 =2.27 GHz. Dash line shows transmission at χ(3) = 0.. losses in NIMs through independently engineered embedded resonant nonlinearity. For the given fields and relaxation parameters, the ratios of the power-dependent energy level populations are: rl ≈ 0.48, rn ≈ 0.18, rg ≈ 0.17, rm ≈ 0.17. Hence, counterintuitively, the simulations show that one can produce transparency of the initially strongly lossy NIM by introducing an additional strong absorption at the frequencies in the vicinity of transition ml through the embedded centers. B. Quasi-resonant control fields Here, the case will be simulated, where the idler corresponds to a higher-frequency transition from the ground state, and the signal corresponds to a lower-frequency transition between the excited states [Fig. 2, (e)]. No incoherent amplification is possible here for the idler, and the dependence of the idler and the signal absorption indices on the control fields cardinally changes. First, the scheme with relatively fast quantum coherence relaxation rates and the case where only. 115.

(10) World Academy of Science, Engineering and Technology 61 2010. Ω =30Γ , G =254.64GHz,. Ω =30Γ , G =254.64GHz,. g/α. 116. 20 2. γ /α. g/α20. 20. γ1/α20. Δk/α. 20. 3 gl 3 3 gl 3 a two-photon, Raman-like resonance for the signal holds is Ω =20Γ , G =108.48GHz Ω =20Γ , G =108.48GHz 4 mn 4 4 mn 4 0.1 0.01 considered; all other one-photon frequency offsets are on the α /α 2 20 order of several tens of the optical transition widths. Then the 0.05 α /α scheme with the same quantum coherence relaxation rates but 1 20 with higher partial spontaneous transition rates is considered, 0 0 in which case population inversion at the coupled optical transitions is impossible. Finally the scheme with longer −0.05 quantum coherence lifetimes will be considered, which still does not allow population inversion at the optical transitions −0.01 −0.1 20 20.05 20.1 y1 0 20 40 y1 nor Raman-like amplification. The fact that all involved op(a) (b) tical transitions are absorptive determines essentially different features of the overall loss-compensation technique in such Ω =30Γ , G =254.64GHz, Ω =30Γ , G =254.64GHz, 3 gl 3 3 gl 3 −3 Ω =20Γ , G =108.48GHz composites in each proposed scheme. In all of the schemes Ω =20Γ , G =108.48GHz x 10 4 mn 4 4 mn 4 0.01 5 outlined above, the linear and nonlinear local parameters can Im Im Re Re be tailored through quantum control by varying the intensities 0 and frequency-resonance offsets for combinations of the two control driving fields. 0 The following model, which is characteristic of ions and −0.01 some molecules embedded in a solid host, has been adopted: energy level relaxation rates Γn = 20, Γg = Γm = 120; partial −5 transition probabilities γgn = 50, γmn = 70, (all in 106 s−1 ); −0.02 20 20.05 20.1 y1 20 20.05 20.1 y1 homogeneous transition half-widths Γlg = 1, Γlm = 1.9, (c) (d) Γng = 1.5, Γnm = 1.8 (all in 1012 s−1 ); Γgm = 5, Γln = 1 (all in 1010 s−1 ); λ1 = 756 nm and λ2 = 480 nm. The densityΩ3=30Γgl, G3=254.64GHz, Ω =30Γ , G =254.64GHz, 3 gl 3 Ω =20Γ , G =108.48GHz Ω =20Γ , G =108.48GHz matrix method [18] is used for calculating the intensity4 mn 4 4 mn 4 0.01 0.01 dependent local parameters while accounting for the quantum nonlinear interference effects. This allows us to investigate the changes in absorption, amplification, and refractive indices as well as in the magnitudes and signs of NLO susceptibilities 0 0 caused by the control fields. These changes depend on the Re population redistribution over the coupled levels, which in Im turn strongly depends on the ratio of the partial transition −0.01 −0.01 probabilities. 20 22 y1 24 19.95 20 20.05 20.1 y1 Figure 15 depicts such modifications at the given res(e) (f) onance offsets and intensities of the control fields. Here, Ω1 = ω1 − ωgn ; other resonance detunings Ωj are defined Fig. 15. Nonlinear spectral structures in local optical quantities produced in a similar way. Coupling Rabi frequencies are introduced by the control fields. y1 = (ω1 − ωgn )/Γgn , ω2 = ω3 + ω4 − ω1 . (a): as G3 = E3 dlg /2 and G4 = E4 dnm /2. The quantity α20 absorption/gain indices for the signal and the idler; (b): phase mismatch; (c)denotes the fully resonant value of absorption introduced by (f): four-wave mixing coupling parameters. Coupling Rabi frequencies and resonance frequency offsets for the control fields are: G3 = 254.64 GHz, the embedded centers at ω2 = ωml with all driving fields Ω3 = 30Γgl , G4 = 108.48 GHz, Ω4 = 20Γmn . turned off. Figure 15(a) displays the modified absorption/gain indices. The nonlinear spectral structures are caused by the modulation of the probability amplitudes, which exhibits itself as an effective splitting of the energy levels coupled with [Fig.15(a)]. Alternatively, two-photon, Raman-like amplificathe driving fields. Figure 15(b) shows the contribution to the tion at Ω1 ≈ 20.05Γgn shown in Fig.15(a) supports coherent, phase mismatch associated with one such spectral structure. parametric energy-conversion from the control fields to the Figure 15(c) and (d) indicate that the real and imaginary signal. Figures 17(a)-(d) display the spectral properties of parts of the NLO susceptibilities become commensurate for the output signal at z = 0 for one of the resonances in the the given susceptibility, but may exceed their counterparts for vicinity of the signal frequency offset ω1 − ωgn ≈ 20.05Γgn the idler by several times. This occurs due to the fact that at different optical densities of the slab at ωml attributed to different population differences contribute in different ways the impurity centers. Assume that the absorption of the host to the NLO susceptibilities [18], and driving fields cause material in the slab at ω1 is fixed at 90% and it is equal significant redistributions of the level populations (Fig. 16). to 88% at ω2 . The density of the embedded centers and At the given partial probabilities of spontaneous transition the slab thickness, and hence, the additional resonant optical between the levels, population inversions at the signal tran- thickness of the slab contributed by these impurities, may vary sition become possible [Fig.16(e),(g)-(i)]. However, for the as shown in the panels. Actual quasi-resonant absorption/gain given frequency offsets of the control fields, corresponding indices depend on the intensities and frequency offsets of amplification contributes negligibly to the energy conversion the control fields, as shown in Fig.15(a). Besides the features.

(11) World Academy of Science, Engineering and Technology 61 2010. rg. r. m. 8 T1. 0.4. 6. 0.2. Ω3=30Γgl, G3=254.64GHz, Ω =20Γ , G =108.48GHz 4. mn. α L=250. 1. 20. 200 150. 0.2. 10. 4. 100. G. 100. 0 0. 4. 200 G3. 300. 0 200 100 G4. (a). 100. 0 0. 200 G3. 2. 300. 0. 10 0 20.02. 20.04. T1. 1. 0.6. 10. 10. 5 α20L=364.8. 0 300. 100 0 200 G 4. 200 100 G 3. (c) Δr1. 2. 100 G 4. 20.02. 20.04. 20.06. y. 20.02 20.04 20.06. 10. 10 T1. 0.5. Ω =30Γ , G =254.64GHz, 3 gl 3 Ω =20Γ , G =108.48GHz 4. mn. 4. 0. 0 G. 0. 100 200 300. 4. 0 0 100 G 200300 0 3. 200. 100 G3. (e) gl. 4. 100. G4. (f). Ω =30Γ ,Ω =20Γ 3. 200. 3. Ω =30Γ ,Ω =20Γ. rl rg. 1. 3. gl. 4. n. rm. 0. 4. 10 360. g n. 100. (h). G4. 200. 1. T. 2. η2. 365 α20L 370. 0 0. 0.5. z/L. 1. (f). m. Fig. 17. Dependence of the transmission of the slab on the resonance frequency offset y1 = (ω1 − ωgn )/Γgn for different optical densities of the slab, (a)-(d), on the resonant optical density of the slab, (e), and the distribution of the signal and the idler along the slab, (f). G3 =254.64 GHz, Ω3 = 30Γgl , G4 =108.48 GHz, Ω4 = 20Γmn .. 0.2 0 0. 4. α20L=365.6 y =20.0396077. r. 0.4. 0.2. mn. y1=20.0396077. (e). r. 0.6. 4. 1. , G =108.48GHz. mn. 4 y1=20.058211521. r. r. 0.4. 10. 100 G 4. rl. 0.8. Ω3=30Γgl, G3=254.64GHz, 11 Ω =20Γ , G =108.48GHz. 8 x 10. 5. 200. (g). , G =254.64GHz. mn. 1. (d). 6. 0 100 200 G 300 0 3. y. 1. (c). 1. 0.5. 0. α20L=365.6. T1. 0. 10. rl−rn. 1. 0.2. Ω3=30Γgl, G3=254.64GHz, Ω4=20Γmn, G4=108.48GHz. 10. 0. 200. (d) Δr. 0.4. 1. 10. 0 0 100 G 200 3 300 0. 0. y. 20.06. 5. 0.5. 0.2. 20.04. (b). Ω3=30Γgl, G3=254.64GHz, Ω4=20Γmn, G4=108.48GHz. 10. 0.4. 20.02. (a). rl. n. y1. 20.06. (b). r. α20L=363.9. 5. 0.1 0 200. Ω =30Γ , G =254.64GHz, 3 gl 3 Ω4=20Γmn, G4=108.48GHz. 10. 10 T. 4. 0 0. 100. 200 G 300 3. (i). Fig. 16. Difference of the energy-level populations and their dependence on the Rabi frequency of the control fields G3 and G4 (given in GHz). Ω3 = 30Γgl , Ω4 = 20Γmn . (h): G3 =254.64 GHz, (i): G4 =108.48 GHz.. imposed by the counter-propagation of the coupled waves, the output magnitudes of the signal at z = 0 and the idler at z = L and their distributions inside the slab are determined by the interplay of several contributing linear and nonlinear processes. They include the phase mismatch, absorption of the signal and the idler, and the parametric gain g, which are all controlled by the driving fields E3 and E4 . The dependence of the overall optimized output signal on the density of the impurities and on the slab thickness (on the resonant optical thickness of the slab) is depicted in Fig. 17(e). Such a behavior is determined by the radically different distributions of the. idler, which propagates from left to right, and the signal, which propagates from right to left, [Fig. 17(f)]. Figures 17(b)-(f) indicate the possibility of mirrorless self-oscillation. Figure 18 shows the role of partial spontaneous transitions between the energy levels. Here, γmn = 9 × 107 sec−1 , which makes both population inversion and two-photon gain impossible [Fig. 18(a)]. At the indicated Rabi frequencies and frequency offsets for the driving control fields, the energylevel populations are: rl ≈ 0.4, rg ≈ 0.2009, rn ≈ 0.2031, rm ≈ 0.2. The magnitude of the four-wave mixing coupling parameters appear comparable with those depicted in Fig. 15(c)-(h). However, the absence of one- and two-photon amplification that would support energy-conversion processes, like in [8] and in Fig. 15(a), dramatically decreases the achievable amplification and increases the required optical thickness of the slab [Fig. 18(g),(h)]. Figure 19 shows that, even in such. 117.

(12) World Academy of Science, Engineering and Technology 61 2010. Ω3=−Ω4=30Γgl, G3=37.5GHz, G4=202GHz. Ω3=3Γgl, G3=0.86GHz, Ω =−2Γ , G =8.99GHz. 0.4. α /α 2. 0. 0 −0.02 20. 20.02. −0.04. 0.1. −0.2. 20. 20.02. 20.04. 0 −5. y1. (b) 4 x 10. Im Re. 4. mn. 4. Im Re. 20. −2. 4. 2. 2.1. y 2.2 1. Ω3=3Γgl, G3=0.86GHz, Ω =−2Γ , G =8.99GHz 4. 0.01. mn. 4. Im Re. 0. 2. γ /α. γ1/α20. 20 2. 0. mn. (b). Ω3=3Γgl, G3=0.86GHz, Ω =−2Γ , G =8.99GHz. 0.04. 2 γ /α. γ1/α20. −0.3 1.9. (a). Ω =−Ω =30Γ , 3 4 gl −3 G =37.5GHz, G4=202GHz 3. Ω3=−Ω4=30Γgl, G3=37.5GHz, G4=202GHz. y1 5. 0. 4. 0. −0.1. (a). 0. α1/α20. 0.3. Ω =3Γ , G =0.86GHz, 3 gl 3 Ω =−2Γ , G =8.99GHz. 0.1. 0.2. y1. 20.04. 4. Δk/α20. 0.02. mn. α2/α20. 20. α1/α20. 0.04. 4. Δk/α20. Ω3=−Ω4=30Γgl, G3=37.5GHz, G4=202GHz. 0.06. 0. −0.01. −4 −0.01 20. 20.02. 20.04. −6 20. y. 1. (c) 3. r. l. n. 20 y1. 10. T1 2. Ω3=−Ω4=30Γgl, α20L=800 G =37.5GHz, G =202GHz 3. 4. T1 2 1.5. 1. 1. 0.5. 0.5 20.02. 20.04. (g). y1. 0 0. 2. 100. G. 4. Im Re. −4. 1.2. 400. 0.5. r. l. 0.4. rg. 0.3. rn rm. −2. 0.1 0. 0 0. 2 y1 4. 5 G4,GHz 10. (f). Ω3=3Γgl, G3=0.86GHz, α20L=150 Ω4=−2Γmn, G4=8.99GHz. T1. 200. Ω3=3Γgl, Ω4=−2Γmn, G3=0.86GHz. (e). Ω3=−Ω4=30Γgl, y1=20.021 G3=37.5GHz, G4=202GHz. 2.1 y1 2.2. 0.2. −0.02. 200. 2. 2.2. 0. (f). 1.5. 0 20. −0.02. g/α. 0 0. (e). 4. 0. 0.02 20. 20.02 20.04. 0. mn. 0.02. 0.2. −0.01 20. 4. r. 0.01. 0. −0.02 1.9. (d). Ω =3Γ , G =0.86GHz, 3 gl 3 Ω =−2Γ , G =8.99GHz. 0.04. rg rm. 0. 2.1 y1 2.2. 2. Ω3=−Ω4=30Γgl, G3=37.5GHz. 4. 0.4. g/α20. 1.9. (c). Im Re. −0.01. y1. 20.04. (d). Ω3=−Ω4=30Γgl, G =37.5GHz, G =202GHz. 0.01. −0.04 20.02. 1.5. Ω3=3Γgl, G3=0.86GHz, y1=2.029 Ω =−2Γ , G =8.99GHz 4 mn 4. T. 1. 0.8. 1. 0.4. 0.5. 600 α20L. 0 2. (h). Fig. 18. Energy-conversion in the scheme with neither population inversion nor two-photon gain possible [γmn = 9 × 107 sec−1 ); all other relaxation parameters are the same as in the previous case]. y1 = (ω1 − ωgn )/Γgn , ω2 = ω3 + ω4 − ω1 . (a): absorption indices for the signal and the idler; (b): phase mismatch; (c)-(f): four-wave mixing coupling parameters; (g) and (h): transmission factor, the dashed line shows transmission at g = 0. Coupling Rabi frequencies and resonance frequency offsets for the control fields are: G3 =37.5 GHz, G4 =202 GHz, Ω3 = −Ω4 = 30Γgl .. cases, the optimized magnitude of the required control field intensities and the slab optical density can be substantially reduced for centers with lower coherence relaxation rates and quasi-resonant coupling. Here, quantum nonlinear interference effects play an important role [18]. At the indicated Rabi. 2.05. (g). y1 2.1. 0 0. 50. 100. α L 20. (h). Fig. 19. Quasi-resonant coupling at lower quantum coherence relaxation rates and at neither population inversion nor two-photon gain possible [Γgl =1.8, Γmn =1.9, Γgn =1, Γml =1.5, Γmg = 5 × 10−2 , Γnl = 5 × 10−3 (in 1011 sec−1 ); all other relaxation parameters are the same as in the previous case]. y1 = (ω1 − ωgn )/Γgn , ω2 = ω3 + ω4 − ω1 . (a): absorption indices for the signal and the idler; (b): phase mismatch; (c)-(e): four-wave mixing coupling parameters; (f): energy level populations; (g) and (h): transmission factor, dash line shows transmission at g = 0. Coupling Rabi frequencies and resonance frequency offsets for the control fields are: G3 =0.86 GHz, Ω3 = 3Γgl ; G4 =8.99 GHz, Ω4 = −2Γmn .. frequencies and frequency offsets for the driving control fields shown in Fig. 19, the energy-level populations are: rl ≈ 0.504,. 118.

(13) World Academy of Science, Engineering and Technology 61 2010. rg ≈ 0.165, rn ≈ 0.167, rm ≈ 0.164. Like in the previous examples, the losses in the host NIM material are taken to be fixed and equal to αN IM 1 L = 2.3 for the signal and αN IM 2 L = 2.1 for the idler. The requirements for the proposed all-optical control of transmission and reflectivity of a doped metamaterial slab are as follow. For the given transitions, the magnitude G=1 GHz corresponds to the control field intensities of I ∼ 1 W/(0.1mm)2 . At a resonance absorption cross-section σ40 ∼ 10−16 cm2 , which is typical for transitions with oscillator strength of about one, and a concentration of the embedded centers N ∼ 1019 cm−3 , one estimates α10 ∼ 103 cm−1 and the required slab thickness in the microscopic range L ∼ (1 − 100)μm. The contribution by the impurities to the refractive index is estimated as Δn < 0.5(λ/4π)α40 ∼ 10−3 , which essentially does not change the negative refractive index.. VI. C ONCLUSION In conclusion, the feasibility of compensation of strong losses in negative-index metamaterials, e.g., in metal-dielectric nanostructured composites is numerically simulated. This is the key problems that determines numerous revolutionary applications of such a novel class of electromagnetic metamaterials. All-optical tailoring of transparency and reflectivity is shown to be possible through coherent nonlinear-optical energy transfer from the ordinary control electromagnetic wave to the negative-index, backward-wave. Backwardness of traveling electromagnetic waves is intrinsic to the negativeindex metamaterials. It shown that besides the nonlinearity attributed to the building blocks of the negative-index host, a strong nonlinear optical response of the composite can be provided by the embedded resonant four-level nonlinearoptical centers. This opens the way to independent nanoengineering and adjustment of negative index and nonlinearity of metamaterials. In addition, the opportunity for quantum control over the local optical parameters of the metamaterial has been shown in this case, which employs constructive and destructive quantum interference tailored by two auxiliary driving control fields. Such a possibility is proven with the aid of a realistic numerical model. Among the possible applications of the proposed technique are a novel class of the miniature frequency-tunable narrow-band filters, quantum switchers, amplifiers, cavity-free microscopic optical parametric oscillators that allow generation of entangled counterpropagating left- and right-handed photons, and all-optical data processing chips. The unique unparalleled features of the underlying processes are revealed, such as the strongly resonant behavior with respect to the material thickness, the density of the embedded resonant centers and the intensities of the control fields, the feasibility of negating the linear phasemismatch introduced by the host material, and the role of absorption or, conversely, the supplementary nonparametric amplification of the idler. .. A PPENDIX A S PECIAL C ASES : M ANLEY-ROWE RELATIONS AND SOLUTIONS TO THE EQUATIONS FOR COUPLED COUNTER - PROPAGATING WAVES. Consider special case where spatial inhomogeneity of the control fields can be neglected. Then Eq. (9)-(10) reduce to the the coupled equation for slowly varying amplitudes a1 and a2 at a3 and a4 taken constant. At α1,2 = 0, g1 = g2 , e.g., for off-resonant coupling, one finds with the aid of equations (2) and (9), (10):  . S2z d  d S1z |a1 |2 + |a2 |2 = 0. − = 0, dz ω1 ω2 dz These equations represent the Manley-Rowe relations [10], [11], which describe the creation of pairs of entangled counterpropagating photons ω1 and ω2 . The second equation predicts that the sum of the terms proportional to the squared amplitudes of the signal and idler remains constant through the sample, which is due to the opposite signs of S1z and S2z and is in contrast with the requirement that the difference of such terms is constant in the analogous case in ordinary nonlinear-optical materials. Taking into account the boundary conditions a1 (z = L) = a1L , and a2 (z = 0) = a20 (L is the slab thickness) and assuming a3 and a4 constant, the solutions to equations (9), (10) can be written as Δk )z] + 2 Δk + A2 exp[(β2 + i )z], 2 Δk a∗2 (z) = κ1 A1 exp[(β1 − i )z] + 2 Δk + κ2 A2 exp[(β2 − i )z], 2 a1 (z) =. A1 exp[(β1 + i. (A1). (A2). where β1,2 = (α1 − α2 )/4 ± iR, κ1,2 = [±R + is]/g,  R = g 2 − s2 , g 2 = g2∗ g1 , s = (α1 + α2 )/4 − iΔk/2, Δk )L]}/D, A1 = {a1L κ2 − a∗20 exp[(β2 + i 2 Δk A2 = −{a1L κ1 − a∗20 exp[(β1 + i )L]}/D, 2 Δk Δk D = κ2 exp[(β1 + i )L] − κ1 exp[(β2 + i )L]. 2 2 At Δk = 0 and Im g = 0, (α1 + α2 )L π (off-resonance), equations (A1) and (A2) reduce to a∗1 (z) ≈ a2 (z) ≈. ia20 a∗1L cos(gz) + sin[g(z − L)], cos(gL) cos(gL) ∗ a20 ia1L sin(gz) + cos[g(z − L)]. cos(gL) cos(gL). The output amplitudes are then given by. 119. a∗10. =. a2L. =. [a∗1L /cos(gL)] − ia20 tan(gL),. ia∗1L tan(gL) + [a20 /cos(gL)]..

(14) World Academy of Science, Engineering and Technology 61 2010. At a20 = 0, the equations for the energy distribution for the 2 backward wave, T1 (z) = |a1 (z)/a1L | , and for the PI idler, ∗ 2 η2 (z) = |a2 (z)/a1L | , across the slab take the form 2. T1 (z) =. |[κ2 exp (β1 z) − κ1 exp (β2 z)]/D| ,. (A3). η2 (z) =. |[exp (β1 z) − exp (β2 z)]/D| .. (A4). 2. R1 = Δr1 (1 + v5 + g5 ) (1 + v5 − g6 )Δr4 g1 (1 + g5 − v6 )Δr3 − v1 1− (1 + v5 + g5 )Δr1 (1 + v5 + g5 )Δr1 , × (1 + g4 + v4 ) + [v5 + v4 (v5 − g6 ) + g5 + g4 (g5 − v6 )] ∗ ∗ g1 = |G3 |2 /P13 P3∗ , g2 = |G3 |2 /P32 P2 , g3 = |G3 |2 /P42 P4∗ ,. Then the transmission factor for the backward-wave signal at z = 0, T10 , and the output idler at z = L, η2L , are given by. a1 (0) 2 exp {− [(α1 /2) − s] L} 2. , (A5). =. T10 =. a1L. cos RL + (s/R) sin RL. 2. a2 (L) 2. (g/R) sin RL. . (A6). =. η2L =. a1L. cos RL + (s/R) sin RL. At a1L = 0, a2 (z = 0) = a20 , the slab serves as an NLO mirror with a reflectivity r10 (output conversion efficiency) at ω1 given by the equation identical to Eq.(A6):. 2. a1 (0) 2. (g/R) sin RL. .. (A7) r10 = ∗ =. a20 cos RL + (s/R) sin RL. g4 = |G3 |2 /P13 P1 , g5 = |G3 |2 /P14 d∗2 , g6 = |G3 |2 P13 d∗2 ,. ∗ ∗ ∗ ∗ g7 = |G3 |2 /P42 d1 , g8 = |G3 |2 /P32 d1 , v1 = |G4 |2 /P14 P4∗ ,. ∗ ∗ v2 = |G4 |2 /P42 P2 , v3 = |G4 |2 /P42 P4∗ , v4 = |G4 |2 /P14 P1 ,. ∗ ∗ v5 = |G4 |2 /P13 d∗2 , v6 = |G4 |2 /P14 d∗2 , v7 = |G4 |2 /P32 d1 , ∗ ∗ v8 = |G4 |2 /P42 d1 .. The populations are described by the formulas below. OPEN CONFIGURATION:. The above given solutions are useful for numerical analysis of basic transmission properties of a NIM slab, their extraordinary features as compared with the ordinary, positive index media and for demonstration the feasibilities of all-optical control the transparency and reflectivity such a slab.. Δr3. =. Δr4. =. Δr2 Δr1. =. (1 + æ4 )Δn3 + b1 æ4 Δn4 , (1 + æ3 )(1 + æ4 ) − a1 æ3 b1 æ4 (1 + æ3 )Δn4 + a1 æ3 Δn3 , (1 + æ3 )(1 + æ4 ) − a1 æ3 b1 æ4 Δn2 − b2 æ4 Δr4 − a2 æ3 Δr3 ,. =. Δn1 − a3 æ3 Δr3 − b3 æ4 Δr4 ,. A PPENDIX B N ONLINEAR SUSCEPTIBILITIES AND ENERGY LEVEL. rm. =. nm + (1 − b2 )æ4 Δr4 ,. POPULATIONS FOR THE CASE WHERE EACH LEVEL IS COUPLED TO ONLY ONE DRIVING FIELD : OPEN AND CLOSED SCHEMES. rg. =. ng + (1 − a3 )æ3 Δr3 ,. rn. = =. nn − b2 æ4 Δr4 + a1 æ3 Δr3 , nl − b1 æ4 Δr4 + a3 æ3 Δr3 ,. rl. With the aid of the solution of a set of equations (11) for the off-diagonal elements of the density matrix up to first order in perturbation theory with respect to the weak fields, the equations for the susceptibilities can be presented as [18]. where æ3 æ03 æ04. Γi Δri χi Γi Ri χi = (i = 3, 4), 0 = (i = 1, 2). χ0i Pi Δni χi Pi Δni Here, χ0i is a resonance value of the susceptibility for all fields turned off, where R1,2 are given below. dml dlg dgn dnm /83 d (1 + v5∗ + g5∗ )  2

(15).

(16)  Δr3 Δr4 1 R1∗ 1 × + + + , ∗ ∗ ∗ ∗ P3 P13 P4 P14 P1∗ P13 P14. χ ˜2 = −iN. dml dlg dgn dnm /83 d (1 + v7∗ + g7∗ )  1

(17).

(18)  Δr3 Δr4 1 R2∗ 1 × + + + ∗ , P3 P32 P4 P42 P2 P32 P42. χ ˜1 = −iN. = æ03 Γ2lg /|P3 |2 ,. æ4 = æ04 Γ2mn /|P4 |2 ,. = 2(Γl + Γg − γgl )|G3 |2 /Γl Γg Γlg , = 2(Γm + Γn − γmn )|G4 |2 /Γm Γn Γmn ,. γgn Γl γgn a2 γgn Γl a3 = , = Γn − γgn Γn (Γg − γgl ) Γn (Γl + Γg − γgl ) γml b3 γml Γn γml Γn b2 = = . b1 = Γl (Γm − γmn ) Γl (Γl − γml ) Γl (Γm + Γn − γmn ) a1 =. CLOSED CONFIGURATION: In this case, the populations of levels are given by the equations Γm rm Γg r g Γn rn. R2 = Δr2 (1 + g7 + v7 ) (1 + v7 − g8 )Δr4 (1 + g7 − v8 )Δr3 − g3 1 − v3 (1 + g7 + v7 )Δr2 (1 + g7 + v7 )Δr2 , × (1 + g2 + v2 ) + [g7 + g2 (g7 − v8 ) + v7 + v2 (v7 − g8 )]. rl. 120. =. wm rl − 2 Re {iG∗4 r4 } ,. = wg rl − 2 Re {iG∗3 r3 } , = wn rl + 2 Re {iG∗4 r4 } + γgn rg + γmn rm , = 1 − rm − rg − rn ,.

(19) World Academy of Science, Engineering and Technology 61 2010. whose solution is. [9] M. W. Klein, M. Wegener, N. Feth, and S. Linden, “Experiments on second- and third-harmonic generation from magnetic metamaterials,” Opt. Express, vol. 15, pp. 5238–5247, Apr. 2007, erratum:ibid, 16, 8055 (2008). [10] J. M. Manley and H. E. Rowe, “Some general properties of nonlinear elements–Part I. General energy relations,” Proceedings of the IRE, vol. 44, no. 7, pp. 904–913, July 1956. [11] ——, “General energy relations in nonlinear reactances,” Proceedings of the IRE, vol. 47, pp. 2115–2116, 1959. [12] A. Yariv, Quantum Electronics, 2d ed. New York: Wiley, 1975. [13] S. E. Harris, “Proposed backward wave oscillation in the infrared,” Appl. Phys. Lett., vol. 9, no. 3, pp. 114–116, Aug. 1966. [Online]. Available: http://link.aip.org/link/?APL/9/114/1 [14] K. I. Volyak and A. S. Gorshkov, “Investigations of a reverse-wave parametric oscillator,” Radiotekhnika i Elektronika (Radiotechnics and Electronics) [in Russian], vol. 18, pp. 2075–2082, 1973. [15] C. Canalias and V. Pasiskevicius, “Mirrorless optical parametric oscillator,” Nat. Photonics, vol. 1, pp. 459–462, Aug. 2007. [16] J. B. Khurgin, “Optical parametric oscillator: Mirrorless magic,” Nat. Photonics, vol. 1, pp. 446–447, Aug. 2007. [17] A. K. Popov and S. A. Myslivets, “Transformable broad-band transparency and amplification in negative-index films,” Appl. Phys. Lett., vol. 93, no. 19, p. 191117(3), 2008. [Online]. Available: http://link.aip.org/link/?APL/93/191117/1 [18] A. K. Popov, S. A. Myslivets, and T. F. George, “Nonlinear interference effects and all-optical switching in optically dense inhomogeneously broadened media,” Phys. Rev. A, vol. 71, no. 4, p. 043811(13), Apr. 2005.. rl = nl (1 + æ4 )(1 + æ3 )/β, rg = (1 + æ4 )[nl (1 + æ3 ) − Δn3 ]/β, rn = {nm (1 + æ4 )(1 + æ3 ) + [Δn4 (1 + æ3 ) + Δn3 γ2 æ3 /Γn ](1 + bæ4 ) } /β, rm = {nm (1 + æ4 )(1 + æ3 ) + [Δn4 (1 + æ3 ) + Δn3 γ2 æ3 /Γn ]bæ4 } /β, Δr4 = rn − rm = [Δn4 (1 + æ3 ) + Δn3 γ2 æ3 /Γn ] /β, Δr3 = rl − rg = Δn3 (1 + æ4 )/β. Here, Δn3 nm nl wn. . b æ3 β. = =. nl − ng , Δn4 = nn − nm , nl wm /Γm , ng = nl wg /Γg , nn = nl wn  /Γn ,. =. (1 + wm /Γm + wg /Γg + wn  /Γn )−1 ,. = wn + wg γgn /Γn + wm γmn /Γn , = Γn /(Γm + Γn − γ4 ).. (2|G3 |2 2|G4 |2 (Γm + Γn − γ4 ) , æ4 = , 2 2 Γ3 Γg )(Γ3 /|P3 | ) Γm Γn Γ4 (Γ24 /|P4 |2 ) = (1 + æ4 )[1 − Δn4 + 2(nl + nm )æ3 ] + (1 + 2bæ4 )[Δn4 (1 + æ3 ) + Δn3 γ2 æ3 /Γn ].. =. The remaining notations are as before. ACKNOWLEDGMENT This material is based upon work supported by by the U. S. Army Research Laboratory and by the U. S. Army Research Office under grant numbers W911NF-0710261 and by the Siberian Division of the Russian Academy of Sciences under Integration Project No 5. The authors thank V. M. Shalaev for fruitful discussions some of the results of this paper. R EFERENCES [1] V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photonics, vol. 1, pp. 41–48, Jan. 2007. [2] C. M. Soukoulis and M. Kafesaki, “Weakly and strongly coupled optical metamaterials,” Invited talk at the Nanometa 2009, The 2nd European Topical Meeting on Nanophotonics and Metamaterials 5 - 8 January, Seefeld, Tirol, Austria, 2009. [3] N. Katsarakis, G. Konstantinidis, A. Kostopoulos, R. S. Penciu, T. F. Gundogdu, M. Kafesaki, E. N. Economou, T. Koschny, and C. M. Soukoulis, “Magnetic response of split-ring resonators in the far-infrared frequency regime,” Opt. Lett., vol. 30, no. 11, pp. 1348–1350, 2005. [Online]. Available: http://ol.osa.org/abstract.cfm?URI=ol-30-11-1348 [4] X. Zhang, “Optical bulk metamaterials,” Plenary talk at the Nanometa 2009, The 2nd European Topical Meeting on Nanophotonics and Metamaterials 5 - 8 January, Seefeld, Tirol, Austria, 2009. [5] J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index ,” Nature, vol. 455, pp. 376–379, Sep. 2008. [6] A. K. Popov and V. M. Shalaev, “Negative-index metamaterials: secondharmonic generation, Manley Rowe relations and parametric amplification,” Appl. Phys. B: Lasers and Optics, vol. 84, pp. 131–137, Jul. 2006. [7] A. K. Popov, S. A. Myslivets, T. F. George, and V. M. Shalaev, “Four-wave mixing, quantum control, and compensating losses in doped negative-index photonic metamaterials,” Opt. Lett., vol. 32, pp. 3044– 3046, 2007. [8] A. K. Popov, S. A. Myslivets, and V. M. Shalaev, “Resonant nonlinear optics of backward waves in negative-index metamaterials,” Appl. Phys. B: Lasers and Optics, vol. 96, no. 2–3, pp. 315–323, Feb. 2009.. 121.

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