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Second harmonic generation and pulse shaping in positively and negatively spatially dispersive nanowaveguides: comparative analysis

Comparative analysis of second harmonic generation in ordinary and backward-wave settings is presented. Extraordinary properties of frequency doubling nonlinear optical reflectivity and pulse shaping through phase matching of ordinary and backward electromagnetic waves in the nanowaveguides with mixed negative/positive spatial dispersion is demonstrated with numerical simulations.

Current mainstream in fabricating bulk NIM slabs relies on engineering of LC nanocircuits -plasmonic mesoatoms with phase shifted, negative electromagnetic response. Extraordinary coherent, nonlinear optical, frequency converting propagation processes predicted in NIMs have been experimentally realized so far only in the microwave [6]. A different paradigm which employs spatial dispersion [7,8] was proposed to realize outlined processes. Basic underlaying idea is nanoengineering negative spatial dispersion which would enable negative group velocity vgr = ∂ω/∂k < 0. Then energy flux, which is directed along the group velocity, appears directed against the wave vector. Such a property makes an analog of the effective negative refractive index. Various particular realizations of negative dispersion were proposed [9,10,11,12,13]. Basically, many hyperbolic metamaterials may support backward-wave electromagnetic modes, see, e.g., [14], as well as specially designed waveguides [15,16]. This opens new avenues for creation of backward-wave photonic devices with unparalleled functional properties. As regards coherent nonlinear-optical propagation processes, critically important is to provide for coexistence of ordinary and backward coupled waves which frequencies and wave vectors satisfy to energy and momentum conservation law (phase matching). Such possibilities were described in ref. [11,12,13]. Corresponding MM slabs can be viewed as the plasmonic nanowaveguides which support both ordinary EMWs with co-directed phase velocity and energy flux (∂ω/∂k > 0) and extraordinary, BW modes with contra-directed phase and group velocities (∂ω/∂k < 0). Some of them exhibit spatial dispersion which changes from positive to negative in different wavelength intervals whereas the others predominantly maintain negative dispersion. Frequencies and gaps between the modes can be tailored by changing shapes, sizes and spacing of the MM nano-building blocks so that phase velocities for the modes matching the photon energy conservation law would become equal as their group velocities remain contra-directed. Basically, different materials and geometries can be utilized to control spatial dispersion and losses in such MM with variable spatial dispersion.
Second harmonic generation (SHG) is the basic fundamental coherent (i.e., phase-dependent) nonlinear optical propagation process which finds many practically important application. Overall conversion efficiency is controlled by the intensity dependent local conversion rate and by the corresponding depletion of the fundamental beam. As shown in [1,2,3,4,5], such process exhibit unusual properties in the BW settings which are significantly different from both BW optical parametrical amplification and SHG in ordinary materials. This paper is to analyze such properties of SHG and fundamental differences between SHG in ordinary and backward-wave settings with focus on the pulse regime which is of particular interest.

Basic Equations
As noted, the properties of SHG will be investigated for the cases where one of the coupled waves is ordinary and the other one is backward wave, in continuous-wave and pulse regimes, and compared with their ordinary counterparts. To achieve phase matching, phase velocities of the fundamental the SH waves must be codirected. This means that in BW setting and pulse regime, the pulse of SH will propagate against the pulse of the fundamental radiation. Such extraordinary NLO coupling scheme dictates exotic properties of the frequency-doubling ultracompact NLO mirror under consideration, which are described below. One of our goals is to investigate the requirements for the magnitude of the metamaterial nonlinearity as well as for intensity of the control field(s) that would enable efficient functional properties of the indicated SHG mirror. Corresponding basic equations are as follows.

SHG in ordinary NLO medium
For the case of ordinary waves s 1 = s 2 = 1 and a 20 = 0 with account for a 10 = 1, one finds from Eq. (10) that Solution to Eqs. (9) is found as [2,3]  S 2,1 ∝ |a 2,1 | 2 across the ordinary, positive index material slab. According to Eqs. (11) and (12), sum of fluxes of photons ω 2 and of the pairs of photons ω 1 is conserved across the medium.  (9) and (10) dictates fundamentally different behavior. Equation (10) predicts where |B| 2 is a constant which, however, depends on the slab thickness and on strength of the input fundamental field. Equations (9) reduce to Besides the fact that the in this case the equations have different signs in the right sides, the boundary conditions for fundamental and SH waves must be applied to opposite edges of the slabs of thickness L: a 10 = 1, a 2L = 0. Indicated differences give rise to fundamental changes in the solution to the equations for the amplitudes and, consequently, to the properties of the SHG as a whole. Solution to Eqs. (14) can be presented as [2,3] As follows from (16), quantity B presents transmitted fundamental wave, a 1L = B. It is found from Eq. (16) at z = 0 as: According to Eqs. (13) and (15) - (16), difference between the numbers of pairs of photons ω 1 and the number of photons ω 2 is conserved so that the number of photons ω 2 at z = L is always equal to zero. The later requirement controls whole process of BWSHG. Such a dependence is in strict contrast with SHG in ordinary NLO materials, where the SH flux may exceed the fundamental one in some area inside the slab. In the case of BWSHG, its energy flux is directed against the fundamental one towards the areas with higher intensity of the fundamental wave whereas remains lower everywhere inside the slab. As seen from Eq. (17), the gap between the two curves, |B| 2 , decreases and, hence, the conversion efficiency grows with gL = L/z 0 → ∞. Like in the preceding case, z 0 represent a characteristic medium length required for significant energy conversion to the SH. It shortens with growth of strength of the pump field.  Figure 3(a) illustrate unparalleled properties of SH generation with backward waves as compared with its ordinary counterpart at similar other parameters for the particular example of gL = 1. As noted, the remarkable property is the fact that SH propagates against the fundamental wave and, therefore, metaslabs operates as a frequency doubling metamirror with reflectivity controlled by the fundamental wave. Further materials which support only ordinary waves and MM that support both ordinary and backward waves will be referred to as PIM and NIM. Figure 3(b) compares output intensity of the SH and transmitted fundamental wave. Major important conclusion are as follows. It appears that the efficiency of SHG in ordinary settings exceeds that in the NIM metaslab at equal other parameters. At that, propagation properties of SH appear fundamentally different which holds promise of extraordinary applications. Intensity of SH is less than that of the fundamental wave across the NIM slab, whereas it can exceed that in the vicinity of exit from PIM slab [ Fig. 3(a)] Quantum conversion grows sharper with increase of intensity of fundamental beam and higher conversion at lower intensities occurs in PIM as compared with NIM.

Comparison of ordinary and BW SHG in short-pulse regime
In this section, comparative analysis of SHG in the ordinary and BW settings will be done for the case where intensity of input fundamental wave varies in time and, therefore, SHG is described by the system of partial differential equations (7) and (8). The input pulse shape is chosen close to a rectangular form where δτ is the duration of the pulse front and tail, and τ 0 is the shift of the front relative to t = 0. Parameters δτ = 0.01 and τ 0 = 0.1 have been selected for numerical simulations. Absorption is neglected (α 1 = α 2 = 0). Phase velocities are equal ∆k = 0. Group velocities are equal (v 12 ) and contra-directed. Unusual properties of SHG in NIMs in the pulsed regime stem from the fact that it occurs only inside the traveling pulse of fundamental radiation. Generation begins on its leading edge, grows towards its trailing edge, and then exits the fundamental pulse with no further changes. Since the fundamental pulse propagates across the slab, the duration of the SH pulse may occur significantly longer than that of the fundamental one. Depletion of the fundamental radiation along the pulse and the overall conversion efficiency depend not only on maximum intensity of the input pulse, on matching of phase and group velocities of the fundamental and second harmonic, but on the ratio of the fundamental pulse length and slab thickness. Such properties are in strict contrast with that of SHG in PIM as illustrated in Figs. 4 and 5. Shape of the input fundamental pulse is given by the function T 1 = |a 1 (τ, z = 0)| 2 /|a 10 | 2 when its leading front enters the medium. The results of numerical simulations for the output fundamental pulse, when its tail reaches the slab boundary, is given by T 1 = |a 1 (τ, z = L)| 2 /|a 10 | 2 . Shape of the output pulse of SH, when its tail passes the slab's edge at z = 0, are given by the function η 2 = |a 2 (τ, z = 0)| 2 /|a 10 | 2 . Pulse energy are represented by the time integrated pulse areas S j which varies across the slab. As seen from Figs. 4 (a) and (b), saturation is homogeneous across the pulse in PIM and shape of both fundamental and SH output pulses remain rectangular. On the contrary, shapes of the output fundamental and SH pulses in PIM are different and change with change of intensity of input fundamental pulse. Moreover, it appears that shapes of the output pulses varies with change of the input pulse length at unchanged other parameters as seen from Figs and 5(a) prove that shapes and width of fundamental and generated SH pulses as well as the energy conversion efficiency to the reflected pulses at doubled frequency can be controlled by changing intensity and ratio of the the input pulse length to the metamaterial thickness (parameter d).

Conclusion
Comparative analysis of second harmonic generation in ordinary and backwardwave settings in continuous and pulse regimes is presented. Backward-wave regime is attributed to the nanomaterials which possess spatial dispersion of opposite signs at fundamental at second harmonic frequencies. This makes possible equal phase Here input pulse duration is decreased in four times as compared with Fig. 4 (a) at the same peak intensity. T 1 is pulse shape for the fundamental radiation, η 2 -for SH, S 1 (z) and S 10 = S 1 (z = 0) are fundamental pulse energy, 2S 2 /S 10 is energy (photon) conversion efficiency per pulse, d = L/l, velocities of the coupled waves whereas their energy fluxes appear contra-directed. Such deliberately engineered nanolayers can be viewed as nanowaveguides. It is shown that properties of second harmonic generation in ordinary and backwardwave settings are fundamentally different. In the latter case, metaslab serves as microscopic frequency doubling nonlinear-optical mirror which properties can be all-optically controlled. In pulse regime, energy conversion property are qualitatively similar to those in continuous wave regime, whereas depend on the ratio of pulse length to the metaslab thickness. It is shown that width and shape of transmitted fundamental and generated second harmonic pulses can be controlled by changing intensity and width of input pulses.