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Abstract
The sign of the Kerr nonlinear coefficient, which determines the nonlinear propagation characteristics of high-power lasers, is usually only sensitive to the operating frequency. Here, we report a method to reverse the sign according to the polarization of the input beam. The sign reversal is realized in the first band of a two-dimensional Kerr photonic crystal (PhC) by using local-field effects. By appropriately adjusting the unit cell, we find that the effective nonlinear coefficient of the PhC is negative for the TM mode while positive for the TE mode. Using the FDTD method, polarization-dependent spectral broadening and narrowing, self-focusing and defocusing are demonstrated in the nearly isotropic PhC.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
For a high-power laser propagating in a Kerr medium, the refractive index is usually intensity-dependent, i.e., n(|E|2) = n0 + n2|E|2, where n0 and n2 are linear refractive index and nonlinear refractive index coefficient, respectively [1]. Generally, the nonlinear propagation characteristics are up to the sign of the nonlinear coefficient n2. For example, for an up-chirped pulse propagating in the nonlinear medium, a positive n2 leads to a spectral broadening while the negative n2 results in a spectral narrowing [1, 2]. On the other hand, self-focusing or self-defocusing, as a spatial manifestation of the intensity-dependent refractive index, is also determined by the sign of n2 [3–6], because almost all of the natural materials possess positive n0. However, for a conventional Kerr medium the sign of n2 is usually only sensitive to the operating frequency [7,8]. As a result, at a given frequency, switching from spectral broadening to narrowing or from self-focusing to defocusing is usually unachievable.
Due to their rich and flexible electromagnetic properties, man-made nanostructures, such as photonic crystals (PhCs) and metamaterials, have demonstrated their unusual abilities to tailor the nonlinear effects [9–22]. With properly designed guiding mode dispersion, Morandotti et al. have experimentally demonstrated a switching from self-focusing to defocusing in planar waveguide arrays by slightly tilting the incident angle [15]. By tuning the interplay between the Kerr nonlinearity and self-collimation effect, enhanced and suppressed self-focusing have been proposed in two-dimensional (2D) PhCs by Yu and Jiang et al. [16,17]. Moreover, self-focusing and defocusing of Gaussian beams were theoretically predicted in hypothetical negative-refractive-index materials many years ago and have been numerically proved in PhCs recently [6,18]. On the other hand, numerous results about surprising nonlinear effects in metamaterials such as giant nonlinear response [19,20], backward phase-matching [21] and phase mismatch-free harmonic generation [22] have been reported.
Nevertheless, most of their controlling mechanisms were based on the abnormal linear spatial dispersion of the man-made nanostructures such as strong anisotropy and negative refraction. More importantly, the sign of the effective n2 that dominates the nonlinear propagation have never been changed yet. Here, based on the asymmetry of the localized field distribution of the TE and TM mode, we report a PhC structure in which the sign of the effective nonlinear index n2 may depends on the polarization of the input beam by using the positive and negative nonlinear refractive index materials. Moreover, the polarization-dependent spectral broadening and narrowing, self-focusing and defocusing have been presented by nonlinear finite-difference time-domain (FDTD) simulations.
2. Theoretical analysis for the sign reversal of effective n2
The total optical response of a nanocomposite is dependent on the local field acting on each individual ingredient besides on the macroscopic average field [23]. Moreover, the local field is usually very sensitive to the geometry and optical parameters of the ingredient and its background. Therefore, by combining the local-field effects with nanostructuring, greatly enhanced nonlinear response and optical amplification have been demonstrated experimentally in a great many of nanocomposites [24–27].
Here, we utilize the local-field effects in a 2D PhC to reverse the sign of its effective n2. Because the photonic band structure always moves down or up when the dielectric constants of the constituents increase or decrease [28,29], two Kerr materials with opposite signs of n2 are taken to construct the 2D PhC to realize the sign reversal. As an example, zinc selenide (ZnSe) and liquid carbon disulfide (CS2) are suitable since at the operating wavelength λ0 = 532 nm the n2 of ZnSe and CS2 are n2_ZnSe = −4.9 × 10−20m2/V2 and n2_CS2 = 1.55 × 10−20m2/V2, respectively [7]. While, the linear refractive index of ZnSe and CS2 are n0_ZnSe = 3.7 and n0_CS2 = 1.63. Note that our method can also be extended to other Kerr materials with opposite signs of n2.
The PhC, as schematically shown in Fig. 1(a), is made of a square-lattice of ZnSe cylinders embedded in liquid CS2 background and the ZnSe cylinder has a radius of r = 0.3a, where a is the lattice constant [30, 31]. For predicting the nonlinear responses from its linear dispersion properties [32–34], we first ignore the nonlinearity of these constituents and calculate the photonic band structures for both TM and TE mode by using the plane wave expansion method. To ensure that the PhC can be characterized with the effective electromagnetic parameters [35], we choose a/λ0 = 0.1 (a = 0.1λ0 = 53.2 nm) as the operating frequency. As plotted in Fig. 1(b), at this frequency, wave vectors of the PhC for both TM and TE mode are symmetrically distributed along the directions of ΓX and ΓM, indicating a good isotropy. It is noted that the operation frequency can also be chosen from 0 to 0.15 due to the good isotropy of the PhC in this range. Moreover, the equal-frequency contours of the PhC are nearly perfect circles for both TM and TE mode as shown in the inset of Fig. 1(b), which are quite different from the flat equal-frequency contours that give rise to self-collimation effects for controlling the beam propagation [16,17]. Additionally, the effective linear refractive indexes of the PhC for the TM and TE mode are about n0_TM = 2.39 and n0_TE = 1.97, respectively.
Fig. 1 (a) The sketch and (b) the photonic band structure of the PhC studied in this paper. The operating frequency nf = 0.1 is denoted by the black dashed line.
Furthermore, the normalized electric field intensities in a unit cell are also obtained and shown in the insets of Fig. 2. Obviously, the electric field is slightly concentrated in the ZnSe cylinder for the TM mode, while it is obviously excluded from ZnSe for the TE mode [33,34]. To quantitatively evaluating the local-field effect, we define two spatial enhancement factors RZnSe and RCS2 as
where , , and are the total electric field intensity in the ZnSe cylinder, the background CS2 and the whole unit cell, respectively. For the TM mode, as shown in Fig. 2(a) with the increment of the cylinder radius, RZnSe increases rapidly from nearly zero to about 0.75, while RCS2 decreases quickly from nearly 1 to about 0.25. However, for the TE mode, RCS2 and RCS2 vary slowly and RCS2 is always much greater than RZnSe as plotted in Fig. 2 (b).Fig. 2 The spatial enhancement factors of (a) TM mode and (b) TE mode in the PhC. The insets are the distributions of the electric field intensity in a unit cell.
Finally, we use a numerical retrieval method to quantitatively get the effective n2 of the PhC. The numerical retrieval method is based on the self-phase-modulation induced spectral changes which can be obtained by analytically solving the nonlinear Schrödinger equation [36] if the group velocity dispersion effect in the PhC is negligible. Therefore, for a given chirped Gaussian pulse propagating through a PhC with a known length, the effective n2 is unambiguously determined by the SPM-induced spectral changes [18]. Through matching the analytically calculated spectra and the spectra obtained numerically by using the nonlinear FDTD simulation, we can retrieve the effective n2.
As plotted in Fig. 3, for the TE mode, the effective n2 is always positive and increases with the increment of the cylinder radius. While, for the TM mode, the n2 drops quickly from positive to negative when r increases from 0.1a to 0.2a. From Fig. 3 we can see that the effective n2 of the PhC possesses opposite signs under different polarizations if the cylinder radius r is larger than about 0.13a. This interesting phenomenon can be explained by the local-field effect analyzed above. As shown in Fig. 2(a), for the TM mode, the electric field intensities distributed in the ZnSe cylinder and background CS2 change greatly with respect to the cylinder radius r and the total nonlinear response of a unit cell, which determines the effective n2, is mainly up to the radius r, i.e., the filling ratios of these two nonlinear components. Thus, with the increment of cylinder radius r, the nonlinear effects arising from the ZnSe cylinder (n2_ZnSe|EZnSe|2 < 0) gradually overcome that from the CS2 background (n2_CS2 |ECS2|2 > 0) and the effective n2 decreases. However, for the TE mode, the electric field is strongly excluded from the ZnSe cylinder so that the nonlinear effects originating from the ZnSe cylinder are offset by that from the CS2 background. As a result, the effective n2 of the PhC is always positive for the TE mode.
3. Numerical simulations for demonstrating the polarization-dependent nonlinear propagation induced by the reversal of n2
As we know, under the slowly varying envelop approximation, the split-step Fourier has been widely used for analyzing nonlinear effects in homogeneous nonlinear medium, in which polarization effects are completely neglected [36]. To verify the polarization-dependent sign reversal in our nonlinear PhC, we utilize the FDTD method that directly solves the full-wave vector Maxwell’s equations in time domain by using central difference approximations to the space and time partial derivatives. The Kerr nonlinearity is provided by incorporating a Lorentz nonlinear material model with the FDTD solver [37,38]. We take the radius of the ZnSe cylinder r = 0.3a as an example. In this situation, the effective n2 of the PhC for the TM and TE mode are n2_TE = 5.6 × 10−20m2/V2 and n2_TM = −6.2 × 10−20m2/V2, respectively. Note that the signs of the effective n2 of the TM and TE mode are opposite as long as the cylinder radius is larger than about 0.13a.
To demonstrate the polarization-dependent self-focusing and defocusing induced by the sign reversal of n2, we apply high-power continuous Gaussian beams of λ = 532 nm with different polarizations normally incident to the PhC. The direction of propagation in the PhC is along the ΓX. The beam waist (half-width at 1/e) and the peak amplitude of the incident Gaussian beams are set as w0 = 16a (about 800 nm) and E0 = 7 × 108 V/m, respectively. As shown in Fig. 4(a), for the TM mode, the incident high-power beam is clearly diverged after propagating through the nonlinear PhC with a length of 80a because of the negative n2 induced nonlinear self-defocusing. However, as shown in Fig. 4(b), for the TE mode the incident Gaussian beam is gradually focused due to the positive n2 induced self-focusing. For comparison, similar simulations are performed with low-power Gaussian beams with E0 = 7 × 104 V/m. It can be seen from Fig. 4(c) and (d) that the incident beams almost do not change after passing the nonlinear PhC, because the nonlinear effects are too weak to influence the wavefront in this short propagation length. Different from previous switchable self-defocusing and focusing based on a negative refractive index and a positive n2 [18], our mechanism is the polarization-dependent sign reversal of n2, which is a result of a properly designed local-field effects.
Fig. 4 The propagations for continuous Gaussian beams: (a) TM mode with E0 = 7 × 108 V/m, (b) TE mode with E0 = 7 × 108 V/m, (c) TM mode with E0 = 7 × 104 V/m, and (d) TE mode with E0 = 7 × 104 V/m. Propagation is from left to right.
The intensity-dependent refractive index not only affects the spatial profiles of the continuous beams but also alters the spectrums of the pulses [1]. In the propagation of chirped Gaussian pulses, the intensity-dependent refractive index usually induces self-phase modulation effects, which broaden or narrow the spectrum of the incident chirped pulse [39]. The sign of n2 plays an important role in determining whether the spectrum is broadened or narrowed. Thus, for a chirped Gaussian pulse propagating through the PhC we designed above, the polarization of the incident pulse will determine the characteristics of the output spectrum. To further prove our polarization-dependent sign reversal of n2, we also simulate high-power chirped Gaussian pulses propagating along the ΓX direction in the PhC. The central frequency, temporal half-width, and chirp parameters of the input Gaussian pulses are taken as f0 = c/λ0 = 563.91 THz, T0 = 2.36 ps, and Cp = ±1, respectively. The input pulses with peak amplitude E0 = 7 × 108 V/m are detected after passing through a 201a-thick (about L = 10 μm) nonlinear PhC. After performing Fourier transformation on the detected pulses, the normalized spectra are eventually obtained and plotted in Fig. 5. Clearly, for the TM mode, because of the negative n2, the spectra of the input up-chirped (Cp = +1) and down-chirped (Cp = −1) Gaussian pulse are narrowed and broadened, respectively. However, for the TE mode shown in Fig. 5(b), the situation is reversed. The output spectrum is broadened for the incident pulse with Cp = +1 while it is narrowed for Cp = −1.
Fig. 5 Normalized spectra of pulses with opposite chirp parameters Cp and different polarizations after passing through a 201a-thick nonlinear PhC: (a) TM mode, (b) TE mode. The blue solid line represents the normalized spectrum with a chirp parameter Cp = +1 and the red is the normalized spectrum with a chirp parameter Cp = −1.
4. Conclusion
In summary, based on the local-field effects, we have proposed a method to reverse the sign of the Kerr nonlinear coefficient, which determines the nonlinear propagation characteristics of high-power lasers while is only sensitive to the operating frequency. The sign reversal is achieved in an isotropic 2D Kerr PhC made of two Kerr media with opposite Kerr nonlinear coefficients at the operating frequency. Take ZnSe cylinders imbedded into the CS2 backgrounds as an example: for the TM mode, the total nonlinear response of the 2D PhC can be switched from positive to negative by simply increasing the radius of ZnSe cylinders because the electric field in each unit cell is relatively well-distributed. However, for the TE mode the electric field is excluded from the ZnSe cylinders greatly and the nonlinear response arising from CS2 always outweighs that from ZnSe. As a result, the total nonlinear response is always positive.
For demonstrating our designed sign reversal of n2, numerical simulations have been performed by using the nonlinear FDTD method. The polarization-dependent self-focusing and defocusing, spectral broadening and narrowing have been clearly presented. Different from previous self-defocusing and focusing in 2D PhCs with negative refractive indexes or strong anisotropy [16–18], our mechanism is based on the polarization-dependent sign reversal of n2, which is a result of a properly designed local-field effects. Our results may provide a new idea for controlling the nonlinear propagation.
Funding
National Natural Science Foundation of China (Grant Nos. 61571186 and 61378002); Key Research and Development Plan of Hunan Province (Grant No. 2017NK2121).
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