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By simultaneously taking field localization and slow light effects into account, in this paper we make use of a field averaging method to calculate the effective nonlinear refractive index coefficient (n 2) of Kerr photonic crystals (PhCs) in the first band. Although the nonlinear PhC is beyond the traditional long-wavelength limit, interestingly, the theoretically calculated effective n 2 agrees well with one numerically measured via the self-phase-modulation induced spectral broadening. Moreover, due to the cooperative influence of field localization and slow light effects, the effective n 2 of the PhC decreases slowly at first and then goes up quickly with increasing frequency. This kind of dispersive nonlinearity is purely induced by the periodic nanostructures because the optical parameters of both components of the PhC we took are frequency-independent. Our results may pave the way for enhancing or limiting nonlinear effects and provide a method for producing the dispersive nonlinearity.
© 2015 Optical Society of America
1. Introduction
Nonlinear optical effects play a key role in many modern optical applications such as laser frequency conversion, all-optical switching and all-optical signal processing [1]. For further improving and enlarging these applications, there is always a need for nonlinear optical materials that have large nonlinearities surpassing existing ones. In recent years, nanostructured photonic materials, such as photonic crystals (PhCs) and metamaterials, have increasingly demonstrated their potential to tailor nonlinear effects to such an unprecedented degree [2–12 ].
To quantitatively estimate the tailored nonlinear effects in composite nanostructures, effective nonlinear parameters are evidently good measures. Based on the effective-medium theory, a great many theoretical expressions have been developed for computing the effective third-order (Kerr) nonlinear susceptibilities of composite nanostructures [13–16 ]. On the other hand, for second-order nonlinear metamaterials Smith et al. have also proposed a few retrieval methods under the assumption of a non-depleted pump approximation [17–20 ]. However, most of these methods are valid only when the unit cells of the composite nanostructure are far smaller than the operating wavelength. Therefore, for PhCs whose unit cells are on the order of the magnitude as the operating wavelength, no methods for evaluating effective nonlinear parameters have been reported yet. In this paper, we propose a method to calculate the effective nonlinear refractive index coefficient (n 2) of Kerr PhCs by simultaneously taking field localization and slow light effects into account. The theoretically calculated n 2 has been validated by numerical simulation. Interestingly, a sort of dispersive nonlinearity induced by the cooperation of these two effects was discovered.
2. Theoretical methods for calculating the effective n2 of the photonic crystals
It has been widely acknowledged that field localization and slow light effects are two main ways to enhance nonlinearity in nanostructured materials [2–11 ]. For nanostructured materials consisting of only pure dielectric constituents, field localization is always induced by constructive coherences in periodic nanostructures such as PhCs [2, 10, 11 ]. However, for most metamaterials and metasurfaces in which metals are the key constituents, the surface plasmonic resonance plays a crucial role in exciting extremely large intensities near the metal surface [7–9 ].
On the other hand, the slow light effect mainly originates either from strong temporal dispersion in homogeneous dispersive media or from strong spatial dispersion in spatial periodic nanostructures [3, 5, 6 ]. It has attracted enormous attention in the past few decades because of its potential applications in buffering optical signals, tuning light-matter interactions, etc [21, 22 ]. Thanks to the development of nanofabrication techniques, nonlinear effects enhanced by field localization or slow light effect have been respectively demonstrated in a great many nanostructured materials theoretically and experimentally [2–12, 21, 22 ].
These two effects are obviously possible to be excited in a periodic nanostructure simultaneously and contribute to nonlinear effect enhancement together. Nevertheless, their joint influence on nonlinearity has not been demonstrated yet. Here, taking a two-dimensional (2D) PhC as an example, we quantitatively demonstrate this joint influence through calculating the effective n 2 of a Kerr PhC in the first band. The PhC, as schematically shown in Fig. 1(a) , is made of a square-lattice of silicon cylinders embedded in a typical liquid Kerr medium carbon disulfide (CS2). The silicon cylinder has a dielectric constant of ε = 12.96 [23] and a diameter of d = 0.5a, where a is the lattice constant, while the refractive index of CS2 is intensity-dependent, i.e., n(|E|2) = n 0 + n 2|E|2, where n 0 = 1.63 and n 2 = 1.67 × 10−20 m2/V2 are the linear refractive index and nonlinear refractive index coefficient of CS2, respectively [24].
Fig. 1 Sketch (a) and photonic band structure (b) of the two-dimensional Kerr PhC made of Si cylinders embedded in CS2 background. Only TM waves propagating along the direction of ГX are considered in this paper. Insets of (b) are the normalized amplitude distributions of the electric field in a unit cell at two frequencies indicated by red dashed lines in the first and the second band, respectively.
For this 2D PhC, we only consider TM waves (the electric field E z is parallel to these pillars) propagating along the ΓX direction. It is noted that the method we used in this paper is also suitable for TE waves, i.e., the magnetic field H z is parallel to the pillars. Under the condition that the light intensity in the PhC is low enough to ignore the nonlinearity of CS2, the photonic band structure is calculated by using the plane wave expansion method (PWEM) and plotted in Fig. 1(b) in which the frequency is normalized as a/λ. Meanwhile, amplitude distributions of the electric field in a unit cell and group velocities of the first band are also obtained. As an example, the normalized amplitude distribution at normalized frequency nf = 0.1 is displayed in the inset of Fig. 1(b). It clearly shows that the electric fields are mainly concentrated in the silicon cylinder denoted by the dotted circle. By the way, the amplitude distribution in the second band, taking nf = 0.35 as an example, is more complicated because in the second band strong electromagnetic resonance has been established.
To quantitatively characterize this field localization, we define two spatial enhancement coefficients RSi and RCS 2 as
where Eav_Si = (∫Si EdS) / SSi, Eav_CS 2 = (∫CS2 EdS) / SCS2, and Eav_un = (∫un EdS) / Sun are the spatial average amplitude of electric field in silicon cylinder, nonlinear background CS2 and the whole unit cell, respectively. As plotted in Fig. 2(a) , RSi increases rapidly while RCS 2 decreases slowly when the frequency approaches the bandgap along ГX. In other words, for higher frequencies of the first band, more fields concentrate in the silicon cylinder while less in the nonlinear background. Different from previous experiments [9–11, 25 ], the field localization effect weakens the nonlinear effect in this nanostructure because the silicon nonlinearity is not considered here. However, on the other hand, group velocities reduce quickly with the increment of frequency as shown in Fig. 2(b), which elongates the interaction time between light and nanostructures and results in enhancing the nonlinear effects [3, 5, 6, 21, 22 ]. Evidently, for this Kerr PhC, field localization and slow light effect should be taken into account simultaneously in the analytic expression of effective n 2.Taking the Kerr PhC as an effective medium, the effective electric displacement field in a unit cell Dav_un can be given by a simple spatial averaging
where fSi = πr2/a 2 and fCS 2 = 1− fSi are the volume fractions of silicon cylinder and nonlinear background CS2, respectively. εSi and εCS 2 are the relative permittivity of silicon and CS2 and εCS 2 is given bywhereSubstituting Eqs. (1), (2) and (4) into Eq. (3), we obtain
Thus, considering the effect of field localization, the effective relative permittivity of the Kerr PhC should be revised asBecause no strong electromagnetic resonances have been developed in the first band, we assume μeff = 1. Then, the intensity-dependent refractive index of the PhC is obtained
whereMoreover, considering that slow light effect saves a factor of ~c/vg in propagation length for achieving the same nonlinear effect [3, 5, 6, 21, 22 ], the final effective n 2 is modified by multiplying a slow-light enhancement factor
where fvg is a dispersionless factor that can be determined by a known n' 2 _ eff at a certain frequency.3. Verification and discussion
To test our theoretical attempt, we first checked the validity of the effective linear refractive index which is an important parameter for determining effective n 2 as shown in Eq. (11). As plotted in Fig. (3) , in the whole first band along ГX the linear refractive index obtained by use of Eq. (9) agrees very well with the one calculated by using the PWEM. This proves that our field averaging method works very well for calculating linear optical parameters of this PhC.
Fig. 3 Comparison of effective linear refractive index of the PhC. The blue line and the red circles are calculated by using plane wave expansion method (PWEM) and Eq. (9), repectively.
To calculate the effective n 2, a prior knowledge of fvg is necessary. As we know, for this PhC, the slow light effect can be neglected and its corresponding enhancement factor will go to unity when the normalized frequency nf approaches 0, i.e.,
As shown in Fig. 2(b), vg /c is about 0.46 when nf approaches to 0, i.e., the dispersionless fvg ≈0.46. Substituting fvg into Eq. (11), we calculated the effective n 2 and plotted it in Fig. 4(a) with a blue line. Furthermore, for validating the theoretically calculated results, we also numerically measured the effective n 2 at several normalized frequencies by using the nonlinear finite-difference time-domain (FDTD) method [26]. The numerical measurements are based on the self-phase-modulation (SPM) induced spectral broadening which can be obtained analytically if the group velocity dispersion effect in the PhC is negligible [27]. Therefore, for a given chirped Gaussian pulse propagating through a PhC with a known length, the effective n 2 is unambiguously determined by the SPM-induced spectral broadening. Using this method, we retrieved the effective n 2 and plotted them with red circles in Fig. 4(a). Amazingly, these two lines coincide with each other very precisely, demonstrating our theoretical attempt Eq. (11) successful in calculating nonlinear parameters.Fig. 4 Comparison of effective n 2 of the Kerr PhC: (a) only the background and (b) only the silicon cylinder responds nonlinearly, respectively. The blue line is obtained by using Eq. (11) and the red circles are retrieved based on the self-phase modulation effect.
More interestingly, as shown in Fig. 4(a), the effective n 2 decreases slowly at first and then goes up quickly as the frequency increases, which directly proves that the field localization and slow light effect have indeed influenced the nonlinear effect together: it clearly shows in Fig. 2(a) that the electric fields in the nonlinear background CS2, which contribute to the nonlinear effect, decrease slowly with the increment of frequency, while the group velocity for light propagating in the PhC, as presented in Fig. 2(b), drops quickly when the frequency approaches the bandgap along ГX. Therefore, the nonlinear effect enhanced by the slow light effect becomes stronger and stronger and at last overcomes the weakening effect rising from field localization. Furthermore, this kind of nonlinear dispersion, which may play a new role in self-steepening of ultrashort pulse, is purely induced by the periodic nanostructures because optical parameters of both components of the PhC we took are frequency-independent.
Additionally, we also calculated the effective n 2 by using Eq. (11) when only the silicon cylinder has an intensity-dependent refractive index (CS2 responses linearly). In order to keep the field distributions and group velocities unaltered, we preserved all linear optical parameters and geometric parameters of the PhC. Fig. 4(b) presents the theoretically calculated and numerically measured n 2 when the silicon cylinder has a nonlinear response of n 2 = 2.67 × 10−20 m2/V2. It is obvious that these two results also agree with each other, but not as well as those in Fig. 4(a). The inhomogeneous field distribution in the silicon cylinder maybe the main reason for this discrepancy because the field averaging method is the key for predicting effective n 2 by using Eq. (11). As clearly shown in the inset of Fig. 1(b), a bell-like electric field distributes in the silicon cylinder while a more homogeneous field in the background. For estimating the nonlinear responses, our simple volume averaging method works better in the background than in the cylinders. As a result, Eq. (11) predicts effective n 2 more precisely when only the background responds nonlinearly.
4. Conclusion
In summary, in this paper we have proposed a method to calculate the effective nonlinear refractive index coefficient (n 2) of Kerr PhCs in the first band which are beyond the traditional long-wavelength limit. Using the PWEM, we obtained the linear properties such as group velocities and field distributions in a unit cell which are helpful for quantitatively characterizing the slow light effect and field localization effect respectively. By simultaneously taking these two effects into account, we have calculated the effective n 2 of Kerr PhCs by using the field averaging method.
To demonstrate the validity of our proposed method, the theoretically calculated n 2 has been checked by numerical simulation according to the self-phase-modulation induced spectral broadening. Interestingly, the cooperation of field localization and slow light effect induces a dispersive nonlinearity: the effective n 2 varies with the frequency. It is noted that this kind of nonlinear dispersion is purely induced by the periodic nanostructures because the optical parameters of both components of the PhC we took are frequency-independent. Our results may pave the way for enhancing or limiting nonlinear effects and provide a method for producing the dispersive nonlinearity.
Acknowledgments
This work was partially supported by the National Natural Science Foundation of China (Grant Nos. 11076011, 61025024, and 61378002), Hunan Provincial Natural Science Foundation of China (Grant No. 12JJ7005), National High Technology Research and Development Program of China (863 Program, Grant No. 2012AA01A301-01), the Fundamental Research Funds for the Central Universities and the visiting scholar program of China Scholarship Council.
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