1. Introduction

The photonic crystals (PCs) have been extensively studied over the last decade [1–2]. The photonic band gaps of the PCs attracted the most attentions because of confining and controlling the propagation of photons in these materials, leading to many potential applications [3]. In the last few years, the study of nonlinear optical properties such as the second harmonic generation (SHG) and the third harmonic generation (THG) is of particular interest in PCs [4]. It is well known that for harmonic generation, the phase matching between pump laser and the harmonics signals is an important condition. Traditionally birefringent crystal or quasi-phase matching structures are used to satisfy the condition [5–7]. However, one-dimension (1-D) PC structures, which use spatial modulation of the linear refractive index, are similar as the quasi-phase matching schemes which use modulation of the nonlinear susceptibility as quasi-phase matching interaction for efficient SHG. In PCs, if the parts of sign reversion of the nonlinear materials are replaced by linear materials to compensate the dispersion of nonlinear materials, the phase matching condition for harmonic generation in terms of an effective index of refraction can be satisfied [8–11]. Thus, any material with large nonlinear susceptibility such as some semiconductors can be used and birefringent and/or ferroelectric effects are dispensable as long as it is transparent in wavelengths of interest. Another remarkable property of the PC structure is associated with the transmission resonance that appears near to the band gap, where the optical field is strongly confined which consequently further enhances the harmonic generation. Thus well-designed semiconductor-based PCs with thickness of a few microns can produce a dramatic enhancement of harmonic signals near the photonic band edge.

The enhancement of second harmonic generation (SHG) or third harmonic generation (THG) have been shown in the theory and the experiments in 1-D PCs for several years[11–15]. The SHG in GaAlAs/AlAs PC structure was reported by Dumeige et al. [11]. However, in this 1-D PC structure the phase matching cannot be realized and the conversion efficiency is only 10^-4 %. The optical field enhancement at fundamental frequency near the photonic band edge enhance the SHG in the visible spectral region was shown in the ZnS/SrF2 periodic structure [14]. More recently, we have demonstrated efficient SHG in ZnSe-based PC due to both good phase matching and optical field enhancement at the photonic band edge [16]. Usually, a cascaded two-step process based on χ⁽²⁾ for THG (i.e., sum frequency generation between the second harmonic light with the fundamental to produce the third harmonic wavelength) was used to get the third harmonic (TH) signal, because of the low conversion efficiency of the direct THG based on χ⁽³⁾ [17]. However, large enhancement of direct THG as the result of perfect phase matching and optical field enhancement near the band edge was recently demonstrated in polymer-based PCs [18]. Theoretically, the simultaneous enhancement of SHG and THG via perfect phase matching has been shown to be possible in 1-D PC [12, 13], however, no convincing experimental confirmation of this effect were demonstrated. Although simultaneous SHG and THG were seen in ZnSe/ZnMgS PC [19], the THG in this case is largely due to field enhancement effect and the strong absorption of electronic bandgap of ZnSe limited the harmonic generations above the green spectral region.

In this paper, we demonstrate a dramatic enhancement of the SHG and THG in a 1-D PC structure. The perfect phase matching condition for SHG and THG are simultaneously satisfied in this PC. In addition to the localization enhancement effect of the electromagnetic field near the photonic band edge, the high conversion efficiency of SHG and THG are realized simultaneously for forward and backward directions. The total conversion efficiency of ~4% was obtained in a PC structure composed of ZnS/YF₃ multilayer of only a few microns in total thickness. The theoretical simulation of the spectral response and conversion efficiency of the SHG and THG from this PC structure by the transfer-matrix method [20–21] are in reasonable agreement with the experimental results.

2. The method of simulation

Consider a 1D nonlinear PC with N periods. One layer in each period is linear, with the refractive index n ₀ and width l ₀. The other layer is nonlinear, with the weak-field refractive index n, a second-order nonlinear susceptibility χ⁽²⁾ and a third-order susceptibility χ⁽³⁾. The width of the nonlinear layer is l. A pump wave Ẽ ₁ of frequency ω is incident ormally upon the sample along z axis. Due to the χ⁽²⁾ and χ⁽³⁾ nonlinearity, a SH wave and a TH wave can be generated. We represent the fundamental, the SH and the TH waves by Ẽ ₁(z, t)=E ₁(z)e ^-iωt+c.c., Ẽ ₂(z, t)=E ₂(z)e ^-2iωt+c.c. and Ẽ ₃(z, t)=E ₃(z)e ^-3iωt+c.c., where E_α (z, t)=$E_{\alpha }^{+}$(z)+$E_{\alpha }^{-}$(z) and $E_{\alpha }^{\pm }\left(z\right)=A_{\alpha }^{\pm }\left(z\right)e^{\pm i{k}_{\alpha }z}$ (α=1, 2, 3). Here $E_{\alpha }^{+}$ and $E_{\alpha }^{-}$ denote the forward and backward propagating components, respectively. Assuming $A_{\alpha }^{\pm }=B_{\alpha }^{\pm }{e}^{i{\kappa }_{\alpha }z}$ (α=1, 2, 3) in a nonlinear layer, the coupled wave equations for the fundamental, the SH and the TH waves are described as follows:

where Δk ₁′=(k ₂-2k ₁)+(κ ₂-2κ ₁), Δk ₂′=(k ₃-k ₁-k ₂)+(κ ₃-κ ₁-κ ₂), Δk ₃′=(k ₃-3k ₁)+(κ ₃-3κ ₁), Δk ₄′=(k ₃+k ₁-2k ₂)+(κ ₃+κ ₁-2κ ₂) denote the phase mismatches in the case of strong field. The parameters κ_α(α=1,2,3) are defined as ${\kappa }_1=\frac{6\pi \omega }{n\left(\omega \right)c}{\chi }_{\mathit{Kerr}}^{\left(3\right)}\left(\omega \right)\left[{\mid E_1\mid }^2+2{\mid E_2\mid }^2+2{\mid E_3\mid }^2\right]$ , ${\kappa }_2=\frac{12\pi \omega }{n\left(2\omega \right)c}{\chi }_{\mathit{Kerr}}^{\left(3\right)}\left(2\omega \right)\left[{2\mid E_1\mid }^2+{\mid E_2\mid }^2+2{\mid E_3\mid }^2\right]$ and ${\kappa }_3=\frac{18\pi \omega }{n\left(3\omega \right)c}{\chi }_{\mathit{Kerr}}^{\left(3\right)}\left(3\omega \right)\left[{2\mid E_1\mid }^2+{2\mid E_2\mid }^2+{\mid E_3\mid }^2\right]$ . Note that ${\chi }_{\mathit{\text{Kerr}}}^{\left(3\right)}$(ω), ${\chi }_{\mathit{\text{Kerr}}}^{\left(3\right)}$(2ω) and ${\chi }_{\mathit{\text{Kerr}}}^{\left(3\right)}$(3ω) are χ ⁽³⁾ susceptibility for Kerr effect of the nonlinear layer, χ ⁽²⁾ and χ ⁽³⁾ are, respectively, the susceptibilities for SH and TH generations. In Eq. (1), we have set χ ⁽²⁾≡χ ⁽²⁾(ω)=χ ⁽²⁾(2ω)=χ ⁽²⁾(3ω) and χ ⁽³⁾≡χ ⁽³⁾(ω)=χ ⁽³⁾(2ω)=χ ⁽³⁾(3ω) as required by conservation of energy flow. Using the Eq. (1) and an iterative procedure described elsewhere²¹, we can obtain the SH and TH of forward and backward waves.

3. The experiment results compared with the simulation

The PC structure we designed and fabricated in this letter consists 18 periods of ZnS/YF₃ for the high/low index layers with layer thickness of 250/83.5nm respectively. The ZnS/YF₃ multilayers were grown commercially using conventional e-beam evaporation technique by Chroma Technology Corp. The sample is polycrystalline in nature as expected grown from evaporation technique. However, it is known that there is a net grain orientation distribution direction and thus the effective χ⁽²⁾ coefficient for polycrystalline material is only a few times smaller that the bulk crystal [22]. Thus, the demonstration of efficient harmonic generations based on 1-D PC grown from e-beam evaporation is a major advantage compared to the slow and expensive MBE technique previously used [11, 16].

We use the refractive index of SiO₂ for YF₃ since they are almost the same in the frequency region of interest [23]. Figure 1 shows the simulation of the transmission, effective index neff and density of modes (DOM) for this PC structure and the experimental result of transmission. Due to the judicious choices of the materials and the parameters of the structure, the realization of near perfect phase matching at the fundamental wavelength about 1530nm for both SHG and THG is shown Fig. 1(b). As the result of the phasing matching effect, the coherence length L_c of the PC sample at the phase matching wavelengths are tens of millimeters, compared with ~6 microns and ~1 microns respectively for SHG and THG in the bulk ZnS. Furthermore, the fundamental and the harmonic wavelengths coincide exactly with very high DOM region at the photonic band edge. The existence of forward and backward propagation simultaneously in the PC permits the efficient generation of both forward and backward SHG and THG. The measured transmission spectrum is in good agreement with the simulation one in Fig. 1 which means our sample is a good 1-D PC with a clear photonic band gap and many transmission resonances.

 

Fig. 1. (a) The experimentally measured (dotted line) and the corresponding theoretically calculated (solid line) transmission. (b) Effective refractive index (solid line) and density of modes (DOM) (dashed line) as a function of wavelengths of the same PC.

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The pump laser comes from an optical parametric amplifier tunable from 0.5 to 1.9µm at 10Hz repetition rate with pulse duration of about 35 ps. The laser splits into two parts. One is used as the reference to remove the fluctuation of the laser power and to normalize the signals at different wavelengths. The other is used as the fundamental wave for SHG and THG. The intensity of fundamental wave is controlled by variable neutral density filter from 200MW/cm² to 3GW/cm². The SH and TH signal are detected by a photomultiplier tube coupled to a monochrometer. Figure 2 shows the measured forward SHG and THG conversion efficiency and the corresponding simulations (dashed curves) as a function of fundamental wavelengths at excitation intensity of 3 GW/cm². In all our calculations χ⁽²⁾=7×10^-9 esu and χ⁽³⁾=1×10^-12 esu, which are close to the values given in references [24] and [25]. The dramatic enhancements of SH and TH signals are shown in Fig. 2 when the fundamental excitation wavelengths are peaks at about 1530nm, 1605nm and 1700nm. Comparing with the Fig. 1 these enhancements coincide with the simulation results of SHG and THG due to the good phase matching and high DOM at these wavelengths near the photonic band edge. This is further verified from SHG (Fig. 2a) and THG spectra where no enhancement peaks are seen at shorter wavelength band edge (i.e., at ~1280nm) where DOM is still fairly high but poor phase matching for fundamental (1280nm) with the SHG and THG wavelengths (see Fig. 1(b)). This result show unambiguously that the enhancement of the SHG and THG is due to both phase matching and high DOM.

Although good agreement is obtained between theory and experiment in the SHG, we also observe some discrepancy in the THG. Despite the excellent agreements found between theory (dashed curve) and experiment in the peak positions of THG shown in Fig. 2(b), there exists a two orders of magnitude difference in the ratio of conversion efficiency at the first peak to that at the second peak. This large discrepancy cannot be reduced by adjusting the value of χ⁽³⁾ in our calculation. A possible source of the discrepancy is due to the presence of randomness in our sample. Both SEM cross-section and manufacturer specifications indicate that variation in layer thickness of a few % does exist in our sample. To study the effects due to layer randomness, we have repeated our calculations for many random configurations. For each configuration, the thickness of each layer is allowed to vary randomly within a given percentage. Our results indicate that a 3.5% randomness in the layer thickness is capable of reducing the above discrepancy significantly. The results of a particular configuration with 3.5% randomness are shown by solid curves in Fig. 2. This configuration is chosen so that the simulation result for the THG fits best the experimental data. It should be mentioned that the behavior of the forward SHG (see Fig. 2(a)) is insensitive to the presence such small randomness in the layer thickness. As will be shown later, this choice of random configuration also reduces the discrepancies between theory and experiments in the backward direction for both SHG and THG.

 

Fig. 2. The experimentally (solid circle) measured and theoretically (dash line without and solid line with the fluctuation of layer thickness) calculated SHG (a) and THG (b) conversion efficiency in the forward direction at excitation intensity of 3 GW/cm².

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Due to very large field enhancement, the maximum conversion efficiency of THG almost equals to the one of SHG even in the low excitation. A cubic dependence of THG on the pump energy verifies that the direct THG is achieved. Theoretical calculation also shows the TH signal obtained from the direct THG base on χ⁽³⁾ is larger than obtained via a cascaded two-step process based on χ⁽²⁾.

Figure 3 shows the theoretically calculated and experimentally measured conversion efficiency of SHG and THG in the backward direction as a function of fundamental wavelengths at excitation intensity of 3 GW/cm². The behavior of SHG and THG in the backward direction is similar as the forward one. The magnitude and peak positions of conversion efficiency for both SHG and THG are found to be in good agreement between theory and experiments taking the randomness of layer thickness into account. The presence of layer randomness in a sample can modify the wave functions of the fundamental wave and higher harmonics within the sample, which can in turn alter the conversion efficiencies for both SHG and THG. Thus, some spectral regions where the simulated harmonic intensities are more enhanced in the sample with 3.5% layer fluctuation than the perfect crystal is the result of this effect. When the incident laser was about 3GW/cm² the maximum conversion efficiencies are 1% and 0.5% for SHG and THG respectively in the forward direction, 1.2% and 1.7% for SHG and THG respectively in the backward direction.

 

Fig. 3. Conversion efficiency of experimentally (solid circle) measured and theoretically calculated SHG(a) and THG(b) without (dash line) and with (solid line) fluctuation in layer thickness in the backward direction at excitation intensity of 3 GW/cm².

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4. Conclusion

In this study we have observed the dramatic simultaneous enhancement for SHG and THG in forward and backward directions from 1-D ZnS/YF₃ PC structure. It is shown that good phase matching and simultaneous availability of the high DOM of fundamental and harmonic waves realize these enhancements near the photonic band gap. The conversion efficiencies for SHG and THG in forward and backward directions are in good agreement between the experimental results and theoretical simulations. The large conversion efficiency for THG is due to direct THG base on χ⁽³⁾ process instead of the cascaded two-step process based on χ⁽²⁾. The total SHG and THG conversion efficiency is 4.4% in a six microns thick sample pumped at 3GW/cm². Successful fabrication of this 1-D PC using conventional e-beam evaporation has considerable advantages compared to more complicated and expensive techniques such as MBE. Overall, our results exhibit a good prospect for PC in nonlinear optics applications in the near-UV to visible spectral region.