1. Introduction

Nonlinear optical interactions in the visible and near-infrared spectral range have attracted significant attention due to the possibility of realizing generation of electromagnetic waves at new frequencies and ultrafast manipulation of light with light, opening many opportunities for molecular absorption spectroscopy, biological and chemical sensing, all-optical switching, and free-space communications. However, to date, due to the inherently weak nonlinear response of natural materials, long interaction lengths (such as those provided by optical fibers) and complex phase-matching techniques are required to enable significant nonlinear interactions. The miniaturization of nonlinear optical devices and systems usually requires reaching extremely high light intensities often leading to material breakdown. For this reason, the quest for novel approaches to satisfying the phase-matching conditions and enhanced optical nonlinearities at moderate input intensities is critically important.

In this paper, we focus on novel regimes of nonlinear light-matter interactions enabled by the unique class of photonic materials, called photonic quasicrystals (PQCs). Quasicrystals are material systems that are aperiodic but deterministic, forming an intermediate structure that lacks translational symmetry but possesses long-range order [1,2]. Natural electronic quasicrystals such as AlPdMn metal alloys have been found to show peculiar electronic, vibrational and physical-chemical properties [3]. The discovery of quasicrystals has already revolutionized solid-state physics and initiated a new direction of research in photonics. Similar to naturally occurring quasicrystals, artificial quasicrystals possess properties that are different from those of both uniform and periodic structures. The first quasiperiodic semiconductor superlattice was fabricated by Merlin et al. by molecular-beam epitaxy in 1985 [4].

The emergence of PQCs at visible and near-infrared frequencies opens unprecedented opportunities in the field of nonlinear optics, where the advantages of using PQCs originate from the possibility of supporting unlimited combinations of wavevectors in their reciprocal lattices. This greatly enhanced flexibility enabled by k-space engineering makes PQCs a promising platform for both fundamental studies of nonlinear light-matter interactions and applications in wavelength conversion, supercontinuum generation [5], and the development of classical and quantum optical sources. Since the integration of multiple photonic functionalities on a chip and further shrinking of their size is of paramount importance, combining multiple nonlinear processes in a single device may be desirable despite possible limitations of the conversion efficiency. However, one of the major challenges preventing the realization of this goal is satisfying the phase-matching condition for multiple nonlinear processes in the same structure.

Indeed, the efficiency of nonlinear optical interactions strongly relies on phase- and frequency-matching conditions [6]. For example, the ideal phase-matching condition for the second harmonic generation process is satisfied when the wave-vector mismatch $\Delta k = 0$, meaning that the refractive index $n(\omega ) = n(2\omega )$, and the phase velocity of the fundamental and the second harmonic waves are equal. However, this condition is difficult to satisfy because of dispersion of the medium. As a result, phase mismatching, $\Delta k \ne 0$, significantly limits the efficiency of nonlinear optical processes. One approach to solving this issue is to use the birefringent materials, where the dispersion of the material is balanced with the difference between the indices of refraction of the ordinary and extraordinary waves. However, there are a limited number of materials that are suitable for birefringence phase-matching. Other limitations of this approach include strong angular dependence, limited interaction length due to walk-off, and dependence of the nonlinear susceptibility on the angle and frequency. Another method widely used to boost the efficiency of nonlinear optical processes is quasi-phase-matching (QPM), achieved by spatially modulating the nonlinear coefficient to generate reciprocal lattices containing the necessary wave vectors [6–9]. Quasi-phase-matching of the second-order nonlinear processes has been experimentally realized using electrical, thermal or optical poling, where the sign of the second-order nonlinear susceptibility, ${\chi ^{(2)}}$, is periodically, spatially switched from positive to negative values. While poling is a viable approach for second-order nonlinear optics, it cannot be straightforwardly extended to the third- and higher-order nonlinear optical interactions. Moreover, most of the prior work on QPM was focused on structures with non-altered linear susceptibility (or refractive index) but modulated nonlinear susceptibility. Also, one-dimensional photonic crystals (PCs) with both the linear and the second-order nonlinear susceptibilities modulated have been theoretically investigated [10]. However, the limitation of the majority of these periodically modulated structures is that they usually enable phase-matching of only one nonlinear process with a vector mismatch $\Delta k$ or, more generally, the processes relying on an integer multiple of $\Delta k$. Therefore, these two conventional phase-matching techniques offer limited design flexibility and usually cannot facilitate efficient phase-matching of multiple nonlinear processes in one structure.

In contrast, engineered nonlinear one-, two-, and three-dimensional PQCs and aperiodic structures [11–13] such as the Fibonacci sequence [14–16], Penrose structures [17], and reconfigurable 3D quasi-crystallographic PCs [18] have been considered as promising candidates for realizing multiple nonlinear light-matter interactions in one nanostructure. One of the significant challenges of designing nonlinear PQCs is that, in contrast to periodic nonlinear PCs, there is no simple transformation between reciprocal space and real space. Currently, nonlinear PQCs, as well as PCs and random photonic structures, are designed using the so-called generalized dual-grid method [19–21] that enables designing wavelength converters that phase-match desired set of interacting electromagnetic waves. This method has been previously used for constructing tiling models of quasicrystals. Despite the versatility of this approach, it nevertheless relies on the assumption that the linear susceptibility is spatially uniform, or dispersionless, significantly limiting the practical realization of multiple phase-matched second-order, and especially, third-order nonlinear processes. Therefore, theoretical approaches necessary for the realization of PQCs that offer the desired flexibility in choosing dispersive linear and nonlinear materials is still missing.

Here, we develop a design methodology for simultaneously phase-matching two (or more) nonlinear optical processes in a nonlinear PQC made of a combination of linear and nonlinearoptical materials. We aim at designing a chalcogenide glass (ChG) based multilayer structure enabling simultaneous phase-matching of the third harmonic generation (THG) at two different pump (fundamental wave) frequencies.

2. Theory

The particular material system we consider here includes arsenic trisulfide (As₂S₃) and zinc sulfide (ZnS), where As₂S₃ acts as a nonlinear material (${\chi ^{(3)}} > 0$) and ZnS acts as a linear material (${\chi ^{(3)}} \approx 0$). The measured refractive indices of As₂S₃ and ZnS thin film that are used in our calculations can be found in Fig. S1 (See Supplement 1). ChG was chosen due to its superior nonlinear properties (${\chi ^{(3)}}$ up to three orders of magnitude larger than that of silica glass), excellent nonlinear figure of merit, defined as the ratio of the real and imaginary parts of the nonlinear refractive index, and its high transparency in the near- and mid-infrared wavelength range [22,23]. In addition, the photo-modification properties of some ChG materials make them suitable for direct laser-writing based recording of ultra-small photonic structures in a single step [24]. As a result, ChGs have found applications ranging from optical biosensors [25], to photonic waveguides [26], and acousto-optics [27].

The goal of this work is to develop a comprehensive approach for designing periodic or quasi-periodic structures made of a combination of realistic dispersive optical materials with different linear refractive indices and nonlinear susceptibilities, as shown in Fig. 1. Let us first consider a single THG process, which can be supported by a standard nonlinear PC in three different configurations: (i) nonlinear PC with spatially uniform linear refractive index ${n_\textrm{L}} = {n_{\textrm{A}{\textrm{s}_\textrm{2}}{\textrm{S}_\textrm{3}}}}$ corresponding to As₂S₃ glass and with a periodically alternating sign of the third-order nonlinear susceptibility ${\chi ^{(3)}}$ denoted as (+/−); (ii) nonlinear PC with the same spatially uniform linear refractive index ${n_{\textrm{A}{\textrm{s}_\textrm{2}}{\textrm{S}_\textrm{3}}}}$ and the third-order nonlinear susceptibility periodically changing from positive to zero, denoted as (+/0), (iii) nonlinear PC with linear refractive index periodically changing between that of As₂S₃ (${n_{\textrm{A}{\textrm{s}_\textrm{2}}{\textrm{S}_\textrm{3}}}}$) and that of ZnS (${n_{\textrm{ZnS}}}$), and the third-order nonlinear susceptibility periodically changing from positive to zero, denoted as (+/ZnS).

 

Fig. 1. Photonic structure design and quasi-phase-matching (QPM). (a) Photonic quasicrystal composed of nonlinear As₂S₃ layers and linear ZnS layers; (b) QPM-based THG as a function of propagation distance for three types of nonlinear periodic structures: red line (+/-) corresponds to the case of spatially unmodulated linear refractive index and nonlinear susceptibility switching between positive and negative values; blue line (+/0) corresponds to the case of spatially unmodulated linear refractive index and nonlinear susceptibility switching between positive and zero values; yellow line (+/ZnS) corresponds to the case of spatially modulated linear refractive index and nonlinear susceptibility switching between positive and zero values. The inset shows a zoomed-in part of the plot in the range of propagation distances from 0 to 0.025 mm. (c) Third harmonic field amplitudes as a function of wavelength after propagation over a distance of 0.5 mm.

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Figure 2 illustrates the design algorithm for realizing simultaneous phase matching of two (or more) THG processes using a quasi-periodic photonic structure with both linear, dispersive refractive index and nonlinear susceptibility modulated along the propagation direction (corresponding to case (iii)). In order to design the two-material system corresponding to case (iii), PC structure corresponding to case (ii) by first replacing the layers with linear refractive index of As₂S₃ to that of ZnS, followed by adjusting the width of the linear layers from ${d_\textrm{L}}$ to ${d_{\textrm{ZnS}}}$ in order to compensate for the difference in linear refractive indices of ZnS and As₂S₃ as shown in Fig. 2(d).

 

Fig. 2. Design methodology for a one-dimensional nonlinear PQC enabling simultaneous phase-matching of two collinear nonlinear processes. (a) Step1: Building the dual grid based on the vector mismatches $\Delta {k_1}$ and $\Delta {k_2}$ of two nonlinear processes at different frequencies. (b) Step 2: Constructing the tilling based on the dual grid. Every line of the dual grid corresponds to the tiling with width ${a_1}$ or ${a_2}$ for $\Delta {k_1}$ and $\Delta {k_2}$, respectively. (c) Step 3: Designing a PQC made of the layers with the same linear refractive index $n = n{_{\textrm{A}{\textrm{s}_\textrm{2}}{\textrm{S}_\textrm{3}}}}$. Each tiling has a corresponding duty cycle ${D_1}$ or ${D_2}$. In this scheme ${D_1} = 1$ and ${D_2} = 0$, and therefore, the “nonlinear” layers with the width ${d_{\textrm{NL}}}$ have nonlinear susceptibility ${\chi ^{(3)}}$ and the “linear” layers with the width ${d_\textrm{L}}$ have negligible nonlinear susceptibility, i.e. ${\chi ^{(3)}} = 0$. (d) Step 4: Replacing the linear refractive index of the “linear” layers with the refractive index ${n_{\textrm{ZnS}}}$, followed by changing the width of the linear layers from ${d_\textrm{L}}$ to ${d_{\textrm{ZnS}}}$. The insert shows the case of non-integer duty cycles ${D_1} = 0.4$ and ${D_2} = 0.5$.

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By adjusting the width of the linear layers, we maintain the differences in optical path lengths of the fundamental wave and third harmonic (TH) in the linear layers made of ZnS identical to those in the case of uniform linear refractive index in a PC made of As₂S₃:

Thus, the new width ${d_{\textrm{ZnS}}}$ obtained from the Eq. (1) is given by

The relations between the normalized amplitude of the TH and the propagation distance are shown in Fig. 1(b). It clearly shows that compared with case (i), in case (ii), when the nonlinear susceptibility of the second segment is zero instead of $- {\chi ^{(3)}}$, the amplitude of the TH decreases by half. The amplitude of the TH in the case (iii) is smaller than that in case (i) and (ii) because the total length of nonlinear material in case (iii) is shorter than that in case (ii) for the same propagation distance. This fact is defined by the parameter $\beta (\omega )$ which is $\beta (\omega ) > 1$ for As₂S₃ and ZnS at the wavelength $\lambda = 700\textrm{nm}$.

Figure 1(c) shows the central wavelength and the amplitude of TH after 0.5mm of propagation from the three cases considered above. As predicted, the nonlinear PC with both linear and nonlinear refractive index modulation (e.g., As₂S₃ and ZnS) supports the THG at the same central wavelength as in the case of unmodulated linear refractive index, although with the reduced conversion efficiency for the same propagation length due to the parameter $\beta (\omega ) > 1$ in our case.

Next, we consider more than one THG process simultaneously supported by a single nonlinear PQC. Let us assume the fundamental frequencies (FFs) are ${\omega _1}$ and ${\omega _2}$ with corresponding wavelengths ${\lambda _1} = 2100\textrm{nm}$ and ${\lambda _2} = 1800\textrm{nm}$. The corresponding TH wavelengths that we aim at generating are ${\lambda _1}/3 = 700\textrm{nm}$ and ${\lambda _2}/3 = 600\textrm{nm}$. Figure 2 shows the design process for the one-dimensional nonlinear PQC for two collinear THG processes, consisting of the following steps. First, the wavevector mismatches between the FF and TH waves $\Delta {k_1}$ and $\Delta {k_2}$ are calculated (Fig. 2(a)). Following the original dual grid method (DGM) developed for the case of unmodulated linear refractive index [19], we build the dual grid and insert the tiling of the real-space line according to the order of the lines in the dual grid (Fig. 2(b)). Next, we associate the duty cycle D with each tiling (Fig. 2(c)), and here we choose $D = 1$ or $D = 0$ as duty cycle since it results in the largest Fourier coefficient [28]. Next, since the goal of this work is to generalize the DGM to the case of the material combination, an additional step, similar to that described by Eq. (4), should be included in order to compensate for the differences between the linear refractive indices of As₂S₃ and ZnS (Fig. 2(d)). This step can be generalized to the case of N simultaneous THG processes with FFs ${\omega _1}\ldots {\omega _N}$ in a single PQC.

We start by calculating the transformation factor $\beta (\omega )$ using the linear, frequency-dependent refractive indices of As₂S₃ and ZnS at one of the desired frequencies from the set ${\omega _1}\ldots {\omega _N}$, for example:

$\omega = {\omega _1}$ (that we will refer to as “design frequency”, and the corresponding wavelength ${\lambda _1}$ will be called “design wavelength”). The calculated $\beta (\omega = {\omega _1})$ is used to find new lengths of the linear layers, ${d_{\textrm{ZnS}}} = \beta ({\omega _1}){d_\textrm{L}}$. Now, since ${d_\textrm{L}}$ and the total length of the QPC are fixed, we investigate how other THG processes at frequencies ${\omega _2},{\omega _3},\ldots {\omega _N}$ can be phase-matched, simultaneously. Since the linear refractive indices of As₂S₃ and ZnS are dispersive, i.e. depend on the frequencies of both the FF and TH, the transformation factor $\beta ({\omega _1})$ is not optimized for other fundamental wave frequencies ${\omega _2}\ldots {\omega _N}$. Let us understand how fixing $\beta (\omega )$ at design frequency ${\omega _1}$ would affect the processes of the THG at ${\omega _2}$ and other frequencies.

Following Ref. [19], in order to calculate the corresponding N collinear tiling vectors, ${a_i}$, $i = 1\ldots N$, we first construct a single $N$-component vector ${{\mathbf k}_1} = (\Delta {k_1}\ldots \Delta {k_N})$ from the N given mismatch values in the nonlinear material (in our case it is As₂S₃). This vector spans a one-dimensional subspace of an abstract $N$-dimensional vector space. We complete an orthogonal basis of the $N$-dimensional space by adding $N - 1$ orthogonal vectors ${{\mathbf q}_2}\ldots {{\mathbf q}_N}$. Let us consider the ${\mathbf K}$ matrix for a range of frequencies ${\omega _1}\ldots {\omega _N}$:

The desired tiling vectors ${a_i}$ can be obtained by calculating matrix ${\mathbf A}$:

In order to match the THGs at exact frequencies ${\omega _1},{\omega _2}\ldots {\omega _N}$, we need to take into account the difference between the transformation factor $\beta ({\omega _1})$ at design frequency ${\omega _1}$ and transformation factors $\beta (\omega )$ at other frequencies ${\omega _2}\ldots {\omega _N}$. The first step is to calculate the ratio $\alpha $ between the total width of all linear and nonlinear segments. The number of layers with thickness ${a_i}$ for the fixed total length of the structure L is proportional to the value $\Delta {k_i}$ (Fig. 2). In addition, we can assign the duty cycles ${D_i}$ for every ${a_i}$, as shown in Fig. 2(c). These duty cycles ${D_i}$ define the thicknesses of linear and nonlinear materials in every layer, which are equal to $(1 - {D_i}){a_i}$ and ${D_i}{a_i}$, respectively. Then the ratio $\alpha $ is given by:

Nonlinear PQCs designed to simultaneously match N harmonic generation processes must possess N corresponding nonlinear vector mismatches $\Delta {k_i} = {k_{3{\omega _i}}} - 3{k_{{\omega _i}}}$. We can associate every $\Delta {k_i}$ with the grating period ${\Lambda _i} = 2\pi /\Delta {k_i}$. However, due to the difference between the factor $\beta ({\omega _1})$ at the design frequency and at other frequencies, the grating periods ${\Lambda _i}$ in the PQC are stretched out compared to the desired one. The ratio between the width of a ZnS layer ${d_{\textrm{ZnS}}} = \beta ({\omega _1}){d_\textrm{L}}$ and the width required for a process at the frequency ${\omega _i} \ne {\omega _1}$ is $\beta ({\omega _i})/\beta ({\omega _1})$. Since the sequence of linear and nonlinear layers in the PQC remains constant with changing the factor $\beta $, we can assume that the stretch of the period ${\Lambda _i}$ is proportional to the change in the total QPC length L.

If we denote the total width of all nonlinear layers as ${L_{\textrm{NL}}}$, the total length of all linear layers ${d_{\textrm{ZnS}}}$ is $\alpha {L_{\textrm{NL}}}$ and $L = {L_{\textrm{NL}}} + \alpha {L_{\textrm{NL}}}$. In that case, the ratio of the desired period ${\Lambda _i}$ to the real period $\Lambda _i^{\textrm{real}}$ are proportional to the ratio of values L as follows:

To match the exact desired vector mismatches $\Delta {k_i}$, i.e. $\Delta k_i^{\textrm{real}} = \Delta {k_i}$, the initial vector mismatches must be shifted so they satisfy the following equation:

This can be achieved by using shifted frequencies ${\omega _1},\omega _2^{\textrm{shifted}},\omega _3^{\textrm{shifted}}\ldots \omega _N^{\textrm{shifted}}$ during the PQC calculation. Required values $\omega _i^{\textrm{shifted}}$ can be found from the equations

To evaluate THG conversion efficiency in the multilayer structure, we investigate continuous wave propagation in a PQC structure. Assuming that the nonlinear conversion efficiency is low (${E_{3\omega }} < < {E_\omega }$), the amplitude of the fundamental wave ${E_\omega }$ can be considered constant throughout the entire crystal length L (undepleted pump approximation ${E_\omega } \approx const$). In this case, the equation for TH wave amplitude ${E_{3\omega }}$ in the crystal with constant linear refractive index can be written in the form

3. Results

Based on the calculation above, we design the nonlinear PQC in a planar configuration using a combination of As₂S₃ and ZnS as nonlinear and linear layers, respectively, surrounded by bulk ZnS on the input and output sides, to phase-match two THG processes at ${\lambda _1} = 2100\textrm{nm}$ and ${\lambda _2} = 1800\textrm{nm}$, separately. Figure 3(a) shows the relation between propagation distance and normalized amplitude of the TH at $600\textrm{nm}$ in the following cases: (i) the nonlinear PQC consisting of uniform linear refractive index corresponding to As₂S₃ and modulated third-order nonlinear susceptibility (+/0) designed using the conventional dual grid method; (ii) the nonlinear PQC consisting of As₂S₃ and ZnS designed using the model described above for the “design wavelength” of $2100\textrm{nm}$, but without taking into account the frequency shift of the second THG process, and (iii) the nonlinear PQC consisting of As₂S₃ and ZnS designed using our model for the “design wavelength” of $2100\textrm{nm}$ taking into account the frequency shift of the second THG process: the shifted second frequency $\omega _2^{\textrm{shifted}}$ in term of wavelength is $1811\textrm{nm}$. As a result, we find that the THG process at the fundamental wave wavelength of ${\lambda _2} = 1800\textrm{nm}$ in cases (i) and (iii) are in good phase-matching condition, but in case (ii), the TH wavelength is shifted. Figure 3(b) shows the spectra of the THG processes after $0.5\textrm{mm}$ of propagation for these three nonlinear PQCs. The amplitudes were normalized to the maximum of the absolute value of the TH field. In all three cases the THs are generated at $700\textrm{nm}$, which corresponds to the design wavelength of $2100\textrm{nm}$, but only in cases (i) and (iii) is the TH generated precisely at $600\textrm{nm}$, while in case (ii) the TH is blue shifted. These results confirm that the proposed approach can be used as a versatile tool for designing PQC structures consisting of two different materials with spatially modulated linear and nonlinear refractive indices to enable two or more phase-matched nonlinear processes simultaneously.

 

Fig. 3. The results of numerical simulations for the nonlinear PQC designed for simultaneous matching processes at 2100 nm and 1800nm wavelengths. The design wavelength is 2100 nm (upper row) and 1800nm (lower row), respectively. The normalized field amplitudes of the THs at 600 nm and 700 nm for the cases (i), (ii), and (iii) versus propagation distance (a), (d) and versus wavelength (b), (e). (c), (f) The transmission spectra of the nonlinear PQC near the fundamental wavelengths. The red dash lines in (c) and (f) indicates 1800nm and 2100 nm wavelengths.

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It should be noted that due to the spatial modulation of linear refractive index along the structure, multiple photonic bandgaps are formed, and therefore, the design procedure should include an additional step to ensure that the fundamental wavelengths are not located in the bandgaps. Figure 3(c) shows the transmission spectrum of case (iii) NPQCs. The dashed-red lines indicate $2100\textrm{nm}$ and $1800\textrm{nm}$ wavelengths. In this example, indeed the bandgap of the PQC structure is located near $2100\textrm{nm}$ wavelength that may bring the bandgap to overlap with the real spectra in potential experiments. Therefore, we need to optimize the structure to move the bandgaps away from our desired fundamental wavelengths. This problem can be addressed by either changing the “design wavelength” or by changing the duty cycle of each segment (see Supplement 1). Let us choose the design wavelength at 1800nm instead of 2100nm, and the shifted wavelength for the second process at 2088nm. Then, Fig. 3(d) shows that in cases (i) and (iii) (defined as above), the phase-matching condition is satisfied to generate the TH at 700nm, while in case (ii), the TH is red-shifted. The spectra of the THG after 0.5mm of propagation for cases (i)-(iii) are shown in Fig. 3(e) confirming that in all three cases the TH is generated at 600nm, as expected. Indeed, 600nm corresponds to the TH of the fundamental wavelength of 1800nm, which is the design wavelength in this case. Figure 3(f) shows the transmittance of the structure from case (iii), and now the bandgaps of the PQC structure located much further away from the frequencies of interest compared to Fig. 3(c). The structure of the NPQCs used for the case described above (Fig. 3(e),(f)) is represented in Fig. 4(a). In Fig. 4(b),(c) there is the transmittance of the same NPQC near 1800nm and 2100nm. The green lines represent the effective refractive index of the structure. Since the bandgaps are narrow and located far enough from interested us frequencies 1800nm and 2100nm, the effective refractive index behaves smoothly and does not disturb the method.

 

Fig. 4. (a) Spatial structure of the PQC enabling simultaneous phase-matching of two THG with fundamental wavelengths $\lambda = 1800\textrm{nm}$ (“designed wavelength”) and $\lambda = 2100\textrm{nm}$. Blue color represents nonlinear layers made of As2S3, yellow color – linear layers of ZnS. Parameters responsible for the thickness of the layers are ${a_1} = \textrm{1}\textrm{.27}\mathrm{\mu}\textrm{m}$ and ${a_2} = \textrm{2}\textrm{.24}\mathrm{\mu}\textrm{m}$, $\beta (\lambda = 1800\textrm{nm}) = 2.124$, ${D_1},{D_2} = 0,1$. Vector mismatches: $\Delta {k_1}(\lambda = 2088\textrm{nm)} = \textrm{1}\textrm{.205}\mathrm{\mu}{\textrm{m}^{ - 1}}$ and $\Delta {k_2}(\lambda = 1800\textrm{nm) = 2}\textrm{.121}\mathrm{\mu}{\textrm{m}^{ - 1}}$. Refractive indexes used to calculate the structure are: ${n_{\textrm{ZnS}}}(\lambda = 2088\textrm{nm}) = \textrm{2}\textrm{.26}$, ${n_{\textrm{ZnS}}}(\lambda = 696\textrm{nm}) = \textrm{2}\textrm{.33}$, ${n_{\textrm{ZnS}}}(\lambda = 1800\textrm{nm}) = \textrm{2}\textrm{.27}$, ${n_{\textrm{ZnS}}}(\lambda = 600\textrm{nm}) = \textrm{2}\textrm{.36}$, ${n_{\textrm{A}{\textrm{s}_\textrm{2}}{\textrm{S}_\textrm{3}}}}(\lambda = 2088\textrm{nm}) = \textrm{2}\textrm{.43}$, ${n_{\textrm{A}{\textrm{s}_\textrm{2}}{\textrm{S}_\textrm{3}}}}(\lambda = 696\textrm{nm}) = \textrm{2}\textrm{.56}$, ${n_{\textrm{A}{\textrm{s}_\textrm{2}}{\textrm{S}_\textrm{3}}}}(\lambda = 1800\textrm{nm}) = \textrm{2}\textrm{.43}$, and ${n_{\textrm{A}{\textrm{s}_\textrm{2}}{\textrm{S}_\textrm{3}}}}(\lambda = 600\textrm{nm}) = \textrm{2}\textrm{.63}$. (b),(c) The transmission spectrum of the nonlinear PQC near the interested us wavelengths. The red dash lines indicate the wavelength of 1800nm and 2100 nm; the green lines show the effective refractive index of the PQC.

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It is noteworthy that the proposed approach is not limited to two THG processes but can be extended for a few simultaneously phase-matched processes. As an example, in Supplement 1, we show the design developed for three simultaneous THG processes (Fig. S3). However, with increasing the number of phase-matched processes, the complexity of the structure is likely to increase to the point when the thickness of the layers may reach subwavelength values. In that case, the proposed method will need to be modified to include the effect of spatial dispersion that becomes of significant importance [29]. Moreover, recent research shows that not only the nonlocal response of the multilayer structure should be considered, but also appropriate boundary conditions should be applied in the case of subwavelength periodicity [30].

4. Conclusions

In conclusion, we developed a comprehensive design method for nonlinear PCs and PQCs, consisting of a combination of one nonlinear material and one linear material, that can simultaneously phase-match one or more nonlinear interactions. As an example, we designed a PC and a PQC consisting of a combination of a nonlinear material (As₂S₃) and a linear material (ZnS) in multilayer configurations. The summary of the proposed design procedure can be described in the following steps: first, one needs to measure the refractive indices of the materials as functions of wavelength. For the THG processes of interest, the structure of the PQC is calculated using the proposed model. Next, the transmission spectrum for light propagating in this structure is calculated to ensure that fundamental waves’ wavelengths are located outside of the bandgaps of the structure. If the fundamental wavelength falls into the bandgap, either the design wavelength or the duty cycles of the structure need to be changed and the procedure needs to be repeated. The performance expected from the developed design procedure is in excellent agreement with the results of numerical simulations. Finally, the proposed multilayer structures can be realized using standard deposition methods. Since the deposition of the multilayer structure might be a sophisticated challenge, we focus on the waveguide configuration in our experimental study. The waveguide structure can and already was fabricated by us (see supplementary Fig. S4). However, optimization of the fabrication process and optical characterization requires additional time, so the experimental results will be reported in a follow-up publication. The proposed design method adds flexibility in the design and realization of the multi-wavelength nonlinear converters based on PQCs.