# Combinatorial Frequency Generation in Quasi-Periodic Stacks of Nonlinear Dielectric Layers

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Problem Statement and Methodology

_{A}and d

_{B}, respectively. Stacks of thicknesses L

_{q}are surrounded by a linear homogeneous medium with the relative permittivity ε

_{a}at z ≤ 0 and z ≥ L

_{q}. The constituent dielectric layers with quadratic nonlinearity and 6 mm class of anisotropy are described by the tensors of relative linear permittivity and nonlinear susceptibility where j = A, B:

- (a)
- Fibonacci type QPS (Figure 1a) of order q ≥ 2 is defined recursively by the recurrence relation: S
_{q}= {S_{q}_{−1}◡ S_{q}_{−2}}, where S_{0}= {B}, S_{1}= {A}; the number of layers in a stack S_{q}equals Fibonacci number Ф_{q}_{+1}and the corresponding stack thickness is L_{q}= L_{q}_{−1}+ L_{q}_{−2}with L_{0}= d_{B}; L_{1}= d_{A}; - (b)
- Thue-Morse type QPS (Figure 1b) of order q ≥ 1 is defined recursively by the recurrence relations: Q
_{q}= {Q_{q}_{-1}◡ Q'_{q}_{-1}} and Q'_{q}= {Q'_{q}_{-1}◡ Q_{q}_{-1}}, where Q_{q}and Q'_{q}are complementary to each other and Q_{0}= {A}, Q'_{0}= {B}; the stacks Q_{q}and Q'_{q}contain 2^{q}layers and have thicknesses L_{q}= L'_{q}= (d_{A}+ d_{B})^{q}at q > 0 and L_{0}= d_{A}, L'_{0}= d_{B}.

_{1}and ω

_{2}incident at angles Θ

_{i}

_{1}and Θ

_{i}

_{2}, respectively. Since the layers are assumed isotropic in the x-y plane, the TE and TM polarised waves with the fields independent of the y-coordinate (∂/∂

_{y}= 0) can be analysed separately. Only TM waves with the field components E

_{x}, E

_{z}, H

_{y}are presented in the paper. The TE waves are treated similarly and even somewhat simpler, because the nonlinear susceptibility given by Equation (1) is isotropic for TE polarised waves.

_{3}= ω

_{1}+ ω

_{2}the TM wave fields in each layer are described by non-homogeneous Helmholtz equation:

_{xj}(ω

_{1,2}) and E

_{zj}(ω

_{1,2}) are the electric field components of pump waves in a layer of type j = A, B; k

_{3}= ω

_{3}/c and ; angle Θ

_{3}defines the emission direction from the stacks at frequency ω

_{3}, which is determined by the phase synchronism condition in the three-wave mixing process [39]:

_{x}

_{3}= k

_{x}

_{1}+ k

_{x}

_{2}

_{1,2}= ω

_{1,2}/c

_{yj}(ω

_{3}) can be represented as a superposition of six plane waves with longitudinal propagation constants and .

_{3}, generated inside this layer, which depend on amplitudes of the pump waves refracted into this layer [40]. are amplitudes of the waves of frequency ω

_{3}generated outside the layer and refracted into it.

_{6}= {AB AAB AB AAB AAB} is comprised of “regular” cells {AB} and “defective” cells {AAB} with an additional A layer which forms a layer A doublet of thickness d'

_{A}= 2d

_{A}. Similarly, Thue-Morse QPS, for example, Q

_{4}= {AB BA BA AB BA AB AB BA} is composed of the same “regular” cells {AB} and “defective” cells {BA} with inverse order of layers but the same cell thickness. Thus once the cell serial number n, and layout and position of the regular and defective cells are known, the conventional TMM can be adapted to obtain amplitude coefficients and in Equation (4) for Fibonacci and Thue-Morse QPSs as detailed in Appendix 1.

_{3}in individual layers are determined, amplitudes F

_{r}(ω

_{3}) and F

_{t}(ω

_{3}) of the field emitted from stacks toward the reverse and forward directions of the z-axis, respectively, can be evaluated by the modified TMM [40] as described in Appendix 2. The obtained analytical solutions provide an insight in the fundamental mechanisms of nonlinear scattering and frequency mixing by the QPSs illuminated by a pair of obliquely incident plane waves. The features of CFG in Fibonacci and Thue-Morse QPSs are further illustrated by examples of numerical simulations and discussed in the next section.

## 3. Simulation Results and Discussion

_{a}= 1 and are illuminated by a pair of pump plane waves of frequencies ω

_{1,2}incident at angles Θ

_{i}

_{1,i2}as shown in Figure 1. The QPS constituent layers are anisotropic dielectric films of CdS and ZnO with the following values of the tensor relative linear permittivity and nonlinear susceptibility (the units [m/V] are omitted below for brevity) [45]:

_{xxA}= 5.382, ε

_{zzA}= 5.457, χ

_{xxzA}= 2.1 × 10

^{−7}, χ

_{zxxA}= 1.92 × 10

^{−7}, χ

_{zzzA}= 3.78 × 10

^{−7},

Layer B (ZnO) ε

_{xxB}= 1.4, ε

_{zzB}= 2.6, χ

_{xxzB}= 2.82 × 10

^{−8}, χ

_{zxxB}= 2.58 × 10

^{−8}, χ

_{zzzB}= 8.58 × 10

^{−8}.

_{A}and d

_{B}are related by the golden ratio: .

#### 3.1. Spectral Efficiency of Frequency Mixing

_{1}and ω

_{2}, incident at dissimilar angles, offers extra degrees of freedom in controlling the frequency mixing process and spectrum.

^{2}of TM waves incident at angles Θ

_{i}= 30° and Θ

_{i}= 45° are displayed in Figure 2 for Fibonacci S

_{8}(34 layers, d

_{B}= 12 µm) and Thue-Morse Q

_{5}(32 layers, d

_{B}= 13 µm) QPSs in comparison with the periodic stack of alternating layers (16 unit cells or 32 layers, d

_{B}= 13 µm). All three stacks have the same thickness, and equal number of A- and B-type layers in the periodic and Thue-Morse stacks, while the Fibonacci QPS contains more layers of smaller thicknesses. Considerable differences in |R(ω)| of the periodic stack and QPSs are evident in Figure 2, especially at higher frequencies. This implies that the frequency mixing efficiency may significantly vary due to disparity of the pump wave magnitudes refracted into the stack. Bandgaps, corresponding to |R(ω)| ≈ 1, exist in both periodic structures and QPSs, albeit more intricate spectra of QPSs provide more flexible conditions for CFG as discussed below.

**Figure 2.**Reflectance of TM wave incident at Θ

_{i}= 30° (dashed black lines) and Θ

_{i}= 45° (solid red lines) on (

**a**) Fibonacci S

_{8}and (

**b**) Thue-Morse Q

_{5}QPSs, and (

**c**) periodic stack with 16 unit cells.

_{r}

_{,t}(ω

_{3})| of the waves of combinatorial frequency ω

_{3}emitted from the stacks have been simulated at fixed frequencies ω

_{2}of a pump wave incident at Θ

_{i}

_{2}= 45°, and swept frequency ω

_{1}of the other pump wave incident at Θ

_{i}

_{1}= 30°. Frequencies ω

_{2}have been chosen individually for each stack configuration so that |R(ω

_{2})|

^{2}be at the minima close to the transparency band edges. Since QPS bands are not clearly defined as in periodic stacks, we refer only to the reflectance minima, which depend on the constituent layer parameters and incidence angles of pump waves as illustrated by Figure 2.

_{r}

_{,t}| and |R(ω)| for each type of QPSs. Indeed, |F

_{r}

_{,t}| reach their peaks at frequencies ω

_{1}corresponding to the minima of the pump wave reflectivity, thus confirming that the pump wave refraction into stacks significantly influences the frequency mixing efficiency. It is also evident here that the CFG efficiency is hardly improved in the proximities of band edges for both QPSs and periodic stacks.

**Figure 3.**Intensity of field radiated from QPSs at frequency ω

_{3}= ω

_{1}+ ω

_{2}in the reverse (|F

_{r}|-solid red lines) and forward (|F

_{t}|-dash-dot black lines) directions of the z-axis: (

**a**) Fibonacci S

_{8}stack at ω

_{2}= 2.01 × 10

^{13}s

^{−1}; (

**b**) Thue-Morse Q

_{5}stack at ω

_{2}= 1.73 × 10

^{13}s

^{−1}and (

**c**) periodic stack containing 16 unit cells at ω

_{2 }= 1.76 × 10

^{13}s

^{−1}. All the stacks are illuminated by pump waves incident at Θ

_{i}

_{1}= 30° and Θ

_{i}

_{2}= 45°.

_{r}

_{,t}| for the stacks of the three types shows that the peak CFG efficiency is higher in the QPSs than in the periodic structure. For example, the |F

_{r}| peak at ω

_{1}= 2.38 × 10

^{13}s

^{−1}in Thue-Morse QPS Q

_{5}(Figure 3b) is about 2.2 times higher than in the periodic stack (Figure 3c). Such enhanced CFG is attributed to more favourable conditions for the phase synchronism in the QPS and local field intensification in individual constituent layers due to the stack composition. Additionally, the internal refraction of pump waves and concurrent constructive interference of the generated combinatorial frequencies ω

_{3}in the entire stack lead to cumulative growth of the peak emission in the forward direction (note that almost all peaks of |F

_{t}| in Figure 3 are higher than |F

_{r}| peaks). Examination of the |F

_{r}

_{,}

_{t}| angular dependences has also proven that |F

_{r}

_{,}

_{t}| reach their maxima at the reflectance |R(ω)| minima of not only pump waves but also the waves of combinatorial frequency ω

_{3}. This implies that the CFG efficiency can be further increased by optimising combinations of the incidence angles, layer parameters, and the stack composition.

#### 3.2. Effect of Stack Composition and Layer Anisotropy

_{r}

_{,}

_{t}| simulated for Fibonacci S

_{7}stack, containing 21 layer with d

_{B}= 19 µm, and Thue-Morse Q

_{4}stack, containing 16 layers with d

_{B}= 26 µm, which have the same overall thicknesses as Fibonacci S

_{8}and Thue-Morse Q

_{5}QPSs in Figure 3 but thicker constituent layers. Comparison of Figure 3 and Figure 4 shows that intensities |F

_{r}

_{,t}| of the combinatorial frequency ω

_{3}emitted from the stacks have almost the same peak magnitudes in both cases but the peaks occur at different frequencies ω

_{1}. Such spectral deviations can be attributed to pump wave redistribution in the stack constituent layers, caused by the internal reflection and refraction at the layer interfaces, and to the changes in the phase coherence between the pump waves and combinatorial frequency waves generated in individual constituent layers.

**Figure 4.**Intensity of field radiated from QPSs at frequency ω

_{3}= ω

_{1}+ ω

_{2}in the reverse (|F

_{r}|-solid red lines) and forward (|F

_{t}|-dash-dot black lines) directions of the z-axis: (

**a**) Fibonacci S

_{7}stack at ω

_{2}= 2.02 × 10

^{13}s

^{−1}; (

**b**) Thue-Morse Q

_{4}stack at ω

_{2}= 1.76 × 10

^{13}s

^{−1}. All the stacks are illuminated by pump waves incident at Θ

_{i}

_{1}= 30° and Θ

_{i}

_{2}= 45°.

_{A}

_{,B}and nonlinear susceptibilities χ

_{A}

_{,B}have been set equal to the tensor components specified above: ε

_{xxA}= ε

_{zzA}= 5.382, χ

_{xxzA}= χ

_{zxxA}= χ

_{zzzA}= 2.1 × 10

^{−7}m/V, ε

_{xxB}= ε

_{zzB}= 1.4, χ

_{xxzB}= χ

_{zxxB}= χ

_{zzzB}= 2.82 × 10

^{−8}m/V. Comparison of the simulation results for the isotropic and anisotropic layers has shown that the peak intensities of CFG are generally higher in isotropic cases, especially in QPSs with lower order q. The effect of the layer anisotropy in Fibonacci and Thue-Morse stacks appeared to be different. For example, in Fibonacci S

_{7}stack with isotropic layers |F

_{r}

_{,t}| peaks are nearly twice as high as in the same stacks with anisotropic layers, whereas in a similar Thue-Morse Q

_{4}configuration, magnitudes of |F

_{r}

_{,t}| peaks have about the same magnitudes. These observations suggest that a combination of the stack composition and the constituent layer anisotropy may noticeably influence CFG, especially in QPSs of low orders q. When q increases at fixed stack thickness, the |F

_{r}

_{,t}| peak magnitudes become closer in the stacks with isotropic and anisotropic layers. The latter trend can be attributed to thinning of the constituent layers that leads to the dominant effect of the spatial anisotropy enhanced by finer stratification of the stacks.

#### 3.3. Effect of Loss

_{8}and Thue-Morse Q

_{5}stacks with the same layer parameters as in Figure 3 have been simulated taking into account the loss tangents of the constituent layers: tan δ

_{xx}

_{,zz}= 0.01, 0.1. Comparison of the simulation results in Figure 3 and Figure 5 (lossless case) shows that the dielectric losses suppress sharp spikes of |F

_{r}

_{,t}| observed in the lossless QPSs and stronger affect emission in the forward direction than in the reverse direction of the z-axis [46].

_{t}| becomes nearly two orders of magnitude smaller than |F

_{r}|. It is also important to note that not only magnitudes but also spectral content of the combinatorial frequencies emitted from the QPSs with imperfect layers qualitatively changes at higher losses.

_{xx}

_{,zz}= 0.01), the spectral distributions of |F

_{r}

_{,t}| in Figure 5a,c still qualitatively resemble those for the corresponding lossless stacks in Figure 3. Conversely, at higher losses (tanδ

_{xx}

_{,zz}= 0.1), |F

_{r}

_{,t}| distinctively differ because the frequency mixing products emitted from the stacks are predominantly generated in a few peripheral layers adjacent to the stack outer interfaces. Therefore |F

_{r}|, primarily determined by CFG at the stack front interface where the pump waves are still weakly attenuated by losses, is more than two orders of magnitude higher than |F

_{t}| of the waves emitted from the other interface. Indeed, both the pump waves, travelling through the stack, and the mixing products, generated inside the stack, are strongly attenuated due to the layer losses. Therefore the |F

_{t}| magnitude is significantly lower than |F

_{r}| as illustrated by Figure 5b,d. In this case, the stack internal composition affects the CFG efficiency primarily through linear reflection, refraction, and pump wave extinction while the role of phase coherence becomes less prominent.

**Figure 5.**Intensity of field radiated at frequency ω

_{3}= ω

_{1}+ ω

_{2}from (

**a**,

**b**) Fibonacci S

_{8}and (

**c**,

**d**) Thue-Morse Q

_{5}QPS in the reverse (|F

_{r}|-solid red lines) and forward (|F

_{t}|-dash-dot black lines) directions of the z-axis at ω

_{2}= 2.01 × 10

^{13}s

^{−1}. The layer parameters are the same as in Figure 3a with added losses (

**a**,

**c**) - tanδ

_{xx}

_{,zz}= 0.01 and (

**b**,

**d**) - tanδ

_{xx}

_{,zz}= 0.1. The stack is illuminated by pump waves incident at Θ

_{i}

_{1}= 30° and Θ

_{i}

_{2}= 45°.

## 4. Conclusions

_{1}and ω

_{2}, have been obtained in the approximation of weak nonlinearity using the harmonic balance method. The modified TMM has been adapted to analyse QPSs similarly to perturbed periodic stacks with defects at the specified positions. It has been shown that Fibonacci and Thue-Morse QPSs can be represented as cascades of “regular” and “defective” primitive cells. The types and locations of the “defective” cells have been determined and described in analytical form. The developed theory is illustrated by the examples of numerical simulations, and the features of CFG by Fibonacci and Thue-Morse QPSs are discussed.

_{r}

_{,t}(ω

_{3})| of combinatorial frequency ω

_{3}= ω

_{1}+ ω

_{2}are strongly correlated with the stack linear reflectance |R(ω)|. The latter effect is attributed to the fact that |R(ω)| determines magnitudes of both the pump waves of frequencies ω

_{1,2}, refracted into the stacks and engaged in the frequency mixing process, and the waves of combinatorial frequency ω

_{3}emitted from the stack. It has been found that the overall peak efficiency of CFG in QPSs is higher than in similar periodic stacks. The enhanced CFG in QPS is facilitated by the stack composition, which provides higher density of states, and improves the phase synchronism and local field intensification of the pump waves in the constituent layers. Comparison of CFG in Fibonacci and Thue-Morse stacks has shown that |F

_{r}

_{,t}(ω

_{3})| peak magnitudes reach about the same levels in both types of QPSs but the peaks occur at different frequencies. At the same time, it has been observed that the layer anisotropy may affect the CFG efficiency in the low order QPSs, but its effect becomes marginal at the higher order QPSs. An extensive analysis of the CFG efficiency in lossy QPSs has shown that dissipation may qualitatively alter the frequency mixing process and change the |F

_{r}(ω

_{3})|/|F

_{t}(ω

_{3})| ratio for several orders of magnitude. The latter effect is attributed to changes in the extinction scales and coherence lengths of pump waves and the CFG products.

## Acknowledgments

## Author Contributions

## Appendix 1

_{3}, it is necessary first to determine amplitudes coefficients in Equation (4) for the waves generated inside each constituent layer of QPS. In the non-depleting wave approximation, this is accomplished by the harmonic balance method, which allows (ω

_{1}, ω

_{2}) to be explicitly related to amplitudes of the pump waves of frequencies ω

_{1}and ω

_{2}refracted into each layer [40]. In contrast to regular periodic stacks, evaluation of in QPSs is not straightforward and usually requires direct multiplication of the transfer matrices of all layers one by one. An alternative approach, based upon the QPS decomposition in “regular” and “defective” unit cells, is outlined in Section 2 and further elaborated here.

_{q}of order q. For example, in Fibonacci QPS S

_{5}= {AB AAB AB A}, the 2nd cell is defective, i.e., it contains a layer A doublet of thickness d'

_{A}= 2d

_{A}. The regular and defective cells have the same layer sequence but different thicknesses: d = d

_{A}+ d

_{B}and d' = d'

_{A}+ d

_{B}, respectively. Additionally, at odd q an extra layer A appears at the stack end, cf. S

_{5}above and S

_{6}in Section 2.

_{q}of primitive cells (regular and defective) in Fibonacci QPS of order q is:

_{q}= Φ

_{q-1}+ Φ

_{q-2}is Fibonacci number and Φ

_{1}= Φ

_{2}= 1; Γ

_{q}is the number of defective cells: Γ

_{q}= 0 at q ≤ 3, and at q ≥ 4.

_{q}

_{+1}, composed of 0’s and 1’s. The 1’s are located only in the columns corresponding to the first A layer of the doublets. At q ≥ 4, is defined by the recurrence relations:

_{q}column only, δ

_{i,i}

_{'}is Kronecker delta; is a square Toeplitz matrix with 1’s only at the secondary diagonal offset for Φ

_{q}from the main diagonal. Then the serial number ν

_{i}(q), i = 1,2, … Γ

_{q}of each defective cell in the stack can be deduced from evaluated recursively in Equation (A3) for Fibonacci QPS of arbitrary order q:

_{i}(q) is the column number of the ith 1’s in .

_{q}= 2

^{q}

^{−1}. The positions of the regular and defective cells, determined by their serial number n in the stack, can be deduced using the following recurrence relation at n ≥ 3:

_{n}= 0 represents the regular cells {AB} and t

_{n}= 1 − defective cells {BA}; t

_{1}= 0, t

_{2}= 1.

_{y}(ω

_{p}, x, z) of TM wave of frequency ω

_{p}outside the stack has the following form:

_{p}, p = 1,2,3; k

_{p}= ω

_{p}/c, c is the speed of light; A

_{p}are amplitudes of the incident waves. are amplitude coefficients of the field in individual layers.

_{p}) and transmission T(ω

_{p}) coefficients are obtained by interrelating the fields at the stack outer interfaces with the aid of the transfer matrix

_{q}(ω

_{p}):

_{q}(ω

_{p}) are evaluated by TMM at each frequency ω

_{p}. For Fibonacci QPS,

_{q}(ω

_{p}) is defined recursively at q ≥ 2.

_{0}(ω

_{p}) =

_{LB}(d

_{B}, ω

_{p}),

_{1}(ω

_{p}) =

_{LA}(d

_{A}, ω

_{p});

_{LA}(d

_{A}, ω

_{p}) and

_{LB}(d

_{B}, ω

_{p}) are the transfer matrices of layers A and B, respectively.

_{0}(ω

_{p}) =

_{LA}(d

_{A}, ω

_{p}), '

_{0}(ω

_{p}) =

_{LB}(d

_{B}, ω

_{p}).

_{p}) and T(ω

_{p}) are determined, amplitude coefficients can be progressively deduced for each layer using the transfer matrices. In contrast to regular periodic structures, the direct evaluation of for QPS is rather involved. The approach developed in this paper, where the QPSs are formed by cascading the regular and defective primitive cells, dramatically simplifies the analysis. The conventional procedure utilised for the periodic stacks can be adopted here as long as the positions of the defective cells are determined. Then the field amplitudes in each layer can be expressed in the following form

## Appendix 2

_{y}(ω

_{3}) of frequency ω

_{3}emitted from the stack at combinatorial frequency ω

_{3}has the form:

_{r,t}in Equation (A12), it is necessary to relate the fields of the nonlinear products generated inside the stack to the field emitted from the QPS. This is accomplished by enforcing the tangential field continuity at the stack external interfaces and applying the modified TMM [40] to find the fields inside the stack. The fields at the stack outer interfaces of Fibonacci QPS are related as follows

_{q}is the number of primitive cells in the stack; and (ω

_{3}) are defined in (A1.11); and at j = A, B are:

_{3}) are defined in Equation (A11); and are as follows

_{r,t}of the waves emitted from the QPS at combinatorial frequency ω

_{3}:

_{3}) are defined in (A.11) and λ

_{1,2}are

- -
- for Fibonacci QPS:
- -
- for Thue-Morse QPS:

## Conflicts of Interest

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**MDPI and ACS Style**

Shramkova, O.; Schuchinsky, A.
Combinatorial Frequency Generation in Quasi-Periodic Stacks of Nonlinear Dielectric Layers. *Crystals* **2014**, *4*, 209-227.
https://doi.org/10.3390/cryst4030209

**AMA Style**

Shramkova O, Schuchinsky A.
Combinatorial Frequency Generation in Quasi-Periodic Stacks of Nonlinear Dielectric Layers. *Crystals*. 2014; 4(3):209-227.
https://doi.org/10.3390/cryst4030209

**Chicago/Turabian Style**

Shramkova, Oksana, and Alexander Schuchinsky.
2014. "Combinatorial Frequency Generation in Quasi-Periodic Stacks of Nonlinear Dielectric Layers" *Crystals* 4, no. 3: 209-227.
https://doi.org/10.3390/cryst4030209