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

Abstract

Three-wave mixing in quasi-periodic structures (QPSs) composed of nonlinear anisotropic dielectric layers, stacked in Fibonacci and Thue-Morse sequences, has been explored at illumination by a pair of pump waves with dissimilar frequencies and incidence angles. A new formulation of the nonlinear scattering problem has enabled the QPS analysis as a perturbed periodic structure with defects. The obtained solutions have revealed the effects of stack composition and constituent layer parameters, including losses, on the properties of combinatorial frequency generation (CFG). The CFG features illustrated by the simulation results are discussed. It is demonstrated that quasi-periodic stacks can achieve a higher efficiency of CFG than regular periodic multilayers.

1. Introduction

The recent theoretical and experimental studies of nonlinear metamaterials (MM) and photonic crystals (PhC) have demonstrated potential of their applications in functional devices operating in diverse frequency ranges, from microwaves to optics. Traditionally, artificial media have periodic order with regular spectral characteristics. In contrast, long-range order of quasi-periodic structures (QPSs) and quasi-crystals provides additional degrees of freedom and enables new unusual properties, which recently attracted considerable interest [1].

QPSs with high order symmetry lie between periodic crystals and random amorphous solids. Since the first realization of Fibonacci and Thue-Morse type superlattices [2], the physical properties of 1-D QPSs have been extensively studied, both experimentally and theoretically, and it has been shown that QPSs can exhibit unique features unattainable in periodic arrangements. In particular, it was found that QPSs create localization of electronic states and their collective response differs from that of their constituents. While the linear wave phenomena in quasi-periodic media were extensively explored during the last few decades [3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24], nonlinear QPSs have been studied only recently [25,26,27,28,29,30,31,32,33,34,35].

The main distinctive features of nonlinear artificial media are usually associated with their ability to simultaneously control the dispersion and nonlinear interactions by altering both the geometry of constituent particles and their arrangements in ensembles. Both periodic and quasi-periodic nonlinear structures have been studied primarily with the aim of improving harmonic generation efficiency. QPSs with additional degrees of freedom due to the higher order group symmetry are particularly attractive for creating higher density of states that can facilitate harmonic generation [36,37,38]. It has been observed in [25] that the efficiency of second-harmonic generation (SHG) in QPS is higher than in random medium, but still remains lower than in regular periodic structures. On the other hand, an eight-fold increase of the third-harmonic generation (THG) efficiency has been attained in the two-step cascaded process in a QPS as compared with a similar periodic multilayer [26].

Combinatorial frequency generation (CFG) by mixing pump waves of different tones enables more versatile spectrum control. However, the dominant three- and four-wave mixing processes are usually demanding to not only the nonlinearity itself but also the linear dispersion and field distribution, which primarily depend on the medium intrinsic properties and composition. Therefore, QPSs with their rich spectral properties may provide favourable conditions for frequency mixing and CFG. To realise this potential, it is important to explore in detail the mechanisms and features of CFG in QPSs.

In this paper, a generic approach to the analysis of CFG by layered QPSs, illuminated by two obliquely incident pump waves, has been developed and applied to the study of the properties of frequency mixing products. The problem statement and solution of the boundary value problem for the Fibonacci and Thue-Morse type QPSs are outlined in Section 2 and elaborated in Appendices. The features of CFG by the QPSs composed of nonlinear anisotropic dielectric multilayers are illustrated by the results of numerical simulations and discussed in Section 3. The main findings are summarised in Conclusion.

2. Problem Statement and Methodology

QPSs shown in Figure 1 are composed of stacked weakly nonlinear anisotropic dielectric layers of two kinds (labelled A and B) with thicknesses d_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:

(1)

The stacked layers are arranged in accordance with the following quasi-periodic sequences:

(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 S₀ = {B}, S₁ = {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 L₀ = d_B; L₁ = 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 Q₀ = {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 L₀ = d_A, L'₀ = d_B.

Figure 1. Geometry of the problem: (a) Fibonacci type QPS; (b) Thue-Morse type QPS.

Figure 1. Geometry of the problem: (a) Fibonacci type QPS; (b) Thue-Morse type QPS.

The stacks are illuminated by two pump plane waves of frequencies ω₁ and ω₂ incident at angles Θ_i1 and Θ_i2, 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.

To evaluate the characteristics of TM waves of combinatorial frequencies generated by a nonlinear QPS, the fields of both pump waves and their mixing products inside the stack are to be determined first. Assuming that nonlinearity of the constituent layers is weak and the three-wave mixing process is dominant, the nonlinear products can be obtained by the harmonic balance method. At combinatorial frequency ω₃ = ω₁ + ω₂ the TM wave fields in each layer are described by non-homogeneous Helmholtz equation:

(2)

where E_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; k₃ = ω₃/c and ; angle Θ₃ defines the emission direction from the stacks at frequency ω₃, which is determined by the phase synchronism condition in the three-wave mixing process [39]:

k_x3 = k_x1 + k_x2

(3)

where , k_1,2 = ω_1,2/c

The solution of Equation (2) for H_yj(ω₃) can be represented as a superposition of six plane waves with longitudinal propagation constants and .

(4)

where , , p = 1,2,3 and superscript n is a serial number of a primitive cell, which identifies the position of a jth type layer in the stack. are amplitudes of the waves of frequency ω₃, generated inside this layer, which depend on amplitudes of the pump waves refracted into this layer [40]. are amplitudes of the waves of frequency ω₃ generated outside the layer and refracted into it.

To calculate the amplitude coefficients in Equation (4), it is necessary first to find the pump wave amplitudes in each constituent layer. In periodic stacks comprised of alternating binary layers this is accomplished by the transfer matrix method (TMM) [41,42,43,44] as the unit cell layout and serial number n fully identify the layer type (A or B) and position in the stack. This approach can be extended to QPS because the stack composition is defined by the deterministic rules. However, evaluation of each layer position and type in a specific sequence is usually a challenging task.

Examination of Fibonacci and Thue-Morse QPSs shows that they contain only two types of primitive cells. Indeed, Fibonacci QPS, for example, S₆ = {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, Q₄ = {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.

Once the fields of combinatorial frequencies ω₃ in individual layers are determined, amplitudes F_r(ω₃) and F_t(ω₃) 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

To elucidate the CFG mechanisms in QPSs, the nonlinear scattering characteristics of TM waves by the stacks of Fibonacci and Thue-Morse sequenced binary anisotropic layers have been analysed numerically. Illustrative examples of the simulation results for the specific QPS arrangements are discussed below in comparison with the corresponding periodic stacks of the same thicknesses.

It is assumed that the stacks are surrounded by air with the relative permittivity ε_a = 1 and are illuminated by a pair of pump plane waves of frequencies ω_1,2 incident at angles Θ_i1,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]:

Layer A (CdS): ε_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.

The layer thicknesses d_A and d_B are related by the golden ratio: .

3.1. Spectral Efficiency of Frequency Mixing

The earlier studies of photonic crystals and periodic stacks of dielectric layers have suggested that the SHG and THG efficiency may be higher at the bandgap edges. However, the concurrent increase of losses considerably reduces the gain. Alternatively, CFG efficiency proved to be higher not at the band edges but in the transparency bands, especially at Wolf-Bragg resonances of the stacks [40]. Additionally, CFG involving two pump waves of different frequencies ω₁ and ω₂, incident at dissimilar angles, offers extra degrees of freedom in controlling the frequency mixing process and spectrum.

The frequency mixing efficiency in the stacks of nonlinear layers essentially depends on the phase coherence, which is controlled by linear dispersion of both pump waves and their nonlinear products. Additionally, magnitudes of interacting pump waves are significantly influenced by the stack composition and linear reflectance R(ω), variable with the incidence angle. Therefore rich spectral content of QPS has potential for assisting CFG enhancement.

To illustrate the effect of the stack composition, |R(ω)|² of TM waves incident at angles Θ_i = 30° and Θ_i = 45° are displayed in Figure 2 for Fibonacci S₈ (34 layers, d_B = 12 µm) and Thue-Morse Q₅ (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 S₈ and (b) Thue-Morse Q₅ QPSs, and (c) periodic stack with 16 unit cells.

Figure 2. Reflectance of TM wave incident at Θ_i = 30° (dashed black lines) and Θ_i = 45° (solid red lines) on (a) Fibonacci S₈ and (b) Thue-Morse Q₅ QPSs, and (c) periodic stack with 16 unit cells.

To assess the effect of the pump wave reflectance on CFG, magnitudes |F_r,t(ω₃)| of the waves of combinatorial frequency ω₃ emitted from the stacks have been simulated at fixed frequencies ω₂ of a pump wave incident at Θ_i2 = 45°, and swept frequency ω₁ of the other pump wave incident at Θ_i1 = 30°. Frequencies ω₂ have been chosen individually for each stack configuration so that |R(ω₂)|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.

Inspection of Figure 2 and Figure 3 demonstrates definite correlation between |F_r,t| and |R(ω)| for each type of QPSs. Indeed, |F_r,t| reach their peaks at frequencies ω₁ 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 ω₃ = ω₁ + ω₂ in the reverse (|F_r|-solid red lines) and forward (|F_t|-dash-dot black lines) directions of the z-axis: (a) Fibonacci S₈ stack at ω₂ = 2.01 × 10¹³ s⁻¹; (b) Thue-Morse Q₅ stack at ω₂ = 1.73 × 10¹³ s⁻¹ and (c) periodic stack containing 16 unit cells at ω₂ = 1.76 × 10¹³ s⁻¹. All the stacks are illuminated by pump waves incident at Θ_i1 = 30° and Θ_i2 = 45°.

Figure 3. Intensity of field radiated from QPSs at frequency ω₃ = ω₁ + ω₂ in the reverse (|F_r|-solid red lines) and forward (|F_t|-dash-dot black lines) directions of the z-axis: (a) Fibonacci S₈ stack at ω₂ = 2.01 × 10¹³ s⁻¹; (b) Thue-Morse Q₅ stack at ω₂ = 1.73 × 10¹³ s⁻¹ and (c) periodic stack containing 16 unit cells at ω₂ = 1.76 × 10¹³ s⁻¹. All the stacks are illuminated by pump waves incident at Θ_i1 = 30° and Θ_i2 = 45°.

Comparison of |F_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 ω₁ = 2.38 × 10¹³ s⁻¹ in Thue-Morse QPS Q₅ (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 ω₃ 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 ω₃. 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

Both the analytical estimations and numerical results shown in Figure 2 and Figure 3 have demonstrated strong effect of the stack internal arrangements on both the reflectance and CFG. To illustrate it further, Figure 4 displays |F_r,t| simulated for Fibonacci S₇ stack, containing 21 layer with d_B = 19 µm, and Thue-Morse Q₄ stack, containing 16 layers with d_B = 26 µm, which have the same overall thicknesses as Fibonacci S₈ and Thue-Morse Q₅ 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 ω₃ emitted from the stacks have almost the same peak magnitudes in both cases but the peaks occur at different frequencies ω₁. 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 ω₃ = ω₁ + ω₂ in the reverse (|F_r|-solid red lines) and forward (|F_t|-dash-dot black lines) directions of the z-axis: (a) Fibonacci S₇ stack at ω₂ = 2.02 × 10¹³ s⁻¹; (b) Thue-Morse Q₄ stack at ω₂ = 1.76 × 10¹³ s⁻¹. All the stacks are illuminated by pump waves incident at Θ_i1 = 30° and Θ_i2 = 45°.

Figure 4. Intensity of field radiated from QPSs at frequency ω₃ = ω₁ + ω₂ in the reverse (|F_r|-solid red lines) and forward (|F_t|-dash-dot black lines) directions of the z-axis: (a) Fibonacci S₇ stack at ω₂ = 2.02 × 10¹³ s⁻¹; (b) Thue-Morse Q₄ stack at ω₂ = 1.76 × 10¹³ s⁻¹. All the stacks are illuminated by pump waves incident at Θ_i1 = 30° and Θ_i2 = 45°.

In order to assess the effect of the dielectric layers’ anisotropy on CFG, Fibonacci and Thue-Morse QPSs of several different orders q have been re-simulated with isotropic constituent layers. The scalar relative permittivities ε_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 S₇ 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 Q₄ 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

To assess the effect of dissipation on CFG in QPSs, Fibonacci S₈ and Thue-Morse Q₅ 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].

The latter effect is particularly evident in Figure 5b,d where |F_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.

These results demonstrate that CFG levels in the lossy QPSs are substantially decreased by dissipation of both pump waves and frequency mixing products. Their decays have different spatial scales, and dissimilar extinction rates of the pump waves may further reduce the CFG efficiency. Therefore, at relatively low losses (tanδ_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 ω₃ = ω₁ + ω₂ from (a,b) Fibonacci S₈ and (c,d) Thue-Morse Q₅ QPS in the reverse (|F_r|-solid red lines) and forward (|F_t|-dash-dot black lines) directions of the z-axis at ω₂ = 2.01 × 10¹³ 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_i1 = 30° and Θ_i2 = 45°.

Figure 5. Intensity of field radiated at frequency ω₃ = ω₁ + ω₂ from (a,b) Fibonacci S₈ and (c,d) Thue-Morse Q₅ QPS in the reverse (|F_r|-solid red lines) and forward (|F_t|-dash-dot black lines) directions of the z-axis at ω₂ = 2.01 × 10¹³ 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_i1 = 30° and Θ_i2 = 45°.

4. Conclusions

Combinatorial frequency generation (CFG) in the three-wave mixing process has been studied in the quasi-periodic stacks of binary nonlinear dielectric layers arranged in accordance with Fibonacci and Thue-Morse sequences. The closed-form solutions of the nonlinear scattering problem for QPSs, illuminated by a pair of obliquely incident pump waves of frequencies ω₁ and ω₂, 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.

The simulation results have demonstrated that the nonlinear scattering coefficients |F_r,t(ω₃)| of combinatorial frequency ω₃ = ω₁ + ω₂ 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 ω₃ 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(ω₃)| 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(ω₃)|/|F_t(ω₃)| 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.