Analysis of planar waveguides with a thin overlayer and nonlinear cladding

The aim of this paper is to analyse a planar optical waveguide with Kerr-type nonlinear cladding and a thin linear overlayer separating the cladding and the guiding core. The results of numerical calculations show that the bistable behaviour of the nonlinear waveguides can be easily controlled by changing the parameters of this additional layer. The analysis is based on the self-consistent iteration procedure which provides relevant dispersion characteristics limited to physically stable solutions.


Introduction
Nonlinear guided waves supported by optical planar waveguides have been extensively studied in recent decades and their unique properties are well known in many configurations (Boardman et al. 1991). They have great potential for all-optical signal processing implemented on integrated photonics platforms (Leuthold et al. 2010;Li et al. 2018;Jung et al. 2021;Pitilakis and Kriezis 2022). In particular, the waveguide geometry with a linear core layer and Kerr-type nonlinear cladding has been analysed by many authors both analytically (Akhmediev 1982;Lederer et al. 1983;Robbins 1983;Seaton et al. 1985;Boardman and Egan 1985) and numerically (Ogusu 1989;Rahman et al. 1990;Obayya et al. 2002;Huang 2012). In this configuration the bistable behaviour of nonlinear guided waves has been demonstrated experimentally (Vach et al. 1984;Valera et al. 1986;Karpierz et al. 1999). Analytical solutions show that in most practical cases, the propagation constant is a multivalued function of the input power P 0 . However, the stability analysis has led to a conclusion that solutions of branches related to dP 0 ∕d < 0 are physically unstable (Jones and Moloney 1986;Tran and Ankiewicz 1992;Akhmediev 1991). This is consistent with the results of the numerical approach which ignores unstable solutions and leads to bistability in the waveguide phase dispersion when the input power is changed. In general, phase dispersion bistability due to power dependent nonlinearity forms a basis of various 1 Laboratory of Optical Radiation, Central Institute for Labour Protection -National Research Institute, Czerniakowska 16, 00-701 Warsaw, Poland bistable resonant devices that exhibit two stable transmission states and can be used for alloptical signal processing (Van et al. 2002;Chiangga et al. 2013;Gu et al. 2014). The aim of this paper is to show that parameters of the hysteresis loop relevant to the phase dispersion bistability in waveguides with nonlinear cladding can be controlled by the presence of a thin dielectric overlayer separating the cladding and the guiding core layer. The analysis is restricted to 1-D waveguide configuration and is intended to provide detailed insight into this physical phenomena. The proposed concept has not been reported so far and can be applied in any practical material system with nonlinearity implemented in the cladding.

Calculation method
The waveguide geometry analysed in this paper is illustrated in Fig. 1. It consists of a guiding core layer of thickness h and refractive index n f , covered by a thin layer of refractive index n o and thickness d o . The multilayer is bounded by a linear substrate ( x > h + d o ) of refractive index n s , and a nonlinear cladding ( x < 0 ) of intensity dependent refractive index n c + n 2c I , where n c is the refractive index in the low power regime, n 2c is the nonlinear coefficient, and I is the local intensity in [W/m 2 ]. The fields of TE modes supported by the waveguide are given by where = k 0 n eff , n eff is the mode effective index of refraction, k 0 = 2 ∕ and is the vacuum wavelength.
The structure can be analysed by the standard transfer matrix method applied in the self-consistent iteration procedure (Dios et al. 1989). This procedure involves a sequence of numerical solutions of the dispersion equation expressing the modal condition for linear waveguides. The starting point corresponds to the linear waveguide structure with negligible power flow. The electric field profile E y (x) corresponding to the resulting propagation constant (0) is used in the next iteration loop to modify the intensity dependent refractive index in the nonlinear cladding. For this purpose, the cladding is divided into a sufficient number of thin homogeneous sublayers along the effective depth d c , as shown in Fig. 1. It has been checked that the power flow is negligible for x < −d c , and the sublayers are thin enough to ignore local changes in refractive index and electric field amplitude. The number where E y (x j ) is the electric field amplitude in the middle of the j-th layer, c = n 2c n 2 c ∕z 0 , z 0 = ( 0 ∕ 0 ) 1∕2 , 0 is the free space permittivity and 0 is the free space permeability. The new refractive index profile is then used to calculate the corresponding propagation constant (1) and relevant electric field distribution. The iteration process is repeated until the successive difference of propagation constants (n) − (n−1) is sufficiently small. It is important to note, that in each iteration loop, the zero-power refractive index n c has to be used on the right side of Eq. (2), rather than that resulting from the previous loop. In addition, each time the resulting field profile E y (x) should be normalized to the input power per unit width P 0 . For TE modes, this is done by selecting the field profile E y (x) to obtain Each linear multilayer problem in the iteration scheme can be solved by using the standard 2 × 2 transfer matrix method (Chilwell and Hodgkinson 1984). The method provides a transcendental algebraic equation that results from the requirement of self-consistent fields at the interfaces within a guiding structure and expresses the modal condition. For TE polarized waves the equation can be written as ∕z 0 , and n j and d j are the refractive index and the thickness of the j-th layer, respectively. The numerical solution of the equation gives values of the propagation constants of the guided modes.
In addition, the method provides analytical expressions for field profiles and power equations that can be used in the field normalization process. For a TE mode the power per unit width in the j-th layer is given by where U j(j−1) = E yj(j−1) and V j(j−1) = −H zj(j−1) are the field amplitudes at the j-th layer interfaces, related by In practice, the fields are calculated by choosing a starting value of U n = E ys = E y (x = h + d o ) , and using (4) c m 11 + c s m 12 + m 21 + s m 22 = 0, The asymptotic behaviour of the fields at x → ±∞ leads to the expression for the power per unit width in the semi-infinite cladding P c and the substrate P s Hence, to normalize the field profile in the iteration process, one has to select a single value of E ys to obtain The method can be easily extended to TM modes by taking c,s) , in relevant formulas. It can also be extended to include conductor/lossy or 2-D graphene layers by applying appropriate complex values of the refractive index.
The number of iterations needed for convergence of the method depends on the starting parameters and accuracy required. To keep the number of iterations below 10 with the precision of the effective index Δn eff < 10 −7 , one has to avoid large power steps between consecutive points of the plot, especially within the range where two solutions are expected. However, in all cases, the method provides physically stable solutions for a given power flow carried by the waveguide mode. The results obtained for single layer waveguides with nonlinear cladding agree very well with the analytical solutions presented in Seaton et al. (1985).

Analysis and discussion
The numerical analysis is limited to the fundamental TE 0 mode guided by a dielectric waveguide of the following parameters: h = 1.0 m, n f = 1.57 , n c = n s = 1.55 , n 2c = 10 −9 m 2 /W, and = 0.515 m. These parameters correspond to the material configuration used in the theoretical analysis presented in Seaton et al. (1985) and in the experimental setup where intensity-dependent guided waves have been observed (Vach et al. 1984). In both cases, the nonlinear cladding has been formed by MBBA (p-methoxybenzylidene-p'-butylaniline) liquid crystal placed on a glass waveguide structure.  Fig. 2a shows the well known dispersion characteristic of a single layer waveguide with nonlinear cladding, where in a certain power range there are two stable solutions. The fundamental mode starts at linear waveguide parameters with negligible power flow and evolves with increasing input power into a surface mode along the core and nonlinear cladding interface. The surface mode transforms back into the mode guided by the core when the power is reduced, however the switching takes place at a lower power level. As shown in Fig. 2b, introduction of a thin overlayer with refractive index lower than that of the core, leads to an increase of the bistability threshold, as well as an increase of the difference between the lower and higher switching power levels. Similar behaviour can be observed in single layer waveguides when the core thickness is increased (Seaton et al. 1985). In both cases, less relative percentage power is concentrated in the cladding, and therefore more total power is needed to generate there a nonlinear self-focusing effect.
Consequently, an increase of the overlayer refractive index above that of the core, results in a decrease of the bistability threshold (Fig. 2c) and finally to the elimination of the bistability effect (Fig. 2d), and clear single-valued dependence (Fig. 2e). Thus, the parameters of a thin overlayer can be used in the design of photonic devices, when one is interested in the selected thickness of the core and the desired nonlinear properties of the waveguide. Parameters of the hysteresis loop relevant to the bistability effect can also be controlled by changing the overlayer thickness while keeping its refractive index constant. For n o > n f , increasing the thickness of the overlayer leads to similar effects as those obtained by increasing its refractive index. This is illustrated in Fig. 3 which shows the guided wave power P 0 as a function of the effective index n eff for several values of the overlayer thickness d o , when the refractive index n o = 1.60 . As can be seen, in comparison to the relevant plot presented in Fig. 2c, increasing d o results in the elimination of the bistability effect and in single-valued behaviour of the function. It is clear, that for the depressed overlayer refractive index n o < n f , increasing the thickness of the overlayer leads to the opposite effect.
It can also be seen that the effective index in the low power regime increases with increasing d o and takes a common value for higher power levels where the mode is guided at the interface between the core and the cladding. Similarly, the effective index corresponding to low power levels increases with increasing refractive index n o . Thus, in both cases, higher values of the bistability threshold correspond to lower values of the effective index. This is in contrast to single layer waveguides, where increasing bistability threshold corresponds to increasing low power effective index (Seaton et al. 1985). Figure 4 illustrates the relevant relationship between the effective index n eff and both parameters n o and d o , for the waveguide operating in the linear regime.  As mentioned before, the overlayer parameters affect the optical power distribution in the multilayer structure, including the field interacting with the nonlinear cladding. This is illustrated in Fig. 5a which shows the fundamental mode intensity profiles for the waveguides of Fig. 2 operating at low power levels. Increasing value of n o moves the intensity peak towards the overlayer so that more relative percentage power is concentrated in the cladding. Similar behaviour can be observed when d 0 is increased while keeping a constant value of n o = 1.60 (Fig. 5b). As can be seen, in all cases the peak is located in the guiding film rather than in the overlayer. This shows that when the parameters of the overlayer are properly selected, the geometry does not affect the coupling efficiency with other optical elements such as optical fibres.
As the input power P 0 increases, the intensity peak moves to the nonlinear cladding due to the self-focusing effect. Figures 6a-c show the relevant field evolution for the waveguides of Figs. 2a-c, respectively. The power levels has been chosen to illustrate the switching phenomenon between the mode guided by the film and that located in the nonlinear cladding. The mode profiles are consistent with the calculated dispersion plots and confirm that for the overlayer with refractive index n o < n f the switching requires a higher value of the input power (Fig. 6b) compared to a single layer waveguide (Fig. 6a). On the

Conclusion
In conclusion, it has been shown that the bistable behaviour of waveguides with a nonlinear cladding can be easily controlled by changing the parameters of an additional thin dielectric layer separating the cladding and the guiding film. High index overlayer leads to decreasing bistability threshold or elimination of the bistability effect, whereas depressed index overlayer enhances the difference between the lower and higher switching power levels. Similar effects can be obtained by changing the overlayer thickness. All these observations may be useful in the design of all-optical photonic devices based on nonlinearity implemented in the cladding. In particular, the optimal design of the buffer layer leads to the bistability effect at lower thickness of the guiding core. The main conclusions of this study should remain valid for various waveguide configurations, including planar waveguides with finite thickness nonlinear films deposited on a buffered core, multilayered microfibres embedded in a nonlinear cladding or side-polished fibre devices coated with nonlinear materials. Although the analysis has been performed for a material system of high nonlinearity relevant to liquid crystals, it could be of potential interest in other emerging material platforms, such as voltage tunable electro-optical nonlinear materials, 2D graphene or polymer based nanocomposites.

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