1. Introduction

Tin thiohypodiphosphate (Sn₂P₂S₆) is a wide bandgap semiconductor ferroelectric with large electro-optical coefficients of more than 170pm/V at λ = 0.633 μm and low dispersion in the infrared (e.g., r ₁₁ ≃ 160pm/V at λ = 1.55μm). It has a wide transparency range extending from λ = 0.53 μm to λ = 8μm which makes it ideally suited for nonlinear optical applications in this wavelength range[1]. In addition, Sn₂P₂S₆ has been demonstrated to be one of the most efficient photorefractive materials for visible and near infrared applications[2, 3].

The realization of planar optical waveguides in Sn₂P₂S₆ is the first step towards the production of integrated optical structures for efficient nonlinear optical frequency conversions and for low voltage electro-optical applications. Ion implantation has already been shown to be an interesting technique for the production of optical waveguides without reducing the nonlinear properties[4], especially in ferroelectric materials where low temperature phase transitions (Sn₂P₂S₆ undergoes a phase transition from ferroelectric to paraelectric phase at T_c = 338K) precludes the application of other methods[5]. Generally, the knowledge of the index profile of the waveguide is a very important requisite for tailoring the correct implantation parameters for the desired applications.

Since Sn₂P₂S₆ is a biaxial crystal with monoclinic point group symmetry m at room temperature, the dielectric axes do not coincide with the crystallographic axes and their position depends on both temperature and wavelength[6]. Consequently, waveguide modes are, in general, neither TE nor TM but are of hybrid nature[7]. This aspect has been fully considered in our analysis.

For the first time, we present the production of planar optical waveguides in Sn₂P₂S₆ by low fluence He⁺ ion implantation and reconstruct the induced refractive index profile along the three dielectric axes with a single implantation experiment, taking into account the hybrid nature of the guided modes.

2. Crystal orientation and ion implantation experiments

Three nominally pure Sn₂P₂S₆ samples grown by vapour-transport technique have been cut perpendicular to the Cartesian coordinate system (x,y, z) with the z-axis parallel to the crystallographic axes c and y-axis normal to the mirror plane (see Fig. 1). The positive direction of x and z axes has been chosen so that the Cartesian system (x,y,z) is close to the crystallographic (a,b,c) system.

 

Fig. 1. Orientation of the implanted Sn₂P₂S₆ crystals. (a,b, c) is the crystallographic system of the monoclinic Sn₂P₂S₆, (x,y,z) is the cartesian system used and (x ₁, x ₂, x ₃) indicate the dielectric coordinate system. Main refractive indices and the rotation α of the major axis of the indicatrix x ₃ to the x-axis are given for T = 298K and wavelengths λ = 0.633 μm and λ = 1.55μm[6].

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In the ferroelectric phase the indicatrix of Sn₂P₂S₆ , labeled by (x ₁,x ₂,x ₃), lies in the mirror plane xz and is rotated by an angle α from the x-axis to the major indicatrix axis x ₃. This rotation angle depends on both temperature and wavelength[6]. The crystal configuration and the relevant optical properties at both λ =0.633μm and λ = 1.55 μm are given in Fig.1.

The crystals have been irradiated in a TANDEM accelerator at ETH Zurich on the polished z-face by 2MeV He⁺ ions at three different fluences of 0.5, 1 and 2 × 10¹⁵ cm^-2, respectively. The implanted sample surfaces were tilted by 7° to avoid channeling. The implantation process has been performed at room temperature with the ion current density being kept to extremely low values (about 15nA/cm²) to prevent excessive heating of the crystals during the process. No annealing of the crystals was performed after the implantation.

3. Measurement of the effective indices of the waveguide modes

3.1. Guided modes in biaxial crystal Sn₂P₂S₆

The orientation of the dielectric tensor greatly affects the behaviour of the guided modes in Sn₂P₂S₆ waveguides as explained thoroughly in Appendix A and summarized in this section.

Two types of waveguides are possible in z-face implanted crystals depending on the direction of the propagation vector k⃗. If propagation is along the x-axis of the crystal, TE and TM modes exist in the waveguide. TE modes consist of waves whose electric field is polarized along the dielectric axis x ₂ which makes them ideally suited to study the change induced on the refractive index n ₂. In contrast, TM modes which are formed by waves whose refractive index is simultaneously influenced by both n ₁ and n ₃, are not useful for the present study.

If light propagates along the y-axis of the crystal, it is not possible to split the guided solutions of the waveguide into TE/TM modes, since the axes of the indicatrix are rotated around y. These modes are called hybrid modes[7] since they consist of a linear combination of four crystal eigenwaves having the same propagation constant β (i.e., the same component of the propagation vector k⃗ along the waveguide axis). Yet, we show in Appendix A that under the general assumption that the induced refractive index change is of the order of ∆n ≃ 0.1, and the waveguide core dimensions are larger than the laser wavelength, it is possible to identify two sets of (nearly) decoupled modes. These modes have electric field vector nearly parallel to the main axes of the indicatrix in the xz plane, so their refractive index is approximatively given by n ₁ and n ₃, respectively. We refer to these modes as hybrid-n ₁ and hybrid-n ₃ throughout the rest of the paper. They can be treated as modes of an isotropic material of refractive index n ₁ and n ₃ with the following remark on their behaviour at the core boundaries: in order to determine the conditions for the existence of a mode in a ray-optics approach, the electric field reflection coefficient at the interface between the core and the adjacent materials must be evaluated. If the core is an isotropic material of refractive index n_core , the reflection coefficient for a mode with a propagation constant β is determined by[8]

where k ₀ = 2π/λ is the wavenumber and n_sub is the refractive index of the adjacent medium. The coefficient ρ is ρ = 0,1 for TE and TM modes, respectively. By approximating our waveguides in Sn₂P₂S₆ crystals by an anisotropic core surrounded by air and an isotropic barrier of lower refractive index, we show in Appendix A that the reflection coefficient can still be described by equation (1) provided that ρ ₁ ≃ 0.33 and ρ ₃ ≃ 0.29 for hybrid-n ₁ and hybrid-n ₃, respectively. Consequently, these modes are ideally suited to determine the impact of the implantation on the refractive indices n ₁ and n ₃ independently of each other. Moreover, they are both of special interest because they allow to employ the highest electro-optic coefficient r ₁₁ along the Cartesian x-axis.

It is therefore clear that exploiting hybrid modes and propagation along two different directions in a single crystal provides enough information to reconstruct the index profile of all three main refractive indices, thus reducing the need for several crystal samples with different implanted surfaces.

3.2. Barrier-coupling method

After the implantation, the x and y side faces of the sample have been optically polished to enable light coupling and waveguide modes have been measured using our recently developed barrier-coupling method[9]. We used this method instead of the prism-coupling technique which cannot be applied because Sn₂P₂S₆ main refractive indices are too high for standard rutile prisms and the sample surface can be easily damaged by mechanical contact. Since the method had previously been demonstrated only for higher symmetry crystals, we extended its application range to strongly anisotropic materials like Sn₂P₂S₆ as well.

Using this method, modes are excited by coupling a laser beam from the substrate to the waveguide core through the optical barrier by an evanescent wave. Excitation occurs if the propagation constant of the beam, β, matches the propagation constant of a waveguide mode, defined by the effective refractive index N_m , i.e. β = k ₀ N_m . This condition can be fulfilled at certain external incident angles θ_e between the beam impinging on the lateral surface of the crystal and the surface normal. By measuring the intensity of the light reflected by the barrier as a function of θ_e , modes are revealed as dips in the reflection curve [9].

To apply this method to biaxial waveguides of general orientation, two remarks should be given: first, the relation between β and the external angle θ_e is obtained by determining the normal surfaces of the crystal, namely the relation between the propagation constant β and the internal transversal propagation constant, i.e. for our geometries along the Cartesian z-axis κ = sinθ_e . However, as explained in Appendix A, for situations of practical interest in this paper, the dispersion relations are simply given by

for hybrid-n ₁, TE and hybrid-n ₃ modes, respectively as if we were dealing with isotropic materials.

Second, in a biaxial crystal for a general external polarization and an arbitrary incident angle two propagating beams can be excited in the crystal (hence two possible β), which do not correspond to TE/TM modes of the waveguide structure. Therefore the input polarization has to be chosen to approximatively coincide with the polarization of hybrid-n ₁ or hybrid-n ₃ modes to investigate a single refractive index profile. To conclude, the refractive index profile for n ₂ in Sn₂P₂S₆ can be reconstructed by measuring TE modes if the input surface for the barrier coupling method is perpendicular to the x-axis of the crystal in the xz plane. To obtain analogous information about n ₁ and n ₃ light has to enter from the y-axis in the yz plane of the crystal and the polarization has to be chosen so that only one beam propagates inside the sample.

A typical example of a mode measurement with the barrier-coupling method is shown in Fig. 2 for TE modes of a Sn₂P₂S₆ crystal which has been He⁺ ion-implanted with fluence Φ = 2 × 10¹⁵ ions/cm² and ion energy E = 2MeV. The laser wavelength was λ = 0.633 μm. This wavelength allows the propagation of a large number of modes, which is advantageous for the profile reconstruction. The reflectivity clearly shows 13 dips at external angles that satisfy the matching condition to excite a mode. A slowly varying reflection background, due to incomplete beam coupling at grazing incidence and Fresnel reflections at the input and output surfaces, has been subtracted for clarity. The resulting envelope of the reflectivity function depends, among other parameters, on the optical barrier shape[9]. Although it would be theoretically possible to fit this envelope to collect information about the refractive index profile, this method is too cumbersome, since it is critically sensitive to the beam size and incomplete beam coupling at grazing incidence.

 

Fig. 2. Measurement of TE modes in a He⁺ ion-implanted waveguide in a Sn₂P₂S₆ crystal by barrier-coupling method. The crystal was implanted with an ion energy of 2MeV and a fluence Φ = 2×10¹⁵ ions/cm².

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4. Reconstruction of the refractive index profile

4.1. Method

To reconstruct the refractive index profiles we used a method originally proposed by Chandler and Lama[10]. The method describes the index profile of the waveguide by a parametrized function. The parameter set of this function is determined by minimizing the sum of the squares of the differences between the effective indices of modes for these parameters and the measured effective indices of the waveguide modes.

While most authors choose an empirical function for the refractive index profile, in our work the functional shape has been derived directly from the energy deposited by the ions inside the material, which is described by the quantity G(z) = dE_ion /dz. The ion energy deposition has been in fact shown to be responsible for the refractive index change[4] and can be divided into two contributions: energy deposition to electrons G_el (z), occurring when the ions are fast, and the nuclear energy deposition (energy to recoils) G_n (z) at the end of the ion track when the ions are slower.

The energy deposition profiles G_el (z) and G_n (z) have been calculated for the implantation energy E = 2MeV using SRIM simulation (Stopping and Range of Ions in Matter, www.srim.org). The results are shown in Fig. 3. The maximum recoil energy transfer amounts to 0.94×10⁴ eV/(μm∙ion) and occurs at a depth of about 6.02μm. Most energy goes into electronic excitations, which are nevertheless considered to play only a minor role for permanent changes in most inorganic crystals, while the main contribution to the refractive index change is usually given by the nuclear energy deposition, which locally decreases the refractive index creating an optical barrier[4].

Following the idea presented in Ref. [5] for the ferroelectric crystal KNbO₃, we describe the refractive index change profile function ∆n(z) caused by the energy deposition G(z) by:

 

Fig. 3. Energy deposition profile of 2MeV He⁺ ions implantated into Sn₂P₂S₆ crystals, evaluated with SRIM code. Maximum recoil energy transfer occurs at a depth of 6.02μm with a peak value of 0.94 × 10⁴ eV/μm, while electronic energy transfer shows a more constant energy deposition profile of about 35×10⁴ eV/μm.

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where ∆n_n (z) and ∆n_el (z) are the contributions due to the ion interaction with electrons and nuclei, respectively. In the preceding expression Φ is the implantation fluence and the parameter set is formed by (∆n _n,0, ∆n _el,0, G _n,0, G _el,0, γ_n ,γ_el ,μ). The parameter μ has been introduced to account for small inaccuracies of the SRIM routine, known to occur when estimating the ion range inside heavy targets.

Each main refractive index profile has a different parameter set, but changes induced by different fluences must be described by the same set. Therefore effective indices of the modes for the same main refractive index (i.e., TE modes for n ₂) measured in samples implanted by different fluences have been fitted simultaneously. The ability to fit data corresponding to different fluences with the same parameter set corroborates the choice of this functional shape for the refractive index profile.

A starting value for each parameter set has been obtained by estimating the profile using the well-known WKB inverse method[11], which is nevertheless inappropriate to completely recover the refractive index profile since it has been shown to be better suited for slowly varying profiles.

4.2. Results

The refractive index profiles for all three main refractive indices at λ = 0.633 μm and at three different fluences Φ = 0.5,1 and 2×10¹⁵ ions/cm² are presented in Fig. 4. In the left column the effective indices of the measured waveguide modes are indicated by the open circles. The main error source in the determination of the effective indices stems from the non-perfect polishing of the lateral surfaces, due to the limited size of the crystals (the implanted surface was about 3×4mm²). In the right column the best-fit refractive index profiles are shown. The calculated effective indices of the modes corresponding to these profiles are shown in the left column for comparison with the experimental data. The calculated effective indices have been connected (dotted lines) for clarity reasons. The parameter sets corresponding to the reconstructed profiles are reported in Table 1.

The refractive index profile n ₁(z) shown in the first row of Fig. 4 clearly resembles the nuclear energy deposition profile. A negative index change at the end of the ion track is created and an optical barrier is formed. The barrier is situated at about z = 64μm. This value is 6% higher than the expected SRIM value, well inside the accuracy of the routine. For the lowest fluence of 0.5 × 10¹⁵ ions/cm², the peak index change is about ∆n ₁ = -0.07 . For higher fluences a larger change is obtained, but this effect saturates for fluences higher than about 10¹⁵ ions/cm². The extreme flatness in the barrier region of the profiles for higher fluences is to be attributed to the model described by equation (3) and it may not be accurate. However, the bottom shape of the barrier is not very important since it only marginally affects the effective indices. This saturation threshold is one order of magnitude lower than what has been observed in other ferroelectric cystals such as KNbO₃[5]. The high sensitivity of Sn₂P₂S₆ may be attributed to the lower phase transition temperature and shows that in Sn₂P₂S₆ waveguides can be achieved with a relatively low fluence ion implantation.

The refractive index in the core region is only slightly increased ( ∆n ≃ 6 × 10^-3) by the implantation, which in our model is mainly due to the electronic contribution. A small positive refractive index change was reported as well for the refractive index n_c of KNbO₃[12]. The two phenomena have a further similarity, since the spontaneous polarization is directed along the crystallographic axes c in KNbO₃ and the main dielectric axis n ₁ of Sn₂P₂S₆ is the closest to the direction of the spontaneous polarization in this material[6]. The small index increase implies that the first guided modes have an effective index higher than the refractive index of the bulk substrate, hence they are not detectable by the barrier-coupling method as they are non-leaky. The fitting procedure is, however, very robust because of the large amount of modes avalaible. The calculated modes of the refractive index profiles are very close to the experimental data (as it can be seen in the left column of Fig. 4), especially for the lowest and highest fluence, while the small discrepancy in the intermediate fluence is to be attributed to the lower quality of the sample surface before the ion implantation experiment.

For the main refractive index n ₂ the results are shown in the second row of Fig. 4. A waveguide is created and the index decrease at the peak position for the lowest fluence Φ = 0.5 × 10¹⁵ ions/cm² has a similar magnitude ∆n ₂ ≃ -0.075. Saturation effects for fluences higher than 10¹⁵ ions/cm² are evident. For this orientation, however, the refractive index change in the waveguide core is slightly negative ∆n ≃ -4.3 × 10^-3. Modes of the calculated refractive index profiles are in very good agreement with experimental data for all three fluences.

The data for the main refractive index n ₃ is shown in the third row of Fig. 4. Reliable data was obtained only for the lowest and highest fluences 0.5 and 2 × 10¹⁵ ions/cm², since the lower quality of the sample in the crystal implanted with intermediate fluence yielded inaccurate results for this polarization. The behavior is anyhow evident: an optical barrier of similar refractive index change is created which saturates for the highest fluence. The electronic effect induces a decrease of refractive index in the waveguide core region of about ∆n≃-5 × 10^-3 for the lowest fluence.

The results are summarized in Table 1. The parameter sets show that the behavior is similar for all three refractive indices, except for the sign of the electronic contribution. The threshold for saturation effects, determined by the parameter G _n,0 also has similar magnitude. The errors show that the values of some parameters should be interpreted with caution. They reflect the fact that the effective indices of the modes in an ion implanted waveguide are strongly affected by the position and the steepness of the optical barrier and by the refractive index change in the core, while only to a much lesser extent by the optical barrier height and thickness.

 

Fig. 4. Left column: measured effective indices for the He⁺ ion-implanted Sn₂P₂S₆ waveguides with ion energy E = 2MeV and fluence Φ = 0.5,1 and 2 × 10¹⁵ ions/cm² in black, blue and red open circles, respectively. The calculated modes of the best-fit profiles have been connected for clarity and are shown by the dotted lines. Right column: the reconstructed refractive index profiles described by equations (3) and the parameter sets given in Table 1. The dashed line indicates the unperturbed refractive indices. The three rows refer to hybrid-n ₁, TE modes (n ₂) and hybrid-n ₃, respectively.

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Table 1. Parameters for the calculation of refractive index profiles of He+ implanted Sn₂P₂S₆ crystals using equations (3).

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5. Waveguiding experiments

Waveguiding experiments have been performed along the y-axis of the planar waveguides implanted at all three fluences. This configuration has been chosen because it is the most attractive for electro-optic applications since both hybrid-n ₁ and hybrid-n ₃ modes can experience the highest electro-optic coefficient r ₁₁ by applying an external electric field along the x-axis.

Light has been launched into the polished side facets of the waveguides by end-fire coupling using laser light at λ = 0.633,0.860 and 1.55μm. Using the sample implanted with the lowest fluence 0.5 × 10¹⁵ ions/cm² a sufficient amount of light has been collected at the output, while for the other two crystals the outcoupled light was too weak to be detected. This can be explained by a significant increase of defects in the waveguide core, which is likely to happen when the implantation fluence is close to the saturation levels. This absorption change contributes too little to the effective indices of the modes to be detected in the profile reconstruction routine, but has been shown to be significant in KNbO₃[5]. Therefore we believe that fluences as low as 0.5 × 10¹⁵ ions/cm² are sufficient to create a suitable optical barrier without excessively increasing the absorption in the waveguide region.

Propagation losses in the crystal implanted at the lowest fluence have been measured by detection of Fabry-Pérot fringes[13]. This method is advantageous since it does not imply any assumption about the coupling efficiency, the number of propagating modes, is non-contact and is especially suitable for high refractive index materials as Sn₂P₂S₆ because of the high reflectivity of the end facets.

Measurements have been performed by detecting interference fringes obtained by scanning the emitted wavelength of a tunable laser source centered at λ = 1.55 μm. The tuning resolution of the laser was δλ = 10pm. A measurement for hybrid-n ₁ modes of a Sn₂P₂S₆ crystal implanted with 0.5 × 10¹⁵ ions/cm² He⁺ ions is shown in Fig. 5. The theoretical curve is the transmission of a Fabry-Perot interferometer assuming the length of the sample L = 2.94 mm and perfect Fresnel reflectivity at the facets (25%). The only free parameter is the loss coefficient α, which yields a value of approximatively α = 10±2dB/cm.

This value is sensibly higher than intrinsic material absorption, which was measured to be below 0.2dB/cm. Losses are probably stemming from the cross-coupling between the hybrid modes n ₁ and n ₃ in the waveguide core, since, as mentioned in Appendix A, hybrid-n ₁ modes can excite propagating waves with the same effective index, but with electric field polarized along the main dielectric axes n ₃. These waves are therefore only weakly or not guided at all, because the effective index N ≪ n ₃. Another possible source of losses is the introduction of defects in the waveguide core region even at the lowest fluence. This contribution may be possibly reduced by properly annealing the samples or by further reducing the implantation fluence. The introduction of a limited number of defects in the core region may even be advantageous for photorefractive applications, because they can be related to an improvement of the photorefractive sensitivity with respect to the bulk material, as demonstrated previously in KNbO₃[14].

 

Fig. 5. Waveguide transmission dependence on input wavelength for hybrid-n ₁ modes of a Sn₂P₂S₆ sample implanted with energy E = 2MeV and fluence 0.5×10¹⁵ ions/cm². Sample length is L = 2.94mm. The line represents the theoretical curve of a Fabry Perot interferometer with Fresnel reflectivity of about 25% and propagation losses of 10±2dB/cm.

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6. Conclusions

For the first time planar optical waveguides have been produced by He⁺ ion implantation in electro-optical and photorefractive Sn₂P₂S₆ crystals, using ion energy E = 2MeV and fluences of 0.5, 1 and 2×10¹⁵ ions/cm². Optical modes have been measured using the barrier-coupling method taking into account the effects of anisotropy. Using a minimal number of crystal samples, the refractive index profiles have been reconstructed for all three main refractive indices, using a parametrized function directly related to the ion energy deposition calculated with SRIM. The effective indices of the modes for the reconstructed index profiles coincide within the experimental error with the measured data for all three fluences simultaneously.

An optical barrier of ∆n ₁ ≃ -0.07, ∆n ₂ = -0.07 and ∆n ₃ = -0.09 at λ = 0.633 μm was created at the lowest fluence. Electronic contributions are relevant in the core region and lead to a slight increase (about 4 × 10^-3) of the refractive index for n ₁ and a decrease of n ₂ (-4.5 × 10^-3) and n ₃ (-6 × 10^-3). Waveguiding could be observed by end-fire coupling into the crystal implanted with the lowest fluence of 0.5 × 10¹⁵ ions/cm² and propagation losses of about 10 ± 2dB/cm have been measured by interferometric technique for hybrid-n ₁ modes. The results clearly show that ion implantation is a suitable technique for the realization of planar waveguides in Sn₂P₂S₆ for fluences lower than about 10¹⁵ ions/cm² to avoid excessive propagation losses. This work represents a step of fundamental importance for the production of integrated optical structures in Sn₂P₂S₆.

Appendix A: Guided modes in biaxial crystal Sn₂P₂S₆

The general problem of waveguide modes in biaxial crystals has already been addressed by several authors[15, 7]. We show here how these results apply to Sn₂P₂S₆ waveguides formed by implanting the surface perpendicular to z-axis.

First we introduce a waveguide coordinate system (x′ ,y′ ,z′), where z′ represents the direction of mode propagation, x′ is the direction where the refractive index is changed (here x′ = z) and y′ is orthogonal to x′ and z′. Since the crystals are cut along (x,y,z) and implanted along the z-axis, two types of planar waveguide are possible: a) the mode propagation direction is parallel to x (z′ = ±x, y′ = ∓y), b) the propagation direction is parallel to y(z′ = ±y, y′ = ±x). The two waveguide configurations are illustrated in Fig. 6 and have different properties as discussed in the following subsections.

 

Fig. 6. Two possible planar waveguide configurations for a z-implanted Sn₂P₂S₆ crystal. (x,y,z) represents the Cartesian coordinate system of the crystal as described in Fig.1. (x′ ,y′ ,z′) is the waveguide reference system. a) Propagation is along the Cartesian x-axis. b) Propagation is along the Cartesian y-axis.

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Light propagation of a plane wave in a lossless anisotropic medium is described by the Helmholtz wave equation[8]

(A1)$$\overrightarrow{k}\times \overrightarrow{k}\times \overrightarrow{E}+k_0^2\epsilon \overrightarrow{E}=0$$

where k⃗, E⃗ are the wavevector and the electric field, ε is the dielectric tensor and k ₀ = 2π/λ is the free-space wavenumber. For any wavevector direction specified by k̂ = k⃗/|k⃗|, equation (A1) is an eigenvalue equation which gives two allowed electric field polarizations and two refractive indices n = |k⃗|/k ₀. The two surfaces n(k̂) are called normal surfaces[8] and are useful for examining the propagation of guided modes.

Since the waveguide is an infinite slab in the y′ direction, we can write the wavevector as k⃗ = κ x̂′ + βẑ′, x̂′ and ẑ′ being the unit vectors in direction x′ and z′, respectively. Therefore $n=\frac{\sqrt{{\kappa }^2+{\beta }^2}}{k_0}$ and the normal surfaces are two curves in the (κ, β) plane.

A waveguide mode inside the core of a biaxial waveguide consists of a linear combination of four plane waves which can be determined by inspecting the normal surfaces. The waves must have the same propagation constant β along z′, but have (in general) different transversal propagation constant κ. Since in these solutions neither E _y′ nor H _y′ vanishes, the waves will be cross-coupled at every reflection occurring at an interface perpendicular to x′-axis, i.e. at the core-air and core-substrate interface. Therefore TE and TM modes do not exist and we refer to the modes as hybrid modes. A general equation for the existence of guided modes in biaxial crystal is given for instance in Ref. [7] by taking into account the fully vectorial nature of the modes.

A1. Propagation along the x-axis

If the propagation direction is parallel to the x-axis the dielectric tensor in the waveguide reference system is

(A2)$$\epsilon =\left(\begin{array}{ccc}{\epsilon }_{x\prime x\prime }& 0& {\epsilon }_{x\prime z\prime }\\ 0& {\epsilon }_{y\prime y\prime }& 0\\ {\epsilon }_{x\prime z\prime }& 0& {\epsilon }_{z\prime z\prime }\end{array}\right)\mathrm{where}\{\begin{array}{c}{\epsilon }_{x\prime x\prime }=n_1^2{\sin }^2\alpha +n_3^2{\cos }^2\alpha \\ {\epsilon }_{y\prime y\prime }=n_2^2\\ {\epsilon }_{z\prime z\prime }=n_1^2{\cos }^2\alpha +n_3^2{\sin }^2\alpha \\ {\epsilon }_{x\prime z\prime }={\epsilon }_{z\prime x\prime }=\left(n_1^2-n_3^2\right)\cos \alpha \sin \alpha \end{array}$$

In the above expression α is the angle between the x-axis and the dielectric x ₃-axis. The non-diagonal element ε _x′z′ has opposite sign if propagation is in the - x direction. The solution of (A1) in this case has been discussed in detail in Ref. [15]. Two decoupled sets of waves TE/TM exist. The TE solution has an electric field polarized along y′ = y, hence for the TE solution the crystal behaves like an isotropic medium of refractive index √ε _y′y′ = n ₂, which means that the normal surface is a circle of radius n ₂ and κ ² + α ² = $k_0^2$ $n_2^2$. These modes are ideally suited to study the refractive index change induced on n ₂.

The TM solution is more complicated since y′z′ (yx) is not a symmetry plane of the crystal. For a given α the two transversal propagation constants κ _1,2 can be obtained by solving (A1):

(A3)$${\kappa }_{\mathrm{1,2}}=-\beta \frac{{\epsilon }_{x\prime z\prime }}{{\epsilon }_{x\prime x\prime }}\pm \sqrt{\frac{{\epsilon }_1{\epsilon }_3}{{\epsilon }_{x\prime x\prime }^2}\left(k_0^2{\epsilon }_{x\prime x\prime }-{\beta }^2\right)}$$

The phase refractive index n of the TM solutions depends on the wavevector direction θ = arctan (κ/β) and has an intermediate value between n ₁ and n ₃. These solutions are therefore not well suited for the characterization of the refractive index profiles induced by ion implantation and will not be further considered in this paper. The normal surfaces and the refractive index for this waveguide configuration are shown in Fig.7.

 

Fig. 7. Normal surfaces (left) and refractive index $n=\frac{\sqrt{{\kappa }^2+{\beta }^2}}{k_0^2}$ of eigensolutions as a function of the internal propagation angle θ (right) for waves propagating in Sn₂P₂S₆ crystals along x-axis. The green curve represents TE waves, the red curve TM waves. The refractive index of the TM waves will also change if the wave propagates along the +x or -x-axis.

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A2. Propagation along the y-axis

Optical waveguide modes propagating along the y-axis in a Sn₂P₂S₆ crystal show interesting features. The dielectric tensor in the waveguide reference system is

(A4)$$\epsilon =\left(\begin{array}{ccc}{\epsilon }_{x\prime x\prime }& {\epsilon }_{x\prime y\prime }& 0\\ {\epsilon }_{x\prime y\prime }& {\epsilon }_{y\prime y\prime }& 0\\ 0& 0& {\epsilon }_{z\prime z\prime }\end{array}\right)\mathrm{where}\{\begin{array}{c}{\epsilon }_{x\prime x\prime }=n_1^2{\cos }^2\alpha +n_3^2{\sin }^2\alpha \\ {\epsilon }_{y\prime y\prime }=n_1^2{\sin }^2\alpha +n_3^2{\cos }^2\alpha \\ {\epsilon }_{x\prime y\prime }={\epsilon }_{y\prime x\prime }=\left(n_1^2-n_3^2\right)\cos \alpha \sin \alpha \\ {\epsilon }_{z\prime z\prime }=n_2^2\end{array}$$

To the best of our knowledge, this particular geometry has not yet been discussed in the literature. The dispersion equation is given by:

(A5)$$\left({\kappa }^2+{\beta }^2-k_0^2\frac{{\epsilon }_1{\epsilon }_3}{{\epsilon }_{x\prime x\prime }}\right)\left({\kappa }^2+{\beta }^2\frac{{\epsilon }_{z\prime z\prime }}{{\epsilon }_{x\prime x\prime }}-k_0^2{\epsilon }_{z\prime z\prime }\right)={\epsilon }_{z\prime z\prime }{k}_0^2{\beta }^2\frac{{\epsilon }_{x\prime y\prime }^2}{{\epsilon }_{x\prime x\prime }}$$

The two normal surfaces are two concentric quadric surfaces. For propagation perfectly parallel to the waveguide axis z′ (κ = 0) the dispersion relation gives β ₂ = $k_0^2$ ε ₁ and β ₂ = $k_0^2$ ε ₃, the refractive index will be n = n ₁ and n ₃ respectively and the electric field polarized along the main dielectric axes. By increasing κ, i.e., by increasing the propagation angle θ between k⃗ and the propagation axis z′ the two eigensolutions will deviate from the pure n ₁ and n ₃ solutions. In Fig. 8 we show that by increasing the propagation angle θ from 0° to 90°, the refractive index of one wave changes from n ₃ to $n=\sqrt{\frac{{\epsilon }_1{\epsilon }_3}{{\epsilon }_{x\prime x\prime }}}\approx \sqrt{\frac{\left({\epsilon }_1+{\epsilon }_3\right)}{2}}$, while the other one varies from n ₁ to n ₂.

 

Fig. 8. Normal surfaces (left) and refractive index $n=\frac{\sqrt{{\kappa }^2+{\beta }^2}}{k_0^2}$ (right) of eigensolutions as a function of the internal propagation angle θ (bottom) for eigenwaves of a Sn₂P₂S₆ crystal propagating at λ = 0.633 μm with the wavevector in the yz plane . The green curve represents hybrid-n ₁ waves, the red curve hybrid-n ₃ waves. For internal angles smaller than 15° the deviation from the indices n ₁ and n ₃ (dashed lines) is negligible.

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Nevertheless, we can expect guided modes to be characterized by small internal propagation angles θ. Indeed confinement occurs only if β > k ₀ n_barrier . If the refractive index change induced is ∆n < 0.1 we have a lower boundary for the propagation constant of a guided wave: β > k ₀(n_core -∆n) so that only internal angles smaller than θ_max ≈ 15° are accessible. Thus in our case we can approximate the two eigensolutions of the biaxial crystal with the eigensolutions at θ = 0, i.e. with waves polarized along n ₃ and n ₁ respectively. The dispersion relation (A5) can therefore be approximated by

(A6)$$\left({\beta }^2+{\kappa }^2-k_0^2{\epsilon }_1\right)\left({\beta }^2+{\kappa }^2-k_0^2{\epsilon }_3\right)=0$$

For every value of β the four solutions have non-vanishing electric E _y′ or magnetic H _y′ field components along y′. When these waves are reflected at the core-air interface, they will be cross-coupled, since they have neither TE nor TM symmetry. Therefore planar waveguide modes in Sn₂P₂S₆ crystals with this geometry stem from a linear combination of all four eigenwaves, i.e. they are hybrid modes. Still we can show that in our waveguides two nearly independent sets of solutions can be identified.

To investigate this aspect we approximated the ion-implanted waveguide with an asymmetric step-index slab waveguide consisting of air, Sn₂P₂S₆ anisotropic core of thickness d and an isotropic barrier of refractive index n_s = 2.9 at λ = 0.633 μm. The angle Ω between the polarization of the electric field of the first waveguide mode and the waveguide axis y′ gives an indication on the nature of the guided mode. This angle is shown in Fig.9 as a function of the relative waveguide thickness d/λ. For a very large core size (d/λ → ∞) the angle converges to the orientation of the eigenwave in the pure bulk crystal (i.e. along the dielectric axes x ₃, Ω → α). Strong deviation from this value can be observed only if d ≈ λ or for smaller waveguide cores, where the role played by the interfaces becomes dominant and the mode becomes TE-like (Ω → 0). While in isotropic planar waveguides the orientation of the guided electric field results from the interfaces, in biaxial waveguides of Sn₂P₂S₆ the anisotropy of the dielectric tensor ε has a stronger influence on the symmetry of the solutions as long as the core is large enough, so that the reflection coefficients at the interfaces play a less important role in the waveguide equation. Since in our experiments d/λ ≈ 10 we can safely assume that Sn₂P₂S₆ waveguide modes are approximately polarized along the main axes of the indicatrix for every relevant propagation constant β. We can therefore refer to these modes as hybrid-n ₁ and hybrid-n ₃ modes.

 

Fig. 9. Dependence of the angle Ω etween the electric field of the first waveguide mode and the y′ = x-axis for a Sn₂P₂S₆ waveguide on the waveguide core of size d and propagation direction along z′ = y-axis. The core is surrounded by air and an isotropic material of refractive index n_s = 2.9 to approximate the behaviour of an ion-implanted waveguide. For d ≫ λ the mode has hybrid nature and is polarized along dielectric axes x ₃.

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In a ray-optics approach, the existence of a waveguide mode is determined by the phase shift due to the propagation and the reflection at the interfaces between the core and the adjacent layers. To completely describe the behaviour of these hybrid modes we therefore determined the reflection coefficients of the hybrid modes using the simplified model introduced above. For an isotropic material the reflection coefficient of a waveguide mode at the core-air interface is known to be[8]

(A7)$$r=e^{i2\varphi }\mathrm{where}\tan \varphi ={\left(\frac{n_{\mathit{core}}}{n_{\mathit{air}}}\right)}^{2\rho }\sqrt{\frac{k_0^2{n}_{\mathit{core}}^2-{\beta }^2}{{\beta }^2-k_0^2{n}_{\mathit{air}}^2}}$$

where ρ = 0,1 for TE/TM modes, respectively. For hybrid modes of Sn₂P₂S₆ waveguides the same relation holds, provided that a proper value for the parameter ρ is introduced. We calculated that, for β > k ₀(n_core - ∆_n), ρ ₃ ≈ 0.29±0.01 for the hybrid-n ₃ mode polarized along x ₃ and ρ ₁ ≈ 0.33± 0.01 for the hybrid-n ₁ mode. These values confirm the hybrid nature of the solutions.

Finally we would like to mention that cross-coupling between these modes is not symmetric. Since n ₃ > n ₁, for the first guided hybrid-n ₃ modes the relation β > n ₁ holds. This means that propagating waves with n = n ₁ cannot be excited at the reflection at the interfaces. Thus these hybrid-n ₃ modes are truly decoupled: in literature[7] such modes are referred to as “hybrid inhomogeneous pure guided modes”. On the contrary, hybrid-n ₁ modes can always excite propagating waves with n = n ₃ having the same constant β, possibly even non-guided ones. This can be interpreted as if there was no total internal reflection: every hybrid-n ₁ mode is inherently leaky because of the anisotropy: these modes are called “hybrid homogeneous guided modes” or “hybrid leaky guided modes” depending on the value of β.

To summarize, we have shown that in planar waveguides in Sn₂P₂S₆ with the propagation direction along the y-axis, even though the dielectric tensor does not allow TE/TM solutions, we are allowed to classify the modes in two nearly decoupled sets, under the assumption that the waveguide core thickness is large compared to the wavelength. We refer to these modes as hybrid-n ₁ and hybrid-n ₃ since they have the electric field mainly polarized along x ₁ and x ₃, respectively, for any propagation constant β of a guided solution. They are only weakly coupled, have a nearly constant refractive index for different propagation angles and their reflection coefficient at the interface with an adjacent layer can be modeled by assuming that in equation (A7) ρ ₁ = 0.29 or ρ ₃ = 0.33. These modes are ideally suited to study the effect of ion implantation, because the two indices can be investigated independently.

Acknowledgments

We thank A. Grabar and Y. Vysochanskii for supplying the crystals, M. Dobeli for the implantation experiments and J. Hajfler for professional polishing of the samples. We also thank L. Mutter and D. Rezzonico for encouraging discussions. We acknowledge the financial support of the Swiss National Foundation.

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