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This paper introduces the design and applications of integrated As_{2}S_{3} sidewall Bragg gratings on LiNbO_{3} substrate. The grating reflectance and bandwidth are analyzed with coupled-mode theory. Coupling coefficients are computed by taking overlap integration. Numerical results for uniform gratings, phase-shifted gratings and grating cavities as well as electro-optic tunable gratings are presented. These integrated As_{2}S_{3} sidewall gratings on LiNbO_{3} substrate provide an approach to the design of a wide range of integrated optical devices including switches, laser cavities, modulators, sensors and tunable filters.

Bragg gratings have been widely used in integrated optical devices such as switches, filters, laser, modulators and wavelength division multiplexing [1-5]. Recently, sidewall (or lateral) Bragg gratings show obvious advantages over surface or volume Bragg gratings, owing to its low insertion loss, compact size and relaxed fabrication tolerance [6-14]. On the one hand, lithium niobate (LiNbO_{3}) has become an attractive material for integrated optical applications because of its outstanding electro-optical, acousto-optical and optical transmission properties [_{3} substrate, high-quality channel waveguide can be made by annealed proton exchange (APE) [_{2}S_{3}) is one type of amorphous chalcogenide glass that can be fabricated into low-loss waveguide structures using conventional silicon lithography techniques for both near-IR and mid-IR applications [18,19]. As_{2}S_{3} also possesses a large refractive index contrast that enables strong optical confinement in narrow waveguides with wide bend radii [_{3} channel waveguide into the As_{2}S_{3} waveguide through tapering structures [_{2}S_{3} waveguide structure and LiNbO_{3} substrate provides a powerful hybrid integrated optical platform for many applications [17,20-22].

In this paper, we explore the spectral properties of integrated As_{2}S_{3} sidewall gratings on LiNbO_{3} substrate and investigate their device applications in integrated optics. Sidewall gratings of the sinusoidal, trapezoidal and square shapes are analyzed with coupled-mode theory [23-25]. Numerical results from uniform sidewall gratings under weak coupling and strong coupling conditions are compared with coupled-mode theory. In simulations, material dispersions are considered by applying Sellmeier equations [26,27]. By introducing multiple phase shifting spacers into uniform sidewall gratings, multi-channel transmission filter was implemented. Besides, electro-optical tunable transmission filter is designed utilizing the unique electro-optical property of LiNbO_{3} substrate. A tuning rate of ~4 pm/V was predicted from a narrowband transmission filter based on single phase-shifted sidewall Bragg gratings. Ultrahigh Q-factor of over 10^{6} is also proposed by adjusting the resonant cavity length formed by two identical uniform gratings. This type of integrated As_{2}S_{3} sidewall Bragg gratings on lithium niobate substrate provide an approach to the design of a wide range of integrated optical devices such as optical switches, laser cavities, modulators, sensors and tunable filters.

General principle of the mode coupling resulting from periodic dielectric perturbation was rigorously derived in Ref. [23,24]. For Bragg grating reflectors, only two contra-directional waves are involved. For y-propagation, the coupled-mode equations are given by

where.

A_{1} and A_{2} are complex amplitudes of the normalized waves with propagation constants of β_{1} and β_{2}, respectively. Δβ is phase mismatch; Λ is grating period; K is grating wavenumber and m is a positive integer indicating grating order; k is the coupling coefficient. The power transfer between the two waves through the grating with a length of L is derived as

where.

The Bragg grating reflectance as a function of phase mismatch ΔβL/2 at different coupling strengths for a 1 cm-long Bragg grating is shown in _{max} = tanh^{2}(kL). The corresponding grating period is determined by where λ_{0} is Bragg wavelength and n_{eff} is the mode effective index of the forward propagating wave β_{1}.

The reflection bandwidth is defined as the wavelength span between the first two minima in reflection spectrum

as described by Equation (4). The bandwidth Δλ is determined by the grating length L under weak coupling condition kL π. For strong coupling condition kL π, Δλ is directly proportional to coupling coefficient k.

As depicted in _{2}S_{3} strips on LiNbO_{3}. The grating period is Λ. Duty cycle (DC) is defined as the fractional width with high index material in one grating period and thus DC = τ/Λ where τ is the width of high index material in one period. The rise/fall width is denoted by τ_{r}. ΔW is the grating depth which is a measure of the index perturbation strength. W and t are the width and thickness of equivalent unperturbed As_{2}S_{3} strip waveguide, respectively.

Arbitrary grating shape along xand z-axis with periodic perturbation of index modulation along y-axis can be expressed in the form of Taylor expansion, as described by

Here, Δε_{n} is the Fourier coefficient of the nth harmonic. The influence of each harmonic can be analyzed individually and total effect is obtained by summing all harmonic terms. The values of Δε_{n} for sinusoidal, square and trapezoidal gratings are given in Equations (6)-(8). Sinusoidal gratings have only the first harmonic. Square and trapezoidal gratings have relatively larger Fourier coefficients of the first order. High order harmonics may be included to achieve the greatest accuracy [

Sinusoidal:

Square:

Trapezoidal:

The coupling coefficients of sidewall Bragg gratings are evaluated by taking overlap integration between material index distribution and electric field distribution over grating regions as illustrated in

where is angular frequency; c is vacuum speed of light; is the electric field distribution of TE_{0} mode supported by the unperturbed As_{2}S_{3} strip.

The coupling coefficients resulting from the first order perturbation for square, sinusoidal and trapezoidal gratings at λ_{0} = 1.55 μm are displayed in ^{−1} due to its largest Fourier coefficient of the first order. As grating depths increases, the coupling coefficient increases. For all the gratings, coupling coefficients are maximized at the duty cycle of 0.5 for a given grating depth.

The reflectance and bandwidth of trapezoidal sidewall gratings with Λ = 360 nm are calculated at λ_{0} = 1.55 μm. Numerical results for weak coupling gratings and strong coupling gratings are plotted in Figures 4 and 5, respectively. Under weak coupling, the bandwidth is inversely proportional to grating length L = NΛ where N indicates the total number of grating periods. To improve the reflectance of weak coupling gratings and maintain the same grating length, larger grating depth is required. For instance, the grating reflectance for N = 600 periods can be increased from 60% to 94% by varying grating depth from ΔW = 0.4 μm to ΔW = 1.0 μm, as shown in ^{−1} is observed in

The scattering matrices of the sidewall Bragg gratings

are calculated with Fimmprop (Photon Design, Inc., Oxford, UK). The LiNbO_{3} channel waveguide is fabricated either by APE process or by Ti diffusion process. For x-cut y-propagation LiNbO_{3}, Ti diffused channel waveguide supports both TE (transverse electric) and TM (transverse magnetic) modes. However, only TE modes are supported by APE channel waveguide because extraordinary refractive index (n_{e}) of LiNbO_{3} increases while ordinary refractive index (n_{o}) decreases during the proton exchanging process [_{3} substrate.

The thickness of As_{2}S_{3} strip waveguide are optimized at t = 280 nm for single mode TE_{0} operation. The effective index of TE_{0} mode for W = 3 μm is n_{eff} = 2.1436 at λ_{0} = 1.55 μm. The Bragg grating period is Λ = λ_{0}/(2n_{eff}) = 361.5 nm. We choose Λ = 360 nm for easy fabrication. Linear tapers with a tip width of 350 nm and length of 500 μm is also designed to transfer optical mode power between APE channel waveguide and As_{2}S_{3} strip waveguide.

The reflectance of trapezoidal Bragg gratings with DC = 0.5 and τ_{r}/Λ = 0.25 under weak coupling and strong coupling condition is shown in Figures 6-8. Generally,

the reflectance spectra of weak coupling gratings exhibits the shape of sinc function while the reflectance of strong coupling gratings have a flat top in the reflection band. Because the effective index is larger for a wider As_{2}S_{3} strip, the Bragg wavelength is proportional to the effective mode index for a given grating period according to Equation (6). So the Bragg wavelength can be located to a desired value by varying the unperturbed As_{2}S_{3} strip width as shown in

For weak coupling gratings, a reflection band of ~5 nm centered at ~1.548 μm (corresponding to Λ = 360 nm) is observed in

The uniform sidewall grating with N = 600 periods can be spitted into halves by inserting a phase-shifting spacer with the length L_{s} = Λ/2 (i.e., the quarter-wavelength length λ/(4n_{eff})). As a result, a Fabry-Perot type of optical resonant cavity with a length of L_{s} is formed by two identical distributed Bragg reflectors (DBRs) [_{0}/Δλ = 3.4 × 10^{3} which has the same order as that of the transmission resonant filter in SOI (silicon on insulator) material system in Ref. [

Besides, multi-channel transmission filters can be implemented by introducing multiple phase-shifting spacers in uniform sidewall gratings [6,12]. For instance,

Similar to the phase-shifted sidewall gratings, an optical resonant cavity with the length of L_{c} can be constructed by two identical DBRs. This optical resonant cavity can be treated as a Fabry-Perot resonator with an effective cavity length L_{eff} [_{c} = 0.3 mm, L_{c} = 0.5 mm and L_{c} = 0.7 mm are given in _{c} = 0.5 mm, the resonant transmission bandwidth is 0.04 nm at 1.5496 μm and thus the Q-factor is ~3.874 × 10^{4}. The FSR is 0.9 nm

(~112 GHz). The effective cavity length is estimated at L_{eff} = 0.62 m which is larger than its physical length. The FSR and Q-factor for the L_{c} = 0.7 mm cavity are 0.63 nm and 9.68 × 10^{4}, respective. It is expected that resonant cavities with a Q-factor over 10^{6} can be realized with the physical cavity length longer than 1 cm. The spectral transmission of one ultrahigh Q-factor resonant cavity with the cavity length of 1.2 cm is plotted in ^{6}. Therefore, such sidewall Bragg gratings are very useful for compact resonant cavities with ultrahigh Q-factors at both near-IR and mid-IR wavelengths.

Electro-optic tunable Bragg gratings engraved on Ti diffused waveguide was reported with tuning efficiency at ~5 pm/V [_{2}S_{3} sidewall gratings on APE LiNbO_{3} channel waveguide shown in _{2}S_{3} strip (~38% at 1.55 μm for W = 3 μm and t = 280 nm) and the rest of mode is confined in APE channel waveguide, as shown in _{3} substrate facilitates electro-optic tuning of such sidewall gratings. On the x-cut y-propagation LiNbO_{3} substrate, the optical electric field along the z-axis will experience optimal EO effect due to its highest value of EO coefficient r_{33} = 30.8 × 10^{−12} m/V. As illustrated in _{a} along z-axis is given by

The corresponding amount of refractive index change along z-axis (i.e., extraordinary refractive index) is determined by

Assume d = 10 μm and V_{a} = 50 V, then Δn_{e} = −8.199 × 10^{−4}. ^{−4}, which is smaller than the theoretical value because only up to 75% of the bulk value of r_{33} can be restored by annealing process after proton exchange [

In summary, the spectral properties of As_{2}S_{3} sidewall gratings integrated on LiNbO_{3} substrate were analyzed with coupled-mode theory. Coupling coefficients were evaluated by performing overlap integration. Numerical results of uniform sidewall gratings agree well with the coupled-mode theory. Single and multi-channel transmission resonant filters based on sidewall Bragg gratings

were discussed. An electro-optic tunable narrowband transmission resonant filter was proposed and a tuning rate of ~4 pm/V was realized. Lithium niobate channel waveguides with low propagation loss can be prepared by thin film Ti diffusion as well as annealed proton exchange process. Such As_{2}S_{3} sidewall gratings with nanoscale grating periods can be easily fabricated by direct electron beam writing or focused ion beam lithography.

This type of integrated chacogenide As_{2}S_{3} sidewall Bragg gratings integrated on LiNbO_{3} substrate provide an approach for the design of a wide range of integrated-optic devices such as switches, laser cavities, modulators, sensors and tunable filters.