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.

A₁ and A₂ are complex amplitudes of the normalized waves with propagation constants of β₁ and β₂, 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 Figure 1. The grating reflectance decreases smoothly as Δβ/2 increases from 0 to k. When Δβ is beyond k, the value of s becomes imaginary and the waves change from exponential to sinusoidal. It also shows that the grating reflectance increases as kL increases under phase match condition Δβ = 0, in which case, the power transfer reaches its maximum value at R_max = tanh²(kL). The corresponding grating period is determined by where λ₀ is Bragg wavelength and n_eff is the mode effective index of the forward propagating wave β₁.

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 Figure 2, the sidewall Bragg gratings are engraved in both sidewalls of As₂S₃ strips on LiNbO₃. 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 As₂S₃ 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 [10].

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 Figure 2(b). The coupling coefficient resulting from the nth Fourier harmonic component is computed by

where is angular frequency; c is vacuum speed of light; is the electric field distribution of TE₀ mode supported by the unperturbed As₂S₃ strip.

The coupling coefficients resulting from the first order perturbation for square, sinusoidal and trapezoidal gratings at λ₀ = 1.55 μm are displayed in Figure 3. Square gratings posses strongest coupling strength at k = 8.75 mm⁻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 λ₀ = 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 Figure 4(a). The sidewall gratings under strong coupling condition have much narrower reflection bandwidth compared with weak coupling gratings. It is also more efficient in reflectance especially at larger values of ΔW. A linear relationship between the bandwidth and the coupling coefficient beyond k > 4 mm⁻¹ is observed in Figure 5(b).

The thickness of As₂S₃ strip waveguide are optimized at t = 280 nm for single mode TE₀ operation. The effective index of TE₀ mode for W = 3 μm is n_eff = 2.1436 at λ₀ = 1.55 μm. The Bragg grating period is Λ = λ₀/(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 As₂S₃ 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 As₂S₃ 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 As₂S₃ strip width as shown in Figure 6. The side-lobe levels can be suppressed by anodizing these sidewall gratings in order to minimize cross talk between channels and improve filter responses for add/drop filters used in dense wavelength division multiplexing (DWDM) [11].

For weak coupling gratings, a reflection band of ~5 nm centered at ~1.548 μm (corresponding to Λ = 360 nm) is observed in Figure 7(a) for N = 600 periods. The reflectance increases with increasing numbers of grating periods, however, the reflection bandwidth gets narrower. As shown in Figure 8(a), the grating depth needs to be sufficiently large to achieve a high reflectance for weak coupling sidewall gratings. Under strong coupling condition, the reflectance is very efficient and the bandwidth is determined only by coupling coefficients, as shown in Figures 7(b) and 8(b).

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) [28]. A transmission peak will be created at the center of the stopband of the uniform sidewall gratings [7,9]. As shown in Figure 9(a), the red line indicates the reflectance spectrum from a uniform trapezoidal sidewall grating with N = 600, W = 3 μm, ΔW = 1.0 μm and DC = 0.5 and the blue line denotes the reflectance from a phase-shifted sidewall grating. A transmission band centered at 1.5472 μm with a bandwidth of 0.45 nm is observed in the center of Δλ = 12 nm stopband of the N = 600 uniform sidewall gratings. The quality factor (Q-factor) of this resonant cavity is Q = λ₀/Δλ = 3.4 × 10³ which has the same order as that of the transmission resonant filter in SOI (silicon on insulator) material system in Ref. [9].

Besides, multi-channel transmission filters can be implemented by introducing multiple phase-shifting spacers in uniform sidewall gratings [6,12]. For instance, Figure 9(b) displays the reflectance spectra from sidewall gratings with single, double and triple phase shifting spacers, where each grating section has N/2 = 300 periods and each phase shifting spacer has the quarter-wavelength length. Add/drop filters for DWDM in telecommunication can be implemented with such phase shifted sidewall gratings [6,28].

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 [24]. The reflectance spectra for three optical resonant cavities with length L_c = 0.3 mm, L_c = 0.5 mm and L_c = 0.7 mm are given in Figure 10(a). The free spectral range (FSR) is inversely proportional to the effective cavity length and is determined by Equation (10). For the cavity with L_c = 0.5 mm, the resonant transmission bandwidth is 0.04 nm at 1.5496 μm and thus the Q-factor is ~3.874 × 10⁴. 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 × 10⁴, respective. It is expected that resonant cavities with a Q-factor over 10⁶ 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 Figure 10(b). It demonstrates the FSR (and Q-factor of this resonant cavity are 42.5 pm and 1.21 × 10⁶. 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 [29]. Here we will investigate the electro-optical tenability of sidewall Bragg gratings on APE waveguides. The electro-optic (EO) effect, or Pockels effect, is exhibited by non-centrosymmetric crystals such as lithium niobate [15]. For the As₂S₃ sidewall gratings on APE LiNbO₃ channel waveguide shown in Figure 11, optical mode is partially confined in As₂S₃ 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 Figure 11(b). This hybrid mode feature along with EO property of LiNbO₃ substrate facilitates electro-optic tuning of such sidewall gratings. On the x-cut y-propagation LiNbO₃ substrate, the optical electric field along the z-axis will experience optimal EO effect due to its highest value of EO coefficient r₃₃ = 30.8 × 10⁻¹² m/V. As illustrated in Figure 11(a), if the gap between the two parallel electrodes is d, the electric field from external applied voltage V_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. Figure 11(c) gives electric field distribution generated from the external applied voltage at 50 V. This electric field induces an amount of refractive index change of −4.8579 × 10⁻⁴, which is smaller than the theoretical value because only up to 75% of the bulk value of r₃₃ can be restored by annealing process after proton exchange [30]. The reflectance of the single phase-shifted sidewall grating in Section 3.2 with different external applied voltages is plotted in Figure 12. EO tuning efficiency at ~4 pm/V is achieved.