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Abstract
Discrete solitons (DS), which could propagate without diffraction in the waveguide array (WA), have attracted great attention. However, its applications are limited to incident light with high power. Here, based on the d.c. Kerr effect in the silicon, we propose and demonstrate theoretically an electrically reconfigurable ridge waveguide array. By applying sech-function bias voltages on the WA, a Kerr-type DS could be mimicked by a low-power incident light. The transmitting paths of DS and low-power light in the WA can be rerouted with great flexibility by changing the local bias voltages applied on the waveguide. Our proposed silicon WA provides new opportunities for electric-controlled optical devices, which may open a gateway towards rerouting light on-chip and designing integrated optics devices.
© 2017 Optical Society of America
1. Introduction
Unlike continuous media, the propagation and diffraction of light in WAs [1] can be engineered, which provides an efficient platform for manipulating light propagation and is widely used to design on-chip optical devices. One typical application of WAs is wavelength division multiplexing (WDM) based on the arrayed waveguide gating (AWG) [2, 3], where the interference among waveguides with different lengths are used to route the light with different wavelengths. Another typical application is to realize the spatial-localization states and diffraction-free propagation of light. Tailoring the couplings between the WAs by modulating their separations, Anderson localization [4, 5] and surface bound state in continuum (BIC) [6, 7] have been observed in the linear WAs. In order to achieve the controllable localization state, nonlinear optical effect has been introduced into WAs. Through a balance of discrete diffraction and nonlinearity of material, light in nonlinear WAs could be self-trapped and keep its shape invariant during propagation. The state is called DS, which is interesting not only for the physics behind its formation, but also for its applications in on-chip beam-splitting, optical inter-connects, and switching networks [1, 8].
Discrete solitons have been realized by employing different types of nonlinear optical materials [8]. For example, using biased photorefractive (PR) crystals, photorefractive DS at the 0.5μm wavelength was experimentally demonstrated in SBN: 75 crystal [9, 10], where the 6mm × 6mm two-dimensional WA was optically induced by plane-wave interference. The formation of DS in PR materials is a consequence of the interference between plane waves and probe beam. Using quadratic nonlinear materials, quadratic DS at the two wavelengths (near 1.55μm for fundamental wave) [11] was observed in 7cm-long periodically poled lithium niobate waveguide arrays fabricated by titanium in diffusion. The energy exchange between the fundamental and second harmonic wave in the WA leads to the appearance of the two quadratic DSs. Due to advantages for fabrication and on-chip integration, Kerr-type DSs realized by semiconductor WAs have attracted great attentions. The first reported observation of Kerr-type DS [12, 13] at the 1.53μm wavelength was in a 6mm-long ridge waveguide array etched onto an AlGaAs substrate, where the input light with sufficient power was self-trapped in the WA. At the edge of a 1cm-long AlGaAs waveguide array, discrete surface soliton [14] was also observed in the experiments.
Recently, electric field-induced second-order nonlinear optical effects including the d.c. Kerr effect and second-harmonic generation have been demonstrated in silicon ridge waveguides fabricated by CMOS-compatible method [15], which opens up new opportunities for the on-chip optical devices. Different from the free carrier effect, d.c. Kerr effect changes the refractive index of silicon by the direct-current electric field applied on the intrinsic silicon regions. To achieve the relatively large d.c Kerr effect, the p-i-n junctions within the silicon waveguides can be designed to be reverse-biased [15], which can efficiently reduce the density of free carriers in the core region of silicon waveguide and decrease the free-carriers-absorption loss. By introducing the d.c. Kerr effect into the silicon ridge waveguide array, we propose and demonstrate theoretically that both controlling the propagation of Kerr-type DSs and routing light-transmitting paths in the silicon WA could be efficiently realized by locally applying d.c. electric-field on the silicon waveguides, which provides a novel method to reconfigure the WA.
2. Theoretical modeling
Figure 1(a) shows our proposed ridge WA etched on the crystal silicon with a light-doped p–i–n structure [15] in the [0, 0, 1] (vertical) direction. The doping thickness is supposed to be 30nm on both sides. Every waveguide in the WA along the [0, 1, 0] direction are identical and can be independently modulated by locally applied reverse bias voltages. The large overlapping area between the electric-fields of the TM fundamental mode (Fig. 1(b)) with the applied electric-field (Fig. 1(c)) enhances the d.c. Kerr effect in the silicon waveguides. At a fixed wavelength of 1.5μm, the relatively large separation (2.2μm) between the waveguides leads to the weak coupling between lights in them. Thus, considering both optical Kerr and d.c. Kerr effect in the silicon ridge WA, the evolution of the electric field amplitude in the nth waveguide can be approximately described by the discrete nonlinear Schrödinger equation (DNLS) [1, 13, 15] as
where is the d.c bias voltages applied on the nth waveguide. κ is coupling coefficient of the proposed silicon WA, which is equal to 810.73m−1. That is, coupling length Z0 is 1.23mm. By using the reported third-order nonlinear susceptibility of silicon [15–17], we can obtain the effective Kerr coefficient γ and d.c Kerr coefficient η of the fundamental TM mode in the ridge waveguide equal to 74.14m−1 and 0.38m−1V−2, respectively.
Fig. 1 Schematic illustration of the proposed silicon waveguide array and mimicking Kerr-type DS by using d.c. Kerr effect in the silicon WA. (a) The WA on the SiO2 substrate is a 2.5 cm long array consisting of 101 rib waveguides etched 0.6μm deep (h2) on the 1.5μm-thickness (h1) silicon layer with p-i-n structure. Period a of the WA and width W of rib waveguide are 3.4μm and 1.2μm. (b) The electric-fields of the TM fundamental mode at the wavelength of 1.5μm supported by the silicon rib waveguide. (c) The d.c. electric-field in the p-i-n structured silicon ridge with 1V voltage applied. The small arrows in (b), (c) denote the directions of the transverse electric-field vectors, and the different color hues denote the relative amplitudes of the electric fields. The propagation of low-power (d) and high-power (e) input light in the WA without d.c. electric-field applied. (f) The propagation of low-power input light in the WA with designed bias voltages (Eq. (4) applied.
For a steady Kerr-type DS propagating along the WA under no bias voltage, the amplitude is invariant during propagation. Thus, when the voltages applied on the silicon WA described by Eq. (1) satisfies
a steady Kerr-type DS could be mimicked by d.c. Kerr effect in the silicon WA under low power input light. Without loss of generality, we select a Gaussian beam as the input light in our calculation, whose electric field amplitude at input port of the WA could be written asWithout d.c. electric-field applied on the WA, for a low-power incident light with power equal to 1.1 × 10−11W (amplitude A = 10−6), central position n0 = 0, waist width w = 3, initial phase φn = 0, Fig. 1(d) shows that as a result of discrete diffraction, the input beam become wider while it propagates along the WAs. When increase the power of input light to 6.7W, the discrete diffraction is balanced by the Kerr effect. The input beam is converted into a steady Kerr-type DS with its shape invariant during propagation (Fig. 1(e)). Alternatively, the similar DS-shaped output beam can be achieved under the condition of a low-power input light (Fig. 1(f)) by applying local d.c. electric-field satisfying Eq. (2) on the WA, that is3. Manipulating discrete solitons
Limited by the linear propagation, an oblique Kerr-type DS could not be mimicked by the d.c. Kerr effect in the silicon WA. However, we can efficiently manipulate an oblique Kerr-type DS by adjusting the electric-field applied on the WA. Different from mimicking Kerr-type DS by applying local electric field to increase the refractive indexes of several ridge waveguides and trap light in them, the efficient method to control Kerr-type DS is to destroy the self-trapping of light in the WA by reducing the local refractive indexes of WA. Thus, we apply the voltage distribution contrary to Eq. (4) to manipulate the oblique DS. That is
For a given high-power incident Gaussian beam described by Eq. (3) with A = 0.78 (Input power equal to 6.7W), n0 = 0, w = 3, and φn = −0.1 × nπ, Fig. 2(a) shows that the incident beam becomes an oblique DS and propagation along a straight line in the silicon WA with no bias voltage. While, when the voltages described by Eq. (5) with parameter (V0, n0) equal to (31.5, 8), (13.95, 8) and (22.05, 3) are applied on the WA, we can observe that the oblique Kerr-type DS is reflected (Fig. 2(b)), captured (Fig. 2(c)) and divided into two beams (Fig. 2(d)), respectively.
Fig. 2 Manipulating oblique Kerr-type DS by d.c. electric-field. When a high-power Gaussian beam is incident, an obliquely incident Kerr-type DS (a) appears in the silicon WA (Fig. 1(a)) without bias voltage applied. The Kerr-type DS could be reflected (b), captured (c) and divided into two beams (d) by applying different bias voltages on the silicon waveguide array.
4. Routing light-transmitting paths
Routing light-transmitting paths is a key and fundamental issue in the integrated optics and on-chip optical devices. The proposed silicon WA could be reconfigured by d.c. electric-field, which provides a effectively platform for on-chip routing light. Here, we only consider shifting the center position of transmitted light.
Without bias voltages applied, the low-power incident light, described by Eq. (3) with parameters (A, w, φn, n0) equal to (10−6, 3, 0, 0), is diffracted in the WA (shown Fig. 1(d)). The waist radius of transmitted light at the output end (z = 20Z0) is about 20a (a is the period of the WA). To confine the diffraction, we apply isosceles-trapezoid bias voltages denoted by parameters (n0, n1, n2, n3) on the WA (Fig. 3(a)). The symmetrical center of the isosceles trapezoid denoted by 0.5(n1 + n2) is equal to 0.5(n0 + nT), where nT is the expected position of the transmitted light center. The linearly increasing (decreasing) bias voltage applied between the n0th (n2th) and n1th (n3th) waveguides works as a reflection mirror, which inhibits the diffracted light between the −50th (50th) and n0th (n3th) waveguides. Keeping electric field in the silicon ridge waveguide smaller than the breakdown field 40Vμm−1 [15], the maximal voltage 45V is applied on the waveguides from the n1th to n2th, which forms a region with relatively larger refractive indexes in the WA and guides the incident light to the expected position.
Fig. 3 Routing the light-transmitting paths in the silicon WA by using d.c. electric-field. (a) The d.c voltages with a trapezoidal distribution denoted by (n0, n1, n2, n3) are applied on the silicon WA. With an invariant input Gaussian light centered at the 0th waveguide, the center of transmitted light through the WAs could shift to the 1st (b), 5th (c), 10th (d), 15th (e), 20th(f) by applying the bias voltages with the trapezoidal distribution (n0, n1, n2, n3) equal to (−16, 0, 1, 17), (−12, 0, 5, 17), (−8, 1, 9, 18), (−6, 2, 13, 21), and (−4, 3, 17, 24) on the silicon waveguide arrays, respectively.
In order to evaluate the quality of the shifted light beam, the overlay coefficient η between the transmitted light and the corresponding target light is calculated as
where,, and are amplitude of input, expected and output beam in the nth waveguide. is obtained by replacing the central position n0 in the expression of (Eq. (3) with the expected position nT. For example, Figs. 3(b)-(f) show that with different isosceles-trapezoid bias voltages applied on the WA, the center positions of transmitted light through the WAs could be shifted to the 1st, 5th, 10th, 15th, 20th waveguide. Meanwhile, the overlay coefficients η in Figs. 3(b)-(f) are 0.98, 0.99, 0.97, 0.95, and 0.93, respectively, which means that the incident light is shifted to the expected position with high fidelity and efficiency. It is worth noticing that the spacing between the 0th and the 20th waveguide is equal to the waist radius of transmitted light at the output end when no bias voltage is applied (Fig. 1(d)). It means that adding the propagation length is an efficient method to increase the center position shift of the transmitted beam. However, with the fixed length of WA and maximal bias-voltage, what is the farthest position that the center positions of transmitted light can be shifted to with high fidelity and efficiency, and what kind of local-distributed bias voltages could be applied to realize the movement are two interesting and open questions.5. Conclusion
In summary, we have proposed a ridge WA etched on the crystal silicon with a p–i–n structure. Benefited from the direct-current (d.c.) Kerr effect of silicon, we demonstrate theoretically the propagations of incident light with high and low power in the WA could be efficiently controlled by locally applying different bias voltages on the waveguides. For example, a low-power incident light could propagate as a Kerr-type DS does. An obliquely incident DS could be reflected, captured and split. Moreover, the center position of the transmitted light could be flexibly altered by changing the isosceles-trapezoid bias voltage on the WA, while the characteristic of the incident light is well preserved. The realized maximum spacing between the center of incident and transmitted light reaches the waist radius of transmitted light without bias voltages applied. Introducing the d.c. Kerr effect into the silicon WA provides a novel method to reconfigure WA, which may be of great significance in the integrated optics and on-chip optical communication.
Funding
National Natural Science Foundation of China (11374367).
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