1. Introduction

Photonic crystals (PCs) have inspired great interest since their first introduction [1, 2]. The complex spatial dispersion properties in PCs provide various mechanisms to control light propagation, such as negative refraction [3], superprism effect [4], and self-collimation [5-12]. Recently, self-collimation effect, by which an incident light can propagate with almost no diffraction along a definite direction in a perfectly periodic PC, has attracted particular attention because of its potential for photonic integrated circuits (PICs). Several works on bending and splitting of incident self-collimated beams have been theoretically suggested [7, 8] and experimentally demonstrated [9, 10] for the purpose of using these phenomena as a basis for PICs. Meanwhile, optical switches and logic gates are considered as key components in future PICs. Therefore, such PC-based optical devices have attracted significant research efforts in recent years. However, most of the reported works were based on nonlinear optics [13-15], which suffered from certain fundamental limitations, such as power consumption and narrow operating frequency range. By contrast, self-collimation phenomenon is independent of light intensity [5] and frequency range in which the phenomenon occurs is sufficient for operating in PICs [7]. Moreover, compared with conventional dielectric waveguides and PC waveguides working within the photonic band gap, self-collimation does not require a lateral confinement to prevent either the beam divergence or diffraction broadening [5]. It can release strict alignment for coupling light into narrow waveguides [10, 11]. Therefore, devices based on self-collimation have intriguing potentials for high-density PICs. To take the advantages of the self-collimation, in this paper, we propose and discuss a device for optical switches and logic gates using line-defect-induced 3dB splitter in two-dimensional (2D) PCs based on the interference between incident self-collimated beams.

2. Operation principle and structure analysis

Self-collimated beams can be totally reflected at a PC-air interface with an incident angle θ>θ_c=arcsin (1/n_H) because of the conservation of momentum components parallel to the interface, where a PC and air corresponds to an optically denser medium (high refractive index, n_H) and an optically thinner medium, respectively [7, 8]. What is more, by reducing the radii of the PC rods (line defect) instead of eliminating them totally (air), the total reflection becomes frustrated and the self-collimated beam is reflected partially due to the tunneling of the evanescent wave [8]. It is expected that there should be a phase shift between the reflected beam and the tunneling beam (transmitted beam). Hence, if another self-collimated beam with an appropriate initial phase is introduced, the reflected and transmitted beams may interfere constructively or destructively to realize the switching and logical functions.

To verify our conjecture, as shown in Fig. 1, we consider a 12√2a×12√2a of 2D square lattice PC composed of silicon (Si) rods in air. The radius and dielectric constant of the host Si rods are r=0.35a and ε=12.0, respectively, where a is the lattice constant (the parameters refer to Ref. [8]). A line defect is created by reducing the radii r_d=0.274a of 25 rods aligned in the Γ-X direction different from those of the host rods, as denoted by the green area in Fig. 1. The band diagram and the equifrequency contours (EFCs) of the first band for the E-polarized mode (electric-field is parallel to the rod axes) are shown in Fig. 2. In the EFCs [see Fig. 2(b)], the curves of the frequencies around 0.194(a/λ), where λ is the wavelength of light in free space, can be identified as squares with round corners centered at the M point. It is known that the light propagation direction in the PC is identical to the direction of group velocity given by v_g=∇_kω(k), where ω is the optical frequency at the wave vector k [16]. It means that the group velocity is perpendicular to the EFCs. So the self-collimation phenomenon occurs when the E-polarized light of the frequencies around 0.194(a/λ) propagate along the Γ-M direction. There are four faces in the device structure, two adjoining faces of them function as input faces (I₁, I₂), and the other two as output faces (O₁, O₂), as shown in Fig. 1. Before launching the self-collimated beams to investigate the switching and logical functions, the phase shift between the reflected and transmitted beams should be known.

 

Fig. 1. Schematic diagram of the proposed optical switches and logic gates. The defect region (green area, the reducing Si rods) and the surrounding region (cyan area, the host rods) corresponds to an optically thinner medium (low refractive index, nL) and an optically denser medium (high refractive index, nH), respectively.

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Fig. 2. (a). Band diagram of the PC structure for E-polarized mode. The inset shows the PC structure, which consists of square lattice of Si rods in air. The frequency 0.194(a/λ) is marked by the orange line. (b) Equifrequency contours of the first band.

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From Fig. 1, it is clear that the average index in the defect region (green area, the reducing Si rods) is lower than that in the surrounding region (cyan area, the host rods). So the proposed device can be regarded as two identical high refractive index (n_H) PC prisms separated by a low refractive index (n_L) gap with a spatially symmetric structure. Note that the dielectric constituents are nonabsorbing (the imaginary parts of the dielectric constants are set to zero), which means that the proposed device is a lossless (conservation of energy) system. According to Refs. [12, 17, 18], there is a π/2 phase difference between the reflected and transmitted beams. This is a general result for a spatially symmetric, lossless beam splitting system. Especially for the splitter, by utilizing the line defect in PC to split the self-collimated beams, if the rod radii of the line defect are smaller than that of the host rods, there will be a π/2 phase lag of reflected beam compared to the transmitted beam [12]. Therefore, for the line defect created by reducing the radii of some Si rods in this work, the reflected beam has a π/2 phase lag compared to the transmitted beam after the self-collimated beam goes through the line defect.

3. Calculations of the structure

Because of the symmetry of the PC device (see Fig. 1), it can be supposed that the low index gap has a transmission amplitude te^iφ, so the reflection amplitude is re^i(φ+π/2) for both the incident beams, where φ, t, and r are real, and

According to Ref. [8], the line defect just makes the PC device to be a 3 dB splitter of the self-collimated beams, as shown in Fig. 3. It means that the transmission and reflection amplitudes are equal. Considering their relationship indicated by Eq. (1), it easily gets t=r=1/√2. So the transmission and the reflection amplitudes are e^iφ/√2 and e^i(φ+π/2)/√2, respectively. The computational simulation is carried out by using a finite-difference time-domain (FDTD) code with perfectly matched layer boundary condition and an E-polarized Gaussian wave with the full width at half maximum 5a is used. A monochromatic wave of the frequency 0.194(a/λ) is launched into the PC device along the Γ-M direction, and then a self-collimated beam will be excited.

 

Fig. 3. Simulated steady-state field distribution of the E-polarized mode at 0.194(a/λ) when incident beams propagate along the Γ-M direction. (a) and (b) for the case that the incident beam is only launched into face I₁ and I₂, respectively.

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Further suppose that the input face I₁ and I₂ sees a beam field (Bloch wave) of $E_1={\mathit{uEe}}^{-i{\phi }_1}$ and $E_2={\mathit{uEe}}^{-i{\phi }_2}$, respectively, where φ₁, and φ₂ are real, E represents a plane wave and u is a function with the same periodicity as the PC. There is only a phase difference between the two incident beams. Then the reflected and transmitted beams can be expressed as:

The beams exiting output face O₁ and O₂ can be written as a linear combination (interference) of the reflected and the transmitted beams:

From Eq. (3) we can find the corresponding intensities,

Aside from the trivial case where u or E is zero (when the phase difference becomes meaningless quantity), we can see that I_o1=2|uE|², I_o2=0 when φ₁-φ₂=2kπ+π/2 and I_o1=0, I₀₂=2|uE|² when φ₁-φ₂=2kπ-π/2, where k is an integer.

4. Results and discussions of switches and logic gates

From the discussion in section 3, it is clear that the switching function can be realized by introducing a certain phase difference between the two beams (incident on the input faces I₁ and I₂), as shown schematically in Fig. 4(a). A phase modulator was introduced at input branch I₂, the beam coupled into which we regard as the control beam. Figures 4(b) and 4(c) show the simulated steady-state field distributions of the switch. It shows that once the control beam is introduced, the output state of the device can be controlled. When the phase difference φ₁−φ₂ sets as 2kπ+π/2, where k is an integer, the input lights will be output from the face O₁, and there is no output light in the face O₂, as shown in Fig. 4(b). If the phase difference is changed to 2kπ-π/2, the light power will be output from the face O₂, and the output face O₁ will be cut off, as shown in Fig. 4(c). So the switching is achieved.

 

Fig. 4. (a). Schematic diagram of the switch. Two beams with different phases are incident on the input faces I₁ and I₂. (b) and (c) Simulated steady-state field distribution of the E-polarized mode at 0.194(a/λ) when incident beams propagate along the Γ-M direction. The phase difference φ₁-φ₂ of the two incident beams particularly sets as π/2 and -π/2, respectively.

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The proposed PC device can also operate as logic OR and XOR gates. Based on the previous simulations, we can find that if the phase difference φ₁-φ₂ sets as 2kπ+π/2, the output face O₁ and O₂ operates as OR and XOR logic gates, respectively, as shown in Fig. 3 and Fig. 4(b). Alternatively, if the phase difference φ₁-φ₂ sets as 2kπ-π/2, the output face O₁ and O₂ operates as XOR and OR logic gates, respectively, as shown in Fig. 3 and Fig. 4(c). The total device functions are shown in Table 1 for both cases. The logic 0 and 1 in the table indicate without and with output signal, respectively.

Table 1. The total device functions.

View Table

To evaluate the characters of the device, considering the symmetry, the normalized output intensity spectra from the faces O₁ and O₂ for Fig. 3(a) and Fig. 4(b) are shown in Fig. 5. From Fig. 5(a), it is clear that, at frequency 0.194(a/λ), the incident power is split equally into the two output faces. In the frequency range 0.188–0.199(a/λ), the fluctuation of the output intensities are within 20% referred to the intensity at frequency 0.194(a/λ), and the sum of the transmitted and reflected intensity is larger than 93%. Moreover, from Fig. 5(b), it can be found that, in the frequency range 0.188–0.199(a/λ), there is nearly no fluctuation of the output intensities from the two output faces and the extinction ratio, defined as 10log(I_O1/I_O2), is larger than 17 dB (maximum 20.1 dB). So we conclude that the switching and logical function of the structure is applicable in the frequency range 0.188–0.199(a/λ).

 

Fig. 5. (a). The normalized intensity spectra of faces O₁ (green line) and O₂ (violet line) for Fig. 3(a). The red line represents the total output efficiency. (b) The normalized intensity and extinction ratio spectra for Fig. 4(b).

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5. Conclusion

A device for optical switches and logic gates, based on line-defect-induced 3dB splitter of self-collimated beams in a 2D PC composed of Si rods in air, is proposed and demonstrated. The switching and logical function, as revealed by both theoretical calculation and FDTD simulations, is based on the interference of the reflected and transmitted self-collimated beams and is applicable in frequency range 0.188–0.199(a/λ). The extinction ratio for the switch within the applicable frequencies is larger than 17 dB (maximum 20.1 dB). The device has simple geometric structure and clear operating principle, which shows that this device could be a strong candidate for future PICs.