We propose a method to design antireflection structures to minimize the reflection of light beams at the interfaces between a two-dimensional photonic crystal and a homogeneous dielectric. The design parameters of the optimal structure to give zero reflection can be obtained from the one-dimensional antireflection coating theory and the finite-difference time-domain simulations. We examine the performance of a Mach-Zehnder interferometer utilizing the self-collimated beams in two-dimensional photonic crystals with and without the optimal antireflection structure introduced. It is shown that the optimal antireflection structure significantly improves the performance of the device.
©2008 Optical Society of America
Recently, unique dispersion properties of photonic crystals (PCs) which give rise to the interesting light propagation phenomena such as self-collimated beam propagation [1, 2], negative refraction [3, 4], and superprism effects  have attracted much attention because they could provide new mechanisms to control the light propagation in PCs. The propagation direction of light in a PC is determined by the direction of the group velocity of light in the PC, v g=∇k ω(k). Thus, the equifrequency contours (EFCs), the cross sections of the dispersion surfaces of the Bloch modes in momentum space, are essential to investigate the propagation properties of lights in PCs and to design the dispersion based PC optical devices such as non-channel waveguides [6, 7, 8], beam splitters [9, 10, 11, 12, 13], super lenses [14, 15, 16], demultiplexers [17, 18] and so on.
In general, the unwanted reflection at the interfaces between a two-dimensional (2D) PC and an outside uniform dielectric has been a crucial problem in realizing PC devices. Several approaches have been proposed to reduce the reflection at the 2D PC interfaces. Baba et al. elongated holes in the first layer  and Witzens et al. added multilayered diffraction grating at the end of a PC . Momeni and Adibi gradually varied hole sizes at the interfaces so that the group velocity and the field profile varied slowly . Here, we present a different compact and systematic method which is suitable for practical applications to the dispersion based PC devices.
In optics, the antireflection coating (ARC) method has been systematically studied and widely used to reduce the reflection at the interfaces . The method is simply to insert an antireflection layer between two media to reduce the reflection. The design parameters for the antiflection layer, the refractive index and the layer thickness, must be optimized so that multiply reflected light beams undergo a total destructive interference. In the previous study, Ushida et al. showed that conventional ARCs can be applied to one-dimensional (1D) PC interfaces . Even though the ARC method is simple and powerful to reduce reflection, to our knowledge, there has been no systematic study on the reflection minimization at 2D PC interfaces by applying the concept of ARCs.
In this paper, we describe a method to minimize the reflection at the interfaces between a 2D PC and a background dielectric material by using the concept of ARC. We show that the ARC structures composed of rods or holes can be optimized for minimum reflection at the interfaces by using the finite-difference time-domain (FDTD) simulations . To stress the effectiveness of the proposed method, we simulate the performance of a Mach-Zehnder interferometer (MZI) for self-collimated beams in 2D PCs with and without the optimized ARC structure introduced. The simulated results show that the performance of the MZI can be significantly improved by the introduction of the ARC structure into the PC.
2. Model and Method
When a light beam is normally incident from region 1 onto region 2 which is placed between two semi-infinite homogenous media (region 1 and 3) as shown in Fig. 1(a), the reflection coefficient is given by
where β is the phase change occurred during the time the light goes across region 2 and rij is the reflection coefficient of light propagating from region i to j . The reflectance of the incident light, the square of the amplitude of the reflection coefficient r given by Eq. (1), becomes zero when the following two conditions are satisfied simultaneously:
where |rij| and δij correspond to the amplitude and the phase factor of the reflection coefficient rij, respectively. In the simple case shown in Fig. 1(a), the optimal ARC parameters, the refractive index and the optical thickness h=λ/4, are easily obtained from Eqs. (2) and (3) by using the reflection coefficients given by the Fresnel equations. When region 3 is replaced by a 1D PC, the ARC parameters can also be optimized by using the r 12 given by the Fresnel equations and r 23 given by the numerical calculations as described in Ref. .
We introduce ARC structures to minimize the reflection at the ends of 2D PCs. In a practical point of view, it is reasonable to apply ARC structures composed of rods or holes at the interfaces between a 2D PC and a homogeneous background medium to reduce the reflection. Thus, the radius Rarc of the rod (hole) and the distance darc between the ARC and the PC truncation are chosen as the design parameters of the ARC structure as depicted in Fig. 1(b) that shows the case for a 2D square lattice PC composed of dielectric rods in air. The ARC structure becomes a part of the host PC when Rarc=R and darc=a/√2. The ARC parameters for this configuration can also be optimized from Eqs. (2) and (3), provided that rij are properly modified. Note that, in this analysis, r 12 is the reflection coefficient of the ARC structure embedded in air and r 23 is that of the semi-infinite PC when the light is incident upon it from the air. In the conventional ARC approach the perfect transmission of incident light is resulted from the resonance in the region 2 of Fig. 1(a), whereas in the 2D PC case this is taking place in the air region located between the end of PC and the ARC structure (see Fig. 1(b)). Therefore, the reflection coefficient r 23 in the classical ARC is replaced by the reflection coefficient of the PC starting from the air. We can optimize the parameters for the specific light frequency of interest by using the FDTD simulations. First, the value of Rarc is found to satisfy the condition given by Eq. (2) and then the value of darc to satisfy Eq. (3) at the optimized value of Rarc.
To see the effects of the ARC structures on the reflection of light beam, both the 2D rod-type and hole-type PCs will be considered. In recent years, the self-collimated light propagation in 2D PCs have inspired great interests due to its potential applications in implementing on-chip photonic integrated circuits [26, 27]. However, there has been no study on the reflection minimization of self-collimated beams at the 2D PC interfaces. Hence, we will mainly focus our attention on the reflection minimization of self-collimated beams at the 2D PC interfaces.
3. Results and Discussion
To give practical examples of the reflection minimization, we first consider a 2D square array of dielectric rods in air. The rod has the dielectric constant ε=12.0 and the radius r=0.35 a, where a is the lattice constant. In our previous work, it was shown that the E-polarized lights, which have the electric field parallel to the rod axis, of frequencies around f=0.194 c/a, where c is the speed of light in vacuum, exhibit the self-collimation phenomenon when they propagate along the ΓM-direction in the PC structure considered in this study . We first calculated |r 23|, and then |r 12| as a function of Rarc for the light of frequency f=0.194 c/a by using the FDTD simulations. It is found that |r 12|=|r 23|=0.573 at Rarc=0.2064 a and 0.4347 a, as drawn in Fig. 2(a). In order to find the optimal value of darc, the total reflectance is calculated as a function of darc when Rarc=0.2064 a and 0.4347 a. As can be seen in Fig. 2(b), there are values of darc at which the reflectance becomes zero and the reflectance curves exhibit periodic oscillations with darc. The period of oscillation is about a half wavelength of the incident beam. This result shows that it is possible to control the total phase of Eq. (3), (2β+δ 23-δ 12), by varying the value of darc, even though we do not know the specific values of δij.
The transmission spectra of the PC are obtained with and without the ARC structure when darc=0.74 a and Rarc=0.2064 a. The minimum value of darc is chosen here to minimize the spreading of light beam which may occur during the propagation through the air layer between the ARC structure and the PC and thereby to improve the coupling efficiency. The FDTD simulations are performed for two different PC samples of sizes 12√2 a and 16√2 a in the ΓM-direction. The ARC structures of the same parameters are introduced at both the input and output PC interfaces. The computational geometry is shown in Fig. 3(a). In the x-direction the Bloch periodic boundary condition is applied and the perfectly matched layer (PML) absorbing boundary condition  is used in the y-direction. Fig. 3(b) shows the calculated transmission spectra of the PC without the ARC structure and one can compare them with those shown in Fig. 3(c) which are obtained with the ARC structure applied. The transmission of light beam through the PC without the ARC structure not only strongly oscillates but also depends on the size of the PC; the period of oscillation becomes shortened as the size of PC increases because the optical path of light is increased. The variations in the transmitted power of light result from the constructive or destructive interferences of multiple beams which are reflected and transmitted at the PC interfaces. Hence, it is reasonably expected that the oscillations in the transmission spectra will disappear if the reflection totally vanishes. One can clearly see that the light beams of frequencies around f=0.194 c/a show almost perfect transmission, irrespective of the PC size. More than 99% of the incident power is transmitted through the PC samples for the the lights in the frequency range from 0.188 to 0.202 c/a.
The ARC structure is also introduced into a 2D square lattice PC which consists of air holes with the hole radius r=0.35 a in a high index material of ε=12.0. According to Ref. , the H-polarized lights, which have the magnetic field parallel to the hole axis, of frequencies around f=0.190 c/a propagate with almost no beam spreading along the ΓM-direction in the PC of the same structure. The optimal values of the ARC parameters are found to be Rarc=0.2565 a and darc=0.58 a for the light of frequency f=0.190 c/a. The transmission spectra are calculated for the cases with and without the optimal ARC structure when the sample size is 16√2 a. Figure 4(b) demonstrates that the reflection at the 2D PC interfaces can be efficiently eliminated by the application of the optimal ARC structure. Comparing Fig. 4(b) with Fig. 3(c), one can notice that the frequency range from 0.170 to 0.205 c/a which exhibit over 99% transmission in the hole-type PC is much wider than that of the rod-type PC. Because the reflection of light beam at the hole-type PC interface is smaller than that at the rod-type PC interface, the frequency range in which the incident light exhibits high transmission gets wider.
The reflection at the ends of PC structures may crucially affect the performance of devices because the PCs are truncated and of finite sizes. Thus, the reflection minimization at a crystal truncation is one of the important issues to implement practical PC devices. To compare the performance of a PC device with and without a ARC structure, we choose a PC MZI based on the self-collimated beams as depicted in Fig. 5(a). Recently, Zhao et al. theoretically demonstrated that the phase difference of the two split self-collimated beams at the line-defect beam splitter is π/2 in the PC with the same structural parameters considered in this study . Hence, the normalized output intensity I 1/I 0 measured at the port 1 is ideally unity and I 2/I 0 at the port 2 is zero due to the constructive and destructive interferences, respectively. We calculate the transmission spectra at the ports 1 and 2 around the frequency f=0.194 c/a when the ARC structure is (Fig. 5(b)) and is not (Fig. 5(c)) introduced. In the simulations, a Gaussian pulse with a waist of w=3 a is launched into the input interface of the MZI. Figure 5(b) shows that the transmission of light at the port 1 exceeds 92% with almost no fluctuation for the lights in the frequency range between 0.190 and 0.198 c/a. On the other hand, Fig. 5(c) shows that the transmitted power at the port 1 strongly fluctuates from 23% to 94% due to the reflection at the input and output interfaces of the device. These results reveal that the performance of the PC MZI is significantly improved by the introduction of the ARC structure into the PC.
Finally, we applied the ARC method to the propagation of light beam of frequency f=0.160 c/a at which the light does not exhibit self-collimated propagation in the larger size rod-type PC structure employed in this study. The obtained results showed more than 99% transmission for the lights in the frequency range from 0.153 to 0.170 c/a with the optimal ARC structure applied.
In this study, the loss of light due to the out-of-plane scattering, which could diminish the coupling efficiency, is not considered as all the FDTD simulations have been performed for 2D geometry. For practical applications, therefore, a detailed three-dimensional study is required and that may be the issue of future work.
In conclusion, we propose an effective method to minimize the reflection at the interfaces between a 2D PC and a homogeneous background material. It is shown that the reflection at the 2D PC interfaces can be efficiently eliminated by optimizing the parameters of the ARC structure such as Rarc and darc. We also show that the performance of a PC MZI based on the self-collimated beams can be significantly improved by the introduction of the optimal ARC structure. The proposed method can be important for implementing other PC devices.
This work was partially supported by Ministry of Science and Technology through QSRC, Photonics 2020, and APRI-Research Program of GIST.
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