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We present a high-efficiency antireflection structure for both TE and TM polarizations in two-dimensional self-collimating square lattice photonic crystal consisting of air holes in silicon. The design parameters of the optimal antireflection structure can be obtained by using the concept of Fresnel coefficients and the finite-difference time-domain simulations. The photonic crystal operating in almost identical self-collimation frequencies for two polarizations exhibits a large reflection coefficient for TE and a very small one for TM polarization. In this case, the antireflection structure for TE can also improve the transmission for TM polarization. To confirm a highly efficient antireflection structure designed, we investigate the transmission data of three finite photonic crystal samples consisting of 36, 38 and 40 unit cells for the cases without and with the antireflection structures through finite-difference time-domain simulations.
©2010 Optical Society of America
1. Introduction
The self-collimation (SC) phenomena of light beam in photonic crystals (PhCs) have attracted much attention as a basis for photonic integrated circuits (PICs) since their discovery in 1999 [1, 2]. Various SC-based PhC devices, such as waveguides, mirrors, beam splitters (BSs) [3–7], polarization BSs (PBSs) [8], Mach-Zehnder interferometers [9, 10], photonic logic gates [11], and ring resonators [12], etc. have been demonstrated. These in-band applications require a negligibly small reflection loss at the PhC interfaces. There have been many attempts to reduce the unwanted reflections, i.e., for the design of antireflection structure (ARC) [7, 13–22]. Among the PhC devices mentioned above, the SC-based PBSs should perform equally well for both TE and TM polarizations, which have the magnetic and the electric field parallel to the air hole axis, respectively. In particular, the primary transmission coefficients for both polarization output channels should be as close to unity as possible [8], and thus the SC-based PBSs require a common ARC for both TE and TM polarizations. In this paper, we propose a high-efficiency ARC for both TE and TM polarizations in two-dimensional (2D) square lattice SC PhC of air holes in silicon (Si). Through finite-difference time-domain (FDTD) simulations [23] with Berenger’s perfectly matched layer boundary conditions [24], we show that the transmission coefficients for both TE and TM polarizations can be significantly improved in 2D SC PhCs of various sizes with the designed ARCs applied at the input and output ends of them. In Section 2, how to find the SC frequency range is described and how to design the ARC parameters is illustrated. Presented in Section 3, are not only the common ARC parameters for both TE and TM polarizations but also their allowed fabrication errors, which is followed by the simulation results, confirming that high-efficiency power transmission for both two polarizations can be achieved via the use of the above-designed common ARCs. Conclusions are presented in Section 4.
2. Method and design
2.1. EFCs analysis
It is well known that SC beams of light can propagate with almost no diffraction in a PhC [1, 2]. As the group velocity of light given by vg = ∇k ω(k) is always normal to the equi-frequency contours (EFCs), SC is achieved when the EFCs are as flat as possible. To find the SC frequency range, the EFCs are calculated using the plane-wave expansion method [25, 26]. In this paper, the SC frequencies are within the second photonic band for both TE and TM polarizations, and the SC beams of light propagate along the ΓX direction.
2.2. Reflectance analysis and ARC parameters
As shown in Fig. 1(a), when a light beam is normally incident from the region 1 into the region 2, which is followed by the region 3 of semi-infinite homogeneous medium, the reflection coefficient is given by
where β is the phase change occurred during the time the light goes across the region 2 and rij is the reflection coefficient of light propagating from region i to j [27]. The reflectance of the incident light, the squared amplitude of reflection coefficient r given by Eq. (1), becomes zero when the following two conditions are satisfied simultaneously:
and
where |rij| and δij correspond to the amplitude and the phase factor of the reflection coefficient rij, respectively. In a one-dimensional (1D) simple case shown in Fig. 1(a), the optimal ARC parameters can be easily found to be and h = λ 0/4n 2, where λ 0 is the wavelength of the incident light in vacuum [14, 27, 28]. When the region 3 is replaced by a 1D PhC, multiple reflections in the semi-infinite 1D PhC, which are represented as plane waves, can be renormalized into a single Fresnel coefficient by using the Bloch wave expansion [14]. Furthermore, in a 2D PhC case shown in Fig. 1(b), the ARC parameters of the radius of air holes rarc and the distance darc between the ARC and the 2D semi-infinite PhC, can also be optimized from Eqs. (2) and (3), provided that rij is properly modified [19] [i.e., r 12 → r 123 and r 23 → r 34; accordingly, δ 12 and δ 23 change into δ 123 and δ 34, respectively, and in this case, β is defined as the phase change across the region 3 shown in Fig. 1(b)].
Fig. 1. Schematics of the ARC structure: (a) In the 1D case, the ARC parameters are the refractive index n 2 and the thickness h of ARC structure. (b) In the 2D PhC case, the ARC parameters are the radius of air holes rarc and the distance darc between the ARC (enclosed by the dark green rectangle) and the 2D semi-infinite PhC. For the 2D PhC case of (b), schematics of the rij modified properly: (c) r 123 is the reflection coefficient of the ARC embedded in Si. (d) r 34 is that of the semi-infinite 2D SC PhC when the light is incident upon it from the Si. In (a)–(d), the thick and thin red arrows indicate the incident and reflected beams, respectively.
Figures 1(c) and 1(d) show the schematics of the properly modified reflection coefficients r 123 and r 34, respectively; r 123 is the reflection coefficient of the ARC embedded in Si and r 34 is that of the semi-infinite 2D SC PhC when the light is incident upon it from the Si. Note that the reflection coefficient r 123 is a net reflection coefficient off a structure consisting of regions 1, 2 and 3, encapsulating information about reflections at the two interfaces and the phase acquired between them and that the reflection coefficient r 123 is not a function of darc but only rarc. In the conventional ARC case, the perfect transmission of incident light is resulted from the resonance in the region 2 of Fig. 1(a), whereas in the 2D PhC case shown in Fig. 1(b), this is taking place in the region 3, i.e., the Si region located between the end of PhC and the ARC structure. Therefore, the reflection coefficient r 23 in the classical ARC is replaced by the reflection coefficient r 34 of the PhC starting from the Si in the 2D PhC case of Fig. 1(b) [19].
The ARC parameters for the specific light frequency of interest can be optimized by using the FDTD simulations as follows [7, 10, 19–22]: first, r 34 is calculated and then the value of rarc is found to satisfy the condition given by Eq. (2) since r 123 is a function of rarc. Next, the optimal value of darc is found to satisfy Eq. (3) at the optimized value of rarc since it is possible to control the total phase of Eq. (3) (i.e., in this case modified properly, (2β + δ 34 − δ 123)), by varying the value of darc, even though the specific values of δij are unknown [19].
2.3. FDTD simulation
The TE (TM)-polarized light from the plane wave source, is launched normally into 2D square lattice SC PhCs of air holes in Si, for which a periodic boundary condition is used in the direction perpendicular to the light propagation and perfectly matched layers which are absorption boundaries [24] are applied at the ends of computational domain in the direction of light propagation. In the FDTD simulations for obtaining the reflection coefficients of the semi-infinite 2D SC PhCs, a sufficiently long (more than 100 a, where a is the lattice constant) structure is employed to suppress the unwanted reflections generated from the output end of the structure [17, 19, 22].
3. Results and discussion
3.1. Analysis of SC frequency and reflectance including ARC parameters
Fig. 2. (a) SC frequencies within the second photonic band for TE and TM polarizations and the corresponding reflectances, as a function of the air hole radius. (b) Values of ARC parameters and the reflectances without and with the designed ARCs applied. Considered here are 2D square lattice SC PhCs of air holes in Si with the refractive index of n = 3.518.
To our knowledge, there has been no study on the design of a common ARC for both TE and TM polarizations in a 2D SC PhC. In this paper, we consider 2D square lattice SC PhCs of air holes in Si, of which the refractive index is n = 3.518 for the wavelength of λ = 1.55 µm [29]. Figure 2(a) shows the SC frequencies within the second photonic band for TE- and TM-polarized lights and the reflectances for the corresponding SC lights, as a function of the air hole radius. The red (blue) line indicates the SC frequencies for TE (TM) polarization, and it is clearly seen that the SC phenomena occur at almost identical frequencies for both TE and TM polarizations [8]. In the figure, the orange (dark cyan) symboled line indicates the reflectances for the TE (TM)-polarized lights of the corresponding SC frequencies in the semi-infinite 2D PhCs. It can be found that the 2D PhCs with the air hole radii of not more than 0.36 a [indicated by the grey vertical line in Fig. 2(a)] exhibit large reflection coefficients for TE and sufficiently small ones for TM polarization, in which only the ARCs for TE polarization may be needed, and thereby enabling one to achieve the design of a common ARC for both TE and TM polarizations in a 2D SC PhC.
Note that the Wood anomaly is expected to occur at the frequency of 0.284 (2πc/a) for the refractive index of n = 3.518. Above this Wood anomaly, PhCs may not be homogenized to a scalar, i.e., more than one Bloch mode is important and matrices are required to describe reflections and transmissions [18, 30]. In this case, Eq. (1) becomes invalid. While, in a 2D SC PhC with the air hole radius of 0.36 a, the SC frequency of 0.286 (2πc/a) (0.283 (2πc/a)) for TE (TM) polarization corresponds almost exactly to that of the above-mentioned Wood anomaly. Below theWood anomaly (i.e., for the SC PhCs with air hole radii of up to 0.36 a), it is possible to design the ARCs for TE and TM polarizations using Eq. (1). What is especially noteworthy is that high-efficiency common ARCs for both TE and TM polarizations can be designed based on the fact that for frequencies lower than the Wood anomaly, the reflection coefficients are large for TE and sufficiently small for TM polarization, which is a highly significant discovery and also very important for the photonic crystal community.
Figure 2(b) shows the ARC parameters designed for TE polarization and the reflectances for two polarizations in the semi-infinite 2D SC PhCs without and with the designed ARCs applied, as a function of the air hole radius. According to the description in Section 2.2, we firstly design the ARC for TE polarization; the region 2 is the ARC which consists of a layer of air holes with the radius rarc, the region 3 is the Si region between the ARC and the semiinfinite 2D PhC considered here, the region 4 is the semi-infinite 2D PhC, and the distance between the ARC and the 2D PhC is darc [see Fig. 1(b)]. Through FDTD simulations, the value of rarc satisfying the condition given by Eq. (2) and that of darc yielding minimum reflection, i.e., satisfying the condition given by Eq. (3) are obtained. Next, the above-designed ARC can be applied to the case for TM polarization, and thereby achieving high-efficiency ARC for both two polarizations.
In this paper, we discuss in detail the case of a 2D square lattice SC PhC with the air hole radius of rsc = 0.34 a, as indicated by the orange vertical line in Fig. 2(a). In this case, both TE- and TM-polarized lights have the same SC frequency of 0.275 (2πc/a), where c is the speed of light in vacuum and the SC frequency is indicated by the dark green horizontal line in Fig. 2(a). For the case without any ARC, the reflection of SC light for TE is |r 34|2 TE = 12.48% and that for TM polarization |r 34|2 TM = 0.25%. As the designed ARC is applied, the reflection of SC light for TE becomes almost zero (R TE = 0.018787%) and that for TM polarization also becomes close to zero (R TM = 0.043%), where the value of rarc is found to be 0.223 a and that of darc is optimized to be 0.939 a. In short, the above-two parameters rarc = 0.223 a and darc = 0.939 a are the common ARC parameters for both TE and TM polarizations in a 2D square lattice SC PhC with the air hole radius of rsc = 0.34 a, where the common SC frequency is 0.275 (2πc/a).
3.2. ERCs for TE and TM polarizations
From the point of view of the fabrication, the allowed fabrication errors of these common ARC parameters should be investigated. In this study, we investigated twelve reflectance maps as a function of a pair of parameters, i.e., (darc, rarc), (ω, r), (rarc, r), (darc, r), (rarc, ω), and (darc, ω) for TE and TM polarizations, as shown in Figs. 3–5. Through these reflectance maps, i.e., the equi-reflectance contours (ERCs), we can evaluate the device sensitivity to the various fabrication errors for 2D SC PhCs with the designed ARCs applied. In these all figures, the dark gray solid circles indicate the reflectances R TE = 0.010827% and R TM = 0.038829% at a specific position consisting of rsc = 0.34 a, ωsc = 0.275 (2πc/a), rarc = 0.225 a and darc = 0.94 a.
Figures 3(a) and 3(b) display the ERCs as a function of the ARC parameters darc and rarc in a 2D square lattice SC PhC with the air hole radius of rsc = 0.34 a and the SC frequency of 0.275 (2πc/a) for TE and TM polarizations, respectively. In the figures, the dark green solid circles indicate the reflectances R TE = 0.018787% and R TM = 0.043% at the above-two common ARC parameters rarc = 0.223 a and darc = 0.939 a for both TE and TM polarizations. From Fig. 3(a), it can be seen that the reflectances for TE polarization are less than R TE = 0.25% (0.5%) in the ranges of |∆rarc| and |∆darc| of less than 0.01 a (0.015 a) at the position of darc = 0.94 a and rarc = 0.225 a. The reflectances for TM polarization are still close to zero, i.e., less than R TM = 0.07% in the ranges of |∆rarc| and |∆darc| of less than 0.015 a at the above-same position, as shown in Fig. 3(b). Figures 3(c) and 3(d) show the ERCs as a function of the frequency ω and the radius r in a 2D square lattice SC PhC with the ARC parameters of rarc = 0.225 a and darc = 0.94 a for TE and TM polarizations, respectively. The reflectances for both two polarizations are less than R TE = 0.25% (0.5%) in the ranges of |∆ωsc| of less than 0.005 (2πc/a) (0.0075 (2πc/a)) and |∆r| of less than 0.01 a (0.015 a) at the position of ωsc = 0.275 (2πc/a) and r = 0.335 a.
Fig. 3. Reflectance maps as a function of the parameters darc and rarc for (a) TE and (b) TM polarizations, where rsc = 0.34 a and ωsc = 0.275 (2πc/a). Here, the dark green solid circles indicate the reflectances at the above-two common ARC parameters for both TE and TM polarizations. Reflectance maps as a function of the parameters ω and r for (c) TE and (d) TM polarizations, where rarc = 0.225 a and darc = 0.94 a. In (a)–(d), the dark gray solid circles indicate the reflectance at a specific position consisting of rsc = 0.34 a, ωsc = 0.275 (2πc/a), rarc = 0.225 a and darc = 0.94 a.
Figures 4(a) and 4(b) show the ERCs as a function of the ARC parameter rarc and the radius r in a 2D square lattice SC PhC with the frequency ωsc = 0.275 (2πc/a) and the ARC parameter darc = 0.94 a for TE and TM polarizations, respectively. In the ranges of |∆rarc| and |∆r| of less than 0.01 a (0.015 a) at the position of rarc = 0.225 a and r = 0.335 a, the reflectances for TE and TM polarizations are less than R TE = 0.25% (0.5%) and less than R TM = 0.2% (0.25%), respectively. Figures 4(c) and 4(d) show the ERCs as a function of the ARC parameter darc and the radius r in a 2D square lattice SC PhC with the frequency ωsc = 0.275 (2πc/a) and the ARC parameter rarc = 0.225 a for TE and TM polarizations, respectively. In the ranges of |∆darc| and |∆r| of less than 0.01 a (0.015 a) at the position of darc = 0.94 a and r = 0.335 a, the reflectances for TE and TM polarizations are less than R TE = 0.25% (0.5%) and less than R TM = 0.1% (0.175%), respectively.
Figures 5(a) and 5(b) show the ERCs as a function of the ARC parameter rarc and the frequency ω in a 2D square lattice SC PhC with the radius rsc = 0.34 a and the ARC parameter darc = 0.94 a for TE and TM polarizations, respectively. In the ranges of |∆rarc| of less than 0.01 a (0.015 a) and |∆ωsc| of less than 0.005 (2πc/a) (0.0075 (2πc/a)) at the position of rarc = 0.225 a and ωsc = 0.275 (2πc/a), the reflectances for both two polarizations are less than R TE = 0.25% (0.5%). Figures 5(c) and 5(d) show the ERCs as a function of the ARC parameter darc and the frequency ω in a 2D square lattice SC PhC with the radius rsc = 0.34 a and the ARC parameter rarc = 0.225 a for TE and TM polarizations, respectively. In the ranges of |∆darc| of less than 0.01 a (0.015 a) and |∆ωsc| of less than 0.005 (2πc/a) (0.0075 (2πc/a)) at the position of darc = 0.94 a and ωsc = 0.275 (2πc/a), the reflectances for TE and TM polarizations are less than R TE = 0.5% (1.0%) and less than R TM = 0.25% (0.5%), respectively.
Fig. 4. Reflectance maps as a function of the parameters rarc and r for (a) TE and (b) TM polarizations, where ωsc = 0.275 (2πc/a) and darc = 0.94 a. Reflectance maps as a function of the parameters darc and r for (c) TE and (d) TM polarizations, where ωsc = 0.275 (2πc/a) and rarc = 0.225 a. In (a)–(d), the dark gray solid circles indicate the reflectances at a specific position consisting of rsc = 0.34 a, ωsc = 0.275 (2πc/a), rarc = 0.225 a and darc = 0.94 a.
Fig. 5. Reflectance maps as a function of the parameters rarc and ω for (a) TE and (b) TM polarizations, where rsc = 0.34 a and darc = 0.94 a. Reflectance maps as a function of the parameters darc and ω for (c) TE and (d) TM polarizations, where rsc = 0.34 a and rarc = 0.225 a. In (a)–(d), the dark gray solid circles indicate the reflectances at a specific position consisting of rsc = 0.34 a, ωsc = 0.275 (2πc/a), rarc = 0.225 a and darc = 0.94 a.
From the twelve ERCs mentioned above, the device sensitivity to the various fabrication errors for 2D SC PhCs with the designed ARCs applied, can be summarized as follows: for the radii within ±0.01 a, the frequencies within ±0.005 (2πc/a), the rarc’s and the darc’s within ±0.01 a at a specific position consisting of rsc = 0.34 a, ωsc = 0.275 (2πc/a), rarc = 0.225 a and darc = 0.94 a, as indicated by the dark gray solid circles in Figs. 3–5, the reflectances for TE and TM polarizations are less than R TE = 0.5% and less than R TM = 0.25%, respectively. In other words, it can be evaluated that this common ARC structure for both TE and TM polarizations shows the high efficiency within the above-mentioned ranges (i.e., ∆rsc/rsc = 5.88%, ∆ωsc/ωsc = 3.64%, ∆rarc/rarc = 8.89% and ∆darc/darc = 2.13%), compared with the case without any ARC which shows the reflectances R TE = 12.48% and R TM = 0.25% at the abovesame position.
3.3. Transmission spectra
Fig. 6. Time-averaged power transmission spectra of three finite PhC samples consisting of 36, 38 and 40 unit cells (ucs) for (a) TE and (b) TM polarizations for the case without any ARC. Time-averaged power transmissions for (c) TE and (d) TM polarizations for the case with the designed ARCs applied at the input and output ends of the PhCs. Here, the condition of simulation parameters is as follows: rsc = 0.34 a, ωsc = 0.275 (2πc/a), rarc = 0.223 a and darc = 0.939 a.
To confirm a highly efficient ARC designed, we investigate the time-averaged power transmission spectra of three finite PhC samples consisting of 36, 38 and 40 unit cells for TE and TM polarizations for the two cases without and with the above-designed ARCs applied at both the input and output ends of the PhCs, as shown in Fig. 6. In the figure, one can observe that in 2D SC PhCs with the ARCs applied, the transmissions for TE and TM polarizations are significantly improved. For the case without ARCs for TE polarization, it can be clearly seen that strong oscillations of the transmitted power of light occur due to the interferences of multiple beams through the PhC interfaces and the free spectral range (FSR) becomes shortened as the size of PhC sample increases, as shown in Fig. 6(a); that is, the transmission spectra strongly oscillate between 50.84% and 99.17% in the frequency range from 0.264 to 0.286 (2πc/a) (see Table 1) and the FSR changes from approximately 0.0036 to 0.0032 (2πc/a) as the size of PhC varies from 36 to 40 unit cells [19, 22]. Whereas, in the case without ARCs for TM polarization, one can observe relatively high transmission spectra due to the very low TM mode reflectance (e.g., 0.25% for the SC frequency of 0.275 (2πc/a)), as shown in Fig. 6(b). By employing the above-designed ARCs, we can not only suppress considerable oscillations of the transmission spectra but also improve the transmissions for both TE and TM polarizations, regardless of the PhC size, as shown in Figs. 6(c) and 6(d); especially in the frequency range from 0.270 to 0.280 (2πc/a) (the light blue shaded area of ∆ωsc/ωsc = 3.64%), the transmission for TE (TM) polarization exceeds 98.10% (99.02%). Listed in Table 1 are the minimum, maximum and mean values of the transmission spectra shown in Fig. 6.
Table 1. Transmitted powers (%).
4. Conclusions
In summary, we have presented a high-efficiency ARC for both TE and TM polarizations in a 2D SC PhC of square lattice air holes in Si. The PhC operating in almost identical SC frequencies for two polarizations exhibits a large reflection coefficient for TE and a very small one for TM polarization. In this case, the ARC for TE can also improve the transmission for TM polarization. We have also investigated the device sensitivity to the various fabrication errors. Furthermore, we have also shown that the transmissions of three finite PhC samples consisting of 36, 38 and 40 unit cells can be significantly improved for the cases with the designed ARC applied. The proposed common ARC for both TE and TM polarizations in 2D SC PhC can bring a significant advantage for SC-based PhC devices and be important especially for designing high-efficiency SC-based PBSs.
Acknowledgments
This work was financially supported by the 2nd stage Brain Korea 21 (BK21) program of the Ministry of Education, Science and Technology (MEST) and in part by the IT R&D program of the Ministry of Knowledge Economy (MKE) and the Korea Evaluation Institute of Industrial Technology (KEIT) (2008-F-022-01).
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