Self-imaging phenomena in periodic dielectric waveguides has been predicted and investigated based on multimode interference effect by using the plane wave expansion method and the finite-difference time-domain method. Asymmetric and symmetric interferences were discussed and respective imaging positions were calculated. As examples of application, a demultiplexer and a filter with ultracompact and simple structures were designed and demonstrated theoretically for optical communication wavelengths.
©2009 Optical Society of America
The concept of self-imaging was firstly proposed for planar optical waveguide applications in 1975 by Ulrich et al. . It is a phenomena that an input optical field being reproduced in single or multiple images at periodic intervals along its propagation direction . Based on the concept of self-imaging, wavelength demultiplexers [3, 4], power splitters , optical attenuator , optical switches [7, 8], etc. have been designed or fabricated in multimode planar/ridge waveguides. However, the devices based on the multimode planar/ridge waveguides are quite large (multimode region: 3670 μm × 18 μm , 3600 μm × 56 μm , 3600 μm × 48 μm ). With a rapid development of photonic crystals (PCs), the concept of self-imaging was studied in multimode PCs [9–11] and was used to design and/or fabricate multimode PC waveguide (PCW) devices [9, 10, 12, 13]. The devices in the multimode PCWs have much smaller size (multimode region: 17.7 μm × 2.3 μm , 45 μm × 4 μm ) than those of the multimode planar/ridge waveguides. But a wide PC background region (at least several lattice constants) is required for device applications. Moreover, the design of the PCW devices must depend upon lattice orientation of the PCs. Therefore, the size of the PCW-based multimode interference devices is relatively large and their flexibilities for design are limited by the PCs lattice orientation.
Recently, much attention has been paid on periodic dielectric waveguides (PDWs) [14–19]. This is attributed to that the optical field was confined in perfect periodic dielectric of the PDWs other than the defect of the PCWs. Therefore, the PDWs can be bent arbitrarily with a high transmission almost remain unchanged  within a certain operating frequency. The PDW-based devices will be much simple than the PCW-based devices. As a result, PDW-based beam splitter , Mach-Zehnder interferometer , wavelength and polarization splitters , and Fabry-Pérot microcavities  have been designed with single row input or output PDW. To verify whether the self-imaging used in the multimode planar/ridge waveguides and the multimode PCWs is applicable in the PDWs, in this work, self-imaging phenomena in the PDWs has been predicted and investigated by using a plane wave expansion method  and a finite-difference time-domain (FDTD) method .
2. Guided modes in PDWs
Figure 1(a) schematically shows a single row of periodic dielectric (PD) rods in air, which was so-called PDW. r is the radius of the PD rods and a is the center-to-center distance between two adjacent PD rods. The refractive index of the PD rods is n = 3.45 and the radius r = 0.46a. Its band structure for TM mode (E-polarization) was calculated by the plane wave expansion method and was depicted in Fig. 1(c). The inset denotes the supercell with a size of a×9a, which was used for calculation. The shaded region is for extended modes, which is not suitable for light guiding. The orange is light line. The solid curve below the light line is guided mode. It can be seen that, in the single row PDW, there is only one guided mode (single mode) at a frequency range of 0.132(a/λ) to 0.156(a/λ). Figure 1(b) shows four parallel PD rows in air [22, 23], the row-to-row space between two adjacent rows was set to be a. Fig. 1(d) shows its band structure for TM mode, which was calculated by using the supercell with a size of a×13a (inset). From Fig. 1(d), we can see that, there are three and four guided modes at frequencies of 0.132(a/λ) and 0.156(a/λ), respectively. Therefore, the four parallel PD rows are multimode PDW.
To further investigate self-imaging phenomena in the multimode PDWs, we changed the radius of the dielectric rods (n = 3.45) from 0.46a to 0.45a and did the calculations. Figure 2(a) shows the single row PDW with r = 0.45a and Fig. 2(c) shows the calculated band structure for TM mode. Figure 2(b) shows the multimode PDW with five parallel rows of PD rods (r = 0.45a) in air. The row-to-row space between two adjacent rows was set to be d = 1.5a. The calculated band structure was plotted in Fig. 2(d). For the multimode PDW formed by the row-to-row space of d = 1.5a, there are five guided modes (0th to 4th) at a frequency range of 0.119(a/λ) to 0.140(a/λ). The insets of Figs. 2(c) and 2(d) represent supercells with size of a×9a and a×16a, respectively, for calculations.
3. Analysis of self-imaging phenomena
For application, the multimode PDWs are usually divided into asymmetric and symmetric structures according to positions of the input/output waveguides. Figure 3(a) shows an asymmetric multimode PDW structure (model-I), which was consisted of a single row PDW (input waveguide) and a multimode region. The input waveguide consists of 13 PD rods and the multimode region consists of four parallel rows of PD rods, which was discussed in Fig. 1. Length and width of the multimode region are 65a and 3a, respectively. For this structure, when an input optical field Ψ(0, y) is introduced into the multimode region through the input waveguide, a mirrored image at x = Lm and a direct image at x = Ld will be reproduced, as shown in Fig. 3(b).
We know, in the PCWs structure, guided waves are confined in the defects of the PCs. But in the PDWs structure, guided waves are confined in the PD rods by total internal reflection. We assume that spatial spectrum of the input optical field is narrow enough without unguided modes excited, its total optical field Ψ(x,y) in the multimode region can be found in Ref.  and the conditions for the mirrored image at x = Lm and the direct image at x = Ld can be expressed as
respectively, where βn is the propagation constant. Therefore, the positions of the mirrored image and the direct image can be obtained by the Eqs. (1) and (2) if appropriate positive integers for each kn are decided.
To explore self-imaging phenomena in the multimode PDWs, numerical simulations with perfectly matched layer boundary condition were run by the two-dimensional (2D) FDTD method. In simulation, the configuration of Fig. 3(a) was transformed to the FDTD computational domain. Figure 4 shows the simulated steady-state electric field distributions and Poynting vector (x-component) distributions for continuous waves at frequencies of 0.132(a/λ) and 0.156(a/λ) in the model-I of Fig. 3(a). From the distributions, asymmetric multimode interference effect and self-imaging phenomena are clearly observed in the multimode region. Especially in the Poynting vector distribution [Fig. 4(b)], it can be seen that, for the operating frequency of 0.132(a/λ), two mirrored images were reproduced at positions A1 and A2, and two direct images were reproduced at positions B1 and B2. From Fig. 4(d), we can see that, for the operating frequency of 0.156(a/λ), there are four mirrored images at positions A1, A2, A3 and A4, and two direct images at positions B1 and B2. The mirrored image reproduced at the position A4 is the clearest one. By comparing Figs. 4(b) and 4(d), at the same position x = 50a along the propagation direction, the clearest direct image at B2 for the frequency of 0.132(a/λ) and the clearest mirrored image at A4 for the frequency of 0.156(a/λ) were reproduced, simultaneously. For potential application, if we choose x = 50a as the length of the multimode region, the structure can be used as a wavelength demultiplexer for the frequencies of 0.132(a/λ) and 0.156(a/λ).
The self-imaging is attributed to multimode interference, the imaging positions are depended on the properties of the guided modes. All the guided modes [Fig. 1(d)] in the multimode PDWs at frequencies of 0.132(a/λ) and 0.156(a/λ) were excited by the input optical fields, therefore, they all contributed to the self-imaging. In calculation, the values of the wave vectors for each modes at frequencies of 0.132(a/λ) and 0.156(a/λ) were taken out from the guided mode curves in Fig. 1(d). The values of the propagation constant βn were derived from the values of the wave vectors, accordingly. The average values of Ld and Lm were calculated by utilizing the self-imaging conditions derived from propagation analysis for the guided modes. In general, there are no exact solutions for Eqs. (1) and (2), so an approximate calculation was considered as follows: first, we found the nearest positive integers for each kn, then calculated the Ld and Lm for each n by substituting the values of kn and βn into the Eqs. (1) and (2), accordingly. Last, the average values of Ld and Lm were obtained. The parameters and the calculated results are summarized in Tables 1 and 2. It is easy to see that, for the model-I, a direct image for 0.132(a/λ) and a mirrored image for 0.156(a/λ) were reproduced at the same position of x = 50a along the propagation direction. Along y axis direction, the direct image is at y = 1.5a and the mirrored image is at y = -1.5a. This is the reason, if the length of the multimode region for the model-I is set to be 50a, a 1 -to-2 wavelength demultiplexer can be achieved.
For symmetric multimode PDW structure showed in Fig. 5(a) (model-II), which was formed by the models showed in Figs. 2(a) and 2(b), its input waveguide was placed at the middle of the left of the multimode region. The length of the multimode region is set to be 50a, and the width is 6a. From the band structure showed in Fig. 2(d), there are five modes (two odd modes and three even modes) at each of frequencies of 0.119(a/λ) and 0.140(a/λ). Therefore, for the model-II, when an input optical field Ψ(0,0) is introduced into the multimode region through the input waveguide, symmetric interference phenomena will be occurred. As a result, three kinds of images will be reproduced, i.e. single image (mirrored image or direct image), two-fold images and three-fold images. For simplicity, only imaging positions of single and two-fold images were depicted in Fig. 5(b) schematically. The single image (mirrored or direct image) was assumed to be reproduced at x = Ls and the two-fold images were reproduced at x = Lf.
To describe imaging positions of the single images and the two-fold images, in the following calculation, all the guided modes are considered. According to Ref. , the condition for the single image at x = Ls can be expressed as
Numerical simulations were performed by transforming the symmetric multimode PDW structure to the FDTD computational domain. The continuous waves at frequencies of 0.119(a/λ) and 0.140(a/λ) were launched into the input waveguide individually, the simulated steady-state electric field distributions and Poynting vector (x-component) distributions in the model-II are shown in Fig. 6. The symmetric interference effect and self-imaging phenomena are obviously observed in the multimode region. In the Poynting vector distribution [Fig. 6(b)], for the operating frequency of 0.119(a/λ), there are six two-fold images reproduced at the positions A1, A2, A3, A4, A5 and A6, four three-fold images reproduced at the positions B1, B2, B3 and B4, and two single images reproduced at the positions C1 and C2. From the Poynting vector distribution [Fig. 6(d)] we can see that, for the operating frequency of 0.140(a/λ), four two-fold images reproduced at the positions A1, A2, A3, and A4, three three-fold images reproduced at the positions B1, B2, and B3, and only one single image reproduced at the position C1. From Figs. 6(b) and 6(d), we further found that, the positions of all three-fold images are between the positions of the two-fold images.
Figures 6(a) and 6(b) further show that, for the operating frequency of 0.119(a/λ), the first single image (mirrored image) was reproduced at x = L s1 = 23a (C1) and the second single image (direct image) was reproduced at x = L s2 = 46a (C2). From the positions of the two-fold images A2 (x = L f1 = 11a) and A5 (x = L f2 = 34a), we got that the interval between the two-fold images is 23a. From Figs. 6(c) and 6(d), it can be seen that, for the operating frequency of 0.140(a/λ), a single image (mirrored image) was reproduced at x = L s1 = 35a (C1) and a twofold image was reproduced at x = L f1 = 17a (A2). We predict that more self-images can be observed if the length of the multimode region is sufficient long. As approximate descriptions, the following two formulas were used to express the inherent relation of L s1 and L fk ,
where L s1 is the imaging position of the first single image, and Lfk is the position of the twofold images.
The average value of L s1 can be calculated by the self-imaging conditions. In calculation, the values of the wave vectors for each modes at frequencies of 0.119(a/λ) and 0.140(a/λ) were taken out from the guided mode curves of Fig. 2(d). The values of the propagation constants βn were derived from the values of the wave vectors, accordingly. First, we tried to find the nearest positive integers for each kn, then calculated L s1 for each n by substituting the values of kn and βn into the Eq. (3), accordingly. Last, the average values of L s1 were obtained. Tables 3 and 4 list the parameters and calculated results of L s1 for the frequencies of 0.119(a/λ) and 0.140(a/λ), respectively. By substituting the average value L s1 = 22.9049a (Table 3) into the Eqs. (4) and (5) for the frequency of 0.119(a/λ), we calculated that L f1 = 11.4525a and L f2 = 34.3574a. L f1 = 17.6452a was also obtained by substituting the average value L s1 = 35.2904a (Table 4) into the Eq. (4) for the frequency of 0.140(a/λ). Simulated results agree well with the theoretical results of the imaging positions. Above analysis shows that a two-fold image for 0.119(a/λ) and a single image for 0.140(a/λ) were reproduced at the same position of x = 35a along the propagation direction in the model-II. Therefore, if the length of the multimode region for the model-II is set to be 35a, a filter can be achieved.
4. Applications and discussions
To illustrate the applications of this work, in this section, a demultiplexer and a filter were designed for wavelengths of 1310 nm and 1550 nm.
Based on the model-I in Fig. 3(a), a demultiplexer was formed by adding two symmetrical bent output PDWs to the right of the multimode region. The schematic diagram of the wavelength demultiplexer is shown in Fig. 7. It consists of an input PDW, a multimode region, and two bent output PDWs. The length of the multimode region was chosen to be L = 49a (close to 50a), which is to reproduce a direct image for 0.132(a/λ) and a mirrored image for 0.156(a/λ). To avoid coupling effect between the two outputs, a bending angle of 90° was designed. It should be emphasized that the center-to-center distance between the two adjacent dielectric rods is still a, the radii of the curvature for each bend is R = 0.5a/(sin5°) = 5.74a, which supports low bending loss . For wavelengths of λ1 = 1550 nm [0.132(a/λ)] and λ 2 = 1310 nm [0.156(a/λ)] applications, we choose a = 204 nm. So the calculated total length of the multimode region is about 10 μm and the width is about 0.6 μm.
Figures 8(a) and 8(b) show the simulated steady-state electric filed distributions for the wavelengths of 1550 nm and 1310 nm, respectively. Figure 9 is the normalized optical power spectrum as a function of wavelength from 1250 to 1600 nm. At λ 1 = 1550 nm, the normalized output powers in the outputs 1 and 2 are P 11 = 83.6% and P 21 = 1.6%, respectively. There is about 15% energy loss (flow into the air) in the propagation. At λ 2 = 1310 nm, the normalized output powers in the outputs 1 and 2 are P 12 = 1.9% and P 22 = 92.5% (5.6% energy loss), respectively. Calculated crosstalks are 10log(P 21/P 11) = -17.2 dB for 1550 nm and 10log(P 12/P 22) = -16.9 dB for 1310 nm. If we change the length of the multimode region a little from 49a to 50a, corresponding normalized optical power spectrums will be a little left shift (dashed lines). It can be seen that the peak power at the output 2 decreased a little while the normalized power at the output 1 decreased to below 80% at 1550 nm. That is the reason why we choose L = 49a as the length of the multimode region for the demultiplexer.
Figure 10 shows the schematic diagram of the wavelength filter. It was formed based on the model-II of Fig. 5(a) and consisted of a straight input PDW, a multimode region, and a straight output PDW. The input/output PDWs were connected with the middle row of the multimode region. From the analysis (Section 3), a two-fold image for 0.119(a/λ) and a single image for 0.140(a/λ) will be reproduced at x = 35a. To filter the wavelength of λ 1 = 1550 nm, the length of the filter was set to be 35a, a is specified as a = 184 nm. So the length of the multimode region is 6.4 μm and the width is 1.1 μm.
Figure 11 shows simulated steady-state electric filed distributions in the filter and Figure 12 is the calculated normalized output optical power. From Fig. 11, we can see that, when the wavelengths of 1550 nm and 1310 nm were launched into the filter simultaneously, the 1310 nm will be outputted while the 1550 nm will be filtered. From Fig. 12 we got to know that, the output powers are P 1 = 5.3% (most energy was flowed into the air and/or reflected back to the input waveguide) and P 2 = 94.0% (only few energy was flowed into the air) for 0.119(a/λ) (λ 1 = 1550 nm) and 0.140(a/λ) (λ 2 = 1310 nm), respectively. The calculated extinction ratio is 12.5 dB.
As a comparison, the multimode region of 10 μm × 0.6 μm of the wavelength demultiplexer based on PDWs in this work is much smaller than those reported in planar waveguide-based (3670 μm × 18 μm)  and slot waveguide-based (119.8 μm × 3.0 μm)  ones. Compared with the PCW-based wavelength demultiplexer, the multimode region of the PDW wavelength demultiplexer here is more compact than that of the PCW-based wavelength demultiplexer (17.7 μm × 2.3 μm) , while avoiding the use of wide background. Because in Ref. , about 20 lattice constants as wide PCs background region were used, which occupied much space in transverse dimension. In addition, the design of the PCW devices must be depended on lattice orientation of the PCs, so the design flexibility is limited. For the designed wavelength filter in PDWs, the multimode region is only 6.4 μm × 1.1 μm. This is because for the PDW-based device, a wide PCs background is not required. As a result, the space in the transverse dimension will be much smaller. In contrast, to confine the light wave, a wide PCs background is required in the PCW-based device, so the device size with a multimode region (38a × 4.3a) for the PCW filter is relatively large . For the PDW-based wavelength demultiplexer, crosstalks (-17.2 dB for 1550 nm and -16.9 dB for 1310 nm) are comparable with those of the reported slot waveguide-based demultiplexer (contrasts: 28.14 dB for 1.55 μm and 26.03 dB for 1.30 μm)  and the PCW-based one (estimated crosstalks: -16.2 dB for 1.5 μm and -13.4 dB for 1.3 μm) . For the PDW-based wavelength filter, extinction ratio (12.5 dB) is close to that of the PCW filter (estimated 12.9 dB) . It should be pointed out that the discussed structures and performed simulations/calculations in this work are done in 2D. The difference of the results in 2D and 3D can refer to the Ref. .
Wavelength demultiplexers and filters based on other principles were also achieved in PCs such as by introducing different defects in a T-junction , using high Q coupling cavities  and cascading multiple FP cavities . These devices were functioned by applying different propagation properties in PCs and, thus, they have relatively complex structures. Compared with those reported in PCs, the structures of the PDW-based wavelength demultiplexer and filter are much simple due to the operation principle of self-imaging. Moreover, the lengths and bending angles of the single row input/output PDW can be adjusted according to practical application requirements.
Self-imaging phenomena in periodic dielectric waveguides was predicted and investigated based on multimode interference effect by using the plane wave expansion method and the finite-difference time-domain method. The conditions of the self-imaging were discussed and the imaging positions were calculated. As examples of application, a demultiplexer and a filter with simple structures were designed with multimode regions of 10 μm × 0.6 μm and 6.4 μm × 1.1 μm, respectively. Compared with those based on planar waveguides or photonic crystal waveguides, the designed devices in periodic dielectric waveguides are ultracompact. Crosstalks of -17.2 dB (for 1550 nm) and -16.9 dB (for 1310 nm) for the periodic dielectric waveguide demultiplexer are acceptable. Extinction ratio for the periodic dielectric waveguide filter is 12.5 dB.
This work was supported by the National Natural Science Foundation of China (Grant Nos. 60625404 and 60577001).
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