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We propose a star-type polarizer with equal-power splitting function for each polarization based on polarization-dependent defects (PDDs) in two-dimensional photonic-crystal waveguides (PCWs). The structure is designed by combining two Y-type PCWs, and two types of PDDs are introduced into the PCWs respectively to provide polarization functions. By using finite-element method and optimizing the parameters of the PDDs, it is demonstrated that different polarizations can only transmit through their own PCWs and output with identical power distributions, i.e., the structure can function as polarizer and equal-power splitter for each polarization at the same time. In addition, by scanning the wavelength of the structure, it is proved that the proposed splitter can work in a wide range of wavelength while keeping high output transmission for both the TE and TM polarizations. Such a structure is useful for polarization-relative multi-channel signal processing for optical communications in the mid- and far-infrared wavelength regions.
© 2016 Optical Society of America
1. Introduction
Photonic crystals (PhCs) [1–4], also known as photonic band-gap (PBG) structures and gained worldwide interest during the past decades, are periodic structures belonging to a new type of artificial materials that allow people to manipulate the flow of light. Due to the unique characteristics of PhCs, such as PBG, localized modes, and surface states, many devices have been fabricated based on PhC structures [5–8].
In signal processing for optical communications, equal-power splitters [9–12] are important devices because identical power outputs are widely required. For example, in light interference devices, two equal-power light beams are required for obtaining optimum interference effect. However, to our best knowledge, most of the power splitters proposed can only operate in a single polarization (TE or TM polarization) [9–18]. For input signals of non-polarized lights (including simultaneous TE and TM polarizations), the power splitters operating only in a single polarization will be out of service. The common approach to solve this problem is to cascade polarizer and power splitters. However, this will increase the size of the system and the insertion loss in the system. Moreover, it may reduce the compactness and stability of the system. Consequently, it is worth to pay attention to design a structure combining the function of polarizer and equal-power splitter.
In this paper, a kind of star-type polarizer integrated with the function of equal power splitting has been proposed in two-dimensional (2D) square-lattice PhC waveguides (PCWs). The structure is designed by combining two Y-type PCWs, each of which contains one input port and two output ports, respectively. Without defects, the two Y-type PCWs can be channels for both TE and TM waves. When different defects are introduced, however, they can be regarded as TE-only and TM-only PCW. A TE (TM) wave can only transmit inside the TE (TM)-only PCW, and block by the TM (TE)-only PCW, i.e., the structure can function as a polarizer. In addition, the output powers for a single polarization (TE or TM) are identical, i.e., the structure also functions as an equal power splitter. Compared with the method by cascading a polarizer and a power splitter to deal with the non-polarized inputs, the structure proposed in this paper are more compact and easier to integrate with other devices. The Nelder-Mead optimization method (NOM) [19] is used to optimize the parameters and the finite-element method (FEM) [20–22] is used to demonstrate the performance of the structure.
2. Physical model
The schematic of the star-type structure is shown in Fig. 1. The structure is based on a 2D square-lattice PhC with the background rods of anisotropic Tellurium material. It is formed by combining two Y-type PCWs, each of which contains one input port and two output ports, respectively. Without defects, the two Y-type PCWs are channels for both TE and TM waves. When different defects are introduced, however, they can be regarded as TE-only and TM-only PCWs. Figure 1(a) shows the input and output ports for these PCWs, where the blue, red and dash lines represent the TE-only, TM-only, and common channels, respectively. The lattice constant and the radius of background rods are assumed to be a and rT, respectively. For the TE-only PCW, an array of 3 × 2 square defect rods (TE defects) is set in the middle of the input and two output channels, respectively. For the TM-only PCW, an array of 3 × 1 circular defect rods (TM defects) is introduced in the middle of the input and two output channels, respectively. Figure 1(b) shows the structure parameters of the defect region, where d and r represent the length of TE defects and the radius of TM defects, respectively, D1 represents the Y-offset of the TE defects in the input channel with port 1 along the Y-axis, and D2 represents the Y-offset of the TM defects in the input channel with port 4 against the Y-axis. For Y-offset of 0, it means that the positions of the defects are the same as the lattice points.
Fig. 1 (a) Schematic of the star-type structure, and (b) magnified view for box 1 and 2 shown in (a), where the structure parameters of the defect region are indicated.
In this paper, the background rods, TE defects, and TM defects were chosen to be appropriate for the anisotropic material Tellurium [23–27]. Anisotropic materials have proven to be more efficient than isotropic materials for obtaining absolute PBG [25,26], which is important for polarization-function devices. Moreover, tellurium has a very low loss in the mid- and far-infrared wavelength regime [27], which has important military or medical applications. In practical applications, material dispersions are unavoidable. Therefore, the dispersion of tellurium is considered as follows [27]:
where is the operating wavelength measured in . In this work, all the anisotropic materials are considered non-magnetic (). The extraordinary axis (e-axis) of the background rods and the TM defects is chosen to be parallel to the Z-axis. The e-axis of the TE defects in “Port 1” input channel is parallel to the X-axis and that in “Port 2 and 3” output channels is parallel to the Y-axis.We note that the electric field for TE polarization is parallel to the Z-axis, while for the TM polarization, the electric field is parallel to the X-Y plane. Specifically, the electric field for TM waves propagating in the x direction is parallel to the Y-axis. Since the extraordinary refractive index is higher than the ordinary refractive index, the above arrangement of the e-axes for the defects results in the TE (TM) wave strongly interacting with the TM (TE) defects and weakly interacting with the TE (TM) defects, so that the TE (TM) wave can only enter the TE (TM)-only PCW, and block by the TM (TE)-only PCW.
In order to investigate the performance of the proposed structure, the degree of polarization (DOP) and polarization extinction ratio (PER) are calculated as follows:
where and are the wave intensity in the TE and TM output channels, respectively.Absolute or complete PBG is crucial to build polarization devices that allow both of the polarization waves to transmit inside the structure. In the model studied in this paper, the largest absolute PBG in perfect 2D square-lattice PhC that has previously been obtained uses [28]. The range of the largest absolute bandgap obtained was . Here, the simulated bandgap map or band structure is omitted as it can be found in [28]. When the lattice constant a is selected to be , the range of operating wavelengths is from to , which is located in the mid infrared band. In this range of wavelength, losses due to the tellurium can be ignored [27].
3. Numerical results and discussions
Firstly, in order to check the performance of the TE and TM defects, we investigate the influence of side length d of the TE defects and radius r of the TM defects on DOPs and PERs, respectively, as shown in Fig. 2, where the red and blue lines represent the PERs and DOPs for the TE and TM PCWs, respectively, and the operating wavelength is 4.058a, i.e., the center wavelength of the absolute PBG.
Fig. 2 The PERs and DOPs versus d for the TE PCW (a) and versus r for the TM PCW (b), where the red and blue lines represent the PER and DOP, respectively.
From Fig. 2, we can see that for the TE PCW, the PER is >70 dB and the DOP is almost 1 for d in the range from 0.513a to 0.562a, with the maximum at d = 0.53a. For the TM PCW, the same PER and DOP are obtained for r in the range from 0.112a to 0.213a, and the maximum occurs at r = 0.156a. The PER and DOP for both the TE and TM PCWs can reach a high value. That is to say, with proper selection of d and r, TE waves can only transmit through the TE PCW, i.e., the TE defects is like a door that opens for TE waves but closes for TM waves. The situation is just reversed for the TM PCW. That is the reason why we call these two PCWs as TE-only and TM-only PCWs. In addition, the results also prove that the arrangement of e-axis for the defect rods is right for the desired polarization functions.
We can understand Fig. 2 as follows. Due to the Bragg scattering effect, the periodic arrangement of defect rods results in interferences of different scattered waves from these defect rods. When the forward scattering waves satisfy the condition of constructive interference, the overall transmitted waves will be prompted and the reflected waves will be suppressed. In the meantime, because the interference condition for different polarizations is different, so that the defect rods can permit a certain polarization but block another. In addition, the condition of this type of interference is highly sensitive to geometrical parameters. As a result, the sharp changes of the PER or DOP can be expected, similar to grating interferences which produce alternate stripes of light and dark.
To further understand the operating mechanism of the structure, field distributions are plotted in Fig. 3. Figure 3 shows that the TE and TM defects play their roles as desired. The TE (TM) wave can only transmit inside the TE (TM) PCW, and is blocked by the TM (TE) PCW. In addition, we also focus on the output powers. For the TE wave, the output powers are obtained to be P2 = 38.05%, P3 = 38.05%, and the total output (P2 + P3) is 76.1% (For simplicity, all output powers in this paper are represented relative to the input power); for the TM wave, the output powers are P5 = 36.9%, P6 = 36.9%, and the total output (P5 + P6) is 73.8%. The output powers are identical for both the TE and TM waves due to the symmetry of the structure, i.e., the structure also functions as an equal power splitter. However, the power transmissions or insertion losses are not good enough to meet the demands for practical applications. So, further optimization should be made to obtain low insertion loss.
Fig. 3 Field distributions of the structure for (a) TE input from port 1, (b) TM input from port 1, (c) TE input from port 4, and (d) TM input from port 4. Here, the side length d of TE defects and radius r of TM defects are 0.53a and 0.156a, respectively.
Since the position of the TE and TM defects in the input channels (the defects indicated in box 1 and 2 as shown in Fig. 1, where D1 and D2 represent the Y-offset of the TE and TM defects, respectively) may affect the output powers, we investigate the variations of total outputs by changing the Y-offset D1 and D2 of the TE and TM defects, as shown in Fig. 4. In these figures, both D1 and D2 are changed from −0.5a to 1.2a along the same X-axis.
Fig. 4 Total outputs versus D1 and D2 for the TE wave in the TE PCW (a) and for the TM wave in the TM PCW (b). The insets in (a) and (b) present the field distributions after optimization for TE and TM waves, respectively.
It can be easily seen from the Fig. 4 that, for TE waves, the total output varies obviously with the change of D2, but slightly with D1; while for TM waves, however, the total output varies obviously with D1, but slightly with D2. These behaviors can be understood as follows. For TE wave in the TE PCW, the E-field vector is parallel to the Z-axis, which is parallel to the e-axis of the TM defects in box 2, while the e-axis of the TE defects in box 1 is in X-axis direction, which is perpendicular to the E-field vector, so that TE wave will interact strongly with TM defects but weakly with TE defects. As a result, the position of the TM defects in the input channel will strongly influence the effective refractive index of the channel, resulting different wave impedance of the channel and therefore the influence of the Y-offset D2 on the output power is much stronger than that of the Y-offset D1 for TE wave in the TE PCW. Similarly, we can understand the behavior for TM wave in the TM PCW.
The above results show that D1 and D2 will simultaneously affect the total outputs for TE and TM waves. In addition, the side length d of TE defects and radius r of TM defects may also influence the total outputs. Hence, we should balance the performances of TE and TM waves and select proper d, r, D1 and D2 to maintain a high output transmission for both of them. Here, we use the NOM optimization method to optimize these parameters which are found to be dopt = 0.5323a, ropt = 0.1501a, D1opt = 0.4880a and D2opt = 0.9856a. The field distributions for TE and TM waves by using the optimized parameters are displayed by the insets in Figs. 4(a) and 4(b), respectively. After optimization, for TE wave, the output powers are P2 = 48%, P3 = 48%, and the total output (P2 + P3) is 96%; for TM wave, the output powers are P5 = 47.5%, P6 = 47.5%, and the total output (P5 + P6) is 95%. The performances have been greatly improved for both the TE and TM waves.
To characterize the operation bandwidth of the structure, a wavelength scan is performed on total outputs at the wavelength range of absolute PBG, as shown in Fig. 5, where the parameters are the same with those of Fig. 4. The stable bandwidth for TE wave is (4.0196a ~4.0776a) and (4.1618a ~4.2634a), at which the total output is over 90%; while for TM wave, the stable bandwidth is (4.0102a ~4.2472a). The operating bandwidth for both TE and TM waves is (4.0196a ~4.0776a) and (4.1618a ~4.2472a). After optimization, the structure can work in a wide range of wavelengths while keeping high output transmission for both TE and TM polarizations.
Fig. 5 Wavelength scan on total outputs at the wavelength range of absolute PBG, where the red and blue lines are calculated for the TE and TM waves, respectively.
It should be pointed out that although the scaling law [29] is no longer valid in PhCs made of dispersive materials, the same method for designing such structure can be applied to other wave bands.
At last, we would like to point out some points on the applications of the device in practice. First of all, it should be pointed out that the difference between the optical axis (or e-axis) of the TE defects and that of the background rods introduces some difficulties for fabricating. We suggest a method as follows. First, prepare the TE defect rods with the desired optical axis by cutting along the corresponding direction from a tellurium crystal. Second, drill four circular holes on a small piece of foam with their radii equal to that of the rods in the background PhC and the distance among them equals to the width of the waveguide. The foam generally has refractive index of approximately 1 and low loss to the operating wave. Third, drill the corresponding holes for the TE defect rods designed. Fourth, insert the defect rods into the holes in the foam. Finally, nest the foam into the defect region through the four circular holes and rods in the background PhC. We can repeat the steps to build the TE defects in other regions. Further discussion for practical fabrications can be found in literature, e.g., Refs [28,30].
Secondly, the proposed device can meet the standard of practical applications. First, it has the size of a few wavelengths, much smaller than conventional polarizers or power splitters. In addition, the device is built based on square lattice, which is compact in size and easier to integrate with other devices. Second, although the operating bandwidth is not continuous, it is still relatively wide. Third, it is known that tellurium has very low loss in the mid- and far-infrared wavelength region [31], and the absorption coefficient is as low as α ≈1 cm−1, which can well meet with the demand of practical applications. So, the device can be applied in optical signal processing for optical communications in the mid- and far-infrared wavelength regions.
4. Conclusion
In summary, we have proposed and demonstrated a star-type polarizer integrated with the function of equal-power splitting for each polarization in 2D square-lattice PCWs. The structure is designed by combining TE-only and TM-only PCWs which are formed by inserting polarization-dependent defects in PCWs. Different polarizations can only transmit through their own PCWs and furthermore each polarization wave is split into two identical parts. After optimization, the proposed structure can work in a wide range of wavelengths with low insertion loss for both TE and TM polarizations. Such structures are useful for polarization-relative multi-channel signal processing for optical communications in the mid- and far-infrared wavelength regions.
Funding
National Natural Science Foundation of China (NSFC) (61307048, 61275043, 11574216, 61505114, 61605128); Guangdong Province NSF (Key Project) (8251806001000004); Department of Education of Guangdong Province (2014KQNCX128); Specialized Research Fund for the Shenzhen Strategic Emerging Industries Development (JCYJ20120613115000529, JCYJ20140828163633988); Shenzhen University Foundation (827-000030).
Acknowledgments
The authors would like to thank Dr. Zixian Liang of Shenzhen University for fruitful discussions and facility support.
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