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Photonic crystal (PhC) structures with polarized-wave-guiding property (PhC polarization waveguides) are proposed, demonstrated and applied to construct several new kinds of compact and efficient micro polarization devices in the mid and far infrared wave bands, including TE polarizers, TM polarizers, TE-downward T-shaped polarization-beam splitters (PBSs), TM-downward T-shaped PBSs and lying-T-shaped PBSs. Theoretical models for the operating mechanism of the structures are presented. The polarization devices built as applications of the PhC polarization waveguides are demonstrated by the finite-element method with the dispersion of materials being considered. Furthermore, optimized parameters are obtained by investigating the extinction ratio (EXR), the degree of polarization (DOP) and insertion loss. Moreover, structures based on PhC slabs derived from the 2D ones, together with woodpile PhC covers and substrates are suggested for the 3D version of the proposed devices for implementation. An example of the 3D-version structures shows a performance as good as that of the 2D structure. The devices proposed have relatively wide ranges of operating wavelength. Meanwhile, they are very compact in their structures and convenient for connection or coupling of signals among different optical elements, so they have the potential for wide applications in mid-and-far infrared optical devices or circuits, which are useful in remote sensing, image and vision, positioning and communications with infrared waves. Furthermore, the principle can be applied to build polarizers and PBSs in other wave bands.
© 2013 Optical Society of America
1. Introduction
Infrared devices are useful in remote sensing, image and vision, positioning and communications with infrared waves. Polarization devices are important in infrared optical devices and systems where polarization is essential, e.g., polarizers, polarization beam splitters (PBSs), polarization analyzers, phase-modulated signal detectors, optical interference devices, and optical logic circuits based on a polarization-logic system. With the development of optical information processing and optical communications, optical integrated devices become one of the focuses of scientists and engineers in micro optics, optical detection, remote sensing, infrared vision, space positioning, optical communication, optical computation, and photonics [1–8].
For instance, infrared polarized detectors are widely used for vision under dim environment, especially for military usage [9]. This is a benefit from the fact that by examining the polarized part of infrared radiations, the ability to observe objects hidden in background noise can be significantly improved [10–12]. Furthermore, infrared polarized detectors and analyzers are also widely used in medical devices [13]. However, up to now, most infrared detectors are made of lenses based on geometric-optics systems. The bulkiness and complexity make them heavy, large in size, expensive and low reliability.
After all, conventional polarization devices, e.g., polarizers and PBSs are bulky and difficult to be used for optical integrated circuits. Fortunately, the development of photonic crystals (PhCs) [14–21] provides a way for creating micro devices suitable for large-scale optical integrated circuits [22–26].
Up to now, several approaches for building polarization devices based on PhCs are reported. For example, one is by shooting a beam of light directly onto a chip of PhC, which has a forbidden band for transverse electric (TE) or transverse magnetic (TM) waves and a conduction band for TM or TE waves, so that the transmitted waves are TM or TE waves, while the reflected waves are TE or TM waves [27–29]. However, waves in this kind of devices are not confined in a waveguide, and thus it is not favorable for applications to optical integrated circuits, for which the transmitivity and reliability will be greatly influenced by the components for transition between the unconfined and confined waves. Furthermore, they have relatively large insertion loss and low degree of polarization (DOP). Another approach is by long-range coupling through PhC waveguides based on parity matching, so that different polarized waves are coupled into waveguides with different coupling length [30, 31]. Although the volume of this kind of polarization device is much smaller than that of conventional ones, it is still unsatisfactory because long coupling waveguides have to be used. In addition, there is another way to accomplish polarization-beam splitting by employing the difference between the effective indexes at the long wavelength limit for TE and TM waves in prism-like PhC structures [32]. Regrettably, this kind of splitter has a size much greater than the operating wavelength, i.e., it is not suitable for applications in micro optical integrated circuits.
We propose and demonstrate photonic-crystal structures with polarized-wave-guiding property, called as polarization waveguides (PWGs) in the following. In the proposed PWGs, one-line defect of circular anisotropic dielectric rods in periodic order or two-line defect of square ones, simply called as defects in the following, are set up inside conventional PhCs. In the proposed PWGs, we demonstrate that the one-line defect with circular rods opens a TM guiding mode, while the two-line defect with square rods opens a TE guiding mode.
As applications of the PWGs, several new kinds of compact and efficient micro polarization devices in the mid and far infrared wave bands, including TE polarizers, TM polarizers, TE-downward, TM-downward and lying-T-shaped PBSs are constructed and demonstrated by the finite-element method (FEM). As the waves are confined in waveguides and long-range interaction is not necessary, the polarization devices proposed are advantageous over that reported previously in the following aspects: (1) great extinction ratio (EXR), large DOP as well as high efficiency or low insertion loss; (2) much less volume and high density of integration; and (3) convenient for connection or coupling of signals among different elements in optical circuits. Although this paper concerns applications mainly in the mid and far infrared wave band, the principle can be applied to build polarizers and PBSs in other wave bands by varying the materials. However, the method of simple scaling of sizes and wavelengths is not applicable to designing devices using dispersive materials, so one needs to re-calculate the operating parameters in a similar process indicated in this paper.
This paper is organized as follows. In Sec. 2, physical models and basic theoretical considerations for the proposed PWGs and devices based on the PWGs are described; meanwhile, optimized operating wave bands for the structures are determined. In Sec. 3, the polarizers and PBSs are studied and demonstrated numerically. Performances of the devices are evaluated by investigating their EXRs, DOPs and insertion losses. In Sec. 4, a consideration of 3D deployment of the devices is given. Woodpile-PhC covers and substrates are proposed for confinement in the third dimension. Moreover, a 3D version of the TE-downward T-shaped PBS is presented and simulated as an example. Besides, major optimizations are shown for the 3D structure. Finally, in Sec. 5, a brief conclusion is given.
2. Physical models and basic theoretical considerations
The PWGs for TE and TM waves are shown in Figs. 1(a) and 1(b), respectively. In Figs. 1(a) and 1(b), two lines of circular rods in the PhCs are replaced respectively by two rows of square rods and one row of small-diameter cylinders in periodic order, i.e., the structures in Fig. 1 are PhCs with two kinds of line defects. We will demonstrate in the following that there would be only TE modes for the structure in Fig. 1(a) and only TM modes in Fig. 1(b) for proper operating parameters. The difference in the guiding modes can also be understood in the following way: regarding the line defect as a line array of periodic rods, we can know that such a line array would have photonic bandgaps in the direction along the line array, so that when there is a TE (TM) bandgap, TM (TE) waves can be guided along the line array. From the knowledge of 2D photonic crystals we can deduce that, for a line array of periodic rods with large (small) filling ratio, there are TM (TE) bandgaps but no TE (TM) bandgaps along the axis of the line array, which may be called as “photonic bandgap along a single axis”. As a result, the structures in Figs. 1(a) and 1(b) can function as a TE PWG and a TM PWG, respectively. In this paper, the width of waveguides is chosen to be triple the crystal constant to reduce the transmission loss.
As applications of the PWGs in Fig. 1, two polarizers and three T-shaped PBSs are proposed, as shown in Figs. 2(a)–2(e), respectively. In Figs. 1 and 2, both the circular and square rods are anisotropic materials, which are chosen in this paper to be Tellurium, a kind of positive uniaxial crystal whose optical axis is the same as the e-axis in the refractive index ellipsoid [33, 34], for obtaining wide absolute photonic bandgaps and defect modes with specific guiding properties. As can be seen, the polarizers and the PBSs are basically made of PWGs indicated in Fig. 1 and ordinary square lattice PhC waveguides (free waveguides).
Fig. 2 Basic structures for a TE polarizer (a), a TM polarizer (b), the TE-downward T-shaped PBS (c), the TM-downward T-shaped PBS (d), and the lying T-shaped PBS (e). All the inputs in (a) to (e) are set at the left-hand sides.
In Figs. 1 and 2, the optical axes of the background cylinders and the circular rods in the defect region are all along the axial direction of the cylinders, i.e., the z-axis indicated in Figs. 1 and 2, while the optical axis of the square rods is in the x-y plane and normal to the corresponding waveguide axis This is found also to be important for the structures in Fig. 1 to operate properly within the operating wave band because the effective index of the material, which influences the guiding wavelength, would be different for different angle between the optical axis and the electric vector.
For convenience in constructing optical circuits and coupling among circuits, square lattice is chosen. The radius of the background circular Tellurium rods is indicated as rB. The distance between two adjacent defect rods is the same as the lattice constant, indicated as a. The side width of each square rod in Figs. 1 and 2 is indicated as. The radius of the circular defect rods is indicated as rd. The width of all the waveguides in Figs. 1 and 2, defined as the distance between the two lines linking the centers of the rods on the waveguide borders, is 3a, which means that the waveguides are formed by removing two lines or rows of rods in the perfect PhCs.
In Figs. 1 and 2, optical waves with mixed polarizations are all incident from the left-hand-side ports. Under proper choice of operating values of s and rd, which will be elaborated later, TE waves, whose electric vector E ( = Ez) is normal to the wave propagating direction and parallel to the z-axis, flow to the right port in the TE polarizer as Fig. 2(a); while TM waves, whose magnetic vector H ( = Hz) is normal to the direction of wave propagation and parallel to the z-axis, transmit to the right port in the TM polarizer as Fig. 2(b); similarly, the TE-downward, TM-downward and lying T-shaped PBS in Figs. 2(c), 2(d) and 2(e) split the input of mixed polarizations into TE waves and TM waves as depicted with the output arrows in Fig. 2, respectively. All the devices have very low insertion losses which will be demonstrated in following sections.
As can be seen, the operating mechanisms and properties of the PWGs in Fig. 1 are the keys to understand and design the devices in Fig. 2. So, we first look at the mechanisms and properties of the PWGs in Fig. 1.
First, since both TE and TM waves are needed to be confined and propagate in the waveguides, the PhCs should at least exhibit one absolute photonic bandgap (PBG). For this requirement, the anisotropic material Tellurium is chosen for the PhCs in the mid and far infrared bands. In general, Tellurium has little loss to waves in the infrared range of 3.5~35, but is dispersive described by the following Sellmeier equations [35]:
where λ is the work wavelength measured in micrometers.In this paper, the material dispersion of Tellurium is considered all over by setting the ordinary refractive index no and extraordinary refractive index ne as variables of wavelength given by Eqs. (1) and (2) instead of using no = 4.8 and ne = 6.2 as constants in many studies [36–38].
We employ FEM with Floquet periodic boundary condition to calculate the band structures and scattering boundary conditions for field distribution and wave transmission properties. In the simulations throughout this paper, we set a = 1μm. It should be noted that the sizes and operating wavelength of the devices has no simple relation of scaling as that for non-dispersive devices in changing the operating wavelength. Rather, simulations have to be done again. However, the principle for the devices is the same for all operating wavelengths.
Through the FEM sweep configuration in the commercial software COMSOL, we can obtain the bandgap map as indicated in Fig. 3. From Fig. 3, the widest bandgap is found at rB = 0.3431a to be in the following normalized frequency range:
The corresponding wavelength range is:It has a bandgap width of 0.038, corresponding to a bandgap ratio of 15.71%, and a normalized bandgap-center frequency of 0.2410.Fig. 3 Bandgap map versus radius in the Tellurium PhC. The blue, red and yellow regions stand for the TE, TM and absolute bandgaps, respectively. (b) is an enlarged view of (a) near r/a = 0.3431, where the widest absolute bandgap exists.
Figure 4 shows the band map of the Tellurium PhC for the radius According to Figs. 3 and 4, we choose rB = 0.3431a for the rods of the background PhCs in the following.
Fig. 4 The band map of the Tellurium PhC at the irreducible Brillion zone for rB = 0.3431a. Convention for the colored lines and regions is: pale blue – TE bandgap, pale red – TM bandgap, yellow – absolute bandgap, blue line – TE modes, red line – TM modes.
For the devices indicated in Fig. 2, we first need to consider the free waveguide formed by removing two rods along the x direction in the PhC that no defects are loaded in the waveguides. Setting the whole free waveguide structure as a super cell and calculating the eigenmodes, we can obtain its band map. From Fig. 5 we can see that both TE and TM modes appear in the bandgap region, which means they can be guided in the free waveguide. We can choose the linear-dispersion region in Fig. 5 for both TE and TM waves [39–42] as the operating frequency band to be:
The center of this frequency band is:From Eqs. (5) and (6), we can write out the operating wavelength band and its center:Fig. 5 The guiding modes in the bandgap (a) and the linear dispersion region (b). Blue lines describe the TE band, red lines define the TM band, and the yellow region is the absolute bandgap.
Throughout in the following investigations, the operating wavelength is chosen as given by Eq. (8) for optimization consideration.
Now we study the properties of the PWGs indicated in Fig. 1. We set the structures as super cells as usual in dealing with defect PhCs, i.e., regard the overall structure in Fig. 1(a) or 1(b) as a unit cell of super PhCs, and calculate the band maps, as shown in Fig. 6. Figure 6(a) shows that there are only TE guiding modes at certain frequencies in the bandgap region for the structure with a defect of square rods. Furthermore, Fig. 6(b) shows that there is only one TM guiding mode in the bandgap region for the structure with a defect of circular rods. This demonstrates that the structures in Figs. 1(a) and 1(b) can be served as PWGs that only guide TE or TM modes.
Fig. 6 The guided modes in the TE PWGS with a defect of square rods (a) and the guided modes in the TM PWGS with a defect of small-radius circular rods (b). The blue and red lines in the yellow region are respectively the guided TE and TM modes.
To investigate the properties of the devices shown in Fig. 2, we first calculate the intensities of the transmitted TE and TM waves, ITE and ITM, which are proportional to the square of the electric field amplitudes. Pin and Pout are the power flows at the input and output. Then we calculate the EXR for the output waves as follows:
the DOP as follows: and the insertion loss as:Through the relations among the quantities defined by Eqs. (9)–(13) and the structure parameters of the devices, optimization of the devices indicated in Fig. 2 can be achieved.
3. Design and analysis of the novel devices proposed
3.1. TE and TM polarizers based on PWGs
In this section, we demonstrate the function of the TE and TM polarizers indicated in Figs. 2(a) and 2(b), respectively. The polarizers work as follows. In Figs. 2(a) and 2(b), a TE PWG and a TM PWG are inserted in the waveguides as defects, respectively. As a result, only TE waves can transmit to the output port in Fig. 2(a). Similarly, only TM waves can transmit to the output port in Fig. 2(b). These polarizers can be applied in optical integrated circuits.
For the parameters of rB and a given in Sec. 2., as well as no(ω) and ne(ω) described by Eqs. (1) and (2), we first set λ = λcenter = 4.050a and then seek the proper value as well as the optimum of s for the TE polarizer and that of rd for the TM polarizer through FEM simulations.
Figure 7 shows the EXR and the DOP versus s and rd. From Fig. 7 we can see that there exist regions with both high EXR and high DOP for:
And the optimum side width and radius of the defect rods can be found from Fig. 7 to be:Fig. 7 The EXR and the DOP versus s and rd for the TE-output wave in the TE polarizer (marked with four squares) (a) and for the TM-output wave in the TM polarizer (marked with three circulars) (b). The red and blue lines indicate the EXR and DOP, respectively.
The optimum values given by Eqs. (16)–(17) correspond to (DOP, EXR) = (0.9998, 39.87dB) for the TE polarizer and (DOP, EXR) = (0.9955, 60.21dB) for the TM polarizer.
To look further for more proper operating wavelength range than that given by Eq. (7), which is just judged from the position of the bandgap and that of the defect modes, we calculated the insertion loss for the polarizers, as shown in Fig. 8.
Fig. 8 Insertion loss for the TE polarizer for TE input (a) and TM polarizer for TM input (b). In (a), all the insertion losses are lower than 0.3dB within the wavelength range of 3.929a to 4.158a; while in (b), the insertion loss is less than 0.17 dB in the range of Eq. (7).
Supposing an insertion loss of 0.3dB is acceptable, which is much lower than that in practical devices, we can obtain the proper operating wavelength bands from Fig. 8 to be:
For a further demonstration, the optical field distributions of Ez and Hz components are simulated for λcenter = 4.050a, as illustrated in Fig. 9, in which waves of TE in (a) and (d) but TM in (b) and (c) are incident from left. As can be seen from Figs. 9(a) and 9(b), the TE wave propagates across the square-rod defect region with low insertion loss while the TM wave is almost completely blocked; meanwhile, the TM wave propagates across the circular-rod defect region with low insertion loss while the TE wave is almost completely stopped as shown in Figs. 9(c) and 9(d). The simulation results demonstrate that the corresponding structures function as perfect TE and TM polarizers.
Fig. 9 Field distributions of Ez (a) and Hz (b) for TE and TM inputs respectively in the TE polarizer, as well as Hz (c) and Ez (d) and for TM and TE inputs respectively in the TM polarizer.
3.2. The TE-downward, TM-downward and lying T-shaped PBSs based on PWGs
T-shaped structures are very useful for optical connects between circuits, so the T-shaped PBSs not only have the function of PBSs, but also the advantage for easy optical connections.
Generally, a PBS can be obtained by combining the TE and TM polarizers studied above together with a circulator placed between the input signal and the input port of the polarizers. In this way, however, the size of the PBS would be a few times larger than that of the polarizers. To minimize the structure, the T-shaped PBSs including the TE-downward, TM-downward and lying T-shaped PBSs based on PWGs as indicated in Figs. 2(c)–2(e), are generated by combining the TE and TM polarizers studied in Sec. 3.1. Therefore, parameters like no(ω), ne(ω), rB and a are taken to be the same as that given in Sec. 2 and Sec. 3.1. Furthermore, it is reasonable to choose (s, rd) = (sopt, rd,opt,).
The PBSs in Figs. 2(c)–2(e) may have different optimum operating parameters since elements put together influence each other. Similar to that in Sec. 3.1, we calculate first the EXR and DOP of the output waves as well as the insertion loss for the T-shaped PBSs. Then we go to find the optimum parameters. For the TE-downward, TM-downward and lying T-shaped PBSs, their EXRs and DOPs of the output waves for the operating parameters stated above are shown in Fig. 10.
Fig. 10 The EXRs (a), (c), (e) and the DOPs (b), (d), (f) versus the operating wavelength for the TE-downward, TM-downward and lying T-shaped PBSs, respectively.
From Figs. 10(a) and 10(b), it can be seen that the EXRs and DOPs for the TE wave at the TE-wave output port and TM wave at the TM-wave output port are all greater than 27dB and 0.996, respectively over the wave band given by Eq. (7). This is excellent comparing with the results for Self-collimating devices [28]. However, the insertion loss depicted in Fig. 11(a) shows that from 3.893a to 4a it has two maximum values for the TE wave: one is 1.5dB; the other is 13dB, of which both cannot be acceptable. Rather, the ranges of operating wavelengths that meet with the standard of insertion loss lower than 0.3dB are as follows:
Nevertheless, the TE-downward T-shaped PBS still has a wide range of continuous operating wavelength over 225nm, much wider than that in cavity devices or mode-selecting devices through waveguide coupling for the same values of EXR and DOP [27–31]. Moreover, once the wavelength is greater than 4.001a the insertion loss of the system is very low: less than 0.125dB, which means more than 97% of the power is transmitted from the input port to the output port. The high loss point in Fig. 11(a) outside the optimum operating range is due to constructive interference of reflections by the two rows of square rods at the specific frequency.Fig. 11 (a), (b) and (c) are the insertion losses versus the operating wavelengths for the TE-downward, TM-downward and lying T-shaped PBSs, respectively.
Similarly, in Figs. 10(c), 10(d) and 11(b), the EXR, DOP and insertion loss for the TM-downward T-shaped PBS are illustrated. The optimum operating wavelength ranges are:
which are different from that for the TE-downward T-shaped PBS. For the optimum operating wavelength band, the insertion losses are less than 0.3dB, the EXRs are greater than 19dB, and the DOPs are all greater than 0.975. This demonstrates that the TM-downward T-shaped structure functions as a perfect PBS. Comparing Eqs. (20) and (21), we find that the range given by Eq. (20) is a little wider than that by Eq. (21).In certain optical circuits, the lying-T-shaped PBS indicated in Fig. 2(e) is needed. Since the structure is different from the T-shaped PBSs shown in Figs. 2(c) and 2(d), it is necessary to study on its properties, especially for the optimization. Similarly, the EXR, DOP and insertion loss are obtained as shown in Figs. 10(e), 10(f) and 11(c), respectively. Thus the optimum operating wavelength range for the lying-T-shaped PBS is:
which is different from that for the former two T-shaped PBSs. For the optimum operating wave band, the insertion losses are lower than 0.3dB, the EXRs are greater than 24dB, and the DOPs are all greater than 0.992.Comparing the values of EXR, DOP and insertion loss among the three kinds of T-shaped PBSs, we can see that the TE-downward T-shaped PBS in Fig. 2(c) is much better than the other two T-shaped PBSs in Figs. 2(d) and 2(e). Nevertheless, all of the three T-shaped PBSs are of high quality.
For a further demonstration, the optical field distributions of Ez and Hz are shown in Fig. 12 for λ = 4.050a and other parameters stated above. Figures 12(a)-12(b), 12(c)-12(d) and 12(e)-12(f) in pair demonstrate the electromagnetic fields of the TE-downward T-shaped PBS, the TM-downward T-shaped PBS and the lying T-shaped PBS, respectively. The input is a TE wave for (a), (c) and (e), but a TM wave for (b), (d) and (f). As can be seen from Fig. 12, the TE and TM waves propagate respectively to their own output ports with very low insertion loss. Since the devices are made of linear material, if the inputs are polarization-mixed waves, their TE and TM polarization parts will go to their own output ports accordingly also, i.e., the T-shaped structures functions as perfect PBSs.
Fig. 12 The field distributions Ez for an incident of TE wave in (a), (c) and (e); Hz for that of TM wave in (b), (d) and (f). Sub-figure groups (a)-(b), (c)-(d), and (e)-(f) stand for the TE-downward, TM-downward and lying T-shaped PBS, respectively.
It should be pointed that the PBSs are not limited to just the forms shown above; other types can be designed based on the key components – the PWGs in Fig. 1 and polarizers in Figs. 2(a) and 2(b). Moreover, the polarizers and PBSs suggested can be conveniently integrated with other optical elements in optical circuits because their input and output ports are waveguides, unlike that in the polarizers and PBSs based on self-collimation effects in bulk PhCs [43, 44].
4. Consideration of 3D deployment of the devices proposed
In the above sections, only 2D structures are considered. However, in the real word, everything is three dimensional. To apply the proposed devices above, we have to limit the size in the z direction rather than set it to be infinitely long. This kind of limitation in size converts the 2D structures to be PhC-slab ones. Then we must consider how to confine the waves in z direction. For PhC slabs with air cylinders or air holes in dielectric materials, one can use the light cone for confining the light in z direction. There is one short point, however, for the PhC slabs with air cylinders: the TE band gap is usually small [19]. For this reason, we choose the structure with dielectric cylinders. But we have to face another problem: the confinement of waves in the z direction is generally poor for PhC slabs with dielectric cylinders. To solve this problem, we propose using substrates and covers of woodpile PhCs with a few periods for devices built on PhC-slabs made of dielectric cylinders, because the woodpile can be fabricated in a sequence of layers deposited and patterned by lithographic techniques developed for semiconductor industry, making them more applicable compared to other 3D PhCs [45, 46]. We find that covers or substrates with 1-2 periods, corresponding to 4-8 layers of logs, of the woodpile PhC are good enough for confinement of waves in the third direction. An example of the 3D deployment of the devices proposed is shown in Fig. 13.
Fig. 13 An example of the 3D deployment of devices proposed: the PhC-slab form of the TE-downward T-shaped PBS with a substrate and a cover of a woodpile PhC in one period. The left part (a) is the overview of the whole device. Part (b) is the device with the cover removed away for a clear inside view. Part (c) is the side-sectional view.
To make the structure stable, we can add a layer of dielectric material on top and bottom sides of the woodpile PhC. So, the substrates and covers can be made as follows: first grow a uniform layer, then grow the woodpile-PhC, and finally grow another uniform layer, all using the same material as that of the woodpile PhC. At last, we can put the cover, the PhC slab and the cover together. Also, foams with small dielectric constant and negligible absorption can be filled into the structure to keep it stronger. While the woodpile PhC has 3D bandgaps, one can rotate the substrate and cover in Fig. 13 to any angle in their surface plane so that the structure can experience further higher outside pressure.
For the woodpile PhC, any materials with high refractive index and low absorption to lights, e.g., PbTe, Si and Ge for the mid and far infrared bands, can be chosen.
As an example, we choose PbTe, which is a widely used infrared material with refractive index of 6 around the wavelength of 4μm, for the woodpile PhCs. We first calculate the bandgap map of the woodpile PhC, as shown in Fig. 14(a). From Fig. 14(a) we can see that the bandgap is very large for a wide range of parameters.
Fig. 14 The bandgap map of the woodpile PhC made of PbTe (a) and the band map of the woodpile PhC at W/a = 0.2618. Here, W is the width of the logs constituting the woodpile and a is the lattice constant of the woodpile PhC.
As required by Eq. (6), the operating frequency or the bandgap center should be fcenter = 0.2469. With this in mind and look at the bandgap map in Fig. 14(a), we can see that the bandgap with its center being fcenter = 0.2469 is relatively large: from ωa/2πc = λ/a = 0.2078 to 0.2859, which corresponds to a bandgap ratio of 31.79%. This bandgap corresponds to the woodpile structure parameter W/a = 0.2618. Here a and W are the lattice constant and the width of the logs in the woodpile, respectively. The whole band map of the woodpile PhC for W/a = 0.2618 is shown in Fig. 14(b). It is noted that Te and PbTe have the same lattice constant, so that the substrate, the PhC slab and the cover can be grown to form a firm monolithic structure.
We find that the height of the PhC slab, equal to that of the rods in the slab, is critical for achieving better operating performance of the structure. This is demonstrated by setting an input wave at the left-hand side of a PhC slab, which consists of 3x3 Tellurium rods with a cover and a substrate of one period of woodpile PhC, and calculating the related output power at the right-hand side. The calculated result is shown in Fig. 15. Figure 15(b) displays part of the curves in Fig. 15(a). Lower value of the related output power in Fig. 15 means higher ability of confining waves in the 3D structure. From Fig. 15, we can see that the optimum heights within the operating wavelength range given by Eq. (7) are 2.78a, 3.04a, or 3.3a. We noted that more values of the height for low transmitted power can be obtained through further simulations.
Fig. 15 The related output power transmitted over the 3D structure consisting of a PhC slab with 3x3 Tellurium rods as well as a cover and a substrate of one period of woodpile PhC for various heights (2a to 5.12a) of the PhC slab (a) and that for three optimum values (2.78a, 3.04a, 3.3a) of the height of the PhC slab (b).
With the considerations taken above, the 3D version of the devices proposed can be obtained with good operating performance. As an example, the simulated result of the 3D deployment of the TE-downward T-shaped PBS is presented in Fig. 16. From Fig. 16 we can see that the splitting function for the 3D structure is as good as that for the 2D one. The insertion loss is also low as expected in the 2D T-shaped PBS.
Fig. 16 The field distribution Ez (a) for an input of TE wave and Hz (b) for that of a TM wave in the 3D TE-downward T-shaped PBS.
At last, we should point out that the difference between the optical axis of the square rods and that of the background introduces some difficulties for fabricating. We suggest a method as follows. First, prepare the square defect rods with desired optical axis orientation. 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. Third, drill four square holes for inserting the defect square rods. Fourth, nest the foam into the defect region through the four circular holes and rods in the background PhC. Finally, insert the square defect rods into the four square holes in the foam. The foam should have refractive index of approximately 1 and low loss to the operating wave. Furthermore, studies on structures with specific defects that can make the fabrication process easier are welcome.
5. Conclusions
We have presented and investigated a TE PWG and a TM PWG with PhCs made of anisotropic materials. Based on the two PWGs, a kind of TE polarizer, a kind of TM polarizer and three kinds of T-shaped PBSs are further proposed and demonstrated through analysis of eigenmodes in the structures by FEM simulations. Key optimum operating parameters, as the side width, radius of the defect rods and the operating wave band, are obtained. Furthermore, consideration for 3D deployment of the polarizers and PBSs is presented. A 3D version of the TE-downward T-shaped PBS with woodpile-PhC cover and substrate is presented as an example for the 3D deployment. Simulation shows that the 3D-version structure has operating performance as good as that for the 2D structure. To mention a few, we have demonstrated that these devices have perfect operating properties as follows: (1) great EXR and high DOP, (2) much wider operating wavelength bandwidth than that of conventional micro polarizers or PBSs, (3) high efficiency or low insertion loss, (4) much less volume than that of conventional polarizers or PBSs and thus higher density of optical integration, and (5) convenient for connection or coupling of signals among different elements in optical integrated circuits. Consequently, they may have great potential for applications in large-scale optical integrated circuits in infrared, e.g., high-density optical-signal detector arrays, micro optical-logic circuits based on a polarization-logic system and optical information-processing chips which can purify the signals from background noise. They are also useful in remote sensing, image and vision, positioning and communications with infrared waves. Furthermore, the principle can be applied to build polarizers and PBSs in other wave bands.
Acknowledgments
This work was supported by the NSFC (Grant No.: 61275043, 61171006, 61107049, 60877034), the Guangdong Province NSF (Key project, Grant No.: 8251806001000004) and the Shenzhen Science Bureau (Grant No. 200805, CXB201105050064A).
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