Structural parameters in the four-channel filter
During the past two decades, there have been great interests in developing ways to manipulate photons at nanoscale, realizing optical integrations, developing smaller, faster, and more efficient optoelectronic devices for the purpose of next-generation optoelectronic technology. Great progresses have been made in exploring photonic crystals (PCs) [1,2], plasmonic structures [3,4], and other nanophotonic devices for applications. However, plasmonic structures always involve some metal cells and are subject to strong energy dissipation and absorption loss in optical frequencies. Since silicon has a large refraction index and low loss in the infrared wavelength, it becomes an important optical material that has been widely used for integrated photonics applications. Meanwhile, silicon dominates microelectronics and this makes the silicon-based optical devices have the advantage to integrate with electronic devices.
Among all the semiconductor-based optical devices, a class of integrated optical devices that are built in the platform of periodically patterned silicon structures (namely, silicon PCs) are now attracting much attention [5–7]. Analogous to real crystal, electromagnetic (EM) wave is strongly modulated in PC by means of periodic Bragg scattering. Photonic band gaps (PBGs), which can prevent light from propagation in certain direction for a certain range of wavelengths, are formed similarly to electron band gaps. If we introduce a line defect or a point defect in PC, a defect state will take place within the PBG, where light is strongly localized around the defect. These defects can serve as a high efficient waveguide channel or as a micro-cavity with a high-quality (high-Q) factor. The mismatch of the PBGs spatial inversion symmetry breaking and could lead to the optical isolation in any device where the forward and backward transmissivity of light is very much different.In addition, the transmission bands also provide remarkable dispersion properties due to strong Bragg scattering, and negative refraction, self-collimation, superprism and many other anomalous transport behaviors [8–13] can be achieved by engineering the unit cell geometry of PCs. In this review, we briefly introduce the theoretical background of the light propagation in PC and show our recent results on design, fabrication, and characterization of several basic integrated optical devices in the platform of infrared silicon PC slab.
2. Theoretical and numerical tools
In this section, we briefly introduce the theoretical background and the numerical methods for our study of PC. The propagation of electromagnetic waves in PC is governed by the Maxwell equations . For the sake of simplicity, we only consider a nonmagnetic linear system. In particular, the dielectric constant is independent of frequency and we neglect any absorption of electromagnetic waves by the material. Furthermore, there are no free charges or currents in our system. With all of the assumptions, the magnetic field within the PC satisfies the following equation that directly originates from the Maxwell equations :
Since our system involves a periodic dielectric function , we can apply Bloch-Floquet theorem to our situation, which means that the solutions can be expressed as:
where is the Bloch wave vector and is a periodic function of position. This type of solutions are periodic as a function of .
Several theoretical methods have been developed to handle different problems for PC structures, such as the plane-wave expansion method (PWEM) [15–18], transfer matrix method (TMM) , finite-difference time-domain (FDTD) method [20,21], and multiple scattering method [22,23]. Each method exhibits its own benefits and drawbacks. The PWEM is the earliest method applied to PC , and has shown its great power in the discovery of three-dimensional (3D) diamond-lattice PCs that have a complete PBG . However, this method has a severe limitation in that it can only deal with the photonic band structures. The TMM is an efficient approach that was designed particularly to calculate the transmission spectra of PC, but it can also be used to solve the photonic band structures. This approach works based on the finite-difference scheme in the real space . Later on, a plane-wave based transfer-matrix method (PWTMM) was developed by Li et al. [24–26]. This method works on the plane-wave space and uses plane wave functions (representing Bragg waves) to describe both the EM fields and dielectric functions. This approach can handle a broad range of general PC problems. In addition to the regular solutions of photonic band structures and transmission/reflection/absorption spectra, this approach can efficiently solve the Bloch wave scattering at the interface between semi-infinite PC structures, because it can also work in the Bloch mode space . So far, other methods have not developed such a peculiar capability. For this reason, it can calculate efficiently modal coupling with multimode PC waveguides , transmission efficiency through general two-dimensional (2D) and 3D waveguide bends , and band diagrams and field profiles of PC surface states . On the other hand, this method has adopted advanced numerical and mathematical analysis tools to enhance numerical convergence and accuracy, and it has shown its superior power in dealing with some metal PC structures compared with other methods . Recently, this method was also extended to solve nonlinear optical problems in ferroelectric PC structures [32,33]. The FDTD method is a very popular and universal approach in numerical simulations of various PC problems. In addition to the regular band structures and optical spectra calculations, this technique can govern the EM field evolution with time in arbitrary PC structures with infinite or finite structural domain. The reason is that the technique works in the time domain and directly solves the Maxwell equations.
Many free software packages and commercial software packages have been developed worldwide and they are widely used in numerical simulations and solutions of different PC problems. Our group has also developed homemade codes based on several methods including the PWEM, PWTMM, FDTD, and multiple scattering method. In addition, we also utilize publicly available free software packages as they are more numerically economic or have better numerical efficiency. In our case, we use MIT Photonic-Bands (MPB) package  to compute the photonic band structures and use MEEP, a free FDTD simulation software package developed at MIT  to calculate transmission spectra and model electromagnetic wave transport features in the 2D PC structures and devices.
3. Sample fabrication and optical characterization
After the discovery of PC, many novel devices have been proposed to control light and implement specific functionality of information processing. 2D air-bridged silicon PC slab [Fig. 1(a)] is an excellent platform to fabricate PC integrated optical devices. This system involves a silicon membrane suspended in air, which confines light by high index contrast in the vertical direction, while the periodic structures in the slab give a strong in-plane confinement of light through PBGs. In most cases, a typical PC structure is a kind of periodic array of air holes etched in a silicon-on-insulator (SOI) wafer by microfabrication techniques. The SOI wafer has a Si/SiO2/Si structure. In our case, it has a 220 nm thick silicon top layer and a 3 μm buried silica layer on top of a 0.5 mm thick single crystal silicon wafer. We directly use focused ion beam (FIB) lithography to drill air holes in the silicon membrane or use electron-beam lithography (EBL) to define PC patterns in a thin film of polymethylmethacrylate (PMMA), and then transfer the patterns into the silicon membrane by inductively coupled plasma (ICP) etching under the atmosphere of SF6 and C4F8 gases. Figures 1(b) and (c) are the top-view scanning electron microscope (SEM) picture of our PC structures fabricated by FIB. By utilizing state-of-the-art microfabrication techniques, the optical properties of the periodic array of air holes can be easily and accurately controlled. For instance, one can change the diameters of certain holes or omit to etch one or several holes at certain places. These procedures can allow for engineering of the linear and point defect characteristics. After the air holes PC structures get done, we use HF acid wet etching to remove the buried oxide layer under the silicon membrane PC structures to form air-bridged structures. Usually, wide silicon wire waveguides (also with the air-bridged geometry) close to the interface of PC structures are used as the input and output infrared light beam channels, as shown in Fig. 1. These wire waveguides are further connected with long adiabatically tapering ridge silicon waveguides (each about 0.2 mm long) to allow easy coupling with external infrared signals from single mode optical fibers. As a result, a typical PC sample has a total length of about 0.5 mm and the input and output ends are carefully polished to enhance the coupling efficiency of input and output infrared signals.
The transmission spectra of a PC structure effectively reflect its optical properties. To get this important physical quantity, we have set up an experimental apparatus that involves several functional components. The overall measurement setup is schematically illustrated in Fig. 2(a), while a picture of the corresponding real system is displayed in Fig. 2(b). As shown in Fig. 2(a), the PC samples are placed in the center of the stage, with its two sides connected with the input and output optical fibers. The input optical signal comes from a continuous wave tunable semiconductor laser with the wavelength ranging from 1500 to 1640 nm, launched into one facet of the ridge waveguide via a single-mode lensed fiber. Power meter is used to detect the optical signals transmitted through the PC structures and emitted from the output side. The measurement is made with TE polarization (electric field parallel to the slab plane) since it has a complete band gap in silicon PC slabs. The measurement data are normalized by a ridge waveguide on silicon with the same length and width to yield the final transmission spectra for a specific PC structure.
In addition to the measurement of transmission spectra, our experimental setup can offer another big power: it allows for easy and convenient direct monitoring of the transport path of infrared light through PC devices. As depicted in Fig. 2(a), a charge-coupled device (CCD) camera is mounted above the sample and it can in situ monitors the transport property by imaging the roughness induced scattering infrared light from the surface of the PC structures. The long-focus microscope objective connected with the CCD camera is shown in the upper part of Fig. 2(b). The ray trace can be directly visualized by the camera to yield images at a personal computer monitor, and this gives the researcher a rough but direct estimate abouthow much the infrared signal has gone into the PC structures. The idea is simple: if the infrared light is coupled into the PC sample with a sufficiently high efficiency, infrared light can transport along the input ridge waveguide, PC devices, and the output ridge waveguide. Significant scattering of infrared signal off the sample can take place and is collected by the CCD camera and visualized in situ by the monitor. The strongest scattering occurs at the discontinuity interface, including the end facets of input and output ridge waveguides and the connection section between ridge waveguides and PC structures. Even within the PC structures, remarkable scattering still takes place because of the inevitable roughness on the surface of silicon slab and within the inner walls of air holes. On the other hand, if the infrared signal coupled into the PC samples is weak, then the overall scattering light that can be collected by the CCD camera is also very weak, and no picture can be visualized in the monitor. Figure 2(c) shows a typical microscopy picture recorded by the CCD camera. Such an experimental setup is very convenient to adjust the precise position of the input and output optical fibers, so that they have a precise point-to-point alignment with about 250 nm thick PC samples to allow for high-efficiency optical coupling. The infrared signal transporting through the PC samples usually has a power level of micro-walt when the input signal from semiconductor laser is on the power level of milli-walt. The signal has been already sufficiently strong to allow for transmission spectrum measurement and CCD camera monitoring with high signal-to-noise ratio.
4. PC band–gap devices: waveguides and cavities
4.1. PC waveguides
PC waveguides are one of the most important elements in PC integrated optics, because they offer efficient channels for light propagating at wavelength scale and connect different devices in integrated optical circuits. Usually, PC waveguides are formed by removing one row of holes in a PC structure. The line defects can generate defect states within the complete PBG and serve as waveguide channels for light to propagate efficiently and freely in PC structures. Among many 2D PC slab structures, the triangular lattice of air holes has a relatively large band gap for TE-like electromagnetic modes, where the magnetic field points in the perpendicular direction while the electric field is dominantly within the lateral plane of the slab. In addition, the structures allow for easy fabrication by standard planar nanofabrication technologies such as FIB and EBL, and have good mechanical stability. For these reasons, they are widely and dominantly adopted in designing and exploring PC based integrated optical devices [35,36].
Figure 3(a) shows the calculated TE-like mode photonic band structures of a particular 2D triangular lattice PC slab, where a wide complete PBG is clearly seen. In the region, light propagation inside the PC is prohibited. When removing one or several rows of air holes in the PC structures, some allowed modes (defect states) appear within the PBG [Fig. 3(b)], and they can be used to create waveguides or cavities. In most works, single-mode or multi-mode optical waveguides are usually made along the Γ−Κ direction in the triangular lattice PC. It has been well established in plenty of literatures that the number of waveguide modes as well as the width of the transmission windows can be controlled by tuning the core width of the line defects. However, waveguides along other directions in the triangular lattice PC were rarely discussed. Just like the Γ−Κ direction, waveguides along the Γ−Μ direction should also be able to guide confined modes due to the existence of a complete PBG in the 2D triangular lattice PC. By removing a line of diamond areas, we can obtain a cluster-like waveguide along the Γ−Μ direction as depicted in Fig. 4(a). This kind of waveguide is called Γ−Μ waveguide .
Figure 4(b) is the SEM picture of the original Γ−Μ waveguide, where the air holes remain to locate at the original lattice site and the radius of all air holes remains the same. The air holes are directly drilled by FIB and the lattice constant is 430 nm. According to our simulation and experiment results, we find that the width of the propagation modes for the original Γ−Μ waveguide [Fig. 4(b)] is only 22 nm. Then we optimize the geometry to improve its transmission characteristics. We shrink the radius of the air holes in the two nearest-neighboring rows around the waveguide and enlarge the radius of the air holes in the two second-nearest-neighboring rows, as shown in Fig. 4(c). The key point is to generate a transport pathway with walls as smooth as possible. According to our simulations, the parameters corresponding to an optimized waveguide are that = 50 nm and = 170 nm, while the radius of the original air holes is = 120 nm. Figures 5(a) and (b) are the calculated dispersion relations of the original and optimized Γ−M waveguide, respectively. It’s shown that the optimized waveguides have a high pass band that is much broader than the original waveguide. We can also obtain the same conclusion from the measured transmission spectra in Figs. 5(c) and (d). Besides, the intensities of the transmission spectra are much higher than the original one. As the Γ−Μ waveguide is perpendicular to the usual Γ−Κ waveguide, it offers an alternative to construct a waveguide interconnection beyond the usual scheme of Γ−Κ with Γ−Κ waveguides. A high-performance wide-band Γ−Μ waveguide should be of great help to build integrated-optical devices, such as interconnection networks, channel-drop filters, and wave division multiplexers, with more flexible geometrical configurations in 2D PC slabs.
Based on the design of an optimized Γ−M waveguide, we combine the Γ−K waveguide and Γ−M waveguide together to form a 90° waveguide bend as schematically depicted in Fig. 6(a) . This is the first design of a 90° waveguide bend in the 2D triangular-lattice PC. The whole waveguide bend system is composed of two Γ−K waveguides as the input and output ports and the center Γ−M waveguide as the connection component. In other words, the structure involves two consecutive segments of 90° waveguide bends. In constructing the waveguide bend, we have used the optimized Γ−M waveguide discussed in the above and depicted in Fig. 4(c). The guided modes in the optimized Γ−M waveguide have better phase-matching and mode profile matching with the normal Γ−K waveguides. In the same time, we modify the bend corner geometry by fabricating smaller air holes in the corner to make the guided modes between the two kinds of waveguides matching better. Based on a serial of simulation and experiment tests, we find the best values for and . Figure 6(b) shows the SEM picture of the waveguide bend with = 50 nm and = 150 nm. After optimization, we get 70 nm pass band width while the transmission efficiency of a single bend is 45%. The proposed 90° waveguide bends can help to construct integrated optical circuits with more flexible and diversified infrastructures.
4.2. Coupled–cavity waveguide
Moreover, we have designed an air-bridged silicon PC coupled-cavity waveguides (PCCCWs)  and mapped its near-field optical distributions at different wavelengths around 1550 nm with the scanning near-field optical microscopy (SNOM) technology. For PCCCWs, the eigenmodes usually have relatively narrow bandwidth with slow group velocity in the whole band range. Previously, slow light propagation in such specific PCCCWs had still not been experimentally studied via SNOM technique. Figures 7(b)−7(f) show the calculated optical field distribution profiles at different wavelengths with a simulation model schematized in Fig. 7(a). We fabricated the PCCCW in the SOI wafers with FIB system. Figure 8(a) displays the SEM image of the element composed of the central PCCCW (encircled by a red square), two identical W1 PC waveguides, and the input/output ridge waveguides. Figure 8(b)−8(f) displays the near-field optical intensity distribution patterns of the PCCCW at different wavelengths. The scanning area is 12 μm × 15 μm with the incident light propagating upwards from the bottom of the image. Straight yellow lines in Figs. 8(b)−8(f) are used to label the position for showing the cross-sectional profiles of the field distribution patterns.
The optical intensity distribution patterns are different at 1550 and 1610 nm, even though both of them mainly appear as a single narrow line along the central PCCCW region with a full width at half maximum (FWHM) of about 350 nm. Precisely speaking, the pattern demonstrates a little bit shoulder as a result of mode superposition at 1550 nm, since it comprises two eigenmodes. At 1610 nm, the calculated result consists well with the experimental one in Fig. 8(f), which presents a single line along the whole waveguide. The pattern appears bright and wide with obvious interference nodes in the PCCCW section. The simulated field distribution profiles in the W1 PC waveguide sections agree well with the detected ones at all these wavelengths, which show a snake-like/single-line profile in the input/output W1 PC waveguide except that of a snake-like profile in the output W1 PC waveguide at 1560 nm. In addition, the simulated field distribution patterns of the snake-like profile in the PCCCW section appear deviating greatly from the detected ones at 1550, 1571, and 1590 nm. However, if we calculate the optical field distribution patterns at 1550, 1560, and 1571 nm with the even-to-odd amplitude ratios of 1:4, 1:1, 1:4, and 1:6, respectively, we can find the simulated results are consistent with the experimental patterns evolving from single-line, to snake-like, and then to double-line structures for the PCCCW section. Combination of the near-field optical detection and theoretical simulation shows that SNOM is an efficient tool to study the optical propagation in the PCCCW and can help to design slow light elements.
4.3. High–Q cavity
Quantum information processing and quantum state manipulation have received great attentions because of their potential revolutionary impact on future network communication. Optical cavities, which can be used to store information, are considered to be one of the most important devices in the quantum communication application. The generation and teleportation of qubits require sufficiently high value of Q/V, where V stands for the mode volume of the optical cavities. As a result, high-Q optical cavities show great potential application in quantum information. Among all the optical cavities, 2D PC slab cavities are the best choice because of their simultaneous high-Q and small mode volume characteristics. It has been reported that Rabi splitting can be observed experimentally when the quantum dots are introduced into the PC cavities with high Q/V [40,41]. Moreover, due to the development of nanofabrication technique, multiple high-Q PC slab cavities can be fabricated at the same time on a single slab by the EBL and ICP etching technique. Once atoms or quantum dots are embedded into the high-Q PC cavities, various quantum phenomena can be demonstrated on chip. Recently, our works on high-Q silicon PC microcavities have achieved great progress after extensive exploration and delicate improvement of nanofabrication techniques and sample processing techniques have been made .
We focus on studying the L3 PC microcavities formed by removing three cylindrical air holes in the Γ−K direction in a triangular lattice [Fig. 9(a)]. The lattice constant is 430 nm, the radius of cylindrical air hole is 120 nm and the thickness of silicon slab is 235 nm. The FDTD calculation results indicate that the L3 PC microcavity possesses a quality factor of about 5300 [Fig. 9(c)]. After trying hundreds of simulations, we find that the positions of air holes at the edges of the microcavities affect the Q factor dramatically. The electric field pattern of the cavity mode can be tuned to be Gaussian-type by displacing the six air holes outwardly at the edges of the microcavities, and this can increase the quality factor significantly [41,42]. The optimal displacement is found to be 73, 10 and 73 nm for the first, second and third air holes at both edges of the microcavities, which is depicted in Fig. 9(b). The maximum quality factor of 127,323 [Fig. 9(d)] can be achieved, which is 20 times larger compared with the unadjusted one.
Based on the optimal parameters, we successfully fabricate the designed high-Q planar L3 PC microcavities in SOI wafer by implementing EBL and ICP . As can be seen in Fig. 10(a), the L3 microcavity is side-coupled to a W1 waveguide with the barrier of three rows of air holes. The samples are measured by our home-made fiber coupling system as described in the above section. When the incident wavelength is off-resonant, light cannot couple with the microcavity, leading to strong output. While, at resonance most energy is tunneled into the microcavity, resulting in weak output. For the case of high-Q microcavity, a sharp transmission dip is expected in the transmission spectrum. The lattice constant, radius of cylindrical air hole and the thickness of silicon slab are 430, 120 and 235 nm, respectively. Limited by the fabrication accuracy of 10 nm, the displacement is adjusted to be 80, 20 and 80 nm for the six air holes at both edges of the microcavities. Figure 10(a) shows the enlarged view of the cavity region. A sharp and narrow transmission dip is observed at the 1567.35 nm in the measured transmission spectrum [Fig. 10(c)]. For the purpose of extracting the quality factor accurately, we finely tune the wavelength between 1565 and 1570 nm. The measured spectrum is illustrated in Fig. 10(d) and the Q-factor as large as 71,243 is obtained. Nevertheless, there are some deviation between the simulation and experiment. For example, the resonant wavelength is 16.75 nm red-shifted from the simulated result and the maximum quality factor is significantly less than the calculated value of 127,323 [Fig. 10(b)]. We believe that the deviation is caused by the imperfection of the cylindrical air holes and the actual radius is not exactly the same as the value in simulation. The success of fabricating high-Q silicon PC slab microcavities enables us to investigate various interesting quantum phenomena, such as strong coupling between light and quantum system, quantum information processing technique, single photon source, all-solid quantum manipulation and high-quality biochemistry sensing devices.
4.4. Channel drop filters
Channel drop filters are key components for extraction of light trapped in a point-defect cavity to a neighboring waveguide and they sit on the basis of wave-division multiplexers and demultiplexers. They have great applications in a wide variety of fields, such as photonic integrated circuits, telecommunications, and quantum informatics. Based on the simulation and experiment experiences about PC waveguides, we design and fabricate an ultra compact three ports filter in 2D air-bridged silicon PC slab by closing the bus waveguide for 100% reflection feedback. Figure 11(a) shows the SEM picture of the three ports filter structure . This filter was fabricated by EBL and ICP techniques. The lattice constant of the PC and the radius of the air hole are 430 and 145 nm, respectively. Port 3 is the input waveguide channel, while ports 1 and 2 are two output waveguide channels, respectively. They are formed by missing one row of air holes along the Γ−K direction of the triangular-lattice PC, the so called W1 waveguide. C1 and C2 are two point-defect cavities. The distance between the center of the defect cavity and the neighboring waveguide is 3 rows of holes in the y direction. As seen in Fig. 11(a), the C1 cavity consists of three missing air holes, and the two air holes at the cavity edges are shifted outward by 10 nm apart from the regular positions. Similarly, those of the C2 cavity are shifted by 20 nm. The slight shift of air holes is conducive to confine light inside the cavity and leads to a higher quality factor. Meanwhile, the different shifts of the two cavities make the resonant wavelengths slightly different. The experiment results [Figs. 11(b) and (c)] show that the resonant wavelengths of C1 and C2 are 1529.5 and 1531 nm, respectively. The wavelength spacing of the two cavities is about 1.5 nm and might be further reduced by continuously changing the size of the cavity. The full widths at half maximum of the peaks are 1.5 and 1.4 nm and the corresponding quality factors are about 1020 and 1090, respectively. To estimate the drop efficiency, a reference straight waveguide of the same parameters is positioned near the three-port filter. By keeping the same intensity of input light, the transmission intensities of the reference waveguide and port 1 are 0.330 and 0.158 μw, respectively, when the input wavelength is set at 1529.5 nm. The drop efficiency of port 1 is roughly estimated to be 48% and a similar result has been obtained at port 2.
It has been well known that structure is the kernel of filter design. Usually, the regulation of microcavity resonant frequency is obtained by changing the size of the cavities. We have proposed a new way to design multi-channel filters by changing the shape of the air holes . When the shape of the air holes changes from circle to ellipse, two parameters, the ellipticity and the orientation angle of the ellipse, in addition to its size can be further explored and they can have a great influence on localized cavity modes. Therefore, we can use this for some special purpose.
Figure 12(a) schematically shows a one-channel PC filter. A horizontal channel (W1 waveguide) serves as the input signal channel, which is created by removing a single line of air holes along the Γ−K direction. A cavity is formed by removing three air holes along the Γ−K direction that is rotated 60° from the W1 waveguide. It is located four rows away from the major channel and is connected with the major channel through an indirect side coupling. Another single-mode waveguide is formed parallel to the cavity and serves as the output signal channel. Figure 12(b) shows an enlarged picture of the filter in the region around the cavity. One of its axes is oriented counterclockwise by an angle to the x axis, namely, the input light propagation direction. The sizes of the axes parallel and perpendicular to this orientation are a and b, respectively. Now we have great structural freedom to tune the optical properties of the new PC filter by changing the parameters of , a and b. To show this point, we design and fabricate a four-channel PC filter by using different cavity parameters as described in Table 1. The SEM picture of the fabricated four-channel filter is displayed in Fig. 13(a). Four cavities are located on the two sides of the central linear W1 waveguide. They are engineered by leaving several air holes unetched in the Γ−K orientation. The input signal propagates upwards from the bottom input ridge waveguide. Each cavity is coupled with another W1 waveguide that is connected to a ridge waveguide, which serves as the output signal channel.
The simulation and experimental results of the transmission spectra for the four channels are displayed in Figs. 13(b) and (c). Although significant noise exists, a resonant peak can be clearly found for each channel. The peaks are located at 1549, 1541, 1567, and 1560 nm for channels 1, 2, 3, and 4, respectively. The results confirm that the air-hole shape has a great influence on the functionality of the PC filter devices. The elliptical air holes can induce a fine tuning of the resonant wavelength by changing the ellipticity of the elliptical air holes.
As described in the preceding section, a high-performance wide-band Γ−Μ waveguide can be formed by modifying the radii of the air holes along the pathway. The combination of Γ−Μ waveguides and Γ−K waveguides can offer a more flexible way to interconnect and couple between different devices. We have proposed a type of PC filter using these two kinds of waveguides . The Γ−Μ waveguide and Γ−K waveguide are used as the input and output signal channels respectively, and they are connected via cavity resonance. Figure 14(a) shows the SEM picture of a four-channel filter structure. We change the size of the cavities by moving the end points of the cavity [marked with “a, b” and “c, d” as shown in the inset of Fig. 14(b)] to change the resonant frequency. Table 2 gives the detailed parameters of the four cavities. The experiment demonstrates that the four resonant peaks are at the wavelengths of 1543, 1545, 1548 and 1551 nm, as shown in Fig. 14(b). In spite of the slight shift in the resonant peak toward higher frequency, which we believe is induced by the uncertainties in the fabrication, the experimental results are in fairly good agreement with the simulation results, where the maximum relative deviation of resonant wavelength is within 2 nm. These results clearly demonstrate that the designed Γ−M waveguides can act together with the usual Γ−K waveguide to construct high-performance multichannel filers with more structural flexibility. In our experiment, we also use the CCD camera to directly monitor the transport of infrared signal within the channel-drop filter. The situation of on-resonance and off-resonance can be clearly visualized and distinguished from the CCD camera images. One typical case is shown in Fig. 14(c).
5. PC band–engineering devices for anomalous transport control
In previous sections, we discuss several PC devices, including waveguides, cavities, and channel-drop filters that are built on the silicon 2D PC platform. These devices work on defects that are brought into PBG and they can be considered as PBG materials. As we have mentioned, PC structures possess another important feature: photonic pass bands. In this section we show several example devices that implement the dispersion and refraction properties of PCs at their transmission bands.
Let’s first make an overview of how band dispersion engineering works. Figure 15(a) shows the typical photonic bands structure of an air-bridge PC slab structure composed of a square-lattice array of air holes etched in silicon slab. The areas labeled in Fig. 15(a) show the unconventional light propagation, self-collimation and negative refraction, in PC. One effective way to understand and exploit desirable light propagation properties in PC is using the equifrequency surface (EFS) contours, as shown in Figs. 15(b) and 15(c). Figure 15(b) shows the EFS contours of the first TE-like band. The EFS contours in the red line frame are flat, meaning that this is the self-collimation region. The reason is that the group velocity, which is parallel to the gradient of the EFS, is pointing in the same direction for all the modes located within the region. As a result, if light propagates along the Γ−Μ direction, it does not suffer any diffraction in the PC. and are high-symmetric points in the first Brillouin zone for square lattice. This kind of PC structures can be used as the channelless waveguide in integrated optic devices. Figure 15(c) shows EFS contours of the second band. The EFS contours are roughly circular around the direction at the reduced frequency () range 0.28−0.31, as indicated in the red solid line frame. When the frequency increases, the EFS contours move toward the point, which indicates the existence of negative refraction in the region. If the incident light is mainly parallel to the direction, the PC will behave like an isotropic medium with a negative index of refraction in that particular frequency range.
5.1. Negative refraction
Based on the above analysis, we designed and fabricated an air-bridged PC structure that exhibited negative refraction of infrared light . The structure is schematically shown in Fig. 16(a). These structures are directly drilled by FIB technique. The input infrared signal channel is a silicon wire waveguide, which is inclined with respect to the surface normal by 10°. The lattice constant a of the square array is 460 nm and the diameter of the air hole is 220 nm. We first use 3D FDTD method to simulate the electromagnetic field intensity distribution at wavelength 1503 nm. The result is displayed in Fig. 16(b). We find strong reflection and scattering at the interface between the input waveguide and the PC structure. This is induced by the serious impedance mismatch at the interface, although the high index contrast air-bridged structure can achieve good optical confinement. To surpass this obstacle, we use a tapered air-holes connection layer at the input surface of PC structure to reduce the reflection and scattering losses. As shown in Fig. 16(b), a large fraction of light power from the input waveguide is coupled into the PC structure and negative refraction of light beam within the PC structure is clearly seen. Besides, the reflection or scattering of light at the input interface of the PC is very much reduced. This clearly indicates that the designed tapered interface can reduce the interface impedance mismatch remarkably. The calculated value of negative refraction angle is −45°.
In our measurement, TE-polarized light from a tunable semiconductor laser (1500−1640 nm) was first launched into a tapered single mode fiber, and then coupled to the silicon wire waveguide, and finally incident on the PC structure. The ordinary way to see the light propagation behavior is to directly observe the pattern of the radiated light from the top of the sample using a conventional microscopy objective and an infrared CCD camera. The result is shown in Fig. 16(c). The light spot at the middle bottom part of the pattern is the radiated light from the input silicon wire waveguide. The big light spot at the center represents the scattered light at the interface between the input wire waveguide and the PC due to impedance mismatch. There is also a small bright spot at the top right corner of the pattern, and it is recognized to result from the radiated light when the negative refraction beam hits the end facet of the PC structure. Because the TE-like modes are strongly confined guided mode on the silicon slab and the surface fields are nonradiative and evanescent with respect to the vertical direction of the PC slab, the far-field pattern observed and recorded by the ordinary optical microscopy is not able to reveal the detailed process about how the negative refraction beam propagates inside the PC structure unless the scattering of light by roughness and irregularity is sufficiently strong on the beam propagation path. This is indeed the case for Fig. 16(c). In fact, the small bright spot could not appear without the intentional introduction of the air slot at the far end of the PC structure. It would not be possible to tell which way the infrared beam would refract if without the aid of this scattering light spot.
In order to observe clearly and tell unambiguously the ray trace of the negative refraction beam in the PC structure, we used the SNOM technology (SNOM-100 Nanonics, Israel). A probe scans in the vicinity of the surface of the PC structure and records the near-field intensity distribution. The tip has a resolution of about 100 nm, i.e., 1/15 of the wavelength. The signal is recorded by an infrared single-photon detector, which allows us to capture very weak infrared signals. The probed near field information directly reflects light propagation properties of the TE-like modes for the PC and enables one to visualize the ray trace of the negative refraction light beam because the near field at the surface is an integral part of the modal profile of the confined guided modes that exponentially decay away from the surface of the slab. In the SNOM picture [Fig. 16(d)], a bright spot also appears at the front interface of the PC structure, but it is much smaller than the one in Fig. 16(c). The ray trace of the incident light beam along the silicon wire waveguide and its propagation along the negative refraction direction inside the PC structure can be clearly seen. The negative refraction angle is about −45°, which is in good agreement with the FDTD simulation presented in Fig. 16(b). The SNOM detection unambiguously discloses the negative refraction property of the designed PC.
On the other hand, ordinary positive refraction only occurs for TM-like confined modes, so the designed PC structure can behave as an efficient beam splitter in an integrated optical circuit. The high-resolution SNOM technology can greatly help one to directly visualize the ray trace and acquire deeper understanding on various anomalous wave propagation behaviors, such as super-prism, superlensing, self-collimation, and slow light in deliberately designed 2D PC slab structures in the optical wavelengths. This in turn can help to explore a wider regime of controlling light behaviors on the nanoscale for future basic science and high technology applications.
5.2. Self–collimation effect
Self-collimation effect is the propagation of light without diffraction along the propagation direction. This phenomenon has been used to construct non-channel waveguides, beam splitters and beam combiners [47,48]. The behaviors of these devices are determined by the performance of the self-collimation effect. Recently we have designed and realized a simple structure composed by a square lattice array of elliptical air-holes where broadband large-angle self-collimation effect is observed for TE-like guided modes in infrared wavelength .
Figure 17(a) shows our PC structure formed by a square lattice of elliptical holes. The calculated TE mode photonic band diagram of the fourth, fifth and sixth bands are shown in Fig. 17(b). The self-collimation effect can be observed at the gray regions within a broad normalized frequency range and . For simplicity, we only consider the EFS contours of the fifth TE band [Fig. 17(c)]. The contours are flat at the normalized frequency between 0.43 and 0.46 for all the values of and not just in the vicinity of . This feature indicates that our structure can support self-collimation for incident light beams with large incident angles. Then the FDTD simulation by using the MEEP package is performed to verify our prediction. A structure with the size of 30 × 45 is considered. A Gaussian beam with a width of 4a propagates into the surface at 0°, 20° and 60° incident angles (Fig. 18). For simplicity, we only consider the minimum (0.36) and the maximum normalized frequency (0.43). Figure 18 shows the electric field intensity distribution in the plane with 0°, 20° and 60° incident angles at normalized frequency 0.36 [Fig. 18(a)] and 0.46 [Fig. 18(b)]. From the simulation results, we find that the light beam is collimated along the propagation direction for each situation. However, the couple efficiency of the incident light becomes lower and lower with the increase of the incident angle. We do not show the field distributions for those incident angles that are larger than 60°. These six situations in Fig. 18 are sufficient to show the broadband large-angle characteristic of the self-collimation effect.
Following our simulation results, we fabricate our PC structures in SOI substrate by EBL and ICP etching process. Figure 19 shows the SEM pictures of the designed PC structures with 0°, 20° or 60° incident waveguides. Ray trace of light beam is observed using IR camera and a high numerical aperture (NA = 0.50) objective. Detailed images of the field intensity of the scattered light are recorded for 0°, 20° and 60° incident angles for different incident wavelengths. Here we only show the patterns of the minimum and maximum wavelengths for each incident angle. They demonstrate strong light confinement along the propagation direction for all the situations. The experimental results are in good agreement with FDTD simulations. We believe that this kind of structure may have potential applications in beam combiners and multiplexers.
6. On–chip wavelength–scale optical diode and isolator
Optical isolation is a long pursued object with fundamental difficulty in integrated photonics. The need to overcome this difficulty is becoming increasingly urgent with the emergence of silicon nano-photonics, which promises to create on-chip large-scale integrated optical systems. Motivated by the one-way effect, considerable effort has been dedicated to the study of unidirectional nonreciprocal transmission of electromagnetic waves, showing important promise in optical communications. Until now, on-chip integration of optical diode still stays in theory, particularly in silicon. These “optical diodes” include fluorescent dyes with a concentration gradient, absorbing multilayer systems, and second harmonic generators with a spatially varying wave vector mismatch. An electro-tunable optical isolator based on liquid-crystal heterojunctions, showing nonreciprocal transmission of circularly polarized light in photonic bandgap regions, has been reported. In another configuration using liquid crystals, linearly polarized light is used. In addition to many attempts on magneto-optical materials, optical isolators have also been fabricated using nonlinear optical processes and electro-absorption modulators.
An efficient routine to create optical diode is via time-reversal symmetry breaking or spatial inversion symmetry breaking , which could lead to the optical isolation in any device where the forward and backward transmissivity of light is very much different. We have reported a method for making unidirectional on-chip optical diodes based on the directional bandgap difference of two 2D square-lattice photonic crystals comprising a heterojunction structure and the break of the spatial inversion symmetry. Simulations confirm the existence of a clear isolation effect in the designed heterojunction structure. We fabricate these on-chip optical diodes in silicon and the near-infrared experiment results show high-performance optical isolation, in good agreement with the theoretical prediction . This device may play the same basic role in photonic circuits as the electrical diode does in electronic circuits. It could further pave the way to achieve on-chip optical logical devices without nonlinearity or magnetism and bloom the photonic network integration.
Figure 20(a) shows the schematic configuration of the original diode structure under study, which consists of two PC slab domains (PC1 and PC2) with the same lattice constant a but different air hole radii (r1 and r2, respectively) comprising a heterojunction structure. These two PC regions stand at a silicon slab [grey area in Fig. 20(a)]. Each PC region has a square-lattice pattern of air holes [white holes in Fig. 20(a)], with the hetero-interface between PC1 and PC2 along the Γ-M direction. We set the two hole radii as a fixed ratio to the lattice constant a, which are r1=0.24a and r2=0.36a in order to simplify our discussion. These two composite PCs would comprise a pure PC region if r1=r2. The light source is placed symmetrically aside the structure with two 4a-wide ridge waveguides connecting the surface of the two PC regions. The whole area is surrounded by a perfectly matched layer.
We simulated the transmission spectra for a TE-like light signal transporting along the forward (from left to right) and backward (from right to left) direction. The refractive index of the dielectric slab was set to 3.4, corresponding to that of silicon at 1,550 nm. The slab thickness was h=0.5a. Figure 20(b) shows the calculated forward (black line) and backward (red line) transmission spectra. The frequency is normalized by a/λ. It is clearly seen that there exists an isolation band ranging from 0.2649 to 0.2958 (a/λ), where the forward transmission forms a peak with a transmissivity of about 6% while the backward transmissivity is down between 0.5% and 1%. The forward peak is located at 0.2793 (a/λ), just in the middle of the isolation band. We define S=(TF-TB)/(TF+TB) as the signal contrast of the diode, where TF and TB denote the forward and backward transmissivity, respectively. The maximum S of this original diode equals 0.846 at the peak. Besides, there exists another isolation region from 0.2196 to 0.2649 (a/λ), where the backward transmissivity is higher than the forward transmissivity. This structure thus shows an extraordinary phenomenon of unidirectional transport property.
We calculated the band diagram of the TE-like modes of these two PC slabs using the 3D-FDTD method. Figures 21(a) and 21(b) show the calculation results. The first band (even mode) in bulk PC2 [Fig. 2(b)] is directional as the top mode frequency in the Γ-Χ direction (x-axis) is 0.2345 (a/λ) but that in the Γ-Μ direction (45°-direction) is 0.3087 (a/λ). Inside the region between 0.2345 (a/λ) and 0.3087 (a/λ), the all-directional transparent region of PC1 needs to be above 0.2633 (a/λ) [Fig. 21(a)] in order to match the bottom mode frequency in the Γ-Χ direction of the second band (odd mode). These two modes in PCs are the basic working mode of the diode structure. Here the even and odd modes are defined with respect to the off-slab mirror-reflection symmetry σz of the field component Ey. In the region between 0.2633 (a/λ) and 0.3087 (a/λ) PC1 is transparent in all directions, while PC2 is transparent along the Γ-Μ direction but opaque along the Γ-Χ direction. Compared with Fig. 20(b), the unidirectional transport region [0.2649 to 0.2958 (a/λ)] just coincides with the overlapped region between the directional bandgap of PC2 and the all-directional pass band of PC1. This simple picture indicates that the current unidirectional transport effect involves two ingredients: (I) directional bandgap of PC2 and (II) all-directional pass band of PC1. Noting that the structure does not obey the spatial inversion symmetry alone the propagating direction, the principle of optical isolation can thus be summarized as follows:
Forward. When light goes across PC1 as the odd mode and reaches the hetero-junction along the Γ-Χ direction, it cannot stay in the Γ-Χ direction in PC2 further because of the Γ-Χ directional gap. But the hetero-junction is along the Γ-Μ direction, so light turns to the hetero-junction and diffracts as the even mode at any Γ-Μ direction into PC2, which passes through PC2 and eventually outputs.
Backward. When light goes directly into PC2 as the even mode, it turns to the two Γ-Μ direction paths which cannot convert to the odd mode of PC1 in the Γ-Χ direction and eventually leak out so that it does not output.
Based on the numerical analysis of the optical diode, we have fabricated the original diode structure as well as a revised diode structure (with better performance) in silicon, whose SEM pictures are displayed in Fig. 22(a) and (c), respectively. Figures 22(b) and 22(d) show the theoretical and experimental results of the transmission spectra of the two diodes in the forward and backward directions. In Fig. 22(b) the theoretical forward peak of the original diode structure is at 1,575 nm [0.2793 (a/λ)] and the maximum transmissivity is 6%. The experimental forward peak is at 1,556 nm and the maximum transmissivity is 7%. In Fig. 22(d) the experimental forward peak is at 1,534 nm and the maximum transmissivity is 10% for the revised diode, whereas the theoretical forward peak is at 1,552 nm [0.2834 (a/λ)] with a transmissivity of 13%. The measured signal contrast S equals 0.718 (the original structure) and 0.831 (the revised structure) at the peak frequency. Both experimental peaks in Figs. 22(b) and 22(d) have a nearly 20 nm shift and 50 nm broadening against the theoretical simulations, which is probably due to the imperfections in fabrication. The experiment confirms the existence of the unidirectional transport effect in agreement with the theoretical prediction. Due to the arbitrariness of the lattice constant a, we can freely adjust the working frequency to anywhere as desired. This could be more convenient for the design of realistic photonic devices.
The principle for optical diode as analyzed in the above is robust as it is based on a simple directional bandgap mismatch effect of photonic crystal heterojunction. Yet, it should be noticed that in Figs. 22(b) and 22(d) the backward transmissions are fluctuating within 1% to 2% and both are higher than the simulation values, as a result, the signal contrast S degrades from 0.846 for the original structure and 0.92 for the revised structure in theory to 0.718 for the original structure and 0.831 for the revised structure in experiment. The performance improvement of the diode relies on how to maximize the peak of the forward transmissivity and minimize the backward transmissivity in experiment. Several means can help improve the forward transmissivity. First, one can change the air hole size of PC1 and PC2 and enlarge the directional bandgap. Calculations show that the forward peak transmissivity of the revised structure with r1 =0.30a and r2 =0.45a grows up dramatically to 29.4% while maintaining the same low level of backward transmissivity. Second, one can change the relative size of the input and output waveguides. Calculations show that by changing the input waveguide width to 2a and keeping the output waveguide width 4a, the forward peak signal increases up to 20.8% in transmissivity.
To reduce the backward transmissivity, one can either enlarge the directional bandgap of PC2 to attenuate the backward signal more strongly, or eliminate the return of leak-out light from the outside of slab or the structure boundary by introducing the absorbing metal dots near the structure, or enlarge the heterojunction structure appropriately so that the leak-out light cannot enter the output waveguide. Following the above general ideas, we further optimize the optical diode structures as illustrated in Fig. 23. The structure has parameters of r1=100 nm and r2=160 nm, and the input and output waveguide width are 2a and 6a, respectively. In experiment, we have got an optical diode with an maximum of 32.8% of the forward peak transmissivity and 0.885 of the signal contrast S at 1,557 nm.
Our PC heterojunction diode has advantages of high signal contrast, wavelength-scale small sizes, and being all-dielectric, linear, and passive. Furthermore, it has a much smaller scale than those based on diffraction gratings and thus greatly facilitates large-scale integration. The high performance on-chip optical diode realized in silicon without nonlinearity or magnetism will stimulate the exploration of other more complex on-chip optical logical devices with ultra-high stability, integration and much less power consumption. Such an optical diode may play the same basic role in photonic circuits as the electrical diodes do in electronic circuits, which have significantly revolutionized fundamental science and advanced technology in various aspects of our routine life due to their capability of rectification of current flux. Furthermore, its large-scale fabrication could be readily achieved by the well-developed CMOS techniques. The realization of high-performance on-chip optical diodes may open up a road toward photonic computers.
Strictly speaking, the existence of unidirectional transport effect of light does not mean automatic achievement of optical isolation. Recently there have appeared hot controversies upon whether isolation of light can be realized via linear and passive photonic structures. Several schemes to realize unidirectional transport of light through linear and passive photonic structures have been proposed, which are essentially based on the principle of spatial-inversion symmetry breaking. Feng et al. reports a passive silicon optical diode based on one-way guided mode conversion . However, whether or not nonreciprocal transport of light can happen in the structure has raised hot controversies [53,54]. Fan et al. made a scattering matrix analysis for relevant forward and backward modes of the structure and argued that the structure is essentially reciprocal and cannot enable optical isolation because it possesses a symmetric coupling scattering matrix. In their response, Feng et al. acknowledge that their structure, as a one-way mode converter with asymmetric mode conversion, is Lorentz reciprocal, which states that the relationship between an oscillating current and the resulting electric field is unchanged if one interchanges the points where the current is placed and where the field is measured, and on its own cannot be used as the basis of an optical isolator. The controversies have thus raised a fundamental question: Can one construct an optical isolator by using a linear and time-independent optical system? The answer to this question by the authors of Ref. [53,54] obviously is no.
But our theoretical and experimental study on the optical isolation performance of our PC heterojunction diode leads to a totally different answer to the above question, namely, the spatial inversion symmetry breaking diode can construct an optical isolator in no conflict with any reciprocal principle . To see whether there is a good isolation effect of the silicon diode, we implement a direct method that is originated from the conventional magneto-optical isolator that has been popularly used in laser devices [Fig. 24(a)]. One places a total reflection mirror after the output port in the forward direction of the isolator device and monitor the reflection signal from the input port. This reflection signal well describes and measures the round-trip transmissivity of light across the isolator device. If the reflection signal is the same as or is comparable with the forward signal, then the structure does not have the desired isolation property. In contrast, if the reflection signal is much smaller than the forward signal, then a good isolation property is implied.
An equivalent way to investigate the optical isolation performance of the diode structure is to adopt a doubled-diode structure with a mirror-symmetrical plane at the forward direction output port of the diode, as depicted in Fig. 25. Obviously this method has set all the forward output signals as the backward input signal of the diode and thus can directly test the isolation property of the diode structure. By implementing this method, we calculate simultaneously the forward transmissivity and the round-trip transmissivity of the diode structure. Comparison of these two quantities would directly measure their isolation properties. Figure 25(a) is the schematic geometry of the doubled-diode structure corresponding to the diode depicted in Fig. 24(b). The parameters of the diode are the same as in Fig. 23. The width of the input and round-trip output waveguides is the same 2a (a=440 nm) and the center connection waveguide is 6a. The length of the center connection waveguide, which measures the distance of the total-reflection mirror away from the output port of the diode, is 10a. The spectra [Fig. 25(b)] show that the round-trip reflection peak is located at 1,582 nm and has a quantity of only 0.3%, which is almost two orders of magnitude smaller than the forward peak [with a maximum transmissivity of 22.9%]. The result indicates that the diode has a significant optical isolation property.
Based on the above numerical analysis, the double-diode structure was fabricated in silicon. Figure 25(c) shows the SEM images of the fabricated double-diode structures along the light path. The lattice constant a was 440 nm, and the radii r1 and r2 of air holes in the two photonic crystals of the heterojunction diode were approximately 110 nm and 160 nm. The length of the output waveguide is 4 μm (≈10a). The slab thickness was 220 nm. Figure 25(d) shows the experimentally measured forward and backward transmission spectra of the diode, as well as the round-trip transmission spectrum of the doubled-diode structure. The forward, backward and round-trip transmissions are optimized and the input/output loss has been removed. The spectra show that the maximum round-trip transmissivity, located at 1,553 nm, is only 0.4%, almost two orders of magnitude smaller than the forward peak [with a maximum transmissivity of 32.9%]. The experiment confirms the existence of a significant isolation effect in agreement with the theoretical prediction. We change the length of the center connection waveguide of the double-diode structure from 4 μm to 3 μm and 5 μm. The measured round-trip transmission spectra for the three structures are displayed in Fig. 25(e). The results show that the round-trip transmission signal decreases remarkably along with the increasing length of the center connection waveguide, and already reaches an extremely low level (below 0.05%) in the whole spectrum range when the output waveguide length increases to 5 μm. This clearly indicates that the mode dispersion in the output waveguide of the diode has no influence to the isolation property of the diode. Due to the arbitrariness of the lattice constant a, we can freely adjust the isolation frequency to anywhere as desired. This could be very convenient for the design of realistic photonic devices.
To better understand the underlying physics, we further make a detailed analyses based on the scattering matrix theory adopted in Ref. [53,54], and find that the above numerical results of optical isolation are in no conflict with the reciprocity theorem involved in our linear and passive silicon optical diode structure. Our diode basically consists of two in-plane information channels (A and B, the input and output waveguide channels for infrared signal, which can be either single mode or multimode channels.) as well as many in-plane side-way and off-plane scattering channels (denoted as C as a whole, which causes dissipation of information away the signal channels). At the two ends of the diode device the fields are written as follows:
in which Ain corresponds to the input signal from port A, Aout to the output signal from port A, Bin to the input signal from port B, Bout to the output signal from port B, Cin to the input signal from port C, and Cout to the output signal from port C. The scattering matrix S transforms the input state of all the channels [the column vector in the right hand of Eq. (3)] into the output state of all the channels [the column vector in the left hand of Eq. (3)]. The scattering equation of the forward transmission is written as:
As the silicon diode structure is linear and passive, the system as a whole is reciprocal in regard to time-reversal symmetry following the Lorentz reciprocity theorem. As a result, the scattering matrix S is symmetric with and further satisfies:
Suppose all the output signals are reversed and come back into the system, then the input at port B for the system is now exactly the same as . The scattering equation is then
which is exactly the same as the initial input from port A. This clearly indicates that there is no isolation behavior in the structure if all information is reversed back into the system, consistent with the reciprocity theorem for a time-reversal symmetric system.
However, the story can be very different when the in-plane signal transport is concerned, as is always the case for 2D silicon PC slab structures. In our structure the information and energy involved in C channels are dissipated permanently against the in-plane channel A and B due to scattering loss (both in-plane and off-plane), and they cannot be reversed back totally and input again into the structure, so in practice, Cin in Eq. (3) can be assumed to be zero. As a result, Eq. (6) should be modified as:
In general, Eq. (7) looks very different from Eq. (6), which indicates that the reciprocal transport of light in regard to the signal channel A and B has been broken. It shows that even if the same forward transmission signal of port B is reversed back and input into the diode, the output signal of port A can be much different from the initial input signal ain0 of port A because no signal is reversed and input back into the channel C. Therefore, the considerable unidirectional transmission behavior can take place for the in-plane signal with no conflict with the reciprocal principle. In other words, the could be much different from when . This justifies the occurrence of a good isolation effect in the silicon optical diode. In ideal structures, both of them are zero, and Eq. (7) becomes
which implies a 100% signal contrast of the isolator.
It is worth saying a few more words here for better drawing a clear picture about the physics discussed in the above. In nature, as time always flows forward and cannot be reversed, one usually uses the term of reciprocal or nonreciprocal transport of light to describe a model system of back transport of light, in many cases to describe the reflection of light back into the considered structure. In this regard, simply consider a point source radiating an outgoing spherical wave front. If time can be reversed, the outgoing spherical wave front is contracted into an ingoing spherical wave front, eventually to a point. This is a very good picture to describe reciprocal transport of light in a linear system. However, to realize in real world such a concept, one needs to place a perfect spherical mirror concentric with the point source of light, which reflects back all information carried by the outgoing expanding spherical wave into the ingoing contracting spherical wave. If, however, one has only a small planar mirror placed at some distance and with a limited solid angle with respect to the source, the reflected signal can never return to the initial state of a point source when it reaches the position where the light source is located. The conventional magneto-optical isolator also works in this category of physical picture. It is used to block down the back-reflection signal of the transmission light, and the underlying physics can be well described by the model of time-reversal symmetry breaking. The same physics picture applies equally well to our optical diode. The fact that there exists information dissipation from the signal channels to other channels in a spatial-inversion symmetry breaking structure is sufficient to induce an optical isolation in regard to the signal.
The above numerical calculations and experimental results have shown that our silicon PC slab heterojunction diode exhibits promising performance of optical isolation, with a round-trip transmissivity two orders of magnitude smaller than the forward transmissivity for in-plane infrared light across the structure. Our scattering matrix analysis indicates that the considerable unidirectional transport of in-plane signal light can be attributed to the information dissipation and selective modal conversion in the multiple-channel spatial-inversion symmetry breaking structure and has no conflict with reciprocal principle for a time-reversal symmetric structure. It is expected that optimized connection interfaces between the input and output waveguides with the heterojunction diode can yield better impedance mismatch and bring higher forward transmission efficiency. That optical isolation can occur in a linear, passive, and time-independent optical structure would stimulate more thinkings and insights on the general transport theory of light in the fundamental side and open up a road towards photonic logics in silicon integrated optical devices and circuits in the application side.
7. Parallel–hetero photonic crystal structures
Photonic crystal heterostructures (PCHs) have attracted increasing interest in optical integrated circuits and cavity quantum electrodynamics (cavity QED) due to their useful photonic band-gap structure and the ability to provide nanocavities with ultra-high quality factor (Q factor). The properties of PCHs have been investigated both in theory and experiment over the past several years. In previous works, the transmission and reflection characteristics of PCHs were revealed only across the hetero-interface between two photonic crystals with different lattice constants. The basic character of PCHs was the shift of the band edge, which results in a transmission gap with approximately 100% efficiency . In comparison, recently we have reported a method for making a parallel-hetero perturbation inside the waveguide and analyze the optical properties of the photonic crystal parallel-hetero perturbation (PHP) structure. It is expected that this new type of PCHs not only contains the band-edge shifting property but also has an additional transmission gap which can be easily regulated in the middle of the transmission band. Simulations and experiments confirm the existence of the additional transmission gap . Our work can help to enlarge the usage of PCHs in the design and fabrication of novel cavities and filters via localized modulation of structural dispersion, which are key components in a photonic network. Based on the PCH structures, we have further proposed and realized a new scheme of cavity, an interface heterostructure cavity without any confinement barrier to confine light. Interestingly, the localized resonant mode lies in the pass band of the waveguide, in comparison of those cavities whose localized modes are always located within the band gap of the structures.
Figure 26(a) shows the geometry of the designed cavity, which consists of two identical parallel-hetero perturbation (PHP) waveguides. Each PHP waveguide consists of two semi-infinite PC (PC1 and PC2) slabs having slightly different lattice constants (a1 and a2, respectively). We assume a2 is 5% larger than a1. Each PC region has a triangular-lattice pattern of air holes, with the W1 waveguide formed by a line of missing holes along the Γ-Κ direction. Without periodicity along any direction, the PHP is essentially a kind of incommensurate PC superlattice along the waveguide direction. The transmission spectra of PHP waveguide structure exhibits an additional gap [0.29-0.30 (a1/λ)] in the middle of the pass band of the two individual W1 waveguides besides the usual band edge shifting [0.26-0.27 (a1/λ)] . Figure 26(b) shows the calculated resonance spectrum of the cavity by using 3D FDTD method in association with the Pade approximation for the TE-like modes of the PC slab . The resonance spectrum shows that a resonant mode surprisingly appears in the pass band region of the two PHP waveguides at 0.2855 (a1/λ). In addition, this resonant mode (called band-pass mode) has a rather high value of Q factor reaching 5,340.
We calculated the Ey field distribution of TE-like mode transporting along the waveguide at the resonance frequency 0.2855 (a1/λ). It is surprising that the interface PHC has no influence on the propagation of waveguide mode [Fig. 27(a)]. To further confirm that the resonant mode really exists, we plot in Fig. 27(b) the calculated Ey field distribution of the resonant mode at the interface PHC at 0.2855 (a1/λ) using a point source located within the interface cavity region. The mode does oscillate as a quadrupole form without barriers along the propagation direction even after the light source is turned off. This indicates that the resonant mode does exist and is not a calculation fault.
Based on the numerical analysis, an interface PHC structure was fabricated [Fig. 28(a)].The experimental results [Fig. 28(b)] confirms the theoretical prediction with the Q factor of the cavity resonant mode decreases to 1,517 (centered at 1,570.2 nm) [Fig. 28(c)]. We then use an infrared CCD camera to monitor the resonant properties on top of the interface PHC region. When the laser wavelength is tuned to 1,570 nm, we can readily see a bright light spot located at the exact interface position [red circle in Fig. 28(d)] besides the other two light spots induced by the coupling between PHP waveguide and ridge waveguide. When the laser wavelength is tuned to 1,560 nm, the light spot in the red circle is gone while the other two light spots remain, indicating that the cavity is off resonance. This confirms the existence of the band-pass resonant mode.
The physics of PHC can be understood from the point of view of slow light effect. We have used a narrow light pulse centered at 0.2855 (a1/λ) to pass through the interface PHC and a pure, same-length PHP waveguide, respectively, and recorded the output time of the pulse. The interface PHC is 599.25 time units, while the pure PHP waveguide is 606.3 time units, slightly longer than the interface PHC [Fig. 29(a)]. This indicates that the interface still contains one PC period length, so the whole interface PHC can be seen as a three-part structure, a combination of “slow light”–“fast light”–“slow light” region [Fig. 29(b)]. The central fast region in the PHC, which is only one period long, could also confine energy between two slow light regions and behaves as a cavity.
The physics underlying the interface PHC can also be analyzed by focusing on the phase shift in the waveguide of the interface PHC during propagation due to the fact that this band-pass mode is near the edge of the additional-gap region. The current PHC, which is made from an incommensurate superlattice structure, is similar to a periodic structure with dislocations. As a result, multiple scattering of light around the dislocation will occur and result in pinning effect of light (energy shifting across the dislocation line). Previous study shows that the PHP structure provides the asymmetric phase condition during the light propagation along the waveguide. The interface connects two same anti-symmetric PHPs and can be seen as a vertical edge dislocation line (phase reversal line), so that the phase condition is reversed to the opposite aside the interface. In material science the dislocation could cause the charge accumulation. Similarly our phase reversal dislocation gives an abrupt phase shift across the interface and may cause the photon accumulation around the phase reversal line. This phase reversal dislocation plays as a defect and forces the light energy distribution to become skew and swinging while crossing the interface [Fig. 27(a)]. The local energy oscillation acts as the cavity resonance and leads to the photon accumulation at the interface.
The PHC has several distinct properties. First, unlike the conventional PC cavities which are formed either by removing one or several air holes or by waveguide-like hetero-cavities, the current PHC does not have any confinement barrier to confine light. Second, the resonant mode is located within the pass band of waveguides, so the localized resonant mode and the continuous waveguide mode can easily co-exist in the same frequency and space regime. Third, the coupling efficiency between waveguide and cavity modes is much higher than the conventional cavities, which strongly depend on the transverse evanescent field profile overlap of the cavity mode and the waveguide mode. As a result, this PHC has nearly no influence on the propagation of waveguide mode despite the presence of localized cavity mode. High coupling efficiency between waveguide and cavity mode can help to reduce the operating power for PC lasers, and make it possible to integrate multiple lasers, photodetectors, and switches on a single chip.
In summary, we have presented recent progresses on infrared 2D air-bridged silicon PC slab devices that were made in our group in the past several years. 2D air-bridged silicon PC slab structures can confine light by the high index contrast in the vertical off-plane direction and manipulate light by photonic band and band gaps in the lateral in-plane direction. In addition, silicon is transparent and has a large refractive index in the infrared wavelengths. Therefore, the air-bridged silicon PC structures become one of the most important elements in integrated optics.
In this review, we have discussed several integrated optical elements and devices in regard to their design, fabrication, and characterization. These devices are based on either the PBG or photonic band structure engineering. To bring these devices into reality, we have made extensive efforts to construct high-efficiency numerical simulation tools for solution of photonic band structures, transmission spectra, light propagation dynamics, light wave patterns, and many others. These simulations allow for design of optimized PC structures for a specific application. The fabrication of these devices strongly depends on state-of-the-art nanofabrication technologies including EBL, FIB, ICP etching, and other wet etching techniques and procedures. We have constructed a convenient experimental setup to measure the transmission spectra and monitor the propagation path of infrared signals within PC structures simultaneously. The setup makes the optical characterization of PC devices accurate, user-friendly, fast, and convenient.
We have proposed Γ−Μ waveguides made in 2D triangular-lattice PC slabs. After a series of optimization and improvement, we find out the optimized geometries for high performance wide band Γ−Μ waveguides. As the Γ−Μ waveguide is perpendicular to the usual Γ−Κ waveguide, we combine the Γ−K waveguide and Γ−M waveguide together to form 90° waveguide bends and channel drop filters. The combination of Γ−Μ waveguide and Γ−K waveguide can offer a flexible way to interconnect and couple between different devices. In addition, we have shown design of two other kinds of channel drop filters. They achieve fine tuning of the resonant wavelengths by changing the size of the cavities or the shape of the air holes around the cavities. We have also designed and realized PC waveguide based on the coupled-cavity scheme. There are a lot of geometric parameters to fine tune the light transport properties in the structure. Moreover, we have successfully fabricated high-Q silicon PC microcavities with a Q-factor up to 70000 and this paves the way to experimentally explore light-matter interaction within the strong-coupling regime in the all-solid platform.
We have investigated optical devices that work on band structure engineering. We have explored PC structures that exhibit interesting and useful dispersion and refraction properties, such as negative refraction and self-collimation effects. We have designed and fabricated a kind of PC structure with negative refraction effect and use SNOM technology to observe the negative refraction ray trace of infrared light beam. In addition, we obtain broadband large-angle self-collimation effect for TE-like guided modes in infrared wavelength.
We have demonstrated the design, fabrication, and characterization of on-chip wavelength-scale optical diodes that are made from the heterojunction between two different silicon 2D square-lattice PC slabs with directional bandgap mismatch and different mode transitions. The measured transmission spectra show considerable unidirectional transmission behavior, in good agreement with numerical simulations. The experimental realization of on-chip optical diodes with wavelength-scale size using all-dielectric, passive, and linear silicon photonic crystal structures may help to construct on-chip optical logical devices without nonlinearity or magnetism, and would open up a road towards photonic computers.
We have demonstrated optical isolation of our diode structure. Both numerical simulations and experimental measurements show that the round-trip transmissivity of our diode could be two orders of magnitudes smaller than the forward transmissivity, indicating good performance of optical isolation. The occurrence of in-plane light isolation is attributed to the information dissipation due to off-plane and side-way scattering and selective modal conversion in the multiple-channel structure and has no conflict with the reciprocal principle. That optical isolation can occur in a linear, passive, and time-independent optical structure would stimulate more thinkings and insights on the general transport theory of light in the fundamental side and open up a road towards photonic logics in silicon integrated optical devices and circuits in the application side.
We have designed and fabricated cavities without confinement barrier by combining two incommensurate PC superlattice waveguides. A resonant mode with a high quality factor shows up in the pass band of waveguides. It has nearly no influence on the propagation of waveguide mode and can be directly coupled with the waveguide mode. The experimental measurement confirms the theoretical prediction of extraordinary coexistence of localized cavity mode and continuous waveguide mode with high coupling efficiency in the same frequency and space regime. The novel type of cavity has a number of unique properties that are advantageous to on-chip information transport and processing. The discovery of cavity without confinement barrier might attract interest in fundamental physics and optical engineering communities.
All of these results show that 2D air-bridged silicon PC structures can control light propagation in many flexible ways and have many potential applications in all-optical integrated circuits and other fields. The efforts that we have made and the experiences that we have accumulated in the past several years will allow us to design and realize PC devices with novel functionalities.
This work was supported by the National Key Basic Research Special Foundation of China (No. 2011CB922002), the National Natural Science Foundation of China, and National Center for Nanoscience and Technology of China. The authors would like to thank the previous members of our group, Cheng Ren, Haihua Tao, Yazhao Liu, and Changzhu Zhou for their contributions to the works presented in this review.