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The novel all-optical unidirectional wavelength filters are studied by the finite-difference time-domain method, which are based on the two-dimensional square-lattice photonic crystal with the rectangular defects. Owing to the modes’ match and mismatch between the defect and the adjacent waveguides, the unidirectional propagation of the fundamental-mode light beam resonant at a certain frequency is obtained. Through merely altering the coupling region between the defect and the input waveguide, a unidirectional dual-branch waveguide filter is designed. This kind of devices has both the abilities of wavelength filtering and unidirectional light propagation, and may be potentially applied in the future all-optical complex integrated circuits.
©2013 Optical Society of America
1. Introduction
Photonic crystals (PCs) are inhomogeneous periodic arrays of dielectric or metal with a lattice constant comparable to the wavelength, which are characterized by the existence of the photonic band gap [1,2]. Different kinds of devices based on PCs have been proposed, including waveguides, couplers, beam splitters, wavelength filters, and so on. The devices based on the two-dimensional (2D) PC can be easily fabricated and are conveniently integrated into other conventional devices, which play the potential important roles in the future optical circuits. By introducing Line defects into an otherwise-perfect PC, straight or bent waveguides can be achieved [3–5]. When the point defects and the line defects are introduced simultaneously into a PC, ultra-compact channel filters with a much smaller size than the conventional optical devices can be obtained [6–9].
It is well known that the electrical diode plays an important role in electronic circuits owing to the capability of unidirectional movement of the current flux. An all-optical diode is a nonreciprocal device that offers unidirectional transmission of optical signals, which pays key roles in the all-optical signal processing. The propagation of light in PC is just like the movement of electron in semiconductors. And great efforts have been dedicated to studying the unidirectional propagation of the electromagnetic waves in the PCs. The vast majority of photonic structures are reciprocal. The efficient routines to breaking the reciprocity or time-reversal symmetry and obtaining the unidirectional light propagation are based on magnetic-optical effect [10–12], optical nonlinearity [13–16], opto-acoustic effect [17,18], indirect interband photonic transitions [19], and so on. Wang et al. [11] realized unidirectional transmission of electromagnetic chiral-edge states in a magneto-optical PC within the microwave region, which works well in a strong magnetic field environment. Unfortunately, the magneto-optical effect is not present in standard optoelectronic materials including most metals and semiconductors. Li et al. [17] designed a sonic-crystal-based acoustic diode that had broken spatial inversion symmetry and experimentally realized sound unidirectional transmission in this acoustic diode. Yu and Fan [19] realized a linear, broadband and nonreciprocal isolation through modulating the spatial-temporal refractive index that simultaneously impart frequency and wavevector shifts during the photonic transition process. Recently, several schemes [20–25] have been proposed to achieve the unidirectional propagation of light beam through the linear and passive photonic structures, which are essentially based on the spatial-inversion symmetry breaking. Feng et al. [20] considered a two-mode waveguide with a spatially varying but time-independent dielectric constant, and numerically and experimentally acquired a one-way modal conversion effect. Fan et al. [26] showed that the structure proposed in [20] cannot enable optical isolation because it possesses a symmetric scattering matrix. And an optical isolator cannot be constructed by incorporating this structure into any linear and time-independent system, which is described by materials with a scalar dielectric function. In their response [27], Feng et al. acknowledged that their structure is Lorentz reciprocal, and on its own cannot be used as the basis of an optical isolator. Wang et al. [21] reported a unidirectional on-chip optical diode in silicon based on the directional bandgap difference of the near-infrared square-lattice PCs comprising a heterojunction structure and the break of the spatial inversion symmetry. Recently, a system of two square PCs of the same lattice constant but different scatterer radii, which are cut and brought into contact along their body diagonals, is demonstrated to facilitate unidirectional transmission of light [22]. In our recent work [23], the unidirectional light propagation for the frequencies within the pass band is obtained owing to the different characters of the self-collimation and negative refraction.
In this paper, we designed a kind of unidirectional wavelength filter based on a 2D square-lattice PC with the rectangular defects and the input and output waveguides. The structure itself is linear and passive, so it is reciprocal essentially. Owing to the match and mismatch of the modes’ symmetry between the defect and the adjacent waveguides, it can be obtained that the propagation of the fundamental-mode light beam at a certain frequency is allowed along one direction and is prohibited along the opposite direction. This kind of devices has both the abilities of wavelength filtering and unidirectional light propagation, and may be utilized in the future complex all-optical circuits.
2. Unidirectional wavelength filters based on PC structures with rectangular defects
At first, we consider a PC structure consisting of silicon rods immersed in air, whose refractive index is set to be 3.45 for the near-infrared light around 1550 nm. The length of the PC’s lattice constant is described as “a”, and the radius of the rods is 0.2a. It can be calculated that there exists a photonic band gap from the frequency 0.281 c/a to 0.416 c/a between the lowest and the second bands for the TM-polarized modes, where “c” is the light velocity in vacuum. Based on above PC, we constructed a rectangular defect, whose sketch map is shown in Fig. 1 . The two side lengths of the rectangle are 1.367a and 0.5a, and the long side is placed along the ГX direction of the PC. It is calculated that there are four defective modes localized by this defect, whose frequencies are 0.3246 c/a, 0.3300 c/a, 0.3758 c/a, and 0.3977 c/a. The corresponding modes’ spatial profiles are calculated and the results are shown in Figs. 2(a) , 2(b), 2(c), and 2(d), respectively. It is shown that the mode symmetries of 0.3246 c/a are both even-symmetric with respect to the horizontal and vertical central lines, and those of 0.3758 c/a are odd along these above two orthotropic directions. The mode symmetry of 0.3300 c/a is even along the horizontal line, and is odd along the vertical line, while the mode symmetry of 0.3977 c/a is odd along the horizontal line and is even along the vertical line. Their mode symmetries are just opposite in each direction.
Fig. 1 Sketch map of the 2D square-lattice PC with a central rectangular defect, the two side lengths of which are 1.367a and 0.5a.
Fig. 2 Electric field profiles of the defective modes for the resonant frequencies. (a) 0.3246 c/a, (b) 0.3300 c/a, (c) 0.3758 c/a, and (d) 0.3977 c/a. Red and blue regions represent the positive and negative values of the electric field.
Based on above PC, we constructed a W1-typed waveguide by removing one row of rods along the ГX direction, whose sketch map is shown in Fig. 3(a) . It is shown in Fig. 3(c) that there exists an even-mode guiding band spanning the frequencies of 0.3020 c/a and 0.4160 c/a. Figure 3(b) shows another waveguide constructed by removing one row of rods and shifting the adjacent two rows of rods outwards for the distances of 0.7a and 0.35a, respectively. Figure 3(d) shows the calculated results that there are an even-mode guiding band within the frequencies of 0.2746 c/a and 0.4160 c/a, and an odd-mode guiding band from 0.3308 c/a to 0.4160 c/a.
Fig. 3 Sketch map of the waveguides composed by removing one row of rods along the ГX direction of the PC (a), and removing one row of rods along the ГX direction and shifting the adjacent two rows on each side outward for the distances of 0.7a and 0.35a. The even modes (shown by the solid dots) and odd modes (shown by the hollow dots) for the waveguide structures shown in (a) and (b) are described in (c) and (d), respectively.
In order to obtain a novel unidirectional wavelength filtering device, we designed a PC structure consisting of above rectangular defect and two waveguides, whose sketch map is shown in Fig. 4(a) . The left W1-typed waveguide only sustains an even-mode guiding band, while the odd-mode and even-mode guiding bands exist together in the under waveguide shown in Fig. 3(b). A rectangular defect is located at the corner, and is connected with two waveguides. When a fundamental-mode light beam is launched at the left side and centered at the central line of the W1-typed waveguide, the transmitted light energy is collected at the bottom of the structure and the calculated transmission spectrum is shown by the solid line in Fig. 4(b). When the light beam is incident from the bottom of the structure, the light energy is collected leftward and the corresponding calculated transmission spectrum is shown by the dotted line in Fig. 4(b). It can be seen that there are two resonant frequencies of 0.3246 c/a and 0.3977 c/a in the left incidence case, while there is only one resonant frequency of 0.3246 c/a on the condition that the light beam is incident upwards.
Fig. 4 (a) Sketch map of the wavelength filter consisting of the rectangular defect and the two waveguides shown in Fig. 3. (b) Light transmittances through the wavelength filter device from a light beam at the normal incidence from the left side (shown by the solid line) and down side (shown by the dotted line) of the waveguide. The solid line represents the transmittance spectrum when the light beam is incident from the left side, and the dotted line shows that in rightward incidence.
In order to have a visual sight of the propagating characteristics of the light waves through the above structure, the spatial electric field distributions of the upward and rightward light beam at the frequencies of 0.3246 c/a and 0.3977 c/a are calculated and the results are shown in Fig. 5 . In Fig. 5(a), the light wave resonant at 0.3246 c/a goes upwards in the under waveguide and reaches the central defect. Owing to the coupling between the defect and the input and output waveguides, the light energy can be finally exported through the left W1-typed waveguide. When the light beam of the same frequency is incident from the left waveguide port, the similar propagation character occurs and the light energy can reach the under waveguide port, as shown in Fig. 5(b). The resonant field profile of 0.3246 c/a is symmetric about two central lines vertical to each other. When the light beam’s frequency is changed to be 0.3977 c/a, it is shown in Fig. 5(c) that the light beam is launched upward and arrived at the defect, and no light energy is exported from the left waveguide. It is obvious that there are no coupling between the input waveguide and the defect owing to the mismatch of mode’s symmetry, which can be understood by analyzing the defective field profile shown in Fig. 2(c). When the light beam at the same frequency is incident from the left side of the structure, Fig. 5(d) shows that the even mode supported by the left W1-typed waveguide can couple with the defect, and finally an odd mode is transferred by the defect and propagates through the under waveguide, which can allow the even and odd modes propagate simultaneously within a certain frequency region. Furthermore, the field distributions of the odd-mode light beams resonant at 0.3977 c/a at upward and rightward incidences are also calculated, and the results are shown in Figs. 5(e) and 5(f), respectively. Figure 5(e) shows that the odd-mode light beam can travel upwards, and is converted into an even-mode light beam within the left W1-typed waveguide by the rectangular defect. It is shown in Fig. 5(f) that the odd-mode light beam cannot propagate rightwards within the left waveguide owing to its odd-mode band gap. It can be clearly seen by comparing Fig. 5(d) with Fig. 5(f) that the structure itself is essentially reciprocal. But the structure allows the unidirectional propagation of the fundamental-mode light beam at certain frequency with a large transmission contrast, so it also has actual application for some particular purposes.
Fig. 5 Spatial distributions of the electric field at the frequencies of 0.3246 c/a and 0.3977 c/a through the structure shown in Fig. 4(a). (a) a fundamental-mode light beam for the frequency 0.3246 c/a at upward incidence, (b) a fundamental-mode light beam for 0.3246 c/a at rightward incidence, (c) a fundamental-mode light beam for 0.3977 c/a at upward incidence, (d) a fundamental-mode light beam for 0.3977 c/a at rightward incidence. (e) an odd-mode light beam for 0.3977 c/a at upward incidence. (f) an odd-mode light beam for 0.3977 c/a at rightward incidence. The arrows point at the directions of light propagation through the structure. The red and blue regions represent the positive and negative values of the electric field.
Finally, we constructed a kind of two-branch waveguide filter consisting of one W1-typed input waveguide and two output waveguides with the same structure shown in Fig. 6(a) . The two defects connecting the input and output waveguides are just the same, except that the coupling regions connecting the input waveguide and the defects are different from each other. Figure 6(b) shows the calculated light transmission spectra collected from the upper (shown by the dotted line) and bottom (shown by the solid line) output waveguides. The incident light beam is launched at the left side of the structure with a beam width of 2a and centered at the W1-typed waveguide. It is shown that the transmitted peak propagating through the upper branch waveguide is located at 0.3758 c/a, while the transmitted peak is changed to be 0.3977 c/a detected from the bottom output waveguide.
Fig. 6 (a) Sketch map of the two-branch wavelength filter consisting of two rectangular defects, one input waveguide, and the two output waveguides. (b) Light transmittances through the wavelength filter from a fundamental-mode light beam at the normal incidence from the left side of the input W1-typed waveguide. The dotted line represents the transmission spectrum collected at the upper waveguide port, and the solid line shows that at the under waveguide port.
Based on above simulated results, the unidirectional light beam propagation and wavelength filtering characters are verified by calculating the electric field distributions through above structure. The spatial electric field distributions through the structure for the rightward and leftward light beams at the frequencies of 0.3758 c/a and 0.3977 c/a are calculated and the results are shown in Fig. 7 . In Fig. 7(a), the light wave resonant at 0.3758 c/a goes straight in the left waveguide and reaches the upward defect. Owing to the coupling between the defect and the input and output waveguides, the light energy can finally reach the output waveguide export. When the light beam of the same frequency is incident from the right waveguide as shown in Fig. 7(b), no light energy can be detected in the left waveguide owing to mismatch of mode profile between the defect and the right waveguide. When the light beam’s frequency is changed to be 0.3977 c/a, it can be seen in Fig. 7(c) that the light beam propagates through the bottom defect and is exported from the under waveguide, while the light beam propagation along the inverse direction is prohibited shown in Fig. 7(d).
Fig. 7 Spatial distributions of the electric field at the frequencies of 0.3758 c/a and 0.3977 c/a through the structure shown in Fig. 6(a) at normal incidence of the fundamental-mode light source. (a) 0.3758 c/a at rightward incidence, (b) 0.3758 c/a at leftward incidence, (c) 0.3977 c/a at rightward incidence, (d) 0.3977 c/a at leftward incidence. The white arrows point at the directions of light propagation.
3. Conclusion
In summary, we have studied the unidirectional wavelength filtering characteristics of the PC structure consisting of the rectangular defect and two waveguides with different symmetric guiding modes. Based on the mode’s match and mismatch between the defect and the adjacent waveguides, the unidirectional light propagation at a certain frequency is obtained, and the resonant frequency can be adjusted by merely altering the coupling region between the defect and the input waveguide. This kind of devices has both the abilities of wavelength filtering and unidirectional light propagation, and may be potentially applied in the future all-optical complex integrated circuits.
Acknowledgments
This work is supported by the National Natural Science Foundation of China with Grant Nos. 10904176 and 11004169, the Fundamental Research Funds for the Central Universities, and the “985 Project” and the “211Project” of the Ministry of Education of China.
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