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Photonic crystal filter based on defect mode and waveguide mode symmetry matching
Accepted Manuscript Photonic crystal filter based on defect mode and waveguide mode symmetry matching Tong Zhang, Jing Sun, Yunxing Yang, Zhixin Li
PII: DOI: Reference:
S0030-4018(18)30631-X https://doi.org/10.1016/j.optcom.2018.07.039 OPTICS 23312
To appear in:
Optics Communications
Received date : 5 April 2018 Revised date : 6 June 2018 Accepted date : 12 July 2018 Please cite this article as: T. Zhang, J. Sun, Y. Yang, Z. Li, Photonic crystal filter based on defect mode and waveguide mode symmetry matching, Optics Communications (2018), https://doi.org/10.1016/j.optcom.2018.07.039 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Manuscript Click here to view linked References
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Photonic Crystal Filter Based on Defect Mode and Waveguide Mode Symmetry Matching Tong Zhang, Jing Sun*, Yunxing Yang, Zhixin Li College of Physics and Mechanical Engineering, Jishou University, Jishou 416000, Hunan, China *Corresponding author. E-mail address: [email protected] (J. Sun). ABSTRACT In this paper, a new type of filter based on symmetry matching between the defect and waveguide modes in a square lattice photonic crystal for 1.31-μm wavelength is designed and evaluated. The filter consists of one elliptical defect and three wire defect waveguides with two single-mode waveguides and one dual-mode waveguide. Because the point defect state has four different resonant frequencies and four corresponding types of mode field symmetries, the proposed filter uses the match or mismatch between the defect and waveguide modes to realize two resonant frequency filtering outputs through a dual channel. To evaluate the device, the transmission characteristics of the electromagnetic waves in the device are simulated using the finite element method. The quality factor and transmission coefficient of the filter is 2,017 and 0.84 at the frequency of 0.4034 c/ɑ (corresponding to 1.31-μm wavelength). This device can be applied to wave division multiplexing, filtering, and has the potential for applications in optical integrated chip and optical communications. Keywords: photonic crystal; defect mode; waveguide mode; symmetry; filter
1.Introduction Photonic crystals have attracted much attention [1,2] since they were first proposed by Yablonovitch [3] and John [4] in 1987 owing to their ability to control the flow of photons in optical fibers [5], resonators [6–9], waveguides [1,6,10], optical switches [11], filters [8,9,12–14], sensors [15], wavelength division multiplexers [16,17], and other devices. The combination of a waveguide and micro-cavity in a photonic crystal is mainly used to realize a filtering function. Two-dimensional (2D) photonic crystals play an important role in highly integrated, efficient, and stable wide-bandwidth optical communication systems because they have a flexible design, simple structure, and strong light transmission control. Multichannel photonic crystal filters at micron scale have good application potential and are receiving wide attention at present. For example, Noda et al. [18] used a combination of a waveguide and micro-cavity to implement an upload/download filter. Qiu [19] produced a filter composed of two waveguides and a dual-mode micro-cavity based on the triangular lattice photonic crystal. Takano et al. [20] designed an efficient multichannel filter based on multiple simple heterostructure filters. Feng et al. [14]
realized a one-way transmission filter based on the symmetry match of defect and waveguide 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
modes in a square lattice photonic crystal using a two-ring cavity. Gomyo et al. [21] realized a filter with a quality factor of up to 45,000 and transmission of 95% at a 1.6-μm wavelength by combining two parallel waveguides and ring cavities. Wang et al. [12] realized an ultra-narrow band filter based on a silicon 2D photonic crystal resonator and reflectors and with a quality factor of up to 1,300 and transmission of 95.3% at a 1.5-μm wavelength. Wu et al. [17] realized a multiform wavelength division multiplexer by adjusting the various dielectric column arrays in the ring cavity; the best effect was obtained when the dielectric cylinder was arranged in a 3×3 grid, yielding a transmission of 92% and quality factor of up to 3,800. In this paper, a photonic crystal filter is proposed based on the symmetry match between defect and waveguide modes in 2D square lattice photonic crystals. Because the point defect is smaller than any ring cavity [8], the device is easier to integrate, while the three waveguides are coupled with a micro-cavity. In addition, the resonant frequency of the coupled cavity can be varied by adjusting the size parameters of the elliptical dielectric cylinder in the point defect to adjust the working frequency of the filter. 2. Principle and analysis In the structure, the Si dielectric cylinder is periodic and arranged in a square in an air background with a refractive index of n = 3.4 and radius r = 0.18ɑ, where ɑ is the lattice constant. The frequency between the photonic band gap is prohibited from propagation, and the optical signals are selectively propagated by controlling the flow of photons. Then, using the plane wave expansion method [22] to solve the band gap, the TM mode (where the electric field is parallel to the axial direction of the dielectric cylinder) band gap is in the range 0.303–0.444 c/ɑ. As shown in Fig. 1, the point defect is established by converting a center circular dielectric into an elliptical dielectric cylinder with a major half axis of 0.72ɑ and minor half axis of 0.289ɑ. Here, the major axis lies along the horizontal. The four defective modes with different symmetric modes of 0.3281, 0.3409, 0.4034, and 0.4171 c/ɑ were calculated using COMSOL Multiphysics finite-element method, which can calculate the field distribution, band structure, and transmission. The corresponding modes’ spatial profiles are shown in Figs. 2(a), 2(b), 2(c), and 2(d), respectively. The red/blue regions represent positive/negative electric field distribution. Therefore, we can determine the waveguide defect modes symmetry according to the electric field distribution. When the electric field distribution on both sides of the center line is the same/different, the waveguide defect mode corresponds to even mode (symmetry)/odd mode (anti-symmetry). The figures show that the defect mode of 0.3281 c/ɑ is anti-symmetric around the major axis but symmetric around the minor axis, the defect mode of 0.3409 c/ɑ is symmetric around both the major and minor axes, the defect mode of 0.4034 c/ɑ is anti-symmetric around both the major and minor axes, and the defect mode of 0.4171 c/ɑ is symmetric around the major axis but anti-symmetric around the minor axis.
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Fig. 1. Structure of a 2D photonic crystal with a central elliptical defect whose major half axis is 0.72ɑ and minor half axis is 0.289ɑ, where the major axis lies along the horizontal of the photonic crystals.
Fig. 2. Spatial profiles of four point defect resonant frequency modes (a) ƒ = 0.3281 c/ɑ, (b) ƒ = 0.3409 c/ɑ, (c) ƒ = 0.4034 c/ɑ, and (d) ƒ = 0.4171 c/ɑ.
Figure 3(a) shows a single mode waveguide constructed by removing a dielectric cylinder row along the ΓΧ direction. The single mode waveguide band structure is shown in Fig. 3(b), in which an even-mode waveguide mode exists at frequencies in the range 0.3116–0.4440 c/ɑ. The defect mode field distribution is shown in Fig. 4(a). Furthermore, after removing a dielectric cylinder row along the ΓΧ direction and shifting the two adjacent dielectric cylinder rows on each side outward by 0.56ɑ, a dual mode waveguide is constructed, as shown in Fig. 3(c). The dual mode waveguide band structure is shown in Fig. 3(d), in which an even-mode waveguide mode exists at frequencies in the range 0.3074–0.4440 c/ɑ and an odd-mode waveguide mode exists at frequencies in the range 0.3699–0.4440 c/ɑ. The defect mode field distributions are shown in Fig. 4(b) and Fig. 4(c).
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Fig. 3. (a) Single mode waveguide 1×9 super cell constructed by removing a dielectric cylinder row in the ΓΧ direction. (b) Even-mode waveguide mode at frequencies in the range 0.3116–0.4440 c/ɑ. (c) Removal of a dielectric cylinder row along the ΓΧ direction and shifting the two adjacent dielectric cylinder rows on each side outward by 0.56ɑ to construct a dual mode waveguide. (d) Even-mode waveguide mode at frequencies in the range 0.3074–0.4440 c/ɑ and odd-mode waveguide mode at frequencies in the range 0.3699–0.4440 c/ɑ.
Fig. 4. (a) Single mode waveguide field distribution at f=0.3831 c/ɑ, k=0.279. (b) Odd-mode waveguide mode field distribution at f=0.3831 c/ɑ, k=0.13. (c) Even-mode waveguide mode field distribution at f=0.3831 c/ɑ, k=0.34.
3. Design simulation of photonic crystal filter To design a filter based on the symmetry match between the defect and waveguide modes, the structure shown in Fig. 5 is used. The structure consists of a point defect and the three waveguides shown in Figs. 1 and 3. Moreover, an even-mode waveguide mode exists in both waveguides 1 and 2, while even-mode and odd-mode waveguide modes exist in waveguide 3. When the defect mode and waveguide mode symmetry are matched, the point defect is excited and the optical signal is coupled to the output waveguide to realize a filtering output.
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Fig. 5. Filter consisting of two single mode waveguides (waveguides 1 and 2), a dual mode waveguide (waveguide 3), and a central elliptical defect. The input port is port 3, and the two output ports are ports 1 and 2.
When an anti-symmetric optical signal at a frequency of 0.4171 c/ɑ is input at port 3 of waveguide 3, waveguide 3 has two modes, the odd-mode waveguide mode is anti-symmetric around the waveguide center vertical line and the defect mode is also anti-symmetric around the direction of the minor axis. Here, the defect mode symmetrically matches with the waveguide mode and the center point defect is excited. Furthermore, the defect mode and even-mode waveguide mode in waveguide 1 are both symmetric around the waveguide center horizontal line. In this case, the defect mode symmetrically matches with the waveguide mode of waveguide 1, so they are coupled. Moreover, the waveguide mode in waveguide 2 is anti-symmetric around the waveguide center vertical line, so the defect mode symmetrically matches with the waveguide mode of waveguide 2, and they are coupled. However, because the coupling capability between the point defect and waveguide 1 is better than that of waveguide 2, ultimately, most of the optical signal with a frequency of 0.4171 c/ɑ is output from port 1. Similarly, when the anti-symmetric optical signal at a frequency of 0.4034 c/ɑ is input at port 3 of waveguide 3, the symmetry matching around the direction of the minor axis between the point defect and waveguide 3 is consistent with the frequency of 0.4171 c/ɑ and the center point defect is hence excited. Furthermore, the even-mode waveguide mode in waveguide 1 is symmetric around the waveguide center horizontal line; however, the defect mode is anti-symmetric around the direction of the major axis, the defect mode does not symmetrically match with the waveguide mode of waveguide 1, and they cannot be coupled. Hence, there is no signal output through output port 1 of waveguide 1, and the transmission is nearly zero. In addition, the defect mode and waveguide mode in waveguide 2 are both anti-symmetric around the direction of the minor axis; thus, the defect mode symmetrically matches with the waveguide mode of waveguide 2 and they are coupled, and a frequency of 0.4034 c/ɑ is output from port 2. To verify the above theoretical analysis, the transmission spectrum for the filter obtained by calculating the ratio of light energy as well as by setting three integral segments near the in/output ports about two single mode waveguides at each output port (ports 1 and 2) and the dual mode waveguide 3 input port 3 are shown in Fig. 6(a). The figure shows that the transmission peak of
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waveguide 1 is at frequency 0.4172 c/ɑ, where the transmission is 0.96 incident from port 1. Moreover, there is a waveguide 2 transmission peak at frequency 0.4034 c/ɑ, where the transmission is 0.84 incident from port 2. The frequency of the two transmission peaks 0.4034 c/ɑ and 0.4172 c/ɑ are the same as the resonant frequencies of the defect mode (0.4034 c/ɑ and 0.4171 c/ɑ); thus, the theoretical analysis is consistent with the verification results.
Fig. 6. (a) Transmission spectrum of the two filter output ports. (b) Spatical profiles at the frequency of 0.4034 c/ɑ. (c) Spatical profiles at the frequency of 0.4172 c/ɑ.
4. Conclusion In this paper, a new type of filter with a structure consisting of three coupled waveguides and one point defect was proposed. It uses symmetry matching of the defect and waveguide modes to realize a filtering function of two resonance frequency optical signals. The structure can adjust the working frequency by varying the parameters of the elliptic column in the point defect, as well as simple modulation. Because it is a photonic crystal, it has high transmission efficiency, good stability, small size, and is easy to integrate. Moreover, the defect modes proposed in this paper can also be applied to devices with one-way optical signal transmission such as photodiodes. Funding This work was supported by the Excellent Youth Program in Education Department of Hunan Province (No. 17B212), Postgraduate Research Projects of Jishou University (No.Jdy17033), and Postgraduate Educational Reform (No. JG2017A01). References [1] Mekis A, Chen J C, Kurland I I, et al. High Transmission through Sharp Bends in Photonic Crystal Waveguides[J]. Physical Review Letters, 1996, 77(18): 3787-3790. [2] Edition S. Photonic crystals:molding the flow of light[M]. Princeton University Press, 1995. [3] Yablonovitch E. Inhibited spontaneous emission in solid-state physics and electronics[J]. Phys Rev Lett, 1987, 58(20): 2059-2062. [4] John S. Strong localization of photons in certain disordered dielectric superlattices[J]. Phys Rev Lett, 1987, 58(23): 2486-2489. [5] Barkou S E, Broeng J, Bjarklev A. Silica-air photonic crystal fiber design that permits waveguiding by a true
photonic bandgap effect[J]. Optics Letters, 1999, 24(1): 46-48.
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[6] Yan P, Vasseur J O, Djafari-Rouhani B, et al. Two-dimensional phononic crystals: Examples and applications[J]. Surface Science Reports, 2010, 65(8): 229-291. [7] Hwang I K, Kim G H, Lee Y H. Optimization of coupling between photonic crystal resonator and curved microfiber[J]. IEEE Journal of Quantum Electronics, 2006, 42(2): 131-136. [8] Li L, Liu G Q. Photonic crystal ring resonator channel drop filter[J]. Optik - International Journal for Light and Electron Optics, 2013, 124(17): 2966-2968. [9] Qiu M, Jaskorzynska B. Design of a channel drop filter in a two-dimensional triangular photonic crystal[J]. Applied Physics Letters, 2003, 83(6): 1074-1076. [10] Tsuji Y, Morita Y, Hirayama K. Photonic Crystal Waveguide Based on 2-D Photonic Crystal With Absolute Photonic Band Gap[J]. IEEE Photonics Technology Letters, 2006, 18(22): 2410-2412. [11] Dawes A M, Illing L, Clark S M, et al. All-optical switching in rubidium vapor[J]. Science, 2005, 308(5722): 672-674. [12] Wang Y, Chen D, Zhang G, et al. A super narrow band filter based on silicon 2D photonic crystal resonator and reflectors[J]. Optics Communications, 2016, 363: 13-20. [13] Kim S, Park I, Lim H, et al. Highly efficient photonic crystal-based multichannel drop filters of three-port system with reflection feedback[J]. Optics Express, 2004, 12(22): 5518-5525. [14] Feng S, Wang Y. Unidirectional reciprocal wavelength filters based on the square-lattice photonic crystal structures with the rectangular defects[J]. Optics Express, 2013, 21(1): 220-228. [15] Beiu R M, Beiu V. Fiber Optic Mechanical Sensor Based on a Triangular-lattice Photonic Crystal[C]. Photonicsglobal@singapore, 2008. Ipgc, 2008: 1-4. [16] Hayran Z, Turduev M, Botey M, et al. Numerical and experimental demonstration of a wavelength demultiplexer design by point-defect cavity coupled to a tapered photonic crystal waveguide[J]. Optics Letters, 2016, 41(1): 119-122. [17] Wu Y D, Shih T T, Lee J J. High-quality-factor filter based on a photonic crystal ring resonator for wavelength division multiplexing applications[J]. Applied Optics, 2009, 48(25): 25-31. [18] Noda S, Tomoda K, Yamamoto N, et al. Full three-dimensional photonic bandgap crystals at near-infrared wavelengths[J]. Science, 2000, 289(5479): 604-606. [19] Qiu M. Ultra-compact optical filter in two-dimensional photonic crystal[J]. Electronics Letters, 2004, 40(9): 539-540. [20] Takano H, Song B S, Asano T, et al. Highly efficient multi-channel drop filter in a two-dimensional hetero photonic crystal[J]. Optics Express, 2006, 14(8): 3491-3496. [21] Gomyo A, Ushida J, Shirane M. Highly drop-efficient channel-drop optical filters with Si-based photonic crystal slabs[J]. Thin Solid Films, 2006, 508(1): 422-425. [22] Leung K M, Liu Y F. Photon band structures: The plane-wave method[J]. Phys.rev.b, 1990, 41(14): 10188-10190.
PII: DOI: Reference:
S0030-4018(18)30631-X https://doi.org/10.1016/j.optcom.2018.07.039 OPTICS 23312
To appear in:
Optics Communications
Received date : 5 April 2018 Revised date : 6 June 2018 Accepted date : 12 July 2018 Please cite this article as: T. Zhang, J. Sun, Y. Yang, Z. Li, Photonic crystal filter based on defect mode and waveguide mode symmetry matching, Optics Communications (2018), https://doi.org/10.1016/j.optcom.2018.07.039 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Manuscript Click here to view linked References
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Photonic Crystal Filter Based on Defect Mode and Waveguide Mode Symmetry Matching Tong Zhang, Jing Sun*, Yunxing Yang, Zhixin Li College of Physics and Mechanical Engineering, Jishou University, Jishou 416000, Hunan, China *Corresponding author. E-mail address: [email protected] (J. Sun). ABSTRACT In this paper, a new type of filter based on symmetry matching between the defect and waveguide modes in a square lattice photonic crystal for 1.31-μm wavelength is designed and evaluated. The filter consists of one elliptical defect and three wire defect waveguides with two single-mode waveguides and one dual-mode waveguide. Because the point defect state has four different resonant frequencies and four corresponding types of mode field symmetries, the proposed filter uses the match or mismatch between the defect and waveguide modes to realize two resonant frequency filtering outputs through a dual channel. To evaluate the device, the transmission characteristics of the electromagnetic waves in the device are simulated using the finite element method. The quality factor and transmission coefficient of the filter is 2,017 and 0.84 at the frequency of 0.4034 c/ɑ (corresponding to 1.31-μm wavelength). This device can be applied to wave division multiplexing, filtering, and has the potential for applications in optical integrated chip and optical communications. Keywords: photonic crystal; defect mode; waveguide mode; symmetry; filter
1.Introduction Photonic crystals have attracted much attention [1,2] since they were first proposed by Yablonovitch [3] and John [4] in 1987 owing to their ability to control the flow of photons in optical fibers [5], resonators [6–9], waveguides [1,6,10], optical switches [11], filters [8,9,12–14], sensors [15], wavelength division multiplexers [16,17], and other devices. The combination of a waveguide and micro-cavity in a photonic crystal is mainly used to realize a filtering function. Two-dimensional (2D) photonic crystals play an important role in highly integrated, efficient, and stable wide-bandwidth optical communication systems because they have a flexible design, simple structure, and strong light transmission control. Multichannel photonic crystal filters at micron scale have good application potential and are receiving wide attention at present. For example, Noda et al. [18] used a combination of a waveguide and micro-cavity to implement an upload/download filter. Qiu [19] produced a filter composed of two waveguides and a dual-mode micro-cavity based on the triangular lattice photonic crystal. Takano et al. [20] designed an efficient multichannel filter based on multiple simple heterostructure filters. Feng et al. [14]
realized a one-way transmission filter based on the symmetry match of defect and waveguide 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
modes in a square lattice photonic crystal using a two-ring cavity. Gomyo et al. [21] realized a filter with a quality factor of up to 45,000 and transmission of 95% at a 1.6-μm wavelength by combining two parallel waveguides and ring cavities. Wang et al. [12] realized an ultra-narrow band filter based on a silicon 2D photonic crystal resonator and reflectors and with a quality factor of up to 1,300 and transmission of 95.3% at a 1.5-μm wavelength. Wu et al. [17] realized a multiform wavelength division multiplexer by adjusting the various dielectric column arrays in the ring cavity; the best effect was obtained when the dielectric cylinder was arranged in a 3×3 grid, yielding a transmission of 92% and quality factor of up to 3,800. In this paper, a photonic crystal filter is proposed based on the symmetry match between defect and waveguide modes in 2D square lattice photonic crystals. Because the point defect is smaller than any ring cavity [8], the device is easier to integrate, while the three waveguides are coupled with a micro-cavity. In addition, the resonant frequency of the coupled cavity can be varied by adjusting the size parameters of the elliptical dielectric cylinder in the point defect to adjust the working frequency of the filter. 2. Principle and analysis In the structure, the Si dielectric cylinder is periodic and arranged in a square in an air background with a refractive index of n = 3.4 and radius r = 0.18ɑ, where ɑ is the lattice constant. The frequency between the photonic band gap is prohibited from propagation, and the optical signals are selectively propagated by controlling the flow of photons. Then, using the plane wave expansion method [22] to solve the band gap, the TM mode (where the electric field is parallel to the axial direction of the dielectric cylinder) band gap is in the range 0.303–0.444 c/ɑ. As shown in Fig. 1, the point defect is established by converting a center circular dielectric into an elliptical dielectric cylinder with a major half axis of 0.72ɑ and minor half axis of 0.289ɑ. Here, the major axis lies along the horizontal. The four defective modes with different symmetric modes of 0.3281, 0.3409, 0.4034, and 0.4171 c/ɑ were calculated using COMSOL Multiphysics finite-element method, which can calculate the field distribution, band structure, and transmission. The corresponding modes’ spatial profiles are shown in Figs. 2(a), 2(b), 2(c), and 2(d), respectively. The red/blue regions represent positive/negative electric field distribution. Therefore, we can determine the waveguide defect modes symmetry according to the electric field distribution. When the electric field distribution on both sides of the center line is the same/different, the waveguide defect mode corresponds to even mode (symmetry)/odd mode (anti-symmetry). The figures show that the defect mode of 0.3281 c/ɑ is anti-symmetric around the major axis but symmetric around the minor axis, the defect mode of 0.3409 c/ɑ is symmetric around both the major and minor axes, the defect mode of 0.4034 c/ɑ is anti-symmetric around both the major and minor axes, and the defect mode of 0.4171 c/ɑ is symmetric around the major axis but anti-symmetric around the minor axis.
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Fig. 1. Structure of a 2D photonic crystal with a central elliptical defect whose major half axis is 0.72ɑ and minor half axis is 0.289ɑ, where the major axis lies along the horizontal of the photonic crystals.
Fig. 2. Spatial profiles of four point defect resonant frequency modes (a) ƒ = 0.3281 c/ɑ, (b) ƒ = 0.3409 c/ɑ, (c) ƒ = 0.4034 c/ɑ, and (d) ƒ = 0.4171 c/ɑ.
Figure 3(a) shows a single mode waveguide constructed by removing a dielectric cylinder row along the ΓΧ direction. The single mode waveguide band structure is shown in Fig. 3(b), in which an even-mode waveguide mode exists at frequencies in the range 0.3116–0.4440 c/ɑ. The defect mode field distribution is shown in Fig. 4(a). Furthermore, after removing a dielectric cylinder row along the ΓΧ direction and shifting the two adjacent dielectric cylinder rows on each side outward by 0.56ɑ, a dual mode waveguide is constructed, as shown in Fig. 3(c). The dual mode waveguide band structure is shown in Fig. 3(d), in which an even-mode waveguide mode exists at frequencies in the range 0.3074–0.4440 c/ɑ and an odd-mode waveguide mode exists at frequencies in the range 0.3699–0.4440 c/ɑ. The defect mode field distributions are shown in Fig. 4(b) and Fig. 4(c).
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Fig. 3. (a) Single mode waveguide 1×9 super cell constructed by removing a dielectric cylinder row in the ΓΧ direction. (b) Even-mode waveguide mode at frequencies in the range 0.3116–0.4440 c/ɑ. (c) Removal of a dielectric cylinder row along the ΓΧ direction and shifting the two adjacent dielectric cylinder rows on each side outward by 0.56ɑ to construct a dual mode waveguide. (d) Even-mode waveguide mode at frequencies in the range 0.3074–0.4440 c/ɑ and odd-mode waveguide mode at frequencies in the range 0.3699–0.4440 c/ɑ.
Fig. 4. (a) Single mode waveguide field distribution at f=0.3831 c/ɑ, k=0.279. (b) Odd-mode waveguide mode field distribution at f=0.3831 c/ɑ, k=0.13. (c) Even-mode waveguide mode field distribution at f=0.3831 c/ɑ, k=0.34.
3. Design simulation of photonic crystal filter To design a filter based on the symmetry match between the defect and waveguide modes, the structure shown in Fig. 5 is used. The structure consists of a point defect and the three waveguides shown in Figs. 1 and 3. Moreover, an even-mode waveguide mode exists in both waveguides 1 and 2, while even-mode and odd-mode waveguide modes exist in waveguide 3. When the defect mode and waveguide mode symmetry are matched, the point defect is excited and the optical signal is coupled to the output waveguide to realize a filtering output.
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Fig. 5. Filter consisting of two single mode waveguides (waveguides 1 and 2), a dual mode waveguide (waveguide 3), and a central elliptical defect. The input port is port 3, and the two output ports are ports 1 and 2.
When an anti-symmetric optical signal at a frequency of 0.4171 c/ɑ is input at port 3 of waveguide 3, waveguide 3 has two modes, the odd-mode waveguide mode is anti-symmetric around the waveguide center vertical line and the defect mode is also anti-symmetric around the direction of the minor axis. Here, the defect mode symmetrically matches with the waveguide mode and the center point defect is excited. Furthermore, the defect mode and even-mode waveguide mode in waveguide 1 are both symmetric around the waveguide center horizontal line. In this case, the defect mode symmetrically matches with the waveguide mode of waveguide 1, so they are coupled. Moreover, the waveguide mode in waveguide 2 is anti-symmetric around the waveguide center vertical line, so the defect mode symmetrically matches with the waveguide mode of waveguide 2, and they are coupled. However, because the coupling capability between the point defect and waveguide 1 is better than that of waveguide 2, ultimately, most of the optical signal with a frequency of 0.4171 c/ɑ is output from port 1. Similarly, when the anti-symmetric optical signal at a frequency of 0.4034 c/ɑ is input at port 3 of waveguide 3, the symmetry matching around the direction of the minor axis between the point defect and waveguide 3 is consistent with the frequency of 0.4171 c/ɑ and the center point defect is hence excited. Furthermore, the even-mode waveguide mode in waveguide 1 is symmetric around the waveguide center horizontal line; however, the defect mode is anti-symmetric around the direction of the major axis, the defect mode does not symmetrically match with the waveguide mode of waveguide 1, and they cannot be coupled. Hence, there is no signal output through output port 1 of waveguide 1, and the transmission is nearly zero. In addition, the defect mode and waveguide mode in waveguide 2 are both anti-symmetric around the direction of the minor axis; thus, the defect mode symmetrically matches with the waveguide mode of waveguide 2 and they are coupled, and a frequency of 0.4034 c/ɑ is output from port 2. To verify the above theoretical analysis, the transmission spectrum for the filter obtained by calculating the ratio of light energy as well as by setting three integral segments near the in/output ports about two single mode waveguides at each output port (ports 1 and 2) and the dual mode waveguide 3 input port 3 are shown in Fig. 6(a). The figure shows that the transmission peak of
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waveguide 1 is at frequency 0.4172 c/ɑ, where the transmission is 0.96 incident from port 1. Moreover, there is a waveguide 2 transmission peak at frequency 0.4034 c/ɑ, where the transmission is 0.84 incident from port 2. The frequency of the two transmission peaks 0.4034 c/ɑ and 0.4172 c/ɑ are the same as the resonant frequencies of the defect mode (0.4034 c/ɑ and 0.4171 c/ɑ); thus, the theoretical analysis is consistent with the verification results.
Fig. 6. (a) Transmission spectrum of the two filter output ports. (b) Spatical profiles at the frequency of 0.4034 c/ɑ. (c) Spatical profiles at the frequency of 0.4172 c/ɑ.
4. Conclusion In this paper, a new type of filter with a structure consisting of three coupled waveguides and one point defect was proposed. It uses symmetry matching of the defect and waveguide modes to realize a filtering function of two resonance frequency optical signals. The structure can adjust the working frequency by varying the parameters of the elliptic column in the point defect, as well as simple modulation. Because it is a photonic crystal, it has high transmission efficiency, good stability, small size, and is easy to integrate. Moreover, the defect modes proposed in this paper can also be applied to devices with one-way optical signal transmission such as photodiodes. Funding This work was supported by the Excellent Youth Program in Education Department of Hunan Province (No. 17B212), Postgraduate Research Projects of Jishou University (No.Jdy17033), and Postgraduate Educational Reform (No. JG2017A01). References [1] Mekis A, Chen J C, Kurland I I, et al. High Transmission through Sharp Bends in Photonic Crystal Waveguides[J]. Physical Review Letters, 1996, 77(18): 3787-3790. [2] Edition S. Photonic crystals:molding the flow of light[M]. Princeton University Press, 1995. [3] Yablonovitch E. Inhibited spontaneous emission in solid-state physics and electronics[J]. Phys Rev Lett, 1987, 58(20): 2059-2062. [4] John S. Strong localization of photons in certain disordered dielectric superlattices[J]. Phys Rev Lett, 1987, 58(23): 2486-2489. [5] Barkou S E, Broeng J, Bjarklev A. Silica-air photonic crystal fiber design that permits waveguiding by a true
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