Abstract

We propose a ring-type Fabry-Pérot filter (RFPF) based on the self-collimation effect in photonic crystals. The transmission characteristics of self-collimated beams are experimentally measured in this structure and compared with the results obtained with the simulations. Bending and splitting mechanisms of light beams by the line defects introduced into the RFPF are used to control the self-collimated beam. Antireflection structures are also employed at the input and output photonic crystal interfaces in order to minimize the coupling loss. Reflectance of the line-defect beam splitters can be controlled by adjusting the radius of defect rods. As the reflectance of the line-defect beam splitters increases, the transmission peaks become sharper and the filter provides a Q-factor as high as 1037. Proposed RFPF can be used as a sharply tuned optical filter or as a spectrum analyzer based on the self-collimation phenomena of photonic crystals. Furthermore, it is suitable for a building block of photonic integrated circuits, as it does not back reflect any of the incoming self-collimated beams owing to the antireflection structure applied.

© 2010 OSA

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2010 (1)

2009 (2)

X. Chen, Z. Qiang, D. Zhao, H. Li, Y. Qiu, W. Yang, and W. Zhou, “Polarization-independent drop filters based on photonic crystal self-collimation ring resonators,” Opt. Express 17(22), 19808–19813 (2009).
[CrossRef] [PubMed]

T.-T. Kim, S.-G. Lee, M.-W. Kim, H. Y. Park, and J.-E. Kim, “Experimental demonstration of reflection minimization at two-dimensional photonic crystal interfaces via antireflection structures,” Appl. Phys. Lett. 95(1), 011119 (2009).
[CrossRef]

2008 (1)

2007 (1)

D. Zhao, J. Zhang, P. Yao, X. Jiang, and X. Chen, “Photonic crystal Mach-Zehnder interferometer based on self-collimation,” Appl. Phys. Lett. 90(23), 231114 (2007).
[CrossRef]

2006 (1)

P. T. Rakich, M. S. Dahlem, S. Tandon, M. Ibanescu, M. Soljacić, G. S. Petrich, J. D. Joannopoulos, L. A. Kolodziejski, and E. P. Ippen, “Achieving centimetre-scale supercollimation in a large-area two-dimensional photonic crystal,” Nat. Mater. 5(2), 93–96 (2006).
[CrossRef] [PubMed]

2005 (2)

S.-G. Lee, S. S. Oh, J.-E. Kim, H. Y. Park, and C.-S. Kee, “Line-defect-induced bending and splitting of self-collimated beams in two-dimensional photonic crystals,” Appl. Phys. Lett. 87(18), 181106 (2005).
[CrossRef]

B. Momeni and A. Adibi, “Adiabatic matching stage for coupling of light to extended Bloch modes of photonic crystals,” Appl. Phys. Lett. 87(17), 171104 (2005).
[CrossRef]

2004 (6)

J. Zimmermann, M. Kamp, A. Forchel, and R. Marz, “Photonic crystal waveguide directional couplers as wavelength selective optical filters,” Opt. Commun. 230(4-6), 387–392 (2004).
[CrossRef]

J. Witzens, M. Hochberg, T. Baehr-Jones, and A. Scherer, “Mode matching interface for efficient coupling of light into planar photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 69(4), 046609 (2004).
[CrossRef] [PubMed]

Z. Li, H. Chen, Z. Song, F. Yang, and S. Feng, “Finite-width waveguide and waveguide intersections for self-collimated beams in photonic crystals,” Appl. Phys. Lett. 85(21), 4834 (2004).
[CrossRef]

D. W. Prather, S. Shi, D. M. Pustai, C. Chen, S. Venkataraman, A. Sharkawy, G. J. Schneider, and J. Murakowski, “Dispersion-based optical routing in photonic crystals,” Opt. Lett. 29(1), 50–52 (2004).
[CrossRef] [PubMed]

F. S. Chien, Y. Hsu, W. Hsieh, and S. Cheng, “Dual wavelength demultiplexing by coupling and decoupling of photonic crystal waveguides,” Opt. Express 12(6), 1119–1125 (2004).
[CrossRef] [PubMed]

S. Kim, I. Park, H. Lim, and C.-S. Kee, “Highly efficient photonic crystal-based multichannel drop filters of three-port system with reflection feedback,” Opt. Express 12(22), 5518–5525 (2004).
[CrossRef] [PubMed]

2003 (3)

D. Chigrin, S. Enoch, C. Sotomayor Torres, and G. Tayeb, “Self-guiding in two-dimensional photonic crystals,” Opt. Express 11(10), 1203–1211 (2003).
[CrossRef] [PubMed]

X. Yu and S. Fan, “Bends and splitters for self-collimated beams in photonic crystals,” Appl. Phys. Lett. 83(16), 3251 (2003).
[CrossRef]

C. Jin, S. Fan, S. Han, and D. Zhang, “Refectionless multichannel wavelength demultiplexer in a transmission resonator configuration,” IEEE J. Quantum Electron. 39(1), 160–165 (2003).
[CrossRef]

2002 (1)

2001 (3)

1999 (2)

T. F. Krauss and R. M. De La Rue, “Photonic crystals in the optical regime-past, present and future,” Prog. Quantum Electron. 23(2), 51–96 (1999).
[CrossRef]

E. Centeno, B. Guizal, and D. Felbacq, “Multiplexing & demultiplexing with photonic crystal,” J. Opt. A, Pure Appl. Opt. 1(5), 103 (1999).
[CrossRef]

1994 (1)

J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114(2), 185–200 (1994).
[CrossRef]

1987 (1)

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58(23), 2486–2489 (1987).
[CrossRef] [PubMed]

Adibi, A.

B. Momeni and A. Adibi, “Adiabatic matching stage for coupling of light to extended Bloch modes of photonic crystals,” Appl. Phys. Lett. 87(17), 171104 (2005).
[CrossRef]

Baehr-Jones, T.

J. Witzens, M. Hochberg, T. Baehr-Jones, and A. Scherer, “Mode matching interface for efficient coupling of light into planar photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 69(4), 046609 (2004).
[CrossRef] [PubMed]

Berenger, J.-P.

J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114(2), 185–200 (1994).
[CrossRef]

Centeno, E.

E. Centeno, B. Guizal, and D. Felbacq, “Multiplexing & demultiplexing with photonic crystal,” J. Opt. A, Pure Appl. Opt. 1(5), 103 (1999).
[CrossRef]

Chen, C.

Chen, H.

Z. Li, H. Chen, Z. Song, F. Yang, and S. Feng, “Finite-width waveguide and waveguide intersections for self-collimated beams in photonic crystals,” Appl. Phys. Lett. 85(21), 4834 (2004).
[CrossRef]

Chen, X.

X. Chen, Z. Qiang, D. Zhao, H. Li, Y. Qiu, W. Yang, and W. Zhou, “Polarization-independent drop filters based on photonic crystal self-collimation ring resonators,” Opt. Express 17(22), 19808–19813 (2009).
[CrossRef] [PubMed]

D. Zhao, J. Zhang, P. Yao, X. Jiang, and X. Chen, “Photonic crystal Mach-Zehnder interferometer based on self-collimation,” Appl. Phys. Lett. 90(23), 231114 (2007).
[CrossRef]

Cheng, S.

Chien, F. S.

Chigrin, D.

Choi, J.-S.

Dahlem, M. S.

P. T. Rakich, M. S. Dahlem, S. Tandon, M. Ibanescu, M. Soljacić, G. S. Petrich, J. D. Joannopoulos, L. A. Kolodziejski, and E. P. Ippen, “Achieving centimetre-scale supercollimation in a large-area two-dimensional photonic crystal,” Nat. Mater. 5(2), 93–96 (2006).
[CrossRef] [PubMed]

De La Rue, R. M.

T. F. Krauss and R. M. De La Rue, “Photonic crystals in the optical regime-past, present and future,” Prog. Quantum Electron. 23(2), 51–96 (1999).
[CrossRef]

Enoch, S.

Fan, S.

C. Jin, S. Fan, S. Han, and D. Zhang, “Refectionless multichannel wavelength demultiplexer in a transmission resonator configuration,” IEEE J. Quantum Electron. 39(1), 160–165 (2003).
[CrossRef]

X. Yu and S. Fan, “Bends and splitters for self-collimated beams in photonic crystals,” Appl. Phys. Lett. 83(16), 3251 (2003).
[CrossRef]

Felbacq, D.

E. Centeno, B. Guizal, and D. Felbacq, “Multiplexing & demultiplexing with photonic crystal,” J. Opt. A, Pure Appl. Opt. 1(5), 103 (1999).
[CrossRef]

Feng, S.

Z. Li, H. Chen, Z. Song, F. Yang, and S. Feng, “Finite-width waveguide and waveguide intersections for self-collimated beams in photonic crystals,” Appl. Phys. Lett. 85(21), 4834 (2004).
[CrossRef]

Forchel, A.

J. Zimmermann, M. Kamp, A. Forchel, and R. Marz, “Photonic crystal waveguide directional couplers as wavelength selective optical filters,” Opt. Commun. 230(4-6), 387–392 (2004).
[CrossRef]

Guizal, B.

E. Centeno, B. Guizal, and D. Felbacq, “Multiplexing & demultiplexing with photonic crystal,” J. Opt. A, Pure Appl. Opt. 1(5), 103 (1999).
[CrossRef]

Han, S.

C. Jin, S. Fan, S. Han, and D. Zhang, “Refectionless multichannel wavelength demultiplexer in a transmission resonator configuration,” IEEE J. Quantum Electron. 39(1), 160–165 (2003).
[CrossRef]

Hochberg, M.

J. Witzens, M. Hochberg, T. Baehr-Jones, and A. Scherer, “Mode matching interface for efficient coupling of light into planar photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 69(4), 046609 (2004).
[CrossRef] [PubMed]

Hsieh, W.

Hsu, Y.

Ibanescu, M.

P. T. Rakich, M. S. Dahlem, S. Tandon, M. Ibanescu, M. Soljacić, G. S. Petrich, J. D. Joannopoulos, L. A. Kolodziejski, and E. P. Ippen, “Achieving centimetre-scale supercollimation in a large-area two-dimensional photonic crystal,” Nat. Mater. 5(2), 93–96 (2006).
[CrossRef] [PubMed]

Ippen, E. P.

P. T. Rakich, M. S. Dahlem, S. Tandon, M. Ibanescu, M. Soljacić, G. S. Petrich, J. D. Joannopoulos, L. A. Kolodziejski, and E. P. Ippen, “Achieving centimetre-scale supercollimation in a large-area two-dimensional photonic crystal,” Nat. Mater. 5(2), 93–96 (2006).
[CrossRef] [PubMed]

Jiang, X.

D. Zhao, J. Zhang, P. Yao, X. Jiang, and X. Chen, “Photonic crystal Mach-Zehnder interferometer based on self-collimation,” Appl. Phys. Lett. 90(23), 231114 (2007).
[CrossRef]

Jin, C.

C. Jin, S. Fan, S. Han, and D. Zhang, “Refectionless multichannel wavelength demultiplexer in a transmission resonator configuration,” IEEE J. Quantum Electron. 39(1), 160–165 (2003).
[CrossRef]

Joannopoulos, J. D.

P. T. Rakich, M. S. Dahlem, S. Tandon, M. Ibanescu, M. Soljacić, G. S. Petrich, J. D. Joannopoulos, L. A. Kolodziejski, and E. P. Ippen, “Achieving centimetre-scale supercollimation in a large-area two-dimensional photonic crystal,” Nat. Mater. 5(2), 93–96 (2006).
[CrossRef] [PubMed]

S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8(3), 173–190 (2001).
[CrossRef] [PubMed]

John, S.

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58(23), 2486–2489 (1987).
[CrossRef] [PubMed]

Johnson, S. G.

Kamp, M.

J. Zimmermann, M. Kamp, A. Forchel, and R. Marz, “Photonic crystal waveguide directional couplers as wavelength selective optical filters,” Opt. Commun. 230(4-6), 387–392 (2004).
[CrossRef]

Kee, C. S.

Kee, C.-S.

Kim, J. E.

Kim, J.-E.

T.-T. Kim, S.-G. Lee, M.-W. Kim, H. Y. Park, and J.-E. Kim, “Experimental demonstration of reflection minimization at two-dimensional photonic crystal interfaces via antireflection structures,” Appl. Phys. Lett. 95(1), 011119 (2009).
[CrossRef]

S.-G. Lee, J.-S. Choi, J.-E. Kim, H. Y. Park, and C.-S. Kee, “Reflection minimization at two-dimensional photonic crystal interfaces,” Opt. Express 16(6), 4270–4277 (2008).
[CrossRef] [PubMed]

S.-G. Lee, S. S. Oh, J.-E. Kim, H. Y. Park, and C.-S. Kee, “Line-defect-induced bending and splitting of self-collimated beams in two-dimensional photonic crystals,” Appl. Phys. Lett. 87(18), 181106 (2005).
[CrossRef]

Kim, M.-W.

T.-T. Kim, S.-G. Lee, M.-W. Kim, H. Y. Park, and J.-E. Kim, “Experimental demonstration of reflection minimization at two-dimensional photonic crystal interfaces via antireflection structures,” Appl. Phys. Lett. 95(1), 011119 (2009).
[CrossRef]

Kim, S.

Kim, T.-T.

T.-T. Kim, S.-G. Lee, H. Y. Park, J. E. Kim, and C. S. Kee, “Asymmetric Mach-Zehnder filter based on self-collimation phenomenon in two-dimensional photonic crystals,” Opt. Express 18(6), 5384–5389 (2010).
[CrossRef] [PubMed]

T.-T. Kim, S.-G. Lee, M.-W. Kim, H. Y. Park, and J.-E. Kim, “Experimental demonstration of reflection minimization at two-dimensional photonic crystal interfaces via antireflection structures,” Appl. Phys. Lett. 95(1), 011119 (2009).
[CrossRef]

Kolodziejski, L. A.

P. T. Rakich, M. S. Dahlem, S. Tandon, M. Ibanescu, M. Soljacić, G. S. Petrich, J. D. Joannopoulos, L. A. Kolodziejski, and E. P. Ippen, “Achieving centimetre-scale supercollimation in a large-area two-dimensional photonic crystal,” Nat. Mater. 5(2), 93–96 (2006).
[CrossRef] [PubMed]

Koshiba, M.

Krauss, T. F.

T. F. Krauss and R. M. De La Rue, “Photonic crystals in the optical regime-past, present and future,” Prog. Quantum Electron. 23(2), 51–96 (1999).
[CrossRef]

Lee, S.-G.

T.-T. Kim, S.-G. Lee, H. Y. Park, J. E. Kim, and C. S. Kee, “Asymmetric Mach-Zehnder filter based on self-collimation phenomenon in two-dimensional photonic crystals,” Opt. Express 18(6), 5384–5389 (2010).
[CrossRef] [PubMed]

T.-T. Kim, S.-G. Lee, M.-W. Kim, H. Y. Park, and J.-E. Kim, “Experimental demonstration of reflection minimization at two-dimensional photonic crystal interfaces via antireflection structures,” Appl. Phys. Lett. 95(1), 011119 (2009).
[CrossRef]

S.-G. Lee, J.-S. Choi, J.-E. Kim, H. Y. Park, and C.-S. Kee, “Reflection minimization at two-dimensional photonic crystal interfaces,” Opt. Express 16(6), 4270–4277 (2008).
[CrossRef] [PubMed]

S.-G. Lee, S. S. Oh, J.-E. Kim, H. Y. Park, and C.-S. Kee, “Line-defect-induced bending and splitting of self-collimated beams in two-dimensional photonic crystals,” Appl. Phys. Lett. 87(18), 181106 (2005).
[CrossRef]

Li, H.

Li, Z.

Z. Li, H. Chen, Z. Song, F. Yang, and S. Feng, “Finite-width waveguide and waveguide intersections for self-collimated beams in photonic crystals,” Appl. Phys. Lett. 85(21), 4834 (2004).
[CrossRef]

Lim, H.

Marz, R.

J. Zimmermann, M. Kamp, A. Forchel, and R. Marz, “Photonic crystal waveguide directional couplers as wavelength selective optical filters,” Opt. Commun. 230(4-6), 387–392 (2004).
[CrossRef]

Momeni, B.

B. Momeni and A. Adibi, “Adiabatic matching stage for coupling of light to extended Bloch modes of photonic crystals,” Appl. Phys. Lett. 87(17), 171104 (2005).
[CrossRef]

Murakowski, J.

Oh, S. S.

S.-G. Lee, S. S. Oh, J.-E. Kim, H. Y. Park, and C.-S. Kee, “Line-defect-induced bending and splitting of self-collimated beams in two-dimensional photonic crystals,” Appl. Phys. Lett. 87(18), 181106 (2005).
[CrossRef]

Park, H. Y.

T.-T. Kim, S.-G. Lee, H. Y. Park, J. E. Kim, and C. S. Kee, “Asymmetric Mach-Zehnder filter based on self-collimation phenomenon in two-dimensional photonic crystals,” Opt. Express 18(6), 5384–5389 (2010).
[CrossRef] [PubMed]

T.-T. Kim, S.-G. Lee, M.-W. Kim, H. Y. Park, and J.-E. Kim, “Experimental demonstration of reflection minimization at two-dimensional photonic crystal interfaces via antireflection structures,” Appl. Phys. Lett. 95(1), 011119 (2009).
[CrossRef]

S.-G. Lee, J.-S. Choi, J.-E. Kim, H. Y. Park, and C.-S. Kee, “Reflection minimization at two-dimensional photonic crystal interfaces,” Opt. Express 16(6), 4270–4277 (2008).
[CrossRef] [PubMed]

S.-G. Lee, S. S. Oh, J.-E. Kim, H. Y. Park, and C.-S. Kee, “Line-defect-induced bending and splitting of self-collimated beams in two-dimensional photonic crystals,” Appl. Phys. Lett. 87(18), 181106 (2005).
[CrossRef]

Park, I.

Petrich, G. S.

P. T. Rakich, M. S. Dahlem, S. Tandon, M. Ibanescu, M. Soljacić, G. S. Petrich, J. D. Joannopoulos, L. A. Kolodziejski, and E. P. Ippen, “Achieving centimetre-scale supercollimation in a large-area two-dimensional photonic crystal,” Nat. Mater. 5(2), 93–96 (2006).
[CrossRef] [PubMed]

Prather, D. W.

Pustai, D. M.

Qiang, Z.

Qiu, Y.

Rakich, P. T.

P. T. Rakich, M. S. Dahlem, S. Tandon, M. Ibanescu, M. Soljacić, G. S. Petrich, J. D. Joannopoulos, L. A. Kolodziejski, and E. P. Ippen, “Achieving centimetre-scale supercollimation in a large-area two-dimensional photonic crystal,” Nat. Mater. 5(2), 93–96 (2006).
[CrossRef] [PubMed]

Scherer, A.

J. Witzens, M. Hochberg, T. Baehr-Jones, and A. Scherer, “Mode matching interface for efficient coupling of light into planar photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 69(4), 046609 (2004).
[CrossRef] [PubMed]

Schneider, G. J.

Sharkawy, A.

Shi, S.

Soljacic, M.

P. T. Rakich, M. S. Dahlem, S. Tandon, M. Ibanescu, M. Soljacić, G. S. Petrich, J. D. Joannopoulos, L. A. Kolodziejski, and E. P. Ippen, “Achieving centimetre-scale supercollimation in a large-area two-dimensional photonic crystal,” Nat. Mater. 5(2), 93–96 (2006).
[CrossRef] [PubMed]

Song, Z.

Z. Li, H. Chen, Z. Song, F. Yang, and S. Feng, “Finite-width waveguide and waveguide intersections for self-collimated beams in photonic crystals,” Appl. Phys. Lett. 85(21), 4834 (2004).
[CrossRef]

Sotomayor Torres, C.

Tandon, S.

P. T. Rakich, M. S. Dahlem, S. Tandon, M. Ibanescu, M. Soljacić, G. S. Petrich, J. D. Joannopoulos, L. A. Kolodziejski, and E. P. Ippen, “Achieving centimetre-scale supercollimation in a large-area two-dimensional photonic crystal,” Nat. Mater. 5(2), 93–96 (2006).
[CrossRef] [PubMed]

Tayeb, G.

Venkataraman, S.

Witzens, J.

J. Witzens, M. Hochberg, T. Baehr-Jones, and A. Scherer, “Mode matching interface for efficient coupling of light into planar photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 69(4), 046609 (2004).
[CrossRef] [PubMed]

Yang, F.

Z. Li, H. Chen, Z. Song, F. Yang, and S. Feng, “Finite-width waveguide and waveguide intersections for self-collimated beams in photonic crystals,” Appl. Phys. Lett. 85(21), 4834 (2004).
[CrossRef]

Yang, W.

Yao, P.

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[CrossRef]

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[CrossRef]

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C. Jin, S. Fan, S. Han, and D. Zhang, “Refectionless multichannel wavelength demultiplexer in a transmission resonator configuration,” IEEE J. Quantum Electron. 39(1), 160–165 (2003).
[CrossRef]

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[CrossRef]

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X. Chen, Z. Qiang, D. Zhao, H. Li, Y. Qiu, W. Yang, and W. Zhou, “Polarization-independent drop filters based on photonic crystal self-collimation ring resonators,” Opt. Express 17(22), 19808–19813 (2009).
[CrossRef] [PubMed]

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[CrossRef]

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Zimmermann, J.

J. Zimmermann, M. Kamp, A. Forchel, and R. Marz, “Photonic crystal waveguide directional couplers as wavelength selective optical filters,” Opt. Commun. 230(4-6), 387–392 (2004).
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[CrossRef]

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[CrossRef]

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[CrossRef]

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[CrossRef]

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[CrossRef]

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[CrossRef]

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http://ab-initio.mit.edu/meep

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Figures (5)

Fig. 1
Fig. 1

(a) Equifrequency contours of the 2D PC. The inset represents the 2D square lattice PC composed of alumina rods (ε = 9.7) in air. (b) Schematic diagram of a ring-type Fabry-Pérot filter composed of two line-defect beam splitters (BS 1, BS 2) and two perfect mirrors (M1 , M2 ). (c) The ARSs are introduced at the input and two output ports. (d) Zoom-ins of the beam splitters and mirrors. Arrows indicate the direction of light propagation.

Fig. 2
Fig. 2

(a) Configuration of the FDTD simulations. Perfectly matched layers (PMLs) are placed at the ends of computational domain in the x- and y-directions. A Gaussian beam of width 3a is launched along the ΓM direction. (b) Schematic diagram of transmission measurement setup for the microwave. Two aluminum plates hold alumina rods vertically.

Fig. 3
Fig. 3

(a) Simulated spatial distribution of the steady-state electric field of the self-collimated beams of f = 12.50 GHZ at the line-defect beam splitter. (b) Simulated reflection and transmission powers at the beam splitter which are normalized with respect to the input power as a function of the radius of defect rods rd in the line-defect.

Fig. 4
Fig. 4

Transmission spectra of the PC-RFPF for (a) rd = 1.75 mm (RS = 0.12), (b) rd = 1.50 mm (RS = 0.49), (c) rd = 1.25 mm (RS = 0.77), (d) rd = 1.00 mm (RS = 0.88). Two dashed lines represent the transmission obtained from the simulations at the drop port (black line) and the through port (red line), respectively, while the solid lines correspond to the experimentally measured results.

Fig. 5
Fig. 5

Simulated spatial distributions of the steady-state electric fields for the self-collimated beams at frequencies (a) 12.44 GHz for the drop port and (b) 12.30 GHz for the through port. (c) The proposed PC-RFPF combines the light beam of the resonance frequency fr = 12.44 GHz which is inserted into the additional input port and a light beam of non-resonance frequencies which are injected into the original input port. Calculations are performed for the case of rd = 1.00 mm.

Equations (2)

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I 1 = ( 1 R S R M ) 2 ( 1 R S R M ) 2 + 4 R S R M sin 2 ( δ / 2 ) I 0 ,
I 2 = 4 R S R M sin 2 ( δ / 2 ) ( 1 R S R M ) 2 + 4 R S R M sin 2 ( δ / 2 ) I 0 ,

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