Abstract

We theoretically investigate the collimation ability of photonic crystals (PhCs) and present a design of rectangular lattice PhC structure which has ultra-high collimation ability, referred to hyper collimation ability of PhC in this work. The competition between the range and the flatness of a “flat segment” on the PhC equi-frequency contour (EFC) is revealed, so that both should be considered simultaneously if we hope to evaluate the collimation ability of a PhC structure. We introduce a new dimensionless value, the normalized collimation length (NCL), to evaluate the collimation ability of a PhC structure. We find that the hyper collimation ability can be achieved by tuning the aspect ratio of a rectangular lattice PhC. It is also demonstrated that our theoretical predictions of the propagation behavior of beams agree very well with numerical experiments. We propose that the PhCs with hyper collimation ability could be widely used in the design of photonic circuits and other devices.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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    [Crossref] [PubMed]
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    [Crossref]
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2018 (1)

M. Gumus, I. H. Giden, O. Akcaalan, M. Turduev, and H. Kurt, “Enhanced superprism effect in symmetry reduced photonic crystals,” Appl. Phys. Lett. 113, 131103 (2018).
[Crossref]

2017 (2)

Y. Li, F. Meng, S. Zhou, M. Lu, and X. Huang, “Broadband all-angle negative refraction by optimized phononic crystals,” Sci. Reports 7, 7445 (2017).
[Crossref]

S. Gao, Y. Dou, Q. Li, and X. Jiang, “Tunable photonic crystal lens with high sensitivity of refractive index,” Opt. Express 25, 7112–7120 (2017).
[Crossref] [PubMed]

2014 (1)

2013 (1)

2012 (1)

J. Arlandis, E. Centeno, R. Pollès, A. Moreau, J. Campos, O. Gauthier-Lafaye, and A. Monmayrant, “Mesoscopic self-collimation and slow light in all-positive index layered photonic crystals,” Phys. Rev. Lett. 108, 037401 (2012).
[Crossref] [PubMed]

2010 (3)

Z. Yang, A. Wu, N. Fang, X. Lin, X. Jiang, S. Zou, and X. Wang, “Millimeter-scale self-collimation in planar photonic crystals fabricated by cmos technology,” Opt. Commun. 283, 604–607 (2010).
[Crossref]

Z.-l. Wang, H.-t. Jiang, Y.-h. Li, and C. Hong, “Enhancement of self-collimated fields in photonic crystals consisting of two kinds of single-negative materials,” Opt. Express 18, 14312–14318 (2010).

H. M. Nguyen, M. A. Dundar, R. W. van der Heijden, E. W. J. M. van der Drift, H. W. M. Salemink, S. Rogge, and J. Caro, “Compact mach-zehnder interferometer based on self-collimation of light in a silicon photonic crystal,” Opt. Express 18, 6437–6446 (2010).
[Crossref] [PubMed]

2009 (3)

2008 (3)

D. Gao, Z. Zhou, and D. S. Citrin, “Self-collimated waveguide bends and partial bandgap reflection of photonic crystals with parallelogram lattice,” J. Opt. Soc. Am. A Opt. Image Sci. Vis. 25, 791–795 (2008).
[Crossref] [PubMed]

Y. Xu, C. Xiaojun, S. Lan, Q. Guo, W. Hu, and W. Lijun, “The all-angle self-collimating phenomenon in photonic crystals with rectangular symmetry,” J. Opt. A Pure Appl. Opt. 10, 085201 (2008).
[Crossref]

F. J. Lawrence, L. C. Botten, K. B. Dossou, and C. M. D. Sterke, “Antireflection coatings for two-dimensional photonic crystals using a rigorous impedance definition,” Appl. Phys. Lett. 93, 171104 (2008).
[Crossref]

2007 (1)

D. W. Prather, S. Shi, J. Murakowski, G. J. Schneider, A. Sharkawy, C. Chen, B. Miao, and R. Martin, “Self-collimation in photonic crystal structures: a new paradigm for applications and device development,” J. Phys. D Appl. Phys. 40, 2635–2651 (2007).
[Crossref]

2006 (1)

P. T. Rakich, M. S. Dahlem, S. Tandon, M. Ibanescu, M. Soljačić, 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, 93–96 (2006).
[Crossref] [PubMed]

2005 (2)

M. J. Steel, R. Zoli, C. Grillet, R. C. Mcphedran, D. Sterke, C. Martijn, A. Norton, P. Bassi, and B. J. Eggleton, “Analytic properties of photonic crystal superprism parameters,” Phys. Rev. E 71, 056608 (2005).
[Crossref]

T. P. White, C. M. D. Sterke, R. C. Mcphedran, and L. C. Botten, “Highly efficient wide-angle transmission into uniform rod-type photonic crystals,” Appl. Phys. Lett. 87, 5920 (2005).
[Crossref]

2004 (1)

2003 (1)

2002 (1)

C. Luo, S. G. Johnson, and J. D. Joannopoulos, “All-angle negative refraction without negative effective index,” Phys. Rev. B 65, 201104 (2002).
[Crossref]

1999 (1)

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys. Lett. 74, 1212–1214 (1999).
[Crossref]

1998 (1)

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, 10096–10099 (1998).
[Crossref]

1993 (1)

E. Yablonovitch, “Photonic band-gap structures,” J. Opt. Soc. Am. B-optical Phys. 10, 283–295 (1993).
[Crossref]

1990 (1)

K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structure,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[Crossref] [PubMed]

1987 (2)

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[Crossref] [PubMed]

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

Akcaalan, O.

M. Gumus, I. H. Giden, O. Akcaalan, M. Turduev, and H. Kurt, “Enhanced superprism effect in symmetry reduced photonic crystals,” Appl. Phys. Lett. 113, 131103 (2018).
[Crossref]

Arlandis, J.

J. Arlandis, E. Centeno, R. Pollès, A. Moreau, J. Campos, O. Gauthier-Lafaye, and A. Monmayrant, “Mesoscopic self-collimation and slow light in all-positive index layered photonic crystals,” Phys. Rev. Lett. 108, 037401 (2012).
[Crossref] [PubMed]

Bassi, P.

M. J. Steel, R. Zoli, C. Grillet, R. C. Mcphedran, D. Sterke, C. Martijn, A. Norton, P. Bassi, and B. J. Eggleton, “Analytic properties of photonic crystal superprism parameters,” Phys. Rev. E 71, 056608 (2005).
[Crossref]

Botten, L. C.

F. J. Lawrence, L. C. Botten, K. B. Dossou, and C. M. D. Sterke, “Antireflection coatings for two-dimensional photonic crystals using a rigorous impedance definition,” Appl. Phys. Lett. 93, 171104 (2008).
[Crossref]

T. P. White, C. M. D. Sterke, R. C. Mcphedran, and L. C. Botten, “Highly efficient wide-angle transmission into uniform rod-type photonic crystals,” Appl. Phys. Lett. 87, 5920 (2005).
[Crossref]

Cabrini, S.

V. Mocella, S. Cabrini, A. S. P. Chang, P. Dardano, L. Moretti, I. Rendina, D. Qlynick, B. Harteneck, and S. Dhuey, “Self-collimation of light over millimeter-scale distance in a quasi-zero-average-index metamaterial,” Phys. Rev. Lett. 102, 133902 (2009).
[Crossref] [PubMed]

Campos, J.

J. Arlandis, E. Centeno, R. Pollès, A. Moreau, J. Campos, O. Gauthier-Lafaye, and A. Monmayrant, “Mesoscopic self-collimation and slow light in all-positive index layered photonic crystals,” Phys. Rev. Lett. 108, 037401 (2012).
[Crossref] [PubMed]

Caro, J.

Centeno, E.

J. Arlandis, E. Centeno, R. Pollès, A. Moreau, J. Campos, O. Gauthier-Lafaye, and A. Monmayrant, “Mesoscopic self-collimation and slow light in all-positive index layered photonic crystals,” Phys. Rev. Lett. 108, 037401 (2012).
[Crossref] [PubMed]

Chan, C. T.

K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structure,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[Crossref] [PubMed]

Chang, A. S. P.

V. Mocella, S. Cabrini, A. S. P. Chang, P. Dardano, L. Moretti, I. Rendina, D. Qlynick, B. Harteneck, and S. Dhuey, “Self-collimation of light over millimeter-scale distance in a quasi-zero-average-index metamaterial,” Phys. Rev. Lett. 102, 133902 (2009).
[Crossref] [PubMed]

Chen, C.

D. W. Prather, S. Shi, J. Murakowski, G. J. Schneider, A. Sharkawy, C. Chen, B. Miao, and R. Martin, “Self-collimation in photonic crystal structures: a new paradigm for applications and device development,” J. Phys. D Appl. Phys. 40, 2635–2651 (2007).
[Crossref]

D. M. Pustai, S. Shi, C. Chen, A. Sharkawy, and D. Prather, “Analysis of splitters for self-collimated beams in planar photonic crystals,” Opt. Express 12, 1823–1831 (2004).
[Crossref] [PubMed]

Chen, L.

Chen, X.

Chigrin, D. N.

Citrin, D. S.

D. Gao, Z. Zhou, and D. S. Citrin, “Self-collimated waveguide bends and partial bandgap reflection of photonic crystals with parallelogram lattice,” J. Opt. Soc. Am. A Opt. Image Sci. Vis. 25, 791–795 (2008).
[Crossref] [PubMed]

Dahlem, M. S.

P. T. Rakich, M. S. Dahlem, S. Tandon, M. Ibanescu, M. Soljačić, 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, 93–96 (2006).
[Crossref] [PubMed]

Dai, Q.

Dardano, P.

V. Mocella, S. Cabrini, A. S. P. Chang, P. Dardano, L. Moretti, I. Rendina, D. Qlynick, B. Harteneck, and S. Dhuey, “Self-collimation of light over millimeter-scale distance in a quasi-zero-average-index metamaterial,” Phys. Rev. Lett. 102, 133902 (2009).
[Crossref] [PubMed]

Dhuey, S.

V. Mocella, S. Cabrini, A. S. P. Chang, P. Dardano, L. Moretti, I. Rendina, D. Qlynick, B. Harteneck, and S. Dhuey, “Self-collimation of light over millimeter-scale distance in a quasi-zero-average-index metamaterial,” Phys. Rev. Lett. 102, 133902 (2009).
[Crossref] [PubMed]

Dossou, K. B.

F. J. Lawrence, L. C. Botten, K. B. Dossou, and C. M. D. Sterke, “Antireflection coatings for two-dimensional photonic crystals using a rigorous impedance definition,” Appl. Phys. Lett. 93, 171104 (2008).
[Crossref]

Dou, Y.

Dundar, M. A.

Eggleton, B. J.

M. J. Steel, R. Zoli, C. Grillet, R. C. Mcphedran, D. Sterke, C. Martijn, A. Norton, P. Bassi, and B. J. Eggleton, “Analytic properties of photonic crystal superprism parameters,” Phys. Rev. E 71, 056608 (2005).
[Crossref]

Enoch, S.

Fang, N.

Z. Yang, A. Wu, N. Fang, X. Lin, X. Jiang, S. Zou, and X. Wang, “Millimeter-scale self-collimation in planar photonic crystals fabricated by cmos technology,” Opt. Commun. 283, 604–607 (2010).
[Crossref]

Gao, D.

D. Gao, Z. Zhou, and D. S. Citrin, “Self-collimated waveguide bends and partial bandgap reflection of photonic crystals with parallelogram lattice,” J. Opt. Soc. Am. A Opt. Image Sci. Vis. 25, 791–795 (2008).
[Crossref] [PubMed]

Gao, S.

Gauthier-Lafaye, O.

J. Arlandis, E. Centeno, R. Pollès, A. Moreau, J. Campos, O. Gauthier-Lafaye, and A. Monmayrant, “Mesoscopic self-collimation and slow light in all-positive index layered photonic crystals,” Phys. Rev. Lett. 108, 037401 (2012).
[Crossref] [PubMed]

Giden, I. H.

M. Gumus, I. H. Giden, O. Akcaalan, M. Turduev, and H. Kurt, “Enhanced superprism effect in symmetry reduced photonic crystals,” Appl. Phys. Lett. 113, 131103 (2018).
[Crossref]

Grillet, C.

M. J. Steel, R. Zoli, C. Grillet, R. C. Mcphedran, D. Sterke, C. Martijn, A. Norton, P. Bassi, and B. J. Eggleton, “Analytic properties of photonic crystal superprism parameters,” Phys. Rev. E 71, 056608 (2005).
[Crossref]

Gumus, M.

M. Gumus, I. H. Giden, O. Akcaalan, M. Turduev, and H. Kurt, “Enhanced superprism effect in symmetry reduced photonic crystals,” Appl. Phys. Lett. 113, 131103 (2018).
[Crossref]

Guo, Q.

Y. Xu, X. Chen, S. Lan, Q. Dai, Q. Guo, and L. Wu, “Polarization-independent self-collimation based on pill-void photonic crystals with square symmetry,” Opt. Express 17, 4903–4912 (2009).
[Crossref] [PubMed]

Y. Xu, C. Xiaojun, S. Lan, Q. Guo, W. Hu, and W. Lijun, “The all-angle self-collimating phenomenon in photonic crystals with rectangular symmetry,” J. Opt. A Pure Appl. Opt. 10, 085201 (2008).
[Crossref]

Hamam, R. E.

Harteneck, B.

V. Mocella, S. Cabrini, A. S. P. Chang, P. Dardano, L. Moretti, I. Rendina, D. Qlynick, B. Harteneck, and S. Dhuey, “Self-collimation of light over millimeter-scale distance in a quasi-zero-average-index metamaterial,” Phys. Rev. Lett. 102, 133902 (2009).
[Crossref] [PubMed]

Ho, K. M.

K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structure,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[Crossref] [PubMed]

Hong, C.

Z.-l. Wang, H.-t. Jiang, Y.-h. Li, and C. Hong, “Enhancement of self-collimated fields in photonic crystals consisting of two kinds of single-negative materials,” Opt. Express 18, 14312–14318 (2010).

Hu, W.

Y. Xu, C. Xiaojun, S. Lan, Q. Guo, W. Hu, and W. Lijun, “The all-angle self-collimating phenomenon in photonic crystals with rectangular symmetry,” J. Opt. A Pure Appl. Opt. 10, 085201 (2008).
[Crossref]

Huang, X.

Y. Li, F. Meng, S. Zhou, M. Lu, and X. Huang, “Broadband all-angle negative refraction by optimized phononic crystals,” Sci. Reports 7, 7445 (2017).
[Crossref]

Ibanescu, M.

R. E. Hamam, M. Ibanescu, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Broadband super-collimation in a hybrid photonic crystal structure,” Opt. Express 17, 8109–8118 (2009).
[Crossref] [PubMed]

P. T. Rakich, M. S. Dahlem, S. Tandon, M. Ibanescu, M. Soljačić, 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, 93–96 (2006).
[Crossref] [PubMed]

Ippen, E. P.

P. T. Rakich, M. S. Dahlem, S. Tandon, M. Ibanescu, M. Soljačić, 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, 93–96 (2006).
[Crossref] [PubMed]

Jiang, H.-t.

Z.-l. Wang, H.-t. Jiang, Y.-h. Li, and C. Hong, “Enhancement of self-collimated fields in photonic crystals consisting of two kinds of single-negative materials,” Opt. Express 18, 14312–14318 (2010).

Jiang, X.

Joannopoulos, J. D.

R. E. Hamam, M. Ibanescu, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Broadband super-collimation in a hybrid photonic crystal structure,” Opt. Express 17, 8109–8118 (2009).
[Crossref] [PubMed]

P. T. Rakich, M. S. Dahlem, S. Tandon, M. Ibanescu, M. Soljačić, 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, 93–96 (2006).
[Crossref] [PubMed]

C. Luo, S. G. Johnson, and J. D. Joannopoulos, “All-angle negative refraction without negative effective index,” Phys. Rev. B 65, 201104 (2002).
[Crossref]

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University Press, 2008).

John, S.

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

Johnson, S. G.

R. E. Hamam, M. Ibanescu, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Broadband super-collimation in a hybrid photonic crystal structure,” Opt. Express 17, 8109–8118 (2009).
[Crossref] [PubMed]

C. Luo, S. G. Johnson, and J. D. Joannopoulos, “All-angle negative refraction without negative effective index,” Phys. Rev. B 65, 201104 (2002).
[Crossref]

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University Press, 2008).

Kawakami, S.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys. Lett. 74, 1212–1214 (1999).
[Crossref]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, 10096–10099 (1998).
[Crossref]

Kawashima, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys. Lett. 74, 1212–1214 (1999).
[Crossref]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, 10096–10099 (1998).
[Crossref]

Kolodziejski, L. A.

P. T. Rakich, M. S. Dahlem, S. Tandon, M. Ibanescu, M. Soljačić, 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, 93–96 (2006).
[Crossref] [PubMed]

Kosaka, H.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys. Lett. 74, 1212–1214 (1999).
[Crossref]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, 10096–10099 (1998).
[Crossref]

Kurt, H.

M. Gumus, I. H. Giden, O. Akcaalan, M. Turduev, and H. Kurt, “Enhanced superprism effect in symmetry reduced photonic crystals,” Appl. Phys. Lett. 113, 131103 (2018).
[Crossref]

Lan, S.

Y. Xu, X. Chen, S. Lan, Q. Dai, Q. Guo, and L. Wu, “Polarization-independent self-collimation based on pill-void photonic crystals with square symmetry,” Opt. Express 17, 4903–4912 (2009).
[Crossref] [PubMed]

Y. Xu, C. Xiaojun, S. Lan, Q. Guo, W. Hu, and W. Lijun, “The all-angle self-collimating phenomenon in photonic crystals with rectangular symmetry,” J. Opt. A Pure Appl. Opt. 10, 085201 (2008).
[Crossref]

Lawrence, F. J.

F. J. Lawrence, L. C. Botten, K. B. Dossou, and C. M. D. Sterke, “Antireflection coatings for two-dimensional photonic crystals using a rigorous impedance definition,” Appl. Phys. Lett. 93, 171104 (2008).
[Crossref]

Li, Q.

Li, W.

Li, Y.

Y. Li, F. Meng, S. Zhou, M. Lu, and X. Huang, “Broadband all-angle negative refraction by optimized phononic crystals,” Sci. Reports 7, 7445 (2017).
[Crossref]

Li, Y.-h.

Z.-l. Wang, H.-t. Jiang, Y.-h. Li, and C. Hong, “Enhancement of self-collimated fields in photonic crystals consisting of two kinds of single-negative materials,” Opt. Express 18, 14312–14318 (2010).

Lijun, W.

Y. Xu, C. Xiaojun, S. Lan, Q. Guo, W. Hu, and W. Lijun, “The all-angle self-collimating phenomenon in photonic crystals with rectangular symmetry,” J. Opt. A Pure Appl. Opt. 10, 085201 (2008).
[Crossref]

Lin, X.

Lu, M.

Y. Li, F. Meng, S. Zhou, M. Lu, and X. Huang, “Broadband all-angle negative refraction by optimized phononic crystals,” Sci. Reports 7, 7445 (2017).
[Crossref]

Luo, C.

C. Luo, S. G. Johnson, and J. D. Joannopoulos, “All-angle negative refraction without negative effective index,” Phys. Rev. B 65, 201104 (2002).
[Crossref]

Martijn, C.

M. J. Steel, R. Zoli, C. Grillet, R. C. Mcphedran, D. Sterke, C. Martijn, A. Norton, P. Bassi, and B. J. Eggleton, “Analytic properties of photonic crystal superprism parameters,” Phys. Rev. E 71, 056608 (2005).
[Crossref]

Martin, R.

D. W. Prather, S. Shi, J. Murakowski, G. J. Schneider, A. Sharkawy, C. Chen, B. Miao, and R. Martin, “Self-collimation in photonic crystal structures: a new paradigm for applications and device development,” J. Phys. D Appl. Phys. 40, 2635–2651 (2007).
[Crossref]

Mcphedran, R. C.

T. P. White, C. M. D. Sterke, R. C. Mcphedran, and L. C. Botten, “Highly efficient wide-angle transmission into uniform rod-type photonic crystals,” Appl. Phys. Lett. 87, 5920 (2005).
[Crossref]

M. J. Steel, R. Zoli, C. Grillet, R. C. Mcphedran, D. Sterke, C. Martijn, A. Norton, P. Bassi, and B. J. Eggleton, “Analytic properties of photonic crystal superprism parameters,” Phys. Rev. E 71, 056608 (2005).
[Crossref]

Meade, R. D.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University Press, 2008).

Meng, F.

Y. Li, F. Meng, S. Zhou, M. Lu, and X. Huang, “Broadband all-angle negative refraction by optimized phononic crystals,” Sci. Reports 7, 7445 (2017).
[Crossref]

Miao, B.

D. W. Prather, S. Shi, J. Murakowski, G. J. Schneider, A. Sharkawy, C. Chen, B. Miao, and R. Martin, “Self-collimation in photonic crystal structures: a new paradigm for applications and device development,” J. Phys. D Appl. Phys. 40, 2635–2651 (2007).
[Crossref]

Mocella, V.

V. Mocella, S. Cabrini, A. S. P. Chang, P. Dardano, L. Moretti, I. Rendina, D. Qlynick, B. Harteneck, and S. Dhuey, “Self-collimation of light over millimeter-scale distance in a quasi-zero-average-index metamaterial,” Phys. Rev. Lett. 102, 133902 (2009).
[Crossref] [PubMed]

Monmayrant, A.

J. Arlandis, E. Centeno, R. Pollès, A. Moreau, J. Campos, O. Gauthier-Lafaye, and A. Monmayrant, “Mesoscopic self-collimation and slow light in all-positive index layered photonic crystals,” Phys. Rev. Lett. 108, 037401 (2012).
[Crossref] [PubMed]

Moreau, A.

J. Arlandis, E. Centeno, R. Pollès, A. Moreau, J. Campos, O. Gauthier-Lafaye, and A. Monmayrant, “Mesoscopic self-collimation and slow light in all-positive index layered photonic crystals,” Phys. Rev. Lett. 108, 037401 (2012).
[Crossref] [PubMed]

Moretti, L.

V. Mocella, S. Cabrini, A. S. P. Chang, P. Dardano, L. Moretti, I. Rendina, D. Qlynick, B. Harteneck, and S. Dhuey, “Self-collimation of light over millimeter-scale distance in a quasi-zero-average-index metamaterial,” Phys. Rev. Lett. 102, 133902 (2009).
[Crossref] [PubMed]

Murakowski, J.

D. W. Prather, S. Shi, J. Murakowski, G. J. Schneider, A. Sharkawy, C. Chen, B. Miao, and R. Martin, “Self-collimation in photonic crystal structures: a new paradigm for applications and device development,” J. Phys. D Appl. Phys. 40, 2635–2651 (2007).
[Crossref]

Nguyen, H. M.

Norton, A.

M. J. Steel, R. Zoli, C. Grillet, R. C. Mcphedran, D. Sterke, C. Martijn, A. Norton, P. Bassi, and B. J. Eggleton, “Analytic properties of photonic crystal superprism parameters,” Phys. Rev. E 71, 056608 (2005).
[Crossref]

Notomi, M.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys. Lett. 74, 1212–1214 (1999).
[Crossref]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, 10096–10099 (1998).
[Crossref]

Petrich, G. S.

P. T. Rakich, M. S. Dahlem, S. Tandon, M. Ibanescu, M. Soljačić, 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, 93–96 (2006).
[Crossref] [PubMed]

Pollès, R.

J. Arlandis, E. Centeno, R. Pollès, A. Moreau, J. Campos, O. Gauthier-Lafaye, and A. Monmayrant, “Mesoscopic self-collimation and slow light in all-positive index layered photonic crystals,” Phys. Rev. Lett. 108, 037401 (2012).
[Crossref] [PubMed]

Prather, D.

Prather, D. W.

D. W. Prather, S. Shi, J. Murakowski, G. J. Schneider, A. Sharkawy, C. Chen, B. Miao, and R. Martin, “Self-collimation in photonic crystal structures: a new paradigm for applications and device development,” J. Phys. D Appl. Phys. 40, 2635–2651 (2007).
[Crossref]

Pustai, D. M.

Qlynick, D.

V. Mocella, S. Cabrini, A. S. P. Chang, P. Dardano, L. Moretti, I. Rendina, D. Qlynick, B. Harteneck, and S. Dhuey, “Self-collimation of light over millimeter-scale distance in a quasi-zero-average-index metamaterial,” Phys. Rev. Lett. 102, 133902 (2009).
[Crossref] [PubMed]

Rakich, P. T.

P. T. Rakich, M. S. Dahlem, S. Tandon, M. Ibanescu, M. Soljačić, 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, 93–96 (2006).
[Crossref] [PubMed]

Rendina, I.

V. Mocella, S. Cabrini, A. S. P. Chang, P. Dardano, L. Moretti, I. Rendina, D. Qlynick, B. Harteneck, and S. Dhuey, “Self-collimation of light over millimeter-scale distance in a quasi-zero-average-index metamaterial,” Phys. Rev. Lett. 102, 133902 (2009).
[Crossref] [PubMed]

Rogge, S.

Sakoda, K.

K. Sakoda, Optical Properties of Photonic Crystals (Springer-Verlag, 2005).

Salemink, H. W. M.

Sato, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys. Lett. 74, 1212–1214 (1999).
[Crossref]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, 10096–10099 (1998).
[Crossref]

Schneider, G. J.

D. W. Prather, S. Shi, J. Murakowski, G. J. Schneider, A. Sharkawy, C. Chen, B. Miao, and R. Martin, “Self-collimation in photonic crystal structures: a new paradigm for applications and device development,” J. Phys. D Appl. Phys. 40, 2635–2651 (2007).
[Crossref]

Sharkawy, A.

D. W. Prather, S. Shi, J. Murakowski, G. J. Schneider, A. Sharkawy, C. Chen, B. Miao, and R. Martin, “Self-collimation in photonic crystal structures: a new paradigm for applications and device development,” J. Phys. D Appl. Phys. 40, 2635–2651 (2007).
[Crossref]

D. M. Pustai, S. Shi, C. Chen, A. Sharkawy, and D. Prather, “Analysis of splitters for self-collimated beams in planar photonic crystals,” Opt. Express 12, 1823–1831 (2004).
[Crossref] [PubMed]

Shi, S.

D. W. Prather, S. Shi, J. Murakowski, G. J. Schneider, A. Sharkawy, C. Chen, B. Miao, and R. Martin, “Self-collimation in photonic crystal structures: a new paradigm for applications and device development,” J. Phys. D Appl. Phys. 40, 2635–2651 (2007).
[Crossref]

D. M. Pustai, S. Shi, C. Chen, A. Sharkawy, and D. Prather, “Analysis of splitters for self-collimated beams in planar photonic crystals,” Opt. Express 12, 1823–1831 (2004).
[Crossref] [PubMed]

Soljacic, M.

Soukoulis, C. M.

K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structure,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[Crossref] [PubMed]

Steel, M. J.

M. J. Steel, R. Zoli, C. Grillet, R. C. Mcphedran, D. Sterke, C. Martijn, A. Norton, P. Bassi, and B. J. Eggleton, “Analytic properties of photonic crystal superprism parameters,” Phys. Rev. E 71, 056608 (2005).
[Crossref]

Sterke, C. M. D.

F. J. Lawrence, L. C. Botten, K. B. Dossou, and C. M. D. Sterke, “Antireflection coatings for two-dimensional photonic crystals using a rigorous impedance definition,” Appl. Phys. Lett. 93, 171104 (2008).
[Crossref]

T. P. White, C. M. D. Sterke, R. C. Mcphedran, and L. C. Botten, “Highly efficient wide-angle transmission into uniform rod-type photonic crystals,” Appl. Phys. Lett. 87, 5920 (2005).
[Crossref]

Sterke, D.

M. J. Steel, R. Zoli, C. Grillet, R. C. Mcphedran, D. Sterke, C. Martijn, A. Norton, P. Bassi, and B. J. Eggleton, “Analytic properties of photonic crystal superprism parameters,” Phys. Rev. E 71, 056608 (2005).
[Crossref]

Tamamura, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys. Lett. 74, 1212–1214 (1999).
[Crossref]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, 10096–10099 (1998).
[Crossref]

Tandon, S.

P. T. Rakich, M. S. Dahlem, S. Tandon, M. Ibanescu, M. Soljačić, 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, 93–96 (2006).
[Crossref] [PubMed]

Tayeb, G.

Tomita, A.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys. Lett. 74, 1212–1214 (1999).
[Crossref]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, 10096–10099 (1998).
[Crossref]

Torres, C. M. S.

Turduev, M.

M. Gumus, I. H. Giden, O. Akcaalan, M. Turduev, and H. Kurt, “Enhanced superprism effect in symmetry reduced photonic crystals,” Appl. Phys. Lett. 113, 131103 (2018).
[Crossref]

van der Drift, E. W. J. M.

van der Heijden, R. W.

Wang, X.

Z. Yang, A. Wu, N. Fang, X. Lin, X. Jiang, S. Zou, and X. Wang, “Millimeter-scale self-collimation in planar photonic crystals fabricated by cmos technology,” Opt. Commun. 283, 604–607 (2010).
[Crossref]

Wang, Z.-l.

Z.-l. Wang, H.-t. Jiang, Y.-h. Li, and C. Hong, “Enhancement of self-collimated fields in photonic crystals consisting of two kinds of single-negative materials,” Opt. Express 18, 14312–14318 (2010).

White, T. P.

T. P. White, C. M. D. Sterke, R. C. Mcphedran, and L. C. Botten, “Highly efficient wide-angle transmission into uniform rod-type photonic crystals,” Appl. Phys. Lett. 87, 5920 (2005).
[Crossref]

Winn, J. N.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University Press, 2008).

Wu, A.

Z. Yang, A. Wu, N. Fang, X. Lin, X. Jiang, S. Zou, and X. Wang, “Millimeter-scale self-collimation in planar photonic crystals fabricated by cmos technology,” Opt. Commun. 283, 604–607 (2010).
[Crossref]

Wu, L.

Xiaojun, C.

Y. Xu, C. Xiaojun, S. Lan, Q. Guo, W. Hu, and W. Lijun, “The all-angle self-collimating phenomenon in photonic crystals with rectangular symmetry,” J. Opt. A Pure Appl. Opt. 10, 085201 (2008).
[Crossref]

Xu, Y.

Y. Xu, X. Chen, S. Lan, Q. Dai, Q. Guo, and L. Wu, “Polarization-independent self-collimation based on pill-void photonic crystals with square symmetry,” Opt. Express 17, 4903–4912 (2009).
[Crossref] [PubMed]

Y. Xu, C. Xiaojun, S. Lan, Q. Guo, W. Hu, and W. Lijun, “The all-angle self-collimating phenomenon in photonic crystals with rectangular symmetry,” J. Opt. A Pure Appl. Opt. 10, 085201 (2008).
[Crossref]

Yablonovitch, E.

E. Yablonovitch, “Photonic band-gap structures,” J. Opt. Soc. Am. B-optical Phys. 10, 283–295 (1993).
[Crossref]

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[Crossref] [PubMed]

Yang, Z.

Z. Yang, A. Wu, N. Fang, X. Lin, X. Jiang, S. Zou, and X. Wang, “Millimeter-scale self-collimation in planar photonic crystals fabricated by cmos technology,” Opt. Commun. 283, 604–607 (2010).
[Crossref]

Zhang, X.

Zhou, S.

Y. Li, F. Meng, S. Zhou, M. Lu, and X. Huang, “Broadband all-angle negative refraction by optimized phononic crystals,” Sci. Reports 7, 7445 (2017).
[Crossref]

Zhou, Z.

D. Gao, Z. Zhou, and D. S. Citrin, “Self-collimated waveguide bends and partial bandgap reflection of photonic crystals with parallelogram lattice,” J. Opt. Soc. Am. A Opt. Image Sci. Vis. 25, 791–795 (2008).
[Crossref] [PubMed]

Zoli, R.

M. J. Steel, R. Zoli, C. Grillet, R. C. Mcphedran, D. Sterke, C. Martijn, A. Norton, P. Bassi, and B. J. Eggleton, “Analytic properties of photonic crystal superprism parameters,” Phys. Rev. E 71, 056608 (2005).
[Crossref]

Zou, S.

Z. Yang, A. Wu, N. Fang, X. Lin, X. Jiang, S. Zou, and X. Wang, “Millimeter-scale self-collimation in planar photonic crystals fabricated by cmos technology,” Opt. Commun. 283, 604–607 (2010).
[Crossref]

Appl. Phys. Lett. (4)

M. Gumus, I. H. Giden, O. Akcaalan, M. Turduev, and H. Kurt, “Enhanced superprism effect in symmetry reduced photonic crystals,” Appl. Phys. Lett. 113, 131103 (2018).
[Crossref]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys. Lett. 74, 1212–1214 (1999).
[Crossref]

F. J. Lawrence, L. C. Botten, K. B. Dossou, and C. M. D. Sterke, “Antireflection coatings for two-dimensional photonic crystals using a rigorous impedance definition,” Appl. Phys. Lett. 93, 171104 (2008).
[Crossref]

T. P. White, C. M. D. Sterke, R. C. Mcphedran, and L. C. Botten, “Highly efficient wide-angle transmission into uniform rod-type photonic crystals,” Appl. Phys. Lett. 87, 5920 (2005).
[Crossref]

J. Opt. A Pure Appl. Opt. (1)

Y. Xu, C. Xiaojun, S. Lan, Q. Guo, W. Hu, and W. Lijun, “The all-angle self-collimating phenomenon in photonic crystals with rectangular symmetry,” J. Opt. A Pure Appl. Opt. 10, 085201 (2008).
[Crossref]

J. Opt. Soc. Am. A Opt. Image Sci. Vis. (1)

D. Gao, Z. Zhou, and D. S. Citrin, “Self-collimated waveguide bends and partial bandgap reflection of photonic crystals with parallelogram lattice,” J. Opt. Soc. Am. A Opt. Image Sci. Vis. 25, 791–795 (2008).
[Crossref] [PubMed]

J. Opt. Soc. Am. B-optical Phys. (1)

E. Yablonovitch, “Photonic band-gap structures,” J. Opt. Soc. Am. B-optical Phys. 10, 283–295 (1993).
[Crossref]

J. Phys. D Appl. Phys. (1)

D. W. Prather, S. Shi, J. Murakowski, G. J. Schneider, A. Sharkawy, C. Chen, B. Miao, and R. Martin, “Self-collimation in photonic crystal structures: a new paradigm for applications and device development,” J. Phys. D Appl. Phys. 40, 2635–2651 (2007).
[Crossref]

Nat. Mater. (1)

P. T. Rakich, M. S. Dahlem, S. Tandon, M. Ibanescu, M. Soljačić, 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, 93–96 (2006).
[Crossref] [PubMed]

Opt. Commun. (1)

Z. Yang, A. Wu, N. Fang, X. Lin, X. Jiang, S. Zou, and X. Wang, “Millimeter-scale self-collimation in planar photonic crystals fabricated by cmos technology,” Opt. Commun. 283, 604–607 (2010).
[Crossref]

Opt. Express (8)

Z.-l. Wang, H.-t. Jiang, Y.-h. Li, and C. Hong, “Enhancement of self-collimated fields in photonic crystals consisting of two kinds of single-negative materials,” Opt. Express 18, 14312–14318 (2010).

S. Gao, Y. Dou, Q. Li, and X. Jiang, “Tunable photonic crystal lens with high sensitivity of refractive index,” Opt. Express 25, 7112–7120 (2017).
[Crossref] [PubMed]

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

D. M. Pustai, S. Shi, C. Chen, A. Sharkawy, and D. Prather, “Analysis of splitters for self-collimated beams in planar photonic crystals,” Opt. Express 12, 1823–1831 (2004).
[Crossref] [PubMed]

Y. Xu, X. Chen, S. Lan, Q. Dai, Q. Guo, and L. Wu, “Polarization-independent self-collimation based on pill-void photonic crystals with square symmetry,” Opt. Express 17, 4903–4912 (2009).
[Crossref] [PubMed]

R. E. Hamam, M. Ibanescu, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Broadband super-collimation in a hybrid photonic crystal structure,” Opt. Express 17, 8109–8118 (2009).
[Crossref] [PubMed]

H. M. Nguyen, M. A. Dundar, R. W. van der Heijden, E. W. J. M. van der Drift, H. W. M. Salemink, S. Rogge, and J. Caro, “Compact mach-zehnder interferometer based on self-collimation of light in a silicon photonic crystal,” Opt. Express 18, 6437–6446 (2010).
[Crossref] [PubMed]

X. Lin, X. Zhang, L. Chen, M. Soljačić, and X. Jiang, “Super-collimation with high frequency sensitivity in 2d photonic crystals induced by saddletype van hove singularities,” Opt. Express 21, 30140–30147 (2013).
[Crossref]

Opt. Lett. (1)

Phys. Rev. B (2)

C. Luo, S. G. Johnson, and J. D. Joannopoulos, “All-angle negative refraction without negative effective index,” Phys. Rev. B 65, 201104 (2002).
[Crossref]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, 10096–10099 (1998).
[Crossref]

Phys. Rev. E (1)

M. J. Steel, R. Zoli, C. Grillet, R. C. Mcphedran, D. Sterke, C. Martijn, A. Norton, P. Bassi, and B. J. Eggleton, “Analytic properties of photonic crystal superprism parameters,” Phys. Rev. E 71, 056608 (2005).
[Crossref]

Phys. Rev. Lett. (5)

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[Crossref] [PubMed]

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

K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structure,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[Crossref] [PubMed]

V. Mocella, S. Cabrini, A. S. P. Chang, P. Dardano, L. Moretti, I. Rendina, D. Qlynick, B. Harteneck, and S. Dhuey, “Self-collimation of light over millimeter-scale distance in a quasi-zero-average-index metamaterial,” Phys. Rev. Lett. 102, 133902 (2009).
[Crossref] [PubMed]

J. Arlandis, E. Centeno, R. Pollès, A. Moreau, J. Campos, O. Gauthier-Lafaye, and A. Monmayrant, “Mesoscopic self-collimation and slow light in all-positive index layered photonic crystals,” Phys. Rev. Lett. 108, 037401 (2012).
[Crossref] [PubMed]

Sci. Reports (1)

Y. Li, F. Meng, S. Zhou, M. Lu, and X. Huang, “Broadband all-angle negative refraction by optimized phononic crystals,” Sci. Reports 7, 7445 (2017).
[Crossref]

Other (2)

K. Sakoda, Optical Properties of Photonic Crystals (Springer-Verlag, 2005).

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University Press, 2008).

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Figures (8)

Fig. 1
Fig. 1 (a) Schematic of the PhC structure formed by 2D rectangular lattices of silicon rods. The rods radius are r=0.3a and the aspect ratio is β=2. (b)The EFCs of second band of the rectangular lattice PhC (β=2) include flat form(red line), curved form(black lines), and separated form by the ΓX axis (blue dashed lines). Flat EFC represents SC according to the direction of group velocities indicated by black arrows.
Fig. 2
Fig. 2 The second band EFCs of the rectangular lattice PhC (β=2) for different calculation precision. (a) δ k x = 1.25 × 10 3 2 π a, (b) δ k x = 2 × 10 4 2 π a, (c) δ k x = 1 × 10 4 2 π a, (d) δ k x = 8 × 10 5 2 π a, (e) δ k x = 5 × 10 5 2 π a, (f) δ k x = 2 × 10 5 2 π a, (g) δ k x = 1 × 10 5 2 π a, (h) δ k x = 5 × 10 6 2 π a. The EFCs with flat segment are indicated by the red lines and the flat segments are signed by black dashed lines. The blue lines correspond to the frequency f = 0.3856(c/a).
Fig. 3
Fig. 3 The half-range Δky of flat segments on EFCs (red solid line) and their corresponding NCLs (black dashed line) under different calculation precisions.
Fig. 4
Fig. 4 The half-range Δky of the flat segments for different lattice aspect ratios. (a) β = 2.2, (b) β = 2.4, (c) β = 2.6, (d) β = 2.8, (e) β = 3, respectively.
Fig. 5
Fig. 5 The values of NCL η versus the normalized frequencies for (a) square lattice PhC (β = 1), (b) rectangular lattice PhCs with β = 2, (c) rectangular lattice PhCs with β = 3. The inset in (c) is an enlarged view of the dashed box.
Fig. 6
Fig. 6 (a) The second band structure (Hz polarization) of the rectangular lattice PhC (β = 2) as shown in Fig. 1(a). The square marker points are simulation frequencies (f = 0.38335c/a, 0.38447c/a, 0.38556c/a) with significant curvature differences. (b) EFCs for the simulation frequencies.
Fig. 7
Fig. 7 The FDTD simulations of field distributions of the Hz component. Frequencies of incident waves are (a) f = 0.38335c/a, (b) f = 0.38447c/a, (c) f = 0.38556c/a. Gaussian beams with waist width W 0 = 4 2 b enter from the left side of PhCs whose rod radius r=0.3a, β=2.
Fig. 8
Fig. 8 The simulated beam width versus the propagation distance. Waist widths are (a) W 0 = 2 2 b, (b) W 0 = 4 2 b, (c) W 0 = 6 2 b. The dotted lines consist of beam widths at each position recorded in the FDTD simulation, and the solid lines are the theory results of Eq. (4).

Equations (7)

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k x = k x 0 + ξ 2 k y 2
| ξ | δ k x Δ k y 2
H z ( x , y ) = H z 0 W 0 W 0 2 4 i ξ x e y 2 σ 2 e i 2 y 2 ξ x W 0 4 + 16 ξ 2 x 2
W ( x ) = 2 σ = W 0 1 + ( x x 0 ) 2
l SC = 3 x 0 = 3 W 0 2 4 | ξ |
η = l SC W 0 = 3 W 0 4 | ξ |
η = 3 Δ k y 2 δ k x