Abstract

The proposed numerical method, “FLAME-slab,” solves electromagnetic wave scattering problems for aperiodic slab structures by exploiting short-range regularities in these structures. The computational procedure involves special difference schemes with high accuracy even on coarse grids. These schemes are based on Trefftz approximations, utilizing functions that locally satisfy the governing differential equations, as is done in the Flexible Local Approximation Method (FLAME). Radiation boundary conditions are implemented via Fourier expansions in the air surrounding the slab. When applied to ensembles of slab structures with identical short-range features, such as amorphous or quasicrystalline lattices, the method is significantly more efficient, both in runtime and in memory consumption, than traditional approaches. This efficiency is due to the fact that the Trefftz functions need to be computed only once for the whole ensemble.

© 2017 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2016 (4)

H.-T. Chen, A. J. Taylor, and N. Yu, “A review of metasurfaces: physics and applications,” Rep. Prog. Phys. 79, 076401 (2016).
[Crossref] [PubMed]

H. Shi, X. Lu, and Y. Y. Lu, “Vertical mode expansion method for numerical modeling of biperiodic structures,” J. Opt. Soc. Am. A 33, 836–844 (2016).
[Crossref]

S. Mansha, Z. Yongquan, Q. J. Wang, and Y. D. Chong, “Optimization of TM modes for amorphous slab lasers,” Opt. Ex. 24, 4890 (2016).
[Crossref]

A. Paganini, L. Scarabosio, R. Hiptmair, and I. Tsukerman, “Trefftz approximations: a new framework for nonreflecting boundary conditions,” IEEE Trans. Mag. 52, 7201604 (2016).
[Crossref]

2015 (4)

S. F. Liew, S. Knitter, W. Xiong, and H. Cao, “Photonic crystals with topological defects,” Phys. Rev. A 91, 023811 (2015).
[Crossref]

S. Knitter, S. F. Liew, W. Xiong, M. I. Guy, G. S. Solomon, and H. Cao, “Topological defect lasers,” J. Opt. 18, 014005 (2015).
[Crossref]

H. Egger, F. Kretzschmar, S. M. Schnepp, I. Tsukerman, and T. Weiland, “Transparent boundary conditions for a discontinuous Galerkin Trefftz method,” Appl. Math. Comp. 267, 42 (2015).
[Crossref]

H. Egger, F. Kretzschmar, S. M. Schnepp, and T. Weiland, “A Space-time Discontinuous Galerkin Trefftz method for time dependent Maxwell’s equations,” SIAM J. Sci. Comp. 37, B689 (2015).
[Crossref]

2014 (3)

X. Lu, H. Shi, and Y. Y. Lu, “Vertical mode expansion method of transmission of light through a single circular hole in a slab,” J. Opt. Soc. Am. A 31, 293 (2014).
[Crossref]

N. Yu and F. Capasso, “Flat optics with designer metasurfaces,” Nat. Mat. 13, 139 (2014).
[Crossref]

F. Kretzschmar, S. Schnepp, I. Tsukerman, and T. Weiland, “Discontinuous Galerkin methods with Trefftz approximations,” J. Comp. Appl. Math. 270, 211 (2014).
[Crossref]

2013 (3)

Z. V. Vardeny, A. Nahata, and A. Agrawal, “Optics of photonic quasicrystals,” Nat. Phot. 7, 177 (2013).
[Crossref]

Ralf Hiptmair, Andrea Moiola, and Ilaria Perugia, “Error analysis of Trefftz-Discontinuous Galerkin methods for the time-harmonic Maxwell equations,” Mathematics of Computation 82, 247–268 (2013).
[Crossref]

H. K. Liang, B. Meng, G. Liang, J. Tao, Y. Chong, Q. J. Wang, and Y. Zhang, “Electrically pumped mid-infrared random lasers,” Adv. Mat. 25, 6589 (2013).
[Crossref]

2012 (1)

V. Liu and S. Fan, “S4: A free electromagnetic solver for layered periodic structures,” Comp. Phys. Comm. 183, 2233 (2012).
[Crossref]

2011 (2)

S.-L. Chua, Y. Chong, A. D. Stone, M. Soljačić, and J. Bravo-Abad, “Low-threshold lasing action in photonic crystal slabs enabled by Fano resonances,” Opt. Ex. 19, 1539 (2011).
[Crossref]

H. Noh, J. K. Yang, S. F. Liew, M. J. Roooks, and H. Cao, “Control of lasing in biomimetic structures with short-range order,” Phys. Rev. Lett. 106, 183901 (2011).
[Crossref] [PubMed]

2010 (2)

J.-K. Yang, S. V. Boriskina, H. Noh, M. J. Rooks, G. S. Solomon, L. Dal Negro, and H. Cao, “Demonstration of laser action in a pseudorandom medium,” Appl. Phys. Lett. 97223101 (2010).
[Crossref]

I. Tsukerman, “Trefftz difference schemes on irregular stencils,” J. Comp. Phys. 229, 2948 (2010).
[Crossref]

2009 (3)

A. Ricciardi, I. Gallina, S. Campopiano, G. Castaldi, M. Pisco, V. Galdi, and A. Cusano, “Guided resonances in photonic quasicrystals,” Opt. Ex. 17, 6335 (2009).

M. Florescu, S. Torquato, and P. Steinhardt, “Designer disordered materials with large, complete photonic band gaps,” Proc. Nat. Acad. Sci. Am. 106, 20658 (2009).
[Crossref]

H. Pinheiro and J. P. Webb, “A FLAME molecule for 3-D electromagnetic scattering,” IEEE Trans. Mag. 45, 1120 (2009).
[Crossref]

2008 (1)

H. Matsubara, S. Yoshimoto, H. Saito, Y. Jianglin, Y. Tanaka, and S. Noda, “GaN photonic-crystal surface-emitting laser at blue-violet wavelengths,” Science 319, 445 (2008).
[Crossref]

2007 (2)

P. D. García, R. Sapienca, Á. Blanco, and C. López, “Photonic glass: a novel random material for light,” Adv. Mat. 19, 2597 (2007).
[Crossref]

W. Steurer and D. Sutter-Widmer, “Photonic and phononic quasicrystals,” J. Phys. D: Appl. Phys. 40, R229 (2007).
[Crossref]

2006 (1)

I. Tsukerman, “A class of difference schemes with flexible local approximation,” J. Comp. Phys. 211, 659–699 (2006).
[Crossref]

2005 (2)

I. Tsukerman, “Electromagnetic applications of a new finite-difference calculus,” IEEE Trans. Mag. 41, 2206 (2005).
[Crossref]

Z. Feng, X. Zhang, Y. Wang, Z.-Y. Li, B. Cheng, and D.-Z. Zhang, “Negative refraction and imaging using 12-fold-symmetry quasicrystals,” Phys. Rev. Lett. 94, 247402 (2005).
[Crossref]

2002 (2)

M. Imada, A. Chutinan, S. Noda, and M. Mochizuki, “Multidirectionally distributed feedback photonic crystal lasers,” Phys. Rev. B 65, 195306 (2002).
[Crossref]

H.-Y. Ryu, S.-H. Kwon, Y.-J. Lee, and Y.-H. Lee, “Very-low-threshold photonic band-edge lasers from free-standing triangular photonic crystal slabs,” Appl. Phys. Lett. 80, 3476 (2002).
[Crossref]

2001 (3)

A. A. Erchak, D. J. Ripin, S. Fan, P. Rakich, J. D. Joannopoulos, E. P. Ippen, G. S. Petrich, and L. A. Kolodziejski, “Enhanced coupling to vertical radiation using a two-dimensional photonic crystal in a semiconductor light-emitting diode,” Appl. Phys. Lett. 78, 563 (2001).
[Crossref]

S. Noda, M. Yokoyama, M. Imada, A. Chutinan, and M. Mochizuki, “Polarization mode control of two-dimensional photonic crystal laser by unit cell structure design,” Science 293, 1123 (2001).
[Crossref] [PubMed]

C. Jin, X. Meng, B. Cheng, Z. Li, and D. Zhang, “Photonic gap in amorphous photonic materials,” Phys. Rev. B 63, 195107 (2001).
[Crossref]

2000 (1)

I. Herrera, “Trefftz method: A general theory,” Num. Meth. for Partial Diff. Eq. 16, 561–580 (2000).
[Crossref]

1999 (3)

D. M. Whittaker and I. S. Culshaw, “Scattering-matrix treatment of patterned multilayer photonic structures,” Phys. Rev. B 60, 2610 (1999).
[Crossref]

J. Ballato, J. Dimaio, and A. James, “Photonic band engineering through tailored microstructural order,” Appl. Phys. Lett. 75, 1497 (1999).
[Crossref]

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284, 1819 (1999).
[Crossref] [PubMed]

1997 (1)

1995 (1)

1994 (1)

A. Boag, A. Boag, R. Mittra, and Y. Leviatan, “A numerical absorbing boundary condition for finite-difference and finite-element analysis of open structures,” Microwave Opt. Tech. Lett. 7, 395 (1994).
[Crossref]

1986 (1)

Y. Saad and M. H. Schultz, “GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems,” SIAM J. Sci. Stat. Comp. 7, 856 (1986).
[Crossref]

1978 (1)

J. Jirousek, “Basis for development of large finite elements locally satisfying all field equations,” Comp. Meth. Appl. Mech. Eng. 14, 65–92 (1978).
[Crossref]

Agrawal, A.

Z. V. Vardeny, A. Nahata, and A. Agrawal, “Optics of photonic quasicrystals,” Nat. Phot. 7, 177 (2013).
[Crossref]

Ahmadi, F.

F. Kretzschmar, S. M. Schnepp, H. Egger, F. Ahmadi, N. Nowak, V. Markel, and I. Tsukerman, The Power of Trefftz Approximations: Finite Difference, Boundary Difference and Discontinuous Galerkin Methods; Nonreflecting Conditions and Non-Asymptotic Homogenization (Springer, 2015), 47–58.

Ballato, J.

J. Ballato, J. Dimaio, and A. James, “Photonic band engineering through tailored microstructural order,” Appl. Phys. Lett. 75, 1497 (1999).
[Crossref]

Bandlow, B.

C. Classen, B. Bandlow, R. Schuhmann, and I. Tsukerman, “FIT & FLAME for sharp edges in electrostatics,” in URSI EMTS 2010 Symposium Digest, Berlin, Germany, 2010.

Blanco, Á.

P. D. García, R. Sapienca, Á. Blanco, and C. López, “Photonic glass: a novel random material for light,” Adv. Mat. 19, 2597 (2007).
[Crossref]

Boag, A.

A. Boag, A. Boag, R. Mittra, and Y. Leviatan, “A numerical absorbing boundary condition for finite-difference and finite-element analysis of open structures,” Microwave Opt. Tech. Lett. 7, 395 (1994).
[Crossref]

A. Boag, A. Boag, R. Mittra, and Y. Leviatan, “A numerical absorbing boundary condition for finite-difference and finite-element analysis of open structures,” Microwave Opt. Tech. Lett. 7, 395 (1994).
[Crossref]

Boriskina, S. V.

J.-K. Yang, S. V. Boriskina, H. Noh, M. J. Rooks, G. S. Solomon, L. Dal Negro, and H. Cao, “Demonstration of laser action in a pseudorandom medium,” Appl. Phys. Lett. 97223101 (2010).
[Crossref]

Bravo-Abad, J.

S.-L. Chua, Y. Chong, A. D. Stone, M. Soljačić, and J. Bravo-Abad, “Low-threshold lasing action in photonic crystal slabs enabled by Fano resonances,” Opt. Ex. 19, 1539 (2011).
[Crossref]

Campopiano, S.

A. Ricciardi, I. Gallina, S. Campopiano, G. Castaldi, M. Pisco, V. Galdi, and A. Cusano, “Guided resonances in photonic quasicrystals,” Opt. Ex. 17, 6335 (2009).

Cao, H.

S. F. Liew, S. Knitter, W. Xiong, and H. Cao, “Photonic crystals with topological defects,” Phys. Rev. A 91, 023811 (2015).
[Crossref]

S. Knitter, S. F. Liew, W. Xiong, M. I. Guy, G. S. Solomon, and H. Cao, “Topological defect lasers,” J. Opt. 18, 014005 (2015).
[Crossref]

H. Noh, J. K. Yang, S. F. Liew, M. J. Roooks, and H. Cao, “Control of lasing in biomimetic structures with short-range order,” Phys. Rev. Lett. 106, 183901 (2011).
[Crossref] [PubMed]

J.-K. Yang, S. V. Boriskina, H. Noh, M. J. Rooks, G. S. Solomon, L. Dal Negro, and H. Cao, “Demonstration of laser action in a pseudorandom medium,” Appl. Phys. Lett. 97223101 (2010).
[Crossref]

Capasso, F.

N. Yu and F. Capasso, “Flat optics with designer metasurfaces,” Nat. Mat. 13, 139 (2014).
[Crossref]

Castaldi, G.

A. Ricciardi, I. Gallina, S. Campopiano, G. Castaldi, M. Pisco, V. Galdi, and A. Cusano, “Guided resonances in photonic quasicrystals,” Opt. Ex. 17, 6335 (2009).

Chen, H.-T.

H.-T. Chen, A. J. Taylor, and N. Yu, “A review of metasurfaces: physics and applications,” Rep. Prog. Phys. 79, 076401 (2016).
[Crossref] [PubMed]

Cheng, B.

Z. Feng, X. Zhang, Y. Wang, Z.-Y. Li, B. Cheng, and D.-Z. Zhang, “Negative refraction and imaging using 12-fold-symmetry quasicrystals,” Phys. Rev. Lett. 94, 247402 (2005).
[Crossref]

C. Jin, X. Meng, B. Cheng, Z. Li, and D. Zhang, “Photonic gap in amorphous photonic materials,” Phys. Rev. B 63, 195107 (2001).
[Crossref]

Chong, Y.

H. K. Liang, B. Meng, G. Liang, J. Tao, Y. Chong, Q. J. Wang, and Y. Zhang, “Electrically pumped mid-infrared random lasers,” Adv. Mat. 25, 6589 (2013).
[Crossref]

S.-L. Chua, Y. Chong, A. D. Stone, M. Soljačić, and J. Bravo-Abad, “Low-threshold lasing action in photonic crystal slabs enabled by Fano resonances,” Opt. Ex. 19, 1539 (2011).
[Crossref]

Chong, Y. D.

S. Mansha, Z. Yongquan, Q. J. Wang, and Y. D. Chong, “Optimization of TM modes for amorphous slab lasers,” Opt. Ex. 24, 4890 (2016).
[Crossref]

Chua, S.-L.

S.-L. Chua, Y. Chong, A. D. Stone, M. Soljačić, and J. Bravo-Abad, “Low-threshold lasing action in photonic crystal slabs enabled by Fano resonances,” Opt. Ex. 19, 1539 (2011).
[Crossref]

Chutinan, A.

M. Imada, A. Chutinan, S. Noda, and M. Mochizuki, “Multidirectionally distributed feedback photonic crystal lasers,” Phys. Rev. B 65, 195306 (2002).
[Crossref]

S. Noda, M. Yokoyama, M. Imada, A. Chutinan, and M. Mochizuki, “Polarization mode control of two-dimensional photonic crystal laser by unit cell structure design,” Science 293, 1123 (2001).
[Crossref] [PubMed]

Classen, C.

C. Classen, B. Bandlow, R. Schuhmann, and I. Tsukerman, “FIT & FLAME for sharp edges in electrostatics,” in URSI EMTS 2010 Symposium Digest, Berlin, Germany, 2010.

Culshaw, I. S.

D. M. Whittaker and I. S. Culshaw, “Scattering-matrix treatment of patterned multilayer photonic structures,” Phys. Rev. B 60, 2610 (1999).
[Crossref]

Cusano, A.

A. Ricciardi, I. Gallina, S. Campopiano, G. Castaldi, M. Pisco, V. Galdi, and A. Cusano, “Guided resonances in photonic quasicrystals,” Opt. Ex. 17, 6335 (2009).

Dal Negro, L.

J.-K. Yang, S. V. Boriskina, H. Noh, M. J. Rooks, G. S. Solomon, L. Dal Negro, and H. Cao, “Demonstration of laser action in a pseudorandom medium,” Appl. Phys. Lett. 97223101 (2010).
[Crossref]

Dapkus, P. D.

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284, 1819 (1999).
[Crossref] [PubMed]

Dimaio, J.

J. Ballato, J. Dimaio, and A. James, “Photonic band engineering through tailored microstructural order,” Appl. Phys. Lett. 75, 1497 (1999).
[Crossref]

Egger, H.

H. Egger, F. Kretzschmar, S. M. Schnepp, I. Tsukerman, and T. Weiland, “Transparent boundary conditions for a discontinuous Galerkin Trefftz method,” Appl. Math. Comp. 267, 42 (2015).
[Crossref]

H. Egger, F. Kretzschmar, S. M. Schnepp, and T. Weiland, “A Space-time Discontinuous Galerkin Trefftz method for time dependent Maxwell’s equations,” SIAM J. Sci. Comp. 37, B689 (2015).
[Crossref]

F. Kretzschmar, S. M. Schnepp, H. Egger, F. Ahmadi, N. Nowak, V. Markel, and I. Tsukerman, The Power of Trefftz Approximations: Finite Difference, Boundary Difference and Discontinuous Galerkin Methods; Nonreflecting Conditions and Non-Asymptotic Homogenization (Springer, 2015), 47–58.

Erchak, A. A.

A. A. Erchak, D. J. Ripin, S. Fan, P. Rakich, J. D. Joannopoulos, E. P. Ippen, G. S. Petrich, and L. A. Kolodziejski, “Enhanced coupling to vertical radiation using a two-dimensional photonic crystal in a semiconductor light-emitting diode,” Appl. Phys. Lett. 78, 563 (2001).
[Crossref]

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V. Liu and S. Fan, “S4: A free electromagnetic solver for layered periodic structures,” Comp. Phys. Comm. 183, 2233 (2012).
[Crossref]

A. A. Erchak, D. J. Ripin, S. Fan, P. Rakich, J. D. Joannopoulos, E. P. Ippen, G. S. Petrich, and L. A. Kolodziejski, “Enhanced coupling to vertical radiation using a two-dimensional photonic crystal in a semiconductor light-emitting diode,” Appl. Phys. Lett. 78, 563 (2001).
[Crossref]

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Z. Feng, X. Zhang, Y. Wang, Z.-Y. Li, B. Cheng, and D.-Z. Zhang, “Negative refraction and imaging using 12-fold-symmetry quasicrystals,” Phys. Rev. Lett. 94, 247402 (2005).
[Crossref]

Florescu, M.

M. Florescu, S. Torquato, and P. Steinhardt, “Designer disordered materials with large, complete photonic band gaps,” Proc. Nat. Acad. Sci. Am. 106, 20658 (2009).
[Crossref]

Galdi, V.

A. Ricciardi, I. Gallina, S. Campopiano, G. Castaldi, M. Pisco, V. Galdi, and A. Cusano, “Guided resonances in photonic quasicrystals,” Opt. Ex. 17, 6335 (2009).

Gallina, I.

A. Ricciardi, I. Gallina, S. Campopiano, G. Castaldi, M. Pisco, V. Galdi, and A. Cusano, “Guided resonances in photonic quasicrystals,” Opt. Ex. 17, 6335 (2009).

García, P. D.

P. D. García, R. Sapienca, Á. Blanco, and C. López, “Photonic glass: a novel random material for light,” Adv. Mat. 19, 2597 (2007).
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Grann, E. B.

Guy, M. I.

S. Knitter, S. F. Liew, W. Xiong, M. I. Guy, G. S. Solomon, and H. Cao, “Topological defect lasers,” J. Opt. 18, 014005 (2015).
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A. Paganini, L. Scarabosio, R. Hiptmair, and I. Tsukerman, “Trefftz approximations: a new framework for nonreflecting boundary conditions,” IEEE Trans. Mag. 52, 7201604 (2016).
[Crossref]

R. Hiptmair, A. Moiola, and I. Perugia, A Survey of Trefftz Methods for the Helmholtz Equation, in: G. R. Barrenechea, A. Cangiani, and E. H. Geogoulis eds., Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations, Lecture Notes in Computational Science and Engineering (LNCSE)114 (Springer, 2016), pp. 237–278.

Hiptmair, Ralf

Ralf Hiptmair, Andrea Moiola, and Ilaria Perugia, “Error analysis of Trefftz-Discontinuous Galerkin methods for the time-harmonic Maxwell equations,” Mathematics of Computation 82, 247–268 (2013).
[Crossref]

Imada, M.

M. Imada, A. Chutinan, S. Noda, and M. Mochizuki, “Multidirectionally distributed feedback photonic crystal lasers,” Phys. Rev. B 65, 195306 (2002).
[Crossref]

S. Noda, M. Yokoyama, M. Imada, A. Chutinan, and M. Mochizuki, “Polarization mode control of two-dimensional photonic crystal laser by unit cell structure design,” Science 293, 1123 (2001).
[Crossref] [PubMed]

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A. A. Erchak, D. J. Ripin, S. Fan, P. Rakich, J. D. Joannopoulos, E. P. Ippen, G. S. Petrich, and L. A. Kolodziejski, “Enhanced coupling to vertical radiation using a two-dimensional photonic crystal in a semiconductor light-emitting diode,” Appl. Phys. Lett. 78, 563 (2001).
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C. Jin, X. Meng, B. Cheng, Z. Li, and D. Zhang, “Photonic gap in amorphous photonic materials,” Phys. Rev. B 63, 195107 (2001).
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A. A. Erchak, D. J. Ripin, S. Fan, P. Rakich, J. D. Joannopoulos, E. P. Ippen, G. S. Petrich, and L. A. Kolodziejski, “Enhanced coupling to vertical radiation using a two-dimensional photonic crystal in a semiconductor light-emitting diode,” Appl. Phys. Lett. 78, 563 (2001).
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J. D. Joannopoulos, S. G. Johnson, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University, 2008).

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O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284, 1819 (1999).
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S. Knitter, S. F. Liew, W. Xiong, M. I. Guy, G. S. Solomon, and H. Cao, “Topological defect lasers,” J. Opt. 18, 014005 (2015).
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S. F. Liew, S. Knitter, W. Xiong, and H. Cao, “Photonic crystals with topological defects,” Phys. Rev. A 91, 023811 (2015).
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A. A. Erchak, D. J. Ripin, S. Fan, P. Rakich, J. D. Joannopoulos, E. P. Ippen, G. S. Petrich, and L. A. Kolodziejski, “Enhanced coupling to vertical radiation using a two-dimensional photonic crystal in a semiconductor light-emitting diode,” Appl. Phys. Lett. 78, 563 (2001).
[Crossref]

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H. Egger, F. Kretzschmar, S. M. Schnepp, and T. Weiland, “A Space-time Discontinuous Galerkin Trefftz method for time dependent Maxwell’s equations,” SIAM J. Sci. Comp. 37, B689 (2015).
[Crossref]

H. Egger, F. Kretzschmar, S. M. Schnepp, I. Tsukerman, and T. Weiland, “Transparent boundary conditions for a discontinuous Galerkin Trefftz method,” Appl. Math. Comp. 267, 42 (2015).
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F. Kretzschmar, S. Schnepp, I. Tsukerman, and T. Weiland, “Discontinuous Galerkin methods with Trefftz approximations,” J. Comp. Appl. Math. 270, 211 (2014).
[Crossref]

F. Kretzschmar, S. M. Schnepp, H. Egger, F. Ahmadi, N. Nowak, V. Markel, and I. Tsukerman, The Power of Trefftz Approximations: Finite Difference, Boundary Difference and Discontinuous Galerkin Methods; Nonreflecting Conditions and Non-Asymptotic Homogenization (Springer, 2015), 47–58.

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H.-Y. Ryu, S.-H. Kwon, Y.-J. Lee, and Y.-H. Lee, “Very-low-threshold photonic band-edge lasers from free-standing triangular photonic crystal slabs,” Appl. Phys. Lett. 80, 3476 (2002).
[Crossref]

Lee, R. K.

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284, 1819 (1999).
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H.-Y. Ryu, S.-H. Kwon, Y.-J. Lee, and Y.-H. Lee, “Very-low-threshold photonic band-edge lasers from free-standing triangular photonic crystal slabs,” Appl. Phys. Lett. 80, 3476 (2002).
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Lee, Y.-J.

H.-Y. Ryu, S.-H. Kwon, Y.-J. Lee, and Y.-H. Lee, “Very-low-threshold photonic band-edge lasers from free-standing triangular photonic crystal slabs,” Appl. Phys. Lett. 80, 3476 (2002).
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C. Jin, X. Meng, B. Cheng, Z. Li, and D. Zhang, “Photonic gap in amorphous photonic materials,” Phys. Rev. B 63, 195107 (2001).
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Z. Feng, X. Zhang, Y. Wang, Z.-Y. Li, B. Cheng, and D.-Z. Zhang, “Negative refraction and imaging using 12-fold-symmetry quasicrystals,” Phys. Rev. Lett. 94, 247402 (2005).
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H. K. Liang, B. Meng, G. Liang, J. Tao, Y. Chong, Q. J. Wang, and Y. Zhang, “Electrically pumped mid-infrared random lasers,” Adv. Mat. 25, 6589 (2013).
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S. F. Liew, S. Knitter, W. Xiong, and H. Cao, “Photonic crystals with topological defects,” Phys. Rev. A 91, 023811 (2015).
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S. Knitter, S. F. Liew, W. Xiong, M. I. Guy, G. S. Solomon, and H. Cao, “Topological defect lasers,” J. Opt. 18, 014005 (2015).
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V. Liu and S. Fan, “S4: A free electromagnetic solver for layered periodic structures,” Comp. Phys. Comm. 183, 2233 (2012).
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P. D. García, R. Sapienca, Á. Blanco, and C. López, “Photonic glass: a novel random material for light,” Adv. Mat. 19, 2597 (2007).
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F. Kretzschmar, S. M. Schnepp, H. Egger, F. Ahmadi, N. Nowak, V. Markel, and I. Tsukerman, The Power of Trefftz Approximations: Finite Difference, Boundary Difference and Discontinuous Galerkin Methods; Nonreflecting Conditions and Non-Asymptotic Homogenization (Springer, 2015), 47–58.

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H. Matsubara, S. Yoshimoto, H. Saito, Y. Jianglin, Y. Tanaka, and S. Noda, “GaN photonic-crystal surface-emitting laser at blue-violet wavelengths,” Science 319, 445 (2008).
[Crossref]

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

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H. K. Liang, B. Meng, G. Liang, J. Tao, Y. Chong, Q. J. Wang, and Y. Zhang, “Electrically pumped mid-infrared random lasers,” Adv. Mat. 25, 6589 (2013).
[Crossref]

Meng, X.

C. Jin, X. Meng, B. Cheng, Z. Li, and D. Zhang, “Photonic gap in amorphous photonic materials,” Phys. Rev. B 63, 195107 (2001).
[Crossref]

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A. Boag, A. Boag, R. Mittra, and Y. Leviatan, “A numerical absorbing boundary condition for finite-difference and finite-element analysis of open structures,” Microwave Opt. Tech. Lett. 7, 395 (1994).
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M. Imada, A. Chutinan, S. Noda, and M. Mochizuki, “Multidirectionally distributed feedback photonic crystal lasers,” Phys. Rev. B 65, 195306 (2002).
[Crossref]

S. Noda, M. Yokoyama, M. Imada, A. Chutinan, and M. Mochizuki, “Polarization mode control of two-dimensional photonic crystal laser by unit cell structure design,” Science 293, 1123 (2001).
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Moiola, A.

R. Hiptmair, A. Moiola, and I. Perugia, A Survey of Trefftz Methods for the Helmholtz Equation, in: G. R. Barrenechea, A. Cangiani, and E. H. Geogoulis eds., Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations, Lecture Notes in Computational Science and Engineering (LNCSE)114 (Springer, 2016), pp. 237–278.

Moiola, Andrea

Ralf Hiptmair, Andrea Moiola, and Ilaria Perugia, “Error analysis of Trefftz-Discontinuous Galerkin methods for the time-harmonic Maxwell equations,” Mathematics of Computation 82, 247–268 (2013).
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H. Matsubara, S. Yoshimoto, H. Saito, Y. Jianglin, Y. Tanaka, and S. Noda, “GaN photonic-crystal surface-emitting laser at blue-violet wavelengths,” Science 319, 445 (2008).
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M. Imada, A. Chutinan, S. Noda, and M. Mochizuki, “Multidirectionally distributed feedback photonic crystal lasers,” Phys. Rev. B 65, 195306 (2002).
[Crossref]

S. Noda, M. Yokoyama, M. Imada, A. Chutinan, and M. Mochizuki, “Polarization mode control of two-dimensional photonic crystal laser by unit cell structure design,” Science 293, 1123 (2001).
[Crossref] [PubMed]

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H. Noh, J. K. Yang, S. F. Liew, M. J. Roooks, and H. Cao, “Control of lasing in biomimetic structures with short-range order,” Phys. Rev. Lett. 106, 183901 (2011).
[Crossref] [PubMed]

J.-K. Yang, S. V. Boriskina, H. Noh, M. J. Rooks, G. S. Solomon, L. Dal Negro, and H. Cao, “Demonstration of laser action in a pseudorandom medium,” Appl. Phys. Lett. 97223101 (2010).
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F. Kretzschmar, S. M. Schnepp, H. Egger, F. Ahmadi, N. Nowak, V. Markel, and I. Tsukerman, The Power of Trefftz Approximations: Finite Difference, Boundary Difference and Discontinuous Galerkin Methods; Nonreflecting Conditions and Non-Asymptotic Homogenization (Springer, 2015), 47–58.

O’Brien, J. D.

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284, 1819 (1999).
[Crossref] [PubMed]

Paganini, A.

A. Paganini, L. Scarabosio, R. Hiptmair, and I. Tsukerman, “Trefftz approximations: a new framework for nonreflecting boundary conditions,” IEEE Trans. Mag. 52, 7201604 (2016).
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Painter, O.

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284, 1819 (1999).
[Crossref] [PubMed]

Perugia, I.

R. Hiptmair, A. Moiola, and I. Perugia, A Survey of Trefftz Methods for the Helmholtz Equation, in: G. R. Barrenechea, A. Cangiani, and E. H. Geogoulis eds., Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations, Lecture Notes in Computational Science and Engineering (LNCSE)114 (Springer, 2016), pp. 237–278.

Perugia, Ilaria

Ralf Hiptmair, Andrea Moiola, and Ilaria Perugia, “Error analysis of Trefftz-Discontinuous Galerkin methods for the time-harmonic Maxwell equations,” Mathematics of Computation 82, 247–268 (2013).
[Crossref]

Petrich, G. S.

A. A. Erchak, D. J. Ripin, S. Fan, P. Rakich, J. D. Joannopoulos, E. P. Ippen, G. S. Petrich, and L. A. Kolodziejski, “Enhanced coupling to vertical radiation using a two-dimensional photonic crystal in a semiconductor light-emitting diode,” Appl. Phys. Lett. 78, 563 (2001).
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A. A. Erchak, D. J. Ripin, S. Fan, P. Rakich, J. D. Joannopoulos, E. P. Ippen, G. S. Petrich, and L. A. Kolodziejski, “Enhanced coupling to vertical radiation using a two-dimensional photonic crystal in a semiconductor light-emitting diode,” Appl. Phys. Lett. 78, 563 (2001).
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A. Ricciardi, I. Gallina, S. Campopiano, G. Castaldi, M. Pisco, V. Galdi, and A. Cusano, “Guided resonances in photonic quasicrystals,” Opt. Ex. 17, 6335 (2009).

Ripin, D. J.

A. A. Erchak, D. J. Ripin, S. Fan, P. Rakich, J. D. Joannopoulos, E. P. Ippen, G. S. Petrich, and L. A. Kolodziejski, “Enhanced coupling to vertical radiation using a two-dimensional photonic crystal in a semiconductor light-emitting diode,” Appl. Phys. Lett. 78, 563 (2001).
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Rooks, M. J.

J.-K. Yang, S. V. Boriskina, H. Noh, M. J. Rooks, G. S. Solomon, L. Dal Negro, and H. Cao, “Demonstration of laser action in a pseudorandom medium,” Appl. Phys. Lett. 97223101 (2010).
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Roooks, M. J.

H. Noh, J. K. Yang, S. F. Liew, M. J. Roooks, and H. Cao, “Control of lasing in biomimetic structures with short-range order,” Phys. Rev. Lett. 106, 183901 (2011).
[Crossref] [PubMed]

Ryu, H.-Y.

H.-Y. Ryu, S.-H. Kwon, Y.-J. Lee, and Y.-H. Lee, “Very-low-threshold photonic band-edge lasers from free-standing triangular photonic crystal slabs,” Appl. Phys. Lett. 80, 3476 (2002).
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Y. Saad and M. H. Schultz, “GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems,” SIAM J. Sci. Stat. Comp. 7, 856 (1986).
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H. Matsubara, S. Yoshimoto, H. Saito, Y. Jianglin, Y. Tanaka, and S. Noda, “GaN photonic-crystal surface-emitting laser at blue-violet wavelengths,” Science 319, 445 (2008).
[Crossref]

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P. D. García, R. Sapienca, Á. Blanco, and C. López, “Photonic glass: a novel random material for light,” Adv. Mat. 19, 2597 (2007).
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Scarabosio, L.

A. Paganini, L. Scarabosio, R. Hiptmair, and I. Tsukerman, “Trefftz approximations: a new framework for nonreflecting boundary conditions,” IEEE Trans. Mag. 52, 7201604 (2016).
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O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284, 1819 (1999).
[Crossref] [PubMed]

Schnepp, S.

F. Kretzschmar, S. Schnepp, I. Tsukerman, and T. Weiland, “Discontinuous Galerkin methods with Trefftz approximations,” J. Comp. Appl. Math. 270, 211 (2014).
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Schnepp, S. M.

H. Egger, F. Kretzschmar, S. M. Schnepp, I. Tsukerman, and T. Weiland, “Transparent boundary conditions for a discontinuous Galerkin Trefftz method,” Appl. Math. Comp. 267, 42 (2015).
[Crossref]

H. Egger, F. Kretzschmar, S. M. Schnepp, and T. Weiland, “A Space-time Discontinuous Galerkin Trefftz method for time dependent Maxwell’s equations,” SIAM J. Sci. Comp. 37, B689 (2015).
[Crossref]

F. Kretzschmar, S. M. Schnepp, H. Egger, F. Ahmadi, N. Nowak, V. Markel, and I. Tsukerman, The Power of Trefftz Approximations: Finite Difference, Boundary Difference and Discontinuous Galerkin Methods; Nonreflecting Conditions and Non-Asymptotic Homogenization (Springer, 2015), 47–58.

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S. Knitter, S. F. Liew, W. Xiong, M. I. Guy, G. S. Solomon, and H. Cao, “Topological defect lasers,” J. Opt. 18, 014005 (2015).
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J.-K. Yang, S. V. Boriskina, H. Noh, M. J. Rooks, G. S. Solomon, L. Dal Negro, and H. Cao, “Demonstration of laser action in a pseudorandom medium,” Appl. Phys. Lett. 97223101 (2010).
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M. Florescu, S. Torquato, and P. Steinhardt, “Designer disordered materials with large, complete photonic band gaps,” Proc. Nat. Acad. Sci. Am. 106, 20658 (2009).
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H. Matsubara, S. Yoshimoto, H. Saito, Y. Jianglin, Y. Tanaka, and S. Noda, “GaN photonic-crystal surface-emitting laser at blue-violet wavelengths,” Science 319, 445 (2008).
[Crossref]

Tao, J.

H. K. Liang, B. Meng, G. Liang, J. Tao, Y. Chong, Q. J. Wang, and Y. Zhang, “Electrically pumped mid-infrared random lasers,” Adv. Mat. 25, 6589 (2013).
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A. Paganini, L. Scarabosio, R. Hiptmair, and I. Tsukerman, “Trefftz approximations: a new framework for nonreflecting boundary conditions,” IEEE Trans. Mag. 52, 7201604 (2016).
[Crossref]

H. Egger, F. Kretzschmar, S. M. Schnepp, I. Tsukerman, and T. Weiland, “Transparent boundary conditions for a discontinuous Galerkin Trefftz method,” Appl. Math. Comp. 267, 42 (2015).
[Crossref]

F. Kretzschmar, S. Schnepp, I. Tsukerman, and T. Weiland, “Discontinuous Galerkin methods with Trefftz approximations,” J. Comp. Appl. Math. 270, 211 (2014).
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I. Tsukerman, “Trefftz difference schemes on irregular stencils,” J. Comp. Phys. 229, 2948 (2010).
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I. Tsukerman, “A class of difference schemes with flexible local approximation,” J. Comp. Phys. 211, 659–699 (2006).
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I. Tsukerman, “Electromagnetic applications of a new finite-difference calculus,” IEEE Trans. Mag. 41, 2206 (2005).
[Crossref]

F. Kretzschmar, S. M. Schnepp, H. Egger, F. Ahmadi, N. Nowak, V. Markel, and I. Tsukerman, The Power of Trefftz Approximations: Finite Difference, Boundary Difference and Discontinuous Galerkin Methods; Nonreflecting Conditions and Non-Asymptotic Homogenization (Springer, 2015), 47–58.

C. Classen, B. Bandlow, R. Schuhmann, and I. Tsukerman, “FIT & FLAME for sharp edges in electrostatics,” in URSI EMTS 2010 Symposium Digest, Berlin, Germany, 2010.

Vardeny, Z. V.

Z. V. Vardeny, A. Nahata, and A. Agrawal, “Optics of photonic quasicrystals,” Nat. Phot. 7, 177 (2013).
[Crossref]

Wang, Q. J.

S. Mansha, Z. Yongquan, Q. J. Wang, and Y. D. Chong, “Optimization of TM modes for amorphous slab lasers,” Opt. Ex. 24, 4890 (2016).
[Crossref]

H. K. Liang, B. Meng, G. Liang, J. Tao, Y. Chong, Q. J. Wang, and Y. Zhang, “Electrically pumped mid-infrared random lasers,” Adv. Mat. 25, 6589 (2013).
[Crossref]

Wang, Y.

Z. Feng, X. Zhang, Y. Wang, Z.-Y. Li, B. Cheng, and D.-Z. Zhang, “Negative refraction and imaging using 12-fold-symmetry quasicrystals,” Phys. Rev. Lett. 94, 247402 (2005).
[Crossref]

Webb, J. P.

H. Pinheiro and J. P. Webb, “A FLAME molecule for 3-D electromagnetic scattering,” IEEE Trans. Mag. 45, 1120 (2009).
[Crossref]

Weiland, T.

H. Egger, F. Kretzschmar, S. M. Schnepp, and T. Weiland, “A Space-time Discontinuous Galerkin Trefftz method for time dependent Maxwell’s equations,” SIAM J. Sci. Comp. 37, B689 (2015).
[Crossref]

H. Egger, F. Kretzschmar, S. M. Schnepp, I. Tsukerman, and T. Weiland, “Transparent boundary conditions for a discontinuous Galerkin Trefftz method,” Appl. Math. Comp. 267, 42 (2015).
[Crossref]

F. Kretzschmar, S. Schnepp, I. Tsukerman, and T. Weiland, “Discontinuous Galerkin methods with Trefftz approximations,” J. Comp. Appl. Math. 270, 211 (2014).
[Crossref]

Whittaker, D. M.

D. M. Whittaker and I. S. Culshaw, “Scattering-matrix treatment of patterned multilayer photonic structures,” Phys. Rev. B 60, 2610 (1999).
[Crossref]

Winn, J. N.

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

Xiong, W.

S. Knitter, S. F. Liew, W. Xiong, M. I. Guy, G. S. Solomon, and H. Cao, “Topological defect lasers,” J. Opt. 18, 014005 (2015).
[Crossref]

S. F. Liew, S. Knitter, W. Xiong, and H. Cao, “Photonic crystals with topological defects,” Phys. Rev. A 91, 023811 (2015).
[Crossref]

Yang, J. K.

H. Noh, J. K. Yang, S. F. Liew, M. J. Roooks, and H. Cao, “Control of lasing in biomimetic structures with short-range order,” Phys. Rev. Lett. 106, 183901 (2011).
[Crossref] [PubMed]

Yang, J.-K.

J.-K. Yang, S. V. Boriskina, H. Noh, M. J. Rooks, G. S. Solomon, L. Dal Negro, and H. Cao, “Demonstration of laser action in a pseudorandom medium,” Appl. Phys. Lett. 97223101 (2010).
[Crossref]

Yariv, A.

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284, 1819 (1999).
[Crossref] [PubMed]

Yokoyama, M.

S. Noda, M. Yokoyama, M. Imada, A. Chutinan, and M. Mochizuki, “Polarization mode control of two-dimensional photonic crystal laser by unit cell structure design,” Science 293, 1123 (2001).
[Crossref] [PubMed]

Yongquan, Z.

S. Mansha, Z. Yongquan, Q. J. Wang, and Y. D. Chong, “Optimization of TM modes for amorphous slab lasers,” Opt. Ex. 24, 4890 (2016).
[Crossref]

Yoshimoto, S.

H. Matsubara, S. Yoshimoto, H. Saito, Y. Jianglin, Y. Tanaka, and S. Noda, “GaN photonic-crystal surface-emitting laser at blue-violet wavelengths,” Science 319, 445 (2008).
[Crossref]

Yu, N.

H.-T. Chen, A. J. Taylor, and N. Yu, “A review of metasurfaces: physics and applications,” Rep. Prog. Phys. 79, 076401 (2016).
[Crossref] [PubMed]

N. Yu and F. Capasso, “Flat optics with designer metasurfaces,” Nat. Mat. 13, 139 (2014).
[Crossref]

Zhang, D.

C. Jin, X. Meng, B. Cheng, Z. Li, and D. Zhang, “Photonic gap in amorphous photonic materials,” Phys. Rev. B 63, 195107 (2001).
[Crossref]

Zhang, D.-Z.

Z. Feng, X. Zhang, Y. Wang, Z.-Y. Li, B. Cheng, and D.-Z. Zhang, “Negative refraction and imaging using 12-fold-symmetry quasicrystals,” Phys. Rev. Lett. 94, 247402 (2005).
[Crossref]

Zhang, X.

Z. Feng, X. Zhang, Y. Wang, Z.-Y. Li, B. Cheng, and D.-Z. Zhang, “Negative refraction and imaging using 12-fold-symmetry quasicrystals,” Phys. Rev. Lett. 94, 247402 (2005).
[Crossref]

Zhang, Y.

H. K. Liang, B. Meng, G. Liang, J. Tao, Y. Chong, Q. J. Wang, and Y. Zhang, “Electrically pumped mid-infrared random lasers,” Adv. Mat. 25, 6589 (2013).
[Crossref]

Adv. Mat. (2)

P. D. García, R. Sapienca, Á. Blanco, and C. López, “Photonic glass: a novel random material for light,” Adv. Mat. 19, 2597 (2007).
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H. K. Liang, B. Meng, G. Liang, J. Tao, Y. Chong, Q. J. Wang, and Y. Zhang, “Electrically pumped mid-infrared random lasers,” Adv. Mat. 25, 6589 (2013).
[Crossref]

Appl. Math. Comp. (1)

H. Egger, F. Kretzschmar, S. M. Schnepp, I. Tsukerman, and T. Weiland, “Transparent boundary conditions for a discontinuous Galerkin Trefftz method,” Appl. Math. Comp. 267, 42 (2015).
[Crossref]

Appl. Phys. Lett. (4)

J.-K. Yang, S. V. Boriskina, H. Noh, M. J. Rooks, G. S. Solomon, L. Dal Negro, and H. Cao, “Demonstration of laser action in a pseudorandom medium,” Appl. Phys. Lett. 97223101 (2010).
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H.-Y. Ryu, S.-H. Kwon, Y.-J. Lee, and Y.-H. Lee, “Very-low-threshold photonic band-edge lasers from free-standing triangular photonic crystal slabs,” Appl. Phys. Lett. 80, 3476 (2002).
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V. Liu and S. Fan, “S4: A free electromagnetic solver for layered periodic structures,” Comp. Phys. Comm. 183, 2233 (2012).
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IEEE Trans. Mag. (3)

I. Tsukerman, “Electromagnetic applications of a new finite-difference calculus,” IEEE Trans. Mag. 41, 2206 (2005).
[Crossref]

H. Pinheiro and J. P. Webb, “A FLAME molecule for 3-D electromagnetic scattering,” IEEE Trans. Mag. 45, 1120 (2009).
[Crossref]

A. Paganini, L. Scarabosio, R. Hiptmair, and I. Tsukerman, “Trefftz approximations: a new framework for nonreflecting boundary conditions,” IEEE Trans. Mag. 52, 7201604 (2016).
[Crossref]

J. Comp. Appl. Math. (1)

F. Kretzschmar, S. Schnepp, I. Tsukerman, and T. Weiland, “Discontinuous Galerkin methods with Trefftz approximations,” J. Comp. Appl. Math. 270, 211 (2014).
[Crossref]

J. Comp. Phys. (2)

I. Tsukerman, “A class of difference schemes with flexible local approximation,” J. Comp. Phys. 211, 659–699 (2006).
[Crossref]

I. Tsukerman, “Trefftz difference schemes on irregular stencils,” J. Comp. Phys. 229, 2948 (2010).
[Crossref]

J. Opt. (1)

S. Knitter, S. F. Liew, W. Xiong, M. I. Guy, G. S. Solomon, and H. Cao, “Topological defect lasers,” J. Opt. 18, 014005 (2015).
[Crossref]

J. Opt. Soc. Am. A (4)

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W. Steurer and D. Sutter-Widmer, “Photonic and phononic quasicrystals,” J. Phys. D: Appl. Phys. 40, R229 (2007).
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Mathematics of Computation (1)

Ralf Hiptmair, Andrea Moiola, and Ilaria Perugia, “Error analysis of Trefftz-Discontinuous Galerkin methods for the time-harmonic Maxwell equations,” Mathematics of Computation 82, 247–268 (2013).
[Crossref]

Microwave Opt. Tech. Lett. (1)

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[Crossref]

Nat. Mat. (1)

N. Yu and F. Capasso, “Flat optics with designer metasurfaces,” Nat. Mat. 13, 139 (2014).
[Crossref]

Nat. Phot. (1)

Z. V. Vardeny, A. Nahata, and A. Agrawal, “Optics of photonic quasicrystals,” Nat. Phot. 7, 177 (2013).
[Crossref]

Num. Meth. for Partial Diff. Eq. (1)

I. Herrera, “Trefftz method: A general theory,” Num. Meth. for Partial Diff. Eq. 16, 561–580 (2000).
[Crossref]

Opt. Ex. (3)

A. Ricciardi, I. Gallina, S. Campopiano, G. Castaldi, M. Pisco, V. Galdi, and A. Cusano, “Guided resonances in photonic quasicrystals,” Opt. Ex. 17, 6335 (2009).

S. Mansha, Z. Yongquan, Q. J. Wang, and Y. D. Chong, “Optimization of TM modes for amorphous slab lasers,” Opt. Ex. 24, 4890 (2016).
[Crossref]

S.-L. Chua, Y. Chong, A. D. Stone, M. Soljačić, and J. Bravo-Abad, “Low-threshold lasing action in photonic crystal slabs enabled by Fano resonances,” Opt. Ex. 19, 1539 (2011).
[Crossref]

Phys. Rev. A (1)

S. F. Liew, S. Knitter, W. Xiong, and H. Cao, “Photonic crystals with topological defects,” Phys. Rev. A 91, 023811 (2015).
[Crossref]

Phys. Rev. B (3)

C. Jin, X. Meng, B. Cheng, Z. Li, and D. Zhang, “Photonic gap in amorphous photonic materials,” Phys. Rev. B 63, 195107 (2001).
[Crossref]

M. Imada, A. Chutinan, S. Noda, and M. Mochizuki, “Multidirectionally distributed feedback photonic crystal lasers,” Phys. Rev. B 65, 195306 (2002).
[Crossref]

D. M. Whittaker and I. S. Culshaw, “Scattering-matrix treatment of patterned multilayer photonic structures,” Phys. Rev. B 60, 2610 (1999).
[Crossref]

Phys. Rev. Lett. (2)

H. Noh, J. K. Yang, S. F. Liew, M. J. Roooks, and H. Cao, “Control of lasing in biomimetic structures with short-range order,” Phys. Rev. Lett. 106, 183901 (2011).
[Crossref] [PubMed]

Z. Feng, X. Zhang, Y. Wang, Z.-Y. Li, B. Cheng, and D.-Z. Zhang, “Negative refraction and imaging using 12-fold-symmetry quasicrystals,” Phys. Rev. Lett. 94, 247402 (2005).
[Crossref]

Proc. Nat. Acad. Sci. Am. (1)

M. Florescu, S. Torquato, and P. Steinhardt, “Designer disordered materials with large, complete photonic band gaps,” Proc. Nat. Acad. Sci. Am. 106, 20658 (2009).
[Crossref]

Rep. Prog. Phys. (1)

H.-T. Chen, A. J. Taylor, and N. Yu, “A review of metasurfaces: physics and applications,” Rep. Prog. Phys. 79, 076401 (2016).
[Crossref] [PubMed]

Science (3)

H. Matsubara, S. Yoshimoto, H. Saito, Y. Jianglin, Y. Tanaka, and S. Noda, “GaN photonic-crystal surface-emitting laser at blue-violet wavelengths,” Science 319, 445 (2008).
[Crossref]

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284, 1819 (1999).
[Crossref] [PubMed]

S. Noda, M. Yokoyama, M. Imada, A. Chutinan, and M. Mochizuki, “Polarization mode control of two-dimensional photonic crystal laser by unit cell structure design,” Science 293, 1123 (2001).
[Crossref] [PubMed]

SIAM J. Sci. Comp. (1)

H. Egger, F. Kretzschmar, S. M. Schnepp, and T. Weiland, “A Space-time Discontinuous Galerkin Trefftz method for time dependent Maxwell’s equations,” SIAM J. Sci. Comp. 37, B689 (2015).
[Crossref]

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Other (6)

R. Hiptmair, A. Moiola, and I. Perugia, A Survey of Trefftz Methods for the Helmholtz Equation, in: G. R. Barrenechea, A. Cangiani, and E. H. Geogoulis eds., Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations, Lecture Notes in Computational Science and Engineering (LNCSE)114 (Springer, 2016), pp. 237–278.

C. Classen, B. Bandlow, R. Schuhmann, and I. Tsukerman, “FIT & FLAME for sharp edges in electrostatics,” in URSI EMTS 2010 Symposium Digest, Berlin, Germany, 2010.

F. Kretzschmar, S. M. Schnepp, H. Egger, F. Ahmadi, N. Nowak, V. Markel, and I. Tsukerman, The Power of Trefftz Approximations: Finite Difference, Boundary Difference and Discontinuous Galerkin Methods; Nonreflecting Conditions and Non-Asymptotic Homogenization (Springer, 2015), 47–58.

Q. H. Qin, The Trefftz Finite and Boundary Element Method (WIT Press, 2000).

Luca Dal Negro (ed.), Optics of Aperiodic Structures: Fundamentals and Device Applications (Pan Stanford, 2013).
[Crossref]

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

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Figures (7)

Fig. 1
Fig. 1 (a) Schematic of electromagnetic wave scattering from a 3D slab. The slab consists of one or more dielectric layers stacked in the z direction, and is patterned (e.g., with pillars) along the x-y plane. A plane wave is incident from the −z side, with a wave vector kin. (b) The 2D version of the problem, where the dielectric function does not vary with y, and the wave is incident along the x-z plane.
Fig. 2
Fig. 2 (a) A coarse Nx × Nz grid for the slab structure, where Nz = 3. (b) Assignment of DoF for patches centered on column m. Left: patch centered at z0, whose DoF are the y-components of the electric field in the nine nodes. Center and right: boundary patches, whose DoF are the electric fields at the six nodes and the x-component of the magnetic field at the boundary nodes.
Fig. 3
Fig. 3 Procedure for generating Trefftz basis functions. (a)–(b) Schematic of “Trefftz cells” defined as unit cells of periodic slabs with one pillar per period. Two periods are used, L1 and L2; for each, the RCWA solver produces four Trefftz functions corresponding to angles of illumination θ1,…, θ4. (c) For a given patch centered at grid coordinate xm, one uses the eight functions precomputed in (a, b) according to the position x′ of the nearest pillar.
Fig. 4
Fig. 4 Structure of the matrix A for Nz = 3. Non-zero entries are indicated in red.
Fig. 5
Fig. 5 (a) Real part of the electric field Es versus horizontal position x, along the middle layer of grid points at z = z0. (b) Real part of the magnetic field Hs versus x, along the outer layer of grid points at z = z. In both cases, blue dots show FLAME-slab results calculated using Nxw/Lx = 1.44, NTw/L1 = 160, and NG = 150; red curves show reference solutions calculated using RCWA with N G ref = 1000. The frequency is f = 0.25, and all other parameters are as stated in the main text. (c) Reflectance R versus frequency. Blue dots show FLAME-slab results, obtained using Nxw/Lx = 1.44, NTw/L1 = 160, and NG = 97; the red curve shows RCWA results calculated with N G ref = 1000.
Fig. 6
Fig. 6 (a) Solution error δ versus the discretization of the local lattice used for computing the Trefftz functions, NTw/L1. (b) Solution error δ versus the discretization of the global lattice, Nxw/Lx. The frequency is f = 0.25, and the reference solution is computed by RCWA using N G ref = 1000.
Fig. 7
Fig. 7 Solution error versus runtime for FLAME-slab and RCWA solutions, with two differently-sized test cases: (i) 10 pillars and Lx = 14, and (ii) 20 pillars and Lx = 28. The FLAME-slab solutions were calculated with Nxw/Lx = 4.30 and NTw/L1 = 160; and NG = 97 for case (i), and NG = 120 for case (ii). The RCWA solutions were calculated with NG = 200, 300, 400, 500. The reference solutions were calculated by RCWA using NG = 1000 for case (i), and NG = 1200 for case (ii).

Equations (23)

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× E = i k 0 H , × H = i k 0 ε E ,
E in ( r ) = exp ( i k r ) , k x = k 0 sin θ in , k z = k 0 cos θ in ,
E ( L x , z ) = E ( 0 , z ) exp ( i k x L x ) , H ( L x , z ) = H ( 0 , z ) exp ( i k x L x ) .
E s ( r ) = E ( r ) E in ( r ) .
E ( r , z ) = G E ˜ G exp [ i q z z ] exp [ i ( k + G ) r ] ,
T RCWA ~ O ( N G 3 ) , M RCWA ~ O ( N G 2 ) .
D ( r ) u ( r ) = 0 ,
u ( r ) u h ( r ) α c α φ α ( r ) = c T φ ( r ) ,
β = 1 m s β l β ( u ) = 0 ,
s Null ( N T ) , where N α β T = l β ( φ α ) .
A FLAME ψ = 0 ,
E ( x = L 2 , z ) = exp ( i k 0 L cos θ i ) E ( x = L 2 , z ) H x ( x = L 2 , z ) = exp ( i k 0 L cos θ i ) H x ( x = L 2 , z )
E = E in + E s H = H in + H s .
A FLAME ψ s = A FLAME ψ in ,
H s ( x , z ) = i ω E s z .
E s ( x , z ) = n c n exp [ i ( k n z z + k n x x ) ] ,
H s ( x , z ) = 1 ω n c n k n z exp [ i ( k n x x + k n z z ) ] .
k n z = ± k 2 0 k 2 n x .
H s ( x m , z ± ) = 1 ω e i m π ( 1 N x 1 ) FT m { k n z FT n { E s ( x m , z ± ) e i m π   ( 1 N x 1 ) } } .
FT m { f n } = n = 0 N 1 e 2 π i m n / N f n .
( A FLAME A BC ) ψ s = ( A FLAME ψ in 0 ) .
S z ( x , z ) = 1 2 Re [ E ( x , z ) H * ( x , z ) ] ,
δ = ( i = 1 M | ψ i Fs ψ i ref | 2 i = 1 M | ψ i ref | 2 ) 1 / 2 .

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