Abstract

There is tremendous demand for numerical methods to perform rigorous analysis of devices that are both large scale and complex throughout their volume. This can arise when devices must be considered with realistic geometry or when they contain artificial materials such as photonic crystals, left-handed materials, nanoparticles, or other metamaterials. The slice absorption method (SAM) was developed to address this need. The method is fully numerical and able to break large problems down into small pieces, or slices, using matrix division or Gaussian elimination instead of eigensystem computations and scattering matrix manipulations. In these regards, the SAM is an attractive alternative to popular techniques like the finite-difference time domain method, rigorous coupled-wave analysis, and the transfer matrix method. To demonstrate the utility of the SAM and benchmark its accuracy, reflection was simulated for a photonic crystal fabricated in SU-8 by multiphoton direct laser writing. Realistic geometry was incorporated into the model by simulating the microfabrication process, which yielded simulation results that matched experimental measurements remarkably well.

© 2007 Optical Society of America

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2007 (1)

2006 (3)

R. C. Rumpf, P. Srinivasan, and E. G. Johnson, "Modeling the fabrication of nano-optical structures," Proc. SPIE 6110, 13-26 (2006).

P. Srinivasan, R. C. Rumpf, and E. G. Johnson, "Fabrication of 3-D photonic crystals by two-step dry etching of layered media," Proc. SPIE 6110, 36-43 (2006).

J. Li, B. Jia, G. Zhou, and M. Gu, "Fabrication of three-dimensional woodpile photonic crystals in a PbSe quantum dot composite material," Opt. Express 14, 10740-10745 (2006).
[CrossRef] [PubMed]

2005 (2)

R. Rumpf and E. Johnson, "Comprehensive modeling of near-field nano-patterning," Opt. Express 13, 7198-7208 (2005).
[CrossRef] [PubMed]

R. C. Rumpf, and E. G. Johnson, "Modeling the formation of photonic crystals by holographic lithography," Proc. SPIE 5720, 18-26 (2005).
[CrossRef]

2004 (4)

R. C. Rumpf and E. G. Johnson, "Fully three-dimensional modeling of the fabrication and behavior of photonic crystals formed by holographic lithography," J. Opt. Soc. Am. A 21, 1703-1713 (2004).
[CrossRef]

R. Dekker, D. J. W. Klunder, A. Borreman, M. B. J. Diemeer, K. Worhoff, A. Driessen, J. W. Stouwdam, and F. C. J. M. Veggel, "Stimulated emission of optical gain in LaF3:Nd nanoparticle-doped polymer-based waveguides," Appl. Phys. Lett. 85, 6104-6106 (2004).
[CrossRef]

J. Pendry, "Manipulating the near field with metamaterials," Opt. Photon. News 15(9), 32-37 (2004).
[CrossRef]

M. Deubel, G. Freymann, M. Wegener, S. Pereira, K. Busch, and C. M. Soukoulis, "Direct laser writing of three-dimensional photonic-crystal templates for telecommunications," Nat. Mater. 4, 444-447 (2004).
[CrossRef]

2003 (2)

S. Guo and S. Albin, "Simple plane wave implementation for photonic crystal calculations," Opt. Express 11, 167-175 (2003).
[CrossRef] [PubMed]

Z. Li and L. Lin, "Photonic band structures solved by a plane-wave-based transfer matrix method," Phys. Rev. E 67, 046607 (2003).
[CrossRef]

2002 (4)

H. A. Jamid and M. N. Akram, "Analysis of deep waveguide gratings: an efficient cascading and doubling algorithm in the method of lines framework," J. Lightwave Technol. 20, 1204-1209 (2002).
[CrossRef]

S. Wu and E. N. Glytsis, "Finite-number-of-periods holographic gratings with finite-width incident beams: analysis using the finite-difference frequency-domain method," J. Opt. Soc. Am. A 19, 2018-2029 (2002).
[CrossRef]

W. Wenseleers, F. Stellacci, T. Meyer-Friedrichsen, T. Mangel, C. Bauer, S. J. K. Pond, S. R. Marder, and J. W. Perry, "Five orders-of-magnitude enhancement of two-photon absorption for dyes on silver nanoparticle fractal clusters," J. Phys. Chem. B 106, 6853-6863 (2002).
[CrossRef]

Q. Cao and P. Lalanne, "Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits," Phys. Rev. Lett. 88, 057403 (2002).
[CrossRef] [PubMed]

2001 (3)

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001).
[CrossRef] [PubMed]

G. Granet, "Rigorous electromagnetic analysis of resonant sub-wavelength metallic gratings by parametric Fourier modal method," Opt. Quantum Electron. 33, 471-483 (2001).
[CrossRef]

S. Johnson and J. Joannopoulos, "Block-iterative frequency-domain methods for Maxwell's equations in a planewave basis," Opt. Express 8, 173-190 (2001).
[CrossRef] [PubMed]

2000 (1)

J. B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966-3969 (2000).
[CrossRef] [PubMed]

1999 (1)

E. A. Marengo, C. M. Rappaport, and E. L. Miller, "Optimum PML ABC conductivity profile in FDFD," IEEE Trans. Magn. 35, 1506-1509 (1999).
[CrossRef]

1997 (1)

1996 (3)

J. Fang, "Generalized perfectly matched layer for the absorption of propagating and evanescent waves in lossless and lossy media," IEEE Trans. Microwave Theory Tech. 44, 2216-2222 (1996).
[CrossRef]

G. Granet and B. Guizal, "Efficient implementation of the coupled-wave method for metallic lamellar gratings in TM polarization," J. Opt. Soc. Am. A 13, 1019-1023 (1996).
[CrossRef]

W. Sun, K. Liu, and C. A. Balanis, "Analysis of singly and doubly periodic absorbers by frequency-domain finite-difference method," IEEE Trans. Antennas Propag. 44, 798-805 (1996).
[CrossRef]

1995 (4)

P. M. Bell, J. B. Pendry, L. M. Moreno, and A. J. Ward, "A program for calculating photonic band structures and transmission coefficients of complex structures," Comput. Phys. Commun. 85, 306-322 (1995).
[CrossRef]

M. G. Moharam, E. B. Grann, and D. A. Pommet, "Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings," J. Opt. Soc. Am. A 12, 1068-1076 (1995).
[CrossRef]

M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, "Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach," J. Opt. Soc. Am. A 12, 1077-1086 (1995).
[CrossRef]

M. Celuch-Marcysiak and W. K. Gwarek, "Spatially looped algorithms for time-domain analysis of periodic structures," IEEE Trans. Microwave Theory Tech. 43, 860-865 (1995).
[CrossRef]

1994 (2)

J. B. Pendry, "Photonic band structures," J. Mod. Opt. 41, 209-229 (1994).
[CrossRef]

K. M. Ho, C. T. Chan, C. M. Soukoulis, R. Biswas, and M. Sigalas, "Photonic band gaps in three dimensions: new layer-by-layer periodic structures," Solid State Commun. 89, 413-416 (1994).
[CrossRef]

1993 (1)

A. C. Cangellaris, M. Gribbons, and G. Sohos, "A hybrid spectral/FDTD method for the electromagnetic analysis of guided waves in periodic structures," IEEE Microw. Guid. Wave Lett. 3, 375-377 (1993).
[CrossRef]

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]

1968 (1)

V. G. Veselago, Sov. Phys. Usp. 10, 509-514 (1968).
[CrossRef]

1966 (1)

K. S. Yee, "Numerical solution of the initial boundary value problems involving Maxwell's equations in isotropic media," IEEE Trans. Antennas Propag. 14, 302-307 (1966).
[CrossRef]

1902 (1)

R. W. Wood, "On remarkable case of uneven distribution of light in diffraction grating problems," Philos. Mag. 4, 396-402 (1902).

Akram, M. N.

Albin, S.

Anton, H.

H. Anton, Elementary Linear Algebra (Wiley, 2000), pp. 8-16.

Balanis, C. A.

W. Sun, K. Liu, and C. A. Balanis, "Analysis of singly and doubly periodic absorbers by frequency-domain finite-difference method," IEEE Trans. Antennas Propag. 44, 798-805 (1996).
[CrossRef]

Bauer, C.

W. Wenseleers, F. Stellacci, T. Meyer-Friedrichsen, T. Mangel, C. Bauer, S. J. K. Pond, S. R. Marder, and J. W. Perry, "Five orders-of-magnitude enhancement of two-photon absorption for dyes on silver nanoparticle fractal clusters," J. Phys. Chem. B 106, 6853-6863 (2002).
[CrossRef]

Bell, P. M.

P. M. Bell, J. B. Pendry, L. M. Moreno, and A. J. Ward, "A program for calculating photonic band structures and transmission coefficients of complex structures," Comput. Phys. Commun. 85, 306-322 (1995).
[CrossRef]

Biswas, R.

K. M. Ho, C. T. Chan, C. M. Soukoulis, R. Biswas, and M. Sigalas, "Photonic band gaps in three dimensions: new layer-by-layer periodic structures," Solid State Commun. 89, 413-416 (1994).
[CrossRef]

Borreman, A.

R. Dekker, D. J. W. Klunder, A. Borreman, M. B. J. Diemeer, K. Worhoff, A. Driessen, J. W. Stouwdam, and F. C. J. M. Veggel, "Stimulated emission of optical gain in LaF3:Nd nanoparticle-doped polymer-based waveguides," Appl. Phys. Lett. 85, 6104-6106 (2004).
[CrossRef]

Busch, K.

M. Deubel, G. Freymann, M. Wegener, S. Pereira, K. Busch, and C. M. Soukoulis, "Direct laser writing of three-dimensional photonic-crystal templates for telecommunications," Nat. Mater. 4, 444-447 (2004).
[CrossRef]

Canale, R. P.

S. C. Chapra and R. P. Canale, Numerical Methods for Engineers With Software and Programming Applications, 4th ed. (McGraw-Hill, 2002), pp. 632-643.

Cangellaris, A. C.

A. C. Cangellaris, M. Gribbons, and G. Sohos, "A hybrid spectral/FDTD method for the electromagnetic analysis of guided waves in periodic structures," IEEE Microw. Guid. Wave Lett. 3, 375-377 (1993).
[CrossRef]

Cao, Q.

Q. Cao and P. Lalanne, "Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits," Phys. Rev. Lett. 88, 057403 (2002).
[CrossRef] [PubMed]

Celuch-Marcysiak, M.

M. Celuch-Marcysiak and W. K. Gwarek, "Spatially looped algorithms for time-domain analysis of periodic structures," IEEE Trans. Microwave Theory Tech. 43, 860-865 (1995).
[CrossRef]

Chan, C. T.

K. M. Ho, C. T. Chan, C. M. Soukoulis, R. Biswas, and M. Sigalas, "Photonic band gaps in three dimensions: new layer-by-layer periodic structures," Solid State Commun. 89, 413-416 (1994).
[CrossRef]

Chapra, S. C.

S. C. Chapra and R. P. Canale, Numerical Methods for Engineers With Software and Programming Applications, 4th ed. (McGraw-Hill, 2002), pp. 632-643.

Dekker, R.

R. Dekker, D. J. W. Klunder, A. Borreman, M. B. J. Diemeer, K. Worhoff, A. Driessen, J. W. Stouwdam, and F. C. J. M. Veggel, "Stimulated emission of optical gain in LaF3:Nd nanoparticle-doped polymer-based waveguides," Appl. Phys. Lett. 85, 6104-6106 (2004).
[CrossRef]

Deubel, M.

M. Deubel, G. Freymann, M. Wegener, S. Pereira, K. Busch, and C. M. Soukoulis, "Direct laser writing of three-dimensional photonic-crystal templates for telecommunications," Nat. Mater. 4, 444-447 (2004).
[CrossRef]

Diemeer, M. B. J.

R. Dekker, D. J. W. Klunder, A. Borreman, M. B. J. Diemeer, K. Worhoff, A. Driessen, J. W. Stouwdam, and F. C. J. M. Veggel, "Stimulated emission of optical gain in LaF3:Nd nanoparticle-doped polymer-based waveguides," Appl. Phys. Lett. 85, 6104-6106 (2004).
[CrossRef]

Driessen, A.

R. Dekker, D. J. W. Klunder, A. Borreman, M. B. J. Diemeer, K. Worhoff, A. Driessen, J. W. Stouwdam, and F. C. J. M. Veggel, "Stimulated emission of optical gain in LaF3:Nd nanoparticle-doped polymer-based waveguides," Appl. Phys. Lett. 85, 6104-6106 (2004).
[CrossRef]

Fang, J.

J. Fang, "Generalized perfectly matched layer for the absorption of propagating and evanescent waves in lossless and lossy media," IEEE Trans. Microwave Theory Tech. 44, 2216-2222 (1996).
[CrossRef]

Freymann, G.

M. Deubel, G. Freymann, M. Wegener, S. Pereira, K. Busch, and C. M. Soukoulis, "Direct laser writing of three-dimensional photonic-crystal templates for telecommunications," Nat. Mater. 4, 444-447 (2004).
[CrossRef]

Gaylord, and T. K.

Glytsis, E. N.

Granet, G.

G. Granet, "Rigorous electromagnetic analysis of resonant sub-wavelength metallic gratings by parametric Fourier modal method," Opt. Quantum Electron. 33, 471-483 (2001).
[CrossRef]

G. Granet and B. Guizal, "Efficient implementation of the coupled-wave method for metallic lamellar gratings in TM polarization," J. Opt. Soc. Am. A 13, 1019-1023 (1996).
[CrossRef]

Grann, E. B.

Gribbons, M.

A. C. Cangellaris, M. Gribbons, and G. Sohos, "A hybrid spectral/FDTD method for the electromagnetic analysis of guided waves in periodic structures," IEEE Microw. Guid. Wave Lett. 3, 375-377 (1993).
[CrossRef]

Gu, and M.

Guizal, B.

Guo, S.

Gwarek, W. K.

M. Celuch-Marcysiak and W. K. Gwarek, "Spatially looped algorithms for time-domain analysis of periodic structures," IEEE Trans. Microwave Theory Tech. 43, 860-865 (1995).
[CrossRef]

Hagness, S. C.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).

Ho, K. M.

K. M. Ho, C. T. Chan, C. M. Soukoulis, R. Biswas, and M. Sigalas, "Photonic band gaps in three dimensions: new layer-by-layer periodic structures," Solid State Commun. 89, 413-416 (1994).
[CrossRef]

Itoh, T.

T. Itoh, Numerical Techniques for Microwave and Millimeter-Wave Passive Structures (Wiley, 1989).

Jamid, H. A.

Jia, B.

Jin, J.

J. Jin, The Finite Element Method in Electromagnetic, 2nd ed. (Wiley, 2002).

Joannopoulos, J.

Joannopoulous, J.

J. Joannopoulous, Photonic Crystals: Molding the Flow of Light (Princeton, 1995).

John, S.

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

Johnson, and E. G.

P. Srinivasan, R. C. Rumpf, and E. G. Johnson, "Fabrication of 3-D photonic crystals by two-step dry etching of layered media," Proc. SPIE 6110, 36-43 (2006).

Johnson, E.

Johnson, E. G.

R. C. Rumpf and E. G. Johnson, "Modeling fabrication to accurately place GMR resonances," Opt. Express 15, 3452-3464 (2007).
[CrossRef] [PubMed]

R. C. Rumpf, P. Srinivasan, and E. G. Johnson, "Modeling the fabrication of nano-optical structures," Proc. SPIE 6110, 13-26 (2006).

R. C. Rumpf, and E. G. Johnson, "Modeling the formation of photonic crystals by holographic lithography," Proc. SPIE 5720, 18-26 (2005).
[CrossRef]

R. C. Rumpf and E. G. Johnson, "Fully three-dimensional modeling of the fabrication and behavior of photonic crystals formed by holographic lithography," J. Opt. Soc. Am. A 21, 1703-1713 (2004).
[CrossRef]

Johnson, S.

Klunder, D. J. W.

R. Dekker, D. J. W. Klunder, A. Borreman, M. B. J. Diemeer, K. Worhoff, A. Driessen, J. W. Stouwdam, and F. C. J. M. Veggel, "Stimulated emission of optical gain in LaF3:Nd nanoparticle-doped polymer-based waveguides," Appl. Phys. Lett. 85, 6104-6106 (2004).
[CrossRef]

Kuebler, S. M.

S. M. Kuebler and M. Rumi, Encyclopedia of Modern Optics, R.D.Guenther, D.G.Steel, and L.Bayvel, eds. (Oxford, 2004), pp. 189.

Lalanne, P.

Q. Cao and P. Lalanne, "Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits," Phys. Rev. Lett. 88, 057403 (2002).
[CrossRef] [PubMed]

P. Lalanne, "Improved formulation of the coupled-wave method for two-dimensional gratings," J. Opt. Soc. Am. A 14, 1592-1598 (1997).
[CrossRef]

Li, J.

Li, Z.

Z. Li and L. Lin, "Photonic band structures solved by a plane-wave-based transfer matrix method," Phys. Rev. E 67, 046607 (2003).
[CrossRef]

Lin, L.

Z. Li and L. Lin, "Photonic band structures solved by a plane-wave-based transfer matrix method," Phys. Rev. E 67, 046607 (2003).
[CrossRef]

Liu, K.

W. Sun, K. Liu, and C. A. Balanis, "Analysis of singly and doubly periodic absorbers by frequency-domain finite-difference method," IEEE Trans. Antennas Propag. 44, 798-805 (1996).
[CrossRef]

Lourtioz, J.

J. Lourtioz, Photonic Crystals: Towards Nanoscale Photonic Devices (Springer, 2005).

Mangel, T.

W. Wenseleers, F. Stellacci, T. Meyer-Friedrichsen, T. Mangel, C. Bauer, S. J. K. Pond, S. R. Marder, and J. W. Perry, "Five orders-of-magnitude enhancement of two-photon absorption for dyes on silver nanoparticle fractal clusters," J. Phys. Chem. B 106, 6853-6863 (2002).
[CrossRef]

Manolakis, D. G.

J. G. Proakis and D. G. Manolakis, Digital Signal Processing: Principles, Algorithms, and Applications, 3rd ed. (Prentice Hall, 1996), pp. 399-403.

Marder, S. R.

W. Wenseleers, F. Stellacci, T. Meyer-Friedrichsen, T. Mangel, C. Bauer, S. J. K. Pond, S. R. Marder, and J. W. Perry, "Five orders-of-magnitude enhancement of two-photon absorption for dyes on silver nanoparticle fractal clusters," J. Phys. Chem. B 106, 6853-6863 (2002).
[CrossRef]

Marengo, E. A.

E. A. Marengo, C. M. Rappaport, and E. L. Miller, "Optimum PML ABC conductivity profile in FDFD," IEEE Trans. Magn. 35, 1506-1509 (1999).
[CrossRef]

Meyer-Friedrichsen, T.

W. Wenseleers, F. Stellacci, T. Meyer-Friedrichsen, T. Mangel, C. Bauer, S. J. K. Pond, S. R. Marder, and J. W. Perry, "Five orders-of-magnitude enhancement of two-photon absorption for dyes on silver nanoparticle fractal clusters," J. Phys. Chem. B 106, 6853-6863 (2002).
[CrossRef]

Miller, E. L.

E. A. Marengo, C. M. Rappaport, and E. L. Miller, "Optimum PML ABC conductivity profile in FDFD," IEEE Trans. Magn. 35, 1506-1509 (1999).
[CrossRef]

Moharam, M. G.

Moreno, L. M.

P. M. Bell, J. B. Pendry, L. M. Moreno, and A. J. Ward, "A program for calculating photonic band structures and transmission coefficients of complex structures," Comput. Phys. Commun. 85, 306-322 (1995).
[CrossRef]

Mosallaei, H.

H. Mosallaei and K. Sarabandi, "Periodic meta-material structures in electromagnetics: concept, analysis, and applications," in Antennas and Propagation Society International Symposium (IEEE, 2002), Vol. 2, pp. 380-383.

Pendry, J.

J. Pendry, "Manipulating the near field with metamaterials," Opt. Photon. News 15(9), 32-37 (2004).
[CrossRef]

Pendry, J. B.

J. B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966-3969 (2000).
[CrossRef] [PubMed]

P. M. Bell, J. B. Pendry, L. M. Moreno, and A. J. Ward, "A program for calculating photonic band structures and transmission coefficients of complex structures," Comput. Phys. Commun. 85, 306-322 (1995).
[CrossRef]

J. B. Pendry, "Photonic band structures," J. Mod. Opt. 41, 209-229 (1994).
[CrossRef]

Pereira, S.

M. Deubel, G. Freymann, M. Wegener, S. Pereira, K. Busch, and C. M. Soukoulis, "Direct laser writing of three-dimensional photonic-crystal templates for telecommunications," Nat. Mater. 4, 444-447 (2004).
[CrossRef]

Perry, J. W.

W. Wenseleers, F. Stellacci, T. Meyer-Friedrichsen, T. Mangel, C. Bauer, S. J. K. Pond, S. R. Marder, and J. W. Perry, "Five orders-of-magnitude enhancement of two-photon absorption for dyes on silver nanoparticle fractal clusters," J. Phys. Chem. B 106, 6853-6863 (2002).
[CrossRef]

Pommet, and D. A.

Pommet, D. A.

Pond, S. J. K.

W. Wenseleers, F. Stellacci, T. Meyer-Friedrichsen, T. Mangel, C. Bauer, S. J. K. Pond, S. R. Marder, and J. W. Perry, "Five orders-of-magnitude enhancement of two-photon absorption for dyes on silver nanoparticle fractal clusters," J. Phys. Chem. B 106, 6853-6863 (2002).
[CrossRef]

Proakis, J. G.

J. G. Proakis and D. G. Manolakis, Digital Signal Processing: Principles, Algorithms, and Applications, 3rd ed. (Prentice Hall, 1996), pp. 399-403.

Quarteroni, A.

A. Quarteroni, R. Sacco, and F. Saleri, Numerical Mathematics (Springer, 2000), pp. 72-83.

Rappaport, C. M.

E. A. Marengo, C. M. Rappaport, and E. L. Miller, "Optimum PML ABC conductivity profile in FDFD," IEEE Trans. Magn. 35, 1506-1509 (1999).
[CrossRef]

Rumi, M.

S. M. Kuebler and M. Rumi, Encyclopedia of Modern Optics, R.D.Guenther, D.G.Steel, and L.Bayvel, eds. (Oxford, 2004), pp. 189.

Rumpf, R.

Rumpf, R. C.

R. C. Rumpf and E. G. Johnson, "Modeling fabrication to accurately place GMR resonances," Opt. Express 15, 3452-3464 (2007).
[CrossRef] [PubMed]

R. C. Rumpf, P. Srinivasan, and E. G. Johnson, "Modeling the fabrication of nano-optical structures," Proc. SPIE 6110, 13-26 (2006).

P. Srinivasan, R. C. Rumpf, and E. G. Johnson, "Fabrication of 3-D photonic crystals by two-step dry etching of layered media," Proc. SPIE 6110, 36-43 (2006).

R. C. Rumpf, and E. G. Johnson, "Modeling the formation of photonic crystals by holographic lithography," Proc. SPIE 5720, 18-26 (2005).
[CrossRef]

R. C. Rumpf and E. G. Johnson, "Fully three-dimensional modeling of the fabrication and behavior of photonic crystals formed by holographic lithography," J. Opt. Soc. Am. A 21, 1703-1713 (2004).
[CrossRef]

R. C. Rumpf, "Design and optimization of nano-optical elements by coupling fabrication to optical behavior," Ph.D. dissertation (University of Central Florida, 2006).

Sacco, R.

A. Quarteroni, R. Sacco, and F. Saleri, Numerical Mathematics (Springer, 2000), pp. 72-83.

Saleri, and F.

A. Quarteroni, R. Sacco, and F. Saleri, Numerical Mathematics (Springer, 2000), pp. 72-83.

Sarabandi, K.

H. Mosallaei and K. Sarabandi, "Periodic meta-material structures in electromagnetics: concept, analysis, and applications," in Antennas and Propagation Society International Symposium (IEEE, 2002), Vol. 2, pp. 380-383.

Schultz, and S.

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001).
[CrossRef] [PubMed]

Shelby, R. A.

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001).
[CrossRef] [PubMed]

Sigalas, and M.

K. M. Ho, C. T. Chan, C. M. Soukoulis, R. Biswas, and M. Sigalas, "Photonic band gaps in three dimensions: new layer-by-layer periodic structures," Solid State Commun. 89, 413-416 (1994).
[CrossRef]

Smith, D. R.

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001).
[CrossRef] [PubMed]

Sohos, and G.

A. C. Cangellaris, M. Gribbons, and G. Sohos, "A hybrid spectral/FDTD method for the electromagnetic analysis of guided waves in periodic structures," IEEE Microw. Guid. Wave Lett. 3, 375-377 (1993).
[CrossRef]

Soukoulis, and C. M.

M. Deubel, G. Freymann, M. Wegener, S. Pereira, K. Busch, and C. M. Soukoulis, "Direct laser writing of three-dimensional photonic-crystal templates for telecommunications," Nat. Mater. 4, 444-447 (2004).
[CrossRef]

Soukoulis, C. M.

K. M. Ho, C. T. Chan, C. M. Soukoulis, R. Biswas, and M. Sigalas, "Photonic band gaps in three dimensions: new layer-by-layer periodic structures," Solid State Commun. 89, 413-416 (1994).
[CrossRef]

Srinivasan, P.

R. C. Rumpf, P. Srinivasan, and E. G. Johnson, "Modeling the fabrication of nano-optical structures," Proc. SPIE 6110, 13-26 (2006).

P. Srinivasan, R. C. Rumpf, and E. G. Johnson, "Fabrication of 3-D photonic crystals by two-step dry etching of layered media," Proc. SPIE 6110, 36-43 (2006).

Stellacci, F.

W. Wenseleers, F. Stellacci, T. Meyer-Friedrichsen, T. Mangel, C. Bauer, S. J. K. Pond, S. R. Marder, and J. W. Perry, "Five orders-of-magnitude enhancement of two-photon absorption for dyes on silver nanoparticle fractal clusters," J. Phys. Chem. B 106, 6853-6863 (2002).
[CrossRef]

Stouwdam, J. W.

R. Dekker, D. J. W. Klunder, A. Borreman, M. B. J. Diemeer, K. Worhoff, A. Driessen, J. W. Stouwdam, and F. C. J. M. Veggel, "Stimulated emission of optical gain in LaF3:Nd nanoparticle-doped polymer-based waveguides," Appl. Phys. Lett. 85, 6104-6106 (2004).
[CrossRef]

Sullivan, D. M.

D. M. Sullivan, Electromagnetic Simulation Using the FDTD Method (IEEE, 2000).
[CrossRef]

Sun, W.

W. Sun, K. Liu, and C. A. Balanis, "Analysis of singly and doubly periodic absorbers by frequency-domain finite-difference method," IEEE Trans. Antennas Propag. 44, 798-805 (1996).
[CrossRef]

Taflove, A.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).

Veggel, F. C. J. M.

R. Dekker, D. J. W. Klunder, A. Borreman, M. B. J. Diemeer, K. Worhoff, A. Driessen, J. W. Stouwdam, and F. C. J. M. Veggel, "Stimulated emission of optical gain in LaF3:Nd nanoparticle-doped polymer-based waveguides," Appl. Phys. Lett. 85, 6104-6106 (2004).
[CrossRef]

Veselago, V. G.

V. G. Veselago, Sov. Phys. Usp. 10, 509-514 (1968).
[CrossRef]

Ward, and A. J.

P. M. Bell, J. B. Pendry, L. M. Moreno, and A. J. Ward, "A program for calculating photonic band structures and transmission coefficients of complex structures," Comput. Phys. Commun. 85, 306-322 (1995).
[CrossRef]

Wegener, M.

M. Deubel, G. Freymann, M. Wegener, S. Pereira, K. Busch, and C. M. Soukoulis, "Direct laser writing of three-dimensional photonic-crystal templates for telecommunications," Nat. Mater. 4, 444-447 (2004).
[CrossRef]

Wenseleers, W.

W. Wenseleers, F. Stellacci, T. Meyer-Friedrichsen, T. Mangel, C. Bauer, S. J. K. Pond, S. R. Marder, and J. W. Perry, "Five orders-of-magnitude enhancement of two-photon absorption for dyes on silver nanoparticle fractal clusters," J. Phys. Chem. B 106, 6853-6863 (2002).
[CrossRef]

Wood, R. W.

R. W. Wood, "On remarkable case of uneven distribution of light in diffraction grating problems," Philos. Mag. 4, 396-402 (1902).

Worhoff, K.

R. Dekker, D. J. W. Klunder, A. Borreman, M. B. J. Diemeer, K. Worhoff, A. Driessen, J. W. Stouwdam, and F. C. J. M. Veggel, "Stimulated emission of optical gain in LaF3:Nd nanoparticle-doped polymer-based waveguides," Appl. Phys. Lett. 85, 6104-6106 (2004).
[CrossRef]

Wu, S.

Yablonovitch, E.

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

Yasumoto, H.

H. Yasumoto, Electromagnetic Theory and Applications for Photonic Crystals (CRC Press, 2006), pp. 299-304.

Yee, K. S.

K. S. Yee, "Numerical solution of the initial boundary value problems involving Maxwell's equations in isotropic media," IEEE Trans. Antennas Propag. 14, 302-307 (1966).
[CrossRef]

Zhou, G.

Appl. Phys. Lett. (1)

R. Dekker, D. J. W. Klunder, A. Borreman, M. B. J. Diemeer, K. Worhoff, A. Driessen, J. W. Stouwdam, and F. C. J. M. Veggel, "Stimulated emission of optical gain in LaF3:Nd nanoparticle-doped polymer-based waveguides," Appl. Phys. Lett. 85, 6104-6106 (2004).
[CrossRef]

Comput. Phys. Commun. (1)

P. M. Bell, J. B. Pendry, L. M. Moreno, and A. J. Ward, "A program for calculating photonic band structures and transmission coefficients of complex structures," Comput. Phys. Commun. 85, 306-322 (1995).
[CrossRef]

IEEE Microw. Guid. Wave Lett. (1)

A. C. Cangellaris, M. Gribbons, and G. Sohos, "A hybrid spectral/FDTD method for the electromagnetic analysis of guided waves in periodic structures," IEEE Microw. Guid. Wave Lett. 3, 375-377 (1993).
[CrossRef]

IEEE Trans. Antennas Propag. (2)

W. Sun, K. Liu, and C. A. Balanis, "Analysis of singly and doubly periodic absorbers by frequency-domain finite-difference method," IEEE Trans. Antennas Propag. 44, 798-805 (1996).
[CrossRef]

K. S. Yee, "Numerical solution of the initial boundary value problems involving Maxwell's equations in isotropic media," IEEE Trans. Antennas Propag. 14, 302-307 (1966).
[CrossRef]

IEEE Trans. Magn. (1)

E. A. Marengo, C. M. Rappaport, and E. L. Miller, "Optimum PML ABC conductivity profile in FDFD," IEEE Trans. Magn. 35, 1506-1509 (1999).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (2)

J. Fang, "Generalized perfectly matched layer for the absorption of propagating and evanescent waves in lossless and lossy media," IEEE Trans. Microwave Theory Tech. 44, 2216-2222 (1996).
[CrossRef]

M. Celuch-Marcysiak and W. K. Gwarek, "Spatially looped algorithms for time-domain analysis of periodic structures," IEEE Trans. Microwave Theory Tech. 43, 860-865 (1995).
[CrossRef]

J. Lightwave Technol. (1)

J. Mod. Opt. (1)

J. B. Pendry, "Photonic band structures," J. Mod. Opt. 41, 209-229 (1994).
[CrossRef]

J. Opt. Soc. Am. A (6)

J. Phys. Chem. B (1)

W. Wenseleers, F. Stellacci, T. Meyer-Friedrichsen, T. Mangel, C. Bauer, S. J. K. Pond, S. R. Marder, and J. W. Perry, "Five orders-of-magnitude enhancement of two-photon absorption for dyes on silver nanoparticle fractal clusters," J. Phys. Chem. B 106, 6853-6863 (2002).
[CrossRef]

Nat. Mater. (1)

M. Deubel, G. Freymann, M. Wegener, S. Pereira, K. Busch, and C. M. Soukoulis, "Direct laser writing of three-dimensional photonic-crystal templates for telecommunications," Nat. Mater. 4, 444-447 (2004).
[CrossRef]

Opt. Express (5)

Opt. Photon. News (1)

J. Pendry, "Manipulating the near field with metamaterials," Opt. Photon. News 15(9), 32-37 (2004).
[CrossRef]

Opt. Quantum Electron. (1)

G. Granet, "Rigorous electromagnetic analysis of resonant sub-wavelength metallic gratings by parametric Fourier modal method," Opt. Quantum Electron. 33, 471-483 (2001).
[CrossRef]

Philos. Mag. (1)

R. W. Wood, "On remarkable case of uneven distribution of light in diffraction grating problems," Philos. Mag. 4, 396-402 (1902).

Phys. Rev. E (1)

Z. Li and L. Lin, "Photonic band structures solved by a plane-wave-based transfer matrix method," Phys. Rev. E 67, 046607 (2003).
[CrossRef]

Phys. Rev. Lett. (4)

J. B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966-3969 (2000).
[CrossRef] [PubMed]

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]

Q. Cao and P. Lalanne, "Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits," Phys. Rev. Lett. 88, 057403 (2002).
[CrossRef] [PubMed]

Proc. SPIE (3)

R. C. Rumpf, P. Srinivasan, and E. G. Johnson, "Modeling the fabrication of nano-optical structures," Proc. SPIE 6110, 13-26 (2006).

R. C. Rumpf, and E. G. Johnson, "Modeling the formation of photonic crystals by holographic lithography," Proc. SPIE 5720, 18-26 (2005).
[CrossRef]

P. Srinivasan, R. C. Rumpf, and E. G. Johnson, "Fabrication of 3-D photonic crystals by two-step dry etching of layered media," Proc. SPIE 6110, 36-43 (2006).

Science (1)

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001).
[CrossRef] [PubMed]

Solid State Commun. (1)

K. M. Ho, C. T. Chan, C. M. Soukoulis, R. Biswas, and M. Sigalas, "Photonic band gaps in three dimensions: new layer-by-layer periodic structures," Solid State Commun. 89, 413-416 (1994).
[CrossRef]

Sov. Phys. Usp. (1)

V. G. Veselago, Sov. Phys. Usp. 10, 509-514 (1968).
[CrossRef]

Other (26)

J. Joannopoulous, Photonic Crystals: Molding the Flow of Light (Princeton, 1995).

J. Lourtioz, Photonic Crystals: Towards Nanoscale Photonic Devices (Springer, 2005).

H. Mosallaei and K. Sarabandi, "Periodic meta-material structures in electromagnetics: concept, analysis, and applications," in Antennas and Propagation Society International Symposium (IEEE, 2002), Vol. 2, pp. 380-383.

R. C. Rumpf, "Design and optimization of nano-optical elements by coupling fabrication to optical behavior," Ph.D. dissertation (University of Central Florida, 2006).

H. Anton, Elementary Linear Algebra (Wiley, 2000), pp. 8-16.

A. Quarteroni, R. Sacco, and F. Saleri, Numerical Mathematics (Springer, 2000), pp. 72-83.

Ref. , p. 65.

S. C. Chapra and R. P. Canale, Numerical Methods for Engineers With Software and Programming Applications, 4th ed. (McGraw-Hill, 2002), pp. 632-643.

Ref. , pp. 58-60.

Ref. , pp. 60-84.

Ref. , pp. 125-152.

T. Itoh, Numerical Techniques for Microwave and Millimeter-Wave Passive Structures (Wiley, 1989).

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).

D. M. Sullivan, Electromagnetic Simulation Using the FDTD Method (IEEE, 2000).
[CrossRef]

Ref. , pp. 553-555.

J. G. Proakis and D. G. Manolakis, Digital Signal Processing: Principles, Algorithms, and Applications, 3rd ed. (Prentice Hall, 1996), pp. 399-403.

Ref. , pp. 132-134.

Ref. , pp. 109-124.

Ref. , pp. 72-74.

H. Yasumoto, Electromagnetic Theory and Applications for Photonic Crystals (CRC Press, 2006), pp. 299-304.

Ref. , pp. 75-78.

J. Jin, The Finite Element Method in Electromagnetic, 2nd ed. (Wiley, 2002).

Ref. , pp. 823-824.

Ref. , pp. 186-212.

Ref. , pp. 147-149.

S. M. Kuebler and M. Rumi, Encyclopedia of Modern Optics, R.D.Guenther, D.G.Steel, and L.Bayvel, eds. (Oxford, 2004), pp. 189.

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Figures (9)

Fig. 1
Fig. 1

Relation between the reordered, or block tridiagonal, wave matrix A ¯ and the problem grid. Each slice through the grid corresponds to a row in the block matrix wave equation that consists of three square matrices [ a , b , c ] and a column vector f. The real-space grid is visualized as an array of discrete cells distributed throughout a Cartesian grid. The Fourier-space grid is visualized as a set of plane waves at different angles in each slice. Evanescent fields are not shown, but are accounted for in the formulation.

Fig. 2
Fig. 2

Concept of slice absorption. Information from an interior slice is absorbed into the immediately surrounding slices and then eliminated from memory.

Fig. 3
Fig. 3

Geometry of the TF/SF framework. The total-field region contains both the source and fields scattered by the device being modeled. The scattered-field region, usually located at the top of the problem space, does not contain the source.

Fig. 4
Fig. 4

Progression of the SAM through a stack of slices. Algorithm consists of calculating data for a slice, absorbing information from the preceding slice, and clearing preceding slice from memory.

Fig. 5
Fig. 5

Cascading and doubling algorithm exploits longitudinal periodicity. First, a unit cell is reduced to its boundary slices by using the basic SAM algorithm. Second, any number of longitudinal periods can be represented by successively stacking the slices onto themselves and absorbing interior slices.

Fig. 6
Fig. 6

Scanning electron microscope images of a face-centered-tetragonal photonic crystal fabricated in SU-8. Left, perspective view of the photonic crystal. Top right, top view normal to the supporting substrate and normal to the [100] crystal plane. Bottom right, side view perpendicular to a [110] plane that shows spacing of the rods.

Fig. 7
Fig. 7

Illumination of photonic crystal by Cassegrain optics in a Fourier transform infrared spectrometer. Hollow cone of incident light must be accounted for in simulation to obtain results that match experimental measurements.

Fig. 8
Fig. 8

Illustration of how the photonic crystal slab was stored in memory. Stored lattice only consisted of one lattice period in the x and y directions and about two lattice periods in the z direction. The cascading and doubling algorithm was applied to the highlighted portion of the lattice to simulate a total of six lattice periods.

Fig. 9
Fig. 9

Comparison of two simulations and experimentally measured infrared reflectance from a photonic crystal created by multiphoton DLW in SU-8. The first simulation was based on perfect geometry, and the second was based on realistic geometry determined by modeling the fabrication process. Simulation results were obtained using a Fourier-space SAM with M = N = 11 spatial harmonics in the x y plane and 150 nm grid resolution along the z axis.

Equations (99)

Equations on this page are rendered with MathJax. Learn more.

× E = k 0 μ ̿ r H ̃ ,
× H ̃ = k 0 ϵ ̿ r E .
E z y ̂ E y z ̂ = μ x x H ̃ x ,
E x z ̂ E z x ̂ = μ y y H ̃ y ,
E y x ̂ E x y ̂ = μ z z H ̃ z ,
H ̃ z y ̂ H ̃ y z ̂ = ϵ x x E x ,
H ̃ x z ̂ H ̃ z x ̂ = ϵ y y E y ,
H ̃ y x ̂ H ̃ x y ̂ = ϵ z z E z .
( E z , i p , q + 1 E z , i p , q ) Δ y ̂ ( E y , i + 1 p , q E y , i p , q ) Δ z ̂ μ x x , i p , q H ̃ x , i p , q ,
( E x , i + 1 p , q E x , i p , q ) Δ z ̂ ( E z , i p + 1 , q E z , i p , q ) Δ x ̂ μ y y , i p , q H ̃ y , i p , q ,
( E y , i p + 1 , q E y , i p , q ) Δ x ̂ ( E x , i p , q + 1 E x , i p , q ) Δ y ̂ μ z z , i p , q H ̃ z , i p , q ,
( H ̃ z , i p , q H ̃ z , i p , q 1 ) Δ y ̂ ( H ̃ y , i p , q H ̃ y , i 1 p , q ) Δ z ̂ ϵ x x , i p , q E x , i p , q ,
( H ̃ x , i p , q H ̃ x , i 1 p , q ) Δ z ̂ ( H ̃ z , i p , q H ̃ z , i p 1 , q ) Δ x ̂ ϵ y y , i p , q E y , i p , q ,
( H ̃ y , i p , q H ̃ y , i p 1 , q ) Δ x ̂ ( H ̃ x , i p , q H ̃ x , i p , q 1 ) Δ y ̂ ϵ z z , i p , q E z , i p , q .
D y ̂ E E z D z ̂ E E y = μ x x H ̃ x ,
D z ̂ E E x D x ̂ E E z = μ y y H ̃ y ,
D x ̂ E E y D y ̂ E E x = μ z z H ̃ z ,
D y ̂ H H ̃ z D z ̂ H H ̃ y = ϵ x x E x ,
D z ̂ H H ̃ x D x ̂ H H ̃ z = ϵ y y E y ,
D x ̂ H H ̃ y D y ̂ H H ̃ x = ϵ z z E z .
C E E = μ ̿ H ̃ ,
C H H ̃ = ϵ ̿ E ,
E = [ E x E y E z ] T , H ̃ = [ H ̃ x H ̃ y H ̃ z ] T ,
C H = [ 0 D z ̂ H D y ̂ H D z ̂ H 0 D x ̂ H D y ̂ H D x ̂ H 0 ] ,
C E = [ 0 D z ̂ E D y ̂ E D z ̂ E 0 D x ̂ E D y ̂ E D x ̂ E 0 ] ,
μ ̿ = [ μ x x 0 0 0 μ y y 0 0 0 μ z z ] , ϵ ̿ = [ ϵ x x 0 0 0 ϵ y y 0 0 0 ϵ z z ] .
A E E = 0 ,
A E = C H μ ̿ 1 C E ϵ ̿ .
( e j k y , n Δ y S z , i m , n S z , i m , n ) Δ y ̂ ( S y , i + 1 m , n S y , i m , n ) Δ z ̂ ( μ x x , i U ̃ x , i ) m , n ,
( S x , i + 1 m , n S x , i m , n ) Δ z ̂ ( e j k x , m Δ x S z , i m , n S z , i m , n ) Δ x ̂ ( μ y y , i U ̃ y , i ) m , n ,
( e j k x , m Δ x S y , i m , n S y , i m , n ) Δ x ̂ ( e j k y , n Δ y S x , i m , n S x , i m , n ) Δ y ̂ ( μ z z , i U ̃ z , i ) m , n ,
( U ̃ z , i m , n e j k y , n Δ y U ̃ z , i m , n ) Δ y ̂ ( U ̃ y , i m , n U ̃ y , i 1 m , n ) Δ z ̂ ( ϵ x x , i S x , i ) m , n ,
( U ̃ x , i m , n U ̃ x , i 1 m , n ) Δ z ̂ ( U ̃ z , i m , n e j k x , m Δ x U ̃ z , i m , n ) Δ x ̂ ( ϵ y y , i S y , i ) m , n ,
( U ̃ y , i m , n e j k x , m Δ x U ̃ y , i m , n ) Δ x ̂ ( U ̃ x , i m , n e j k y , n Δ y U ̃ x , i m , n ) Δ y ̂ ( ϵ z z , i S z , i ) m , n .
( ϵ S ) m , n = p q w m p , n q S p , q ,
w m , n = 1 Λ x Λ y Λ x Λ y ϵ ( x , y ) e j 2 π [ ( m x Λ x ) + ( n y Λ y ) ] d x d y ,
( μ U ̃ ) m , n = p q v m p , n q U ̃ p , q ,
v m , n = 1 Λ x Λ y Λ x Λ y μ ( x , y ) e j 2 π [ ( m x Λ x ) + ( n y Λ y ) ] d x d y .
k x , m = k x , inc 2 π m Λ x ,
k y , n = k y , inc 2 π n Λ y .
k ̃ y S S z D z ̂ S S y = μ x x U ̃ x ,
D z ̂ S S x k ̃ x S S z = μ y y U ̃ y ,
k ̃ x S S y k ̃ y S S x = μ z z U ̃ z ,
k ̃ y U U ̃ z D z ̂ U U ̃ y = ϵ x x S x ,
D z ̂ U U ̃ x k ̃ x U U ̃ z = ϵ y y S y ,
k ̃ x U U ̃ y k ̃ y U U ̃ x = ϵ z z S z .
k ̃ x , m S = [ exp ( j k x , m Δ x ) 1 ] ( k 0 Δ x ) ,
k ̃ y , n S = [ exp ( j k y , n Δ y ) 1 ] ( k 0 Δ y ) ,
k ̃ x , m U = [ 1 exp ( j k x , m Δ x ) ] ( k 0 Δ x ) ,
k ̃ y , n U = [ 1 exp ( j k y , n Δ y ) ] ( k 0 Δ y ) .
C S S = μ ̿ U ̃ ,
C U U ̃ = ϵ ̿ S ,
S = [ S x S y S z ] T , U ̃ = [ U ̃ x U ̃ y U ̃ z ] T ,
C U = [ 0 D z ̂ U k ̃ y U D z ̂ U 0 k ̃ x U k ̃ y U k ̃ x U 0 ] ,
C S = [ 0 D z ̂ S k ̃ y S D z ̂ S 0 k ̃ x S k ̃ y S k ̃ x S 0 ] ,
μ ̿ = [ μ x x 0 0 0 μ y y 0 0 0 μ z z ] , ϵ ̿ = [ ϵ x x 0 0 0 ϵ y y 0 0 0 ϵ z z ] .
A S S = 0 ,
A S = C U ( μ ̿ ) 1 C S ϵ ̿ .
A ¯ = reorder ( A ) .
A ¯ p ¯ , q ¯ = A p , q ,
p = 1 + G mod 3 ( p ¯ 1 ) + ( p ¯ 1 ) 3 ,
q = 1 + G mod 3 ( q ¯ 1 ) + ( q ¯ 1 ) 3 .
a i 1 E i 2 + b i 1 E i 1 + c i 1 E i = f i 1 ,
a i E i 1 + b i E i + c i E i + 1 = f i ,
a i + 1 E i + b i + 1 E i + 1 + c i + 1 E i + 2 = f i + 1 .
a i 1 E i 2 + b i 1 E i 1 c i 1 E i + 1 = f i 1 ,
a i + 1 E i 1 + b i + 1 E i + 1 c i + 1 E i + 2 = f i + 1 ,
a i 1 = a i 1 , a i + 1 = a i + 1 b i 1 a i ,
b i 1 = b i 1 c i 1 b i 1 a i , b i + 1 = b i + 1 a i + 1 b i 1 c i ,
c i 1 = c i 1 b i 1 c i , c i + 1 = c i + 1 ,
f i 1 = f i 1 c i 1 b i 1 f i , f i + 1 = f i + 1 a i + 1 b i 1 f i ,
a 1 E 0 + b 1 E 1 + c 1 ( E 2 f 2 , src ) = 0 ,
a 2 ( E 1 + f 1 , src ) + b 2 E 2 + c 2 E 3 = 0.
a 1 E 0 + b 1 E 1 + c 1 E 2 = c 1 f 2 , src ,
a 2 E 1 + b 2 E 2 + c 2 E 3 = a 2 f 1 , src .
r eal s pace: f 1 , src = reorder ( [ p x e j ( k x , inc x + k y , inc y ) p y e j ( k x , inc x + k y , inc y ) p z e j ( k x , inc x + k y , inc y ) ] ) ,
Fourier space: f 1 , src = reorder ( [ p x δ 0 , 0 p y δ 0 , 0 p z δ 0 , 0 ] ) .
k z , m , n ref = ( k 0 2 μ ref ε ref k x , m 2 k y , n 2 ) * ,
k z , m , n trn = ( k 0 2 μ trn ε trn k x , m 2 k y , n 2 ) * .
Z ref = [ e j k z , 1 , 1 ref Δ z 0 0 e j k z , M , N ref Δ z ] ,
Z trn = [ e j k z , 1 , 1 trn Δ z 0 0 e j k z , M , N trn Δ z ] .
Z ref = reorder ( [ Z ref 0 0 0 Z ref 0 0 0 Z ref ] ) ,
Z trn = reorder ( [ Z trn 0 0 0 Z trn 0 0 0 Z trn ] ) .
P ref E = T 1 F 1 Z ref F T = ( F T ) 1 Z ref ( F T ) ,
P trn E = T 1 F 1 Z trn F T = ( F T ) 1 Z trn ( F T ) .
T = exp [ j ( k x , inc X + k y , inc Y ) ] .
T = reorder ( [ T 0 0 0 T 0 0 0 T ] ) .
H ( m , n ) = p q h ( p , q ) e j 2 π [ ( m p M ) + ( n q N ) ] .
F = reorder ( [ F 0 0 0 F 0 0 0 F ] ) .
E 0 = P ref E 1 ,
E N + 1 = P trn E N .
b 1 E 1 + c 1 E 2 = f 1 ,
a N E N 1 + b N E N = f N ,
b 1 = b 1 + a 1 P ref ,
b N = b N + c N P trn .
[ b 1 c 1 a N b N ] [ E 1 E N ] = [ f 1 f N ] .
[ E 1 E N ] = [ b 1 c 1 a N b N ] 1 [ f 1 f N ] .
[ E x , i E y , i E z , i ] = reorder 1 ( E i ) .
P m , n = S m , n 2 Re [ k z , m , n k z inc μ r inc μ r ] .

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