Abstract

In this paper, we study tunable holographic lithography using an electrically addressable spatial light modulator as a programmable phase mask. We control the phases of interfering beams diffracted from the phase pattern displayed in the spatial light modulator. We present a calculation method for the assignment of phases in the laser beams and validate the phases of the interfering beams in phase-sensitive, dual-lattice, and two-dimensional patterns formed by a rotationally non-symmetrical configuration. A good agreement has been observed between fabricated holographic structures and simulated interference patterns. The presented method can potentially help design a gradient phase mask for the fabrication of graded photonic crystals or metamaterials.

© 2013 Optical Society of America

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2013 (2)

J. Lutkenhaus, D. George, D. Garrett, K. Ohlinger, H. Zhang, and Y. Lin, “Holographic formation of compound photonic crystal and nano-antenna templates through laser interference,” J. Appl. Phys.113(10), 103103 (2013).
[CrossRef]

M. Kumar and J. Joseph, “Embedding a nondiffracting defect site in helical lattice wave-field by optical phase engineering,” Appl. Opt.52(23), 5653–5658 (2013).
[CrossRef] [PubMed]

2012 (4)

J. Xavier, R. Dasgupta, S. Ahlawat, J. Joseph, and P. K. Gupta, “Three dimensional optical twisters-driven helically stacked multi-layered microrotors,” Appl. Phys. Lett.100(12), 121101 (2012).
[CrossRef]

R. C. Rumpf and J. Pazos, “Synthesis of spatially variant lattices,” Opt. Express20(14), 15263–15274 (2012).
[CrossRef] [PubMed]

M. Boguslawski, A. Kelberer, P. Rose, and C. Denz, “Multiplexing complex two-dimensional photonic superlattices,” Opt. Express20(24), 27331–27343 (2012).
[CrossRef] [PubMed]

B. Arigong, K. Ohlinger, H. S. Kim, Y. Lin, and H. Zhang, “Transformation optics designed general optical luneburg lens with flattened shapes,” Proc. SPIE8376, 83760R–1, 83760R-6 (2012).
[CrossRef]

2011 (5)

2010 (1)

J. Xavier, M. Boguslawski, P. Rose, J. Joseph, and C. Denz, “Reconfigurable optically induced quasicrystallographic three-dimensional complex nonlinear photonic lattice structures,” Adv. Mater.22(3), 356–360 (2010).
[CrossRef] [PubMed]

2009 (1)

D. Xu, K. P. Chen, A. Harb, D. Rodriguez, K. Lozano, and Y. Lin, “Phase tunable holographic fabrication for three-dimensional photonic crystal templates by using a single optical element,” Appl. Phys. Lett.94(23), 231116 (2009).

2008 (1)

2006 (3)

T. Y. M. Chan, O. Toader, and S. John, “Photonic band-gap formation by optical-phase-mask lithography,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.73(4), 046610 (2006).
[CrossRef] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science312(5781), 1780–1782 (2006).
[CrossRef] [PubMed]

U. Leonhardt, “Optical conformal mapping,” Science312(5781), 1777–1780 (2006).
[CrossRef] [PubMed]

2005 (1)

Y. Lin, P. R. Herman, and K. Darmawikarta, “Design and holographic fabrication of tetragonal and cubic photonic crystals with phase mask: toward the mass-production of three-dimensional photonic crystals,” Appl. Phys. Lett.86(7), 071117 (2005).
[CrossRef]

2004 (1)

M. Deubel, G. von Freymann, M. Wegener, S. Pereira, K. Busch, and C. M. Soukoulis, “Direct laser writing of three-dimensional photonic-crystal templates for telecommunications,” Nat. Mater.3(7), 444–447 (2004).
[CrossRef] [PubMed]

2003 (1)

I. Divliansky, T. S. Mayer, K. S. Holliday, and V. H. Crespi, “Fabrication of three-dimensional polymer photonic crystal structures using single diffraction element interference lithography,” Appl. Phys. Lett.82(11), 1667–1669 (2003).
[CrossRef]

2002 (2)

K. P. Chen, P. R. Herman, and R. Tam, “Strong fiber Bragg grating fabrication by hybrid 157- and 248-nm laser exposure,” IEEE Photon. Technol. Lett.14(2), 170–172 (2002).
[CrossRef]

S. Yang, M. Megens, J. Aizenberg, P. Wiltzius, P. M. Chaikin, and W. B. Russel, “Creating periodic three-dimensional structures by multibeam interference of visible laser,” Chem. Mater.14(7), 2831–2833 (2002).
[CrossRef]

2001 (1)

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,” Science293(5532), 1123–1125 (2001).
[CrossRef] [PubMed]

2000 (2)

A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lopez, F. Meseguer, H. Miguez, J. P. Mondia, G. A. Ozin, O. Toader, A. Geoffrey, O. Toader, and H. M. van Driel, “Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres,” Nature405(6785), 437–440 (2000).
[CrossRef] [PubMed]

M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, and A. J. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature404(6773), 53–56 (2000).
[CrossRef] [PubMed]

1994 (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(5), 413–416 (1994).
[CrossRef]

1987 (2)

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

E. Yablonovitch, “Inhibited spontaneous emission in Solid-State Physics and Electronics,” Phys. Rev. Lett.58(20), 2059–2062 (1987).
[CrossRef] [PubMed]

Abbate, G.

Ahlawat, S.

J. Xavier, R. Dasgupta, S. Ahlawat, J. Joseph, and P. K. Gupta, “Three dimensional optical twisters-driven helically stacked multi-layered microrotors,” Appl. Phys. Lett.100(12), 121101 (2012).
[CrossRef]

Aizenberg, J.

S. Yang, M. Megens, J. Aizenberg, P. Wiltzius, P. M. Chaikin, and W. B. Russel, “Creating periodic three-dimensional structures by multibeam interference of visible laser,” Chem. Mater.14(7), 2831–2833 (2002).
[CrossRef]

Arakawa, Y.

A. Tandaechanurat, S. Ishida, D. Guimard, M. Nomura, S. Iwamoto, and Y. Arakawa, “Lasing oscillation in a three-dimensional photonic crystal nanocavity with a complete bandgap,” Nat. Photonics5(2), 91–94 (2011).
[CrossRef]

Arigong, B.

B. Arigong, K. Ohlinger, H. S. Kim, Y. Lin, and H. Zhang, “Transformation optics designed general optical luneburg lens with flattened shapes,” Proc. SPIE8376, 83760R–1, 83760R-6 (2012).
[CrossRef]

Arrizón, V.

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(5), 413–416 (1994).
[CrossRef]

Blanco, A.

A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lopez, F. Meseguer, H. Miguez, J. P. Mondia, G. A. Ozin, O. Toader, A. Geoffrey, O. Toader, and H. M. van Driel, “Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres,” Nature405(6785), 437–440 (2000).
[CrossRef] [PubMed]

Boguslawski, M.

M. Boguslawski, A. Kelberer, P. Rose, and C. Denz, “Multiplexing complex two-dimensional photonic superlattices,” Opt. Express20(24), 27331–27343 (2012).
[CrossRef] [PubMed]

J. Xavier, M. Boguslawski, P. Rose, J. Joseph, and C. Denz, “Reconfigurable optically induced quasicrystallographic three-dimensional complex nonlinear photonic lattice structures,” Adv. Mater.22(3), 356–360 (2010).
[CrossRef] [PubMed]

Busch, K.

M. Deubel, G. von Freymann, M. Wegener, S. Pereira, K. Busch, and C. M. Soukoulis, “Direct laser writing of three-dimensional photonic-crystal templates for telecommunications,” Nat. Mater.3(7), 444–447 (2004).
[CrossRef] [PubMed]

Campbell, M.

M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, and A. J. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature404(6773), 53–56 (2000).
[CrossRef] [PubMed]

Chaikin, P. M.

S. Yang, M. Megens, J. Aizenberg, P. Wiltzius, P. M. Chaikin, and W. B. Russel, “Creating periodic three-dimensional structures by multibeam interference of visible laser,” Chem. Mater.14(7), 2831–2833 (2002).
[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(5), 413–416 (1994).
[CrossRef]

Chan, T. Y. M.

T. Y. M. Chan, O. Toader, and S. John, “Photonic band-gap formation by optical-phase-mask lithography,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.73(4), 046610 (2006).
[CrossRef] [PubMed]

Chen, K. P.

K. Ohlinger, H. Zhang, Y. Lin, D. Xu, and K. P. Chen, “A tunable three layer phase mask for single laser exposure 3D photonic crystal generations: bandgap simulation and holographic fabrication,” Opt. Mater. Express1(5), 1034–1039 (2011).
[CrossRef]

D. Xu, K. P. Chen, A. Harb, D. Rodriguez, K. Lozano, and Y. Lin, “Phase tunable holographic fabrication for three-dimensional photonic crystal templates by using a single optical element,” Appl. Phys. Lett.94(23), 231116 (2009).

K. P. Chen, P. R. Herman, and R. Tam, “Strong fiber Bragg grating fabrication by hybrid 157- and 248-nm laser exposure,” IEEE Photon. Technol. Lett.14(2), 170–172 (2002).
[CrossRef]

Chomski, E.

A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lopez, F. Meseguer, H. Miguez, J. P. Mondia, G. A. Ozin, O. Toader, A. Geoffrey, O. Toader, and H. M. van Driel, “Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres,” Nature405(6785), 437–440 (2000).
[CrossRef] [PubMed]

Chutinan, A.

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,” Science293(5532), 1123–1125 (2001).
[CrossRef] [PubMed]

Crespi, V. H.

I. Divliansky, T. S. Mayer, K. S. Holliday, and V. H. Crespi, “Fabrication of three-dimensional polymer photonic crystal structures using single diffraction element interference lithography,” Appl. Phys. Lett.82(11), 1667–1669 (2003).
[CrossRef]

Darmawikarta, K.

Y. Lin, P. R. Herman, and K. Darmawikarta, “Design and holographic fabrication of tetragonal and cubic photonic crystals with phase mask: toward the mass-production of three-dimensional photonic crystals,” Appl. Phys. Lett.86(7), 071117 (2005).
[CrossRef]

Dasgupta, R.

J. Xavier, R. Dasgupta, S. Ahlawat, J. Joseph, and P. K. Gupta, “Three dimensional optical twisters-driven helically stacked multi-layered microrotors,” Appl. Phys. Lett.100(12), 121101 (2012).
[CrossRef]

de-la-Llave, D. S.

Denning, R. G.

M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, and A. J. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature404(6773), 53–56 (2000).
[CrossRef] [PubMed]

Denz, C.

M. Boguslawski, A. Kelberer, P. Rose, and C. Denz, “Multiplexing complex two-dimensional photonic superlattices,” Opt. Express20(24), 27331–27343 (2012).
[CrossRef] [PubMed]

J. Xavier, M. Boguslawski, P. Rose, J. Joseph, and C. Denz, “Reconfigurable optically induced quasicrystallographic three-dimensional complex nonlinear photonic lattice structures,” Adv. Mater.22(3), 356–360 (2010).
[CrossRef] [PubMed]

Deubel, M.

M. Deubel, G. von Freymann, M. Wegener, S. Pereira, K. Busch, and C. M. Soukoulis, “Direct laser writing of three-dimensional photonic-crystal templates for telecommunications,” Nat. Mater.3(7), 444–447 (2004).
[CrossRef] [PubMed]

Divliansky, I.

I. Divliansky, T. S. Mayer, K. S. Holliday, and V. H. Crespi, “Fabrication of three-dimensional polymer photonic crystal structures using single diffraction element interference lithography,” Appl. Phys. Lett.82(11), 1667–1669 (2003).
[CrossRef]

Garrett, D.

J. Lutkenhaus, D. George, D. Garrett, K. Ohlinger, H. Zhang, and Y. Lin, “Holographic formation of compound photonic crystal and nano-antenna templates through laser interference,” J. Appl. Phys.113(10), 103103 (2013).
[CrossRef]

Geoffrey, A.

A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lopez, F. Meseguer, H. Miguez, J. P. Mondia, G. A. Ozin, O. Toader, A. Geoffrey, O. Toader, and H. M. van Driel, “Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres,” Nature405(6785), 437–440 (2000).
[CrossRef] [PubMed]

George, D.

J. Lutkenhaus, D. George, D. Garrett, K. Ohlinger, H. Zhang, and Y. Lin, “Holographic formation of compound photonic crystal and nano-antenna templates through laser interference,” J. Appl. Phys.113(10), 103103 (2013).
[CrossRef]

Grabtchak, S.

A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lopez, F. Meseguer, H. Miguez, J. P. Mondia, G. A. Ozin, O. Toader, A. Geoffrey, O. Toader, and H. M. van Driel, “Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres,” Nature405(6785), 437–440 (2000).
[CrossRef] [PubMed]

Guimard, D.

A. Tandaechanurat, S. Ishida, D. Guimard, M. Nomura, S. Iwamoto, and Y. Arakawa, “Lasing oscillation in a three-dimensional photonic crystal nanocavity with a complete bandgap,” Nat. Photonics5(2), 91–94 (2011).
[CrossRef]

Gupta, P. K.

J. Xavier, R. Dasgupta, S. Ahlawat, J. Joseph, and P. K. Gupta, “Three dimensional optical twisters-driven helically stacked multi-layered microrotors,” Appl. Phys. Lett.100(12), 121101 (2012).
[CrossRef]

Harb, A.

D. Xu, K. P. Chen, A. Harb, D. Rodriguez, K. Lozano, and Y. Lin, “Phase tunable holographic fabrication for three-dimensional photonic crystal templates by using a single optical element,” Appl. Phys. Lett.94(23), 231116 (2009).

Harrison, M. T.

M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, and A. J. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature404(6773), 53–56 (2000).
[CrossRef] [PubMed]

Herman, P. R.

Y. Lin, P. R. Herman, and K. Darmawikarta, “Design and holographic fabrication of tetragonal and cubic photonic crystals with phase mask: toward the mass-production of three-dimensional photonic crystals,” Appl. Phys. Lett.86(7), 071117 (2005).
[CrossRef]

K. P. Chen, P. R. Herman, and R. Tam, “Strong fiber Bragg grating fabrication by hybrid 157- and 248-nm laser exposure,” IEEE Photon. Technol. Lett.14(2), 170–172 (2002).
[CrossRef]

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(5), 413–416 (1994).
[CrossRef]

Holliday, K. S.

I. Divliansky, T. S. Mayer, K. S. Holliday, and V. H. Crespi, “Fabrication of three-dimensional polymer photonic crystal structures using single diffraction element interference lithography,” Appl. Phys. Lett.82(11), 1667–1669 (2003).
[CrossRef]

Ibisate, M.

A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lopez, F. Meseguer, H. Miguez, J. P. Mondia, G. A. Ozin, O. Toader, A. Geoffrey, O. Toader, and H. M. van Driel, “Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres,” Nature405(6785), 437–440 (2000).
[CrossRef] [PubMed]

Imada, 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,” Science293(5532), 1123–1125 (2001).
[CrossRef] [PubMed]

Ishida, S.

A. Tandaechanurat, S. Ishida, D. Guimard, M. Nomura, S. Iwamoto, and Y. Arakawa, “Lasing oscillation in a three-dimensional photonic crystal nanocavity with a complete bandgap,” Nat. Photonics5(2), 91–94 (2011).
[CrossRef]

Iwamoto, S.

A. Tandaechanurat, S. Ishida, D. Guimard, M. Nomura, S. Iwamoto, and Y. Arakawa, “Lasing oscillation in a three-dimensional photonic crystal nanocavity with a complete bandgap,” Nat. Photonics5(2), 91–94 (2011).
[CrossRef]

John, S.

T. Y. M. Chan, O. Toader, and S. John, “Photonic band-gap formation by optical-phase-mask lithography,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.73(4), 046610 (2006).
[CrossRef] [PubMed]

A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lopez, F. Meseguer, H. Miguez, J. P. Mondia, G. A. Ozin, O. Toader, A. Geoffrey, O. Toader, and H. M. van Driel, “Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres,” Nature405(6785), 437–440 (2000).
[CrossRef] [PubMed]

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

Joseph, J.

M. Kumar and J. Joseph, “Embedding a nondiffracting defect site in helical lattice wave-field by optical phase engineering,” Appl. Opt.52(23), 5653–5658 (2013).
[CrossRef] [PubMed]

J. Xavier, R. Dasgupta, S. Ahlawat, J. Joseph, and P. K. Gupta, “Three dimensional optical twisters-driven helically stacked multi-layered microrotors,” Appl. Phys. Lett.100(12), 121101 (2012).
[CrossRef]

J. Xavier and J. Joseph, “Tunable complex photonic chiral lattices by reconfigurable optical phase engineering,” Opt. Lett.36(3), 403–405 (2011).
[CrossRef] [PubMed]

J. Xavier, M. Boguslawski, P. Rose, J. Joseph, and C. Denz, “Reconfigurable optically induced quasicrystallographic three-dimensional complex nonlinear photonic lattice structures,” Adv. Mater.22(3), 356–360 (2010).
[CrossRef] [PubMed]

Kelberer, A.

Kim, H. S.

B. Arigong, K. Ohlinger, H. S. Kim, Y. Lin, and H. Zhang, “Transformation optics designed general optical luneburg lens with flattened shapes,” Proc. SPIE8376, 83760R–1, 83760R-6 (2012).
[CrossRef]

Kumar, M.

Leonard, S. W.

A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lopez, F. Meseguer, H. Miguez, J. P. Mondia, G. A. Ozin, O. Toader, A. Geoffrey, O. Toader, and H. M. van Driel, “Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres,” Nature405(6785), 437–440 (2000).
[CrossRef] [PubMed]

Leonhardt, U.

U. Leonhardt, “Optical conformal mapping,” Science312(5781), 1777–1780 (2006).
[CrossRef] [PubMed]

Li, Z.-Y.

Lin, Y.

J. Lutkenhaus, D. George, D. Garrett, K. Ohlinger, H. Zhang, and Y. Lin, “Holographic formation of compound photonic crystal and nano-antenna templates through laser interference,” J. Appl. Phys.113(10), 103103 (2013).
[CrossRef]

B. Arigong, K. Ohlinger, H. S. Kim, Y. Lin, and H. Zhang, “Transformation optics designed general optical luneburg lens with flattened shapes,” Proc. SPIE8376, 83760R–1, 83760R-6 (2012).
[CrossRef]

K. Ohlinger, H. Zhang, Y. Lin, D. Xu, and K. P. Chen, “A tunable three layer phase mask for single laser exposure 3D photonic crystal generations: bandgap simulation and holographic fabrication,” Opt. Mater. Express1(5), 1034–1039 (2011).
[CrossRef]

D. Xu, K. P. Chen, A. Harb, D. Rodriguez, K. Lozano, and Y. Lin, “Phase tunable holographic fabrication for three-dimensional photonic crystal templates by using a single optical element,” Appl. Phys. Lett.94(23), 231116 (2009).

Y. Lin, P. R. Herman, and K. Darmawikarta, “Design and holographic fabrication of tetragonal and cubic photonic crystals with phase mask: toward the mass-production of three-dimensional photonic crystals,” Appl. Phys. Lett.86(7), 071117 (2005).
[CrossRef]

K. Ohlinger, J. Lutkenhaus, H. Zhang, and Y. Lin, “Spatially addressable design of Luneburg lens through spatial light modulator based holographic lithography,” J. Appl. Phys.submitted.

Liu, Y.

Lopez, C.

A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lopez, F. Meseguer, H. Miguez, J. P. Mondia, G. A. Ozin, O. Toader, A. Geoffrey, O. Toader, and H. M. van Driel, “Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres,” Nature405(6785), 437–440 (2000).
[CrossRef] [PubMed]

Lozano, K.

D. Xu, K. P. Chen, A. Harb, D. Rodriguez, K. Lozano, and Y. Lin, “Phase tunable holographic fabrication for three-dimensional photonic crystal templates by using a single optical element,” Appl. Phys. Lett.94(23), 231116 (2009).

Lutkenhaus, J.

J. Lutkenhaus, D. George, D. Garrett, K. Ohlinger, H. Zhang, and Y. Lin, “Holographic formation of compound photonic crystal and nano-antenna templates through laser interference,” J. Appl. Phys.113(10), 103103 (2013).
[CrossRef]

K. Ohlinger, J. Lutkenhaus, H. Zhang, and Y. Lin, “Spatially addressable design of Luneburg lens through spatial light modulator based holographic lithography,” J. Appl. Phys.submitted.

Mao, Q.-H.

Marino, A.

Mayer, T. S.

I. Divliansky, T. S. Mayer, K. S. Holliday, and V. H. Crespi, “Fabrication of three-dimensional polymer photonic crystal structures using single diffraction element interference lithography,” Appl. Phys. Lett.82(11), 1667–1669 (2003).
[CrossRef]

Megens, M.

S. Yang, M. Megens, J. Aizenberg, P. Wiltzius, P. M. Chaikin, and W. B. Russel, “Creating periodic three-dimensional structures by multibeam interference of visible laser,” Chem. Mater.14(7), 2831–2833 (2002).
[CrossRef]

Méndez, G.

Meng, Z.-M.

Meseguer, F.

A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lopez, F. Meseguer, H. Miguez, J. P. Mondia, G. A. Ozin, O. Toader, A. Geoffrey, O. Toader, and H. M. van Driel, “Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres,” Nature405(6785), 437–440 (2000).
[CrossRef] [PubMed]

Miguez, H.

A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lopez, F. Meseguer, H. Miguez, J. P. Mondia, G. A. Ozin, O. Toader, A. Geoffrey, O. Toader, and H. M. van Driel, “Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres,” Nature405(6785), 437–440 (2000).
[CrossRef] [PubMed]

Mochizuki, 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,” Science293(5532), 1123–1125 (2001).
[CrossRef] [PubMed]

Mondia, J. P.

A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lopez, F. Meseguer, H. Miguez, J. P. Mondia, G. A. Ozin, O. Toader, A. Geoffrey, O. Toader, and H. M. van Driel, “Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres,” Nature405(6785), 437–440 (2000).
[CrossRef] [PubMed]

Noda, S.

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,” Science293(5532), 1123–1125 (2001).
[CrossRef] [PubMed]

Nomura, M.

A. Tandaechanurat, S. Ishida, D. Guimard, M. Nomura, S. Iwamoto, and Y. Arakawa, “Lasing oscillation in a three-dimensional photonic crystal nanocavity with a complete bandgap,” Nat. Photonics5(2), 91–94 (2011).
[CrossRef]

Ohlinger, K.

J. Lutkenhaus, D. George, D. Garrett, K. Ohlinger, H. Zhang, and Y. Lin, “Holographic formation of compound photonic crystal and nano-antenna templates through laser interference,” J. Appl. Phys.113(10), 103103 (2013).
[CrossRef]

B. Arigong, K. Ohlinger, H. S. Kim, Y. Lin, and H. Zhang, “Transformation optics designed general optical luneburg lens with flattened shapes,” Proc. SPIE8376, 83760R–1, 83760R-6 (2012).
[CrossRef]

K. Ohlinger, H. Zhang, Y. Lin, D. Xu, and K. P. Chen, “A tunable three layer phase mask for single laser exposure 3D photonic crystal generations: bandgap simulation and holographic fabrication,” Opt. Mater. Express1(5), 1034–1039 (2011).
[CrossRef]

K. Ohlinger, J. Lutkenhaus, H. Zhang, and Y. Lin, “Spatially addressable design of Luneburg lens through spatial light modulator based holographic lithography,” J. Appl. Phys.submitted.

Ozin, G. A.

A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lopez, F. Meseguer, H. Miguez, J. P. Mondia, G. A. Ozin, O. Toader, A. Geoffrey, O. Toader, and H. M. van Driel, “Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres,” Nature405(6785), 437–440 (2000).
[CrossRef] [PubMed]

Pazos, J.

Pendry, J. B.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science312(5781), 1780–1782 (2006).
[CrossRef] [PubMed]

Pereira, S.

M. Deubel, G. von Freymann, M. Wegener, S. Pereira, K. Busch, and C. M. Soukoulis, “Direct laser writing of three-dimensional photonic-crystal templates for telecommunications,” Nat. Mater.3(7), 444–447 (2004).
[CrossRef] [PubMed]

Piccirillo, B.

Qin, F.

Rodriguez, D.

D. Xu, K. P. Chen, A. Harb, D. Rodriguez, K. Lozano, and Y. Lin, “Phase tunable holographic fabrication for three-dimensional photonic crystal templates by using a single optical element,” Appl. Phys. Lett.94(23), 231116 (2009).

Rose, P.

M. Boguslawski, A. Kelberer, P. Rose, and C. Denz, “Multiplexing complex two-dimensional photonic superlattices,” Opt. Express20(24), 27331–27343 (2012).
[CrossRef] [PubMed]

J. Xavier, M. Boguslawski, P. Rose, J. Joseph, and C. Denz, “Reconfigurable optically induced quasicrystallographic three-dimensional complex nonlinear photonic lattice structures,” Adv. Mater.22(3), 356–360 (2010).
[CrossRef] [PubMed]

Ruiz, U.

Rumpf, R. C.

Russel, W. B.

S. Yang, M. Megens, J. Aizenberg, P. Wiltzius, P. M. Chaikin, and W. B. Russel, “Creating periodic three-dimensional structures by multibeam interference of visible laser,” Chem. Mater.14(7), 2831–2833 (2002).
[CrossRef]

Santamato, E.

Schurig, D.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science312(5781), 1780–1782 (2006).
[CrossRef] [PubMed]

Sharp, D. N.

M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, and A. J. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature404(6773), 53–56 (2000).
[CrossRef] [PubMed]

Sigalas, 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(5), 413–416 (1994).
[CrossRef]

Smith, D. R.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science312(5781), 1780–1782 (2006).
[CrossRef] [PubMed]

Soukoulis, C. M.

M. Deubel, G. von Freymann, M. Wegener, S. Pereira, K. Busch, and C. M. Soukoulis, “Direct laser writing of three-dimensional photonic-crystal templates for telecommunications,” Nat. Mater.3(7), 444–447 (2004).
[CrossRef] [PubMed]

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(5), 413–416 (1994).
[CrossRef]

Tam, R.

K. P. Chen, P. R. Herman, and R. Tam, “Strong fiber Bragg grating fabrication by hybrid 157- and 248-nm laser exposure,” IEEE Photon. Technol. Lett.14(2), 170–172 (2002).
[CrossRef]

Tandaechanurat, A.

A. Tandaechanurat, S. Ishida, D. Guimard, M. Nomura, S. Iwamoto, and Y. Arakawa, “Lasing oscillation in a three-dimensional photonic crystal nanocavity with a complete bandgap,” Nat. Photonics5(2), 91–94 (2011).
[CrossRef]

Tkachenko, V.

Toader, O.

T. Y. M. Chan, O. Toader, and S. John, “Photonic band-gap formation by optical-phase-mask lithography,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.73(4), 046610 (2006).
[CrossRef] [PubMed]

A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lopez, F. Meseguer, H. Miguez, J. P. Mondia, G. A. Ozin, O. Toader, A. Geoffrey, O. Toader, and H. M. van Driel, “Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres,” Nature405(6785), 437–440 (2000).
[CrossRef] [PubMed]

A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lopez, F. Meseguer, H. Miguez, J. P. Mondia, G. A. Ozin, O. Toader, A. Geoffrey, O. Toader, and H. M. van Driel, “Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres,” Nature405(6785), 437–440 (2000).
[CrossRef] [PubMed]

Turberfield, A. J.

M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, and A. J. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature404(6773), 53–56 (2000).
[CrossRef] [PubMed]

van Driel, H. M.

A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lopez, F. Meseguer, H. Miguez, J. P. Mondia, G. A. Ozin, O. Toader, A. Geoffrey, O. Toader, and H. M. van Driel, “Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres,” Nature405(6785), 437–440 (2000).
[CrossRef] [PubMed]

von Freymann, G.

M. Deubel, G. von Freymann, M. Wegener, S. Pereira, K. Busch, and C. M. Soukoulis, “Direct laser writing of three-dimensional photonic-crystal templates for telecommunications,” Nat. Mater.3(7), 444–447 (2004).
[CrossRef] [PubMed]

Wegener, M.

M. Deubel, G. von Freymann, M. Wegener, S. Pereira, K. Busch, and C. M. Soukoulis, “Direct laser writing of three-dimensional photonic-crystal templates for telecommunications,” Nat. Mater.3(7), 444–447 (2004).
[CrossRef] [PubMed]

Wiltzius, P.

S. Yang, M. Megens, J. Aizenberg, P. Wiltzius, P. M. Chaikin, and W. B. Russel, “Creating periodic three-dimensional structures by multibeam interference of visible laser,” Chem. Mater.14(7), 2831–2833 (2002).
[CrossRef]

Xavier, J.

J. Xavier, R. Dasgupta, S. Ahlawat, J. Joseph, and P. K. Gupta, “Three dimensional optical twisters-driven helically stacked multi-layered microrotors,” Appl. Phys. Lett.100(12), 121101 (2012).
[CrossRef]

J. Xavier and J. Joseph, “Tunable complex photonic chiral lattices by reconfigurable optical phase engineering,” Opt. Lett.36(3), 403–405 (2011).
[CrossRef] [PubMed]

J. Xavier, M. Boguslawski, P. Rose, J. Joseph, and C. Denz, “Reconfigurable optically induced quasicrystallographic three-dimensional complex nonlinear photonic lattice structures,” Adv. Mater.22(3), 356–360 (2010).
[CrossRef] [PubMed]

Xu, D.

K. Ohlinger, H. Zhang, Y. Lin, D. Xu, and K. P. Chen, “A tunable three layer phase mask for single laser exposure 3D photonic crystal generations: bandgap simulation and holographic fabrication,” Opt. Mater. Express1(5), 1034–1039 (2011).
[CrossRef]

D. Xu, K. P. Chen, A. Harb, D. Rodriguez, K. Lozano, and Y. Lin, “Phase tunable holographic fabrication for three-dimensional photonic crystal templates by using a single optical element,” Appl. Phys. Lett.94(23), 231116 (2009).

Yablonovitch, E.

E. Yablonovitch, “Inhibited spontaneous emission in Solid-State Physics and Electronics,” Phys. Rev. Lett.58(20), 2059–2062 (1987).
[CrossRef] [PubMed]

Yang, S.

S. Yang, M. Megens, J. Aizenberg, P. Wiltzius, P. M. Chaikin, and W. B. Russel, “Creating periodic three-dimensional structures by multibeam interference of visible laser,” Chem. Mater.14(7), 2831–2833 (2002).
[CrossRef]

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,” Science293(5532), 1123–1125 (2001).
[CrossRef] [PubMed]

Zhang, H.

J. Lutkenhaus, D. George, D. Garrett, K. Ohlinger, H. Zhang, and Y. Lin, “Holographic formation of compound photonic crystal and nano-antenna templates through laser interference,” J. Appl. Phys.113(10), 103103 (2013).
[CrossRef]

B. Arigong, K. Ohlinger, H. S. Kim, Y. Lin, and H. Zhang, “Transformation optics designed general optical luneburg lens with flattened shapes,” Proc. SPIE8376, 83760R–1, 83760R-6 (2012).
[CrossRef]

K. Ohlinger, H. Zhang, Y. Lin, D. Xu, and K. P. Chen, “A tunable three layer phase mask for single laser exposure 3D photonic crystal generations: bandgap simulation and holographic fabrication,” Opt. Mater. Express1(5), 1034–1039 (2011).
[CrossRef]

K. Ohlinger, J. Lutkenhaus, H. Zhang, and Y. Lin, “Spatially addressable design of Luneburg lens through spatial light modulator based holographic lithography,” J. Appl. Phys.submitted.

Zhou, F.

Zito, G.

Adv. Mater. (1)

J. Xavier, M. Boguslawski, P. Rose, J. Joseph, and C. Denz, “Reconfigurable optically induced quasicrystallographic three-dimensional complex nonlinear photonic lattice structures,” Adv. Mater.22(3), 356–360 (2010).
[CrossRef] [PubMed]

Appl. Opt. (1)

Appl. Phys. Lett. (4)

J. Xavier, R. Dasgupta, S. Ahlawat, J. Joseph, and P. K. Gupta, “Three dimensional optical twisters-driven helically stacked multi-layered microrotors,” Appl. Phys. Lett.100(12), 121101 (2012).
[CrossRef]

D. Xu, K. P. Chen, A. Harb, D. Rodriguez, K. Lozano, and Y. Lin, “Phase tunable holographic fabrication for three-dimensional photonic crystal templates by using a single optical element,” Appl. Phys. Lett.94(23), 231116 (2009).

I. Divliansky, T. S. Mayer, K. S. Holliday, and V. H. Crespi, “Fabrication of three-dimensional polymer photonic crystal structures using single diffraction element interference lithography,” Appl. Phys. Lett.82(11), 1667–1669 (2003).
[CrossRef]

Y. Lin, P. R. Herman, and K. Darmawikarta, “Design and holographic fabrication of tetragonal and cubic photonic crystals with phase mask: toward the mass-production of three-dimensional photonic crystals,” Appl. Phys. Lett.86(7), 071117 (2005).
[CrossRef]

Chem. Mater. (1)

S. Yang, M. Megens, J. Aizenberg, P. Wiltzius, P. M. Chaikin, and W. B. Russel, “Creating periodic three-dimensional structures by multibeam interference of visible laser,” Chem. Mater.14(7), 2831–2833 (2002).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

K. P. Chen, P. R. Herman, and R. Tam, “Strong fiber Bragg grating fabrication by hybrid 157- and 248-nm laser exposure,” IEEE Photon. Technol. Lett.14(2), 170–172 (2002).
[CrossRef]

J. Appl. Phys. (2)

K. Ohlinger, J. Lutkenhaus, H. Zhang, and Y. Lin, “Spatially addressable design of Luneburg lens through spatial light modulator based holographic lithography,” J. Appl. Phys.submitted.

J. Lutkenhaus, D. George, D. Garrett, K. Ohlinger, H. Zhang, and Y. Lin, “Holographic formation of compound photonic crystal and nano-antenna templates through laser interference,” J. Appl. Phys.113(10), 103103 (2013).
[CrossRef]

Nat. Mater. (1)

M. Deubel, G. von Freymann, M. Wegener, S. Pereira, K. Busch, and C. M. Soukoulis, “Direct laser writing of three-dimensional photonic-crystal templates for telecommunications,” Nat. Mater.3(7), 444–447 (2004).
[CrossRef] [PubMed]

Nat. Photonics (1)

A. Tandaechanurat, S. Ishida, D. Guimard, M. Nomura, S. Iwamoto, and Y. Arakawa, “Lasing oscillation in a three-dimensional photonic crystal nanocavity with a complete bandgap,” Nat. Photonics5(2), 91–94 (2011).
[CrossRef]

Nature (2)

M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, and A. J. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature404(6773), 53–56 (2000).
[CrossRef] [PubMed]

A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lopez, F. Meseguer, H. Miguez, J. P. Mondia, G. A. Ozin, O. Toader, A. Geoffrey, O. Toader, and H. M. van Driel, “Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres,” Nature405(6785), 437–440 (2000).
[CrossRef] [PubMed]

Opt. Express (5)

Opt. Lett. (1)

Opt. Mater. Express (1)

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (1)

T. Y. M. Chan, O. Toader, and S. John, “Photonic band-gap formation by optical-phase-mask lithography,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.73(4), 046610 (2006).
[CrossRef] [PubMed]

Phys. Rev. Lett. (2)

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

E. Yablonovitch, “Inhibited spontaneous emission in Solid-State Physics and Electronics,” Phys. Rev. Lett.58(20), 2059–2062 (1987).
[CrossRef] [PubMed]

Proc. SPIE (1)

B. Arigong, K. Ohlinger, H. S. Kim, Y. Lin, and H. Zhang, “Transformation optics designed general optical luneburg lens with flattened shapes,” Proc. SPIE8376, 83760R–1, 83760R-6 (2012).
[CrossRef]

Science (3)

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science312(5781), 1780–1782 (2006).
[CrossRef] [PubMed]

U. Leonhardt, “Optical conformal mapping,” Science312(5781), 1777–1780 (2006).
[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,” Science293(5532), 1123–1125 (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(5), 413–416 (1994).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Scheme of experimental optics setup using the SLM for multi-beam interference lithography. Phase pattern was displayed on the SLM. Imaging lenses f1 and f2 were arranged in a 4f configuration.

Fig. 2
Fig. 2

(a) An enlarged view of designed pattern. The gray levels in six equilateral triangles inside a hexagon are assigned digitally. The diffractions are formed in directions having periodic patterns. The diffraction pattern is predicted following the diffraction direction. (b) Experimentally recorded diffraction pattern. (c) A scheme of an enlarged view of the pattern. The red hexagon indicates a unit cell of the pattern. The phases of the diffracted beams (1-6) are determined by the averaged gray levels of kite-type four-side polygons inside the red hexagon.

Fig. 3
Fig. 3

(a1, b1, c1) A scheme of an enlarged view of designed patterns with a gray level in a set of triangles changed from 153 to 102 and to 51. The hexagon indicates the unit cell of the designed pattern. (a2, b2, c2) CCD recorded images for four-beam interference pattern from beams 1, 3, 4 and 5 diffracted from phase patterns in (a1, b1, c1), respectively. (a3, b3, c3) Simulated four-beam interference pattern formed by beams 1, 3, 4 and 5 with phases assignments from the gray levels in (a1, b1, c1), respectively. Labels a and b in (a3) indicates the lattice constants and red arrows in (b3) indicate the pattern shifting directions. (a4, b4, c4) 3D view of simulated four-beam interference patterns given in (a3, b3, c3).

Fig. 4
Fig. 4

(a1,a2,a3) CCD images of interference patterns recorded at different locations longitudinally translated along z-direction by 0, 127, and 254 microns. (b) SEM image of fabricated holographic structures in DPHPA. (c) An enlarged view of the SEM in (b). (d) Atomic force microscope image of the fabricated holographic structures in DPHPA. (e) A surface profile measured along the line in the AFM image in (d).

Fig. 5
Fig. 5

(a) A scheme of an enlarged view of a designed pattern with different gray levels in triangles from a to f. The gray levels are labeled in the right side of the pattern. The hexagon indicates the unit cell of the designed pattern. (b) Simulated four-beam interference pattern using the phase information in (a). Inset is a 3D view. (c) CCD recorded images for four-beam interference pattern from beams diffracted from phase patterns in (a). (d) SEM image of fabricated holographic structures in DPHPA. (e, f, g) Simulated four-beam interference patterns formed by beams 1, 3, 4 and 5 with phases assignments from the gray levels shown in Fig. 3(a1, b1, c1), respectively. The patterns were obtained by setting the iso-intensity surface to be a half of maximum intensity.

Equations (5)

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

1 3  light gray+ 8 18  black+ 1 18  black+  1 6  black,  
1 3  dark grey+ 8 18  black+ 1 18  black+  1 6  black,  
I(r)= I 0 +ΔI(r)=< i=1 4 E i 2 (r,t)>+ i<j 4 E i E j e i e j cos[( k j k i )r+( δ j δ i )] ,
ΔI(r)= E 1 E 3 e 1 e 3 cos[2π((3/a)i(1/b)j)(x+y)+( δ 3 δ 1 )] + E 1 E 5 e 1 e 5 cos[2π((3/a)i+(1/b)j)(x+y)+( δ 5 δ 1 )] + E 1 E 4 e 1 e 4 cos[2π((4/a)i+0j)(x+y)+( δ 4 δ 1 )] + E 3 E 5 e 3 e 5 cos[2π((0i+(2/b)j)(x+y)+( δ 5 δ 3 )] + E 3 E 4 e 3 e 4 cos[2π((1/a)i+(1/b)j)(x+y)+( δ 4 δ 3 )] + E 5 E 4 e 5 e 4 cos[2π((1/a)i(1/b)j)(x+y)+( δ 4 δ 5 )],
ΔI(relatedtobeam4)= E 1 E 4 e 1 e 4 cos[2π((4/a)i+0j)(x+y)mπ] + E 3 E 4 e 3 e 4 cos[2π((1/a)i+(1/b)j)(x+y)mπ] + E 5 E 4 e 5 e 4 cos[2π((1/a)i(1/b)j)(x+y)mπ] = E 1 E 4 e 1 e 4 cos[2π((4/a)i+0j)(x+y+m( 1 8 a± 3 8 b))] + E 3 E 4 e 3 e 4 cos[2π((1/a)i+(1/b)j)(x+y+m( 1 8 a 3 8 b))] + E 5 E 4 e 5 e 4 cos[2π((1/a)i(1/b)j)(x+y+m( 1 8 a+ 3 8 b))],

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