Abstract

We demonstrate that an anisotropic photonic crystal can modify the shape of a highly convergent incident optical beam. The beam shape engineering is relatively easy, and the photonic crystal is less alignment demanding than beam shapers that incorporate several optical systems. The shape of the output beam can be controlled by an appropriate choice of the angular divergence of the beam, the number of periods and the birefringence values and layer widths of the constituent materials.

© 2010 Optical Society of America

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References

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2009 (3)

M. Ojima, N. Numata, Y. Ogawa, K. Murata, H. Kubo, A. Fujii, and M. Ozaki, “Electric field tuning of plasmonic absorption of metallic grating with twisted nematic liquid crystal,” Appl. Phys. Express 2086001 (2009).

Z. Tian, M. Nix, and S. S.-H. Yam, “Laser beam shaping using a single-mode fiber abrupt taper,” Opt. Lett. 34, 229-231(2009).
[CrossRef]

M. Fratz, S. Sinzinger, and D. Giel, “Design and fabrication of polarization-holographic elements for laser beam shaping,” Appl. Opt. 48, 2669-2677 (2009).
[CrossRef]

2008 (4)

2007 (3)

T. R. Wolinski, A. Czapla, S. Ertman, M. Tefelska, A. W. Domanski, E. Nowinowski-Kruszelnicki, and R. Dabrowski, “Tunable highly birefringent solid-core photonic liquid crystal fibers,” Opt. Quantum Electron. 39, 1021-1032 (2007).
[CrossRef]

T. R. Wolinski, S. Ertman, A. Czapla, P. Lesiak, K. Nowecka, A. W. Domanski, E. Nowinowski-Kruszelnicki, R. Dabrowski, and J. Wojcik, “Polarization effects in photonic liquid crystal fibers,” Meas. Sci. Technol. 183061-3069 (2007).
[CrossRef]

R. Hamdi, B.-E. Benkelfat, Q. Zou, and Y. Gottesman, “Bandwidth tuning of hybrid liquid-crystal Šolc filters based on an optical cancelling technique,” Opt. Commun. 269, 64-68(2007).
[CrossRef]

2006 (1)

2005 (3)

2002 (2)

F. J. García-Vidal and L. Martín-Moreno, “Transmission and focusing of light in one-dimensional periodically nanostructured metals,” Phys. Rev. B 66, 155412 (2002).

H. F. Gleeson, A. J. Murray, E. Fraser, and A. Zoro, “An electrically addressed liquid crystal filter for tunable lasers,” Opt. Commun. 212, 165-168 (2002).
[CrossRef]

2000 (2)

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1-7 (2000).
[CrossRef]

A. M. Weiner, “Femtosecond pulse shaping using spatial light modulators,” Rev. Sci. Instrum. 71, 1929-1960 (2000).
[CrossRef]

1998 (1)

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, R10096-R10099 (1998).

1995 (1)

R. Martínez-Herrero, G. Piquero, and P. M. Mejías, “On the propagation of the kurtosis of general beams,” Opt. Commun. 115, 225-232 (1995).
[CrossRef]

Audouard, E.

Benkelfat, B.-E.

R. Hamdi, B.-E. Benkelfat, Q. Zou, and Y. Gottesman, “Bandwidth tuning of hybrid liquid-crystal Šolc filters based on an optical cancelling technique,” Opt. Commun. 269, 64-68(2007).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, 1970).

Choi, Y.

Y.-M. Lee, K.-H. Lee, Y. Choi, and J.-H. Kim, “Fast bistable microlens arrays based on a birefringent layer and ferroelectric liquid crystals,” Jpn. J. Appl. Phys. 47, 6343-6346 (2008).

Czapla, A.

T. R. Wolinski, S. Ertman, A. Czapla, P. Lesiak, K. Nowecka, A. W. Domanski, E. Nowinowski-Kruszelnicki, R. Dabrowski, and J. Wojcik, “Polarization effects in photonic liquid crystal fibers,” Meas. Sci. Technol. 183061-3069 (2007).
[CrossRef]

T. R. Wolinski, A. Czapla, S. Ertman, M. Tefelska, A. W. Domanski, E. Nowinowski-Kruszelnicki, and R. Dabrowski, “Tunable highly birefringent solid-core photonic liquid crystal fibers,” Opt. Quantum Electron. 39, 1021-1032 (2007).
[CrossRef]

Dabrowski, R.

T. R. Wolinski, A. Czapla, S. Ertman, M. Tefelska, A. W. Domanski, E. Nowinowski-Kruszelnicki, and R. Dabrowski, “Tunable highly birefringent solid-core photonic liquid crystal fibers,” Opt. Quantum Electron. 39, 1021-1032 (2007).
[CrossRef]

T. R. Wolinski, S. Ertman, A. Czapla, P. Lesiak, K. Nowecka, A. W. Domanski, E. Nowinowski-Kruszelnicki, R. Dabrowski, and J. Wojcik, “Polarization effects in photonic liquid crystal fibers,” Meas. Sci. Technol. 183061-3069 (2007).
[CrossRef]

Domanski, A. W.

T. R. Wolinski, S. Ertman, A. Czapla, P. Lesiak, K. Nowecka, A. W. Domanski, E. Nowinowski-Kruszelnicki, R. Dabrowski, and J. Wojcik, “Polarization effects in photonic liquid crystal fibers,” Meas. Sci. Technol. 183061-3069 (2007).
[CrossRef]

T. R. Wolinski, A. Czapla, S. Ertman, M. Tefelska, A. W. Domanski, E. Nowinowski-Kruszelnicki, and R. Dabrowski, “Tunable highly birefringent solid-core photonic liquid crystal fibers,” Opt. Quantum Electron. 39, 1021-1032 (2007).
[CrossRef]

Dorn, R.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1-7 (2000).
[CrossRef]

Doyle, C.

Eberler, M.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1-7 (2000).
[CrossRef]

Ertman, S.

T. R. Wolinski, S. Ertman, A. Czapla, P. Lesiak, K. Nowecka, A. W. Domanski, E. Nowinowski-Kruszelnicki, R. Dabrowski, and J. Wojcik, “Polarization effects in photonic liquid crystal fibers,” Meas. Sci. Technol. 183061-3069 (2007).
[CrossRef]

T. R. Wolinski, A. Czapla, S. Ertman, M. Tefelska, A. W. Domanski, E. Nowinowski-Kruszelnicki, and R. Dabrowski, “Tunable highly birefringent solid-core photonic liquid crystal fibers,” Opt. Quantum Electron. 39, 1021-1032 (2007).
[CrossRef]

Fauchet, P. M.

Fraser, E.

H. F. Gleeson, A. J. Murray, E. Fraser, and A. Zoro, “An electrically addressed liquid crystal filter for tunable lasers,” Opt. Commun. 212, 165-168 (2002).
[CrossRef]

Fratz, M.

Fujii, A.

M. Ojima, N. Numata, Y. Ogawa, K. Murata, H. Kubo, A. Fujii, and M. Ozaki, “Electric field tuning of plasmonic absorption of metallic grating with twisted nematic liquid crystal,” Appl. Phys. Express 2086001 (2009).

García-Vidal, F. J.

F. J. García-Vidal and L. Martín-Moreno, “Transmission and focusing of light in one-dimensional periodically nanostructured metals,” Phys. Rev. B 66, 155412 (2002).

Giel, D.

Gleeson, H. F.

H. F. Gleeson, A. J. Murray, E. Fraser, and A. Zoro, “An electrically addressed liquid crystal filter for tunable lasers,” Opt. Commun. 212, 165-168 (2002).
[CrossRef]

Glöckl, O.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1-7 (2000).
[CrossRef]

Glückstad, J.

Gottesman, Y.

R. Hamdi, B.-E. Benkelfat, Q. Zou, and Y. Gottesman, “Bandwidth tuning of hybrid liquid-crystal Šolc filters based on an optical cancelling technique,” Opt. Commun. 269, 64-68(2007).
[CrossRef]

Graugnard, E.

E. Graugnard, J. S. King, S. Jain, C. J. Summers, Y. Zhang-Williams, and I. C. Khoo, “Electric-field tuning of the Bragg peak in large-pore TiO2 inverse shell opals,” Phys. Rev. B 72, 233105 (2005).

Hamdi, R.

R. Hamdi, B.-E. Benkelfat, Q. Zou, and Y. Gottesman, “Bandwidth tuning of hybrid liquid-crystal Šolc filters based on an optical cancelling technique,” Opt. Commun. 269, 64-68(2007).
[CrossRef]

Huang, Y.

Huignard, J.-P.

Huot, N.

Jabbour, T. G.

Jain, S.

E. Graugnard, J. S. King, S. Jain, C. J. Summers, Y. Zhang-Williams, and I. C. Khoo, “Electric-field tuning of the Bragg peak in large-pore TiO2 inverse shell opals,” Phys. Rev. B 72, 233105 (2005).

Kawakami, S.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, R10096-R10099 (1998).

Kawashima, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, R10096-R10099 (1998).

Khoo, I. C.

E. Graugnard, J. S. King, S. Jain, C. J. Summers, Y. Zhang-Williams, and I. C. Khoo, “Electric-field tuning of the Bragg peak in large-pore TiO2 inverse shell opals,” Phys. Rev. B 72, 233105 (2005).

Kim, J.-H.

Y.-M. Lee, K.-H. Lee, Y. Choi, and J.-H. Kim, “Fast bistable microlens arrays based on a birefringent layer and ferroelectric liquid crystals,” Jpn. J. Appl. Phys. 47, 6343-6346 (2008).

King, J. S.

E. Graugnard, J. S. King, S. Jain, C. J. Summers, Y. Zhang-Williams, and I. C. Khoo, “Electric-field tuning of the Bragg peak in large-pore TiO2 inverse shell opals,” Phys. Rev. B 72, 233105 (2005).

Kosaka, H.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, R10096-R10099 (1998).

Kubo, H.

M. Ojima, N. Numata, Y. Ogawa, K. Murata, H. Kubo, A. Fujii, and M. Ozaki, “Electric field tuning of plasmonic absorption of metallic grating with twisted nematic liquid crystal,” Appl. Phys. Express 2086001 (2009).

Kuebler, S. M.

Larat, C.

Lee, K.-H.

Y.-M. Lee, K.-H. Lee, Y. Choi, and J.-H. Kim, “Fast bistable microlens arrays based on a birefringent layer and ferroelectric liquid crystals,” Jpn. J. Appl. Phys. 47, 6343-6346 (2008).

Lee, Y.-M.

Y.-M. Lee, K.-H. Lee, Y. Choi, and J.-H. Kim, “Fast bistable microlens arrays based on a birefringent layer and ferroelectric liquid crystals,” Jpn. J. Appl. Phys. 47, 6343-6346 (2008).

Lesiak, P.

T. R. Wolinski, S. Ertman, A. Czapla, P. Lesiak, K. Nowecka, A. W. Domanski, E. Nowinowski-Kruszelnicki, R. Dabrowski, and J. Wojcik, “Polarization effects in photonic liquid crystal fibers,” Meas. Sci. Technol. 183061-3069 (2007).
[CrossRef]

Leuchs, G.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1-7 (2000).
[CrossRef]

Loiseaux, B.

Martínez-Herrero, R.

R. Martínez-Herrero, G. Piquero, and P. M. Mejías, “On the propagation of the kurtosis of general beams,” Opt. Commun. 115, 225-232 (1995).
[CrossRef]

Martín-Moreno, L.

F. J. García-Vidal and L. Martín-Moreno, “Transmission and focusing of light in one-dimensional periodically nanostructured metals,” Phys. Rev. B 66, 155412 (2002).

Mejías, P. M.

R. Martínez-Herrero, G. Piquero, and P. M. Mejías, “On the propagation of the kurtosis of general beams,” Opt. Commun. 115, 225-232 (1995).
[CrossRef]

Murata, K.

M. Ojima, N. Numata, Y. Ogawa, K. Murata, H. Kubo, A. Fujii, and M. Ozaki, “Electric field tuning of plasmonic absorption of metallic grating with twisted nematic liquid crystal,” Appl. Phys. Express 2086001 (2009).

Murray, A. J.

H. F. Gleeson, A. J. Murray, E. Fraser, and A. Zoro, “An electrically addressed liquid crystal filter for tunable lasers,” Opt. Commun. 212, 165-168 (2002).
[CrossRef]

Nix, M.

Notomi, M.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, R10096-R10099 (1998).

Nowecka, K.

T. R. Wolinski, S. Ertman, A. Czapla, P. Lesiak, K. Nowecka, A. W. Domanski, E. Nowinowski-Kruszelnicki, R. Dabrowski, and J. Wojcik, “Polarization effects in photonic liquid crystal fibers,” Meas. Sci. Technol. 183061-3069 (2007).
[CrossRef]

Nowinowski-Kruszelnicki, E.

T. R. Wolinski, S. Ertman, A. Czapla, P. Lesiak, K. Nowecka, A. W. Domanski, E. Nowinowski-Kruszelnicki, R. Dabrowski, and J. Wojcik, “Polarization effects in photonic liquid crystal fibers,” Meas. Sci. Technol. 183061-3069 (2007).
[CrossRef]

T. R. Wolinski, A. Czapla, S. Ertman, M. Tefelska, A. W. Domanski, E. Nowinowski-Kruszelnicki, and R. Dabrowski, “Tunable highly birefringent solid-core photonic liquid crystal fibers,” Opt. Quantum Electron. 39, 1021-1032 (2007).
[CrossRef]

Numata, N.

M. Ojima, N. Numata, Y. Ogawa, K. Murata, H. Kubo, A. Fujii, and M. Ozaki, “Electric field tuning of plasmonic absorption of metallic grating with twisted nematic liquid crystal,” Appl. Phys. Express 2086001 (2009).

Ogawa, Y.

M. Ojima, N. Numata, Y. Ogawa, K. Murata, H. Kubo, A. Fujii, and M. Ozaki, “Electric field tuning of plasmonic absorption of metallic grating with twisted nematic liquid crystal,” Appl. Phys. Express 2086001 (2009).

Ojima, M.

M. Ojima, N. Numata, Y. Ogawa, K. Murata, H. Kubo, A. Fujii, and M. Ozaki, “Electric field tuning of plasmonic absorption of metallic grating with twisted nematic liquid crystal,” Appl. Phys. Express 2086001 (2009).

Ouyang, H.

Ozaki, M.

M. Ojima, N. Numata, Y. Ogawa, K. Murata, H. Kubo, A. Fujii, and M. Ozaki, “Electric field tuning of plasmonic absorption of metallic grating with twisted nematic liquid crystal,” Appl. Phys. Express 2086001 (2009).

Palima, D.

Piquero, G.

R. Martínez-Herrero, G. Piquero, and P. M. Mejías, “On the propagation of the kurtosis of general beams,” Opt. Commun. 115, 225-232 (1995).
[CrossRef]

Quabis, S.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1-7 (2000).
[CrossRef]

Sakoda, K.

K. Sakoda, Optical Properties of Photonic Crystals (Springer, 2001).

Sanner, N.

Sato, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, R10096-R10099 (1998).

Sinzinger, S.

Summers, C. J.

E. Graugnard, J. S. King, S. Jain, C. J. Summers, Y. Zhang-Williams, and I. C. Khoo, “Electric-field tuning of the Bragg peak in large-pore TiO2 inverse shell opals,” Phys. Rev. B 72, 233105 (2005).

Tamamura, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, R10096-R10099 (1998).

Tefelska, M.

T. R. Wolinski, A. Czapla, S. Ertman, M. Tefelska, A. W. Domanski, E. Nowinowski-Kruszelnicki, and R. Dabrowski, “Tunable highly birefringent solid-core photonic liquid crystal fibers,” Opt. Quantum Electron. 39, 1021-1032 (2007).
[CrossRef]

Tian, Z.

Tomita, A.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, R10096-R10099 (1998).

Weiner, A. M.

A. M. Weiner, “Femtosecond pulse shaping using spatial light modulators,” Rev. Sci. Instrum. 71, 1929-1960 (2000).
[CrossRef]

Weiss, S. M.

Wojcik, J.

T. R. Wolinski, S. Ertman, A. Czapla, P. Lesiak, K. Nowecka, A. W. Domanski, E. Nowinowski-Kruszelnicki, R. Dabrowski, and J. Wojcik, “Polarization effects in photonic liquid crystal fibers,” Meas. Sci. Technol. 183061-3069 (2007).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, 1970).

Wolinski, T. R.

T. R. Wolinski, A. Czapla, S. Ertman, M. Tefelska, A. W. Domanski, E. Nowinowski-Kruszelnicki, and R. Dabrowski, “Tunable highly birefringent solid-core photonic liquid crystal fibers,” Opt. Quantum Electron. 39, 1021-1032 (2007).
[CrossRef]

T. R. Wolinski, S. Ertman, A. Czapla, P. Lesiak, K. Nowecka, A. W. Domanski, E. Nowinowski-Kruszelnicki, R. Dabrowski, and J. Wojcik, “Polarization effects in photonic liquid crystal fibers,” Meas. Sci. Technol. 183061-3069 (2007).
[CrossRef]

Wu, S.-T.

Yam, S. S.-H.

Zhang, J.

Zhang-Williams, Y.

E. Graugnard, J. S. King, S. Jain, C. J. Summers, Y. Zhang-Williams, and I. C. Khoo, “Electric-field tuning of the Bragg peak in large-pore TiO2 inverse shell opals,” Phys. Rev. B 72, 233105 (2005).

Zhou, Y.

Zoro, A.

H. F. Gleeson, A. J. Murray, E. Fraser, and A. Zoro, “An electrically addressed liquid crystal filter for tunable lasers,” Opt. Commun. 212, 165-168 (2002).
[CrossRef]

Zou, Q.

R. Hamdi, B.-E. Benkelfat, Q. Zou, and Y. Gottesman, “Bandwidth tuning of hybrid liquid-crystal Šolc filters based on an optical cancelling technique,” Opt. Commun. 269, 64-68(2007).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Express (1)

M. Ojima, N. Numata, Y. Ogawa, K. Murata, H. Kubo, A. Fujii, and M. Ozaki, “Electric field tuning of plasmonic absorption of metallic grating with twisted nematic liquid crystal,” Appl. Phys. Express 2086001 (2009).

Jpn. J. Appl. Phys. (1)

Y.-M. Lee, K.-H. Lee, Y. Choi, and J.-H. Kim, “Fast bistable microlens arrays based on a birefringent layer and ferroelectric liquid crystals,” Jpn. J. Appl. Phys. 47, 6343-6346 (2008).

Meas. Sci. Technol. (1)

T. R. Wolinski, S. Ertman, A. Czapla, P. Lesiak, K. Nowecka, A. W. Domanski, E. Nowinowski-Kruszelnicki, R. Dabrowski, and J. Wojcik, “Polarization effects in photonic liquid crystal fibers,” Meas. Sci. Technol. 183061-3069 (2007).
[CrossRef]

Opt. Commun. (4)

H. F. Gleeson, A. J. Murray, E. Fraser, and A. Zoro, “An electrically addressed liquid crystal filter for tunable lasers,” Opt. Commun. 212, 165-168 (2002).
[CrossRef]

R. Hamdi, B.-E. Benkelfat, Q. Zou, and Y. Gottesman, “Bandwidth tuning of hybrid liquid-crystal Šolc filters based on an optical cancelling technique,” Opt. Commun. 269, 64-68(2007).
[CrossRef]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1-7 (2000).
[CrossRef]

R. Martínez-Herrero, G. Piquero, and P. M. Mejías, “On the propagation of the kurtosis of general beams,” Opt. Commun. 115, 225-232 (1995).
[CrossRef]

Opt. Express (5)

Opt. Lett. (2)

Opt. Quantum Electron. (1)

T. R. Wolinski, A. Czapla, S. Ertman, M. Tefelska, A. W. Domanski, E. Nowinowski-Kruszelnicki, and R. Dabrowski, “Tunable highly birefringent solid-core photonic liquid crystal fibers,” Opt. Quantum Electron. 39, 1021-1032 (2007).
[CrossRef]

Phys. Rev. B (3)

F. J. García-Vidal and L. Martín-Moreno, “Transmission and focusing of light in one-dimensional periodically nanostructured metals,” Phys. Rev. B 66, 155412 (2002).

E. Graugnard, J. S. King, S. Jain, C. J. Summers, Y. Zhang-Williams, and I. C. Khoo, “Electric-field tuning of the Bragg peak in large-pore TiO2 inverse shell opals,” Phys. Rev. B 72, 233105 (2005).

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, R10096-R10099 (1998).

Rev. Sci. Instrum. (1)

A. M. Weiner, “Femtosecond pulse shaping using spatial light modulators,” Rev. Sci. Instrum. 71, 1929-1960 (2000).
[CrossRef]

Other (3)

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, 1970).

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Figures (8)

Fig. 1
Fig. 1

Schematic representation of the periodic structure.

Fig. 2
Fig. 2

TM transmissivity of (a) an anisotropic structure with ten periods and k 0 d 1 = π / 2 , k 0 d 2 = π / 4 (solid curve), k 0 d 1 = π / 2 , k 0 d 2 = π (dashed curve), k 0 d 1 = 3 π / 2 , k 0 d 2 = π (dotted curve), and k 0 d 1 = 3 π / 4 , k 0 d 2 = π / 2 (dashed–dotted curve).

Fig. 3
Fig. 3

(a) TM transmissivity for a ten-period anisotropic structure with k 0 d 1 = π / 2 , k 0 d 2 = π / 4 , and n e 1 = 1.7 (solid curve, n e 1 = 1.65 (dashed curve), n e 1 = 1.6 (dotted curve), and n e 1 = 1.55 (dashed–dotted curve) for the same ordinary refractive index n o 1 = 1.5 . (b) TM transmissivity of an anisotropic structure with k 0 d 1 = 3 π / 4 , k 0 d 2 = π / 2 and 6 (dotted curve), 10 (solid curve), and 14 (dashed curve) periods.

Fig. 4
Fig. 4

Intensity of the output beam with a = 5 (solid curves), a = 10 (dotted curves), and a = 5 but n e 1 = 1.6 (dashed curves) emerging from a ten-period structure with (a)  k 0 d 1 = 3 π / 4 , k 0 d 2 = π / 2 , (b)  k 0 d 1 = 3 π / 2 , k 0 d 2 = π , and (c)  k 0 d 1 = π , k 0 d 2 = π / 4 .

Fig. 5
Fig. 5

Intensity of the output beam with a = 5 emerging from a 6-period (dashed curve), a 10-period (solid curve), and a 14-period structure (dotted curve) with k 0 d 1 = 3 π / 4 and k 0 d 2 = π / 2 .

Fig. 6
Fig. 6

Kurtosis dependence on a for ten-period structures with k 0 d 1 = π / 2 , k 0 d 2 = π / 4 (solid curve), k 0 d 1 = π / 2 , k 0 d 2 = π (dashed curve), k 0 d 1 = 3 π / 2 , k 0 d 2 = π (dotted curve), and k 0 d 1 = 3 π / 4 , k 0 d 2 = π / 2 (dashed–dotted curve).

Fig. 7
Fig. 7

Kurtosis dependence on a for an anisotropic structure with k 0 d 1 = 3 π / 4 , k 0 d 2 = π / 2 composed of 6 (dotted curve), 10 (solid curve), and 14 periods (dashed curve). The dashed–dotted curve represents the kurtosis for the ten-period structure, but with a smaller extraordinary refractive index of n e 1 = 1.65 .

Fig. 8
Fig. 8

Two examples of output beam intensity (solid curves) and desired intensity distribution (dotted curves) for anisotropic photonic crystals with best-fit parameters (top), and the corresponding error function (bottom).

Equations (10)

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n 1 = 1 / cos 2 ϑ 1 / n o 1 2 + sin 2 ϑ 1 / n e 1 2 ,
t = 2 p in ( m 11 + m 12 p out ) p in + ( m 21 + m 22 p out ) ,
M j = ( cos ( k 0 L j ) i sin ( k 0 L j ) / p j i p j sin ( k 0 L j ) cos ( k 0 L j ) ) ,
φ in ( x ) = exp ( x 2 / 2 x 0 2 ) .
ψ in ( k ) = ( 2 π ) 1 / 2 φ in ( x ) exp ( i k x ) d x = x 0 exp ( k 2 x 0 2 / 2 ) ,
φ out ( x ) = ( 2 π ) 1 / 2 t ( k ) ψ in ( k ) exp ( i k x ) d k .
m i = x i I ( x ) d x I ( x ) d x ,
K = m 4 / ( m 2 ) 2 .
φ HG ( x ; s , g ) = g [ 2 ( s x ) 2 1 ] exp [ ( s x ) 2 / 2 ] , φ disc ( x ; s , g ) = g [ 1 | s x / 2 | ] exp [ ( s x ) 2 / 2 ] .
Δ I / I max = ( | φ des ( x ) | 2 | φ out ( x ) | 2 ) / | φ des , max | 2

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