Abstract

We introduce graded photonic quasicrystals and investigate properties of such structures on the example of a Luneburg lens based on a dodecagonal photonic quasicrystal. It is shown that the graded photonic quasicrystal lens has better focusing properties as compared with the graded photonic crystal lens in a frequency range suitable for experimental realization. The proposed graded photonic quasicrystals can be used in optical systems where compact and powerful focusing elements are required.

© 2012 Optical Society of America

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  1. J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, Nat. Mater. 8, 568 (2009).
    [CrossRef]
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    [CrossRef]
  3. H.-W. Wang and L.-W. Chen, J. Appl. Phys. 109, 103104 (2011).
    [CrossRef]
  4. J. Hunt, T. Tyler, S. Dhar, Y.-J. Tsai, P. Bowen, S. Larouche, N. M. Jokerst, and D. R. Smith, Opt. Express 20, 1706 (2012).
    [CrossRef]
  5. H. Gao, B. Zhang, S. G. Johnson, and G. Barbastathis, Opt. Express 20, 1617 (2012).
    [CrossRef]
  6. E. Centeno and D. Cassagne, Opt. Lett. 30, 2278 (2005).
    [CrossRef]
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    [CrossRef]
  8. H.-T. Chien and C.-C. Chen, Opt. Express 14, 10759 (2006).
    [CrossRef]
  9. F. Roux and I. De Leon, Phys. Rev. B 74, 113103 (2006).
    [CrossRef]
  10. H. Kurt, E. Colak, O. Cakmak, H. Caglayan, and E. Ozbay, Appl. Phys. Lett. 93, 171108 (2008).
    [CrossRef]
  11. Q. Wu, J. M. Gibbons, and W. Park, Opt. Express 16, 16941 (2008).
    [CrossRef]
  12. B. Vasić, G. Isić, R. Gajić, and K. Hingerl, Opt. Express 18, 20321 (2010).
    [CrossRef]
  13. E. Cassan and K.-V. Do, J. Opt. Soc. Am. B 28, 1905 (2011).
    [CrossRef]
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    [CrossRef]
  15. K.-V. Do, X. Le Roux, D. Marris-Morini, L. Vivien, and E. Cassan, Opt. Express 20, 4776 (2012).
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    [CrossRef]
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    [CrossRef]
  18. P. N. Dyachenko, N. D. Kundikova, and Y. V. Miklyaev, Phys. Rev. B 79, 233102 (2009).
    [CrossRef]
  19. T. P. Rose, E. D. Gennaro, G. Abbate, and A. Andreone, Phys. Rev. B 84, 125111 (2011).
    [CrossRef]
  20. M. Oxborrow and C. L. Henley, Phys. Rev. B 48, 6966 (1993).
    [CrossRef]
  21. A. Sihvola, Electromagnetic Mixing Formulas and Applications (IEE Publishing, 1999).
  22. R. K. Luneburg, Mathematical Theory of Optics (Brown U. P., 1944).
  23. A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, Comput. Phys. Commun. 181, 687 (2010).
    [CrossRef]
  24. A. Farjadpour, D. Roundy, A. Rodriguez, M. Ibanescu, P. Bermel, J. D. Joannopoulos, S. G. Johnson, and G. W. Burr, Opt. Lett. 31, 2972 (2006).
    [CrossRef]

2012

2011

H.-W. Wang and L.-W. Chen, J. Appl. Phys. 109, 103104 (2011).
[CrossRef]

E. Cassan and K.-V. Do, J. Opt. Soc. Am. B 28, 1905 (2011).
[CrossRef]

B. Vasić and R. Gajić, J. Appl. Phys. 110, 053103 (2011).
[CrossRef]

T. P. Rose, E. D. Gennaro, G. Abbate, and A. Andreone, Phys. Rev. B 84, 125111 (2011).
[CrossRef]

2010

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, Comput. Phys. Commun. 181, 687 (2010).
[CrossRef]

B. Vasić, G. Isić, R. Gajić, and K. Hingerl, Opt. Express 18, 20321 (2010).
[CrossRef]

2009

P. N. Dyachenko, N. D. Kundikova, and Y. V. Miklyaev, Phys. Rev. B 79, 233102 (2009).
[CrossRef]

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, Nat. Mater. 8, 568 (2009).
[CrossRef]

L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, Nat. Photon. 3, 461 (2009).
[CrossRef]

2008

H. Kurt, E. Colak, O. Cakmak, H. Caglayan, and E. Ozbay, Appl. Phys. Lett. 93, 171108 (2008).
[CrossRef]

Q. Wu, J. M. Gibbons, and W. Park, Opt. Express 16, 16941 (2008).
[CrossRef]

E. D. Gennaro, C. Miletto, S. Savo, A. Andreone, D. Morello, V. Galdi, G. Castaldi, and V. Pierro, Phys. Rev. B 77, 193104 (2008).
[CrossRef]

2007

2006

2005

1993

M. Oxborrow and C. L. Henley, Phys. Rev. B 48, 6966 (1993).
[CrossRef]

Abbate, G.

T. P. Rose, E. D. Gennaro, G. Abbate, and A. Andreone, Phys. Rev. B 84, 125111 (2011).
[CrossRef]

Albert, J.-P.

E. Centeno, D. Cassagne, and J.-P. Albert, Phys. Rev. B 73, 235119 (2006).
[CrossRef]

Andreone, A.

T. P. Rose, E. D. Gennaro, G. Abbate, and A. Andreone, Phys. Rev. B 84, 125111 (2011).
[CrossRef]

E. D. Gennaro, C. Miletto, S. Savo, A. Andreone, D. Morello, V. Galdi, G. Castaldi, and V. Pierro, Phys. Rev. B 77, 193104 (2008).
[CrossRef]

Barbastathis, G.

Bartal, G.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, Nat. Mater. 8, 568 (2009).
[CrossRef]

Bermel, P.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, Comput. Phys. Commun. 181, 687 (2010).
[CrossRef]

A. Farjadpour, D. Roundy, A. Rodriguez, M. Ibanescu, P. Bermel, J. D. Joannopoulos, S. G. Johnson, and G. W. Burr, Opt. Lett. 31, 2972 (2006).
[CrossRef]

Bowen, P.

Burr, G. W.

Caglayan, H.

H. Kurt, E. Colak, O. Cakmak, H. Caglayan, and E. Ozbay, Appl. Phys. Lett. 93, 171108 (2008).
[CrossRef]

Cakmak, O.

H. Kurt, E. Colak, O. Cakmak, H. Caglayan, and E. Ozbay, Appl. Phys. Lett. 93, 171108 (2008).
[CrossRef]

Cardenas, J.

L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, Nat. Photon. 3, 461 (2009).
[CrossRef]

Cassagne, D.

E. Centeno, D. Cassagne, and J.-P. Albert, Phys. Rev. B 73, 235119 (2006).
[CrossRef]

E. Centeno and D. Cassagne, Opt. Lett. 30, 2278 (2005).
[CrossRef]

Cassan, E.

Castaldi, G.

E. D. Gennaro, C. Miletto, S. Savo, A. Andreone, D. Morello, V. Galdi, G. Castaldi, and V. Pierro, Phys. Rev. B 77, 193104 (2008).
[CrossRef]

Centeno, E.

E. Centeno, D. Cassagne, and J.-P. Albert, Phys. Rev. B 73, 235119 (2006).
[CrossRef]

E. Centeno and D. Cassagne, Opt. Lett. 30, 2278 (2005).
[CrossRef]

Chen, C.-C.

Chen, L.-W.

H.-W. Wang and L.-W. Chen, J. Appl. Phys. 109, 103104 (2011).
[CrossRef]

Cheng, B.

Chien, H.-T.

Colak, E.

H. Kurt, E. Colak, O. Cakmak, H. Caglayan, and E. Ozbay, Appl. Phys. Lett. 93, 171108 (2008).
[CrossRef]

De Leon, I.

F. Roux and I. De Leon, Phys. Rev. B 74, 113103 (2006).
[CrossRef]

Dhar, S.

Do, K.-V.

Dyachenko, P. N.

P. N. Dyachenko, N. D. Kundikova, and Y. V. Miklyaev, Phys. Rev. B 79, 233102 (2009).
[CrossRef]

Farjadpour, A.

Gabrielli, L. H.

L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, Nat. Photon. 3, 461 (2009).
[CrossRef]

Gajic, R.

Galdi, V.

E. D. Gennaro, C. Miletto, S. Savo, A. Andreone, D. Morello, V. Galdi, G. Castaldi, and V. Pierro, Phys. Rev. B 77, 193104 (2008).
[CrossRef]

Gao, H.

Gennaro, E. D.

T. P. Rose, E. D. Gennaro, G. Abbate, and A. Andreone, Phys. Rev. B 84, 125111 (2011).
[CrossRef]

E. D. Gennaro, C. Miletto, S. Savo, A. Andreone, D. Morello, V. Galdi, G. Castaldi, and V. Pierro, Phys. Rev. B 77, 193104 (2008).
[CrossRef]

Gibbons, J. M.

Henley, C. L.

M. Oxborrow and C. L. Henley, Phys. Rev. B 48, 6966 (1993).
[CrossRef]

Hingerl, K.

Hunt, J.

Ibanescu, M.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, Comput. Phys. Commun. 181, 687 (2010).
[CrossRef]

A. Farjadpour, D. Roundy, A. Rodriguez, M. Ibanescu, P. Bermel, J. D. Joannopoulos, S. G. Johnson, and G. W. Burr, Opt. Lett. 31, 2972 (2006).
[CrossRef]

Isic, G.

Joannopoulos, J. D.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, Comput. Phys. Commun. 181, 687 (2010).
[CrossRef]

A. Farjadpour, D. Roundy, A. Rodriguez, M. Ibanescu, P. Bermel, J. D. Joannopoulos, S. G. Johnson, and G. W. Burr, Opt. Lett. 31, 2972 (2006).
[CrossRef]

Johnson, S. G.

Jokerst, N. M.

Kundikova, N. D.

P. N. Dyachenko, N. D. Kundikova, and Y. V. Miklyaev, Phys. Rev. B 79, 233102 (2009).
[CrossRef]

Kurt, H.

H. Kurt, E. Colak, O. Cakmak, H. Caglayan, and E. Ozbay, Appl. Phys. Lett. 93, 171108 (2008).
[CrossRef]

Larouche, S.

Le Roux, X.

Li, J.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, Nat. Mater. 8, 568 (2009).
[CrossRef]

Li, Z.

Lipson, M.

L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, Nat. Photon. 3, 461 (2009).
[CrossRef]

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (Brown U. P., 1944).

Marris-Morini, D.

Miklyaev, Y. V.

P. N. Dyachenko, N. D. Kundikova, and Y. V. Miklyaev, Phys. Rev. B 79, 233102 (2009).
[CrossRef]

Miletto, C.

E. D. Gennaro, C. Miletto, S. Savo, A. Andreone, D. Morello, V. Galdi, G. Castaldi, and V. Pierro, Phys. Rev. B 77, 193104 (2008).
[CrossRef]

Morello, D.

E. D. Gennaro, C. Miletto, S. Savo, A. Andreone, D. Morello, V. Galdi, G. Castaldi, and V. Pierro, Phys. Rev. B 77, 193104 (2008).
[CrossRef]

Oskooi, A. F.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, Comput. Phys. Commun. 181, 687 (2010).
[CrossRef]

Oxborrow, M.

M. Oxborrow and C. L. Henley, Phys. Rev. B 48, 6966 (1993).
[CrossRef]

Ozbay, E.

H. Kurt, E. Colak, O. Cakmak, H. Caglayan, and E. Ozbay, Appl. Phys. Lett. 93, 171108 (2008).
[CrossRef]

Park, W.

Pierro, V.

E. D. Gennaro, C. Miletto, S. Savo, A. Andreone, D. Morello, V. Galdi, G. Castaldi, and V. Pierro, Phys. Rev. B 77, 193104 (2008).
[CrossRef]

Poitras, C. B.

L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, Nat. Photon. 3, 461 (2009).
[CrossRef]

Rodriguez, A.

Rose, T. P.

T. P. Rose, E. D. Gennaro, G. Abbate, and A. Andreone, Phys. Rev. B 84, 125111 (2011).
[CrossRef]

Roundy, D.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, Comput. Phys. Commun. 181, 687 (2010).
[CrossRef]

A. Farjadpour, D. Roundy, A. Rodriguez, M. Ibanescu, P. Bermel, J. D. Joannopoulos, S. G. Johnson, and G. W. Burr, Opt. Lett. 31, 2972 (2006).
[CrossRef]

Roux, F.

F. Roux and I. De Leon, Phys. Rev. B 74, 113103 (2006).
[CrossRef]

Savo, S.

E. D. Gennaro, C. Miletto, S. Savo, A. Andreone, D. Morello, V. Galdi, G. Castaldi, and V. Pierro, Phys. Rev. B 77, 193104 (2008).
[CrossRef]

Sihvola, A.

A. Sihvola, Electromagnetic Mixing Formulas and Applications (IEE Publishing, 1999).

Smith, D. R.

Tsai, Y.-J.

Tyler, T.

Valentine, J.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, Nat. Mater. 8, 568 (2009).
[CrossRef]

Vasic, B.

Vivien, L.

Wang, H.-W.

H.-W. Wang and L.-W. Chen, J. Appl. Phys. 109, 103104 (2011).
[CrossRef]

Wu, Q.

Zentgraf, T.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, Nat. Mater. 8, 568 (2009).
[CrossRef]

Zhang, B.

Zhang, D.-Z.

Zhang, X.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, Nat. Mater. 8, 568 (2009).
[CrossRef]

X. Zhang, Z. Li, B. Cheng, and D.-Z. Zhang, Opt. Express 15, 1292 (2007).
[CrossRef]

Appl. Phys. Lett.

H. Kurt, E. Colak, O. Cakmak, H. Caglayan, and E. Ozbay, Appl. Phys. Lett. 93, 171108 (2008).
[CrossRef]

Comput. Phys. Commun.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, Comput. Phys. Commun. 181, 687 (2010).
[CrossRef]

J. Appl. Phys.

H.-W. Wang and L.-W. Chen, J. Appl. Phys. 109, 103104 (2011).
[CrossRef]

B. Vasić and R. Gajić, J. Appl. Phys. 110, 053103 (2011).
[CrossRef]

J. Opt. Soc. Am. B

Nat. Mater.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, Nat. Mater. 8, 568 (2009).
[CrossRef]

Nat. Photon.

L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, Nat. Photon. 3, 461 (2009).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. B

E. Centeno, D. Cassagne, and J.-P. Albert, Phys. Rev. B 73, 235119 (2006).
[CrossRef]

F. Roux and I. De Leon, Phys. Rev. B 74, 113103 (2006).
[CrossRef]

E. D. Gennaro, C. Miletto, S. Savo, A. Andreone, D. Morello, V. Galdi, G. Castaldi, and V. Pierro, Phys. Rev. B 77, 193104 (2008).
[CrossRef]

P. N. Dyachenko, N. D. Kundikova, and Y. V. Miklyaev, Phys. Rev. B 79, 233102 (2009).
[CrossRef]

T. P. Rose, E. D. Gennaro, G. Abbate, and A. Andreone, Phys. Rev. B 84, 125111 (2011).
[CrossRef]

M. Oxborrow and C. L. Henley, Phys. Rev. B 48, 6966 (1993).
[CrossRef]

Other

A. Sihvola, Electromagnetic Mixing Formulas and Applications (IEE Publishing, 1999).

R. K. Luneburg, Mathematical Theory of Optics (Brown U. P., 1944).

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Figures (4)

Fig. 1.
Fig. 1.

Schematic of Luneburg lens based on (a) the dodecagonal PQ and (b) the square lattice with the graded radii of the rods.

Fig. 2.
Fig. 2.

Transmission coefficients for the GPQ lens (green), the GPC lens (red), and the original Luneburg lens (blue) for TE polarization.

Fig. 3.
Fig. 3.

Distributions of Hz field for (a) the GPQ lens and (c) the GPC lens. The magnetic field intensity distributions for (b) the GPQ lens and (d) the GPC lens. In both cases, the frequency is ω=0.376.

Fig. 4.
Fig. 4.

Time-averaged |Hz|2 field intensity distributions along (a) the longitudinal (X) direction and (b) the transverse (Y) direction in the image plane for the GPQ lens (green) and the GPC lens (red).

Equations (4)

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

εplane=εhost+fεhost(εrodsεhost)εhost+0.5(1f)(εrodsεhost),
εz=(1f)εhost+fεrods,
rte=S(εhostn2)(εhost+εrods)πN(εhost+n2)(εhostεrods).
rtm=S(εhostn2)πN(εhostεrods).

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