Abstract

Opposite directionality of the Poynting vector and the wave vector, an inherent property of negative index metamaterials (NIMs), was predicted to enable backward phase-matching condition for a second harmonic generation (SHG) process. As a result, such a nonlinear negative index slab acts as a nonlinear mirror. In this Letter, we predict that SHG with structured light carrying orbital angular momentum (OAM) and propagating in NIMs results in a possibility of generating a backward propagating beam with simultaneously doubled frequency, OAM, and reversed rotation direction of the wavefront. These results may find applications in high-dimensional communication systems, quantum information processing, and optical manipulation on a nanoscale.

© 2013 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  7. S. E. Harris, Appl. Phys. Lett. 9, 114 (1966).
    [CrossRef]
  8. V. G. Veselago, Sov. Phys. Usp. 10, 509 (1968).
    [CrossRef]
  9. D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, Science 305, 788 (2004).
    [CrossRef]
  10. V. M. Shalaev, Nat. Photonics 1, 41 (2007).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  17. N. M. Litchinitser, Science 337, 1054 (2012).
    [CrossRef]
  18. A. K. Popov and V. M. Shalaev, Appl. Phys. B 84, 131 (2006).
    [CrossRef]
  19. A. Rose, D. Huang, and D. R. Smith, Phys. Rev. Lett. 107, 063902 (2011).
    [CrossRef]
  20. F. Tamburini, E. Mari, A. Sponselli, B. Thidé, A. Bianchini, and F. Romanato, New J. Phys. 14, 033001 (2012).
    [CrossRef]
  21. J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, Nat. Photonics 6, 488 (2012).
    [CrossRef]
  22. A. E. Willner, J. Wang, and H. Huang, Science 337, 655 (2012).
    [CrossRef]
  23. G. Molina-Terriza, J. P. Torres, and L. Torner, Phys. Rev. Lett. 88, 013601 (2001).
    [CrossRef]
  24. G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, Opt. Express 12, 5448 (2004).
    [CrossRef]
  25. K. Dholakia, N. B. Simpson, M. J. Padgett, and L. Allen, Phys. Rev. A 54, R3742 (1996).
    [CrossRef]
  26. H. Luo, Z. Ren, W. Shu, and S. Wen, Phys. Rev. A 77, 023812 (2008).
    [CrossRef]
  27. J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, Science 321, 930 (2008).
    [CrossRef]
  28. Y. M. Liu, G. Bartal, and X. Zhang, Opt. Express 16, 15439 (2008).
    [CrossRef]

2012

F. Tamburini, E. Mari, A. Sponselli, B. Thidé, A. Bianchini, and F. Romanato, New J. Phys. 14, 033001 (2012).
[CrossRef]

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, Nat. Photonics 6, 488 (2012).
[CrossRef]

A. E. Willner, J. Wang, and H. Huang, Science 337, 655 (2012).
[CrossRef]

N. M. Litchinitser, Science 337, 1054 (2012).
[CrossRef]

2011

A. Rose, D. Huang, and D. R. Smith, Phys. Rev. Lett. 107, 063902 (2011).
[CrossRef]

C. M. Soukoulis and M. Wegener, Nat. Photonics 5, 523 (2011).
[CrossRef]

2008

M. B. Nasr, S. Carrasco, B. E. A. Saleh, A. V. Sergienko, M. C. Teich, J. P. Torres, L. Torner, D. S. Hum, and M. M. Fejer, Phys. Rev. Lett. 100, 183601 (2008).
[CrossRef]

Y. M. Liu, G. Bartal, and X. Zhang, Opt. Express 16, 15439 (2008).
[CrossRef]

H. Luo, Z. Ren, W. Shu, and S. Wen, Phys. Rev. A 77, 023812 (2008).
[CrossRef]

J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, Science 321, 930 (2008).
[CrossRef]

2007

V. M. Shalaev, Nat. Photonics 1, 41 (2007).
[CrossRef]

2006

A. K. Popov, V. V. Slabko, and V. M. Shalaev, Laser Phys. Lett. 3, 293 (2006).
[CrossRef]

A. K. Popov and V. M. Shalaev, Appl. Phys. B 84, 131 (2006).
[CrossRef]

I. V. Shadrivov, A. A. Zharov, and Y. S. Kivshar, J. Opt. Soc. Am. B 23, 529 (2006).
[CrossRef]

2005

M. Scalora, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, J. W. Haus, and A. M. Zheltikov, Phys. Rev. Lett. 95, 013902 (2005).
[CrossRef]

2004

V. M. Agranovich, Y. R. Shen, R. H. Baughman, and A. A. Zakhidov, Phys. Rev. B 69, 165112 (2004).
[CrossRef]

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, Science 305, 788 (2004).
[CrossRef]

G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, Opt. Express 12, 5448 (2004).
[CrossRef]

2001

G. Molina-Terriza, J. P. Torres, and L. Torner, Phys. Rev. Lett. 88, 013601 (2001).
[CrossRef]

1998

A. Fiore, V. Berger, E. Rosencher, P. Bravetti, and J. Nagle, Nature 391, 463 (1998).
[CrossRef]

1996

Y. J. Ding and J. B. Khurgin, IEEE J. Quantum Electron. 32, 1574 (1996).
[CrossRef]

K. Dholakia, N. B. Simpson, M. J. Padgett, and L. Allen, Phys. Rev. A 54, R3742 (1996).
[CrossRef]

1984

R. Eckhardt and J. Reintjes, IEEE J. Quantum Electron. 20, 1178 (1984).
[CrossRef]

1968

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

1966

S. E. Harris, Appl. Phys. Lett. 9, 114 (1966).
[CrossRef]

1961

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, Phys. Rev. Lett. 7, 118 (1961).
[CrossRef]

Agranovich, V. M.

V. M. Agranovich, Y. R. Shen, R. H. Baughman, and A. A. Zakhidov, Phys. Rev. B 69, 165112 (2004).
[CrossRef]

Ahmed, N.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, Nat. Photonics 6, 488 (2012).
[CrossRef]

Allen, L.

K. Dholakia, N. B. Simpson, M. J. Padgett, and L. Allen, Phys. Rev. A 54, R3742 (1996).
[CrossRef]

Barnett, S. M.

Bartal, G.

Y. M. Liu, G. Bartal, and X. Zhang, Opt. Express 16, 15439 (2008).
[CrossRef]

J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, Science 321, 930 (2008).
[CrossRef]

Baughman, R. H.

V. M. Agranovich, Y. R. Shen, R. H. Baughman, and A. A. Zakhidov, Phys. Rev. B 69, 165112 (2004).
[CrossRef]

Berger, V.

A. Fiore, V. Berger, E. Rosencher, P. Bravetti, and J. Nagle, Nature 391, 463 (1998).
[CrossRef]

Bianchini, A.

F. Tamburini, E. Mari, A. Sponselli, B. Thidé, A. Bianchini, and F. Romanato, New J. Phys. 14, 033001 (2012).
[CrossRef]

Bloemer, M. J.

M. Scalora, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, J. W. Haus, and A. M. Zheltikov, Phys. Rev. Lett. 95, 013902 (2005).
[CrossRef]

Boyd, R. W.

R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic, 2008).

Bravetti, P.

A. Fiore, V. Berger, E. Rosencher, P. Bravetti, and J. Nagle, Nature 391, 463 (1998).
[CrossRef]

Carrasco, S.

M. B. Nasr, S. Carrasco, B. E. A. Saleh, A. V. Sergienko, M. C. Teich, J. P. Torres, L. Torner, D. S. Hum, and M. M. Fejer, Phys. Rev. Lett. 100, 183601 (2008).
[CrossRef]

Courtial, J.

D’Aguanno, G.

M. Scalora, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, J. W. Haus, and A. M. Zheltikov, Phys. Rev. Lett. 95, 013902 (2005).
[CrossRef]

Dholakia, K.

K. Dholakia, N. B. Simpson, M. J. Padgett, and L. Allen, Phys. Rev. A 54, R3742 (1996).
[CrossRef]

Ding, Y. J.

Y. J. Ding and J. B. Khurgin, IEEE J. Quantum Electron. 32, 1574 (1996).
[CrossRef]

Dolinar, S.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, Nat. Photonics 6, 488 (2012).
[CrossRef]

Eckhardt, R.

R. Eckhardt and J. Reintjes, IEEE J. Quantum Electron. 20, 1178 (1984).
[CrossRef]

Fazal, I. M.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, Nat. Photonics 6, 488 (2012).
[CrossRef]

Fejer, M. M.

M. B. Nasr, S. Carrasco, B. E. A. Saleh, A. V. Sergienko, M. C. Teich, J. P. Torres, L. Torner, D. S. Hum, and M. M. Fejer, Phys. Rev. Lett. 100, 183601 (2008).
[CrossRef]

Fiore, A.

A. Fiore, V. Berger, E. Rosencher, P. Bravetti, and J. Nagle, Nature 391, 463 (1998).
[CrossRef]

Franke-Arnold, S.

Franken, P. A.

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, Phys. Rev. Lett. 7, 118 (1961).
[CrossRef]

Gabitov, I. R.

N. M. Litchinitser, I. R. Gabitov, A. I. Maimistov, and V. M. Shalaev, in Progress in Optics, E. Wolf, ed. (Elsevier, 2008), Vol. 51, Chap. 1, pp. 1–68.

Gibson, G.

Harris, S. E.

S. E. Harris, Appl. Phys. Lett. 9, 114 (1966).
[CrossRef]

Haus, J. W.

M. Scalora, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, J. W. Haus, and A. M. Zheltikov, Phys. Rev. Lett. 95, 013902 (2005).
[CrossRef]

Hill, A. E.

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, Phys. Rev. Lett. 7, 118 (1961).
[CrossRef]

Huang, D.

A. Rose, D. Huang, and D. R. Smith, Phys. Rev. Lett. 107, 063902 (2011).
[CrossRef]

Huang, H.

A. E. Willner, J. Wang, and H. Huang, Science 337, 655 (2012).
[CrossRef]

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, Nat. Photonics 6, 488 (2012).
[CrossRef]

Hum, D. S.

M. B. Nasr, S. Carrasco, B. E. A. Saleh, A. V. Sergienko, M. C. Teich, J. P. Torres, L. Torner, D. S. Hum, and M. M. Fejer, Phys. Rev. Lett. 100, 183601 (2008).
[CrossRef]

Khurgin, J. B.

Y. J. Ding and J. B. Khurgin, IEEE J. Quantum Electron. 32, 1574 (1996).
[CrossRef]

Kivshar, Y. S.

Litchinitser, N. M.

N. M. Litchinitser, Science 337, 1054 (2012).
[CrossRef]

N. M. Litchinitser, I. R. Gabitov, A. I. Maimistov, and V. M. Shalaev, in Progress in Optics, E. Wolf, ed. (Elsevier, 2008), Vol. 51, Chap. 1, pp. 1–68.

Liu, Y.

J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, Science 321, 930 (2008).
[CrossRef]

Liu, Y. M.

Liu, Z.

J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, Science 321, 930 (2008).
[CrossRef]

Luo, H.

H. Luo, Z. Ren, W. Shu, and S. Wen, Phys. Rev. A 77, 023812 (2008).
[CrossRef]

Maimistov, A. I.

N. M. Litchinitser, I. R. Gabitov, A. I. Maimistov, and V. M. Shalaev, in Progress in Optics, E. Wolf, ed. (Elsevier, 2008), Vol. 51, Chap. 1, pp. 1–68.

Mari, E.

F. Tamburini, E. Mari, A. Sponselli, B. Thidé, A. Bianchini, and F. Romanato, New J. Phys. 14, 033001 (2012).
[CrossRef]

Mattiucci, N.

M. Scalora, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, J. W. Haus, and A. M. Zheltikov, Phys. Rev. Lett. 95, 013902 (2005).
[CrossRef]

Molina-Terriza, G.

G. Molina-Terriza, J. P. Torres, and L. Torner, Phys. Rev. Lett. 88, 013601 (2001).
[CrossRef]

Nagle, J.

A. Fiore, V. Berger, E. Rosencher, P. Bravetti, and J. Nagle, Nature 391, 463 (1998).
[CrossRef]

Nasr, M. B.

M. B. Nasr, S. Carrasco, B. E. A. Saleh, A. V. Sergienko, M. C. Teich, J. P. Torres, L. Torner, D. S. Hum, and M. M. Fejer, Phys. Rev. Lett. 100, 183601 (2008).
[CrossRef]

Padgett, M. J.

Pas’ko, V.

Pendry, J. B.

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, Science 305, 788 (2004).
[CrossRef]

Peters, C. W.

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, Phys. Rev. Lett. 7, 118 (1961).
[CrossRef]

Popov, A. K.

A. K. Popov and V. M. Shalaev, Appl. Phys. B 84, 131 (2006).
[CrossRef]

A. K. Popov, V. V. Slabko, and V. M. Shalaev, Laser Phys. Lett. 3, 293 (2006).
[CrossRef]

Reintjes, J.

R. Eckhardt and J. Reintjes, IEEE J. Quantum Electron. 20, 1178 (1984).
[CrossRef]

Ren, Y.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, Nat. Photonics 6, 488 (2012).
[CrossRef]

Ren, Z.

H. Luo, Z. Ren, W. Shu, and S. Wen, Phys. Rev. A 77, 023812 (2008).
[CrossRef]

Romanato, F.

F. Tamburini, E. Mari, A. Sponselli, B. Thidé, A. Bianchini, and F. Romanato, New J. Phys. 14, 033001 (2012).
[CrossRef]

Rose, A.

A. Rose, D. Huang, and D. R. Smith, Phys. Rev. Lett. 107, 063902 (2011).
[CrossRef]

Rosencher, E.

A. Fiore, V. Berger, E. Rosencher, P. Bravetti, and J. Nagle, Nature 391, 463 (1998).
[CrossRef]

Saleh, B. E. A.

M. B. Nasr, S. Carrasco, B. E. A. Saleh, A. V. Sergienko, M. C. Teich, J. P. Torres, L. Torner, D. S. Hum, and M. M. Fejer, Phys. Rev. Lett. 100, 183601 (2008).
[CrossRef]

Scalora, M.

M. Scalora, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, J. W. Haus, and A. M. Zheltikov, Phys. Rev. Lett. 95, 013902 (2005).
[CrossRef]

Sergienko, A. V.

M. B. Nasr, S. Carrasco, B. E. A. Saleh, A. V. Sergienko, M. C. Teich, J. P. Torres, L. Torner, D. S. Hum, and M. M. Fejer, Phys. Rev. Lett. 100, 183601 (2008).
[CrossRef]

Shadrivov, I. V.

Shalaev, V. M.

V. M. Shalaev, Nat. Photonics 1, 41 (2007).
[CrossRef]

A. K. Popov, V. V. Slabko, and V. M. Shalaev, Laser Phys. Lett. 3, 293 (2006).
[CrossRef]

A. K. Popov and V. M. Shalaev, Appl. Phys. B 84, 131 (2006).
[CrossRef]

N. M. Litchinitser, I. R. Gabitov, A. I. Maimistov, and V. M. Shalaev, in Progress in Optics, E. Wolf, ed. (Elsevier, 2008), Vol. 51, Chap. 1, pp. 1–68.

Shen, Y. R.

V. M. Agranovich, Y. R. Shen, R. H. Baughman, and A. A. Zakhidov, Phys. Rev. B 69, 165112 (2004).
[CrossRef]

Shu, W.

H. Luo, Z. Ren, W. Shu, and S. Wen, Phys. Rev. A 77, 023812 (2008).
[CrossRef]

Simpson, N. B.

K. Dholakia, N. B. Simpson, M. J. Padgett, and L. Allen, Phys. Rev. A 54, R3742 (1996).
[CrossRef]

Slabko, V. V.

A. K. Popov, V. V. Slabko, and V. M. Shalaev, Laser Phys. Lett. 3, 293 (2006).
[CrossRef]

Smith, D. R.

A. Rose, D. Huang, and D. R. Smith, Phys. Rev. Lett. 107, 063902 (2011).
[CrossRef]

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, Science 305, 788 (2004).
[CrossRef]

Soukoulis, C. M.

C. M. Soukoulis and M. Wegener, Nat. Photonics 5, 523 (2011).
[CrossRef]

Sponselli, A.

F. Tamburini, E. Mari, A. Sponselli, B. Thidé, A. Bianchini, and F. Romanato, New J. Phys. 14, 033001 (2012).
[CrossRef]

Stacy, A. M.

J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, Science 321, 930 (2008).
[CrossRef]

Sun, C.

J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, Science 321, 930 (2008).
[CrossRef]

Tamburini, F.

F. Tamburini, E. Mari, A. Sponselli, B. Thidé, A. Bianchini, and F. Romanato, New J. Phys. 14, 033001 (2012).
[CrossRef]

Teich, M. C.

M. B. Nasr, S. Carrasco, B. E. A. Saleh, A. V. Sergienko, M. C. Teich, J. P. Torres, L. Torner, D. S. Hum, and M. M. Fejer, Phys. Rev. Lett. 100, 183601 (2008).
[CrossRef]

Thidé, B.

F. Tamburini, E. Mari, A. Sponselli, B. Thidé, A. Bianchini, and F. Romanato, New J. Phys. 14, 033001 (2012).
[CrossRef]

Torner, L.

M. B. Nasr, S. Carrasco, B. E. A. Saleh, A. V. Sergienko, M. C. Teich, J. P. Torres, L. Torner, D. S. Hum, and M. M. Fejer, Phys. Rev. Lett. 100, 183601 (2008).
[CrossRef]

G. Molina-Terriza, J. P. Torres, and L. Torner, Phys. Rev. Lett. 88, 013601 (2001).
[CrossRef]

Torres, J. P.

M. B. Nasr, S. Carrasco, B. E. A. Saleh, A. V. Sergienko, M. C. Teich, J. P. Torres, L. Torner, D. S. Hum, and M. M. Fejer, Phys. Rev. Lett. 100, 183601 (2008).
[CrossRef]

G. Molina-Terriza, J. P. Torres, and L. Torner, Phys. Rev. Lett. 88, 013601 (2001).
[CrossRef]

Tur, M.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, Nat. Photonics 6, 488 (2012).
[CrossRef]

Vasnetsov, M.

Veselago, V. G.

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

Wang, J.

A. E. Willner, J. Wang, and H. Huang, Science 337, 655 (2012).
[CrossRef]

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, Nat. Photonics 6, 488 (2012).
[CrossRef]

Wang, Y.

J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, Science 321, 930 (2008).
[CrossRef]

Wegener, M.

C. M. Soukoulis and M. Wegener, Nat. Photonics 5, 523 (2011).
[CrossRef]

Weinreich, G.

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, Phys. Rev. Lett. 7, 118 (1961).
[CrossRef]

Wen, S.

H. Luo, Z. Ren, W. Shu, and S. Wen, Phys. Rev. A 77, 023812 (2008).
[CrossRef]

Willner, A. E.

A. E. Willner, J. Wang, and H. Huang, Science 337, 655 (2012).
[CrossRef]

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, Nat. Photonics 6, 488 (2012).
[CrossRef]

Wiltshire, M. C. K.

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, Science 305, 788 (2004).
[CrossRef]

Yan, Y.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, Nat. Photonics 6, 488 (2012).
[CrossRef]

Yang, J.-Y.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, Nat. Photonics 6, 488 (2012).
[CrossRef]

Yao, J.

J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, Science 321, 930 (2008).
[CrossRef]

Yue, Y.

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Figures (3)

Fig. 1.
Fig. 1.

SHG with optical vortices in NIMs. (a) Phase-matching geometry for the FF and generated SH waves and (b) helical wavefront for the FF and SH for an incident Laguerre–Gaussian beam with topological charge 1. Arrows show the direction of rotation of the wavefront for the FF in PIM (top left) and in NIM (top right), for the SH (bottom).

Fig. 2.
Fig. 2.

Profiles of FF and generated SH wave. Three cases were studied: (a) collimated, (b) divergent, or (c) convergent incident Laguerre–Gaussian beam propagating from linear PIM into nonlinear NIM. Absolute values of FF and SH field in cross section obtained (d)–(f) analytically and (g)–(i) numerically when depletion of incident field is negligible and assuming that each plot normalized to its own maximum.

Fig. 3.
Fig. 3.

Analytically and numerically calculated energy conversion efficiency of FF to the SH field for lossless (green dashed and blue solid lines) and lossy (red dash-dotted line) cases as a function of incident field intensity in logarithmic scale. Inset shows the comparison between numerical and analytical results at low intensities (linear scale).

Equations (5)

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122E1+ik1E1z+ik1α12E1+4πω12μ1χ(2)c2E1*E2=0,
122E2+ik2E2z+ik2α22E2+2πω22μ2χ(2)c2E12=0,
ELG(m)=E0w0w(rw)|m|er2w2+i[(|m|+1)arctan(z/LR)+kr2/(2R)+mϕ+kz],
dθdz=m1k1r2=m2k2r2,
A(z)=gln(1iz2/zR1iz/zR),

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