Abstract

An enhanced method is developed for analysis of third-order nonlinearities in optical nanostructures with a scalar magnetic field frequency-domain formulation; it is shown to produce fast and accurate results for 2D problems without a superfluous vector electric field formalism. While a standard TM representation using cubic nonlinear susceptibility results in an intractable implicit equation, our technique alleviates this problem. In addition to a universal approach, simpler, more efficient solutions are proposed for media having solely either a real (lossless Kerr-type medium) or an imaginary (nonlinear absorbing medium) nonlinearity. Combining these solutions with a finite-element method, we show simulation examples validated with alternative approaches.

© 2009 Optical Society of America

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  1. R. L. Fante, J. Appl. Phys. 42, 4202 (1971).
    [Crossref]
  2. O. V. Baghdasaryan and V. A. Permyakov, Radiophys. Quantum Electron. 21, 940 (1978).
    [Crossref]
  3. J. H. Marburger and F. S. Felber, Phys. Rev. A 17, 335 (1978).
    [Crossref]
  4. W. Chen and D. L. Mills, Phys. Rev. B 36, 6269 (1987).
    [Crossref]
  5. U. Langbein, F. Lederer, T. Peschel, U. Trutschel, and D. Mihalache, Phys. Rep. 194, 325 (1990).
    [Crossref]
  6. L. M. Brekhovskikh, Waves in Layered Media (Academic, 1960).
  7. M. Born and E. Wolf, Principles of Optics (MacMillan, 1964).
  8. H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984).
  9. J. Jin, The Finite Element Method in Electromagnetics (Wiley, 1993).
  10. G. Cevini, G. Oliveri, and M. Raffetto, Microwave Opt. Technol. Lett. 48, 2524 (2006).
    [Crossref]
  11. G. Birkhoff and S. Mac Lane, Survey of Modern Algebra, 4th ed. (Macmillan, 1977), pp. 102-103.
  12. See, for example, , where u=2z and Δ=2C.
  13. H. V. Baghdasaryan and T. M. Knyazyan, Opt. Quantum Electron. 31, 1059 (1999).
    [Crossref]
  14. W. Cai, U. K. Chettiar, Y. Hsiao-Kuan, V. C. de Silva, A. V. Kildishev, V. P. Drachev, and V. M. Shalaev, Opt. Express 15, 3333 (2007).
    [Crossref] [PubMed]
  15. S. Xiao, U. K. Chettiar, A. V. Kildishev, V. P. Drachev, I.-C. Khoo, and V. M. Shalaev, Appl. Phys. Lett. 95, 033115 (2009).
    [Crossref]
  16. A. A. Zharov, I. V. Shadrivov, and Y. S. Kivshar, Phys. Rev. Lett. 91, 037401 (2003).
    [Crossref] [PubMed]
  17. S. O'Brien, D. McPeake, S. A. Ramakrishna, and J. B. Pendry, Phys. Rev. B 69, 241101 (2004).
    [Crossref]
  18. A. V. Kildishev, U. K. Chettiar, Z. Liu, V. M. Shalaev, D.-H. Kwon, Z. Bayraktar, and D. H. Werner, J. Opt. Soc. Am. B 24, A34 (2007).
    [Crossref]
  19. Q. Chen, L. Kuang, E. H. Sargent, and Z. Y. Wang, Appl. Phys. Lett. 83, 2115 (2003).
    [Crossref]
  20. S. J. Mathews, S. C. Kumar, L. Giribabu, and S. V. Rao, Mater. Lett. 61, 4426 (2007).
    [Crossref]
  21. A. V. Kildishev and N. M. Litchinitser, “Efficient simulation of non-linear effects in 2D optical nanostructures to TM waves,” Opt. Commun., submitted for publication.
  22. C. Min, P. Wang, X. Jiao, Y. Deng, and H. Ming, Opt. Express 15, 9541 (2009).
    [Crossref]

2009 (2)

S. Xiao, U. K. Chettiar, A. V. Kildishev, V. P. Drachev, I.-C. Khoo, and V. M. Shalaev, Appl. Phys. Lett. 95, 033115 (2009).
[Crossref]

C. Min, P. Wang, X. Jiao, Y. Deng, and H. Ming, Opt. Express 15, 9541 (2009).
[Crossref]

2007 (3)

2006 (1)

G. Cevini, G. Oliveri, and M. Raffetto, Microwave Opt. Technol. Lett. 48, 2524 (2006).
[Crossref]

2004 (1)

S. O'Brien, D. McPeake, S. A. Ramakrishna, and J. B. Pendry, Phys. Rev. B 69, 241101 (2004).
[Crossref]

2003 (2)

Q. Chen, L. Kuang, E. H. Sargent, and Z. Y. Wang, Appl. Phys. Lett. 83, 2115 (2003).
[Crossref]

A. A. Zharov, I. V. Shadrivov, and Y. S. Kivshar, Phys. Rev. Lett. 91, 037401 (2003).
[Crossref] [PubMed]

1999 (1)

H. V. Baghdasaryan and T. M. Knyazyan, Opt. Quantum Electron. 31, 1059 (1999).
[Crossref]

1990 (1)

U. Langbein, F. Lederer, T. Peschel, U. Trutschel, and D. Mihalache, Phys. Rep. 194, 325 (1990).
[Crossref]

1987 (1)

W. Chen and D. L. Mills, Phys. Rev. B 36, 6269 (1987).
[Crossref]

1978 (2)

O. V. Baghdasaryan and V. A. Permyakov, Radiophys. Quantum Electron. 21, 940 (1978).
[Crossref]

J. H. Marburger and F. S. Felber, Phys. Rev. A 17, 335 (1978).
[Crossref]

1971 (1)

R. L. Fante, J. Appl. Phys. 42, 4202 (1971).
[Crossref]

Baghdasaryan, H. V.

H. V. Baghdasaryan and T. M. Knyazyan, Opt. Quantum Electron. 31, 1059 (1999).
[Crossref]

Baghdasaryan, O. V.

O. V. Baghdasaryan and V. A. Permyakov, Radiophys. Quantum Electron. 21, 940 (1978).
[Crossref]

Bayraktar, Z.

Birkhoff, G.

G. Birkhoff and S. Mac Lane, Survey of Modern Algebra, 4th ed. (Macmillan, 1977), pp. 102-103.

Born, M.

M. Born and E. Wolf, Principles of Optics (MacMillan, 1964).

Brekhovskikh, L. M.

L. M. Brekhovskikh, Waves in Layered Media (Academic, 1960).

Cai, W.

Cevini, G.

G. Cevini, G. Oliveri, and M. Raffetto, Microwave Opt. Technol. Lett. 48, 2524 (2006).
[Crossref]

Chen, Q.

Q. Chen, L. Kuang, E. H. Sargent, and Z. Y. Wang, Appl. Phys. Lett. 83, 2115 (2003).
[Crossref]

Chen, W.

W. Chen and D. L. Mills, Phys. Rev. B 36, 6269 (1987).
[Crossref]

Chettiar, U. K.

de Silva, V. C.

Deng, Y.

Drachev, V. P.

S. Xiao, U. K. Chettiar, A. V. Kildishev, V. P. Drachev, I.-C. Khoo, and V. M. Shalaev, Appl. Phys. Lett. 95, 033115 (2009).
[Crossref]

W. Cai, U. K. Chettiar, Y. Hsiao-Kuan, V. C. de Silva, A. V. Kildishev, V. P. Drachev, and V. M. Shalaev, Opt. Express 15, 3333 (2007).
[Crossref] [PubMed]

Fante, R. L.

R. L. Fante, J. Appl. Phys. 42, 4202 (1971).
[Crossref]

Felber, F. S.

J. H. Marburger and F. S. Felber, Phys. Rev. A 17, 335 (1978).
[Crossref]

Giribabu, L.

S. J. Mathews, S. C. Kumar, L. Giribabu, and S. V. Rao, Mater. Lett. 61, 4426 (2007).
[Crossref]

Haus, H. A.

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984).

Hsiao-Kuan, Y.

Jiao, X.

Jin, J.

J. Jin, The Finite Element Method in Electromagnetics (Wiley, 1993).

Khoo, I.-C.

S. Xiao, U. K. Chettiar, A. V. Kildishev, V. P. Drachev, I.-C. Khoo, and V. M. Shalaev, Appl. Phys. Lett. 95, 033115 (2009).
[Crossref]

Kildishev, A. V.

S. Xiao, U. K. Chettiar, A. V. Kildishev, V. P. Drachev, I.-C. Khoo, and V. M. Shalaev, Appl. Phys. Lett. 95, 033115 (2009).
[Crossref]

A. V. Kildishev, U. K. Chettiar, Z. Liu, V. M. Shalaev, D.-H. Kwon, Z. Bayraktar, and D. H. Werner, J. Opt. Soc. Am. B 24, A34 (2007).
[Crossref]

W. Cai, U. K. Chettiar, Y. Hsiao-Kuan, V. C. de Silva, A. V. Kildishev, V. P. Drachev, and V. M. Shalaev, Opt. Express 15, 3333 (2007).
[Crossref] [PubMed]

A. V. Kildishev and N. M. Litchinitser, “Efficient simulation of non-linear effects in 2D optical nanostructures to TM waves,” Opt. Commun., submitted for publication.

Kivshar, Y. S.

A. A. Zharov, I. V. Shadrivov, and Y. S. Kivshar, Phys. Rev. Lett. 91, 037401 (2003).
[Crossref] [PubMed]

Knyazyan, T. M.

H. V. Baghdasaryan and T. M. Knyazyan, Opt. Quantum Electron. 31, 1059 (1999).
[Crossref]

Kuang, L.

Q. Chen, L. Kuang, E. H. Sargent, and Z. Y. Wang, Appl. Phys. Lett. 83, 2115 (2003).
[Crossref]

Kumar, S. C.

S. J. Mathews, S. C. Kumar, L. Giribabu, and S. V. Rao, Mater. Lett. 61, 4426 (2007).
[Crossref]

Kwon, D.-H.

Langbein, U.

U. Langbein, F. Lederer, T. Peschel, U. Trutschel, and D. Mihalache, Phys. Rep. 194, 325 (1990).
[Crossref]

Lederer, F.

U. Langbein, F. Lederer, T. Peschel, U. Trutschel, and D. Mihalache, Phys. Rep. 194, 325 (1990).
[Crossref]

Litchinitser, N. M.

A. V. Kildishev and N. M. Litchinitser, “Efficient simulation of non-linear effects in 2D optical nanostructures to TM waves,” Opt. Commun., submitted for publication.

Liu, Z.

Mac Lane, S.

G. Birkhoff and S. Mac Lane, Survey of Modern Algebra, 4th ed. (Macmillan, 1977), pp. 102-103.

Marburger, J. H.

J. H. Marburger and F. S. Felber, Phys. Rev. A 17, 335 (1978).
[Crossref]

Mathews, S. J.

S. J. Mathews, S. C. Kumar, L. Giribabu, and S. V. Rao, Mater. Lett. 61, 4426 (2007).
[Crossref]

McPeake, D.

S. O'Brien, D. McPeake, S. A. Ramakrishna, and J. B. Pendry, Phys. Rev. B 69, 241101 (2004).
[Crossref]

Mihalache, D.

U. Langbein, F. Lederer, T. Peschel, U. Trutschel, and D. Mihalache, Phys. Rep. 194, 325 (1990).
[Crossref]

Mills, D. L.

W. Chen and D. L. Mills, Phys. Rev. B 36, 6269 (1987).
[Crossref]

Min, C.

Ming, H.

O'Brien, S.

S. O'Brien, D. McPeake, S. A. Ramakrishna, and J. B. Pendry, Phys. Rev. B 69, 241101 (2004).
[Crossref]

Oliveri, G.

G. Cevini, G. Oliveri, and M. Raffetto, Microwave Opt. Technol. Lett. 48, 2524 (2006).
[Crossref]

Pendry, J. B.

S. O'Brien, D. McPeake, S. A. Ramakrishna, and J. B. Pendry, Phys. Rev. B 69, 241101 (2004).
[Crossref]

Permyakov, V. A.

O. V. Baghdasaryan and V. A. Permyakov, Radiophys. Quantum Electron. 21, 940 (1978).
[Crossref]

Peschel, T.

U. Langbein, F. Lederer, T. Peschel, U. Trutschel, and D. Mihalache, Phys. Rep. 194, 325 (1990).
[Crossref]

Raffetto, M.

G. Cevini, G. Oliveri, and M. Raffetto, Microwave Opt. Technol. Lett. 48, 2524 (2006).
[Crossref]

Ramakrishna, S. A.

S. O'Brien, D. McPeake, S. A. Ramakrishna, and J. B. Pendry, Phys. Rev. B 69, 241101 (2004).
[Crossref]

Rao, S. V.

S. J. Mathews, S. C. Kumar, L. Giribabu, and S. V. Rao, Mater. Lett. 61, 4426 (2007).
[Crossref]

Sargent, E. H.

Q. Chen, L. Kuang, E. H. Sargent, and Z. Y. Wang, Appl. Phys. Lett. 83, 2115 (2003).
[Crossref]

Shadrivov, I. V.

A. A. Zharov, I. V. Shadrivov, and Y. S. Kivshar, Phys. Rev. Lett. 91, 037401 (2003).
[Crossref] [PubMed]

Shalaev, V. M.

Trutschel, U.

U. Langbein, F. Lederer, T. Peschel, U. Trutschel, and D. Mihalache, Phys. Rep. 194, 325 (1990).
[Crossref]

Wang, P.

Wang, Z. Y.

Q. Chen, L. Kuang, E. H. Sargent, and Z. Y. Wang, Appl. Phys. Lett. 83, 2115 (2003).
[Crossref]

Werner, D. H.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (MacMillan, 1964).

Xiao, S.

S. Xiao, U. K. Chettiar, A. V. Kildishev, V. P. Drachev, I.-C. Khoo, and V. M. Shalaev, Appl. Phys. Lett. 95, 033115 (2009).
[Crossref]

Zharov, A. A.

A. A. Zharov, I. V. Shadrivov, and Y. S. Kivshar, Phys. Rev. Lett. 91, 037401 (2003).
[Crossref] [PubMed]

Appl. Phys. Lett. (2)

S. Xiao, U. K. Chettiar, A. V. Kildishev, V. P. Drachev, I.-C. Khoo, and V. M. Shalaev, Appl. Phys. Lett. 95, 033115 (2009).
[Crossref]

Q. Chen, L. Kuang, E. H. Sargent, and Z. Y. Wang, Appl. Phys. Lett. 83, 2115 (2003).
[Crossref]

J. Appl. Phys. (1)

R. L. Fante, J. Appl. Phys. 42, 4202 (1971).
[Crossref]

J. Opt. Soc. Am. B (1)

Mater. Lett. (1)

S. J. Mathews, S. C. Kumar, L. Giribabu, and S. V. Rao, Mater. Lett. 61, 4426 (2007).
[Crossref]

Microwave Opt. Technol. Lett. (1)

G. Cevini, G. Oliveri, and M. Raffetto, Microwave Opt. Technol. Lett. 48, 2524 (2006).
[Crossref]

Opt. Express (2)

Opt. Quantum Electron. (1)

H. V. Baghdasaryan and T. M. Knyazyan, Opt. Quantum Electron. 31, 1059 (1999).
[Crossref]

Phys. Rep. (1)

U. Langbein, F. Lederer, T. Peschel, U. Trutschel, and D. Mihalache, Phys. Rep. 194, 325 (1990).
[Crossref]

Phys. Rev. A (1)

J. H. Marburger and F. S. Felber, Phys. Rev. A 17, 335 (1978).
[Crossref]

Phys. Rev. B (2)

W. Chen and D. L. Mills, Phys. Rev. B 36, 6269 (1987).
[Crossref]

S. O'Brien, D. McPeake, S. A. Ramakrishna, and J. B. Pendry, Phys. Rev. B 69, 241101 (2004).
[Crossref]

Phys. Rev. Lett. (1)

A. A. Zharov, I. V. Shadrivov, and Y. S. Kivshar, Phys. Rev. Lett. 91, 037401 (2003).
[Crossref] [PubMed]

Radiophys. Quantum Electron. (1)

O. V. Baghdasaryan and V. A. Permyakov, Radiophys. Quantum Electron. 21, 940 (1978).
[Crossref]

Other (7)

L. M. Brekhovskikh, Waves in Layered Media (Academic, 1960).

M. Born and E. Wolf, Principles of Optics (MacMillan, 1964).

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984).

J. Jin, The Finite Element Method in Electromagnetics (Wiley, 1993).

G. Birkhoff and S. Mac Lane, Survey of Modern Algebra, 4th ed. (Macmillan, 1977), pp. 102-103.

See, for example, , where u=2z and Δ=2C.

A. V. Kildishev and N. M. Litchinitser, “Efficient simulation of non-linear effects in 2D optical nanostructures to TM waves,” Opt. Commun., submitted for publication.

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Figures (1)

Fig. 1
Fig. 1

Simulation results for coupled nanostrip pairs covered with a n 0 = 1.53 Kerr material. (a) Transmission T 0 , 2 and reflection R 0 , 2 spectra for the linear ( T 0 , R 0 , solid) and NL ( T 2 , R 2 , dashed, n 2 I inc = 0.0013 ) cases. (b) Same as (a); enlargement of the resonance range. (c) Average enhancement ρ 2 over the spacer boundary.

Equations (5)

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

ε r = ε l + 3 χ ( 3 ) | E | 2 = ε l ( 1 + 4 n 2 η 1 | E | 2 ) .
E = ι ( ω ε 0 ε r ) 1 h × z ̂ ,
ε r = ε l + f e ι φ | ε r | 2 ,
ε r = v + 2 v cosh ( 1 3 cosh 1 Δ 2 ) = ( 1 + a + a 1 ) v ,
ε r = ε l + ι 2 v sinh ( 1 3 sinh 1 Δ 2 ) = ε l + ι ( b b 1 ) v .

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