Abstract

We study the second-harmonic generation (SHG) in left-handed metamaterials with a quadratic nonlinear response. We demonstrate a novel type of the exact phase matching between the backward-propagating wave of the fundamental frequency and the forward-propagating wave of the second harmonic (SH). We show that this novel parametric process can convert a surface of the left-handed metamaterial into an effective mirror that totally reflects the SH generated by an incident wave. We derive and analyze theoretically the coupled-mode equations for a semi-infinite nonlinear metamaterial. We also numerically study the SHG by a metamaterial slab of a finite thickness and reveal the existence of multistable nonlinear effects.

© 2006 Optical Society of America

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  1. V. G. Veselago, "Properties of materials having simultaneously negative values of dielectric (xi) and magnetic (mu) susceptibilities," Sov. Phys. Solid State 8, 2854-2856 (1967).
  2. V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of epsilon and mu," Usp. Fiz. Nauk 92, 517-526 (1967) (in Russian) [Phys. Usp. 10, 509-514].
  3. D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
    [CrossRef] [PubMed]
  4. D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, "Metamaterials and negative refractive index," Science 305, 788-792 (2004).
    [CrossRef] [PubMed]
  5. R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001).
    [CrossRef] [PubMed]
  6. A. A. Zharov, I. V. Shadrivov, and Yu. S. Kivshar, "Nonlinear properties of left-handed metamaterials," Phys. Rev. Lett. 91, 037401-4 (2003).
    [CrossRef] [PubMed]
  7. N. A. Zharova, I. V. Shadrivov, A. A. Zharov, and Yu. S. Kivshar, "Nonlinear transmission and spatiotemporal solitons in metamaterials with negative refraction," Opt. Express 13, 1291-1298 (2005).
    [CrossRef] [PubMed]
  8. M. Lapine, M. Gorkunov, and K. H. Ringhofer, "Nonlinearity of a metamaterial arising from diode insertions into resonant conductive elements," Phys. Rev. E 67, 065601-4 (2003).
    [CrossRef]
  9. V. M. Agranovich, Y. R. Shen, R. H. Baughman, and A. A. Zakhidov, "Linear and nonlinear wave propagation in negative refraction metamaterials," Phys. Rev. B 69, 165112-165117 (2004).
    [CrossRef]
  10. M. Lapine and M. Gorkunov, "Three-wave coupling of microwaves in metamaterial with nonlinear resonant conductive elements," Phys. Rev. E 70, 66601-8 (2004).
    [CrossRef]
  11. A. A. Zharov, "Parametric interaction of surface electromagnetic waves at the edge of a plasma-like medium with a thin transition layer," Fiz. Plazmy 17, 20-27 (1991).
  12. A. A. Zharov, N. A. Zharova, I. V. Shadrivov, and Yu. S. Kivshar, "Subwavelength imaging with opaque nonlinear left-handed lenses," Appl. Phys. Lett. 87, 091104-3 (2005).
    [CrossRef]
  13. L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, 1963).

2005 (2)

A. A. Zharov, N. A. Zharova, I. V. Shadrivov, and Yu. S. Kivshar, "Subwavelength imaging with opaque nonlinear left-handed lenses," Appl. Phys. Lett. 87, 091104-3 (2005).
[CrossRef]

N. A. Zharova, I. V. Shadrivov, A. A. Zharov, and Yu. S. Kivshar, "Nonlinear transmission and spatiotemporal solitons in metamaterials with negative refraction," Opt. Express 13, 1291-1298 (2005).
[CrossRef] [PubMed]

2004 (3)

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, "Metamaterials and negative refractive index," Science 305, 788-792 (2004).
[CrossRef] [PubMed]

V. M. Agranovich, Y. R. Shen, R. H. Baughman, and A. A. Zakhidov, "Linear and nonlinear wave propagation in negative refraction metamaterials," Phys. Rev. B 69, 165112-165117 (2004).
[CrossRef]

M. Lapine and M. Gorkunov, "Three-wave coupling of microwaves in metamaterial with nonlinear resonant conductive elements," Phys. Rev. E 70, 66601-8 (2004).
[CrossRef]

2003 (2)

A. A. Zharov, I. V. Shadrivov, and Yu. S. Kivshar, "Nonlinear properties of left-handed metamaterials," Phys. Rev. Lett. 91, 037401-4 (2003).
[CrossRef] [PubMed]

M. Lapine, M. Gorkunov, and K. H. Ringhofer, "Nonlinearity of a metamaterial arising from diode insertions into resonant conductive elements," Phys. Rev. E 67, 065601-4 (2003).
[CrossRef]

2001 (1)

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001).
[CrossRef] [PubMed]

2000 (1)

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

1991 (1)

A. A. Zharov, "Parametric interaction of surface electromagnetic waves at the edge of a plasma-like medium with a thin transition layer," Fiz. Plazmy 17, 20-27 (1991).

1967 (2)

V. G. Veselago, "Properties of materials having simultaneously negative values of dielectric (xi) and magnetic (mu) susceptibilities," Sov. Phys. Solid State 8, 2854-2856 (1967).

V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of epsilon and mu," Usp. Fiz. Nauk 92, 517-526 (1967) (in Russian) [Phys. Usp. 10, 509-514].

Agranovich, V. M.

V. M. Agranovich, Y. R. Shen, R. H. Baughman, and A. A. Zakhidov, "Linear and nonlinear wave propagation in negative refraction metamaterials," Phys. Rev. B 69, 165112-165117 (2004).
[CrossRef]

Baughman, R. H.

V. M. Agranovich, Y. R. Shen, R. H. Baughman, and A. A. Zakhidov, "Linear and nonlinear wave propagation in negative refraction metamaterials," Phys. Rev. B 69, 165112-165117 (2004).
[CrossRef]

Gorkunov, M.

M. Lapine and M. Gorkunov, "Three-wave coupling of microwaves in metamaterial with nonlinear resonant conductive elements," Phys. Rev. E 70, 66601-8 (2004).
[CrossRef]

M. Lapine, M. Gorkunov, and K. H. Ringhofer, "Nonlinearity of a metamaterial arising from diode insertions into resonant conductive elements," Phys. Rev. E 67, 065601-4 (2003).
[CrossRef]

Kivshar, Yu. S.

N. A. Zharova, I. V. Shadrivov, A. A. Zharov, and Yu. S. Kivshar, "Nonlinear transmission and spatiotemporal solitons in metamaterials with negative refraction," Opt. Express 13, 1291-1298 (2005).
[CrossRef] [PubMed]

A. A. Zharov, N. A. Zharova, I. V. Shadrivov, and Yu. S. Kivshar, "Subwavelength imaging with opaque nonlinear left-handed lenses," Appl. Phys. Lett. 87, 091104-3 (2005).
[CrossRef]

A. A. Zharov, I. V. Shadrivov, and Yu. S. Kivshar, "Nonlinear properties of left-handed metamaterials," Phys. Rev. Lett. 91, 037401-4 (2003).
[CrossRef] [PubMed]

Landau, L. D.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, 1963).

Lapine, M.

M. Lapine and M. Gorkunov, "Three-wave coupling of microwaves in metamaterial with nonlinear resonant conductive elements," Phys. Rev. E 70, 66601-8 (2004).
[CrossRef]

M. Lapine, M. Gorkunov, and K. H. Ringhofer, "Nonlinearity of a metamaterial arising from diode insertions into resonant conductive elements," Phys. Rev. E 67, 065601-4 (2003).
[CrossRef]

Lifshitz, E. M.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, 1963).

Nasser, S. C.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

Padilla, W. J.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

Pendry, J. B.

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, "Metamaterials and negative refractive index," Science 305, 788-792 (2004).
[CrossRef] [PubMed]

Ringhofer, K. H.

M. Lapine, M. Gorkunov, and K. H. Ringhofer, "Nonlinearity of a metamaterial arising from diode insertions into resonant conductive elements," Phys. Rev. E 67, 065601-4 (2003).
[CrossRef]

Schultz, S.

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001).
[CrossRef] [PubMed]

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

Shadrivov, I. V.

A. A. Zharov, N. A. Zharova, I. V. Shadrivov, and Yu. S. Kivshar, "Subwavelength imaging with opaque nonlinear left-handed lenses," Appl. Phys. Lett. 87, 091104-3 (2005).
[CrossRef]

N. A. Zharova, I. V. Shadrivov, A. A. Zharov, and Yu. S. Kivshar, "Nonlinear transmission and spatiotemporal solitons in metamaterials with negative refraction," Opt. Express 13, 1291-1298 (2005).
[CrossRef] [PubMed]

A. A. Zharov, I. V. Shadrivov, and Yu. S. Kivshar, "Nonlinear properties of left-handed metamaterials," Phys. Rev. Lett. 91, 037401-4 (2003).
[CrossRef] [PubMed]

Shelby, R. A.

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001).
[CrossRef] [PubMed]

Shen, Y. R.

V. M. Agranovich, Y. R. Shen, R. H. Baughman, and A. A. Zakhidov, "Linear and nonlinear wave propagation in negative refraction metamaterials," Phys. Rev. B 69, 165112-165117 (2004).
[CrossRef]

Smith, D. R.

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, "Metamaterials and negative refractive index," Science 305, 788-792 (2004).
[CrossRef] [PubMed]

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001).
[CrossRef] [PubMed]

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

Veselago, V. G.

V. G. Veselago, "Properties of materials having simultaneously negative values of dielectric (xi) and magnetic (mu) susceptibilities," Sov. Phys. Solid State 8, 2854-2856 (1967).

V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of epsilon and mu," Usp. Fiz. Nauk 92, 517-526 (1967) (in Russian) [Phys. Usp. 10, 509-514].

Vier, D. C.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

Wiltshire, M. C.

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, "Metamaterials and negative refractive index," Science 305, 788-792 (2004).
[CrossRef] [PubMed]

Zakhidov, A. A.

V. M. Agranovich, Y. R. Shen, R. H. Baughman, and A. A. Zakhidov, "Linear and nonlinear wave propagation in negative refraction metamaterials," Phys. Rev. B 69, 165112-165117 (2004).
[CrossRef]

Zharov, A. A.

N. A. Zharova, I. V. Shadrivov, A. A. Zharov, and Yu. S. Kivshar, "Nonlinear transmission and spatiotemporal solitons in metamaterials with negative refraction," Opt. Express 13, 1291-1298 (2005).
[CrossRef] [PubMed]

A. A. Zharov, N. A. Zharova, I. V. Shadrivov, and Yu. S. Kivshar, "Subwavelength imaging with opaque nonlinear left-handed lenses," Appl. Phys. Lett. 87, 091104-3 (2005).
[CrossRef]

A. A. Zharov, I. V. Shadrivov, and Yu. S. Kivshar, "Nonlinear properties of left-handed metamaterials," Phys. Rev. Lett. 91, 037401-4 (2003).
[CrossRef] [PubMed]

A. A. Zharov, "Parametric interaction of surface electromagnetic waves at the edge of a plasma-like medium with a thin transition layer," Fiz. Plazmy 17, 20-27 (1991).

Zharova, N. A.

N. A. Zharova, I. V. Shadrivov, A. A. Zharov, and Yu. S. Kivshar, "Nonlinear transmission and spatiotemporal solitons in metamaterials with negative refraction," Opt. Express 13, 1291-1298 (2005).
[CrossRef] [PubMed]

A. A. Zharov, N. A. Zharova, I. V. Shadrivov, and Yu. S. Kivshar, "Subwavelength imaging with opaque nonlinear left-handed lenses," Appl. Phys. Lett. 87, 091104-3 (2005).
[CrossRef]

Appl. Phys. Lett. (1)

A. A. Zharov, N. A. Zharova, I. V. Shadrivov, and Yu. S. Kivshar, "Subwavelength imaging with opaque nonlinear left-handed lenses," Appl. Phys. Lett. 87, 091104-3 (2005).
[CrossRef]

Fiz. Plazmy (1)

A. A. Zharov, "Parametric interaction of surface electromagnetic waves at the edge of a plasma-like medium with a thin transition layer," Fiz. Plazmy 17, 20-27 (1991).

Opt. Express (1)

Phys. Rev. B (1)

V. M. Agranovich, Y. R. Shen, R. H. Baughman, and A. A. Zakhidov, "Linear and nonlinear wave propagation in negative refraction metamaterials," Phys. Rev. B 69, 165112-165117 (2004).
[CrossRef]

Phys. Rev. E (2)

M. Lapine and M. Gorkunov, "Three-wave coupling of microwaves in metamaterial with nonlinear resonant conductive elements," Phys. Rev. E 70, 66601-8 (2004).
[CrossRef]

M. Lapine, M. Gorkunov, and K. H. Ringhofer, "Nonlinearity of a metamaterial arising from diode insertions into resonant conductive elements," Phys. Rev. E 67, 065601-4 (2003).
[CrossRef]

Phys. Rev. Lett. (2)

A. A. Zharov, I. V. Shadrivov, and Yu. S. Kivshar, "Nonlinear properties of left-handed metamaterials," Phys. Rev. Lett. 91, 037401-4 (2003).
[CrossRef] [PubMed]

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

Science (2)

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, "Metamaterials and negative refractive index," Science 305, 788-792 (2004).
[CrossRef] [PubMed]

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001).
[CrossRef] [PubMed]

Sov. Phys. Solid State (1)

V. G. Veselago, "Properties of materials having simultaneously negative values of dielectric (xi) and magnetic (mu) susceptibilities," Sov. Phys. Solid State 8, 2854-2856 (1967).

Usp. Fiz. Nauk (1)

V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of epsilon and mu," Usp. Fiz. Nauk 92, 517-526 (1967) (in Russian) [Phys. Usp. 10, 509-514].

Other (1)

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, 1963).

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Figures (6)

Fig. 1
Fig. 1

(Color online) Frequency-dependent magnetic permeability μ (solid curve) and electric permittivity ϵ (dashed curve) of the composite. Two types of the regions (LHM or RHM) where the material is transparent are shaded; for other frequencies, it is opaque. Characteristic frequencies ω 0 , ω M , and ω p are defined in Eqs. (1, 2, 3). Inset shows the unit cell of the metamaterial.

Fig. 2
Fig. 2

(Color online) Dispersion of plane waves k ( ω ) in the metamaterial. Arrows indicate the parameters of the FF and SH waves corresponding to the exact spatiotemporal phase matching.

Fig. 3
Fig. 3

(Color online) Geometry of the SHG problem. Thick arrows show the direction of the energy flow; thin arrows show the direction of the wavevectors. Indices i, r, and tr stand for incident, reflected, and transmitted waves, respectively.

Fig. 4
Fig. 4

(Color online) Geometry of the SHG process for a finite-width slab of a nonlinear quadratic metamaterial (shaded).

Fig. 5
Fig. 5

(Color online) Reflection coefficients of the generated SH wave (solid curve) and transmitted FF (dotted curve), as well as the transmission coefficient of the FF wave (dashed curve) versus the normalized field intensity. Normalized slab thickness is L = 10 .

Fig. 6
Fig. 6

(Color online) Transmission coefficient of the FF wave (dashed curve) and reflection coefficient of the SH wave (solid curve) versus the slab thickness L for a fixed amplitude of the transmitted wave, H 1 ( L ) = 10 2 . The right plot shows a blow up of the region depicted by a dashed box.

Equations (30)

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ϵ ( ω ) = 1 ω p 2 ω 2 ,
μ ( ω ) = 1 + F ω 2 ( ω 0 2 ω 2 ) ,
ω 0 < ω < min { ω p , ω M } , ω M = ω 0 1 F ,
δ = ( k 2 ϵ μ ω 2 c 2 ) 1 2 < λ 17 ,
D ( ω , k ) = [ k 2 ϵ ( ω ) μ ( ω ) ω 2 c 2 ] = 0 .
I = U R d ( 1 + U U c ) ,
Δ H 1 + ϵ ( ω ) μ ( ω ) ω 2 c 2 H 1 = σ 1 H 1 * H 2 ,
Δ H 2 + 4 ϵ ( 2 ω ) μ ( 2 ω ) ω 2 c 2 H 2 = σ 2 H 1 2 ,
σ 1 = κ 2 R ( ω ) , σ 2 = κ R * ( ω ) ,
κ = 6 π ( π a 2 ) 3 d 3 c 5 [ ω 0 4 ω 2 U c R d R ( ω ) R ( 2 ω ) ] ,
H 1 , 2 ( t , z ) = a 1 , 2 ( t , z ) exp ( i k 1 , 2 z ) + c . c . ,
a 1 t + v g 1 a 1 z = i σ 1 a 1 * a 2 ν 1 a 1 ,
a 2 t + v g 2 a 2 z = i σ 2 a 1 2 + ν 2 a 2 i Ω a 2 ,
Ω = q 2 D ( 2 ω , 2 k ) 2 σ 2 ,
q 1 = σ 1 [ D ( ω , k ) ω ] 1 , q 2 = σ 2 [ D ( 2 ω , 2 k ) ω ] 1 .
b 1 t + v g 1 b 1 z = q 1 b 1 * b 2 ν 1 b 1 ,
b 2 t + v g 2 b 2 z = q 2 b 1 2 + ν 2 b 2 i Ω b 2 ,
b 2 = i q 2 Ω b 1 2 .
b 1 t + v g 1 b 1 z = i q 1 q 2 Ω b 1 2 b 1 .
b 1 = A ( t , z ) exp [ i Φ ( t , z ) ] ,
b 1 t + v g 1 b 1 z = q 1 b 1 * b 2 , b 2 t v g 2 b 2 z = q 2 b 1 2 .
v g 1 q 1 b 1 2 v g 2 q 2 b 2 2 = C .
b 1 = ( v g 1 v g 2 q 1 q 2 ) 1 2 1 ( z + z 0 ) , b 2 = v g 1 q 1 1 ( z + z 0 ) ,
b 1 ( t , z ) = A 1 sech [ ( t z ζ ) T ] , b 2 ( t , z ) = A 2 tanh [ ( t z ζ ) T ] ,
A 1 = [ ( v g 2 + ζ ) ( v g 1 ζ ) ζ 2 T 2 q 1 q 2 ] 1 2 , A 2 = ( v g 1 ζ ) ζ T q 1 ,
v g 2 v g 1 [ q 1 b 2 2 + 2 ν 1 b 2 ] = q 2 b 1 2 .
b 1 ( z ) = ν 1 ( v g 2 q 1 q 2 v g 1 ) 1 2 csch [ ν 1 v g 1 ( z + z 0 ) ] ,
b 2 ( z ) = 2 ν 1 q 1 ( exp { 2 ν 1 [ ( z + z 0 ) v g 1 ] } 1 ) 1 .
d 2 H 1 d z 2 + [ ϵ ( ω ) μ ( ω ) k x 2 ] H 1 = H 2 H 1 * ,
d 2 H 2 d z 2 + 4 [ ϵ ( 2 ω ) μ ( 2 ω ) k x 2 ] H 2 = Q H 1 2 ,

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