Abstract

An analytic solution for a magnetic line source imaged through a uniaxially anisotropic slab is derived for electric fields, and example results are supported by finite element simulations. The analytical result provides a convenient means to evaluate the effect of material parameters in applications. We introduce a spectrometer that can be realized by illuminating a slab composed of a Ag/ZnO multilayer stack through a small aperture, and we use this example to evaluate the performance of the analytic model.

© 2010 Optical Society of America

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  1. S. A. Ramakrishna, J. B. Pendry, M. C. K. Wiltshire, and W. J. Stewart, J. Mod. Opt. 50, 1419 (2003).
  2. B. Wood, J. B. Pendry, and D. P. Tsai, Phys. Rev. B 74, 115116 (2006).
    [CrossRef]
  3. H. Liu, Shivanand, and K. J. Webb, Opt. Lett. 33, 2568 (2008).
    [PubMed]
  4. A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, Nat. Mater. 6, 946 (2007).
    [CrossRef] [PubMed]
  5. H. Liu, Shivanand, and K. J. Webb, Phys. Rev. B 79, 094203 (2009).
    [CrossRef]
  6. D. R. Smith and D. Schurig, Phys. Rev. Lett. 90, 077405 (2003).
    [CrossRef] [PubMed]
  7. A. Ishimaru, Electromagnetic Wave Propagation, Radiation and Scattering (Prentice-Hall, 1991).
  8. COMSOL: Products, http://www.comsol.com/products/.
  9. J. K. H. Wong, K. G. Balmain, and G. V. Eleftheriades, IEEE Trans. Antennas Propag. 54, 2742 (2006).
    [CrossRef]
  10. E.D.Palik, ed., Handbook of Optical Constants of Solids(Academic, 1998).
  11. H. Yoshikawa and S. Adachi, Jpn. J. Appl. Phys. 36, 6237 (1997).
    [CrossRef]

2009

H. Liu, Shivanand, and K. J. Webb, Phys. Rev. B 79, 094203 (2009).
[CrossRef]

2008

2007

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, Nat. Mater. 6, 946 (2007).
[CrossRef] [PubMed]

2006

B. Wood, J. B. Pendry, and D. P. Tsai, Phys. Rev. B 74, 115116 (2006).
[CrossRef]

J. K. H. Wong, K. G. Balmain, and G. V. Eleftheriades, IEEE Trans. Antennas Propag. 54, 2742 (2006).
[CrossRef]

2003

D. R. Smith and D. Schurig, Phys. Rev. Lett. 90, 077405 (2003).
[CrossRef] [PubMed]

S. A. Ramakrishna, J. B. Pendry, M. C. K. Wiltshire, and W. J. Stewart, J. Mod. Opt. 50, 1419 (2003).

1997

H. Yoshikawa and S. Adachi, Jpn. J. Appl. Phys. 36, 6237 (1997).
[CrossRef]

Adachi, S.

H. Yoshikawa and S. Adachi, Jpn. J. Appl. Phys. 36, 6237 (1997).
[CrossRef]

Alekseyev, L.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, Nat. Mater. 6, 946 (2007).
[CrossRef] [PubMed]

Balmain, K. G.

J. K. H. Wong, K. G. Balmain, and G. V. Eleftheriades, IEEE Trans. Antennas Propag. 54, 2742 (2006).
[CrossRef]

Eleftheriades, G. V.

J. K. H. Wong, K. G. Balmain, and G. V. Eleftheriades, IEEE Trans. Antennas Propag. 54, 2742 (2006).
[CrossRef]

Franz, K. J.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, Nat. Mater. 6, 946 (2007).
[CrossRef] [PubMed]

Gmachl, C.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, Nat. Mater. 6, 946 (2007).
[CrossRef] [PubMed]

Hoffman, A. J.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, Nat. Mater. 6, 946 (2007).
[CrossRef] [PubMed]

Howard, S. S.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, Nat. Mater. 6, 946 (2007).
[CrossRef] [PubMed]

Ishimaru, A.

A. Ishimaru, Electromagnetic Wave Propagation, Radiation and Scattering (Prentice-Hall, 1991).

Liu, H.

H. Liu, Shivanand, and K. J. Webb, Phys. Rev. B 79, 094203 (2009).
[CrossRef]

H. Liu, Shivanand, and K. J. Webb, Opt. Lett. 33, 2568 (2008).
[PubMed]

Narimanov, E. E.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, Nat. Mater. 6, 946 (2007).
[CrossRef] [PubMed]

Pendry, J. B.

B. Wood, J. B. Pendry, and D. P. Tsai, Phys. Rev. B 74, 115116 (2006).
[CrossRef]

S. A. Ramakrishna, J. B. Pendry, M. C. K. Wiltshire, and W. J. Stewart, J. Mod. Opt. 50, 1419 (2003).

Podolskiy, V. A.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, Nat. Mater. 6, 946 (2007).
[CrossRef] [PubMed]

Ramakrishna, S. A.

S. A. Ramakrishna, J. B. Pendry, M. C. K. Wiltshire, and W. J. Stewart, J. Mod. Opt. 50, 1419 (2003).

Schurig, D.

D. R. Smith and D. Schurig, Phys. Rev. Lett. 90, 077405 (2003).
[CrossRef] [PubMed]

Shivanand,

H. Liu, Shivanand, and K. J. Webb, Phys. Rev. B 79, 094203 (2009).
[CrossRef]

H. Liu, Shivanand, and K. J. Webb, Opt. Lett. 33, 2568 (2008).
[PubMed]

Sivco, D. L.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, Nat. Mater. 6, 946 (2007).
[CrossRef] [PubMed]

Smith, D. R.

D. R. Smith and D. Schurig, Phys. Rev. Lett. 90, 077405 (2003).
[CrossRef] [PubMed]

Stewart, W. J.

S. A. Ramakrishna, J. B. Pendry, M. C. K. Wiltshire, and W. J. Stewart, J. Mod. Opt. 50, 1419 (2003).

Tsai, D. P.

B. Wood, J. B. Pendry, and D. P. Tsai, Phys. Rev. B 74, 115116 (2006).
[CrossRef]

Wasserman, D.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, Nat. Mater. 6, 946 (2007).
[CrossRef] [PubMed]

Webb, K. J.

H. Liu, Shivanand, and K. J. Webb, Phys. Rev. B 79, 094203 (2009).
[CrossRef]

H. Liu, Shivanand, and K. J. Webb, Opt. Lett. 33, 2568 (2008).
[PubMed]

Wiltshire, M. C. K.

S. A. Ramakrishna, J. B. Pendry, M. C. K. Wiltshire, and W. J. Stewart, J. Mod. Opt. 50, 1419 (2003).

Wong, J. K. H.

J. K. H. Wong, K. G. Balmain, and G. V. Eleftheriades, IEEE Trans. Antennas Propag. 54, 2742 (2006).
[CrossRef]

Wood, B.

B. Wood, J. B. Pendry, and D. P. Tsai, Phys. Rev. B 74, 115116 (2006).
[CrossRef]

Yoshikawa, H.

H. Yoshikawa and S. Adachi, Jpn. J. Appl. Phys. 36, 6237 (1997).
[CrossRef]

IEEE Trans. Antennas Propag.

J. K. H. Wong, K. G. Balmain, and G. V. Eleftheriades, IEEE Trans. Antennas Propag. 54, 2742 (2006).
[CrossRef]

J. Mod. Opt.

S. A. Ramakrishna, J. B. Pendry, M. C. K. Wiltshire, and W. J. Stewart, J. Mod. Opt. 50, 1419 (2003).

Jpn. J. Appl. Phys.

H. Yoshikawa and S. Adachi, Jpn. J. Appl. Phys. 36, 6237 (1997).
[CrossRef]

Nat. Mater.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, Nat. Mater. 6, 946 (2007).
[CrossRef] [PubMed]

Opt. Lett.

Phys. Rev. B

H. Liu, Shivanand, and K. J. Webb, Phys. Rev. B 79, 094203 (2009).
[CrossRef]

B. Wood, J. B. Pendry, and D. P. Tsai, Phys. Rev. B 74, 115116 (2006).
[CrossRef]

Phys. Rev. Lett.

D. R. Smith and D. Schurig, Phys. Rev. Lett. 90, 077405 (2003).
[CrossRef] [PubMed]

Other

A. Ishimaru, Electromagnetic Wave Propagation, Radiation and Scattering (Prentice-Hall, 1991).

COMSOL: Products, http://www.comsol.com/products/.

E.D.Palik, ed., Handbook of Optical Constants of Solids(Academic, 1998).

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Figures (5)

Fig. 1
Fig. 1

(a) Imaging a magnetic line source through an anisotropic medium. (b) 2D bulk anisotropic slab from a metal– insulator stack.

Fig. 2
Fig. 2

Contour integration in the complex k x plane. The integral along Re ( k x ) is denoted C. C b denotes the branch cut, which is drawn along Im ( k z 0 ) = 0 from the branch point, k x = k 0 , by assuming a small amount of loss in free space. C r denotes the semicircle at infinity. The transmission poles k x n shown were calculated with ϵ x = 1.0 + i 0.3 , ϵ z = 5.0 + i 0.1 , and λ = 650 nm .

Fig. 3
Fig. 3

(a) Comparison between analytical and numerical results for | E ( z = d ) | with z = 0 , for a homogenized Ag/ZnO multilayer stack, using published material data [10, 11], with D = 0.5 and d = 100 nm . (b) Comparison of field amplitudes at λ = 746.8 nm between the COMSOL simulation, the analytical expression in Eq. (7), a direct numerical integration of Eqs. (2, 3), and numerical integration of Eqs. (2, 3) over the subwavelength regime | k x | 6 k 0 .

Fig. 4
Fig. 4

| E ( z = d ) | , with z = 0 and different sets of material parameters. (a) Effect of κ due to material absorption. ϵ x 0 = 9.95 + i 0.735 and ϵ z 0 = 9.09 + i 0.215 are bulk parameters at wavelength λ 0 = 746.8 nm , with other parameters as given in Fig. 3. (b) Effect of κ . κ 0 is the Ag/ZnO bulk parameter case at wavelength λ 0 = 746.8 nm . ϵ x = 39.1 + i 6.73 and ϵ z = ϵ z 0 give κ = 0.5 κ 0 , κ = κ 0 . κ = 1.5 κ 0 , κ = κ 0 correspond to ϵ x = 4.43 + i 0.18 and ϵ z = ϵ z 0 .

Fig. 5
Fig. 5

Variation of the (a) position and (b) FWHM of the principal peak as a function of wavelength for a homogenized Ag/ZnO stack slab, with D = 0.5 . The source plane and image plane are located at z = 0 and z = d , respectively.

Equations (8)

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H y = ω ϵ 0 4 π T ( k x ) exp ( i k z 0 z D + i k x x ) k z 0 d k x ,
E x = 1 4 π T ( k x ) exp ( i k z 0 z D + i k x x ) d k x ,
E z = 1 4 π k x T ( k x ) exp ( i k z 0 z D + i k x x ) k z 0 d k x .
C = C b + 2 π i n Res [ Integrand , k x n ] ,
k x n n κ + Δ , κ = π ϵ z / ϵ x / d , Δ = 1 d ϵ z ϵ x arctan ( 2 ϵ x ϵ z 1 + ϵ x ϵ z ) + π d ϵ z ϵ x ,
E x i ϵ z ( 1 + ϵ x ϵ z ) d exp [ i Δ ( | x | + i z D ) ] 1 + exp ( i κ | x | ) i E z .
| E | exp ( Δ | x | ) 1 + 2 exp ( κ x ) cos ( κ x ) + exp ( 2 κ x ) ,
δ x π / κ [ 1 exp ( κ x ) ] 1 1 2 = 2 π κ tanh [ π κ 2 κ ] ,

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