Abstract

The light propagation properties of a slab that consists of periodic multilayers of two materials with different refractive indices are studied. Optical transmittance properties against various wavelengths are calculated by using an orthodox method of a characteristic matrix of a stratified medium. By using a finite-difference-time-domain method, light propagation properties of the slab, especially in the near-field, are simulated. Generally speaking, the presence of the slab promotes light propagation. Extraordinary light propagation, including splitting into two branches, can be observed repeatedly, and in some specific cases a focusing effect can be realized. Realization of negative refraction is checked.

© 2004 Optical Society of America

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References

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  1. V. G. Veselago, “The electrodynamics of substances with simultaneous negative values of ε and μ,” Sov. Phys. Usp. 10, 509–514 (1968).
    [CrossRef]
  2. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
    [CrossRef] [PubMed]
  3. J. B. Pendry, “Negative refractive makes a perfect lens,” Phys. Rev. Lett. 30, 3966–3969 (2000).
    [CrossRef]
  4. M. G. Moharam and T. K. Gayload, “Rigorous coupled-wave analysis of planar grating diffraction,” J. Opt. Soc. Am. 71, 811–818 (1981).
    [CrossRef]
  5. M. Notomi, “Theory of light propagation in strongly-modulated photonic crystals: Refractionlike behavior in the vicinity of photonic band gap,” Phys. Rev. B 62, 10696–10705 (2000).
    [CrossRef]
  6. C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “Subwavelength imaging in photonic crystals,” Phys. Rev. B 68, 045115 (2003).
    [CrossRef]
  7. For example, M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon Oxford, UK, 1970), pp. 55–59.

2003 (1)

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “Subwavelength imaging in photonic crystals,” Phys. Rev. B 68, 045115 (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 (2)

J. B. Pendry, “Negative refractive makes a perfect lens,” Phys. Rev. Lett. 30, 3966–3969 (2000).
[CrossRef]

M. Notomi, “Theory of light propagation in strongly-modulated photonic crystals: Refractionlike behavior in the vicinity of photonic band gap,” Phys. Rev. B 62, 10696–10705 (2000).
[CrossRef]

1981 (1)

1968 (1)

V. G. Veselago, “The electrodynamics of substances with simultaneous negative values of ε and μ,” Sov. Phys. Usp. 10, 509–514 (1968).
[CrossRef]

Gayload, T. K.

Joannopoulos, J. D.

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “Subwavelength imaging in photonic crystals,” Phys. Rev. B 68, 045115 (2003).
[CrossRef]

Johnson, S. G.

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “Subwavelength imaging in photonic crystals,” Phys. Rev. B 68, 045115 (2003).
[CrossRef]

Luo, C.

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “Subwavelength imaging in photonic crystals,” Phys. Rev. B 68, 045115 (2003).
[CrossRef]

Moharam, M. G.

Notomi, M.

M. Notomi, “Theory of light propagation in strongly-modulated photonic crystals: Refractionlike behavior in the vicinity of photonic band gap,” Phys. Rev. B 62, 10696–10705 (2000).
[CrossRef]

Pendry, J. B.

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “Subwavelength imaging in photonic crystals,” Phys. Rev. B 68, 045115 (2003).
[CrossRef]

J. B. Pendry, “Negative refractive makes a perfect lens,” Phys. Rev. Lett. 30, 3966–3969 (2000).
[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]

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]

Smith, D. R.

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

Veselago, V. G.

V. G. Veselago, “The electrodynamics of substances with simultaneous negative values of ε and μ,” Sov. Phys. Usp. 10, 509–514 (1968).
[CrossRef]

J. Opt. Soc. Am. (1)

Phys. Rev. B (2)

M. Notomi, “Theory of light propagation in strongly-modulated photonic crystals: Refractionlike behavior in the vicinity of photonic band gap,” Phys. Rev. B 62, 10696–10705 (2000).
[CrossRef]

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “Subwavelength imaging in photonic crystals,” Phys. Rev. B 68, 045115 (2003).
[CrossRef]

Phys. Rev. Lett. (1)

J. B. Pendry, “Negative refractive makes a perfect lens,” Phys. Rev. Lett. 30, 3966–3969 (2000).
[CrossRef]

Science (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]

Sov. Phys. Usp. (1)

V. G. Veselago, “The electrodynamics of substances with simultaneous negative values of ε and μ,” Sov. Phys. Usp. 10, 509–514 (1968).
[CrossRef]

Other (1)

For example, M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon Oxford, UK, 1970), pp. 55–59.

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Figures (15)

Fig. 1
Fig. 1

From the Kramers–Kronig equation, material with a sharp band edge should have a refractive index dispersion curve with a very sharp downward spike at the shorter wavelengths from the lower band edge. The bottom of the spike can be expected to cross the zero level into the negative area.

Fig. 2
Fig. 2

Explanation of the imaging property of the Lippmann–Denisyuk-type volume hologram. A hologram that records interference fringes by two light sources placed mutually at opposite sides of the hologram is considered and illuminated by one of the light sources. Reflected light by Bragg reflection forms a virtual image, and, at the shifted conditions from the Bragg reflection, light toward the opposite direction of the Bragg reflection can be diffracted and form a real image.

Fig. 3
Fig. 3

Illustration of a periodic multilayer with two components. n2 is defined as the higher refractive index of the two components, n1 the lower. For realism, we assume ZnS as the high-refractive-index material of n2=2.35 and MgF2 as the low-refractive-index-material of n1=1.38.

Fig. 4
Fig. 4

Optical transmittance properties of the periodic multilayer of two materials are plotted against various wavelengths. The profile of the periodic unit is defined by n2d2:n1d1 and n2:n1 values, with n2:n1=1.703 and d1, d2 the thicknesses of the two materials. (a) n2d2:n1d1=10.0. (b) n2d2:n1d1=1.0; the bandwidth becomes very wide and even-numbered high-order bands disappear. (c) n2d2:n1d1=0.1. In any case, sharp band edges can be obtained from a periodic multilayer.

Fig. 5
Fig. 5

Position and width of the lowest band in Fig. 4 are plotted as the wavelengths of the upper and lower band edges. By changing n2d2:n1d1 of the periodic unit materials, the position and width of the band are shifted. If the periodic unit approaches a sinusoidal profile, the bandwidth becomes narrowest and higher bands will disappear. The outermost curve, upper and lower, has n2:n1=2.0. The inner remaining curves follow the same order of n2:n1 as in the legend.

Fig. 6
Fig. 6

Example of the focusing effect of a periodic multilayer-type slab lens. A multilayer of 12 periods (cycles) is used. A silver pinhole of 10-nm diameter is placed in front of a slab lens (bottom of this image) at a spacing of 150 nm. The white bars show the wavelength of the light λ and all images, including those in figures following display an area of 1410 nm by 3010 nm. (a) Thicknesses of the periodic unit materials are d1=5 nm (MgF2) and d2=45 nm (ZnS); then n2d2:n1d1=15.33, the average refractive index is 2.253, and the corresponding wavelength λ0 is 225.3 nm. Wavelength of the light is 208.6 nm(=0.926λ0). These conditions of the calculation are represented by point a in Fig. 5. (b) d1=45 nm and d2=5 nm; then n2d2:n1d1=0.189, the average refractive index is 1.477, and the corresponding wavelength λ0 is 147.7 nm; wavelength of the light is 126.0 nm(=0.853λ0). The conditions of the calculation are represented by point b in Fig. 5. (c) The case without a periodic multilayer.

Fig. 7
Fig. 7

Effects of change in wavelength. (a) The conditions of and the center image are the same as for Fig. 6(a). (b) The conditions of and the center image are the same as for Fig. 6(b).

Fig. 8
Fig. 8

Effects of the spacing between the pinhole and slab. The periodic multilayer and wavelength are the same as those for (a) Fig. 6(a), (b) Fig. 6(b).

Fig. 9
Fig. 9

Effects of number of periods (cycles). The periodic multilayer and wavelength are the same as those for (a) Fig. 6(a), (b) Fig. 6(b).

Fig. 10
Fig. 10

Effects of number of periods. The periodic multilayer is the same as that for Fig. 9(a), but the spacing of pinhole and slab is 20 nm, and the wavelength of the light is 221.1 nm(=0.981λ0).

Fig. 11
Fig. 11

(a) Effects of the wavelength; the spacing between pinhole and slab is 20 nm. (b) Effects of the spacing of pinhole and slab; the wavelength of the light is 221.1 nm(=0.981λ0). In both cases n2d2:n1d1=15.33 and the number of periods (cycles) is four.

Fig. 12
Fig. 12

(a) Effects of the wavelength; the spacing of the pinhole and slab is 20 nm. (b) Effects of the spacing of pinhole and slab; the wavelength of the light is 133.7 nm(=0.905λ0). In both cases n2d2:n1d1=0.189 and the number of periods (cycles) is four.

Fig. 13
Fig. 13

Extraordinary light propagation in which the light splits into two branches, right and left. In this case wavelength is 240.9 nm(=1.069λ0) and the other conditions are the same as those for Fig. 6(a).

Fig. 14
Fig. 14

Frequency of the lower band edge ωe against transverse wave vector k for various values of n2d2:n1d1. The refractive indices are fixed as n1=1.38 and n2=2.35. The transverse wave vector k and the structure of the periodic multilayer are shown in the inset.

Fig. 15
Fig. 15

Light propagation of tilted incident light. The incident light is assumed to have a Gaussian profile and inclination of 1/10 to the normal. Other conditions are the same as those for Fig. 6(a). To emphasize the refractive angle, the horizontal scale is doubled. Dotted curve indicates the line of ridge plots. It is confirmed that light refracts to the same side as the incident light.

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