Abstract

A thin film comprising parallel tilted nanorods was deposited by directing silver vapor obliquely towards a plane substrate. The reflection and transmission coefficients of the thin film were measured at three wavelengths in the visible regime for normal-illumination conditions, using ellipsometry and walk-off interferometry. The thin film was found to display a negative real refractive index. Since vapor deposition is a well-established industrial technique to deposit thin films, this finding is promising for large-scale production of negatively refracting metamaterials.

© 2009 Optical Society of America

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References

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  1. D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, "Metamaterials and negative refractive index," Science 305, 788-792 (2004).
    [CrossRef] [PubMed]
  2. V. M. Shalaev, "Optical negative-index metamaterials," Nature Photon. 1, 41-48 (2007).
    [CrossRef]
  3. R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001).
    [CrossRef] [PubMed]
  4. V. M. Shalaev, W. Cai, U. K. Chettiar, H.-K. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, "Negative index of refraction in optical metamaterials," Opt. Lett. 30,3356-3358 (2005).
    [CrossRef]
  5. S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, "Experimental demonstration of near-infrared negative-index metamaterials," Phys. Rev. Lett. 95,137404 (2005).
    [CrossRef] [PubMed]
  6. J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, "Optical negative refraction in bulk metamaterials of nanowires," Science 321, 930 (2008).
    [CrossRef] [PubMed]
  7. D. M. Mattox, The Foundations of Vacuum Coating Technology (Noyes Publications, Norwich, NY, USA, 2003).
  8. H. A. Macleod, Thin-Film Optical Filters (Institute of Physics, Bristol, United Kingdom, 2001), Chap. 9.
    [CrossRef]
  9. P. W. Baumeister, Optical Coating Technology (SPIE, Bellingham, WA, USA, 2004), Chap. 9.
    [CrossRef]
  10. H. van Kranenburg and C. Lodder, "Tailoring growth and local composition by oblique-incidence deposition: a review and new experimental data," Mater. Sci. Eng. R 11,295-354 (1994).
    [CrossRef]
  11. H. König and G. Helwig, "Über die Struktur schräg aufgedampfter Schichten und ihr Einfluß auf die Entwicklung submikroskopischer Oberflächenrauhigkeiten," Optik 6, 111-124 (1950).
  12. R. Messier, "The nano-world of thin films," J. Nanophoton. 2, 021995 (2008).
    [CrossRef]
  13. A. Kundt, "Ueber die electromagnetische Drehung der Polarisationsebene des Lichtes im Eisen," Ann. Phys. Chem. Lpz. 27, 191-202 (1886).
    [CrossRef]
  14. J. F. Nye, Physical Properties of Crystals (Clarendon Press, Oxford, United Kingdom, 1985).
  15. Y.-P. Zhao, S. B. Chaney, and Z.-Y. Zhang, "Absorbance spectra of aligned Ag nanorod arrays prepared by oblique angle deposition," J. Appl. Phys. 100, 063527 (2007).
    [CrossRef]
  16. A. Lakhtakia and R. Messier, Sculptured Thin Films: Nanoengineered Morphology and Optics (SPIE Press, Bellingham, WA, USA, 2004), Chaps. 2 and 7.
  17. D. R. Smith and S. Schultz, "Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients," Phys. Rev. B 65, 195104 (2002).
    [CrossRef]
  18. P. Markos and C. M. Soukoulis, "Transmission properties and effective electromagnetic parameters of double negative metamaterials," Opt. Express 11, 649-661 (2003).
    [CrossRef] [PubMed]
  19. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (Elsevier, Amsterdam, The Netherlands, 1977).
  20. A. V. Kildishev, W. Cai, U. K. Chettiar, H.-K. Yuan, A. K. Sarychev, V. P. Drachev, and V. M. Shalaev, "Negative refractive index in optics of metal-dielectric composites," J. Opt. Soc. Am. B 23, 423-433 (2006).
    [CrossRef]
  21. J. B. Pendry, "Extremely low frequency plasmons in metallic mesostructures," Phys. Rev. Lett. 76, 4773-4776 (1996).
    [CrossRef] [PubMed]
  22. J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Low frequency plasmons in thin-wire structures," J. Phys.: Condens. Matter 10, 4758-4809 (1998).
    [CrossRef]
  23. R. A. Depine and A. Lakhtakia, "A new condition to identify isotropic dielectric-magnetic materials displaying negative phase velocity," Microwave Opt. Technol. Lett. 41, 315-317 (2004).
    [CrossRef]
  24. S. A. Ramakrishna and J. B. Pendry, "Removal of absorption and increase in resolution in a near-field lens via optical gain," Phys. Rev. B 67, 201101 (2003).
    [CrossRef]

2008 (2)

J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, "Optical negative refraction in bulk metamaterials of nanowires," Science 321, 930 (2008).
[CrossRef] [PubMed]

R. Messier, "The nano-world of thin films," J. Nanophoton. 2, 021995 (2008).
[CrossRef]

2007 (2)

Y.-P. Zhao, S. B. Chaney, and Z.-Y. Zhang, "Absorbance spectra of aligned Ag nanorod arrays prepared by oblique angle deposition," J. Appl. Phys. 100, 063527 (2007).
[CrossRef]

V. M. Shalaev, "Optical negative-index metamaterials," Nature Photon. 1, 41-48 (2007).
[CrossRef]

2006 (1)

2005 (2)

V. M. Shalaev, W. Cai, U. K. Chettiar, H.-K. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, "Negative index of refraction in optical metamaterials," Opt. Lett. 30,3356-3358 (2005).
[CrossRef]

S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, "Experimental demonstration of near-infrared negative-index metamaterials," Phys. Rev. Lett. 95,137404 (2005).
[CrossRef] [PubMed]

2004 (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. Depine and A. Lakhtakia, "A new condition to identify isotropic dielectric-magnetic materials displaying negative phase velocity," Microwave Opt. Technol. Lett. 41, 315-317 (2004).
[CrossRef]

2003 (2)

S. A. Ramakrishna and J. B. Pendry, "Removal of absorption and increase in resolution in a near-field lens via optical gain," Phys. Rev. B 67, 201101 (2003).
[CrossRef]

P. Markos and C. M. Soukoulis, "Transmission properties and effective electromagnetic parameters of double negative metamaterials," Opt. Express 11, 649-661 (2003).
[CrossRef] [PubMed]

2002 (1)

D. R. Smith and S. Schultz, "Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients," Phys. Rev. B 65, 195104 (2002).
[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]

1998 (1)

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Low frequency plasmons in thin-wire structures," J. Phys.: Condens. Matter 10, 4758-4809 (1998).
[CrossRef]

1996 (1)

J. B. Pendry, "Extremely low frequency plasmons in metallic mesostructures," Phys. Rev. Lett. 76, 4773-4776 (1996).
[CrossRef] [PubMed]

1994 (1)

H. van Kranenburg and C. Lodder, "Tailoring growth and local composition by oblique-incidence deposition: a review and new experimental data," Mater. Sci. Eng. R 11,295-354 (1994).
[CrossRef]

1950 (1)

H. König and G. Helwig, "Über die Struktur schräg aufgedampfter Schichten und ihr Einfluß auf die Entwicklung submikroskopischer Oberflächenrauhigkeiten," Optik 6, 111-124 (1950).

1886 (1)

A. Kundt, "Ueber die electromagnetische Drehung der Polarisationsebene des Lichtes im Eisen," Ann. Phys. Chem. Lpz. 27, 191-202 (1886).
[CrossRef]

Bartal, G.

J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, "Optical negative refraction in bulk metamaterials of nanowires," Science 321, 930 (2008).
[CrossRef] [PubMed]

Brueck, S. R. J.

S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, "Experimental demonstration of near-infrared negative-index metamaterials," Phys. Rev. Lett. 95,137404 (2005).
[CrossRef] [PubMed]

Cai, W.

Chaney, S. B.

Y.-P. Zhao, S. B. Chaney, and Z.-Y. Zhang, "Absorbance spectra of aligned Ag nanorod arrays prepared by oblique angle deposition," J. Appl. Phys. 100, 063527 (2007).
[CrossRef]

Chettiar, U. K.

Depine, R. A.

R. A. Depine and A. Lakhtakia, "A new condition to identify isotropic dielectric-magnetic materials displaying negative phase velocity," Microwave Opt. Technol. Lett. 41, 315-317 (2004).
[CrossRef]

Drachev, V. P.

Fan, W.

S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, "Experimental demonstration of near-infrared negative-index metamaterials," Phys. Rev. Lett. 95,137404 (2005).
[CrossRef] [PubMed]

Helwig, G.

H. König and G. Helwig, "Über die Struktur schräg aufgedampfter Schichten und ihr Einfluß auf die Entwicklung submikroskopischer Oberflächenrauhigkeiten," Optik 6, 111-124 (1950).

Holden, A. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Low frequency plasmons in thin-wire structures," J. Phys.: Condens. Matter 10, 4758-4809 (1998).
[CrossRef]

Kildishev, A. V.

König, H.

H. König and G. Helwig, "Über die Struktur schräg aufgedampfter Schichten und ihr Einfluß auf die Entwicklung submikroskopischer Oberflächenrauhigkeiten," Optik 6, 111-124 (1950).

Kundt, A.

A. Kundt, "Ueber die electromagnetische Drehung der Polarisationsebene des Lichtes im Eisen," Ann. Phys. Chem. Lpz. 27, 191-202 (1886).
[CrossRef]

Lakhtakia, A.

R. A. Depine and A. Lakhtakia, "A new condition to identify isotropic dielectric-magnetic materials displaying negative phase velocity," Microwave Opt. Technol. Lett. 41, 315-317 (2004).
[CrossRef]

Liu, Y.

J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, "Optical negative refraction in bulk metamaterials of nanowires," Science 321, 930 (2008).
[CrossRef] [PubMed]

Liu, Z.

J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, "Optical negative refraction in bulk metamaterials of nanowires," Science 321, 930 (2008).
[CrossRef] [PubMed]

Lodder, C.

H. van Kranenburg and C. Lodder, "Tailoring growth and local composition by oblique-incidence deposition: a review and new experimental data," Mater. Sci. Eng. R 11,295-354 (1994).
[CrossRef]

Malloy, K. J.

S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, "Experimental demonstration of near-infrared negative-index metamaterials," Phys. Rev. Lett. 95,137404 (2005).
[CrossRef] [PubMed]

Markos, P.

Messier, R.

R. Messier, "The nano-world of thin films," J. Nanophoton. 2, 021995 (2008).
[CrossRef]

Osgood, R. M.

S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, "Experimental demonstration of near-infrared negative-index metamaterials," Phys. Rev. Lett. 95,137404 (2005).
[CrossRef] [PubMed]

Panoiu, N. C.

S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, "Experimental demonstration of near-infrared negative-index metamaterials," Phys. Rev. Lett. 95,137404 (2005).
[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]

S. A. Ramakrishna and J. B. Pendry, "Removal of absorption and increase in resolution in a near-field lens via optical gain," Phys. Rev. B 67, 201101 (2003).
[CrossRef]

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Low frequency plasmons in thin-wire structures," J. Phys.: Condens. Matter 10, 4758-4809 (1998).
[CrossRef]

J. B. Pendry, "Extremely low frequency plasmons in metallic mesostructures," Phys. Rev. Lett. 76, 4773-4776 (1996).
[CrossRef] [PubMed]

Ramakrishna, S. A.

S. A. Ramakrishna and J. B. Pendry, "Removal of absorption and increase in resolution in a near-field lens via optical gain," Phys. Rev. B 67, 201101 (2003).
[CrossRef]

Robbins, D. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Low frequency plasmons in thin-wire structures," J. Phys.: Condens. Matter 10, 4758-4809 (1998).
[CrossRef]

Sarychev, A. K.

Schultz, S.

D. R. Smith and S. Schultz, "Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients," Phys. Rev. B 65, 195104 (2002).
[CrossRef]

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

Shalaev, V. M.

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.

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

D. R. Smith and S. Schultz, "Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients," Phys. Rev. B 65, 195104 (2002).
[CrossRef]

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

Soukoulis, C. M.

Stacy, A. M.

J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, "Optical negative refraction in bulk metamaterials of nanowires," Science 321, 930 (2008).
[CrossRef] [PubMed]

Stewart, W. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Low frequency plasmons in thin-wire structures," J. Phys.: Condens. Matter 10, 4758-4809 (1998).
[CrossRef]

Sun, C.

J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, "Optical negative refraction in bulk metamaterials of nanowires," Science 321, 930 (2008).
[CrossRef] [PubMed]

van Kranenburg, H.

H. van Kranenburg and C. Lodder, "Tailoring growth and local composition by oblique-incidence deposition: a review and new experimental data," Mater. Sci. Eng. R 11,295-354 (1994).
[CrossRef]

Wang, Y.

J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, "Optical negative refraction in bulk metamaterials of nanowires," Science 321, 930 (2008).
[CrossRef] [PubMed]

Wiltshire, M. C. K.

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

Yao, J.

J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, "Optical negative refraction in bulk metamaterials of nanowires," Science 321, 930 (2008).
[CrossRef] [PubMed]

Yuan, H.-K.

Zhang, S.

S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, "Experimental demonstration of near-infrared negative-index metamaterials," Phys. Rev. Lett. 95,137404 (2005).
[CrossRef] [PubMed]

Zhang, X.

J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, "Optical negative refraction in bulk metamaterials of nanowires," Science 321, 930 (2008).
[CrossRef] [PubMed]

Zhang, Z.-Y.

Y.-P. Zhao, S. B. Chaney, and Z.-Y. Zhang, "Absorbance spectra of aligned Ag nanorod arrays prepared by oblique angle deposition," J. Appl. Phys. 100, 063527 (2007).
[CrossRef]

Zhao, Y.-P.

Y.-P. Zhao, S. B. Chaney, and Z.-Y. Zhang, "Absorbance spectra of aligned Ag nanorod arrays prepared by oblique angle deposition," J. Appl. Phys. 100, 063527 (2007).
[CrossRef]

Ann. Phys. Chem. Lpz. (1)

A. Kundt, "Ueber die electromagnetische Drehung der Polarisationsebene des Lichtes im Eisen," Ann. Phys. Chem. Lpz. 27, 191-202 (1886).
[CrossRef]

J. Appl. Phys. (1)

Y.-P. Zhao, S. B. Chaney, and Z.-Y. Zhang, "Absorbance spectra of aligned Ag nanorod arrays prepared by oblique angle deposition," J. Appl. Phys. 100, 063527 (2007).
[CrossRef]

J. Nanophoton. (1)

R. Messier, "The nano-world of thin films," J. Nanophoton. 2, 021995 (2008).
[CrossRef]

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

J. Phys.: Condens. Matter (1)

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Low frequency plasmons in thin-wire structures," J. Phys.: Condens. Matter 10, 4758-4809 (1998).
[CrossRef]

Mater. Sci. Eng. R (1)

H. van Kranenburg and C. Lodder, "Tailoring growth and local composition by oblique-incidence deposition: a review and new experimental data," Mater. Sci. Eng. R 11,295-354 (1994).
[CrossRef]

Microwave Opt. Technol. Lett. (1)

R. A. Depine and A. Lakhtakia, "A new condition to identify isotropic dielectric-magnetic materials displaying negative phase velocity," Microwave Opt. Technol. Lett. 41, 315-317 (2004).
[CrossRef]

Nature Photon. (1)

V. M. Shalaev, "Optical negative-index metamaterials," Nature Photon. 1, 41-48 (2007).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Optik (1)

H. König and G. Helwig, "Über die Struktur schräg aufgedampfter Schichten und ihr Einfluß auf die Entwicklung submikroskopischer Oberflächenrauhigkeiten," Optik 6, 111-124 (1950).

Phys. Rev. B (2)

D. R. Smith and S. Schultz, "Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients," Phys. Rev. B 65, 195104 (2002).
[CrossRef]

S. A. Ramakrishna and J. B. Pendry, "Removal of absorption and increase in resolution in a near-field lens via optical gain," Phys. Rev. B 67, 201101 (2003).
[CrossRef]

Phys. Rev. Lett. (2)

J. B. Pendry, "Extremely low frequency plasmons in metallic mesostructures," Phys. Rev. Lett. 76, 4773-4776 (1996).
[CrossRef] [PubMed]

S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, "Experimental demonstration of near-infrared negative-index metamaterials," Phys. Rev. Lett. 95,137404 (2005).
[CrossRef] [PubMed]

Science (3)

J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, "Optical negative refraction in bulk metamaterials of nanowires," Science 321, 930 (2008).
[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, J. B. Pendry, and M. C. K. Wiltshire, "Metamaterials and negative refractive index," Science 305, 788-792 (2004).
[CrossRef] [PubMed]

Other (6)

A. Lakhtakia and R. Messier, Sculptured Thin Films: Nanoengineered Morphology and Optics (SPIE Press, Bellingham, WA, USA, 2004), Chaps. 2 and 7.

D. M. Mattox, The Foundations of Vacuum Coating Technology (Noyes Publications, Norwich, NY, USA, 2003).

H. A. Macleod, Thin-Film Optical Filters (Institute of Physics, Bristol, United Kingdom, 2001), Chap. 9.
[CrossRef]

P. W. Baumeister, Optical Coating Technology (SPIE, Bellingham, WA, USA, 2004), Chap. 9.
[CrossRef]

J. F. Nye, Physical Properties of Crystals (Clarendon Press, Oxford, United Kingdom, 1985).

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (Elsevier, Amsterdam, The Netherlands, 1977).

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Figures (2)

Fig. 1.
Fig. 1.

(a) Schematic indicating the normal illumination by linearly polarized light of a silver thin film comprising parallel tilted nanorods. The electric field of the illuminating light either has a component parallel to the nanorods (E⃗p ).or is perpendicular to the nanorods (E⃗s ). (b) Two scanning-electron-microscopic (SEM) images of the thin film.

Fig. 2.
Fig. 2.

Measured spectra of the transmittances Tp,s = ∣τp,s 2 when the silver thin film is illuminated normally.

Tables (2)

Tables Icon

Table 1. Measured data. The ratio τp / τs was measured using an ellipsometer. The coefficients τp , τs , and rs were measured by walk-off interferometry.

Tables Icon

Table 2. Equivalent refractive indices, relative intrinsic impedances, relative permittivities, and relative permeabilities for a silver thin film comprising parallel tilted nanorods.

Equations (7)

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

ην=±(1+rν)2τν2(1rν)2τν2 , ην>0,
nν=λ2πd cos1 (1rν2+τν22τν) .
ε=[sinβ0cosβ010cosβ0sinβ][ε1000ε2000ε3][sinβ0cosβ010cosβ0sinβ],
μ=[sinβ0cosβ010cosβ0sinβ][μ1000μ2000μ3][sinβ0sinβ010cosβ0sinβ].
dExdz=μ0μpHy(i)ppolarizationstatedHydz=ε0εpEyEz=Exε1ε32ε1ε3εpsin2β } ,
dEydz=μ0μsHx(i)spolarizationstatedHxdz=ε0εsEyHz=Hxμ1μ32μ1μ3μssin2β } .
μp=μ2,εp=ε1ε3ε1cos2β+ε3sin2β,μs=μ1μ2μ1cos2β+μ3sin2β,andεs=ε2.

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