Abstract

The perfectly conducting stacked fishnet metamaterial is studied in this paper. The analysis is based on a combination of the mode matching method together with the generalized eigenvalue problem, and takes into account wave propagation along all three Cartesian axes. The analysis has been developed for a fishnet of square lateral periodicity and for two particular polarizations, namely TE and TM, corresponding to the two most common excitations. The 1D and 2D dispersion characteristics are calculated for both polarizations, showing that the TM waves undergo negative refraction in a narrow frequency band just below Wood’s anomaly, whereas TE polarized waves exhibit ordinary positive refraction. Finally, possible homogenization of the fishnet metamaterial is considered, showing that only for small angles of incidence and in the immediate vicinity of Wood’s anomaly can the fishnet be seen as homogenizable uniaxial medium.

© 2010 Optical Society of America

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References

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  1. S. Zhang,W. Fan, K. J. Malloy, and S. R. J. Brueck, "Near-infrared double negative metamaterials," Opt. Express 13, 4922-4929 (2005).
    [CrossRef] [PubMed]
  2. 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]
  3. G. Dolling, Ch. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, "Low-loss negative-index metamaterial at telecommunication wavelengths," Opt. Lett. 31, 1800-1802 (2005).
    [CrossRef]
  4. M. Beruete, M. Sorolla, and I. Campillo, "Left-handed extraordinary optical transmission through a photonic crystal of subwavelength hole arrays," Opt. Express 14, 5445-5455 (2006).
    [CrossRef] [PubMed]
  5. G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, "Negative-index metamaterial at 780 nm wavelength," Opt. Lett. 32, 53-55 (2007).
    [CrossRef]
  6. M. Beruete, I. Campillo, M. Navarro-Cia, F. Falcone, and M. S. Ayza, "Molding left- or right-handed metamaterials by stacked cutoff metallic hole arrays," IEEE Trans. Antennas Propag. 55, 1514-1521 (2007).
    [CrossRef]
  7. M. Navarro-Cia, M. Beruete, M. Sorolla, and I. Campillo, "Negative refraction in a prism made of stacked subwavelength hole arrays," Opt. Express 16, 560-566 (2008).
    [CrossRef] [PubMed]
  8. J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, "Three-dimensional optical metamaterial with a negative refractive index," Nature (London) 455, 376-379 (2008).
    [CrossRef] [PubMed]
  9. A. Mary, S. G. Rodrigo, F. J. Garcia-Vidal, and L. Martin-Moreno, "Theory of negative-refractive-index response of double-fishnet structures," Phys. Rev. Lett. 101, 103902 (2008).
    [CrossRef] [PubMed]
  10. M. Beruete, M. Navarro-Cia, M. Sorolla, and I. Campillo, "Negative refraction through an extraordinary transmission left-handed metamaterial slab," Phys. Rev. B 79, 195107 (2009).
    [CrossRef]
  11. R. Marques, L. Jelinek, F. Mesa, and F. Medina, "Analytical theory of wave propagation through stacked Fishnet metamaterials," Opt. Express 17, 11582-11593 (2009).
    [CrossRef] [PubMed]
  12. R. Marques, F. Mesa, L. Jelinek, and F. Medina, "Analytical theory of extraordinary transmission through metallic diffraction screens perforated by small holes," Opt. Express 17, 5571-5579 (2009).
    [CrossRef] [PubMed]
  13. C. Menzel, T. Paul, C. Rockstuhl, T. Pertsch, S. Tretyakov, and F. Lederer, "Validity of effective material parameters for optical fishnet metamaterials," Phys. Rev. B 81, 035320 (2010).
    [CrossRef]
  14. M. Beruete, M. Navarro-Cia and M. Sorolla, "Strong lateral displacement in polarization anisotropic extraordinary transmission metamaterial," New J. Phys. 12, 063037 (2010).
    [CrossRef]
  15. C. Garcia-Meca, R. Ortuno, F. J. Rodriguez-Fortuno, J. Marti, and A. Martinez, "Negative refractive index metamaterials aided by extraordinary optical transmission," Opt. Express 17, 6026-6031 (2009).
    [CrossRef] [PubMed]
  16. L. D. Landau, E. M. Lifshitz and L. P. Pitaevskii, Electrodynamics of Continuous Media, 2nd ed. (Pergamon Press, New York, 1984).

2010 (2)

C. Menzel, T. Paul, C. Rockstuhl, T. Pertsch, S. Tretyakov, and F. Lederer, "Validity of effective material parameters for optical fishnet metamaterials," Phys. Rev. B 81, 035320 (2010).
[CrossRef]

M. Beruete, M. Navarro-Cia and M. Sorolla, "Strong lateral displacement in polarization anisotropic extraordinary transmission metamaterial," New J. Phys. 12, 063037 (2010).
[CrossRef]

2009 (4)

2008 (3)

M. Navarro-Cia, M. Beruete, M. Sorolla, and I. Campillo, "Negative refraction in a prism made of stacked subwavelength hole arrays," Opt. Express 16, 560-566 (2008).
[CrossRef] [PubMed]

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, "Three-dimensional optical metamaterial with a negative refractive index," Nature (London) 455, 376-379 (2008).
[CrossRef] [PubMed]

A. Mary, S. G. Rodrigo, F. J. Garcia-Vidal, and L. Martin-Moreno, "Theory of negative-refractive-index response of double-fishnet structures," Phys. Rev. Lett. 101, 103902 (2008).
[CrossRef] [PubMed]

2007 (2)

G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, "Negative-index metamaterial at 780 nm wavelength," Opt. Lett. 32, 53-55 (2007).
[CrossRef]

M. Beruete, I. Campillo, M. Navarro-Cia, F. Falcone, and M. S. Ayza, "Molding left- or right-handed metamaterials by stacked cutoff metallic hole arrays," IEEE Trans. Antennas Propag. 55, 1514-1521 (2007).
[CrossRef]

2006 (1)

2005 (3)

Ayza, M. S.

M. Beruete, I. Campillo, M. Navarro-Cia, F. Falcone, and M. S. Ayza, "Molding left- or right-handed metamaterials by stacked cutoff metallic hole arrays," IEEE Trans. Antennas Propag. 55, 1514-1521 (2007).
[CrossRef]

Beruete, M.

M. Beruete, M. Navarro-Cia and M. Sorolla, "Strong lateral displacement in polarization anisotropic extraordinary transmission metamaterial," New J. Phys. 12, 063037 (2010).
[CrossRef]

M. Beruete, M. Navarro-Cia, M. Sorolla, and I. Campillo, "Negative refraction through an extraordinary transmission left-handed metamaterial slab," Phys. Rev. B 79, 195107 (2009).
[CrossRef]

M. Navarro-Cia, M. Beruete, M. Sorolla, and I. Campillo, "Negative refraction in a prism made of stacked subwavelength hole arrays," Opt. Express 16, 560-566 (2008).
[CrossRef] [PubMed]

M. Beruete, I. Campillo, M. Navarro-Cia, F. Falcone, and M. S. Ayza, "Molding left- or right-handed metamaterials by stacked cutoff metallic hole arrays," IEEE Trans. Antennas Propag. 55, 1514-1521 (2007).
[CrossRef]

M. Beruete, M. Sorolla, and I. Campillo, "Left-handed extraordinary optical transmission through a photonic crystal of subwavelength hole arrays," Opt. Express 14, 5445-5455 (2006).
[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]

S. Zhang,W. Fan, K. J. Malloy, and S. R. J. Brueck, "Near-infrared double negative metamaterials," Opt. Express 13, 4922-4929 (2005).
[CrossRef] [PubMed]

Campillo, I.

M. Beruete, M. Navarro-Cia, M. Sorolla, and I. Campillo, "Negative refraction through an extraordinary transmission left-handed metamaterial slab," Phys. Rev. B 79, 195107 (2009).
[CrossRef]

M. Navarro-Cia, M. Beruete, M. Sorolla, and I. Campillo, "Negative refraction in a prism made of stacked subwavelength hole arrays," Opt. Express 16, 560-566 (2008).
[CrossRef] [PubMed]

M. Beruete, I. Campillo, M. Navarro-Cia, F. Falcone, and M. S. Ayza, "Molding left- or right-handed metamaterials by stacked cutoff metallic hole arrays," IEEE Trans. Antennas Propag. 55, 1514-1521 (2007).
[CrossRef]

M. Beruete, M. Sorolla, and I. Campillo, "Left-handed extraordinary optical transmission through a photonic crystal of subwavelength hole arrays," Opt. Express 14, 5445-5455 (2006).
[CrossRef] [PubMed]

Dolling, G.

Enkrich, Ch.

Falcone, F.

M. Beruete, I. Campillo, M. Navarro-Cia, F. Falcone, and M. S. Ayza, "Molding left- or right-handed metamaterials by stacked cutoff metallic hole arrays," IEEE Trans. Antennas Propag. 55, 1514-1521 (2007).
[CrossRef]

Fan, W.

S. Zhang,W. Fan, K. J. Malloy, and S. R. J. Brueck, "Near-infrared double negative metamaterials," Opt. Express 13, 4922-4929 (2005).
[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]

Garcia-Meca, C.

Garcia-Vidal, F. J.

A. Mary, S. G. Rodrigo, F. J. Garcia-Vidal, and L. Martin-Moreno, "Theory of negative-refractive-index response of double-fishnet structures," Phys. Rev. Lett. 101, 103902 (2008).
[CrossRef] [PubMed]

Jelinek, L.

Lederer, F.

C. Menzel, T. Paul, C. Rockstuhl, T. Pertsch, S. Tretyakov, and F. Lederer, "Validity of effective material parameters for optical fishnet metamaterials," Phys. Rev. B 81, 035320 (2010).
[CrossRef]

Linden, S.

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]

S. Zhang,W. Fan, K. J. Malloy, and S. R. J. Brueck, "Near-infrared double negative metamaterials," Opt. Express 13, 4922-4929 (2005).
[CrossRef] [PubMed]

Marques, R.

Marti, J.

Martinez, A.

Martin-Moreno, L.

A. Mary, S. G. Rodrigo, F. J. Garcia-Vidal, and L. Martin-Moreno, "Theory of negative-refractive-index response of double-fishnet structures," Phys. Rev. Lett. 101, 103902 (2008).
[CrossRef] [PubMed]

Mary, A.

A. Mary, S. G. Rodrigo, F. J. Garcia-Vidal, and L. Martin-Moreno, "Theory of negative-refractive-index response of double-fishnet structures," Phys. Rev. Lett. 101, 103902 (2008).
[CrossRef] [PubMed]

Medina, F.

Menzel, C.

C. Menzel, T. Paul, C. Rockstuhl, T. Pertsch, S. Tretyakov, and F. Lederer, "Validity of effective material parameters for optical fishnet metamaterials," Phys. Rev. B 81, 035320 (2010).
[CrossRef]

Mesa, F.

Navarro-Cia, M.

M. Beruete, M. Navarro-Cia and M. Sorolla, "Strong lateral displacement in polarization anisotropic extraordinary transmission metamaterial," New J. Phys. 12, 063037 (2010).
[CrossRef]

M. Beruete, M. Navarro-Cia, M. Sorolla, and I. Campillo, "Negative refraction through an extraordinary transmission left-handed metamaterial slab," Phys. Rev. B 79, 195107 (2009).
[CrossRef]

M. Navarro-Cia, M. Beruete, M. Sorolla, and I. Campillo, "Negative refraction in a prism made of stacked subwavelength hole arrays," Opt. Express 16, 560-566 (2008).
[CrossRef] [PubMed]

M. Beruete, I. Campillo, M. Navarro-Cia, F. Falcone, and M. S. Ayza, "Molding left- or right-handed metamaterials by stacked cutoff metallic hole arrays," IEEE Trans. Antennas Propag. 55, 1514-1521 (2007).
[CrossRef]

Ortuno, R.

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]

Paul, T.

C. Menzel, T. Paul, C. Rockstuhl, T. Pertsch, S. Tretyakov, and F. Lederer, "Validity of effective material parameters for optical fishnet metamaterials," Phys. Rev. B 81, 035320 (2010).
[CrossRef]

Pertsch, T.

C. Menzel, T. Paul, C. Rockstuhl, T. Pertsch, S. Tretyakov, and F. Lederer, "Validity of effective material parameters for optical fishnet metamaterials," Phys. Rev. B 81, 035320 (2010).
[CrossRef]

Rockstuhl, C.

C. Menzel, T. Paul, C. Rockstuhl, T. Pertsch, S. Tretyakov, and F. Lederer, "Validity of effective material parameters for optical fishnet metamaterials," Phys. Rev. B 81, 035320 (2010).
[CrossRef]

Rodrigo, S. G.

A. Mary, S. G. Rodrigo, F. J. Garcia-Vidal, and L. Martin-Moreno, "Theory of negative-refractive-index response of double-fishnet structures," Phys. Rev. Lett. 101, 103902 (2008).
[CrossRef] [PubMed]

Rodriguez-Fortuno, F. J.

Sorolla, M.

M. Beruete, M. Navarro-Cia and M. Sorolla, "Strong lateral displacement in polarization anisotropic extraordinary transmission metamaterial," New J. Phys. 12, 063037 (2010).
[CrossRef]

M. Beruete, M. Navarro-Cia, M. Sorolla, and I. Campillo, "Negative refraction through an extraordinary transmission left-handed metamaterial slab," Phys. Rev. B 79, 195107 (2009).
[CrossRef]

M. Navarro-Cia, M. Beruete, M. Sorolla, and I. Campillo, "Negative refraction in a prism made of stacked subwavelength hole arrays," Opt. Express 16, 560-566 (2008).
[CrossRef] [PubMed]

M. Beruete, M. Sorolla, and I. Campillo, "Left-handed extraordinary optical transmission through a photonic crystal of subwavelength hole arrays," Opt. Express 14, 5445-5455 (2006).
[CrossRef] [PubMed]

Soukoulis, C. M.

Tretyakov, S.

C. Menzel, T. Paul, C. Rockstuhl, T. Pertsch, S. Tretyakov, and F. Lederer, "Validity of effective material parameters for optical fishnet metamaterials," Phys. Rev. B 81, 035320 (2010).
[CrossRef]

Ulin-Avila, E.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, "Three-dimensional optical metamaterial with a negative refractive index," Nature (London) 455, 376-379 (2008).
[CrossRef] [PubMed]

Valentine, J.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, "Three-dimensional optical metamaterial with a negative refractive index," Nature (London) 455, 376-379 (2008).
[CrossRef] [PubMed]

Wegener, M.

Zentgraf, T.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, "Three-dimensional optical metamaterial with a negative refractive index," Nature (London) 455, 376-379 (2008).
[CrossRef] [PubMed]

Zhang, S.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, "Three-dimensional optical metamaterial with a negative refractive index," Nature (London) 455, 376-379 (2008).
[CrossRef] [PubMed]

S. Zhang,W. Fan, K. J. Malloy, and S. R. J. Brueck, "Near-infrared double negative metamaterials," Opt. Express 13, 4922-4929 (2005).
[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]

IEEE Trans. Antennas Propag. (1)

M. Beruete, I. Campillo, M. Navarro-Cia, F. Falcone, and M. S. Ayza, "Molding left- or right-handed metamaterials by stacked cutoff metallic hole arrays," IEEE Trans. Antennas Propag. 55, 1514-1521 (2007).
[CrossRef]

Nature (London) (1)

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, "Three-dimensional optical metamaterial with a negative refractive index," Nature (London) 455, 376-379 (2008).
[CrossRef] [PubMed]

New J. Phys. (1)

M. Beruete, M. Navarro-Cia and M. Sorolla, "Strong lateral displacement in polarization anisotropic extraordinary transmission metamaterial," New J. Phys. 12, 063037 (2010).
[CrossRef]

Opt. Express (6)

Opt. Lett. (2)

Phys. Rev. B (2)

M. Beruete, M. Navarro-Cia, M. Sorolla, and I. Campillo, "Negative refraction through an extraordinary transmission left-handed metamaterial slab," Phys. Rev. B 79, 195107 (2009).
[CrossRef]

C. Menzel, T. Paul, C. Rockstuhl, T. Pertsch, S. Tretyakov, and F. Lederer, "Validity of effective material parameters for optical fishnet metamaterials," Phys. Rev. B 81, 035320 (2010).
[CrossRef]

Phys. Rev. Lett. (2)

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]

A. Mary, S. G. Rodrigo, F. J. Garcia-Vidal, and L. Martin-Moreno, "Theory of negative-refractive-index response of double-fishnet structures," Phys. Rev. Lett. 101, 103902 (2008).
[CrossRef] [PubMed]

Other (1)

L. D. Landau, E. M. Lifshitz and L. P. Pitaevskii, Electrodynamics of Continuous Media, 2nd ed. (Pergamon Press, New York, 1984).

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Figures (7)

Fig. 1.
Fig. 1.

Panels (a) and (b) show the front and side view of the analyzed unit cell. Panel (c) shows the dispersion diagram for normal incidence on a structure with p 1 = p 2, wx = wy = p 1/3, p 3 = p 1/9, t = p 1/30 and some host material (lines come from the presented method, while dots come from CST). The lateral boundaries are indicated in all the panels and are either a periodic boundary (PB), a perfect magnetic conductor (PMC) or a perfect electric conductor (PEC). The gray regions in all the panels represent a perfect electric conductor.

Fig. 2.
Fig. 2.

Band diagram cut for qz = qx = 0 corresponding to the TM case (a) and band diagram for qz = qy = 0 corresponding to the TE case (b). The structural parameters are identical with Fig. 1c. The lines come from the presented method, while dots come from CST.

Fig. 3.
Fig. 3.

Iso-frequency contours for TM polarization. The structural parameters are the same as in Fig. 1c.

Fig. 4.
Fig. 4.

Iso-frequency contours for (a) TM and (b) TE polarization. The structural parameters are the same as in Fig. 1c. The interval of allowed q is restricted by (8).

Fig. 5.
Fig. 5.

Graphical representation of TM polarized plane wave refraction on the planar boundary between free space and fishnet. Group velocity is signed as v g, with black arrows corresponding to the waves propagating along the z-axis and gray arrows corresponding to the waves in the opposite direction. Note that the phase velocity in z direction v fz =ω/kz or v fz =ω/qz is positive for the wave on the left panel and negative for the right panel.

Fig. 6.
Fig. 6.

Least-square fitted parameters n TM 0, n TM 0, r TE, and r TE corresponding to (9). The structural parameters are the same as in Fig. 1c.

Fig. 7.
Fig. 7.

Refraction angle versus incidence angle and frequency for (a) TM polarization and (b) TE polarization. The n h = 1 is assumed. As input data the curves from Fig. 6 have been used. Note that (11) only gives absolute value of refraction angle, however, using (10) the negative sign has to be assigned to TM case, while positive sign has to be assigned to TE case.

Equations (18)

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

b 3 = s ¯ 11 a 3 + s ¯ 12 a 4
b 4 = s ¯ 21 a 3 + s ¯ 22 a 4
a 2 = { L ¯ 1 + x ¯ 11 L ¯ 1 + } b 2 + { L ¯ 1 + x ¯ 12 L ¯ 1 + } b 7
a 7 = { L ¯ 1 + x ¯ 21 L ¯ 1 + } b 2 + { L ¯ 1 + x ¯ 22 L ¯ 1 + } b 7 ,
x ¯ = [ s ¯ 11 + s ¯ 12 L ¯ 2 + s ¯ 22 x ¯ 2 L ¯ 2 + s ¯ 21 s ¯ 12 L ¯ 2 + ( s ¯ 22 x ¯ 2 x ¯ 1 s ¯ 21 + s ¯ 21 ) s ¯ 12 x ¯ 2 L ¯ 2 + s ¯ 21 s ¯ 12 x ¯ 2 x ¯ 1 s ¯ 21 + s ¯ 11 ]
L ¯ 1 + = diag [ e j k z 1 ( p 3 t ) 2 ]
L ¯ 2 + = diag [ e j k z 2 t ]
x ¯ 1 = L ¯ 2 + s ¯ 22 L ¯ 2 +
x ¯ 2 = { 1 ¯ x ¯ 1 s ¯ 22 } 1 .
k z 1 2 = ε h μ h ( ω c 0 ) 2 + ( 2 m π p 1 + q x ) 2 + ( 2 n π p 2 + q y ) 2 ; m = M . . M , n = N . . N
k z 2 2 = ε h μ h ( ω c 0 ) 2 + ( r π w x ) 2 + ( s π w y ) 2 ; r = 0 . . R , s = 0 . . S .
a 7 = e j q z p 3 b 2
b 7 = e j q z p 3 a 2
[ 1 ¯ s ¯ 11 tot 0 ¯ s ¯ 21 tot ] [ a 2 b 2 ] = e j q z p 3 [ s ¯ 12 tot 0 ¯ s ¯ 22 tot 1 ¯ ] [ a 2 b 2 ]
q x · p 1 2 + q y · p 2 2 + q z · p 3 2 < ( π 5 ) 2 ,
q z 2 = n 2 k 0 2 + rq 2 ,
tan ( θ r ) = S S z = r q q z ,
sin 2 ( θ r ) = r 2 sin 2 ( θ i ) n 2 + ( r 2 + r ) sin 2 ( θ i ) ,

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