Abstract

We address the problem of the transmission through subwavelength dielectric gratings. Following Maurel et al. [Phys. Rev. B 88, 115416 (2013)], the problem is reduced to the transmission by an homogeneous slab, either anisotropic (for transverse magnetic waves, TM) or isotropic (for transverse electric waves, TE), and an explicit expression of the transmission coefficient is derived. The optimum angle realizing perfect impedance matching (Brewster angle) is shown to depend on the contrasts of the dielectric layers with respect to the air. Besides, we show that the Fabry–Perot resonances may be dependent on the incident angle, in addition to the dependence on the frequency. These facts depart from the case of metallic gratings usually considered; they are confirmed experimentally both for TE and TM waves in the microwave regime.

© 2014 Optical Society of America

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References

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  1. X.-R. Huang, R.-W. Peng, and R.-H. Fan, Phys. Rev. Lett. 105, 243901 (2010).
    [CrossRef]
  2. A. Alù, G. D’Aguanno, N. Mattiucci, and M. J. Bloemer, Phys. Rev. Lett. 106, 123902 (2011).
    [CrossRef]
  3. N. Aközbek, N. Mattiucci, D. de Ceglia, R. Trimm, A. Alù, G. D’Aguanno, M. A. Vincenti, M. Scalora, and M. J. Bloemer, Phys. Rev. B 85, 205430 (2012).
    [CrossRef]
  4. C. Argyropoulos, G. D. Aguanno, N. Mattiucci, N. Aköztek, M. J. Bloemer, and A. Alù, Phys. Rev. B 85, 024304 (2012).
    [CrossRef]
  5. C. Qiu, R. Hao, F. Li, S. Xu, and Z. Liu, Appl. Phys. Lett. 100, 191908 (2012).
    [CrossRef]
  6. D.-X. Qi, R.-H. Fan, R.-W. Peng, S. R. Huang, M.-H. Lu, X. Ni, Q. Hu, and M. Wang, Appl. Phys. Lett. 101, 061912 (2012).
    [CrossRef]
  7. G. D. Aguanno, K. Q. Le, R. Trimm, A. Alù, N. Mattiucci, A. D. Mathias, N. Aközbek, and M. J. Bloemer, Sci. Rep. 2, 340 (2012).
  8. A. Maurel, S. Félix, and J.-F. Mercier, Phys. Rev. B 88, 115416 (2013).
  9. Y. Q. Ye and Y. Jin, Phys. Rev. E 80, 036606 (2009).
    [CrossRef]
  10. M. J. Lockyear, A. P. Hibbins, and J. R. Sambles, Appl. Phys. Lett. 91, 251106 (2007).
    [CrossRef]
  11. Y. Wang, Y. Wang, Y. Zhang, and S. Liu, Opt. Express 17, 5014 (2009).
    [CrossRef]
  12. Y. Wang, J. Ye, Y. Zhang, and S. Liu, Opt. Commun. 284, 877 (2011).
    [CrossRef]
  13. Z. Liu, H. Li, S. Xie, H. Xu, S. Fu, X. Zhou, and C. Wu, Opt. Express 17, 5014 (2011).
  14. C. Qiu, S. Li, R. Chen, B. Hou, F. Li, and Z. Liu, Phys. Rev. B 87, 205129 (2013).
    [CrossRef]
  15. D. Felbacq, J. Math. Phys. 43, 52 (2002).
  16. J. Shin, J. Shen, P. B. Catrysse, and S. Fan, IEEE J. Quantum Electron. 12, 1116 (2006).
    [CrossRef]
  17. J. T. Shen, P. B. Catrysse, and S. Fan, Phys. Rev. Lett. 94, 197401 (2005).
    [CrossRef]

2013 (2)

A. Maurel, S. Félix, and J.-F. Mercier, Phys. Rev. B 88, 115416 (2013).

C. Qiu, S. Li, R. Chen, B. Hou, F. Li, and Z. Liu, Phys. Rev. B 87, 205129 (2013).
[CrossRef]

2012 (5)

N. Aközbek, N. Mattiucci, D. de Ceglia, R. Trimm, A. Alù, G. D’Aguanno, M. A. Vincenti, M. Scalora, and M. J. Bloemer, Phys. Rev. B 85, 205430 (2012).
[CrossRef]

C. Argyropoulos, G. D. Aguanno, N. Mattiucci, N. Aköztek, M. J. Bloemer, and A. Alù, Phys. Rev. B 85, 024304 (2012).
[CrossRef]

C. Qiu, R. Hao, F. Li, S. Xu, and Z. Liu, Appl. Phys. Lett. 100, 191908 (2012).
[CrossRef]

D.-X. Qi, R.-H. Fan, R.-W. Peng, S. R. Huang, M.-H. Lu, X. Ni, Q. Hu, and M. Wang, Appl. Phys. Lett. 101, 061912 (2012).
[CrossRef]

G. D. Aguanno, K. Q. Le, R. Trimm, A. Alù, N. Mattiucci, A. D. Mathias, N. Aközbek, and M. J. Bloemer, Sci. Rep. 2, 340 (2012).

2011 (3)

A. Alù, G. D’Aguanno, N. Mattiucci, and M. J. Bloemer, Phys. Rev. Lett. 106, 123902 (2011).
[CrossRef]

Y. Wang, J. Ye, Y. Zhang, and S. Liu, Opt. Commun. 284, 877 (2011).
[CrossRef]

Z. Liu, H. Li, S. Xie, H. Xu, S. Fu, X. Zhou, and C. Wu, Opt. Express 17, 5014 (2011).

2010 (1)

X.-R. Huang, R.-W. Peng, and R.-H. Fan, Phys. Rev. Lett. 105, 243901 (2010).
[CrossRef]

2009 (2)

2007 (1)

M. J. Lockyear, A. P. Hibbins, and J. R. Sambles, Appl. Phys. Lett. 91, 251106 (2007).
[CrossRef]

2006 (1)

J. Shin, J. Shen, P. B. Catrysse, and S. Fan, IEEE J. Quantum Electron. 12, 1116 (2006).
[CrossRef]

2005 (1)

J. T. Shen, P. B. Catrysse, and S. Fan, Phys. Rev. Lett. 94, 197401 (2005).
[CrossRef]

2002 (1)

D. Felbacq, J. Math. Phys. 43, 52 (2002).

Aguanno, G. D.

C. Argyropoulos, G. D. Aguanno, N. Mattiucci, N. Aköztek, M. J. Bloemer, and A. Alù, Phys. Rev. B 85, 024304 (2012).
[CrossRef]

G. D. Aguanno, K. Q. Le, R. Trimm, A. Alù, N. Mattiucci, A. D. Mathias, N. Aközbek, and M. J. Bloemer, Sci. Rep. 2, 340 (2012).

Aközbek, N.

G. D. Aguanno, K. Q. Le, R. Trimm, A. Alù, N. Mattiucci, A. D. Mathias, N. Aközbek, and M. J. Bloemer, Sci. Rep. 2, 340 (2012).

N. Aközbek, N. Mattiucci, D. de Ceglia, R. Trimm, A. Alù, G. D’Aguanno, M. A. Vincenti, M. Scalora, and M. J. Bloemer, Phys. Rev. B 85, 205430 (2012).
[CrossRef]

Aköztek, N.

C. Argyropoulos, G. D. Aguanno, N. Mattiucci, N. Aköztek, M. J. Bloemer, and A. Alù, Phys. Rev. B 85, 024304 (2012).
[CrossRef]

Alù, A.

C. Argyropoulos, G. D. Aguanno, N. Mattiucci, N. Aköztek, M. J. Bloemer, and A. Alù, Phys. Rev. B 85, 024304 (2012).
[CrossRef]

G. D. Aguanno, K. Q. Le, R. Trimm, A. Alù, N. Mattiucci, A. D. Mathias, N. Aközbek, and M. J. Bloemer, Sci. Rep. 2, 340 (2012).

N. Aközbek, N. Mattiucci, D. de Ceglia, R. Trimm, A. Alù, G. D’Aguanno, M. A. Vincenti, M. Scalora, and M. J. Bloemer, Phys. Rev. B 85, 205430 (2012).
[CrossRef]

A. Alù, G. D’Aguanno, N. Mattiucci, and M. J. Bloemer, Phys. Rev. Lett. 106, 123902 (2011).
[CrossRef]

Argyropoulos, C.

C. Argyropoulos, G. D. Aguanno, N. Mattiucci, N. Aköztek, M. J. Bloemer, and A. Alù, Phys. Rev. B 85, 024304 (2012).
[CrossRef]

Bloemer, M. J.

C. Argyropoulos, G. D. Aguanno, N. Mattiucci, N. Aköztek, M. J. Bloemer, and A. Alù, Phys. Rev. B 85, 024304 (2012).
[CrossRef]

G. D. Aguanno, K. Q. Le, R. Trimm, A. Alù, N. Mattiucci, A. D. Mathias, N. Aközbek, and M. J. Bloemer, Sci. Rep. 2, 340 (2012).

N. Aközbek, N. Mattiucci, D. de Ceglia, R. Trimm, A. Alù, G. D’Aguanno, M. A. Vincenti, M. Scalora, and M. J. Bloemer, Phys. Rev. B 85, 205430 (2012).
[CrossRef]

A. Alù, G. D’Aguanno, N. Mattiucci, and M. J. Bloemer, Phys. Rev. Lett. 106, 123902 (2011).
[CrossRef]

Catrysse, P. B.

J. Shin, J. Shen, P. B. Catrysse, and S. Fan, IEEE J. Quantum Electron. 12, 1116 (2006).
[CrossRef]

J. T. Shen, P. B. Catrysse, and S. Fan, Phys. Rev. Lett. 94, 197401 (2005).
[CrossRef]

Chen, R.

C. Qiu, S. Li, R. Chen, B. Hou, F. Li, and Z. Liu, Phys. Rev. B 87, 205129 (2013).
[CrossRef]

D’Aguanno, G.

N. Aközbek, N. Mattiucci, D. de Ceglia, R. Trimm, A. Alù, G. D’Aguanno, M. A. Vincenti, M. Scalora, and M. J. Bloemer, Phys. Rev. B 85, 205430 (2012).
[CrossRef]

A. Alù, G. D’Aguanno, N. Mattiucci, and M. J. Bloemer, Phys. Rev. Lett. 106, 123902 (2011).
[CrossRef]

de Ceglia, D.

N. Aközbek, N. Mattiucci, D. de Ceglia, R. Trimm, A. Alù, G. D’Aguanno, M. A. Vincenti, M. Scalora, and M. J. Bloemer, Phys. Rev. B 85, 205430 (2012).
[CrossRef]

Fan, R.-H.

D.-X. Qi, R.-H. Fan, R.-W. Peng, S. R. Huang, M.-H. Lu, X. Ni, Q. Hu, and M. Wang, Appl. Phys. Lett. 101, 061912 (2012).
[CrossRef]

X.-R. Huang, R.-W. Peng, and R.-H. Fan, Phys. Rev. Lett. 105, 243901 (2010).
[CrossRef]

Fan, S.

J. Shin, J. Shen, P. B. Catrysse, and S. Fan, IEEE J. Quantum Electron. 12, 1116 (2006).
[CrossRef]

J. T. Shen, P. B. Catrysse, and S. Fan, Phys. Rev. Lett. 94, 197401 (2005).
[CrossRef]

Felbacq, D.

D. Felbacq, J. Math. Phys. 43, 52 (2002).

Félix, S.

A. Maurel, S. Félix, and J.-F. Mercier, Phys. Rev. B 88, 115416 (2013).

Fu, S.

Hao, R.

C. Qiu, R. Hao, F. Li, S. Xu, and Z. Liu, Appl. Phys. Lett. 100, 191908 (2012).
[CrossRef]

Hibbins, A. P.

M. J. Lockyear, A. P. Hibbins, and J. R. Sambles, Appl. Phys. Lett. 91, 251106 (2007).
[CrossRef]

Hou, B.

C. Qiu, S. Li, R. Chen, B. Hou, F. Li, and Z. Liu, Phys. Rev. B 87, 205129 (2013).
[CrossRef]

Hu, Q.

D.-X. Qi, R.-H. Fan, R.-W. Peng, S. R. Huang, M.-H. Lu, X. Ni, Q. Hu, and M. Wang, Appl. Phys. Lett. 101, 061912 (2012).
[CrossRef]

Huang, S. R.

D.-X. Qi, R.-H. Fan, R.-W. Peng, S. R. Huang, M.-H. Lu, X. Ni, Q. Hu, and M. Wang, Appl. Phys. Lett. 101, 061912 (2012).
[CrossRef]

Huang, X.-R.

X.-R. Huang, R.-W. Peng, and R.-H. Fan, Phys. Rev. Lett. 105, 243901 (2010).
[CrossRef]

Jin, Y.

Y. Q. Ye and Y. Jin, Phys. Rev. E 80, 036606 (2009).
[CrossRef]

Le, K. Q.

G. D. Aguanno, K. Q. Le, R. Trimm, A. Alù, N. Mattiucci, A. D. Mathias, N. Aközbek, and M. J. Bloemer, Sci. Rep. 2, 340 (2012).

Li, F.

C. Qiu, S. Li, R. Chen, B. Hou, F. Li, and Z. Liu, Phys. Rev. B 87, 205129 (2013).
[CrossRef]

C. Qiu, R. Hao, F. Li, S. Xu, and Z. Liu, Appl. Phys. Lett. 100, 191908 (2012).
[CrossRef]

Li, H.

Li, S.

C. Qiu, S. Li, R. Chen, B. Hou, F. Li, and Z. Liu, Phys. Rev. B 87, 205129 (2013).
[CrossRef]

Liu, S.

Y. Wang, J. Ye, Y. Zhang, and S. Liu, Opt. Commun. 284, 877 (2011).
[CrossRef]

Y. Wang, Y. Wang, Y. Zhang, and S. Liu, Opt. Express 17, 5014 (2009).
[CrossRef]

Liu, Z.

C. Qiu, S. Li, R. Chen, B. Hou, F. Li, and Z. Liu, Phys. Rev. B 87, 205129 (2013).
[CrossRef]

C. Qiu, R. Hao, F. Li, S. Xu, and Z. Liu, Appl. Phys. Lett. 100, 191908 (2012).
[CrossRef]

Z. Liu, H. Li, S. Xie, H. Xu, S. Fu, X. Zhou, and C. Wu, Opt. Express 17, 5014 (2011).

Lockyear, M. J.

M. J. Lockyear, A. P. Hibbins, and J. R. Sambles, Appl. Phys. Lett. 91, 251106 (2007).
[CrossRef]

Lu, M.-H.

D.-X. Qi, R.-H. Fan, R.-W. Peng, S. R. Huang, M.-H. Lu, X. Ni, Q. Hu, and M. Wang, Appl. Phys. Lett. 101, 061912 (2012).
[CrossRef]

Mathias, A. D.

G. D. Aguanno, K. Q. Le, R. Trimm, A. Alù, N. Mattiucci, A. D. Mathias, N. Aközbek, and M. J. Bloemer, Sci. Rep. 2, 340 (2012).

Mattiucci, N.

G. D. Aguanno, K. Q. Le, R. Trimm, A. Alù, N. Mattiucci, A. D. Mathias, N. Aközbek, and M. J. Bloemer, Sci. Rep. 2, 340 (2012).

C. Argyropoulos, G. D. Aguanno, N. Mattiucci, N. Aköztek, M. J. Bloemer, and A. Alù, Phys. Rev. B 85, 024304 (2012).
[CrossRef]

N. Aközbek, N. Mattiucci, D. de Ceglia, R. Trimm, A. Alù, G. D’Aguanno, M. A. Vincenti, M. Scalora, and M. J. Bloemer, Phys. Rev. B 85, 205430 (2012).
[CrossRef]

A. Alù, G. D’Aguanno, N. Mattiucci, and M. J. Bloemer, Phys. Rev. Lett. 106, 123902 (2011).
[CrossRef]

Maurel, A.

A. Maurel, S. Félix, and J.-F. Mercier, Phys. Rev. B 88, 115416 (2013).

Mercier, J.-F.

A. Maurel, S. Félix, and J.-F. Mercier, Phys. Rev. B 88, 115416 (2013).

Ni, X.

D.-X. Qi, R.-H. Fan, R.-W. Peng, S. R. Huang, M.-H. Lu, X. Ni, Q. Hu, and M. Wang, Appl. Phys. Lett. 101, 061912 (2012).
[CrossRef]

Peng, R.-W.

D.-X. Qi, R.-H. Fan, R.-W. Peng, S. R. Huang, M.-H. Lu, X. Ni, Q. Hu, and M. Wang, Appl. Phys. Lett. 101, 061912 (2012).
[CrossRef]

X.-R. Huang, R.-W. Peng, and R.-H. Fan, Phys. Rev. Lett. 105, 243901 (2010).
[CrossRef]

Qi, D.-X.

D.-X. Qi, R.-H. Fan, R.-W. Peng, S. R. Huang, M.-H. Lu, X. Ni, Q. Hu, and M. Wang, Appl. Phys. Lett. 101, 061912 (2012).
[CrossRef]

Qiu, C.

C. Qiu, S. Li, R. Chen, B. Hou, F. Li, and Z. Liu, Phys. Rev. B 87, 205129 (2013).
[CrossRef]

C. Qiu, R. Hao, F. Li, S. Xu, and Z. Liu, Appl. Phys. Lett. 100, 191908 (2012).
[CrossRef]

Sambles, J. R.

M. J. Lockyear, A. P. Hibbins, and J. R. Sambles, Appl. Phys. Lett. 91, 251106 (2007).
[CrossRef]

Scalora, M.

N. Aközbek, N. Mattiucci, D. de Ceglia, R. Trimm, A. Alù, G. D’Aguanno, M. A. Vincenti, M. Scalora, and M. J. Bloemer, Phys. Rev. B 85, 205430 (2012).
[CrossRef]

Shen, J.

J. Shin, J. Shen, P. B. Catrysse, and S. Fan, IEEE J. Quantum Electron. 12, 1116 (2006).
[CrossRef]

Shen, J. T.

J. T. Shen, P. B. Catrysse, and S. Fan, Phys. Rev. Lett. 94, 197401 (2005).
[CrossRef]

Shin, J.

J. Shin, J. Shen, P. B. Catrysse, and S. Fan, IEEE J. Quantum Electron. 12, 1116 (2006).
[CrossRef]

Trimm, R.

N. Aközbek, N. Mattiucci, D. de Ceglia, R. Trimm, A. Alù, G. D’Aguanno, M. A. Vincenti, M. Scalora, and M. J. Bloemer, Phys. Rev. B 85, 205430 (2012).
[CrossRef]

G. D. Aguanno, K. Q. Le, R. Trimm, A. Alù, N. Mattiucci, A. D. Mathias, N. Aközbek, and M. J. Bloemer, Sci. Rep. 2, 340 (2012).

Vincenti, M. A.

N. Aközbek, N. Mattiucci, D. de Ceglia, R. Trimm, A. Alù, G. D’Aguanno, M. A. Vincenti, M. Scalora, and M. J. Bloemer, Phys. Rev. B 85, 205430 (2012).
[CrossRef]

Wang, M.

D.-X. Qi, R.-H. Fan, R.-W. Peng, S. R. Huang, M.-H. Lu, X. Ni, Q. Hu, and M. Wang, Appl. Phys. Lett. 101, 061912 (2012).
[CrossRef]

Wang, Y.

Wu, C.

Xie, S.

Xu, H.

Xu, S.

C. Qiu, R. Hao, F. Li, S. Xu, and Z. Liu, Appl. Phys. Lett. 100, 191908 (2012).
[CrossRef]

Ye, J.

Y. Wang, J. Ye, Y. Zhang, and S. Liu, Opt. Commun. 284, 877 (2011).
[CrossRef]

Ye, Y. Q.

Y. Q. Ye and Y. Jin, Phys. Rev. E 80, 036606 (2009).
[CrossRef]

Zhang, Y.

Y. Wang, J. Ye, Y. Zhang, and S. Liu, Opt. Commun. 284, 877 (2011).
[CrossRef]

Y. Wang, Y. Wang, Y. Zhang, and S. Liu, Opt. Express 17, 5014 (2009).
[CrossRef]

Zhou, X.

Appl. Phys. Lett. (3)

C. Qiu, R. Hao, F. Li, S. Xu, and Z. Liu, Appl. Phys. Lett. 100, 191908 (2012).
[CrossRef]

D.-X. Qi, R.-H. Fan, R.-W. Peng, S. R. Huang, M.-H. Lu, X. Ni, Q. Hu, and M. Wang, Appl. Phys. Lett. 101, 061912 (2012).
[CrossRef]

M. J. Lockyear, A. P. Hibbins, and J. R. Sambles, Appl. Phys. Lett. 91, 251106 (2007).
[CrossRef]

IEEE J. Quantum Electron. (1)

J. Shin, J. Shen, P. B. Catrysse, and S. Fan, IEEE J. Quantum Electron. 12, 1116 (2006).
[CrossRef]

J. Math. Phys. (1)

D. Felbacq, J. Math. Phys. 43, 52 (2002).

Opt. Commun. (1)

Y. Wang, J. Ye, Y. Zhang, and S. Liu, Opt. Commun. 284, 877 (2011).
[CrossRef]

Opt. Express (2)

Phys. Rev. B (4)

C. Qiu, S. Li, R. Chen, B. Hou, F. Li, and Z. Liu, Phys. Rev. B 87, 205129 (2013).
[CrossRef]

A. Maurel, S. Félix, and J.-F. Mercier, Phys. Rev. B 88, 115416 (2013).

N. Aközbek, N. Mattiucci, D. de Ceglia, R. Trimm, A. Alù, G. D’Aguanno, M. A. Vincenti, M. Scalora, and M. J. Bloemer, Phys. Rev. B 85, 205430 (2012).
[CrossRef]

C. Argyropoulos, G. D. Aguanno, N. Mattiucci, N. Aköztek, M. J. Bloemer, and A. Alù, Phys. Rev. B 85, 024304 (2012).
[CrossRef]

Phys. Rev. E (1)

Y. Q. Ye and Y. Jin, Phys. Rev. E 80, 036606 (2009).
[CrossRef]

Phys. Rev. Lett. (3)

X.-R. Huang, R.-W. Peng, and R.-H. Fan, Phys. Rev. Lett. 105, 243901 (2010).
[CrossRef]

A. Alù, G. D’Aguanno, N. Mattiucci, and M. J. Bloemer, Phys. Rev. Lett. 106, 123902 (2011).
[CrossRef]

J. T. Shen, P. B. Catrysse, and S. Fan, Phys. Rev. Lett. 94, 197401 (2005).
[CrossRef]

Sci. Rep. (1)

G. D. Aguanno, K. Q. Le, R. Trimm, A. Alù, N. Mattiucci, A. D. Mathias, N. Aközbek, and M. J. Bloemer, Sci. Rep. 2, 340 (2012).

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Figures (4)

Fig. 1.
Fig. 1.

Schematic view of the experimental setup; the two insets show the geometry of the dielectric grating and an image of the grating 1 (epoxy/air).

Fig. 2.
Fig. 2.

Transmission spectra Texp experimentally measured (left) and T=|t|2 from Eq. (6) with Eq. (7) (right), as a function of the incident angle θ and of the frequency f for incident TE waves. (a) Grating 1 (epoxy-air) and (b) grating 2 (epoxy-teflon). Dotted lines show the theoretical positions of the Fabry–Perot resonances, Eq. (8), with m=3 and 4.

Fig. 3.
Fig. 3.

Same representation as in Fig. 2, for incident TM waves. Dotted lines show the theoretical positions of the Fabry–Perot resonances, Eq. (10), here m=3 for (a) and m=3 and 4 for (b). Solid lines show the theoretical positions of the Brewster angles, Eq. (12).

Fig. 4.
Fig. 4.

Transmission spectra |t|2 as a function of the incident angle θ and of the frequency f for TM waves (a) in a metal-air grating and (b) in a metal-teflon grating. Dotted lines show the positions of the Fabry–Perot resonances k=mπ, here for (a) m=2 and (b) m=3. Plain lines show the positions of the Brewster angle.

Equations (13)

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

·(1ϵ(r)H(r))+k2H(r)=0,
·[(1/ϵ001/ϵ)H]+k2H=0,
{1ϵ=w1(w1+w2)1ϵ1+w2(w1+w2)1ϵ2,ϵ=w1(w1+w2)ϵ1+w2(w1+w2)ϵ2.
·E(r)+k2ϵ(r)E(r)=0,
·E+k2ϵeffE=0,withϵeff=ϵ.
t=4Zreikcosθ(1+Zr)2eik(1Zr)2eik,
{k2ϵ+k2ϵsin2θ=k2,Zr=kkϵcosθ,forTMwaves,k2+k2sin2θ=k2ϵeff,Zr=kkcosθ,forTEwaves.
fFP=mc2(ϵeffsin2θ)1/2,
Zr=cosθϵeffsin2θ,
fFP=mc21ϵ(1sin2θϵ)1/2.
Zr=cosθϵsin2θϵϵ.
θopt=cos1ϵ1ϵϵ1,
Zr=cosθϵ2w2+w1w2,

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