Abstract

Graphene can be used as a platform for tunable absorbers for its tunability of conductivity. In this paper, we proposed an “uneven dielectric slab structure” for the terahertz (THz) tunable absorber based on graphene. The absorber consists of graphene-dielectric stacks and an electric conductor layer, which is easy to fabricate in the manufacturing technique. Fine tuning of the absorption resonances can be conveniently achieved by adjusting the bias voltage. Both narrowband and broadband tunable absorbers made of this structure are demonstrated without using a patterned graphene. In addition, this type of graphene-based absorber exhibits stable resonances with a wide range angles of obliquely incident electromagnetic waves.

© 2013 Optical Society of America

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    [CrossRef]
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    [CrossRef] [PubMed]
  27. R. R. Nair, P. Blake, A. N. Grigorenko, K. S. Novoselov, T. J. Booth, T. Stauber, N. M. R. Peres, A. K. Geim, “Fine structure constant defines visual transparency of graphene,” Science 320, 1308 (2008).
    [CrossRef] [PubMed]
  28. A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys 81, 109–162 (2009)
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  29. H. Yan, X. Li, B. Chandra, G. Tulevski, Y. Wu, M. Freitag, W. Zhu, P. Avouris, F. Xia, “Tunable infrared plasmonic devices using graphene/insulator stacks,” Nat. Nanotech. 7, 330–334 (2012).
    [CrossRef]
  30. D. Drew, X. Cai, A. Sushkov, G. Jenkins, M. Fuhrer, L. Nyakiti, V. Wheeler, R. L. Myers-Ward, N. Y. Garces, C. R. Eddy-Jr, D. K. Gaskill, “Single layer graphene plasmonic detector for broadband THz spectroscopy,” Bull. APS. 58, (2013).
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    [CrossRef]
  32. V. P. Gusynin, S. G. Sharapov, J. P. Carbotte, “Magneto-optical conductivity in graphene,” J. Phys.: Condens. Matter 19, 026222 (2007).
    [CrossRef]
  33. Z. Q. Li, E. A. Henriksen, Z. Jiang, Z. Hao, M. C. Martin, P. Kim, H. L. Stormer, D. N. Basov, “Dirac charge dynamics in graphene by infrared spectroscopy,” Nature Physics 4, 532–535 (2008)
    [CrossRef]
  34. C. Lee, J. Y. Kim, S. Bae, K. S. Kim, B. H. Hong, E. J. Choi, “Optical response of large scale single layer graphene,” Appl. Phys. Lett. 98, 071905 (2011).
    [CrossRef]
  35. J. Y. Kim, C. Lee, S. Bae, K. S. Kim, B. H. Hong, E. J. Choi, “Far-infrared study of substrate-effect on large scale graphene,” Appl. Phys. Lett. 98, 201907 (2011).
    [CrossRef]
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    [CrossRef]

2013 (6)

A. Andryieuski, A. Lavrinenko, “Graphene metamaterials based tunable terahertz absorber: effective surface conductivity approach,” Opt. Express 21, 9144–9155 (2013).
[CrossRef] [PubMed]

M. A. K. Othman, C. Guclu, F. Capolino, “Graphene-based tunable hyperbolic metamaterials and enhanced near-field absorption,” Opt. Express 21, 7614–7632 (2013).
[CrossRef] [PubMed]

Y. R. Padooru, A. B. Yakovlev, C. S. Kaipa, G. W. Hanson, F. Medina, F. Mesa, “Dual capacitive-inductive nature of periodic graphene patches: Transmission characteristics at low-terahertz frequencies,” Phys. Rev. B 87, 115401 (2013).
[CrossRef]

D. Drew, X. Cai, A. Sushkov, G. Jenkins, M. Fuhrer, L. Nyakiti, V. Wheeler, R. L. Myers-Ward, N. Y. Garces, C. R. Eddy-Jr, D. K. Gaskill, “Single layer graphene plasmonic detector for broadband THz spectroscopy,” Bull. APS. 58, (2013).

M. Pu, P. Chen, Y. Wang, Z. Zhao, C. Wang, C. Huang, C. Hu, X. Luo, “Strong enhancement of light absorption and highly directive thermal emission in graphene,” Opt. Express 21, 11618–11627 (2013).
[CrossRef] [PubMed]

X. Wang, W. S. Zhao, J. Hu, “A Novel Tunable Antenna at Thz Frequencies Using Graphene-Based Artificial Magnetic Conductor (Amc),” Progress in Electromagnetics Research Letters 41, 29–38 (2013).

2012 (12)

H. Yan, X. Li, B. Chandra, G. Tulevski, Y. Wu, M. Freitag, W. Zhu, P. Avouris, F. Xia, “Tunable infrared plasmonic devices using graphene/insulator stacks,” Nat. Nanotech. 7, 330–334 (2012).
[CrossRef]

R. Alaee, M. Farhat, C. Rockstuhl, F. Lederer, “A perfect absorber made of a graphene micro-ribbon metamaterial,” Opt. Express 20, 28017–28024 (2012).
[CrossRef] [PubMed]

C. S. Kaipa, A. B. Yakovlev, G. W. Hanson, Y. R. Padooru, F. Medina, F. Mesa, “Enhanced transmission with a graphene-dielectric microstructure at low-terahertz frequencies,” Phys. Rev. B 85, 245407 (2012).
[CrossRef]

S. Thongrattanasiri, F. H. Koppens, F. J. Garcia de Abajo, “Complete optical absorption in periodically patterned graphene,” Phys. Rev. Lett. 108, 047401 (2012).
[CrossRef] [PubMed]

A. Ferreira, N. M. R. Peres, “Complete light absorption in graphene-metamaterial corrugated structures,” Phys. Rev. B 86, 205401 (2012).
[CrossRef]

A. Y. Nikitin, F. Guinea, L. Martin-Moreno, “Resonant plasmonic effects in periodic graphene antidot arrays,” Appl. Phys. Lett. 101, 151119 (2012).
[CrossRef]

A. Y. Nikitin, F. Guinea, F. J. Garcia-Vidal, L. Martin-Moreno, “Surface plasmon enhanced absorption and suppressed transmission in periodic arrays of graphene ribbons,” Phys. Rev. B 85, 081405 (2012).
[CrossRef]

A. Fallahi, J. Perruisseau-Carrier, “Design of tunable biperiodic graphene metasurfaces,” Phys. Rev. B 86, 195408 (2012).
[CrossRef]

H. J. Xu, W. B. Lu, Y. Jiang, Z. G. Dong, “Beam-scanning planar lens based on graphene,” Appl. Phys. Lett. 100, 051903 (2012).
[CrossRef]

H. J. Xu, W. B. Lu, W. Zhu, Z. G. Dong, T. J. Cui, “Efficient manipulation of surface plasmon polariton waves in graphene,” Appl. Phys. Lett. 100, 243110 (2012).
[CrossRef]

Q. Bao, K. P. Loh, “Graphene photonics, plasmonics, and broadband optoelectronic devices.” ACS Nano 6, 3677–3694 (2012).
[CrossRef] [PubMed]

A. N. Grigorenko, M. Polini, K. S. Novoselov, “Graphene plasmonics,” Nature Photon. 487, 749–758 (2012).
[CrossRef]

2011 (5)

A. Vakil, N. Engheta, “Transformation optics using graphene,” Science 332, 1291–1294 (2011).
[CrossRef] [PubMed]

F. H. L. Koppens, D. E. Chang, F. J. G. de Abajo, “Graphene plasmonics: a platform for strong light-matter interactions,” Nano Lett. 11, 3370–3377 (2011).
[CrossRef] [PubMed]

P. Y. Chen, A. Alu, “Atomically thin surface cloak using graphene monolayers,” ACS nano 5(7), 5855–5863 (2011)
[CrossRef]

C. Lee, J. Y. Kim, S. Bae, K. S. Kim, B. H. Hong, E. J. Choi, “Optical response of large scale single layer graphene,” Appl. Phys. Lett. 98, 071905 (2011).
[CrossRef]

J. Y. Kim, C. Lee, S. Bae, K. S. Kim, B. H. Hong, E. J. Choi, “Far-infrared study of substrate-effect on large scale graphene,” Appl. Phys. Lett. 98, 201907 (2011).
[CrossRef]

2010 (1)

D. Y. Shchegolkov, A. K. Azad, J. F. OHara, E. I. Simakov, “Perfect subwavelength fishnetlike metamaterial-based film terahertz absorbers.” Phys. Rev. B 82, 205117 (2010).
[CrossRef]

2009 (6)

A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys 81, 109–162 (2009)
[CrossRef]

N. I. Landy, C. M. Bingham, T. Tyler, N. Jokerst, D. R. Smith, W. J. Padilla, “Design, theory, and measurement of a polarization-insensitive absorber for terahertz imaging.” Phys. Rev. B 79, 125104 (2009).
[CrossRef]

Q. Y. Wen, H. W. Zhang, Y. S. Xie, Q. H. Yang, Y. L. Liu, “Dual band terahertz metamaterial absorber: Design, fabrication, and characterization.” Appl. Phys. Lett. 95, 241111 (2009).
[CrossRef]

M. Jablan, H. Buljan, M. Soljacic, “Plasmonics in graphene at infrared frequencies,” Phys. Rev. B 80, 245435 (2009).
[CrossRef]

A. K. Geim, “Graphene: status and prospects,” Science 324, 1530–1534 (2009).
[CrossRef] [PubMed]

K. S. Kim, Y. Zhao, H. Jang, S. Y. Lee, J. M. Kim, K. S. Kim, J. H. Ahn, P. Kim, J. J. Choi, B. H. Hong, “Large-scale pattern growth of graphene films for stretchable transparent electrodes,” Nature 457, 706–710 (2009).
[CrossRef] [PubMed]

2008 (4)

R. R. Nair, P. Blake, A. N. Grigorenko, K. S. Novoselov, T. J. Booth, T. Stauber, N. M. R. Peres, A. K. Geim, “Fine structure constant defines visual transparency of graphene,” Science 320, 1308 (2008).
[CrossRef] [PubMed]

H. Tao, N. I. Landy, C. M. Bingham, X. Zhang, R. D. Averitt, W. J. Padilla, “A metamaterial absorber for the terahertz regime: design, fabrication and characterization.” Opt. Express 16, 7181–7188 (2008).
[CrossRef] [PubMed]

Z. Q. Li, E. A. Henriksen, Z. Jiang, Z. Hao, M. C. Martin, P. Kim, H. L. Stormer, D. N. Basov, “Dirac charge dynamics in graphene by infrared spectroscopy,” Nature Physics 4, 532–535 (2008)
[CrossRef]

G. W. Hanson, “Dyadic Greens functions and guided surface waves for a surface conductivity model of graphene,” J. Appl. Phys. 103, 064302 (2008).
[CrossRef]

2007 (4)

V. P. Gusynin, S. G. Sharapov, J. P. Carbotte, “Magneto-optical conductivity in graphene,” J. Phys.: Condens. Matter 19, 026222 (2007).
[CrossRef]

M. Tonouchi, “Cutting-edge terahertz technology,” Nature Photon. 1, 97–105 (2007).
[CrossRef]

A. K. Geim, K. S. Novoselov, “The rise of graphene,” Nature Mat. 6, 183–191 (2007).
[CrossRef]

S. Mikhailov, K. Ziegler, “New electromagnetic mode in graphene,” Phys. Rev. Lett. 99, 016803 (2007).
[CrossRef] [PubMed]

Ahn, J. H.

K. S. Kim, Y. Zhao, H. Jang, S. Y. Lee, J. M. Kim, K. S. Kim, J. H. Ahn, P. Kim, J. J. Choi, B. H. Hong, “Large-scale pattern growth of graphene films for stretchable transparent electrodes,” Nature 457, 706–710 (2009).
[CrossRef] [PubMed]

Alaee, R.

Alu, A.

P. Y. Chen, A. Alu, “Atomically thin surface cloak using graphene monolayers,” ACS nano 5(7), 5855–5863 (2011)
[CrossRef]

Andryieuski, A.

Averitt, R. D.

Avouris, P.

H. Yan, X. Li, B. Chandra, G. Tulevski, Y. Wu, M. Freitag, W. Zhu, P. Avouris, F. Xia, “Tunable infrared plasmonic devices using graphene/insulator stacks,” Nat. Nanotech. 7, 330–334 (2012).
[CrossRef]

Azad, A. K.

D. Y. Shchegolkov, A. K. Azad, J. F. OHara, E. I. Simakov, “Perfect subwavelength fishnetlike metamaterial-based film terahertz absorbers.” Phys. Rev. B 82, 205117 (2010).
[CrossRef]

Bae, S.

C. Lee, J. Y. Kim, S. Bae, K. S. Kim, B. H. Hong, E. J. Choi, “Optical response of large scale single layer graphene,” Appl. Phys. Lett. 98, 071905 (2011).
[CrossRef]

J. Y. Kim, C. Lee, S. Bae, K. S. Kim, B. H. Hong, E. J. Choi, “Far-infrared study of substrate-effect on large scale graphene,” Appl. Phys. Lett. 98, 201907 (2011).
[CrossRef]

Bao, Q.

Q. Bao, K. P. Loh, “Graphene photonics, plasmonics, and broadband optoelectronic devices.” ACS Nano 6, 3677–3694 (2012).
[CrossRef] [PubMed]

Basov, D. N.

Z. Q. Li, E. A. Henriksen, Z. Jiang, Z. Hao, M. C. Martin, P. Kim, H. L. Stormer, D. N. Basov, “Dirac charge dynamics in graphene by infrared spectroscopy,” Nature Physics 4, 532–535 (2008)
[CrossRef]

Bingham, C. M.

N. I. Landy, C. M. Bingham, T. Tyler, N. Jokerst, D. R. Smith, W. J. Padilla, “Design, theory, and measurement of a polarization-insensitive absorber for terahertz imaging.” Phys. Rev. B 79, 125104 (2009).
[CrossRef]

H. Tao, N. I. Landy, C. M. Bingham, X. Zhang, R. D. Averitt, W. J. Padilla, “A metamaterial absorber for the terahertz regime: design, fabrication and characterization.” Opt. Express 16, 7181–7188 (2008).
[CrossRef] [PubMed]

Blake, P.

R. R. Nair, P. Blake, A. N. Grigorenko, K. S. Novoselov, T. J. Booth, T. Stauber, N. M. R. Peres, A. K. Geim, “Fine structure constant defines visual transparency of graphene,” Science 320, 1308 (2008).
[CrossRef] [PubMed]

Booth, T. J.

R. R. Nair, P. Blake, A. N. Grigorenko, K. S. Novoselov, T. J. Booth, T. Stauber, N. M. R. Peres, A. K. Geim, “Fine structure constant defines visual transparency of graphene,” Science 320, 1308 (2008).
[CrossRef] [PubMed]

Buljan, H.

M. Jablan, H. Buljan, M. Soljacic, “Plasmonics in graphene at infrared frequencies,” Phys. Rev. B 80, 245435 (2009).
[CrossRef]

Cai, X.

D. Drew, X. Cai, A. Sushkov, G. Jenkins, M. Fuhrer, L. Nyakiti, V. Wheeler, R. L. Myers-Ward, N. Y. Garces, C. R. Eddy-Jr, D. K. Gaskill, “Single layer graphene plasmonic detector for broadband THz spectroscopy,” Bull. APS. 58, (2013).

Capolino, F.

Carbotte, J. P.

V. P. Gusynin, S. G. Sharapov, J. P. Carbotte, “Magneto-optical conductivity in graphene,” J. Phys.: Condens. Matter 19, 026222 (2007).
[CrossRef]

Castro Neto, A. H.

A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys 81, 109–162 (2009)
[CrossRef]

Chandra, B.

H. Yan, X. Li, B. Chandra, G. Tulevski, Y. Wu, M. Freitag, W. Zhu, P. Avouris, F. Xia, “Tunable infrared plasmonic devices using graphene/insulator stacks,” Nat. Nanotech. 7, 330–334 (2012).
[CrossRef]

Chang, D. E.

F. H. L. Koppens, D. E. Chang, F. J. G. de Abajo, “Graphene plasmonics: a platform for strong light-matter interactions,” Nano Lett. 11, 3370–3377 (2011).
[CrossRef] [PubMed]

Chen, P.

Chen, P. Y.

P. Y. Chen, A. Alu, “Atomically thin surface cloak using graphene monolayers,” ACS nano 5(7), 5855–5863 (2011)
[CrossRef]

Choi, E. J.

C. Lee, J. Y. Kim, S. Bae, K. S. Kim, B. H. Hong, E. J. Choi, “Optical response of large scale single layer graphene,” Appl. Phys. Lett. 98, 071905 (2011).
[CrossRef]

J. Y. Kim, C. Lee, S. Bae, K. S. Kim, B. H. Hong, E. J. Choi, “Far-infrared study of substrate-effect on large scale graphene,” Appl. Phys. Lett. 98, 201907 (2011).
[CrossRef]

Choi, J. J.

K. S. Kim, Y. Zhao, H. Jang, S. Y. Lee, J. M. Kim, K. S. Kim, J. H. Ahn, P. Kim, J. J. Choi, B. H. Hong, “Large-scale pattern growth of graphene films for stretchable transparent electrodes,” Nature 457, 706–710 (2009).
[CrossRef] [PubMed]

Cui, T. J.

H. J. Xu, W. B. Lu, W. Zhu, Z. G. Dong, T. J. Cui, “Efficient manipulation of surface plasmon polariton waves in graphene,” Appl. Phys. Lett. 100, 243110 (2012).
[CrossRef]

de Abajo, F. J. G.

F. H. L. Koppens, D. E. Chang, F. J. G. de Abajo, “Graphene plasmonics: a platform for strong light-matter interactions,” Nano Lett. 11, 3370–3377 (2011).
[CrossRef] [PubMed]

Dong, Z. G.

H. J. Xu, W. B. Lu, W. Zhu, Z. G. Dong, T. J. Cui, “Efficient manipulation of surface plasmon polariton waves in graphene,” Appl. Phys. Lett. 100, 243110 (2012).
[CrossRef]

H. J. Xu, W. B. Lu, Y. Jiang, Z. G. Dong, “Beam-scanning planar lens based on graphene,” Appl. Phys. Lett. 100, 051903 (2012).
[CrossRef]

Drew, D.

D. Drew, X. Cai, A. Sushkov, G. Jenkins, M. Fuhrer, L. Nyakiti, V. Wheeler, R. L. Myers-Ward, N. Y. Garces, C. R. Eddy-Jr, D. K. Gaskill, “Single layer graphene plasmonic detector for broadband THz spectroscopy,” Bull. APS. 58, (2013).

Eddy-Jr, C. R.

D. Drew, X. Cai, A. Sushkov, G. Jenkins, M. Fuhrer, L. Nyakiti, V. Wheeler, R. L. Myers-Ward, N. Y. Garces, C. R. Eddy-Jr, D. K. Gaskill, “Single layer graphene plasmonic detector for broadband THz spectroscopy,” Bull. APS. 58, (2013).

Engheta, N.

A. Vakil, N. Engheta, “Transformation optics using graphene,” Science 332, 1291–1294 (2011).
[CrossRef] [PubMed]

Fallahi, A.

A. Fallahi, J. Perruisseau-Carrier, “Design of tunable biperiodic graphene metasurfaces,” Phys. Rev. B 86, 195408 (2012).
[CrossRef]

Farhat, M.

Ferreira, A.

A. Ferreira, N. M. R. Peres, “Complete light absorption in graphene-metamaterial corrugated structures,” Phys. Rev. B 86, 205401 (2012).
[CrossRef]

Freitag, M.

H. Yan, X. Li, B. Chandra, G. Tulevski, Y. Wu, M. Freitag, W. Zhu, P. Avouris, F. Xia, “Tunable infrared plasmonic devices using graphene/insulator stacks,” Nat. Nanotech. 7, 330–334 (2012).
[CrossRef]

Fuhrer, M.

D. Drew, X. Cai, A. Sushkov, G. Jenkins, M. Fuhrer, L. Nyakiti, V. Wheeler, R. L. Myers-Ward, N. Y. Garces, C. R. Eddy-Jr, D. K. Gaskill, “Single layer graphene plasmonic detector for broadband THz spectroscopy,” Bull. APS. 58, (2013).

Garces, N. Y.

D. Drew, X. Cai, A. Sushkov, G. Jenkins, M. Fuhrer, L. Nyakiti, V. Wheeler, R. L. Myers-Ward, N. Y. Garces, C. R. Eddy-Jr, D. K. Gaskill, “Single layer graphene plasmonic detector for broadband THz spectroscopy,” Bull. APS. 58, (2013).

Garcia de Abajo, F. J.

S. Thongrattanasiri, F. H. Koppens, F. J. Garcia de Abajo, “Complete optical absorption in periodically patterned graphene,” Phys. Rev. Lett. 108, 047401 (2012).
[CrossRef] [PubMed]

Garcia-Vidal, F. J.

A. Y. Nikitin, F. Guinea, F. J. Garcia-Vidal, L. Martin-Moreno, “Surface plasmon enhanced absorption and suppressed transmission in periodic arrays of graphene ribbons,” Phys. Rev. B 85, 081405 (2012).
[CrossRef]

Gaskill, D. K.

D. Drew, X. Cai, A. Sushkov, G. Jenkins, M. Fuhrer, L. Nyakiti, V. Wheeler, R. L. Myers-Ward, N. Y. Garces, C. R. Eddy-Jr, D. K. Gaskill, “Single layer graphene plasmonic detector for broadband THz spectroscopy,” Bull. APS. 58, (2013).

Geim, A. K.

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Figures (6)

Fig. 1
Fig. 1

Geometry of a stack of periodic graphene patches separated by dielectric slabs with a plane wave of incidence, the thickness of the graphene sheet is neglected. (a) 3D view and (b) cross-section view.

Fig. 2
Fig. 2

(a) Real and (b) imaginary parts of the graphene surface impedance (Ω) in terms of frequency for various bias chemical potential (from μc = 0 to 1.0 eV) in the terahertz frequency regime. T = 300 K, τ = 1 ps is considered throughout the paper.

Fig. 3
Fig. 3

The equivalent circuit model of the absorber formed by graphene sheet.

Fig. 4
Fig. 4

(a) Absorption as a function of frequency and dielectric slab t3 for the absorber formed by graphene monolayer with μc = 0.5 eV. (b) Absorption as a function of frequency and chemical potential for the absorber with t1=14.1 nm, t2=1.5 μm, t3 = 38 μm. (c) Magnitude of the absorption versus frequency for different chemical potential are plotted. The analytical predictions (symbols) are in a good correspondence with full-wave simulation results (dash line). And the relationship between the bias voltage and the chemical potential is depicted in subfigure.

Fig. 5
Fig. 5

(a) Schematic of uneven ground plane underneath the graphene layer. The gap distance between the silicon and graphene sheet is filled up with a regular dielectric space (SiO2). The lattice parameter is p = 5 μm, and the gap w1 = 0.25 μm, w2 = 1.175 μm. The thickness of dielectric slabs is t′3 = 12 μm and t3 = 3.5 μm. (b) Absorption of the structure is tuned by the bias voltage from 0 to 12 V.

Fig. 6
Fig. 6

Simulation of full-wave CST results of the absorption for the graphene-based absorber at oblique angles of incidence for (a) TE and (b) TM polarization (Vg = 12 V, μc = 0.45 eV in sα, μc = 0.07 eV in sβ). (c) Absorption of another type of structure is tuned by the bias voltage from 0 to 12 V. The lattice parameter is p = 6 μm.

Equations (7)

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σ g r ( ω , τ , T , μ c ) = σ g r + j σ g r = σ 0 1 + j ω τ ,
σ 0 = e 2 τ π h ¯ 2 ( μ c + 2 k B T ln ( e μ c k B T + 1 ) ) ,
n s = 2 π h ¯ 2 ν F 2 0 ε [ f d ( ε ) f d ( ε + 2 μ c ) ] d ε ,
n s = ε SiO 2 ε 0 V d c / et 1 .
R = | S 11 | 2 = | A + B / Z 0 C Z 0 D A + B / Z 0 + C Z 0 + D | 2 ,
[ A B C D ] = D g r D SiO 2 D Si D d D e ,
D g r = [ 1 0 σ g r 1 ] D SiO 2 = [ cos ( β SiO 2 t 1 ) j Z SiO 2 sin ( β SiO 2 t 1 ) j sin ( β SiO 2 t 1 ) / Z SiO 2 cos ( β SiO 2 t 1 ) ] D Si = [ cos ( β Si t 2 ) j Z Si sin ( β Si t 2 ) j sin ( β Si t 2 ) / Z Si cos ( β Si t 2 ) ] D d = [ cos ( β r t 3 ) j Z d sin ( β r t 3 ) j sin ( β r t 3 ) / Z d cos ( β r t 3 ) ] D e = [ 1 0 σ z 1 ] .

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