Abstract

In this work, we study the influence of disorder on the omnidirectional bandgap in a one dimensional stack of alternating positive and dispersive negative index materials. We achieve this through using the transfer matrix method to study wave propagation properties. In the case where the number of periods becomes infinitely large, the limit of the transmittance is derived from the trace of the matrix, and thus reducing the calculation complexity. The origin of the transmission resonances and their relation with the field localization for random systems are analyzed and compared with that of the periodic case. Our result shows that the zero average refractive index bandgap is not affected by small disorders in layer thickness or refractive index, and thus the multilayer stack is robust against fabrication. The finding is expected to achieve potential applications in optoelectronic sensor devices such as omnidirectional reflectors in airplane radomes. We also show that a random mixture of positive and dispersive negative index materials in an equal ratio always possesses a zero average refractive index bandgap.

© 2010 Optical Society of America

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  1. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966-3969 (2000).
    [CrossRef] [PubMed]
  2. V. G. Veselago, “The electrodynamics of substances with simultaneously negative value of ε and μ,” Sov. Phys. Usp. 10, 509-514 (1968).
    [CrossRef]
  3. D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184-4187 (2000).
    [CrossRef] [PubMed]
  4. R. Aylo, P. P. Banerjee, and G. Nehmetallah, “ Perturbation of multilayered structures of positive and negative index materials,” Proc. SPIE 7392, 7392Q1 (2009).
  5. R. Aylo, P. P. Banerjee, and G. Nehmetallah, “Optical propagation through a homogeneous mixture of positive and negative index materials,” Proc. SPIE 7029, 702917 (2008).
    [CrossRef]
  6. H. Jiang, H. Chen, H. Li, Y. Zhang, and S. Zhu, “Omnidirectional gap and defect mode of one-dimensional photonic crystals containing negative index materials,” Appl. Phys. Lett. 83, 5386-5438 (2003).
    [CrossRef]
  7. J. Li, L. Zhou, C. T. Chan, and P. Sheng, “Photonic band gap from a stack of positive and negative index materials,” Phys. Rev. Lett. 90, 083901 (2003).
    [CrossRef] [PubMed]
  8. Y. Yuan, L. Ran, J. Huangfu, and H. Chen, “Experimental verification of zero order bandgap in a layered stack of left-handed and right-handed materials,” Opt. Express 14, 2220-2227 (2006).
    [CrossRef] [PubMed]
  9. P. Han, C. T. Chan, and Z. Q. Zhang, “Wave localization in one-dimensional random structures composed of single-negative metamaterials,” Phys. Rev. B 77, 115332 (2008).
    [CrossRef]
  10. Y. Weng, Z. Wang, and H. Chen, “Band structures of 1D wavelength photonic crystals containing metamaterials,” Phys. Rev. E 75, 046601 (2007).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2009

R. Aylo, P. P. Banerjee, and G. Nehmetallah, “ Perturbation of multilayered structures of positive and negative index materials,” Proc. SPIE 7392, 7392Q1 (2009).

W. H. Sun, Y. Lu, R. Peng, L. Bao, D. Li, X. Wu, and M. Wang, “Omnidirectional transparency induced by matched impedance in disordered metamaterials,” J. Appl. Phys. 106, 013104 (2009).
[CrossRef]

2008

P. Han, C. T. Chan, and Z. Q. Zhang, “Wave localization in one-dimensional random structures composed of single-negative metamaterials,” Phys. Rev. B 77, 115332 (2008).
[CrossRef]

R. Aylo, P. P. Banerjee, and G. Nehmetallah, “Optical propagation through a homogeneous mixture of positive and negative index materials,” Proc. SPIE 7029, 702917 (2008).
[CrossRef]

2007

Y. Weng, Z. Wang, and H. Chen, “Band structures of 1D wavelength photonic crystals containing metamaterials,” Phys. Rev. E 75, 046601 (2007).
[CrossRef]

A. A. Asatryan, L. C. Botten, M. Byrne, V. Freilkher, S. Gredeskul, I. V. Shardrivov, R. C. McPhedran, and Y. Kishvar, “Suppression of Anderson localization in disordered metamaterials,” Phys. Rev. Lett. 99, 193902 (2007).
[CrossRef]

2006

Y. Dong and X. Zhang, “Unusual transmission properties of wave in one-dimensional random system containing left-handed material,” Phys. Lett. A 359, 542-546 (2006).
[CrossRef]

Y. Yuan, L. Ran, J. Huangfu, and H. Chen, “Experimental verification of zero order bandgap in a layered stack of left-handed and right-handed materials,” Opt. Express 14, 2220-2227 (2006).
[CrossRef] [PubMed]

2003

H. Jiang, H. Chen, H. Li, Y. Zhang, and S. Zhu, “Omnidirectional gap and defect mode of one-dimensional photonic crystals containing negative index materials,” Appl. Phys. Lett. 83, 5386-5438 (2003).
[CrossRef]

J. Li, L. Zhou, C. T. Chan, and P. Sheng, “Photonic band gap from a stack of positive and negative index materials,” Phys. Rev. Lett. 90, 083901 (2003).
[CrossRef] [PubMed]

2000

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966-3969 (2000).
[CrossRef] [PubMed]

1998

D. Felbacq and B. Guizal, “Limit analysis of the diffraction of a plane wave by a one-dimensional periodic medium,” J. Math. Phys. 39, 4604-4607 (1998).
[CrossRef]

1994

1968

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

Asatryan, A. A.

A. A. Asatryan, L. C. Botten, M. Byrne, V. Freilkher, S. Gredeskul, I. V. Shardrivov, R. C. McPhedran, and Y. Kishvar, “Suppression of Anderson localization in disordered metamaterials,” Phys. Rev. Lett. 99, 193902 (2007).
[CrossRef]

Aylo, R.

R. Aylo, P. P. Banerjee, and G. Nehmetallah, “ Perturbation of multilayered structures of positive and negative index materials,” Proc. SPIE 7392, 7392Q1 (2009).

R. Aylo, P. P. Banerjee, and G. Nehmetallah, “Optical propagation through a homogeneous mixture of positive and negative index materials,” Proc. SPIE 7029, 702917 (2008).
[CrossRef]

Banerjee, P. P.

R. Aylo, P. P. Banerjee, and G. Nehmetallah, “ Perturbation of multilayered structures of positive and negative index materials,” Proc. SPIE 7392, 7392Q1 (2009).

R. Aylo, P. P. Banerjee, and G. Nehmetallah, “Optical propagation through a homogeneous mixture of positive and negative index materials,” Proc. SPIE 7029, 702917 (2008).
[CrossRef]

Bao, L.

W. H. Sun, Y. Lu, R. Peng, L. Bao, D. Li, X. Wu, and M. Wang, “Omnidirectional transparency induced by matched impedance in disordered metamaterials,” J. Appl. Phys. 106, 013104 (2009).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999).

Botten, L. C.

A. A. Asatryan, L. C. Botten, M. Byrne, V. Freilkher, S. Gredeskul, I. V. Shardrivov, R. C. McPhedran, and Y. Kishvar, “Suppression of Anderson localization in disordered metamaterials,” Phys. Rev. Lett. 99, 193902 (2007).
[CrossRef]

Byrne, M.

A. A. Asatryan, L. C. Botten, M. Byrne, V. Freilkher, S. Gredeskul, I. V. Shardrivov, R. C. McPhedran, and Y. Kishvar, “Suppression of Anderson localization in disordered metamaterials,” Phys. Rev. Lett. 99, 193902 (2007).
[CrossRef]

Chan, C. T.

P. Han, C. T. Chan, and Z. Q. Zhang, “Wave localization in one-dimensional random structures composed of single-negative metamaterials,” Phys. Rev. B 77, 115332 (2008).
[CrossRef]

J. Li, L. Zhou, C. T. Chan, and P. Sheng, “Photonic band gap from a stack of positive and negative index materials,” Phys. Rev. Lett. 90, 083901 (2003).
[CrossRef] [PubMed]

Chen, H.

Y. Weng, Z. Wang, and H. Chen, “Band structures of 1D wavelength photonic crystals containing metamaterials,” Phys. Rev. E 75, 046601 (2007).
[CrossRef]

Y. Yuan, L. Ran, J. Huangfu, and H. Chen, “Experimental verification of zero order bandgap in a layered stack of left-handed and right-handed materials,” Opt. Express 14, 2220-2227 (2006).
[CrossRef] [PubMed]

H. Jiang, H. Chen, H. Li, Y. Zhang, and S. Zhu, “Omnidirectional gap and defect mode of one-dimensional photonic crystals containing negative index materials,” Appl. Phys. Lett. 83, 5386-5438 (2003).
[CrossRef]

Dong, Y.

Y. Dong and X. Zhang, “Unusual transmission properties of wave in one-dimensional random system containing left-handed material,” Phys. Lett. A 359, 542-546 (2006).
[CrossRef]

Felbacq, D.

D. Felbacq and B. Guizal, “Limit analysis of the diffraction of a plane wave by a one-dimensional periodic medium,” J. Math. Phys. 39, 4604-4607 (1998).
[CrossRef]

Freilkher, V.

A. A. Asatryan, L. C. Botten, M. Byrne, V. Freilkher, S. Gredeskul, I. V. Shardrivov, R. C. McPhedran, and Y. Kishvar, “Suppression of Anderson localization in disordered metamaterials,” Phys. Rev. Lett. 99, 193902 (2007).
[CrossRef]

Gredeskul, S.

A. A. Asatryan, L. C. Botten, M. Byrne, V. Freilkher, S. Gredeskul, I. V. Shardrivov, R. C. McPhedran, and Y. Kishvar, “Suppression of Anderson localization in disordered metamaterials,” Phys. Rev. Lett. 99, 193902 (2007).
[CrossRef]

Guizal, B.

D. Felbacq and B. Guizal, “Limit analysis of the diffraction of a plane wave by a one-dimensional periodic medium,” J. Math. Phys. 39, 4604-4607 (1998).
[CrossRef]

Han, P.

P. Han, C. T. Chan, and Z. Q. Zhang, “Wave localization in one-dimensional random structures composed of single-negative metamaterials,” Phys. Rev. B 77, 115332 (2008).
[CrossRef]

Huangfu, J.

Jiang, H.

H. Jiang, H. Chen, H. Li, Y. Zhang, and S. Zhu, “Omnidirectional gap and defect mode of one-dimensional photonic crystals containing negative index materials,” Appl. Phys. Lett. 83, 5386-5438 (2003).
[CrossRef]

Kishvar, Y.

A. A. Asatryan, L. C. Botten, M. Byrne, V. Freilkher, S. Gredeskul, I. V. Shardrivov, R. C. McPhedran, and Y. Kishvar, “Suppression of Anderson localization in disordered metamaterials,” Phys. Rev. Lett. 99, 193902 (2007).
[CrossRef]

Lekner, J.

Li, D.

W. H. Sun, Y. Lu, R. Peng, L. Bao, D. Li, X. Wu, and M. Wang, “Omnidirectional transparency induced by matched impedance in disordered metamaterials,” J. Appl. Phys. 106, 013104 (2009).
[CrossRef]

Li, H.

H. Jiang, H. Chen, H. Li, Y. Zhang, and S. Zhu, “Omnidirectional gap and defect mode of one-dimensional photonic crystals containing negative index materials,” Appl. Phys. Lett. 83, 5386-5438 (2003).
[CrossRef]

Li, J.

J. Li, L. Zhou, C. T. Chan, and P. Sheng, “Photonic band gap from a stack of positive and negative index materials,” Phys. Rev. Lett. 90, 083901 (2003).
[CrossRef] [PubMed]

Lu, Y.

W. H. Sun, Y. Lu, R. Peng, L. Bao, D. Li, X. Wu, and M. Wang, “Omnidirectional transparency induced by matched impedance in disordered metamaterials,” J. Appl. Phys. 106, 013104 (2009).
[CrossRef]

Macleod, H. A.

H. A. Macleod, Thin Film Optical filters (Taylor & Francis, 2001).
[CrossRef]

McPhedran, R. C.

A. A. Asatryan, L. C. Botten, M. Byrne, V. Freilkher, S. Gredeskul, I. V. Shardrivov, R. C. McPhedran, and Y. Kishvar, “Suppression of Anderson localization in disordered metamaterials,” Phys. Rev. Lett. 99, 193902 (2007).
[CrossRef]

Nasser, S. C.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

Nehmetallah, G.

R. Aylo, P. P. Banerjee, and G. Nehmetallah, “ Perturbation of multilayered structures of positive and negative index materials,” Proc. SPIE 7392, 7392Q1 (2009).

R. Aylo, P. P. Banerjee, and G. Nehmetallah, “Optical propagation through a homogeneous mixture of positive and negative index materials,” Proc. SPIE 7029, 702917 (2008).
[CrossRef]

Padilla, W. J.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

Pendry, J. B.

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966-3969 (2000).
[CrossRef] [PubMed]

Peng, R.

W. H. Sun, Y. Lu, R. Peng, L. Bao, D. Li, X. Wu, and M. Wang, “Omnidirectional transparency induced by matched impedance in disordered metamaterials,” J. Appl. Phys. 106, 013104 (2009).
[CrossRef]

Ran, L.

Schultz, S.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

Shardrivov, I. V.

A. A. Asatryan, L. C. Botten, M. Byrne, V. Freilkher, S. Gredeskul, I. V. Shardrivov, R. C. McPhedran, and Y. Kishvar, “Suppression of Anderson localization in disordered metamaterials,” Phys. Rev. Lett. 99, 193902 (2007).
[CrossRef]

Sheng, P.

J. Li, L. Zhou, C. T. Chan, and P. Sheng, “Photonic band gap from a stack of positive and negative index materials,” Phys. Rev. Lett. 90, 083901 (2003).
[CrossRef] [PubMed]

Smith, D. R.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

Sun, W. H.

W. H. Sun, Y. Lu, R. Peng, L. Bao, D. Li, X. Wu, and M. Wang, “Omnidirectional transparency induced by matched impedance in disordered metamaterials,” J. Appl. Phys. 106, 013104 (2009).
[CrossRef]

Veselago, V. G.

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

Vier, D. C.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

Wang, M.

W. H. Sun, Y. Lu, R. Peng, L. Bao, D. Li, X. Wu, and M. Wang, “Omnidirectional transparency induced by matched impedance in disordered metamaterials,” J. Appl. Phys. 106, 013104 (2009).
[CrossRef]

Wang, Z.

Y. Weng, Z. Wang, and H. Chen, “Band structures of 1D wavelength photonic crystals containing metamaterials,” Phys. Rev. E 75, 046601 (2007).
[CrossRef]

Weng, Y.

Y. Weng, Z. Wang, and H. Chen, “Band structures of 1D wavelength photonic crystals containing metamaterials,” Phys. Rev. E 75, 046601 (2007).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999).

Wu, X.

W. H. Sun, Y. Lu, R. Peng, L. Bao, D. Li, X. Wu, and M. Wang, “Omnidirectional transparency induced by matched impedance in disordered metamaterials,” J. Appl. Phys. 106, 013104 (2009).
[CrossRef]

Yuan, Y.

Zhang, X.

Y. Dong and X. Zhang, “Unusual transmission properties of wave in one-dimensional random system containing left-handed material,” Phys. Lett. A 359, 542-546 (2006).
[CrossRef]

Zhang, Y.

H. Jiang, H. Chen, H. Li, Y. Zhang, and S. Zhu, “Omnidirectional gap and defect mode of one-dimensional photonic crystals containing negative index materials,” Appl. Phys. Lett. 83, 5386-5438 (2003).
[CrossRef]

Zhang, Z. Q.

P. Han, C. T. Chan, and Z. Q. Zhang, “Wave localization in one-dimensional random structures composed of single-negative metamaterials,” Phys. Rev. B 77, 115332 (2008).
[CrossRef]

Zhou, L.

J. Li, L. Zhou, C. T. Chan, and P. Sheng, “Photonic band gap from a stack of positive and negative index materials,” Phys. Rev. Lett. 90, 083901 (2003).
[CrossRef] [PubMed]

Zhu, S.

H. Jiang, H. Chen, H. Li, Y. Zhang, and S. Zhu, “Omnidirectional gap and defect mode of one-dimensional photonic crystals containing negative index materials,” Appl. Phys. Lett. 83, 5386-5438 (2003).
[CrossRef]

Appl. Phys. Lett.

H. Jiang, H. Chen, H. Li, Y. Zhang, and S. Zhu, “Omnidirectional gap and defect mode of one-dimensional photonic crystals containing negative index materials,” Appl. Phys. Lett. 83, 5386-5438 (2003).
[CrossRef]

J. Appl. Phys.

W. H. Sun, Y. Lu, R. Peng, L. Bao, D. Li, X. Wu, and M. Wang, “Omnidirectional transparency induced by matched impedance in disordered metamaterials,” J. Appl. Phys. 106, 013104 (2009).
[CrossRef]

J. Math. Phys.

D. Felbacq and B. Guizal, “Limit analysis of the diffraction of a plane wave by a one-dimensional periodic medium,” J. Math. Phys. 39, 4604-4607 (1998).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Express

Phys. Lett. A

Y. Dong and X. Zhang, “Unusual transmission properties of wave in one-dimensional random system containing left-handed material,” Phys. Lett. A 359, 542-546 (2006).
[CrossRef]

Phys. Rev. B

P. Han, C. T. Chan, and Z. Q. Zhang, “Wave localization in one-dimensional random structures composed of single-negative metamaterials,” Phys. Rev. B 77, 115332 (2008).
[CrossRef]

Phys. Rev. E

Y. Weng, Z. Wang, and H. Chen, “Band structures of 1D wavelength photonic crystals containing metamaterials,” Phys. Rev. E 75, 046601 (2007).
[CrossRef]

Phys. Rev. Lett.

J. Li, L. Zhou, C. T. Chan, and P. Sheng, “Photonic band gap from a stack of positive and negative index materials,” Phys. Rev. Lett. 90, 083901 (2003).
[CrossRef] [PubMed]

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966-3969 (2000).
[CrossRef] [PubMed]

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

A. A. Asatryan, L. C. Botten, M. Byrne, V. Freilkher, S. Gredeskul, I. V. Shardrivov, R. C. McPhedran, and Y. Kishvar, “Suppression of Anderson localization in disordered metamaterials,” Phys. Rev. Lett. 99, 193902 (2007).
[CrossRef]

Proc. SPIE

R. Aylo, P. P. Banerjee, and G. Nehmetallah, “ Perturbation of multilayered structures of positive and negative index materials,” Proc. SPIE 7392, 7392Q1 (2009).

R. Aylo, P. P. Banerjee, and G. Nehmetallah, “Optical propagation through a homogeneous mixture of positive and negative index materials,” Proc. SPIE 7029, 702917 (2008).
[CrossRef]

Sov. Phys. Usp.

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

Other

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999).

H. A. Macleod, Thin Film Optical filters (Taylor & Francis, 2001).
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

Schematic of a stack of differing refractive index materials on top of a substrate.

Fig. 2
Fig. 2

(a) Permittivity and permeability of the NIM as given by Eq. (9); (b) transmittance of the alternating periodic layered structure in Fig. 1 where the length of each layer is d = d 1 = d 2 = 1   cm , the numbers of periods are N = 4 (dashed line), N = 25 (dotted line), and N (solid line), and for normal incidence. The zero n bandgap is between 2 and 3 GHz while the Bragg gap is between 6.5 and 7.5 GHz.

Fig. 3
Fig. 3

Photonic band structures of the periodic PIM/NIM stack in terms of the frequency and incident angle θ. The black areas represent forbidden gaps (transmittivity is less than 0.001). The top figure is for TE polarization and the bottom is for TM polarization.

Fig. 4
Fig. 4

(a) Tr ¯ [ T ] for the case of d = d 1 = d 2 = 1   cm , PIM is air, NIM index of refraction from Eq. (9), and for increasing values of perturbation parameter ε = 0 , 1 20 , 1 2 , 1 3 . Inset shows the standard deviation σ ( Tr ¯ [ T ] ) . (b) Localization length as a function of frequency for the periodic stack (solid line) and random stack (dashed line).

Fig. 5
Fig. 5

Average transmission for a transparent mode as a function of the disorder strength ε at f = 9   GHz . The dashed line is the mixed structure (PIM/NIM) and the solid line is the purely positive index structure (PIM/PIM).

Fig. 6
Fig. 6

Localization length as a function of frequency a random stack of 200 layers.

Fig. 7
Fig. 7

Photonic band structures of a disordered PIM/NIM stack in terms of the frequency and incident angle θ. The black area represents forbidden gap (transmittivity is less than 0.001). The top figure is for TE polarization and the bottom is for TM polarization.

Fig. 8
Fig. 8

Photonic band structure for a random stack of 200 layers. The black area represents forbidden gap (transmittivity is less than 0.001). The top figure is for TE polarization and the bottom is for TM polarization.

Fig. 9
Fig. 9

Localization length as a function of frequency for a random (position) stack of 100 layers.

Equations (19)

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

M r = ( cos ( δ r ) i η r sin ( δ r ) i η r   sin ( δ r ) cos ( δ r ) ) ,
[ B C ] = { r = 1 q ( cos   δ r i η r sin   δ r i η r   sin   δ r cos   δ r ) } [ 1 η s ] ,
I i = Re ( B C ) E q E q 2 ( 1 R ) ,
R = ( η 0 B C η 0 B + C ) ( η 0 B C η 0 B + C ) .
T = I q I i = Re ( η s ) ( 1 R ) Re ( B C ) = 4 η 0   Re ( η s ) ( η 0 B + C ) ( η 0 B + C ) .
A = ( 1 R ) [ 1 Re ( η s ) Re ( B C ) ] = 4 η 0   Re ( B C η s ) ( η 0 B + C ) ( η 0 B + C ) .
M p N = [ m 11 m 12 m 21 m 22 ] N = [ m 11 S N S N 1 m 12 S N m 21 S N m 22 S N S N 1 ] ,
M p = r = 1 p ( cos   δ r i η r sin   δ r i η r   sin   δ r cos   δ r ) ,
lim N T = 4 η 0   Re ( η s ) sin 2 ϕ | η 0 ( m 11 cos   ϕ + η s m 12 ) + ( m 21 + η s m 22 η s   cos   ϕ ) | 2 ,
ε r ( f ) = 1 + 5 2 0.9 2 f 2 + 10 2 11.5 2 f 2 ,     μ r ( f ) = 1 + 3 2 0.902 2 f 2 ,
M r ± = ( cos ( 2 π λ n r d ( 1 ± ε Z ̃ r ) ) i η r sin ( 2 π λ n r d ( 1 ± ε Z ̃ r ) ) i η r   sin ( 2 π λ n r d ( 1 ± ε Z ̃ r ) ) cos ( 2 π λ n r d ( 1 ± ε Z ̃ r ) ) ) .
P ¯ r = P ¯ = ( P ¯ 1 , 1 P ¯ 1 , 2 P ¯ 2 , 1 P ¯ 2 , 2 ) .
ς = L / ln   T ¯ ,
ς d = L / ln ( T ) ,
n ( f ) = ( 1 + 5 2 0.9 2 f 2 + 10 2 11.5 2 f 2 ) ( 1 + 3 2 0.902 2 f 2 ) + ξ r .
T = 4 η 0   Re ( η s ) | η 0 ( m 11 S N S N 1 + η s m 12 S N ) + m 21 S N + η s m 22 S N η s S N 1 | 2 .
S N = sin   N ϕ sin   ϕ 1 sin   ϕ ,
S N 1 = sin ( N 1 ) ϕ sin   ϕ = sin ( N ϕ ) cos   ϕ sin   ϕ   cos ( N ϕ ) sin   ϕ cos   ϕ sin   ϕ .
T = 4 η 0   Re ( η s ) sin 2 ϕ | η 0 ( m 11 cos   ϕ + η m m 12 ) + m 21 + η m m 22 η m   cos   ϕ | 2 ,

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