Abstract

The transmission of evanescent waves in a gain compensated perfect lens is discussed. In particular, the impact of gain saturation is included in the analysis, and a method for calculating the fields of such nonlinear systems is developed. Gain compensation clearly improves the resolution; however, a number of nonideal effects arise as a result of gain saturation. The resolution associated with the lens is strongly dependent on the saturation constant of the active medium.

© 2010 Optical Society of America

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2010

Z. G. Dong, H. Liu, T. Li, Z. H. Zhu, S. M. Wang, J. X. Cao, S. N. Zhu, and X. Zhang, “Optical loss compensation in a bulk left-handed metamaterial by the gain in quantum dots,” Appl. Phys. Lett. 96, 044104 (2010).
[CrossRef]

2009

Ø. Lind-Johansen, K. Seip, and J. Skaar, “The perfect lens on a finite bandwidth,” J. Math. Phys. 50, 012908 (2009).
[CrossRef]

J. Zhang, H. Jiang, B. Gralak, S. Enoch, G. Tayeb, and M. Lequime, “Compensation of loss to approach −1 effective index by gain in metal-dielectric stacks,” Eur. Phys. J.: Appl. Phys. 46, 32603–32608 (2009).
[CrossRef]

A. Fang, Th. Koschny, M. Wegener, and C. M. Soukoulis, “Self-consistent calculation of metamaterials with gain,” Phys. Rev. B 79, 241104 (2009).
[CrossRef]

Y. Sivan, S. Xiao, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Frequency-domain simulations of a negative-index material with embedded gain,” Opt. Express 17, 24060–24074 (2009).
[CrossRef]

M. A. Vincenti, D. de Ceglia, V. Rondinone, A. Ladisa, A. D’Orazio, M. J. Bloemer, and M. Scalora, “Loss compensation in metal-dielectric structures in negative-refraction and super-resolving regimes,” Phys. Rev. A 80, 053807 (2009).
[CrossRef]

2008

B. Nistad and J. Skaar, “Causality and electromagnetic properties of active media,” Phys. Rev. E 78, 036603 (2008).
[CrossRef]

2007

V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photonics 1, 41–48 (2007).
[CrossRef]

2006

2005

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, Th. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. 95, 203901 (2005).
[CrossRef] [PubMed]

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]

2004

N. M. Lawandy, “Localized surface plasmon singularities in amplifying media,” Appl. Phys. Lett. 85, 5040–5042 (2004).
[CrossRef]

2003

S. Anantha 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]

2002

S. Anantha Ramakrishna, J. B. Pendry, D. Schurig, D. R. Smith, and S. Schultz, “The asymmetric lossy near-perfect lens,” J. Mod. Opt. 49, 1747–1762 (2002).
[CrossRef]

2000

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

1992

1968

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

Adegoke, J.

Bahoura, M.

Bloemer, M. J.

M. A. Vincenti, D. de Ceglia, V. Rondinone, A. Ladisa, A. D’Orazio, M. J. Bloemer, and M. Scalora, “Loss compensation in metal-dielectric structures in negative-refraction and super-resolving regimes,” Phys. Rev. A 80, 053807 (2009).
[CrossRef]

Brueck, S. R. J.

Burger, S.

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, Th. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. 95, 203901 (2005).
[CrossRef] [PubMed]

Cai, W.

Cao, J. X.

Z. G. Dong, H. Liu, T. Li, Z. H. Zhu, S. M. Wang, J. X. Cao, S. N. Zhu, and X. Zhang, “Optical loss compensation in a bulk left-handed metamaterial by the gain in quantum dots,” Appl. Phys. Lett. 96, 044104 (2010).
[CrossRef]

Chettiar, U. K.

D’Orazio, A.

M. A. Vincenti, D. de Ceglia, V. Rondinone, A. Ladisa, A. D’Orazio, M. J. Bloemer, and M. Scalora, “Loss compensation in metal-dielectric structures in negative-refraction and super-resolving regimes,” Phys. Rev. A 80, 053807 (2009).
[CrossRef]

de Ceglia, D.

M. A. Vincenti, D. de Ceglia, V. Rondinone, A. Ladisa, A. D’Orazio, M. J. Bloemer, and M. Scalora, “Loss compensation in metal-dielectric structures in negative-refraction and super-resolving regimes,” Phys. Rev. A 80, 053807 (2009).
[CrossRef]

Destro, M. G.

Dolling, G.

Dong, Z. G.

Z. G. Dong, H. Liu, T. Li, Z. H. Zhu, S. M. Wang, J. X. Cao, S. N. Zhu, and X. Zhang, “Optical loss compensation in a bulk left-handed metamaterial by the gain in quantum dots,” Appl. Phys. Lett. 96, 044104 (2010).
[CrossRef]

Drachev, V. P.

Enkrich, C.

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Low-loss negative-index metamaterials at telecommunication wavelengths,” Opt. Lett. 31, 1800–1802 (2006).
[CrossRef] [PubMed]

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, Th. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. 95, 203901 (2005).
[CrossRef] [PubMed]

Enoch, S.

J. Zhang, H. Jiang, B. Gralak, S. Enoch, G. Tayeb, and M. Lequime, “Compensation of loss to approach −1 effective index by gain in metal-dielectric stacks,” Eur. Phys. J.: Appl. Phys. 46, 32603–32608 (2009).
[CrossRef]

Fan, W.

Fang, A.

A. Fang, Th. Koschny, M. Wegener, and C. M. Soukoulis, “Self-consistent calculation of metamaterials with gain,” Phys. Rev. B 79, 241104 (2009).
[CrossRef]

Gralak, B.

J. Zhang, H. Jiang, B. Gralak, S. Enoch, G. Tayeb, and M. Lequime, “Compensation of loss to approach −1 effective index by gain in metal-dielectric stacks,” Eur. Phys. J.: Appl. Phys. 46, 32603–32608 (2009).
[CrossRef]

Jiang, H.

J. Zhang, H. Jiang, B. Gralak, S. Enoch, G. Tayeb, and M. Lequime, “Compensation of loss to approach −1 effective index by gain in metal-dielectric stacks,” Eur. Phys. J.: Appl. Phys. 46, 32603–32608 (2009).
[CrossRef]

Kildishev, A. V.

Y. Sivan, S. Xiao, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Frequency-domain simulations of a negative-index material with embedded gain,” Opt. Express 17, 24060–24074 (2009).
[CrossRef]

T. A. Klar, A. V. Kildishev, V. P. Drachev, and V. M. Shalaev, “Negative-index metamaterials: Going optical,” IEEE J. Sel. Top. Quantum Electron. 12, 1106–1115 (2006).
[CrossRef]

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]

D. H. Kwon, D. H. Werner, A. V. Kildishev, and V. M. Shalaev, “Dual-band negative-index metamaterials in the near-infrared frequency range,” in Proceedings of IEEE Antennas and Propagation Society International Symposium (IEEE, 2007), pp. 2861–2864.

Klar, T. A.

T. A. Klar, A. V. Kildishev, V. P. Drachev, and V. M. Shalaev, “Negative-index metamaterials: Going optical,” IEEE J. Sel. Top. Quantum Electron. 12, 1106–1115 (2006).
[CrossRef]

Koschny, Th.

A. Fang, Th. Koschny, M. Wegener, and C. M. Soukoulis, “Self-consistent calculation of metamaterials with gain,” Phys. Rev. B 79, 241104 (2009).
[CrossRef]

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, Th. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. 95, 203901 (2005).
[CrossRef] [PubMed]

Kwon, D. H.

D. H. Kwon, D. H. Werner, A. V. Kildishev, and V. M. Shalaev, “Dual-band negative-index metamaterials in the near-infrared frequency range,” in Proceedings of IEEE Antennas and Propagation Society International Symposium (IEEE, 2007), pp. 2861–2864.

Ladisa, A.

M. A. Vincenti, D. de Ceglia, V. Rondinone, A. Ladisa, A. D’Orazio, M. J. Bloemer, and M. Scalora, “Loss compensation in metal-dielectric structures in negative-refraction and super-resolving regimes,” Phys. Rev. A 80, 053807 (2009).
[CrossRef]

Lawandy, N. M.

N. M. Lawandy, “Localized surface plasmon singularities in amplifying media,” Appl. Phys. Lett. 85, 5040–5042 (2004).
[CrossRef]

Leonhardt, U.

U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006).
[CrossRef] [PubMed]

U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” New J. Phys. 8, paper 247 (2006).
[CrossRef]

Lequime, M.

J. Zhang, H. Jiang, B. Gralak, S. Enoch, G. Tayeb, and M. Lequime, “Compensation of loss to approach −1 effective index by gain in metal-dielectric stacks,” Eur. Phys. J.: Appl. Phys. 46, 32603–32608 (2009).
[CrossRef]

Li, T.

Z. G. Dong, H. Liu, T. Li, Z. H. Zhu, S. M. Wang, J. X. Cao, S. N. Zhu, and X. Zhang, “Optical loss compensation in a bulk left-handed metamaterial by the gain in quantum dots,” Appl. Phys. Lett. 96, 044104 (2010).
[CrossRef]

Linden, S.

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Low-loss negative-index metamaterials at telecommunication wavelengths,” Opt. Lett. 31, 1800–1802 (2006).
[CrossRef] [PubMed]

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, Th. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. 95, 203901 (2005).
[CrossRef] [PubMed]

Lind-Johansen, Ø.

Ø. Lind-Johansen, K. Seip, and J. Skaar, “The perfect lens on a finite bandwidth,” J. Math. Phys. 50, 012908 (2009).
[CrossRef]

Liu, H.

Z. G. Dong, H. Liu, T. Li, Z. H. Zhu, S. M. Wang, J. X. Cao, S. N. Zhu, and X. Zhang, “Optical loss compensation in a bulk left-handed metamaterial by the gain in quantum dots,” Appl. Phys. Lett. 96, 044104 (2010).
[CrossRef]

Malloy, K. J.

Nistad, B.

B. Nistad and J. Skaar, “Causality and electromagnetic properties of active media,” Phys. Rev. E 78, 036603 (2008).
[CrossRef]

Noginov, M. A.

Osgood, R. M.

Panoiu, N. C.

Pendry, J. B.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[CrossRef] [PubMed]

S. Anantha 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]

S. Anantha Ramakrishna, J. B. Pendry, D. Schurig, D. R. Smith, and S. Schultz, “The asymmetric lossy near-perfect lens,” J. Mod. Opt. 49, 1747–1762 (2002).
[CrossRef]

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

Philbin, T. G.

U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” New J. Phys. 8, paper 247 (2006).
[CrossRef]

Popov, A. K.

Ramakrishna, S. Anantha

S. Anantha 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]

S. Anantha Ramakrishna, J. B. Pendry, D. Schurig, D. R. Smith, and S. Schultz, “The asymmetric lossy near-perfect lens,” J. Mod. Opt. 49, 1747–1762 (2002).
[CrossRef]

Ritzo, B. A.

Rondinone, V.

M. A. Vincenti, D. de Ceglia, V. Rondinone, A. Ladisa, A. D’Orazio, M. J. Bloemer, and M. Scalora, “Loss compensation in metal-dielectric structures in negative-refraction and super-resolving regimes,” Phys. Rev. A 80, 053807 (2009).
[CrossRef]

Sarychev, A. K.

Scalora, M.

M. A. Vincenti, D. de Ceglia, V. Rondinone, A. Ladisa, A. D’Orazio, M. J. Bloemer, and M. Scalora, “Loss compensation in metal-dielectric structures in negative-refraction and super-resolving regimes,” Phys. Rev. A 80, 053807 (2009).
[CrossRef]

Schmidt, F.

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, Th. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. 95, 203901 (2005).
[CrossRef] [PubMed]

Schultz, S.

S. Anantha Ramakrishna, J. B. Pendry, D. Schurig, D. R. Smith, and S. Schultz, “The asymmetric lossy near-perfect lens,” J. Mod. Opt. 49, 1747–1762 (2002).
[CrossRef]

Schurig, D.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[CrossRef] [PubMed]

S. Anantha Ramakrishna, J. B. Pendry, D. Schurig, D. R. Smith, and S. Schultz, “The asymmetric lossy near-perfect lens,” J. Mod. Opt. 49, 1747–1762 (2002).
[CrossRef]

Scully, M. O.

M. O. Scully and M. S. Zubairy, “Atom-field interaction—semiclassical theory,” in Quantum Optics (Cambridge Univ. Press, 1997), pp. 145–192.

Seip, K.

Ø. Lind-Johansen, K. Seip, and J. Skaar, “The perfect lens on a finite bandwidth,” J. Math. Phys. 50, 012908 (2009).
[CrossRef]

Shalaev, V. M.

Sivan, Y.

Skaar, J.

Ø. Lind-Johansen, K. Seip, and J. Skaar, “The perfect lens on a finite bandwidth,” J. Math. Phys. 50, 012908 (2009).
[CrossRef]

B. Nistad and J. Skaar, “Causality and electromagnetic properties of active media,” Phys. Rev. E 78, 036603 (2008).
[CrossRef]

Small, C. E.

Smith, D. R.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[CrossRef] [PubMed]

S. Anantha Ramakrishna, J. B. Pendry, D. Schurig, D. R. Smith, and S. Schultz, “The asymmetric lossy near-perfect lens,” J. Mod. Opt. 49, 1747–1762 (2002).
[CrossRef]

Soukoulis, C. M.

A. Fang, Th. Koschny, M. Wegener, and C. M. Soukoulis, “Self-consistent calculation of metamaterials with gain,” Phys. Rev. B 79, 241104 (2009).
[CrossRef]

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Low-loss negative-index metamaterials at telecommunication wavelengths,” Opt. Lett. 31, 1800–1802 (2006).
[CrossRef] [PubMed]

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, Th. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. 95, 203901 (2005).
[CrossRef] [PubMed]

Svelto, O.

O. Svelto, Principles of Lasers (Plenum, 1998).

Tayeb, G.

J. Zhang, H. Jiang, B. Gralak, S. Enoch, G. Tayeb, and M. Lequime, “Compensation of loss to approach −1 effective index by gain in metal-dielectric stacks,” Eur. Phys. J.: Appl. Phys. 46, 32603–32608 (2009).
[CrossRef]

Veselago, V. G.

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

Vincenti, M. A.

M. A. Vincenti, D. de Ceglia, V. Rondinone, A. Ladisa, A. D’Orazio, M. J. Bloemer, and M. Scalora, “Loss compensation in metal-dielectric structures in negative-refraction and super-resolving regimes,” Phys. Rev. A 80, 053807 (2009).
[CrossRef]

Wang, S. M.

Z. G. Dong, H. Liu, T. Li, Z. H. Zhu, S. M. Wang, J. X. Cao, S. N. Zhu, and X. Zhang, “Optical loss compensation in a bulk left-handed metamaterial by the gain in quantum dots,” Appl. Phys. Lett. 96, 044104 (2010).
[CrossRef]

Wegener, M.

A. Fang, Th. Koschny, M. Wegener, and C. M. Soukoulis, “Self-consistent calculation of metamaterials with gain,” Phys. Rev. B 79, 241104 (2009).
[CrossRef]

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Low-loss negative-index metamaterials at telecommunication wavelengths,” Opt. Lett. 31, 1800–1802 (2006).
[CrossRef] [PubMed]

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, Th. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. 95, 203901 (2005).
[CrossRef] [PubMed]

Werner, D. H.

D. H. Kwon, D. H. Werner, A. V. Kildishev, and V. M. Shalaev, “Dual-band negative-index metamaterials in the near-infrared frequency range,” in Proceedings of IEEE Antennas and Propagation Society International Symposium (IEEE, 2007), pp. 2861–2864.

Xiao, S.

Yuan, H. K.

Zhang, J.

J. Zhang, H. Jiang, B. Gralak, S. Enoch, G. Tayeb, and M. Lequime, “Compensation of loss to approach −1 effective index by gain in metal-dielectric stacks,” Eur. Phys. J.: Appl. Phys. 46, 32603–32608 (2009).
[CrossRef]

Zhang, S.

Zhang, X.

Z. G. Dong, H. Liu, T. Li, Z. H. Zhu, S. M. Wang, J. X. Cao, S. N. Zhu, and X. Zhang, “Optical loss compensation in a bulk left-handed metamaterial by the gain in quantum dots,” Appl. Phys. Lett. 96, 044104 (2010).
[CrossRef]

Zhou, J. F.

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, Th. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. 95, 203901 (2005).
[CrossRef] [PubMed]

Zhu, G.

Zhu, S. N.

Z. G. Dong, H. Liu, T. Li, Z. H. Zhu, S. M. Wang, J. X. Cao, S. N. Zhu, and X. Zhang, “Optical loss compensation in a bulk left-handed metamaterial by the gain in quantum dots,” Appl. Phys. Lett. 96, 044104 (2010).
[CrossRef]

Zhu, Z. H.

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Figures (6)

Fig. 1
Fig. 1

Perfect lens in vacuum. The parameter d is the thickness of the lens, and a and b are the distances from the source to the lens, and from the lens to the image plane, respectively. The parameters are governed by the equation d = a + b . The numbers 1 through N indicate the different slices. The lens is considered to be infinite in the x y -plane.

Fig. 2
Fig. 2

The absolute value of the transmission coefficient when ω 0 / c = 1 (normalized), ω 0 d / c = 2 π / 10 , a = b = d / 2 , Im   χ p ( ω 0 ) = 0.05 , and N = 20 : (a) Noncompensated lens; (b) E s = 10 , Δ χ = 0 ; (c) E s = 4 , Δ χ = 0 ; (d) E s = 10 , Δ χ = 0.015 ; (e) E s = 10 , Δ χ = 0 , two waves, k x and k x , both having amplitude of 1/2.

Fig. 3
Fig. 3

The resolution of the lens as a function of the saturation constant. The resolution is defined as the k x -value where the transmission equals 1/2. Parameters: ω 0 / c = 1 , ω 0 d / c = 2 π / 10 , Im   χ p ( ω 0 ) = 0.05 , Δ χ = 0 , a = b = d / 2 , and N = 20 .

Fig. 4
Fig. 4

The absolute value of the reflection coefficient at the source plane after convergence, for the same cases as those in Fig. 2.

Fig. 5
Fig. 5

The distribution of the two components of the evanescent field in the lens, for one wave with k x = 5.1408 . The distance is normalized with respect to lens thickness d. Parameters: ω 0 / c = 1 , ω 0 d / c = 2 π / 10 , Im   χ p ( ω 0 ) = 0.05 , Δ χ = 0 , E s = 10 , a = b = d / 2 , and N = 20 . The solid line shows the absolute value of the nonzero component of h + , and the dotted line shows the absolute value of the nonzero component of h .

Fig. 6
Fig. 6

The absolute value of the transmitted magnetic field at the image plane, when the source consists of two slits. Parameters: ω 0 / c = 1 , ω 0 d / c = 2 π / 10 , Im   χ p ( ω 0 ) = 0.05 , Δ χ = 0 , a = b = d / 2 , and N = 20 : (a) The incident magnetic field at the source, (b) E s = 0.1 , and (c) E s = 50 .

Equations (27)

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ϵ ( ω ) = 1 + χ p ( ω ) + χ a ( ω ) ,
χ a ( ω ) = A ( ω ) ( ω ω 0 γ i ) 1 + | E | 2 E s ( ω ) 2 .
χ a ( ω 0 ) = i A ( ω 0 ) 1 + | E | 2 E s ( ω 0 ) 2 .
Δ χ = Im   χ p ( ω 0 ) A ( ω 0 ) .
E ( x , z ) = 1 i ω ϵ ( x , z ) ϵ 0 × H ( x , z ) .
H ( x , z ) = H ( x , z ) y ̂ = m h m ( z ) exp ( i k x m x ) y ̂ ,
ϵ ( x , z ) = m e m ( z ) exp ( i k x m x ) ,
2 H + ϵ μ k 2 H 1 ϵ ϵ x H x = 0 ,
1 ϵ ( x ) = m Q m   exp ( i k x m x ) ,
1 ϵ ( x ) ϵ ( x ) x = m F m   exp ( i k x m x ) .
F m = m Q m m i k x m e m ,
F = i Q k x e ,
d 2 h m ( z ) d z 2 k x m 2 h m ( z ) + k 2 μ m ε m m ( z ) h m ( z ) m i F m m k x m h m = 0 ,
d 2 h ( z ) d z 2 + [ k z 2 + V ] h ( z ) = 0 ,
V = k 2 I + k 2 μ G F .
d h + d z = i k z h + + i ( 2 k z ) 1 V ( h + + h ) ,
d h d z = i k z h i ( 2 k z ) 1 V ( h + + h ) ,
d h d z = i k z ( h + h ) ,
Ψ = [ h + h ] ,     C = [ i k z + i ( 2 k z ) 1 V i ( 2 k z ) 1 V i ( 2 k z ) 1 V i k z i ( 2 k z ) 1 V ] ,
d Ψ d z = C Ψ .
Ψ ( z b ) = exp { ( z b z a ) C } Ψ ( z a ) ,
Ψ ( z j + Δ j ) = exp { Δ j C j } Ψ ( z j ) .
M j = exp { Δ j C j } .
H j + 1 ( z j + 1 ) z = ϵ j + 1 ϵ j H j ( z j + 1 ) z .
d h j d z ( z j + 1 ) = P j ( z j + 1 ) d h j + 1 d z ( z j + 1 ) ,
[ h j + 1 + h j + 1 ] = 1 2 [ I + k z 1 P j k z I k z 1 P j k z I k z 1 P j k z I + k z 1 P j k z ] [ h j + h j ] .
[ T 0 ] = M N + 1 j = N 0 ( P j M j ) [ I R ] .

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