Abstract

A new route toward a lossless superlens has been proposed recently. It relies on the association of two phase-conjugating sheets. The aim of this study is to show how such a lens can be implemented experimentally at optical frequencies. Because efficient phase conjugation of evanescent waves is illusory with the current technology, only the case of propagating waves is considered here. Four wave mixing in BaTiO3 is shown to provide efficient backward and forward phase conjugation over a major part of the angular spectrum, taking advantage of internal reflections inside the non-linear slab. However, phase distortions arise for high spatial frequencies and limit the resolving power of the device. The addition of a second phase-conjugator automatically compensates for these phase distortions. The wave field is then perfectly translated through the system. Actually, such a device performs even better than a negative refracting lens since the association of two phase-conjugating mirrors behaves like a resonant cavity. An amplification of the wave field by a factor of 102 in intensity is predicted, despite the important absorption in BaTiO3.

© 2009 Optical Society of America

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2008 (1)

J. B. Pendry, “Time reversal and negative refraction,” Science 322, 71-73 (2008).
[CrossRef] [PubMed]

2006 (1)

C. M. Soukoulis, M. Kafesaki, and E. N. Economou, “Negative-index materials: new frontiers in optics,” Adv. Mater. 18, 1941-1952 (2006).
[CrossRef]

2005 (2)

S. A. Ramakrishna, “Physics of negative refractive index materials,” Rep. Prog. Phys. 68, 449-521 (2005).
[CrossRef]

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308, 534-537 (2005).
[CrossRef] [PubMed]

2004 (1)

A. Grbic and G. V. Eleftheriades, “Overcoming the diffraction limit with a planar left-handed transmission-line lens,” Phys. Rev. Lett. 92, 117403 (2004).
[CrossRef] [PubMed]

2003 (4)

A. L. Pokrovsky and A. L. Efros, “Diffraction theory and focusing of light by a slab of left-handed material,” Physica B 338, 333-337 (2003).
[CrossRef]

S. Maslovski and S. Tretyakov, “Phase conjugation and perfect lensing,” J. Appl. Phys. 94, 4241-4243 (2003).
[CrossRef]

D. R. Smith, D. Schurig, M. Rosenbluth, S. Schultz, S. A. Ramakrishna, and J. B. Pendry, “Limitations on subdiffraction imaging with a negative refractive index slab,” Appl. Phys. Lett. 82, 1506-1508 (2003).
[CrossRef]

Z. Ye, “Optical transmission and reflection of perfect lenses by left handed materials,” Phys. Rev. B 67, 193106 (2003).
[CrossRef]

2002 (2)

J. T. Shen and P. M. Platzman, “Near field imaging with negative dielectric constant lenses,” Appl. Phys. Lett. 80, 3286-3288 (2002).
[CrossRef]

G. S. He, “Optical phase conjugation: principles, techniques, and applications,” Prog. Quantum Electron. 26, 131-191 (2002).
[CrossRef]

2000 (1)

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

1998 (1)

B. Vohnsen and S. I. Bozhevolnyi, “Near-field optical microscopy with a phase-conjugating mirror,” Opt. Commun. 148, 331-337 (1998).
[CrossRef]

1997 (3)

K. S. Syed, G. J. Crofts, R. P. M. Green, and M. J. Damzen, “Vectorial phase conjugation via four-wave mixing in isotropic saturable-gain media,” J. Opt. Soc. Am. B 14, 2067-2078 (1997).
[CrossRef]

F. Wang, L. Liu, and G. Li, “Self-pumped forward and backward phase conjugator with Cu-doped KNSBN crystal: caused by scattering-oscillation-amplification,” Quantum Opt. 134, 195-198 (1997).
[CrossRef]

M. Saito, A. Okamoto, K. Sato, and Y. Takayama, “Phase matching property of cross polarization four wave mixing in BaTiO3 crystal,” Opt. Rev. 4, 686-690 (1997).
[CrossRef]

1996 (1)

1995 (3)

1994 (2)

S. I. Bozhevolnyi, O. Keller, and I. I. Smolyaninov, “Phase conjugation of an optical near field,” Opt. Lett. 19, 1601-1603 (1994).
[CrossRef] [PubMed]

M. Zgonik, P. Bernasconi, M. Duelli, R. Schlesser, P. Günter, M. H. Garrett, D. Rytz, Y. Zhu, and X. Wu, “Dielectric, elastic, piezoelectric, electro-optic, and elasto-optic tensors of BaTiO3 crystals,” Phys. Rev. B 50, 5941-5949 (1994).
[CrossRef]

1992 (2)

1991 (1)

S. H. Tang, X. H. He, and H.-Y. Zhang, “Experimental study of the forward phase-conjugate wave in degenerate four-wave mixing in LiNbO3:Fe,” J. Eur. Opt. Soc. Part B 3, 179-183 (1991).
[CrossRef]

1987 (1)

A. Bledowski, W. Krolikowski, and A. Kujawski, “Forward phase-conjugate wave in four-wave mixing in photorefractive media,” Opt. Commun. 61, 71-74 (1987).
[CrossRef]

1985 (1)

1984 (2)

M. Cronin-Golomb, B. Fisher, J. O. White, and A. Yariv, “Theory and applications of four-wave mixing in photorefractive media,” IEEE J. Quantum Electron. 20, 12-30 (1984).
[CrossRef]

A. Khyznika, V. Kondilenko, Y. Kucherov, S. Lesnik, S. Odoulov, and M. Soskin, “Phase conjugation by degenerate forward four-wave mixing,” J. Opt. Soc. Am. A 1, 169-175 (1984).
[CrossRef]

1982 (1)

J. O. White, M. Cronin-Golomb, B. Fischer, and A. Yariv, “Coherent oscillation by self-induced gratings in the photorefractive crystal BaTiO3,” Appl. Phys. Lett. 40, 450-452 (1982).
[CrossRef]

1981 (1)

Y. Silberberg and I. Bar-Joseph, “Low power phase conjugation in thin films of saturable absorbers,” Opt. Commun. 39, 265-268 (1981).
[CrossRef]

1980 (3)

1979 (4)

J. P. Huignard, J. P. Herriau, P. Aubourg, and E. Spitz, “Phase-conjugate wavefront generation via real-time holography in Bi12SiO20 crystals,” Opt. Lett. 4, 21-23 (1979).
[CrossRef] [PubMed]

A. Tomita, “Phase conjugation using gain saturation of a Nd:YAG laser,” Appl. Phys. Lett. 34, 463-464 (1979).
[CrossRef]

R. A. Fisher and B. J. Feldman, “On-resonant phase-conjugate reflection and amplification at 10.6 μm in inverted CO2,” Opt. Lett. 4, 140-142 (1979).
[CrossRef] [PubMed]

N. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskinand, and V. L. Vinetskii, “Holographic storage ion electro-optic crystals. I. Steady-state,” Ferroelectrics 22, 949-960 (1979).
[CrossRef]

1978 (4)

R. L. Abrams and R. C. Lind, “Degenerate four wave mixing in absorbing media,” Opt. Lett. 2, 94-96 (1978).
[CrossRef] [PubMed]

D. M. Bloom, P. F. Liao, and N. P. Economou, “Observation of amplified reflection by degenerate four-wave mixing in atomic sodium vapor,” Opt. Lett. 2, 58-60 (1978).
[CrossRef] [PubMed]

D. M. Pepper, D. Fekete, and A. Yariv, “Observation of amplified phase-conjugate reflection and optical parametric oscillation by degenerate four-wave mixing in a transparent medium,” Appl. Phys. Lett. 33, 41-44 (1978).
[CrossRef]

A. Yariv, “Phase conjugation,” IEEE J. Quantum Electron. 14, 650-660 (1978).
[CrossRef]

1977 (3)

D. M. Bloom and G. C. Bjorklund, “Conjugate wave-front generation and image reconstruction by four-wave mixing,” Appl. Phys. Lett. 31, 592-594 (1977).
[CrossRef]

R. W. Hellwarth, “Generation of time-reversed wave fronts by nonlinear refraction,” J. Opt. Soc. Am. 67, 1-3 (1977).
[CrossRef]

P. V. Avizonis, F. A. Hopf, W. D. Bomberger, S. F. Jacobs, A. Tomita, and K. H. Womack, “Optical phase conjugation in a lithium formate crystal,” Appl. Phys. Lett. 31, 435-437 (1977).
[CrossRef]

1968 (1)

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

Abrams, R. L.

Aubourg, P.

AuYeung, J. C.

Avizonis, P. V.

P. V. Avizonis, F. A. Hopf, W. D. Bomberger, S. F. Jacobs, A. Tomita, and K. H. Womack, “Optical phase conjugation in a lithium formate crystal,” Appl. Phys. Lett. 31, 435-437 (1977).
[CrossRef]

Balmain, K. G.

G. V. Eleftheriades and K. G. Balmain, Negative-Refraction Metamaterials: Fundamental Principles and Applications (Wiley-IEEE, 2005).
[CrossRef]

Bar-Joseph, I.

Y. Silberberg and I. Bar-Joseph, “Low power phase conjugation in thin films of saturable absorbers,” Opt. Commun. 39, 265-268 (1981).
[CrossRef]

Barthelemy, A.

Bernasconi, P.

M. Zgonik, P. Bernasconi, M. Duelli, R. Schlesser, P. Günter, M. H. Garrett, D. Rytz, Y. Zhu, and X. Wu, “Dielectric, elastic, piezoelectric, electro-optic, and elasto-optic tensors of BaTiO3 crystals,” Phys. Rev. B 50, 5941-5949 (1994).
[CrossRef]

Bjorklund, G. C.

D. M. Bloom and G. C. Bjorklund, “Conjugate wave-front generation and image reconstruction by four-wave mixing,” Appl. Phys. Lett. 31, 592-594 (1977).
[CrossRef]

Bledowski, A.

A. Bledowski, W. Krolikowski, and A. Kujawski, “Forward phase-conjugate wave in four-wave mixing in photorefractive media,” Opt. Commun. 61, 71-74 (1987).
[CrossRef]

Bloom, D. M.

D. M. Bloom, P. F. Liao, and N. P. Economou, “Observation of amplified reflection by degenerate four-wave mixing in atomic sodium vapor,” Opt. Lett. 2, 58-60 (1978).
[CrossRef] [PubMed]

D. M. Bloom and G. C. Bjorklund, “Conjugate wave-front generation and image reconstruction by four-wave mixing,” Appl. Phys. Lett. 31, 592-594 (1977).
[CrossRef]

Bomberger, W. D.

P. V. Avizonis, F. A. Hopf, W. D. Bomberger, S. F. Jacobs, A. Tomita, and K. H. Womack, “Optical phase conjugation in a lithium formate crystal,” Appl. Phys. Lett. 31, 435-437 (1977).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principle of Optics (Pergamon, 1980).

Bozhevolnyi, S. I.

B. Vohnsen and S. I. Bozhevolnyi, “Near-field optical microscopy with a phase-conjugating mirror,” Opt. Commun. 148, 331-337 (1998).
[CrossRef]

S. I. Bozhevolnyi, O. Keller, and I. I. Smolyaninov, “Scattered light enhancement near a phase conjugating mirror,” Opt. Commun. 115, 115-120 (1995).
[CrossRef]

S. I. Bozhevolnyi, O. Keller, and I. I. Smolyaninov, “Phase conjugation of an optical near field,” Opt. Lett. 19, 1601-1603 (1994).
[CrossRef] [PubMed]

Crofts, G. J.

Cronin-Golomb, M.

M. Cronin-Golomb, B. Fisher, J. O. White, and A. Yariv, “Theory and applications of four-wave mixing in photorefractive media,” IEEE J. Quantum Electron. 20, 12-30 (1984).
[CrossRef]

J. O. White, M. Cronin-Golomb, B. Fischer, and A. Yariv, “Coherent oscillation by self-induced gratings in the photorefractive crystal BaTiO3,” Appl. Phys. Lett. 40, 450-452 (1982).
[CrossRef]

Cruz, S. -C. D. L.

Damzen, M. J.

Duelli, M.

M. Zgonik, P. Bernasconi, M. Duelli, R. Schlesser, P. Günter, M. H. Garrett, D. Rytz, Y. Zhu, and X. Wu, “Dielectric, elastic, piezoelectric, electro-optic, and elasto-optic tensors of BaTiO3 crystals,” Phys. Rev. B 50, 5941-5949 (1994).
[CrossRef]

Economou, E. N.

C. M. Soukoulis, M. Kafesaki, and E. N. Economou, “Negative-index materials: new frontiers in optics,” Adv. Mater. 18, 1941-1952 (2006).
[CrossRef]

Economou, N. P.

Efros, A. L.

A. L. Pokrovsky and A. L. Efros, “Diffraction theory and focusing of light by a slab of left-handed material,” Physica B 338, 333-337 (2003).
[CrossRef]

Eleftheriades, G. V.

A. Grbic and G. V. Eleftheriades, “Overcoming the diffraction limit with a planar left-handed transmission-line lens,” Phys. Rev. Lett. 92, 117403 (2004).
[CrossRef] [PubMed]

G. V. Eleftheriades and K. G. Balmain, Negative-Refraction Metamaterials: Fundamental Principles and Applications (Wiley-IEEE, 2005).
[CrossRef]

Fang, N.

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308, 534-537 (2005).
[CrossRef] [PubMed]

Feinberg, J.

Fekete, D.

D. Fekete, J. C. AuYeung, and A. Yariv, “Phase-conjugate reflection by degenerate four-wave mixing in a nematic liquid crystal in the isotropic phase,” Opt. Lett. 5, 51-53 (1980).
[CrossRef] [PubMed]

D. M. Pepper, D. Fekete, and A. Yariv, “Observation of amplified phase-conjugate reflection and optical parametric oscillation by degenerate four-wave mixing in a transparent medium,” Appl. Phys. Lett. 33, 41-44 (1978).
[CrossRef]

Feldman, B. J.

Fischer, B.

J. O. White, M. Cronin-Golomb, B. Fischer, and A. Yariv, “Coherent oscillation by self-induced gratings in the photorefractive crystal BaTiO3,” Appl. Phys. Lett. 40, 450-452 (1982).
[CrossRef]

Fisher, B.

M. Cronin-Golomb, B. Fisher, J. O. White, and A. Yariv, “Theory and applications of four-wave mixing in photorefractive media,” IEEE J. Quantum Electron. 20, 12-30 (1984).
[CrossRef]

Fisher, R.

R. Fisher, Optical Phase Conjugation (Academic, 1984).

Fisher, R. A.

Garrett, M. H.

M. Zgonik, P. Bernasconi, M. Duelli, R. Schlesser, P. Günter, M. H. Garrett, D. Rytz, Y. Zhu, and X. Wu, “Dielectric, elastic, piezoelectric, electro-optic, and elasto-optic tensors of BaTiO3 crystals,” Phys. Rev. B 50, 5941-5949 (1994).
[CrossRef]

Grbic, A.

A. Grbic and G. V. Eleftheriades, “Overcoming the diffraction limit with a planar left-handed transmission-line lens,” Phys. Rev. Lett. 92, 117403 (2004).
[CrossRef] [PubMed]

Green, R. P. M.

Günter, P.

M. Zgonik, K. Nakagawa, and P. Günter, “Electro-optic and dielectric properties of photorefractive BaTiO3 and KNbO3,” J. Opt. Soc. Am. B 12, 1416-1421 (1995).
[CrossRef]

M. Zgonik, P. Bernasconi, M. Duelli, R. Schlesser, P. Günter, M. H. Garrett, D. Rytz, Y. Zhu, and X. Wu, “Dielectric, elastic, piezoelectric, electro-optic, and elasto-optic tensors of BaTiO3 crystals,” Phys. Rev. B 50, 5941-5949 (1994).
[CrossRef]

P. Günter and J.-P. Huignard, Photorefractive Materials and Their Applications I (Springer-Verlag, 1988).

He, G. S.

G. S. He, “Optical phase conjugation: principles, techniques, and applications,” Prog. Quantum Electron. 26, 131-191 (2002).
[CrossRef]

He, Q. B.

He, X. H.

S. H. Tang, X. H. He, and H.-Y. Zhang, “Experimental study of the forward phase-conjugate wave in degenerate four-wave mixing in LiNbO3:Fe,” J. Eur. Opt. Soc. Part B 3, 179-183 (1991).
[CrossRef]

Heiman, D.

J. Feinberg, D. Heiman, J. A. R. Tanguay, and R. W. Hellwarth, “Photorefractive effects and light-induced charge migration in barium titanate,” J. Appl. Phys. 51, 1297-1305 (1980).
[CrossRef]

Hellwarth, R. W.

Herriau, J. P.

Hopf, F. A.

P. V. Avizonis, F. A. Hopf, W. D. Bomberger, S. F. Jacobs, A. Tomita, and K. H. Womack, “Optical phase conjugation in a lithium formate crystal,” Appl. Phys. Lett. 31, 435-437 (1977).
[CrossRef]

Huignard, J. P.

Huignard, J. -P.

P. Günter and J.-P. Huignard, Photorefractive Materials and Their Applications I (Springer-Verlag, 1988).

Jacobs, S. F.

P. V. Avizonis, F. A. Hopf, W. D. Bomberger, S. F. Jacobs, A. Tomita, and K. H. Womack, “Optical phase conjugation in a lithium formate crystal,” Appl. Phys. Lett. 31, 435-437 (1977).
[CrossRef]

Kafesaki, M.

C. M. Soukoulis, M. Kafesaki, and E. N. Economou, “Negative-index materials: new frontiers in optics,” Adv. Mater. 18, 1941-1952 (2006).
[CrossRef]

Keller, O.

S. I. Bozhevolnyi, O. Keller, and I. I. Smolyaninov, “Scattered light enhancement near a phase conjugating mirror,” Opt. Commun. 115, 115-120 (1995).
[CrossRef]

S. I. Bozhevolnyi, O. Keller, and I. I. Smolyaninov, “Phase conjugation of an optical near field,” Opt. Lett. 19, 1601-1603 (1994).
[CrossRef] [PubMed]

Khyznika, A.

Kondilenko, V.

Krolikowski, W.

A. Bledowski, W. Krolikowski, and A. Kujawski, “Forward phase-conjugate wave in four-wave mixing in photorefractive media,” Opt. Commun. 61, 71-74 (1987).
[CrossRef]

Kucherov, Y.

Kujawski, A.

A. Bledowski, W. Krolikowski, and A. Kujawski, “Forward phase-conjugate wave in four-wave mixing in photorefractive media,” Opt. Commun. 61, 71-74 (1987).
[CrossRef]

Kukhtarev, N.

N. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskinand, and V. L. Vinetskii, “Holographic storage ion electro-optic crystals. I. Steady-state,” Ferroelectrics 22, 949-960 (1979).
[CrossRef]

Lee, H.

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308, 534-537 (2005).
[CrossRef] [PubMed]

Lefort, L.

Lesnik, S.

Li, G.

F. Wang, L. Liu, and G. Li, “Self-pumped forward and backward phase conjugator with Cu-doped KNSBN crystal: caused by scattering-oscillation-amplification,” Quantum Opt. 134, 195-198 (1997).
[CrossRef]

Liao, P. F.

Lind, R. C.

Liu, H. -K.

Liu, L.

F. Wang, L. Liu, and G. Li, “Self-pumped forward and backward phase conjugator with Cu-doped KNSBN crystal: caused by scattering-oscillation-amplification,” Quantum Opt. 134, 195-198 (1997).
[CrossRef]

MacCormack, S.

Markov, V. B.

N. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskinand, and V. L. Vinetskii, “Holographic storage ion electro-optic crystals. I. Steady-state,” Ferroelectrics 22, 949-960 (1979).
[CrossRef]

Maslovski, S.

S. Maslovski and S. Tretyakov, “Phase conjugation and perfect lensing,” J. Appl. Phys. 94, 4241-4243 (2003).
[CrossRef]

Nakagawa, K.

Nieto-Vesperinas, M.

Odoulov, S.

Odulov, S. G.

N. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskinand, and V. L. Vinetskii, “Holographic storage ion electro-optic crystals. I. Steady-state,” Ferroelectrics 22, 949-960 (1979).
[CrossRef]

Okamoto, A.

M. Saito, A. Okamoto, K. Sato, and Y. Takayama, “Phase matching property of cross polarization four wave mixing in BaTiO3 crystal,” Opt. Rev. 4, 686-690 (1997).
[CrossRef]

Pendry, J. B.

J. B. Pendry, “Time reversal and negative refraction,” Science 322, 71-73 (2008).
[CrossRef] [PubMed]

D. R. Smith, D. Schurig, M. Rosenbluth, S. Schultz, S. A. Ramakrishna, and J. B. Pendry, “Limitations on subdiffraction imaging with a negative refractive index slab,” Appl. Phys. Lett. 82, 1506-1508 (2003).
[CrossRef]

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

Pepper, D. M.

D. M. Pepper, D. Fekete, and A. Yariv, “Observation of amplified phase-conjugate reflection and optical parametric oscillation by degenerate four-wave mixing in a transparent medium,” Appl. Phys. Lett. 33, 41-44 (1978).
[CrossRef]

Platzman, P. M.

J. T. Shen and P. M. Platzman, “Near field imaging with negative dielectric constant lenses,” Appl. Phys. Lett. 80, 3286-3288 (2002).
[CrossRef]

Pokrovsky, A. L.

A. L. Pokrovsky and A. L. Efros, “Diffraction theory and focusing of light by a slab of left-handed material,” Physica B 338, 333-337 (2003).
[CrossRef]

Ramakrishna, S. A.

S. A. Ramakrishna, “Physics of negative refractive index materials,” Rep. Prog. Phys. 68, 449-521 (2005).
[CrossRef]

D. R. Smith, D. Schurig, M. Rosenbluth, S. Schultz, S. A. Ramakrishna, and J. B. Pendry, “Limitations on subdiffraction imaging with a negative refractive index slab,” Appl. Phys. Lett. 82, 1506-1508 (2003).
[CrossRef]

Rosenbluth, M.

D. R. Smith, D. Schurig, M. Rosenbluth, S. Schultz, S. A. Ramakrishna, and J. B. Pendry, “Limitations on subdiffraction imaging with a negative refractive index slab,” Appl. Phys. Lett. 82, 1506-1508 (2003).
[CrossRef]

Rytz, D.

M. Zgonik, P. Bernasconi, M. Duelli, R. Schlesser, P. Günter, M. H. Garrett, D. Rytz, Y. Zhu, and X. Wu, “Dielectric, elastic, piezoelectric, electro-optic, and elasto-optic tensors of BaTiO3 crystals,” Phys. Rev. B 50, 5941-5949 (1994).
[CrossRef]

Saito, M.

M. Saito, A. Okamoto, K. Sato, and Y. Takayama, “Phase matching property of cross polarization four wave mixing in BaTiO3 crystal,” Opt. Rev. 4, 686-690 (1997).
[CrossRef]

Sato, K.

M. Saito, A. Okamoto, K. Sato, and Y. Takayama, “Phase matching property of cross polarization four wave mixing in BaTiO3 crystal,” Opt. Rev. 4, 686-690 (1997).
[CrossRef]

Schlesser, R.

M. Zgonik, P. Bernasconi, M. Duelli, R. Schlesser, P. Günter, M. H. Garrett, D. Rytz, Y. Zhu, and X. Wu, “Dielectric, elastic, piezoelectric, electro-optic, and elasto-optic tensors of BaTiO3 crystals,” Phys. Rev. B 50, 5941-5949 (1994).
[CrossRef]

Schultz, S.

D. R. Smith, D. Schurig, M. Rosenbluth, S. Schultz, S. A. Ramakrishna, and J. B. Pendry, “Limitations on subdiffraction imaging with a negative refractive index slab,” Appl. Phys. Lett. 82, 1506-1508 (2003).
[CrossRef]

Schurig, D.

D. R. Smith, D. Schurig, M. Rosenbluth, S. Schultz, S. A. Ramakrishna, and J. B. Pendry, “Limitations on subdiffraction imaging with a negative refractive index slab,” Appl. Phys. Lett. 82, 1506-1508 (2003).
[CrossRef]

Shen, J. T.

J. T. Shen and P. M. Platzman, “Near field imaging with negative dielectric constant lenses,” Appl. Phys. Lett. 80, 3286-3288 (2002).
[CrossRef]

Silberberg, Y.

Y. Silberberg and I. Bar-Joseph, “Low power phase conjugation in thin films of saturable absorbers,” Opt. Commun. 39, 265-268 (1981).
[CrossRef]

Smith, D. R.

D. R. Smith, D. Schurig, M. Rosenbluth, S. Schultz, S. A. Ramakrishna, and J. B. Pendry, “Limitations on subdiffraction imaging with a negative refractive index slab,” Appl. Phys. Lett. 82, 1506-1508 (2003).
[CrossRef]

Smolyaninov, I. I.

S. I. Bozhevolnyi, O. Keller, and I. I. Smolyaninov, “Scattered light enhancement near a phase conjugating mirror,” Opt. Commun. 115, 115-120 (1995).
[CrossRef]

S. I. Bozhevolnyi, O. Keller, and I. I. Smolyaninov, “Phase conjugation of an optical near field,” Opt. Lett. 19, 1601-1603 (1994).
[CrossRef] [PubMed]

Soskin, M.

Soskinand, M. S.

N. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskinand, and V. L. Vinetskii, “Holographic storage ion electro-optic crystals. I. Steady-state,” Ferroelectrics 22, 949-960 (1979).
[CrossRef]

Soukoulis, C. M.

C. M. Soukoulis, M. Kafesaki, and E. N. Economou, “Negative-index materials: new frontiers in optics,” Adv. Mater. 18, 1941-1952 (2006).
[CrossRef]

Spitz, E.

Sun, C.

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308, 534-537 (2005).
[CrossRef] [PubMed]

Syed, K. S.

Takayama, Y.

M. Saito, A. Okamoto, K. Sato, and Y. Takayama, “Phase matching property of cross polarization four wave mixing in BaTiO3 crystal,” Opt. Rev. 4, 686-690 (1997).
[CrossRef]

Tang, S. H.

S. H. Tang, X. H. He, and H.-Y. Zhang, “Experimental study of the forward phase-conjugate wave in degenerate four-wave mixing in LiNbO3:Fe,” J. Eur. Opt. Soc. Part B 3, 179-183 (1991).
[CrossRef]

Tanguay, J. A. R.

J. Feinberg, D. Heiman, J. A. R. Tanguay, and R. W. Hellwarth, “Photorefractive effects and light-induced charge migration in barium titanate,” J. Appl. Phys. 51, 1297-1305 (1980).
[CrossRef]

Tomita, A.

A. Tomita, “Phase conjugation using gain saturation of a Nd:YAG laser,” Appl. Phys. Lett. 34, 463-464 (1979).
[CrossRef]

P. V. Avizonis, F. A. Hopf, W. D. Bomberger, S. F. Jacobs, A. Tomita, and K. H. Womack, “Optical phase conjugation in a lithium formate crystal,” Appl. Phys. Lett. 31, 435-437 (1977).
[CrossRef]

Tretyakov, S.

S. Maslovski and S. Tretyakov, “Phase conjugation and perfect lensing,” J. Appl. Phys. 94, 4241-4243 (2003).
[CrossRef]

Veselago, V. G.

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

Vinetskii, V. L.

N. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskinand, and V. L. Vinetskii, “Holographic storage ion electro-optic crystals. I. Steady-state,” Ferroelectrics 22, 949-960 (1979).
[CrossRef]

Vohnsen, B.

B. Vohnsen and S. I. Bozhevolnyi, “Near-field optical microscopy with a phase-conjugating mirror,” Opt. Commun. 148, 331-337 (1998).
[CrossRef]

Wang, F.

F. Wang, L. Liu, and G. Li, “Self-pumped forward and backward phase conjugator with Cu-doped KNSBN crystal: caused by scattering-oscillation-amplification,” Quantum Opt. 134, 195-198 (1997).
[CrossRef]

White, J. O.

M. Cronin-Golomb, B. Fisher, J. O. White, and A. Yariv, “Theory and applications of four-wave mixing in photorefractive media,” IEEE J. Quantum Electron. 20, 12-30 (1984).
[CrossRef]

J. O. White, M. Cronin-Golomb, B. Fischer, and A. Yariv, “Coherent oscillation by self-induced gratings in the photorefractive crystal BaTiO3,” Appl. Phys. Lett. 40, 450-452 (1982).
[CrossRef]

Wolf, E.

Womack, K. H.

P. V. Avizonis, F. A. Hopf, W. D. Bomberger, S. F. Jacobs, A. Tomita, and K. H. Womack, “Optical phase conjugation in a lithium formate crystal,” Appl. Phys. Lett. 31, 435-437 (1977).
[CrossRef]

Wu, X.

M. Zgonik, P. Bernasconi, M. Duelli, R. Schlesser, P. Günter, M. H. Garrett, D. Rytz, Y. Zhu, and X. Wu, “Dielectric, elastic, piezoelectric, electro-optic, and elasto-optic tensors of BaTiO3 crystals,” Phys. Rev. B 50, 5941-5949 (1994).
[CrossRef]

Yariv, A.

M. Cronin-Golomb, B. Fisher, J. O. White, and A. Yariv, “Theory and applications of four-wave mixing in photorefractive media,” IEEE J. Quantum Electron. 20, 12-30 (1984).
[CrossRef]

J. O. White, M. Cronin-Golomb, B. Fischer, and A. Yariv, “Coherent oscillation by self-induced gratings in the photorefractive crystal BaTiO3,” Appl. Phys. Lett. 40, 450-452 (1982).
[CrossRef]

D. Fekete, J. C. AuYeung, and A. Yariv, “Phase-conjugate reflection by degenerate four-wave mixing in a nematic liquid crystal in the isotropic phase,” Opt. Lett. 5, 51-53 (1980).
[CrossRef] [PubMed]

A. Yariv, “Phase conjugation,” IEEE J. Quantum Electron. 14, 650-660 (1978).
[CrossRef]

D. M. Pepper, D. Fekete, and A. Yariv, “Observation of amplified phase-conjugate reflection and optical parametric oscillation by degenerate four-wave mixing in a transparent medium,” Appl. Phys. Lett. 33, 41-44 (1978).
[CrossRef]

A. Yariv, Quantum Electronics (Wiley, 1989).

Ye, Z.

Z. Ye, “Optical transmission and reflection of perfect lenses by left handed materials,” Phys. Rev. B 67, 193106 (2003).
[CrossRef]

Yeh, P.

Zgonik, M.

M. Zgonik, K. Nakagawa, and P. Günter, “Electro-optic and dielectric properties of photorefractive BaTiO3 and KNbO3,” J. Opt. Soc. Am. B 12, 1416-1421 (1995).
[CrossRef]

M. Zgonik, P. Bernasconi, M. Duelli, R. Schlesser, P. Günter, M. H. Garrett, D. Rytz, Y. Zhu, and X. Wu, “Dielectric, elastic, piezoelectric, electro-optic, and elasto-optic tensors of BaTiO3 crystals,” Phys. Rev. B 50, 5941-5949 (1994).
[CrossRef]

Zhang, H. -Y.

S. H. Tang, X. H. He, and H.-Y. Zhang, “Experimental study of the forward phase-conjugate wave in degenerate four-wave mixing in LiNbO3:Fe,” J. Eur. Opt. Soc. Part B 3, 179-183 (1991).
[CrossRef]

Zhang, X.

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308, 534-537 (2005).
[CrossRef] [PubMed]

Zhu, Y.

M. Zgonik, P. Bernasconi, M. Duelli, R. Schlesser, P. Günter, M. H. Garrett, D. Rytz, Y. Zhu, and X. Wu, “Dielectric, elastic, piezoelectric, electro-optic, and elasto-optic tensors of BaTiO3 crystals,” Phys. Rev. B 50, 5941-5949 (1994).
[CrossRef]

Adv. Mater. (1)

C. M. Soukoulis, M. Kafesaki, and E. N. Economou, “Negative-index materials: new frontiers in optics,” Adv. Mater. 18, 1941-1952 (2006).
[CrossRef]

Appl. Phys. Lett. (7)

J. T. Shen and P. M. Platzman, “Near field imaging with negative dielectric constant lenses,” Appl. Phys. Lett. 80, 3286-3288 (2002).
[CrossRef]

D. R. Smith, D. Schurig, M. Rosenbluth, S. Schultz, S. A. Ramakrishna, and J. B. Pendry, “Limitations on subdiffraction imaging with a negative refractive index slab,” Appl. Phys. Lett. 82, 1506-1508 (2003).
[CrossRef]

D. M. Bloom and G. C. Bjorklund, “Conjugate wave-front generation and image reconstruction by four-wave mixing,” Appl. Phys. Lett. 31, 592-594 (1977).
[CrossRef]

D. M. Pepper, D. Fekete, and A. Yariv, “Observation of amplified phase-conjugate reflection and optical parametric oscillation by degenerate four-wave mixing in a transparent medium,” Appl. Phys. Lett. 33, 41-44 (1978).
[CrossRef]

A. Tomita, “Phase conjugation using gain saturation of a Nd:YAG laser,” Appl. Phys. Lett. 34, 463-464 (1979).
[CrossRef]

J. O. White, M. Cronin-Golomb, B. Fischer, and A. Yariv, “Coherent oscillation by self-induced gratings in the photorefractive crystal BaTiO3,” Appl. Phys. Lett. 40, 450-452 (1982).
[CrossRef]

P. V. Avizonis, F. A. Hopf, W. D. Bomberger, S. F. Jacobs, A. Tomita, and K. H. Womack, “Optical phase conjugation in a lithium formate crystal,” Appl. Phys. Lett. 31, 435-437 (1977).
[CrossRef]

Ferroelectrics (1)

N. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskinand, and V. L. Vinetskii, “Holographic storage ion electro-optic crystals. I. Steady-state,” Ferroelectrics 22, 949-960 (1979).
[CrossRef]

IEEE J. Quantum Electron. (2)

M. Cronin-Golomb, B. Fisher, J. O. White, and A. Yariv, “Theory and applications of four-wave mixing in photorefractive media,” IEEE J. Quantum Electron. 20, 12-30 (1984).
[CrossRef]

A. Yariv, “Phase conjugation,” IEEE J. Quantum Electron. 14, 650-660 (1978).
[CrossRef]

J. Appl. Phys. (2)

J. Feinberg, D. Heiman, J. A. R. Tanguay, and R. W. Hellwarth, “Photorefractive effects and light-induced charge migration in barium titanate,” J. Appl. Phys. 51, 1297-1305 (1980).
[CrossRef]

S. Maslovski and S. Tretyakov, “Phase conjugation and perfect lensing,” J. Appl. Phys. 94, 4241-4243 (2003).
[CrossRef]

J. Eur. Opt. Soc. Part B (1)

S. H. Tang, X. H. He, and H.-Y. Zhang, “Experimental study of the forward phase-conjugate wave in degenerate four-wave mixing in LiNbO3:Fe,” J. Eur. Opt. Soc. Part B 3, 179-183 (1991).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

J. Opt. Soc. Am. B (3)

Opt. Commun. (4)

Y. Silberberg and I. Bar-Joseph, “Low power phase conjugation in thin films of saturable absorbers,” Opt. Commun. 39, 265-268 (1981).
[CrossRef]

S. I. Bozhevolnyi, O. Keller, and I. I. Smolyaninov, “Scattered light enhancement near a phase conjugating mirror,” Opt. Commun. 115, 115-120 (1995).
[CrossRef]

B. Vohnsen and S. I. Bozhevolnyi, “Near-field optical microscopy with a phase-conjugating mirror,” Opt. Commun. 148, 331-337 (1998).
[CrossRef]

A. Bledowski, W. Krolikowski, and A. Kujawski, “Forward phase-conjugate wave in four-wave mixing in photorefractive media,” Opt. Commun. 61, 71-74 (1987).
[CrossRef]

Opt. Lett. (10)

L. Lefort and A. Barthelemy, “Revisiting optical phase conjugation by difference-frequency generation,” Opt. Lett. 21, 848-850 (1996).
[CrossRef] [PubMed]

S. I. Bozhevolnyi, O. Keller, and I. I. Smolyaninov, “Phase conjugation of an optical near field,” Opt. Lett. 19, 1601-1603 (1994).
[CrossRef] [PubMed]

G. J. Crofts, R. P. M. Green, and M. J. Damzen, “Investigation of multipass geometries for efficient degenerate four-wave mixing in Nd:YAG,” Opt. Lett. 17, 920-922 (1992).
[CrossRef] [PubMed]

M. J. Damzen, R. P. M. Green, and G. J. Crofts, “High-reflectivity four-wave mixing by gain saturation of nanosecond and microsecond radiation in Nd:YAG,” Opt. Lett. 17, 1331-1333 (1992).
[CrossRef] [PubMed]

R. A. Fisher and B. J. Feldman, “On-resonant phase-conjugate reflection and amplification at 10.6 μm in inverted CO2,” Opt. Lett. 4, 140-142 (1979).
[CrossRef] [PubMed]

J. P. Huignard, J. P. Herriau, P. Aubourg, and E. Spitz, “Phase-conjugate wavefront generation via real-time holography in Bi12SiO20 crystals,” Opt. Lett. 4, 21-23 (1979).
[CrossRef] [PubMed]

J. Feinberg and R. W. Hellwarth, “Phase-conjugating mirror with continuous-wave gain,” Opt. Lett. 5, 519-521 (1980).
[CrossRef] [PubMed]

D. Fekete, J. C. AuYeung, and A. Yariv, “Phase-conjugate reflection by degenerate four-wave mixing in a nematic liquid crystal in the isotropic phase,” Opt. Lett. 5, 51-53 (1980).
[CrossRef] [PubMed]

R. L. Abrams and R. C. Lind, “Degenerate four wave mixing in absorbing media,” Opt. Lett. 2, 94-96 (1978).
[CrossRef] [PubMed]

D. M. Bloom, P. F. Liao, and N. P. Economou, “Observation of amplified reflection by degenerate four-wave mixing in atomic sodium vapor,” Opt. Lett. 2, 58-60 (1978).
[CrossRef] [PubMed]

Opt. Rev. (1)

M. Saito, A. Okamoto, K. Sato, and Y. Takayama, “Phase matching property of cross polarization four wave mixing in BaTiO3 crystal,” Opt. Rev. 4, 686-690 (1997).
[CrossRef]

Phys. Rev. B (2)

M. Zgonik, P. Bernasconi, M. Duelli, R. Schlesser, P. Günter, M. H. Garrett, D. Rytz, Y. Zhu, and X. Wu, “Dielectric, elastic, piezoelectric, electro-optic, and elasto-optic tensors of BaTiO3 crystals,” Phys. Rev. B 50, 5941-5949 (1994).
[CrossRef]

Z. Ye, “Optical transmission and reflection of perfect lenses by left handed materials,” Phys. Rev. B 67, 193106 (2003).
[CrossRef]

Phys. Rev. Lett. (2)

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

A. Grbic and G. V. Eleftheriades, “Overcoming the diffraction limit with a planar left-handed transmission-line lens,” Phys. Rev. Lett. 92, 117403 (2004).
[CrossRef] [PubMed]

Physica B (1)

A. L. Pokrovsky and A. L. Efros, “Diffraction theory and focusing of light by a slab of left-handed material,” Physica B 338, 333-337 (2003).
[CrossRef]

Prog. Quantum Electron. (1)

G. S. He, “Optical phase conjugation: principles, techniques, and applications,” Prog. Quantum Electron. 26, 131-191 (2002).
[CrossRef]

Quantum Opt. (1)

F. Wang, L. Liu, and G. Li, “Self-pumped forward and backward phase conjugator with Cu-doped KNSBN crystal: caused by scattering-oscillation-amplification,” Quantum Opt. 134, 195-198 (1997).
[CrossRef]

Rep. Prog. Phys. (1)

S. A. Ramakrishna, “Physics of negative refractive index materials,” Rep. Prog. Phys. 68, 449-521 (2005).
[CrossRef]

Science (2)

J. B. Pendry, “Time reversal and negative refraction,” Science 322, 71-73 (2008).
[CrossRef] [PubMed]

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308, 534-537 (2005).
[CrossRef] [PubMed]

Sov. Phys. Usp. (1)

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

Other (5)

M. Born and E. Wolf, Principle of Optics (Pergamon, 1980).

G. V. Eleftheriades and K. G. Balmain, Negative-Refraction Metamaterials: Fundamental Principles and Applications (Wiley-IEEE, 2005).
[CrossRef]

R. Fisher, Optical Phase Conjugation (Academic, 1984).

A. Yariv, Quantum Electronics (Wiley, 1989).

P. Günter and J.-P. Huignard, Photorefractive Materials and Their Applications I (Springer-Verlag, 1988).

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Figures (13)

Fig. 1
Fig. 1

Imaging properties of a device combining (a) two convergent lenses and (b) two PCs’s placed at z = 0 and z = d , assuming geometrical optics. The focal length of the lenses is d / 4 .

Fig. 2
Fig. 2

Experimental configuration considered for FWM.

Fig. 3
Fig. 3

Scheme describing the paths taken by the four partial waves propagating inside the non-linear slab. E 4 + can be produced either by reflection of E 4 or by PC of E 3 . It results in two images located at z = z o + L ( 1 ± n 1 ) .

Fig. 4
Fig. 4

Notations and conventions used to describe the four partial waves propagating inside the non-linear slab. k 0 ± = k z 0 u z ± k u x and k ± = k z u z ± k u x .

Fig. 5
Fig. 5

Evolution of the coupling coefficient γ ( m 1 ) as a function of θ 0 and α 0 considering the physical properties of BaTiO 3 and experimental conditions given in Table 1.

Fig. 6
Fig. 6

Evolution of the PC (a) transmission and (b) reflection coefficients as a function of α 0 and θ 0 , for L = 10   mm .

Fig. 7
Fig. 7

Experimental setup simulated ( α 0 = 60 ° , L = 10   mm , Δ L = 0.1   mm ). Both PR slabs are pumped by not shown counter-propagating pump beams (see Fig. 2).

Fig. 8
Fig. 8

(a) Transmittance o ( x , y ) of the opaque screen placed at z = z 0 . The white line represents 20 λ . (b) Transmitted image by one PCs at z = z 0 + L ( 1 ± n 1 ) [Eqs. (29, 30)]. (c) Transmitted image by one PCs at z = z 0 + L ( 1 + n 1 ) [Eq. (30)] in the presence of an anti-reflective coating and taking into account absorption losses. (d) Transmitted image by two PCs’s at z = F + d z 0 + n 1 Δ L [Eq. (54)] taking into account absorption losses and in the presence of anti-reflective coatings.

Fig. 9
Fig. 9

Evolution of T 0 a c as a function of L and θ 0 (a) neglecting or (b) taking into account absorption. Evolution of R a c as a function of L and θ 0 (c) neglecting or (d) taking into account absorption.

Fig. 10
Fig. 10

Scheme describing the different waves propagating when two PCs’s are placed in front of each other.

Fig. 11
Fig. 11

Evolution of T 2 as a function of L and θ 0 for two PCs’s of the same thickness ( L = 10   mm , α 0 = 60 ° ).

Fig. 12
Fig. 12

Scheme describing the paths taken by the partial waves inside the non-linear slabs when two PCs’s are associated.

Fig. 13
Fig. 13

Grating vectors k I and k I I .

Tables (1)

Tables Icon

Table 1 Physical Properties of BaTiO 3 and Experimental Conditions

Equations (83)

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E 1 ± ( r , t ) = A 1 e ̂ ± α e i ( ω t q ± r ) + c .c . ,
E 2 ± ( r , t ) = A 2 e ̂ ± α e i ( ω t + q ± r ) + c .c . ,
e ̂ ± α = cos   α u x sin   α u z ,
q ± = ± sin   α u x + cos   α u z ,
E 3 ± ( r , t ) = A 3 ± ( r ) e i ω t + c .c . ,
E 4 ± ( r , t ) = A 4 ± ( r ) e i ω t + c .c .
A 3 ± ( r ) = { A 3 ± ( k , z ) e ̂ ± θ 0 e i k r e ± i k z o z d k if   r Ω A 3 ± ( k , z ) e ̂ ± θ e i k r e ± i k z z d k if   r Ω , }
A 4 ± ( r ) = { A 4 ± ( k , z ) e ̂ θ 0 e i k r e i k z o z d k , if   r Ω A 4 ± ( k , z ) e ̂ θ e i k r e i k z z d k , if   r Ω , }
k z o = ω 2 c 0 2 k 2 ,     k z = n 2 ω 2 c 0 2 k 2 ,
θ 0 = arctan [ k k z o ] ,     θ = arctan [ k k z ] .
d A 3 + d z γ ( θ ) 2 A 3 + + a 2 A 3 + = γ ( θ ) 2 A 4 ,
d A 4 d z γ ( θ ) 2 A 4 a 2 A 4 = γ ( θ ) 2 A 3 + ,
d A 3 d z + γ ( θ ) 2 A 3 a 2 A 3 = γ ( θ ) 2 A 4 + ,
d A 4 + d z + γ ( θ ) 2 A 4 + + a 2 A 4 + = γ ( θ ) 2 A 3 ,
γ ( θ ) = γ ( θ ) .
d A 3 d z γ ( θ ) 2 A 3 a 2 A 3 = γ ( θ ) 2 A 4 + ,
d A 4 + d z γ ( θ ) 2 A 4 + + a 2 A 4 + = γ ( θ ) 2 A 3 .
d 2 A 4 ± d z 2 γ d A 4 ± d z a 2 4 A 4 ± = 0.
A 4 + ( k , z ) = e γ z / 2 [ C e η γ z / 2 + D e η γ z / 2 ] ,
A 4 ( k , z ) = e γ z / 2 [ E e η γ z / 2 + F e η γ z / 2 ] ,
A 3 ( k , z ) = e γ z / 2 [ C ( η + 1 η 2 ) e η γ z / 2 + D ( η + 1 η 2 ) e η γ z / 2 ] ,
A 3 + ( k , z ) = e γ z / 2 [ E ( η + 1 η 2 ) e η γ z / 2 + F ( η + 1 η 2 ) e η γ z / 2 ] .
ρ ( θ ) = ρ ( θ ) = tan ( θ θ 0 ) tan ( θ + θ 0 ) ,
τ ( θ ) = 2   cos   θ 0   sin   θ sin ( θ + θ 0 ) cos ( θ 0 θ ) ,
τ ( θ ) = 2   cos   θ   sin   θ 0 sin ( θ + θ 0 ) cos ( θ θ 0 ) .
A 3 + ( k , z = 0 + ) = τ A 3 + ( k , z = 0 ) + ρ A 3 ( k , z = 0 + ) ,
A 4 + ( k , z = 0 + ) = ρ A 4 ( k , z = 0 + ) ,
e i k z L A 3 ( k , z = L ) = ρ e i k z L A 3 + ( k , z = L ) ,
e i k z L A 4 ( k , z = L ) = ρ e i k z L A 4 + ( k , z = L ) .
R = E 4 ( k , z = 0 ) E 3 + ( k , z = 0 ) = ( 1 ρ 2 ) 2 sinh ( γ L / 2 ) cosh ( γ L / 2 ) ( 1 ρ 2 ) 2 cosh 2 ( γ L / 2 ) + 4 ρ 2 sin 2 ( k z L ) ,
lim γ L 1 T 0 = 2 ρ ( 1 ρ 2 ) exp ( γ L ) 0 ,
lim γ L 1 T 0 = 2 ρ ( 1 ρ 2 ) = T m a x 0.
lim | γ | L 1 R = tanh ( γ L / 2 ) .
E 3 + ( r , z < 0 ) = O ( k ) e i k r e i k z o ( z + z 0 ) d k ,
E 4 + ( r , z > L ) = O ( k ) T 0 ( k ) e i k r e i ( k z o ( z L z 0 ) + k z L ) d k
O ( k ) T 0 ( k ) e i k z L e i k r e i ( k z o ( z L z 0 ) k z L ) d k .
k z = ( n ω c ) 2 k 2 n ω c k 2 2 n ω / c + ( k 4 4 ( n ω / c ) 3 + ) ,
k z o = ( ω c ) 2 k 2 ω c k 2 2 ω / c + ( k 4 4 ( ω / c ) 3 + ) .
k z o ( z L z 0 ) ± k z L = ω c ( z z 0 L ( 1 ± n ) )
k 2 2 ω / c ( z z 0 L ( 1 ± n 1 ) )
+ ( k 4 4 ( ω / c ) 3 ( z z 0 L ( 1 ± n 3 ) ) + ) .
θ l i m = arcsin [ ( 2 λ L ( n 1 n 3 ) ) 1 / 4 ] .
Δ ( λ 3 L ( n 1 n 3 ) 2 ) 1 / 4 .
A 3 + ( k , z = 0 + ) = τ 0 A 3 + ( k , z = 0 ) ,
A 4 + ( k , z = 0 + ) = 0.
R a c = [ ( 1 ρ 2 ) η   cosh ( η γ L / 2 ) + ( 1 + ρ 2 ) 1 η 2 sinh ( η γ L / 2 ) ] sinh ( η γ L / 2 ) ( 1 ρ 2 ( 2 η 2 1 ) ) sinh 2 ( η γ L / 2 ) + η 1 η 2 sinh ( η γ L ) + η 2 ,
lim γ L 1 R a c a γ η 1 + 2 ( 1 η 2 ) 1 ρ 2 = β < 1 ,
lim γ L 1 T 0 a c a γ τ τ 0 ρ ( 1 ρ 2 ) β e γ ( 1 η ) L = T m a x a c .
Ψ 0 ( k , z < 0 ) = ψ 0 ( k ) e i k z o z e i k r e ̂ θ 0 .
Ψ 1 ( k , L a < z < d + L a ) = ψ 1 ( k ) e i k z o ( z L a ) e i k r e ̂ θ 0 ,
Ψ 3 ( k , L a < z < d + L a ) = ψ 3 ( k ) e i k z o ( z L a ) e i k r e ̂ θ 0 .
Ψ 2 ( k , z > F ) = ψ 2 ( k ) e i k z o ( z F ) e i k r e ̂ θ 0 ,
1 st   PCs :     ψ 1 ( k ) = T a ( k ) ψ 0 ( k ) + R a ( k ) ψ 3 ( k ) ,
2 nd   PCs :     ψ 3 ( k ) e i k z o d = R b ( k ) [ ψ 1 ( k ) e i k z o d ] ,
ψ 2 ( k ) = T b ( k ) [ ψ 1 ( k ) e i k z o d ] .
Ψ 2 ( k , z > F ) = T 2 ( k ) ψ 0 ( k ) e i k r e i k z o ( z F d ) e ̂ θ 0 ,
T 2 ( k ) = T a ( k ) T b ( k ) 1 R a ( k ) R b ( k ) .
T 2 ( k ) = | T ( k ) | 2 1 | R ( k ) | 2 .
lim γ L 1 R ( k ) = 1 lim γ L 1 T 2 ( k ) = .
ψ 2 ( r , z > F ) = O ( k ) T 2 ( k ) e i k r e i k z o ( z + z 0 F d ) d k .
T a , b ( k ) = T 0 , a , b ( k ) e i k z L a , b .
ψ 2 ( r , z > d + F ) = O ( k ) T 0 , 2 ( k ) e i k r e i k z o ( z + z 0 F d ) e i k z Δ L d k .
θ l i m arcsin [ ( 2 λ Δ L ( n 1 n 3 ) ) 1 / 4 ] ,
Δ ( λ 3 Δ L ( n 1 n 3 ) 2 ) 1 / 4 .
I ( r ) = I 0 + [ I I   exp ( i k I r ) + c .c . ] + [ I I I   exp ( i k I I r ) + c .c . ] ,
I I = ( A 1 A 3 + + A 4 A 2 ) cos ( α θ ) .
I I I = ( A 4 A 1 + A 2 A 3 + ) cos ( α θ ) .
E s c ( r ) = [ E I s c   exp ( i k I r ) + c .c . ] u I + [ E I I s c   exp ( i k I I r ) + c .c . ] u I I .
E I , I I s c = i I I , I I I 0 E I , I I d 1 + E I , I I d / E I , I I p .
E I , I I d = k B T k I , I I e ,     E I , I I p = e N A ϵ e f f , I , I I S k I , I I ,
Δ ϵ i j = ϵ i i r i j k E k s c ϵ j j ,
Δ ϵ = ( n o 4 r 13 E z s c n o 2 n e 2 r 42 E x s c n o 2 n e 2 r 42 E x s c n e 4 r 33 E z s c ) .
Δ ϵ = Δ ϵ I e i k I r + Δ ϵ I I e i k I I r + c .c .
Δ ( E 3 + + E 4 ) + ( n ω c ) 2 ( E 3 + + E 4 ) = ω 2 c 2 [ e ̂ θ Δ ϵ e ̂ α ] ( E 1 + + E 2 + ) .
d A 3 + d z = γ I + I 0 ( A 1 A 3 + + A 2 A 4 ) A 1 + γ I I + I 0 ( A 1 A 4 + A 2 A 3 + ) A 2 ,
d A 4 d z = γ I + I 0 ( A 1 A 3 + + A 2 A 4 ) A 2 + γ I I + I 0 ( A 1 A 4 + A 2 A 3 + ) A 1 .
γ I , I I + = ω 2 n c   cos   θ r I , I I E I , I I d 1 + E I , I I d / E I , I I p cos ( α θ ) ,
r I = n o 4 r 13   sin ( α + θ 2 ) cos   α   cos   θ n o 2 n e 2 r 42   cos ( α + θ 2 ) sin ( α + θ ) n e 4 r 33   sin ( α + θ 2 ) sin   α   sin   θ ,
r I I = n o 4 r 13   cos ( α + θ 2 ) cos   α   cos   θ + n o 2 n e 2 r 42   sin ( α + θ 2 ) sin ( α + θ ) n e 4 r 33   cos ( α + θ 2 ) sin   α   sin   θ .
d A 3 + d z γ + 2 A 3 + = γ + 2 e i ϕ A 4 ,
d A 4 d z γ + 2 A 4 = γ + 2 e i ϕ A 3 + ,
γ ( θ , α ) = γ + ( θ , α ) .
γ ( θ , α ) = γ + ( θ , α ) + γ ( θ , α ) = γ + ( θ , α ) + γ + ( θ , α ) .

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