Abstract

We study the reflection properties of squeezing devices based on transformation optics. An analytical expression for the angle-dependent reflection coefficient of a generic three-dimensional squeezer is derived. In contrast with previous studies, we find that there exist several conditions that guarantee no reflections so it is possible to build transformation-optics-based reflectionless squeezers. Moreover, it is shown that the design of antireflective coatings for the non-reflectionless case can be reduced to matching the impedance between two dielectrics. We illustrate the potential of these devices by proposing two applications in which a reflectionless squeezer is the key element: an ultra-short perfect coupler for high-index nanophotonic waveguides and a completely flat reflectionless hyperlens. We also apply our theory to the coupling of two metallic waveguides with different cross-section. Finally, we show how the studied devices can be implemented with non-magnetic isotropic materials by using a quasi-conformal mapping technique.

© 2011 OSA

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    [CrossRef]
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    [CrossRef]
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2010 (3)

2009 (2)

B. Vasić, G. Isic, R. Gajic, and K. Hingerl, “Coordinate transformation based design of confined metamaterial structures,” Phys. Rev. B 79(8), 85103 (2009).
[CrossRef]

Y. Xiong, Z. Liu, and X. Zhang, “A simple design of flat hyperlens for lithography and imaging with half-pitch resolution down to 20 nm,” Appl. Phys. Lett. 94(20), 203108 (2009).
[CrossRef]

2008 (7)

V. M. Shalaev, “Physics. Transforming light,” Science 322(5900), 384–386 (2008).
[CrossRef] [PubMed]

G. Roelkens, D. Vermeulen, D. Van Thourhout, R. Baets, S. Brision, P. Lyan, P. Gautier, and J. M. Fedeli, “High efficiency diffractive grating couplers for interfacing a single mode optical fiber with a nanophotonic silicon-on-insulator waveguide circuit,” Appl. Phys. Lett. 92(13), 131101 (2008).
[CrossRef]

M. Rahm, D. A. Roberts, J. B. Pendry, and D. R. Smith, “Transformation-optical design of adaptive beam bends and beam expanders,” Opt. Express 16(15), 11555–11567 (2008).
[CrossRef] [PubMed]

R. Yang, M. A. Abushagur, and Z. Lu, “Efficiently squeezing near infrared light into a 21 nm-by-24 nm nanospot,” Opt. Express 16(24), 20142–20148 (2008).
[CrossRef] [PubMed]

M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, and D. R. Smith, “Optical design of reflectionless complex media by finite embedded coordinate transformations,” Phys. Rev. Lett. 100(6), 063903 (2008).
[CrossRef] [PubMed]

D. P. Gaillot, C. Croënne, F. Zhang, and D. Lippens, “Transformation optics for the full dielectric electromagnetic cloak and metal–dielectric planar hyperlens,” N. J. Phys. 10(11), 115039 (2008).
[CrossRef]

J. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett. 101(20), 203901 (2008).
[CrossRef] [PubMed]

2007 (1)

2006 (2)

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[CrossRef] [PubMed]

U. Leonhardt and T. G. Philbin, “General Relativity in Electrical Engineering,” N. J. Phys. 8(10), 247 (2006).
[CrossRef]

2005 (2)

T. Tsuchizawa, K. Yamada, H. Fukuda, T. Watanabe, J. Takahashi, M. Takahashi, T. Shoji, E. Tamechika, S. Itabashi, and H. Morita, “Microphotonics devices based on silicon microfabrication technology,” IEEE J. Sel. Top. Quantum Electron. 11(1), 232–240 (2005).
[CrossRef]

T. M. Grzegorczyk, X. Chen, J. Pacheco, J. Chen, B. I. Wu, and J. A. Kong, “Reflection coefficients and Goos-Hanchen shifts in anisotropic and bianisotropic left-handed metamaterials,” Prog. Electromagn. Res. 51, 83–113 (2005).
[CrossRef]

2003 (1)

2002 (1)

D. Taillaert, W. Bogaerts, P. Bienstman, T. F. Krauss, P. Van Daele, I. Moerman, S. Verstuyft, K. De Mesel, and R. Baets, “An out-of-plane grating coupler for efficient butt-coupling between compact planar waveguides and single-mode fibers,” IEEE J. Quantum Electron. 38(7), 949–955 (2002).
[CrossRef]

Abushagur, M. A.

Baets, R.

G. Roelkens, D. Vermeulen, D. Van Thourhout, R. Baets, S. Brision, P. Lyan, P. Gautier, and J. M. Fedeli, “High efficiency diffractive grating couplers for interfacing a single mode optical fiber with a nanophotonic silicon-on-insulator waveguide circuit,” Appl. Phys. Lett. 92(13), 131101 (2008).
[CrossRef]

D. Taillaert, W. Bogaerts, P. Bienstman, T. F. Krauss, P. Van Daele, I. Moerman, S. Verstuyft, K. De Mesel, and R. Baets, “An out-of-plane grating coupler for efficient butt-coupling between compact planar waveguides and single-mode fibers,” IEEE J. Quantum Electron. 38(7), 949–955 (2002).
[CrossRef]

Bienstman, P.

D. Taillaert, W. Bogaerts, P. Bienstman, T. F. Krauss, P. Van Daele, I. Moerman, S. Verstuyft, K. De Mesel, and R. Baets, “An out-of-plane grating coupler for efficient butt-coupling between compact planar waveguides and single-mode fibers,” IEEE J. Quantum Electron. 38(7), 949–955 (2002).
[CrossRef]

Bogaerts, W.

D. Taillaert, W. Bogaerts, P. Bienstman, T. F. Krauss, P. Van Daele, I. Moerman, S. Verstuyft, K. De Mesel, and R. Baets, “An out-of-plane grating coupler for efficient butt-coupling between compact planar waveguides and single-mode fibers,” IEEE J. Quantum Electron. 38(7), 949–955 (2002).
[CrossRef]

Brision, S.

G. Roelkens, D. Vermeulen, D. Van Thourhout, R. Baets, S. Brision, P. Lyan, P. Gautier, and J. M. Fedeli, “High efficiency diffractive grating couplers for interfacing a single mode optical fiber with a nanophotonic silicon-on-insulator waveguide circuit,” Appl. Phys. Lett. 92(13), 131101 (2008).
[CrossRef]

Burokur, S. N.

Cassan, E.

Chang, Z.

Chen, J.

T. M. Grzegorczyk, X. Chen, J. Pacheco, J. Chen, B. I. Wu, and J. A. Kong, “Reflection coefficients and Goos-Hanchen shifts in anisotropic and bianisotropic left-handed metamaterials,” Prog. Electromagn. Res. 51, 83–113 (2005).
[CrossRef]

Chen, X.

T. M. Grzegorczyk, X. Chen, J. Pacheco, J. Chen, B. I. Wu, and J. A. Kong, “Reflection coefficients and Goos-Hanchen shifts in anisotropic and bianisotropic left-handed metamaterials,” Prog. Electromagn. Res. 51, 83–113 (2005).
[CrossRef]

Croënne, C.

D. P. Gaillot, C. Croënne, F. Zhang, and D. Lippens, “Transformation optics for the full dielectric electromagnetic cloak and metal–dielectric planar hyperlens,” N. J. Phys. 10(11), 115039 (2008).
[CrossRef]

Cummer, S. A.

M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, and D. R. Smith, “Optical design of reflectionless complex media by finite embedded coordinate transformations,” Phys. Rev. Lett. 100(6), 063903 (2008).
[CrossRef] [PubMed]

de Lustrac, A.

De Mesel, K.

D. Taillaert, W. Bogaerts, P. Bienstman, T. F. Krauss, P. Van Daele, I. Moerman, S. Verstuyft, K. De Mesel, and R. Baets, “An out-of-plane grating coupler for efficient butt-coupling between compact planar waveguides and single-mode fibers,” IEEE J. Quantum Electron. 38(7), 949–955 (2002).
[CrossRef]

Fedeli, J. M.

G. Roelkens, D. Vermeulen, D. Van Thourhout, R. Baets, S. Brision, P. Lyan, P. Gautier, and J. M. Fedeli, “High efficiency diffractive grating couplers for interfacing a single mode optical fiber with a nanophotonic silicon-on-insulator waveguide circuit,” Appl. Phys. Lett. 92(13), 131101 (2008).
[CrossRef]

Fukuda, H.

T. Tsuchizawa, K. Yamada, H. Fukuda, T. Watanabe, J. Takahashi, M. Takahashi, T. Shoji, E. Tamechika, S. Itabashi, and H. Morita, “Microphotonics devices based on silicon microfabrication technology,” IEEE J. Sel. Top. Quantum Electron. 11(1), 232–240 (2005).
[CrossRef]

Gaillot, D. P.

D. P. Gaillot, C. Croënne, F. Zhang, and D. Lippens, “Transformation optics for the full dielectric electromagnetic cloak and metal–dielectric planar hyperlens,” N. J. Phys. 10(11), 115039 (2008).
[CrossRef]

Gajic, R.

B. Vasić, G. Isic, R. Gajic, and K. Hingerl, “Coordinate transformation based design of confined metamaterial structures,” Phys. Rev. B 79(8), 85103 (2009).
[CrossRef]

Gautier, P.

G. Roelkens, D. Vermeulen, D. Van Thourhout, R. Baets, S. Brision, P. Lyan, P. Gautier, and J. M. Fedeli, “High efficiency diffractive grating couplers for interfacing a single mode optical fiber with a nanophotonic silicon-on-insulator waveguide circuit,” Appl. Phys. Lett. 92(13), 131101 (2008).
[CrossRef]

Grzegorczyk, T. M.

T. M. Grzegorczyk, X. Chen, J. Pacheco, J. Chen, B. I. Wu, and J. A. Kong, “Reflection coefficients and Goos-Hanchen shifts in anisotropic and bianisotropic left-handed metamaterials,” Prog. Electromagn. Res. 51, 83–113 (2005).
[CrossRef]

Hingerl, K.

B. Vasić, G. Isic, R. Gajic, and K. Hingerl, “Coordinate transformation based design of confined metamaterial structures,” Phys. Rev. B 79(8), 85103 (2009).
[CrossRef]

Hu, G.

Hu, J.

Isic, G.

B. Vasić, G. Isic, R. Gajic, and K. Hingerl, “Coordinate transformation based design of confined metamaterial structures,” Phys. Rev. B 79(8), 85103 (2009).
[CrossRef]

Itabashi, S.

T. Tsuchizawa, K. Yamada, H. Fukuda, T. Watanabe, J. Takahashi, M. Takahashi, T. Shoji, E. Tamechika, S. Itabashi, and H. Morita, “Microphotonics devices based on silicon microfabrication technology,” IEEE J. Sel. Top. Quantum Electron. 11(1), 232–240 (2005).
[CrossRef]

Jiang, C.

Kildishev, A. V.

Kong, J. A.

T. M. Grzegorczyk, X. Chen, J. Pacheco, J. Chen, B. I. Wu, and J. A. Kong, “Reflection coefficients and Goos-Hanchen shifts in anisotropic and bianisotropic left-handed metamaterials,” Prog. Electromagn. Res. 51, 83–113 (2005).
[CrossRef]

Krauss, T. F.

D. Taillaert, W. Bogaerts, P. Bienstman, T. F. Krauss, P. Van Daele, I. Moerman, S. Verstuyft, K. De Mesel, and R. Baets, “An out-of-plane grating coupler for efficient butt-coupling between compact planar waveguides and single-mode fibers,” IEEE J. Quantum Electron. 38(7), 949–955 (2002).
[CrossRef]

Laval, S.

Le Roux, X.

Leonhardt, U.

U. Leonhardt and T. G. Philbin, “General Relativity in Electrical Engineering,” N. J. Phys. 8(10), 247 (2006).
[CrossRef]

Li, J.

J. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett. 101(20), 203901 (2008).
[CrossRef] [PubMed]

Lippens, D.

D. P. Gaillot, C. Croënne, F. Zhang, and D. Lippens, “Transformation optics for the full dielectric electromagnetic cloak and metal–dielectric planar hyperlens,” N. J. Phys. 10(11), 115039 (2008).
[CrossRef]

Liu, Z.

Y. Xiong, Z. Liu, and X. Zhang, “A simple design of flat hyperlens for lithography and imaging with half-pitch resolution down to 20 nm,” Appl. Phys. Lett. 94(20), 203108 (2009).
[CrossRef]

Lu, Z.

Lyan, P.

G. Roelkens, D. Vermeulen, D. Van Thourhout, R. Baets, S. Brision, P. Lyan, P. Gautier, and J. M. Fedeli, “High efficiency diffractive grating couplers for interfacing a single mode optical fiber with a nanophotonic silicon-on-insulator waveguide circuit,” Appl. Phys. Lett. 92(13), 131101 (2008).
[CrossRef]

Moerman, I.

D. Taillaert, W. Bogaerts, P. Bienstman, T. F. Krauss, P. Van Daele, I. Moerman, S. Verstuyft, K. De Mesel, and R. Baets, “An out-of-plane grating coupler for efficient butt-coupling between compact planar waveguides and single-mode fibers,” IEEE J. Quantum Electron. 38(7), 949–955 (2002).
[CrossRef]

Morita, H.

T. Tsuchizawa, K. Yamada, H. Fukuda, T. Watanabe, J. Takahashi, M. Takahashi, T. Shoji, E. Tamechika, S. Itabashi, and H. Morita, “Microphotonics devices based on silicon microfabrication technology,” IEEE J. Sel. Top. Quantum Electron. 11(1), 232–240 (2005).
[CrossRef]

Narimanov, E. E.

Pacheco, J.

T. M. Grzegorczyk, X. Chen, J. Pacheco, J. Chen, B. I. Wu, and J. A. Kong, “Reflection coefficients and Goos-Hanchen shifts in anisotropic and bianisotropic left-handed metamaterials,” Prog. Electromagn. Res. 51, 83–113 (2005).
[CrossRef]

Pascal, D.

Pendry, J. B.

M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, and D. R. Smith, “Optical design of reflectionless complex media by finite embedded coordinate transformations,” Phys. Rev. Lett. 100(6), 063903 (2008).
[CrossRef] [PubMed]

M. Rahm, D. A. Roberts, J. B. Pendry, and D. R. Smith, “Transformation-optical design of adaptive beam bends and beam expanders,” Opt. Express 16(15), 11555–11567 (2008).
[CrossRef] [PubMed]

J. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett. 101(20), 203901 (2008).
[CrossRef] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[CrossRef] [PubMed]

Philbin, T. G.

U. Leonhardt and T. G. Philbin, “General Relativity in Electrical Engineering,” N. J. Phys. 8(10), 247 (2006).
[CrossRef]

Rahm, M.

M. Rahm, D. A. Roberts, J. B. Pendry, and D. R. Smith, “Transformation-optical design of adaptive beam bends and beam expanders,” Opt. Express 16(15), 11555–11567 (2008).
[CrossRef] [PubMed]

M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, and D. R. Smith, “Optical design of reflectionless complex media by finite embedded coordinate transformations,” Phys. Rev. Lett. 100(6), 063903 (2008).
[CrossRef] [PubMed]

Roberts, D. A.

Roelkens, G.

G. Roelkens, D. Vermeulen, D. Van Thourhout, R. Baets, S. Brision, P. Lyan, P. Gautier, and J. M. Fedeli, “High efficiency diffractive grating couplers for interfacing a single mode optical fiber with a nanophotonic silicon-on-insulator waveguide circuit,” Appl. Phys. Lett. 92(13), 131101 (2008).
[CrossRef]

Schurig, D.

M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, and D. R. Smith, “Optical design of reflectionless complex media by finite embedded coordinate transformations,” Phys. Rev. Lett. 100(6), 063903 (2008).
[CrossRef] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[CrossRef] [PubMed]

Shalaev, V. M.

V. M. Shalaev, “Physics. Transforming light,” Science 322(5900), 384–386 (2008).
[CrossRef] [PubMed]

Shoji, T.

T. Tsuchizawa, K. Yamada, H. Fukuda, T. Watanabe, J. Takahashi, M. Takahashi, T. Shoji, E. Tamechika, S. Itabashi, and H. Morita, “Microphotonics devices based on silicon microfabrication technology,” IEEE J. Sel. Top. Quantum Electron. 11(1), 232–240 (2005).
[CrossRef]

Smith, D. R.

M. Rahm, D. A. Roberts, J. B. Pendry, and D. R. Smith, “Transformation-optical design of adaptive beam bends and beam expanders,” Opt. Express 16(15), 11555–11567 (2008).
[CrossRef] [PubMed]

M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, and D. R. Smith, “Optical design of reflectionless complex media by finite embedded coordinate transformations,” Phys. Rev. Lett. 100(6), 063903 (2008).
[CrossRef] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[CrossRef] [PubMed]

Taillaert, D.

D. Taillaert, W. Bogaerts, P. Bienstman, T. F. Krauss, P. Van Daele, I. Moerman, S. Verstuyft, K. De Mesel, and R. Baets, “An out-of-plane grating coupler for efficient butt-coupling between compact planar waveguides and single-mode fibers,” IEEE J. Quantum Electron. 38(7), 949–955 (2002).
[CrossRef]

Takahashi, J.

T. Tsuchizawa, K. Yamada, H. Fukuda, T. Watanabe, J. Takahashi, M. Takahashi, T. Shoji, E. Tamechika, S. Itabashi, and H. Morita, “Microphotonics devices based on silicon microfabrication technology,” IEEE J. Sel. Top. Quantum Electron. 11(1), 232–240 (2005).
[CrossRef]

Takahashi, M.

T. Tsuchizawa, K. Yamada, H. Fukuda, T. Watanabe, J. Takahashi, M. Takahashi, T. Shoji, E. Tamechika, S. Itabashi, and H. Morita, “Microphotonics devices based on silicon microfabrication technology,” IEEE J. Sel. Top. Quantum Electron. 11(1), 232–240 (2005).
[CrossRef]

Tamechika, E.

T. Tsuchizawa, K. Yamada, H. Fukuda, T. Watanabe, J. Takahashi, M. Takahashi, T. Shoji, E. Tamechika, S. Itabashi, and H. Morita, “Microphotonics devices based on silicon microfabrication technology,” IEEE J. Sel. Top. Quantum Electron. 11(1), 232–240 (2005).
[CrossRef]

Tichit, P. H.

Tsuchizawa, T.

T. Tsuchizawa, K. Yamada, H. Fukuda, T. Watanabe, J. Takahashi, M. Takahashi, T. Shoji, E. Tamechika, S. Itabashi, and H. Morita, “Microphotonics devices based on silicon microfabrication technology,” IEEE J. Sel. Top. Quantum Electron. 11(1), 232–240 (2005).
[CrossRef]

Van Daele, P.

D. Taillaert, W. Bogaerts, P. Bienstman, T. F. Krauss, P. Van Daele, I. Moerman, S. Verstuyft, K. De Mesel, and R. Baets, “An out-of-plane grating coupler for efficient butt-coupling between compact planar waveguides and single-mode fibers,” IEEE J. Quantum Electron. 38(7), 949–955 (2002).
[CrossRef]

Van Thourhout, D.

G. Roelkens, D. Vermeulen, D. Van Thourhout, R. Baets, S. Brision, P. Lyan, P. Gautier, and J. M. Fedeli, “High efficiency diffractive grating couplers for interfacing a single mode optical fiber with a nanophotonic silicon-on-insulator waveguide circuit,” Appl. Phys. Lett. 92(13), 131101 (2008).
[CrossRef]

Vasic, B.

B. Vasić, G. Isic, R. Gajic, and K. Hingerl, “Coordinate transformation based design of confined metamaterial structures,” Phys. Rev. B 79(8), 85103 (2009).
[CrossRef]

Vermeulen, D.

G. Roelkens, D. Vermeulen, D. Van Thourhout, R. Baets, S. Brision, P. Lyan, P. Gautier, and J. M. Fedeli, “High efficiency diffractive grating couplers for interfacing a single mode optical fiber with a nanophotonic silicon-on-insulator waveguide circuit,” Appl. Phys. Lett. 92(13), 131101 (2008).
[CrossRef]

Verstuyft, S.

D. Taillaert, W. Bogaerts, P. Bienstman, T. F. Krauss, P. Van Daele, I. Moerman, S. Verstuyft, K. De Mesel, and R. Baets, “An out-of-plane grating coupler for efficient butt-coupling between compact planar waveguides and single-mode fibers,” IEEE J. Quantum Electron. 38(7), 949–955 (2002).
[CrossRef]

Vivien, L.

Watanabe, T.

T. Tsuchizawa, K. Yamada, H. Fukuda, T. Watanabe, J. Takahashi, M. Takahashi, T. Shoji, E. Tamechika, S. Itabashi, and H. Morita, “Microphotonics devices based on silicon microfabrication technology,” IEEE J. Sel. Top. Quantum Electron. 11(1), 232–240 (2005).
[CrossRef]

Wu, B. I.

T. M. Grzegorczyk, X. Chen, J. Pacheco, J. Chen, B. I. Wu, and J. A. Kong, “Reflection coefficients and Goos-Hanchen shifts in anisotropic and bianisotropic left-handed metamaterials,” Prog. Electromagn. Res. 51, 83–113 (2005).
[CrossRef]

Xiong, Y.

Y. Xiong, Z. Liu, and X. Zhang, “A simple design of flat hyperlens for lithography and imaging with half-pitch resolution down to 20 nm,” Appl. Phys. Lett. 94(20), 203108 (2009).
[CrossRef]

Yamada, K.

T. Tsuchizawa, K. Yamada, H. Fukuda, T. Watanabe, J. Takahashi, M. Takahashi, T. Shoji, E. Tamechika, S. Itabashi, and H. Morita, “Microphotonics devices based on silicon microfabrication technology,” IEEE J. Sel. Top. Quantum Electron. 11(1), 232–240 (2005).
[CrossRef]

Yang, R.

Zang, X.

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Appl. Phys. Lett. (2)

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Y. Xiong, Z. Liu, and X. Zhang, “A simple design of flat hyperlens for lithography and imaging with half-pitch resolution down to 20 nm,” Appl. Phys. Lett. 94(20), 203108 (2009).
[CrossRef]

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D. Taillaert, W. Bogaerts, P. Bienstman, T. F. Krauss, P. Van Daele, I. Moerman, S. Verstuyft, K. De Mesel, and R. Baets, “An out-of-plane grating coupler for efficient butt-coupling between compact planar waveguides and single-mode fibers,” IEEE J. Quantum Electron. 38(7), 949–955 (2002).
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IEEE J. Sel. Top. Quantum Electron. (1)

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N. J. Phys. (2)

D. P. Gaillot, C. Croënne, F. Zhang, and D. Lippens, “Transformation optics for the full dielectric electromagnetic cloak and metal–dielectric planar hyperlens,” N. J. Phys. 10(11), 115039 (2008).
[CrossRef]

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Opt. Express (5)

Opt. Lett. (1)

Phys. Rev. B (1)

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Figures (7)

Fig. 1
Fig. 1

Sketch of the problem. Cartesian coordinate mesh in the original media is “seen” distorted by the fields in the transformed media.

Fig. 2
Fig. 2

Electric field distribution in the squeezer. The working wavelength is λ = 0.2 m. The relative permittivity of medium 3 is (a) ε r d = F 1 2 . (b) ε r d = 1 .

Fig. 3
Fig. 3

(a) Gaussian beam propagating in free space. (b) The squeezer couples the beam to a nanophotonic dielectric waveguide.

Fig. 4
Fig. 4

(a) Simulation of a 2D Gaussian beam propagating in free space. (b) The beam in (a) is squeezed and perfectly coupled to the nanophotonic waveguide.

Fig. 5
Fig. 5

Lens power flow distribution. The working wavelength is λ = 1.5 μm.

Fig. 6
Fig. 6

Several solutions for coupling two metallic waveguides of different transverse size. In this example, a=0.4 m and d=0.5 m. The free space wavelength is λ = 0.37 m, below cut-off in all waveguides. |E| distribution (a) without coupler, (b) with coupler and n 3 = 1, and (c) with coupler and n 3=2. (d) S 11 parameter as a function of n 3 when the coupler is used.

Fig. 7
Fig. 7

Implementation of the squeezer with an isotropic spatially varying refractive index. (a) |E| field distribution. (b) Refractive index profile of the squeezer. For this particular implementation we obtained an anisotropy factor, as defined in [13], of 1.025. We employed a smooth cosine-like profile for the squeezer.

Equations (34)

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f i : = { x i , < z 0 { h i | h ( i ) , ( i ) ( z = 0 ) = 1 , h ( i ) , ( i ) ( z = d 1 ) = F i } , 0 < z d 1 x k D i k , d 1 < z d 2 x i ( d 2 d 2 ) δ i 1 , d 2 < z < ,
ξ i ' j ' : = ε 2 r i ' j ' = μ 2 r i ' j ' = F 1 F 2 F 3 d i a g ( 1 / F 1 2 , 1 / F 2 2 , 1 / F 3 2 ) .
R = α ρ 3 ω 2 ε 0 μ 0 F 1 2 k x 2 F 2 ω 2 ε 3 μ 3 k x 2 ρ 3 ω 2 ε 0 μ 0 F 1 2 k x 2 + F 2 ω 2 ε 3 μ 3 k x 2 ,
× E = i ω B × H = i ω B
D = ε . E B = μ . H
ε = ( ε 11 ε 12 ε 13 ε 21 ε 22 ε 23 ε 31 ε 32 ε 33 ) μ = ( μ 11 μ 12 μ 13 μ 21 μ 22 μ 23 μ 31 μ 32 μ 33 )
ε t t = ( ε 11 ε 12 0 ε 21 ε 22 0 0 0 0 ) ε t z = ( 0 0 ε 13 0 0 ε 23 0 0 0 ) ε z t = ( 0 0 0 0 0 0 ε 31 ε 32 0 )
i k t × E z z ^ + z ^ × E t z = i ω μ t t H t + i ω μ t z H z z ^
i k t × E t = i ω μ z t H t + i ω μ 33 H z z ^
i k t × H z z ^ + z ^ × H t z = i ω ε t t E t i ω ε t z E z z ^
i k t × H t = i ω ε z t E t i ω ε 33 E z z ^
H z z ^ = 1 ω μ 33 k t × E t 1 μ 33 μ z t H t
E z z ^ = 1 ω ε 33 k t × H t 1 ε 33 ε z t E t
E t z = ( i μ 33 z ^ × I μ t z k t × I i ε 33 z ^ × I k t × I ε z t ) E t + + ( i ω z ^ × I μ t t + i ω μ 33 z ^ × I μ t z μ z t i ω ε 33 z ^ × I k t × I k t × I ) H t
H t z = ( i ω z ^ × I ε t t i ω ε 33 z ^ × I ε t z ε z t + i ω μ 33 z ^ × I k t × I k t × I ) E t + + ( i ε 33 z ^ × I ε t z k t × I i μ 33 z ^ × I k t × I μ z t ) H t
i k z ( E t H t ) = A ( E t H t )
A = ( 0 0 0 i k x 2 ε 33 ω + i ω μ 22 0 0 i ω μ 11 0 0 i k x 2 μ 33 ω i ω ε 22 0 0 i ω ε 11 0 0 0 )
TE: k z 1 , 2 = ± μ 11 μ 33 ( ω 2 ε 22 μ 33 k x 2 ) E t = E y ^ H t = E k Z 1 , 2 ω μ 11 x ^
TM: k z 3 , 4 = ± ε 11 ε 33 ( ω 2 μ 22 ε 33 k x 2 ) E t = E x ^ H t = E ω ε 11 k Z 3 , 4 y ^
ε 2 i j = ε 0 ξ i j μ 2 i j = μ 0 ξ i j
ξ i j = F 1 F 2 F 3 ( 1 F 1 0 0 0 1 F 2 0 0 0 1 F 3 ) T ( 1 F 1 0 0 0 1 F 2 0 0 0 1 F 3 ) = ( F 2 F 3 F 1 0 0 0 F 1 F 3 F 2 0 0 0 F 1 F 2 F 3 )
ε 3 i j = ε 3 δ i j μ 3 i j = μ 3 δ i j
TE: k z , a u x 1 , 2 = ± ξ 11 ξ 33 ( ω 2 ε 0 μ 0 ξ 22 ξ 33 k x 2 ) E t = E y ^ H t = E k z , a u x 1 , 2 ω μ 0 ξ 11 x ^
TM: k z , a u x 3 , 4 = ± ξ 11 ξ 33 ( ω 2 ε 0 μ 0 ξ 22 ξ 33 k x 2 ) E t = E x ^ H t = E ω ε 0 ξ 11 k z , a u x 3 , 4 y ^
TE: k z , o u t 1 , 2 = ± ω 2 ε 3 μ 3 k x 2 E t = E y ^ H t = E k z , o u t 1 , 2 ω μ 3 x ^
TM: k z , o u t 3 , 4 = ± ω 2 ε 3 μ 3 k x 2 E t = E x ^ H t = E ω ε 3 k z , o u t 3 , 4 y ^
( 0 1 k z , a u x 1 ω μ 0 ξ 11 0 ) + R 11 ( 0 1 k z , a u x 1 ω μ 0 ξ 11 0 ) + R 21 ( 1 0 0 ω ε 0 ξ 11 k z , a u x 1 ) = T 11 ( 0 1 k z , o u t 1 ω μ 3 0 ) + T 21 ( 1 0 0 ω ε k z , o u t 1 )
R 21 = T 21 = 0
R 11 = μ 3 k z , a u x 1 μ 0 ξ 11 k z , o u t 1 μ 3 k z , a u x 1 + μ 0 ξ 11 k z , o u t 1 T 11 = 2 μ 3 k z , a u x 1 μ 3 k z , a u x 1 + μ 0 ξ 11 k z , o u t 1
( 1 0 0 ω ε 0 ξ 11 k z , a u x 1 ) + R 12 ( 0 1 k z , a u x 1 ω μ 0 ξ 11 0 ) + R 22 ( 1 0 0 ω ε 0 ξ 11 k z , a u x 1 ) = T 12 ( 0 1 k z , o u t 1 ω μ 3 0 ) + T 22 ( 1 0 0 ω ε k z , o u t 1 )
R 12 = T 12 = 0
R 22 = ξ 11 ε 0 k z , o u t 1 ε 3 k z , a u x 1 ξ 11 ε 0 k z , o u t 1 + ε 3 k z , a u x 1 T 22 = 2 ξ 11 ε 0 k z , o u t 1 ξ 11 ε 0 k z , o u t 1 + ε 3 k z , a u x 1
R T E = R 11 = μ r 3 ω 2 ε 0 μ 0 F 1 2 k x 2 F 2 ω 2 ε 3 μ 3 k x 2 μ r 3 ω 2 ε 0 μ 0 F 1 2 k x 2 + F 2 ω 2 ε 3 μ 3 k x 2
R T M = R 22 = F 2 ω 2 ε 3 μ 3 k x 2 ε r 3 ω 2 ε 0 μ 0 F 1 2 k x 2 F 2 ω 2 ε 3 μ 3 k x 2 + ε r 3 ω 2 ε 0 μ 0 F 1 2 k x 2

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