Abstract

We report simulations of nonlinear optical transmission of an optical beam through heterogeneous metallo dielectric stacks under the action of nonlinear absorption. We use the finite element method (FEM) with two- dimensional transverse effects and transfer matrix method simulation techniques as complementary methods to validate the FEM approach. We find a significant nonlinear absorption effect across spectral regimes where transmission is high. We compare the results with energy and group velocity results, but the enhancement of the nonlinear response is attributed to field confinement in the metal layers.

© 2011 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef]

2010 (4)

D. T. Owens, C. Fuentes-Hernandez, J. M. Hales, J. W. Perry, and B. Kippelen, “Nonlinear optical properties of induced transmission filters,” Opt. Express 18, 19101–19113 (2010).
[CrossRef] [PubMed]

D. T. Owens, C. Fuentes-Hernandez, J. M. Hales, J. W. Perry, and B. Kippelen, “A comprehensive analysis of the contributions to the nonlinear optical properties of thin Ag films,” J. Appl. Phys. 107, 123114 (2010).
[CrossRef]

J. W. Haus, M. Scalora, N. Katte, and J. B. Serushema, “Metallodielectrics as metamaterials,” Proc. SPIE 7756, 77560F (2010).
[CrossRef]

C. Fuentes-Hernandez and B. Kippelen, “Nonlinear optical properties of copper-based photonic bandgap structures at the onset of interband transitions” Nonlinear Optics Quantum Optics 40, 69–82 (2010).

2009 (1)

2008 (3)

D. de Ceglia, M. A. Vincenti, M. G. Cappeddu, M. Centini, N. Akozbek, A. D’Orazio, J. W. Haus, M. J. Bloemer, and M. Scalora, “Tailoring metallodielectric structures for super-resolution and superguiding application in the visible and near IR ranges,” Phys. Rev. A 77, 033848 (2008).
[CrossRef]

J. Wei and M. Xiao, “Z-scan model for optical nonlinear nanometric films,” J. Opt. A 10, 115102 (2008).
[CrossRef]

Z.-M. Meng, H.-Y. Liu, Q.-F. Dai, L.-J. Wu, Q. Guo, W. Hu, S.-H. Liu, S. Lan, and V. A. Trofimov, “Dependence of nonlinearity enhancement on power density in photonic crystals characterized by numerical Z-scan experiments based on the finite-difference time-domain technique,” J. Opt. Soc. Am. B 25, 555–563 (2008).
[CrossRef]

2007 (3)

M. Bloemer, G. D’Aguanno, N. Mattiucci, M. Scalora, and N. Akozbek, “Broadband super-resolving lens with high transparency in the visible range,” Appl. Phys. Lett. 90, 174113 (2007).
[CrossRef]

M. Scalora, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, D. de Ceglia, M. Centini, A. Mandatori, C. Sibilia, N. Akozbek, M. G. Cappeddu, M. Fowler, and J. W. Haus, “Negative refraction and subwavelength focusing in the visible range using transparent metallodielectric stacks,” Opt. Express 15, 508–523(2007).
[CrossRef] [PubMed]

G. J. Lee, Y. P. Lee, S. G. Jung, B. Y. Jung, C. K. Hwangbo, S. Kim, and I. Park, “Design, fabrication, linear, and nonlinear optical properties of metal-dielectric photonic bandgap structures,” J. Korean Phys. Soc. 51, 431–437 (2007).
[CrossRef]

2006 (1)

2004 (1)

N. N. Lepeshkin, A. Schweinsberg, G. Piredda, R. S. Bennink, and R. W. Boyd, “Enhanced nonlinear optical response of one-dimensional metal-dielectric photonic crystals,” Phys. Rev. Lett. 93, 123902 (2004).
[CrossRef] [PubMed]

2003 (1)

M. C. Larciprete, C. Sibilia, S. Paoloni, M. Bertolotti, F. Sarto, and M. Scalora, “Accessing the optical limiting properties of metallodielectric photonic bandgap structures,” J. Appl. Phys. 93, 5013–5017 (2003).
[CrossRef]

2001 (2)

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, “Group velocity, energy velocity, and superluminal propagation in finite photonic bandgap structures,” Phys. Rev. E 63, 036610 (2001).
[CrossRef]

R. A. Ganeev, A. I. Ryasnyanskii, M. K. Kodirov, S. R. Kamalov, and T. Usmanov, “Nonlinear optical characteristics of colloidal solutions of metals,” Opt. Spectrosc. 90, 568–573 (2001).
[CrossRef]

1999 (1)

1996 (1)

R. J. Gehr and R. W. Boyd, “Optical properties of nanostructured optical materials,” Chem. Mater. 8, 1807–1819 (1996).
[CrossRef]

1994 (1)

1990 (3)

M. J. Bloemer, J. W. Haus, and P. R. Ashley, “Degenerate four-wave mixing in colloidal gold as a function of particle size,” J. Opt. Soc. Am. B 7, 790–795 (1990).
[CrossRef]

M. J. Bloemer, P. R. Ashley, J. W. Haus, N. Kalyaniwalla, and C. R. Christensen, “Third-Order optical nonlinearities in waveguide geometry,” IEEE J. Quantum Electron. 26, 1075–1080 (1990).
[CrossRef]

M. Sheik-Bahae, A. A. Said, T. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
[CrossRef]

1989 (1)

J. W. Haus, R. Inguva, and C. M. Bowden, “Effective-Medium theory of nonlinear ellipsoidal composites,” Phys. Rev. A 40, 5729–5734 (1989).
[CrossRef] [PubMed]

1985 (1)

Akozbek, N.

D. de Ceglia, M. A. Vincenti, M. G. Cappeddu, M. Centini, N. Akozbek, A. D’Orazio, J. W. Haus, M. J. Bloemer, and M. Scalora, “Tailoring metallodielectric structures for super-resolution and superguiding application in the visible and near IR ranges,” Phys. Rev. A 77, 033848 (2008).
[CrossRef]

M. Scalora, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, D. de Ceglia, M. Centini, A. Mandatori, C. Sibilia, N. Akozbek, M. G. Cappeddu, M. Fowler, and J. W. Haus, “Negative refraction and subwavelength focusing in the visible range using transparent metallodielectric stacks,” Opt. Express 15, 508–523(2007).
[CrossRef] [PubMed]

M. Bloemer, G. D’Aguanno, N. Mattiucci, M. Scalora, and N. Akozbek, “Broadband super-resolving lens with high transparency in the visible range,” Appl. Phys. Lett. 90, 174113 (2007).
[CrossRef]

Alfano, R. R.

Ashley, P. R.

M. J. Bloemer, J. W. Haus, and P. R. Ashley, “Degenerate four-wave mixing in colloidal gold as a function of particle size,” J. Opt. Soc. Am. B 7, 790–795 (1990).
[CrossRef]

M. J. Bloemer, P. R. Ashley, J. W. Haus, N. Kalyaniwalla, and C. R. Christensen, “Third-Order optical nonlinearities in waveguide geometry,” IEEE J. Quantum Electron. 26, 1075–1080 (1990).
[CrossRef]

Becker, K.

Bennink, R. S.

N. N. Lepeshkin, A. Schweinsberg, G. Piredda, R. S. Bennink, and R. W. Boyd, “Enhanced nonlinear optical response of one-dimensional metal-dielectric photonic crystals,” Phys. Rev. Lett. 93, 123902 (2004).
[CrossRef] [PubMed]

R. S. Bennink, Y. K. Yoon, R. W. Boyd, and J. E. Sipe, “Accessing the optical nonlinearity of metals with metal-dielectric photonic bandgap structures,” Opt. Lett. 24, 1416–1418 (1999).
[CrossRef]

Bertolotti, M.

M. C. Larciprete, C. Sibilia, S. Paoloni, M. Bertolotti, F. Sarto, and M. Scalora, “Accessing the optical limiting properties of metallodielectric photonic bandgap structures,” J. Appl. Phys. 93, 5013–5017 (2003).
[CrossRef]

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, “Group velocity, energy velocity, and superluminal propagation in finite photonic bandgap structures,” Phys. Rev. E 63, 036610 (2001).
[CrossRef]

M. Bertolotti, C. M. Bowden, and C. Sibilia, “Nanoscale linear and nonlinear optics: International School on Quantum Electronics,” in Proceedings of the International School on Quantum Electronics (AIP, 2001).

Bloemer, M.

M. Bloemer, G. D’Aguanno, N. Mattiucci, M. Scalora, and N. Akozbek, “Broadband super-resolving lens with high transparency in the visible range,” Appl. Phys. Lett. 90, 174113 (2007).
[CrossRef]

Bloemer, M. J.

D. de Ceglia, M. A. Vincenti, M. G. Cappeddu, M. Centini, N. Akozbek, A. D’Orazio, J. W. Haus, M. J. Bloemer, and M. Scalora, “Tailoring metallodielectric structures for super-resolution and superguiding application in the visible and near IR ranges,” Phys. Rev. A 77, 033848 (2008).
[CrossRef]

M. Scalora, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, D. de Ceglia, M. Centini, A. Mandatori, C. Sibilia, N. Akozbek, M. G. Cappeddu, M. Fowler, and J. W. Haus, “Negative refraction and subwavelength focusing in the visible range using transparent metallodielectric stacks,” Opt. Express 15, 508–523(2007).
[CrossRef] [PubMed]

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, “Group velocity, energy velocity, and superluminal propagation in finite photonic bandgap structures,” Phys. Rev. E 63, 036610 (2001).
[CrossRef]

M. J. Bloemer, J. W. Haus, and P. R. Ashley, “Degenerate four-wave mixing in colloidal gold as a function of particle size,” J. Opt. Soc. Am. B 7, 790–795 (1990).
[CrossRef]

M. J. Bloemer, P. R. Ashley, J. W. Haus, N. Kalyaniwalla, and C. R. Christensen, “Third-Order optical nonlinearities in waveguide geometry,” IEEE J. Quantum Electron. 26, 1075–1080 (1990).
[CrossRef]

Bowden, C. M.

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, “Group velocity, energy velocity, and superluminal propagation in finite photonic bandgap structures,” Phys. Rev. E 63, 036610 (2001).
[CrossRef]

J. W. Haus, R. Inguva, and C. M. Bowden, “Effective-Medium theory of nonlinear ellipsoidal composites,” Phys. Rev. A 40, 5729–5734 (1989).
[CrossRef] [PubMed]

M. Bertolotti, C. M. Bowden, and C. Sibilia, “Nanoscale linear and nonlinear optics: International School on Quantum Electronics,” in Proceedings of the International School on Quantum Electronics (AIP, 2001).

Boyd, R. W.

N. N. Lepeshkin, A. Schweinsberg, G. Piredda, R. S. Bennink, and R. W. Boyd, “Enhanced nonlinear optical response of one-dimensional metal-dielectric photonic crystals,” Phys. Rev. Lett. 93, 123902 (2004).
[CrossRef] [PubMed]

R. S. Bennink, Y. K. Yoon, R. W. Boyd, and J. E. Sipe, “Accessing the optical nonlinearity of metals with metal-dielectric photonic bandgap structures,” Opt. Lett. 24, 1416–1418 (1999).
[CrossRef]

R. J. Gehr and R. W. Boyd, “Optical properties of nanostructured optical materials,” Chem. Mater. 8, 1807–1819 (1996).
[CrossRef]

Bristow, A. D.

Cappeddu, M. G.

D. de Ceglia, M. A. Vincenti, M. G. Cappeddu, M. Centini, N. Akozbek, A. D’Orazio, J. W. Haus, M. J. Bloemer, and M. Scalora, “Tailoring metallodielectric structures for super-resolution and superguiding application in the visible and near IR ranges,” Phys. Rev. A 77, 033848 (2008).
[CrossRef]

M. Scalora, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, D. de Ceglia, M. Centini, A. Mandatori, C. Sibilia, N. Akozbek, M. G. Cappeddu, M. Fowler, and J. W. Haus, “Negative refraction and subwavelength focusing in the visible range using transparent metallodielectric stacks,” Opt. Express 15, 508–523(2007).
[CrossRef] [PubMed]

Centini, M.

D. de Ceglia, M. A. Vincenti, M. G. Cappeddu, M. Centini, N. Akozbek, A. D’Orazio, J. W. Haus, M. J. Bloemer, and M. Scalora, “Tailoring metallodielectric structures for super-resolution and superguiding application in the visible and near IR ranges,” Phys. Rev. A 77, 033848 (2008).
[CrossRef]

M. Scalora, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, D. de Ceglia, M. Centini, A. Mandatori, C. Sibilia, N. Akozbek, M. G. Cappeddu, M. Fowler, and J. W. Haus, “Negative refraction and subwavelength focusing in the visible range using transparent metallodielectric stacks,” Opt. Express 15, 508–523(2007).
[CrossRef] [PubMed]

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, “Group velocity, energy velocity, and superluminal propagation in finite photonic bandgap structures,” Phys. Rev. E 63, 036610 (2001).
[CrossRef]

Chen, S.

Christensen, C. R.

M. J. Bloemer, P. R. Ashley, J. W. Haus, N. Kalyaniwalla, and C. R. Christensen, “Third-Order optical nonlinearities in waveguide geometry,” IEEE J. Quantum Electron. 26, 1075–1080 (1990).
[CrossRef]

D’Aguanno, G.

M. Scalora, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, D. de Ceglia, M. Centini, A. Mandatori, C. Sibilia, N. Akozbek, M. G. Cappeddu, M. Fowler, and J. W. Haus, “Negative refraction and subwavelength focusing in the visible range using transparent metallodielectric stacks,” Opt. Express 15, 508–523(2007).
[CrossRef] [PubMed]

M. Bloemer, G. D’Aguanno, N. Mattiucci, M. Scalora, and N. Akozbek, “Broadband super-resolving lens with high transparency in the visible range,” Appl. Phys. Lett. 90, 174113 (2007).
[CrossRef]

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, “Group velocity, energy velocity, and superluminal propagation in finite photonic bandgap structures,” Phys. Rev. E 63, 036610 (2001).
[CrossRef]

D’Orazio, A.

D. de Ceglia, M. A. Vincenti, M. G. Cappeddu, M. Centini, N. Akozbek, A. D’Orazio, J. W. Haus, M. J. Bloemer, and M. Scalora, “Tailoring metallodielectric structures for super-resolution and superguiding application in the visible and near IR ranges,” Phys. Rev. A 77, 033848 (2008).
[CrossRef]

Dai, Q.-F.

de Ceglia, D.

D. de Ceglia, M. A. Vincenti, M. G. Cappeddu, M. Centini, N. Akozbek, A. D’Orazio, J. W. Haus, M. J. Bloemer, and M. Scalora, “Tailoring metallodielectric structures for super-resolution and superguiding application in the visible and near IR ranges,” Phys. Rev. A 77, 033848 (2008).
[CrossRef]

M. Scalora, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, D. de Ceglia, M. Centini, A. Mandatori, C. Sibilia, N. Akozbek, M. G. Cappeddu, M. Fowler, and J. W. Haus, “Negative refraction and subwavelength focusing in the visible range using transparent metallodielectric stacks,” Opt. Express 15, 508–523(2007).
[CrossRef] [PubMed]

Dorsinville, R.

Flytzanis, C.

Fowler, M.

Fuentes-Hernandez, C.

D. T. Owens, C. Fuentes-Hernandez, J. M. Hales, J. W. Perry, and B. Kippelen, “A comprehensive analysis of the contributions to the nonlinear optical properties of thin Ag films,” J. Appl. Phys. 107, 123114 (2010).
[CrossRef]

C. Fuentes-Hernandez and B. Kippelen, “Nonlinear optical properties of copper-based photonic bandgap structures at the onset of interband transitions” Nonlinear Optics Quantum Optics 40, 69–82 (2010).

D. T. Owens, C. Fuentes-Hernandez, J. M. Hales, J. W. Perry, and B. Kippelen, “Nonlinear optical properties of induced transmission filters,” Opt. Express 18, 19101–19113 (2010).
[CrossRef] [PubMed]

C. Fuentes-Hernandez, L. A. Padilha, J. M. Hales, D. Owens, J. Kim, S. Webster, S. R. Marder, J. W. Perry, D. J. Hagan, E. W. Van Stryland, and B. Kippelen, “Nonlinear optical response of transparent metal-dielectric multilayer structures,” in Integrated Photonics and Nanophotonics Research and Applications, OSA Technical Digest (CD) (Optical Society of America, 2009), paper JTuA2.

Ganeev, R. A.

R. A. Ganeev, A. I. Ryasnyanskii, M. K. Kodirov, S. R. Kamalov, and T. Usmanov, “Nonlinear optical characteristics of colloidal solutions of metals,” Opt. Spectrosc. 90, 568–573 (2001).
[CrossRef]

Gehr, R. J.

R. J. Gehr and R. W. Boyd, “Optical properties of nanostructured optical materials,” Chem. Mater. 8, 1807–1819 (1996).
[CrossRef]

Guo, Q.

Hagan, D. J.

M. Sheik-Bahae, A. A. Said, T. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
[CrossRef]

C. Fuentes-Hernandez, L. A. Padilha, J. M. Hales, D. Owens, J. Kim, S. Webster, S. R. Marder, J. W. Perry, D. J. Hagan, E. W. Van Stryland, and B. Kippelen, “Nonlinear optical response of transparent metal-dielectric multilayer structures,” in Integrated Photonics and Nanophotonics Research and Applications, OSA Technical Digest (CD) (Optical Society of America, 2009), paper JTuA2.

Haglund, R. F.

Hales, J. M.

D. T. Owens, C. Fuentes-Hernandez, J. M. Hales, J. W. Perry, and B. Kippelen, “A comprehensive analysis of the contributions to the nonlinear optical properties of thin Ag films,” J. Appl. Phys. 107, 123114 (2010).
[CrossRef]

D. T. Owens, C. Fuentes-Hernandez, J. M. Hales, J. W. Perry, and B. Kippelen, “Nonlinear optical properties of induced transmission filters,” Opt. Express 18, 19101–19113 (2010).
[CrossRef] [PubMed]

C. Fuentes-Hernandez, L. A. Padilha, J. M. Hales, D. Owens, J. Kim, S. Webster, S. R. Marder, J. W. Perry, D. J. Hagan, E. W. Van Stryland, and B. Kippelen, “Nonlinear optical response of transparent metal-dielectric multilayer structures,” in Integrated Photonics and Nanophotonics Research and Applications, OSA Technical Digest (CD) (Optical Society of America, 2009), paper JTuA2.

Haus, J. W.

J. W. Haus, M. Scalora, N. Katte, and J. B. Serushema, “Metallodielectrics as metamaterials,” Proc. SPIE 7756, 77560F (2010).
[CrossRef]

D. de Ceglia, M. A. Vincenti, M. G. Cappeddu, M. Centini, N. Akozbek, A. D’Orazio, J. W. Haus, M. J. Bloemer, and M. Scalora, “Tailoring metallodielectric structures for super-resolution and superguiding application in the visible and near IR ranges,” Phys. Rev. A 77, 033848 (2008).
[CrossRef]

M. Scalora, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, D. de Ceglia, M. Centini, A. Mandatori, C. Sibilia, N. Akozbek, M. G. Cappeddu, M. Fowler, and J. W. Haus, “Negative refraction and subwavelength focusing in the visible range using transparent metallodielectric stacks,” Opt. Express 15, 508–523(2007).
[CrossRef] [PubMed]

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, “Group velocity, energy velocity, and superluminal propagation in finite photonic bandgap structures,” Phys. Rev. E 63, 036610 (2001).
[CrossRef]

M. J. Bloemer, J. W. Haus, and P. R. Ashley, “Degenerate four-wave mixing in colloidal gold as a function of particle size,” J. Opt. Soc. Am. B 7, 790–795 (1990).
[CrossRef]

M. J. Bloemer, P. R. Ashley, J. W. Haus, N. Kalyaniwalla, and C. R. Christensen, “Third-Order optical nonlinearities in waveguide geometry,” IEEE J. Quantum Electron. 26, 1075–1080 (1990).
[CrossRef]

J. W. Haus, R. Inguva, and C. M. Bowden, “Effective-Medium theory of nonlinear ellipsoidal composites,” Phys. Rev. A 40, 5729–5734 (1989).
[CrossRef] [PubMed]

N. Katte, J. W. Haus, J. B. Serushema, and M. Scalora, “Super-resolving properties of metallodielectric stacks,” presented at COMSOL Conference, Boston, Massachusetts, USA (7–9 October 2010), http://www.dtic.mil/cgi-bin/GetTRDoc?Location=U2&doc=GetTRDoc.pdf&AD=ADA540223.

Hu, W.

Hubner, J.

Hwangbo, C. K.

G. J. Lee, Y. P. Lee, S. G. Jung, B. Y. Jung, C. K. Hwangbo, S. Kim, and I. Park, “Design, fabrication, linear, and nonlinear optical properties of metal-dielectric photonic bandgap structures,” J. Korean Phys. Soc. 51, 431–437 (2007).
[CrossRef]

Inguva, R.

J. W. Haus, R. Inguva, and C. M. Bowden, “Effective-Medium theory of nonlinear ellipsoidal composites,” Phys. Rev. A 40, 5729–5734 (1989).
[CrossRef] [PubMed]

Jung, B. Y.

G. J. Lee, Y. P. Lee, S. G. Jung, B. Y. Jung, C. K. Hwangbo, S. Kim, and I. Park, “Design, fabrication, linear, and nonlinear optical properties of metal-dielectric photonic bandgap structures,” J. Korean Phys. Soc. 51, 431–437 (2007).
[CrossRef]

Jung, S. G.

G. J. Lee, Y. P. Lee, S. G. Jung, B. Y. Jung, C. K. Hwangbo, S. Kim, and I. Park, “Design, fabrication, linear, and nonlinear optical properties of metal-dielectric photonic bandgap structures,” J. Korean Phys. Soc. 51, 431–437 (2007).
[CrossRef]

Kalyaniwalla, N.

M. J. Bloemer, P. R. Ashley, J. W. Haus, N. Kalyaniwalla, and C. R. Christensen, “Third-Order optical nonlinearities in waveguide geometry,” IEEE J. Quantum Electron. 26, 1075–1080 (1990).
[CrossRef]

Kamalov, S. R.

R. A. Ganeev, A. I. Ryasnyanskii, M. K. Kodirov, S. R. Kamalov, and T. Usmanov, “Nonlinear optical characteristics of colloidal solutions of metals,” Opt. Spectrosc. 90, 568–573 (2001).
[CrossRef]

Katte, N.

J. W. Haus, M. Scalora, N. Katte, and J. B. Serushema, “Metallodielectrics as metamaterials,” Proc. SPIE 7756, 77560F (2010).
[CrossRef]

N. Katte, J. W. Haus, J. B. Serushema, and M. Scalora, “Super-resolving properties of metallodielectric stacks,” presented at COMSOL Conference, Boston, Massachusetts, USA (7–9 October 2010), http://www.dtic.mil/cgi-bin/GetTRDoc?Location=U2&doc=GetTRDoc.pdf&AD=ADA540223.

Kim, J.

C. Fuentes-Hernandez, L. A. Padilha, J. M. Hales, D. Owens, J. Kim, S. Webster, S. R. Marder, J. W. Perry, D. J. Hagan, E. W. Van Stryland, and B. Kippelen, “Nonlinear optical response of transparent metal-dielectric multilayer structures,” in Integrated Photonics and Nanophotonics Research and Applications, OSA Technical Digest (CD) (Optical Society of America, 2009), paper JTuA2.

Kim, S.

G. J. Lee, Y. P. Lee, S. G. Jung, B. Y. Jung, C. K. Hwangbo, S. Kim, and I. Park, “Design, fabrication, linear, and nonlinear optical properties of metal-dielectric photonic bandgap structures,” J. Korean Phys. Soc. 51, 431–437 (2007).
[CrossRef]

Kippelen, B.

C. Fuentes-Hernandez and B. Kippelen, “Nonlinear optical properties of copper-based photonic bandgap structures at the onset of interband transitions” Nonlinear Optics Quantum Optics 40, 69–82 (2010).

D. T. Owens, C. Fuentes-Hernandez, J. M. Hales, J. W. Perry, and B. Kippelen, “A comprehensive analysis of the contributions to the nonlinear optical properties of thin Ag films,” J. Appl. Phys. 107, 123114 (2010).
[CrossRef]

D. T. Owens, C. Fuentes-Hernandez, J. M. Hales, J. W. Perry, and B. Kippelen, “Nonlinear optical properties of induced transmission filters,” Opt. Express 18, 19101–19113 (2010).
[CrossRef] [PubMed]

C. Fuentes-Hernandez, L. A. Padilha, J. M. Hales, D. Owens, J. Kim, S. Webster, S. R. Marder, J. W. Perry, D. J. Hagan, E. W. Van Stryland, and B. Kippelen, “Nonlinear optical response of transparent metal-dielectric multilayer structures,” in Integrated Photonics and Nanophotonics Research and Applications, OSA Technical Digest (CD) (Optical Society of America, 2009), paper JTuA2.

Kirkpatrick, S.

R. L. Sutherland, D. G. McLean, and S. Kirkpatrick, Handbook of Nonlinear Optics2nd ed. (CRC Press, 2003).
[CrossRef]

Kodirov, M. K.

R. A. Ganeev, A. I. Ryasnyanskii, M. K. Kodirov, S. R. Kamalov, and T. Usmanov, “Nonlinear optical characteristics of colloidal solutions of metals,” Opt. Spectrosc. 90, 568–573 (2001).
[CrossRef]

Lan, S.

Larciprete, M. C.

M. C. Larciprete, C. Sibilia, S. Paoloni, M. Bertolotti, F. Sarto, and M. Scalora, “Accessing the optical limiting properties of metallodielectric photonic bandgap structures,” J. Appl. Phys. 93, 5013–5017 (2003).
[CrossRef]

Lee, G. J.

G. J. Lee, Y. P. Lee, S. G. Jung, B. Y. Jung, C. K. Hwangbo, S. Kim, and I. Park, “Design, fabrication, linear, and nonlinear optical properties of metal-dielectric photonic bandgap structures,” J. Korean Phys. Soc. 51, 431–437 (2007).
[CrossRef]

Lee, T. K.

Lee, Y. P.

G. J. Lee, Y. P. Lee, S. G. Jung, B. Y. Jung, C. K. Hwangbo, S. Kim, and I. Park, “Design, fabrication, linear, and nonlinear optical properties of metal-dielectric photonic bandgap structures,” J. Korean Phys. Soc. 51, 431–437 (2007).
[CrossRef]

Lepeshkin, N. N.

N. N. Lepeshkin, A. Schweinsberg, G. Piredda, R. S. Bennink, and R. W. Boyd, “Enhanced nonlinear optical response of one-dimensional metal-dielectric photonic crystals,” Phys. Rev. Lett. 93, 123902 (2004).
[CrossRef] [PubMed]

Liu, H.-Y.

Liu, S.-H.

Liu, X.

Magruder, R. H.

Mandatori, A.

Marder, S. R.

C. Fuentes-Hernandez, L. A. Padilha, J. M. Hales, D. Owens, J. Kim, S. Webster, S. R. Marder, J. W. Perry, D. J. Hagan, E. W. Van Stryland, and B. Kippelen, “Nonlinear optical response of transparent metal-dielectric multilayer structures,” in Integrated Photonics and Nanophotonics Research and Applications, OSA Technical Digest (CD) (Optical Society of America, 2009), paper JTuA2.

Mattiucci, N.

McLean, D. G.

R. L. Sutherland, D. G. McLean, and S. Kirkpatrick, Handbook of Nonlinear Optics2nd ed. (CRC Press, 2003).
[CrossRef]

Meng, Z.-M.

Moloney, J. V.

Owens, D.

C. Fuentes-Hernandez, L. A. Padilha, J. M. Hales, D. Owens, J. Kim, S. Webster, S. R. Marder, J. W. Perry, D. J. Hagan, E. W. Van Stryland, and B. Kippelen, “Nonlinear optical response of transparent metal-dielectric multilayer structures,” in Integrated Photonics and Nanophotonics Research and Applications, OSA Technical Digest (CD) (Optical Society of America, 2009), paper JTuA2.

Owens, D. T.

D. T. Owens, C. Fuentes-Hernandez, J. M. Hales, J. W. Perry, and B. Kippelen, “A comprehensive analysis of the contributions to the nonlinear optical properties of thin Ag films,” J. Appl. Phys. 107, 123114 (2010).
[CrossRef]

D. T. Owens, C. Fuentes-Hernandez, J. M. Hales, J. W. Perry, and B. Kippelen, “Nonlinear optical properties of induced transmission filters,” Opt. Express 18, 19101–19113 (2010).
[CrossRef] [PubMed]

Padilha, L. A.

C. Fuentes-Hernandez, L. A. Padilha, J. M. Hales, D. Owens, J. Kim, S. Webster, S. R. Marder, J. W. Perry, D. J. Hagan, E. W. Van Stryland, and B. Kippelen, “Nonlinear optical response of transparent metal-dielectric multilayer structures,” in Integrated Photonics and Nanophotonics Research and Applications, OSA Technical Digest (CD) (Optical Society of America, 2009), paper JTuA2.

Palik, E. D.

E. D. Palik, Handbook of Optical Constants of Solids(Academic, 1985).

Paoloni, S.

M. C. Larciprete, C. Sibilia, S. Paoloni, M. Bertolotti, F. Sarto, and M. Scalora, “Accessing the optical limiting properties of metallodielectric photonic bandgap structures,” J. Appl. Phys. 93, 5013–5017 (2003).
[CrossRef]

Park, I.

G. J. Lee, Y. P. Lee, S. G. Jung, B. Y. Jung, C. K. Hwangbo, S. Kim, and I. Park, “Design, fabrication, linear, and nonlinear optical properties of metal-dielectric photonic bandgap structures,” J. Korean Phys. Soc. 51, 431–437 (2007).
[CrossRef]

Perry, J. W.

D. T. Owens, C. Fuentes-Hernandez, J. M. Hales, J. W. Perry, and B. Kippelen, “A comprehensive analysis of the contributions to the nonlinear optical properties of thin Ag films,” J. Appl. Phys. 107, 123114 (2010).
[CrossRef]

D. T. Owens, C. Fuentes-Hernandez, J. M. Hales, J. W. Perry, and B. Kippelen, “Nonlinear optical properties of induced transmission filters,” Opt. Express 18, 19101–19113 (2010).
[CrossRef] [PubMed]

C. Fuentes-Hernandez, L. A. Padilha, J. M. Hales, D. Owens, J. Kim, S. Webster, S. R. Marder, J. W. Perry, D. J. Hagan, E. W. Van Stryland, and B. Kippelen, “Nonlinear optical response of transparent metal-dielectric multilayer structures,” in Integrated Photonics and Nanophotonics Research and Applications, OSA Technical Digest (CD) (Optical Society of America, 2009), paper JTuA2.

Peyghambarian, N.

Piredda, G.

N. N. Lepeshkin, A. Schweinsberg, G. Piredda, R. S. Bennink, and R. W. Boyd, “Enhanced nonlinear optical response of one-dimensional metal-dielectric photonic crystals,” Phys. Rev. Lett. 93, 123902 (2004).
[CrossRef] [PubMed]

Ricard, D.

Roussignol, P.

Ryasnyanskii, A. I.

R. A. Ganeev, A. I. Ryasnyanskii, M. K. Kodirov, S. R. Kamalov, and T. Usmanov, “Nonlinear optical characteristics of colloidal solutions of metals,” Opt. Spectrosc. 90, 568–573 (2001).
[CrossRef]

Said, A. A.

M. Sheik-Bahae, A. A. Said, T. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
[CrossRef]

Sarto, F.

M. C. Larciprete, C. Sibilia, S. Paoloni, M. Bertolotti, F. Sarto, and M. Scalora, “Accessing the optical limiting properties of metallodielectric photonic bandgap structures,” J. Appl. Phys. 93, 5013–5017 (2003).
[CrossRef]

Scalora, M.

J. W. Haus, M. Scalora, N. Katte, and J. B. Serushema, “Metallodielectrics as metamaterials,” Proc. SPIE 7756, 77560F (2010).
[CrossRef]

D. de Ceglia, M. A. Vincenti, M. G. Cappeddu, M. Centini, N. Akozbek, A. D’Orazio, J. W. Haus, M. J. Bloemer, and M. Scalora, “Tailoring metallodielectric structures for super-resolution and superguiding application in the visible and near IR ranges,” Phys. Rev. A 77, 033848 (2008).
[CrossRef]

M. Scalora, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, D. de Ceglia, M. Centini, A. Mandatori, C. Sibilia, N. Akozbek, M. G. Cappeddu, M. Fowler, and J. W. Haus, “Negative refraction and subwavelength focusing in the visible range using transparent metallodielectric stacks,” Opt. Express 15, 508–523(2007).
[CrossRef] [PubMed]

M. Bloemer, G. D’Aguanno, N. Mattiucci, M. Scalora, and N. Akozbek, “Broadband super-resolving lens with high transparency in the visible range,” Appl. Phys. Lett. 90, 174113 (2007).
[CrossRef]

M. C. Larciprete, C. Sibilia, S. Paoloni, M. Bertolotti, F. Sarto, and M. Scalora, “Accessing the optical limiting properties of metallodielectric photonic bandgap structures,” J. Appl. Phys. 93, 5013–5017 (2003).
[CrossRef]

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, “Group velocity, energy velocity, and superluminal propagation in finite photonic bandgap structures,” Phys. Rev. E 63, 036610 (2001).
[CrossRef]

N. Katte, J. W. Haus, J. B. Serushema, and M. Scalora, “Super-resolving properties of metallodielectric stacks,” presented at COMSOL Conference, Boston, Massachusetts, USA (7–9 October 2010), http://www.dtic.mil/cgi-bin/GetTRDoc?Location=U2&doc=GetTRDoc.pdf&AD=ADA540223.

Schülzgen, A.

Schweinsberg, A.

N. N. Lepeshkin, A. Schweinsberg, G. Piredda, R. S. Bennink, and R. W. Boyd, “Enhanced nonlinear optical response of one-dimensional metal-dielectric photonic crystals,” Phys. Rev. Lett. 93, 123902 (2004).
[CrossRef] [PubMed]

Serushema, J. B.

J. W. Haus, M. Scalora, N. Katte, and J. B. Serushema, “Metallodielectrics as metamaterials,” Proc. SPIE 7756, 77560F (2010).
[CrossRef]

N. Katte, J. W. Haus, J. B. Serushema, and M. Scalora, “Super-resolving properties of metallodielectric stacks,” presented at COMSOL Conference, Boston, Massachusetts, USA (7–9 October 2010), http://www.dtic.mil/cgi-bin/GetTRDoc?Location=U2&doc=GetTRDoc.pdf&AD=ADA540223.

Sheik-Bahae, M.

M. Sheik-Bahae, A. A. Said, T. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
[CrossRef]

Sibilia, C.

M. Scalora, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, D. de Ceglia, M. Centini, A. Mandatori, C. Sibilia, N. Akozbek, M. G. Cappeddu, M. Fowler, and J. W. Haus, “Negative refraction and subwavelength focusing in the visible range using transparent metallodielectric stacks,” Opt. Express 15, 508–523(2007).
[CrossRef] [PubMed]

M. C. Larciprete, C. Sibilia, S. Paoloni, M. Bertolotti, F. Sarto, and M. Scalora, “Accessing the optical limiting properties of metallodielectric photonic bandgap structures,” J. Appl. Phys. 93, 5013–5017 (2003).
[CrossRef]

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, “Group velocity, energy velocity, and superluminal propagation in finite photonic bandgap structures,” Phys. Rev. E 63, 036610 (2001).
[CrossRef]

M. Bertolotti, C. M. Bowden, and C. Sibilia, “Nanoscale linear and nonlinear optics: International School on Quantum Electronics,” in Proceedings of the International School on Quantum Electronics (AIP, 2001).

Sipe, J. E.

Smith, F. M.

Sutherland, R. L.

R. L. Sutherland, D. G. McLean, and S. Kirkpatrick, Handbook of Nonlinear Optics2nd ed. (CRC Press, 2003).
[CrossRef]

Tian, J.

Trofimov, V. A.

Usmanov, T.

R. A. Ganeev, A. I. Ryasnyanskii, M. K. Kodirov, S. R. Kamalov, and T. Usmanov, “Nonlinear optical characteristics of colloidal solutions of metals,” Opt. Spectrosc. 90, 568–573 (2001).
[CrossRef]

van Driel, H. M.

Van Stryland, E. W.

M. Sheik-Bahae, A. A. Said, T. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
[CrossRef]

C. Fuentes-Hernandez, L. A. Padilha, J. M. Hales, D. Owens, J. Kim, S. Webster, S. R. Marder, J. W. Perry, D. J. Hagan, E. W. Van Stryland, and B. Kippelen, “Nonlinear optical response of transparent metal-dielectric multilayer structures,” in Integrated Photonics and Nanophotonics Research and Applications, OSA Technical Digest (CD) (Optical Society of America, 2009), paper JTuA2.

Vincenti, M. A.

D. de Ceglia, M. A. Vincenti, M. G. Cappeddu, M. Centini, N. Akozbek, A. D’Orazio, J. W. Haus, M. J. Bloemer, and M. Scalora, “Tailoring metallodielectric structures for super-resolution and superguiding application in the visible and near IR ranges,” Phys. Rev. A 77, 033848 (2008).
[CrossRef]

Webster, S.

C. Fuentes-Hernandez, L. A. Padilha, J. M. Hales, D. Owens, J. Kim, S. Webster, S. R. Marder, J. W. Perry, D. J. Hagan, E. W. Van Stryland, and B. Kippelen, “Nonlinear optical response of transparent metal-dielectric multilayer structures,” in Integrated Photonics and Nanophotonics Research and Applications, OSA Technical Digest (CD) (Optical Society of America, 2009), paper JTuA2.

Wei, J.

J. Wei and M. Xiao, “Z-scan model for optical nonlinear nanometric films,” J. Opt. A 10, 115102 (2008).
[CrossRef]

Wei, T.

M. Sheik-Bahae, A. A. Said, T. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
[CrossRef]

Wu, L.-J.

Xiao, M.

J. Wei and M. Xiao, “Z-scan model for optical nonlinear nanometric films,” J. Opt. A 10, 115102 (2008).
[CrossRef]

Yang, L.

Yeh, P.

P. Yeh, Optical Waves in Layered Media, J.W.Goodman, ed. (Wiley, 1988).

Yoon, Y. K.

Zang, W.

Zuhr, R. A.

Appl. Phys. Lett. (1)

M. Bloemer, G. D’Aguanno, N. Mattiucci, M. Scalora, and N. Akozbek, “Broadband super-resolving lens with high transparency in the visible range,” Appl. Phys. Lett. 90, 174113 (2007).
[CrossRef]

Chem. Mater. (1)

R. J. Gehr and R. W. Boyd, “Optical properties of nanostructured optical materials,” Chem. Mater. 8, 1807–1819 (1996).
[CrossRef]

IEEE J. Quantum Electron. (2)

M. Sheik-Bahae, A. A. Said, T. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
[CrossRef]

M. J. Bloemer, P. R. Ashley, J. W. Haus, N. Kalyaniwalla, and C. R. Christensen, “Third-Order optical nonlinearities in waveguide geometry,” IEEE J. Quantum Electron. 26, 1075–1080 (1990).
[CrossRef]

J. Appl. Phys. (2)

D. T. Owens, C. Fuentes-Hernandez, J. M. Hales, J. W. Perry, and B. Kippelen, “A comprehensive analysis of the contributions to the nonlinear optical properties of thin Ag films,” J. Appl. Phys. 107, 123114 (2010).
[CrossRef]

M. C. Larciprete, C. Sibilia, S. Paoloni, M. Bertolotti, F. Sarto, and M. Scalora, “Accessing the optical limiting properties of metallodielectric photonic bandgap structures,” J. Appl. Phys. 93, 5013–5017 (2003).
[CrossRef]

J. Korean Phys. Soc. (1)

G. J. Lee, Y. P. Lee, S. G. Jung, B. Y. Jung, C. K. Hwangbo, S. Kim, and I. Park, “Design, fabrication, linear, and nonlinear optical properties of metal-dielectric photonic bandgap structures,” J. Korean Phys. Soc. 51, 431–437 (2007).
[CrossRef]

J. Opt. A (1)

J. Wei and M. Xiao, “Z-scan model for optical nonlinear nanometric films,” J. Opt. A 10, 115102 (2008).
[CrossRef]

J. Opt. Soc. Am. B (4)

Nonlinear Optics Quantum Optics (1)

C. Fuentes-Hernandez and B. Kippelen, “Nonlinear optical properties of copper-based photonic bandgap structures at the onset of interband transitions” Nonlinear Optics Quantum Optics 40, 69–82 (2010).

Opt. Express (2)

Opt. Lett. (3)

Opt. Spectrosc. (1)

R. A. Ganeev, A. I. Ryasnyanskii, M. K. Kodirov, S. R. Kamalov, and T. Usmanov, “Nonlinear optical characteristics of colloidal solutions of metals,” Opt. Spectrosc. 90, 568–573 (2001).
[CrossRef]

Phys. Rev. A (2)

J. W. Haus, R. Inguva, and C. M. Bowden, “Effective-Medium theory of nonlinear ellipsoidal composites,” Phys. Rev. A 40, 5729–5734 (1989).
[CrossRef] [PubMed]

D. de Ceglia, M. A. Vincenti, M. G. Cappeddu, M. Centini, N. Akozbek, A. D’Orazio, J. W. Haus, M. J. Bloemer, and M. Scalora, “Tailoring metallodielectric structures for super-resolution and superguiding application in the visible and near IR ranges,” Phys. Rev. A 77, 033848 (2008).
[CrossRef]

Phys. Rev. E (1)

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, “Group velocity, energy velocity, and superluminal propagation in finite photonic bandgap structures,” Phys. Rev. E 63, 036610 (2001).
[CrossRef]

Phys. Rev. Lett. (1)

N. N. Lepeshkin, A. Schweinsberg, G. Piredda, R. S. Bennink, and R. W. Boyd, “Enhanced nonlinear optical response of one-dimensional metal-dielectric photonic crystals,” Phys. Rev. Lett. 93, 123902 (2004).
[CrossRef] [PubMed]

Proc. SPIE (1)

J. W. Haus, M. Scalora, N. Katte, and J. B. Serushema, “Metallodielectrics as metamaterials,” Proc. SPIE 7756, 77560F (2010).
[CrossRef]

Other (6)

E. D. Palik, Handbook of Optical Constants of Solids(Academic, 1985).

P. Yeh, Optical Waves in Layered Media, J.W.Goodman, ed. (Wiley, 1988).

M. Bertolotti, C. M. Bowden, and C. Sibilia, “Nanoscale linear and nonlinear optics: International School on Quantum Electronics,” in Proceedings of the International School on Quantum Electronics (AIP, 2001).

N. Katte, J. W. Haus, J. B. Serushema, and M. Scalora, “Super-resolving properties of metallodielectric stacks,” presented at COMSOL Conference, Boston, Massachusetts, USA (7–9 October 2010), http://www.dtic.mil/cgi-bin/GetTRDoc?Location=U2&doc=GetTRDoc.pdf&AD=ADA540223.

C. Fuentes-Hernandez, L. A. Padilha, J. M. Hales, D. Owens, J. Kim, S. Webster, S. R. Marder, J. W. Perry, D. J. Hagan, E. W. Van Stryland, and B. Kippelen, “Nonlinear optical response of transparent metal-dielectric multilayer structures,” in Integrated Photonics and Nanophotonics Research and Applications, OSA Technical Digest (CD) (Optical Society of America, 2009), paper JTuA2.

R. L. Sutherland, D. G. McLean, and S. Kirkpatrick, Handbook of Nonlinear Optics2nd ed. (CRC Press, 2003).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Optical properties of the TiO 2 and Cu multilayer system called MDS1. The sequence is a 40 nm thick TiO 2 layer followed by three and one-half periods of 20 nm thick Cu film and 80 nm thick TiO 2 film and finally a 40 nm thick TiO 2 film at the other end. This design is a transparent metal and it is denoted by the following notation describing the thickness of each layer: TiO 2 ( 40 nm ) / [ Cu ( 20 nm ) / TiO 2 ( 80 nm ) ] 3.5 / TiO 2 ( 40 nm ) . (a) Transmittance and absorbance versus wavelength, (b) group and energy indices versus wavelength.

Fig. 2
Fig. 2

Optical properties of the ZnS and Ag multilayer system called MDS2. The sequence of layers using the same notation given in Fig. 1 is ZnS ( 40 nm ) / [ Ag ( 20 nm ) / ZnS ( 80 nm ) ] 2.5 ZnS ( 40 nm ) . (a) Transmittance and absorbance versus wavelength, (b) group and energy indices versus wavelength.

Fig. 3
Fig. 3

Field squared at a wavelength of 650 nm throughout the MDS1 sample. The vertical lines are positions of the metal/dielectric interfaces. (a) TMM plot of field squared for TE polarization at different incident angles: 0 deg , solid; 30 deg , long dash; and 45 deg , dotted. (b) FEM and TMM plot of the field at plane-wave normal incidence.

Fig. 4
Fig. 4

(a) Z-scan for a 400 nm thick material whose refractive index at 532 nm is 1.2 and with a nonlinear refractive index n 2 = 9 × 10 16 m 2 / W . (b) Z-scan setup. The sample is moved a distance z from the beam focus while recording the transmittance at a detector placed behind a closed (i.e., small) or open aperture in the far-field regime.

Fig. 5
Fig. 5

Electric field amplitude squared versus position in the sample MDS1. The vertical lines mark the interfaces between the TiO 2 and Cu films.

Fig. 6
Fig. 6

Open aperture Z-scan curves for MDS1. The position of the sample is normalized by the Rayleigh range, z 0 .

Fig. 7
Fig. 7

Electric field amplitude squared versus position in the sample MDS2. The vertical lines mark the interfaces between the ZnS and Ag films.

Fig. 8
Fig. 8

Open aperture Z-scan curves for MDS2. The position of the sample is normalized by the Rayleigh range, z 0 .

Equations (14)

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t = | t | e i ϕ t , r = | r | e i ϕ r .
T = n t cos θ t n 0 cos θ i | t | 2 , R = | r | 2 ;
A = 1 T R .
ϕ t = k e L ,
N g = c k e ω = c L ϕ t ω .
U = 1 4 ( ( ω ε ) ω | E | 2 + ( ω μ ) ω | H | 2 ) = U E + U M ;
S av = 1 2 Re ( E × H * ) .
V E = 0 L S av · d z 0 L U d z .
N E = c / V E .
× ( 1 μ r × E φ ) ε r k 0 2 E φ = 0 .
ε r = ( n L + γ S av , z ) 2 ;
E ( Z , r ) = E 0 w 0 w ( z ) exp ( r 2 w 2 ( Z ) + i π r 2 λ R ( z ) + i ϕ ) .
T ( z ) = [ 0 r d r d r | E N L ( z , r ) | 2 0 r d r d r | E L ( z , r ) | 2 ] .
T r = 1 + ( 4 Z Φ ( Z 2 + 9 ) ( Z 2 + 1 ) ) ,

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