Abstract

We study fundamental and higher-order (HO) localized Bloch surface plasmons (BSPs) in a 1D subwavelength-periodic thin metal film with silicon and glass claddings. The film is a lamellar bimetallic grating where the real part of permittivity of the slit is also negative. We show that transmission enhancement due to HO BSPs is much weaker compared with the case when the first-order or fundamental BSP (FBSP) mediates resonant transmission. We also identify parameters of the structure corresponding to the mixed-order double resonance, i.e., a resonance between the FBSP on one surface of the film and a HO BSP on the other surface. We show that, unlike the double resonance between two FBSPs, this condition does not affect the transmission enhancement. Finally, we observe strong suppression of transmittance for a structure with a silicon superstrate and glass substrate. This effect occurs only when all higher-order harmonics in the substrate are evanescent. In the regime of transmission suppression, the propagating zero-order diffracted wave and evanescent first-order diffracted harmonic at the metal–glass interface are phase shifted by π.

© 2008 Optical Society of America

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2008

2007

X. D. Xoa, A. G. Kirk, and M. Tabrizian, “Towards integrated and sensitive surface plasmon resonance biosensors: a review of recent progress,” Biosens. Bioelectron. 23, 151-160 (2007).
[CrossRef]

S. Shen, E. Forsberg, Z. Han, and S. He, “Strong resonance coupling of surface plasmon polaritons to radiation modes through a thin metal slab with dielectric gratings,” J. Opt. Soc. Am. A 24, 225-230 (2007).
[CrossRef]

2006

P. G. Etchegoin, E. C. Le Ru, and M. Meyer, “An analytic model for the optical properties of gold,” J. Chem. Phys. 125, 164705 (2006).
[CrossRef] [PubMed]

2005

W. Cai, D. A. Genov, and V. M. Shalaev, “Superlens based metal-dielectric composites,” Phys. Rev. B 72, 193101 (2005).
[CrossRef]

K. G. Lee and Q.-H. Park, “Coupling of surface plasmon polaritons and light in metallic nanoslits,” Phys. Rev. Lett. 95, 103902 (2005).
[CrossRef] [PubMed]

S.-H. Chang, S. K. Gray, and G. C. Schatz, “Surface plasmon generation and light transmission by isolated nanoholes and arrays of nanoholes in thin metal films,” Opt. Express 13, 3150-3165 (2005).
[CrossRef] [PubMed]

A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. 408, 131-314 (2005).
[CrossRef]

Y. Xie, A. R. Zakharian, J. V. Moloney, and M. Mansuripur, “Transmission of light through a periodic array of slits in a thick metallic film,” Opt. Express 13, 4485-4491 (2005).
[CrossRef] [PubMed]

2004

W. L. Barnes, W. A. Murray, J. Dintinger, E. Devaux, and T. W. Ebbesen, “Surface plasmon polaritons and their role in the enhanced transmission of light through periodic arrays of subwavelength holes in a metal film,” Phys. Rev. Lett. 92, 107401 (2004).
[CrossRef] [PubMed]

H. Lezec and T. Thio, “Diffracted evanescent wave model for enhanced and suppressed optical transmission through subwavelength hole arrays,” Opt. Express 12, 3629-3651 (2004).
[CrossRef] [PubMed]

S. A. Darmanyan, M. Nevière, and A. V. Zayats, “Analytical theory of optical transmission through periodically structured metal films via tunnel-coupled surface polariton modes,” Phys. Rev. B 70, 075103 (2004).
[CrossRef]

X. Luo and T. Ishihara, “Subwavelength photolithography based on surface-plasmon polariton resonance,” Opt. Express 12, 3055-3065 (2004).
[CrossRef] [PubMed]

A. V. Kats and A. Y. Nikitin, “Analytical treatment of anomalous transparency of a modulated metal film due to surface plasmon-polariton excitation,” Phys. Rev. B 70, 235412 (2004).
[CrossRef]

2003

N. Bonod, S. Enoch, L. Li, E. Popov, and M. Nevière, “Resonant optical transmission through thin metallic films with and without holes,” Opt. Express 11, 482-490 (2003).
[CrossRef] [PubMed]

S. A. Darmanyan and A. V. Zayats, “Light tunneling via resonant surface plasmon polariton states and the enhanced transmission of periodically nanostructured metal films: an analytical study,” Phys. Rev. B 67, 035424 (2003).
[CrossRef]

A. M. Dykhne, A. K. Sarychev, and V. M. Shalaev, “Resonant transmission through metal films with fabricated and light-induced modulation,” Phys. Rev. B 67, 195402 (2003).
[CrossRef]

F. J. García-Vidal, H. J. Lezec, T. W. Ebbesen, and L. Martín-Moreno, “Multiple paths to enhance optical transmission through a single subwavelength slit,” Phys. Rev. Lett. 90, 213901 (2003).
[CrossRef] [PubMed]

2002

Q. Cao and P. Lalanne, “Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits,” Phys. Rev. Lett. 88, 057403 (2002).
[CrossRef] [PubMed]

2001

A. Krishnan, T. Thio, T. J. Kim, H. Lezec, T. W. Ebbesen, P. A. Wolff, J. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, “Evanescently coupled resonance in surface plasmon enhanced transmission,” Opt. Commun. 200, 1-7 (2001).
[CrossRef]

T. Thio, K. M. Pellerin, R. A. Linke, H. J. Lezec, and T. W. Ebbesen, “Enhanced light transmission through a single subwavelength aperture,” Opt. Lett. 26, 1972-1974 (2001).
[CrossRef]

2000

1999

J. A. Porto, F. J. García-Vidal, and J. B. Pendry, “Transmission resonances on metallic gratings with very narrow slits,” Phys. Rev. Lett. 83, 2845-2848 (1999).
[CrossRef]

1984

N. E. Glass and A. A. Maradudin, “Diffraction of light by a periodically modulated dielectric half-space,” Phys. Rev. B 29, 1840-1847 (1984).
[CrossRef]

1981

B. Laks, D. L. Mills, and A. A. Maradudin, “Surface polaritons on large-amplitude gratings,” Phys. Rev. B 23, 4965-4976 (1981).
[CrossRef]

1965

Avrutsky, I.

Barnes, W. L.

W. L. Barnes, W. A. Murray, J. Dintinger, E. Devaux, and T. W. Ebbesen, “Surface plasmon polaritons and their role in the enhanced transmission of light through periodic arrays of subwavelength holes in a metal film,” Phys. Rev. Lett. 92, 107401 (2004).
[CrossRef] [PubMed]

Bonod, N.

Cai, W.

W. Cai, D. A. Genov, and V. M. Shalaev, “Superlens based metal-dielectric composites,” Phys. Rev. B 72, 193101 (2005).
[CrossRef]

Cao, Q.

Q. Cao and P. Lalanne, “Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits,” Phys. Rev. Lett. 88, 057403 (2002).
[CrossRef] [PubMed]

Chang, S.-H.

Darmanyan, S. A.

A. Kobyakov, A. R. Zakharian, A. Mafi, and S. A. Darmanyan, “Semi-analytical method for light interaction with 1D-periodic nanoplasmonic structures,” Opt. Express 16, 8938-8957 (2008).
[CrossRef] [PubMed]

S. A. Darmanyan, M. Nevière, and A. V. Zayats, “Analytical theory of optical transmission through periodically structured metal films via tunnel-coupled surface polariton modes,” Phys. Rev. B 70, 075103 (2004).
[CrossRef]

S. A. Darmanyan and A. V. Zayats, “Light tunneling via resonant surface plasmon polariton states and the enhanced transmission of periodically nanostructured metal films: an analytical study,” Phys. Rev. B 67, 035424 (2003).
[CrossRef]

Devaux, E.

W. L. Barnes, W. A. Murray, J. Dintinger, E. Devaux, and T. W. Ebbesen, “Surface plasmon polaritons and their role in the enhanced transmission of light through periodic arrays of subwavelength holes in a metal film,” Phys. Rev. Lett. 92, 107401 (2004).
[CrossRef] [PubMed]

Dintinger, J.

W. L. Barnes, W. A. Murray, J. Dintinger, E. Devaux, and T. W. Ebbesen, “Surface plasmon polaritons and their role in the enhanced transmission of light through periodic arrays of subwavelength holes in a metal film,” Phys. Rev. Lett. 92, 107401 (2004).
[CrossRef] [PubMed]

Dykhne, A. M.

A. M. Dykhne, A. K. Sarychev, and V. M. Shalaev, “Resonant transmission through metal films with fabricated and light-induced modulation,” Phys. Rev. B 67, 195402 (2003).
[CrossRef]

Ebbesen, T. W.

W. L. Barnes, W. A. Murray, J. Dintinger, E. Devaux, and T. W. Ebbesen, “Surface plasmon polaritons and their role in the enhanced transmission of light through periodic arrays of subwavelength holes in a metal film,” Phys. Rev. Lett. 92, 107401 (2004).
[CrossRef] [PubMed]

F. J. García-Vidal, H. J. Lezec, T. W. Ebbesen, and L. Martín-Moreno, “Multiple paths to enhance optical transmission through a single subwavelength slit,” Phys. Rev. Lett. 90, 213901 (2003).
[CrossRef] [PubMed]

A. Krishnan, T. Thio, T. J. Kim, H. Lezec, T. W. Ebbesen, P. A. Wolff, J. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, “Evanescently coupled resonance in surface plasmon enhanced transmission,” Opt. Commun. 200, 1-7 (2001).
[CrossRef]

T. Thio, K. M. Pellerin, R. A. Linke, H. J. Lezec, and T. W. Ebbesen, “Enhanced light transmission through a single subwavelength aperture,” Opt. Lett. 26, 1972-1974 (2001).
[CrossRef]

Enoch, S.

Etchegoin, P. G.

P. G. Etchegoin, E. C. Le Ru, and M. Meyer, “An analytic model for the optical properties of gold,” J. Chem. Phys. 125, 164705 (2006).
[CrossRef] [PubMed]

Forsberg, E.

Garcia-Vidal, F. J.

A. Krishnan, T. Thio, T. J. Kim, H. Lezec, T. W. Ebbesen, P. A. Wolff, J. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, “Evanescently coupled resonance in surface plasmon enhanced transmission,” Opt. Commun. 200, 1-7 (2001).
[CrossRef]

García-Vidal, F. J.

F. J. García-Vidal, H. J. Lezec, T. W. Ebbesen, and L. Martín-Moreno, “Multiple paths to enhance optical transmission through a single subwavelength slit,” Phys. Rev. Lett. 90, 213901 (2003).
[CrossRef] [PubMed]

J. A. Porto, F. J. García-Vidal, and J. B. Pendry, “Transmission resonances on metallic gratings with very narrow slits,” Phys. Rev. Lett. 83, 2845-2848 (1999).
[CrossRef]

Genov, D. A.

W. Cai, D. A. Genov, and V. M. Shalaev, “Superlens based metal-dielectric composites,” Phys. Rev. B 72, 193101 (2005).
[CrossRef]

Glass, N. E.

N. E. Glass and A. A. Maradudin, “Diffraction of light by a periodically modulated dielectric half-space,” Phys. Rev. B 29, 1840-1847 (1984).
[CrossRef]

Gray, S. K.

Han, Z.

He, S.

Hessel, A.

Ishihara, T.

Kats, A. V.

A. V. Kats and A. Y. Nikitin, “Analytical treatment of anomalous transparency of a modulated metal film due to surface plasmon-polariton excitation,” Phys. Rev. B 70, 235412 (2004).
[CrossRef]

Kihm, J. E.

J. E. Kihm, Y. C. Yoon, K. G. Yee, D. J. Park, D. S. Kim, C. Ropers, C. Lienau, J. W. Park, J. Kim, and Q.-H. Park, “Positive and negative band gaps, Rayleigh-Wood's anomalies in plasmonic band-gap structures,” in Proceedings of the Quantum Electronics and Laser Science Conference 2005 (QELS 2005), Technical Digest (CD) (Optical Society of America, 2005), paper QMK6.
[CrossRef]

Kim, D. S.

J. E. Kihm, Y. C. Yoon, K. G. Yee, D. J. Park, D. S. Kim, C. Ropers, C. Lienau, J. W. Park, J. Kim, and Q.-H. Park, “Positive and negative band gaps, Rayleigh-Wood's anomalies in plasmonic band-gap structures,” in Proceedings of the Quantum Electronics and Laser Science Conference 2005 (QELS 2005), Technical Digest (CD) (Optical Society of America, 2005), paper QMK6.
[CrossRef]

Kim, J.

J. E. Kihm, Y. C. Yoon, K. G. Yee, D. J. Park, D. S. Kim, C. Ropers, C. Lienau, J. W. Park, J. Kim, and Q.-H. Park, “Positive and negative band gaps, Rayleigh-Wood's anomalies in plasmonic band-gap structures,” in Proceedings of the Quantum Electronics and Laser Science Conference 2005 (QELS 2005), Technical Digest (CD) (Optical Society of America, 2005), paper QMK6.
[CrossRef]

Kim, T. J.

A. Krishnan, T. Thio, T. J. Kim, H. Lezec, T. W. Ebbesen, P. A. Wolff, J. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, “Evanescently coupled resonance in surface plasmon enhanced transmission,” Opt. Commun. 200, 1-7 (2001).
[CrossRef]

Kirk, A. G.

X. D. Xoa, A. G. Kirk, and M. Tabrizian, “Towards integrated and sensitive surface plasmon resonance biosensors: a review of recent progress,” Biosens. Bioelectron. 23, 151-160 (2007).
[CrossRef]

Kobyakov, A.

Kochergin, V.

Krishnan, A.

A. Krishnan, T. Thio, T. J. Kim, H. Lezec, T. W. Ebbesen, P. A. Wolff, J. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, “Evanescently coupled resonance in surface plasmon enhanced transmission,” Opt. Commun. 200, 1-7 (2001).
[CrossRef]

Laks, B.

B. Laks, D. L. Mills, and A. A. Maradudin, “Surface polaritons on large-amplitude gratings,” Phys. Rev. B 23, 4965-4976 (1981).
[CrossRef]

Lalanne, P.

Q. Cao and P. Lalanne, “Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits,” Phys. Rev. Lett. 88, 057403 (2002).
[CrossRef] [PubMed]

Le Ru, E. C.

P. G. Etchegoin, E. C. Le Ru, and M. Meyer, “An analytic model for the optical properties of gold,” J. Chem. Phys. 125, 164705 (2006).
[CrossRef] [PubMed]

Lee, K. G.

K. G. Lee and Q.-H. Park, “Coupling of surface plasmon polaritons and light in metallic nanoslits,” Phys. Rev. Lett. 95, 103902 (2005).
[CrossRef] [PubMed]

Lezec, H.

H. Lezec and T. Thio, “Diffracted evanescent wave model for enhanced and suppressed optical transmission through subwavelength hole arrays,” Opt. Express 12, 3629-3651 (2004).
[CrossRef] [PubMed]

A. Krishnan, T. Thio, T. J. Kim, H. Lezec, T. W. Ebbesen, P. A. Wolff, J. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, “Evanescently coupled resonance in surface plasmon enhanced transmission,” Opt. Commun. 200, 1-7 (2001).
[CrossRef]

Lezec, H. J.

F. J. García-Vidal, H. J. Lezec, T. W. Ebbesen, and L. Martín-Moreno, “Multiple paths to enhance optical transmission through a single subwavelength slit,” Phys. Rev. Lett. 90, 213901 (2003).
[CrossRef] [PubMed]

T. Thio, K. M. Pellerin, R. A. Linke, H. J. Lezec, and T. W. Ebbesen, “Enhanced light transmission through a single subwavelength aperture,” Opt. Lett. 26, 1972-1974 (2001).
[CrossRef]

Li, L.

Lienau, C.

J. E. Kihm, Y. C. Yoon, K. G. Yee, D. J. Park, D. S. Kim, C. Ropers, C. Lienau, J. W. Park, J. Kim, and Q.-H. Park, “Positive and negative band gaps, Rayleigh-Wood's anomalies in plasmonic band-gap structures,” in Proceedings of the Quantum Electronics and Laser Science Conference 2005 (QELS 2005), Technical Digest (CD) (Optical Society of America, 2005), paper QMK6.
[CrossRef]

Linke, R. A.

Luo, X.

Mafi, A.

Maier, S. A.

S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007).

Mansuripur, M.

Maradudin, A. A.

A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. 408, 131-314 (2005).
[CrossRef]

N. E. Glass and A. A. Maradudin, “Diffraction of light by a periodically modulated dielectric half-space,” Phys. Rev. B 29, 1840-1847 (1984).
[CrossRef]

B. Laks, D. L. Mills, and A. A. Maradudin, “Surface polaritons on large-amplitude gratings,” Phys. Rev. B 23, 4965-4976 (1981).
[CrossRef]

Martin-Moreno, L.

A. Krishnan, T. Thio, T. J. Kim, H. Lezec, T. W. Ebbesen, P. A. Wolff, J. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, “Evanescently coupled resonance in surface plasmon enhanced transmission,” Opt. Commun. 200, 1-7 (2001).
[CrossRef]

Martín-Moreno, L.

F. J. García-Vidal, H. J. Lezec, T. W. Ebbesen, and L. Martín-Moreno, “Multiple paths to enhance optical transmission through a single subwavelength slit,” Phys. Rev. Lett. 90, 213901 (2003).
[CrossRef] [PubMed]

Meyer, M.

P. G. Etchegoin, E. C. Le Ru, and M. Meyer, “An analytic model for the optical properties of gold,” J. Chem. Phys. 125, 164705 (2006).
[CrossRef] [PubMed]

Mills, D. L.

B. Laks, D. L. Mills, and A. A. Maradudin, “Surface polaritons on large-amplitude gratings,” Phys. Rev. B 23, 4965-4976 (1981).
[CrossRef]

Moloney, J. V.

Murray, W. A.

W. L. Barnes, W. A. Murray, J. Dintinger, E. Devaux, and T. W. Ebbesen, “Surface plasmon polaritons and their role in the enhanced transmission of light through periodic arrays of subwavelength holes in a metal film,” Phys. Rev. Lett. 92, 107401 (2004).
[CrossRef] [PubMed]

Nevière, M.

S. A. Darmanyan, M. Nevière, and A. V. Zayats, “Analytical theory of optical transmission through periodically structured metal films via tunnel-coupled surface polariton modes,” Phys. Rev. B 70, 075103 (2004).
[CrossRef]

N. Bonod, S. Enoch, L. Li, E. Popov, and M. Nevière, “Resonant optical transmission through thin metallic films with and without holes,” Opt. Express 11, 482-490 (2003).
[CrossRef] [PubMed]

Nikitin, A. Y.

A. V. Kats and A. Y. Nikitin, “Analytical treatment of anomalous transparency of a modulated metal film due to surface plasmon-polariton excitation,” Phys. Rev. B 70, 235412 (2004).
[CrossRef]

Oliner, A. A.

Park, D. J.

J. E. Kihm, Y. C. Yoon, K. G. Yee, D. J. Park, D. S. Kim, C. Ropers, C. Lienau, J. W. Park, J. Kim, and Q.-H. Park, “Positive and negative band gaps, Rayleigh-Wood's anomalies in plasmonic band-gap structures,” in Proceedings of the Quantum Electronics and Laser Science Conference 2005 (QELS 2005), Technical Digest (CD) (Optical Society of America, 2005), paper QMK6.
[CrossRef]

Park, J. W.

J. E. Kihm, Y. C. Yoon, K. G. Yee, D. J. Park, D. S. Kim, C. Ropers, C. Lienau, J. W. Park, J. Kim, and Q.-H. Park, “Positive and negative band gaps, Rayleigh-Wood's anomalies in plasmonic band-gap structures,” in Proceedings of the Quantum Electronics and Laser Science Conference 2005 (QELS 2005), Technical Digest (CD) (Optical Society of America, 2005), paper QMK6.
[CrossRef]

Park, Q.-H.

K. G. Lee and Q.-H. Park, “Coupling of surface plasmon polaritons and light in metallic nanoslits,” Phys. Rev. Lett. 95, 103902 (2005).
[CrossRef] [PubMed]

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[CrossRef] [PubMed]

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[CrossRef]

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Figures (12)

Fig. 1
Fig. 1

(a) Schematic of the planar bimetallic grating with dielectric claddings and (b) the real part of the permittivity of the film ϵ as a function of the transverse coordinate x. The incident plane TM wave has nonzero field components E x and H y : ϵ and ϵ + are the dielectric permittivity of the substrate and the superstrate, respectively. The grating has period Λ, duty cycle ρ = w Λ , contrast Δ ϵ , and thickness h.

Fig. 2
Fig. 2

Estimate of the k th -order resonant wavelength of a BSP for a glass–SM interface (thin curves, ϵ s = 2.3 , dispersion in the glass is neglected) and a silicon–SM interface (thick curves, ϵ s ( λ ) is taken from [32]). The results for the first three resonant orders are shown. The dependence ϵ met ( λ ) is given by Eq. (1).

Fig. 3
Fig. 3

Transmittance in decibels ( 10 log 10 T ) versus the free-space wavelength of incident light for a symmetric structure with both the substrate and the superstrate made of either glass, ϵ + = ϵ = 2.56 , or silicon, ϵ + = ϵ = 13.2 . The film thickness is 100 nm . Solid curves are obtained with the semianalytical method with N = 15 . Scattered data show FEM results for a true rectangular grating. Dispersion of the dielectric in the wavelength window of interest is neglected, i.e., ϵ ± ( λ ) = constant .

Fig. 4
Fig. 4

Field components E x , E z (V/m), and H y (A/m) plotted over the period Λ = 500 nm in the structured film ( h < z < 0 ) , h = 100 nm , and in glass ( ϵ + = ϵ = 2.56 ) due to a normally incident TM plane wave with A x = 1 V m ; log 10 S and S ( x , z ) are shown in the rightmost plot. The wavelength λ = 844.6 nm corresponds to point A in Fig. 3.

Fig. 5
Fig. 5

Same as Fig. 4 for Si substrates ϵ + = ϵ = 13.2 and the wavelength λ = 854.1 nm corresponding to point B in Fig. 3.

Fig. 6
Fig. 6

Transmittance of the structured metal film with a glass superstrate and a silicon substrate ( ϵ = 13.2 ) . Scattered data show FEM results. The film thickness is 100 nm .

Fig. 7
Fig. 7

Field structure for a glass superstrate ( ϵ + = 2.56 ) and a silicon substrate ( ϵ = 13.2 ) . The wavelength λ = 841.8 nm corresponds to point C in Fig. 6. Bottom plots show the magnified region near the SM–Si interface z = 100 nm .

Fig. 8
Fig. 8

Transmission (dB) through the structured metal film for a silicon superstrate ( ϵ + = 13.2 ) and a glass substrate. Scattered data show FEM calculations. The film thickness is 100 nm .

Fig. 9
Fig. 9

Electromagnetic field in the structure with a silicon superstrate ( ϵ + = 13.2 ) and a glass substrate ( ϵ = 2.56 ) at the wavelength λ = 841.8 nm corresponding to point D in Fig. 8.

Fig. 10
Fig. 10

Electromagnetic field in the structure with a silicon superstrate ( ϵ + = 13.2 ) and a glass substrate ( ϵ = 2.625 ) at the wavelength λ = 852.2 nm corresponding to point E in Fig. 8.

Fig. 11
Fig. 11

(a) Amplitudes and (b) phases of the propagating zero-order diffracted wave T x 0 and the evanescent wave T x 1 in the glass substrate ( ϵ = 2.56 ) . The superstrate is silicon with ϵ + = 13.2 . The dotted curve shows the phase difference Δ ϕ = arg ( T x 1 ) arg ( T x 0 ) .

Fig. 12
Fig. 12

Electromagnetic field in the structure with a silicon superstrate ( ϵ + = 13.2 ) and a glass substrate ( ϵ = 2.56 ) at the wavelength λ = 839.8 nm corresponding to point F in Fig. 8.

Equations (6)

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ϵ 1 , 2 ( λ ) = ϵ 1 , 2 [ ( λ p λ ) 2 + i λ p 2 ( γ p λ ) ] 1 ,
λ k Λ = ϵ met ( λ ) ϵ s ( λ ) ϵ met ( λ ) + ϵ s ( λ ) ,
E x + = A x e i K + z + R x 0 e i K + z + k = 1 N R x k cos ( 2 π k x Λ ) e s + η k + z ,
E x = T x 0 e i K z + k = 1 N T x k cos ( 2 π k x Λ ) e s η k ( z + h ) ,
η k ± = 2 π k 2 Λ 2 ϵ ± λ 2 ,
T = 1 A x 2 ϵ ϵ + [ T x 0 2 + 1 2 k = 1 M T x k 2 ( 1 k 2 λ 2 ϵ Λ 2 ) 1 2 ] ,

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