Abstract

Penetration depth defines the measurable range in evanescent-wave-based sensing techniques such as surface plasmon resonance (SPR). We investigate penetration depth variation implemented with dielectric layers in a SPR sensing structure. The results show that the penetration depth can be controlled to increase or decrease depending on a specific configuration. Effective medium theory was introduced to describe the field penetration in dielectric multilayer designs. Comparison was made with the field penetration of a localized SPR structure based on periodic nanowires. The penetration depth variation in response to environmental changes was also explored.

© 2007 Optical Society of America

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2007

2006

2005

A. W. Wark, H. J. Lee, and R. M. Corn, "Long-range surface plasmon resonance imaging for bioaffinity sensors," Anal. Chem. 77, 3904-3907 (2005).
[CrossRef] [PubMed]

M. R. Malone, J.-F. Masson, S. Beaudoin, and K. S. Booksh, "Novel surface coatings for antibody attachment to surface plasmon resonance sensors," Proc. SPIE 6007, 36-45 (2005).

2003

K. Stock, R. Sailer, W. S. L. Strauss, M. Lyttek, R. Steiner, and H. Schneckenburger, "Variable-angle total internal reflection fluorescence microscopy (VA-TIRFM): realization and application of a compact illumination device," J. Microsc. 211, 19-29 (2003).
[CrossRef] [PubMed]

A. J. A. El-Haija, "Effective medium approximation for the effective optical constants of a bilayer and a multilayer structure based on the characteristic matrix technique," J. Appl. Phys. 93, 2590-2594 (2003).
[CrossRef]

2002

M. Hide, T. Tsutsui, H. Sato, T. Nishimura, K. Morimoto, S. Yamamoto, and K. Yoshizato, "Real-time analysis of ligand-induced cell surface and intracellular reactions of living mast cells using a surface plasmon resonance-based biosensor," Anal. Biochem. 302, 28-37 (2002).
[CrossRef] [PubMed]

2001

J. A. Steyer and W. Almers, "A real-time view of life within 100 nm of the plasma membrane," Nat. Rev. Mol. Cell Biol. 2, 268-275 (2001).
[CrossRef] [PubMed]

1999

K.-F. Giebel, C. Bechinger, S. Herminghaus, M. Riedel, P. Leiderer, U. Weiland, and M. Bastmeyer, "Imaging of cell/substrate contacts of living cells with surface plasmon resonance microscopy," Biophys. J. 76, 509-516 (1999).
[CrossRef] [PubMed]

1997

P. C. Ke, X. S. Gan, J. Szajman, S. Schilders, and M. Gu, "Optimizing the strength of an evanescent wave generated from a prism coated with a double-layer thin-film stack," Bioimaging 5, 1-8 (1997).
[CrossRef]

C.-H. Liao, C.-M. Lee, L.-B. Chang, and J.-H. Tsai, "Effects of a metal film and prism dielectric on properties of surface plasmon resonance in a multilayer system," Jpn. J. Appl. Phys., Part 1 36, 1105-1111 (1997).
[CrossRef]

1994

W. J. H. Bender, R. E. Dessy, M. S. Miller, and R. O. Claus, "Feasibility of a chemical microsensor based on surface plasmon resonance on fiber optics modified by multilayer vapor deposition," Anal. Chem. 66, 963-970 (1994).
[CrossRef]

R. Kaiser, Y. Levy, N. Vansteenkiste, A. Aspect, W. Seifert, D. Leipold, and J. Mlynek, "Resonant enhancement of evanescent waves with a thin dielectric waveguide," Opt. Commun. 104, 234-240 (1994).
[CrossRef]

1990

1983

1982

1972

Y. Levy, "Étude du champ inhomogène obtenu par la réflexion totale d'une onde plane sur un système de couches minces," Nouv. Rev. Opt. Appl. 3, 25-30 (1972).
[CrossRef]

1956

S. M. Rytov, "Electromagnetic properties of a finely stratified medium," Sov. Phys. JETP 2, 466-475 (1956).

Almers, W.

J. A. Steyer and W. Almers, "A real-time view of life within 100 nm of the plasma membrane," Nat. Rev. Mol. Cell Biol. 2, 268-275 (2001).
[CrossRef] [PubMed]

Armelles, G.

Aspect, A.

R. Kaiser, Y. Levy, N. Vansteenkiste, A. Aspect, W. Seifert, D. Leipold, and J. Mlynek, "Resonant enhancement of evanescent waves with a thin dielectric waveguide," Opt. Commun. 104, 234-240 (1994).
[CrossRef]

Bastmeyer, M.

K.-F. Giebel, C. Bechinger, S. Herminghaus, M. Riedel, P. Leiderer, U. Weiland, and M. Bastmeyer, "Imaging of cell/substrate contacts of living cells with surface plasmon resonance microscopy," Biophys. J. 76, 509-516 (1999).
[CrossRef] [PubMed]

Beaudoin, S.

M. R. Malone, J.-F. Masson, S. Beaudoin, and K. S. Booksh, "Novel surface coatings for antibody attachment to surface plasmon resonance sensors," Proc. SPIE 6007, 36-45 (2005).

Bechinger, C.

K.-F. Giebel, C. Bechinger, S. Herminghaus, M. Riedel, P. Leiderer, U. Weiland, and M. Bastmeyer, "Imaging of cell/substrate contacts of living cells with surface plasmon resonance microscopy," Biophys. J. 76, 509-516 (1999).
[CrossRef] [PubMed]

Bender, W. J. H.

W. J. H. Bender, R. E. Dessy, M. S. Miller, and R. O. Claus, "Feasibility of a chemical microsensor based on surface plasmon resonance on fiber optics modified by multilayer vapor deposition," Anal. Chem. 66, 963-970 (1994).
[CrossRef]

Booksh, K. S.

M. R. Malone, J.-F. Masson, S. Beaudoin, and K. S. Booksh, "Novel surface coatings for antibody attachment to surface plasmon resonance sensors," Proc. SPIE 6007, 36-45 (2005).

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999), pp. 837-840.

Calle, A.

Chang, L.-B.

C.-H. Liao, C.-M. Lee, L.-B. Chang, and J.-H. Tsai, "Effects of a metal film and prism dielectric on properties of surface plasmon resonance in a multilayer system," Jpn. J. Appl. Phys., Part 1 36, 1105-1111 (1997).
[CrossRef]

Claus, R. O.

W. J. H. Bender, R. E. Dessy, M. S. Miller, and R. O. Claus, "Feasibility of a chemical microsensor based on surface plasmon resonance on fiber optics modified by multilayer vapor deposition," Anal. Chem. 66, 963-970 (1994).
[CrossRef]

Corn, R. M.

A. W. Wark, H. J. Lee, and R. M. Corn, "Long-range surface plasmon resonance imaging for bioaffinity sensors," Anal. Chem. 77, 3904-3907 (2005).
[CrossRef] [PubMed]

Cottier, K.

R. E. Kunz and K. Cottier, "Optimizing integrated optical chips for label-free (bio-)chemical sensing," Anal. Bioanal. Chem. 384, 180-190 (2006).
[CrossRef]

Dessy, R. E.

W. J. H. Bender, R. E. Dessy, M. S. Miller, and R. O. Claus, "Feasibility of a chemical microsensor based on surface plasmon resonance on fiber optics modified by multilayer vapor deposition," Anal. Chem. 66, 963-970 (1994).
[CrossRef]

El-Haija, A. J. A.

A. J. A. El-Haija, "Effective medium approximation for the effective optical constants of a bilayer and a multilayer structure based on the characteristic matrix technique," J. Appl. Phys. 93, 2590-2594 (2003).
[CrossRef]

Gan, X. S.

P. C. Ke, X. S. Gan, J. Szajman, S. Schilders, and M. Gu, "Optimizing the strength of an evanescent wave generated from a prism coated with a double-layer thin-film stack," Bioimaging 5, 1-8 (1997).
[CrossRef]

Gaylord, T. K.

Gerritsma, G. J.

Ghatak, A.

A. Ghatak and K. Thyagarajan, Optical Electronics (Cambridge U. Press, 1989), Chap. 2.

Giebel, K.-F.

K.-F. Giebel, C. Bechinger, S. Herminghaus, M. Riedel, P. Leiderer, U. Weiland, and M. Bastmeyer, "Imaging of cell/substrate contacts of living cells with surface plasmon resonance microscopy," Biophys. J. 76, 509-516 (1999).
[CrossRef] [PubMed]

Gu, M.

P. C. Ke, X. S. Gan, J. Szajman, S. Schilders, and M. Gu, "Optimizing the strength of an evanescent wave generated from a prism coated with a double-layer thin-film stack," Bioimaging 5, 1-8 (1997).
[CrossRef]

Herminghaus, S.

K.-F. Giebel, C. Bechinger, S. Herminghaus, M. Riedel, P. Leiderer, U. Weiland, and M. Bastmeyer, "Imaging of cell/substrate contacts of living cells with surface plasmon resonance microscopy," Biophys. J. 76, 509-516 (1999).
[CrossRef] [PubMed]

Hide, M.

M. Hide, T. Tsutsui, H. Sato, T. Nishimura, K. Morimoto, S. Yamamoto, and K. Yoshizato, "Real-time analysis of ligand-induced cell surface and intracellular reactions of living mast cells using a surface plasmon resonance-based biosensor," Anal. Biochem. 302, 28-37 (2002).
[CrossRef] [PubMed]

Homola, J.

Kaiser, R.

R. Kaiser, Y. Levy, N. Vansteenkiste, A. Aspect, W. Seifert, D. Leipold, and J. Mlynek, "Resonant enhancement of evanescent waves with a thin dielectric waveguide," Opt. Commun. 104, 234-240 (1994).
[CrossRef]

Kawata, S.

Ke, P. C.

P. C. Ke, X. S. Gan, J. Szajman, S. Schilders, and M. Gu, "Optimizing the strength of an evanescent wave generated from a prism coated with a double-layer thin-film stack," Bioimaging 5, 1-8 (1997).
[CrossRef]

Kim, D.

Kim, K.

Kreuwel, H. J. M.

Kunz, R. E.

R. E. Kunz and K. Cottier, "Optimizing integrated optical chips for label-free (bio-)chemical sensing," Anal. Bioanal. Chem. 384, 180-190 (2006).
[CrossRef]

Lambeck, P. V.

Lechuga, L. M.

Lee, C.-M.

C.-H. Liao, C.-M. Lee, L.-B. Chang, and J.-H. Tsai, "Effects of a metal film and prism dielectric on properties of surface plasmon resonance in a multilayer system," Jpn. J. Appl. Phys., Part 1 36, 1105-1111 (1997).
[CrossRef]

Lee, H. J.

A. W. Wark, H. J. Lee, and R. M. Corn, "Long-range surface plasmon resonance imaging for bioaffinity sensors," Anal. Chem. 77, 3904-3907 (2005).
[CrossRef] [PubMed]

Leiderer, P.

K.-F. Giebel, C. Bechinger, S. Herminghaus, M. Riedel, P. Leiderer, U. Weiland, and M. Bastmeyer, "Imaging of cell/substrate contacts of living cells with surface plasmon resonance microscopy," Biophys. J. 76, 509-516 (1999).
[CrossRef] [PubMed]

Leipold, D.

R. Kaiser, Y. Levy, N. Vansteenkiste, A. Aspect, W. Seifert, D. Leipold, and J. Mlynek, "Resonant enhancement of evanescent waves with a thin dielectric waveguide," Opt. Commun. 104, 234-240 (1994).
[CrossRef]

Levy, Y.

R. Kaiser, Y. Levy, N. Vansteenkiste, A. Aspect, W. Seifert, D. Leipold, and J. Mlynek, "Resonant enhancement of evanescent waves with a thin dielectric waveguide," Opt. Commun. 104, 234-240 (1994).
[CrossRef]

Y. Levy, "Étude du champ inhomogène obtenu par la réflexion totale d'une onde plane sur un système de couches minces," Nouv. Rev. Opt. Appl. 3, 25-30 (1972).
[CrossRef]

Liao, C.-H.

C.-H. Liao, C.-M. Lee, L.-B. Chang, and J.-H. Tsai, "Effects of a metal film and prism dielectric on properties of surface plasmon resonance in a multilayer system," Jpn. J. Appl. Phys., Part 1 36, 1105-1111 (1997).
[CrossRef]

Lyttek, M.

K. Stock, R. Sailer, W. S. L. Strauss, M. Lyttek, R. Steiner, and H. Schneckenburger, "Variable-angle total internal reflection fluorescence microscopy (VA-TIRFM): realization and application of a compact illumination device," J. Microsc. 211, 19-29 (2003).
[CrossRef] [PubMed]

Malone, M. R.

M. R. Malone, J.-F. Masson, S. Beaudoin, and K. S. Booksh, "Novel surface coatings for antibody attachment to surface plasmon resonance sensors," Proc. SPIE 6007, 36-45 (2005).

Masson, J.-F.

M. R. Malone, J.-F. Masson, S. Beaudoin, and K. S. Booksh, "Novel surface coatings for antibody attachment to surface plasmon resonance sensors," Proc. SPIE 6007, 36-45 (2005).

Matsubara, K.

Miller, M. S.

W. J. H. Bender, R. E. Dessy, M. S. Miller, and R. O. Claus, "Feasibility of a chemical microsensor based on surface plasmon resonance on fiber optics modified by multilayer vapor deposition," Anal. Chem. 66, 963-970 (1994).
[CrossRef]

Minami, S.

Mlynek, J.

R. Kaiser, Y. Levy, N. Vansteenkiste, A. Aspect, W. Seifert, D. Leipold, and J. Mlynek, "Resonant enhancement of evanescent waves with a thin dielectric waveguide," Opt. Commun. 104, 234-240 (1994).
[CrossRef]

Moharam, M. G.

Moon, S.

Morimoto, K.

M. Hide, T. Tsutsui, H. Sato, T. Nishimura, K. Morimoto, S. Yamamoto, and K. Yoshizato, "Real-time analysis of ligand-induced cell surface and intracellular reactions of living mast cells using a surface plasmon resonance-based biosensor," Anal. Biochem. 302, 28-37 (2002).
[CrossRef] [PubMed]

Nishimura, T.

M. Hide, T. Tsutsui, H. Sato, T. Nishimura, K. Morimoto, S. Yamamoto, and K. Yoshizato, "Real-time analysis of ligand-induced cell surface and intracellular reactions of living mast cells using a surface plasmon resonance-based biosensor," Anal. Biochem. 302, 28-37 (2002).
[CrossRef] [PubMed]

Popma, T. J. A.

Quail, J. C.

Raether, H.

H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, 1988).

Rako, J. G.

Reinhoudt, D. N.

Riedel, M.

K.-F. Giebel, C. Bechinger, S. Herminghaus, M. Riedel, P. Leiderer, U. Weiland, and M. Bastmeyer, "Imaging of cell/substrate contacts of living cells with surface plasmon resonance microscopy," Biophys. J. 76, 509-516 (1999).
[CrossRef] [PubMed]

Rytov, S. M.

S. M. Rytov, "Electromagnetic properties of a finely stratified medium," Sov. Phys. JETP 2, 466-475 (1956).

Sailer, R.

K. Stock, R. Sailer, W. S. L. Strauss, M. Lyttek, R. Steiner, and H. Schneckenburger, "Variable-angle total internal reflection fluorescence microscopy (VA-TIRFM): realization and application of a compact illumination device," J. Microsc. 211, 19-29 (2003).
[CrossRef] [PubMed]

Sato, H.

M. Hide, T. Tsutsui, H. Sato, T. Nishimura, K. Morimoto, S. Yamamoto, and K. Yoshizato, "Real-time analysis of ligand-induced cell surface and intracellular reactions of living mast cells using a surface plasmon resonance-based biosensor," Anal. Biochem. 302, 28-37 (2002).
[CrossRef] [PubMed]

Schilders, S.

P. C. Ke, X. S. Gan, J. Szajman, S. Schilders, and M. Gu, "Optimizing the strength of an evanescent wave generated from a prism coated with a double-layer thin-film stack," Bioimaging 5, 1-8 (1997).
[CrossRef]

Schneckenburger, H.

K. Stock, R. Sailer, W. S. L. Strauss, M. Lyttek, R. Steiner, and H. Schneckenburger, "Variable-angle total internal reflection fluorescence microscopy (VA-TIRFM): realization and application of a compact illumination device," J. Microsc. 211, 19-29 (2003).
[CrossRef] [PubMed]

Seifert, W.

R. Kaiser, Y. Levy, N. Vansteenkiste, A. Aspect, W. Seifert, D. Leipold, and J. Mlynek, "Resonant enhancement of evanescent waves with a thin dielectric waveguide," Opt. Commun. 104, 234-240 (1994).
[CrossRef]

Sepúlveda, B.

Simon, H. J.

Slavík, R.

Steiner, R.

K. Stock, R. Sailer, W. S. L. Strauss, M. Lyttek, R. Steiner, and H. Schneckenburger, "Variable-angle total internal reflection fluorescence microscopy (VA-TIRFM): realization and application of a compact illumination device," J. Microsc. 211, 19-29 (2003).
[CrossRef] [PubMed]

Steyer, J. A.

J. A. Steyer and W. Almers, "A real-time view of life within 100 nm of the plasma membrane," Nat. Rev. Mol. Cell Biol. 2, 268-275 (2001).
[CrossRef] [PubMed]

Stock, K.

K. Stock, R. Sailer, W. S. L. Strauss, M. Lyttek, R. Steiner, and H. Schneckenburger, "Variable-angle total internal reflection fluorescence microscopy (VA-TIRFM): realization and application of a compact illumination device," J. Microsc. 211, 19-29 (2003).
[CrossRef] [PubMed]

Strauss, W. S. L.

K. Stock, R. Sailer, W. S. L. Strauss, M. Lyttek, R. Steiner, and H. Schneckenburger, "Variable-angle total internal reflection fluorescence microscopy (VA-TIRFM): realization and application of a compact illumination device," J. Microsc. 211, 19-29 (2003).
[CrossRef] [PubMed]

Sudholter, E. J. R.

Szajman, J.

P. C. Ke, X. S. Gan, J. Szajman, S. Schilders, and M. Gu, "Optimizing the strength of an evanescent wave generated from a prism coated with a double-layer thin-film stack," Bioimaging 5, 1-8 (1997).
[CrossRef]

Thyagarajan, K.

A. Ghatak and K. Thyagarajan, Optical Electronics (Cambridge U. Press, 1989), Chap. 2.

Tsai, J.-H.

C.-H. Liao, C.-M. Lee, L.-B. Chang, and J.-H. Tsai, "Effects of a metal film and prism dielectric on properties of surface plasmon resonance in a multilayer system," Jpn. J. Appl. Phys., Part 1 36, 1105-1111 (1997).
[CrossRef]

Tsutsui, T.

M. Hide, T. Tsutsui, H. Sato, T. Nishimura, K. Morimoto, S. Yamamoto, and K. Yoshizato, "Real-time analysis of ligand-induced cell surface and intracellular reactions of living mast cells using a surface plasmon resonance-based biosensor," Anal. Biochem. 302, 28-37 (2002).
[CrossRef] [PubMed]

van Gent, J.

Vansteenkiste, N.

R. Kaiser, Y. Levy, N. Vansteenkiste, A. Aspect, W. Seifert, D. Leipold, and J. Mlynek, "Resonant enhancement of evanescent waves with a thin dielectric waveguide," Opt. Commun. 104, 234-240 (1994).
[CrossRef]

Wark, A. W.

A. W. Wark, H. J. Lee, and R. M. Corn, "Long-range surface plasmon resonance imaging for bioaffinity sensors," Anal. Chem. 77, 3904-3907 (2005).
[CrossRef] [PubMed]

Weiland, U.

K.-F. Giebel, C. Bechinger, S. Herminghaus, M. Riedel, P. Leiderer, U. Weiland, and M. Bastmeyer, "Imaging of cell/substrate contacts of living cells with surface plasmon resonance microscopy," Biophys. J. 76, 509-516 (1999).
[CrossRef] [PubMed]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999), pp. 837-840.

Yamamoto, S.

M. Hide, T. Tsutsui, H. Sato, T. Nishimura, K. Morimoto, S. Yamamoto, and K. Yoshizato, "Real-time analysis of ligand-induced cell surface and intracellular reactions of living mast cells using a surface plasmon resonance-based biosensor," Anal. Biochem. 302, 28-37 (2002).
[CrossRef] [PubMed]

Yoon, S. J.

Yoshizato, K.

M. Hide, T. Tsutsui, H. Sato, T. Nishimura, K. Morimoto, S. Yamamoto, and K. Yoshizato, "Real-time analysis of ligand-induced cell surface and intracellular reactions of living mast cells using a surface plasmon resonance-based biosensor," Anal. Biochem. 302, 28-37 (2002).
[CrossRef] [PubMed]

Anal. Bioanal. Chem.

R. E. Kunz and K. Cottier, "Optimizing integrated optical chips for label-free (bio-)chemical sensing," Anal. Bioanal. Chem. 384, 180-190 (2006).
[CrossRef]

Anal. Biochem.

M. Hide, T. Tsutsui, H. Sato, T. Nishimura, K. Morimoto, S. Yamamoto, and K. Yoshizato, "Real-time analysis of ligand-induced cell surface and intracellular reactions of living mast cells using a surface plasmon resonance-based biosensor," Anal. Biochem. 302, 28-37 (2002).
[CrossRef] [PubMed]

Anal. Chem.

A. W. Wark, H. J. Lee, and R. M. Corn, "Long-range surface plasmon resonance imaging for bioaffinity sensors," Anal. Chem. 77, 3904-3907 (2005).
[CrossRef] [PubMed]

W. J. H. Bender, R. E. Dessy, M. S. Miller, and R. O. Claus, "Feasibility of a chemical microsensor based on surface plasmon resonance on fiber optics modified by multilayer vapor deposition," Anal. Chem. 66, 963-970 (1994).
[CrossRef]

Appl. Opt.

Bioimaging

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Figures (5)

Fig. 1
Fig. 1

Schematics of a thin-film-based SPR structure using N dielectric multilayers deposited (a) on top of a gold thin film and (b) between the thin film and a dielectric substrate. (c) LSPR structure based on periodic nanowires.

Fig. 2
Fig. 2

Penetration depths with respect to resonance angles ( θ SPR ) for an air and water environment on a gold film and a substrate of BK7 or SF10. No dielectric layers are assumed between gold film and environment, i.e., N = 0 . Refractive indices were taken from Sellmeier models at λ = 0.6328 μ m .

Fig. 3
Fig. 3

Penetration depths at SPR in the presence of a single dielectric layer ( N = 1 ) at d 2 = 10 , 30, and 50 nm for Case I: (a) BK7 substrate and air environment (BK7/air) and (b) SF10 substrate and water environment (SF10/water). Solid curves represent the results calculated by propagation matrix analysis. Filled and open squares correspond to TE and TM effective medium-based calculation given by Eqs. (8, 9), respectively.

Fig. 4
Fig. 4

Penetration depths at SPR in the presence of a single dielectric layer ( N = 1 ) for Case II: BK7/air and SF10/water.

Fig. 5
Fig. 5

Penetration depths at SPR of a LSPR structure based on periodic nanowires (Case III). The presented data refer the penetration depths calculated at the center of nanowire tops. d g is a variable that denotes nanowire thickness. The thickness of the film below the nanowires is fixed at 40 nm . The dotted horizontal lines represent the penetration depth of a thin-film-based SPR structure for comparison, where gold film thickness is provided as d f .

Tables (1)

Tables Icon

Table 1 Per Unit Resonance Angle Shift, Penetration Depth Variation ( Δ δ Δ θ SPR ) , and Normalized Penetration Depth Modulation ( Δ δ δ Δ θ SPR ) with Typical Experimental Parameters for a 40 - nm -Thick Gold Film a

Equations (15)

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δ = λ 4 π n 0 2 sin 2 θ i n 1 2 ,
S j = 1 t j , j + 1 [ e j Δ j + 1 r j , j + 1 e j Δ j + 1 r j , j + 1 e j Δ j + 1 e j Δ j + 1 ] ,
δ = λ 4 π n 0 2 sin 2 θ SPR n N + 2 2 ,
n 0 sin θ SPR = ( ϵ 1 ϵ N + 2 ϵ 1 + ϵ N + 2 ) 1 2 .
δ δ θ SPR = ( π n 0 δ λ ) 2 sin 2 θ SPR .
δ λ SPR = 1 4 π n 0 2 sin 2 θ n N + 2 2 λ SPR 4 π ( n 0 2 sin 2 θ n N + 2 2 ) ( n 0 n 0 λ SPR n N + 2 n N + 2 λ SPR ) ,
n 0 sin θ SPR = ( ϵ 1 ϵ eff ϵ 1 + ϵ eff ) 1 2 ( ϵ 1 ϵ eff ϵ 1 + ϵ eff ) 1 2 ,
ϵ eff TE = { ϵ 2 N = 0 ϵ 2 ( 1 e d 2 δ ) + ϵ 3 e d 2 δ N = 1 ϵ 2 ( 1 e d 2 δ ) + i = 2 N + 1 ϵ i + 1 e j = 2 i d j δ ( 1 e d i + 1 δ ) + ϵ N + 2 e j = 2 N + 2 d j δ N > 1 } ,
1 ϵ eff TM = { 1 ϵ 2 N = 0 1 ϵ 2 ( 1 e d 2 δ ) + 1 ϵ 3 e d 2 δ N = 1 1 ϵ 2 ( 1 e d 2 δ ) + i = 2 N + 1 1 ϵ i + 1 e j = 2 i d j δ ( 1 e d i + 1 δ ) + 1 ϵ N + 2 e j = 2 N + 2 d j δ N > 1 } .
ln ϵ 3 ϵ 2 ϵ eff TE ϵ 2 = 4 π d 2 λ ( ϵ 1 ϵ eff TE ϵ 1 + ϵ eff TE ϵ 3 ) 1 2 ,
λ 4 π δ = { ϵ 1 [ ϵ 2 + ( ϵ 3 ϵ 2 ) e d 2 δ ] ϵ 1 + ϵ 2 + ( ϵ 3 ϵ 2 ) e d 2 δ ϵ 3 } 1 2 .
ln 1 ϵ 3 1 ϵ 2 1 ϵ eff TM 1 ϵ 2 = 4 π d 2 λ ( ϵ 1 ϵ eff TM ϵ 1 + ϵ eff TM ϵ 3 ) 1 2 ,
λ 4 π δ = { ϵ 1 1 + ϵ 1 [ ( 1 e d 2 δ ) ϵ 2 + e d 2 δ ϵ 3 ] ϵ 3 } 1 2 .
n eff sin θ SPR = ( ϵ N + 1 ϵ N + 2 ϵ N + 1 + ϵ N + 2 ) 1 2 ,
δ = λ 4 π n eff 2 sin 2 θ SPR n N + 2 2 .

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