Effects of the equivalent coupling layer on ultra-long-range surface-plasmon-polariton waves

doi:10.1364/OE.18.007982

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Effects of the equivalent coupling layer on ultra-long-range surface-plasmon-polariton waves

Ching-Wei Yu and Yi-Jun Jen »View Author Affiliations

Optics Express, Vol. 18, Issue 8, pp. 7982-7993 (2010)
http://dx.doi.org/10.1364/OE.18.007982

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▸ Article Outline
– Abstract
1. Introduction
2. Locus of a metal film for ultra-long-range SPP propagation in the NAD
3. Equivalent coupling layer for p- and s-polarized ultra-long-range SPP propagation
4. Ultra-long-range SPP waves for p- and s-polarization states
5. Concluding remarks
– References and links

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Abstract

The mechanism and design of p- and s- polarized ultra-long-range surface-plasmon-polariton (SPP) propagation in the configuration {prism/ equivalent coupling layer (ECL)/ silver film (20 nm)/ equivalent substrate (ES)} are investigated using a normalized admittance diagram (NAD). The excitation of ultra-long-range SPP waves is characterized as a huge open loop of the NAD of the metal film at a designated angle of incidence. We propose three kinds of ECLs to complete the multilayer ultra-long-range SPP design: the normalized admittance of the ECL is (i) real (ii) infinite (iii) imaginary. The ultra-long propagation lengths in the three designs are compared at a wavelength of 632.8 nm for p- and s-polarization states.

1. Introduction

Surface-plasmon-polaritons (SPPs) are waves that propagate along a metal-dielectric interface [1

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

–3

E. N. Economu, “Surface plasmons in thin films,” Phys. Rev. 182(2), 539–554 (1969). [CrossRef]

]. As the metal film thickness is sufficiently thin, SPP waves on both sides of the metal are coupled, then long-range surface-plasmon-polariton (LRSPP) propagation occurs [4

D. Sarid, “Long-range surface-plasma waves on very thin metal films,” Phys. Rev. Lett. 47(26), 1927–1930 (1981). [CrossRef]

–8

R. Adato and J. Guo, “Characteristics of ultra-long range surface plasmon waves at optical frequencies,” Opt. Express 15(8), 5008–5017 (2007). [CrossRef] [PubMed]

]. The research on LRSPP propagation provides an important way for manipulating lights in novel sensing applications and other active photonic devices [9

A. Boltasseva, T. Nikolajsen, K. Leosson, K. Kjaer, M. Larsen, and S. Bozhevolnyi, “Integrated optical components utilizing long-range surface plasmon polaritons,” J. Lightwave Technol. 23(1), 413–422 (2005). [CrossRef]

–11

G. G. Nenninger, P. Tobiska, J. Homola, and S. S. Yee, “Long-range surface plasmons for high-resolution surface plasmon resonance sensors,” Sens. Act. B 74(1-3), 145–151 (2001). [CrossRef]

]. Sarid proposed the general multilayer device for typical LRSPP propagation [4

D. Sarid, “Long-range surface-plasma waves on very thin metal films,” Phys. Rev. Lett. 47(26), 1927–1930 (1981). [CrossRef]

]. It is a very thin metal film sandwiched between two dielectric layers with low-refractive-indices in the prism-coupling configuration. The propagation length of the LRSPP wave is increased to 300

μ m

at the wavelength of 632.8 nm when a silver film with a thickness of 20 nm is used.

It has been demonstrated that periodically nonhomogeneous sculptured nematic thin films (SNTF) arranged next to a metal film can lead to multiple SPPs excitation including s-polarization and p-polarization states [12

A. Lakhtakia, Y.-J. Jen, and C.-F. Lin, “Multiple trains of same-color surface plasmon-polaritons guided by the planar interface of a metal and a sculptured nematic thin film. Part III: Experimental evidence,” J. Nanophoton. 3(1), 033506 (2009). [CrossRef]

]. The SNTF acts like a symmetrical film stack that provides a special equivalent admittance in the system to achieve the multiple SPPs. In our previous work, the normalized admittance diagram (NAD) is introduced to assist in designing multilayer structures for the excitation of LRSPP waves of either the p- or the s-polarization state [13

Y.-J. Jen, A. Lakhtakia, C.-W. Yu, and T.-Y. Chan, “Multilayered structures for p- and s-polarized long-range surface-plasmon-polariton propagation,” J. Opt. Soc. Am. A 26(12), 2600–2606 (2009). [CrossRef]

]. The periodic multilayer structure, comprising symmetrical film stacks with a real or imaginary equivalent admittance, is utilized as a coupling layer, which is inserted between the prism and the metal film. Such a scheme has been applied to excite p- and s-polarized LRSPP modes in a single device.

Consider a p- or s-polarized plane wave that is incident on the planar interface of two semi-infinite media. One is the dielectric medium with a real refractive index

N_{i}

and the other is the medium with a complex refractive index

N = n - i k

. Based on an admittance diagram for the design of a thin-film optical filter [14

H. A. Macleod, Thin-Film Optical Filters , 2nd ed. (Adam Hilger, Bristol, 1986).

], the admittances of the two media are

{\tilde{η}}_{i}^{c h a r} = N_{i} \sqrt{ε_{0} / μ_{0}}

and

{\tilde{η}}^{c h a r} = N \sqrt{ε_{0} / μ_{0}}

, respectively, where

ε_{0}

is the permittivity and

μ_{0}

is the permeability, of free space. Therefore, the reflection coefficient at the planar interface can be written as

r = ({\tilde{η}}_{i} - \tilde{η}) / ({\tilde{η}}_{i} + \tilde{η})

, where

{\tilde{η}}_{i} = {\begin{matrix} {\tilde{η}}_{i}^{c h a r} / \cos θ_{i} \\ {\tilde{η}}_{i}^{c h a r} \cos θ_{i} \end{matrix}, \tilde{η} = {\begin{matrix} {\tilde{η}}_{}^{c h a r} / \cos θ \\ {\tilde{η}}_{}^{c h a r} \cos θ \end{matrix}, p o l a r i z a t i o n = {\begin{matrix} p \\ s \end{matrix} .

(1)

θ_{i}

and θ denote the real-valued angle of incidence and the complex-valued angle of refraction, respectively. If the normalized admittance is defined as

η = {\begin{matrix} (\tilde{η} \cos θ_{i}) / \sqrt{ε_{0} / μ_{0}} \\ (\tilde{η} / \cos θ_{i}) / \sqrt{ε_{0} / μ_{0}} \end{matrix}, p o l a r i z a t i o n = {\begin{matrix} p \\ s \end{matrix},

(2)

then the reflection coefficient is given by

r = (N_{i} - η) / (N_{i} + η)

. As the normalized admittance η approaches the refractive index

N_{i}

in the NAD, the reflection coefficient goes to zero.

In this work, ultra-long-range surface-plasmon-polariton (SPP) waves for either the p- or the s-polarization state are excited in the configuration {prism/ equivalent coupling layer (ECL) / metal film/ equivalent substrate (ES)}. Three ECL paths are proposed for ultra-long-range SPP propagation at a designated angle of incidence in the NAD. The NAD in Section 2 describes a huge open loop for p- and s-polarized ultra-long-range SPP propagation. Section 3 discusses the three ECL paths and the associated equivalent admittance of the metal film in the NAD. Section 4 addresses the effects of coupling on the excitation of either the p- or the s-polarized ultra-long-range SPP wave for a 20nm-thick silver film at the wavelength of 632.8 nm.

2. Locus of a metal film for ultra-long-range SPP propagation in the NAD

The normalized admittance diagram (NAD) supports the multilayered structure design for ultra-long-range SPP waves in a prism-coupling system {prism/ ECL/ metal film/ ES}. The ECL and the ES are the equivalent coupling layer and the equivalent substrate, respectively, one on each of the two sides of the metal film. The ES is set to provide a positive and imaginary admittance i γ when total reflection of either the p- or the s-polarization state occurs, as shown in Fig. 1 . The terminal point

η_{m}

is the intrinsic admittance of a metal film, which is defined as the normalized admittance when the thickness of the film goes to infinity. If the time dependence

\exp (i w t)

is implicit with w as the angular frequency, then

η_{m}

is given by

η_{m} = {\begin{matrix} {(n - i k)}^{2} \cos θ_{i} / \sqrt{n^{2} - k^{2} - {(N_{i} \sin θ_{i})}^{2} - 2 i n k} \\ \sqrt{n^{2} - k^{2} - {(N_{i} \sin θ_{i})}^{2} - 2 i n k} / \cos θ_{i} \end{matrix}, p o l a r i z a t i o n = {\begin{matrix} p \\ s \end{matrix},

(3)

where

n - i k

is the complex refractive index of the metal film;

θ_{i}

is the angle of the incidence, and

N_{i}

is the refractive index of the prism.

Fig. 1 Locus of a metal film for ultra-long-range SPP propagation in the NAD.

If the initial imaginary part γ is higher than the imaginary part of the virtual point

Im (- η_{m})

, then the locus of a metal film is a huge open loop in the NAD. As the thickness of the metal increases, the locus moves toward the terminal point

η_{m}

and eventually intersects the real axis of the NAD at a positive value of ξ. A larger initial imaginary part γ is associated with a larger ξ.

Consider the thickness of the metal film,

d_{m}

. Using the transfer admittance matrix [14

H. A. Macleod, Thin-Film Optical Filters , 2nd ed. (Adam Hilger, Bristol, 1986).

], the equivalent admittance

η_{m}'

at the end of the metal locus can be expressed as

η_{m}' = \frac{i γ \cos (2 π d_{m} α / λ) + i η_{m} \sin (2 π d_{m} α / λ)}{\cos (2 π d_{m} α / λ) - γ η_{m}^{- 1} \sin (2 π d_{m} α / λ)},

(4)

where

α = \sqrt{n^{2} - k^{2} - {(N_{i} \sin θ_{i})}^{2} - 2 i n k};

λ,

N_{i}

and

θ_{i}

are the wavelength of incidence, refractive index of the prism and the angle of incidence, respectively; γ is the initial imaginary part associated with the equivalent substrate (ES).

For ultra-long range SPP propagation of either the p- or the s-polarization state, the ECL interposed between the prism and a metal film is used to connect the equivalent admittance of the metal film

η_{m}'

with the refractive index of the prism

N_{i}

in a prism-coupling system {prism/ ECL/ metal film/ ES}, causing a sharp reflection dip in the angular spectrum. The relationship between the refractive index of the prism and the real part of the equivalent admittance

Re (η_{m}')

determines three ECL paths. The coupling effects of the ECLs are governed by the real part of

η_{m}'

which can be (i) larger, (ii) equal, or (iii) less than the refractive index of the prism.

Figures 2(a)-(c) plot the required thickness of the metal film

d_{r e q}

against the initial imaginary part γ for the three cases, (i)

Re (η_{m}') = 2 N_{i}

, (ii)

Re (η_{m}') = N_{i}

, and (iii)

Re (η_{m}') = Re (η_{m})

, where λ is 632.8 nm and

θ_{i}

41.31 °

which slightly exceeds the total reflection angle at the prism/air interface. The metal is silver with a complex refractive index 0.06656-i4.04520 and the index of

N_{i}

1.51511

. The intrinsic admittance

η_{m}

for p- and s-polarization states are 0.05133-i2.94974 and 0.08602-i5.54748 at

θ_{i} = 41.31 °

, respectively. The imaginary part γ should be higher than the imaginary part of

Re (- η_{m})

to cause a huge open loop for ultra-long-range SPP propagation in the NAD [13

]. A larger initial part γ is associated with a lower thickness of the silver film

d_{r e q}

. When γ increases to 20 for the p-polarization state, only

d_{r e q} = 4.04

nm is required to yield

Re (η_{m}') = 2 N_{i}

which exceeds the refractive index of the prism at

θ_{i} = 41.31 °

Fig. 2 The required thickness of the silver film

d_{r e q}

against the initial part γ for the p-polarization state and the s-polarization state at

θ_{i} = 41.31 °

when (a)

Re (η_{m}') = 2 N_{i}

, (b)

Re (η_{m}') = N_{i}

, and (c)

Re (η_{m}') = Re (η_{m})

. The refractive index of the prism is 1.51511 at

λ = 632.8

nm.

Figure 3 plots large values ξ against γ for p- and s-polarization states at

θ_{i} = 41.31 °

when the wavelength is 632.8 nm. The metal is made of silver. Let

ξ_{p}

and

ξ_{s}

be the value ξ for p- and s-polarization states, respectively. For the p-polarization state, the intrinsic admittance

η_{m}

is 0.05133-i2.94974. The initial imaginary part γ should be higher than the imaginary part of the virtual point

Im (- η_{m}) = 2
                     .94974

in order to cause the locus of the silver film to be projected through a large and positive real point

ξ_{p}

from a purely positive imaginary point iγ in the NAD. The value

ξ_{p}

is 9201 when γ is increased to 20.

Fig. 3 ξ against γ for p- and s-polarization states for a silver film at

θ_{i} = 41.31 °

.The wavelength is 632.8 nm

For the s-polarization state, the intrinsic admittance of the silver film is 0.08602-i5.54748. The initial part γ provided by the ES should be higher than 5.54748, which is the imaginary part of the virtual point of

η_{m}

for the excitation of s-polarized ultra-long range SPP waves at

θ_{i} = 41.31 °

. When the initial part γ is increased to 20,

ξ_{s}

becomes a huge value, 22703, which exceeds

ξ_{p}

. Additionally, the values of

ξ_{s}

and

ξ_{p}

both equal to 4780 when γ is at the critical value of 12.50027. When γ is less than 12.50027,

ξ_{s}

is smaller than

ξ_{p}

3. Equivalent coupling layer for p- and s-polarized ultra-long-range SPP propagation

As stated in Sec. 2, the locus of a metal film for exciting ultra-long range SPP waves is a huge open loop in the NAD. The equivalent admittance

η_{m}'

at the end of the loop is determined by the initial imaginary admittance γ that is supplied by the equivalent substrate (ES). For the excitation of either the p- or the s-polarization state in the prism-coupling configuration {prism/ ECL/ metal film/ ES}, the equivalent coupling layer (ECL) with an appropriate intrinsic admittance

η_{c}

must be interposed between the metal film and the prism to connect the admittance

η_{m}'

with the refractive index of the prism

N_{i}

, causing a sharp reflection dip in the angular spectrum. For that reason, we proposes three kinds of the ECL paths- (i) clockwise, (ii) near vertical, and (iii) counter-clockwise loci, which are associated with (i)

Re (η_{m}') = 2 N_{i}

, (ii)

Re (η_{m}') = N_{i}

, and (iii)

Re (η_{m}') = Re (η_{m})

, respectively, for ultra-long-range SPP waves in the NAD. The required intrinsic admittance of the ECL is determined by the real part of the equivalent admittance

η_{m}'

, which is associated with the three coupling paths.

Assume that

d_{c}

is the thickness of the ECL; λ and

θ_{i}

are the wavelength and the angle of incidence, respectively. The required intrinsic admittance

η_{c}

for the p- and s-polarization states is a solution to the equation

\frac{η_{c} [i η_{c} \sin δ + η_{m}^{'} \cos δ]}{η_{c} \cos δ + i η_{m}^{'} \sin δ} ≃ N_{i}, where δ = {\begin{matrix} 2 π η_{c} d_{c} \cos^{2} θ / (\cos θ_{i} λ) \\ 2 π η_{c} d_{c} \cos θ_{i} / λ \end{matrix}, p o l a r i z a t i o n = {\begin{matrix} p \\ s \end{matrix} .

(5)

If the real part of the equivalent admittance of the metal film is given by

Re (η_{m}^{'}) = 2 N_{i},

(6)

then the real part of

η_{m}^{'}

is larger than the refractive index of the prism. As the thickness

d_{c}

increases, the path of the ECL follows a clockwise locus to make the equivalent admittance of the metal film

η_{m}^{'}

approach the refractive index

N_{i}

, as shown in Fig. 4 . The real and positive intrinsic admittance

η_{c}

must be used to establish this relationship in the prism-coupling configuration {prism/ ECL/ metal film/ ES}.If the real part of the equivalent admittance of the metal film is given by

Re (η_{m}^{'}) = N_{i},

(7)

then the real part of

η_{m}^{'}

equals to the refractive index of the prism. As the thickness

d_{c}

increases, the path of the ECL approaches a vertical locus and connects the equivalent admittance of the metal film

η_{m}^{'}

with the refractive index

N_{i}

, as shown in Fig. 5 . The intrinsic admittance

η_{c}

with a large and positive imaginary or real part must be used.If the real part of the equivalent admittance of the metal film is given by

Re (η_{m}^{'}) = Re (η_{m}),

(8)

then the real part of

η_{m}^{'}

is less than the refractive index of the prism. As the thickness

d_{c}

increases, the path of the ECL follows a counter-clockwise locus to connect the equivalent admittance

η_{m}^{'}

with the refractive index

N_{i}

, as shown in Fig. 6 . The intrinsic admittance

η_{c}

with a positive imaginary value must therefore be used.

Fig. 4 Clockwise locus of the equivalent coupling layer for ultra-long-range SPP propagation in the NAD.

Fig. 5 Near vertical locus of the equivalent coupling layer for ultra-long-range SPP propagation in the NAD.

Fig. 6 Counter-clockwise locus of the equivalent coupling layer for ultra-long-range SPP propagation in the NAD.

To achieve the ultra-long range SPP multilayer design, the ECL with the required intrinsic admittance

η_{c}

at a designated angle of incidence cannot be produced from an existing bulk material. A symmetrical film stack in which a material with a low-refractive-index is inserted between a pair of materials with high-refractive-indices as the ECL can be employed to realize the three coupling paths in the NAD.

4. Ultra-long-range SPP waves for p- and s-polarization states

Based on the ECL design that is proposed in Sec. 3, ultra-long-range SPP propagation at a designated angle that exceeds the total reflection angle in the prism-coupling system {prism/ ECL/ metal film/ ES} can be excited. Suppose we wish to excite ultra-long range SPP waves for a 20 nm-thick film of silver at an angle of incidence

θ_{i} = 41.31 °

λ = 632.8

nm. The proper initial imaginary part γ provided by the ES is required for the ultra-long-range SPP design, as shown in Fig. 2. The intrinsic admittances of the ECL

η_{c}

can also be determined for the three ECL paths using Eq. (5).

4.1 Clockwise coupling (p-polarization state)

When

Re (η_{m}^{'}) = 2 N_{i}

, the structure {prism/ ECL (

η_{c} = 39
                        .92855

) / silver film (20 nm)/ ES (

γ = 4
                        .66291

)} is employed to excite the p-polarized ultra-long-range SPP wave at

θ_{i} = 41.31 °

, where the refractive index of the prism

N_{i}

is 1.51511 at

λ = 632.8

. The locus of the silver film yields a large contour and the path of the ECL is clockwise in the NAD.

To realize the multilayer design for ultra-long-range SPP propagation, a Ta₂O₅ layer above the substrate (air) is used as the ES to provide the required initial imaginary part γ. The periodic multilayer structure that comprises the symmetrical film stack [Ta₂O₅ (λ/8)/SiO₂ (λ/4) /Ta₂O₅ (λ/8)] is used as the ECL to yield the required intrinsic admittance

η_{c}

, where a quarter-wavelength-thick SiO₂ layer is interposed between a pair of eighth-wavelength-thick Ta₂O₅ layers. The refractive indices of Ta₂O₅ and SiO₂ are 2.13338 and 1.45705 at

λ = 632.8

nm, respectively.

Figure 7 plots the p-polarization reflectance

{| r_{p} |}^{2}

as a function of

θ_{i}

for the structure {prism/ [Ta₂O₅ (X*37.077 nm)/SiO₂ (X*108.58 nm)/Ta₂O₅ (X*37.077 nm)] ⁴³/ silver film (20 nm)/ [Ta₂O₅ (150.48 nm)/ air]} with a clockwise coupling path at

λ = 632.8

nm, where X=1.31496. The required γ and

η_{c}

are obtained using a {Ta₂O₅ (150.48 nm)/ air} and the periodic multilayer structure [Ta₂O₅ (48.75 nm)/SiO₂ (142.78 nm)/Ta₂O₅ (48.75 nm)] with 43 periods, respectively. The reflection dip occurs at

θ_{i} = 41.31 °

and the half-width of the reflection dip,

θ_{H W}

, is

0.00306 °

Fig. 7

{| r_{p} |}^{2}

against

θ_{i}

for the structure {prism/ [Ta₂O₅ (48.75 nm)/ SiO₂ (142.78 nm)/ Ta₂O₅ (48.75 nm)]⁴³/ silver film (20 nm)/ [Ta₂O₅ (150.48 nm)/ air]} with a clockwise coupling path at

λ = 632.8

nm.

4.2 Near vertical coupling (p-polarization state)

When

Re (η_{m}^{'}) = N_{i}

, the structure {prism/ ECL (

η_{c} = 116
                        .82044i

) / silver film (20 nm)/ ES (

γ = 4
                        .79576

)} is obtained to excite the ultra-long range SPP waves at

θ_{i} = 41.31 °

, where the refractive index of the prism

N_{i}

is 1.51511 at

λ = 632.8

nm. The locus of the silver film produces a large contour and the path of the ECL is near vertical in the NAD.

Figure 8 plots

{| r_{p} |}^{2}

against

θ_{i}

for the structure {prism/ [Ta₂O₅ (X*37.077 nm)/SiO₂ (X*108.58 nm)/Ta₂O₅ (X*37.077 nm)] ⁴⁰/ silver film (20 nm)/ [Ta₂O₅ (150.98 nm)/ air]} with a near vertical coupling locus at

λ = 632.8

nm, where X = 1.31461. The required γ and

η_{c}

are obtained using {Ta₂O₅ (150.98 nm)/ air} and the periodic multilayer structure [Ta₂O₅ (48.74 nm)/SiO₂ (142.74 nm)/Ta₂O₅ (48.74 nm)] with 40 periods, respectively. The reflection dip occurs at

θ_{i} = 41.31 °

and the half-width of the reflection dip,

θ_{H W}

, is

0.00300 °

Fig. 8

{| r_{p} |}^{2}

against

θ_{i}

for the structure {prism/ [Ta₂O₅ (48.74 nm)/ SiO₂ (142.73 nm)/ Ta₂O₅ (48.74 nm)] ⁴⁰/ silver film (20 nm)/ [Ta₂O₅ (150.98 nm)/ air]} with a near vertical coupling path at

λ = 632.8

nm.

4.3 Counter-clockwise coupling (p-polarization state)

When

Re (η_{m}^{'}) = Re (η_{m})

, the structure {prism/ ECL (

η_{c} = 7
                        .53359i

) / silver film (20 nm)/ ES (

γ = 7
                        .65128

)} is utilized to excite the ultra-long-range SPP waves at

θ_{i} = 41.31 °

, where the refractive index of the prism

N_{i}

is 1.51511 at

λ = 632.8

nm. The locus of the silver film generates a large contour and the path of the ECL goes counter-clockwise in the NAD. The equivalent admittance

η_{m}'

at the end of the metal locus is 0.05133-7.41623i in the NAD. As the thickness of the metal film is increased, the quantities of real and imaginary parts of

η_{m}'

become smaller. As the thickness of the metal film up to infinity,

η_{m}'

ends at the point of the intrinsic admittance of silver

η_{m} = 0.05133 - 2.94974 i

θ_{i} = 41.31 °

λ = 632.8

nm.

Figure 9 plots

{| r_{p} |}^{2}

against

θ_{i}

for the structure {prism/ [Ta₂O₅ (X*37.077 nm)/SiO₂ (X*108.58 nm)/Ta₂O₅ (X*37.077 nm)]²⁸/ silver film (20 nm)/ [Ta₂O₅ (157.86 nm)/ air]} with a counter-clockwise coupling path at

λ = 632.8

nm, where X=1.30643. The required γ and

η_{c}

are obtained using {Ta₂O₅ (157.86 nm)/ air} and the periodic multilayer structure [Ta₂O₅ (48.44 nm)/SiO₂ (141.85 nm)/Ta₂O₅ (48.44 nm)] with 28 periods, respectively. The reflection dip occurs at

θ_{i} = 41.31 °

and the half-width of the reflection dip,

θ_{H W}

, is

0.00230 °

Fig. 9

{| r_{p} |}^{2}

against

θ_{i}

for the structure {prism/ [Ta₂O₅ (48.44 nm)/ SiO₂ (141.85 nm)/ Ta₂O₅ (48.44 nm)] ²⁸/ silver film (20 nm)/ [Ta₂O₅ (157.86 nm)/ air]} with a counter-clockwise coupling path at

λ = 632.8

nm.

4.4 Propagation lengths of p- and s-polarized ultra-long range SPP waves

The propagation lengths of the ultra-long-range SPP waves in the configuration {prism/ ECL/ silver film (20 nm)/ ES} are calculated using the reflection pole method [15

E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, “Determination of guided and leaky modes in lossless and lossy planar multilayer optical waveguides: reflection pole method and wavevector density method,” J. Lightwave Technol. 17(5), 929–941 (1999). [CrossRef]

,16

J. Guo and R. Adato, “Extended long range plasmon waves in finite thickness metal film and layered dielectric materials,” Opt. Express 14(25), 12409–12418 (2006). [CrossRef] [PubMed]

]. The guided plasmon wave modes have the complex propagation constants β and thus effective mode indices

n_{e} = β / k_{0}

, where

k_{0} = 2 π / λ

and λ is the wavelength of incidence. Table 1 shows the design parameters γ and

η_{c}

, large values ξ, effective mode indices

n_{e}

, half-widths

θ_{H W}

, and propagation lengths L of the p-polarization state for the three kinds of ECL paths at

θ_{i} = 41.31 °

λ = 632.8

nm. The periodic multilayer structures [Ta₂O₅/SiO₂/Ta₂O₅] with 43, 40, and 28 periods are used as the ECLs for (i) clockwise, (ii) near vertical, and (iii) counter-clockwise coupling paths, respectively. The refractive indices of Ta₂O₅ and SiO₂ materials are 2.13338 and 1.45705 at

λ = 632.8

nm, respectively. A Ta₂O₅ layer above the substrate (air) is used as the ES to provide the required initial imaginary part γ. The thicknesses of the Ta₂O₅ layers used in the cases (i), (ii), and (iii) are 150.48 nm, 150.98 nm, and 157.86 nm, respectively. The reflectance spectra for the three kinds of coupling paths in Table 1 have been shown previously in Fig. 7, Fig. 8, and Fig. 9. A larger initial value of γ yields a larger ξ. The propagation length increases from 1664.43

μ m

to 2188.45

μ m

as the real part of

η_{m}'

declines from 3.03022 to 0.05133. The counter-clockwise path provides the longest propagation length in the multilayer design for p-polarized ultra-long range SPP propagation.

Table 1 Design parameters γ and

η_{c}

, real parts of

η_{m}'

, large values ξ, effective mode indices

n_{e}

, half-widths

θ_{H W}

, and propagation lengths L of the p-polarization state for the three types of coupling at

θ_{i} = 41.31 °

and

λ = 632.8

nm.

γ	$η_{c}$	$Re (η_{m}')$	ξ	$n_{e}$ ( mode indices)	$θ_{H W}$ (° )	L ( $μ m$ )	ECL
4.66291	39.92855	3.03022	435	1.000173774-3.0255×0^-5i	0.00306	1664.43	(i)
4.79576	116.82044i	1.51511	484	1.000173774-2.9585×0^-5i	0.00300	1702.10	(ii)
7.65128	7.53359i	0.05133	1872	1.000173774-2.3010×0^-5i	0.00233	2188.45	(iii)

The real parts of the effective mode indices in Table 1 are equal to 1.000173774. The Ta₂O₅ layer of the ES supplies the required initial part of γ to the metal/ES interface and supports the enhanced SPP propagation in the multilayer configuration [16

J. Guo and R. Adato, “Extended long range plasmon waves in finite thickness metal film and layered dielectric materials,” Opt. Express 14(25), 12409–12418 (2006). [CrossRef] [PubMed]

,17

F. Y. Kou and T. Tamir, “Range extension of surface plasmons by dielectric layers,” Opt. Lett. 12(5), 367–369 (1987). [CrossRef] [PubMed]

]. Figure 10(a) shows the absolute value of the parallel electric field distribution

| E_{z} |

for the counter-clockwise coupling case at

θ_{i} = 41.31 °

and

λ = 632.8

nm. As mentioned in Sec. 2, the huge open loop of the NAD describes a symmetric field distribution with respect to the center of the silver film. The parallel electric filed has an undulating variation that decays away from the metal/ECL interface. None of the modes in Table 1 is a waveguide mode that the ECL could act as. More energy is located in the Ta₂O₅ layer of the ES. The electric filed diminishes monotonically from the Ta₂O₅/air interface and penetrates far away into air. The reduction of field confinement in the metal acts as a trade-off effect [18

P. Berini, “Figures of merit for surface plasmon waveguides,” Opt. Express 14(26), 13030–13042 (2006). [CrossRef] [PubMed]

] to enhance the propagation length.

Fig. 10 The absolute value of the parallel electric field distribution for (a) the p-polarization state and (b) the s-polarization state for the counter-clockwise coupling case in Table 1 and Table 2 at

θ_{i} = 41.31 °

λ = 632.8

nm, respectively.

Three coupling paths are unitized to excite ultra-long-range SPP waves for the s-polarization state in a prism-coupling system {prism/ ECL / silver film (20 nm)/ ES}. An SiO₂ layer above the substrate (air) is used as the ES to provide the required initial imaginary part γ. The periodic multilayer structure, comprising the symmetrical film stack [Ta₂O₅ (λ/8)/SiO₂ (λ/4) /Ta₂O₅ (λ/8)], is used as the ECL to generate the required intrinsic admittance

η_{c}

. Table 2 presents the design parameters γ and

η_{c}

, large values ξ, effective mode indices

n_{e}

, half-widths

θ_{H W}

, and propagation lengths L for the three kinds of ECL paths at

θ_{i} = 41.31 °

λ = 632.8

nm. The periodic multilayer structures [Ta₂O₅/SiO₂/Ta₂O₅] with 16, 13, and 13 periods are used as the ECLs for (i) clockwise, (ii) near vertical, and (iii) counter-clockwise coupling paths, respectively. The thicknesses of the SiO₂ layer above the substrate (air) are 135.96 nm, 136.39 nm, and 139.87 nm in the cases (i), (ii), and (iii), respectively. The propagation length increases from 2124.05

μ m

to 4222.38

μ m

as the real part of

η_{m}'

changes from 3.03022 to 0.08602. The counter-clockwise path of the ECL is the best for increasing the length of propagation of s-polarized ultra-long-range SPP waves.

Table 2 Design parameters γ and

η_{c}

, real parts of

η_{m}'

, large values ξ, effective mode indices

n_{e}

, half-widths

θ_{H W}

, and propagation lengths L of the s-polarization state for the three types of coupling at

θ_{i} = 41.31 °

and

λ = 632.8

nm.

γ	$η_{c}$	$Re (η_{m}')$	ξ	$n_{e}$ ( mode indices)	$θ_{H W}$ (° )	L ( $μ m$ )	ECL
8.85101	64.22461	3.03022	1208	1.000173774-2.3708×0^-5i	0.00238	2124.05	(i)
9.11889	116.92408i	1.51511	1379	1.000173774-2.0614×0^-5i	0.00210	2442.85	(ii)
12.01175	18.00478i	0.08602	4093	1.000173774-1.1926×0^-5i	0.00120	4222.38	(iii)

The real parts of the effective mode indices in the three coupling cases for the s-polarization state are equal to 1.000173774 that is less than the refractive index of a SiO₂ layer of the ES. Figure 10(b) shows the absolute value of the parallel electric field distribution

| E_{y} |

for the counter-clockwise ECL path in Table 2 at

θ_{i} = 41.31 °

and

λ = 632.8

nm. The electric field has a symmetric field distribution with respect to the center of the silver film and an undulating variation that decays away from the metal/ECL interface. The energy diminishes monotonically from a maximum at the SiO₂/air interface and penetrates far away into air.

Figure 11 plots the s-polarization reflectance

{| r_{s} |}^{2}

as a function of

θ_{i}

for the structure {prism/ [Ta₂O₅ (157.78 nm)/SiO₂ (53.88 nm)/Ta₂O₅ (157.78 nm)]¹³/ silver film (20 nm)/ [SiO₂ (139.87 nm)/ air]} for the counter-clockwise coupling path in Table 2 at

λ = 632.8

nm. The reflection dip occurs at

θ_{i} = 41.31 °

and the half-width of the reflection dip,

θ_{H W}

, is

0.00120 °

Fig. 11

{| r_{s} |}^{2}

as a function of

θ_{i}

for the structure {prism/ [Ta₂O₅ (157.78 nm)/SiO₂ (53.88 nm)/Ta₂O₅ (157.78 nm)]¹³/ silver film (20 nm)/ [SiO₂ (139.87 nm)/ air]} for the counter-clockwise coupling case in Table 2. The wavelength is 632.8 nm.

5. Concluding remarks

We have proposed three kinds of equivalent coupling layers for the excitation of either p- or s-polarized ultra-long-range SPP waves in a normalized admittance diagram of the prism-coupling configuration {prism/ equivalent coupling layer (ECL) / metal film/ equivalent substrate (ES)}. For a typical silver film with a thickness of 20nm, p- or s-polarized ultra-long-range SPP waves can be excited at a designated angle of incidence. The counter-clockwise path is associated with the longest LRSPP propagation length. General optical coating materials can be applied to form the multilayer to realize our idea. Compare with previous results [4

D. Sarid, “Long-range surface-plasma waves on very thin metal films,” Phys. Rev. Lett. 47(26), 1927–1930 (1981). [CrossRef]

–8

R. Adato and J. Guo, “Characteristics of ultra-long range surface plasmon waves at optical frequencies,” Opt. Express 15(8), 5008–5017 (2007). [CrossRef] [PubMed]

], the configuration we proposed can reach an ultra-long propagation length of LRSPP. With a propagation length of greater than 1 mm, the LRSPP will be observable in the visible range in the near future.

Acknowledgements

CWY and YJJ thank the National Science Council of the Republic of China, Taiwan, for financially supporting this research under Contract No. NSC 96-2221-E- 027-051-MY3.

References and links

1.	H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, Berlin Heidelberg, 1988).
2.	A. Lakhtakia, Y.-J. Jen, and C.-F. Lin, “Multiple trains of same-color surface plasmon-polaritons guided by the planar interface of a metal and a sculptured nematic thin film. Part III: Experimental evidence,” J. Nanophoton. 3(1), 033506 (2009). [CrossRef]
3.	E. N. Economu, “Surface plasmons in thin films,” Phys. Rev. 182(2), 539–554 (1969). [CrossRef]
4.	D. Sarid, “Long-range surface-plasma waves on very thin metal films,” Phys. Rev. Lett. 47(26), 1927–1930 (1981). [CrossRef]
5.	L. Wedler and R. Haupt, “Long-range surface plasmon-polaritons in asymmetric layer structures,” J. Appl. Phys. 59(9), 3289–3291 (1986). [CrossRef]
6.	F. Yang, J. R. Sambles, and G. W. Bradberry, “Long-range modes supported by thin films,” Phys. Rev. B 44(11), 5855–5872 (1991). [CrossRef]
7.	P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of asymmetric structures,” Phys. Rev. B 63(12), 125417 (2001). [CrossRef]
8.	R. Adato and J. Guo, “Characteristics of ultra-long range surface plasmon waves at optical frequencies,” Opt. Express 15(8), 5008–5017 (2007). [CrossRef] [PubMed]
9.	A. Boltasseva, T. Nikolajsen, K. Leosson, K. Kjaer, M. Larsen, and S. Bozhevolnyi, “Integrated optical components utilizing long-range surface plasmon polaritons,” J. Lightwave Technol. 23(1), 413–422 (2005). [CrossRef]
10.	R. Charbonneau, C. Scales, I. Breukelaar, S. Fafard, N. Lahoud, G. Mattiussi, and P. Berini, “Passive integrated optics elements based on long-range surface plasmon polaritons,” J. Lightwave Technol. 24(1), 477–494 (2006). [CrossRef]
11.	G. G. Nenninger, P. Tobiska, J. Homola, and S. S. Yee, “Long-range surface plasmons for high-resolution surface plasmon resonance sensors,” Sens. Act. B 74(1-3), 145–151 (2001). [CrossRef]
12.	A. Lakhtakia, Y.-J. Jen, and C.-F. Lin, “Multiple trains of same-color surface plasmon-polaritons guided by the planar interface of a metal and a sculptured nematic thin film. Part III: Experimental evidence,” J. Nanophoton. 3(1), 033506 (2009). [CrossRef]
13.	Y.-J. Jen, A. Lakhtakia, C.-W. Yu, and T.-Y. Chan, “Multilayered structures for p- and s-polarized long-range surface-plasmon-polariton propagation,” J. Opt. Soc. Am. A 26(12), 2600–2606 (2009). [CrossRef]
14.	H. A. Macleod, Thin-Film Optical Filters , 2nd ed. (Adam Hilger, Bristol, 1986).
15.	E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, “Determination of guided and leaky modes in lossless and lossy planar multilayer optical waveguides: reflection pole method and wavevector density method,” J. Lightwave Technol. 17(5), 929–941 (1999). [CrossRef]
16.	J. Guo and R. Adato, “Extended long range plasmon waves in finite thickness metal film and layered dielectric materials,” Opt. Express 14(25), 12409–12418 (2006). [CrossRef] [PubMed]
17.	F. Y. Kou and T. Tamir, “Range extension of surface plasmons by dielectric layers,” Opt. Lett. 12(5), 367–369 (1987). [CrossRef] [PubMed]
18.	P. Berini, “Figures of merit for surface plasmon waveguides,” Opt. Express 14(26), 13030–13042 (2006). [CrossRef] [PubMed]

OCIS Codes
(240.6680) Optics at surfaces : Surface plasmons
(310.2790) Thin films : Guided waves
(310.4165) Thin films : Multilayer design

ToC Category:
Optics at Surfaces

History
Original Manuscript: February 3, 2010
Revised Manuscript: February 26, 2010
Manuscript Accepted: March 22, 2010
Published: March 31, 2010

Citation
Ching-Wei Yu and Yi-Jun Jen, "Effects of the equivalent coupling layer on ultra-long-range surface-plasmon-polariton waves," Opt. Express 18, 7982-7993 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-8-7982

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References

H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, Berlin Heidelberg, 1988).
A. Lakhtakia, Y.-J. Jen, and C.-F. Lin, “Multiple trains of same-color surface plasmon-polaritons guided by the planar interface of a metal and a sculptured nematic thin film. Part III: Experimental evidence,” J. Nanophoton. 3(1), 033506 (2009). [CrossRef]
E. N. Economu, “Surface plasmons in thin films,” Phys. Rev. 182(2), 539–554 (1969). [CrossRef]
D. Sarid, “Long-range surface-plasma waves on very thin metal films,” Phys. Rev. Lett. 47(26), 1927–1930 (1981). [CrossRef]
L. Wedler and R. Haupt, “Long-range surface plasmon-polaritons in asymmetric layer structures,” J. Appl. Phys. 59(9), 3289–3291 (1986). [CrossRef]
F. Yang, J. R. Sambles, and G. W. Bradberry, “Long-range modes supported by thin films,” Phys. Rev. B 44(11), 5855–5872 (1991). [CrossRef]
P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of asymmetric structures,” Phys. Rev. B 63(12), 125417 (2001). [CrossRef]
R. Adato and J. Guo, “Characteristics of ultra-long range surface plasmon waves at optical frequencies,” Opt. Express 15(8), 5008–5017 (2007). [CrossRef] [PubMed]
A. Boltasseva, T. Nikolajsen, K. Leosson, K. Kjaer, M. Larsen, and S. Bozhevolnyi, “Integrated optical components utilizing long-range surface plasmon polaritons,” J. Lightwave Technol. 23(1), 413–422 (2005). [CrossRef]
R. Charbonneau, C. Scales, I. Breukelaar, S. Fafard, N. Lahoud, G. Mattiussi, and P. Berini, “Passive integrated optics elements based on long-range surface plasmon polaritons,” J. Lightwave Technol. 24(1), 477–494 (2006). [CrossRef]
G. G. Nenninger, P. Tobiska, J. Homola, and S. S. Yee, “Long-range surface plasmons for high-resolution surface plasmon resonance sensors,” Sens. Act. B 74(1-3), 145–151 (2001). [CrossRef]
A. Lakhtakia, Y.-J. Jen, and C.-F. Lin, “Multiple trains of same-color surface plasmon-polaritons guided by the planar interface of a metal and a sculptured nematic thin film. Part III: Experimental evidence,” J. Nanophoton. 3(1), 033506 (2009). [CrossRef]
Y.-J. Jen, A. Lakhtakia, C.-W. Yu, and T.-Y. Chan, “Multilayered structures for p- and s-polarized long-range surface-plasmon-polariton propagation,” J. Opt. Soc. Am. A 26(12), 2600–2606 (2009). [CrossRef]
H. A. Macleod, Thin-Film Optical Filters, 2nd ed. (Adam Hilger, Bristol, 1986).
E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, “Determination of guided and leaky modes in lossless and lossy planar multilayer optical waveguides: reflection pole method and wavevector density method,” J. Lightwave Technol. 17(5), 929–941 (1999). [CrossRef]
J. Guo and R. Adato, “Extended long range plasmon waves in finite thickness metal film and layered dielectric materials,” Opt. Express 14(25), 12409–12418 (2006). [CrossRef] [PubMed]
F. Y. Kou and T. Tamir, “Range extension of surface plasmons by dielectric layers,” Opt. Lett. 12(5), 367–369 (1987). [CrossRef] [PubMed]
P. Berini, “Figures of merit for surface plasmon waveguides,” Opt. Express 14(26), 13030–13042 (2006). [CrossRef] [PubMed]

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