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.

© 2010 OSA

1. Introduction

Surface-plasmon-polaritons (SPPs) are waves that propagate along a metal-dielectric interface [13]. 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 [48]. The research on LRSPP propagation provides an important way for manipulating lights in novel sensing applications and other active photonic devices [911]. Sarid proposed the general multilayer device for typical LRSPP propagation [4]. 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 μm 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]. 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]. 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 Ni and the other is the medium with a complex refractive index N=nik. Based on an admittance diagram for the design of a thin-film optical filter [14], the admittances of the two media are η˜ichar=Niε0/μ0 and η˜char=Nε0/μ0, respectively, where ε0 is the permittivity and μ0 is the permeability, of free space. Therefore, the reflection coefficient at the planar interface can be written as r=(η˜iη˜)/(η˜i+η˜), where

η˜i={η˜ichar/cosθiη˜icharcosθi,η˜={η˜char/cosθη˜charcosθ,polarization={ps.
θi 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={(η˜cosθi)/ε0/μ0(η˜/cosθi)/ε0/μ0,polarization={ps,
then the reflection coefficient is given by r=(Niη)/(Ni+η). As the normalized admittance η approaches the refractive index Ni 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 ηm 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(iwt) is implicit with w as the angular frequency, then ηm is given by

ηm={(nik)2cosθi/n2k2(Nisinθi)22inkn2k2(Nisinθi)22ink/cosθi,polarization={ps,
where nik is the complex refractive index of the metal film; θi is the angle of the incidence, and Ni is the refractive index of the prism.

 

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

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If the initial imaginary part γ is higher than the imaginary part of the virtual point Im(ηm), 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 ηm 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, dm. Using the transfer admittance matrix [14], the equivalent admittance ηm' at the end of the metal locus can be expressed as

ηm'=iγcos(2πdmα/λ)+iηmsin(2πdmα/λ)cos(2πdmα/λ)γηm1sin(2πdmα/λ),
where α=n2k2(Nisinθi)22ink; λ, Ni and θi 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 ηm' with the refractive index of the prism Ni 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(ηm') determines three ECL paths. The coupling effects of the ECLs are governed by the real part of ηm' 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 dreq against the initial imaginary part γ for the three cases, (i) Re(ηm')=2Ni, (ii) Re(ηm')=Ni, and (iii) Re(ηm')=Re(ηm), where λ is 632.8 nm and θi is 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 Ni is 1.51511. The intrinsic admittance ηm for p- and s-polarization states are 0.05133-i2.94974 and 0.08602-i5.54748 at θi=41.31°, respectively. The imaginary part γ should be higher than the imaginary part of Re(ηm) 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 dreq. When γ increases to 20 for the p-polarization state, only dreq=4.04 nm is required to yield Re(ηm')=2Ni which exceeds the refractive index of the prism at θi=41.31°.

 

Fig. 2 The required thickness of the silver film dreq against the initial part γ for the p-polarization state and the s-polarization state at θi=41.31° when (a) Re(ηm')=2Ni, (b) Re(ηm')=Ni, and (c) Re(ηm')=Re(ηm). The refractive index of the prism is 1.51511 at λ=632.8 nm.

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Figure 3 plots large values ξ against γ for p- and s-polarization states at θi=41.31° when the wavelength is 632.8 nm. The metal is made of silver. Let ξp and ξs be the value ξ for p- and s-polarization states, respectively. For the p-polarization state, the intrinsic admittance ηm is 0.05133-i2.94974. The initial imaginary part γ should be higher than the imaginary part of the virtual point Im(ηm)=2 .94974 in order to cause the locus of the silver film to be projected through a large and positive real point ξp from a purely positive imaginary point iγ in the NAD. The value ξp is 9201 when γ is increased to 20.

 

Fig. 3 ξ against γ for p- and s-polarization states for a silver film at θi=41.31°.The wavelength is 632.8 nm

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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 ηm for the excitation of s-polarized ultra-long range SPP waves at θi=41.31°. When the initial part γ is increased to 20, ξs becomes a huge value, 22703, which exceeds ξp. Additionally, the values of ξs and ξp both equal to 4780 when γ is at the critical value of 12.50027. When γ is less than 12.50027, ξs is smaller than ξp.

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 ηm' 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 ηc must be interposed between the metal film and the prism to connect the admittance ηm' with the refractive index of the prism Ni, 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(ηm')=2Ni, (ii) Re(ηm')=Ni, and (iii) Re(ηm')=Re(ηm), 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 ηm', which is associated with the three coupling paths.

Assume that dc is the thickness of the ECL; λ and θi are the wavelength and the angle of incidence, respectively. The required intrinsic admittance ηc for the p- and s-polarization states is a solution to the equation

ηc[iηcsinδ+ηm'cosδ]ηccosδ+iηm'sinδNi,whereδ={2πηcdccos2θ/(cosθiλ)2πηcdccosθi/λ,polarization={ps.
If the real part of the equivalent admittance of the metal film is given by
Re(ηm')=2Ni,
then the real part of ηm' is larger than the refractive index of the prism. As the thickness dc increases, the path of the ECL follows a clockwise locus to make the equivalent admittance of the metal film ηm' approach the refractive index Ni, as shown in Fig. 4 . The real and positive intrinsic admittance ηc 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(ηm')=Ni,
then the real part of ηm' equals to the refractive index of the prism. As the thickness dc increases, the path of the ECL approaches a vertical locus and connects the equivalent admittance of the metal film ηm' with the refractive index Ni, as shown in Fig. 5 . The intrinsic admittance ηc 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(ηm')=Re(ηm),
then the real part of ηm' is less than the refractive index of the prism. As the thickness dc increases, the path of the ECL follows a counter-clockwise locus to connect the equivalent admittance ηm' with the refractive index Ni, as shown in Fig. 6 . The intrinsic admittance ηc 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.

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

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

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To achieve the ultra-long range SPP multilayer design, the ECL with the required intrinsic admittance ηc 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 θi=41.31° at 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 ηc can also be determined for the three ECL paths using Eq. (5).

4.1 Clockwise coupling (p-polarization state)

When Re(ηm')=2Ni, the structure {prism/ ECL (ηc=39 .92855) / silver film (20 nm)/ ES (γ=4 .66291)} is employed to excite the p-polarized ultra-long-range SPP wave at θi=41.31°, where the refractive index of the prism Ni 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 ηc, 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 |rp|2 as a function of θi 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=632.8nm, where X=1.31496. The required γ and ηc 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 θi=41.31° and the half-width of the reflection dip, θHW, is 0.00306°.

 

Fig. 7 |rp|2 against θi 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.

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4.2 Near vertical coupling (p-polarization state)

When Re(ηm')=Ni, the structure {prism/ ECL (ηc=116 .82044i) / silver film (20 nm)/ ES (γ=4 .79576)} is obtained to excite the ultra-long range SPP waves at θi=41.31°, where the refractive index of the prism Ni is 1.51511 at λ=632.8nm. 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 |rp|2 against θi 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 ηc 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 θi=41.31° and the half-width of the reflection dip, θHW, is 0.00300°.

 

Fig. 8 |rp|2 against θi 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.

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4.3 Counter-clockwise coupling (p-polarization state)

When Re(ηm')=Re(ηm), the structure {prism/ ECL (ηc=7 .53359i) / silver film (20 nm)/ ES (γ=7 .65128)} is utilized to excite the ultra-long-range SPP waves at θi=41.31°, where the refractive index of the prism Ni is 1.51511 at λ=632.8nm. 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 ηm' 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 ηm' become smaller. As the thickness of the metal film up to infinity, ηm' ends at the point of the intrinsic admittance of silver ηm=0.051332.94974i at θi=41.31° at λ=632.8nm.

Figure 9 plots |rp|2 against θi 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 ηc 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 θi=41.31° and the half-width of the reflection dip, θHW, is 0.00230°.

 

Fig. 9 |rp|2 against θi 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.

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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 [15,16]. The guided plasmon wave modes have the complex propagation constants β and thus effective mode indices ne=β/k0, where k0=2π/λ and λ is the wavelength of incidence. Table 1 shows the design parameters γ and ηc, large values ξ, effective mode indices ne, half-widths θHW, and propagation lengths L of the p-polarization state for the three kinds of ECL paths at θi=41.31° at λ=632.8nm. 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 μm to 2188.45 μm as the real part of ηm' 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.

Tables Icon

Table 1. Design parameters γ and ηc, real parts of ηm', large values ξ, effective mode indices ne, half-widths θHW, and propagation lengths L of the p-polarization state for the three types of coupling at θi=41.31° and λ=632.8nm.

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 [16,17]. Figure 10(a) shows the absolute value of the parallel electric field distribution |Ez| for the counter-clockwise coupling case at θi=41.31° and λ=632.8nm. 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] 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 θi=41.31° at λ=632.8nm, respectively.

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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 ηc. Table 2 presents the design parameters γ and ηc, large values ξ, effective mode indices ne, half-widths θHW, and propagation lengths L for the three kinds of ECL paths at θi=41.31° at λ=632.8nm. 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 μm to 4222.38 μm as the real part of ηm' 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.

Tables Icon

Table 2. Design parameters γ and ηc, real parts of ηm', large values ξ, effective mode indices ne, half-widths θHW, and propagation lengths L of the s-polarization state for the three types of coupling at θi=41.31° and λ=632.8nm.

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 |Ey| for the counter-clockwise ECL path in Table 2 at θi=41.31° and λ=632.8nm. 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 |rs|2 as a function of θi 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=632.8nm. The reflection dip occurs at θi=41.31° and the half-width of the reflection dip, θHW, is 0.00120°.

 

Fig. 11 |rs|2 as a function of θi 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.

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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 [48], 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.

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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]  

References

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  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]

2009 (3)

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]

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]

2007 (1)

2006 (3)

2005 (1)

2001 (2)

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]

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]

1999 (1)

1991 (1)

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]

1987 (1)

1986 (1)

L. Wedler and R. Haupt, “Long-range surface plasmon-polaritons in asymmetric layer structures,” J. Appl. Phys. 59(9), 3289–3291 (1986).
[Crossref]

1981 (1)

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

1969 (1)

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

Adato, R.

Anemogiannis, E.

Berini, P.

Boltasseva, A.

Bozhevolnyi, S.

Bradberry, G. W.

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]

Breukelaar, I.

Chan, T.-Y.

Charbonneau, R.

Economu, E. N.

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

Fafard, S.

Gaylord, T. K.

Glytsis, E. N.

Guo, J.

Haupt, R.

L. Wedler and R. Haupt, “Long-range surface plasmon-polaritons in asymmetric layer structures,” J. Appl. Phys. 59(9), 3289–3291 (1986).
[Crossref]

Homola, J.

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]

Jen, Y.-J.

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]

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]

Kjaer, K.

Kou, F. Y.

Lahoud, N.

Lakhtakia, A.

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]

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]

Larsen, M.

Leosson, K.

Lin, C.-F.

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]

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]

Mattiussi, G.

Nenninger, G. G.

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]

Nikolajsen, T.

Sambles, J. R.

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]

Sarid, D.

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

Scales, C.

Tamir, T.

Tobiska, P.

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]

Wedler, L.

L. Wedler and R. Haupt, “Long-range surface plasmon-polaritons in asymmetric layer structures,” J. Appl. Phys. 59(9), 3289–3291 (1986).
[Crossref]

Yang, F.

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]

Yee, S. S.

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]

Yu, C.-W.

J. Appl. Phys. (1)

L. Wedler and R. Haupt, “Long-range surface plasmon-polaritons in asymmetric layer structures,” J. Appl. Phys. 59(9), 3289–3291 (1986).
[Crossref]

J. Lightwave Technol. (3)

J. Nanophoton. (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]

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]

J. Opt. Soc. Am. A (1)

Opt. Express (3)

Opt. Lett. (1)

Phys. Rev. (1)

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

Phys. Rev. B (2)

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]

Phys. Rev. Lett. (1)

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

Sens. Act. B (1)

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]

Other (2)

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

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

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Figures (11)

Fig. 1
Fig. 1

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

Fig. 2
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.

Fig. 3
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

Fig. 4
Fig. 4

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

Fig. 5
Fig. 5

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

Fig. 6
Fig. 6

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

Fig. 7
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.

Fig. 8
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.

Fig. 9
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.

Fig. 10
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 ° at λ = 632.8 nm, respectively.

Fig. 11
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.

Tables (2)

Tables Icon

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.

Tables Icon

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.

Equations (8)

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η ˜ i = { η ˜ i c h a r / cos θ i η ˜ i c h a r cos θ i , η ˜ = { η ˜ c h a r / cos θ η ˜ c h a r cos θ , p o l a r i z a t i o n = { p s .
η = { ( η ˜ cos θ i ) / ε 0 / μ 0 ( η ˜ / cos θ i ) / ε 0 / μ 0 , p o l a r i z a t i o n = { p s ,
η m = { ( n i k ) 2 cos θ i / n 2 k 2 ( N i sin θ i ) 2 2 i n k n 2 k 2 ( N i sin θ i ) 2 2 i n k / cos θ i , p o l a r i z a t i o n = { p s ,
η m ' = 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 ) ,
η c [ i η c sin δ + η m ' cos δ ] η c cos δ + i η m ' sin δ N i , where δ = { 2 π η c d c cos 2 θ / ( cos θ i λ ) 2 π η c d c cos θ i / λ , p o l a r i z a t i o n = { p s .
Re ( η m ' ) = 2 N i ,
Re ( η m ' ) = N i ,
Re ( η m ' ) = Re ( η m ) ,

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