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Optics Express

  • Editor: Michael Duncan
  • Vol. 14, Iss. 26 — Dec. 25, 2006
  • pp: 13030–13042
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Figures of merit for surface plasmon waveguides

Pierre Berini  »View Author Affiliations


Optics Express, Vol. 14, Issue 26, pp. 13030-13042 (2006)
http://dx.doi.org/10.1364/OE.14.013030


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Abstract

Three figures of merit are proposed as quality measures for surface plasmon waveguides. They are defined as benefit-to-cost ratios where the benefit is confinement and the cost is attenuation. Three different ways of measuring confinement are considered, leading to three figures of merit. One of the figures of merit is connected to the quality factor. The figures of merit were then used to assess and compare the wavelength response of three popular 1-D surface plasmon waveguides: the single metal-dielectric interface, the metal slab bounded by dielectric and the dielectric slab bounded by metal. Closed form expressions are given for the figures of merit of the single metal-dielectric interface.

© 2006 Optical Society of America

1. Introduction

It has long been known that a metal plane can support a surface plasmon-polariton (SPP) mode at optical wavelengths [1

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

]. This waveguide consists of optically semi-infinite metallic (y<0) and dielectric (y>0) half-spaces, as shown in cross-sectional view in Fig. 1(a), and the SPP mode is guided along the plane as an evanescent wave with fields that peak at the interface. Barnes has recently produced a very lucid review of the single-interface SPP, emphasizing the characteristic lengths associated with the mode [2

2. W.L. Barnes, “Surface plasmon-polariton length scales: a route to sub-wavelength optics,” J. Opt. A: Pure Appl. Opt. 8, S87–S93 (2006). [CrossRef]

].

Guiding SPPs means confining them along the plane (or an axis) transverse to the direction of propagation, preferably with low attenuation. The problem with SPP waveguides is that confinement and attenuation are in general correlated, meaning that increased confinement is accompanied by increased attenuation. This correlation was discussed in some detail for the case of the metal stripe [7

7. P. Berini, “Plasmon polariton waves guided by thin lossy metal films of finite width: bound modes of symmetric structures,” Phys. Rev. B 61, 10484–10503 (2000). [CrossRef]

] and for 1-D metal/dielectric structures [17

17. R. Zia, M.D. Selker, P.B. Catrysse, and M. L. Brongersma, “Geometries and materials for subwavelength surface plasmon modes,” J. Opt. Soc. Am. A 21, 2442–2446 (2006). [CrossRef]

]. It is also apparent from the expressions for the penetration depth and propagation length of the single-interface SPP [2

2. W.L. Barnes, “Surface plasmon-polariton length scales: a route to sub-wavelength optics,” J. Opt. A: Pure Appl. Opt. 8, S87–S93 (2006). [CrossRef]

]. This correlation necessarily leads to a confinement-attenuation trade-off in the design of a waveguide. Hence a “good” waveguide is one that optimises this trade-off for an intended application.

The desire to optimise the confinement-attenuation trade-off drives the requirement for a set of SPP waveguide quality measures or “figures of merit”. Global or local optimality could then be defined as maximizing (globally or locally) a figure of merit. Figures of merit would also be useful for comparing different architectures and modes of operation, or different designs, material choices and operating wavelengths. Steps in this direction were taken by Zia et. al. in [17

17. R. Zia, M.D. Selker, P.B. Catrysse, and M. L. Brongersma, “Geometries and materials for subwavelength surface plasmon modes,” J. Opt. Soc. Am. A 21, 2442–2446 (2006). [CrossRef]

] by proposing the use of “descriptors” like mode size, confinement factor [7

7. P. Berini, “Plasmon polariton waves guided by thin lossy metal films of finite width: bound modes of symmetric structures,” Phys. Rev. B 61, 10484–10503 (2000). [CrossRef]

] and propagation length for exploring the trade-off. Some preliminary work on figures of merit for SPP waveguides was also recently reported in [18

18. L. Thylén and E. Berglind, “Nanophotonics and negative ε materials”, J. Zheijiang University: Science A 7, 41–44 (2006). [CrossRef]

]. Within the context of SPP nano-resonators, a figure of merit for single nanoparticle chemical sensors was defined by Sherry et. al. [19

19. L. J. Sherry, S.-H. Chang, G. C. Schatz, and R. P. Van Duyne, “Localized Surface Plasmon Resonance Spectroscopy of Single Silver Nanocubes,” Nanoletters 5, 2034–2038 (2005). [CrossRef]

], and Maier defined a figure of merit for SPP cavities as the ratio Q/Veff , where Q is the quality factor and Veff is an effective mode volume [20

20. S. A. Maier, “Effective mode volume of nanoscale plasmon cavities,” Opt. Quant. Elec. 38, 257–267 (2006). [CrossRef]

, 21

21. S. A. Maier, “Plasmonic field enhancement and SERS in the effective mode volume picture,” Opt. Express 14, 1957–1964 (2006). http://www.opticsexpress.org/abstract.cfm?URI=oe-14-5-1957. [CrossRef] [PubMed]

]. Other figures of merit for cavities in general are Q/V, Q2 /V and Q/V1/2 where V is the mode volume [22

22. D. Englund, I. Fushman, and J Vučković, “General Recipe for designing photonic crystal cavities,” Opt. Express 13, 5961–5975 (2005). http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-16-5961. [CrossRef] [PubMed]

].

2. Definition of the figures of merit

The confinement of a purely bound (non-radiative, non-leaky) mode can be measured in a number of ways, leading to different definitions for the figure of merit. Three ways of measuring confinement and three figures of merit are discussed in the following sub-sections. The intended waveguide application then determines which figure of merit is most relevant.

Henceforth, an e +jωt time dependence is assumed and modes propagate in the +z direction according to eγzz . The complex propagation constant γz expands as γz =αz +jβz where αz and βz are the attenuation and phase constants, respectively. The complex effective index of the mode Neff is given by Neff =γz /β0 =αz /β0 +jβz /β0 =keff +jneff where β0 =/λ0 is the phase constant of plane waves in free space and λ0 is the free-space wavelength. In Fig. 1, εr,m =-εR -jεI is the relative permittivity of the metal and εr,1 =n12 is the relative permittivity of the dielectric with n1 its refractive index.

2.1 Definition of the M11D figure of merit

A direct measure of confinement is the inverse mode size. But the measure used for the mode size depends on the dimensionality of the structure: for a 1-D waveguide the mode size is described by a linear measure such as width, for a 2-D waveguide it's an area, and for a 3-D structure such as a cavity it's a volume. Hence, a figure of merit defined on the basis of mode size must take into account the dimensionality of the structure. The definition of such a figure of merit for the 1-D case M11D is treated in detail here while the 2-D case M12D is relegated to future work.

In a 1-D structure, the mode width δw decreases with confinement so the inverse of the mode width (1/δw ) is adopted as the confinement measure. In keeping with convention, δw is determined by the 1/e mode field magnitude decay points and the main transverse electric field component of the mode is used. Hence the figure of merit M11D for 1-D waveguides, defined as a “benefit-to-cost” ratio, is:

M11D=1δwαz
(1)

where M11D is dimensionless. M11D is a field-based figure of merit and hence is useful for comparing waveguides in applications where achieving a prescribed or small mode width is important, such as end-fire coupling to a source or dense optical interconnects. The mode width δw can be identified directly from the spatial distribution of |Ey (y)| associated with the SPP mode of interest. However, useful closed-form expressions for δw can be obtained for 1-D structures and so they are derived in the following subsections for the three example waveguides of Fig. 1.

Fig. 1. 1-D SPP waveguides considered: (a) single-interface, (b) symmetric dielectric-cladded metal slab, and (c) symmetric metal-cladded dielectric slab.

2.2 Mode size and M11D of the single-interface SPP

Consider first the single-interface structure of Fig. 1(a). A sketch of Re{Ey (y)} associated with the SPP mode is superposed onto the structure, and two field points are identified: Ey,1 (0+) and Ey,m (0-) which are the values of Re{Ey (y)} in the dielectric and metal regions, respectively, on either side of the interface. In the lossless case, δD and δm are the 1/e field penetration depths into the dielectric and metal regions relative to Ey,1 (0+) and Ey,m (0-), respectively. When losses are taken into account, |Ey,1 (0+)| and |Ey,m (0-)| are used to define δD and δm .

The mode width δw is the distance between the 1/e field magnitude points relative to the global maximum which is |Ey,1 (0+)|. A 1/e point relative to |Ey,1 (0+)| may or may not exist in the metal depending on how far the operating wavelength is from the SPP resonance. The condition that determines whether a 1/e point exists in the metal is obtained by inspecting the boundary condition applicable to the normal fields on either side of the discontinuity [23

23. R.E. Collin, Field theory of Guided Waves (IEEE Press, Piscataway, New Jersey, 1991).

]: Dy,m (0-)=Dy,1 (0+) or Ey,m (0-)=Ey,1 (0+)εr,1 /εr,m . Now a 1/e field magnitude point occurs in the metal if |Ey,m (0-)|>|Ey,1 (0+)|/e, which yields the following condition on the permittivities upon application of the boundary condition:

εr,m<eεr,1
(2)

Eq. (2) is satisfied near the SPP resonance, where it is known that the field magnitude in the metal can indeed be large. When Eq. (2) is satisfied, the location of the 1/e point in the metal can easily be determined by assuming the usual exponential form for the field magnitude:

Ey,m(y)=Ey,1(0)eαy,my=εr,1εr,mEy,1(0+)eαy,myfory<0
(3)

where αy,m =1/δm , and from which the mode width δw is derived:

δw={δDforεr,meεr,1δD+δmln(eεr,1εr,m)forεr,m<eεr,1
(4)

Expressions for the penetration depths δD and δm are easily derived from the constraint equations [23

23. R.E. Collin, Field theory of Guided Waves (IEEE Press, Piscataway, New Jersey, 1991).

], which must hold independently within each region of the waveguide. The constraint equation in the dielectric is γy,12 + γz2=γ12 where γ12=-β02n12 =-β12, yielding γy,1=β12γz2 . The field magnitude follows an eαy,1y dependence in the +y direction where αy,1 =Re{γy,1 } so the field penetration depth into the dielectric δD =1/αy,1 is:

δD=1Re{β12γz2}
(5)

Likewise, the constraint equation in the metal is γy,m2+γz2=γm2 where γm2=-β02 ε r,m, yielding γm=β02εr,mγz2 . The field magnitude follows an eαy,my dependence along -y where αy,m =Re{γy,m } so the field penetration depth δm =1/αy,m is:

δm=1Re{β02εr,mγz2}
(6)

The figure of merit M11D is thus readily computed analytically for the single-interface waveguide (Fig. 1(a)) using Eq. (1) with (4)–(6), requiring only the propagation constant γz of the SPP [7

7. P. Berini, “Plasmon polariton waves guided by thin lossy metal films of finite width: bound modes of symmetric structures,” Phys. Rev. B 61, 10484–10503 (2000). [CrossRef]

]:

βzβ0=Re{εr,1εr,mεr,1+εr,m}andαzβ0=Im{εr,1εr,mεr,1+εr,m}
(7)

In the low-loss case (αzβz , εIεR ) and away from resonance where |εr,m |≥r,1 holds, Eqs. (5) and (6) can be simplified, and M11D works out to:

M11Dβz2β12αz=neff2n12keff
(8)

2.3 Mode size and M11D of the metal slab bounded by dielectric (IMI)

δw=t+2δD
(9)

δw does not depend on the magnitude of the mode fields in the metal nor on whether 1/e points exist therein. The figure of merit is then readily computed using Eqs. (1), (5) and (9) with the propagation constant γz determined from an appropriately defined boundary-value problem representing the waveguide [4

4. J. J. Burke, G. I. Stegeman, and T. Tamir, “Surface-polariton-like waves guided by thin, lossy metal films,” Phys. Rev. B , 33, 5186–5201 (1986). [CrossRef]

, 7

7. P. Berini, “Plasmon polariton waves guided by thin lossy metal films of finite width: bound modes of symmetric structures,” Phys. Rev. B 61, 10484–10503 (2000). [CrossRef]

].

In the low-loss case (αzβz , εIεR ) and away from resonance M11D works out to:

M11D1αz[t+2βz2β12]1=1keff[β0t+2neff2n12]1
(10)

For δDt the above further simplifies to:

M11D12βz2β12αz=12neff2n12keff
(11)

For an asymmetric structure (different top and bottom dielectric claddings), δw is determined directly from the computed field distribution.

2.4 Mode size and M11D of the dielectric slab bounded by metal (MIM)

Consider as the last example the symmetric MIM of thickness t shown in Fig. 1(c). A sketch of Re{Ey (y)} associated with the symmetric mode is superposed onto the structure and δm is the 1/e field penetration depth into the metal claddings given by Eq. (6). The width of the mode δw is taken as:

δw={tforεr,meεr,1t+2δmln(eεr,1εr,m)forεr,m<eεr,1
(12)

which does not depend on the magnitude of the mode fields in the dielectric nor on whether 1/e points exist therein. However, it is possible for 1/e points to extend into the metal claddings, as in the single-interface case. The figure of merit is readily computed using Eqs. (1), (6) and (12) with the propagation constant γz determined from an appropriately defined boundary-value problem representing the waveguide [5

5. J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Plasmon slot waveguides: Towards chipscale propagation with subwavelength-scale localization” Phys. Rev. B 73, 035407 (2006). [CrossRef]

, 7

7. P. Berini, “Plasmon polariton waves guided by thin lossy metal films of finite width: bound modes of symmetric structures,” Phys. Rev. B 61, 10484–10503 (2000). [CrossRef]

].

In the low-loss case (αzβz , εIεR ) and away from resonance where |εr,m |≥r,1 holds, M11D works out to:

M11D1αzt=1keffβ0t
(13)

For an asymmetric structure (different top and bottom metal claddings), δw is determined directly from the computed field distribution.

2.5 Role of the confinement factor

The confinement factor [7

7. P. Berini, “Plasmon polariton waves guided by thin lossy metal films of finite width: bound modes of symmetric structures,” Phys. Rev. B 61, 10484–10503 (2000). [CrossRef]

, 17

17. R. Zia, M.D. Selker, P.B. Catrysse, and M. L. Brongersma, “Geometries and materials for subwavelength surface plasmon modes,” J. Opt. Soc. Am. A 21, 2442–2446 (2006). [CrossRef]

] measures the fraction of mode complex power flowing through a prescribed area in the cross-section of the waveguide. Traditionally, this prescribed area is the core of a conventional dielectric waveguide, and the confinement factor varies from 0 to 1 with 1 indicating fields completely confinement to the core and 0 indicating an unconfined mode with fields extending infinitely into a cladding(s). In this structure the confinement factor is a clear measure of confinement.

However, SPPs are surface waves and many SPP waveguides do not have a core within which most of the mode is confined. For instance, no core is identifiable in the singleinterface waveguide of Fig. 1(a), making it awkward to define a confinement factor for this structure. In the case of the IMI of Fig. 1(b), the metal region might be considered the core, and hence the confinement factor clearly defined, but the mode fields are mostly in the dielectric claddings so the confinement factor remains close to 0 even for tightly bound (highly confined) modes. In the case of the MIM of Fig. 1(c), the dielectric region might be considered the core, and hence the confinement factor clearly defined, but the mode fields are mostly in the dielectric so the confinement factor remains close to 1 even for weakly bound (low confinement) modes.

Given these difficulties the inverse of the mode size was adopted in M1 as the field-based measure of confinement. The confinement factor, however, remains useful if one is interested in assessing the fraction of mode power carried within the metal region [7

7. P. Berini, “Plasmon polariton waves guided by thin lossy metal films of finite width: bound modes of symmetric structures,” Phys. Rev. B 61, 10484–10503 (2000). [CrossRef]

], or within another clearly defined region of the structure such as a layer with gain [24

24. C. Sirtori, C. Gmachl, F. Capasso, J. Faist, D. L. Sivco, A. L. Hutchinson, and A. Y. Cho, “Longwavelength (λ≈8-11.5 µm) semiconductor lasers with waveguides based on surface plasmons,” Opt. Lett. 23, 1366–1368 (1998). [CrossRef]

].

2.6 Definition of the M2 figure of merit

Another direct measure of confinement is the distance between the mode's phase constant and the light line. For the structures of Fig. 1, the light line is defined by β1 =ωn1 /c0 where c0 is the velocity of light in free-space. The distance then is given by βz -β1 , or in terms of the effective index, by neff -n1 . Hence the figure of merit M2 , defined as a “benefit-to-cost” ratio, is:

M2=βzβ1αz=neffn1keff
(14)

where M2 is dimensionless. Advantageously, and contrary to M1D1 , M2 holds for 1- and 2-D waveguide structures. M2 is useful for comparing waveguides in applications where achieving a prescribed or large distance from the light line is important, such as in the design of Bragg gratings [25

25. S. Jetté-Charbonneau, R. Charbonneau, N. Lahoud, G. Mattiussi, and P. Berini, “Demonstration of Bragg gratings based on long-ranging surface plasmon polariton waveguides,” Opt. Express 13, 4674–4682 (2005). http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-12-4674. [CrossRef] [PubMed]

].

2.7 Definition of the M3 figure of merit and connection to the quality factor

When the confinement increases, the guided wavelength λg =/βz generally decreases. Hence the figure of merit M3 , defined as a “benefit-to-cost” ratio, using the inverse guided wavelength 1/λg as the measure of confinement, is:

M3=1λgαz=12πβzαz=12πneffkeff
(15)

where M3 is dimensionless and neff >n1 . Advantageously, and contrary to M11D , M3 holds for 1- and 2-D waveguide structures. M3 is useful for comparing waveguides in applications where achieving a small guided wavelength is important, such as in the design of nano-resonators or nanoscale grating couplers.

An intimate connection exists between the unloaded quality factor Q of a resonant mode and M3 . The Q of a lossy dispersive waveguide mode, resonating due to perfect reflectors placed at the input and output of the waveguide, is [23

23. R.E. Collin, Field theory of Guided Waves (IEEE Press, Piscataway, New Jersey, 1991).

]: Q=ω(Energy stored in the mode)/(Mode power dissipated). This works out to:

Q=ωτspp
(16)

where τspp is the SPP lifetime [26

26. M. Fukui, V. C. Y. So, and R. Normandin, “Lifetimes of Surface Plasmons in thin Silver Films”, Phys. Stat. Sol. (b) 91, K61–K64 (1979). [CrossRef]

]:

τspp=1vg2αz
(17)

and vg is the group velocity (velocity of energy transport) [23

23. R.E. Collin, Field theory of Guided Waves (IEEE Press, Piscataway, New Jersey, 1991).

]:

vg1=βzω=1c0(neffλ0neffλ0)
(18)

For negligible dispersion in the metal and dielectric, vgvp =λgω/() where vp is the phase velocity. Using vp for vg in Eq. (17), and comparing the resulting expression for Q obtained via Eq. (16) with the definition for M3 given by Eq. (15), reveals that Q=πM3 . Hence, although M3 and Q are independent quality measures, they become the same to within a factor of π when dispersion is negligible. Of course Q itself can be used as a waveguide figure of merit but M3 is easier to compute.

3. Expressions for the single-interface SPP

Useful approximate expressions for the figures of merit can be derived for the single-interface SPP of Fig. 1(a) using the following approximation to Eq. (7) for neff and keff [1

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

, 2

2. W.L. Barnes, “Surface plasmon-polariton length scales: a route to sub-wavelength optics,” J. Opt. A: Pure Appl. Opt. 8, S87–S93 (2006). [CrossRef]

]:

neff=βzβ0(εr,1εRεRεr,1)12andkeff=αzβ0=εI2εR2(εr,1εRεRεr,1)32
(19)

Substituting the above into Eq. (8), which holds in the low-loss case (αzβz , εIεR ) and away from resonance where |εr,m |≥r,1 holds, yields for M11D :

M11D2εr,112εR12εI(εRεr,1)andM11D2εr,112εR32εIforεRεr,1
(20)

Assuming the Drude model for the relative permittivity of the metal [7

7. P. Berini, “Plasmon polariton waves guided by thin lossy metal films of finite width: bound modes of symmetric structures,” Phys. Rev. B 61, 10484–10503 (2000). [CrossRef]

]:

εr,m=εRjεI=1ωp2ω2+1τD2jωp2τDω(ω2+1τD2)
(21)

where ωp is the plasma frequency and τD the relaxation time, and substituting this into Eq. (20) for εRεr,1 , yields the following relationship under the conditions ω2ωp2 and ω2 ≫1/τD2 :

M11D2εr,112ωpτD
(22)

From Eq. (22), it is noted that M11D does not depend on the wavelength of operation in the Drude region (neglecting dispersion in εr,1 ). This implies that any increase in confinement, measured as the inverse mode size (1/δw ), is perfectly balanced by an increase in attenuation as λ0 decreases. It is also noted that M11D increases as εr,1 decreases, implying that the confinement (1/δw ) decreases less rapidly than the attenuation. Finally, it is observed that maximizing M11D means choosing a metal that maximizes the product ωpτD .

Following the same development for M2 (Eq. (19) into Eq. (14)) yields:

M22εRεI1εr,1(εRεr,1(εRεr,1)3εR)andM2εRεIforεRεr,1
(23)

Substituting the Drude model (Eq. (21)) into Eq. (23) for εRεr,1 yields the following relationship under the conditions ω2ωp2 and ω2 ≫1/τD2 :

M2ωτD=2πc0τDλ0
(24)

From Eq. (24), it is observed that M2 is inversely proportional to the wavelength of operation in the Drude region. Hence the confinement, measured as the distance from the light line (βz -β1 ), increases more rapidly than the attenuation with decreasing λ0 . It is also noted that M2 increases with τD , which is consistent with M11D .

Finally, following the same development for M3 (Eq. (19) into Eq. (15)) yields:

M31πεr,1εRεI(εRεr,1)andM31πεr,1εR2εIforεRεr,1
(25)

Substituting the Drude model (Eq. (21)) into Eq. (25) for εRεr,1 , yields the following relationship under the conditions ω2ωp2 and ω2 ≫1/τD2 :

M31πεr,1ωp2τDω=1πεr,1ωp2τDλ0(2πc0)
(26)

4. Wavelength response of 1-D structures

The wavelength response and figures of merit were computed directly via the analytical solution for the single-interface SPP (Eq. (7)), while those of modes in the IMI and MIM were computed using the method of lines [7

7. P. Berini, “Plasmon polariton waves guided by thin lossy metal films of finite width: bound modes of symmetric structures,” Phys. Rev. B 61, 10484–10503 (2000). [CrossRef]

].

Ag and SiO2 were adopted as the materials and measured optical parameters (n, k) were used [27–30

27. E.D. Palik (Editor), Handbook of Optical Constants of Solids, (Academic Press, Orlando, Florida, 1985).

]. The n and k values were splined and interpolated at the desired wavelengths, and then used to compute the relative permittivities, which are plotted in Fig. 2 (a) for both materials. The plasma frequency and relaxation time of Ag were obtained by fitting Eq. (21) to the relative permittivity in the Drude region (following [31

31. D. J. Nash and J. R. Sambles, “Surface plasmon-polariton study of the optical dielectric function of silver,” J. Mod. Opt. 43, 81–91 (1996).

]), yielding ωp =1.26×1016 rad/s and τD =8.40×10-15 s. The Drude region was taken as λ0 ≥725 nm and λ0 =2000 nm was taken the upper limit of the wavelength range considered.

Figure 2(c) and (d) give the computed values of neff and keff , respectively, for the single-interface SPP, the ab and sb modes of the IMI, and the symmetric (henceforth denoted sb ) mode of the MIM. t of the IMI structure was taken as 20 nm, since for this thickness, the sb and ab modes remain distinct from each other and from the single-interface SPP. From Fig. 2(d) it is noted that these modes are longer- and shorter-range, respectively, than the single-interface SPP. t of the MIM structure was taken as 50 nm, ensuring that the mode maintains sub-wavelength confinement (δw <λ0 /(2n1 )) over the wavelength range of analysis. From Fig. 2(d) it is noted that this mode is shorter-range than the single-interface SPP. The short wavelength limit to the range of analysis of each mode was taken as the wavelength where neffn1 , so the energy asymptote and the non-radiative portion of the bend-back curve are included in the analysis.

Fig. 2. (a) Relative permittivity of Ag (εr,m =-εR -jεI ) and SiO2 (εr,1 =n12 ) over the range 230≤λ0 ≥2000 nm; the inset zooms-in on the range 230≤λ0 ≤500 nm. (b) Dispersion curve of the single-interface SPP. (c) and (d) neff and keff of modes supported by the structures of Fig. 1, respectively.

Figure 3(a) shows the wavelength response of the mode width δw computed via Eqs. (4), (9) and (12). At any given wavelength, δw of the sb mode in the IMI is much greater than that of the others, while δw of the sb mode in the MIM structure is much smaller. δw of the ab mode in the IMI follows very closely δw of the single-interface SPP, despite the attenuation being greater by about one order of magnitude. For λ0 <445 nm, the inequality |εr,m |<r,1 is satisfied and δw of the sb mode in the MIM begins to increase as shown and prescribed by Eq. (12). Also, δw of the single-interface SPP decreases less rapidly with λ0 below 445 nm as observed and prescribed by Eq. (4).

Figure 3(b) shows the wavelength response of the figure of merit M11D computed via Eq. (1). The approximation given by Eq. (20) (εRεr,1 ) for M11D of the single-interface SPP is also plotted, along with Eq. (22) for λ0 >725 nm. For all structures, it is noted that M11D remains approximately flat for λ0 >725 nm, implying that any change in confinement (1/δw ) is balanced by a change in attenuation as λ0 is varied in this region. M11D decreases rapidly for λ0 <600 nm as the modes head towards their energy asymptote, indicating that the attenuation is increasing more rapidly than the confinement (1/δw ). At ant given wavelength, M11D of the sb mode in the IMI is larger than that of the single-interface SPP, while M11D of the ab mode in the same structure is smaller. Hence, the attenuation decreased more rapidly than the confinement (1/δw ) in the case of the sb mode as the thickness was reduced to t=20 nm, while the attenuation increased more rapidly than the confinement (1/δw ) in the case of the ab mode. It is also noted that at a given wavelength the sb mode in the MIM has a slightly better M11D than the single-interface SPP indicating that the attenuation increased less rapidly than the confinement (1/δw ) as the thickness was reduced to t=50 nm.

Figure 3(c) shows the wavelength response of the figure of merit M2 computed via Eq. (14). The approximation given by Eq. (23) (εRεr,1 ) for M2 of the single-interface SPP is also plotted, along with Eq. (24) for λ0 >725 nm. For all structures, it is noted that M2 increases with decreasing λ0 up to a maximum value from which M2 then decreases rapidly as the modes tend toward their asymptote. Hence an optimal wavelength of operation exists that maximizes M2 . Clearly, the confinement (βz -β1 ) increases more rapidly than the attenuation with decreasing λ0 on the long-wavelength side of the maximum, and vise versa on the short wavelength side. For the materials and thicknesses selected the wavelengths that maximize M2 are: 830 nm for the ab mode in the IMI, 840 nm for the single-interface SPP and 850 nm for the sb mode in the MIM; the sb mode in the IMI has two peaks at 870 and 1120 nm. The observations made with respect to the thickness of the IMI and MIM in the case of M11D also hold in this case.

Figure 3(d) shows the wavelength response of the figure of merit M3 computed via Eq. (15). The approximation given by Eq. (25) (εRεr,1 ) for M3 of the single-interface SPP is also plotted, along with Eq. (26) for λ0 > 725 nm. Figure 3(e) shows the group velocity vg of the modes computed via Eq. (18) and Fig. 3(f) plots the quality factor Q computed via Eq. (16). As discussed in Section 2.7, when dispersion is low then M3 and Q converge to essentially the same measure (Q=πM3 ). This is indeed observed in the Drude region for all cases, especially the less dispersive ones such as the sb mode in the IMI and the singleinterface SPP. It is noteworthy that the sb mode in the MIM does not follow the same trend with decreasing wavelength as the other modes, in that it exhibits a maximum in M3 and Q near 840 nm. The other modes follow the same trend in that M3 and Q increase linearly with λ0 . The observations made with respect to the thickness of the IMI in the case of M11D also hold in this case, but not for the thickness of the MIM. It is noted that at a given wavelength the sb mode in the MIM has a significantly worse M3 and Q than the single-interface SPP indicating that the attenuation increased more rapidly than the confinement (1/λg ) as the thickness was reduced to t=50 nm. Finally, it is noted that reasonably high Q values (10,000) are achievable with the sb mode in the IMI.

As already noted, M11D , M2 , M3 and Q eventually tend to zero with decreasing λ0 , as the modes tend toward their energy asymptote. Clearly then, the attenuation increases more rapidly than the confinement in this region, regardless of how the confinement is measured. This is caused by the rapidly increasing fraction of mode power within the metal(s). Also, the modes are nearing the band edge of Ag (~310 nm), so the onset and increase of interband absorption prevents εI from becoming small while doing nothing to increase the confinement.

The simple functions of εR and εI given by Eqs. (20), (23) and (25) for εRεr,1 are good approximations to M11D , M2 , M3 and Q of the single-interface SPP and reasonable predictors of features, but not trends, in the wavelength response of other modes.

Fig. 3. (a) δw , (b) M11D , (c) M2 , (d) M3 , (e) vg , and (f) Q of modes supported by the structures of Fig. 1.

5. Summary and conclusions

Three methods of measuring confinement were discussed and three figures of merit (M11D , M2 , M3 ) were defined as benefit-to-cost ratios with the benefit being confinement and the cost being attenuation. M11D is defined using a field-based measure of confinement, the inverse mode width (1/δw ), and is limited to 1-D waveguides. M11D is useful for assessing and optimizing waveguides intended for application in, for example, end-fire coupling and optical interconnects. M2 and M3 are defined using the distance of the mode from the light line (βz -β1 ) and the inverse guided wavelength (1/λg ) as their measures of confinement, respectively, and they are applicable to 1- and 2-D waveguides. M2 and M3 are useful for assessing and optimizing waveguides intended for application in, for example, Bragg gratings and resonators, respectively. The intimate connection between M3 and Q (the unloaded quality factor) was also discussed.

References and links

1.

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

2.

W.L. Barnes, “Surface plasmon-polariton length scales: a route to sub-wavelength optics,” J. Opt. A: Pure Appl. Opt. 8, S87–S93 (2006). [CrossRef]

3.

E. N. Economou, “Surface Plasmons in thin Films,” Phys. Rev. 182, 539–554 (1969). [CrossRef]

4.

J. J. Burke, G. I. Stegeman, and T. Tamir, “Surface-polariton-like waves guided by thin, lossy metal films,” Phys. Rev. B , 33, 5186–5201 (1986). [CrossRef]

5.

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Plasmon slot waveguides: Towards chipscale propagation with subwavelength-scale localization” Phys. Rev. B 73, 035407 (2006). [CrossRef]

6.

J.-C. Weeber, A. Dereux, C. Girard, J. R. Krenn, and J.-P. Goudonnet, “Plasmon polaritons of metallic nanowires for controlling submicron propagation of light,” Phys. Rev. B 60, 9061–9068 (1999). [CrossRef]

7.

P. Berini, “Plasmon polariton waves guided by thin lossy metal films of finite width: bound modes of symmetric structures,” Phys. Rev. B 61, 10484–10503 (2000). [CrossRef]

8.

R. Charbonneau, P. Berini, E. Berolo, and E. Lisicka-Shrzek, “Experimental observation of plasmonpolariton waves supported by a thin metal film of finite width,” Opt. Lett. 25, 844–846 (2000). [CrossRef]

9.

B. Lamprecht, J. R. Krenn, G. Schider, H. Ditlbacher, M. Salerno, N. Felidj, A. Leitner, and F.R. Aussenegg, “Surface plasmon propagation in microscale metal stripes,” Appl. Phys. Lett. , 7951–53 (2001). [CrossRef]

10.

J.-C. Weeber, J. R. Krenn, A. Dereux, B. Lamprecht, Y. Lacroute, and J. P. Goudonnet, “Near-field observation of surface plasmon polariton propagation on thin metal stripes,” Phys. Rev. B 64, 045411 (2001). [CrossRef]

11.

R. Nikolajsen, K. Leosson, I. Salakhutdinov, and S.I. Bozhevolnyi, “Polymer-based surface-plasmonpolariton stripe waveguides at telecommunication wavelengths,” Appl. Phys. Lett. 82, 668–670 (2003). [CrossRef]

12.

I. V. Novikov and A.A. Maradudin, “Channel polaritons,” Phys. Rev. B 66, 035403 (2002). [CrossRef]

13.

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonators,” Nat. 440, 508–511 (2006). [CrossRef]

14.

D. F. P. Pile, T. Ogawa, D. K. Gramotnev, Y. Matsuzaki, K. C. Vernon, K. Yamaguchi, T. Okamoto, M. Haraguchi, and M. Fukui, “Two-dimensionally localized modes of a nanoscale gap plasmon waveguide,” Appl. Phys. Lett. , 87261114 (2005). [CrossRef]

15.

W.L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nat. 424, 824–830 (2003). [CrossRef]

16.

S. A. Maier and H. A. Atwater, “Plasmonics: Localization and guiding of electromagnetic energy in metal/dielectric structures,” J. Appl. Phys. 98, 011101 (2005). [CrossRef]

17.

R. Zia, M.D. Selker, P.B. Catrysse, and M. L. Brongersma, “Geometries and materials for subwavelength surface plasmon modes,” J. Opt. Soc. Am. A 21, 2442–2446 (2006). [CrossRef]

18.

L. Thylén and E. Berglind, “Nanophotonics and negative ε materials”, J. Zheijiang University: Science A 7, 41–44 (2006). [CrossRef]

19.

L. J. Sherry, S.-H. Chang, G. C. Schatz, and R. P. Van Duyne, “Localized Surface Plasmon Resonance Spectroscopy of Single Silver Nanocubes,” Nanoletters 5, 2034–2038 (2005). [CrossRef]

20.

S. A. Maier, “Effective mode volume of nanoscale plasmon cavities,” Opt. Quant. Elec. 38, 257–267 (2006). [CrossRef]

21.

S. A. Maier, “Plasmonic field enhancement and SERS in the effective mode volume picture,” Opt. Express 14, 1957–1964 (2006). http://www.opticsexpress.org/abstract.cfm?URI=oe-14-5-1957. [CrossRef] [PubMed]

22.

D. Englund, I. Fushman, and J Vučković, “General Recipe for designing photonic crystal cavities,” Opt. Express 13, 5961–5975 (2005). http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-16-5961. [CrossRef] [PubMed]

23.

R.E. Collin, Field theory of Guided Waves (IEEE Press, Piscataway, New Jersey, 1991).

24.

C. Sirtori, C. Gmachl, F. Capasso, J. Faist, D. L. Sivco, A. L. Hutchinson, and A. Y. Cho, “Longwavelength (λ≈8-11.5 µm) semiconductor lasers with waveguides based on surface plasmons,” Opt. Lett. 23, 1366–1368 (1998). [CrossRef]

25.

S. Jetté-Charbonneau, R. Charbonneau, N. Lahoud, G. Mattiussi, and P. Berini, “Demonstration of Bragg gratings based on long-ranging surface plasmon polariton waveguides,” Opt. Express 13, 4674–4682 (2005). http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-12-4674. [CrossRef] [PubMed]

26.

M. Fukui, V. C. Y. So, and R. Normandin, “Lifetimes of Surface Plasmons in thin Silver Films”, Phys. Stat. Sol. (b) 91, K61–K64 (1979). [CrossRef]

27.

E.D. Palik (Editor), Handbook of Optical Constants of Solids, (Academic Press, Orlando, Florida, 1985).

28.

P. Winsemius, F. F. van Kampen, H. P. Lengkeek, and C. G. van Went, “Temperature dependence of the optical properties of Au, Ag and Cu,” J. Phys. F: Metal Phys. 6, 1583–1606 (1976). [CrossRef]

29.

G. Leveque, C. G. Olson, and D. W. Lynch, “Reflectance spectra and dielectric functions for Ag in the region of interband transitions,” Phys. Rev. B 27, 4654–4660 (1983). [CrossRef]

30.

B. Brixner, “Refractive-index interpolation for fused silica,” J. Opt. Soc. Am. 57, 674–676 (1967). [CrossRef]

31.

D. J. Nash and J. R. Sambles, “Surface plasmon-polariton study of the optical dielectric function of silver,” J. Mod. Opt. 43, 81–91 (1996).

32.

E. T. Arakawa, M. W. Williams, R. N. Hamm, and R. H. Ritchie, “Effect of Damping on Surface Plasmon Dispersion,” Phys. Rev. Lett. 31, 1127–1129 (1973 ). [CrossRef]

OCIS Codes
(130.2790) Integrated optics : Guided waves
(240.6680) Optics at surfaces : Surface plasmons

ToC Category:
Optics at Surfaces

History
Original Manuscript: October 2, 2006
Manuscript Accepted: December 11, 2006
Published: December 22, 2006

Citation
Pierre Berini, "Figures of merit for surface plasmon waveguides," Opt. Express 14, 13030-13042 (2006)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-26-13030


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References

  1. H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer, Berlin, 1988).
  2. W. L. Barnes, "Surface plasmon-polariton length scales: a route to sub-wavelength optics," J. Opt. A: Pure Appl. Opt. 8, S87-S93 (2006). [CrossRef]
  3. E. N. Economou, "Surface Plasmons in thin Films," Phys. Rev. 182, 539-554 (1969). [CrossRef]
  4. J. J. Burke, G. I. Stegeman and T. Tamir, "Surface-polariton-like waves guided by thin, lossy metal films," Phys. Rev. B 33, 5186-5201 (1986). [CrossRef]
  5. J. A. Dionne, L. A. Sweatlock, H. A. Atwater and A. Polman, "Plasmon slot waveguides: towards chip-scale propagation with subwavelength-scale localization," Phys. Rev. B 73, 035407 (2006). [CrossRef]
  6. J.-C. Weeber, A. Dereux, C. Girard, J. R. Krenn and J.-P. Goudonnet, "Plasmon polaritons of metallic nanowires for controlling submicron propagation of light," Phys. Rev. B 60, 9061-9068 (1999). [CrossRef]
  7. P. Berini, "Plasmon polariton waves guided by thin lossy metal films of finite width: bound modes of symmetric structures," Phys. Rev. B 61, 10484-10503 (2000). [CrossRef]
  8. R. Charbonneau, P. Berini, E. Berolo and E. Lisicka-Shrzek, "Experimental observation of plasmon-polariton waves supported by a thin metal film of finite width," Opt. Lett. 25, 844-846 (2000). [CrossRef]
  9. B. Lamprecht, J. R. Krenn, G. Schider, H. Ditlbacher, M. Salerno, N. Felidj, A. Leitner and F. R. Aussenegg, "Surface plasmon propagation in microscale metal stripes," Appl. Phys. Lett.,  7951-53 (2001). [CrossRef]
  10. J.-C. Weeber, J. R. Krenn, A. Dereux, B. Lamprecht, Y. Lacroute, and J. P. Goudonnet, "Near-field observation of surface plasmon polariton propagation on thin metal stripes," Phys. Rev. B 64, 045411 (2001). [CrossRef]
  11. R. Nikolajsen, K. Leosson, I. Salakhutdinov and S. I. Bozhevolnyi, "Polymer-based surface-plasmon-polariton stripe waveguides at telecommunication wavelengths," Appl. Phys. Lett. 82, 668-670 (2003) [CrossRef]
  12. I. V. Novikov and A. A. Maradudin, "Channel polaritons," Phys. Rev. B 66, 035403 (2002). [CrossRef]
  13. S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, "Channel plasmon subwavelength waveguide components including interferometers and ring resonators," Nature 440, 508-511 (2006). [CrossRef]
  14. D. F. P. Pile, T. Ogawa, D. K. Gramotnev, Y. Matsuzaki, K. C. Vernon, K. Yamaguchi, T. Okamoto, M. Haraguchi and M. Fukui, "Two-dimensionally localized modes of a nanoscale gap plasmon waveguide," Appl. Phys. Lett.,  87261114 (2005). [CrossRef]
  15. W. L. Barnes, A. Dereux and T. W. Ebbesen, "Surface plasmon subwavelength optics," Nature 424, 824-830 (2003). [CrossRef]
  16. S. A. Maier and H. A. Atwater, "Plasmonics: Localization and guiding of electromagnetic energy in metal/dielectric structures," J. Appl. Phys. 98, 011101 (2005). [CrossRef]
  17. R. Zia, M. D. Selker, P. B. Catrysse and M. L. Brongersma, "Geometries and materials for subwavelength surface plasmon modes," J. Opt. Soc. Am. A 21, 2442-2446 (2006). [CrossRef]
  18. L. Thylén and E. Berglind, "Nanophotonics and negative ε materials," J. Zheijiang University: Science A 7, 41-44 (2006). [CrossRef]
  19. L. J. Sherry, S.-H. Chang, G. C. Schatz and R. P. Van Duyne, "Localized surface plasmon resonance Spectroscopy of single silver nanocubes," Nanoletters 5, 2034-2038 (2005). [CrossRef]
  20. S. A. Maier, "Effective mode volume of nanoscale plasmon cavities," Opt. Quantum Electron. 38, 257-267 (2006). [CrossRef]
  21. S. A. Maier, "Plasmonic field enhancement and SERS in the effective mode volume picture," Opt. Express 14, 1957-1964 (2006). [CrossRef] [PubMed]
  22. D. Englund, I. Fushman and J Vučković, "General recipe for designing photonic crystal cavities," Opt. Express 13, 5961-5975 (2005). [CrossRef] [PubMed]
  23. R.E. Collin, Field theory of Guided Waves (IEEE Press, Piscataway, New Jersey, 1991).
  24. C. Sirtori, C. Gmachl, F. Capasso, J. Faist, D. L. Sivco, A. L. Hutchinson and A. Y. Cho, "Long-wavelength (λ ≈ 8 - 11.5 µm) semiconductor lasers with waveguides based on surface plasmons," Opt. Lett. 23, 1366-1368 (1998). [CrossRef]
  25. S. Jetté-Charbonneau, R. Charbonneau, N. Lahoud, G. Mattiussi and P. Berini, "Demonstration of Bragg gratings based on long-ranging surface plasmon polariton waveguides," Opt. Express 13, 4674-4682 (2005). [CrossRef] [PubMed]
  26. M. Fukui, V. C. Y. So, and R. Normandin, "Lifetimes of Surface Plasmons in thin Silver Films," Phys. Stat. Sol.(b) 91,K61-K64 (1979). [CrossRef]
  27. E. D. Palik, ed., Handbook of Optical Constants of Solids, (Academic Press, Orlando, Florida, 1985).
  28. P. Winsemius, F. F. van Kampen, H. P. Lengkeek and C. G. van Went, "Temperature dependence of the optical properties of Au, Ag and Cu," J. Phys. F: Met. Phys. 6, 1583-1606 (1976). [CrossRef]
  29. G. Leveque, C. G. Olson and D. W. Lynch, "Reflectance spectra and dielectric functions for Ag in the region of interband transitions," Phys. Rev. B 27, 4654-4660 (1983). [CrossRef]
  30. B. Brixner, "Refractive-index interpolation for fused silica," J. Opt. Soc. Am. 57, 674-676 (1967). [CrossRef]
  31. D. J. Nash and J. R. Sambles, "Surface plasmon-polariton study of the optical dielectric function of silver," J. Mod. Opt. 43, 81-91 (1996).
  32. E. T. Arakawa, M. W. Williams, R. N. Hamm and R. H. Ritchie, "Effect of damping on Surface Plasmon Dispersion," Phys. Rev. Lett. 31, 1127-1129 (1973). [CrossRef]

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