Surface Plasmon-polaritons on thin metal films Dielectric ε 3 Metal ε 2 Dielectric ε 1
References Surface plasmons in thin films, E.N. Economou, Phy. Rev. Vol.182, 539-554 (1969) Surface-polariton-like waves guided by thin, lossy metal films, J.J. Burke, G. I. Stegeman, T. Tamir, Phy. Rev. B, Vol.33, 5186-5201 (1986) Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of symmetric structures, P. Berini, Phy. Rev. B, Vol.61, 10484 (2000) Geometries and materials for subwavelength surface plasmon modes, Rashid Zia, Mark D. Selker, Peter B. Catrysse, and Mark L. Brongersma, J. Opt. Soc. Am. A, Vol. 21, 2442-2446 (2004) Plasmon slot waveguides: Towards chip-scale propagation with subwavelength-scale localization, J. A. Dionne,* L. A. Sweatlock, and H. A. Atwater, Phy. Rev. B, Vol.73, 035407 (2006)
Introduction: When the film thickness becomes finite. mode overlap
Introduction: Possibility of Propagation Range Extension Long-Range SP: weak surface confinement, low loss frequency Short-Range SP: strong surface confinement, high loss in-plane wavevector
Introduction: Extremely long-range SP? Symmetrically coupled LRSP frequency Anti-symmetrically coupled LRSP in-plane wavevector
Introduction: Dependence of dispersion on film thickness 1 1 0.75 0.75 0.5 0.5 0.25 0.25 250 500 750 1000 1250 1500 200 400 600 800 250 500 750 1000 1250 1500 200 400 600 800-0.25-0.25-0.5-0.75-1 h = 10 60nm -0.5-0.75-1 practically forbidden
Surface plasmons in thin films E.N. Economou, Phy. Rev. Vol.182, 539-554 (1969) LOCAL THEORY FOR MULTIPLE-FILM SYSTEM Maxwell equations with a local current-field relation as follows: Local approximation to the current-field relation with The local approximation satisfies the dielectric function in the metal, For electric (or TM) waves that H z = H x = E y = 0 in all of the media, the solution for any component of the fields can thus be represented in the form
A. Single Metal-Dielectric Interface The solution for E x that remains finite at infinity is The continuity of E and H fields across the boundary gives the dispersion relation as If the dielectric constant for the insulator ε i =1, ( ) ( ) / 1 / metal insulator K R K ε ε = m d x m d k c ε ε ω ε ε = + 2 2 2 2 ) (1 ) ( p d d p sp x c k k ω ω ε ε ω ω ω + = =
Retardation (radiative loss) It is obvious that retardation effects are important for q = (k/k p ) < 1 and that they do not play any role for q>>1. Therefore, we are interested in region IV. Dispersion of bulk plasmon: 2 c c ω p ω = ω =±, ω =± ω + if εm = 1 2 ω 2 2 2 2 2 2 k k p c k ε m ε m
B. Insulating Film between Two Semi-Infinite Metals Branches I and II are adequately described by the longitudinal electrostatic theory when k >> k p, If ε i =1, the low-frequency k < k p, part of I, If k p d i << l, as is usually the case, branch III is If k p d i << l, the k << k p portion of II is and, at k = 0
C. Metal Film in Vacuum R ( K / ε ) ( K / ε ) metal insulator K 2 2 ω εim, im, = k 2 c * Burke, PRB 1986 : 2 2 S = k ε k 2, n= 1, 2, 3. n n o When k < k p, for branch II (antisymmetric oscillation) When k < k p, for branch I (symmetric oscillation) When k > k p, both I and II approach
D. Swihart's Geometry When k << k p, for branch I Branch II just R=1, corresponds to oscillations on the external interface (insulator/metal). Branch III, The intersection of branch III with the line ω=ck occurs, when Branch IV behaves for small k as
E. Two Metal Films of Different Thicknesses Where, When k << k p, branch II, III
The branch II and III Where, Branch II Branch III Branch II Branch III
F. Two Dielectric Films For k<<k p branches I and II are given by In the special case when d 1 =d 2 =d i they correspond to symmetric and antisymmetric solutions, And, in the limit when Branch III starts, when k= 0,
Branch III starts, when k= 0, For k<k p Branch IV is, for small k, It corresponds to an oscillation which couples the two junctions, and, when one of the thicknesses of the dielectric films becomes large, the coupling is broken and the oscillation is confined to one of them.
G. Three Metal Films When k<<k p branches I and II are given by When k<<k p branches III and IV: When k<<k p branches V and VI
H. Periodic structure of alternating metal and insulating films Periodicity of alternating thicknesses d m and d i, implies that the eigensolutions obey the Floquet-Bloch theorem; namely, The secular equation is Solutions are found in the shaded regions. When k<<k p curves III and IV can be taken as straight lines with phase velocities, Any intermediate solution has phase velocity
The upper region within which solutions lie is bounded by I, a portion of IIa, and IIb. If (k p 2 d i d m )<< 1, Curve I is while curve IIa is Solutions near I are given by When k>>k p the secular equation is
Conclusions of Economou Modes of SPO in multiple-film systems can be classified into two main groups. One group contains those modes whose dispersion curves start from zero frequency at k=o, increase as k increases, but remain below the line ω= ck. The other group starts at k= 0 from ω= ω p, or a value slightly less than ω p, and remains close to the line ω= ω p. For very large k, all the dispersion curves of both groups converge asymptotically to the classical surface plasmon frequency ω p /root(2). In addition to these two groups, some uninteresting modes may appear with dispersion curves that lie just below the curve ω 2 = ω p 2 + c 2 k 2 which corresponds to the trivial solution of zero fields. For normal metals this description is valid only for high enough frequencies so that oscillation damping is negligible. On the other hand, for superconducting metals, the picture is valid not only for the high-frequency region but also for low frequencies. In Multiple-film structures radiative SPO exists, which should have observable effects in the radiation properties of these structures. In particular, there seems to be a possibility of obtaining intense radiation as the number of the films increases.
Surface-polariton-like waves guided by thin, lossy metal films J.J. Burke, G. I. Stegeman, T. Tamir, Phy. Rev. B, Vol.33, 5186-5201 (1986). Dispersion relations for waves guided by a thin, lossy metal film surrounded by dielectric media Characteristic of "spatial transients" : Usual symmetric and antisymmetric branches each split into a pair of waves one radiative (leaky waves) and the other nonradiative (bound waves). Symmetric modes : the transverse electric field does not exhibit a zero inside the metal film Antisymmetric modes : the transverse electric field has a zero inside the film. h ε 1 ε m = - ε R iε I ε 3 z x ε 1 > ε 3 Burke, PRB 1986
Dispersion relation for thin metal films (3 layers) obtained from the Maxwell equations Hi( x, z, t) = e yh0 fi( z)exp i( βx ωt) [ 3 ] ( ) [ 2 ] 0 [ 2 ] ( ) [ sz] ( z ) Bexp s ( z h) in medium 3 z h fi( z) = Ahexp s ( z h) + A exp s z in medium 2 0 z h exp 1 in medium 1 0 s = β ε k 2 2 2 j j j 0 i H y i df j ( z) Ex = = H0 exp i x ωε z ωε dz i E= H Ey = 0 ωε β β Ez = Hy = H0 f j ( z) exp[ iβ x] ωε ωε [ β ] s3 B exp [ s 3( z h )] ( z h ) ε3 i s2 Ex = H0exp[ iβ x] A exp s2( z h) A0exp s2z 0 z h ωε ε 2 s1 exp[ sz 1 ] ( z 0) ε1 { h [ ] [ ]} ( )
From the boundary conditions, ( ) ( ) z = 0: Hx 1 = Hx2 exp[ s2h] Ah + A0 = 1 z = h: Hx2 = Hx3 Ah + exp[ s2h] A0 = B ε 2s1 z = 0 : Ex 1 = Ex2 exp[ s2h] Ah A0 = ε1s2 ε s = = = 2 3 z h: Ex2 Ex3 Ah exp[ s2h] A0 B ε3s2 From the equations at z = 0, A h, A o, and B can be determined by, ε 2s 1 cosh[ sh 2 ] + sinh[ sh 2 ] exp [ s3( z h) ] ( j= 3: z h) ε1s2 ε s f j ( z) = cosh s2z + sinh s2z j = 2 : 0 z h ε1s2 exp[ sz 1 ] ( j= 1: z 0) 2 1 [ ] [ ] ( ) Therefore, anther equations at z = h gives the dispersion relation, 2 2 ( εε 1 3s2 ε2s1s3) [ s2h] ε2s2( ε3s1 ε1s3) + tanh + + = 0 s = β ε k 2 2 2 j j j 0
Dispersion relation when h When h >> c/ω p (classical skin depth), tanh(s 2 h) 1, ( ) ω εiε m βi= 1,3 h = c ε + ε i m The solutions consist of decoupled surface-plasmon polaritons (SPP) : SP 1 : propagating along the ε 1 -ε m interface SP 3 : propagating along the ε 3 -ε m interface If we assume that β >> β and ε = ε + iε R I m R I ε m 2 ω p = 1 2 ω Burke, PRB 1986
If β > ε ω/ c (light line), i S = β ε k > 0 2 2 2 i i o [ 3 ] ( ) [ 2 ] 0 [ 2 ] ( ) [ sz] ( z ) Bexp s ( z h) in medium 3 z h fi( z) = Ahexp s ( z h) + A exp s z in medium 2 0 z h exp 1 in medium 1 0 Hence S i is real. But, there are two types of solutions for the semi-infinite media SPPs : (1) S i > 0, SPPs are nonradiative, or bound (2) S i < 0, SPPs are grow exponentially with distance from the interface, which are physically rejected because of their non-guiding property Therefore, SPPs, are bound at the semi-infinite media only when S i are real and positive. For a finite film thickness, the two allowed semi-infinite SPPs are coupled. One of the SPPs could become leaky (radiative) in the ε 3 medium when ε 1 > ε 3 If β < ε ω/ c (light line), i S < 0 2 i Hence S i is imaginary. The field is a plane wave radiating away form the metal boundary. Burke, PRB 1986
Waves guided by symmetric structures ( ε = ε ) 1 3 There are four types of solutions satisfying : S = β ε k > 0 2 2 2 i i o We can estimate the properties of the solutions as follows: h h 0 ± S1 =± S3 (1) : S1 > 0 & S3 > 0 S1 S3 (2): < 0 & < 0 ± S1 = S3 (3) : S1 > 0 & S3 < 0 S1 S3 (4): < 0 & > 0 (1) : S > 0 & S > 0 1 3 The fields decay exponentially into both ε 1 and ε 3, and the wave fronts are tilted in towards the metal film. -> nonradiative waves There are two types of solutions for S i > 0 : symmetric (s), antisymmetric (a) (2): S < 0 & S < 0 1 3 2 SP solutions No solution The fields grow exponentially with wave front tilted to carry energy away from the metal ε 1 ε 3 ε m H y Leaky (ε 1 ) Leaky (ε 3 ) ε 1 ε m ε 3 H y Bound (s) ε 1 ε m ε 3 H y Bound (a) Two nonradiative, Fano modes (3) : S > 0 & S < 0 1 3 1 SP solution The field in ε 1 and the metal is guided by the interface, the filed in ε 3 grows exponentially (leaky). ε 1 ε m ε 3 Bound (ε 1 ) Leaky (ε 3 ) Four SPP modes (4): S < 0 & S > 0 1 3 1 SP solution ε 1 H y Leaky (ε 1 ) H y The field in ε 3 and the metal is guided by the interface, the filed in ε 1 grows exponentially (leaky). ε m ε 3 Bound (ε 3 ) Two radiative, leaky modes Burke, PRB 1986
Two nonradiative, Fano modes ε 1 H y ε 1 H y (1) : S > 0 & S > 0 1 3 ε m ε 3 Bound (s) ε m ε 3 Bound (a) Symmetric bound (s b ) Asymmetric bound (a b )
Surface plasmon dispersion for thin films Two modes appear Drude model ε 1 (ω)=1-(ω p /ω) 2 L - (asymmetric) L + (symmetric) Thinner film: Shorter SP wavelength Example: λ HeNe = 633 nm λ SP = 60 nm L - Propagation lengths: cm!!! (infrared)
Ex) λ = 1.55 μm, ε = 118 + i11.58, ε = ε = 2.25 2 1 3 ( 2 2 εε ) 1 3s2 ε2s1s3 [ s2h] ε2s2( ε3s1 ε1s3) + tanh + + = 0 s b a b s = β ε k 2 2 2 j j j 0 ε 1 ε m H y ε 1 ε m H y ε 3 Bound (s) ε 3 Bound (a) 2.0 1 1.9 a b 0.1 a b β r /k 0 1.8 1.7 s b β i /k 0 0.01 1E-3 1E-4 s b 1.6 1.5 0 50 100 150 200 thickness (h : nm) 1E-5 1E-6 1E-7 0 50 100 150 200 thickness (h : nm)
Waves guided by asymmetric structures There are also four types of solutions satisfying ( ε ε ) 1 3 ± S =± S & ± S = S 1 3 1 3 One antisymmetric mode is always obtained. The "symmetric" solutions are of two types, nonradiative (bound) and nonradiative (leaky): S = β ε k > 0 2 2 2 i i o 2 2 2 Bound and Leaky modes when Si β εiko 0 = > Growing modes ε 1 H y (1) : S > 0 & S > 0 1 3 ε 1 H y For example, S < 0 & β > 0 1R 3I when S = S - is and β = β -iβ i ir ii i ir ii H y ε m ε m ε 1 Growing (z) ε 3 Bound (s) ε 3 Bound (a) ε m z x ε 3 Growing (x) (2): S < 0 & S < 0 (3) : S1 > 0 & S3 < 0 (4): S1 < 0 & S3 > 0 1 3 ε 1 H y Leaky (ε 1 ) ε 1 Bound (ε 1 ) ε 1 H y Leaky (ε 1 ) ε m ε m ε m ε 3 Leaky (ε 3 ) ε 3 Leaky (ε 3 ) ε 3 Bound (ε 3 ) Hy Burke, PRB 1986
Waves guided by asymmetric structures ( ε ε ) 1 3 Nonradiative mode the fields in the dielectric decay exponentially away from the film and the wave fronts are tilted into the metal film, in order to remove energy from the dielectric media (for dissipation in the metal) as the wave attenuates with propagation distance. Leaky (radiative) mode The wave energy is localized in one of the dielectrics, say ε 1, at that dielectric-metal interface. The wave amplitude decays exponentially across the film, and then grows exponentially into the other dielectric medium, ε 3 in this case. In the ε 1 medium, the wave fronts are tilted towards the film to supply energy from ε 1 for both dissipation in the metal, and radiation into ε 3. Leaky (radiative) mode The analogous case of localization in ε 3, and radiation into ε 1. Growing mode The field amplitude grows both with propagation distance as exp(β I x), and into one of the dielectrics as exp(s R z). Since the wave-front tilt is into the film (as opposed to away from it for leaky waves), these waves are dependent on externally incident fields supplying energy to make the total wave amplitude grow. Burke, PRB 1986
Leaky waves They only have meaning in a limited region of space above the film and require some transverse plane (say x=o) containing an effective source that launches a localized wave in one dielectric near its metal-dielectric boundary. The field decays across the metal and couples to radiation fields in the opposite dielectric. The ε 1 field amplitude grows exponentially for only a finite distance z, In this sense, the solutions do not violate boundary conditions as z - infinite. For fields radiated at an angle θ relative to the surface, the angular spectrum of the radiated plane waves is Thus, the radiated power is The wave attenuation due to radiation loss can be estimated from the solutions by calculating the Poynting vector for energy leaving, for example, the ε 1 -ε m boundary, per meter of wave front, θ Burke, PRB 1986
Dispersion relation for thin metal strips with finite widths metal strip dielectric
Finite film thickness and width Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of symmetric structures, P. Berini, Phy. Rev. B, Vol.61, 10484 (2000)
E( xyzt,,, ) = E ( xye, ) H( xyzt,,, ) = H ( xye, ) 0 0 i( βz ωt) i( β z ωt) E o, H o : polarization direction z-axis : propagation direction From the Maxwell equations ( ) = k 1 2 ε r H 0 H Assume that all media be isotropic. The magnetic field on x-y (transverse plane) satisfies ( ε H ) ε ( H ) ( k β ε ) H = 0 1 1 2 2 1 t r t t r t t t 0 r t where = t i + j x y H = ( H i + H j) e i β z ωt t x y ( ) This eigenvalue problem can be solved by a numerical method with proper boundary conditions, such as one of FDM, FEM, MoL, Here, we use the FDM (finite difference method).
y
Ex) λ = 1.55 μm, ε = 118 + i11.58, ε = ε = 2.25, w = 5μm 2 1 3 FDM z y x x y 1.64 1.62 1.60 sa b o ss b o 0.01 1E-3 β r /k o 1.58 1.56 1.54 1.52 β i /k 0 1E-4 1E-5 1E-6 sa b o ss b o 1.50 0 50 100 150 200 thickness of metal (nm) 1E-7 0 50 100 150 200 thickness of metal (nm)
Plasmon slot waveguides : Metal-Insulator-Metal (MIM) Plasmon slot waveguides: Towards chip-scale propagation with subwavelength-scale localization J. A. Dionne,* L. A. Sweatlock, and H. A. Atwater Phy. Rev. B, Vol.73, 035407 (2006) z Metal ε 2 y x d Insulator ε 1 Metal ε 2
z Metal ε 2 y x d Insulator ε 1 Metal ε 2
Mode L+ Mode L- Tangential (E x ) Electric Field Profiles
(infinite d) (infinite d) Asymmetric bound mode (L+) : a b Symmetric bound mode (L-) : s b The curves for decoupled SP (infinite d) exhibit exact agreement
(infinite d) (infinite d)
D = 250 nm (infinite d) SP modes conventional waveguide modes In S i O 2 core SP modes
D = 100 nm (infinite d) S b conventional waveguide modes within ΔE ~ 1 ev SP modes
s b SP : D = 50 nm 30 nm 25 nm 12 nm y z x d Metal ε 2 Insulator ε 1 Metal ε 2 The dispersion of the 50-nm-thick sample lies completely to the left of the decoupled SP mode. Low-energy asymptotic behavior follows a light line of n = 1.5. It suggests that polariton modes of MIM more highly sample the imaginary dielectric component. In the low energy limit, the S b SP truly represent a photon trapped on the metal surface.
a b SP : D = 50 nm 30 nm 25 nm 12 nm Purely plasmonic nature of the mode The cutoff frequencies remains essentially unchanged, possibly by the Goos-Hanchen effect. As waveguide dimensions are decreased, energy densities are more highly concentrated at the metal surface. This enhanced field magnifies Goss-Hanchen contributions significantly. In the limit of d << s (skin depth), complete SP dephasing could result.
MIM (Ag/SiO 2 /Ag) TM-polarized propagation and skin depth ( D = 250 nm ) Forbidden band a b s b a b s b Note that only a slight relation correlation between propagation distance and skin depth (σ). The metal absorption is not the limiting loss mechanism in MIM structures.
MIM (Ag/SiO 2 /Ag) TM-polarized propagation and skin depth ( D = 12 nm, 20 nm, 35 nm, 50 nm, and 100 nm ) a b s b σ ~ 20 nm Approximately constant in the Ag cladding. Thus, MIM can achieve micron-scale propagation with nanometer-scale confinement. Evanescent within 10 nm for all wavelength Local minima corresponding to the transition between quasibound modes and radiation modes Unlike IMI, extinction (prop. distance) is determined not by ohmic loss (metal absorption) but by field interference upon phase shifts induced by the metal.
TE modes in MIM structures (~ 4 ev: ~300 nm)
EM energy density profiles of MIM structures (Ag/SiO2/Ag) d = 250 nm d = 100 nm
Geometries and materials for subwavelength surface plasmon modes Rashid Zia, Mark D. Selker, Peter B. Catrysse, and Mark L. Brongersma J. Opt. Soc. Am. A/Vol. 21, No. 12/December 2004, 2442-2446. We demonstrate that, to achieve subwavelength pitches, a metal insulator metal geometry is required with higher confinement factors and smaller spatial extent than conventional insulator metal insulator structures. The resulting trade-off between propagation and confinement for surface plasmons is discussed, and optimization by materials selection is described.
Consider the isotropic wave equation for a generic three-layer plasmonic slab waveguide with metallic and dielectric regions, where z is the propagation direction and thus k z is the conserved quantity. For a guided surface-plasmon mode to exist, If the radiation is unconfined in the y dimension (i.e., k y = 0),
Ultimate confinement of the IMI structure is limited by the decay length into the dielectric cladding. For confinement below the limit of a conventional dielectric waveguide (λ/2n), 2π x (1/k x,dielectric ) < (λ/2n) Note that this condition is met only near the surface plasmon resonance frequency. Confinement of the MIM structure is limited by the decay length into the metallic regions, which can be approximated as follows for metals below the surface-plasmon resonance: k xmetal, 1/2 2 ω ε metal ω = > ε d c εmetal + εd c ( ) 1/2
Power confinement factor (Γ) of field-symmetric TM modes - MIM and IMI plasmonic waveguides - (Au air, λ = 1.55 μm) 99.4% 2% If plasmonic waveguides are intended to propagate light in subwavelength modes, MIM geometries with higher confinement factors and shorter spatial extents are much better suited for this purpose.
Appendix
Dispersion relation and Attenuation damping of surface plasmons at Ag/air Ag/glass interface 2 2 2 2 ωτ ' " p ωτ p εm = εm + iεm = εb + i 2 2 3 3 1+ ω τ ωτ + ω τ