Surface Plasmon Polaritons on Structured Surfaces. Alexei A. Maradudin and Tamara A. Leskova

Transcription

1 Surface Plasmon Polaritons on Structured Surfaces Alexei A. Maradudin and Tamara A. Leskova Department of Physics and Astronomy and Institute for Surface and Interface Science, University of California, Irvine, California, 9697, U.S.A.

2 Surface Plasmon Polaritons on Structured Surfaces Negative Refraction. H. Shin and S. Fan, Phys. Rev. Lett. 96, (006). A. A. Maradudin and T. A. Leskova, Physica B. 405, 97 (00). metal: ɛ (ω) dielectric metal: ɛ (ω)

3 Talbot Effect. M. R. Dennis, N. I. Zheludev, and F. J. García de Abajo, Opt. Express 5, 969 (007). A. A. Maradudin and T. A. Leskova, New J. Phys., (009).

4 Cloaking. I. I. Smolyaninov, Y. J. Hung, and C. C. Davis, A Opt. Lett. 33, 34 (008). r B r r 0 µm 4.µm C B. Baumeier, T. A. Leskova, and A. A. Maradudin, Phys. Rev. Lett. 03, (009) x [λ] x [λ] 0

5 Lasing. A. Tredicucci, C. Gmachl, F. Capasso, A. L. Hutchinson, D. L. Sivco, and A. Y. Cho, Appl. Phys. Lett. 76, 64 (000) R. Li, A. Banerjee, and H. Grebel, Opt. Express 7, 6 (009) Young s double-slit experiment. R. Zia and M. L. Brongersma, Nature Nanotechnology, 46 (007). F. Huerkamp, A. A. Maradudin, and T. A. Leskova, Proc. SPIE 7467, 7467OH (009). L (d )

6 Outline. Introduction. Transverse Localization of a Surface Plasmon Polariton 3. Wannier Stark Ladder for a Surface Plasmon Polariton 4. Conclusions

7 . Introduction Impedance boundary conditions ɛ (ω) x p (")! k (")! t ɛ (ω) x J E (x ω) i = K (0) ij (x ω)j H (x ω) j i, j =, J E (x ω) = ˆx 3 E(x ω) x3 =0 J H (x ω) = ˆx 3 H(x ω) x3 =0 E > (x ω) ( H > (x ω)) is the total electric (magnetic ) field in the vacuum region x 3 > 0. The only nonzero elements of the surface impedance tensor K (0) (x ω) are K (0) (x ω) = κ (ω) + [κ (ω) κ (ω)]s(x ) = K (0) (x ω), where S(x ) is the characteristic function of the region of the surface occupied by the metal whose dielectric function is ɛ (ω), and κ j (ω) = i/( ɛ j (ω)).

8 The total electric fields in the vacuum region where E > (x ω) = ê p (k ) exp[ik (ω) x β 0 (ω)x 3 ] + d { q (π) exp[iq A p (q ) x β 0 (q )x 3 ] ê p (q ) β 0 (q ) + i(ω/c)κ (ω) } A s (q ) +ê s (q ) (ω/c) iκ (ω)β 0 (q ) ê p (q ) = c ω [ iβ0 (q ) ˆq ˆx 3 q ], ê s (q ) = ˆx 3 ˆq k = k (ω)(cos θ, sin θ, 0) k (ω) = ω [ ] c ɛ (ω) [ ( ω ) ] β 0 (q ) = q, Reβ 0 (q ) > 0, Imβ 0 (q ) < 0 c β 0 (ω) = β 0 (k (ω)) = ω c ɛ (ω).

9 The equations satisfied by A p (q ) and A s (q ) are A p (p ) + i ω d { c [κ q A p (q ) (ω) κ (ω)] (π) Ŝ(p q ) (ˆp ˆq ) β 0 (q ) + i(ω/c)κ (ω) } c i(ˆp ˆq ) 3 ω β A s (q ) 0(q ) (ω/c) iκ (ω)β 0 (q ) = i ω c [κ (ω) κ (ω)](ˆp ˆk (ω))ŝ(p k (ω)) A s (p ) ω c [κ (ω) κ (ω)] d { q A p (q ) (π) Ŝ(p q ) (ˆp ˆq ) 3 β 0 (q ) + i(ω/c)κ (ω) } A s (q ) +i(ˆp ˆq ) c ω β 0(q ) (ω/c) iκ (ω)β 0 (q ) = ω c [κ (ω) κ (ω)](ˆp ˆk (ω)) 3 Ŝ(p k (ω)), where Ŝ(Q ) = d x S(x ) exp( iq x ). We solve these equations numerically and, where we can, analytically by the Wiener Hopf method.

10 In the region x > 0 beyond the surface defect the electric field of the transmitted surface plasmon polariton is given by E > (x ω) tr,spp = ê p (k ) exp[ik (ω) x β 0 (ω)x 3 ] + ω c exp[ β 0(ω)x 3 ] dq π ( ) A p (k q ), q ( ) (k q ) ê p (k q ), q exp [i(k q ) x ] + iq x.

11 . Transverse Localization of Surface Plasmon Polaritons x ε (ω) ε (ω) ε (ω) 3b + a/ + d 3 3b a/ + d 3 ε (ω) b + a / + d b a / + d x ε (ω) b + a / + d b a / + d a ε (ω) ε (ω) 3b + a / + d 3 3b a / + d 3 S(x ) = θ(x ) n= θ ( [ x (n + )b a ]) ([ + d n+ θ (n + )b + a ] ) + d n+ x, where {d m+ } are independent identically distributed random deviates drawn from a uniform distribution in the interval ( d/, d/), where d < b a/. Ŝ(Q ) = ( a ) i(q iη) asinc Q exp { iq [(n + )b + d n+ ]}. n=

12 Parameters of the system: silver, λ = 630 nm, ɛ (ω) =.59 + i.8, ɛ = 6.3, b = 00 nm, a = 30 nm, d = 55nm, w = λ.

13 A model system x x The only nonzero elements of the surface impedance tensor K (0) (x ω) are K (0) (x ω) = κ (ω) + i ω c S(x ) = K (0) (x ω), S(x ) = θ(x )ζ(x ) Ŝ(Q ) = i(q iη) ζ(q ), ζ(q ) = dx ζ(x ) exp( iq x ), where ζ(x ) is assumed to be a single-valued function of x stationary, zero-mean, Gaussian random process defined by ζ(x )ζ(x ) = δ W ( x x ) and to constitute a δ = ζ (x ), and the angle brackets denote an average over the ensemble of realizations of ζ(x ). For the surface height autocorrelation function W ( x ) we assume the Gaussian form W ( x ) = exp( x /a ).

14 In the model calculation we neglect the scattering into s polarized radiation. The p polarized component of the electric field in the vacuum has the form E > (x ω) p = ê p (k ) exp[ik (ω) x β 0 (ω)x 3 ] + d q (π) G 0(q )A p (q )ê p (q ) exp [ iq x β 0 (q )x 3 ], where we have introduced the Green s function of surface plasmon polaritons on a planar metal surface in the impedance approximation, β 0 (q ) + i(ω/c)κ (ω) = G 0(q ). Then the equation for A p (p ) can be rewritten in the form d q A p (p ) (π) V (p q )G 0 (q )A p (q ) = V (p k ), where V (p q ) = ( ω c ) i(p k iη) ζ(p q ). ζ(q ) = dx ζ(x ) exp( iq x ),

15 We use the standard procedure for a one-dimensional randomly rough surface. Introduce the exact Green function G(p k ) by G 0 (q )A p (q k )G 0 (k ) = G(q k ) G 0 (k )(π) δ(p q ). It satisfies the equation or identically G = G 0 + G 0 V G, G = G + G t G, where the averaged Green s function G satisfies the Dyson equation the operator t satisfies G = G 0 + G 0 M G t = (V M ) + ((V M ) G t, t = 0, and the self-energy operator M satisfies M = V + V G 0 (M M ). Note that all the averaging is one dimensional.

16 It is convenient to rewrite the expression for the electric field in the region x > 0 in terms of the exact Green s function, d E > q (x ω) p = (π) G(q )t(q k )ê p (q ) exp [ ] iq x β 0 (q )x 3, where G(p )(π) δ(p q ) = G(p q ), and the incident field is canceled by the δ function in the passage from G 0 A p G. Then E > (x ω) tr,spp = i ω c κ (ω) dq [ ] ) π exp iq x + i(p q ) x β 0 (q )x 3 ê p ( p q, q t ((p p q ), ) q k, q where p = p (ω) is the wavenumber of surface plasmon polaritons of frequency ω supported by the rough surface at x > 0, i.e. it is the pole of the function G(p ).

17 We employ the paraxial approximation (p q ) = p (q /p ) to calculate the field of the transmitted surface plasmon polariton in the far zone E > (x ω) p = i ω c κ (ω) exp ( ) ip x [ ] dq ( ) π exp iq x i q x ê p p p q, q t ((p p q ), ) q k, q To calculate the intensity of the transmitted beam we need to calculate t(q k )t (q k ) = τ(q, k q, k ), where τ is the reducible vertex function, then multiply the result by (k w/) π exp{ [k w/] θ } exp{ [k w/] (θ ) }, and integrate the product with respect to θ and θ in the interval π/ θ π/, π/ θ π/.

18 We use the results for τ obtained in: A. R. McGurn and A. A. Maradudin, Weak transverse localization of light scattered incoherently from a one-dimensional random metal surface, J. Opt. Soc. Am. B 0, 539 (993) In the case where a w, the intensity of the transmitted surface polaritons I(x, x ) = E(x ω) tr,spp breaks into two contributions [ I(x, x ) = I x ( + sp w /x ) l (ω) exp[ sp(x /x ] ) ( + sp w /x ) ( ) ( ) + x ( + 4w p /x ) I mc (ω) exp sp x + w p exp x sp + wp /x /x x + w p, /x where I l (ω) and I mc (ω) are independent of spatial variables, and sp = /l sc, l sc is the scattering length of surface plasmon polaritons on the rough surface.

19 3. Wannier Stark Ladder of a Surface Plasmon Polariton One-Dimensional Schrödinger Equation ] [ h d m dx + V p(x) + F x ψ(x) = Eψ(x). V p (x) is a periodic function of x with period a: ( ) V p (x + a) = V p (x). We replace x by x na, where n is an integer in the Schrödinger equation: ] [ h d m dx + V p(x) + F (x na) ψ(x na) = Eψ(x na). Define ψ(x) = ψ(x na). Then ] [ h d m dx + V p(x) + F x ψ(x) = (E + naf ) ψ(x). From these results we see that if ψ(x) is an eigenfunction of Eq. ( ) with eigenvalue E, then ψ(x na) is also an eigenfunction with an eigenvalue E + nf a. This set of eigenvalues separated in energy by the constant value E = F a is called a Wannier Stark ladder. Calculations show that there are discrete values of E for which Eq. ( ) has solutions, on which a Wannier Stark ladder can be built.

20 x ɛ (ω) ɛ ɛ (ω) ɛ ɛ (ω) ɛ ɛ (ω) ɛ ɛ (ω) ɛ z 0 z z z 3 z 4 x d 0 d d d 3 d 4 L 0 L L L 3 z n = z 0 η ln( nηl 0) L n = z n+ z n = ( η ln ηl ) 0 nηl 0 S(x ) = n m=0 θ(x z m )θ(z m + d m x ) Ŝ(Q ) = πδ(q ) n m=0 d m exp [ iq ( z m + d m )] ( ) d n sinc Q.

21 η=0.050 η=0.045 η=0.040 η=0 T(ω) ω [0 5 s - ] Parameters of the system: The first medium is silver, with ɛ (ω) = ωp /ω, λ p = πc/ω p = 57 nm. The second medium is silicon, with ɛ (ω) =. L 0 = 300 nm, d 0 = 00 nm.

22 ω [0 4 s - ] η [µm - ]

23 4. Conclusions The diffractive spreading of a surface plasmon polariton beam propagating through a parallel array of semi-infinite strips of width a formed from a metal whose dielectric function is ɛ (ω) deposited on the planar surface of a metal whose dielectric function is ɛ (ω), saturates when the separation between consecutive strips are randomized. This is the transverse localization of surface plasmon polaritons. The transmissivity as a function of the frequency of a surface plasmon polariton propagating on a suitably chirped grating displays equally spaced peaks. These peaks are the signature of a surface plasmon polariton Wannier Stark ladder.