Surface-waves generated by nanoslits Philippe Lalanne Jean Paul Hugonin Jean Claude Rodier INSTITUT d'OPTIQUE, Palaiseau - France Acknowledgements : Lionel Aigouy, Béziers

Basic diffraction problem

Motivation : providing a microscopic description of the interaction between nanoslits

 = 750 nm 320-nm-thick Ag film Ebbesen et al., Nature 391, 667 (1998)  = /3 Motivation : ET

Motivation : beaming effect H. Lezec et al., Science 297, 820 (2002) Garcia-Vidal et al., APL 83, 4500 (2003) Gay et al., Appl. Phys. B 81, (2005) 20 µm calculation measurements

Outline Nature of the surface waves SPP? Try to answer basic questions

Outline Nature of the surface waves SPP? – other waves? Try to answer basic questions

Outline Nature of the surface waves SPP? – other waves? Influence of the geometrical parameter Try to answer basic questions slit width w

Outline Nature of the surface waves SPP? – other waves? Influence of the geometrical parameter Influence of the metal dielectric property Try to answer basic questions visible or IR illumination silver or gold

Outline Nature of the surface waves SPP? – other waves? Influence of the geometrical parameter Influence of the metal dielectric property Experimental validation Try to answer basic questions slit groove experiment d Young’s experiment d

SPP generation n2n2 n1n1

 + (x)  - (x) r0r0 SPP generation n2n2 n1n1

n2n2 n1n1  + (x)  - (x) t0t0 SPP generation

S =  + [t 0  exp(2ik 0 n eff h)] / [1-r 0 exp(2ik 0 n eff h)] Easy generalization SS

(a) (b) General theoretical formalism 1) Calculate the transverse (E z,H y ) near-field 2) make use of the completeness theorem for the normal modes of waveguides H y = E z = 3) Use orthogonality of normal modes P. Lalanne, J.P. Hugonin and J.C. Rodier, PRL 95, (2005)

Analytical model -The SP excitation probability |   | 2 scales as |  | -1/2 -Immersing the sample in a dielectric enhances the SP excitation (  n 2 /n 1 ) n2n2 n1n1  + (x)  - (x) 1) assumption : the near-field distribution in the immediate vicinity of the slit is weakly dependent on the dielectric properties 2) Calculate this field for the PC case 3) Use orthogonality of normal modes describe material properties P. Lalanne, J.P. Hugonin and J.C. Rodier, JOSAA 23, 1608 (2006)

n2n2 n1n1  + (x)  - (x) 1) assumption : the near-field distribution in the immediate vicinity of the slit is weakly dependent on the dielectric properties 2) Calculate this field for the PC case 3) Use orthogonality of normal modes describe material properties describe geometrical properties -A universal dependence of the SPP excitation that peaks at a value w= For w=0.23 and for visible frequency, |   | 2 can reach 0.5, which means that of the power coupled out of the slit half goes into heat Analytical model

total SP excitation probability Results obtained for gold Total SP excitation efficiency model : solid curves vectorial theory : marks

SPP? - other waves?

z H(x,x’,z=0) x’=0 Green function (1D) x H = H SP + H c H SP = H c = Integral over a single real variable

z Green function (1D) =0.633 µm =1 µm =9 µm =3 µm x/ |H| (a.u.) x/ (result for silver) H SP H c (x/ ) -1/2 P. Lalanne and J.P. Hugonin, Nature Phys. 2, 556 (2006)

w=100 nm w=352 nm 0 1 =0.852 µm =3 µm plane wave illumination ( ) x/ P. Lalanne and J.P. Hugonin, Nature Phys. 2, 556 (2006)

d S  = 852 nm S0S0 d/ |S/S 0 | 2  m  m  m  m  m PC ….. SPP only computational results (d/   P. Lalanne and J.P. Hugonin, Nature Phys. 2, 556 (2006)

w=100 nm 0 1 =0.852 µm plane wave illumination ( ) w=785 nm =0.852 µm w=1 µm 0 1 =0.852 µm x/

Outline Nature of the surface waves SPP? – other waves? Influence of the geometrical parameter Influence of the metal dielectric property Experimental validation Try to answer basic questions slit groove experiment d Young’s experiment d

Validation : Young’s slit experiment H.F. Schouten et al., PRL. 94, (2005). d gold glass TM incident light d=4.9 µm d=9.9 µm d=14.8 µm d=19.8 µm

P. Lalanne, J.P. Hugonin and J.C. Rodier, PRL 95, (2005) S = |t 0 +  -  + exp[ik SP d)]| 2 S ++  - d gold *** semi-analytical model o o o Schouten’s experiment numerical results d=4.9µm d=9.9µm S S Validation : Young’s slit experiment

d |S| 2 |S 0 | 2 |S/S 0 | 2 d (µm) Slit-Groove experiment G. Gay et al. Nature Phys. 2, 262 (2006) promote an other model than SPP d Fall off for d < 5 frequency = 1.05 k 0 k SP =k 0 [1-1/(2  Ag )]  1.01k 0 silver =852 nm

v =  + ru exp(ik SP d) u =  +  b exp(ikn eff h)+ rv exp(ik SP d) b = r m a exp(ikn eff h) a = t 0 +  v exp(ik SP d) + r 0 b exp(ikn eff h) S = t 0 +  u exp(ik SP d) S = t 0 +  exp(ik SP d) +   r m exp(ik SP d)exp(2ikn eff h) [ (t 0 +  exp(ik SP d)) / (1-r m r 0 exp(2ikn eff h))] S uv a b d h

|S/S 0 | 2 d (µm) d S  = 852 nm S0S0 computational results o o o experiment ….. SPP theory SPP theory and computational results are in perfect agreement P. Lalanne and J.P. Hugonin, Nature Phys. 2, 556 (2006)

d S  = 852 nm S0S0 d/ |S/S 0 | 2  m  m  m  m  m PC ….. SPP only computational results (d/   P. Lalanne and J.P. Hugonin, Nature Phys. 2, 556 (2006)

Lionel Aigouy, Laboratoire ‘Spectroscopie en Lumière Polarisée’ ESPCI, Paris 2 µm Gwénaelle Julié, Véronique Mathet Institut d’Electronique Fondamentale, Orsay, France Near field validation gold TM incident light slit

E z = A SP sin(k SP x) + A c [ik 0 -m/(x+d)] - A c [ik 0 +m/(x-d)] standing SPP right-traveling creeping wave left-traveling creeping wave fitted parameter A SP (real) A c (complex) (m=0.5) slit

Real partImaginary part Real partImaginary part distance extracted from fit computational results total field creeping wave ONLY Field at a single aperture

Conclusion The surface wave is a combination of SPP [exp(ik SP x)] and a creeping wave with a free space character [exp(ik 0 x)]/x 1/2 SPP is predominant at optical frequency for noble metal The creeping wave is dominant for >1.5 µm and for noble metals The SPP generation probability can be surprisingly high for subwavelength slits (  50%) at optical wavelengths The probability scales as |  | -1/2 The probability is enhanced when immersing the sample Experimental validation is difficult but on a good track