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1.The emblematic example of the EOT -extraordinary optical transmission (EOT)

1.The emblematic example of the EOT -extraordinary optical transmission (EOT) -limitation of classical "macroscopic" grating theories -a microscopic pure-SPP model of the EOT 2.SPP generation by 1D sub- l indentation -rigorous calculation (orthogonality relationship)

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1.The emblematic example of the EOT -extraordinary optical transmission (EOT)

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  1. 1.The emblematic example of the EOT -extraordinary optical transmission (EOT) -limitation of classical "macroscopic" grating theories -a microscopic pure-SPP model of the EOT 2.SPP generation by 1D sub-l indentation -rigorous calculation (orthogonality relationship) -the important example of slit -scaling law with the wavelength 3.The quasi-cylindrical wave -the importance of the quasi-CW -definition & properties -scaling law with the wavelength 4.Microscopic theory of sub-l surfaces -definition of scattering coefficients for the quasi-CW -dual wave picture microscopic model

  2. What is due to SPP in the EOT RCWA SPP model a=0.68 µm q=0° a=0.94 µm 0.2 Transmittance a=2.92 µm 0.1 l/a 0 0.95 1 1.05 1.1 1.15

  3. Tableau : *mention that far away from the surface the wave is cylindrical with (1/r)1/2 dependence. *mention on the surface, it is different, and this is why we will call that a quasi-cylindrical wave. *Tableau : plot a hole, says that the SPP has a (1/r)1/2 damping with a exp(ikspr), and that the quasi cylindrical wave becomes a quasi-spherical wave. But this is another story. Dual wave picture l = 940 nm =exp(ikSPx)  x-m exp(ik0x) P. Lalanne et J.P. Hugonin, Nature Phys. 2, 556 (2006).

  4. Tableau : *mention that far away from the surface the wave is cylindrical with (1/r)1/2 dependence. *mention on the surface, it is different, and this is why we will call that a quasi-cylindrical wave. *Tableau : plot a hole, says that the SPP has a (1/r)1/2 damping with a exp(ikspr), and that the quasi cylindrical wave becomes a quasi-spherical wave. But this is another story. l = 940 nm Dual wave picture =exp(ikSPx)  x-m exp(ik0x) P. Lalanne et J.P. Hugonin, Nature Phys. 2, 556 (2006).

  5. l = 940 nm Dual wave picture Tableau : *mention that far away from the surface the wave is cylindrical with (1/r)1/2 dependence. *mention on the surface, it is different, and this is why we will call that a quasi-cylindrical wave. *Tableau : plot a hole, says that the SPP has a (1/r)1/2 damping with a exp(ikspr), and that the quasi cylindrical wave becomes a quasi-spherical wave. But this is another story. =exp(ikSPx) quasi- Why calling it a quasi-CW? P. Lalanne et J.P. Hugonin, Nature Phys. 2, 556 (2006).

  6. Young’s slit experiment L1 F PBS λ/2 L2 S CCD laser Au titanium glass N. Kuzmin et al., Opt. Lett. 32, 445 (2007).

  7. Young’s slit experiment q 0 10 20 30 40 l=850 nm 0 10 20 30 40 q (°) in air N. Kuzmin et al., Opt. Lett. 32, 445 (2007). S. Ravets et al., JOSA B 26, B28 (2009).

  8. Computational results l=0.6 µm SPP mainly l=1 µm Quasi-CW & SPP l=3 µm Far-field intensity (a.u.) l=10 µm Quasi-CW mainly Quasi-CW only PC q (°) S. Ravets et al., JOSA B 26, B28 (2009).

  9. 1.The emblematic example of the EOT -extraordinary optical transmission (EOT) -limitation of classical "macroscopic" grating theories -a microscopic pure-SPP model of the EOT 2.SPP generation by 1D sub-l indentation -rigorous calculation (orthogonality relationship) -the important example of slit -scaling law with the wavelength 3.The quasi-cylindrical wave -the importance of the quasi-CW -definition & properties -scaling law with the wavelength 4.Microscopic theory of sub-l surfaces -definition of scattering coefficients for the quasi-CW -dual wave picture microscopic model

  10. Es, Hs= scattered field E actual field Es = jwµ0Hs Hs = -jwe(r)Es –jwDeE Hinc 1/Hypothesis : the sub-l indentation can be replaced by an effective dipole p=pxx+pyy. When is it reliable? Polarizability tensor px=(axxEx,inc + axzEz,inc)x pz=(azxEx,inc+azzEz,inc)z

  11. Es, Hs= scattered field E actual field Es = jwµ0Hs Hs = -jwe(r)Es –jwDeE Hinc 1/Hypothesis : the sub-l indentation can be replaced by an effective dipole p=pxx+pyy. 2/The effective dipoles px and py are unknown. They probably depend on many parameters especially for sub-l indentation that are not much smaller than l (such as resonant grooves)

  12. Es, Hs= scattered field E actual field Es = jwµ0Hs Hs = -jwe(r)Es–jwDeE Hinc 1/Hypothesis : the sub-l indentation can be replaced by an effective dipole p=pxx+pyy. 2/The effective dipoles px and py are unknown. 3/ We solve Maxwell's equation for both dipole source E = jwµ0H H = -jwe(r)E +(Es,xx+ Es,yy) d(x,y)

  13. The bad scenario

  14. H (a.u.) x/l x/l l=800 nm, gold The actual scenario The two dipole sources approximately generate the same field

  15. Es, Hs= scattered field E actual field Es = jwµ0Hs Hs = -jwe(r)Es–jwDeE Hinc 1/Hypothesis : the sub-l indentation can be replaced by an effective dipole p=pxx+pyy. 2/The effective dipoles px and py are unknown. 3/ The field scattered (in the vicinity of the surface for a given frequency) has always the same shape : F(x,y)= [aSPEinc] FSP(x,y) + [aCWEinc] FCW(x,y)

  16. HSP = Analytical expression for the quasi-CW Cauchy theorem H = HSP + HCW HCW= Integral over a single real variable Dominated by the branch-point singularity A single pole singularity : SPP contribution Two Branch-cut singularities : Gm and Gd

  17. Maths have been initially developped for the transmission theory in wireless telegraphy with a Hertzian dipole radiating over the earth I. Zenneck, Propagation of plane electromagnetic waves along a plane conducting surface and its bearing on the theory of transmission in wireless telegraphy, Ann. Phys (1907) 23, 846-866. R.W.P. King and M.F. Brown, Lateral electromagnetic waves along plane boundaries: a summarizing approach, Proc. IEEE (1984) 72, 595-611. R. E. Collin, Hertzian dipole radiating over a lossy earth or sea: some early and late 20th-century controversies, IEEE Antennas Propag. Mag. (2004) 46, 64-79. Sommerfeld

  18. 0/H. J. Lezec and T. Thio, "Diffracted evanescent wave model for enhanced and suppressed optical transmission through subwavelength hole arrays", Opt. Exp. 12, 3629-41 (2004). 1/G. Gay, O. Alloschery, B. Viaris de Lesegno, C. O'Dwyer, J. Weiner, H. J. Lezec, "The optical response of nanostructured surfaces and the composite diffracted evanescent wave model", Nature Phys. 2, 262-267 (2006). 2/PL and J.P. Hugonin, Nature Phys. 2, 556 (2006). 3/B. Ung, Y.L. Sheng, "Optical surface waves over metallo-dielectric nanostructures", Opt. Express (2008) 16, 90739086. 4/Y. Ravel, Y.L. Sheng, "Rigorous formalism for the transient surface Plasmon polariton launched by subwavelength slit scattering", Opt. Express (2008) 16, 2190321913. 5/W. Dai & C. Soukoulis, "Theoretical analysis of the surface wave along a metal-dielectric interface", PRB accepted for publication (private communication). 6/L. Martin Moreno, F. Garcia-Vidal, SPP4 proceedings 2009. 7/PL, J.P. Hugonin, H. Liu and B. Wang, "A microscopic view of the electromagnetic properties of sub-l metallic surfaces", Surf. Sci. Rep. (review article under proof corrections, see ArXiv too)

  19. Analytical expression for the quasi-CW • Maxwell's equations • E = jwµ0H • H = -jwe(r)E +(Es,xx+ Es,yy) d(x,y) • Analytical solution • Hz,CW • [cm/emEs,x + nd/eSEs,y] Ex,CW • Ey,CW • eS is either ed or em, whether the Dirac source is located in the dielectric material or in the metallic medium • Under the Hypothesis that • |em| >> ed • z < l • x > l/2p

  20. quasi-CW for gold at l=940 nm Independant of the indentation, as long as it is subwavelength and can be considered as an effective dipole Independant of the incident illumination. You could illuminate by a plane wave at normal incidence, at oblique incidence, by a SPP, you always get this, just the complex amplitude is varying!! dominant Hy z Ex x dominant Ez 0 10 20 30 x/λ

  21. Intrinsic properties of quasi-CW Hy,CW z x • Hy,CW Hy,0 • Ex,CW = F(x)Ex,0 • Ez,CW Ez,0 • F(x) is a slowly-varying envelop • [Hy,0, Ex,0, Ez,0] is the normalized field associated to the limit case of the reflection of a plane-wave at grazing incidence • Hypothesis • z < l • x > l/2p PL et al., Surf. Sci. Rep. (review article under production, 2009)

  22. Grazing plane-wave field Hy,CW Hy,0 Ex,CW = F(x)Ex,0 Ez,CW Ez,0 Hy,CW z x • linear z-dependence for z < l • Main fields are almost null for z (l/2p)|em|1/2 • nearly an exp(ik0x) x-dependence for x >> l PL et al., Surf. Sci. Rep. (review article under production, 2009)

  23. 1/x 1/x3 Closed-form expression for F(x) 100 10-2 |F(x)| (a.u.) 10-4 10-6 silver @ l=1 µm 100 102 x/l PL et al., Surf. Sci. Rep. (review article under production, 2009)

  24. Closed-form expression for F(x) Highly accurate form for any x F(x) = exp(ik0x) W[2p(nSP-nd)x/l] (x/l)3/2with W(t) an Erf-like function Highly accurate for x < 10l F(x) = exp(ik0x) (x/l)-m m varies from 0.9 in the visible to 0.5 in the far IR m=0.83 for silver @ 852 nm

  25. 1.The emblematic example of the EOT -extraordinary optical transmission (EOT) -limitation of classical "macroscopic" grating theories -a microscopic pure-SPP model of the EOT 2.SPP generation by 1D sub-l indentation -rigorous calculation (orthogonality relationship) -the important example of slit -scaling law with the wavelength 3.The quasi-cylindrical wave -the importance of the quasi-CW -definition & properties -scaling law with the wavelength 4.Microscopic theory of sub-l surfaces -definition of scattering coefficients for the quasi-CW -dual wave picture microscopic model

  26. HSP HCW (x/l)-1/2 (PC) 100 10-1 |H| (a.u.) |H| (a.u.) 10-2 l=1 µm l=0.633 µm 10-3 x/l x/l 100 10-1 |H| (a.u.) |H| (a.u.) 10-2 l=3 µm l=9 µm 10-3 100 101 102 100 101 102 x/l x/l Scaling law (result for silver) PL and J.P. Hugonin, Nature Phys. 2, 556 (2006)

  27. HSP HCW (x/l)-1/2 (PC) 100 10-1 |H| (a.u.) |H| (a.u.) 10-2 l=1 µm l=0.633 µm 10-3 x/l x/l 100 10-1 |H| (a.u.) |H| (a.u.) 10-2 l=3 µm l=9 µm 10-3 |em| 100 101 102 100 101 102 x/l x/l 2pnd3 Scaling law (result for silver) PL and J.P. Hugonin, Nature Phys. 2, 556 (2006)

  28. 1.The emblematic example of the EOT -extraordinary optical transmission (EOT) -limitation of classical "macroscopic" grating theories -a microscopic pure-SPP model of the EOT 2.SPP generation by 1D sub-l indentation -rigorous calculation (orthogonality relationship) -the important example of slit -scaling law with the wavelength 3.The quasi-cylindrical wave -the importance of the quasi-CW -definition & properties -scaling law with the wavelength -experimental evidence 4.Microscopic theory of sub-l surfaces -definition of scattering coefficients for the quasi-CW -dual wave picture microscopic model

  29. d |S/S0|2 d (µm) Slit-Groove experiment Fall off for d < 5l silver l=852 nm d frequency = 1.05 k0 kSP=k0 [1-1/(2eAg)] 1.01k0 |S|2 |S0|2 promote an other model than SPP & quasi-CW (CDEW model) G. Gay et al. Nature Phys. 2, 262 (2006)

  30. 2 µm 2 µm Near field validation l=975 nm gold experiment G. JULIE, V. MATHET IEF, Orsay computation L. AIGOUY ESPCI, Paris slit slit TM

  31. Field recorded on the surface E E// gold l=974 nm

  32. slit slit Ez = ASP sin(kSPx) + Ac [ik0-m/(x+d)] - Ac [ik0+m/(x-d)] standing SPP right-traveling cylindrical wave left-traveling cylindrical wave fitted parameter ASP (real) Ac (complex) (m=0.5)

  33. total field SPP cylindrical wave z x L. Aigouy et al., PRL 98, 153902 (2007)

  34. A direct observation? If one controls the two beam intensity and phase (or the separation distance) so that there is no SPP generated on the right side, do I observe a pure quasi-CW on the right side of the doublet?

  35. 1.The emblematic example of the EOT -extraordinary optical transmission (EOT) -limitation of classical "macroscopic" grating theories -a microscopic pure-SPP model of the EOT 2.SPP generation by 1D sub-l indentation -rigorous calculation (orthogonality relationship) -the important example of slit -scaling law with the wavelength 3.The quasi-cylindrical wave -the importance of the quasi-CW -definition & properties -scaling law with the wavelength -experimental evidence 4.Microscopic theory of sub-l surfaces -definition of scattering coefficients for the quasi-CW -dual wave picture microscopic model

  36. CW Defining scattering coefficients for CWs The SPP is a normal mode It takes almost one year to solve that problem. SPP  elastic scattering coefficients like the SPP transmission may be easily defined. You may also define inelastic scattering coefficients with other modes like the radiated plane waves You may use mode orthogonality and reciprocity relationships. ?

  37. CW-to-SPP cross-conversion tc CW SPP SPP rc CW

  38. Other scattering coefficients bCW bSP CW SPP aCW aSP bCW = bSP aCW = aSP X. Yang et al., Phys. Rev. Lett. 102, 153903 (2009)

  39. Cross-conversion scattering coefficients Because the SPP mode and the CW have similar characteristics at the metal surface (nearly identical propagation constants, similar penetration depth in the metal …) The same causes produce the same effects. We refer here to a form of causality principle where equal causes have equal effects CW SPP Ansatz :THE SCATTERED FIELDS ARE IDENTICAL. (if the two waves are normalized so that they have identical amplitudes at the slit) X. Yang et al., Phys. Rev. Lett. 102, 153903 (2009)

  40. 0.2 0.2 |tc|2 |rc|2 0 0 -0.2 -0.2 -0.4 -0.4 0 5 10 0 5 10 Scaling law for rc and tc rc tc |rc|2 & |tc|2 -1 Im(rc) Im(tc) 10 Re(rc) Re(tc) -2 10 |em|-1 -4 10 0 1 10 10 l (µm) l (µm) l (µm)

  41. Other scattering coefficients bCW bSP CW SPP aCW aSP bCW = bSP aCW = aSP X. Yang et al., Phys. Rev. Lett. 102, 153903 (2009)

  42. 2a2 2ab rA= tA= t + u-1- (r+t) u-1- (r+t) Mixed SPP-CW model for the extraordinary optical transmission pure SPP model CW mixed SPP-CW model 2a2 2ab rA= tA= t + SPP (P-1+1)- (r+t) (P-1+1)- (r+t) P = 1/(u-1 - 1) + Sn=1, HCW(na) H. Liu et al. (submitted)

  43. Pure SPP-model prediction of the EOT RCWA SPP model a=0.68 µm q=0° a=0.94 µm 0.2 Transmittance a=2.92 µm 0.1 l/a 0 0.95 1 1.05 1.1 1.15 H. Liu & P. Lalanne, Nature 452, 448 (2008).

  44. RCWA SPP model SPP-CW model Mixed SPP-CW model for the extraordinary optical transmission 0.3 a=0.68 µm a=0.94 µm 0.2 Transmittance q=0° a=2.92 µm 0.1 0 l/a 0.95 1 1.05 1.1 1.15

  45. RCWA CW-SPP model a=900 nm CW SPP

  46. SPP+quasi-CW coupled-mode equations It is not necessary to be periodic It is not necessary to deal with the same indentations An-1 An Bn-1 Bn

  47. Conclusion Two different microscopic waves, the SPP mode and the quasi-CW, are at the essence of the rich physics of metallic sub-l surfaces Their relative weights strongly vary with the metal permittivity They echange their energy through a cross-conversion process, whose efficiency scales as |em|-1 The local SPP elastic or inelastic scattering coefficients are essential to understand the optical properties, since they apply to both the SPP and the quasi-CW

  48. Quasi-Spherical Wave Quasi-SW Cylindrical SPP Cylindrical SPP SPP, quasi-CW, C-SPP, quasi-SP ld=.8;k0=2*pi/ld;nh=1;nb=retindice(ld,2);ns=nb;S=[1,0,0,0,0,0]; st=3*ld;st1=1/10*ld;x=[linspace(-st,-st1,31) linspace(st1,st,31)];y=x;z=0; [e,e_oc,e_res]=retoc([nh,nb,ns],S,[0,0,0],x,y,z,k0,struct('z0_varie',0)); Ez=squeeze(e_oc(1,:,:,3)); figure(1);retcolor(x/ld,y/ld,real(Ez)),shading interp, axis equal % caxis(caxis/10); Ez=squeeze(e_res(1,:,:,3)); figure(2);retcolor(x/ld,y/ld,real(Ez)),shading interp,axis equal % caxis(caxis/10); figure(3); st=10*ld;st1=1/10*ld;x=linspace(st1,st,101);y=0;z=0; [e,e_oc,e_res]=retoc([nh,nb,ns],S,[0,0,0],x,y,z,k0,struct('z0_varie',0)); Ez1=squeeze(e_oc(1,:,:,3));Ez2=squeeze(e_res(1,:,:,3)); plot(x/ld,real(Ez1),'r','linewidth',3),hold on plot(x/ld,real(Ez2),'g --','linewidth',3) E (normal) y/l gold @ l=800 nm

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