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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- 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- surfaces -definition of scattering coefficients for the quasi-CW -dual wave picture microscopic model

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=0° 0.9511.051.11.15 0 0.1 0.2 a=0.68 µm a=0.94 µm a=2.92 µm Transmittance /a RCWA SPP model What is due to SPP in the EOT

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Dual wave picture x -m exp(ik 0 x ) =exp(ik SP x ) = 940 nm 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. 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. P. Lalanne et J.P. Hugonin, Nature Phys. 2, 556 (2006).

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Dual wave picture x -m exp(ik 0 x ) =exp(ik SP x ) = 940 nm 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. 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. P. Lalanne et J.P. Hugonin, Nature Phys. 2, 556 (2006).

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Dual wave picture =exp(ik SP x ) = 940 nm 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. 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. P. Lalanne et J.P. Hugonin, Nature Phys. 2, 556 (2006). quasi- Why calling it a quasi-CW?

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Young’s slit experiment laser F L1 L2PBS SCCD λ/2 titanium glass Au N. Kuzmin et al., Opt. Lett. 32, 445 (2007).

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Young’s slit experiment 010203040 010203040 (°) in air =850 nm N. Kuzmin et al., Opt. Lett. 32, 445 (2007). S. Ravets et al., JOSA B 26, B28 (2009).

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

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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- 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- surfaces -definition of scattering coefficients for the quasi-CW -dual wave picture microscopic model

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H inc E s, H s = scattered field E actual field E s = j µ 0 H s H s = -j (r)E s –j E 1/Hypothesis : the sub- indentation can be replaced by an effective dipole p=p x x+p y y. When is it reliable? Polarizability tensor p x =( xx E x,inc + xz E z,inc )x p z =( zx E x,inc + zz E z,inc ) z

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H inc E s, H s = scattered field E actual field E s = j µ 0 H s H s = -j (r)E s –j E 1/Hypothesis : the sub- indentation can be replaced by an effective dipole p=p x x+p y y. 2/The effective dipoles p x and p y are unknown. They probably depend on many parameters especially for sub- indentation that are not much smaller than (such as resonant grooves)

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H inc E s, H s = scattered field E actual field E s = j µ 0 H s H s = -j (r)E s –j E 1/Hypothesis : the sub- indentation can be replaced by an effective dipole p=p x x+p y y. 2/The effective dipoles p x and p y are unknown. 3/ We solve Maxwell's equation for both dipole source E = j µ 0 H H = -j (r)E +(E s,x x+ E s,y y) (x,y)

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The bad scenario

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

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H inc E s, H s = scattered field E actual field E s = j µ 0 H s H s = -j (r)E s –j E 1/Hypothesis : the sub- indentation can be replaced by an effective dipole p=p x x+p y y. 2/The effective dipoles p x and p y are unknown. 3/ The field scattered (in the vicinity of the surface for a given frequency) has always the same shape : (x,y)= [ SP E inc ] SP (x,y) + [ CW E inc ] CW (x,y)

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Cauchy theorem H = H SP + H CW H SP = H CW = Integral over a single real variable A single pole singularity : SPP contribution Two Branch-cut singularities : m and d Dominated by the branch-point singularity Analytical expression for the quasi-CW

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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

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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- metallic surfaces", Surf. Sci. Rep. (review article under proof corrections, see ArXiv too)

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Analytical expression for the quasi-CW Maxwell's equations E = j µ 0 H H = -j (r)E +(E s,x x+ E s,y y) (x,y) Analytical solution H z,CW [ m / m E s,x + n d / S E s,y ] E x,CW E y,CW S is either d or m, whether the Dirac source is located in the dielectric material or in the metallic medium Under the Hypothesis that | m | >> d z < x > /2

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HyHy ExEx EzEz x/λ 0102030 z x quasi-CW for gold at =940 nm dominant 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!! 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!!

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

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linear z-dependence for z < Main fields are almost null for z ( /2 m nearly an exp(ik 0 x) x-dependence for x >> H y,CW H y,0 E x,CW = F(x) E x,0 E z,CW E z,0 Grazing plane-wave field z x H y,CW PL et al., Surf. Sci. Rep. (review article under production, 2009)

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10 2 10 0 10 -2 10 0 10 -4 10 -6 |F(x)| (a.u.) x/ 1/ x 3 1/ x silver @ =1 µm Closed-form expression for F(x) PL et al., Surf. Sci. Rep. (review article under production, 2009)

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Closed-form expression for F(x) Highly accurate form for any x F(x) = exp(ik 0 x) W[2 n SP -n d x/ ] (x/ ) 3/2 with W(t) an Erf like function Highly accurate for x < 10 F(x) = exp(ik 0 x) (x/ ) -m m varies from 0.9 in the visible to 0.5 in the far IR m=0.83 for silver @ 852 nm

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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- 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- surfaces -definition of scattering coefficients for the quasi-CW -dual wave picture microscopic model

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Scaling law (result for silver) 10 -1 10 -2 10 0 10 -3 10 1 10 2 10 0 10 1 10 2 10 0 =0.633 µm =1 µm =9 µm =3 µm x/ |H| (a.u.) x/ 10 -1 10 -2 10 0 10 -3 H SP H CW (x/ ) -1/2 ( PC) PL and J.P. Hugonin, Nature Phys. 2, 556 (2006)

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Scaling law (result for silver) 10 -1 10 -2 10 0 10 -3 10 1 10 2 10 0 10 1 10 2 10 0 =0.633 µm =1 µm =9 µm =3 µm x/ |H| (a.u.) x/ 10 -1 10 -2 10 0 10 -3 H SP H CW (x/ ) -1/2 ( PC) PL and J.P. Hugonin, Nature Phys. 2, 556 (2006) m 2nd32nd3

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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- 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- surfaces -definition of scattering coefficients for the quasi-CW -dual wave picture microscopic model

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d |S| 2 |S 0 | 2 |S/S 0 | 2 d (µm) Slit-Groove experiment G. Gay et al. Nature Phys. 2, 262 (2006) 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 promote an other model than SPP & quasi-CW (CDEW model)

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

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E // EE Field recorded on the surface =974 nm gold

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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 cylindrical wave left-traveling cylindrical wave fitted parameter A SP (real) A c (complex) (m=0.5) slit

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

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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?

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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- 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- surfaces -definition of scattering coefficients for the quasi-CW -dual wave picture microscopic model

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SPP CW Defining scattering coefficients for CWs 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. ? It takes almost one year to solve that problem. The SPP is a normal mode

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CW-to-SPP cross-conversion CW SPP tctc CW SPP rcrc

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Other scattering coefficients CWSPP CW = SP CW = SP CW CW SP SP X. Yang et al., Phys. Rev. Lett. 102, 153903 (2009)

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Cross-conversion scattering coefficients CWSPP Ansatz : THE SCATTERED FIELDS ARE IDENTICAL. (if the two waves are normalized so that they have identical amplitudes at the slit) 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 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 X. Yang et al., Phys. Rev. Lett. 102, 153903 (2009)

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Scaling law for r c and t c -0.4 -0.2 0 0.2 05105 0 -0.4 -0.2 0 0.2 10 0 1 -4 10 -2 10 (µm) |m|1|m|1 Im(r c ) Re(r c ) Im(t c ) Re(t c ) tctc (µm) |r c | 2 & |t c | 2 |t c | 2 |r c | 2 rcrc

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Other scattering coefficients CWSPP CW = SP CW = SP CW CW SP SP X. Yang et al., Phys. Rev. Lett. 102, 153903 (2009)

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Mixed SPP-CW model for the extraordinary optical transmission pure SPP model mixed SPP-CW model t A = t + 2 (P -1 +1) ( + ) r A = 22 (P -1 +1) ( + ) P = 1/(u -1 - 1) + n=1, H CW (na) SPP CW H. Liu et al. (submitted) t A = t + 2 u -1 ( + ) rA =rA = 2222 u -1 ( + )

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=0° 0.9511.051.11.15 0 0.1 0.2 a=0.68 µm a=0.94 µm a=2.92 µm Transmittance /a RCWA SPP model Pure SPP-model prediction of the EOT H. Liu & P. Lalanne, Nature 452, 448 (2008).

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=0° 0.9511.051.11.15 0 0.1 0.2 0.3 a=0.68 µm a=0.94 µm a=2.92 µm Transmittance /a RCWA SPP model SPP-CW model Mixed SPP-CW model for the extraordinary optical transmission

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RCWACW-SPP model SPP CW a=900 nm

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SPP+quasi-CW coupled-mode equations AnAn BnBn A n-1 B n-1 It is not necessary to be periodic It is not necessary to deal with the same indentations

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Conclusion Two different microscopic waves, the SPP mode and the quasi- CW, are at the essence of the rich physics of metallic sub- surfaces Their relative weights strongly vary with the metal permittivity They echange their energy through a cross-conversion process, whose efficiency scales as | m | 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

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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) gold @ =800 nm Quasi-SW Cylindrical SPP Quasi-Spherical Wave Cylindrical SPP y/ E (normal) SPP, quasi-CW, C-SPP, quasi-SP

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