Long wavelength Near Field Microscopy
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TeraHertz Gap Wavelength Wavelength 3000 m m 3000 m m - 15 m m 15 m m Frequency Frequency 10 10 11 11 -10 -10 13 13 Hz Hz Energy Energy E E 0.4meV-80meV 0.4meV-80meV Evanescentbeam generated by the nanocollettor Object surface Evanescentbeam generated the object surface Propagating beam reemitted by the nanocollettor Propagating w ave Incident lightbeam Nanocollettor O ne wavelength Near Field Microscopy: Nearfield m icroscopy isbased on thedetection of evanescentfields. evanescentfields. They are confined on the object surfaceand related with its nanostructure. Fortheir non non propagating propagating nature nature , it’snot possibleto detectthem faraw ay from the sam ple (i.e.farfield region). It’snecessary to put very nearto theobject’ssurface (i.e. near field region )a sm allscattering element,the nanocollettor nanocollettor wich is able to capture these fieldsand convertthem into propagating ones. The field detected preservesthe informationsabouttheobject subwavelength structure. The existing nearfield m icroscopy techniquesw ork in the V isible-Infrared region. The generalschem e ofsuch microscopesisillustrated above:the sam ple can be illum inated by a laserin adirectw ay orthrough a m etallicor dielectric tip;the evanescentfield so generated justoverthe sam ple is collected by the tip. Thanksto a scanning system the tip ism oved along the objectsurface and m aintained ata certain distance from the sam ple (typically about10 nm ). The finalim age’sresolution isdictated by thetip’s dimensions. Scanning driver D istance m odule control laser object tip Piezo tube Optical fibre photom ultiplier 1 4 3 2 Transm ission in collection m ode Internalreflection in collection m ode Externalreflection in collection m ode Transm ission in illum ination m ode 1 2 3 4 On theleftthe evanescent field iscaptured by am etallic collector;the nearfield isprojected onto thenorm almodesofthistapered waveguide and so converted in a propagating field. The resolution isdictated by hole’sdim ensions. Thisconfiguration hasbeen adopted formillim etric wavelenght. On theright theevanescentfield isproduced by am etallic tip working in thecut-offregion. Asin the Far-Infrared m icroscopetechnique, the resolution isgiven by tip’sdim ensions. laser waveguide Subwavelength hole Hole evanescent field Object evanescent field collector Object far field laser Object evanescent field tip collector Long wavelength Near Field Microscopy In 1942 the physicist H.A.Bethe H.A.Bethe solved with am athem aticaltheory, the problem ofdiffraction from aholewhich dim ensionsare negligible respectto the incident radiation wavelength. He wanted to estim atetheeffectsofa sm allholem ade on a com mon conductivewalloftwo coupled waveguides. He supposed thatin orderto preserveboundary conditionson thehole and on theconductive plane, fictitiouselectric and m agnetic dipoles haveto existinstead ofthe hole.From thispointofview thefield diffused by the aperture isequivalentto the field irradiated by these dipoles;forthisreason they are called equivalentdipoles equivalentdipoles . ? ? ? ? d ? ? Subwavelength aperture: Bethe’s theory Theprinciplesstatesthatthefield irradiated on the side2 ofthe waveguidesby asource situated on the side1 isequivalentto thatgenerated by an electric (transversalcoupling)or m agnetic (longitudinalcoupling) current on the aperture.The equivalent currentsare 0 H n J e 0 E n J m Z Transversalcoupling 1 2 Source n Longitudinalcoupling Ourconfiguration Z Source 1 2 n W here and arethe electricand m agneticfieldsproduced by thesourcesituated in region 1 when the apertureisclosed by a perfectconductivewall. 0 E 0 H Equivalent principles In awaveguidean arbitrary electrom agnetic field can bewritten asanorm almodesguide serieswith index through the expansion coefficientsA Let’srem em bera Poynting theorem ’sform The integration overtheguide’sw allsiszero;that overthe lateral w allsgivesan expression forthe coefficientsA is thewaveguideimpedance E E AE H H AH S + S - _ n J(x,t) V + 3 2 V Z A J E dx c 3 ( ) S V n E H E H da J E dx Ifon the guide’swalls, between S + and S - , there’san aperture, (longitudinal coupling), wehaveto add thisterm in the rightsideoftheexpression forA Ifwe suppose that hole’sdim ensionsare negligible respectto the wavelenght, we can expand in Taylorseriesthefield H ; unlikely from w hat happenswhen thefield E isexpanded, in thiscase thezero orderterm is responsible ofam agnetic coupling and the firstorderterm ofan electric one. So both thedipole’sm omentshave the magnetic and electric interactionsexchanged. Using theexpression forJ m , w e finally can w ritethe expressionsfortheequivalentdipoles. 3 0 0 0 0 2 ( ) 3 P r nEn Holeradius 3 0 4 3 t M rH Trasversalfield H m hole H J da Equivalent electric and magnetic dipoles These expressionshave been derived in the nearfield lim it(r<< ).Let’s notice thatthey are exactly the sam e expressionsknown forthe fieldsdue to static static electric and m agnetic dipoles. So ifwe calculate the Poynting vectorofthese fieldswe obtain exactly zero because they don’tcarry outenergy don’tcarry outenergy ;they only have a stored energy stored energy ( E 2 , B 2 ). Electricand m agneticfieldsproduced by electricdipole;thefieldsproduced by magnetic dipole areobtained substituting E , B, P with B, -E, M . 2 0 0 0 4 ) ) , ( ( ) , , ( r c P n ik r B 3 0 4 1 )) , ( 3 ( ) , ( ) , , ( r P n P n r E Y X Z P M r n Dipoles’ electromagnetic field: stored energy Stored energy and electromagnetic field produced by both the dipole Z (mm) X (mm) W e calculatethe electrom agnetic fieldsin a tapered waveguidesby solving a system oftwo coupled differentialequationsw hich describe the localreflection ofthew aveguidem ode as the diam eterchanges gradually. The m odelcan be applied both in the cut-offand in the propagating region. z H b d d i K 1 z () H b 1 az () z az ()H b d d 1 2K 1 z () z K 1 z () H b H f d d z H f d d iK 1 z () H f 1 az () z az ()H f d d 1 2K 1 z () z K 1 z () H f H b d d Propagationterm Decrease (increase) ofm agneticfield in alarger(narrower) structure M utual reflectivecoupling offorward and backw ardw aves Electromagnetic fields in tapered metallic waveguide TeraH ertz configuration:the m etallic tip isused in the cut-offregion in orderto produce an evanescentfiled (ploton the left). M illim etric configuration:the collectorw orksin the propagating region;in these calculationsdielectric lossesare notincluded (ploton the right). forw ard backw ard field W avelenght 700 700 m Tip radius 70 70 m m \ 10 10 Tip fullangle 20 degree 20 degree Tip length 5.5 m m 5.5 m m Coupling coefficientbetw een near field and w aveguide m odesversus m odesindex. Collector radius 1.05 m m 1.05 m m Taperfullangle 10 degree 10 degree Collectorlength 4.5 cm 4.5 cm W avelenght 2.6 m m 2.6 m m 0 2 4 6 8 0 0.5 1 m 40 140 240 340 440 540 640 1 10 8 1 10 7 1 10 6 1 10 5 1 10 4 1 10 3 0.01 0.1 1 10 100 Results Distance from the waveguide horizontal plane 0.22mm 0.22mm References References H.A.Bethe, Theory of diffraction by small holes Phy. Rev. Vol. 66, pp 163-182, 1944 Collin, Foundation for microwave engennering. S.A.Shelkunoff, Field equivalence Theorems, Comm. On Pure and Appl. Math, vol.4 pp. 43-59, 1951 Electromagnetic Field in the cutoff regime of tapered metallic waveguides. B.Knoll,F.Keilmann. Opt. Comm. 162 Physical principles, mathematical treatment and Physical principles, mathematical treatment and realization of a new near field microscope in the realization of a new near field microscope in the THz region THz region (10 (10 11 11 -10 -10 13 13 Hz) Hz) D.Coniglio, A.Doria ENEA, FIS-ACC via Enrico Fermi 45 D.Coniglio, A.Doria ENEA, FIS-ACC via Enrico Fermi 45 , , 00044 00044 Frascati (Rome) ITALY Frascati (Rome) ITALY Visible-Infrared near field microscopy Waveguide with a localised source Y (mm) Z (mm) Wavelenght 2.6mm 2.6mm Hole radius 0.13mm 0.13mm \ \ 20 20 Waveguide dimensions 1mm-4mm 1mm-4mm Waveguide fundamental mode TE 10 TE 10 d << d << physical principles The detection of evanescent fields makes possible beating the Abbe Abbe diffraction limit diffraction limit , which states that in optics it’s impossible to get a subwaveleng subwaveleng th th resolution. resolution. This detection is based on the optical optical frustration principle frustration principle : onto a prism the total internal reflection can be avoided if a second prism is brought very near to the first one.This phenomenon is called optical tunnel effect optical tunnel effect : on the surface of the first prism exists an evanescent field; if a suitable dielectric material is immersed in it, this one will be converted into a propagating field in order to respect the continuity conditions at the interface. A general structure of such non-propagating fields is this one Propagation term Time dependence Uxy x t ( ) Axy z ( )e jk x x k y y e z e j t Amplitude field Evanescent term Fields Energy
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principles. physical. Amplitude field. Evanescent term. Distance from the waveguide horizontal plane 0.22mm. The detection of evanescent fields. makes possible beating the. Abbe. diffraction limit. , which states that in. diffraction limit. optics it’s impossible to get a. subwaveleng. - PowerPoint PPT Presentation
Transcript of Long wavelength Near Field Microscopy
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Long wavelength Near Field MicroscopySubwavelength aperture: Bethes theoryEquivalent principlesEquivalent electric and magnetic dipolesDipoles electromagnetic field: stored energyStored energy and electromagnetic field produced by both the dipole Electromagnetic fields in tapered metallic waveguideResultsReferences
H.A.Bethe, Theory of diffraction by small holes Phy. Rev. Vol. 66, pp 163-182, 1944
Collin, Foundation for microwave engennering.
S.A.Shelkunoff, Field equivalence Theorems, Comm. On Pure and Appl. Math, vol.4 pp. 43-59, 1951
Electromagnetic Field in the cutoff regime of tapered metallic waveguides. B.Knoll,F.Keilmann. Opt. Comm. 162
Physical principles, mathematical treatment and realization of a new near field microscope in the THz region (1011-1013 Hz) D.Coniglio, A.Doria ENEA, FIS-ACC via Enrico Fermi 45, 00044 Frascati (Rome) ITALY
Visible-Infrared near field microscopyWaveguide with a localised sourceWavelenght 2.6mmHole radius 0.13mm (l\20)Waveguide dimensions 1mm-4mmWaveguide fundamental modeTE 10FieldsEnergy
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TeraHertz Gap
Wavelength l 3000 mm-15 mm
Frequency n 1011-1013 Hz
Energy E 0.4meV-80meV
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Near Field Microscopy:
Near field microscopy is based on the detection of evanescent fields. They are confined on the object surface and related with its nanostructure. For their non propagating nature, its not possible to detect them far away from the sample (i.e. far field region). Its necessary to put very near to the objects surface (i.e.near field region) a small scattering element, the nanocollettor wich is able to capture these fields and convert them into propagating ones.The field detected preserves the informations about the object subwavelength structure.
Evanescent beamgenerated by the nanocollettor
Object surface
Evanescent beam generated the object surface
Propagating beam reemitted by the nanocollettor
Propagating wave
Incident light beam
Nanocollettor
One wavelength
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The existing near field microscopy techniques work in the Visible-Infrared region. The general scheme of such microscopes is illustrated above: the sample can be illuminated by a laser in a direct way or through a metallic or dielectric tip; the evanescent field so generated just over the sample is collected by the tip. Thanks to a scanning system the tip is moved along the object surface and maintained at a certain distance from the sample (typically about 10 nm). The final images resolution is dictated by the tips dimensions.
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On the left the evanescent field is captured by a metallic collector; the near field is projected onto the normal modes of this tapered waveguide and so converted in a propagating field. The resolution is dictated by holes dimensions. This configuration has been adopted for millimetric wavelenght.On the right the evanescent field is produced by a metallic tip working in the cut-off region. As in the Far-Infrared microscope technique, the resolution is given by tips dimensions.
laser
waveguide
Subwavelength hole
Hole evanescent field
collector
Object far field
laser
tip
collector
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In 1942 the physicist H.A.Bethe solved with a mathematical theory, the problem of diffraction from a hole which dimensions are negligible respect to the incident radiation wavelength.
He wanted to estimate the effects of a small hole made on a common conductive wall of two coupled waveguides.
He supposed that in order to preserve boundary conditions on the hole and on the conductive plane, fictitious electric and magnetic dipoles have to exist instead of the hole. From this point of view the field diffused by the aperture is equivalent to the field irradiated by these dipoles; for this reason they are called equivalent dipoles.
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The principles states that the field irradiated on the side 2 of the waveguides by a source situated on the side 1 is equivalent to that generated by an electric (transversal coupling) or magnetic (longitudinal coupling) current on the aperture. The equivalent currents are
Where and are the electric and magnetic fields produced by the source situated in region 1 when the aperture is closed by a perfect conductive wall.
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In a waveguide an arbitrary electromagnetic field can be written as a normal modes guide series with index l through the expansion coefficients Al
Lets remember a Poynting theorems form
The integration over the guides walls is zero; that over the lateral walls gives an expression for the coefficients Al ( Zl is the waveguide impedance)
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S-
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If on the guides walls, between S+ and S-, theres an aperture, (longitudinal coupling), we have to add this term in the right side of the expression for Al
If we suppose that holes dimensions are negligible respect to the wavelenght, we can expand in Taylor series the field H; unlikely from what happens when the field E is expanded, in this case the zero order term is responsible of a magnetic coupling and the first order term of an electric one. So both the dipoles moments have the magnetic and electric interactions exchanged. Using the expression for Jm , we finally can write the expressions for the equivalent dipoles.
Hole radius
Trasversal field H
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These expressions have been derived in the near field limit (r
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We calculate the electromagnetic fields in a tapered waveguides by solving a system of two coupled differential equations which describe the local reflection of the waveguide mode as the diameter changes gradually. The model can be applied both in the cut-off and in the propagating region.
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TeraHertz configuration:the metallic tip is used in the cut-off region in order to produce an evanescent filed (plot on the left).
Millimetric configuration: the collector works in the propagating region; in these calculations dielectric losses are not included (plot on the right).
forward backward field
Coupling coefficient between near field and waveguide modes versus modes index.
