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Physical principles, mathematical treatment and realization of a new near field microscope in the 11 13 THz region (10 -10 Hz) D.Coniglio, A.Doria ENEA, FIS-ACC via Enrico Fermi 45, 00044 Frascati (Rome) ITALY TeraHertz Gap Near Field Microscopy: Propagating beam reemitted by the nanocollettor Propagating wave Incident light beam Wavelength l 3000 mmmm 15 mm Frequency n 1011-1013 Hz Energy E 0.4meV-80meV Evanescent beam generated the object surface Nanocollettor Object surface Evanescent beam generated by the nanocollettor One wavelength 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 nature, it’s not possible to detect them far away from the sample (i.e. far field region). It’s necessary to put very near to the object’s 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. physical principles The detection of evanescent fields makes possible beating the Abbe diffraction limit, which states that in optics it’s impossible to get a subwavelength subwavelength resolution. This detection is based on the optical 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: 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. d << l A general structure of such non-propagating fields is this one U(x y x t) Subwavelength aperture: Bethe’s theory ( ) e j kxxkyy z jt A(x y z) e Amplitude field Long wavelength Near Field Microscopy Visible-Infrared near field microscopy e Evanescent term Propagation term Time dependence Equivalent principles Waveguide with a localised source ?? l d ?? ?? 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. n (E H He wanted to estimate the effects of a small hole made on a common conductive wall of two coupled waveguides. S 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. dipoles Equivalent electric and magnetic dipoles Hl J m da hole If we suppose that hole’s 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 dipole’s 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 rr 2 3 rr P r0 e0 (n0 E)n0 3 rr 4 3 M r0 Ht 3 Stored energy and electromagnetic field produced by both the dipole Z (mm) Fields r q n Z f M X Electromagnetic fields in tapered metallic waveguide d Hb dz d Hf dz ( 1 d 1 d iK1(z) Hb a (z) Hb K1 (z) Hb Hf a(z)d z 2 K1(z) d z ( ) ) 1 d 1 d a (z) Hf K1(z) Hf Hb iK1(z) Hf a(z)d z 2K1 (z)d z Propagation term Decrease (increase) of magnetic field in Mutual reflective coupling of forward and a larger (narrower) structure backward waves Energy P These expressions have been derived in the near field limit (r << l). Let’s notice that they are exactly the same expressions known for the fields due to static electric and magnetic dipoles. So if we calculate the Poynting vector of these fields we obtain exactly zero because they don’t carry out energy; energy they only have a stored energy (E2, B2). 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. V Y 1 E(J,f,r) [n(J,f)(3Pn(J,f))P] 3 4e0r Electric and magnetic fields produced by electric dipole; the fields produced by magnetic dipole are obtained substituting E, B, P with B, -E, M. El H )da J El d x 3 2 Z l 3 Al J El d x V c Y (mm) cm0e0 B(J,f , r) ik0 (n(J,f ) P) 2 4r l Dipoles’ electromagnetic field: stored energy If on the guide’s walls, between S+ and S-, there’s an aperture, (longitudinal coupling), we have to add this term in the right side of the expression for A l Distance from the waveguide horizontal plane 0.22mm X (mm) Z (mm) Wavelenght 2.6mm Hole radius 0.13mm (l\20) Waveguide dimensions 1mm-4mm Waveguide fundamental mode TE 10 Results 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 100 10 1 0.1 0.01 1103 1104 1105 1106 1107 1108 Wavelenght 700 m m Wavelenght 2.6 mm 1 Tip radius 70 m m (l\ (l\10) 10)0.5 Collector radius 1.05 mm Taper full angle 10 degree Tip full angle 20 degree 0 Tip length 5.5 mm 640 540 440 340 240 140 40 mm 0 2 4 6 8 Coupling coefficient between near field and waveguide modes versus modes index. Collector length 4.5 cm 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