Asymmetric fluid-structure dynamics in nanoscale imprint lithography

R( λ)( 4πnd/ λ)2 2 −2αd−αdρ1,2+ ρ2,3e− 2ρ1,2ρ2,3ecos= [4.1]1−2 −2αd−αd( ρ ρ ) e + 2ρρ e cos( 4πnd/ λ)1,22,31,2where ρ, + 1are the reflectivity coefficients at the interface of thei ii −1andi interface, n index of refraction, d is the thickness of the film, and α isthe absorption coefficient of the film. Generally speaking, the index of refractionn varies with wavelength λ.Equation 4.1 was derived by assuming that the substrate exhibits perfectreflection and no absorption. It can be shown that equation 4.1 can be reduced tothe general form2,3( ) = A + B cos( knd )R λ 2[4.2]To better understand equation 4.2, take a simple optical thin filmcomposed of the template, air, and the substrate. Since the template is very thick,the light passing through it does not see the effect of the template. Figure 4.1illustrates this example with a thin film of air above the substrate. Incident light,with energy E I , is reflected off the template-gap interface with energy E R0 andpartially transmitted into the gap and this is in turn reflected off the substrate.The light reflected from the substrate is then partially reflected off and transmittedthrough the interface through the template. When the light E R0 and E R1 are inphase, they will add constructively. When the light E R0 and E R1 are out of phaseby they will add destructively.54