Phase change of light reflected by a discontinuity in the derivatives ...
Phase change of light reflected by a discontinuity in the derivatives ...
Phase change of light reflected by a discontinuity in the derivatives ...
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R. Diamant, M. Fernández-Guasti / Optics Communications 294 (2013) 64–72 65where k 2 0 ¼ po2 m 0 E 0 and <strong>the</strong> refractive <strong>in</strong>dex is n ¼ E=E ffiffiffiffiffiffiffiffiffi0,E0 is <strong>the</strong> To model a s<strong>in</strong>gle <strong>in</strong>terface, n(z) must be a monotonic cont<strong>in</strong>uousA 2 d ¼ 1 The parameter D is a measure <strong>of</strong> <strong>the</strong> <strong>in</strong>terface thickness, correspond<strong>in</strong>g to 90% <strong>of</strong>n : ð12Þ <strong>the</strong> <strong>in</strong>dex <strong>change</strong>. The overall refractive <strong>in</strong>dex <strong>change</strong> is labeled Dn and <strong>the</strong> halfvalue<strong>by</strong> n avg. The figure shows a pr<strong>of</strong>ile with D ¼ 2l.electric permittivity <strong>of</strong> vacuum. Now, consider a complex E x ,namely E x ¼ Ae iq , where <strong>the</strong> amplitude A and phase q depend onz. We restrict <strong>the</strong> problem to non absorb<strong>in</strong>g media, so we willassume <strong>the</strong> refractive <strong>in</strong>dex to be a real quantity. Substitution <strong>in</strong>function evolv<strong>in</strong>g from n i to n t . Far from <strong>the</strong> <strong>in</strong>terface n should bealmost constant. To evaluate <strong>the</strong> reflectivity, <strong>the</strong> convenient <strong>in</strong>itialcondition is a s<strong>in</strong>gle transmitted wave through <strong>the</strong> second medium,so <strong>the</strong> <strong>in</strong>cident <strong>light</strong> is assumed to come only from <strong>the</strong> first mediumEq. (3) and separation <strong>of</strong> real and imag<strong>in</strong>ary parts render:side. This implies that <strong>the</strong> solution <strong>in</strong> <strong>the</strong>psecond medium, far fromd 2 AA dq 2<strong>the</strong> <strong>in</strong>terface, is almost constant A d ¼ffiffiffiffiffiffiffiffiffiffi1=n t.Under this condition,¼ k 2dz 20dzA, ð4Þ <strong>the</strong> amplitude oscillations <strong>in</strong> <strong>the</strong> first medium, far from <strong>the</strong> <strong>in</strong>terface,will exhibit <strong>the</strong> medium’s reflectivity: R ¼ r 2 . Given <strong>the</strong> <strong>in</strong>dices n i2 dA and n t , we expect <strong>the</strong> reflectivity to depend on <strong>the</strong> <strong>in</strong>terfacedqþA d2 qdz dz dz ¼ 0:ð5Þ abruptness. This property can be characterized <strong>by</strong> <strong>the</strong> distance D2 through which <strong>the</strong> <strong>in</strong>dex varies from n i þð1=20ÞDn to n t ð1=20ÞDn,Eq. (5) can be readily <strong>in</strong>tegrated to obta<strong>in</strong> an <strong>in</strong>variant quantitygiven <strong>by</strong>so that parameter D can be thought as <strong>the</strong> <strong>in</strong>terface thickness,correspond<strong>in</strong>g to 90% <strong>of</strong> <strong>the</strong> <strong>in</strong>dex <strong>change</strong> as shown <strong>in</strong> Fig. 1.In an earlier paper [13] studies <strong>of</strong> <strong>the</strong> <strong>in</strong>terface reflectivity,Q ¼ A 2 dqdz :ð6Þ given different pr<strong>of</strong>iles with vary<strong>in</strong>g thicknesses, have been done.All pr<strong>of</strong>iles were cont<strong>in</strong>uous, but some were piecewise def<strong>in</strong>edA nonl<strong>in</strong>ear ord<strong>in</strong>ary differential equation for <strong>the</strong> amplitude isobta<strong>in</strong>ed upon substitution <strong>of</strong> this result <strong>in</strong> Eq. (4)and <strong>the</strong>ir <strong>derivatives</strong> were discont<strong>in</strong>uous. For ‘‘hard’’ <strong>in</strong>terfaces,mean<strong>in</strong>g D l=100, reflectivity was close to <strong>the</strong> Fresnel resultd 2 A Q 2dz 2 A ¼ 3 k2 0 n2 A: ð7Þ R ¼ n t n 2i,n t þn iThis is an Ermakov–Milne–P<strong>in</strong>ney type equation. In order to work regardless <strong>of</strong> <strong>the</strong> pr<strong>of</strong>ile type. For ‘‘s<strong>of</strong>ter’’ <strong>in</strong>terfaces, D l=2, <strong>the</strong>withpa dimensionless amplitude function let us <strong>in</strong>troduceA d ¼ Affiffiffiffiffiffiffiffiffiffiffireflectivity <strong>of</strong> almost all n(z) pr<strong>of</strong>iles fell to less than 6% <strong>of</strong> <strong>the</strong> Fresnelk 0 =Q, <strong>the</strong>n Eq. (7) can be rewrittenresult. The reflectivity for <strong>the</strong> analytic pr<strong>of</strong>iles dropped monotonicallyd 2 1for D4l. In contrast, for <strong>the</strong> piecewise def<strong>in</strong>ed pr<strong>of</strong>iles, <strong>the</strong> reflectivityoscillates as a function <strong>of</strong> thickness. Every piecewise def<strong>in</strong>edA d¼ n 2 Ak 2 0 dz2 A 3 d : ð8Þdpr<strong>of</strong>ile had two attachment planes, at z 1 and z 2 , <strong>in</strong> order to keep <strong>the</strong>This is <strong>the</strong> ord<strong>in</strong>ary differential equation for <strong>the</strong> electric fieldamplitude. It is not common to turn a l<strong>in</strong>ear differential equation<strong>in</strong>terface symmetry and a bounded refractive <strong>in</strong>dex. The reflectivityoscillations <strong>in</strong> <strong>the</strong>se cases were <strong>in</strong> accordance with th<strong>in</strong> film <strong>in</strong>terference,<strong>in</strong>to a nonl<strong>in</strong>ear ODE, but Eq. (8) poses no challenge to be solvedwith a film thickness <strong>of</strong> z 2 z 1 . Evidently discont<strong>in</strong>uities <strong>in</strong>numerically if A d is real and n is bounded. Also, <strong>in</strong>itial conditionsare easily imposed hav<strong>in</strong>g a clear physical mean<strong>in</strong>g and f<strong>in</strong>ally,<strong>in</strong>terpretation <strong>of</strong> <strong>the</strong> solutions is straightforward. The f<strong>in</strong>itedifference method for numerically solv<strong>in</strong>g ord<strong>in</strong>ary differentialequations is used, with s<strong>in</strong>gle precision float<strong>in</strong>g po<strong>in</strong>t numbers.<strong>the</strong> pr<strong>of</strong>ile <strong>derivatives</strong> were caus<strong>in</strong>g reflections. However, it was notclear, only from <strong>the</strong> <strong>in</strong>terference data, which type <strong>of</strong> phase <strong>change</strong>was <strong>the</strong> <strong>reflected</strong> wave undergo<strong>in</strong>g at <strong>the</strong> z 1 and z 2 boundaries. Wecould only <strong>in</strong>fer <strong>the</strong> relative phase between <strong>the</strong> two reflections. Ourtask now is to evaluate <strong>the</strong> phase <strong>change</strong> upon reflection, based on<strong>the</strong> <strong>in</strong>terpretation <strong>of</strong> <strong>the</strong> amplitude equation numerical solutions.1.2. Interpretation <strong>of</strong> <strong>the</strong> solutionsA good example <strong>of</strong> how <strong>the</strong> junctions <strong>of</strong> a cont<strong>in</strong>uous butpiecewise n(z) function still generate reflection, even when <strong>the</strong><strong>in</strong>terface is gradual, is shown <strong>in</strong> Fig. 2. The bottom graph displaysA constant n represents a homogeneous medium, <strong>in</strong> this casereflectivity R versus <strong>in</strong>terface thickness D, <strong>in</strong> wavelength units, for<strong>the</strong> solutions <strong>of</strong> Eq. (8) are known and must be <strong>of</strong> <strong>the</strong> formtwo n(z) pr<strong>of</strong>ile types. Both pr<strong>of</strong>iles are cont<strong>in</strong>uous but one <strong>of</strong>[12–14,20]:<strong>the</strong>m, labeled as ‘‘tanh’’, is analytic and <strong>the</strong> o<strong>the</strong>r, ‘‘l<strong>in</strong>ear’’, isqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA d ¼ A 2 1 þA2 2 þ2A 1A 2 cosð2k 0 nzþdÞ: ð9ÞThe field amplitude A d is produced <strong>by</strong> <strong>the</strong> superposition <strong>of</strong> twocounter propagat<strong>in</strong>g waves, with <strong>in</strong>dividual constant amplitudesA 1 and A 2 . The constant d is <strong>the</strong> phase difference between bothwaves at z¼0. There is a restriction for <strong>the</strong>se amplitudes if Eq. (8)is to be satisfied:ðA 2 1 A 2 2 Þ2 ¼ 1 n : 2 ð10ÞThis homogeneous medium solution A d (z) oscillates periodically ifboth, A 1 and A 2 , are nonzero. Maxima A max and m<strong>in</strong>ima A m<strong>in</strong> occurwhen <strong>the</strong> <strong>in</strong>com<strong>in</strong>g and outgo<strong>in</strong>g waves are <strong>in</strong> or out <strong>of</strong> phase,respectively. These extrema can also be related to <strong>the</strong> ratior ¼ A 2 =A 1 [12,13]:r ¼ A max A m<strong>in</strong>:A max þA m<strong>in</strong>ð11ÞIf <strong>the</strong>re is only one wave propagat<strong>in</strong>g, A 1 or A 2 is zero and A d isconstant, particularlyFig. 1. To model a s<strong>in</strong>gle <strong>in</strong>terface, n(z) must be a monotonic cont<strong>in</strong>uous functionthat evolves from n i to n t . Far from <strong>the</strong> <strong>in</strong>terface n should be practically constant.