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 69amplitude equation solutions we arrive at similar outcomes asabridged for all four C 0 profiles in Table 2.4.1.1. Type C 1 profilesTo build some C 1 type profiles, in a similar fashion as with theC 0 type, a constant function is piecewise joined with a squaredhyperbolic secant at its critical point in four different ways,shown in Table 3. Plots of the numerical solutions for theamplitude equation (8), taking D ¼ 2l, are illustrated in Fig. 6.Reflectivity for type C 1 profiles, estimated with Eq. (11), andphase difference between incident and reflected waves at thereflection plane are reported in Table 4. The reflectivity is evenlower for these profiles. Incident and reflected waves are in phaseTable 3Type C 1 profiles. The piecewise junction is always at z¼0. Herepa 2 ¼ 1=2 lnð19Þþln½ð1þffiffiffiffiffiffiffiffiffiffi p0:95Þ=ð1þffiffiffiffiffiffiffiffiffiffi0:05ÞŠ so that D is the interface thickness,the distance needed for n to change 90% of the total Dn ¼ n g n a.Function label n(z) d 2 nat the junction2dz8lin&sechþ < n a zo0n g ðn g n aÞ sech 2 a 2:D z zZ0sech&linþ8n a þðn g n aÞ sech 2 a
70R. Diamant, M. Fernández-Guasti / Optics Communications 294 (2013) 64–72Fig. 7. Amplitude equation solutions for type C 2 profiles, a close look at the solutions in the oscillating regions. (a) Solutions for ‘‘lin&cubexpþ’’ and ‘‘cubexp&linþ’’ and(b) solutions for ‘‘lin&cubexp ’’ and ‘‘cubexp&lin ’’.Table 6Calculated reflectivity and approximate phase difference d between incident and reflected waves at the reflection plane, z¼0, related to the type C 2 profiles. The symbol s Rstands for the reflectivity standard deviation.C 2 function Reflectivity (%) R Reflectivity relativestandard deviation (%) s R 100Rlin&cubexpþ 1:33978 10 6 0.0045 p2cubexp&linþ 5:15290 10 8 1.2 p2lin&cubexp 5:18475 10 8 0.020 p2cubexp&lin 1:33984 10 6 0.014 p2Approximate phase changeat reflection plane z¼0Offset from the exact p 2 orpvalue in p units20:00170:0030:00470:0040:01570:0050:01170:005Table 7Type C 3 profiles. The piecewise junction is always at z¼0. Herepa 4 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi44lnð0:95Þ lnð0:05Þ so that D is the interface thickness, the distance neededfor n to change 90% of the total Dn ¼ n g n a.Label n(z) d 4 nat the junction4dz8lin&quartexpþ < n a zo0a 4:4n g ðn g n aÞ expD z zZ0quartexp&linþ8
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