Phase change of light reflected by a discontinuity in the derivatives ...

R. Diamant, M. Fernández-Guasti / Optics Communications 294 (2013) 64–72 65where k 2 0 ¼ po2 m 0 E 0 and the refractive index is n ¼ E=E ffiffiffiffiffiffiffiffiffi0,E0 is the To model a single interface, n(z) must be a monotonic continuousA 2 d ¼ 1 The parameter D is a measure of the interface thickness, corresponding to 90% ofn : ð12Þ the index change. The overall refractive index change is labeled Dn and the halfvalueby n avg. The figure shows a profile with D ¼ 2l.electric permittivity of vacuum. Now, consider a complex E x ,namely E x ¼ Ae iq , where the amplitude A and phase q depend onz. We restrict the problem to non absorbing media, so we willassume the refractive index to be a real quantity. Substitution infunction evolving from n i to n t . Far from the interface n should bealmost constant. To evaluate the reflectivity, the convenient initialcondition is a single transmitted wave through the second medium,so the incident light is assumed to come only from the first mediumEq. (3) and separation of real and imaginary parts render:side. This implies that the solution in thepsecond medium, far fromd 2 AA dq 2the interface, is almost constant A d ¼ffiffiffiffiffiffiffiffiffiffi1=n t.Under this condition,¼ k 2dz 20dzA, ð4Þ the amplitude oscillations in the first medium, far from the interface,will exhibit the medium’s reflectivity: R ¼ r 2 . Given the indices n i2 dA and n t , we expect the reflectivity to depend on the interfacedqþA d2 qdz dz dz ¼ 0:ð5Þ abruptness. This property can be characterized by the distance D2 through which the index varies from n i þð1=20ÞDn to n t ð1=20ÞDn,Eq. (5) can be readily integrated to obtain an invariant quantitygiven byso that parameter D can be thought as the interface thickness,corresponding to 90% of the index change as shown in Fig. 1.In an earlier paper [13] studies of the interface reflectivity,Q ¼ A 2 dqdz :ð6Þ given different profiles with varying thicknesses, have been done.All profiles were continuous, but some were piecewise definedA nonlinear ordinary differential equation for the amplitude isobtained upon substitution of this result in Eq. (4)and their derivatives were discontinuous. For ‘‘hard’’ interfaces,meaning D l=100, reflectivity was close to the 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–Pinney type equation. In order to work regardless of the profile type. For ‘‘softer’’ interfaces, D l=2, thewithpa dimensionless amplitude function let us introduceA d ¼ Affiffiffiffiffiffiffiffiffiffiffireflectivity of almost all n(z) profiles fell to less than 6% of the Fresnelk 0 =Q, then Eq. (7) can be rewrittenresult. The reflectivity for the analytic profiles dropped monotonicallyd 2 1for D4l. In contrast, for the piecewise defined profiles, the reflectivityoscillates as a function of thickness. Every piecewise definedA d¼ n 2 Ak 2 0 dz2 A 3 d : ð8Þdprofile had two attachment planes, at z 1 and z 2 , in order to keep theThis is the ordinary differential equation for the electric fieldamplitude. It is not common to turn a linear differential equationinterface symmetry and a bounded refractive index. The reflectivityoscillations in these cases were in accordance with thin film interference,into a nonlinear ODE, but Eq. (8) poses no challenge to be solvedwith a film thickness of z 2 z 1 . Evidently discontinuities innumerically if A d is real and n is bounded. Also, initial conditionsare easily imposed having a clear physical meaning and finally,interpretation of the solutions is straightforward. The finitedifference method for numerically solving ordinary differentialequations is used, with single precision floating point numbers.the profile derivatives were causing reflections. However, it was notclear, only from the interference data, which type of phase changewas the reflected wave undergoing at the z 1 and z 2 boundaries. Wecould only infer the relative phase between the two reflections. Ourtask now is to evaluate the phase change upon reflection, based onthe interpretation of the amplitude equation numerical solutions.1.2. Interpretation of the solutionsA good example of how the junctions of a continuous butpiecewise n(z) function still generate reflection, even when theinterface is gradual, is shown in Fig. 2. The bottom graph displaysA constant n represents a homogeneous medium, in this casereflectivity R versus interface thickness D, in wavelength units, forthe solutions of Eq. (8) are known and must be of the formtwo n(z) profile types. Both profiles are continuous but one of[12–14,20]:them, labeled as ‘‘tanh’’, is analytic and the other, ‘‘linear’’, isqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA d ¼ A 2 1 þA2 2 þ2A 1A 2 cosð2k 0 nzþdÞ: ð9ÞThe field amplitude A d is produced by the superposition of twocounter propagating waves, with individual constant amplitudesA 1 and A 2 . The constant d is the phase difference between bothwaves at z¼0. There is a restriction for these 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 minima A min occurwhen the incoming and outgoing waves are in or out of phase,respectively. These extrema can also be related to the ratior ¼ A 2 =A 1 [12,13]:r ¼ A max A min:A max þA minð11ÞIf there is only one wave propagating, A 1 or A 2 is zero and A d isconstant, particularlyFig. 1. To model a single interface, n(z) must be a monotonic continuous functionthat evolves from n i to n t . Far from the interface n should be practically constant.