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

Optics Communications 294 (2013) 64–72Contents lists available at SciVerse ScienceDirectOptics Communicationsjournal homepage: www.elsevier.com/locate/optcomPhase change of light reflected by a discontinuity in the derivativesof the refractive indexR. Diamant n , M. Fernández-GuastiLab. de Óptica Cuántica, Depto. de Física, Universidad A. Metropolitana - Iztapalapa, 09340 México D.F., Ap. 55-534, Mexicoarticle infoArticle history:Received 22 November 2012Received in revised form15 December 2012Accepted 22 December 2012Available online 9 January 2013Keywords:Light propagationStratified mediaRugate filtersabstractThe amplitude and phase representation of classic plane electromagnetic waves is used to model lightpropagating through transparent stratified media, with a continuous refractive index profile. Numericsolutions for the nonlinear amplitude equation at normal incidence are obtained. Discontinuities of therefractive index derivatives exhibit reflection enhancement. Depending on the order of the discontinuity,the phase change upon reflection obtained from the numerical results can be 7p=2 in additionto the usual 0, or 7p.& 2012 Elsevier B.V. All rights reserved.1. IntroductionThe problem of inhomogeneous thin films has been approachedby many authors. The potentialities of these coatings for optics haveintrigued scientists for over a century and they are still a matter ofresearch both experimentally and theoretically [1–11]. Discontinuitiesin the refractive index derivatives influence strongly the reflectivityof continuously varying refractive index films, this fact has beennoticed since early works [4,6,7]. In these works the discontinuityplanes were not treated individually as localized reflectivity sources.Two discontinuity planes were always inserted for each stratifiedmedium, merely to model the two borders of a thin film.By introducing only one derivative discontinuity, in an otherwisesmooth and gradual refractive index profile, we may find itsproperties as a reflection plane. The amplitude and phase representationwill be used to model light propagation throughtransparent stratified media [12,13]. Numeric solutions for thenonlinear amplitude equation at normal incidence will beobtained, from these solutions reflectivity and phase change uponreflection will be inferred. From the analysis of discontinuousderivatives at different orders as well as many different profilefunctions, a conjecture can be established.1.1. Amplitude equationLet us take a close look to this physical phenomenon, regardingreflection of light on a plane surface with discontinuous refractiveindex or discontinuous index derivatives. Important works tryingn Corresponding author. Tel.: þ52 55 58 04 49 12.E-mail address: ruth@xanum.uam.mx (R. Diamant).to describe propagation of electromagnetic waves through stratifiedmedia have already been published. Somehow, these worksuse approximate expressions, most of them being for either aslowly or abruptly varying refractive index [3–5,7]. These approximationsallow analytical expressions to be written for the fieldsand the reflectivity. Note that these early works were publishedbefore commercial computers were widely used by scientists andregular population, numerical analysis was not as easily performedas it is today. In the present work, no approximations of this kindare intended, just those inherent to the use of numerical methods.For that purpose a differential equation for the electric fieldamplitude will be convenient [12–15]. It is a nonlinear ordinarydifferential equation of the Ermakov–Milne–Pinney type [16–18].Consider Maxwell’s equations, in an isotropic, transparent,dielectric medium with a linear response and no free charges, letthe electric permittivity and magnetic permeability vary spatially.The second order differential equation for the electric field isrðE r ln eÞþr ln m ðr EÞ¼me @2 Er 2 E, ð1Þ@t 2where E and m represent the electric permittivity and magneticpermeability, respectively. Let z be the direction of stratification,so that E and m will depend only on z. Consider a monochromaticplane wave with frequency o, and linear TE polarization, say, inthe x direction E ¼ E x ðy,zÞe iot^i. Eq. (1) leads to [19]@ 2 E x@y þ @2 E x2 @z 2 þo2 EmE x ¼dðln mÞ @E xdz @z :For normal incidence and non magnetic media m ¼ m 0 . This PDEbecomes the non autonomous ODE@ 2 E x@z 2 þk2 0 n2 E x ¼ 0, ð3Þð2Þ0030-4018/$ - see front matter & 2012 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.optcom.2012.12.036

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.

R. Diamant, M. Fernández-Guasti / Optics Communications 294 (2013) 64–72 67Fig. 3. Amplitude equation solutions for (a) ‘‘stepþ’’, n i ¼ 1, n t ¼ 1:5 and (b) ‘‘step’’, n i ¼ 1:5, n t ¼ 1. The junction is at z¼0.Fig. 4. (a) Plot of the profile in Eq. (17) ‘‘lin&tanh’’, it is constructed by joining a constant function with a hyperbolic tangent. It models a gradual interface with D ¼ 2l andone discontinuity in the first derivative at z¼0, where this derivative is increasing. The amplitude equation solution A d for ‘‘lin&tanhþ’’ profile is also shown and (b) acloser look at the solution in the oscillating region.these waves point in opposite directions; if there is a maximum, itmeans the fields point in the same direction. If maxima occur atL max ¼ ð ð1=4Þ ðm=2ÞÞl and minima at L min ¼ ðm=2Þl, form ¼ 0,1,2,3 ..., the electric field waves must have phase differenceof d ¼ p between them at z¼0 (the discontinuity site), thusrevealing a phase change of p due to reflection at the junctionplane. If maxima and minima positions are interchanged, wavesmust be in phase at z¼0, revealing no phase change due toreflection at the junction plane. For the former interpretation tobe valid, the refractive index n(z) must be reasonably constantalong the interval where the local extrema are found.The location of the critical points for the ‘‘stepþ’’ case, showsan approximate phase change of p upon reflection at z¼0. Thereis a mean offset of ð0:004Þp from the exact d ¼ p value with verysmall deviation (sop=1000). For ‘‘step ’’, maxima fall close toL max ¼ ðm=2Þl and minima at L min ¼ ð ð1=4Þ m=2Þl, revealingan almost null phase change due to reflection at the junction.However, there is a mean offset of ð0:004Þp from the exact d ¼ 0value with standard deviation of s ¼ð0:002Þp. These resultsmatch very well with Fresnel predictions.4. Gradual and continuous interfaces4.1. Type C 0 profilesA type C 0 profile with only one piecewise junction can beconstructed by joining a constant function with a hyperbolictangent at z¼0:(n lin&tanh þ ðzÞ¼ n a zo0n a þðn g n a Þ tanh a 1Dz ð17ÞzZ0,Herea 1 ¼ 1 1:95ln ln 1:05 ,2 0:05 0:95so that D is the interface thickness, the distance needed for n tochange 90% of the total Dn ¼ n g n a . The function n lin&tanh þ ðzÞ iscontinuous, at the junction it is increasing while its derivativedn=dz is discontinuous and is also increasing. Interface thicknessis chosen to be reasonably soft D ¼ 2l to discard any significantsource of reflection except at the junction. Fig. 4(a) shows a plot ofthis profile. Light is always assumed to be incident from the left.The numerical solution of the corresponding amplitude equation(8) is also plotted in Fig. 4.In the case of ‘‘lin&tanhþ’’ profile, applying Eq. (11) for zo0,the reflectivity is found to be R lin&tanh þ ¼ 3:15565 10 2 %, with astandard deviation of s R ¼ 1:8 10 6 %. This reflectivity is muchsmaller than the abrupt interface result 4%, since it is related to avery gradual interface (D ¼ 2l). Yet, it is still far greater than thereflectivity computed for the analytic ‘‘tanh’’ profile stated in Eq.(13), with the same interface thickness, R tanh ¼ 3:16840 10 10 %.The first derivative discontinuity in n(z), thus enhances thereflectivity substantially.To analyze phase relationship between incident and reflectedwaves let us take as reference the reflection plane (always drawnat z¼0 in the figures). In this case, it lies where the first orderderivative is discontinuous, although the refractive index itself iscontinuous. The phase between incident and reflected waves maybe evaluated at any such region with constant refractive index.However, it is easier to consider a plane located at a multiple ofthe half-integer wavelength, L ¼ mðl=2Þ where m is an integer.From any of these planes, the incident wave travels to thereflection plane and then the reflected part travels back an integer

68R. Diamant, M. Fernández-Guasti / Optics Communications 294 (2013) 64–72number of wavelengths, and thus no phase needs to be added dueto the extra path traveled by this wave. If no phase change occursby reflection at the derivative discontinuity (L ¼ 0), the reflectedwave must be in phase with the incident wave after m wavelengthstravel. However, consider the plane L ¼ 1:0l in Fig. 4(b).The incident and reflected waves are not in phase. In the middlebetween one amplitude maximum and a contiguous minimum,these waves must have a relative phase difference of d ¼ 7ðp=2Þ.The numerical solution shown at L ¼ 1:0l in Fig. 4(b), reveals aphase difference of p=2. Therefore, there must be a phase changeof d ¼ p=2 introduced by the reflection at the plane where thederivative is discontinuous. A detailed analysis of the numericalsolution shows that there is only a small offset ofð 0:02470:004Þp=2 from the exact d ¼ p=2 value. This is certainlya very surprising result, since we know from Fresnel equations,that a phase change of p is introduced between two transparentdielectric media when the refractive index is discontinuous andincreasing. The present result shows that a phase change of p=2can be introduced between two transparent dielectrics whoserefractive index is equal at the interface but its first derivative isdiscontinuous. The former ‘‘lin&tanh’’ profile is increasing at thejunction plane as well as the derivative dn=dz, to see what wouldbe the case for an increasing profile n(z) but decreasing firstderivative dn=dz let us devise another C 0 type profile. Let us againpiecewise join a hyperbolic tangent with a constant function atz¼0, but this time placing the hyperbolic tangent first:(n tanh&lin þ ðzÞ¼ n g þðn g n a Þ tanh a 1Dz zo0ð18ÞzZ0:n gOnce more, the interface thickness is D ¼ 2l to discard anysignificant source of reflection except for the junction.Fig. 5(a) shows a plot of this profile. Again, light is assumed tobe incident from the left. Plots for the amplitude equation (8)numerical solution are shown in Fig. 5.In the case of ‘‘tanh&linþ’’, Eq. (11) must be applied tooscillations at the far left, for Lt 4l, since the medium is notsufficiently homogeneous closer to L ¼ 0. Oscillations are distortedwhen the refractive index varies considerably as a functionof position. Computed reflectivity is R tanh&lin þ ¼ 6:27498 10 3 %with a standard deviation of s R ¼ 7:115 10 5 . The maxima andminima are approximately located at L max ¼ð1=8 m=2Þl andL min ¼ð 1=8 m=2Þl, as seen by looking at the inset in Fig. 5(b).The phase difference between incident and reflected waves is veryclose to d ¼ p=2 atL ¼ m=2 revealing an approximate phasechange of p=2 due to reflection at the junction plane. There is anoffset of ð 0:01070:002Þp from the exact d ¼ p=2 value. Thesign in d ¼ 7ðp=2Þ depends on whether the discontinuous derivativeincreases or decreases, rather than on the refractive indexvalues at the reflection plane.By interchanging the n a and n b refractive indices for the‘‘lin&tanhþ’’ and ‘‘tanh&linþ’’ profiles, one can generate anothertwo C 0 type profiles:(n lin&tanh ðzÞ¼ n g zo0n g þðn a n g Þ tanh a 1Dz ð19ÞzZ0(n tanh&lin ðzÞ¼ n a þðn a n g Þ tanh a 1Dz n azo0zZ0:ð20ÞWhile the refractive index profile is decreasing for both profiles,the first derivative is increasing in one case and decreasing in theother. Performing the same analysis to the correspondingFig. 5. (a) Plot of the profile in Eq. (18) ‘‘tanh&lin’’, it is constructed by joining a hyperbolic tangent with a constant function. It models a gradual interface with D ¼ 2l andone discontinuity in the first derivative at z¼0, where this derivative is decreasing. The amplitude equation solution A d for ‘‘tanh&linþ’’ profile is also shown and (b) acloser look at the solution in the oscillating region.Table 2Calculated reflectivity and approximate phase difference d between incident and reflected waves at the reflection plane, z¼0, related to the type C 0 profiles. The symbol s Rstands for the reflectivity standard deviation.C 0 function Reflectivity (%) R Reflectivity relative standarddeviation (%) s R 100Rlin&tanhþ 3:15565 10 2 0.0057 p2tanh&linþ 6:27498 10 3 1.1 p2lin&tanh 6:24063 10 3 0.013 p2tanh&lin 3:16123 10 2 0.48 p2Approximate phasechange at reflection plane z¼0Offset from the exact p 2 orpvalue in p units20:02470:0030:01470:0030:00570:0060:01970:005

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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