Refractive-index-profiledeterminations by using Lloyd's mirage
B. E. Allman, A. G. Klein, K. A. Nugent, and G. 1. Opat
A method is presented for the experimental determination of refractive-index profiles for planar media ofmonotonically decreasing refractive index, such as those used for optical waveguides. The technique isbased on a generalization of the classical experiment of Lloyd's mirror, involving the interference patternformed by a point source and its mirage, i.e., its reflection in such a graded planar medium.
Key words: Graded index optics, integrated optics, optical waveguides, mirage.
Introduction
The fields of gradient-index optics and integratedoptics are based on processes by which the refractiveindex of a transparent substrate may be modified.Such processes include ion exchange, diffusion, hydra-tion, and evaporation. The proper and controlleddevelopment of devices using these techniques re-quires a method of experimentally determining therefractive index profile. In this paper we reviewexisting approaches and then propose and test amethod for refractive-index-profile measurement.
Consider the important case of a planar integratedoptical waveguide. This device is formed by creationof a planar region of refractive index higher than thatof the surrounding medium such that propagatingmodes of light may be supported by total internalreflection. If the planar region of higher index isformed by diffusion, then the concentration profileresults in a monotonically decreasing refractive indexprofile.' To characterize these devices properly, wefind it important to know this refractive index profile.
Present methods of refractive-index-profile determi-nation may be broadly grouped into optical andnonoptical techniques. Nonoptical techniques arebased on the fact that the refractive-index profile isproportional to the diffusant concentration profile';the problem is thus reduced to finding the diffusionprofile. The diffusion profile may be achieved by use
The authors are with the School of Physics, The University ofMelbourne, Parkville, Victoria 3052, Australia.
Received 21 December 1992; revised manuscript received 12 July1993.
of concentration-sensitive analytical tools, e.g., elec-tron and ion microprobes,2 scanning electron micro-scopes,3 or atomic absorption spectrophotometry.4Although accurate, these techniques require inten-sive (often destructive) sample preparation and aredependent on the availability of sophisticated equip-ment. Related techniques require the solution ofthe diffusion equation for ion exchange.5 6 All thesemethods give only the overall shape of the profile butno information about the refractive indices at the endpoints. Thus an independent determination of theindex at the extremes is needed.
An alternative approach is to carry out opticalmeasurements of refractive-index-dependent param-eters, e.g., the index dependence of polarization (ellip-sometry), the index dependence of reflectivity,7 or theindex dependence of phase (interferometry).11 El-lipsometry and reflectivity measurements require thesample to be scanned with a spot of light, and thespatial resolution of these is a function of the spot sizeof the incident beam. This leads to an averaging ofthe profile. Interferometry is potentially the mostdirect and accurate of all the techniques, but itrequires the device under investigation to be sec-tioned to the tolerance of interferometry and placedwithin the interferometer. The so-called near-fieldand refracted near-field" techniques use the knownrelationship between the light intensity in the nearfield at the end of a waveguide and the refractive-index profile as a way to recover the profile shape.With these approaches the index at either extreme ofthe waveguide still requires independent measure-ment.
Finally, there are other commonly used methodsapplied to waveguides that require little preparation,are easy to implement, and are nondestructive.
These involve numerical recursions of simple equa-tions derived from either a W-KB approximation tothe measured mode indices of the waveguide 2"13 or aray-optical approximation to the guided-beam track-ing of modes in the waveguide.14 The profile shape isapproximated by use of straight-line segments joiningthe results of these recursions. The accuracy ofthese methods is then dependent on the number ofpoints in the recursion, i.e., the number of modesthe waveguide supports. However, for small indexchanges this is a small number, and hence the resultsare poor. Furthermore, waveguides of most interestsupport only a single mode and thus these methodsare of no practical use. Our technique for recoveringthe refractive-index profile, described below, is basedon observing the Lloyd's mirage interference patternand overcomes the deficiencies of other methods.
Lloyd's Mirage and Its Interference Pattern
In the classical Lloyd's mirror experiment, an interfer-ence pattern is produced between the light comingdirectly from a point source (or slit) and indirectlyfrom its reflection in a plane mirror, as shown in Fig.1(a). If the plane mirror is replaced by a mirage-producing layer, i.e., by a stratified medium with a
ISIa
I
monotonically decreasing refractive index, we obtaina situation that we have called Lloyd's mirage.'5The interference is between the light from the sourceand its mirage, as shown in Fig. 1(b). In the paraxialapproximation this interference exhibits fringes analo-gous to those of Lloyd's mirror, except the phase shiftbecomes a more complicated function of angle andhence the fringe spacing is no longer uniform butvaries with height. Not surprisingly, this variationof the fringe spacing is intimately related to the actualrefractive-index profile of the stratified layer thatgives rise to the mirage. This forms the basis for ourmethod of determining the refractive-index profile ofsuch planar media. We first consider the problem ofderiving the fringe spacing given the index profile.
Consider the situation shown in Fig. 1(b) in whichthe mirage-producing boundary has a refractive-index profile given by n(z). It is possible to map anyray path in the mirage by use of the ray (or eikonal)approximation, and the generalization of Snell's lawapplicable to a graded medium:
n(z)cos 0(z) = n(O)cos 0(0) = n(ztp). (1)
The quantity n(z)cos 0(z), termed the ray invariant,see inset of Fig. 1(b), is constant at all points along aray, from the surface value n(O)cos 0(0) to the value ofn(ztp), which is the refractive index at the turningpoint.
Inside the graded medium, the optical path lengthLo along the (curved) ray may be shown to be given bythe integral
L = 2 Pn2(Z) - n2(Z )]1/2 ' (2)
and the phase of the wave along this path Up is relatedto L by
C0
00-
InsetI S- x, .~ III R ((0) I medium 1
L - _-- __ '
(b)
Fig. 1. Schematic of (a) Lloyd's mirror experiment, (b) theanalogous Lloyd's mirage experiment, with the mirage profile.The changing position of the virtual source (and mirror) is indi-cated to explain the variation in the fringe spacing. The curvedray path obeying the generalized Snell's law is shown in the inset.
(lop = 2 2r , +- (3)
(The extra phase shift of rr/2 is a correction to the rayapproximation required because of phase change atthe turning point.'6) The horizontal displacement xtraversed by the (curved) ray is given by
z p dzx= 2n(zt,,) J~2(Z) -
2(Ztp)]1/2 (4
Equations (1)-(4) are derived in Ref. 17. Given themeasured quantities a, the height of the source abovethe mirage surface, and s, the horizontal displace-ment to the observation plane, we see that the aboveexpressions completely determine the path of any ray.Other phase shifts, along the straight sections of thedirect and indirect rays, are easily computed from theray path, and hence the overall phase difference as afunction of position in the observation plane is calcu-
lable. Thus for a given refractive-index profile n(z),it is possible to compute the fringe pattern numeri-cally.
Inverse Problem
In practice, we are faced with the inverse problem ofdeducing the index profile from an observed interfer-ence pattern. This deduction requires two steps:(a) determining the phase of the light leaving theprofile region and (b) the use of this phase informa-tion to determine the index profile. The first ofthese steps is quite straightforward for interferomet-ric data, and accurate methods for recovering thephase have been extensively discussed in the litera-ture.' 8 An outline of this method will now be given.
As we noted above, unlike the constant fringespacing of Lloyd's mirror interference patterns,Lloyd's mirage interference patterns exhibit a vari-able fringe frequency, because the distance 2a to thevirtual source (or, alternatively, the virtual mirrorplane) increases. A general form of the intensitydistribution in the plane of observation is
I(z) = B(z) + V(z)cos[27r(o0z + +(z)], (5)
where +(z) is the required phase information respon-sible for the chirping of some fundamental frequencyw0, B(z) is the background variation, and V(z) is thevariation in fringe visibility. An example of such adistribution is shown in Fig. 2.
In order to extract the function b(z), we take theFourier transform of the intensity distribution of Eq.(5), obtaining a result that looks like Fig. 3. Thepositive sidelobe of the spatial frequency representa-tion is then shifted to be centered on zero frequencywhile the other sidelobe and the dc component, thebackground factor, B(z), are suppressed. The remain-ing data are inverse transformed to yield V'(z), fromwhich the phase function +'(z) is obtained as
+'(z) = Im[ln V'(z)] (mod 2rr), (6)
which is independent of both B(z) and V(z). Thephase information and hence the fringe distributionat any point of the interference pattern is now knownas a function of position.
200
-1 150-
Ti)
a 100 _
-i 50
-0.008 -0.004 0.000 0.004 0.008
Spatial FrequencyFig. 3. Fourier transform of the apodized fringes of Fig. 2.
The remaining problem is the recovery of the indexprofile of the transition region. The continuousprofile may be approximated by a set of points on theprofile corresponding to a set of depths in the wave-guiding layer and the refractive indices at thosedepths. That is, for each of m chosen ray pathsthrough the waveguide, we find the depth of theturning point ztp and the refractive index at thatdepth n(z4p). We achieve this by modeling the refrac-tion of the symmetrically curved ray through thetransition region as a reflection at a virtual mirror[see Fig. 1(b)]. Using the fringe separation (obtainedfrom the phase function) at m sampled points, wemay define the m direct and reflected rays responsiblefor the interference.' 9 The angle of incidence of themth reflected ray gives the index at the turning point,nim defined in Eq. (1). The mth reflected ray pathalso gives xm, the horizontal displacement from thesource to where the ray again reaches the height ofthe source. Thus for each of the m sampled raypaths we have a pair of values nm and xm. All thatremains is the application of Eq. (4) to the pairs ofnm's and xm's in order to recover the turning-pointdepth zm. We then have n = n(zm), a number ofdata points describing the refractive-index profile.Clearly, the greater the sampling rate of the interfer-ence phase map, the more accurate the reclaimedprofile; hence the application of the Fourier trans-form method.
In order to find the zm's we make a finite-differenceapproximation to Eq. (4). Following Ref. 12, we maywrite Eq. (4) as a sum of integrals:
.4
c-
1.0
0.8
0.6
0.4
0.2
0.00 200 400 800 800 1000 1200 1400
Position (m)Fig. 2. Theoretical Lloyd's mirage fringe pattern for an erfcdiffusion profile. The chirping of the fringes is clearly seen. Theapodization of the fringes is also shown (dotted curve).
Xm m jZk dz
2nm = ,7 1 [n2(z) - 2]1/2(7)
Assume that n(z) is a piecewise linear function con-necting the turning-point indices, i.e.,
n(z) =nk + (nk1 -nk ) (Z(Zk - Zk-)
for Zk-l < Z < Zk-
(8)
Then if n(z) + nm is replaced by a midpoint value of[(nk-1 + nk)/2] + n,, forZk l < Z < Zk, the term in the
The integral now represents a known function of zand is easily solved. The solution for z is
`zm =zmi + [1 (nnm- + 3nm)1/22(
XZ -2 m1- +k-1 + 2 k +|2nm k= 1 2
n1 - nM) 1/2
1 |2 Zk - Z- 1
- m nk- -nk}X [(nk-1 - nm)"/2 (nk - nm)1/2
form = 2,3, ... , M, (10)
X1 no + 3n, /21 4n, 2 (no - )/ 2 . (11)
This solution represents a simple recursive algorithmfor evaluating the zm based on Xn, the nm's andpreviously calculated zm's starting with z1, where z0 =0 (i.e., the surface) and no is the surface index.
Simulated Results
To test the above approach, we produced a simulatedinterference pattern in which we assumed that therefractive-index profile is produced by diffusion of alow-index material into a higher-index substrate.The expected form of the index profile is20
n(z) = ni - (n - r2 )erfc(z), (12)
where erfc is a normalized complementary errorfunction with n, the substrate index, and n2, theindex at the diffusion boundary. Given the geometri-cal constants of a Lloyd's mirage experiment that wassuch a medium, it is straightforward to calculate theresultant interference pattern. Figure 2 shows the re-sults of just such a calculation in which we used a = 10[um, s = 40 mm, nr = 1.501, and n2 = 1.5, and the erfcwas obtained from a Gaussian with a FWVHM of 10 jim.
We recovered the phase information in the interfer-ence pattern by apodizing the data and applying theinterferogram analysis techniques described above.This analysis can yield a large number of data pointsquickly. The phase information is then used torecover the refractive-index profile. A comparison ofthe recovered profile and the initial profile is shown inFig. 4. The agreement between the two is excellent.As an example, a waveguide with the same profile asthat treated above supports only four modes. The
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32
Depth (lLm)
Fig. 4. Complement of error function used in calculating thetheoretical interference pattern of Fig. 2 (dashed curve) and thereclaimed profile. For the erfc, depth 0 lm defines the sourceposition, and the transition layer starts at 10 plm. Produced froma normalized Gaussian with a FWHM of 10 plm, the erfc extends afurther 21 m to the diffusion surface. The reclaimed profile wasreproduced entirely relative to the source position at 0 11m.
mode indices for this simulation were determined2 land the values used in the recursive index recoveryalgorithm described in Refs. 12 and 13. Figure 5gives a comparative illustration of the profiles derived.It is clear that the small number of effective modeindices would present a severe limitation to the accu-cacy of techniques based on mode-index measurements.
Other monotonically decreasing functions have alsobeen accurately reproduced by, our technique. Inparticular, the procedure is highly suited to profiles ofsmall index change and whose variation of index withdepth is slow. The mirage then occurs at shallowglancing angles, and the reflected ray is accuratelyrepresented by the virtual ray. The optical pathdifference in this case is due to the separation of thereal and virtual sources. Such cases represent awaveguide that would support few or no modes,whereas large index changes would represent wave-guides that are highly multimode and fall outside thiscategory. Hence they would be amenable to theother approaches reviewed in the Introduction.
Experimental Results
Experimental verifications were carried out with amirage-forming diffusion layer prepared with gelatin.'5
8)a)
- -
.)
Cc
'4-8)
1.50 10
1.5008
1.5006
1.5004
1.5002
1.5000
0 2 4 6 8 10 12 14 16 18 20 22 24 2 28 30 32
Depth (m)
Fig. 5. Comparison of the erfc profile with the fits produced by themethods of Refs. 12 and 13 using the mode indices appropriate to awaveguide with such a profile.
--------------------------------------. . . . I - . . I . . I I I
I I . I . . I I . . I I I I I 11
Commercially available gelatin crystals were dis-solved in a mixture of glycerine (n = 1.4686) andwater (n = 1.333). Water was then added to thesurface of the set gelatin. Diffusion of the water intothe gelatin displaced some of the glycerine and estab-lished a region of decreasing refractive index to thesurface of the gelatin, i.e., the mirage-forming layer.
Fig. 6. Experimental fringe pattern achieved with the gelatinmirage.
l, .l | BIl
X 1.4150 -CD
10 1.4145-0 t.424-Exp Profile
(1 .4140- - Erfc FitX> 1.4135C 1.4130-
I... 1.4125 -4-.
pL 1.4120 -
1.4115
0 200 400 6oo 800 1000 1200
Depth (um)
Fig. 7. Refractive-index profile of the experimental diffused layer,fitted with an erfc.
The resultant interference pattern is shown in Fig. 6,in which the variable fringe spacing is quite evident.(The intensity of the observed fringes matches that ofa more rigorous solution based on an Airy function, asshown by Raman and Pancharatnam.22 )
We then repeated the procedure on a set of experi-mentally achieved fringes from the gelatin mirage,e.g., Fig. 6. First a horizontal average of the CCDimage of the interferogram was taken. (For thisreason the fringe position must display little lateralvariation across the interferogram.) This horizontalaverage was then processed, and the refractive-indexdistribution was recovered by the technique describedhere. The result is shown in Fig. 7. Diffusiontheory predicts a profile described by a complemen-tary error function, and a fit to the data with such aprofile is also shown in this figure. The agreement isexcellent.
Conclusion
After a brief review of existing methods, in this paperwe have presented, to the best of our knowledge, anew approach to the determination of refractive-index profiles in planar substrates. The techniquearises from a generalization of the familiar Lloyd'smirror experiment. It is believed that, in solving theinverse scattering problem, the technique offers analternative to other techniques for refractive-index-profile measurements and may find application inoptical waveguides that are not otherwise amenableto nondestructive analysis, particularly the single-mode case.
References1. R. V. Ramaswamy and R. Srivastava, "Ion-exchanged glass
waveguides: a review," J. Lightwave Technol. 6, 984-1002(1988).
2. G. Stewart, C. A. Millar, P. J. R. Laybourn, C. D. W. Wilkinson,and R. M. DeLaRue, "Planar optical waveguides formed bysilver-ion migration in glass," IEEE J. Quantum Electron.QE-13, 192-200 (1977).
3. R. V. Ramaswamy and S. I. Najafi, "Planar, buried, ion-exchanged glass waveguides: diffusion characteristics," IEEEJ. Quantum Electron. QE-22, 883-891 (1986).
4. P. Chludzinski, R. V. Ramaswamy, and T. J. Anderson,"Silver-sodium ion-exchange in soda-lime silicate glass," Phys.Chem. Glasses 28, 169-173 (1987).
5. M. Guntau, A. Brauer, W. Karthe, and T. Pol3ner, "Numericalsimulation of ion-exchange in glass for integrated opticalcomponents," J. Lightwave Technol. 10, 312-315 (1992).
6. R. K. Lagu and R. V. Ramaswamy, "A variational finite-difference method for analysing channel waveguides witharbitrary index profiles," IEEE J. Quantum Electron. QE-22,968-976 (1986).
7. H. J. Lilienhof, E. Vorges, D. Ritter, and B. Pantschew,"Field-induced index profiles of multimode ion-exchangedstrip waveguides," IEEE J. Quantum Electron. QE-18, 1877-1883 (1982).
8. W. E. Martin, "Refractive index profile measurements ofdiffused optical waveguides," Appl. Opt. 13,2112-2116 (1974).
9. E. Okuda, I. Tanaka, and T. Yamasaki, "Planar gradient-indexglass waveguide and its applications to a 4-port branchedcircuit and star coupler," Appl. Opt. 23, 1745-1748 (1984).
10. T. Izawa and H. Nakagome, "Optical waveguide formed byelectrically induced migration of ions in glass plates," Appl.Phys. Lett. 21, 584-586 (1972).
11. R. Goring and M. Rothhardt, "Application of the refractednear-field technique to multimode planar and channel wave-guides in glass," J. Opt. Commun. 7, 82-85 (1986).
12. J. W. White and P. F. Heidrich, "Optical waveguide refractiveindex profiles determined from measurement of mode indices:a simple analysis," Appl. Opt. 15, 151-155 (1976).
13. K. S. Chiang, "Construction of refractive-index profiles ofplanar dielectric waveguides from the distribution of effectiveindices," J. Lightwave Technol. LT-3, 385-391 (1985).
14. J. Ctyroky, J. Janta, and J. Schrofel, "Refractive-index profilemeasurement of highly multimode planar waveguides byguided-beam tracking," Opt. Lett. 7, 552-554 (1982).
15. B. E. Allman, A. G. Klein, K. A. Nugent, and G. I. Opat,"Lloyd's mirage," Rep. UM-P-92/10 (University of Mel-bourne, to be published).
16. See, for example, R. H. Dicke and J. P. Wittke, Introduction toQuantum Mechanics (Addison-Wesley, Reading, Mass., 1960),pp.245-248.
17. A. W. Snyder and J. D. Love, Optical Waveguide Theory(Chapman and Hall, London, 1983), pp. 13-20.
18. K. A. Nugent, "Interferogram analysis using an accurate fullyautomatic algorithm," Appl. Opt. 24, 3101-3105 (1985).
19. See, for example, E. Hecht, Optics, 2nd ed. (Addison-Wesley,Reading, Mass., 1987), pp. 339-346.
20. J. Crank, The Mathematics of Diffusion (Oxford U. Press,London, 1956), p. 32.
21. G. B. Hocker and W. K. Burns, "Modes in diffused opticalwaveguides of arbitrary index profile," IEEE J. QuantumElectron. QE-l, 270-276 (1975).
22. C. V. Raman and S. Pancharatnam, "The optics of mirages,"Proc. Indian Acad. Sci. Sect. A 49, 251-261 (1959).
