Direct measurement of substrate refractiveindices and determination of layer indices inslab-guiding structures

Nicole A. Paraire, Nathalie Moresmau, Shufen Chen, Pierre Dansas,and Florent Bertrand

We present a new method for accurate and nondestructive measurement of the refractive indices ofsubstrates and guiding layers in slab waveguides. This method is based on the excitation of leaky wavesin substrates and guided waves in guiding layers owing to the etching of grating couplers on the top ofstructures. It is particularly applicable to high refractive-index materials and to in situ measurementsnear the energy band gap of semiconductor waveguides. We present results that were obtained for anInP substrate with an InGaAsP epitaxial layer with regard to their refractive indices and temperaturecoefficients. © 1997 Optical Society of America

Key words: Direct refractive-index measurements, grating waveguide, near-band-gap semiconductorindex, InP, refractive-index temperature coefficients.

1. Introduction

III–V-based optical structures have been investi-gated extensively because of their possible use aslight sources, optical switches, or modulators and inintegrated optics. To optimize the design and fabri-cation of these waveguide devices, accurate knowl-edge of the various layer optical characteristics, inparticular, refractive indices at the operating wave-length, is necessary. Currently, this wavelength isnear the absorption band edge of the active layermaterial, where the latter properties are sensitive togrowth conditions and a direct determination of thelayer indices in the device itself would be helpful.The precision needed for this determination is all thegreater as the index differences between the variouslayers of a heterostructure laser or a waveguide aresmall: usually to within 65 3 1023.1 Moreover, insemiconductors at such an operating wavelength, re-

When this research was done all the authors were with theInstitut d’Electronique Fondamentale, Centre National de la Re-cherche Scientifique, Unite de Recherche Associee 22, UniversiteParis-Sud, Batiment 220, 91405 Orsay cedex, France. S. Chen isnow with the Department of Optical Engineering, Beijing Instituteof Technology, Laboratory 405, Beijing 100081, China.Received 12 February 1996; revised manuscript received 22 July

1996.0003-6935y97y122545-09$10.00y0© 1997 Optical Society of America

fractive indices are relatively high and most classicaltechniques are not easy to implement.Indeed, refractive-index direct measurementmeth-

ods are few: ~i! prism deviation2 needs bulk trans-parent materials and cannot be used in situ, ~ii!Brewster angle measurements3,4 give direct determi-nation provided a single well-controlled interface isinvolved and absorption is negligible, otherwise,modeling is necessary.5,6Until now, most experimental results were ob-

tained by indirect methods7: either polarimetric orinterferometric. Reflectometry was used for opticalconstant determination of InP ~Refs. 8 and 9! andGaAs.1 For this method one needs absolute mea-surements and a single interface; otherwise modelingbecomes necessary.Spectroscopic ellipsometry, which was used to de-

termine the AlGaAs refractive index,10 also necessi-tates a model. Interferometric techniques provideaccess to optical thicknesses but cannot be used insitu or for absorbing or multilayered structures.In integrated optics, various techniques have been

developed to determine effective ~or modal! indicesusing excitation of guided or leaky modes with aprism coupler,11–13 a grating,14,15 or other phenome-na.16 Numerical techniques are then necessary17–20to calculate the corresponding refractive indices.Recently, these methods of modal excitation by use ofa prism21 or a grating15 coupler have allowed obser-vation of leaky waves in the substrate.

20 April 1997 y Vol. 36, No. 12 y APPLIED OPTICS 2545

Here we present a method for direct and precisemeasurement of the refractive index of a substrate,included in a multilayered guiding structure, underor near the energy gap of its active layer. Thismethod is based on the possible excitation of suchleaky waves in the substrate when one uses a gratingcoupler etched on the structure surface. Based onobservations of diffracted or guided beams, this directmeasurement also allows one to determine the opticalcharacteristics of the active layer. We present theresults that were obtained in a nonintentionallydoped InP substrate with a quaternary InGaAsP epi-taxial layer making up an all-optical switch. Therefractive indices of these materials and their tem-perature coefficients have been determined near theoperating wavelength of the device. This techniquecould be a useful tool in grating optimization for dis-tributed feedback lasers or optical waveguide devices.

2. Theory

A. Main Outline

The technique we describe here is valid for any sub-strate, multilayered or not, provided that the trans-mission of the upper part of the guiding structure islarge enough to allow large changes in the reflectionor transmission coefficient of the whole structure asthe substrate–guiding layer interface transmissioncoefficient varies.Let nO, nS, and nG be the refractive indices of the

outside medium, substrate, and guiding layer, re-spectively. For a plane wave with frequency v ~\v #Eg if Eg is the energy gap of the guiding layer!, wedenote k 5 vyC 5 2pyl as the wavevector amplitudeof this wave in air; kL is the wavevector in layer L,where kL 5 nLk 5 nONLk; NL is the normalizedwavevector in layer L ~NO 5 1 whatever the outsidemedium and the frequency!; and b is its normalizedcomponent parallel to the interfaces ~the same in allmedia!.If b , 1, the plane wave can propagate in any

medium. If

1 , b , NS, (1)

the wave can propagate in the substrate and can becalled a half-guided wave21 or a leaky wave; it is amode of the whole structure.22If

NS , b , NG, (2)

the wave is guided.If b . NG, no wave can propagate in the structure.If a grating ~with period p! is etched at the struc-

ture surface, a plane wave, with an incident angle u~and b 5 bi! and an incident plane perpendicular tothe grating grooves, will be diffracted with b 5 bd:

bi 5 sin u,

bdm 5 bi 1 mLyp,

2546 APPLIED OPTICS y Vol. 36, No. 12 y 20 April 1997

where L5lynO is the wavelength in the outside me-dium. In the following we assume that u . 0 andm isan integer.The reflected and transmitted beams in the outside

medium correspond to the 0th diffracted order. Form Þ 0, one can obtain ubd

mu . 1 and excite eithermode of the whole structure @relation ~1!# or guidedmodes @relation ~2!#. When 1 , ubd

mu , NS, the am-plitude of the wave that propagates in the wholestructure depends on u.For a transparent substrate with parallel faces,

modes of the whole structure can propagate only forquantified values of b. However, the angular spac-ing between two modes is so small ~Du is approxi-mately a few 1025 rad for a 300-mm-thick substrate!that it cannot be observed experimentally. For anabsorbing or a semi-infinite substrate ~this can ac-count for a transparent substrate whose rear face isunpolished or polished with an angle between its fac-es!, the possible values of b that correspond to prop-agating waves with noticeable amplitude are notquantified: a mode of the whole structure exists forany u.In both cases, if, for a givenm, the ubd

mu amplitudepasses through NS as u varies, a diffracted beam ap-pears in the substrate for uS with

NS 5 usin uS 1 mLypu. (3)

When the substrate is transparent, this diffractedbeam can propagate and be transmitted in the out-side medium through a cross section perpendicular tothe incidence plane and the interfaces. The beampropagates along the interface substrate–guidinglayers for u 5 uS and its emergence angle increaseswith uu 2 uSu. Excitation of this diffracted beam isassociated with a sudden variation in the reflected 5and ~possibly! transmitted 7 intensities for u 5 uS.Angle uS, theoretically defined with a precision bet-

ter than the whole structure intermode angular spac-ing, and experimentally noticeable by a discontinuityon d5ydu, and, in the case of a transparent substrate,a discontinuity on d7ydu and the appearance of adiffracted beam propagating in the substrate, allowsone to calculate NS with Eq. ~3!. Note that this di-rect measurement does not depend on the intermedi-ate layer characteristics or on the grating coupler, buton the incident wavelength and grating period, bothquantities that can be determined with a high degreeof accuracy.When NS , ubd

mu , NG, a diffracted wave canpropagate in the guiding structure but has only no-ticeable amplitude for bd

m 5 bGq, where bG

q is theeffective index of guided mode q. Because the guid-ing layer is usually thin ~of the order of l!, the modalindex has a finite number of values that depend onthe whole structure ~characteristics of each layer andof the coupler!. For

u 5 uGq with usin uG

q 1 mLypu 5 bGq, (4)

a guided mode propagates. It can emerge in theoutside medium ~for a nonabsorbing guiding layer!

and induces rapid changes on the reflected and ~pos-sibly! transmitted intensities. Observation ofguided light for specific values of u allows for modalindex determination. A model is necessary to de-duce precisely the guiding-layer refractive index fromthe reflection 5~u! or transmission 7~u! measure-ments, which can be completed only if the experimen-tal results allow one to determine unambiguouslyeach occurring parameter.

B. Method Implementation

The above-mentioned direct measurement method al-lows one to determine a substrate high refractive indexwith great accuracy. However, the theoretical preci-sion of NS determination suggested by Eq. ~3! can beachieved only if a number of conditions are fulfilled:

NS is precisely defined by Eq. ~3! as long as it is areal quantity; otherwise uS is not so easily deter-mined ~all the more so because uS is not associatedwith a resonance but with a discontinuity on thevarious diffracted intensities!.Observation of a discontinuity on d5ydu or d7ydu

will be much easier if the epitaxial layers havesmaller absorption. Indeed, as soon as absorptionoccurs, modeling is necessary for accurate index de-termination, as in the Brewster angle method.3Finally, NS is unambiguously and easily deter-

mined if the grating period is conveniently chosen,because L is usually quasi-fixed a priori. p is usu-ally selected to have ~i! only one order propagating inthe outside medium ~m 5 0! and ~ii! orders m 5 61propagating in the waveguide or the substrate, whichsimplifies the analysis. This implies that

p #L

max@2, ~NG 1 1!y2#.

Equation ~3! can then be rewritten as

NS 5 Lyp 6 sin uS, (5)

Fig. 1. Diagram of the first-order wave propagation in the sub-strate based on relative values of NS and Lyp.

which shows that uS exists only if

NS 2 1 , Lyp , NS 1 1. (6)

According to the respective values for NS and Lyp,various phenomena can be observed ~see Fig. 1!:For Lyp , NS 2 1 ~or Lyp . NS 1 1!, uS does not

exist as 61 diffracted orders propagate ~or do notpropagate! for any value of u.For NS 2 1 , Lyp , NS, the 21 diffracted order

propagates in the substrate for any value of u,whereas the11 diffracted order exists only for u , uS.In this case, as u varies, the number of diffractedorders that propagate in the substrate varies. Ex-perimentally, as 5 and 7 depend on all other dif-fracted orders, d5ydu and d7ydu present adiscontinuity for u 5 uS. Moreover, light associatedwith the diffracted beam emerges from the structureon the side corresponding to the 11 diffracted orderpropagation direction for u , uS. Actually, it couldhave been observed by Martin et al.15 but was con-sidered as a light peak that is not a priori justified.The observed peak could be accidentally due to thedetector geometric selectivity.For NS , Lyp , NS 1 1, the 11 diffracted order

never propagates, whereas the 21 diffracted orderpropagates only for u . uS. This configuration is themost convenient because, in this case, 5 and 7 coef-ficients that present a quasi-constant value for u , uSshow a sharp change for u 5 uS. Moreover, light canbe observed outside the device, at the edge that cor-responds to the 21 diffracted order propagation di-rection for u . uS.The preceding remarks are valid for either a simple

air–substrate interface or multilayered structures.For the latter, another phenomenon is superimposed:a guided-mode excitation. For guiding structuresLyp is usually used to verify NG 2 1 , Lyp , NG 11, amore restrictive condition thanLyp$max@2, ~NG1 1!y2# as soon as NG is larger than 3, which isusually the case for semiconductor devices. Then, asu varies, waves can propagate either in the guidinglayer for u 5 uG

q with bGq 5 uLyp 6 sin uG

qu @from Eq.

Fig. 2. Diagram of the first-order wave propagation in the sub-strate and waveguide based on the relative values of NS, NG, andLyp.

20 April 1997 y Vol. 36, No. 12 y APPLIED OPTICS 2547

~4!# or in the substrate, the latter exists in a contin-uous range of u, limited by uS.Again, the observed phenomena depend on the rel-

ative values ofNS, bGq,LypwithNS , bG

q ,NG. InFig. 2, we have assumed that NG 2 NS , 1, which ismost often met in semiconductor heterostructures.As already quoted, for NG 2 1 , Lyp , NS, the 21diffracted order propagates in the substrate for any u,whereas the 11 diffracted order propagates in thesubstrate for 0 , u , uS and in the waveguide for u 5uG

q . uS. uGq and uS can be easily determined from

observations of the emerging light associated withthe 11 diffracted order when the guiding layer isnonabsorbing. Measurements of 5 or 7 are not soaccurate.ForNS , Lyp , NG, the 11 diffracted order cannot

propagate in the substrate whereas the 21 diffractedorder will propagate for u . uS. In this configura-tion, nothing can be said about the relative values ofLyp and bG

q and, consequently, about the diffractedorder that excites the guided mode. Then uS ~andNS! and uG

q ~and bGq! can be defined precisely by the

measurements of 5 or7, the precision onNG depend-ing on the relative values of uS and uG

q.For NG , Lyp , NS 1 1, the 11 diffracted order

cannot propagate. The21 diffracted order can prop-agate in the waveguide for u 5 uG

q and in the sub-strate for u . uS . uG

q. In this case, the guided andleaky waves do not exist in the same angular range.uS and uG

q can bemeasured easily andNS andNG canbe determined accurately with 5~u! or 7~u! measure-ments.To summarize, it can be said that the method we

present here to measure directly a substrate refrac-tive index from in situ determination of a leaky angleis all the more precise because absorption in the var-ious layers of the structure is smaller and only onediffracted order, beside the 0th diffracted order, i.e.,the well-known transmitted beam, can propagate inthe substrate. The same experimental technique al-lows for accurate determination of the effective indexof the guiding structure. Determination of the ac-tive layer refractive index is then that much easier

Fig. 3. ~a! Schematic of the multilayered structure under study.~b! Experimental arrangement that was used to obtain angularreflection or transmission curves for various wavelengths.

2548 APPLIED OPTICS y Vol. 36, No. 12 y 20 April 1997

because the uuu existing range for leaky and guidedwaves is distinct.

3. Experiment

The guiding structure under study is made of anIn1–xGaxAsyP1–y layer ~with thickness eQ! epitaxiallygrown on a nonintentionally doped InP substrate andcovered with an InP layer ~with thickness e! in whicha diffraction grating is etched @Fig. 3~a!#. An Al2O3antireflection coating was deposited on the substraterear face, which is parallel to the structure entranceface. Such a structure has been grown to operate asan all-optical switch by the absorption edge of theguiding layer lg, switching being obtained on either areflected or transmitted beam. The quaternary al-loy composition was chosen to obtain lg # 1.035 mmand, for this wavelength, measurements were per-formed on the transmitted beam. Determination ofnS and nG ~and not only of bG

q! is necessary to opti-mize the grating coupler, i.e., the guided-wave con-finement and switching contrast.23

A. Experimental Setup

An incident beam, delivered by a Spectra-Physics Ti:sapphire laser, pumped with an Ar1-ion laser, is sentto a sample through an attenuator @made up of ahalf-wave plate and a Glan–Taylor polarizer; see Fig.3~b!#. The sample can rotate around a vertical axissituated in its plane and parallel to the diffractiongrating grooves. Its temperature can be stabilizedwith a Peltier cell and can be known to within 0.01 K.The incident beam is TE polarized and the incidentangle u is defined to within 0.01°. We inserted thebeam waist into the sample plane using a long-focal-length spherical lens and controlled the beam char-acteristics. The beam intensity was measuredsimultaneously before ~D1! and after the sample ~D2!,so that the ratio D2yD1 is independent of the sourceintensity fluctuations. Detection cells consist of abeam splitter and a silicon photodetector. The ratioD2yD1~l! was first measured without a sample forcalibration. We then took measurements betweentwo symmetric values of u ~6uL! to eliminate possibleautocollimation uncertainties. Angles were thenknown to within 0.01°. The linewidth of the incidentbeam is 40 GHz and the grating period has beenmeasured using a Littrow mount and the 4579.35-Åline of an argon laser. All the measurements de-scribed in the following were taken on any of twosamples, issued from the same wafer, but with twodifferent grating couplers ~different period and mod-ulation depth!:

device A p 5 ~3105.0 6 0.5! Å,

device B p 5 ~3097.0 6 0.5! Å.

B. Determination of the InP Substrate Refractive Index

Within the studied wavelength range ~1.015 mm , l, 1.055 mm!, with nS ' 3.3, observation of a leakageangle uS requires a grating period of 0.24 mm # p #0.45 mm, which is not difficult to implement. We

report the first results that we obtained with device Aat room temperature ~TR 5 25 °C! without makinguse of a Peltier cell. Figures 4~a! and 4~b! show twotypical transmission curves obtained for differentwavelengths: ~a! l 5 1019.7 nm, ~b! l 5 1040.0 nm.Figure 5 shows plots of angles uS and uG versus wave-length. ~uG corresponds to the transmissivity mini-mum associated with the fundamental guided-modeexcitation.!Because nO 5 1, nS 5 NS as given by Eq. ~5!.

From a theoretical point of view, the accuracy is quitehigh ~possibly limited by angular spacing of thetransparent substrate modes: DnS is less than orequal to a few 1025!, but depends on the wavelength.As has already been pointed out and as can be seen inFig. 4~a! for lyp , nS, the 21 diffracted order prop-agates for any u. We can observe only slowly vary-

Fig. 4. Transmission curves obtained with device A ~InGaAsPlayer on an InP substrate, grating coupler period p 5 310.5 nm! fortwo wavelengths: ~a! l 5 1019.7 nm, ~b! l 5 1040.0 nm.

Fig. 5. Experimental dispersion curves obtained for the substrateleakage angle uS~l! and the minimum transmission angle uG~l!associated with guided-wave excitation in device A. Error barsindicate the angular uncertainty that is noticeable for uS at l ,1028.4 nm.

ing changes in 7~u! because of changes in leakageintensity. Eventually, the uncertainty of the uS lo-cation is around 0.1° in Fig. 4~a!, which yields DnS '2.5 3 1023 with our experimental setup. This un-certainty increases with absorption, i.e., at shorterwavelengths. Computing is necessary for greaterprecision.For lyp $ nS ~i.e., l $ 1028.4 nm in device A!, the

11 diffracted order never propagates and the 21 dif-fracted order propagates only in the substrate for u .uS. uS can then be read with a maximum precisionbecause d7ydu ' 0 for u , uS @see Fig. 4~b!#. UsingEq. ~5! and the uncertainties for l, p, and u evaluatedin Section 2, we get DnS ' 1023, i.e., DnSynS 5 3 31024. In this wavelength range, the proposedmethod allows direct index determination with highaccuracy.Figure 6~a! shows nS versus l from Eq. ~5! and uS

data obtained for device A ~reported in Fig. 5! anddevice B; the entiremeasurement set is coherent. Inthe study range and according to measurement accu-racy, it leads to a linear fit for nS~l!:

nS~l! 5 3.8809–0.5531l ~mm! for TR 5 25 °C. (7)

These results are in agreement with those given byStone and Whalen8 for InP samples—with free car-

Fig. 6. Experimental results obtained for ~a! InP substrate and~b! quaternary epitaxial layer refractives indices versus wave-length ~open squares, results for device A; filled squares, results fordevice B!. The error bars were calculated from angular uncer-tainty and represent the estimated accuracy of the measurements.The solid curve indicates ~a! linear fit or ~b! Sellmeier-type fit toexperimental data.

20 April 1997 y Vol. 36, No. 12 y APPLIED OPTICS 2549

rier concentration n ' 1016 cm23—within 1.5 3 1023,and those compiled by Seraphin and Bennett.24 Inagreement with the Sellmeier formula given by Pettitand Turner,25 within 6 3 1023, they are noticeablyhigher ~10.04! than those calculated by Adachi.26Note the slight discrepancy between data obtained indevices A and B that led us to stabilize the sampletemperature.

C. Application to Experimental Determination of aGuiding Layer Refractive Index near the Material EnergyBand Gap

Near the absorption edge of a direct band-gap semi-conductor, its refractive index is a complex quantity,which is sensitive to growth conditions and to impu-rities ~especially in the case of thin layers!. It maybe important to determine this index in situ, i.e., inthe multilayer structure in which it is included, tooptimize the optical or electro-optical component thatwill be made from the structure.Near this energy band gap, absorption is usually

too high to allow propagation of a guided wave in thelayer far from the excited area.27 However, if thestructure exhibits a grating coupler, wheneverguided modes are excited by incident light, they canbe located by 5~u! or 7~u!measurements; these quan-tities vary rapidly around uG

q. Whenever the struc-ture allows only one output propagating beam ~thespecular reflected one!, it has been shown23 that the

Fig. 7. Transmission curves obtained with device A for variouswavelengths: ~a! 1013.9 nm, ~b! 1025.5 nm, ~c! 1031.3 nm, ~d!1040.0 nm; left-hand side, experiment; right-hand side, theoryusing a plane wave model.

2550 APPLIED OPTICS y Vol. 36, No. 12 y 20 April 1997

5~u! curve has a Lorentzian shape, the minimum ofwhich takes place for uG

q when the qth guided modeis resonantly excited. Then, one can easily deducethe modal index from uG

q: bGq 5 Lyp 6 sin uG

q,which is known with great accuracy ~as is nS!.Generally, 5~u! and 7~u! are not Lorentz curves.23

The relation between uGq ~or nG! and the minimum

~here uG! or the maximum of those curves, wheneverthey exist, is not straightforward but must be com-puted. Diffracted intensities have to be calculatedtaking into account the physical parameters of thevarious layers of the grating coupler and of the inci-dent beam. These parameters are then adjusted sothat the theoretical curves fit the experimentalcurves. To do this we developed a model28 that canbe used to describe the diffraction of a Gaussian beamby a multilayered structure including a diffractiongrating of any profile. Experimentally, we have ob-served one minimum, uG, in the transmission curves7~u!, in agreement with the simulations. uG~l! isreported in Fig. 5.Using nS given by Eq. ~7!, we calculated 7~u! as-

suming plane wave excitation, a semi-infinite sub-strate, a sine profile grating coupler, and variouslayer thicknesses that are approximations from otherexperiments.23 nQ9, the real part of the quaternarylayer refractive index, was chosen to fit the positionand shape of resonance curves 7~u! for the entirewavelength range. A few results are reported in Fig.7. Note the good agreement between the shape ofthe experimental ~left! and theoretical ~right! curvesin the entire wavelength range, which confirms theassumptions quoted above. The fit is all the moreaccurate as the guided and leaky waves do not existin the same u range. This is the case for any l .1037 nm in device A and typically in Fig. 7~d! ~l 51040.0 nm!. Calculations show that uG and uG

0 dif-fer by only a few 1022 deg.In Fig. 6~b! we reported nQ9 data relative to sam-

ples A and B. The whole data set can be representedby a Sellmeier law:

nQ92 5 A 1Bl2

l2 2 C2 ,

as shown in Fig. 6~b!. The parameters were cal-culated using a nonlinear least-squares fit29 thatgives A 5 11.1283, B 5 4.835 3 1022, C 5 0.9752mm2.From Fig. 7 it can be seen that, for high damping in

the guiding material, diffracted efficiencies dependmainly on absorption @Figs. 7~a! and 7~b!#, whereas,when damping becomes small @Fig. 7~d!#, a planewave approximation can no longer properly describeexperimental results. For example, the resonancelinewidth depends on the epitaxial layer absorption~nQ0!, the grating modulation depth ~h!, and the in-cident beam divergence ~characterized by a half-waist at 1ye2 maximum intensity, v0! altogether.For a given device, these three parameters were ad-justed simultaneously from a priori approximate val-ues, taking into account the whole data set obtained

with this device. Indeed, the determination is notsingle valued, because identical resonance curve line-widths can be obtained for different ~nQ0, h, v0! sets ata given wavelength.As an example, we adjusted the parameters that

describe device A study conditions and obtained h 5~760 6 10! Å, v0 5 ~225 6 5! mm. Using these val-ues, one can fit the experimental transmission curve7~u! obtained for l 5 1040.0 nm and T 5 23.30 °C byusing the following parameters: nS 5 3.3033, nQ9 53.4051, nQ0 5 ~3.5 6 0.3! 3 1024 ~with e 5 118 nmand eQ 5 700 nm!. Figure 8 shows the experimentalcurve and points computed with extreme values ofthe validity domain: v0 5 220 mm, nQ0 5 3.2 31024; the resonance amplitude is still respected.

D. Application to Experimental Determination of theSubstrate and Epitaxial Layer Refractive-IndexTemperature Coefficients near the Guiding Material EnergyBand Gap

In a device that operates near the energy band gap Egof the guiding layer, it is interesting to determine insitu, not only the refractive indices, but also the tem-perature dependence of these quantities. Indeed,propagation of an electromagnetic wave in a guidinglayer, with photon energy close to the layer band gap,is associated with absorption that can, in turn, inducetemperature changes. The guide refractive indexcan then be modified and nonlinear effects propor-tional to ]nGy]T can appear. Moreover, as the struc-ture is heated, the substrate refractive index itself

Fig. 8. Transmission curve obtained with device A for l 5 1040.0nm ~solid line, experiment; dots, computed results assuming aGaussian beam with a waist v0 5 220 mm, a grating with modu-lation depth h5 76 nm, and absorption in the quaternary layer nQ05 3.2 3 1024.

Fig. 9. Transmission curves obtained with device A for l 5 1040.0nm and various temperatures: 1, T 5 16.65°; 2, T 5 28.40°; 3,T 5 34.50°; 4, T 5 46.70°.

changes, which modifies the confinement efficiency ofthe guiding device. Both quantities ]nGy]T and]nSy]T can be determined easily if the guiding layertransmission is large enough.Experimentally, this has been carried out in device

A ~described above! at the operating wavelength of l05 1040.0 nm at a frequency range where uS and uG~'uG

0! can be determined accurately ~nS , lyp ,bG

0!. The results are plotted in Figs. 9 and 10 for atemperature range from 288 to 318 K and tempera-ture coefficients have been determined around theoperating temperature of T0 5 295 K. Determina-tion of ]nSy]T~l0, T0! is quite straightforward. uScan be defined completely by the relationship sin uS 5Lyp 2 nS. As the temperature changes, uS variesbecause of changes in p and nS. One can then write

DuS 5 S]uS]nSUp ]nS

]T Ul0

1]nS]p U

nS

]p]TDDT

so that

DuS 5 S21

cos uS

]nS]T U

l0

2l

p cos uSaDDT, (8)

where a is the linear expansion coefficient of sub-strate a 5 4.75 3 1026 K21.30 Because curve uS~T!is well defined,

duSdT

5 limDT30

SDuSDTD

can be determined from Fig. 10 all themore precisely.This is the limiting factor for ]nSy]T accuracy, be-cause linear expansion takes over only as a correctivefactor.

Fig. 10. Variation of the substrate leakage angle uS and the min-imum transmission angle uG associated with guided-wave excita-tion versus temperature for l 5 1040.0 nm obtained for device A.Error bars indicate the uncertainty of angle and temperature mea-surements during recording. Solid curves indicate linear fit ~uS!or quadratic fit ~uG! to experimental data.

20 April 1997 y Vol. 36, No. 12 y APPLIED OPTICS 2551

In the studied temperature range, uS~T! is linearand a least-squares fit gives duSydT~l0! 5 2~3.12 60.15! 3 1024 K21, independent of T0, and uS~T0! 52.645°. Using Eq. ~8!, we finally obtain ]nSy]T~l0! 5~2.95 6 0.15! 3 1024 K21, independent of T0 aroundT0. This is in fairly good agreement with the Ber-tolotti et al. results,31 although our results exhibitgreater precision. Determination of ]nGy]T~l0, T0!is not so accurate; indeed, uG depends on all the phys-ical parameters that describe the structure layersand coupler. Experimental data uG~T! can be de-scribed by a quadratic law: uG~°! 5 0.559 1 6.63 31023 T 1 5.10 3 1024 T2 ~T °C!, as reported in Fig.10. Then, duGydT~l0! depends linearly on tempera-ture and can be evaluated around T0: duGydT~l0,T0! 5 ~4.9 6 0.2! 3 1024 K21. Changes in the uGposition that are due to the grating linear expansioncan be evaluated by computing, which gives ]uGy]p '1.1 3 1023 Å21, and by using the linear expansioncoefficient a of InP:

]uG]p

]p]T

~l0, T0! < 1.6 3 1025 K21 < 3%duGdT

~l0, T0!.

If we assume that the linear expansion of the var-ious layers and the grating modulation depth andperiod has a similar corrective influence, we can ob-tain ]nQy]T to within 10% precision from

duG~l0!

dT<

]uG]nQ9

]nQ9

]T.

One can deduce ]uGy]nQ9 by computing ~]uGy]nQ9 552.6°!. From this relation we observe that, near theenergy band gap of the guiding layer, ]nQ9y]T~l0! is aquasi-linear function of temperature and, for T0, weeventually obtain ]nQ9y]T~l0! 5 ~5.4 6 0.5! 3 1024

K21.For the same wavelength, it is clear from Fig. 9

that the imaginary part of nQ varies rapidly withtemperature. Because T 5 T0, nQ0 is quite small.

Fig. 11. Variation of the imaginary part of the quaternary layernQ0 versus temperature for l 5 1040.0 nm. Computed valueswere obtained for a pumping beam waist v0 5 ~230 6 3! mm and agrating coupler modulation depth h 5 76 nm. Error bars werecalculated from beam waist uncertainty. The solid curve indi-cates cubic equation fit to experimental data.

2552 APPLIED OPTICS y Vol. 36, No. 12 y 20 April 1997

Then one can observe a significant influence of thepumping beam divergence on the resonance curvelinewidth. To describe the data set properly, we as-sumed a Gaussian beam with a waist v0 5 ~230 6 3!mm. The corresponding values for nQ0~T! are re-ported in Fig. 11. These results show that, when asemiconductor is heated, either the real or the imag-inary part of its refractive index, measured near theabsorption edge ~l ' lg!, does not vary linearly withtemperature.

4. Conclusion

We have proposed an experimental method that al-lows direct and precise measurements of substraterefractive indices even for high-index materials.This method, based on excitation of substrate leakywaves owing to a grating coupler etched at the top ofmultilayered structures, is interesting for guiding de-vices because it allows the substrate refractive-indexdetermination in situ and is valid as long as superfi-cial layers allow light transmission to the substrate.Moreover, the substrate refractive-index determina-tion allows that of the guiding layer. This quantityhas to be redefined accurately in situ for most cases.We have used themethod to determine the refractive-index dispersion of an InP substrate and its epitaxialInGaAsP layer just below the absorption edge ofthe latter. We were also able to determine therefractive-index temperature coefficients of both lay-ers in the vicinity of the operating condition.

The authors acknowledge Alcatel-Alsthom Re-cherche for supplying the samples, and France Tele-com for its financial support.

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