966 OPTICS LETTERS / Vol. 23, No. 12 / June 15, 1998

Simultaneous measurement of the phase and groupindices and the thickness of

transparent plates by low-coherence interferometry

Masamitsu Haruna and Masato Ohmi

School of Allied Health Sciences, Faculty of Medicine, Osaka University 1-7, Yamada-Oka, Suita, Osaka 565-0871, Japan

Teruki Mitsuyama and Hideyuki Tajiri

Course of Electronic Engineering, Graduate School of Engineering, Osaka University 2-1, Yamada-Oka, Suita, Osaka 565-0871, Japan

Hideki Maruyama

Material and Component Research Laboratory, Kyushu Matsushita Electric Company, Ltd.,1-62, 4-chome, Minoshima, Hakata-ku, Fukuoka 812-8531, Japan

Masahiro Hashimoto

Department of Lightwave Sciences, Osaka Electro-Communication University, Neyagawa 572-8530, Japan

Received February 2, 1998

We propose and demonstrate a novel technique for simultaneous measurement of the phase index, np , the groupindex, ng , and the thickness, t, of transparent plates by use of a low-coherence interferometer. The outputlight from a superluminescent diode is focused upon the front plane of a transparent plate that is used asthe sample. The sample stage is subsequently moved until the light is focused upon the rear plane of theplate. Measurement of the stage movement distance and the corresponding optical path difference allows usto determine both np and ng . By placing the sample between two glass plates, we measured np , ng , and tsimultaneously, with an error of 0.3% or less, for nearly 1-mm-thick transparent plates, including glass andelectro-optic crystals. 1998 Optical Society of America

OCIS codes: 030.1640, 290.3030, 260.3160, 230.0250.

Low-coherence interferometry is useful for opticalranging with an accuracy of the order of a micro-meter.1 – 3 In existing methods of interferometry themeasured value is not the real thickness but the opticalthickness of n 3 t, where n and t are the refractiveindex and the thickness of a sample, respectively. Itis therefore necessary to measure n and t separately.Recently, low-coherence interferometry was applied toin vitro measurement of n and t of human tissue byTearney et al.4 Fukano and Yamaguchi demonstratedthe measurement of n and t of multiple layers on thebasis of low-coherence interferometry with confocalmicroscopy.5 The measurement accuracy was alsodiscussed by Ohmi et al.6 In all these reports, how-ever, there was no discussion of distinguishing betweenthe phase index np and the group index ng, althoughlow-coherence light forms a wave packet that sees ngof the sample. In fact, it was already pointed out, e.g.,by Hopler and Rogers7 and by de Groot and Deck,8 thatthe group-velocity optical thickness was measured inlow-coherence interferometers, resulting in ng, whenthe thickness t was known. np was then derived froma curve f itted over several measured values of ng fordifferent interference filters.7 A tunable laser diodewas also used as the light source of the interferometerto measure the group-velocity optical thickness, andnp was measured by conventional laser interferome-try.9 In addition, it was found that Fourier transformspectroscopy is useful for determination of ng.10

0146-9592/98/120966-03$15.00/0

Here we propose and demonstrate a novel tech-nique for simultaneous measurement of np, ng, andt, by use of a low-coherence Michelson interferome-ter. The advantage of our technique over the existingmethods described above is that np and ng are mea-sured simultaneously with an accuracy of the orderof 0.1%.

The low-coherence Michelson interferometer de-scribed here is shown in Fig. 1. The light sourceis a superluminescent diode (SLD) with spectralwidth Dl ­ 24 nm (FWHM) at center wavelengthlc ­ 850 nm. The coherence length Dlc is 12 mm. Inthe measurement procedure collimated light is focusedupon the front plane of a transparent plate that is usedas the sample, and the position of a reference mirror isthen adjusted so that there is no optical path differencebetween the reference and the sample arms in theinterferometer. Subsequently, we move the samplestage a distance Dz to focus a 203 objective upon therear plane of the plate. As shown in Fig. 1, it can befound from Snell’s law that Dz satisfies the equation

Dz ­ tµ

1 2 z 2

np2 2 z 2

∂1/2

or

np ­

"z 2 1 s1 2 z 2d

µt

Dz

∂2#1/2

, (1)

where z s­ sin ud is the numerical aperture of thefocusing lens. Note that the refractive index must be

1998 Optical Society of America

June 15, 1998 / Vol. 23, No. 12 / OPTICS LETTERS 967

Fig. 1. Low-coherence interferometer with precise trans-lation stages. Light focusing on the front and rear planesof the sample and the corresponding positions of the ref-erence mirror are shown schematically. PZT, piezoelectrictransducer; A/D, analog–digital converter.

np in Eq. (1) because one derives Snell’s law by takingrefraction and ref lection of the wave front of a planewave into account.11 When light is focused upon therear plane, the reference mirror position is shifted by adistance DL so that the null optical path difference be-tween two arms is again determined. Obviously, thesum of Dz and DL equals the optical thickness of theplate. In the low-coherence interferometer shown inFig. 1, the SLD light propagates along the plate in theform of a wave packet because of imperfect monochro-matic light.11 Accordingly, the optical thickness is de-termined by ng, resulting in

DL 1 Dz ­ ngt or ng ­ sDL 1 Dzdyt. (2)

The relationship between ng and np is given by

ng ­ np 2 lcsdnpydldlc , (3)

where dnpydl is usually negative.12 From Eqs. (1)and (2), one can see that two measurable values, Dzand DL, yield np and ng under the condition that t isknown.

Moreover, to measure np, ng, and t simultaneously,we place a transparent plate between two glass plates,as shown in Fig. 2. The SLD light is focused uponref lection planes 1, 2, 3, and 4 in turn. The null opticalpath difference is determined for light focusing uponeach plane. Dz and DL are then measured in a sampleof thickness t with two air gaps, g1 and g2. In theair gaps, it is easily found from Eq. (1) that Dz ­ g1or g2 because np ­ 1. Similarly, we measure the gapg0 between two glass plates before setting the sample.We can then find t:

t ­ g0 2 sg1 1 g2d . (4)

Once t is determined, np and ng are derived fromEqs. (1) and (2), respectively, by use of the measuredvalues Dz and DL.

The measurement was performed with a low-coherence interferometer, as shown in Fig. 1, in which

both the sample and the reference-mirror-chippedpiezoelectric transducer were placed upon translationstages with 0.1-mm resolution. The piezoelectrictransducer was driven by a 500-Hz ac voltage, and theenvelope of the interference fringes was then detected.The sample was also set in the same manner as shownin Fig. 2, where the gap g0 between the two glassplates was measured to be 2292 mm before settingof the sample. Dz and DL were measured by theso-called sample-scanning method.6 We scanned thesample repeatedly by changing the reference-mirrorposition in steps of 2 to 5 mm. Examples of detectedsignal patterns when a z-cut sapphire plate was usedas the sample are shown in Fig. 3. In the signalpattern for each ref lection plane, the signal profilewith the maximum peak was obtained when the nulloptical path difference was performed only for lightfocusing upon the ref lection plane. Such a specif icprofile yielded a suitable combination (xk and zk,where k ­ 1, 2, 3, 4) of the positions of two translationstages. DL s­x3 2 x2d and Dz s­z3 2 z2d were ob-tained for the sample, and we had Dzg1 ­ g1s­z2 2 z1dand Dzg2 ­ g2s­z4 2 z3d, with DLg1 and DLg2 ­ 0in two air gaps. The measured thickness t s­tmd ofthe sample was then calculated from Eq. (4). Besidesz-cut sapphire, the measurement was made for electro-optic crystals, such as x-cut LiNbO3 and z-cut LiTaO3plates, fused quartz, f lint glass (HOYA FD60), andCrown glass (HOYA BaCD14). The birefringence

Fig. 2. Setting of a transparent plate used as the samplefor simultaneous measurement of np , ng, and t.

Fig. 3. Detected signal patterns for the ref lection planesin the case of a z-cut sapphire plate. The signal prof ilewith the maximum peak for each plane is shown by a boldcurve.

968 OPTICS LETTERS / Vol. 23, No. 12 / June 15, 1998

Table 1. Simultaneous Measurement of np, ng, and t at l 5 850 nm

Measured Values smmd Values Calculated from Eqs. (1), (2), and (4) with DLand Dz

Optical Path Thickness Phase Index Group IndexCompared Values Difference

ObjectMoving

Distance tm Error Error ErrorMaterial nps

a ngsa ts smmdb DL Dz smmdc Dt s%dd npm

e Dp s%dd ngme Dg s%dd

Fused 1.4525 1.4657 1026 800 702 1025 0.1 – – 1.465(3) 0.0quartzf

z-cut 1.7589 1.7793 997 1210 565 999 0.2 1.756(3) 0.2 1.776(8) 0.1sapphire no

x-cutLiNbO3

no 2.2494 2.3411 1022 1940 452 1024 0.2 2.247(5) 0.1 2.335(9) 0.2ne 2.1706 2.2497 1022 1830 468 1024 0.2 2.171(0) 0.0 2.244(1) 0.3

z-cut 2.1501 2.2147 496 865 228 494 0.2 2.149(9) 0.0 2.216(1) 0.1LiTaO3 no

Optical glassBaCD14 1.5947 1.6123 1008 995 628 1008 0.0 1.595(4) 0.0 1.610(1) 0.1

FD60 1.7816 1.8250 1060 1345 590 1060 0.0 1.784(4) 0.2 1.825(5) 0.0

aTheoretical values from proper Sellmeier equations.bMeasured with a micrometer gauge.cOptically measured value determined by Eq. (4).dMeasurement error: Dt ­ j

tm2tsts j, Dp ­ j

npm2npsnps j, Dg ­ j

ngm2ngsngs j.

eDerived from Eqs. (1) and (2), respectively, by use of the measured values DL and Dz. The numbers in parentheses are uncertainties.f The numerical aperture z was calibrated for fused quartz by use of Eq. (5): z ­ sin u ­ 0.140.

of x-cut LiNbO3 was measured without any polari-zation control because of random polarization of theSLD light. For all the materials used as samples, wefound the Sellmeier equations in the literature thatgive the theoretical values of the phase and group in-dices, nps and ngs, respectively. The measured valuesof tm, Dz, and DL are summarized in Table 1.

Before we calculated the phase and group indiceswith Eqs. (1) and (2), it was necessary to calibratethe numerical aperture z of the focusing lens. Equa-tion (1) was rewritten as

z ­

µt2 2 np

2Dz2

t2 2 Dz2

∂1/2. (5)

The calibration was made for a fused-quartz plate bysubstitution of the measured thickness tm, Dz, andthe theoretical value of nps into Eq. (5), resulting inz ­ 0.140. The measured phase and group indices,npm and ngm, respectively, were then calculated fromEqs. (1) and (2) by use of Dz and DL for other materials.The measurement error Dt of the thickness was evalu-ated by comparison of tm and the measured value tswith a micrometer gauge, and the index measurementerrors Dnp and Dng were determined by differences ofjnpm 2 nps j and jngm 2 ngsj, respectively, as shown inTable 1. All the errors are less than 0.3%.

In conclusion, we have demonstrated simultaneousmeasurement of the phase and group indices andthe thickness of transparent plates, with an error of0.3% or less, with low coherence interferometry. Webelieve that this method will be widely used for precise

measurement of the indices of all-optical materials,including crystals, glass, and polymers.

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