Ellipsometry with rotating plane-polarized light

Joseph Shamir and Aurel Klein

The polarization plane of a laser beam is made to rotate at high frequency with the help of a special setupcontaining wave plates and an acoustooptic modulator. The application of this beam for ellipsometricmeasurements is investigated and a number of applications are proposed. We describe some novel approach-es for the analysis of thin films and optical surfaces and for measurements on static and time-varyinganisotropic phenomena such as the electrooptic effect, optical activity, and strain analysis using the photo-elastic effect.

1. Introduction

One of the most useful measuring methods for theanalysis of thin films and optical surfaces is ellipso-metry.1 The precision of most ellipsometric systems islimited by the requirement of an accurate registrationof the physical orientation of an anisotropic element(polarizer or wave plate). This requirement existswhether this element is continuously rotating or mustbe adjusted for a specific orientation.

The initial purpose of this work was the applicationof a recently developed method of producing polarizedlight in a rotating plane2 to improve the performance ofmeasuring techniques for thin film and surface analy-sis. During previous research3 the application of po-larization switching for interferometric measurementof the ellipsometric phase shift was found very useful,and in Ref. 4 a modified form of the polarization rota-tor was incorporated. Although good results were ob-tained, the measuring procedure proved to be compli-cated as the sample itself was an integral part of aninterferometric setup.

In Sec. II we describe the system used here for theproduction of the rotating plane-polarized light beam.This will be followed by the theoretical analysis of anovel ellipsometric method. In Sec. IV we describesome measuring procedures where most of the infor-mation about the sample can be deduced from elec-tronic phase measurements. Since mechanical rota-tions or adjustments do not have to be involved in the

measuring process, this method is also applicable tofast, time-varying phenomena. In the approach de-scribed in Sec. V only the amplitude of the ac photode-tector signal is recorded while an analyzer (polarizer)rotates in front of the detector. The advantage of themethod over conventional ellipsometry is that the ac-tual orientation of the polarizer as a function of time isnot required for parameter determination, and eventhe uniformity of the mechanical rotation has no effecton measurement precision.

11. Rotating Plane-Polarized Light

One possible configuration for producing a rotatingplane-polarized light beam is shown schematically inFig. 1. The acoustooptic transducer splits the laserbeam into two plane-polarized beams with a frequencydifference 2 X . After reflection off mirrors M, thepolarization plane of one beam is rotated by 900 withthe help of a halfwave plate. After recombination inbeam splitter B, each outgoing beam may be represent-ed by its Jones vector.5 Thus one may write the com-plex amplitude of the beam E in the form

E = { expj[(w + )t + qp] = expjotFexpj(Qt + ,)expj[(w - Q)t - j exp -j(Qt + 0)I (1)

where X is the original laser frequency, the phase dif-ference between the two beams introduced by the opti-cal system is 2 X , and we assume the two beams tohave unit amplitudes. After traversing a quarterwaveplate with its axis at 450 to the xy axis (see also Fig. 2),the Jones vector will become

E2 = WE1, (2)

whereThe authors are with Technion-Israel Institute of Technology,

Department of Electrical Engineering, Haifa 32 000, Israel.Received 20 July 1985.0003-6935/86/091476-05$02.00/0.© 1986 Optical Society of America.

W= 1 ( 1 -j). (3)

After some algebraic manipulations this may bebrought into the form

1476 APPLIED OPTICS / Vol. 25, No. 9 / 1 May 1986

ml M

Fig. 1. Diagram of the system for production of rotating plane-polarized light beams: L, laser; AO, acoustooptic modulator; M,mirrors; B, beam splitter, and corresponding wave plates. Themeasuring beam is usually E2 while the other beam may serve as a

reference.

Fig. 2. Configuration of the quarterwave plate and polarizer lead-ing to Eq. (9).

E2S

%la PR'D

Fig. 3. Sample (S) positioning in the measuring beam: P is apolarizer; D is a detector for reflecting samples, while D' is an

alternative position for analyzing transparent samples.

E= g expji( - r/2)[sin(t + o - -r/4)J (4)

The x component of the vector varies as sinQt while they component varies as cos~t. This kind of variationmay be considered a plane-polarized beam with itspolarization plane rotating at a frequency U. Actuallyhalf of a rotation corresponds to a whole cycle andtherefore the frequency of the polarization rotation is2Q. Detecting this beam by an isotropic photodetec-tor will indicate a stable dc output. However, if apolarizer is placed in front of the detector we obtain anac signal. To find the field amplitude transmitted by arotated polarizer we denote the Jones matrix of a po-larizer oriented in the y direction by

PY= (o ) (5)

and then rotate it using the rotation matrix (see alsoFig. 2)

R(y) = cosy siny) (6)v-iney cosy)

where 'y is the angle of rotation. Thus the field trans-mitted by the polarizer will be represented in the rotat-ed frame of reference by the vector

E = PYR(y)E2 (7)

Substitution of Eqs. (4)-(6) yields

= 2 expi(wt - 7r/4) - r/4 + s +(cos(Ot - r4+ 0+Y) (8)

To find the intensity detected by an isotropic photode-tector we do not have to return to the original frame ofreference. Neglecting some constant factors, the de-tected intensity will be just the squared magnitude ofthe field:

I, = 2cos2 (Qt-7/4 + v + y) = 1 + sin2(Qt + + ). (9)

This is an ac signal of frequency Q and a phase 2(so + -y),the importance of which for measuring applicationswill be indicated in the next section.

111. Ellipsometry

Beam E2 of Fig. 1 [and Eq. (4)] may be utilized forellipsometric measurements in a number of ways.One possibility is shown in Fig. 3: The rotating plane-polarized beam is incident on the sample at some arbi-trary measured angle and reflected toward a detector.Representing the sample by its Jones matrix,

s=('r' °)'0o r.y

(10)

where r and rp are the complex amplitude reflectioncoefficients for the s and p components, respectively,and the y coordinate was chosen in the s direction.

1 May 1986 Vol. 25, No. 9 / APPLIED OPTICS 1477

The reflected beam is now represented by the Jonesvector,

E 3 = SE 2. (11)

Inserting a polarizer with orientation y as for Eq. (7)will yield the field component

E, = PyR(y)E 3 = PyR(y)SE 2 (12)

to be detected by the photodetector. After perform-ing the indicated algebraic calculations, the detectedbeam in the rotated frame of reference will be

E = 4g exp(wt - r/4)

[ 01X r, cosy cos(Ut + o - r/4) - rp siny sin(Ut + so - r/4).

(13)

The intensity detected by a photodetector is, as for Eq.(9), the squared magnitude:

I, = IE I2 = 2[ IrJ2 coss cos2(pt + <,-7r/4)

+ r, 2sin 2 y sin2(9t + p - r/4)

+ (r,*rp + rsrp*) siny cos-y sin(Ut + --7r/4)

X cos(Qt + p - r/4)]. (14)

Defining the ellipsometric parameters by the relations

-- = tanq/ exp -jA (15)rp

and using some trigonometrical identities, we maywrite

Ir, 2[tan2 q/ cos2-y + sin%2

+ (tank cos2-y - sin2-Y) sin2(Qt + wp)+ tanp cosA sin2-y cos2(Qt + 0)]. (16)

A more useful expression is obtained by defining aphase shift by the relation

tankl =tan sin2-y cosA = 2 tani tany cosA (17)tan 2 Y cos2y - sin y tan2 q, - tan2

7y

to obtain

I, = Irp12[tan2,p cos2 y + sin2'y

+ (tan 2 ' cos- sin2 -y)2 + (tani cosA sin2y) 2

X cos(2Qt + 2so + 3)]. (18)

The two first terms in the brackets constitute a dccomponent while the third term is an ac componentwith frequency 2g and a phase 2s + . In the next twosections this relation will be utilized for a number ofmeasuring procedures.

IV. Direct Measurement of Ellipsometric Parameters

As should be expected, tan/ can be directly mea-sured from the ratio of the two orthogonal compo-nents, i.e., for y = 0 and 7r/2 in Eq. (16), either for the dcor the ac components. In principle it should be theeasiest procedure to use the dc term but we performedall measurements on the ac term since, with our experi-mental setup, the dc term was more susceptible toenvironmental noise (stray light) and measurementshad to be made on the ac component anyway. We also

see from Eq. (17) that for these two angles we havespecial values for l(,y):

3(0) = 0, 3(Ir/2) = 7r,

with the additional relation

1 = r/2 for y = ,.

For y = 450 we obtain from Eq. (17)

tan# 4 5 = 2 tan - tan2 COS.

(19)

(20)

(21)

We see that the measurement of the phase shift isvery useful: One can measure AL by finding the value of,y for which A = (7r/2) and then determine A from themeasurement of a at a different angle, such as 450.For many applications it is adequate to measure al for apreset angle (such as 0, 450, or/and 900); this meansthat the whole measurement is electronic, indepen-dent of calibration, and may be performed withoutmechanically moving parts. Therefore, if we work athigh acoustic frequencies of the modulator, the mea-surement is applicable to fast time-varying processes.

In conventional ellipsometric methods the opticalaxes (usually the s and p orientations) are prealignedwith the assumption that they are known. However,for anisotropic samples, the determination of the axesmay have great importance. For example, in photo-elastic measurements these axes give the directions ofthe strain. Using the present method the optical axescan be determined by rotating the polarizer to find theorientations that yield the phase readings 0 and 7r.

A. Experiment

To test the above-described measuring procedures anumber of preliminary experiments have been per-formed. In the first set of experiments three measure-ments were made on a glass sample at a number ofincidence angles and the measurements were com-pared to the calculated values. Tank/ was measured bythe ratio of the ac components in the s and p directionsand also by finding y for a phase shift 7r/2. The valueof 45 was then used to determine A [Eq. (21)].While the determination of intensity ratios is straight-forward, there is difficulty in the phase measurements.This difficulty stems from the fact that the phase ofthe ac signal also includes the phase angle so [Eq. (18)]which depends on, apart from constant contributionsfrom the optical components, a fluctuating part in-duced by environmental path difference variation inan interferometric configuration. The solution of thisdifficulty is the addition of a polarizer and detector tothe second output of the beam splitter in Fig. 1 to serveas a reference for phase measurement. The zero phaseof the reference signal may be adjusted by rotating thepolarizer [Eq. (9)] to cancel constant differences in so ofthe two signals and also for the determination of theoptical axes for anisotropic samples.

All measured values were within 1.5% of the calcu-lated ones. This is very encouraging as measurementswere quite crude, intensities and phases were deter-mined only by visual observation of the oscilloscope

1478 APPLIED OPTICS / Vol. 25, No. 9 / 1 May 1986

00 15 30 45 60 75 90 105 120 135 150 165 180

10

9

8

7

6

5

4

3

2

00 15 30 45 60 75 90 105 120 135 150 165 180 0 15 30 45 60 75 90 105 120 135 150 165 180

Fig. 4. Curves showing the variation of A as a function of y with 4 and A as parameters: (a) 4 = 30°, (b) ' = 450, (c) 4 = 60°, and (d) 4 = 700.

For all curves A increases downward with the values 00, 30°, 600, and 90°.

traces, angle measurements were of relatively low ac-curacy, and the sample parameters were not exactlyknown.

In the second set of experiments anisotropic phaseobjects were used in transmission measurements. Forthese objects the matrix S [Eq. (10)] has only phaseelements and tano = 1. In one of these experimentsthe axes and the retardation of wave plates were deter-mined and in the second experiment photoelasticstrains in Plexiglas cantilever were observed in realtime. Here too the results were good and demonstrat-ed the effectiveness of the method.

V. Rotating Analyzer Ellipsometry

The measuring procedures described in the previoussection require the setting of an analyzer at knownangles and the measurement accuracy depends on theprecision of the angle setting. This kind of limitationalso exists in conventional ellipsometry, in particular ifthe analyzer or polarizer have to be rotated duringmeasurement and its orientation has to be accuratelyregistered.

In this section we describe a novel approach where arotating analyzer is used as in some conventional ellip-someters but here its exact orientation at each momentis irrelevant for a proper measurement.

Returning to Eq. (18) we define a function

A(-y) = V(tan21p cos2-Y - sin2-y)

2 + tan 24/ cos2 A sin2 2,y (22)

that describes the variation of the ac signal amplitudeas a function of the orientation of the analyzer -y, with 4and A as parameters. A representative set of thesefunctions is shown in Fig. 4. With a continuouslyrotating analyzer, each of these curves corresponds tothe temporal variation of the ac signal as a function oftime for a given sample and is unique for each set ofellipsometric parameters. Thus, the ellipsometric pa-rameters of the sample can be determined from onecycle of the rotating analyzer (one such curve) in aunique way.

To evaluate the characteristics of these curves wedifferentiate A with respect to y and find the extremaof the curve from the relation

1 May 1986 / Vol. 25, No. 9 / APPLIED OPTICS 1479

(tan q/ cos2 y - sin2 -)(tan 2 4' + 1)

X sin2y - 2 tan 24 cos2 A cos2-y sin2y = 0. (23)

It is evident that all curves will have extrema for y = 0,7r/2, and 7r. Furthermore, returning to Eq. (22) we seethat

A(7r/2) = 1. (24)

Thus the measured value of A(7r/2) may serve as anormalization factor for all other measurements. Asindicated earlier, tan/ can be determined from theratio A(O)/A(7r/2). The important point here is thatthese values correspond to two extrema of the curvethat can be easily and accurately measured. The valueof A can be determined from any other measurementon the curve, such as A(7r/4)/A(7r/2) or, if secondaryextrema exists, their position can be used with Eq.(23).

A. Experiment

As in the previous section, here too some preliminaryexperiments were done using relatively crude measur-ing instrumentation. Measurements of tani/ and Awere made for a SiO2 film on Si. Comparison withmeasurements by a conventional ellipsometer yieldeda correspondence better than 1%.

VI. Conclusions

A rotating plane-polarized light beam was appliedfor ellipsometric measurements in a number of ways.The main advantages of this novel approach are therelaxation of the requirements for highly accurate me-chanical components and their replacement by elec-

tronic single-frequency ac signal analysis. In general,the method is quite flexible: It can be used for ellipso-metric analysis of transmitting and reflecting samples,the sample may be isotropic or anisotropic (with thepossibility of also determining the optical axes). Un-like most ellipsometric methods the present techniqueis also applicable for time-resolved measurements onrelatively fast processes. In addition to ellipsometricmeasurements the present method may also be utilizedin polarimetric and photoelastic analytical procedures.

This work was supported by the Technion V.P.R.Fund, Barsky fund for optics research, and by a grantfrom the National Council for Research & Develop-ment, Israel, and the Heinrich Hertz Institute, Berlin,Germany.

References1. See, for example, J. M. Bennett and H. E. Bennett, "Polarization"

in Handbook of Optics, W. G. Driscoll, Ed. (McGraw-Hill, NewYork, 1978); R. M. A. Azzam and N. M. Bashara, Ellipsometryand Polarized Light (North-Holland, Amsterdam, 1977); A. R.M. Zaghloul and R. M. A. Azzam, "Single-Element Rotating-Polarizer Ellipsometer for Film-Substrate Systems," J. Opt. Soc.Am. 67, 1286 (1977).

2. J. Shamir and Y. Fainman, "Rotating Linearly Polarized LightSource," Appl. Opt. 21, 365 (1982).

3. H. Rosen and J. Shamir," Interferometric Determination of El-lipsometric Parameters," J. Phys. E 11, 1 (1978).

4. J. Shamir, "Interferometer with Rotating Linearly PolarizedLight," in Conference Digest, Optics in Modern Science andTechnology, Sapporo, 20-24 Aug 1984, pp. 494 and 495.

5. R. C. Jones, "A New Calculus for the Treatment of OpticalSystems: I, II, III," J. Opt. Soc. Am 31, 488 (1941); "A NewCalculus for the Treatment of Optical Systems: IV," 32, 486(1942).

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1480 APPLIED OPTICS / Vol. 25, No. 9 / 1 May 1986