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Calibration of Two Difrerent Types of Modulators for an Application in Spectroscopic Phase Modulated
Ellipsometry
S. Kouko, C. Llinares
To cite this version:
S. Kouko, C. Llinares. Calibration of Two Difrerent Types of Modulators for an Application in Spectroscopic Phase Modulated Ellipsometry. Journal de Physique III, EDP Sciences, 1995, 5 (8), pp.1211-1227. �10.1051/jp3:1995187�. �jpa-00249374�
J. Phys. III Iiance 5 (1995) 1211-1227 AUGUST 1995, PAGE 1211
Classification Physics Abstracts
07.60F
Calibration of Two Different Types of Modulators for an Appli-
cation in Spectroscopic Phase Modulated Ellipsometry
S.L. Kouko and C. Llinares
C-E-M- Universit6 Montpellier II, Place Eugbne Bataillon, 34095 Montpellier cedex 5, France
(Received 6 January 1995, accepted 9 May 1995)
R4sumd. Dans le but de choisir un modulateur photo61astique pour une application en ellipsomdtrie spectroscopique h modulation de phase, le calibrage de deux modulateurs a At6 r6alisA. Dans un premier temps, le modble ddcrivant chacun des modulateurs
a 6td ddtermind avec pr4cision vu que la proc4dure de calibrage d4pend de celui-ci les deux modulateurs fonctionnent
suivant le modble avec bir4fringence statique ayant un d4phasage de la forme 6
= 60+A sin wt. En
utilisant les proc4dures de calibrage approprides, la tension de pilotage Vmod et la bir6fringence statique 60 ant 6t6 d4termin6es
avec §r6cision
en fonction de la longueur d'onde dans la gamme spectrale s'4tendant de 0,25 ~lm h 0,75 ~lm. Une m4thode pour rdduire les erreurs dues au
calibrage de la tension de modulation et h la birdfringence r6siduelle est dgalement proposde. Un test de la prdcision du calibrage fait en comparant les indices de rdfraction de deux prismes de silice obtenus
avec notre ellipsomAtre et avec la mdthode de la ddviation minimum a montr6 un
trbs bon accord entre les indices mesurds.
Abstract. In order to choose a photoelastic modulator for
an application in spectroscopic phase modulated ellipsometry a complete calibration of two different types of photoelastic mod- ulators is achieved. First the model that describes the behavior of each of them is accurately
determined since the calibration procedure depends on it. Both modulators behave
as a static strain model with a phase shift of the form 6
= 60 + Asinwt. Using the suitable procedures, the driving voltage Vm~d and the static strain 60 of both devices are accurately determined as
a function of the wavelength from 0.25 ~lm to 0.75 ~lm. A method to reduce the errors due to incorrect settings of the modulation voltage and the static birefringence is also proposed.
An accuracy check of the calibration done by comparing the indexes of refraction of two silica
priims obtained with
our ellipsometer and with the high precision goniometer method shows a very good agreement.
1. Introduction
Polarization Modulated Spectroscopic Ellipsometry (PMSE) is an optical non destructive tech- nique to study sample surfaces and thin films. It gives informations on dielectric functions and layer thickness and morphology iii.
A PMSE ellipsometer consists of a light source, a monochromator, a polarizer, a photoelastic modulator (PEM) and an analyzer that analyzes the light beam after reflection on a sample.
The first PMSE ellipsometer was introduced by Jasperson and Schatterly [2]. Since, many studies on refinement of the instrument have been done. The calibration of the photoelastic
modulator which is the heart of the intrument is one of the most important problem to deal with in phase modulated ellipsometry. The PEM is a fused silica bar in which a time varying birefringence is used to modulate the polarization of the linearly polarized light beam.
Three main types of modulators differing from the way the periodic stress is applied in the fused silica bar to achieve the modulated birefringence, were proposed. In the first type, the bar is sandwiched between a fix support and a rod excited by a piezoelectric transducer [3].
This type of modulator may have an important static birefringence with its axes different from the modulation axes. This fact makes it unusable in phase modulated ellipsometry [4]. The second type consists of a piezoelectric transducer and a bar placed against each other and have the same resonant frequency [5]. In the third type of modulator, the bar is excited by shear strain induced by piezoelectric ceramics. Such modulator was investigated by l3adoz et al. who noted a lowering of the static birefringence and an adequate resonant frequency of the device by dividing the monolithic bar of the fused silica in two parts [6].
In all types of modulators the modulated birefringence is achieved through a periodical
stress applied to a fused silica bar. This birefringence generates a periodic relative phase shift b between orthogonal amplitude components in the transmitted light beam. Till now, several different models where used to account for the behavior of the modulator and determine the
theoretical expression of the modulation b. However there is no standard model that suits to all types of modulators so, before operating a modulator in PMSE, one have to determine with
precision its model since the calibration procedure depends on it.
In this paper, we achieve a calibration of two modulators of the second and third type for
an application in phase modulated ellipsometry. The two devices were first tested in order to find out the model that suits to each of them. Then the driving voltage as a function of the wavelength, the static birefringence and the gain factors of the electronic system of
both modulators were determined. The interest of such an experimental investigation is to make a comparison between the models, driving powers and static birefringences which are
determinant parameters in the choice of a modulator for an application in PMSE.
2. Experimental Details
The principle of Polarization Modulated Spectroscopic Ellipsometer (PMSE) has been de- scribed elsewhere [7]. The schematic diagram of the optical and data acquisition systems of the PMSE built in our laboratory is shown in Figure 1. The light source is a 75 W Xe short
arc lamp. The emerging light beam goes through a grating monochromator that scans the
wavelength during the measurements. Calcite prism polarizers are used to polarize the inci- dent beam and to analyze the light reflected by the sample. This light beam is detected by a
photomultiplier tube. The wavelength domain ranges from o.25 ~lm to 0.75 ~lm.
The data acquisition system uses three lock-in amplifiers that measure the Fourier compo-
nents at 1200 Hz, 50 kHz and 100 kHz frequencies of the optical signal. These components
are transferred to a computer that controls the whole set-up. In particular it adjusts the modulation amplitude of the modulator according to the scanned wavelength.
The photoelastic modulator is the most important component of the instrument. It consists of a fused silica bar submitted to a periodical stress that induces a time varying birefringence;
the optical index for light beam with its polarization parallel to the strain differs from the index of a beam with its polarization orthogonal to the stress. When a linearly polarized light passes
N°8 CALIBRATION OF TWO MODULATORS AND APPLICATION IN P-M-S-E. 1213
Monochrcmator Chopper Moclulator
filter
pojanzer a
Xeitght
°4/b~~
source
'er
(picialkk
Comyuw
Fig. 1. Schematic diagram of the ellipsometer.
through the modulator with its direction different from the strain eigen axes, the emerging light
beam has an elliptic polarization. Since the strain is modulated, the emerging beam has its
polarization modulated. The periodic phase shift b induced by the modulator between the two
eigenmodes has been thoroughly investigated [4]. In the earlier papers [2,7] b(t) was assumed to be b(t) = A sin uJt, where uJ is the resonant angular frequency of the modulator unit and A the modulation amplitude. However, this no longer holds if the device possesses imperfections.
Some other models were then proposed to describe more accurately the modulator behavior.
G-E- Jellison et al. [8] propose a model in which the modulator imperfections are included in
a static stress-induced retardation bo so that the phase shift has the form b
= bo + A sin wt.
A more complex model with higher harmonics has also been described. It takes mainly into account the discrepencies in the electrical excitation signal [4]. The general form of the phase shift is then;
b(t) = A sin uJt + A2 sin 2uJt + A3 sin 3uJt +
All these models were investigated assuming that the static and modulated birefringence have the same eigen axes. This assumption is ruled out in the model of the photoelastic modulator with distributed birefringence proposed by Badoz et al. [9]; the principal axis of the static
birefringence is inclined to the modulation axis by an angle ~f.
Our purpose, here, is to find out which of these models suits to the behavior of our modu- lators.
We studied two modulators of two different types. The modulator A manufactured by HINDS
(PEM-80) belongs to the second type [5]: it consists of a precisely cut crystalline quartz (-18°X
cut parallelepiped glued to a fused quartz parallelepiped. The piezoelectric crystalline quartz is driven at its resonant frequency (50 kHz) by an applied ac voltage as shown in Figure 2a
main
~XY
Z fused Silica bar
(photcelasfic medium) beam path
a)
and bkeffingence eigen axes
fused Silica bar
(photcelastic medium) beam path
a
i
ais the maximun s~ess or bilefnngence as a function of the position alongthe bar.
b)
Fig. 2. Schematic diagram of the two types of modulators. a) Modulator "Hinds" excited by longitudinal stress, b) Modular "Microcontrole" excited by shear stress.
Monochromator Chopper Modulator
ii
Xe light ~~~~~ P°'3'lZer ~~~Y~~~
p~~~~
~°~~~~ ~°~"~~~~~
multiplier Fig. 3. Straight-through configuration.
which will set up an oscillating strain in the material. Mechanical coupling induces a strain in the fused quartz which is tuned to the same resonant frequency. The strain is preliminary in the X-direction, resulting in a time-dependent stress; therefore, the refractive index difference
between the X- and Y-directions, which is induced by the photoelastic effect, is a function of time. A time varying birefringence created thus modulates the polarization of the linearly
polarized incident light.
N°8 CALIBRATION OF TWO MODULATORS AND APPLICATION IN P-M-S-E. 1215
The modulator B manufactured by MICRO-CONTROLE is a parallelepiped photoelastic
rod (Fig. 2b). This modulator is one of the third type [6]. It is driven to oscillate by a
ceramic piezoelectric transducer bonded approximately at a 1/4 distance from the end of the
photoelastic bar. The transducer driven by an electric field applied along Y oscillates along X
which induces longitudinal standing vibrations in the photoelastic rod at the proper frequency f. The vibrating photoelastic medium is thus a birefringence plate. The induced birefringence
is controlled by the voltage which appears across the pickup transducer.
The two modulators were studied on a straight optical bench (Fig. 3). It corresponds
to a configuration of the ellipsometer with no sample, or with an equivalent sample whose ellipsometric angles are 4l
= 45° and A
= 0°. Having thus eliminated the sample as an
unknown element, it is possible to focus on the modulator's behavior, examine with which model it can be described and determine all the calibration parameters.
3. Theoretical Considerations
The intensity of the incident light on the detector in a PMSE experiment has been calculated by many workers [2,7, 8,10] using Jones or Muller matrices. It takes the general form
IpM(t)
= I(Io + lasinjbjt)j + Iccosjbjt)j) ii)
where I, lo, Is, and Ic as a function of the azimuthal settings of the polarizer (P), the modulator
(M), the analyzer (A), and the ellipsometric angles il and A of the sample, are given by [4]
I j~2~TP~~ + ~Ts(~
° 4
(2) lo = II cos 2il cos 2A
+ cos 2(P M) cos 2M(cos 2A cos 2il)
+ sin 2A cos A cos 2(P M) sin 2il sin 2M] (3)
la = sin 2(P M) sin 2A sin 2il sin A (4)
Ic = sin 2(P M [(cos 2il cos 2A) sin 2M
+ sin2Acos2Msin2ilcosA] (5)
where Eo is the amplitude of the electric field of the incident light beam. rp (ra) are the
complex Fresnel reflection coefficients for light polarized parallel (perpendicular) to the plane
of incidence. The ellipsometric angles il and A are defined by the complex reflectance ratio:
p = rp/ra = tan il exp(iA).
In straight-through configuration il = 45°, A
= 0° and whatever the aximuthal settings A, M and P, la
= 0. The detected intensity is then
ip~(t)
= i(io + i~cosjb(t)j) (6)
where lo = I + cos 2(P M) cos 2(A M) and Ic = sin 2(P M) sin 2(A M). To relate the measured harmonics of the signal to I, lo and Ic, one has to know the Fourier development of
cos[b(t)] to the sixth harmonic (See Section 4.1).
3.I. HARMONIC MODEL. When the modulator is considered as an ideal device, the retar- dation angle introduced is [2,7]
b(t) = A sin wt. (7)
Using equation (6) and Fourier development of cosb(t) the detected intensity becomes [4]
IpM(t)
= I(Io + Ic[Jo (A) + 2J2(A) cos 2uJt + 2J4(A) cos 4uJt + 2J6(A) cos 6uJt + ) (8)
where Jn(A) is the nth Bessel function of A.
It can be seen that each even harmonic of the signal at the frequency nuJ is proportional to the Bessel function of order n. The odd harmonics are all null.
3.2. RESIDUAL STRAIN MODEL. A residual static strain bo can be used to account for the imperfections in the modulator [5]. The phase shift is then :
b = bo + A sin wt. (9)
Using again equation (6) and Fourier development of cosb(t) the expression of the detected
signal is:
IpM(t)
= I(Io + Ic[Jo(A) 2boJi (A) cos uJt + 2J2(A) cos 2uJt 2boJ3(A) cos 3uJt + 2J4(A) cos 4uJt 2boJ5(A) cos 5uJt + 2J6(A) cos 6uJt + ]) (10)
In this case it appears that the amplitude of each odd harmonic at frequency 2n +1 where
n is an integer is proportional to the Bessel function of order 2n +1 and the static strain bo.
The even harmonics of the signal are only proportional Bessel functions as above.
3.3. MULTIPLE HARMONICS MODEL. This model is used to describe the modulator be- havior when its electrical excitation signal is not exactly sinusoidal. The higher harmonics it contains are likely to be present in the stress, and therefore in the optical modulation. The
modulation signal is then [4]:
b(t) = A sin uJt + A2 sin 2uJt + A3 sin 3uJt + Ill
A2 and A3 are generally supposed to be small enough to legitimate a first order development.
The expression of the light intensity detected is then:
~PM(1) ~(~0 + ic(J0(-~) -~3~3(-~) -~2(~i(W4) + ~3(W4)j C°Sldl
+ (2J2 IA) A3[Ji(A) + J5IA)]) cos 2uJt A2[J5(A) Ji(A)] cos 3uJt + (2J4(A) A3(J7(A) Ji(A)]) cos 4uJt A2(J7(A) J3(A)] cos 5uJt
+ (2J6(A) A3(J~(A) J3(A)]) cos 6uJt + ]). (12)
The amplitude of the harmonics are proportional to combinations of Bessel functions.
3.4. CONSEQUENCES. It can be seen in straight-through configuration that the theoretical expressions of the harmonics of the light intensity depend on the model of the modulator.
Therefore a study of the detected harmonics can be used to determine the model which describes precisely the modulator's behavior. Once this model is known, the proper theoretical expression of IpM(t) in a PMSE experiment with a sample can be established.
In the residual strain model the detected intensity of the light reflected by a sample is given by equation II). Where b
= bo + A sin wt. Using the Fourier development of cosb(t) and sin b(t) [4] the light intensity can be expressed as
N°8 CALIBRATION OF TWO MODULATOBS AND APPLICATION IN P-M-S-E- 1217
IpM(t)
= I(Io + Jo(A)[Ic + bold] + 2Ji(A)[Ia bole] sin uJt + 2J2(A)[Ic + bola cos 2uJt +
= I(Idc + IUJ sin uJt + I2w cos 2uJt + ) (13)
The simplifications of the expressions lo, la and Ic (Eqs. 3, 4, 5) are generally obtained using
two measurement configurations described elsewhere [4,8].
The experimentally determined parameters are generally R~ and R~~, defined by:
Rw = ([ =
() (14)
~~~ ~~ ~
~~~~
Where So " aide, S~
= bI~ and S2w " cI2w are the amplitudes of the detected harmonics
and Idc, I~ and I2w the theoretical expressions deduced from equation (13). The factors
a
and ~ are considered as terms of the form
= kill and ~
= k212. k, and I, are described
a a a
elsewhere [8]. The two ellipsometric parameters il and A can be extracted from R~ and R~~.
For example in configuration II (P M
= 45°, M
= 0°, A
= 45° ), using expressions (14), (Is)
and assuming that Jo(A)
= 0, the relations used are:
il - Arcsin (i /
~i
lib)~ + lhl~j ) (16)
~mi ~
jfi fl~ + bo
A = Arctan ~ ~) (17)
) fibo
Where Rm~ = 2 Ji(A) and R~f
= 2~ J2(A) depend only on the detection and data acquisition
a a
system.
4. Calibration Procedure
In a PMSE experiment, the determination of all the calibration factors including the modu- lation voltage required to have Jo(A)
= 0 at each wavelength, the modulation static strain bo, the factors R[~~ and R[~ as a function of the wavelength and the azimuthal angles of the
optical elements with respect to the incident plane, is essential for the determination of the
ellipsometric parameters il and A.
4.I. EXPERIMENTAL DETERMINATION OF THE MODELS OF THE MODULATORS. In order
to find experimentally the models of the modulators, the ellipsometer is set in straight-through configuration without any sample. The polarizer P and the analyser A are set so that the
extinction occurs. In this situation, the modulator head which is not excited is inserted between the two optical components. For the two modulators tested the extinction no longer occurs
due to the presence of a static birefringence. This means that the two devices do not behave as the harmonic model modulator. The static birefringence eigen axes are determined by rotating
