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Fast three-dimensional ellipsometry
Surface Science 0 North-Holland
96 (1980) 149-155 Publishing Company
FAST THREE-DIMENSIONAL
ELLIF’SOMETRY
G.R. BOYER, B.F. LAMOUROUX Ecole Polytechnique, d’optique Appliqute,
Received
20 August
and B.S. PRADE
Ecole Nationale Supkrieure de Techniques AvancCes, Lahoratoire Batterie de I’Yvette, F-91 120 Palaiseau, France
1979
An automatic scattering method of three-dimensioal ellipsometry is described. It provides the Mueller matrix of an elementary volume within a birefringent medium. The method requires a piezo-optic modulator, four lock-in amplifiers, and an on-line micro-computer for data processing. Typical performance parameters such as response time, accuracy, and spatial resolution are given.
1. Introduction The optical methods allowing stress determination in a volume use 3 polarimetric measurements along 3 orthogonal directions [ 1,2] with respect to a reference direction, from which the azimuth of the neutral lines and retardation of the associated birefringence are measured at each point. Many methods previously used necessitate a rotation of the sample, and thus are inadequate for fast measurements. The present paper describes a three-dimensional scattered-light automatic technique for measurement of the stresses inside a transparent volume. Since the birefringence associated with an elementary volume can be measured without rotation of the sample, fast analysis of the stresses is possible. This method can also be applied to anisotropic media presenting no dichroism, such as stressed photo-elastic solids, or flow-birefringence liquids. Each measurement conducted on any elementary volume inside the sample is made by analyzing the polarization state of the light scattered along two directions making a n/4 angle and in a plane perpendicular to the incident light beam and the associated Mueller matrix of each elementary volume is calculated in real time with a micro-computer.
2. Principle of measurement Fig. 1 shows the schematic arrangement. The incident light traverses successively the polarizer P, defining the reference direction, a photo-elastic modulator M with principal axes at n/4 to the reference directions and the sample from N to N’. 149
G.R. Boyer et al. / Fast 30 eilipsometry
150
Fig. 1. Principle
of fast three-dimensional
stress analysis.
Let us consider an elementary volume bounded by points P and Q, the scattered light issuing from these two points, and following the directions Dr //P and Dz at n/4 to P, is collected by two objectives giving the image of P on the detectors Dr, Dz and the image of Q on the detectors D;, Di. It will be shown later that the simultaneous measurements related to Dr, D2 permit computation of the Mueller matrix M(xe) corresponding to the path NP, while Di, Di give the matrix M(x, + A_x) corresponding to the path NQ. These matrices represent generally elliptical birefringence. The elementary matrix B(xe) is obtained by the equation M(x, + ax) = B(x,) M(x,)
.
(1)
3. Analysis of the method The Stokes vector of a light beam emerging from an optical element characterized by a 4 X 4 Mueller matrix is the product of this matrix and the incident Stokes vector Vi. So we have for Dr : AeM(x)DPVi
= RI 9
(2)
where A0 is the matrix of the scattering process, that is assumed to be a Rayleigh scattering, and can be represented by a linear polarizer parallel to the reference direction. M(x) is the matrix to be measured, D represents the modulator and P the linear polarizer. R is the Stokes vector of the collected light and the intensity is the first component of R. R i is obtained by replacing M(x) by M(x + dx) in (2). Similarly Rz is expressed by A,/,MDPVi
= R2)
(3)
151
G.R. Boyer et al. /Fast 30 ellipsometry
and RI, is obtained matrix [3] : 1 1 p=-
0
by substituting
M(x + dx) for M(x). We express each Mueller
I
0
11100
10
0
0
0 cos .$
0
-sin
1 $
D= 200 00
0
0
00
1
0
0
0
0 sin 5
0
cos g
where t = Go sin Clt, and J10 and 0 are the amplitude modulator, respectively,.
and the frequency
1
0
0
0
0
D2-E2-F2+G2
2(DE + FG)
2(DF - EG)
0
2(DE - FG)
-D2 tE2 - F2 +G2
2(DG+EF)
0
2(DF +EG)
-2(DG
-D2-E2tF2tG2
M(x) = - EF)
of the
(4)
where D=Msin6/2,E=Csin6/2,F=Ssin6/2,andG=cos6/2. Here,histhe retardation angle of the birefringence, and M, C, S define the fast eigenpolarization of the elliptical birefringence. The scattering of the incident light, along the reference direction is described by the matrix P, while in the n/4 direction it is represented by: \ 1 0 1 0 1 0 A
n/4=j1
0
0
0
0
o
1
o’
0
?
0
Let Ir,zi, Iz,I; be the intensities D2, DL respectively: I,
= $,
[ 1 + M,,(x)
I2
=
[I
;I,,
+
of the incident
light on the detectors Dr ,D;,
cos t + M24(x) sin $1 ,
(5)
M32(~) cos ,$t Ms4(x) sin ,$] ,
(6)
Zi = iI0 [l +PvJ~~(x t dx) cos t + M,,(x
+ dx) sin g] ,
(7)
Zi = $I,, [ 1 + M,,(x
t d.x) sin t] ,
(8)
t dx) cos ,$t M,,(x
where I0 is the incident polarization-modulated light intensity, and Mji are the elements of M. Using the expansions for cos $ and sin ,$ with t = $e sin at and $e = 2.405 rad, SO that ge($e) = 0 (where gfl(x) is the Bessel function of order n), we obtain for
1.52
G.R. Boyer et al. /Fast 30 ellipsortletry
11312: 1, = ;I, 11 + 2 ~l($,>M2,(x)
sin Rt + 2 9z($O)M22(~) cm 2fit t ,..I,
(9)
1, = ;1,[1
sin 52t + 2 92($O)M32(~) cos 2CLtt ...I.
(10)
+ 2 HIM,,
We have limited (9) and (10) to the frequencies s1 and 2R. Measurements amplitudes of components of frequencies 0, L?, 2R give ;&I, 1, 9I(tiO)M24(X)>
1” 9T(G,)M,,(X)
of the
for I,,
RIO)1, 9, (tie )M34(~),Jo 3 Cti,)M32(~) for I2 which are related, after suitable normalization,
to the 4 elements
of the matrix of
eq. (4). Since M(x) is a unitary matrix, it is formed of unit and orthogonal rows and columns vectors, yielding a sufficient number of equations to retrieve the other elements. Moreover, we note that Ma3 = M2&s2 -M3&f22. We must express MD(X), M&), &Z(X) and M4&) as functions ofM&), M&x)M&), and Md3(x). By using the fact that the second row is a unit vector, we obtain M,, = k(l -M&
- M;,)‘n,
and by using the orthogonal Ma, = -lM,,
M&),
M34
property:
+M~2M~2lK~>
42 = -[Mm43 +MdfJlMu, Mat,,= --Pf,&fz4 +Mdf~JlK,~. Since the sign of Mz3 is undeterminated, there are two sets of solutions M(+)(x), M(-)(X). By the same process applied to M(x + dx) we obtain two other sets of solutions M(+)(x + dx), MC-)(x t dx). The elementary matrix B(x) is defined by (1). Because M-r (X) = IV(X)) we obtain B(x) = M(x +
(11)
dX)M’(x).
The two-fold
solutions
for M(x + dx) and M(x) lead to a set of 4 solutions
for
B(x). We now assume that B(x) corresponds to a linear birefringence. In this case we must find some relations between the elements Bii(x) of the solutions B(x). These relations are easily obtained by annulling F in eq. (4). A numerical simulation has demonstrated that only two solutions for the matrix B(x) verify these relations; this leads to an ambiguity on the sign of retardation of the linear birefringent retarder.
G.R. Boyer et al. /Fast 30 ellipsometry
153
4. Realization A scattered-light analysis method requires an intense incident flux; the light source is a 10 W Ar2+ laser associated with a dye laser, with an output of 300 mW in the 6200 ,& region. An optical system focuses the beam in the desirable region. The cell is a Couette cylinder with a static outer transparent cylinder, filled with an aqueous solution of Milling Yellow [4,5] which has a large flow-birefringence effect. The incident light beam is modulated by a Morviie photo-elastic modulator (a = 50 kHz). The light scattered in two directions is collected by two microscope objectives focusing on two RTC XP 1117 photomultipliers. A stop on the photomultipliers materializes the object field represented by points P and Q on fig. 1. The method would have needed two pairs of detectors, but we used only one pair. The detectors are translated along the X axis between the measurements of region P and region Q. Such an apparatus is no longer automatic, but is simple and more convenient for a feasibility experiment. The signal given by each detector is fed to two lock-in amplifiers at R and 2R,
Fig. 2. Experimental arrangement.
154
G. R. Bover et al. /Fast 3D e~li~sor~~~try
and a dc amplifier. The 3 analog signals are processed by an ECTRON 6000 dataacquisition system. The matrices M(x), M(x f dx), and B(x) are computed in real time by a HP 9825 A calculator. The experimental set up is shown on fig. 2.
Adjustment process The gain of each channel is adjusted until the output matrix coincides with some known input matrix. The identity matrix permits adjustment ofMZ2. The matrix of a quarter-wave plate retarder with fast axis in the reference direction permits adjustment of MSG. The same retarder in the 45” orientation gives MZ4; MS2 is adjusted with a 22.5” orientated retarder.
5. Performance The principal factors limiting the precision are the noise caused by the inhomogeneities of the liquid and the intensity and direction fluctuations of the laser beam. In order to reduce the noise, an average of ten measurements is made. IJnder this condition, each term of the Mueller matrix is determined with a precision of 3%. The response time is 0.3 s. In order to evaluate the accuracy of the system we have measured the Mueller matrix when a quarter-wave pIate is present between the modulator D and the point N. By rotating the principal axes of the retarder, some discrepancies appear between the true values of the retardation angle 6 and the azimuth B of the principal axes and the measured values of 6 and 0. 6 remains constant with an accuracy of 1’ and 6 is determined with an error of 1’. The spatial resolution of the system is Ax = 0.3 mm; it depends upon the aperture of the optical system collecting the scattered light.
6. Conclusion The system which we have described is suitable for ellipsometric measurement in a volume. The response time and accuracy can be reduced by using an intensity and direction-stab~ized laser beam and by filtering the aqueous solution of ~~l~g Yellow.
[l] A. Robert and J. Royer, Compt. Rend. (Paris) B281 (1975) 373. Matrix Theory of Photoelasticity [2] P.S. Theocaris and E.E. Gdoutas, 1979).
(Springer,
Berlin,
G.R. Boyer et al. /Fast 30 ellipsometry
155
[3] W.A. Shurcliff, Polarized Light (Harvard Univ. Press, Cambridge, MA, 1962). [4] A.R. Krishnamurthy and J.T. Pindera, in: Foundations of Scattered Light Techniques for Flow Analysis Using Real Liquids, SESA Spring Meeting, Chicago, IL, 1975 (Sot. for Experimental Stress Analysis, 21 Bridge Square, Westport, CT). [5] P. Randria, Birefringence d’Ecoulement en Mecanique des Fluides, These presentee i l’lnstitut National Polytechnique de Toulouse (38me cycle, 1977).
Discussion 0. Hunderi (University of Trondheim): Could this technique also be used to study the Rayleigh Benard instability? B. Prade: I think you can observe and measure birefringence. Deducing the velocity-gradient seems difficult. You need a theory connecting the dielectric tensor e with the velocity gradient aular;for instance: Ejj = A(auj/axj + auj/axi) + e(O)Sjj , where h is the dynamic-optical constant. The Rayleigh-Binard instability is due to change in the viscosity as a function of the temperature so that the dynamo-optical constant remains no longer constant and becomes temperature dependent. R.M.A. Azzam (University of Nebraska Medical Center): Please explain why or how information concerning the effect of the anisotropic medium on the polarization of a propagating light beam can be obtained from the 90” light scattering. The mathematical entity that concisely describes the effect of an elemental volume of the medium on the state of polarization of light is what I call the differential propagation Mueller matrix (see R.M.A. Azzam, J. Opt. Sot. Am. 68 (1978)). Aren’t you actually attempting to measure this matrix? B. Prade: We assume that the scattering light follows the Rayleigh law of scattering. In this case, the scattering process acts as a polarizer. The measurement of the differential propagation Mueller matrix would give more information. In our experiment we assume that the medium presents no dichroism nor depolarization. Our experiment concerns only the measurement of the Jones differential matrix for our birefringence.
96 (1980) 149-155 Publishing Company
FAST THREE-DIMENSIONAL
ELLIF’SOMETRY
G.R. BOYER, B.F. LAMOUROUX Ecole Polytechnique, d’optique Appliqute,
Received
20 August
and B.S. PRADE
Ecole Nationale Supkrieure de Techniques AvancCes, Lahoratoire Batterie de I’Yvette, F-91 120 Palaiseau, France
1979
An automatic scattering method of three-dimensioal ellipsometry is described. It provides the Mueller matrix of an elementary volume within a birefringent medium. The method requires a piezo-optic modulator, four lock-in amplifiers, and an on-line micro-computer for data processing. Typical performance parameters such as response time, accuracy, and spatial resolution are given.
1. Introduction The optical methods allowing stress determination in a volume use 3 polarimetric measurements along 3 orthogonal directions [ 1,2] with respect to a reference direction, from which the azimuth of the neutral lines and retardation of the associated birefringence are measured at each point. Many methods previously used necessitate a rotation of the sample, and thus are inadequate for fast measurements. The present paper describes a three-dimensional scattered-light automatic technique for measurement of the stresses inside a transparent volume. Since the birefringence associated with an elementary volume can be measured without rotation of the sample, fast analysis of the stresses is possible. This method can also be applied to anisotropic media presenting no dichroism, such as stressed photo-elastic solids, or flow-birefringence liquids. Each measurement conducted on any elementary volume inside the sample is made by analyzing the polarization state of the light scattered along two directions making a n/4 angle and in a plane perpendicular to the incident light beam and the associated Mueller matrix of each elementary volume is calculated in real time with a micro-computer.
2. Principle of measurement Fig. 1 shows the schematic arrangement. The incident light traverses successively the polarizer P, defining the reference direction, a photo-elastic modulator M with principal axes at n/4 to the reference directions and the sample from N to N’. 149
G.R. Boyer et al. / Fast 30 eilipsometry
150
Fig. 1. Principle
of fast three-dimensional
stress analysis.
Let us consider an elementary volume bounded by points P and Q, the scattered light issuing from these two points, and following the directions Dr //P and Dz at n/4 to P, is collected by two objectives giving the image of P on the detectors Dr, Dz and the image of Q on the detectors D;, Di. It will be shown later that the simultaneous measurements related to Dr, D2 permit computation of the Mueller matrix M(xe) corresponding to the path NP, while Di, Di give the matrix M(x, + A_x) corresponding to the path NQ. These matrices represent generally elliptical birefringence. The elementary matrix B(xe) is obtained by the equation M(x, + ax) = B(x,) M(x,)
.
(1)
3. Analysis of the method The Stokes vector of a light beam emerging from an optical element characterized by a 4 X 4 Mueller matrix is the product of this matrix and the incident Stokes vector Vi. So we have for Dr : AeM(x)DPVi
= RI 9
(2)
where A0 is the matrix of the scattering process, that is assumed to be a Rayleigh scattering, and can be represented by a linear polarizer parallel to the reference direction. M(x) is the matrix to be measured, D represents the modulator and P the linear polarizer. R is the Stokes vector of the collected light and the intensity is the first component of R. R i is obtained by replacing M(x) by M(x + dx) in (2). Similarly Rz is expressed by A,/,MDPVi
= R2)
(3)
151
G.R. Boyer et al. /Fast 30 ellipsometry
and RI, is obtained matrix [3] : 1 1 p=-
0
by substituting
M(x + dx) for M(x). We express each Mueller
I
0
11100
10
0
0
0 cos .$
0
-sin
1 $
D= 200 00
0
0
00
1
0
0
0
0 sin 5
0
cos g
where t = Go sin Clt, and J10 and 0 are the amplitude modulator, respectively,.
and the frequency
1
0
0
0
0
D2-E2-F2+G2
2(DE + FG)
2(DF - EG)
0
2(DE - FG)
-D2 tE2 - F2 +G2
2(DG+EF)
0
2(DF +EG)
-2(DG
-D2-E2tF2tG2
M(x) = - EF)
of the
(4)
where D=Msin6/2,E=Csin6/2,F=Ssin6/2,andG=cos6/2. Here,histhe retardation angle of the birefringence, and M, C, S define the fast eigenpolarization of the elliptical birefringence. The scattering of the incident light, along the reference direction is described by the matrix P, while in the n/4 direction it is represented by: \ 1 0 1 0 1 0 A
n/4=j1
0
0
0
0
o
1
o’
0
?
0
Let Ir,zi, Iz,I; be the intensities D2, DL respectively: I,
= $,
[ 1 + M,,(x)
I2
=
[I
;I,,
+
of the incident
light on the detectors Dr ,D;,
cos t + M24(x) sin $1 ,
(5)
M32(~) cos ,$t Ms4(x) sin ,$] ,
(6)
Zi = iI0 [l +PvJ~~(x t dx) cos t + M,,(x
+ dx) sin g] ,
(7)
Zi = $I,, [ 1 + M,,(x
t d.x) sin t] ,
(8)
t dx) cos ,$t M,,(x
where I0 is the incident polarization-modulated light intensity, and Mji are the elements of M. Using the expansions for cos $ and sin ,$ with t = $e sin at and $e = 2.405 rad, SO that ge($e) = 0 (where gfl(x) is the Bessel function of order n), we obtain for
1.52
G.R. Boyer et al. /Fast 30 ellipsortletry
11312: 1, = ;I, 11 + 2 ~l($,>M2,(x)
sin Rt + 2 9z($O)M22(~) cm 2fit t ,..I,
(9)
1, = ;1,[1
sin 52t + 2 92($O)M32(~) cos 2CLtt ...I.
(10)
+ 2 HIM,,
We have limited (9) and (10) to the frequencies s1 and 2R. Measurements amplitudes of components of frequencies 0, L?, 2R give ;&I, 1, 9I(tiO)M24(X)>
1” 9T(G,)M,,(X)
of the
for I,,
RIO)1, 9, (tie )M34(~),Jo 3 Cti,)M32(~) for I2 which are related, after suitable normalization,
to the 4 elements
of the matrix of
eq. (4). Since M(x) is a unitary matrix, it is formed of unit and orthogonal rows and columns vectors, yielding a sufficient number of equations to retrieve the other elements. Moreover, we note that Ma3 = M2&s2 -M3&f22. We must express MD(X), M&), &Z(X) and M4&) as functions ofM&), M&x)M&), and Md3(x). By using the fact that the second row is a unit vector, we obtain M,, = k(l -M&
- M;,)‘n,
and by using the orthogonal Ma, = -lM,,
M&),
M34
property:
+M~2M~2lK~>
42 = -[Mm43 +MdfJlMu, Mat,,= --Pf,&fz4 +Mdf~JlK,~. Since the sign of Mz3 is undeterminated, there are two sets of solutions M(+)(x), M(-)(X). By the same process applied to M(x + dx) we obtain two other sets of solutions M(+)(x + dx), MC-)(x t dx). The elementary matrix B(x) is defined by (1). Because M-r (X) = IV(X)) we obtain B(x) = M(x +
(11)
dX)M’(x).
The two-fold
solutions
for M(x + dx) and M(x) lead to a set of 4 solutions
for
B(x). We now assume that B(x) corresponds to a linear birefringence. In this case we must find some relations between the elements Bii(x) of the solutions B(x). These relations are easily obtained by annulling F in eq. (4). A numerical simulation has demonstrated that only two solutions for the matrix B(x) verify these relations; this leads to an ambiguity on the sign of retardation of the linear birefringent retarder.
G.R. Boyer et al. /Fast 30 ellipsometry
153
4. Realization A scattered-light analysis method requires an intense incident flux; the light source is a 10 W Ar2+ laser associated with a dye laser, with an output of 300 mW in the 6200 ,& region. An optical system focuses the beam in the desirable region. The cell is a Couette cylinder with a static outer transparent cylinder, filled with an aqueous solution of Milling Yellow [4,5] which has a large flow-birefringence effect. The incident light beam is modulated by a Morviie photo-elastic modulator (a = 50 kHz). The light scattered in two directions is collected by two microscope objectives focusing on two RTC XP 1117 photomultipliers. A stop on the photomultipliers materializes the object field represented by points P and Q on fig. 1. The method would have needed two pairs of detectors, but we used only one pair. The detectors are translated along the X axis between the measurements of region P and region Q. Such an apparatus is no longer automatic, but is simple and more convenient for a feasibility experiment. The signal given by each detector is fed to two lock-in amplifiers at R and 2R,
Fig. 2. Experimental arrangement.
154
G. R. Bover et al. /Fast 3D e~li~sor~~~try
and a dc amplifier. The 3 analog signals are processed by an ECTRON 6000 dataacquisition system. The matrices M(x), M(x f dx), and B(x) are computed in real time by a HP 9825 A calculator. The experimental set up is shown on fig. 2.
Adjustment process The gain of each channel is adjusted until the output matrix coincides with some known input matrix. The identity matrix permits adjustment ofMZ2. The matrix of a quarter-wave plate retarder with fast axis in the reference direction permits adjustment of MSG. The same retarder in the 45” orientation gives MZ4; MS2 is adjusted with a 22.5” orientated retarder.
5. Performance The principal factors limiting the precision are the noise caused by the inhomogeneities of the liquid and the intensity and direction fluctuations of the laser beam. In order to reduce the noise, an average of ten measurements is made. IJnder this condition, each term of the Mueller matrix is determined with a precision of 3%. The response time is 0.3 s. In order to evaluate the accuracy of the system we have measured the Mueller matrix when a quarter-wave pIate is present between the modulator D and the point N. By rotating the principal axes of the retarder, some discrepancies appear between the true values of the retardation angle 6 and the azimuth B of the principal axes and the measured values of 6 and 0. 6 remains constant with an accuracy of 1’ and 6 is determined with an error of 1’. The spatial resolution of the system is Ax = 0.3 mm; it depends upon the aperture of the optical system collecting the scattered light.
6. Conclusion The system which we have described is suitable for ellipsometric measurement in a volume. The response time and accuracy can be reduced by using an intensity and direction-stab~ized laser beam and by filtering the aqueous solution of ~~l~g Yellow.
[l] A. Robert and J. Royer, Compt. Rend. (Paris) B281 (1975) 373. Matrix Theory of Photoelasticity [2] P.S. Theocaris and E.E. Gdoutas, 1979).
(Springer,
Berlin,
G.R. Boyer et al. /Fast 30 ellipsometry
155
[3] W.A. Shurcliff, Polarized Light (Harvard Univ. Press, Cambridge, MA, 1962). [4] A.R. Krishnamurthy and J.T. Pindera, in: Foundations of Scattered Light Techniques for Flow Analysis Using Real Liquids, SESA Spring Meeting, Chicago, IL, 1975 (Sot. for Experimental Stress Analysis, 21 Bridge Square, Westport, CT). [5] P. Randria, Birefringence d’Ecoulement en Mecanique des Fluides, These presentee i l’lnstitut National Polytechnique de Toulouse (38me cycle, 1977).
Discussion 0. Hunderi (University of Trondheim): Could this technique also be used to study the Rayleigh Benard instability? B. Prade: I think you can observe and measure birefringence. Deducing the velocity-gradient seems difficult. You need a theory connecting the dielectric tensor e with the velocity gradient aular;for instance: Ejj = A(auj/axj + auj/axi) + e(O)Sjj , where h is the dynamic-optical constant. The Rayleigh-Binard instability is due to change in the viscosity as a function of the temperature so that the dynamo-optical constant remains no longer constant and becomes temperature dependent. R.M.A. Azzam (University of Nebraska Medical Center): Please explain why or how information concerning the effect of the anisotropic medium on the polarization of a propagating light beam can be obtained from the 90” light scattering. The mathematical entity that concisely describes the effect of an elemental volume of the medium on the state of polarization of light is what I call the differential propagation Mueller matrix (see R.M.A. Azzam, J. Opt. Sot. Am. 68 (1978)). Aren’t you actually attempting to measure this matrix? B. Prade: We assume that the scattering light follows the Rayleigh law of scattering. In this case, the scattering process acts as a polarizer. The measurement of the differential propagation Mueller matrix would give more information. In our experiment we assume that the medium presents no dichroism nor depolarization. Our experiment concerns only the measurement of the Jones differential matrix for our birefringence.