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An error evaluation technique for the angle of incidence in a rotating element ellipsometer
using a quartz crystal
View the table of contents for this issue, or go to the journal homepage for more 1999 J. Phys. D: Appl. Phys. 32 2246
(http://iopscience.iop.org/0022-3727/32/17/315)
An error evaluation technique for the
angle of incidence in a rotating
element ellipsometer using a quartz
crystal
Y F Chao, M W Wang and Z C Ko
National Chiao Tung University, Electro-optical Engineering, Hsinchu, Taiwan E-mail: yschao@cc.nctu.edu.tw
Received 28 August 1998, in final form 14 May 1999
Abstract.The error in the angle of incidence for a rotating element ellipsometer is evaluated using a uniaxial quartz crystal. At a fixed angle of incidence with respect to the surface of reflection, the ratio of reflectance in the parallel to that in the perpendicular electromagnetic field is measured by rotating the quartz crystal through a full cycle. We determine the deviation in the angle of incidence by comparing the experimentally measured reflectance ratio to its calculated value.
1. Introduction
The reflection technique has been widely used to determine the refractive indices of materials, even in powder form [1–4]. It was proved [4] that the ratio of reflectance in the parallel (Rp) to that in the perpendicular (Rs) electromagnetic field is
insensitive to the condition of the surface. The ellipsometric parameter tan ψ is the square root of the reflectance ratio [5]; it can be easily obtained using ellipsometric measurements. Ellipsometry is a powerful technique for determining the optical properties of materials, such as isotropic multilayer thin films [5] and anisotropic crystals [6]. Rotating-element ellipsometry [7] is now known to be more amendable to automation than the conventional null ellipsometry. It has been known [8] that the beam deviation in the rotating-element ellipsometry can cause serious errors in the angle of incidence (AI), which is one of the crucial angles in ellipsometric measurements. Although both McCrackin et al [9] and Chao et al [10] have systematically align the azimuthal angles of the polarizer and analyser to the incident plane in an ellipsometric system with a high degree of accuracy, no one has dealt with AI deviation.
Because the analytical Fresnel reflection coefficients for uniaxial crystals have been developed explicitly [11], it is possible to calculate the ellipsometric parameter ψ as a function of the azimuthal angle for a known uniaxial crystal at a given incident angle. For aligning an incident angle in a PSA ellipsometry [12], we measure the ellipsometric parameter ψ of a quartz crystal by rotating it around its normal line for a full cycle. A deviation in AI can be clearly observed by comparing the improved values of ψ with its calculated values. Thus AI deviation can be determined by fitting the measured parameter to its calculated value. Because of this
evaluation, we are able to use a dichroic sheet polarizer to substitute a prism polarizer. Taking the AI deviation into consideration, the deduced refractive index of glass (BK7) is comparable to that specified in the vendor’s catalogue.
According to our analysis, only a uniaxial crystal with small difference between the ordinary (no) and extraordinary
(ne) refractive indices can resolve the small error of AI. This
conclusion is borne out numerically in this paper. We have also applied this technique to measure the angle between the normal to a cleavage plane and the optical axis of a yttrium orthovanadate (YVO4) crystal to explore the feasibility of this PSA system.
2. The reflectance ratio of a uniaxial crystal
The reflected χrand incident χipolarization states are related
by [6]
χr=
(rsp/rss)+ χi
(rpp/rss)+ (rps/rss)χi
(1) where rxyis the Fresnel reflection coefficient for the parallel
(p, i.e. x) and perpendicular (s, i.e. y) polarizations. According to [11], we summarize the analytical expressions for these Fresnel reflection coefficients for uniaxial crystals in appendix. The complex pseudoreflectance ratio was defined [6] ashρi = χi/χrfor anisotropic media, while in general ρ
is defined as [5] ρ= tan ψ ei1 thus tan2ψ= χi χr 2. (2)
The angle of incidence in a rotating element ellipsometer
Figure 1.The reflection geometry: θiis the incident angle, the xy
plane is the reflecting face of the crystal, the zx plane is the incident plane, the z-axis is the normal line. OA is the optical axis of the crystal.
Since the cross terms vanish in an isotropic medium, tan ψ [5] equals|rpp/rss|, which is the conventional expression
for the ellipsometric parameter. The reflection geometry for a uniaxial crystal is shown in figure 1. A simple model for anisotropic crystals was proposed by Aspnes [13]: the measured ellipsometric parameters for a particular θa equal
those of the effective isotropic sample whose refractive index is given by its dielectric tensor projection onto the sample surface along the incident direction. This implies that
tan2ψ= Irp
Irs
(3) where Irp represents the reflected intensity parallel to the
incident plane and Irs represents the reflected intensity
perpendicular to the incident plane,for P= 45◦, i.e. χi= 1.
According to equation (3), one can obtain tan ψ simply by measuring the reflected intensities Irp and Irs. If the
optical axis of a non-absorbent uniaxial crystal is parallel to the reflection surface, i.e. θc= 90◦, then the ellipsometric
parameter ψ can be characterized by a twofold symmetry with respect to the azimuthal angle θa. Since we are only
interested in determining the AI in a PSA ellipsometry, we simulate the ellipsometric parameter function ψ(θa)
for a uniaxial crystal with no and ne as its ordinary and
extraordinary refractive indices, respectively. Furthermore, we assume the optical axis of the sample crystal is parallel to the reflection surface so as to obtain the twofold symmetry for comparison.
Two types of crystal are simulated to examine their resolving power in AI. The difference of one crystal’s ordinary and extraordinary refractive indices is about one order of magnitude lower than that of the other crystal. The function ψ(θa)is simulated for χi = 1, i.e. P = 45◦, and
optimized [12] by χi= −1, i.e. P = −45◦, to eliminate the
error caused by the misalignment of the polarizer; according to equation (3), one can obtain
tan ψ= Irp Irs p=45◦ Irp Irs p=−45◦ 1/4 . (4)
The angle of incidence θiin figure 1 is set to be 45◦and 44.94◦
for quartz crystal (no= 1.544 and ne= 1.553) and 45◦and
44.5◦for yttrium orthovanadate crystal (YVO4, no= 1.9929,
ne= 2.2154), as shown in figures 2(a) and 2(b), respectively.
Figure 2.The numerically simulated ψ as a function of azimuthal angle of θawhile θc= 90◦: (a) quartz crystal, no= 1.544,
ne= 1.553 at incident angles of 45◦(full curve) and 44.94◦
(dashed curve); (b) yttrium orthovanadate crystal, n0= 1.9929, ne= 2.2154 at the incident angles of 45◦(full curve) and 44.5◦
(dashed curve).
Figure 3.The numerically simulated ψ as a function of azimuthal angle of θawhile θc= 90◦, for quartz crystal of no= 1.544,
ne= 1.553 at incident angles of 57◦(full curve) and 56.94◦
(dashed curve), respectively.
These two figures indicate that the resolving power of quartz crystal is about one order of magnitude higher than that of YVO4 crystal. The resolving power is even higher when the AI is at the Brewster angle (the numerical simulated curves for θi = 57◦are shown in figure 3). However, the reflectance
of the parallel electromagnetic field is too low to be practical for measurement by this intensity ratio technique, especially for a non-absorbent material.
3. Experimental procedures
Figure 4 depicts the experimental set-up. The light (L: HeNe laser) passes through a polarizer (P: dichroic sheet polarizer of extinction ratio 10−4) whose azimuthal angle is set to be
Figure 4.A schematic set-up of the PSA ellipsometer: L, light source (He–Ne laser); P, polarizer; A, analyser; D, detector.
Figure 5.ψversus θafor quartz crystal of θc= 90◦with
no= 1.544, ne= 1.553: (a) θi= 45◦(full curve, calculated;
•, measured) and θa= 0◦; (b) θi= 44.94◦(full curve, calculated;
•, measured) and θa= −1.78◦. Insert: portion of the main plot,
the error bars show the standard deviation of the measured value to its calculated value.
45◦with respect to the incident plane of the sample. The AI (θi) is taken to be 45◦. The analyser (A) is mounted on a
stepping motor controlled rotator. A sample (quartz crystal, surface flatness λ/4; BK7 glass, surface flatness λ/4; and YVO4 crystal-CASIX, surface flatness 2λ) is mounted on a rotatable holder and measured at 10◦intervals. All intensities are measured using a power meter (D) (Newport 818-SL), digitized by a multimeter (Keithley 195A), and stored in a PC for calculating the ellipsometric parameters. The reflectance ratios are obtained by the ratio of intensity at A= 0◦to that at
Figure 6.ψversus θafor BK7. The standard deviation is 0.02◦.
The line indicates the mean value which is 17.29◦.
Figure 7.ψversus θafor YVO4. θc= 136.01◦with no= 1.9929
and ne= 2.2154, θi= 44.94◦(full curve, calculated;•, measured)
and θa= 7.24◦. Insert: portion of the main plot, the error bars
show the standard deviation of the measured value to its calculated value.
A= 90◦. Prior to the measurements, all the azimuthal angles of the polarizer and the analyser are systematically aligned according to [10] using an optical flat thick platinum plate. Moreover, the reference zeros of the polarizer and analyzer are confirmed using the technique of [12].
4. Results
In parallel to our experiment, we numerically analyse the improved value of ψ , using equation (4) incorporated with the analytical solution of reflection coefficients for uniax-ial crystals. The improved ψ is found to be free from the misalignment of the polarizer and differs by only 0.002◦if there is a 0.5◦misalignment in the analyser. However, the azimuthal deviations of the polarizer and analyser with re-spect to the incident plane can be as low as 0.005◦ in the PSA ellipsometric system [12]. Comparing the measured ψ of the quartz crystal with its calculated value, as shown in figure 5(a), we conclude that the systematic error is mainly caused by the deviation of AI. The deviation is found to be
−0.06 ± 0.01◦ by fitting the experimental data to the
The angle of incidence in a rotating element ellipsometer
optical axis is found to be 1.78± 0.01◦to the incident plane; as shown in figure 5(b), the standard deviation between the measured value and calculated value is 0.02◦after the adjust-ment of the azimuthal angle. The refractive index of BK7 deduced [5] from tan ψ (figure 6) is 1.517± 0.002 at an inci-dent angle of 44.94◦and is 1.521±0.002 at an incident angle of 45◦. Since the refractive index of BK7 is 1.515 (Schott: optical glass No 1000), this error evaluation can improve the measurement of glass. The angle (θc) between the normal
to the cleavage plane and the optical axis of YVO4 crystal was specified as 135◦by the vendor (CASIX). Under the cor-rected incident angle 44.94◦, θcwas obtained to be 136.01◦
and θa(figure 1) to be 7.24± 0.01◦, as shown in figure 7.
5. Concluding remarks
The small difference between the ordinary and extraordinary refractive indices of quartz crystal allows us to resolve the deviation in the angle of incident for rotating-element ellip-sometry. The other two primary errors in a rotating PSA el-lipsometric system, the azimuthal misalignment of polarizer and degradedness of a polarizer, can be reduced by the inten-sity ratio technique [12]. Since all three primary errors in a rotating PSA ellipsometry can be reduced, we can employ a low cost dichroic sheet polarizer in place of a prism polarizer without loosing its accuracy in a PSA ellipsometric system.
In addition to determining the deviation of incident angle in a rotating element ellipsometry, the following three param-eters can be obtained by fitting the measured tan ψ to the ana-lytic solution of uniaxial crystals: the absolute value of no, ne
and the directions of optical axis (θaand θc) in the laboratory
frame. Since the resolving power of the system can be in-creased as the incident angle moves closer to the Brewster an-gle (the reflected intensity at 50◦will be about 0.4% of the in-cident intensity), the system can be improved by using a sen-sitive detector or a higher power light source. It is our interest to extend the current experimental system to measure a mate-rial which consists of both linear and circular birefringence.
Acknowledgment
This work is supported by National Science Council of the Republic of China under grant NSC 86-2112-M-009-026.
Appendix. Fresnel reflection coefficients of uniaxial crystals
The reflection geometry is shown in figure 1. The direction of the optical axis is specified by angles θaand θcrelative to
the laboratory xyz; ifEc is the unit vector of optical axis, we can express it as
Ec = (α, β, γ )
where α = cos θasin θc, β = sin θasin θc and
γ = cos θc. Let the incident wavevector be KEi + q1Ek, where
K= k nisin θi, q1= k nicos θi, for a wavenumber k= ω/c
at an incident angle θi. According to [11], we summarized the
Fresnel reflection coefficients for uniaxial crystals of ordinary refractive index no= √εoand extraordinary refractive index
ne = √εeas follows: rss = [(q1− qe)AEey− (q1− qe)BEyo]/D rsp= 2nik(AExo− BE o x)/D rpp= 2qt[(q1+ qe)ExoEye− (q1+ qo)ExoEyo]/D− 1 rps= 2nik(qe− qo)EyoE e y/D.
The ordinary and extraordinary modes have wavevector normal components qo, and qerelated to the medium as
qe = ( √ d− αγ K1ε)/(εo+ γ21ε) qo= εok2− K2 qt= q1+ K tan θi where 1ε= εe− εoand d= εo[εe(εo+ γ21ε)k2− (εe− β21ε)K2].
The corresponding electric field vectorsEoandEeare noted as
Eo= No(−βqo, αqo− γ K, βK)
Ee= Ne(αqo2− γ qeK, βεok2, γ (εok2− qe2)− αqeK)
where Noand Neare the normalization factors, respectively.
For simplicity, we also state the collective parameters as follows: A= (qo+ q1+ K tan θi)Exo− KE o z B= (qe+ q1+ K tan θi)Eex− KE e z D= (q1+ qe)AEye− (q1+ qo)BEyo. References
[1] Beatie J R 1955 Phil. Mag. 46 235
[2] Taft E A and Philipp H R 1965 Phys. Rev. A 138 197 [3] Tomaselli V P, Rivera R, Edeweard D C and Moeller K D
1981 Appl. Opt. 20 3961
[4] Stagg B J and Charalampopoulos T T 1991 Appl. Opt. 30 4113
[5] Azzam R M A and Bashara N M 1992 Ellipsometry and
Polarized Light (Amsterdam: North-Holland)
[6] Alonso M I and Garriga M 1995 Appl. Phys. Lett. 67 596 [7] Collins R W 1990 Rev. Sci. Instrum. 61 2029
[8] Zeidler J R, Kohles R B and Bashara N M 1974 Appl. Opt.
13 1938
[9] McCrankin F L, Passaglia E, Stromberg R R and Steinberg H L 1963 J. Res. Natl Bur. Stand. A 67 363 [10] Chao Y F, Wei C S, Lee W C, Lin S C and Chao T S 1995
Japan. J. Appl. Phys. 34 5016
[11] Lekner J 1991 J. Phys.: Condens. Matter 3 6121
[12] Chao Y F, Lee W C, Hung C S and Lin J J 1998 J. Phys. D:
Appl. Phys. 31 1968
[13] Aspnes D E 1980 J. Opt. Soc. Am. 70 1275
