橢圓偏光儀之異向晶體量測

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(6) !"#$%& '()' *+,-./ 012345 6789:;<= >?@0ABC* Abstract. In this study, we proved that this PSA ellipsometry not only can measure the isotropic material it also can be used as rotating element ellipsometry for measure the optical properties of uniaxial crystals. Theory, experiment and result. The reflected χr and incident χi polarization states are related by [2]. This report continues our previous work on measuring the optical properties of a uniaxial crystal. We corrected the beam deviation using a Quartz crystal and measure the ordinary and extraordinary refractive indices of YVO4 and compared with vendor’s specifications. We also measured the orientation of its optical axis.. χr =. (r. (r. sp. pp. . . . . . . . ). rss + χ i. ) (. ). rss + rps rss ⋅ χ i. . . . . (1). where rxy is the Fresnel reflection coefficient for the parallel (p, i.e. x) and perpendicular(s, i.e. y) polarizations. The analytical expressions of these Fresnel reflection coefficients for uniaxial crystals in the Appendix. The complex pseudoreflectance ratio was defined [2] as for anisotropic media, while in general ρ is defined as [3] i ∆ ρ = tanψe ,. Keywords: Ellipsometry, birefringence . Introduction. According to our previous study [1], we are able to measure the incident beam deviation in a rotating element ellipsometry by a quartz crystal. Because this calibration, we are able to construct a system to measure the optical axis and refractive indices of a uniaxial crystal by two sheet polarizers. Besides the ordinary and extraordinary refractive indices, we also applied this technique to measure the angle between the normal to the cleavage plane and optic axis of a Yttrium Or-thovanadate (YVO4).. χ ψ = χ. . thus (2) Since the cross terms vanish in an isotropic medium, tan ψ [3] equals |rpp/rss|, which is the conventional expression for the ellipsometric 1.

(7) parameter. The reflection geometry for a uniaxial crystal is shown in Fig. 1.. axis of a nonabsorbent uniaxial crystal is parallel to the reflection surface, i.e. θc= 90o, then the ellipsometric parameter ψ can be characterized by a twofold symmetry with respect to θa, the azimuthal angle. Since we were only interested in determining the AI in a PSA ellipsometry, such as shown in Fig. 2, we simulated 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.. Figure 1. The reflection geometry: θi is the incident angle, xy plane is the θ. X. θc. OA Y. θ. θa reflecting face of the crystal, zx plane is the incident plane, the z axis is the normal line. OA is the optical axis of the crystal.. L. A simple model for anisotropic crystals was proposed by Aspnes [4]: 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. ψ =. θ. A A. Fig. 2 A schematic set-up of the PSA ellipsometer: L, light source (He-Ne laser); P, polarizer; A, analyzer; D, detector. The function ψ(θa) is simulated for χ = , i.e. P = 45o, and optimized [5] by χ = − , i.e. P = -45o, to eliminate the error caused by the misalignment of the polarizer, according to equation (3), one can obtain. . .     ψ =       = Į  =− Į . (3). . (4). where Irp represents the reflected intensity parallel to the incident plane and Irs represents the reflected intensity perpendicular to the incident plane, for P = 45o, 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. The measured and simulated values are compared in Fig. 2. 2.

(8) Acknowledgement 27 26 25 24 23 22. calculated at AI 44.94. We like to thank NSC for grating the research. This research and last one has been combined and published in Journal of Physics: D [8]. ex perim ent al 0. 200. 400. 600. Fig. 3 ψ verses θa : YVO4 θc=136.01o for YVO4 with no=1.9929 and ne=2.2154,θi = 44.94o (line: calculated •: measured) and θa = 7.24o .. Appendix This appendix is cited from reference [6] and [7]. The reflection geometry is shown in Fig. 1. The direction of optical axis is specified by angles θa and θc relative to the r laboratory xyz, if is the unit vector of optical axis, we can express it as. ɜƋConclusion and discussion. r = α β γ ,. The angle (θc) between the normal to the cleavage plane and the optic axis of YVO4 crystal at θi = 44.94o was obtained to be 136.01o and θa (Fig. 1) to be 7.24± 0.01o, as shown in Fig. 3, while θc was specified as 135o from the vendor (CASIX). In addition to determining the deviation of incident angle in a rotating element ellipsometry, the following three parameters can be obtained by fitting the measured tanψ to the analytic solution of uniaxial crystals: the absolute value of no, ne and directions of optical axis (θa and θc) in the laboratory frame. Since the resolving power of the system can be increased as the incident angle moves closer to the Brewster angle (the reflected intensity at 50o will be about 0.4% of the incident intensity), the system can be improved by using a sensitive detector or a higher power light source. It is our interest to extend the current experimental system to measure a material which consists of both linear and circular birefringence.. where α = θ θ , β = θ θ and. γ = θ . Let the incident wave vector be r r + , where K= k ni sinθI, q1= k ni cosθi,. for a wave number =. ω. at incident angle According to reference 11, we θi. summarized the Fresnel reflection coefficients for uniaxial crystals of ordinary refractive index no= ε and extraordinary refractive index ne= ε as follows;

(9) = − − − = −

(10) = + − + − = − The ordinary and extraordinary modes have wave vector normal components qo, and qe related to the medium as = − αγ ∆ ε ε + γ ∆ε , = ε − , = + θ. where ∆ε = ε − ε , and = ε ε ε + γ ∆ ε . 3. . − ε − β ∆ε . . .

(11) the corresponding electric field vectors Eo and Ee noted as Eo = No(-βqo, αqo-γK, βK) , Ee = Ne(αqo2-γqeK, βεok2, γ(εok2-qe2)-αqeK), where No, Ne are the normalization factor, respectively. For simplicity, we also state the collective parameters as follows; = + + θ − . = + + θ − = + − + . 1. 2. 3.. 4. 5. 6. 7.. 8.. NSC 87-2112-M-009-032 (1998) Alonso M I and Garriga M 1995 Appl. Phys. Lett. 67 596 Azzam R M A and Bashara N M, Ellipsometry and Polarized light 1992 (Amsterdam: North-Holland) Aspnes D E 1980 J. Opt. Soc. Am. 70 1275 Chao Y F, Lee W C, Hung C S and Lin J J 1998 J. Phys. D.: Appl. Phys. 31 1968 Lekner J 1991 J. Phys. : Condens. Matter 3 6121. ,

(12) , !"# 85 $%&' Y. F. Chao, M.W. Wang and Z. C. Ko (in press: Journal of Physics D: applied physics ). 4.

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