This study presents a simple method for determining the optical constants of an anisotropic thin film. The sensitivity of enhanced polarization conversion reflectance to optical constants is also calculated and analyzed. Based on the sensitivity calculation, the principal indices and columnar tilt angle can be derived from the polarization conversion reflectance angular spectrum.
© 2007 Optical Society of America
Optical thin films with uniaxial or biaxial property made by oblique angle deposition technique have been widely applied in optical devices such as polarization-selective antireflection, retardation, high-reflection filters [1–2]. Those filters are designed by the equivalent index of refraction, which depends on the polarization direction of incident light. Therefore, the more obvious uniaxial or biaxial property results in more apparent polarization-selective function. The bideposition technique  has been developed to have enhanced uniaxial property and optical activity of anisotropic thin films. Recently, enhanced polarization conversion phenomenon was found from a typical tilt-columnar thin film. It is interesting to observe that more than 90% polarization conversion reflectance (PCR) occurs from a weak anisotropic thin film in a prism/anisotropic thin film/air configuration . By arranging one isotropic film in the system, narrow-band and broad-band polarization conversion reflection filters can be derived . The first and most important application of polarization conversion is optical constant determination of an anisotropic film. According to Horowitz’s model , one of the principal axes is in the direction of columnar growth, and the other two principal axes are perpendicular and parallel to the deposition plane respectively. For a typical anisotropic thin film with a tilt columnar structure, the difference between principal indices appears to be small (approximately 0.02). The anisotropic optical constant determination, including principal indices, thickness, and tilt angle, can be determined from the simple and sensitive polarization conversion reflectance.
Directly measuring reflectance and transmittance of the air/anisotropic film/glass system has difficulty in distinguishing anisotropic thin film from isotropic thin film . The small difference between principal indices of anisotropic thin film requires extremely high intensity sensitivity to accurately resolve and detect these indexes. The guided-wave method (m-line technique) has shown the measurements of transmission in polarized light at normal incidence and the guided-mode propagation constants determine the optical constants of TiO2 columnar films. However, the method requires the film to be sufficiently thick to excite guided modes . The feasibility of the SPR (surface plasmon resonance) method  is limited by two factors: first, the refractive index of the anisotropic film must be lower than that of the prism; second, the detected film must be coated on the metal film. Ellipsometry technique [9–10] requires complex optical components including phase modulator, rotatable polarizer and analyzer to measure parameters corresponding intensity and phase of reflected light from the sample: the elliptical parameters Δ and Ψ refer respectively to the phase difference and the ratio of the amplitude between the p- and s- components of the reflected electric field. In our method, only intensity measurement can reach high sensitive determination of anisotropic optical constants.
This study applies enhanced polarization conversion to measure the optical constants of an anisotropic thin film. At a particular incident angle that exceeds the critical angle in the prism/anisotropic thin film/air system, the total internal reflection enhances the e-ray and o-ray coupling  and significant polarization conversion is expected. The reflection coefficient of polarization conversion rsp (the ratio of the incident p-polarized electric field to the reflected s-polarized electric field) is represented in Eq. (1).
The superscripts (a) and (b) in Eq. (1) represent the interfaces (prism/anisotropic thin film) and (anisotropic thin film/air) respectively. The phase terms Λ+ and Λ- are related to the multi-reflected waves in an upward direction (+) and in a downward direction (-)respectively. From the previous results , it is easy to show that the reflectance Rsp(=∣rsp∣2) at incident plane orientation δ is identical to Rps(=∣rps∣2) at δ+π. As shown in Fig. 1, the incident plane makes an angle δ with the deposition plane. The columnar tilt angle is φ, and the incident angle is α. According to a previous study , the typical low refractive anisotropic thin film with optical constants (n1=1.361, n2=1.360, n3=1.380, φ=45° and d1= 960.0 nm) in the BK7 prism/anisotropic thin film/air system causes strong PCR Rsp (the fraction of the incident s-polarized light converted to p-polarized reflected light). At incident angle α=62.49°, the reflectance Rsp is enhanced to be 0.99, as shown in Fig. 2. There are two peaks in the angular spectrum. The global reflectance maximum Rmax occurs at the incident angle αmax, and the angular distance between the two peaks is denoted as αpp.
In this study, the sensitivities of the angular spectrum of PCR to variation of optical constants are calculated and analyzed in Sec. 2. Based on the sensitivity analysis, the accuracy of detected optical constants can be derived by considering the resolution ability in angle and intensity measurement. In Sec. 3, an anisotropic MgF2 film is prepared by oblique angle deposition technique, and the PCR angular spectrum of the MgF2 film is measured. The triangle convergence method is introduced to show that the possible values of n2 and n3 form a triangle on (n2-n3) plane in (n2, n3, φ) coordinates. The accurate principal indices n2, n3 and tilt angle φ are derived from the convergent triangle area.
2. Sensitivity calculation
The PCR is calculated with respect to the prism/anisotropic thin film/air configuration, as shown in Fig. 1. The Berreman’s matrix  calculation is applied here to obtain the reflection coefficients for various incident angles and columnar tilt angles. To achieve an obvious polarization conversion, the incident plane δ is assumed to be vertical to the deposition plane of the film  (δ=90° in Fig. 1).
The sensitivity of PCR for biaxial thin films is calculated and analyzed by simulating angular spectrum curves of PCR for wavelength λ=632.8 nm versus various columnar tilt angles φ(10° -80°) and thicknesses d1(200–1100 nm). The sensitivities of the parameters αmax, Rmax and αpp to thickness and tilt angle are calculated on condition that the principal indices are n1=1.321, n2=1.320 and n3= 1.336 for a typical low-refractive MgF2 film. Although the tilt angle variation indicates the slight variation of the refractive index, the sensitivities of αmax, Rmax and αpp to principal indices n2 and n3 are calculated for a fixed thickness and tilt angle.
Figures 3(a), 3(b) and 3(c) show the calculated αmax, Rmax and αpp, as a function of the tilt angle φ and of the thickness d1 of MgF2 film. The figures demonstrate that αmax, Rmax and αpp require the resolutions of approximately 5.06×10−4 deg, 1.69×10−4 and 1.27×10−3deg respectively to resolve the thickness variation 0.1 nm between 700 nm and 1000 nm. Furthermore, for thickness 900 nm, αmax, Rmax and αpp require resolutions of 1.36×10−2deg, 1.81×10−2 and 3.60×10−3deg to resolve the tilt angle variation 1° between 20° and 45°. The sensitivity of αmax for the tilt angle increases with the thickness d1.
Next, the sensitivities of αmax, Rmax and αpp to principal indices n2 and n3 are calculated in Fig. 4. The thickness and tilt angle are 900 nm and 30° respectively. The αmax, Rmax and αpp require 5.45×10−2 deg, 5.47×10−2 and 1.27×10−2 deg respectively to resolve the principal index variation Δn2=0.001. Moreover, αmax, Rmax and αpp require 1.80×10−2 deg, 4.32×10−2 and 9.50×10−3 deg to resolve the principal index variation Δn3=0.001. Table 1 lists the sensitivity requirement for the PCR measurement to detect a typical low-refractive thin film with optical constants (nl=1.321, n2=1.320, n3=1.336, φ= 30° and d1= 900 nm).
3. Triangle convergence method
From the sensitivity calculation, it is possible to detect an anisotropic thin film using the direct intensity measurement method. The principal index n1 and thickness d1 can be determined by s-polarized light incident on the film provided that the incident plane is coincident with the deposition plane (δ=0°). Other optical constants, including principal indices n2, n3 and tilt angle δ, are determined by fitting the PCR angular spectrum with deposition plane (δ=90°).
In the fitting procedure, the parameters αmax, Rmax and αpp can not be simultaneously fitted the measured angular spectrum when the assumed tilt angle deviates from the exact angle. For an initial value of the tilt angle, the optimized fitting of both αmax and Rmax is used to obtain the optical constants, which is one point on the plane n2-n3. The optimized fittings of (Rmax and αpp) and (αmax and αpp) give the other two solutions (points) of optical constants (n2 and n3) on the plane n2-n3. Those three points form a triangle on the plane n2-n3. Points in the triangular area represent the possible solution of the optical constants. There are different triangular areas with respect to different tilt angles. By varying the tilt angle, the triangle would converge to a minimum area. When the minimum area is less than the criterion area that caused by the measurement resolution, the optical constants become a certainty. The criterion area is 5×10−7 for the resolution of principal index 10−3.
An anisotropic MgF2 film is prepared by oblique deposited fabrication. The sample is coated by thermal evaporation with obliquely incident vapor flux at 66° to the substrate surface normal direction. The deposition rate is controlled at 0.31 nm/s and the base pressure of vacuum system is hold below 4.2×10−6 Torr. In the polarization conversion measurement as shown in Fig. 5, a helium-neon laser with the stabilized frequency ±5MHz and the signal detection lead to reflectance accuracy of 0.001 by the lock-in amplifier technique are used to measure the reflectance angular spectrum. The thickness 877.1 nm and n1=1.323 of the sample are derived from the s-polarized reflectance spectrum at incident plane orientation δ=0°.
The three characteristics of Rsp angular spectrum are measured as αmax =58.40°, Rmax =0.687 and αpp =7.31°. Moreover, the initial value of the tilt angle is determined by the tangent rule  to be 32°. The triangles on the n2-n3 plane for the tilt angle from 27° to 35° are shown in Fig. 6. The area of the triangle converges to 3.2×10−9 at tilt angle φ=30°. The fitting results for n2, n3 and φ can reasonably be presented as n2=1.322, n3=1.337 and φ=30°. The measured tilt angle φ is in agreement with the cross section SEM image of the MgF2 film in Fig. 7.
The wavelength-dependent principal indices n2(λ) and n3(λ) are also derived based on the above method. The detected results between wavelength 400 nm to 800 nm are shown in Fig. 8. The Cauchy dispersion model, given as: ni(λ)=Ai+Bi/λ2+Ci/λ4 is used to fit the dispersion relation. Therefore, the dispersive n2(λ) and n3(λ) are represented as n2(λ)=1.318+1.701/λ2−0.001/λ4 and n3(λ)=1.334+1.530/λ2+0/λ4 respectively. Both indices n2(λ) and n3(λ) decrease with wavelength increasing: n2(λ=440.0 nm)=1.327 decreases to n2(λ=790.0 nm)=1.320 and n3(λ=440.0 nm)=1.342 decreases to n3(λ=790.0 nm)=1.336. The difference between the principal indices also increases with wavelength.
This study applied enhanced polarization conversion reflection to determine the optical constants of a biaxial thin film. The sensitivity calculation demonstrated that the direct intensity measurement with a stage angular resolution of 0.01° and reflectance accuracy of 0.01 can resolve the principal indices variation Δn=0.001 and tilt angle resolution Δφ=1°. Based on the sensitivity analysis, the triangle area convergent method is developed for measuring optical constants. Without an expensive and complex optical setup, the anisotropic optical constants can easily be derived for low refractive anisotropic thin films.
References and links
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