## Abstract

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

## 1. Introduction

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 [3] 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 [4]. By arranging one isotropic film in the system, narrow-band and broad-band polarization conversion reflection filters can be derived [4]. The first and most important application of polarization conversion is optical constant determination of an anisotropic film. According to Horowitz’s model [5], 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 [6]. 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 TiO_{2} columnar films. However, the method requires the film to be sufficiently thick to excite guided modes [7]. The feasibility of the SPR (surface plasmon resonance) method [8] 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 [11] and significant polarization conversion is expected. The reflection coefficient of polarization conversion r_{sp} (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 [12], it is easy to show that the reflectance R_{sp}(=∣r_{sp}∣^{2}) at incident plane orientation δ is identical to R_{ps}(=∣r_{ps}∣^{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 [11], the typical low refractive anisotropic thin film with optical constants (n_{1}=1.361, n_{2}=1.360, n_{3}=1.380, φ=45° and d_{1}= 960.0 nm) in the BK7 prism/anisotropic thin film/air system causes strong PCR R_{sp} (the fraction of the incident s-polarized light converted to p-polarized reflected light). At incident angle α=62.49°, the reflectance R_{sp} is enhanced to be 0.99, as shown in Fig. 2. There are two peaks in the angular spectrum. The global reflectance maximum R_{max} occurs at the incident angle α_{max}, and the angular distance between the two peaks is denoted as α_{pp}.

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 MgF_{2} film is prepared by oblique angle deposition technique, and the PCR angular spectrum of the MgF_{2} film is measured. The triangle convergence method is introduced to show that the possible values of n_{2} and n_{3} form a triangle on (n_{2}-n_{3}) plane in (n_{2}, n_{3}, φ) coordinates. The accurate principal indices n_{2}, n_{3} 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 [13] 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 [11] (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 d_{1}(200–1100 nm). The sensitivities of the parameters α_{max}, R_{max} and α_{pp} to thickness and tilt angle are calculated on condition that the principal indices are n_{1}=1.321, n_{2}=1.320 and n_{3}= 1.336 for a typical low-refractive MgF_{2} film. Although the tilt angle variation indicates the slight variation of the refractive index, the sensitivities of α_{max}, R_{max} and α_{pp} to principal indices n_{2} and n_{3} are calculated for a fixed thickness and tilt angle.

Figures 3(a), 3(b) and 3(c) show the calculated α_{max}, R_{max} and α_{pp}, as a function of the tilt angle φ and of the thickness d_{1} of MgF_{2} film. The figures demonstrate that α_{max}, R_{max} and α_{pp} require the resolutions of approximately 5.06×10^{−4} deg, 1.69×10^{−4} and 1.27×10^{−3}deg respectively to resolve the thickness variation 0.1 nm between 700 nm and 1000 nm. Furthermore, for thickness 900 nm, α_{max}, R_{max} and α_{pp} require resolutions of 1.36×10^{−2}deg, 1.81×10^{−2} and 3.60×10^{−3}deg to resolve the tilt angle variation 1° between 20° and 45°. The sensitivity of α_{max} for the tilt angle increases with the thickness d_{1}.

Next, the sensitivities of α_{max}, R_{max} and α_{pp} to principal indices n_{2} and n_{3} are calculated in Fig. 4. The thickness and tilt angle are 900 nm and 30° respectively. The α_{max}, R_{max} and α_{pp} require 5.45×10^{−2} deg, 5.47×10^{−2} and 1.27×10^{−2} deg respectively to resolve the principal index variation Δn_{2}=0.001. Moreover, α_{max}, R_{max} and α_{pp} require 1.80×10^{−2} deg, 4.32×10^{−2} and 9.50×10^{−3} deg to resolve the principal index variation Δn_{3}=0.001. Table 1 lists the sensitivity requirement for the PCR measurement to detect a typical low-refractive thin film with optical constants (n_{l}=1.321, n_{2}=1.320, n_{3}=1.336, φ= 30° and d_{1}= 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 n_{1} and thickness d_{1} 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 n_{2}, n_{3} and tilt angle δ, are determined by fitting the PCR angular spectrum with deposition plane (δ=90°).

In the fitting procedure, the parameters α_{max}, R_{max} and α_{pp} 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 α_{max} and R_{max} is used to obtain the optical constants, which is one point on the plane n_{2}-n_{3}. The optimized fittings of (R_{max} and α_{pp}) and (α_{max} and α_{pp}) give the other two solutions (points) of optical constants (n_{2} and n_{3}) on the plane n_{2}-n_{3}. Those three points form a triangle on the plane n_{2}-n_{3}. 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 MgF_{2} 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 n_{1}=1.323 of the sample are derived from the s-polarized reflectance spectrum at incident plane orientation δ=0°.

The three characteristics of R_{sp} angular spectrum are measured as α_{max} =58.40°, R_{max} =0.687 and α_{pp} =7.31°. Moreover, the initial value of the tilt angle is determined by the tangent rule [14] to be 32°. The triangles on the n_{2}-n_{3} 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 n_{2}, n_{3} and φ can reasonably be presented as n_{2}=1.322, n_{3}=1.337 and φ=30°. The measured tilt angle φ is in agreement with the cross section SEM image of the MgF_{2} film in Fig. 7.

The wavelength-dependent principal indices n_{2}(λ) and n_{3}(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: n_{i}(λ)=A_{i}+B_{i}/λ^{2}+C_{i}/λ^{4} is used to fit the dispersion relation. Therefore, the dispersive n_{2}(λ) and n_{3}(λ) are represented as n_{2}(λ)=1.318+1.701/λ^{2}−0.001/λ^{4} and n_{3}(λ)=1.334+1.530/λ^{2}+0/λ^{4} respectively. Both indices n_{2}(λ) and n_{3}(λ) decrease with wavelength increasing: n_{2}(λ=440.0 nm)=1.327 decreases to n_{2}(λ=790.0 nm)=1.320 and n_{3}(λ=440.0 nm)=1.342 decreases to n_{3}(λ=790.0 nm)=1.336. The difference between the principal indices also increases with wavelength.

## 4. Conclusion

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

**1. **I. J. Hodgkinson and Q. H. Wu, *Birefringent Thin Films and Polarizing Elements* (World Scientific, USA, 1997). [CrossRef]

**2. **I. J. Hodgkinson and Q. H. Wu, “Anisotropic antireflection coatings: design and fabrication,” Opt. Lett. **23**, 1553–1555 (1998). http://www.opticsinfobase.org/abstract.cfm?id=36983 [CrossRef]

**3. **I. J. Hodgkinson, Q. H. Wu, L. D. Silva, and M. Arnold, “Inorganic positive uniaxial films fabricated by serial bideposition,” Opt. Express **12**, 3840–3847 (2004). [CrossRef]

**4. **Y. J. Jen and C.Y. Peng, “Narrow-band and broad-band polarization conversion reflection filters,” Appl. Phys. Lett. **89**, 041128 (2006). http://apl.aip.org/apl/top.jsp [CrossRef]

**5. **F. Horowitz, *Structure-Induced Optical Anisotropy in Thin Film* (PhD dissertation, University of Arizona, Optical Sciences Center, 1983).

**6. **H. Wang, “Reflection/transmission measurements of anisotropic films with one of the principal axes in the direction of columnar growth,” J. Mod. Opt. **42**, 497–505 (1995). http://web.ebscohost.com/ehost/detail?vid=1&hid=6&sid=cda59d34-b4ff-4fd6-ae6a-8a417dca0258%40sessionmgr7 [CrossRef]

**7. **F. Flory, D. Endelema, E. Pelletier, and I. Hodgkinson, “Anisotropy in thin films: modeling and measurement of guided and nonguided optical properties: application to TiO_{2} films,” Appl. Opt. **32**, 5649–5659 (1993). http://www.opticsinfobase.org/abstract.cfm?id=60976 [CrossRef]

**8. **Y. J. Jen, C. H. Hsieh, and T. S. Lo, “Optical constant determination of an anisotropic thin film via surface plasmon resonance: analyzed by sensitivity calculation,” Opt. Commun. **224**, 269–277 (2005). http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TVF-4DGD4JP-1&_user=4462129&_coverDate=01%2F03%2F2005&_alid=514382637&_rdoc=1&_fmt=full&_orig=sear ch&_cdi=5533&_sort=d&_st=4&_docanchor=&_acct=C000053837&_version=1&_urlVersion=0&_useri d=4462129&md5=af91299e67249853b95193f09c220403 [CrossRef]

**9. **M. Schubert and W. Dollase, “Generalized ellipsometry for biaxial absorbing materials: determination of crystal orientation and optical constants of Sb_2 S_3,” Opt. Lett. **27**, 2073–2075 (2002). http://www.opticsinfobase.org/abstract.cfm?id=70604 [CrossRef]

**10. **R. M. A. Azzam and N. M. Bashara, *Ellipsometry and Polarized Light* (North Holland, Amsterdam, 1979).

**11. **Y. J. Jen and C. L. Chiang, “Optical thickness and anisotropic orientation of a birefringent thin film: analysis from explicit expressions for reflection and transmission coefficients,” Opt. Eng. **45**, 023802 (2006). http://scitation.aip.org/journals/doc/OPEGAR-ft/vol_45/iss_2/023802_1.html [CrossRef]

**12. **F. Yang and J. R. Sambles, “Optical characterization of a uniaxial material by the polarization-conversion reflectivity technique,” J. Opt. Soc. Am. B **11**, 605–617 (1994). http://www.opticsinfobase.org/abstract.cfm?id=7357 [CrossRef]

**13. **D. W. Berreman, “Optics in Stratified and Anisotropic Media: 4²4-Matrix Formulation,” J. Opt. Soc. Am. **62**, 502–510 (1972). http://www.opticsinfobase.org/abstract.cfm?id=54639 [CrossRef]

**14. **I. Hodgkinson, Q. H. Wu, and J. Hazel, “Empirical Equations for the Principal Refractive Indices and Column Angle of Obliquely Deposited Films of Tantalum Oxide, Titanium Oxide, and Zirconium Oxide,” Appl. Opt. **37**, 2653–2659 (1998). http://www.opticsinfobase.org/abstract.cfm?id=43685 [CrossRef]