Spatially polarization-modulated ellipsometry based on the vectorial optical field and image processing

Chao Gao; Bing Lei

doi:10.1364/AO.395234

Applied Optics
Vol. 59,
Issue 18,
pp. 5377-5384
(2020)
•https://doi.org/10.1364/AO.395234

Spatially polarization-modulated ellipsometry based on the vectorial optical field and image processing

Chao Gao and Bing Lei

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Chao Gao and Bing Lei, "Spatially polarization-modulated ellipsometry based on the vectorial optical field and image processing," Appl. Opt. 59, 5377-5384 (2020)

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Abstract

We report a new (to the best of our knowledge) ellipsometric measurement scheme called spatially polarization-modulated ellipsometry (SPME), which is based on the vectorial optical field and digital image processing. A zero-order vortex half-wave retarder (ZVHR) is employed to generate the vectorial optical field and analyze the elliptically polarized light reflected by a thin film; further, an analyzer is set after the ZVHR to form an hourglass intensity pattern due to the spatially polarization modulation; then, the film’s ellipsometric angles can be obtained by processing the hourglass intensity image. By analyzing the working principle of SPME, we have found that the film’s ellipsometric angles are determined by the bright areas’ azimuth angle and contrast of the modulated images, and their mathematical relationships have been deduced and provided. To reduce the adverse effects of image noise and improve the measurement precision, an improved operation mode of SPME is presented by processing the modulated images with their bright areas’ azimuth locating at near 45° or 135° and 0° or 90°. Numerical analyzing studies have been carried out on the ${{\rm SiO}_2}$ films to validate the feasibility of SPME, and the simulation experiments indicate that the SPME can operate well, even though obvious noise has been added to the modulated intensity image and the measuring error of film thickness and refractive index were less than 0.1 nm and 0.001, respectively.

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Film Thickness $d$	Refractive Index of Air $n_{1}$	Refractive Index of ${S i O}_{2} n_{2}$	Refractive Index of $S i$ Wafer $n_{3}$	Wavelength $λ$	Incident Angle $α$
$135.00 n m$	1.000	1.460	$4.051 - 0.027 i$	$632.8 n m$	70°

Theoretical $d$ (nm) and $n_{2}$	Theoretical $Ψ (^{\circ})$ and $Δ (^{\circ})$	Polarizer $P_{1} (^{\circ})$ and $P_{2} (^{\circ})$	Measured $Ψ (^{\circ})$ and $Δ (^{\circ})$	Measured $d$ (nm) and $n_{2}$
$120.00, 1.460$	$57.02, 87.22$	$33.0, 43.6$	$57.02, 87.23$	$120.00, 1.460$
$100.00, 1.460$	$41.54, 81.13$	$48.5, 40.6$	$41.54, 81.14$	$99.99, 1.460$
$80.00, 1.460$	$33.09, 82.81$	$56.9, 41.4$	$33.09, 82.80$	$80.01, 1.460$
$60.00, 1.460$	$26.69, 90.82$	$63.3, 45.4$	$26.70, 90.82$	$60.02, 1.460$
$40.00, 1.460$	$20.56, 105.93$	$69.4, 53.0$	$20.57, 105.93$	$40.07, 1.459$
$20.00, 1.460$	$14.74, 132.74$	$75.3, 66.4$	$14.74, 132.75$	$19.99, 1.460$

Noise Level	Measured $Ψ (^{\circ})$ and $Δ (^{\circ})$	Measured Error of $Ψ (^{\circ})$ and $Δ (^{\circ})$	Measured $d$ (nm) and $n_{2}$	Measured error of $d$ (nm) and $n_{2}$
$μ = σ^{2} = 0.00$	$77.921, 106.233$	$0.000, 0.000$	$135.000, 1.460$	$0.000, 0.000$
$μ = σ^{2} = 0.02$	$77.917, 106.222$	$0.004, 0.011$	$134.997, 1.460$	$0.003, 0.000$
$μ = σ^{2} = 0.04$	$77.914, 106.229$	$0.007, 0.004$	$134.999, 1.460$	$0.001, 0.000$
$μ = σ^{2} = 0.06$	$77.913, 106.238$	$0.008, 0.005$	$135.001, 1.460$	$0.001, 0.000$
$μ = σ^{2} = 0.08$	$77.911, 106.221$	$0.010, 0.012$	$134.996, 1.460$	$0.004, 0.000$
$μ = σ^{2} = 0.10$	$77.906, 106.211$	$0.015, 0.022$	$134.993, 1.460$	$0.007, 0.000$

Error of $φ_{max} (^{\circ})$	Measured $Ψ (^{\circ})$ and $Δ (^{\circ})$	Measured Error of $Ψ (^{\circ})$ and $Δ (^{\circ})$	Measured $d$ (nm) and $n_{2}$	Measured Error of $d$ (nm) and $n_{2}$
0.00	$77.921, 106.233$	$0.000, 0.000$	$135.00, 1.460$	$0.00, 0.000$
0.01	$77.921, 106.177$	$0.000, 0.056$	$134.98, 1.460$	$0.02, 0.000$
0.05	$77.923, 106.082$	$0.002, 0.151$	$134.96, 1.460$	$0.04, 0.000$
0.10	$77.926, 106.024$	$0.005, 0.209$	$134.94, 1.460$	$0.06, 0.000$
0.20	$77.930, 105.805$	$0.009, 0.428$	$134.87, 1.461$	$0.13, 0.001$
0.30	$77.938, 105.583$	$0.017, 0.650$	$134.81, 1.461$	$0.19, 0.001$

Film Thickness $d$	Refractive Index of Air $n_{1}$	Refractive Index of ${S i O}_{2} n_{2}$	Refractive Index of $S i$ Wafer $n_{3}$	Wavelength $λ$	Incident Angle $α$
$135.00 n m$	1.000	1.460	$4.051 - 0.027 i$	$632.8 n m$	70°

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Applied Optics