Simultaneous microscopic imaging of thickness and refractive index of thin layers by heterodyne interferometric reflectometry (HiRef)

Suppose a homogeneous, isotropic layer of a material with thickness d and refractive index n₁ is sandwiched between two homogeneous semi-infinite media with refractive indices $n_0\gt n_1$ and $n_2\lt n_1$. Neglecting absorption, a plane light wave incident on the layer (for example, from the n₀ side, as shown in figure 1) can be either reflected or transmitted, with the relative amplitudes of the reflected and transmitted waves given by the Fresnel coefficients [23]:

$$\begin{align} r_{jl}^\parallel & = \frac{n_j\cos({\theta_j})-n_l\cos({\theta_l})}{n_j\cos({\theta_j})+n_l\cos{(\theta_l)}}, \end{align} \tag{ 1 }$$

$$\begin{align} t_{jl}^\parallel & = \frac{2n_j\cos{(\theta_j)}}{n_j\cos{(\theta_j)}+n_l\cos{(\theta_l)}}, \end{align} \tag{ 2 }$$

$$\begin{align} r_{jl}^\perp & = \frac{n_l\cos{(\theta_j)}-n_j\cos{(\theta_l)}}{n_l\cos{(\theta_j)}+n_j\cos{(\theta_l)}}, \end{align} \tag{ 3 }$$

$$\begin{align} t_{jl}^\perp & = \frac{2n_j\cos{(\theta_j)}}{n_l\cos{(\theta_j)}+n_j\cos{(\theta_l)}}, \end{align} \tag{ 4 }$$

where the subscript jl indicates that the light is incident on the planar boundary between media with refractive indices n_j and n_l from the n_j side, θ_j is the angle of incidence, θ_l is the angle of transmission (given by Snell's law, $n_j\sin{(\theta_j)} = n_l\sin{(\theta_l)}$), and the superscripts $\parallel$ and $\perp$ denote the polarisation components parallel and perpendicular, respectively, to the plane of incidence.

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If we place a detector on the n₀ side, we can collect the light reflected by the layer. The amplitude of the reflected field is given by interference between all the reflection orders (where the jth reflection order refers to the light that has been reflected exactly j times at the n₁–n₂ interface before reaching the detector). Light incident from the n₀ side is reflected at the first boundary with a relative amplitude $s_0 = r_{01}$, where the superscript is omitted for clarity in the following calculations. Light transmitted into the second semi-infinite medium is not measured. Light transmitted back into the first semi-infinite medium after having been reflected a total of j times at the n₁–n₂ interface and j − 1 times at the n₀–n₁ interface has the relative complex amplitude

$$\begin{align} s_1 = t_{01}t_{10}\sum_{j = 1}^\infty r_{10}^{\,\,j-1}r_{12}{}^{\,\,j} e^{ij\Delta}, \end{align} \tag{ 5 }$$

where $\Delta = 2kdn_1\cos(\theta_1)$ is the phase due to propagation through the layer and back, $k = 2\pi/\lambda$ is the wave number of the light of wavelength λ, and θ₁ is the propagation angle in the layer. The relative complex amplitude of the detected wave is then

$$\begin{align} s & = s_0+s_1 = r_{01}+t_{01}r_{12}t_{10}e^{i\Delta}\sum_{j = 0}^\infty\left(r_{10}r_{12}e^{i\Delta}\right)^j\nonumber\\ & = r_{01}+\frac{t_{01}r_{12}t_{10}e^{i\Delta}}{1-r_{10}r_{12}e^{i\Delta}}\,, \end{align} \tag{ 6 }$$

where $|r_{10}r_{12}e^{i\Delta}|\lt1$ for θ₀ (the angle of incidence) outside the total internal reflection (TIR) regime. Note that if there is no layer (i.e. d = 0) then the right side of equation (6) reduces to r₀₂, the reflection coefficient at the boundary between the two semi-infinite media, as can be seen by using equations (1)–(4) and performing some straightforward, albeit somewhat lengthy, algebra.

For normal incidence ($\theta_0 = 0$), equation (6) can be written as

$$\begin{align} s & = \frac{(n_0+n_1)(n_1-n_2)e^{i\Delta}+(n_0-n_1)(n_1+n_2)}{(n_0-n_1)(n_1-n_2)e^{i\Delta}+(n_0+n_1)(n_1+n_2)}\,. \end{align} \tag{ 7 }$$

Figure 2 shows the resulting amplitude and phase of s for $n_0 = 1.518$, $n_2 = 1.333$, λ = 550 nm, layer thicknesses between 0 and 600 nm and layer refractive indices between n₂ and n₀. The aforementioned values of n₀ and n₂ have been chosen to match microscope-slide glass [24] and water [23], respectively. It is evident that s is periodic in d with a period that is decreasing with n₁, as expected from the term $e^{i\Delta}$ in equation (6); the thickness range was chosen to show three such periods. It is also clear that $|s|\leqslant r_{02}$ for any d as long as $n_2\leqslant n_1\leqslant n_0$.

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HiRef makes use of the heterodyne-detected interference between two beams (see section 3.1), one of which (called the probe beam) interacts with the sample and one of which (called the reference beam) acts as an external reference, to obtain the reflected amplitude and phase. This allows the determination of two properties of the sample, such as thickness and refractive index, simultaneously.

We consider here an aplanatic objective of numerical aperture $\textrm{NA}$. The light field incident on the sample has a wide angular distribution over the polar angle θ (equal to the angle of incidence $\theta_0$), and the maximum angle of incidence is given by $\theta_{\textrm{m}} = \arcsin{\left(\textrm{NA}/n_0\right)}$, where n₀ is the refractive index of the semi-infinite medium the light is incident from. The probe beam entering the objective back-focal plane is chosen to be circularly polarised (see section 3.1), so the probe field in cartesian polarisation basis is given by

$$\begin{align} \mathbf{E}_{\textrm{p}}(\theta) = \frac{E(\theta)}{\sqrt{2}}{\begin{pmatrix}1\\ i\end{pmatrix}}e^{-i\omega t} \end{align} \tag{ 8 }$$

with

$$\begin{align} E(\theta ) = {E_0}\sqrt {\cos (\theta )} {\mkern 1mu} {e^{ - {{\left( {\zeta \sin (\theta )/\sin ({\theta _{\textrm{m}}})} \right)}^2}}}, \end{align} \tag{ 9 }$$

where we assume a gaussian beam profile in the back-focal plane of the objective with inverse objective fill factor (that is, the ratio between the objective back-focal plane radius and the radius at which the input-beam intensity is $1/e^2$) ζ [25]. The fields parallel and perpendicular to the plane of incidence, which are not mixed by the reflection, are then given by

$$\begin{align} E_{\textrm{p}}^\parallel(\theta,\varphi) & = \frac{1}{\sqrt{2}} E(\theta) e^{i(\varphi-\omega t)}, \end{align} \tag{ 10 }$$

$$\begin{align} E_{\textrm{p}}^\perp(\theta,\varphi) & = \frac{i}{\sqrt{2}}E(\theta) e^{i(\varphi-\omega t)}, \end{align} \tag{ 11 }$$

where ϕ is the azimuthal angle. The probe beam is reflected by the sample and recollimated by the objective, and it then interferes with a frequency-shifted reference beam, which does not interact with the sample, in an image plane of the objective back-focal plane. The reference beam is also circularly polarised and has the same beam profile as the probe beam. The reflection inverts the circular polarisation relative to the propagation direction, so we use a reference beam which has opposite circular polarisation to the beam entering the objective. We can therefore assume that its field distribution in the image plane of the back-focal plane of the objective is equal to that of the probe beam, so, up to a constant phase shift, $\mathbf{E}_{\textrm{r}}(\theta) = e^{-i\Omega t}\mathbf{E}_{\textrm{p}}(\theta)$, where Ω is the heterodyne frequency shift (see section 3.1). We thus have, for each incidence direction $(\theta, \varphi)$, the interference term

$$\begin{array}{*{20}{l}} {I(\theta ,\varphi )}&{ = 2{\textrm{Re}}\left( {E_{\textrm{r}}^{\parallel * }{s^\parallel }E_{\textrm{p}}^\parallel + E_{\textrm{r}}^{ \bot * }{s^ \bot }E_{\textrm{p}}^ \bot } \right)}\\ {}&{ = - |E(\theta ){|^2}{\textrm{Re}}\left( {\left( {{s^\parallel }(\theta ) + {s^ \bot }(\theta )} \right){e^{i\Omega t}}} \right),} \end{array} \tag{ 12 }$$

where $^\ast$ indicates the complex conjugate. The interference does not depend on ϕ because we use co-circular polarisations (that is, circular polarisations with the same handedness) for the reflected probe beam and the reference beam at the beam-splitter (see section 3.1). A dual-channel lock-in detection recovers both the in-phase and in-quadrature components at the frequency Ω, so we can detect the complex signal

$$\begin{align} S = S_0\int_{0}^{\theta_{\textrm{m}}}{|E(\theta)|^2\left(s^\parallel(\theta)+s^\perp(\theta)\right)\sin{(\theta)}}\,\textrm{d}{\theta}, \end{align} \tag{ 13 }$$

where S₀ is a constant which depends on the laser powers used, the detection efficiency and other setup-specific constants. The parameters used here are such that TIR does not occur, as the NA of the objective is below the lowest refractive index of the layer structure considered. Therefore, the expressions derived in section 2.1 for s are valid. Expressions for s including the case of TIR can be derived in a similar fashion.

Figure 3 shows the amplitude and phase of $S/S_{02}$, where S₀₂ is the signal detected in the absence of the thin layer, for which $s^\parallel = r_{02}^\parallel$ and $s^\perp = r_{02}^\perp$. Note that S₀₂ is real. The wavelength and refractive indices used are the same as in figure 2; $\theta_{\textrm{m}} = 56.8^\circ$, as determined by $\textrm{NA} = 1.27$; and ζ = 1. We find that the qualitative pattern seen with normal incidence remains, but it is distorted, with features occurring for smaller thicknesses. The distortion increases as the thickness increases, and eventually the features fade into each other because of the suppression of the signal from the second interface by destructive interference which is due to the variation of the phase with the angle of incidence. Another way to understand this is to note that the focal depth is so short that the reflection from the second interface is out of focus and thus suppressed in the confocal detection of the interference with the reference beam. The signal still satisfies $|S|\leqslant S_{02}$.

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We note that using objectives of lower NA will cause the results to tend towards the normal-incidence case shown in figure 3. Already for $\textrm{NA} = 0.75$, shown in figure 4, the deviations from normal incidence are minor.

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For normal incidence, equation (6) can be solved analytically for d, yielding

$$\begin{align} d = \frac{i}{2kn_1}\,\textrm{log}\left(\frac{n_2-n_1}{n_2+n_1}\,\frac{n_0+n_1-(n_0-n_1)s}{n_0-n_1-(n_0+n_1)s}\right). \end{align} \tag{ 14 }$$

Now, the argument of the logarithm in equation (14) has a magnitude of 1, since $\textrm{log}\left(\alpha e^{i\beta}\right) = \ln{(\alpha)}+i\beta$ for $\alpha,\beta\in\mathbb{R}$ and we require $\ln{(\alpha)} = 0$ for d to be real, which in turn requires α = 1 and thus $|\alpha e^{i\beta}| = 1$. This condition can be used to obtain

$$\begin{align} n_1 = \sqrt{n_0~n_2~\frac{n_0-n_2-2n_0\textrm{Re}\!\left(s\right)+(n_0+n_2)|s|^2}{n_0-n_2-2n_2\textrm{Re}\!\left(s\right)-(n_0+n_2)|s|^2}}\,. \end{align} \tag{ 15 }$$

Figure 5 shows the resulting thickness and refractive index of the layer as a function of $s/r_{02}$, where we have chosen the solution (the branch of the complex logarithm) having the smallest thickness.

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We can see that around the circumference, changing the phase of $s/r_{02}$ while keeping its magnitude close to 1, the thickness of the layer changes while its refractive index remains close to n₀. Going instead radially, changing the real part of $s/r_{02}$ starting from 1, the refractive index increases from n₂ for s = 1 to $\sqrt{n_2~n_0}$ for s = 0 and further to n₀ for $s = -1$.

In the case of reflection of a focussed beam (instead of a plane wave at normal incidence), with the signal given by equation (13) and shown in figure 3, the refractive index and thickness of the layer can be determined numerically for the angular distribution considered (given by the objective NA and ζ). Figure 6 shows the resulting d and n₁ as functions of $S/S_{02}$.

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We find that the qualitative behaviour is similar to the one at normal incidence shown in figure 5, apart from a region with no solutions (white) given by $|S/S_{02}|\gt R(\arg{(S/S_{02})})$. The radius R of this region decreases with increasing angle $\arg{\left(S/S_{02}\right)}$, down to about 0.7 in the fourthquadrant, before increasing back to 1, joining the $S/S_{02} = 1$ point of zero thickness. This behaviour reflects the angular averaging of the reflection coefficient, which for a finite thickness d reduces $|S/S_{02}|$ below its maximum of 1.

Measuring $S/S_{02}$ therefore allows the determination of the refractive index and the thickness with high spatial resolution given by the sub-micron focus size created by the objective.

The experimental setup is shown schematically in figure 7 and described below, with more detail given elsewhere [26]. A titanium-sapphire laser (Spectra-Physics Mai Tai) emits 100-fs pulses centred at a wavelength of 820 nm with 80-MHz repetition rate, which pump an optical parametric oscillator (OPO, Inspire Radiantis). The OPO emits a beam of 150-fs pulses centred at 550 nm with horizontal (in the plane of the schematic) linear polarisation. This beam enters an acousto-optic modulator (AOM, IntraAction ASM-802B67) driven at 82 MHz. The zeroth- and first-order deflected beams are used as reference and probe beams, respectively, for HiRef. The probe beam is reflected by a 20R/80T beam-splitter (Edmund Optics NT68-367) and transmitted through a quarter-wave plate (Casix WPA400-550-4) and a half-wave plate (Casix WPA400-550-2), which allow full control of its polarisation state. After the wave plates, the beam is reflected by a dichroic mirror (Eksma Optics, HR415-550nm/HT630-1300nm), transmitted through a 10-mm-thick SF2 glass block used to adjust the beam centring in the back-focal plane of the objective by tip and tilt, and focussed by a lens with a focal length of 100 mm (Edmund Optics T49-360) onto the image plane of an inverted microscope (Nikon Ti-U). The beam enters the right port of the microscope, is reflected upwards by the port prism, is collimated by a 1.5× tube lens with a focal length of 300 mm, and is focussed onto the sample by a 60× 1.27-NA water-immersion objective (Nikon MRD70650). The power at the sample is about 10 µW.

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The beam reflected by the sample travels back through the optics and is transmitted by the 20R/80T beam-splitter. We note that r₀₂ is about 6.5% for the samples we have studied, corresponding to a reflected power of only 0.42%, or 42 nW for an incident power of 10 µW at the sample. The beam is then recombined with the reference beam by a low-polarising 45R/45T beam-splitter (BS, Optosigma 039-0235). The reference beam's optical path length is matched to the probe beam's path length by an optical delay line using a linear stage (Physik Instrumente M-403.6DG) and a fused-silica corner cube (Eksma Optics #340-1217M+3217), and its group velocity dispersion is matched to the probe beam's using 142 mm of SF2 glass (Schott). Note that the probe makes a double pass through the microscope optics described above, accumulating the corresponding amount of dispersion. A linear polariser at 45^∘ creates a well-defined polarisation state of the reference beam with fields of equal amplitude and phase in the horizontal and vertical polarisations.

The two outputs of the BS are separated into horizontal and vertical polarisations by a Wollaston prism and are focussed onto silicon diodes (Hamamatsu S5973-02) to allow balanced detection of the horizontal and vertical polarisation components separately. The reference beam's power was chosen to be around 0.5 mW per diode to ensure shot-noise-limited detection. The differential diode current corresponding to each polarisation is amplified with a transimpedance of 100 kΩ and analysed by a dual-channel lock-in amplifier (Zurich Instruments HF2) locked to the difference of 2 MHz between the AOM upshift and the pulse repetition rate. The lock-in amplifier extracts the interference between the probe and reference beams in amplitude and phase.

The dual-polarisation detection is used to adjust the polarisation at the sample to be circular on axis as follows: A circular polarisation changes helicity upon reflection on an in-plane-isotropic surface, resulting in a cross-circularly-polarised reflected beam emerging from the sample, which after returning through the wave plates is cross-polarised to the horizontal input polarisation. Therefore, minimising the detected signal in the horizontal polarisation using the wave plates ensures a circular polarisation at the sample. Typically, a horizontal field amplitude below 1% of the vertical field amplitude can be stably achieved.

The reflected probe beam $E_\textrm{p}$, with power $|E_{\textrm{p}}|^2$, and the reference beam $E_\textrm{r}$, with power $|E_{\textrm{r}}|^2$, interfere at the BS, yielding two beams with powers $S_1 = |E_\textrm{p}+E_\textrm{r}|^2/2$ and $S_2 = |E_\textrm{p}-E_\textrm{r}|^2/2$. The difference $S_1-S_2 = 2\textrm{Re}\!\left(E_{\textrm{p}}E_{\textrm{r}}^{\ast}\right)$ is measured by the balanced detector and analysed by the lock-in amplifier to obtain the amplitude and phase of the combined field. Because the amplitude and phase of E_r are independent of the sample and ideally constant, the amplitude and phase of the reflected probe beam can be determined.

Due to the high NA used, the focal depth of the focussed probe beam is of the order of 1 µm, much smaller than the thickness of the coverslip and the depth of the well formed by the imaging spacer (see section 3.2), so the glass and oil layers enclosing the sample layer can be considered infinitely thick for the purpose of interference of reflected beams. Furthermore, the short coherence length (around 30 µm) of the 100-fs pulses suppresses interference with reflections occuring at other surfaces.

The sample is moved by an xyz piezoelectric stage with nanometric position accuracy (MadCityLabs NanoLP200) and scanned during image acquisition. Images were acquired with a pixel size $\delta_x = 108$ nm at 0.2 ms per pixel. Each image was a square 80 µm on a side and took around 110 s to acquire.

Immediately before each field of view was imaged with HiRef, the same field of view was imaged with quantitative differential interference contrast (qDIC), which provides an accurate measurement of the thickness under the assumption that the refractive index is known [27].

Three different samples were used.

One sample consisted of a flat layer of polyvinylacetate (PVA) several tens of nanometres thick on a #1 glass coverslip (figure 8). The coverslip was washed with acetone and etched with a 3:1 solution of sulphuric acid to hydrogen peroxide at 95 ^∘C to remove contaminants. 0.26 g of PVA and 4.33 g of water were mixed to form a 6% mass/mass PVA solution. The solution was spin-coated on the coverslip at 3000 rpm for 30 s with 6 s of constant acceleration and deceleration before and after the 30-s constant-speed period. Cuts were made with a sharp razor at different angles in the spin-coated PVA layer in order to create gaps without material. A 120-µm-thick square imaging spacer (Grace BioLabs, OR, USA) with a circular hole 13 mm in diameter was adhered to the coverslip to form a shallow well. The well was filled with water-immersion oil ($n_2 = 1.3339$, Zeiss Immersol W 2010) and covered with a glass microscope slide. Because the immersion oil reduced the adhesiveness of the imaging spacer, nail varnish was used to create a tight seal around the chamber.

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Each of the other two samples consisted of a SLB on a #1 glass coverslip (figure 9). The coverslip was washed and etched as described above. Either pure dipentadecanoylphosphatidylcholine (DC₁₅PC) or a ternary lipid solution consisting of dioleoylphosphatidylcholine (DOPC), chicken-egg sphingomyelin and cholesterol in an 11:5:4 molar ratio (DOPC:sphingomyelin:cholesterol) was diluted in isopropanol to a lipid concentration of 1 mg ml⁻¹. 150 µl of this solution were deposited on the coverslip and spin-coated as described above; the spin-coating parameters were the same as for the PVA sample. In order to avoid rapid absorption of moisture in the later steps of sample preparation, which would have destroyed the structure of the bilayer, the coverslip was placed in a nitrogen-filled centrifuge tube with a small piece of moist tissue paper and incubated at 37 ^∘C for 1 h. Finally, an imaging spacer with the aformentioned dimensions was adhered to the coverslip to form a shallow well. The well was filled with degassed phosphate-buffered saline and covered with a glass microscope slide. Prior to imaging, the ternary SLB sample was stored at 4 ^∘C for a few hours to allow liquid-ordered and liquid-disordered lipid domains to form.

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The HiRef phase images of the SLB samples were filtered to remove low-frequency noise, which obscured the structural features owing to the low contrast from a single lipid bilayer. This filtering consisted of multiplying the Fourier spectrum of the phase of the image data by $1-f(\nu)$, where f is a sum of gaussian functions, each centred at a frequency ν and having a frequency width of 1 Hz, to remove frequencies below 5 Hz (phase drift), a collection of frequency peaks near 10 Hz (likely electronic noise from nearby equipment) and 15 harmonics of the fast-axis scan frequency, given by $1/(2N_x t_\textrm{e})$, where $t_\textrm{e} = 200~\mu$s is the pixel dwell time and N_x is the number of pixels in one row of the image.

Because the position of the sample stage at any given time during image acquisition is not exactly the commanded position, the position x', y' measured by the stage sensor is recorded at each pixel together with the signal S' at that pixel, forming a time trace of the measurement with the time points t'. This time trace is converted into a regular grid of pixels in a post-processing procedure called regularisation. For every pixel position (x, y) on the grid, the signal S is calculated as

$$\begin{align} S(x,y) = \sum_{t^{^{\prime}}}S^{^{\prime}}(t^{^{\prime}})\,W(t^{^{\prime}})\left/\sum_{t^{^{\prime}}}W(t^{^{\prime}})\right.\,, \end{align} \tag{ 16 }$$

where

$$\begin{align} W(t^{^{\prime}}) = \exp\left(-\left(\frac{x-x^{^{\prime}}}{\Delta_x}\right)^2-\left(\frac{y-y^{^{\prime}}}{\Delta_y}\right)^2\right) \end{align} \tag{ 17 }$$

and the sum is taken over all the time points t' for which $|x-x^{^{\prime}}|\lt2\Delta_x$ and $|y-y^{^{\prime}}|\lt2\Delta_y$ for computational efficiency. We used $\Delta_x = \Delta_y = \delta_x$. Regularisation improves the quality of the images in terms of accurate representation of the contents of the sample.

After regularisation, line traces with a length of 121 pixels (about 13 µm) for the PVA sample or 61 pixels (about 7.5 µm) for the SLB samples were taken along the fast scan axis, the x axis. The positions of the traces were chosen so they were centred along the edge of the PVA or lipid (depending on the sample) and thus roughly half of the pixels on any given trace were on PVA or lipid and the rest were on an empty region; the line traces thus had a step in the middle. The step function

$$\begin{array}{*{20}{l}} {\Theta (x)}&{ = \frac{{{a_1}}}{2}{\mkern 1mu} \tanh \left( {\frac{{x - {a_2}}}{{{a_3}}}} \right) + \frac{{{a_4}}}{2}{\mkern 1mu} {\textrm{sech}}\left( {\frac{{x - {a_5}}}{{{a_3}}}} \right)}\\ {}&{\quad + \;{a_6}(x - {a_2}) + {a_7}} \end{array} \tag{ 18 }$$

was simultaneously fitted to the amplitude and phase traces, sharing the spatial parameters (a₂, a₃ and a₅), since features are expected to be at the same locations in the amplitude and phase data. The hyperbolic tangent produces the step, while the hyperbolic secant describes irregularities in the edges, caused, for example, by the nature of the cuts with the razor (PVA sample) or folds in the lipid (SLB samples). The linear component fits slow drift due to sample tilt and to thermal drifts in the beam paths and in the axial (z) position of the sample stage.

Let us assume for the following discussion that with increasing x the trace moves from the glass-water interface onto the investigated layer. For the phase traces, the constant term a₇ represents a phase offset, and the fitted step height a₁ is given by

$$\begin{align} \arg{\left(\frac{S}{S_{02}}\right)} = a_1 \end{align} \tag{ 19 }$$

(for comparison with calculations, see also figure 6). For the amplitude traces, we have

$$\begin{align} \left|\frac{S}{S_{02}}\right| = \frac{a_7-a_1/2}{a_7+a_1/2}\,, \end{align} \tag{ 20 }$$

fully determining $S/S_{02}$. Figure 10 shows an example of a line trace from the PVA sample.

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The thickness and refractive index were computed numerically from the measured value of $S/S_{02}$ at each line trace. For the computation, the region of $(d,n_1)$ space given by $0\leqslant d\leqslant120$ nm and $n_2\leqslant n_1\leqslant n_0$ was partitioned finely and the difference between the quantity $S/S_{02}$ calculated with equation (13) and the quantity calculated from the line trace fit was minimised; the values of d and n₁ assigned to the layer at the line trace were those which minimised this difference.