1. Introduction

The broad need for determining the optical properties of thin films in a multitude of applications is usually served by ellipsometry [1]. Practical application of ellipsometry generally requires prior constraints, typically in the form of a frequency-dependent model, to provide for a suitable solution of the inverse problem, where film parameters are determined from a set of optical measurements. We present motion in structured illumination as a means to obtain additional information and hence avoid the need for a material response model. Using this approach, inversion for multiple parameters at each wavelength becomes possible, and in a mutual information sense, this is achieved by taking intensity measurements at a known set of displacements in a cavity.

Ellipsometry measures the amplitude ratio and the phase difference between polarized light reflected from the surface of a film and determines the refractive index or thickness by fitting the experimental data to an optical model that represents an approximated sample structure [2]. Generally, a model of the frequency-dependent dielectric constant is used for successful parameter extraction, in order to constrain the inversion. For example, such a model may represent a Lorentzian resonance or impose a Drude model. While simplifying the extraction, this imposes an approximate but not necessarily the correct description. Otherwise, a careful choice of the initial variables is needed in ellipsometry [3].

There is a long history of using interferometers to determine the relative position of a surface, and to determine the refractive index of gases [4], and in fuel cells [5], including water content changes in membrane fuel cells [6]. White-light interferometry has been used to retrieve the thickness of thin films [7], under the assumption that the frequency-dependent dielectric constant is known.

We present the concept of an interferometer arrangement where intensity measurements as a function of controlled position of the sample, as could be achieved with a piezoelectric positioner, allow extraction of both the thickness and dielectric constant based on transmission measurements. The simple intensity-based measurement required avoids the alignment and multiple polarization data typical of ellipsometry. Here, the film is moved in a structured background field in steps, and the total power due to the background and scattered fields is measured. The method relies on cost-function minimization using a forward model to compare the measurements to a set of forward model data corresponding to different sample structures rather than repeated corrections to the theoretical dielectric function and initial values in order to fit the experimental data.

2. Concept and simulated results

An illustration of the arrangement used to obtain simulated data is shown in Fig. 1(a). The 1D object to be characterized is located and scanned within a cavity having a low quality ($Q$) factor that provides the structured field, as illustrated in Fig. 1(b). Two dielectric slabs forming the partially reflective mirrors have a refractive index of 1.5 (simulating crown glass) and a thickness of $\lambda /5$, with $\lambda$ being the free-space wavelength, $1.5 \mu$m. The mirrors are separated by 2.7$\lambda$ (inner face-to-face distance). Note that the length of the cavity was not tuned to resonance. An object of total thickness $\lambda /5$, is comprised of two layers of different materials: a slab with a known refractive index of 1.5 and a thin film on top with a thickness $L$ and refractive index $n$. Both $L$ and $n$ are to be determined simultaneously at the single frequency of the measurement, at a free-space wavelength of $\lambda$.

 

Fig. 1. (a) The simulated measurement arrangement has a plane wave incident from the top, with the free-space wavelength as $\lambda = 1.5 \mu$m. Two dielectric slabs act as partially reflecting mirrors and form a low-$Q$ cavity with a length of 2.7$\lambda$ (inner face-to-face distance). An object comprised of a thin film on top of a substrate, and a total thickness of $T = \lambda /5$, is located in this cavity and moved vertically upwards in nm-scale increments. As the object is translated in the cavity to a set of positions, the power is measured at the detector plane, located $0.4\lambda$ below the bottom surface of the lower mirror. (b) An illustration of the magnitude of the background electric field in which the film and substrate are placed and moved (not drawn to scale).

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Referring to Fig. 1(a), the numerical finite element method [8] simulations we used have a normally-incident plane wave coming in from the top, with polarization in the $\hat {\mathbf {z}}$ direction (out of the page). A port boundary condition was used at the top of the domain to set up the incident field, and periodic boundary conditions were applied to the sides. In the scattered field solution employed, a perfectly matched layer was used at the top and bottom, in order to simulate unbounded space. The sum of the incident and scattered field produces the total field, and hence the time-average power at the detector plane can be determined.

In Fig. 1(a), as the object moves vertically within the structured field with subwavelength steps to a set of positions (achievable with a piezoelectric positioner), we measure the time-average Poynting vector magnitude, $|\mathbf {S}_\textrm {av}| = S$, in the transmission direction at the detector plane, as a function of the position ($\Delta y$) of the object. In other words, when the object is scanned over $K$ positions in the cavity, the measurement array will consist of $K$ individual data values, each representing the measured $S$ at each object position. For this simulation, the detector plane is located $0.4\lambda$ below the bottom surface of the lower mirror, although for the single (separable) plane wave problem considered, this distance is arbitrary. The object to be characterized is initially placed with the bottom surface $0.8\lambda$ above the top surface of the bottom mirror and scanned in small steps upwards over a range of $0.5\lambda$, giving 21 positions and measurements in total.

We assume a Gaussian noise model, such that the measurements are normally distributed with a mean equal to the noiseless measurement, $S$, and a standard deviation $\sigma$ proportional to the noiseless data, giving us a measure of the variability of the measured signal. We choose a conservative signal-to-noise ratio (SNR) of 30 dB, which is achievable with an appropriate input power source and integration time, for all results. As a reference, the noise equivalent power (the minimum optical power required for an SNR of 1 in a 1 Hz bandwidth) for commercially available avalanche photodiodes has a reported range of $10^{-14}$—$10^{-16}$ (W/$\sqrt {\mbox {Hz}}$) [9]. The standard deviation, $\sigma$, is determined from the SNR, given by $\text {SNR}_{dB} = 10\log _{10}(S/\sigma )$. We can then numerically generate independent noisy measurements as $S + \sigma \times N(0,1)$, where $N(0,1)$ is a normalized normal (Gaussian) density function with zero mean and unit variance.

In Figs. 2(a) and (b), we plot noiseless measured data $S(\Delta y; L, n)$ with simulated noise represented as error bars against the positions of different slab configurations. In Fig. 2(a), the two plotted curves represent measurements of two different film refractive indices ($n = 2.00$ and $n = 1.95$) with a fixed thickness ($L = 0.005\lambda$), while Fig. 2(b) shows measurements of two different film thicknesses ($L = 0.007\lambda$ and $L = 0.005\lambda$) with a fixed refractive index ($n = 2.00$). In Figs. 2(a) and (b), we show error bars at each measured data point, where the end-to-end length of the error bars is equal to $4\sigma$, representing the range of $95\%$ of the noisy measurements.

Noisy measurements of two slab configurations with minuscule differences in the film’s refractive index $n$ and thickness $L$ can be separated with higher confidence if their error bars do not overlap. This is shown in the magnified data in Figs. 2(a) and (b). In other words, separability is achieved when the difference of the noiseless data is larger than the sum of half of the respective error bar lengths for each measurement. This is further demonstrated in Figs. 2(c) and (d), where we calculate the difference of the two curves in Figs. 2(a) and (b), with the same error bars superimposed at each data point. In Fig. 2(c), the red curve sets $S(\Delta y; L, 1.95)$ to zero as a reference, and the blue curve represents $[S(\Delta y; L, 2.00) - S(\Delta y; L, 1.95)]$. The same is repeated in Fig. 2(d), where the red curve sets $S(\Delta y; 0.005\lambda, n)$ to zero, and the blue curve represents $[S(\Delta y; 0.007\lambda, n) - S(\Delta y; 0.005\lambda, n)]$. From Figs. 2(c) and (d), we see that there exist multiple data points where there are non-overlapping error bars, which can then be used as leverage to distinguish between two different noisy measurements, thus demonstrating sensitivity to small differences in film refractive indices and film thicknesses. For example, in Fig. 2(c), placing the object in positions in the cavity where the measured data can be separable gives more information for estimating the correct parameters when a cost function is employed to compare small differences of the object. Moving the object in the cavity ensures that separable data points from various positions will also be measured and included into the estimation method. Note that the data shown in Fig. 2 is a periodic function of the film’s displacement, with the results for a single period being displayed here. In the following sections, we propose a simple cost function for parameter estimation and investigate the accuracy and detectability limits due to variable noise levels and incorporating different numbers of data points into the cost function by adjusting the motion step size.

 

Fig. 2. Measured power flow against object position for different film parameters. The end-to-end length of the error bars is equal to $4\sigma$, calculated with an SNR of 30 dB. Non-overlapping error bars indicate distinguishability between noisy measurements, demonstrating sensitivity to minuscule differences in film parameters. (a) Film with $L = 0.005\lambda$ and varying refractive indices, $n$. (b) Film with $n = 2.00$ and different thicknesses, $L$. (c) Expanding the scale in (a), the red curve uses $S(\Delta y; L, 1.95)$ as a reference by setting it to zero, and the blue curve gives $[S(\Delta y; L, 2.00) - S(\Delta y; L, 1.95)]$. (d) Expanding the scale in (b), the red curve shows $S(\Delta y; 0.005\lambda, n)$ as a reference (zero), and the blue curve $[S(\Delta y; 0.007\lambda, n) - S(\Delta y; 0.005\lambda, n)]$. Data points exist for when the blue curve is non-overlapping with the red curve, indicating separability and sensitivity to small differences of refractive indices and subwavelength differences of the film thickness.

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3. Film characterization

We demonstrate a straightforward application for the sensitivity provided with motion in structured illumination by reconstructing the film’s thickness $(L)$ and refractive index $(n)$ on top of a slab with known optical properties using the system in Fig. 1(a). Utilizing the measurements obtained this way, we can use a cost function to find parameters of interest by comparing the measurement with forward calculations over a range of different combinations of $L$ and $n$.

We denote $\mathbf {f}(L, n)$ as the array of calculated data, where each entry represents the data at each $\Delta y$ location obtained from the forward model, $S(\Delta y;L,n)$, and $\mathbf {y}$ as the array of simulated noisy measurements with presumably unknown true thickness, $L_t$, and refractive index, $n_t$. Both $\mathbf {f}(L, n)$ and $\mathbf {y}$ have dimensions equal to the total number of positions, $K$, that the object has moved in the structured field. We denote $ {y}_{k}$ as the $k$th element in array $\mathbf {y}$, and ${f}_{k}(L, n)$ as the $k$th element in $\mathbf {f}(L, n)$. Note that each entry of $\mathbf {y}$ can be generated using the noise model described previously, with ${y}_{k} = {f}_{k}(L_t, n_t) + \sigma \times N(0,1)$, where $N(0,1)$ represents a normal distribution with zero mean and unit standard deviation. We can then determine the estimated values of $L$ and $n$ from

Here we chose an L1-norm as our cost function to illustrate the potential of this application due to its robustness and resistance to outliers in data. However, an L2-norm cost function combined with other optimization algorithms can be considered using the data obtained from this method. The forward calculations were made for a range of possible thicknesses ($L$) and refractive indices ($n$) that encompasses $L_{t}$ and $n_{t}$. Consider a hypothetical experiment from a film structure with $L_{t} = 0.006\lambda$ and $n_{t} = 1.72$, with a $\text {SNR} = 30$ dB. The calculated costs from (1) for each combination of $L \in [0.002\lambda, 0.022\lambda ]$ with step increments of $0.002\lambda$, and $n \in [1.62, 1.98]$ with step increments of 0.04 (resulting in an 11x11 grid), is shown in the top left plot of Fig. 3. The cost is at the minimum when $\hat {L} = 0.006\lambda$ and $\hat {n} = 1.72$, as indicated by the red circle on the grid, demonstrating a correct estimation of the thickness and refractive index from noisy data. This accurate reconstruction is possible if the true thickness and refractive index is assumed to fall within the range of calculated configurations in the grid.

 

Fig. 3. Calculated costs from (1) for a thin film substrate by comparing the simulated noisy measurements with forward calculations of different film configurations without multiresolution (top left), and with multiresolution (starting from top right and following the arrows). The film substrate used in the simulated experiment has a film thickness $L_t = 0.006\lambda$ and refractive index $n_t = 1.72$. Without multiresolution, forward calculations were made for different combinations of film thicknesses $L \in [0.002\lambda, 0.022\lambda ]$ with step increments of $0.002\lambda$, and refractive indices $n \in [1.62, 1.98]$ with step increments of 0.04, resulting in an 11x11 grid. The cost is minimized at the correct parameters, where $\hat {L}=0.006\lambda$ and $\hat {n}=1.72$. When using a multiresolution approach, forward calculations were made on a coarse 5x5 grid with a significantly increased range of values of $L \in [0.002\lambda, 0.13\lambda ]$ and $n \in [1,3.56]$. The cost is calculated iteratively on zoomed in regions of interest (following the arrows) that encompasses the the point of minimum cost.

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A multi-resolution inversion approach can be applied, significantly decreasing the calculation time while maintaining the same final resolution, when scanning over a larger search range of $L_{t}$ and $n_{t}$. This is demonstrated in Fig. 3, where we start from the top right plot. The cost is calculated and minimized on a coarse 5x5 grid with an increased range of parameters, where $L \in [0.002\lambda, 0.13\lambda ]$, roughly half of the entire sample, and $n \in [1,3.56]$. The cost is then calculated iteratively on a smaller region of interest and on a finer grid that zooms in at the point of minimum cost and extends a distance equal to the grid spacing from the previous iteration, as shown by following the arrows in Fig. 3. This procedure is repeated five times until the step increments of the film thickness on the grid reaches $0.002\lambda$, which is where the measurements remain separable given a 30 dB SNR.

Through this multi-resolution approach, we are able to reach the same resolution as the top left plot in Fig. 3 with around the same calculation time (multi-resolution: [5 iterations x (5x5) grid] = 125 calculations) vs without multi-resolution: [11x11 grid] = 121 calculations), but with an increase of range in terms of thickness: from $L \in [0.002\lambda, 0.022\lambda ]$ to $L \in [0.002\lambda, 0.13\lambda ]$, and refractive index, from $n \in [1.62, 1.98]$ to $n \in [1,3.56]$.

Since the simulated measurements are noisy, the reconstructed film thickness and refractive index will fall within a distribution. To evaluate the efficacy of this characterization method we use the same film substrate structure, and generate 500 noisy independent simulated measurements for a given SNR. The 500 sets of reconstructed thicknesses and refractive indices were then obtained from these independent measurements, which allows us to obtain a statistical distribution of the reconstructed values for the given SNR. This is repeated for different values of SNR to demonstrate the performance at different SNR values and is shown in Fig. 4. Figure 4 shows box plots of the distribution of reconstructed thickness and refractive index at different values of SNR. For each SNR, the top and bottom edges of the box represent the upper and lower quartile of reconstructed values, the whiskers extend to the minimum and the maximum values and the red dots represent outliers (defined as less than the first quartile or greater than the third quartile by more than 1.5 times the interquartile range). The red dashed line in both plots represents the median of the reconstructed values at each SNR and is identical to the true film thickness, $L_t$, and refractive index, $n_t$. In Fig. 4(a), the length of the box is skewed upwards and asymmetric about the median. This is due to the boundary limits of the possible reconstructed values of the thickness, i.e., the reconstructed thickness cannot go below $0.002\lambda$. In Fig. 4, it is shown that for the reconstructed values of film thickness and refractive index, as the SNR increases, with the exception of several outliers, the size of the boxes and whiskers gradually decrease and converge to the median at around 32 dB, depicting a convergence to $L_t$ and $n_t$.

 

Fig. 4. 500 independent measurements were made at different SNR values to calculate a distribution of reconstructed values of $L/1000\lambda$ and $n$, representing uncertainty in the reconstruction of thin film parameters. (a) Box plots of the distribution of reconstructed film thicknesses for different SNR values. Note that the $y$-axis is on the scale of $10^{-3}$. (b) Box plots of the distribution of reconstructed refractive indices. The top edge of the box represents the upper quartile of the reconstructed values, and the bottom edge represents the lower quartile. The whiskers extend to the upper and lower extremes, and the red dots represent outliers. In both plots, the median (red dashed line) obtained from the set of reconstructed values is equal to the true film thickness and refractive index, $L_t$ and $n_t$.

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4. Sensitivity

The limitations of the approach are further investigated in terms of sensitivity for the cases of very thin films and films with low refractive index contrast relative to the substrate. We consider the same film-substrate structure shown in Fig. 1(a), but for various sets of combinations of thin film thickness and differences in refractive index ($\Delta n$), when compared to the case of a slab with the film having the same material as the substrate. For a given SNR, we generated $10^5$ sets of measurement data for each unique film parameter combination with $L \in [0.001\lambda, 0.01\lambda ]$ (increments of $0.001\lambda$) and $\Delta n \in [0.02, 0.2]$ (increments of 0.02). We use (1) to determine whether the estimated parameters detects the presence of a thin film or returns a slab with no film present due to measurement noise. This would give us the detectability for each film parameter combination (percentage of correct reconstructions showing the thin film present). Contours can then be obtained for when the detectability is at least 99.99%. The contours were then fitted to a two-term power series model ($y = ax^b+c$, in this case, $y$ and $x$ are $L/\lambda$ and $\Delta n$, respectively, $a$ and $b$ are the fitted coefficients). The results of the sensitivity analysis are plotted in Fig. 5. As expected, higher SNR results in greater accuracy and distinguishability of thinner films and lower optical contrast, translating to higher sensitivity. Also, by taking more measurements and moving the slab more times within the cavity, higher sensitivity can be achieved.

 

Fig. 5. Minimum detectability of very thin films with low refractive index contrast relative to the optical properties of the substrate. The region to the right and above of each curve represents detection of the thin film with greater than a 99.99% success rate at: (a) different noise levels, and (b) different numbers of positions, K, incorporated into the cost function at SNR = 30 dB.

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In Fig. 5(a), fitted contours were drawn for SNR values of 35 dB, 30 dB, and 25 dB, which are represented as the dashed black line, solid red line, and dashed-dotted blue line, respectively. In other words, the area to the right and above the curve represents the parameter space of the thin film that can be reconstructed and distinguished from measurements of a film-less slab. It can be seen from Fig. 5(a) that with a higher SNR, it is possible to detect a thinner film with lower optical contrast compared to the slab. For example, in the case of a 30 dB SNR, $0.002\lambda$ is the lower limit for reconstruction and resolution in terms of film thickness for particular values of $n$, while $0.04$ is the lower limit for contrast in refractive index for particular values of $L$. In Fig. 5(b), contours for different number of positions, $K$, are plotted to investigate how $K$ affects the efficacy of the method while fixing the SNR to 30 dB. In Figs. 5(a) and (b), the solid red curve represents identical parameters ($K = 21$, SNR = 30 dB). The solid red curve, the dashed black curve, and the dashed-dotted blue line are for $K$ equals 21, 11, and 5, respectively. Increasing the number of steps that the slab moves within the cavity gives more information and in turn higher detectability.

5. Perspective and implementation

We understand the concept of measurements as a function of known translated position from mutual information (see, for the example, the appendix of [10]). If we know the background optical arrangement, i.e., the cavity and source (wavelength), and measurements are made at a known set of positions (perhaps just the relative change in position), then useful information about the object (the film) can be obtained. Sensitivity (spatial resolution for thickness and the dielectric constant) is limited by noise. The contributors to this noise are: (i) laser amplitude and phase noise (dictating noise in the incident or background field); (ii) object positioning noise (or estimation error), such as with a piezoelectric positioner; and (iii) detector noise (from sensing the intensity at a set of locations). Deterministic sources of error can in principle be removed by signal processing or through experiment design. This leaves the aforementioned random noise sources. Estimating the mean result can thus be improved arbitrarily, with the error in estimating the mean given by the standard deviation of an underlying density function divided by the square root of the number of measurements (so the measurement points lie on a Gaussian density function, and the goal of a measurement is to estimate the mean). In practical terms, there will of course be limits based on the equipment and situation.

Imaging methods based on object motion in structured illumination have been proposed for achieving far-subwavelength resolution using far-field measurements [11]. The film characterization approach described here is a 1D implementation where it is shown that both the dimension and the dielectric constant of a film can be determined using a forward model. Also, measured intensity correlations over object position with motion in a speckled field have shown that both macroscopic and microscopic information are available [12], although in this case extraction is through statistical averaging using intensity data, yielding normalized geometric information about the object, and a forward model is not plausible.

In our presented method, the structured background field is generated to allow small changes in material and geometry information to be detected through motion in a structured field. This is fundamentally different from structured illumination microscopy [13], where the purpose of the structured background fields is to illuminate a sample (embedded with fluorophores), allowing higher resolution by extending the range of observed spatial frequencies and doubling the resolution. On the other hand, our method relies on added information obtained from measurements as a function of object position in a spatially varying background field. The resulting spatial resolution is practically only limited by noise, and deep-subwavelength precision should be possible.

The multi-layer planar film problem is separable using a plane wave basis set, so a laser beam can be decomposed into its plane wave (Fourier) spectrum, and each plane wave propagated through single or multiple film layers using a transmission matrix representation for interfaces and regions of homogeneous material. This approach would be useful when extracting results from experimental data for intensity as a function of translated distance. It does assume that surface roughness and deformation are negligible. Also, in forming the forward model, the laser beam and associated optics (namely the mirrors) need to be adequately characterized for use in the forward model. In this way, with appropriate data, the various layer thicknesses and dielectric constants could be determined through optimization [14]. In this regard, multi-resolution approaches may be useful, both in terms of computational effort but also possible avoidance of local minima [15,16].

This method has the potential to be extended to retrieve the imaginary index of refraction by introducing a third unknown variable into the cost function. This work presents a starting point for investigating the resolution limit for small defects in a slab. With sensitivity to differential changes in parameters, this approach could be extended to detecting defects located within a stack of multiple layers by comparing measurements with that of a defect-free stack. It may thus be possible to determine the presence of a defect in a semiconductor. The ability to characterize defects still depends on the SNR, size, and contrast between the defect and the background layer. We have shown that, given an SNR of 30 dB, a resolution of $L = 0.002\lambda$ and $\Delta n = 0.04$ can be achieved for detection of a thin film. The influence of surface roughness also warrants investigation.

In an implementation, the cavity length can be essentially arbitrarily long, although consideration of the laser and the free spectral range would be needed. The laser beam would need to be compact relative to the size of the membrane. For example, SiN membranes have a dimension in the millimeter range, so tens or a few hundred micron beam size would be appropriate. Stepper motor stages routinely have sub-100-nm steps, and this may be sufficient. Piezoelectric stages are available with a step/precision of about 1 nm. Thus, an experiment could be comprised of two partially reflecting mirrors, a translation stage, a laser diode (those having an external cavity provide high coherence), and a photodiode detector. Other configurations may simplify the experiment. As such, the instrumentation costs are modest.

Two dimensional materials such as transition metal dichalcogenides [17–19] and single [20] and multi-layer graphene [21] are important in optoelectronic applications. Characterizing the sheet dielectric constants for single and multiple layers of such 2D materials is generally required, and this can be facilitated using the approach we present involving motion in structured illumination.

6. Conclusions

We have presented a method to obtain the optical parameters and thickness of a thin film on a substrate using motion in structured illumination. Various implementations are envisioned. One form is insertion in a Fabry-Pérot cavity, as described in detail. However, a Mach-Zhender interferometer and a reflection mode could be employed. The films considered were transmissive, and the resonance condition is modified with a change in position within the cavity. Non-transmissive samples could also be characterized. This is equivalent to using the planar structure as a mirror, but where information on, for example, the impedance boundary condition, becomes available from a modified resonance condition (as a sample is translated). Also, the standing wave could be scanned by controlling a bi-directional incident field with a phase shift operation and making measurements with a fixed detector. Other ways of generating structured illumination can also be applied, such as spatial light modulators, therefore removing the need for a cavity.