Mapping spectroscopic micro-ellipsometry with sub-5 microns lateral resolution and simultaneous broadband acquisition at multiple angles

Ellipsometry is a powerful optical technique for thin film characterization, based on measuring the change in light polarization upon oblique reflection. Its non-destructive nature, high accuracy, simplicity, and availability make ellipsometry an essential tool in various fields of industry and research, such as semiconductors,^1–3 photovoltaics,^4,5 materials characterization,^6,7 optical coatings,^8,9 two-dimensional materials,^10,11 flat panel displays,^12–14 organic films and surfaces,^15,16 antifouling coatings,^17,18 biological materials,^19,20 and many more.

In ellipsometry, measured data are fitted to a relevant model for extracting optical properties and/or thickness information. This makes it crucial to increase the amount of measurement data points by spectroscopic and angle-resolved acquisition for minimizing the local-minima problem^21–23 and increasing the sensitivity and accuracy of the fit parameters. Spectroscopic ellipsometry (SE) acquires ellipsometric data as a function of the light wavelength, and was completely automated in 1975 by Aspnes and Studna,²⁴ which significantly improved not only the measurement time but also the measurement precision.²⁵ Utilization of broadband illumination and a photodiode array for simultaneous measurement at multiple wavelengths was reported in 1990 by Kim et al.,²⁶ and is still the mainstream method for most modern SE instruments. Besides providing spectral information, SE improves the accuracy and data acquisition rate when compared to single-wavelength ellipsometry.²⁵ This significant advantage makes SE a standard method among the polarization-dependent optical techniques for the investigation of optical properties.²⁷ In modern SE instruments, the angle of incidence (AOI) can be varied between measurements by manual or automatic alignment of the mechanical arms. Adding multiple-angle information is specifically important for multi-layered structures as each angle provides new information by traversing a different optical path, optimizing sensitivity to the unknown parameters.^21,28 Although spectroscopic acquisition can be performed on any type of sample, angle-resolved acquisition by conventional ellipsometers is mostly possible for laterally homogeneous samples since the size and shape of the measurement spot vary in dependence on the incidence angle, causing different areas of a laterally inhomogeneous sample being probed.

On the flip side, SE suffers from low lateral resolution due to its off-axis configuration.²⁵ Conventional SE uses quasi-collimated beams, resulting in elliptical and millimeter-scale beam shapes on the sample. For improving the lateral resolution, in 1986, Erman and Theeten²⁹ demonstrated a convergent beam approach, achieving a lateral resolution on the order of 10 μm (with a catadioptric focusing system and monochromatic light) and discussing the modifications and the limitations due to the use of such a nonplanar wave for an ellipsometric measurement. Today, modern focused-beam SE instruments use convergent beams by adding low numerical aperture (NA) objective lenses before and after the sample, reducing the spot size down to tens-of-microns.^30–32 However, focused light on the sample makes it tedious for the mechanical variation of the AOI, and hence, focused-beam SE is usually optimized for a single AOI. This limits the amount of data for acquisition, decreasing the sensitivity of the fit parameters.

Another technique to improve the lateral resolution is imaging SE, which integrates an objective lens and a two-dimensional detector array to its hardware, achieving a lateral resolution down to a few micrometers.³³ In principle, imaging SE merges optical microscopy into a conventional ellipsometer configuration. However, this technique limits the ellipsometric measurement to recording one wavelength at a time (at a single AOI) after mechanical rotation of its polarization components, making it operate with inordinately long measurements times³⁴ for a spectrally resolved response. In addition, since imaging SE performs optical microscopy at a tilted observation angle, only one strip of the overall image can be in focus. This necessitates the longitudinal scan of the objective lens for recording different strips in focus, and then combining them into a focused image of the sample area. All these add to the hardware complexity and measurement times for an imaging SE instrument, and require a very stable sample. In contrast, conventional and focused-beam SE measure broadband responses at a single AOI in a single measurement of a short time frame. This fast data acquisition rate of SE holds significant importance for making it a more practical tool in industry and research.

To overcome mechanical and technical limitations of ellipsometry, an approach called “micro-ellipsometry” has been suggested and experimented.^22,35–45 Micro-ellipsometry makes use of the NA of an on-axis objective lens for oblique reflection and collection of light. The reflected light with the same AOI is focused on a specific location on the Fourier plane, which is imaged by using a camera. This approach simplifies the collection of angle-resolved ellipsometric information by eliminating the need for mechanical arms, while allowing a micrometer-scale spot size with its on-axis configuration. Despite its potential advantages, micro-ellipsometry is still not used widely and is not commercially available. One main factor behind this is the lack of an accurate and simple system characterization and calibration. Micro-ellipsometry introduces new system unknowns to the measurement: the AOI values and the instrumental polarization caused by added reflection/transmission interfaces to the optical path. Theoretical mapping of AOI values on the Fourier plane does not take into account the very possible misalignment and errors (especially with manufacturing and assembly errors of high-NA objective lenses), which can have an effect on the AOI locations. This is critical due to the high sensitivity of ellipsometry to AOI values. In parallel, the instrumental polarization at each AOI needs to be accurately determined for post-measurement calibration.

Building an accurate, simple, spectroscopic micro-ellipsometer is still an ongoing scientific challenge.^22,41,42,44 Although the importance of accurate AOI characterization has been realized lately,⁴³ still no method has been proposed for an accurate, simple, and complete system characterization to unlock all the advantages of micro-ellipsometry simultaneously, namely, fast and accurate acquisition of angle-resolved broadband ellipsometry data in a single measurement with high lateral resolution.

In this paper, we present a spectroscopic micro-ellipsometer with sub-5 μm lateral resolution and simultaneous broadband acquisition at multiple angles in a single measurement of a few seconds. Its high accuracy is based on our system calibration method⁴⁶ that provides a simple, accurate, and complete characterization of the optical system. Our instrument is realized with only a few and standard hardware additions to a generic optical microscope, allowing its capabilities to be integrated into standard optical imaging systems in a simple, compact, and low-cost manner. In contrast to the mentioned imaging SE technique, which integrates optical microscopy into conventional ellipsometry configuration, our technique integrates spectroscopic ellipsometry into a generic optical microscopy setup.

We demonstrate the accuracy of our method by comparing the results of complex refractive index and thin film thicknesses with a commercial spectroscopic ellipsometer. The high lateral resolution performance of our instrument is demonstrated by mapping local variations in the film thickness and complex refractive index over micrometer-scale areas. Finally, we show how the combination of accuracy and high lateral resolution allows for new capabilities by demonstrating measurements of the complex refractive index and the layer number of atomically thin exfoliated materials with micrometer-scale lateral dimensions.

The Spectroscopic Micro-Ellipsometer (SME) consists of a generic optical imaging system (a microscope) with a few added standard optical components and a spectrograph with a two-dimensional detector array. The SME can be integrated into any commercial optical microscope without disturbing its capabilities via the phototube port, which is primarily designed for the integration of digital imaging cameras to the microscope, as illustrated in Fig. 1(a).

The SME uses a standard, incoherent, quasi-collimated, and broadband light source with the desired spectral range, operating mostly in the visible spectra at present. Apart from an external source as currently used, the integral broadband illumination of any optical microscope can also be used for SME measurements as long as the required optical components can be inserted into the built-in optical path of the microscope. A field stop follows the light source, composed of two lenses and a pinhole at their shared focal plane. The pinhole is imaged on the sample with a certain magnification; therefore, modulation of the pinhole aperture gives the SME control over its spot size and, hence, its lateral resolution. All lenses in the SME are achromatic doublets, optimized for broadband performance. Next is the PSG (polarization state generator), consisting of a Glan–Thompson linear polarizer (Thorlabs GTH10M) followed by a Fresnel rhomb quarter-wave (λ/4) retarder (Thorlabs FR600QM), which together modulate the input polarization by mechanical rotations to be linear or circular, as required for a conventional ellipsometry measurement (explained in  Appendix A). These elements are selected for their high broadband extinction ratio and retardance, respectively. Since the rotation of the Fresnel rhomb quarter-wave retarders causes the displacement of the optical path [not illustrated in Fig. 1(a) for the purpose of simplicity], it is kept fixed and, instead, the polarizer is rotated with respect to the fast axis of the retarder, as illustrated with a black arrow around the polarizer in Fig. 1(a).

The broadband and polarized input illumination is then directed by reflection to pass freely over the D-shaped (pickoff) mirror (Thorlabs PFD05-03-P01), reaching the objective lens and finally the sample with multiple AOIs, confined in a micro-spot. The D-shaped (also known as knife-edge) mirror is a precisely cut mirror having a straight edge, designed to enable the separation of closely spaced beams. In the SME, it is used to allow for the free passing of the input beam toward the objective lens and to reflect the collected beam from the objective lens on the slightly shifted output optical path. The D-shaped mirror preserves the light intensity while refraining from the interference effects that might be caused by some beam splitters. Although it results in imaging only one-half of the Fourier plane [as seen in Fig. 1(c)], it does not cause any loss of data since the Fourier plane has reflection (bilateral) symmetry.

In order to have a large spread of AOIs and a small spot size, high-NA objective lenses are preferred. The SME is successfully operated with standard 0.65 and 0.90 NA objective lenses. Currently, the SME microscope uses a semi-apochromat objective lens (Olympus MPLFLN100xBDP) with 0.9 NA, 100× magnification, 1 mm working distance, and ∼1.8 mm focal length. The sample is placed on a scanning stage with three linear degrees of freedom (X, Y, and Z directions). All mechanical movement and rotation in the SME are performed using computer-controlled motorized stages (Zaber Technologies).

The reflected light from the sample is collected by the same objective lens, and this time is reflected by the D-shaped mirror toward the output optical path. The Fourier plane (FP) lens is positioned on a flip mount [as shown by in-and-out arrows in Fig. 1(a)], allowing for its entrance to and exit from the optical path. When taken out, the real image of the sample surface can be imaged by using the camera (detector of the spectrograph), and the optical microscope works like a camera is connected to its phototube. When inserted, the objective lens Fourier plane is imaged and then dispersed through the spectrograph for ellipsometric measurements, as seen in Figs. 1(c) and 1(d). Flipping in-and-out of the FP lens interchanges the SME between microscopy and ellipsometry modes. The following imaging lens is positioned with its focus at the spectrograph slit. After passing through the PSA (polarization state analyzer) consisting of a rotatable analyzer (Glan–Thompson polarizer, Thorlabs GTH10M), the output light enters the spectrograph (Teledyne Princeton Instruments, IsoPlane-160) and reaches the connected detector (Basler acA1920-40um). The SME can practically operate with any standard spectrograph and two-dimensional detector array.

It should be noted that the amount of mirrors in the SME is not represented correctly by the schematic illustration in Fig. 1(a). The SME uses more mirrors for convenient and accurate optical path alignment wherever needed. The location and amount of mirrors do not affect the functioning of the SME. All mirrors used in the SME have protected silver coatings (Thorlabs PF10-03-P01).

The SME system control, data acquisition, measurement, and data processing are performed by using a home-built MATLAB graphical user interface (GUI).

The SME uses conventional static photometric ellipsometry, which measures the spectral intensity of the reflected light at predetermined linear polarization angles when the input illumination is either linearly or circularly polarized.⁴⁷ These intensity values are functions of wavelength and AOI, and are used to calculate the ellipsometric angles Ψ and Δ from their mathematical relations to the Stokes parameters,^25,48 as explained in  Appendix A. Using this conventional ellipsometry algorithm, the SME records four exposures of intensity images at different polarization settings, as one shown in Fig. 1(d), in order to calculate the ellipsometric angles Ψ and Δ.

The Fourier (back focal) plane of the objective lens is the Fourier transform of the spot image on the sample, corresponding to spatially resolved broadband light intensity information at each reflection angle, limited by the NA of the objective lens, as illustrated in Fig. 1(b). The zero-order image of the Fourier plane, seen in Fig. 1(c), provides broadband reflection intensity information at multiple AOIs corresponding to different radii (r) values, and at multiple azimuth angles (φ) with respect to the plane of incidence (marked with red dashed lines), with no spectral resolution. Narrowing the spectrograph slit and dispersing the slit image by a diffraction grating yield a spectrally resolved first-order intensity distribution of the reflected light at multiple AOIs in the plane of incidence (φ = 0°), as seen in Fig. 1(d). Here, the x-axis is the wavelength and the y-axis (y) is the vertical pixel values of the detector (r at φ = 0°), corresponding to the range of AOIs provided by the NA of the objective lens.

In order to extract the accurate ellipsometric response of the sample from the SME measurement, a complete system characterization and calibration are needed. The AOI values on the Fourier plane must be determined with high accuracy, meaning each y value in Fig. 1(d) must be assigned to its experimental AOI correspondence. In addition, instrumental polarization effects included in the output data must be determined as functions of wavelength and AOI. The proposed method in this work, which is explained in  Appendix A, characterizes the system by measuring the AOI values and the instrumental polarization (further elaborated in  Appendix B) with high accuracy by using only experimental ellipsometric calibration measurements. The direct measurement results from the SME can then be calibrated by using these system data for sample-only ellipsometric information.

Following the characterization and calibration process⁴⁶ explained in  Appendix A, the SME is capable of recording angularly and spectrally resolved ellipsometric data (Ψ and Δ) from a spot area of a few micrometers in a few seconds.

The ellipsometric data measurement capability of the calibrated SME in a single measurement is demonstrated in Fig. 2. These data are recorded by using the SME with a spot size of 5 μm on a silicon wafer having nominal 285 nm thick oxide. The Ψ and Δ values at 601 wavelengths between 475 and 775 nm and at 78 different AOIs between 20.0° and 62.5° demonstrate the SME’s simultaneous broadband data acquisition capability at multiple angles of incidence. The measurement time down to around 10 seconds, consisting of four exposures at different polarization settings (explained in  Appendix A), is dependent on the system hardware, mostly limited by the mechanical rotation of polarization elements and the exposure time at each intensity recording. This, to the best of our knowledge, makes the SME the fastest in ellipsometric data acquisition. In addition, the substantial amount of spectrally and angularly resolved data acquired in this short time frame practically gives the SME better sensitivity and accuracy in data analysis, specifically for multi-layered structures.^21,28

The ellipsometric and angular accuracy of the SME are elaborated and compared to conventional ellipsometers in  Appendixes C and  D, respectively. The SME demonstrates promising performance similar to typical conventional ellipsometers having millimeter-scale spot sizes. This is encouraging since it shows that the added advantages of high lateral resolution, high data acquisition rate, and simple integration to optical microscopes do not compromise ellipsometric accuracy.

In its present configuration, the SME achieves fine resolution in both spectra and AOIs with a spectral resolution of ∼0.5 nm and an AOI resolution of ∼0.5° (further elaborated in  Appendix D). The spectral and angular resolutions can be increased or varied as needed depending on the system hardware.

The accuracy of the results obtained by the SME is tested by comparing its complex refractive index and film thickness results to a commercial spectroscopic ellipsometer. Then, the high lateral resolution of the SME is demonstrated by mapping film thickness and complex refractive index variations over micrometer-scale areas with a spot size down to 2 μm in the visible spectral range. Finally, the combined strength of high accuracy and high lateral resolution of the SME is used to make measurements on micrometer-scale flakes of exfoliated atomically thin van der Waals materials for the extraction of optical properties and ellipsometric differentiation among different layer numbers, and the results are compared to works in the literature.

For modeling and fitting, WVASE^® and CompleteEASE^® ellipsometry data analysis software from the J.A. Woollam Co., Inc. are used.

The calibrated SME is used for the pseudo-complex refractive index measurement of an optically thick palladium, and the result is compared with a commercial spectroscopic ellipsometer (J.A. Woollam alpha-SE) having a 3 × 9 mm² spot size, as seen in Fig. 3. Optically thick noble metals allow for a good approximation of complex refractive indices between their pseudo-n (⟨n⟩) and pseudo-k (⟨k⟩) [calculated directly from ellipsometric parameters, as in Eq. (A2) of  Appendix A] and intrinsic n and k values.²⁸ Therefore, the measured values in Fig. 3 can be labeled as intrinsic values.

The measurement performed by the SME has a spot diameter of 5 μm, and the results from multiple AOIs are averaged as the pseudo-complex refractive index of such bulk-like materials should be constant for all angles of incidence.⁴⁹ For an explicit demonstration of the instrument-only result, no data fitting or smoothing is applied to the SME result. Good agreement between the two instruments demonstrates the performance of the SME in measuring pseudo-complex refractive indices. It is important to note that slight variations in the results of the two instruments are expected due to the six orders-of-magnitude difference in the measurement areas.

The accuracy of the SME in pseudo-complex refractive index measurements with comparison to a conventional ellipsometer is elaborated in  Appendix C.

For the demonstration of film thickness measurement accuracy of the SME, silicon wafers with various SiO₂ thicknesses are used. The ellipsometry data from a single SME measurement on a silicon wafer with nominal 285 nm SiO₂ can be seen in Fig. 2.

In ellipsometry data analysis, parameter uniqueness plots show the variation of the error (in RMSE, root-mean-square error) between the model and the data when the model is scanned for a selected fit parameter. The parameter uniqueness plots of nominal 285, 90, 60, and 25 nm SiO₂ measurements by the SME can be seen in Figs. 4(a)–4(d) where the models are scanned for SiO₂ thickness values from zero to 600 nm. In all the measurements, the SME results are obtained from the well-defined global minima. The same samples are also measured by using a commercial spectroscopic ellipsometer (J. A. Woollam alpha-SE) having a spot size of 3 × 9 mm². Very similar results are obtained (written at the top-right corner of each plot) up to slight variations due to the significant difference between the spot sizes. In both SME and commercial data analysis, the same models and material parameters are used. Figure 4(e) plots and fits the SiO₂ thickness results measured by using the commercial ellipsometer vs the SME and obtains excellent linear agreement with a fitted slope parameter of ∼1 (0.999 ± 0.002) and goodness-of-fit measure R² = ∼1, confirming and visually displaying high agreement between the commercial ellipsometer and the SME results over the whole measured thickness range. As the error bars of the fitted thickness results are too small to be visible on the plot, a table is added as the inset of Fig. 4(e), displaying the sub-angstrom scale error values in the thicknesses obtained by the fitting algorithm.

In order to confirm the repeatability of the measurement devices, each measurement by using the commercial spectroscopic ellipsometer and the SME is repeated multiple times, and similar results are received consistently.

The high lateral resolution of the SME is demonstrated by thickness and refractive index maps over micrometer-scale areas. The sample under measurement is placed on a two-axis computer-controlled stage with micrometer movement resolution [as illustrated in Fig. 1(a)], providing the SME with mapping capability. The local thickness profile of a nominal 285 ± 14 nm oxide layer on a silicon wafer is mapped over an area of 35 × 35 µm² with a 5 µm spot diameter and 5 µm step size. The sample illustration and thickness profile result can be seen in Figs. 5(a) and 5(b). The thickness across the scanned area varies by ±0.17 nm. In order to understand if this variation is, indeed, from the oxide landscape, thickness repeatability measurements are performed on the same wafer by ten consecutive measurements on the same spot, resulting in a variation of ±0.04 nm. This sub-angstrom instrumental accuracy is in line with the expected accuracy of typical ellipsometers²¹ and is four times smaller than the observed variation in the thickness mapping measurement, proving that the SME is, indeed, mapping the minute changes in the oxide thickness.

To further demonstrate the SME’s high lateral resolution capability, a micro-structured area of 14 × 14 µm² is laterally scanned with a spot size of 2 µm and a step size of 1 µm. The vicinity of a gold disk with a radius of 5 µm on a silicon substrate is mapped for local variations in pseudo-optical constants. The structure illustration and the mapped ⟨n⟩ and ⟨k⟩ values at 632 nm wavelength are shown in Figs. 5(c) and 5(d). Here, the pseudo-complex refractive indices are calculated directly from the measured ellipsometric data, as stated in Eq. (A2) of  Appendix A. There is good agreement in values of both silicon and gold with the literature;^50,51 however, high variations are observed especially around the transition between the two materials due to the SME spot area not covering a single material and the scattering effects from the thickness step at the edge of the gold disk (∼100 nm).

The combined accuracy and high lateral resolution of the SME allow for measurements of the complex refractive index and layer number of micrometer-scale flakes of atomically thin exfoliated materials. These single- and few-layer two-dimensional van der Waals materials and their heterostructures are promising candidates for future photonic and electronic devices⁵² and, hence, are a widely studied research topic. Mechanical exfoliation of these materials is simple, of low-cost, and results in single- and few-layer flakes with very high purity but mostly up to around 20 μm in lateral dimensions. This makes the ellipsometric characterization of these materials a still ongoing scientific challenge. Focused-beam spectroscopic ellipsometers cannot address these flakes easily because of their low lateral resolution of tens-of-microns at most. On the other hand, imaging spectroscopic ellipsometry with micrometer-scale lateral resolution is slow in spectroscopic data acquisition. Despite this disadvantage, in recent years, there has been a significant body of work^53–58 on exfoliated 2D materials by imaging spectroscopic ellipsometers from Accurion GmbH company.

For testing the measurement sensitivity of the proposed method with atomically thin layers and addressing a modern scientific challenge, SME measurements are made on mechanically exfoliated molybdenum disulfide (MoS₂) and graphene, and the results are compared with the literature.

The SME is used for the complex refractive index measurement of a mechanically exfoliated MoS₂ monolayer (confirmed by Raman spectroscopy,  Appendix E) on a silicon substrate with nominal 285 nm SiO₂, as seen in Figs. 6(a)–6(c). For this measurement, first, the SiO₂ thickness is measured by using the SME in the proximity of the MoS₂ monolayer. Then, a measurement on the monolayer is performed. The structure is modeled from top to bottom as Air/MoS₂/SiO₂/Si, and the previously measured SiO₂ thickness value is entered into the model with the assumption that it remains the same under the monolayer. A thickness of 0.63 nm⁵⁴ is assigned to the MoS₂ layer, and its complex refractive index is calculated by the wavelength-by-wavelength (point-by-point) fit in the spectral region of its A and B excitons around 1.90 eV (652.5 nm) and 2.05 eV (604.8 nm), respectively.⁵⁴ The wavelength-by-wavelength fit directly calculates the optical constants at each spectral point independent of the neighboring spectra in order to find the best match to the experimental data. By this fitting method, the purpose is to demonstrate explicitly the SME’s sensitivity to the optical properties of a single layer of molecules. The wavelength range with the highest relative signal-to-noise ratio in the measurement is selected for obtaining a less noisy result. Figure 6(c) plots the SME result together with experimental data from the literature⁵⁹ on an exfoliated MoS₂ monolayer for a visual comparison. Good agreement proves the sensitivity and accuracy of the SME in extracting the complex refractive index down to a single molecule thick material. Some discrepancy with the literature is expected due to the difference in samples and methods. Here, the physics of this result is not further analyzed as it is out of the scope of this paper.

Next, the SME is used for measurements on exfoliated graphene, a single atomic layer material.⁶⁰ Spectroscopic ellipsometry measurements were previously performed on exfoliated monolayer, bilayer, and trilayer graphene flakes, and it was shown to be an effective method for quantitatively distinguishing between different layer numbers of graphene flakes.^55,61 Clearly discernible Ψ and Δ values at 45° (and 50°) AOI for the substrate (∼300 nm SiO₂ on Si), monolayer, bilayer, and trilayer graphene around the wavelength of 500 nm were reported,^55,61 which can be used as an indicator for the layer number of graphene flakes. Similar measurements are performed by using the SME on exfoliated micrometer-scale flakes of monolayer, bilayer, and trilayer graphene (confirmed by Raman spectroscopy,  Appendix E) on a silicon substrate with 286.12 nm SiO₂, as shown in Figs. 6(e)–6(g), respectively. Distinguishable Ψ and Δ values are observed on different layer numbers of graphene flakes and the substrate in a very similar manner to the literature,^55,61 as seen in Fig. 6(d), proving the sensitivity of the SME to single atomic layers.

The SME performance on small-area exfoliated two-dimensional materials demonstrates the strength of the proposed method and its high potential for future applications owing to its combined ellipsometric accuracy, sensitivity, and high lateral resolution. As a simple and affordable system, which can be integrated into generic optical microscopes, the SME holds the potential to be the new practical tool for the characterization and layer number identification of two-dimensional materials (which is discussed in a separate publication⁶²).

A fast mapping spectroscopic micro-ellipsometer (SME) with a lateral resolution down to 2 μm in the visible spectral range is demonstrated. In a single measurement of a few seconds comprising four exposures at different polarization settings, the SME records ellipsometric data with fine spectral and angular resolution (limited by the choice of system hardware) and measures the broadband optical properties and thicknesses of thin films.

The simultaneous multiple-angle capability of the SME substantially increases the amount of new information collected from multi-layered structures, yielding higher sensitivity to unknown parameters for these structures, which are abundant in today’s technology and research. In addition, unlike conventional ellipsometers, the SME probes the same micro-spot area with all angles of incidence, allowing for the collection of multiple-angle data even from samples that are laterally inhomogeneous.

The accuracy of a microscope-integrated ellipsometer is demonstrated to be significantly improved using the proposed method of full system characterization and calibration. This effectively eliminates uncertainties in the angle of incidence and system response, allowing for performance at a level of conventional ellipsometers together with a high lateral resolution and high data acquisition rate.

The accurate performance of the SME is demonstrated by benchmarking it against a standard commercial ellipsometer, with excellent agreement in both complex refractive index and thickness measurements. The high lateral resolution of the SME is displayed by thickness and complex refractive index mapping over micrometer-scale areas. Finally, the combined power of accuracy and high lateral resolution is used to demonstrate highly sensitive measurements on exfoliated micrometer-scale two-dimensional van der Waals materials.

The similar instrumental accuracy both in ellipsometric parameters and in angle of incidence for the SME prototype compared to conventional ellipsometers demonstrates its promising potential in industry and research for the characterization of micro-structures.

Currently, the measurement time is mostly limited by the mechanical rotation of optics for polarization modulation and total exposure time. Polarization modulation time can be significantly decreased by eliminating mechanical rotations and using electro-optic or photoelastic modulators instead. The exposure time can also be decreased by using a higher intensity light source or a more sensitive detector.

The SME includes only standard optical components and can easily be integrated into new or existing generic optical imaging systems (or microscopes) as an add-on unit, thus allowing for in situ high lateral resolution spectroscopic ellipsometry measurements in various optical measurement systems (photoluminescence, optical pump–probe, etc.).

Compared to focused-beam spectroscopic ellipsometers with relatively larger spot sizes, the SME boosts the lateral resolution by an order-of-magnitude. As of data acquisition rate, the SME collects spectrally resolved ellipsometric data from multiple angles in a single measurement of a few seconds, outperforming both focused-beam and imaging spectroscopic ellipsometers by at least one and three orders-of-magnitude, respectively.

The spectroscopic micro-ellipsometer demonstrated in this work opens up opportunities for future research to investigate the performance of the proposed method in other configurations of ellipsometry and on various sample types, such as rotating compensator/analyzer/polarizer ellipsometry and Mueller matrix ellipsometry on anisotropic and/or depolarizing materials.

We thank Professor Harald Giessen for enlightening discussions. We thank Dr. Pradheesh Ramachandran and Professor Hadar Steinberg for the preparation of 2D material samples. We acknowledge the support from Israel Innovation Authority under Grant Nos. 60523 and 64157.

Ralfy Kenaz and Ronen Rapaport are the inventors on Patent US11262293B2.

Ralfy Kenaz: Conceptualization (equal); Data curation (lead); Formal analysis (lead); Investigation (lead); Methodology (lead); Project administration (supporting); Resources (lead); Software (lead); Validation (equal); Visualization (lead); Writing – original draft (lead); Writing – review & editing (equal). Ronen Rapaport: Conceptualization (equal); Funding acquisition (lead); Methodology (supporting); Project administration (lead); Supervision (lead); Validation (equal); Writing – original draft (supporting); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

The SME system characterization and calibration method⁴⁶ does not require any prior knowledge on system specifications. This fully experimental technique allows for highly accurate ellipsometric measurements in a generic optical imaging configuration.

Ellipsometry measures the complex reflectance ratio (ρ) of a system, which is the ratio of the complex Fresnel reflection coefficients for the p- and s-polarization (r_p and r_s). ρ may be written in terms of the amplitude ratio component Ψ and the phase difference component Δ, as seen in Eq. (A1). Ψ and Δ are, by definition, functions of wavelength (λ) and angle of incidence (AOI),

The first step of the SME characterization and calibration method is measuring the pseudo-complex refractive indices (⟨N⟩) of two different reference samples by using a commercial spectroscopic ellipsometer (J.A. Woollam alpha-SE). These pseudo-n (⟨n⟩) and pseudo-k (⟨k⟩) values can be calculated directly by the mathematical transformation of ellipsometric information as follows:

where ⟨N⟩ is the pseudo-complex refractive index of the material under investigation, N₀ is the complex refractive index of the incident medium (which is usually air with N₀ = 1), θ is the AOI value, and ρ is the complex reflectance ratio.

Single layer, optically thick, homogeneous, and isotropic materials, such as noble metals (that resist oxidation and corrosion), allow for good approximation between their pseudo- and intrinsic-complex refractive indices (⟨N⟩ ≈ N).²⁸ In addition, for these bulk-like materials, the pseudo-complex refractive index is constant for all angles of incidence.⁴⁹ Because of these reasons, they constitute the preferred reference sample types for the SME calibration. Currently, the SME uses sputtered (or evaporated), highly flat, and freshly prepared gold and platinum as the reference samples. Although not experimented yet, in principle, any other two stable and contrasting (in their ellipsometric response) samples can be used as references, including SiO₂/Si wafers, which are commonly used for calibration among conventional ellipsometers.

The complex reflectance ratio (ρ), given in Eq. (A1), can be written in terms of the complex refractive index (N) and the AOI (θ) by rearranging and combining Snell’s law and the equations for complex Fresnel reflection coefficients,⁴⁷ as in Eqs. A3(a) and A3(b). This allows, together with Eq. (A1), for using the measured complex refractive indices of the reference samples (⁠$Nref1,2$⁠) to calculate their experimental ellipsometric parameters at any AOI, namely, $Ψref1,2$ and $Δref1,2$⁠, where subscripts 1 and 2 represent gold and platinum, respectively,

where

The same reference materials are then measured by using the SME. Four exposures, as one shown in Fig. 1(d), are recorded at different analyzer positions (A°) with either 45° linear (I_L) or circular (I_C) input polarization with respect to the plane of incidence. These broadband intensity values [I_L,C(A°)] at a yet-unknown AOI value [y value in Fig. 1(d)] are used for the calculation of the following Stokes parameters s₀, s₁, and s₃:^25,48

The raw experimental results $Ψexp1,2$ and $Δexp1,2$ are then calculated from the Stokes parameters^25,48 using the following equations:

This process is repeated for a desired range of y values and for both reference samples. It should be noted that these raw experimental ellipsometric parameters are not representative of the reference samples because of the included yet-undetermined instrumental polarization caused by the non-normal reflection and transmission from the optical system elements (e.g., mirrors, lenses, objective lens, and spectrograph grating). In addition, they cannot be used for extracting any useful physical information because of the yet-undetermined AOI values. These measurements of the reference samples by the commercial and SME instruments are the input data needed for the complete system characterization.

Our method assumes the properties of a stable optical system to be constant and independent of the measured sample and, hence, presumes the above-mentioned system unknowns to be identical in the two reference measurements performed by using the SME. This sample-independent system information requires the system components and alignment to be identical—and not necessarily ideal—for all performed measurements. The system is treated collectively as a virtual sample, and the wavelength-and-AOI-dependent instrumental polarization is represented by ellipsometric parameters Ψ_ins and Δ_ins.

The final reading from the SME (raw experimental parameters) comprises multiplication of the input polarization by the Jones matrices of the sample and the optical elements in the instrument. In its current configuration, the SME consists of only optically isotropic surfaces for reflection and transmission (including the sample) in its optical path between the PSG (polarization state generator) and PSA (polarization state analyzer) components. This makes the Jones matrices of all the elements diagonal²⁵ and their multiplication commutative. Therefore, the polarization response of the instrument components and the sample can be represented directly by their complex reflectance ratios [ρ, Eq. (A1)]. This allows the measured raw complex reflectance ratio by the SME (ρ_exp) to be written as the multiplication of complex reflectance ratios of the reference sample (ρ_ref) and collectively the instrument (ρ_ins) in the two reference measurements,

Thus, the instrumental Ψ and Δ parameters can be written as functions of the raw experimental and the reference parameters,

Here, the instrumental parameters are defined separately for the two reference measurements (as $Ψins1,2$ and $Δins1,2$⁠), which are indeed identical to a specific instrumental polarization for the system (Ψ_ins and Δ_ins), independent of the measured sample. This representation is temporarily used for the purpose of characterization. Our method is based on searching for the condition of this unique and system-specific instrumental polarization.

For a single y value [as in Fig. 1(d)], the raw experimental parameters (⁠$Ψexp1,2$⁠, $Δexp1,2$⁠) are calculated in the two reference measurements. Next, these raw experimental data are assigned to a range of possible θ (AOI) values with a desired angle resolution (depending on the limits of the optical system). At each θ assignment, the instrumental parameter candidates (⁠$Ψins1,2$ and $Δins1,2$⁠) are calculated via the input of calculated reference parameters (⁠$Ψref1,2$⁠, $Δref1,2$⁠) at that assigned θ into Eqs. A7(a)–A7(b).

Then, the instrumental polarization candidates from the two reference measurements are compared for their dissimilarity sequentially as a function of the assigned θ values (⁠$Ψins1$ vs $Ψins2$ and $Δins1$ vs $Δins2$⁠). For a quantitative comparison of the instrumental polarization spectra, normalized root-mean-square error (N-RMSE) is used,

where A is either Ψ_ins or Δ_ins, subscripts 1 and 2 are the gold and platinum reference measurements, respectively, p is the sequence number of a single wavelength in the spectral range, and N is the total number of wavelength points.

Plotting the calculated N-RMSE_Ψins and N-RMSE_Δins as functions of assigned θ values results in well-defined common minima at a specific θ, as seen in Fig. 7(b), pointing to where the instrumental polarization candidates from two different reference measurements are nearly identical. This overlap of the instrumental responses at a single θ proves the assumption of sample-independent system information, revealing the experimental AOI value and the corresponding broadband instrumental polarization at the selected position on the plane of incidence [y in Fig. 1(d)] simultaneously.

Repeating this process for the desired range of y values characterizes the AOIs, as seen in Fig. 7(a), and their corresponding instrumental polarization. In cases where slight discrepancies between the θ values at minima of N-RMSE_Ψins and N-RMSE_Δins are observed (probably due to some random errors), their average is taken to conclude the final AOI values, represented by the green dashed line in Fig. 7(a). Similarly, the matching instrumental polarizations from the two reference measurements are also averaged at the final AOI value for a unique system response.

As seen in Fig. 7(a), the maximum AOI available in the system is measured to be 62.5°, which is in good agreement with the used NA of 0.9, theoretically allowing until around 64°. As the AOI gets smaller, the minima agreement gets noisier in an expected manner with the decreasing degree of polarization variation.

After the AOI and corresponding instrumental polarization characterization, the calibrated results can be calculated for any sample by rearranging Eqs. A7(a) and A7(b) and replacing reference parameters with the sample ones,

The instrumental, raw experimental, and reference Ψ and Δ values at y = 113 corresponding to AOI = 52.5° [as shown in Fig. 7(b)] are plotted in Fig. 8. Ψ_sample-Au,Pt and Δ_sample-Au,Pt for reference gold (Au) and platinum (Pt) samples are the reconstructed values from Eqs. A9(a) and A9(b), with the input of raw experimental results (Ψ_exp-Au,Pt and Δ_exp-Au,Pt) and the extracted system parameters (Ψ_ins, Δ_ins). These reconstructed parameters are then compared to the calculated reference parameters from the commercial ellipsometer results at the extracted AOI value (Ψ_ref-Au,Pt, Δ_ref-Au,Pt). The achieved agreement visually demonstrates that only a unique AOI and instrumental polarization at a single y value will calibrate the raw experimental results of both materials to be in good agreement with the reference parameters.

An illustrative summary for the characterization of a plane of incidence position (y) is demonstrated in Fig. 9.

Once the SME is characterized, the extracted system data can be stored to calibrate future measurements for accurate ellipsometric results.

Since the method consists of a general AOI and polarization characterization for an optical imaging system, it is expected to work with other arrangements of micro-ellipsometry (including transmission-mode micro-ellipsometry) or with any other measurement where AOI and instrumental polarization information are critical. In case azimuthal angles of reflection are of interest, the method can be implemented on a monochromatic basis without a spectrograph. Although these schemes are not experimented yet, they are theoretically feasible.

For the visualization of the instrumental polarization as functions of wavelength and angle of incidence, the experimentally extracted broadband Ψ_ins and Δ_ins values are plotted for 20 different AOIs between 41.75° and 52.5° (y values from 113 to 132) in Fig. 10. Both parameters show significant yet unpredictable dependence on wavelength and AOI as they are outcomes of complex collective responses of non-normal reflections and transmissions from the optical elements in the system (e.g., mirrors, lenses, objective lens, and spectrograph grating). Each AOI information follows a slightly different optical path in the system, interacting at different locations and angles with the optical elements.

For a conventional ellipsometer, the instrumental polarization is expected to be independent of wavelength and AOI with constant values of Ψ_ins = 45° and Δ_ins = 0° or 180° as no additional elements—other than the measured sample—that can alter the generated or analyzed polarization exist in the optical path (perhaps with some exceptions, such as focusing lenses for focused-beam and imaging ellipsometers).

The values in Fig. 10 are specific to the current hardware and alignment of the SME and are definitely subject to change with variation and/or realignment of the system. Although these values may seem complex, their accuracy and reliability are hard to doubt as these experimentally extracted instrumental polarization values have consistently corrected the raw experimental parameters (as seen in Fig. 8) to highly accurate ellipsometric data of various measured samples (as one seen in Fig. 2). These otherwise difficult to predict or extract instrumental polarization values demonstrate the strength and accuracy of the SME characterization method.⁴⁶ 

A commonly used method to define ellipsometric accuracy is by straight-through measurements where the mechanical arms of the ellipsometer instrument are aligned horizontally and the measurement is performed in the absence of a sample reflection.⁴⁹ This method is used to measure the accuracy in Ψ and Δ parameters since measuring the empty air should result in spectrally constant values of Ψ = 45° and Δ = 0°. Unfortunately, the SME is not capable of such a measurement since its optical path can only be completed by reflection from a sample [see Fig. 1(a)]. In order to realize a similar approach of comparing ellipsometric parameters to their known values, the Ψ and Δ values of gold and platinum by the SME are compared to their reference values obtained by using a commercial ellipsometer (J.A. Woollam alpha-SE), as plotted in Figs. 8(b) and 8(d) with subscripts “sample” and “ref” for a single AOI. The RMSE of Ψ and Δ values for the whole wavelength range of 475–775 nm are calculated and averaged over 20 different angles of incidence from 41.75° to 52.5° (y values from 113 to 132).

This approach results in mean accuracy $(Δ̄)$ values of the ellipsometric parameters as

As stated in the literature,²¹ a typical conventional ellipsometer should accurately measure Ψ and Δ to be better than 0.02° and 0.1°, respectively. Although the SME uses a different geometry and, therefore, approach for defining its ellipsometric accuracy, it can still demonstrate similar and same order-of-magnitude values of accuracy in ellipsometric parameters compared to conventional systems with millimeter-scale lateral resolution.

Another approach to quantify the accuracy of an ellipsometer is pseudo-complex refractive index measurements on bulk-like materials as for these materials, the pseudo-complex refractive index should be essentially constant for all angles of incidence.⁴⁹ For an averaged, sample-independent measure and comparison with a commercial ellipsometer, the standard deviation (σ) of pseudo-complex refractive indices in three different angles of incidence is averaged over three noble metal (gold, platinum, and palladium) measurements as a function of wavelength (475–775 nm) for both instruments. The results are plotted in Fig. 11.

The commercial ellipsometer (J.A. Woollam alpha-SE, 3 × 9 mm² spot size) has three fixed angles of incidence −65°, 70°, and 75°—specifically chosen to be close to the Brewster’s angle of most materials for a higher ellipsometric signal-to-noise ratio. The current version of the SME can measure up to 64°, so three different angles of incidence are chosen below this number toward the higher angles (to be at par with the commercial ellipsometer). As seen in Fig. 11, the error in SME measurements is similar but slightly noisier than the commercial ellipsometer, probably due to the lower angles of incidence and lower amount of light intensity per angle of incidence when compared to the commercial ellipsometer and the relatively low-level specifications of the detector array used. The SME errors also show an increasing trend above 725 nm of wavelength (and slightly below 500 nm), corresponding to the decreasing light intensity of the white light source at those wavelengths [as seen in Fig. 1(d)]. When these factors are taken into consideration, it is safe to say that the SME demonstrates a promising similar performance compared to a commercial ellipsometer also in this approach of ellipsometric accuracy quantification.

To conclude, both approaches to quantify the ellipsometric accuracy of the SME demonstrate that even in its prototype stage, the SME, indeed, shows similar and same order-of-magnitude performance compared to conventional ellipsometers having millimeter-scale spot sizes.

The accuracy of the angle of incidence in the SME is limited by the system optics, such as the spectrograph modulation transfer function (MTF), NA and magnification of the objective lens, and the magnification of the lenses for imaging the Fourier plane. In the current version of the SME, this was calculated to be corresponding to an angular resolution of around 0.5°. The angle of incidence accuracy of commercial ellipsometers is limited by standard mechanical tolerances in alignment, which is on the order of tenths of a degree. In addition, commercial ellipsometers have an option for calibrating the angle of incidence based on the measurement of a reference sample and also of using it as a fit parameter. This way, the angle accuracy can be increased by an order-of-magnitude. Similar procedures can be carried out in the case of the SME for increased angular accuracy.

The SME makes use of the NA of an objective lens for oblique angles of incidence on the sample. This should lead to the variation of incidence angle with wavelength as a result of chromatic aberration. The variation can be estimated by assuming that the semi-apochromat microscope objective lens used in the current version of the SME (Olympus MPLFLN100xBDP, 100×, 0.9 NA) is near diffraction-limited. In this case, the transverse chromatic aberration is on the order of the Airy radius. Based on this, the variation in incidence angle caused by the chromatic aberration is calculated to be about 0.013°. As mentioned, the current angular resolution of the SME resulting from the system optics is around 0.5°. Therefore, the angle of incidence variation resulting from chromatic aberration is negligible.

The measured Raman spectra of the MoS₂ and graphene flakes in the paper are demonstrated in Fig. 12. The Raman measurements are performed in a backscattering geometry by using the Renishaw InVia™ Confocal Raman Microscope instrument. The wavelength of the excitation laser is 514.5 nm, and the laser spot size is around 1 μm with a 50× objective lens. All measured flakes are on silicon substrates with nominal 285 nm thick SiO₂.

Figure 12(a) demonstrates the Raman spectrum obtained on the MoS₂ flake, which is measured by using the SME for its complex refractive index [Figs. 6(a)–6(c)]. The in-plane $E2g1$ vibrational mode at 384.5 cm⁻¹ and out-of-plane A_1g vibrational mode at 404.1 cm⁻¹ show a frequency difference of Δω = 19.6 cm⁻¹, which is in excellent agreement with the literature⁶³ on monolayer MoS₂, confirming that the flake is, indeed, a monolayer.

Figure 12(b) shows the Raman spectra of the monolayer, bilayer, and trilayer graphene flakes, which are measured by using the SME for the demonstration of its ellipsometric signal sensitivity to single-atom thick layers [Figs. 6(d)–6(g)]. The 2D- to G-band peak intensity ratios of ∼2.5, ∼1.1, and ∼0.67, together with the 2D-band full width at half-maximum (FWHM) values of ∼33, ∼53, and ∼62 cm⁻¹, confirm the monolayer, bilayer, and trilayer natures of the graphene flakes, respectively.^64–67