WO2024069173A1

WO2024069173A1 - Optical path length characterisation

Info

Publication number: WO2024069173A1
Application number: PCT/GB2023/052508
Authority: WO
Inventors: Silke Weinfurtner; August GEELMUYDEN; Vitor BARROSO SILVEIRA
Original assignee: The University Of Nottingham
Priority date: 2022-09-30
Filing date: 2023-09-27
Publication date: 2024-04-04
Also published as: GB202214343D0

Abstract

A method is provided, for measuring an optical path length of a sample having a first face (126a) and a second face (126b) on an opposing side of the sample to the first face. The method comprises receiving at a detector (150) interference data comprising a plurality of interferograms (RF, RBn, FBn), each of the interferograms comprising a reflected light beam resulting from reflection of a light beam from at least one of the first face and the second face and each of the interferograms comprising a reflected light beam with reflections from a different combination and/or sequence of surfaces. The method further comprises performing a spatial frequency transform of the interference data to generate transformed data in a spatial frequency domain and analysing features of the transformed data to determine the optical path length, wherein each feature in the transformed data corresponds to one of the interferograms in the interference data and at least some of the features are at least partially separated in the spatial frequency domain.

Description

OPTICAL PATH LENGTH CHARACTERISATION FIELD OF THE INVENTION The present invention relates to characterising materials using interferometry, in particular measuring the optical path length of a layer. BACKGROUND The optical characterisation of material layers is an important process in a wide range of industrial and scientific fields. In particular, it is desirable to be able to characterise the optical path length of films and layers. This optical path length contains information about the physical dimensions of a material (notably its thickness) and also its optical properties (for example, refractive index). If optical path length is measured across an area of a material sample, this may be used to analyse the sample’s surf ace profile and any local deformations in the material. For example, in semiconductor manufacturing, it may be desirable to measure the thickness of a deposited layer to ensure that it is within an expected distribution. Alternatively, one may be required to measure the thickness of a chemical layer (for example, a varnish or clearcoat) that is applied to a material to ensure that it will provide adequate protection. Further still, one may need to measure and understand the interface between two fluid mediums (such as a liquid and a gas) and how this changes over time. These examples vary greatly, but all relate to characterising an optical path length in a material. One technique for characterising optical path length within a material is off-axis digital holography. This technique involves obtaining a two-dimensional hologram (an interference image) of a sample, and using a measured phase obtained from the hologram to reconstruct a three-dimensional profile of the material. Multiple holograms can be obtained simultaneously by multiplexing different interference patterns, each containing different information about the profile. The different holograms may be separated in the spatial-frequency domain, and their respective images reconstructed. However, such techniques essentially involve using multi-beam interferometers to generate the different interference patterns. This requires the use of many optical elements (beam splitters, mirrors, diffraction gratings, etc.) to produce multiple interference arms that each probe the sample at a slightly different angle, or the use of beams with different frequencies. As such, present techniques are complex to build and require careful calibration and operation. Hence, there is a need to develop improved and more simple techniques that are capable of characterising optical path length with sufficient accuracy. SUMMARY According to a first aspect, there is provided a method of measuring an optical path length of a sample, the sample having a first face and a second face on an opposing side of the sample to the first face. The method comprises: illuminating the first face of the sample with a light beam, and receiving at a detector interference data comprising a plurality of interferograms. Each of the interferograms comprises a reflected beam resulting from reflection of the light beam from at least one of the first face and the second face; and each of the interferograms comprises a reflected light beam from a different combination and/or sequence of faces. Each interferogram is the superposition, at the detector, of: i) the reflected beam; and ii) a different reflected beam (also resulting from reflection of the light beam from at least one of the first face and the second face but reflected from a different combination and/or sequence of faces than the reflected beam) or a reference beam. The method further comprises performing a spatial frequency transform of the interference data to generate transformed data and analysing features of the transformed data. Each feature in the transformed data corresponds to one of the interferograms in the interference data and at least some of the features are at least partially separated in the spatial frequency domain. The beam may be an expanded beam. The expanded beam may enable the spatial frequency transform. The expanded beam may have beam diameter (FWHM) of at least 1mm, 5mm, 10mm, 20mm or at least 30mm at the first face. The sample may be a fluid. The sample may be a layer, such as a thin film or a deposition on the surface of a material. The sample layer may be one layer of a plurality of layers. The sample may comprise a plurality of layers. The plurality of layers may comprise two or more materials, wherein adjacent layers have different refractive indices such that an optical interface is created at the boundary of two layers. The plurality of interferograms may comprises at least one of: a first group of interferograms, wherein each interferogram is generated by interference between: a reflected light beam, ^^, that is reflected from the first face without traversing the sample; and a reflected light beam, ^^ _^^ , that is reflected from the second face ^^ times and the first face ^^ − 1 times, wherein ^^ = 1, ⋯ , ^^ ; and a second group of interferograms, wherein each interferogram is generated by interference between: a reflected light beam, ^^ _^^ , that is reflected from the second face ^^ times and the first face ^^ − 1 times, wherein ^^ = 1, ⋯ , ^^ wherein ^^ ≠ ^^; and the reflected light beam, ^^ _^^. The method may further comprise providing a reference beam that does not interact with the sample. At least some of the interferograms may comprise interference between the reference beam and the reflected light beam (that is reflected from at least one of the first face and the second face). The plurality of interferograms may comprise at least one of: an interferogram generated by interference between the reference beam, R, and a reflected light beam, ^^, that is reflected from the first surface without traversing the sample; and a third group of interferograms, wherein each interferogram is generated by interference between: a reference beam, R; and a reflected light beam, ^^ _^^ , that is reflected from the second face ^^ times and the interface ^^ − 1 times, wherein ^^ = 1, ⋯ , ^^. A reflective surface may be provided at the second face of the sample in order to generate the ^^ _^^ reflected light beams. The reflective surface may comprise at least one of a mirror, a metal layer or a Bragg reflector. The reflective surface may be comprised in, or form part of, a holder that retains the sample. For example, in the case that the sample is a fluid, the bottom of a receptacle may be mir rored. In some embodiments, a reflective surface may be provided within a liquid sample (offset from the bottom of a receptacle retaining the fluid). In the case that the sample is a solid layer, the solid layer may be disposed on a reflective surface. The sample may be fixed to the reflective surface. For example, a thin-film may be applied to a reflective substrate using a physical or chemical deposition technique (e.g., vapour deposition) or wetting. The substate to which the sample is applied may generate the ^^ _^^ reflections. Alternatively, the ^^ _^^ reflected light beams may be generated by an interface between layers having different refractive indices. The ^^ _^^ reflections may be generated by reflections at the interfaces between of at least partially transparent layers. For example, the sample layer may be deposited on a transparent glass substrate, and reflections generated at the interface of the sample and the glass. The sample may comprise a plurality of layers and a plurality of interfaces be tween layers with different refractive indices. There may be a first set of reflections generated at an interface between the first and second layers; and second set of reflections generated at an interface of the second and third layers; etc. up to the total number of layers in the sample. The reference beam and the light beam may be generated by splitting a source light beam from a common light source. The light source may be a laser. The source light beam may be directed at a beam splitter, configured to split the source light beam into the light beam (that illuminates the sample) and a reference beam. The beam splitter may split the power of the source light beam equally between the light beam (illuminating the sample) and the reference beam. The method may further comprise providing a second reflective surface configured to reflect the reference beam. The second face and second reflective surfaces may be configured such that the reflected light beam and reference beam interfere at the detector. The method may further comprise increasing a separation between at least some of the features in the transformed data are moved relative to each other , wherein increasing the separation comprises: adjusting an angle of the second face relative to the light beam at the first face and/or adjusting an angle of the second reflective surface relative to the reference beam at the second reflective surface. The increasing of separation may be automatic, performed as part of a computer implemented method, or manual performed by a person (potentially at the same time as viewing a representation of transformed data on a computer implemented graphical user interface). The interference data may be two-dimensional image data comprising a superposition of the plurality of interferograms. The interference data may encode a phase difference for each of the interferograms. The phase difference may comprise a spatial distribution of phase difference for each of the interferograms across the two-dimensional image data. The phase difference in each interferogram may be generated by a phase shift in the illuminating light beam resulting from at least one of: a bulk variation in the sample; a bulk variation in a medium that is not the sample through which the light beam propagates to and/or from the sample; the refractive index of the sample; the refractive index of the medium that is not the sample; and a distance that the light beam travels while transiting the sample and/or the medium that is not the sample. Performing a transform of the interference data may comprise performing a spatial Fourier transform of the interference data. The transformed data may comprise a map of spatial frequency amplitude in an x and y spatial frequency direction. The features of the transformed data may comprise a plurality of intensity peaks in the spatial frequency domain. Each peak may be located at a wavevector (defining a spatial frequency in x and y) corresponding to one of the interferograms (i.e. each peak may be centred on a location in the spatial frequency domain that corresponds with one of the interferograms). Analysing the features may comprise selecting at least one of the intensity peaks in the spatial frequency domain and applying a filter to each of the selected intensity peaks to extract the peak in the spatial frequency domain (and exclude other data, which may comprise other peaks corresponding with different interferograms comprising combinations of reflected light beam(s) and or the reference beam) . The filter may be a top-hat filter that is configured to extract the intensity peak. Other suitable filters, such as a raised-cosine filter, may be used instead of a top-hat filter. The filter may be configured to have dimensions so that an area around each intensity that is extracted is sufficient to encapsulate a phase modulation of the interferogram corresponding to said peak. For example, a Gaussian function may be fitted to the peak, and the filter may be selected with a cutoff corresponding a preselected width of the Gaussian (e.g. a 3-sigma width or a different multiple of sigma). The filter may comprise an adaptive filter, which takes account of a noise floor of the transformed data, and/or the presence of any adjacent peaks that may at least partially overlap with the peak to be extracted. The method may further comprise performing an inverse spatial frequency transform of the transformed data of a plurality of the extracted intensity peaks. The inverse spatial frequency transform generates reconstructed interference data for each of the plurality of extracted peaks, wherein the reconstructed interference data of each extracted peak encodes a phase difference of an interferogram corresponding to that extracted peak. If each peak is extracted appropriately (i.e., a filter used to extract the data in the spatial frequency domain is configured such that phase modulation around each peak is encompassed by the filter, and each peak is sufficiently separated from each other in the spatial frequency domain), the reconstructed interference data may be the interference data of an interferogram corresponding to that extract ed peak. The plurality of peaks may be selected manually. A user may be able to select a peak using a graphical user interface displaying transformed data in the spatial frequency domain. The user may be able to select a point in the displayed transformed data (e.g., at or near to a desired peak), and the graphical user interface may automatically determine and display an area around the selected point that is to be extracted by a filter. An automated peak finding algorithm may be used to identify and select intensity pe aks of transformed data. The automated peak finding algorithm may be configured to identify and extract a desired number of intensity peaks. The desired intensity peaks may be the peaks with the greatest maximum value, or the greatest average value across a filter area fitted around each peak. The automated peak finding algorithm may be implemented with an automatic peak separating algorithm that controls separation of the peaks in order to achieve an appropriate separation between peaks in the transformed data. A peak prediction algorithm may be used to infer the peak positions given sufficient information about the optical alignment (for example, the tilt of the reflective surfaces) is known. Conversely, provided there is calibration, the positions of t he peaks may be used to infer the angular orientation of the reflective surfaces. The inverse spatial frequency transform may be an inverse Fourier transform. The inverse Fourier transform may be performed on two-dimensional spatial frequency data. The method may further comprise generating a covariance matrix for the reconstructed interference data of the plurality of extracted peaks . The eigenvectors of the covariance matrix may be calculated. The method may further comprise extracting a principal component of the covariance matrix, the principal component corresponding to the mutual spatial dependence of the reconstructed interference data of the extracted intensity peaks. The principal component may be an eigenvector of the covariance matrix with the largest eigenvalue. The eigenvalue of the principal component eigenvector may be larger (e.g., at least 50% larger, or at least 100% larger, or at least 200% larger) than the other components of the covariance matrix in the case the phase differences of each of the interferograms are dominated by the optical path length of the sample (relative to noise and any effects not in the sample). The covariance matrix of the reconstructed interference data may be analysed using a variety of different statistical methods. For example, independent component analysis (ICA) may be used in addition to or instead of principle component analysis (PCA). The method may further comprise determining an interferogram that corresponds with each of the extracted intensity peaks by comparing the extracted principal component with an expected phase difference for each interferogram. The expected phase difference for each interferogram may be based on a derived mathemati cal model of the interference of each of the interferograms. The method may further comprise using the expected phase difference corresponding to one of the determined interferograms and the extracted principal component to determine a height of the sample, the height corresponding to a thickness of the sample between the first face and second face and/or the deformation or variation of thickness . The method may further comprise using the expected phase difference for more than one of the determined interferograms and the extracted principal component to determine a refractive index of the sample. At least one preferred determined interferogram may be used to determine the sample height and/or the refractive index. The preferred interferogram may be selec ted based on knowledge of the reflected and/or reference beams that comprise the interferogram. For example, an interferogram may be selected because it is known not to comprise a reference beam. The use of an interferogram that does not comprise the reference beam may improve the accuracy of the height and/or refractive index measurement. Interferograms that do not comprise the reference beam share a common path from the light source to the first face. Any variations in path length between the light source and first face will consequently not lead to a variation in phase difference so will not produce any change in the resulting interferogram. The preferred determined interferogram may be selected based on measured properties of the reconstructed interference data and/or transformed data. The preferred determined interferogram may be the interferogram with an intensity peak with the greatest maximum value in the transformed data or the lowest noise in the reconstructed interference data, for example. The interference data may comprise first interference data received at a first time and second interference data received at a subsequent, second time . The method may further comprise identifying the intensity peak from the first and second interference da ta corresponding to the same interferogram and measuring the change in the phase difference of the peak between the first interference data and the second interference data. The phase difference of at least one interferogram may be used to calculate: the height of the sample interface above the second face; variations in height across an area of the first face that is illuminated by the light beam; and/or the refractive index of the sample. According to a second aspect of the invention, there is provided an apparatus for measuring an optical path length of a sample having a first face and a second face on an opposing side of the sample to the first face . The apparatus comprises: a light source configured to illuminate the first face of the sample with a light beam, and a detector configured to receive interference data generated by reflection of the light beam from the sample; and a processor. The processor is configured to: perform a transform of the interference data to generate transformed data, the transformed data being in a spatial frequency domain; and analyse features of the transformed data in the spatial frequency domain . The interference data comprises a plurality of interferograms, each of the interferograms generated by the reflection of the light beam from at least one of the first face and the second face; and each of the interferograms corresponds with reflections from a different combination or sequence of faces. The light source may comprise a beam expander, configured to illuminate a region of the sample with a collimated light beam via a beam splitter. The apparatus may further comprise a beam splitter configured to split a source light beam generated by the light source into the light beam that is used to illuminate the sample and a reference beam that does not interact with the sample . At least some of the interferograms may comprise interference between the reference beam and the light beam reflected from at least one of the interfaces and the second face. The apparatus may further comprise a second reflective surface configured to reflect the reference beam. The light beam and the reference beam may be perpendicular to one another, such that the apparatus is configured as a Michelson interferometer. According to a third aspect, there is provided a non-transitory machine readable medium, provided with instructions for configuring a computer to: perform a spatial frequency transform on interference data to generate transformed data in a spatial frequency domain; analyse features of the transformed data, wherein each feature in the transformed data corresponds to one of the interferograms in the interference data and at least some of the features are at least partially separated in the spatial frequency domain. The interference data may be obtained according to the illuminating and receiving steps of the first aspect. Features described with reference to the first aspect may be applied to the third aspect. The features of each aspect may be combined with those of any other aspect. Features of the first aspect, including optional features, may be combined with those of the second aspect, or any other aspect. Feature of the second aspect may similarly be combined with those of the first aspect, or any o ther aspect. DETAILED DESCRIPTION Embodiments of the invention will be described, purely by way of example, with reference to the accompanying drawings, in which: Figure 1 shows a schematic diagram of an apparatus according to an embodiment of the invention; Figure 2 shows an example of phase difference generated by bulk disruptions; Figure 3 shows an illustration of expected intensity peaks of interference data in the spatial frequency domain; Figure 4 shows simulated interference data and the corresponding peaks in the spatial frequency domain; Figure 5 shows an example of interference data in the spatial frequency domain; Figure 6 shows further example of interference data and transformed interference data in the spatial frequency domain; and Figure 7 shows an example of principle component analysis performed on a plurality of interferograms; Figure 8 shows a phase reconstruction performed on example data and the phase corresponding variation observed across a sample surface; and Figure 9 shows example data showing evaporation from a sample . Referring to Figure 1, an example apparatus 100 is shown that is configured for characterising a partially transparent sample 124. The apparatus 100 comprises a light source 110 that generates a source light beam 112. The light source 110 may be a collimated, coherent light source (e.g., a laser). The source light beam 112 may be visible light. The source light beam 112 may be passed through a beam splitter 130, such that the source light beam 112 is split into a light beam 114 and a reference beam 116. The beam splitter 130 may split the power of the source light beam 112 equally between the light beam 114 and reference beam 116. The light beam 114 is directed towards the sample 124. The sample 124 is contained on or in a sample holder 120. The sample 124 may be a fluid such as a liquid, in which case the holder 120 may be a receptacle as shown in Figure 1. Alternatively, the sample 124 may be a solid (such as a thin-film, a layer or a coating), in which case the holder 120 may be a platform or stage similar to a microscope stage, for example. The sample 124 may be one of a plurality of layers. For example, inhomogeneous layers (layers of differing material and/or differing refractive index) may be stacked on top of one another, and the sample 124 may be the topmost of these layers. Alternatively, the sample may comprise a number of inhomogeneous layers, wherein each layer has a first face and a second face, with faces being shared at the interfac es of adjacent layers. The sample 124 has a first face 126a that faces the incoming light beam 114. The first face 126a in this example is an interfacial surface between the sample 124 and the surrounding medium. The surrounding medium may be air in the case that the apparatus is operated in atmospheric conditions. Equally however, the apparatus (or at least the part of the apparatus in which the optical effects discussed later occur) may be operated, for example, in a vacuum or submerged in a fluid medium. As such, the surrounding medium may be a vacuum or water, for example. The sample 124 has a second face 126b that is on an opposing side of the sample 124 to the incoming light beam 114. In the example of Figure 1, the first face 126a is a top face of the sample 124 and the second face 126b is a bottom face of the sample 124. A reflective surface 122 may be provided at the second face 126a, such that that the light beam 114 passing from the first face 126a to the second face 126b is reflected. The reflective surface 122 may be a mirror, a metal surface or a Bragg reflector, for example. The reflective surface may be situated in or on a base of the holder 120. A fluid sample may then be poured into the holder 120 such that it covers the mirror 122. Alternatively, the reflective surface 122 may be a reflective subst rate, for example a silicon wafer, which itself acts as the holder 120. A semi-transparent sample may then be applied to the substrate. For example, a thin-film or a fluid droplet may be deposited on the substrate. In the example embodiment, the reflective surface is a t the bottom of the holder 120, but in some embodiments the reflective surface 122 may be spaced apart from the bottom of the holder 120. In such embodiments it may be more straightforward to adjust an angle of the reflective surface 122 relative to the first face of the sample 124. Reflections from the second face 126b may be created by an interface with another material with a different refractive index. Two different layers may be provided as a stack, with the topmost of these forming the sample layer. If the refractive index of the layers is different, light may be reflected from the interface of the two layers, thus eliminating the need for a mirror surface ‘underneath’ the sample. The incoming light beam 114 may be reflected by the interfacial surface 126a and /or the reflective surface 122 to generate a plurality of reflected beams. Each of the reflected beams may be generated by reflections form a different combination or sequence of the surfaces. A first reflected beam, F, may be generated by the reflection of the incoming light beam 114 from the interfacial surface 126a. A proportion of the incoming light beam may not be reflected from the interfacial surface 126a, and instead transit the sample 124. This may result in a plurality of further reflected beams. The first of these further reflected beams, B1, may be generated by the incoming beam being reflected from the reflective surface 122 before passing through the interfacial surface 126a. The second of the further reflected beams, B2, may be generated by the incoming beam being reflected from the reflective surface 122, the int erfacial surface 126a, and the reflective surface 122 a second time, before passing through the interfacial surface 126a. In general, the incoming light beam 114 that transits the sample 124 may generate a plurality of further reflected beams, ^^ _^^ , due to multiple reflections within the sample 124. The incoming beam is reflected from the reflective surface ^^ times and the interface ^^ − 1 times, wherein ^^ = 1, ⋯ , ^^. The different reflections F and B1, B2, …, BJ may form a superposition of reflected light beams. The reference beam 116 generated by the beam splitter 130 may be directed towards a second reflective surface 140, for example a mirror. The mirror 140 may reflect a reflected reference beam, R, back towards the beam splitter 130. The apparatus 100 shown in Figure 1 constitutes a Michelson interferometer with two perpendicular legs (the light beam 114 and the reference beam 116). This configuration may require fewer optical elements than other arrangements. The apparatus 100 may be implemented in a different configuration. The reflected reference beam, R, and the superposition of reflected light beams (comprising F and plurality of reflected beams B_j) may be recombined at the beam splitter 130 to generate an interference beam 118. The interference beam 118 may be received at a detector 150 such as a CCD camera. The interference beam 118 is a supposition of a plurality of interferograms resulting from the various components of the reflected light and reference beam. There are four families of interferograms (assuming a non-reflecting beam splitter): i) RF – interference between the reflected reference beam, R, and the reflection of a light beam from the interfacial surface of the sample, F; ii) RBj – interference between the reflected reference beam R, and a light beam that has transited the sample and been reflected from the reflective surface j times, Bj; iii) FBj – interference between the reflection of a light beam from the interfacial surface of the sample, F, and a light beam that has transited the sample and been reflected from the reflective surface j times, Bj; and iv) BlBj – interference between two different light beams that have transited the sample and been reflected from the reflective surface l and j times respectively. Alternatively, the apparatus may omit a reference arm. In this case a subset of the interferograms may still be obtained, which may be sufficient for certain embodiments. In such a case, only the FBj BlBj groups of interferograms will be generated. The nature and subsequent processing of these interferograms may be treated in the same way as for embodiments that include a reference arm. Generating and/or obtaining a subset of the discussed interferograms may be sufficient to characterise the sample 124. As will be discussed in further detail later, the relative alignment of the first 122 and second 140 reflective surfaces affects the interference data. The sample holder 120/first reflective surface 122 and/or the second reflective surface 140 may be movably mounted such that the angle of the reflective surfaces relative to the incoming light/reference beam can be adjusted. The surfaces may be mounted on a 3-axis adjustable optics mount, for example. Alternatively, the orientation of the light beam 114 may be adjustable relative to the sample 124, thus allowing the angle between light beam 114 and the first reflective surface 122 to be adjusted. For example, the apparatus 100 may be configured as a moveable tool. The light source 110, beam splitter 130, second reflective surface 140 and detector 150 may be comprised in a tool and fixed relative to one another. The light beam 114 may then be directed at the sample 124, and the position of the tool adjusted so that the light beam 114 illuminates the sample and is reflected back into the tool (as shown in Figure 1). The tool may then be moved and rotated, thereby adjusting the angle between the light beam and the first reflective surface. The tool may be a ‘desktop tool’, wherein the apparatus is mounted on a frame and the optical elements are repositionable within the frame. Alternatively, the tool may be a handheld tool, wherein a user directs the light beam 114 at the sample 124 similarly to operating a laser pointer. A moveable tool may be advantageous for characterising samples in situ (i.e., without the need to prepare a sample within the apparatus). A user could, for example, use such a handheld tool to characterise thin -films applied to objects during chemical and industrial processes. The apparatus 100 may comprise further optical elements. For example: a beam expander 160 may be provided such that the light beam 112 is expanded to a larger surface area (which may enable a larger surface area of the sample to be characterised) . The power of the light beam 112 may be modified by neutral-density filters or other forms of optical attenuation. The interference beam 118 may be focused on the detector 150 via a lens 170. The detector 150 may comprise or be connected to a computer/processor 190 that is configured to analyse the interference beam 118, the interference beam 118 containing interference data that describes characteristics of the sample, as discussed below. The computer may comprise a user interface for a user to interact with and control the apparatus, and/or a display which provides data and/or analysis results to a user. Modelling of the interference beam An electric field ^^ can be modelled under a Wentzel–Kramers–Brillouin (WKB) ansatz using the form

The directional energy density flux of an electromagnetic wave with electric field strength E and magnetic field strength B is given by Poynting vector ^^ ≡ ^^⁻¹ ^^ × ^^. The intensity observed at a sheet that is perpendicular to a ray of light is therefore given by ^^ = ^^ ^^^| ^^^|2. Visible light, however, oscillates at very high frequencies, so it makes more sense to consider the temporally averaged intensity 〈 ^^〉 _^^. A superposition of rays { ^^ _^^} of the form shown in equation 1 may be considered, with ^{| ^^ _^^ ^|} _^^ = ^^₀ ^^ _^^ for some vector ^| ^^ _^^ ^| ∈ ^[0,1^]. The result is

1 where ^^₀ ≡ 2 ^^ ^^ ^^₀ ² is the time-averaged intensity of the incoming ray, and ^^ _{^^ ^^} ≡ ^^ _^^ ∙ ^^ _^^ and − Φ _^^ is the phase difference of two superimposed rays . Here, the conservation of squared field amplitudes has been assumed, i.e., ^∑ _^^ ^| ^^^|2 ^_^ = 1. Note that for a single ray, the sum in equation 2 is empty and the initial intensity 〈 ^^〉 _^^ = ^^₀ is recovered. According to equation 2, any two superimposed rays (or reflected light beams) create an interference pattern. A ray traversing a distance ^^ ^^ in a homogeneous medium with refractive index ^^ may be regarded as attaining a phase

= ^^₀ ^^ ^^ ^^ where ^^₀ = 2 ^^/ ^^₀ is the wave-vector of the ray in vacuum. The refractive indices of the surrounding medium and the sample 124 may be denoted by ^^₁ and ^^₂, respectively. Projected normal vectors ^^ _^^ = − ^^ _^^ ^^ _^^/ ^^ _^^ − ^^ _^^ ^^ _^^/ ^^ _^^ and ^^ _^^ = − ^^ _^^ ^^ _^^/ ^^ _^^ − ^^ _^^ ^^ _^^/ ^^ _^^, where ( ^^ _^^, ^^ _^^, ^^ _^^) and ( ^^ _^^, ^^ _^^, ^^ _^^) are the normal vectors of the second reflective surface 140 (the reference mirror) and the first reflective surface 122 (e.g. a submerged mirror) respectively, may be introduced. The interfacial surface 126a of the sample 124 may be located at ^^ = ℎ( ^^, ^^) with zero-thickness. The interface may be assumed to be perfectly non-absorbing, such that if the ratio of the field amplitude in the surrounding medium that is reflected at the interfacial surface 126a of the sample 124 is ^^ _{^^ ^^}, then the transmitted ratio is √1 − ^^ _^ ² ^_^^ . The reflection coefficient ^^ _{^^ ^^} is assumed to be well described by the Fresnel relation for normal incidence :

Assuming all rays to be un-deflected from the optical axis, the possible phase differences for the four families of interferogram are

where ∆ ^^ ≡ ^^ _^^ − ^^ is the difference in arm length of the reference beam ^^ _^^ and the light beam ^^. The terms

above account for contributions from the medium's bulk and any other next-to-leading-order ray deflection terms. Writing ^^ ≡ ^^₁₂, the corresponding amplitudes takes the form ^^ _{^^ ^^} = ^^ cos( ^^ _^^) (5a) ^^ _{^^ ^^ ^^} = ⁽−1^{) ^^−1(}1 − ^^²⁾ ^^ ^{^^−1} cos( ^^ _^^) ⁽5b⁾ ^^ _{^^ ^^ ^^} = ⁽−1^{) ^^−1(}1 − ^^²⁾ ^^ ^{^^ (}5c⁾

where ^^ _^^ is the angle between the polarization of the light beam and the reference beam. In the absence of polarising optical components when using unpolarised light source, the suppression factor cos( ^^ _^^) may be set to one. An overall factor of ^^ _^^ ^^ _^^ may be applied to the amplitudes if the intensity ^^₀ is to refer to the intensity prior to passing a beam splitter with transmission coefficient ^^ _^^ and reflection coefficient ^^ _^^, i.e., ^^ _^^ ^^ _^^ = 1/4 for a 50% — 50% non-polarizing beam splitter. Because the magnitude of the interfacial reflection coefficient, ^| ^^^|, is typically small (| ^^| ~ 14.16% for air-water interfaces), the ^^ ^^₁ interferogram typically has the largest amplitude. The sensitivity of the phase to variations in the height, ^^_ℎΦ _{^^ ^^1} = 2 ^^₀( ^^₁ − ^^ ^^₂) however, is the smallest. The most phase-sensitive of the ^^ = 1 interferograms is ^^ ^^₁, with ^^_ℎΦ _{^^ ^^1} = 2 ^^₀ ^^₂. Contributions in intensity from the three interferograms ^^ ^^, ^^ ^^₁ and ^^ ^^₁ have distinct external influences. For example, inhomogeneities in the bulk of the surrounding medium, from changes in the composition of the surrounding medium, the temperature, or from sound waves, are picked up by ^^ ^^ and ^^ ^^₁, but not by ^^ ^^₁, which only probes the sample bulk. For transparent samples, ^^ ^^₁ is expected to have the largest signal-to- noise ratio, provided the reflection coefficient ^^ is not too small. It may be preferable to consider different interferograms (or different combinations of interferograms) during analysis to account for or eliminate these different factors. Ray deflections and bulk effects The derivations above provide insight to the different interferograms that constitute the interference data. In these derivations, it was assumed that the refracted beams remained aligned with the optical axis, which may not be true when surface gradients become large. Additionally, all beams experience variations in refractive index caused by bulk effects in both the sample and the surrounding medium. Referring to Figure 2, an example of phase distortion in the bulk of the surrounding medium is shown. A jet of compressed air 201 is directed through the light beam, which causes variations in temperature, pressure and composition of the bulk. The white region 202 in the bottom left of the image is the tip of the compressed air cannister. In this data shown, there is no fluid sample 124. Image (a) of Figure 2 shows the interference fringes 203, with the insert image (c) showing the interference data at the edge of the jet. The jet modifies the local composition { ^^ _^^} of the air, modifying the phase of the light beam passing through the jet compared to the portion of the beam passing from the surrounding air that is not in the jet. Using the methods described in detail below, the RB₁ interferogram may be extracted from the interference data, and the phase difference of the interferogram across the image so obtained is shown in (b), illustrating that the jet 201 clearly changes the relative beam phase. The bulk effects in a homogeneous gas mixture or liquid solution with one or more species, where the Lorentz-Lorenz formula^1,2,3 specifies the relation between the refractive index ^^, the mixture composition and the optical properties of each species, may be regarded as follows: ^^² − 1 = ∑ ^^ ^^² + 2 _^^ ^^ _^^ ^{, (6)} ^^ where ^^ _^^ is number density of species ^^ and ^^ _^^ = ^^ _^^/3 ^^₀, where ^^ _^^ is the molecular polarizability of the species. In the case of a gas mixture, thermal effects may be accounted for by approximating the species with ideal gases, i.e., ^^ _^^ = ^^/ ^^ _^^ ^^, where ^^ is pressure, ^^ _^^ is the Boltzmann factor, and ^^ is temperature. Non-ideal behaviour may be included by considering the compressibility factor, which can be obtained from a virial expansion for each species^1,2. Under the ideal gas assumption, the dependence of the refractive index on temperature is ^^² + 2 ^^ 1 ^^ _^^ ^^ = 2 ^^ ^^ _.

^^ − ^^ _^^ ^^ 〈 _{^^ ^^}〉 _^^ It then follows from the phase response ^^Φ = ^^₀ ^^ ^^ ^^ that the contribution to the phase resulting from bulk variations in the (ideal) gas is

¹ K. P. Birch, "Precise determination of refractometric parameters for atmospheric gases," JOSA A 8, 647-651 (1991) ² L. Pendrill, "Refractometry and gas density," Metrologia 41, S40 (2004) ³ N. An, B. Zhuang, M. Li, Y. Lu, and Z.-G. Wang, "Combined theoretical and experimental study of refractive indices of water—acetonitrile—salt systems," The J. Phys. Chem. B 119, 10701-10709 (2015) where ^^ _^^ is relative abundance of the species ^^ to the total, i.e.,

≡ ^^ _^^/ ∑ _^^ ^^ _^^ . There are two main effects that contribute to the phase shift of the bulk equation 8. One is the spatial, or temporal, variations in the composition of the bulk, i.e., at ^^ _^^ ^^ _^^ ≠ 0. The other is the difference in thermal state, i.e., ^^ _^^ ^^ ≠ 0 and ^^ _^^ ^^ ≠ 0. In the derivation of equations 4, only differences in phase resulting from changes in the propagating media, i.e., in the refractive indices , were considered. A more precise description comes from understanding that the refraction and reflection of light at the interfacial surface 126a may cause deflections in the rays, whose direction of propagation gets modified by the surface gradient ∇ℎ. By considering small slopes, i.e., ^|∇ℎ^| ≪ 1, one can perturbativelly expand the optical path length of each ray in powers of ^|∇ℎ^|, and obtain the corresponding phase Φ₀. The next-to-leading-order expansion of the two most relevant ray types, F and B₁, may be expressed as Φ _^^ = 2 ^^₀ ^^₁ ⁽ ^^ − ℎ⁾⁽1 + 2^|∇ℎ^|2)_, (₉)

Equations 9 and 10 show that the phase differences in equations 4 are only reliable when ^|∇ℎ^| ≪ 1, as expected. The B1 beams receive contributions at order ∇ℎ and hence are more prone to display the coupling with gradients. The remaining B_j rays may display similar behaviour. These various couplings with the surface gradient are expected to introduce different biases into the signal of each phase difference Φ_nk, as discussed further below. Demodulation of noisy carriers The interference data may be regarded an image comprising interference data from across the surface area of the sample 124 that is interrogated by the light beam 112. Consider an image ^^ _{^^ ^^} created from intensities of the form shown in equation 2, but with the average, ^^₀, subtracted. That is

+ ^^ ^^ _{^^ ^^}, where ^^ ^^ _{^^ ^^} is a random variable that quantifies the noise. All phase differences Φ _{^^ ^^} = Φ _^^ −

may be assumed to have small variations ^^^{( ^^ ^^)} around a stationary planar phase ^^ _{^^ ^^} ∙ ^^ as is the case of equations 4. That is, Φ _{^^ ^^}( ^^, ^^) = ^^^{( ^^ ^^)}( ^^, ^^) + ^^ _{^^ ^^} ∙ ^^, so that ^^ _{^^ ^^} = ∑ ^^ _^^ cos ( ^^ _^^ ∙ ^^ _{^^ ^^} + ^^ _^ ⁽ ^_^ ^{^} ^^{^)}) + ^^ ^^ _{^^ ^^} ≡ Re[ ^^̃ _{^^ ^^}] + ^^ ^^ _{^^ ^^} ⁽11⁾ ^^ for ^^ _^^ ≡ ^^₀ ^^₀ and ^^₀ ≡ Γ( ^^₀). Here, the sum runs over all ordered pairs of distinct labels indexed by ^^, e.g., ^^ = ^^ ^^ or ^^ = ^^ ^^₁. Equation 11 shows that the Fourier spectrum of the image

displays intensity peaks around the ^^ _^^ vectors. The positions in the spatial frequency domain are defined in accordance with the wavevectors in the planar phases of equations 4. The peaks may be sufficiently separated such that individual peaks may be identified. Since ^^ _{^^ ^^} is real, each non-zero carrier ^^ _^^ must have a so-called twin at − ^^ _^^. The collection of all carrier locations may be denoted as ^^ ≡ ^{ ^^ _^^ ^{} ^} _^^ ^{^} ₌₁ , with one of the twins excluded. That is, if, ^^ _^^ ∈ ^^, then − ^^ _^^ ∉ ^^. Such a collection may be constructed by locating all peaks ^^ _^^ =

of Fourier spectrum of ^^ _{^^ ^^} that lie in the upper half-plane, i.e., ^^ _{^^, ^^} ≥ 0. Referring to Figure 3, a diagram is shown illustrating the expected separation of the interferogram intensity peaks in the spatial frequency domain. The alignment of the light/reference beams to the respective reflective surfaces has been configured such t hat at least some of the peaks are separated. Figure 3 shows each peak at location ^^, with the − ^^ peaks being omitted. In this theoretical case, the FB_j peaks overlap with B_jB_j+1 peaks as shown. In practise, there will also be harmonics of each peak, i.e ., peaks at integer multiples of ^^. In the FBj and BjBj+1 cases, and only in these, harmonics may lead to further overlap of carriers. Referring to Figure 4, simulated interference data is shown. Image (a) of Figure 4 shows interference data emulating a digital signal containing an idealised height deformation consisting of orthogonal standing waves in a fictitious squared receptacle. The data was simulated using ray-tracing. Interference fringes 401 are clearly visible inset. The contribution from each hologram to a spatial frequency ^^ is generally given by the local linearisation ∇Φ of the phase Φ. If the phase Φ is of the form of equations 4, then the k-space region of excited amplitudes near each carrier peak ^^ _^^ is given by ^^ _^^ + ^^∇ℎ for some constant ^^. The corresponding areas, found from considering maxima of ∇ℎ = cos( ^^₁ ∙ ^^) + cos( ^^₂ ∙ ^^), are shown for the three holograms RB₁, RF and FB₁ as white rectangles in image (b) of Figure 4. These are in good agreement with the simulated data. Note that the strongest signal, which is the RB₁ hologram (solid white line), has the smallest modulation area. This is due to the prefactor | ^^₂ − ^^₁| < 1 in image (b), which reduces the response ^^_ℎΦ _{^^ ^^1} of the phase

to excitations of the interface ℎ. The true phases from the ray-tracing algorithm are shown in the four images labelled (RB₁), (RF), (FB₁) and (RB₂) in Figure 4. The phase from the RF hologram is anti- correlated with all the other holograms and that the all the phases respond differently to the height field ℎ( ^^). Here, the RB₁ phase has the lowest amplitude, and the RB₂ has the largest. Referring to Figure 5, an example is shown of transformed data, comprising interference data transformed to the spatial frequency domain. As discussed above, intensity peaks are visible corresponding to the different interferograms. Each of the peaks is smeared over a given area, corresponding to modulation of the phase difference of the interferogram. Peaks 501 in the transformed data may be automatically identified, for example using a peak finding algorithm that identifies local maxima in the amplitude of the transformed data. A threshold prominence for the local maxima may be defined, so that only relatively prominent peaks 501 are identified. The search for peaks may be limited to positive k_y, to avoid double processing of transforms with vectors at − ^^. Once the peaks 501 are identified, a peak region 502 may be identified corresponding with each peak 501. The peak region 502 be defined by a circle centred on the peak, with a radius that is determined based on the spread of the peak. Determining the peak region 502 may be performed by fitting a 2D Gaussian function centred at the peak 501. The radius of the peak region 502 may be defined based on a multiple of the sigma value of the Gaussian. For example, the peak region may be defined as a circle, centred on the peak 501 with a radius of 3 x sigma, or a peak corresponding with the full width half maxima of the fitted Gaussian (i.e. 2.355 x sigma). A peak 501 may be defined as overlapping based on whether the peak regions 502 overlap. A degree of overlap between a selected peak region and any neighbouring peak regions may be defined, based on the percentage area of the selected peak that is overlapped by neighbouring peak regions. An algorithm may be used that seeks to minimise a degree of overlap between peaks in the transformed data. Conversely, an algorithm may be used to extract the particular shape of spatial frequency modulations of each peak to identify and distinguish between the plurality of peaks. Note that the shape of the modulations may vary from those displayed in Figure 4 depending on the experimental setup. An overall figure of merit for peak separation may be defined. For example, the figure of merit may comprise a weighted sum of the degree of separation between a preselected subset of the peaks. The subset of the peaks may be those of selected interferences (which may be identified as discussed below), or simply peaks with highest intensity in the transformed data. In some embodiments, a sequence of interferograms may be obtained, for example at a frame rate of at least 10Hz (for example at more than 30Hz, or more than 100Hz). If a handheld tool is employed, the orientation of the light beam 114 on the sample will tend to vary between frames. A search may be performed on the sequence of interferograms for transformed data that comprises relatively well separated peaks (for example, based on the overall figure of merit). Subsequent analysis may be based on the data that comprises well separated peaks. In some embodiments, an automatic orientation of a reflective surface may be performed in order to produce a suitably high figure of merit for peak separation. The orientation of the reflective surface may be adjusted in an iterative manner to increase the figure of merit for peak separation. The adjustment may be manual by a user, prompted by a graphical user interface that displays the transformed data and/or the figure of merit for peak separation. The adjustment may be automatic, by a controller operating at least one actuator that controls an orientation of the reflective surface in response to the figure of merit. The separation or demodulation of the peaks corresponding with different interferograms may facilitate improved analysis of the optical path length. Referring to Figure 6, a further example is shown of interference data 620 and transformed data 610. At the origin in the transformed data, the mean intensity peak 601 of the interferogram is circled (by hand, for indication only) . The RB1 interferogram/peak region 602 is labelled in the transformed data 610, and the orientation of the fringes 622 corresponding with the RB1 interferogram is indicated on the interference data 620. The RF interferogram/peak region 603 is labelled in the transformed data 610, and the orientation of the fringes 623 corresponding with the RF interferogram is indicated on the interference data 620. The FB₁ interferogram/peak region 603 is labelled in the transformed data 610, and the orientation of the fringes 623 corresponding with the FB₁ interferogram is indicated on the interference data 620. The degree of separation between peaks may be determined by the experimental setup. Changes in the off-axis alignment of both the reference and/or light mirrors shift the plane phases ^^ _^^ ∙ ^^ and ^^ _^^ ∙ ^^ in equations 4, respectively, thus moving their positions in the frequency domain. Adjustments on the reference mirror, and hence in the optical path length of the R beam, can only affect peaks under the RF and RBj categories. Hence, the interferograms and corresponding peaks that remain static when the reference mirror is adjusted can be identified as FB_j -type. Conversely, adjustments in the sample mirror will shift the position of all peaks. As previously mentioned, the RB₁ -type interferogram may have the largest of the combined amplitude ^^ _{^^ ^^} in equations 5 and may appear as the brightest peak in the Fourier spectrum. From the positions of both RB1 and FB1, the values for ^^ _^^ and ^^ _^^ may be determined. The model in equations 4 may then be used to predict the location of the remaining peaks and identify them in the Fourier spectrum accordingly. A transform may be applied to the interference image to transform the interference data into the spatial frequency domain. The transformed data may be considered to comprise carrier spatial frequencies, characteristic of the centre of each interferogram. T he modulation of the carrier spatial frequency by variations in path length over the region of the sample that provides the reflected beams provides information about the distribution of optical path length over the region of the sample. Reference to a “ca rrier peak” may be understood as analogous to an interferogram peak – the peak centred on the carrier spatial frequency and modulated by the spatial distribution of optical path length (and/or refractive index variations) . Let ^^ _^^ be a Fourier filter around a single carrier peak ^^ _^^, i.e., ^^ _^^ ≡ ℱ ⁻¹ ^^ _^^( ^^)ℱ, where ℱ is a two-dimensional spatial Fourier transform with inverse ℱ⁻¹, and ^^ _^^ is a window function that is only non-zero in the vicinity of the n'th carrier ^^ _^^. That is, ^^ _^^( ^^) is such that

wherein [ ^^̃] _{^^ ^^} is a portion of the image within the window function and [ ^^̃^∗] _{^^ ^^} is a portion of the image excluded by the window function, such that

are n extracted interferogram peaks. The Fourier filter takes the interference data, performs a spatial Fourier transform to determine transformed data, filters that transformed data with a window function to select a particular interferogram/carrier, and then inverse transforms the windowed transformed data The window function ^^ _^^( ^^) may be a two-dimensional tophat-filter centred at ^^ _^^, i.e., ^^ _^^ ⁽ ^^⁾ = 1 if ^| ^^ − ^^ _^^ ^| ≤ ^^ _^^ with ^^ _^^ ⁽ ^^⁾ = 0 otherwise, for some appropriate choice of filter size ^^ _^^. In general, the larger the phase modulations ^^⁽ _^^ ⁾ ^_{^ ^^} , the larger ^^ _^^ is required. The filter size may be automatically determined, for example as discussed above with reference to Figure 5. If all carrier peaks ^{ ^^ _^^ ^{} ^} _^^ ^{^} ₌₁ are sufficiently isolated, and no modulation exceeds the filter size ^^ ( ^^) ^_^, then the isolated ^^̃ _{^^ ^^} adds up to the signal, i.e.,

If the noise is assumed to be much smaller than the signal in each peak, i.e., ^^ _^^ ^[ ^^ ^^^] _{^^ ^^} ≪ ^^̃ , equation 12 may be used to observ ( ^^) ^_{^ ^^} e that each phase modulation ^^ _{^^ ^^} can be recovered using ^^ _^ ⁽ ^_^ ^{^} ^^{^)} = Im log(

for some integer field ℓ⁽ ^_{^ ^} ^{^} ^^{^)} ∈ ℤ, enumerating the number of phase-wrappings at the pixel location ⁽ ^^ , ^^⁾. Note that most of the quantities on the right side of equation 14 are accessible from the image ^^ _{^^ ^^}. The carrier peak ^^ _^^ may be found from the peaks of |ℱ ^^| (i.e. peaks in the transformed data) and may be used to construct the filter ^^ _^^, and the phase wrapping can be redu ⁴

ced to a single integer ℓ _^^ ∈ ℤ. This leaves only one scalar quantity, in addition to the noise ^^ ^^( ^^) ^_{^ ^^} , that is yet to be determined. That is the global phase wrapping ℓ _^^. One way to avoid the issue of phase wrappings is to consider changes in phase from some reference time ^^₀ to the time ^^ in question. If the carrier does not drift from ^^₀ to ^^, i.e., ^^ _^^( ^^₀) = ^^ _^^( ^^), then, neglecting the noise, the reconstructed phase difference may be expressed as ⁴ M. A. Herraez, D. R. Burton, M. J. Lalor, and M. A. Gdeisat, "Fast two-dimensional phase- unwrapping algorithm based on sorting by reliability following a noncontinuous path," Appl. Opt. 41, 7437-7444 (2002) Δ ^^ _^ ⁽ ^_^ ^{^} ^^{^)} ≡ ^^ _^ ⁽ ^_^ ^{^} ^^{^)}( ^^) − ^^ _^ ⁽ ^_^ ^{^} ^^{^)}( ^^₀) = Im ln( ^^ _^^[ ^^( ^^) ^^^∗( ^^₀)] _{^^ ^^}) + 2 ^^(ℓ _^^( ^^) − ℓ _^^( ^^₀)). (15) Given a sequence of images

taken at equidistant times Δ ^^ ≡ ^^ _^^+1 − ^^ _^^, there are two canonical choices for the reference time ^^₀. First, one may choose ^^₀ to be a constant reference, e.g., the initial frame ^^₀ = ^^₁. This approach may be referred to as the absolute reconstruction. Alternatively, one can choose the previous image as reference, i.e., ^^₀ = ^^ − Δ ^^. This approach may be referred to as relative reconstruction. Lastly, the fully reconstructed phase ^^⁽ _^^ ⁾ ^_{^ ^^} using equation 14 may be referred to as the synthetic reconstruction, which is always determined up to a global phase wrapping ℓ _^^. Automatised multi-peak identification Following one of the reconstruction procedures outlined above, each of the ^^ = 1, ⋯ , ^^ carrier peaks ^^ ( ^^) ^_^ permits recovering phases Δ ^^ _{^^ ^^} from, in the case of synthetic reconstruction, a single image or, in the case of absolute or relative reconstruction, any pair of images. Notably, each reconstructed phase fields Δ ^^( ^^) ^_{^ ^^} should correspond to one of the equations 4. In particular, all phases should be proportional to the interface height ℎ _{^^ ^^}. Spatial correlations between reconstructed phases may be computed. Principal component analysis (PCA) may then be used to extract the common profile that contributes to all of the spatial correlations. Additionally, the proportionality prefactors between measured phases and the profile of the fluid surface may be recovered. Using the set of equations 4, the computed prefactors can be used to estimate the refractive index ratio ^^₂/ ^^₁. Here, PCA is described as being performed on spatial correlations. Equally however, PCA may be used on temporal correlations. One can compute the correlation matrix using the time average over a given time frame and use PCA on this time average data. This will allow extraction of any dominating effects over the time frame. Referring to Figure 7, the results of PCA analysis performed on a number of interferograms (holograms) are shown. In this example dataset, t he PCA here was performed over time, (i.e., the correlation matrix between holograms was computed by averaging in time the time-dependent amplitudes of each carrier peak). On the left axis, the average principal eigenvector is shown. On the right axis, the corresponding refractive indices are shown. Referring to Figure 8, experimental data is shown illustrating a reconstruction of a number of interferograms. Image (a) of Figure 8 shows captured interference data, with interference fringes 801 clearly visible inset. Image (b) of Figure 8 shows peaks in the spatial frequency domain, with the corresponding reconstructed phases shown in the side images (1) to (4). The peaks are labelled: (1) = RB₁, (3) = FB₁, and (4) = RF. Peaks (2) and (5) are possible harmonics of the remaining peaks. A zoomed in image of the RB₁ peak is shown inset in image (c) of Figure 8. Writing Δ ^^⁽ ^_{^ ^} ^{^} ^^{^)} = ^^ _^^ℎ _{^^ ^^} + ^^( ^^) ^_{^ ^^} , where ^^( ^^) ^_{^ ^^} contains all the noise, the bias from e.g., ray deflection and bulk contribution, and the constant shift, identi fication of the n'th phase ^^( ^^) ^_{^ ^^} can be reduced to finding a single N-dimensional real vector ^^ = ( ^^₁, ⋯ , ^^ _^^). In particular, the ratios of phases from equations 4 should be contained in ^^ (for example ^^ _{^^ ^^}/ ^^ _{^^ ^^1} = ^^₁/ ^^₂). If the only spatial variation in the system is the height ℎ _{^^ ^^}, then the vectors

will be distributed along a line in ℝ ^{^^} that is parallel to ^^. More generally, if the phases Δ ^^( ^^) ^_{^ ^^} all depend significantly on the height ℎ _{^^ ^^}, as expected from equations 4 for small bias fields δΦ _{^^ ^^} compared to the excitation of the interface (as shown in Figure 5), then the (spatial) covariance matrix ^^ with indices

is highly elongated in the direction of ^^. That is, if

is the orthogonal eigenspectrum of ^^, i.e., ^^ ^^ _^^ = ^^ _^^ ^^ _^^, then there is one principal component ^^₀ such that | ^^ _^^0| ≫ | ^^ _^^| for all ^^ ≠ ^^₀. The dominant component ^^ _^^0, can be interpreted as the direction of mutual (spatial) dependence. From the equations 4, ^^ ≈ ^^ ^^ _^^0 for some scalar factor ^^ ∈ ℝ . Thus, the problem of peak identification may be reduced to the determination of a single scalar ^^ translating variations in the phase to variations in the height. Observing that all the height pre- factors in equations 4 are of the form 2 ^^₀ ⁽ ^^ ^^₁ + ^^ ^^₂ ⁾ for integers ^^, ^^ ∈ ℤ, where 2 ^^₀ = 2 ^^/ ^^₀ is a constant that depends only on the angular frequency ^^ of the laser, it is clear that if one knows one refractive index, i.e. that of the surrounding air ^^₁, then all conversion factors are specified, and the physical height may be obtained from any one of the reconstructed phase fields. If one has more than two carrier peaks that match the expressions in equations 4, then the system is over-specified, and the numerical values for both refractive indices ^^₁ and ^^₂ may be obtained. The processes described above may be performed automatically by a computer connected to the detector 150. Following the calculation of the phase differences discussed above, this information may be used to characterise the sample. Using the relevant equations 4, the phase differences may be used to calculate: the average thickness of the sample; variations in the height of the sample relative to the reflective surface across the imaging area (the profile of the sample surface); and the refractive index of the sample and/or the surrounding medium. These measurements may be repeated over a given time period such that these characteristics may be tracked temporally. Figure 9 illustrates a measurement of evaporation rate of a liquid sample performed using an embodiment. This is an early demonstration of the method, and is provide d to illustrate the sensitivity of the method to very small variations in optical path length. The measurement shows the change in inferred depth ℎ − ℎ₀ of a liquid sample, measured using the interferometric methods discussed above. Interferogram data was obtained and demodulated according to the method described above, to produce independent estimates of layer thickness based on different interferograms. Image (a) of Figure 9 shows the measured thickness of the liquid sample over a given time period, wherein each of the plots corresponds to a measurement from one interferogram. Image (b) of Figure 9 shows the evaporation rate extracted from the time varying thickness measurement of each interferogram. The rate of change of thickness of the liquid corresponds with an evaporation of the sample over time, and the measurements from the different interferograms are in good agreement. The rate of ~ 35 nm/s is detected over time frames of less than 300 seconds, which demonstrates very high accuracy over a short measurement time – each frame of the measurement is reliably determining a spatial map of thickness with an accuracy in the nanometre range, and the frame time is <100ms. Although a ~ 300 second time frame is shown in Figure 9, similar results have been obtained over much shorter time frames, such as < 20 seconds. Where the thickness measurements in this example do not agree, the variations may be due to variations in refractive index in the air or of the sample. The demodulated interferograms enable different sources of measurement error to be identified and potentially modelled (as discussed above). For example, the FB interferogram has a common mode rejection of variations in optical path length through the air, whereas the RF and RB₁ interferograms will be sensitive to variations in the optical path length through the air that are different between the reference beam and the light beam (illuminating the sample). Although example embodiments have been described, these are not intended to limit the scope of the invention, which should be determined with reference to the accompanying claims.

Claims

CLAIMS 1. A method of measuring an optical path length of a sample having a first face and a second face on an opposing side of the sample to the first face , the method comprising: illuminating the first face of the sample with a light beam; receiving at a detector interference data comprising a plurality of interferograms, each of the interferograms comprising a reflected light beam resulting from reflection of the light beam from at least one of the first face and the second face; and each of the interferograms comprising a reflected light beam with reflections from a different combination and/or sequence of surfaces; performing a spatial frequency transform of the interference data to generate transformed data in a spatial frequency domain; analysing features of the transformed data to determine the optical path length, wherein each feature in the transformed data corresponds to one of the interferograms in the interference data and at least some of the features are at least partially separated in the spatial frequency domain.

2. The method of claim 1, wherein the plurality of interferograms comprises at least one of: a first group of interferograms, wherein each interferogram is generated by interference between: a reflected light beam, ^^, that is reflected from first face without traversing the sample; and a reflected light beam, ^^ _^^ , that is reflected from the second face ^^ times and the first face ^^ − 1 times, wherein ^^ = 1, ⋯ , ^^ ; and a second group of interferograms, wherein each interferogram is generated by interference between: a reflected light beam, ^^ _^^ , that is reflected from the second face ^^ times and the first face ^^ − 1 times, wherein ^^ = 1, ⋯ , ^^ , wherein ^^ ≠ ^^; and the reflected light beam, ^^ _^^.

3. The method of claim 1 or claim 2, further comprising providing a reference beam that does not interact with the sample, wherein at least some of the interferograms comprise interference between the reference beam and a reflected light beam.

4. The method of claim 3, wherein the plurality of interferograms comprises at least one of: an interferogram generated by interference between a reference beam, R, and a reflected light beam, ^^, that is reflected from the first surface without traversing the sample; and a third group of interferograms, wherein each interferogram is generated by interference between: the reference beam, R; and a reflected light beam, ^^ _^^ , that is reflected from the second face ^^ times and the first surface ^^ − 1 times, wherein ^^ = 1, ⋯ , ^^.

5. The method of claim 3 or 4, wherein the reference beam and the illuminating light beam are generated by splitting a source light beam from a common light source.

6. The method of any of claims 3 to 5, further comprising providing a second reflective surface configured to reflect the reference beam, wherein the second face and second reflective surface are configured such that the reflected light beam and reflected reference beam interfere at the detector.

7. The method of claim 6, further comprising: increasing the separation between at least some of the features in the transformed data relative to each other, wherein the increasing the separation comprises: adjusting an angle of the second face relative to the light beam at the second face; and/or adjusting an angle of the second reflective surface relative to the incoming reference beam at the second reflective surface.

8. The method of any preceding claim, wherein the interference data is two - dimensional image data comprising a superposition of the plurality of interferograms.

9. The method of any preceding claim, wherein the interference data encodes a phase difference for each of the interferograms.

10. The method of claim 9, wherein each of the phase differences is generated by a phase shift in the illuminating light beam resulting from at least one of: a bulk variation in the sample; a bulk variation in a medium that is not the sample through which the light beam propagates to and/or from the sample; the refractive index of the sample; the refractive index of the medium that is not the sample; and a distance that the light beam travels while transiting the sample.

11. The method of any preceding claim, wherein performing a transform of the interference data comprises performing a spatial Fourier transform of the interference data.

12. The method of any preceding claim, wherein the features of the transformed data comprise a plurality of intensity peaks in the spatial frequency domain and each peak is located at a wavevector corresponding to one of the interferograms.

13. The method of claim 12, wherein analysing the features comprises: selecting at least one of the intensity peaks in the spatial frequency domain; and applying a filter to each of the selected intensity peaks to extract the peak.

14. The method of claim 13, further comprising performing an inverse spatial frequency transform of the transformed data of a plurality of the extracted intensity peaks to generate reconstructed interference data for each of the plurality of extracted peaks, wherein the reconstructed interference data of each extracted peak encodes a phase difference of an interferograms corresponding to that extracted peak.

15. The method of claim 14, further comprising: generating a covariance matrix for the reconstructed interference data of the plurality of extracted peaks; extracting a principal component of the covariance matrix, the principal component corresponding to the mutual spatial dependence of the reconstructed interference data of the extracted intensity peaks.

16. The method of claim 15, further comprising determining an interferogram that corresponds with each of the extracted intensity peaks by comparing the extracted principal component with an expected phase difference for each interferogram.

17. The method of claim 16, further comprising using the expected phase differe nce corresponding to one of the determined interferograms and the extracted principal component to determine a height of the sample, the height corresponding to a thickness of the sample between the first face and second face.

18. The method of claim 16 or 17, further comprising using the expected phase difference for more than one of the determined interferograms and the extracted principal component to determine a refractive index of the sample.

19. The method of any previous claim, wherein the interference data comprises first interference data received at a first time and second interference data received at a subsequent, second time, and further comprising: identifying the intensity peak from the first and second interference data corresponding to the same interferogram; and measuring the change in the phase difference of the peak between the first interference data and the second interference data.

20. The method of any of claims 9 to 19, wherein the phase difference of at least one interferogram is used to calculate: the height of the sample interface above the second face; variations in height across an area of the first sample interface that is illuminated by the light beam; and/or the refractive index of the sample.

21. An apparatus for measuring an optical path length of sample having a first face and a second face on an opposing side of the sample to the first face , the apparatus comprising: a light source configured to illuminate the first face of the sample with a light beam; a detector configured to receive interference data generated by reflection of the light beam from the sample; and a processor configured to: perform a transform of the interference data to generate transformed data, the transformed data being in a spatial frequency domain; and analyse features of the transformed data to determine the optical path length wherein the interference data comprises a plurality of interferograms, each of the interferograms comprising a reflected light beam resulting from reflection of the light beam from at least one of the first face and the second face; and each of the interferograms comprising a reflected light beam with reflections from a different combination and/or sequence of surfaces; and each feature in the transformed data corresponds to one of the interferograms in the interference data and at least some of the features are at least partially separated in the spatial frequency domain .

22. The apparatus of claim 21, further comprising a reflective surface at the second face of the sample.

23. The apparatus of claim 21 or claim 22, further comprising: a beam splitter configured to split the light beam generated by the light source into a light beam that is used to illuminate the sample and a reference beam that does not interact with the sample; wherein at least some of the interferograms comprise inte rference between the reference beam and the light beam reflected from at least one of the interfaces and the second face.

24. The apparatus of claim 23, further comprising a second reflective surface configured to reflect the reference beam, wherein the light beam and the reference beam are perpendicular to one another, such that the apparatus is configured as a Michelson interferometer.