Method, system, and sensor for analyzing a sample, and process for manufacturing an electrode

Abstract

A method for analyzing a sample including a layer having a first interface and a second interface, the method comprising: irradiating the sample with a pulse of terahertz radiation, the pulse including a plurality of frequencies in the range of 0.01 THz to 10 THz; detecting the radiation reflected from the sample to generate a sample waveform; obtaining a first reflected waveform from the sample waveform, the first reflected waveform corresponding to reflection from the first interface; obtaining a second reflected waveform from the sample waveform, the second reflected waveform corresponding to reflection from the second interface; comparing the first reflected waveform with the second reflected waveform to generate an estimated value of the thickness and complex refractive index of the layer; generating a synthetic signal using the estimated values of the thickness and complex refractive index; changing at least one of the thickness and complex refractive index to reduce an error between the sample waveform and the synthetic signal; and outputting the thickness of the layer.

Description

Technical Field

[0001] Embodiments of the present invention relate to a method, a system, and a sensor for analyzing a sample, and a process for manufacturing an electrode. In particular, the method, the system, the sensor, and the process use terahertz radiation.

Background Art

[0002] Terahertz radiation is a non-invasive method for determining the internal structure of an object and the thickness of its layers. For example, terahertz radiation can be used to measure the properties of a sample including one or more layers.

[0003] When a terahertz beam interacts with a sample, the beam can change. The properties of the sample can be determined from the changed beam.

[0004] Terahertz time-domain spectroscopy is a technique in which a terahertz pulse is applied to a sample and waveform data of the signal form as a function of optical delay is obtained.

[0005] There is a need for improved methods and systems for measuring the properties of a sample using terahertz radiation. In particular, improved methods and systems are needed when the sample has a high absorption rate and / or a high refractive index.

Summary of the Invention

[0006] According to a first aspect, there is provided a method for analyzing a sample including a layer having a first interface and a second interface, the method comprising: irradiating the sample with a pulse of terahertz radiation, the pulse including a plurality of frequencies in the range of 0.01 THz to 10 THz; detecting the radiation reflected from the sample to generate a sample waveform; A step of obtaining a first reflected waveform from the sample waveform, wherein the first reflected waveform corresponds to the reflection from the first interface A step of obtaining a second reflected waveform from the sample waveform, wherein the second reflected waveform corresponds to the reflection from the second interface A step of comparing the first reflected waveform with the second reflected waveform to generate estimated values of the thickness and complex refractive index of the layer A step of generating a composite signal using the estimated values of the thickness and complex refractive index A step of changing at least one of the thickness and complex refractive index in order to reduce the error between the sample waveform and the composite signal Including a step of outputting the thickness of the layer

[0007] The sample includes a layer having a first interface and a second interface. The layer may be provided on a substrate. The first interface refers to the interface between the outer surface of the layer and the external environment. For example, the external environment is air. The second interface is the interface between the substrate and the layer

[0008] In this method, estimated values of the complex refractive index and thickness of the layer are obtained. This estimation is obtained by irradiating the sample with a pulse of terahertz radiation and detecting the reflected radiation. These estimated values are used as starting points for an iterative procedure to generate the complex refractive index and thickness of the layer

[0009] The thickness of the layer is output

[0010] Also, the complex refractive index may be output

[0011] This method enables obtaining information regarding reflections from a layer by a first reflected waveform and obtaining information regarding transmissions through the layer by a second reflected waveform. This information is obtained using one measurement. The thickness and complex refractive index can be estimated from said information. Using the estimated values, a synthetic signal is obtained. The synthesized signal is compared with the sample waveform and an error is determined. At least one of the thickness and complex refractive index is varied so that the error is reduced. This method provides estimated values with improved accuracy. The estimated values act as starting points for the variation of parameters for reducing the error. By having a more accurate starting point, the error can be reduced more effectively (e.g., more rapidly and / or more accurately with respect to the final solution). The reduction of the error indicates that the parameters accurately describe the layer.

[0012] Varying the parameters to reduce the error between the sample waveform and the synthetic signal is sometimes referred to as optimization.

[0013] The error can be determined in the time domain. In the time domain, the synthetic signal is a time-domain signal. Alternatively, the error may be determined in the frequency domain. In this case, the error may be frequency-dependent (i.e., an error spectrum is obtained). To obtain the error spectrum, the sample waveform may be converted to the frequency domain to generate a sample spectrum and the synthetic signal is a spectrum.

[0014] In one example, the error spectrum may be weighted and one of the thickness and complex refractive index is varied to reduce the weighted error spectrum. Thereby, the error can be reduced more effectively (e.g., more rapidly).

[0015] For example, the parameters (thickness and complex refractive index) are varied until the error is minimized.

[0016] The initial estimated values are obtained from the sample being measured without the need for a separate calibration sample.

[0017] In one example, the range in which the parameter changes is determined in advance. This range can be determined by experiments.

[0018] In one embodiment, the method includes a step of outputting at least one of the density and conductivity of the layer, wherein the density and conductivity are determined from the thickness and / or complex refractive index.

[0019] From the determined thickness and / or complex refractive index, the thickness, density and / or conductivity of the layer are determined and output. This method enables obtaining one or more of the thickness, density, and conductivity using a non-contact, non-destructive method using a single measurement. This method can also be used to extract two or all three of these quantities. The present method is applicable to a production line environment for monitoring the production of layers.

[0020] In one embodiment, the method includes a step of obtaining a reference waveform, wherein the reference waveform is obtained by irradiating a reference sample with a pulse of terahertz radiation including a plurality of frequencies in the range of 0.01 THz to 10 THz, and a step of detecting the radiation reflected from the reference sample to generate the reference waveform.

[0021] The reference sample may be a sample that has not yet been coated. The reference sample may be a substrate before deposition of the layer. The reference sample may be an uncoated substrate. Alternatively, the reference sample may be a plane mirror. The reference waveform may be obtained in advance by measuring the reference sample in advance.

[0022] In one embodiment, obtaining the first reflected waveform and the second reflected waveform from the sample waveform includes using time gating. Time gating is applied in the time domain. The purpose of time gating is to select segments from the sample waveform. Time gate control enables analyzing the reflection from the first interface and the reflection from the second interface separately.

[0023] In one embodiment, the first reflected waveform is converted to the frequency domain to obtain a first spectrum, and the second reflected waveform is converted to the frequency domain to obtain a second spectrum. Thereby, the frequency dependence of the reflection can be analyzed.

[0024] When the first spectrum and the second spectrum are obtained, generating estimated values of the layer thickness and the complex refractive index by comparing the first reflected waveform with the second reflected waveform includes comparing the first spectrum with the second spectrum.

[0025] In one embodiment, the sample waveform is deconvolved with the reference waveform to generate a deconvolved waveform, the deconvolved waveform is time gated to obtain the first reflected waveform and the second reflected waveform, the first reflected waveform is converted to the frequency domain to obtain a first spectrum, and the second reflected waveform is converted to the frequency domain to obtain a second spectrum.

[0026] In an example, deconvolving the sample waveform with the reference waveform includes converting the reference waveform to the frequency domain to obtain a reference spectrum, converting the sample waveform to the frequency domain to obtain a sample spectrum, and dividing the sample spectrum by the reference spectrum. Before converting the reference waveform and the sample waveform into the frequency domain, the background waveform can be subtracted from the reference waveform and the sample waveform, and the background waveform is obtained by performing measurement in a state where there is no sample in the path of terahertz radiation.

[0027] An apodization function can be applied to remove or suppress the edge effect in the sample waveform. For example, the apodization function f(t) can be converted into the frequency domain and multiplied by the sample spectrum. By removing or suppressing the edge effect, the signal-to-noise ratio can be improved.

[0028] The deconvolution of the sample waveform using the reference waveform in the time domain is equivalent to dividing the sample spectrum by the reference spectrum.

[0029] In one embodiment, the method includes the step of determining estimated values of thickness and complex refractive index from the first spectrum and / or the second spectrum, where the complex refractive index is frequency-dependent. Thereby, the frequency dependence of the thickness and the complex refractive index can be captured. Thereby, more accurate estimated values of the thickness and the complex refractive index can be obtained.

[0030] In one embodiment, the method includes the steps of converting the reference waveform into the frequency domain to obtain a reference spectrum, and determining an estimated value of the real part of the complex refractive index from the reference spectrum and the first spectrum.

[0031] In one embodiment, the method includes the steps of acquiring a second reflection spectrum, correcting the second reflection spectrum, determining the imaginary part of the refractive index from the corrected second reflection spectrum, and determining the thickness from the corrected second reflection spectrum.

[0032] In one example, the step of correcting the second reflection spectrum comprises the step of correcting the reflection from the first interface, and / or the step of correcting the reflection from the second interface.

[0033] In one embodiment, the method comprises the step of fitting an estimated value of the complex refractive index to a physical model to generate a model of the layer.

[0034] The complex refractive index is frequency-dependent. Fitting enables representing the layer by a physically realistic model. Having a physical model also reduces the number of parameters that have to be fitted in subsequent optimization steps. This enables the optimization to be more effective (e.g., faster and / or more accurate).

[0035] In one embodiment, the complex refractive index is varied to reduce the error between the sample waveform and the synthetic signal. This includes varying the parameters of the model, where the parameters of the model are related to the complex refractive index.

[0036] In one example, generating an estimated value of the layer thickness and / or the complex refractive index includes averaging. The thickness and the complex reactivity index are frequency-dependent. By taking an average, a single value of the estimated thickness and the complex refractive index in the time or frequency domain can be obtained. The average refers to the average over frequency. Using a single value of the thickness and / or the complex refractive index for the estimation can simplify the fitting procedure.

[0037] In one example, the method comprises the step of applying a filter to the first and / or second spectrum, and the step of converting the filtered first and / or second spectrum to the time domain to generate a filtered sample waveform. changing at least one of thickness and complex refractive index to reduce the error between the filtered sample waveform and the composite signal, This advantage is that the noise in the sample waveform (the measured signal) is reduced and the error can be more effectively reduced.

[0038] In one embodiment, the method includes determining the magnitude of a first reflected waveform, comparing the magnitude of the first reflected waveform with the magnitude of a reference waveform to generate a first ratio, and estimating the real part of the complex refractive index using the first ratio.

[0039] For example, the magnitude of the first reflected waveform is the magnitude of the peak resulting from reflection from the first interface. For example, the magnitude of the reference waveform is the magnitude of the peak resulting from reflection from a reference sample.

[0040] In one embodiment, the method includes determining the magnitude of the second reflected waveform, comparing the magnitude of the first reflected waveform with the magnitude of the second reflected waveform to generate a second ratio, and estimating the imaginary part of the complex refractive index using the second ratio. For example, the magnitude of the second reflected waveform is the magnitude of the peak resulting from reflection from the second interface.

[0041] In one embodiment, the method includes comparing the first reflected waveform and the second reflected waveform to obtain a time delay, and estimating the thickness using the time delay or the time delay combined with refractive index information.

[0042] According to a second aspect, a system for analyzing a sample including a layer having a first interface and a second interface is provided, the system A pulse source of terahertz radiation adapted to irradiate the sample with pulses of terahertz radiation, wherein the pulses are a plurality of frequencies in the range of 0.01 THz to 10 THz, a pulse source, a detector for detecting the reflected radiation and generating a sample waveform derived from the reflected radiation, and a sensor comprising: An analysis unit, the analysis unit comprising a processor and a memory, the processor Obtaining a first reflected waveform from the sample waveform, wherein the first reflected waveform corresponds to the reflection from the first interface, obtaining; Obtaining a second reflected waveform from the sample waveform, wherein the second reflected waveform corresponds to the reflection from the second interface, obtaining; Comparing the first reflected waveform with the second reflected waveform to generate estimated values of the thickness and complex refractive index of the layer; Generating a composite signal using the estimated values of the thickness and complex refractive index; Changing at least one of the thickness and complex refractive index to reduce the error between the sample waveform and the composite signal; Outputting the thickness of the layer, and an analysis unit adapted to perform, including.

[0043] According to a third aspect, a system for analyzing a sample including a layer having a first interface and a second interface is provided, the system A pulse source of terahertz radiation adapted to irradiate the sample with pulses of terahertz radiation, wherein the pulses are a plurality of frequencies in the range of 0.01 THz to 10 THz, a pulse source, a detector for detecting the reflected radiation and generating a sample waveform, the sample waveform being derived from the reflected radiation, and a sensor comprising: the sensor includes an optical element adapted to resolve reflected radiation from both the first interface and the second interface.

[0044] In one embodiment, the optical element has an f-number of 3 or more.

[0045] In one embodiment, the optical element has an f-number of 10 or more.

[0046] For example, the sensor of the second or third aspect further includes a focusing element configured to direct the pulse of the terahertz radiation towards the sample using a first path and to direct the pulse of the terahertz radiation towards an internal mirror using a second path, wherein the sample waveform includes the radiation reflected from the sample via the first path and the radiation reflected from the internal mirror via the second path.

[0047] According to a further aspect, there is provided a method for adapting a system for analyzing a sample including a layer having a first interface and a second interface, the system including a sensor, the sensor including a pulse source of terahertz radiation adapted to irradiate the sample with pulses of terahertz radiation at a plurality of frequencies within the range of 0.01 THz to 10 THz, a detector for detecting the reflected radiation, and an optical element, the method comprises obtaining an estimated value of the refractive index of the layer, obtaining an estimated value of the thickness of the layer, and determining the f-number of the optical element such that the confocal parameter scaled by the estimated value of the refractive index is greater than the estimated value of the thickness of the layer.

[0048] The confocal parameter is given by b = 2z R = π·w02 / λ, where λ = λ0 / n and w0 is the beam waist.

[0049] The optical element is adapted to resolve the reflected radiation from the first and second interfaces.

[0050] According to a further example, a method of analyzing a sample is provided, the sample including a layer having a first interface and a second interface, the method comprising: obtaining an estimated value of the refractive index of the layer; obtaining an estimated value of the thickness of the layer; determining an f-number of an optical element such that a confocal parameter scaled by the estimated value of the refractive index is greater than the estimated value of the thickness of the layer; performing a measurement using a system comprising a sensor, a detector, and an optical element having an f-number greater than or equal to the determined f-number, the performing the measurement comprising: irradiating the sample with a pulse of terahertz radiation, the pulse including a plurality of frequencies in the range of 0.01 THz to 10 THz; detecting the reflected radiation.

[0051] According to a further aspect, a process for manufacturing an electrode for a battery is provided, the process comprising: coating a substrate with a layer; drying the layer; calendering the dried layer. The process further comprises: analyzing the layer using the method of the present specification, analyzing the layer at any one or more of before drying the layer, after drying the layer, before calendering the dried layer, and after calendering the dried layer; adjusting process conditions for any one or more of coating the substrate with a layer, drying the coated layer, and calendering the dried layer.

[0052] According to another aspect, a sensor for analyzing a sample including a layer having a first interface and a second interface is provided, the sensor comprising: A pulse source of terahertz radiation adapted to generate a pulse of terahertz radiation, wherein the pulse includes a plurality of frequencies in the range of 0.01 THz to 10 THz, a pulse source, A focusing element configured to direct the generated pulse of terahertz radiation towards a sample using a first path and to direct the pulse of terahertz radiation towards an internal mirror using a second path, A detector for detecting the reflected radiation to generate a sample waveform, wherein the sample waveform includes the radiation reflected from the sample via the first path and the radiation reflected from the internal mirror via the second path, including a detector.

[0053] The sample may include one or more layers.

[0054] The first path may be referred to as a transmission path, and the second path may be referred to as a reflection path.

[0055] In the sample waveform, the radiation reflected via the first path is temporally separated from the radiation reflected via the second path. This makes it possible to detect the reflection from the internal mirror independently of the reflection from the sample, while also making it possible to obtain the reflected radiation from the internal mirror and the reflected radiation from the sample together (i.e., in the same measurement).

[0056] The arrangement of the focusing element and the internal mirror sets the second path. In use, the arrangement of the focusing element and the sample sets the first path.

[0057] The first path and the second path each refer to the optical path length.

[0058] The optical path length is the product of the geometric path length and the refractive index of the propagation medium.

[0059] The second path (reflection path) may be shorter than the first path (transmission path) such that radiation reflected from the internal mirror reaches the detector before the radiation reflected from the sample. This enables the reflected radiation from the internal mirror to be detected independently of the reflected radiation from the sample.

[0060] In one example, the geometric transmission (sample) path may be shorter, but the optical path length to the sample may be longer due to the high refractive index of the silicon lens. Since the reflection path does not pass through the silicon lens, it does not have such a difference.

[0061] In one example, the focusing element and the internal mirror are movable relative to each other such that the optical path length of the second path is adjustable. By adjusting the optical path length of the second path, it becomes possible to adjust the temporal separation between the reflected radiation from the first path and the reflected radiation from the second path.

[0062] In one example, the focusing element has a front surface and a back surface, wherein the front surface includes a convex surface and the back surface includes a flat surface, In use, the front surface faces the sample and the back surface faces away from the sample.

[0063] In use, the front surface may be closer to the sample than the back surface.

[0064] The purpose of the convex surface on the front is to focus the beam. For example, the convex surface defines the focal length of the lens.

[0065] The purpose of the flat surface on the back is to reflect the beam.

[0066] In one example, the normal vector of the back surface forms an angle with the optical axis defined by the front surface.

[0067] This angle is a non-zero angle. The surface normal vector of the plane of the focusing element is not aligned with the optical axis defined by the convex surface and the beam path to the sample focus. The normal vector of the plane is off-axis with respect to the optical axis.

[0068] The purpose of the angle between the normal vector of the back surface and the optical axis is to avoid interfering with the incident beam and the outgoing beam. The orientation of the plane deflects the reflected beam away from the optical axis and away from the terahertz unit.

[0069] The focusing element enables obtaining a reference signal from the internal mirror without introducing additional losses. The focusing element combines the beam splitting function with focusing into a single element.

[0070] In one example, the front surface of the focusing element is aspherical. The purpose of the aspherical configuration is to achieve an optimal focus.

[0071] In one example, the focusing element includes silicon.

[0072] In one example, the silicon is high-resistivity silicon.

[0073] In one example, the focusing element has an f-number such that the confocal parameter scaled by an estimated refractive index is greater than the estimated layer thickness.

[0074] In one example, the f-number is 3 or greater.

[0075] The focusing element has a focal length. The sensor may also have an aperture. The aperture and the focal length set the f-number of the focusing element.

[0076] According to another aspect, a system for analyzing a sample including a layer having a first interface and a second interface is provided, the system including a sensor and an analysis unit.

Brief Description of the Drawings

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Best Mode for Carrying Out the Invention

[0078] FIG. 1(a) shows a schematic diagram for analyzing sample 100 using a reflected beam. Sample 100 includes a layer (l) provided on a substrate (s). Sample 100 includes a second interface between the substrate and the layer, and a first interface between the outer surface (l) of the layer and air (a). Although the first interface is described as being between the outer surface and air, it will be understood that a vacuum or another gas environment may be used instead of air. For example, the gas environment includes nitrogen gas. Optionally, the sample is placed in a purge box to control the gas environment.

[0079] The sample is irradiated with terahertz radiation. At the first interface, a part of the incident terahertz beam is reflected. The characteristics of the reflection depend on the characteristics of the layer (l). The reflected beam is detected and analyzed. From the reflected beam, the refractive index n of the layer can be determined.

[0080] A part of the incident beam can be transmitted into the layer (l) across the first interface. This is indicated by the dashed arrow in FIG. 1(a).

[0081] Another example of measuring a sample using a reflected terahertz beam is provided in WO2018138523. In WO2018138523, a calibration sample is required to obtain an estimated value of a parameter.

[0082] Another example of measuring a sample using a reflected terahertz beam is provided in S. Krimi, J. Klier, J. Jonuscheit, G. von Freymann, R. Urbansky, R. Beigang, 2016, "High-precision thickness measurement of multilayer automotive paints using terahertz technology", Applied Physics Letters, 109(2), p. 021105. Separate calibrations are performed, and the thickness of the layer is estimated using these calibrations. The exact value of the complex refractive index is not given, and the density or conductivity is not determined either.

[0083] Another example of measuring a sample using a reflected terahertz beam is provided in U.S. Patent No. 10,076,261 B2. In U.S. Patent No. 10,076,261 B2, anomalies in the sample are detected along with an image of the layer thickness. The density, conductivity, and the real and imaginary parts of the refractive index are not measured.

[0084] WO2017051579A1 describes an example of measuring film thickness using a reflected terahertz wave. WO2017051579A1 does not describe the measurement of the complex refractive index. Information on density and conductivity is not determined.

[0085] FIG. 1(b) shows a schematic diagram of analyzing sample 100 using a transmitted beam. Sample 100 is the same as in FIG. 1(a). The sample is irradiated with terahertz radiation. At the first interface, a part of the incident terahertz beam is reflected, and another part of the beam is transmitted through layer (l) towards the second interface. At the second interface, the beam passes through the substrate and exits the sample. The beam exiting the sample is called the transmitted beam. The transmitted beam has traveled through air (a), layer (l), and the substrate. The characteristics of the transmitted beam depend on the characteristics of layer (l) and the substrate (s). The transmitted beam is detected and analyzed. From the transmitted beam, the combined effect of layer (l) and substrate (s) can be determined. For example, the combined effect of the refractive index and thickness of layer (l) and substrate (s) can be derived.

[0086] Further measurements have to be made to determine the refractive index or thickness of layer (l) or substrate (s). Further, if the substrate is lossy and / or thick, the transmitted beam is highly attenuated.

[0087] Figure 2 shows a schematic diagram of analyzing sample 200 by irradiating the sample with terahertz radiation and measuring the reflection. Sample 200 includes a layer (l) having a first interface and a second interface. The layer (l) has a thickness d. The first interface is formed between the outer surface of the layer and air. The first interface is described as an air-layer interface and is represented by the subscript "al". As described in connection with Fig. 1(a), instead of air, a vacuum or another gas environment may exist.

[0088] The second interface is formed between another surface of the layer (l) and the substrate (s). The second interface is on the opposite side of the first interface. The second interface is called the layer-substrate interface and is represented by the subscript "ls".

[0089] Sample 200 is irradiated with terahertz radiation (THz beam). At the first interface, a part of the incident terahertz beam R0 is reflected. The reflected beam R0 is also represented by r al r al may be called the reflection coefficient. r al represents what fraction of the incident beam is reflected. The reflected beam R0 is called the first reflection.

[0090] The characteristics of R0 depend on the layer (l). In particular, R0 depends on the refractive index n of the layer (l).

[0091] Another part of the THz beam is transmitted from the first interface towards the second interface. The ratio of the transmitted THz beam is represented by the transmission coefficient t al

[0092] At the second interface, a part (a small part) of the beam is reflected. The ratio of the beam reflected at the layer-substrate interface (the second interface) is represented by the reflection coefficient r ls The reflected beam travels through the layer (l) of thickness d until it reaches the layer-air interface. The layer-air interface corresponds to the first interface (air-layer interface). The transmission and / or reflection of the beam at the first interface depends on whether the beam is moving from air to the layer or from the layer to air.​

[0093] In the first interface, a part of the beam is transmitted through the sample. The transmitted part is represented by the transmission coefficient t la and is denoted as R1. The beam is represented by R1, which is called the second reflection.

[0094] The characteristics of R1 depend on the characteristics of the layer (l). In particular, R1 depends on the refractive index of the layer (l) and its thickness d. The real part of R1 is R1 = r ls t la t al is represented by exp(−iκ·ω·x / c), where i = √(−1), κ is the imaginary part of the refractive index, ω is the angular frequency, x represents the distance traveled by the beam in the layer (l), and c is the speed of light in vacuum. The imaginary part κ of the refractive index is related to the absorption coefficient.

[0095] The magnitude of the R1 reflection depends on the transmittance through the air-layer interface, the reflectance at the layer-substrate interface, and the absorption rate in the layer.

[0096] Optionally, this can be simplified since the transmittance through the air-layer interface is (1 + r al ), i.e., 1 + R0. As will be described later, R0 can be measured and the transmittance through the air-layer interface (t al = 1 + r al ) can be derived, thereby removing the dependence on t al .

[0097] Similarly, the transmittance t la at the layer-air interface may be derived from the measured R0 (= r al ), and thus the dependence on t la can be removed.

[0098] Optionally, it may be assumed that the air-substrate reflection (r as ) is fixed. For example, it may be assumed that the substrate is a metal substrate. Alternatively, the air-substrate reflection (r as) can be obtained by measurement on an uncoated substrate. The air - substrate reflection (r as ) can then be corrected to obtain the layer - substrate reflection (r ls ).

[0099] By performing the above simplification, the magnitude of the second peak R1 depends on the absorption in the layer. Thus, the measurement of the magnitude of R0, the delay between R0 and R1, and the magnitude of R1 enables the estimation of the optical properties and the thickness of the layer.

[0100] The reflection coefficient r al , the transmission coefficient t al , and the transmission coefficient t la depend on the refractive indices of air and the layer (l). The reflection coefficient r ls depends on the refractive indices of the layer (l) and the substrate (s).

[0101] The terahertz beam incident on the sample encounters the first interface, travels through the layer (l), and then encounters the second interface.

[0102] The first reflection R0 and the second reflection R1 are temporally separated. R0 and R1 undergo different optical delays. R0 and R1 can be measured using terahertz time - domain spectroscopy.

[0103] The optical delay between the R0 reflection and the R1 reflection is proportional to the refractive index and the sample thickness (~nd).

[0104] The R0 reflection and the R1 reflection can be temporally separated using terahertz time - domain spectroscopy. The R0 and R1 reflections may be obtained in a single measurement.

[0105] In the configuration of Figure 2, the reflections R0 and R1 can be obtained in a single measurement. The measurements of R0 and R1 provide simultaneously the information contained in the reflection measurement shown in Figure 1(a) and the information contained in the transmission measurement shown in Figure 1(b). From a single measurement, the thickness and the complex refractive index of the layer can be obtained.

[0106] The measurement is also a non-contact measurement. Such measurements may be performed on production equipment to monitor the properties of the layer during the manufacturing process.

[0107] Other properties such as conductivity or density can be derived from the thickness and complex refractive index of the layer. Conventionally, the optical properties (e.g., thickness) of the layer have been measured using laser interferometry, the conductivity has been measured using a four-point probe measurement, and the density has been measured using ultrasonic measurement or by weighing the sample and measuring the coating independently. These methods have the limitation of requiring calibration and multiple measurements (e.g., weighing and thickness separately to determine density) that can increase errors. Some of these measurement techniques (e.g., beta measurement of weight) are also slow and not compatible with high-speed production lines.

[0108] The embodiments described herein enable one or more of these properties to be determined by one measurement.

[0109] FIG. 3 shows a schematic diagram of the system 1 for analyzing the sample 2. The system has a sensor 3 and an analysis unit 5. The sensor 3 is connected to the analysis unit 5. The sensor comprises an emitter and a detector and is configured to emit and detect terahertz radiation. Examples of sensors are described in International Publication No. WO 2018 / 138523, which is incorporated herein by reference. Another example of a sensor is provided in U.S. Patent No. 10,076,261B2, which is incorporated herein by reference.

[0110] The analysis unit 5 is configured to process the information collected by the sensor 3. The analysis unit 5 will be further described below. The analysis unit may be adapted to implement any of the methods described herein.

[0111] Sensor 3 is configured to emit a broadband pulse of terahertz radiation 7 towards sample 2. The terahertz radiation pulse includes a plurality of frequencies. For example, the radiation is in the range of 0.01 THz to 10 THz. However, in some embodiments, depending on the transmission / absorption characteristics of the coating under study, the range is narrower, such as in the range of 0.06 THz to 4 THz, or in some cases within a lower frequency range.

[0112] Sensor 3 is further configured to detect the terahertz radiation 7 reflected from the sample.

[0113] System 1 can implement any of the methods described herein.

[0114] Optionally, sample 2 may correspond to sample 200 described in connection with FIG. 2.

[0115] Additionally or alternatively, sample 2 may correspond to a coating on an electrode for a battery. For example, sample 2 may correspond to a coating on the anode or cathode of a battery. Sample 2 includes a layer on a metal substrate. The metal substrate may be, for example, copper or aluminum. The metal substrate is coated with a layer of material. The coating includes a mixture of components. For example, in the case of an anode of a lithium-ion battery, the coating includes, for example, an active material, a conductive additive, and a binder. The coating may be porous.

[0116] At least one of the thickness, electrical conductivity (referred to as conductivity), or density of the coating may be monitored by system 1.

[0117] As described herein, the complex refractive index and thickness of the coating can be derived from the reflected terahertz beam.

[0118] The conductivity of the coating can be derived from the complex refractive index.

[0119] The relationship between the optical constant and the conductivity can also be approximated as follows.

[0120] The refractive index is

Equation

Equation

[0121] Next, the dielectric constant can be related to the conductivity through the following equation. ε = ε L + 4πiσ(ω) / ω Here, ω is the angular frequency, ε L is the dielectric constant due to the lattice, and σ(ω) is the frequency-dependent conductivity given by the following equation. σ(ω) = σ0 / (1 - iωτ) Substituting the free electron frequency-dependent conductivity into the above equation gives the following. ε = ε L + 4πiσ0 / ω / (1 - iωτ)

[0122] Note that the model used to fit n and k is a model of ε, which is the Drude model for free carriers. This can be used to fit n and k. Therefore, the conductivity model is incorporated into the refractive index model. The choice of model depends on the material (depending on what approximations are made to simplify the model).

[0123] For example, the imaginary part of the refractive index k of a material at terahertz is related to its high-frequency conductivity σ using the equation σ(ω) = 2.n.κε0ω.

[0124] The density can be determined from the real part of the refractive index. The density of the bulk material is proportional to the terahertz refractive index (see Journal of Pharmaceutical Sciences On-line DOI 10.1002 / jps.23560 (2013)), and thus the density can be determined from the measured real part of the refractive index using the correlation between the two.

[0125] The density (ρ) of the coating can be determined from the porosity.

[0126] The density ρ can be related to the refractive index n by the effective medium theory. Given a medium with refractive index n and assuming the medium is 5% porous, from the effective medium theory, the effective refractive index is the sum of two weighted by their volumes. That is, 0.95*n + 0.05*n0. Here, n0 represents the density of air in the porous material. For example, n0 = 1. This relationship assumes that the pores are much smaller than the wavelength of terahertz radiation.

[0127] Figure 4 shows a schematic diagram of a method for analyzing a sample according to one embodiment. The sample includes a layer having a first interface and a second interface. Optionally, the sample corresponds to sample 200 shown in FIG. 2. The method of FIG. 4 is implemented by the system 1 of FIG. 3.

[0128] In step S101, the reflection from the sample is acquired. The reflection may be obtained, for example, by the system 1 shown in FIG. 3.

[0129] In step S103, an initial estimated value of the parameters of the layer is obtained. The parameters may be, for example, the thickness of the layer and / or the complex refractive index. The real part n of the complex refractive index is sometimes called the refractive index. The imaginary part κ of the complex refractive index is sometimes called the absorption coefficient. How the initial estimated value is obtained will be further described below.

[0130] In step S105, the initial estimate is refined to produce a more accurate value of the layer thickness and / or complex refractive index. How the values of the parameters are refined will be further described below. Briefly, a synthetic signal is generated using the initial estimate, the synthetic signal is compared to the received reflection, and one or more parameters are adjusted to reduce the difference between the synthetic signal and the received reflection.

[0131] Step S105 may be referred to as optimization.

[0132] In step 113, an output is generated from the parameters (layer thickness and / or complex refractive index). The output includes at least one of layer thickness, conductivity, or density.

[0133] FIG. 5 shows a schematic diagram of obtaining an initial estimate of the layer thickness and / or complex refractive index. FIG. 5 shows step S103 of FIG. 4 in more detail. The method of FIG. 5 can be understood as a method of obtaining an initial estimate of the layer thickness and / or complex refractive index in the time domain.

[0134] In step S301, a reference signal is received. The reference signal represents the reflection from a reference sample. The reference signal may be a waveform (such as a reference waveform) or a value (a reference value). For example, the reference sample includes an uncoated sample. The uncoated sample is similar to sample 200 of FIG. 2 except that layer (l) is absent. In other words, the uncoated sample corresponds to the substrate. The reference sample may be, for example, a sample before coating with layer (l). The reference signal is related to the terahertz radiation reflected by the reference sample when irradiated with an incident terahertz beam. The reference signal includes a reflection waveform indicating the intensity of the reflected beam as a function of time delay. When the terahertz beam is incident on the uncoated substrate, the reflected beam is related to the characteristics of the substrate. The substrate may be a reflective metal substrate. The reflected beam represents the reflectivity of the substrate.

[0135] In one example, a reference value is derived from a reference waveform. For example, the reference value may be the magnitude of a peak resulting from reflection from the substrate. The reference value may correspond to peak S R described in connection with FIG. 6(a).

[0136] In step S303, a signal from the sample is received. The signal from the sample is related to terahertz radiation reflected by the sample when irradiated with an incident terahertz beam. The reflected signal includes a reflection waveform showing the intensity of the reflected beam as a function of time delay. The received signal is used to generate a sample waveform. The sample waveform is further described herein.

[0137] In step S305, the sample waveform is compared with the reference value.

[0138] FIG. 6(a) shows a plot of signals obtained from a reference sample and a sample including a layer. The sample having this layer corresponds to sample 200.

[0139] The vertical axis represents the magnitude of the signal in arbitrary units. The horizontal axis indicates the optical delay in picoseconds (ps). The signal from the reference sample is shown in blue (dashed line), and the signal from the sample including the layer is shown in orange (solid line). The signal from the reference sample has a single peak (S R ) resulting from reflection from the substrate. The signal from the sample having the layer includes two peaks. The first peak S L corresponds to reflection from the surface of the layer (l). The second peak S S corresponds to reflection from the substrate.

[0140] In FIG. 6(a), the difference between the reflectance from the layer surface S L (the first peak) and the reflection from the reference sample S R is indicated by ΔR. The time delay between the reflection from the layer surface (S L ) and the reflection from the substrate (S S ) is indicated by Δt. The reflectance from the layer surface S L (the first peak) and the substrate surface SS The difference from the reflection from S (the second peak) is indicated by ΔA.

[0141] Returning to FIG. 5, in step S307, ΔR is determined. ΔR is obtained by comparing the magnitudes of S L and S R For example, peaks may be identified and their magnitudes determined. ΔR provides an approximation of the real part of the layer refractive index. ΔR is related to the ratio of S L / S R = 1 - ΔR / S R As such, S L versus S R is related to the ratio. The ratio of S L and S R can be represented as "r" (S L / S R = r) for simplicity. The relationship between r and n is obtained from Fresnel's equation. For example, for an interface between air (refractive index 1) and a material (refractive index n), the reflection r is given by r = (1 - n) / (1 + n). This can be rearranged to give n = (1 - r) / (1 + r). The reflection r is measured, and then n can be calculated from r. This expression is for normal incidence, but can also be adapted for non-normal incidence.

[0142] In S313, an estimated value of the real part of the refractive index is obtained from the ratio of S L and S R The ratio of S L and S R is also called the first ratio. S R corresponds to the reference value in step S305.

[0143] In step S309, the time delay Δt between the first peak and the second peak is acquired. The reflection from the layer surface (S L ) and the reflection from the substrate (S S) The time delay between them, i.e., (Δt), is a measure of the time (doubled) required to propagate through the layer. This time is proportional to the layer thickness, the speed of light, and the refractive index. Since the speed of light is known and an approximation of the refractive index is known from S313, an initial estimate of the layer thickness can be obtained in step S315.

[0144] The thickness of the layer can be estimated as follows. d ~ Δt / n

[0145] The thickness of the layer can be obtained from d = cΔt / (2n), where c is the speed of light and the factor of 2 is obtained from the fact that the beam passes through the layer twice.

[0146] In step S311, a second ratio is obtained by comparing the magnitudes of S S and S L As shown in FIG. 6(a), the peaks at S S and S L can differ by an amount ΔA. ΔA is obtained by comparing the magnitudes of S L and S S Since S L is associated with S R from the first ratio, the second ratio can alternatively be obtained by comparing the magnitude of S S with S R . In S317, an estimate of the absorption of layer (l) can be determined from the second ratio and the estimated value of the layer thickness from S315. The absorption of layer (l) corresponds to the imaginary part of the refractive index κ.

[0147] Propagation through a layer of thickness d is

Equation

Equation

[0148] Referring to the plot shown in FIG. 6(a), the plot corresponds to the reflected waveform from the sample (sample waveform) and the reflected waveform from the reference sample (reference waveform). The reflected waveform is the raw signal. The sample waveform includes the contribution of the instrument and the contribution of the sample. The reference waveform includes the contribution of the instrument and the contribution of the reference sample.

[0149] The contribution of the instrument is sometimes referred to as the instrument response. The instrument response is a measured value of the contribution of the sensor components used to measure the reflected beam. Optionally, the instrument response is removed from the sample waveform. The sample waveform from which the instrument response has been removed may be called the sample response. The sample response can be obtained in several ways. For example, the sample response is derived from the reflected waveform by deconvolving the reflected waveform with the reference waveform. Alternatively, the sample response can be derived by subtracting the reference waveform from the sample waveform.

[0150] The instrument response can be derived from a mirror. The mirror gives a reflection of ~1. The reference waveform is determined by detecting the THz beam reflected from the mirror. The mirror may be a planar mirror. The mirror may be, for example, a gold mirror. The planar mirror is provided at the focus of sensor 3 such that the terahertz beam is reflected back to the sensor. The terahertz beam is reflected from the reference surface and detected by system 1. The reflected beam describes the instrument response. The determination of the instrument response is further described in WO2018138523.

[0151] The response from the mirror can be used in two ways. First, if the substrate is known to be reflective, the reflectivity of the substrate can be considered the same as that of the mirror. Next, assume that the reference waveform of the substrate corresponds to that of the mirror. Next, an uncoated substrate is measured. The uncoated substrate is similar to sample 200 of FIG. 2 except that layer (l) is not provided. The reflection from the uncoated substrate is measured, compared with the reflection from the mirror, and the reflectivity of the substrate can be derived.

[0152] Alternatively, instead of using a plane mirror, an uncoated substrate is used to derive the device response. The uncoated substrate may include a metal substrate (which is reflective). When a terahertz beam is incident on the uncoated substrate, the reflected beam is related to the characteristics of the substrate including its reflectivity. The reflected beam also describes the device response. A reference waveform can be derived from the reflected beam. The response of the mirror can also be measured. The reference waveform may be further multiplied by (mirror) / (substrate) to convert the substrate measurement to the mirror measurement. The reference waveform may correspond to the reference signal mentioned in steps S301 and S305 of FIG. 5.

[0153] If the substrate is highly reflective, the device response can be obtained using an uncoated substrate, avoiding the use of an additional mirror.

[0154] Additionally, and optionally, the reflectivity of the substrate is obtained before measuring the sample. The reflectivity may be measured on the substrate before deposition of the layer. Alternatively, if the substrate has a consistent reflectivity value, the value may be measured once and stored for use. Further alternatively, a value of the reflectivity may be assumed. The reflectivity of the substrate is related to the received reference signal described in relation to step S301 and FIG. 5. The reflectivity of the substrate is also related to coefficient r described in relation to FIG. 2. ls is also related.

[0155] Note that reflections from the entire structure tend to be reflections from the substrate at low frequencies, so there is a useful check of the substrate reflectivity.

[0156] Figure 6(a) shows the reflection waveform of the THz beam in the time domain. The time domain representation shows that different reflections from the sample reach the detector at different times. The time domain representation also shows the difference in the peaks (S R , S S , S L ) of the reflected signal.

[0157] As described in connection with Figure 5, Δt, S R , S S , and S L are obtained from the plot shown in Figure 6(a) and can be used to determine the layer thickness and / or complex refractive index.

[0158] The method of Figure 5 involves analyzing the waveform shown in Figure 6(a) in the time domain to obtain an estimate. The method of Figure 5 is based on determining the peak height and the time separation between the peaks. The advantage of this method is that it is simpler (requiring less processing) and faster. Such a method is useful for industrial monitoring applications.

[0159] Optionally, the accuracy of the method of Figure 5 is improved by filtering. For dispersive materials, the peak shape changes and the peak separation / height may no longer be separable. The method of Figure 5 can optionally include a frequency filtering step for selecting a narrow range of frequencies. This can reduce the effect of the dispersive material and improve the accuracy of the approximation. The filtering step is 1) the step of acquiring the sample waveform and the reference waveform, and 2) the step of converting the sample waveform and the reference waveform into the frequency domain (e.g., using FFT) to generate the sample spectrum and the reference spectrum, and 3) the step of obtaining the sample / reference ratio and performing deconvolution, and 4) multiplying by a frequency selection filter (an example of the filter is a band-pass); 5) performing an inverse transform to obtain a filtered waveform; 6) analyzing the peak of the filtered waveform.

[0160] For example, analyzing the peak of the filtered waveform includes performing steps S313, S309, S315, and S317 in FIG. 5.

[0161] Note that this peak analysis method does not give a direct measure of conductivity, but gives a measure of the complex refractive index at a single frequency. The measured value of the complex refractive index can be calibrated to provide conductivity.

[0162] The determination of thickness and / or complex refractive index in the frequency domain is described below in connection with FIGS. 6(b) to 6(i). The frequency dependence of the complex refractive index can be incorporated into the initial estimate by considering the frequency response of the reflection waveform.

[0163] FIG. 6(b) shows a schematic diagram of time-gating a time-domain signal. FIG. 6(b) also shows the conversion of the time-gated time-domain signal to the frequency domain.

[0164] Time-gating

[0165] With reference to the reflection waveform (solid line) of the sample shown in FIG. 6(b), time-gating will be described. In time-gating, a first window w1 is applied to the waveform to select only the first reflection S L only. A second window w2 is applied to the waveform to select only the second reflection S S only.

[0166] The purpose of time-gating is to select segments of the waveform in the time domain. In other words, time-gating can be used to separate segments of the waveform.

[0167] In a non-limiting example, the first and / or second window may be a boxcar function.

[0168] The window can be defined by identifying the peak and then selecting an appropriate window width.

[0169] Non-limiting examples of defining the window are as follows. 1) Find the time t0 of the R0 peak. 2) Find the time t1 of the R1 peak. 3) Predetermine the peak width w. 4) The window width is given by t1 - t0 - w. 5) The R0 window is centered on the R0 peak. 6) The R1 window is centered on the R1 peak. The window regions may overlap. However, the window function is weighted towards the center of the window. In this example, both windows have the same width. However, different widths may be used.

[0170] Note that other window functions may be used instead of the boxcar function.

[0171] Time gating generates a first reflected waveform represented by w1 in FIG. 6(b) representing a first reflection S L and a separate second reflected waveform represented by w2 in FIG. 6(b) representing a second reflection S S as shown in FIG. 6(b).

[0172] Frequency domain

[0173] As shown in FIG. 6(b), the time-gated reflected waveforms w1 and w2 are separately converted to the frequency domain. For example, to convert the time-gated waveforms to the frequency domain, a fast Fourier transform (FFT) is performed on each of the first and second reflected waveforms. Two frequency domain reflection spectra are obtained. The first spectrum is the first reflection S in the frequency domain Lis described. The second spectrum is the second reflection S in the frequency domain S is described. The first and second spectra are the first reflection S L and the second reflection S S respectively, including magnitude and phase as functions of frequency.

[0174] The first spectrum may be called R0. The second spectrum may be called R1. R0 and R1 are frequency-dependent.

[0175] In an alternative example (not shown), time gating is performed as follows. - Obtain a time-domain waveform. - Convert to the frequency domain. - Process in the frequency domain. The processing may include any one or more of filtering, deconvolution, or correction. - Convert to the time domain. - Apply a window for time gating as described above.

[0176] The first and second spectra may then be used to derive initial estimates of the complex refractive index and / or thickness. This derivation is similar to the derivation described in connection with FIG. 5, except that the ratio, and thus the complex refractive index, is frequency-dependent.

[0177] Next, an explanation of how initial estimates of the complex refractive index and thickness are obtained by converting a time-domain waveform to a spectrum is provided. The processing in the frequency domain is presented in connection with the ratios presented in FIG. 5 (relating to processing in the time domain) for ease of understanding. Further below, how initial estimates of the complex refractive index and thickness are obtained will be described in more detail below in connection with the figures of FIGS. 6(c) to 6(i).

[0178] To estimate the first ratio, the magnitude of the first spectrum is obtained and compared with the received reference signal. The relationship between the refractive index and the ratio is obtained as r(ω) = (1 - n(ω)) / (1 + n(ω)).

[0179] When comparing the first spectrum with the reference, the reference waveform is converted to generate the reference spectrum. r(ω) corresponds to the first spectrum / reference spectrum.

[0180] The ratio of the sample spectrum / reference spectrum is independent of the device and contains only information about the sample.

[0181] To estimate the time delay, the phase difference between two spectra is obtained. The phase shift is obtained by subtracting the phase of the reference spectrum from the second spectrum. The ratio R1 / Reference in the frequency domain (i.e., the ratio spectrum) performs the phase subtraction (complex number property). Thus, the phase in the ratio spectrum provides the delay. The delay can be related to the imaginary part of the exponential term exp(-i.n.ω.d / c). Due to the coefficient i, this is the real part of the refractive index, i.e., the phase shift = -n·ω·d / c, and (since n is known), d can be calculated.

[0182] To estimate the second ratio, the magnitude of the reference spectrum (Ref) is taken and compared with the magnitude of the second spectrum (R1). Refer to Figure 2. R1 / Ref is given by t 12 ×t 21 ×r ls ×exp(), where t 12 and t 21 are the transmittances through the interface and are equal to (1 - r0) and -(1 - r0), respectively. The subscript 1 refers to air and the subscript 2 refers to the layer (l). Since R0 is measured, these can be corrected. The substrate reflection is calculated by prior measurement or a known value is assumed. When these terms are removed, only the real part of the exponential function (attenuation) related to k remains. This relationship is R1 / Ref = t 12 ×t 21 ×rls can be expressed as ×exp(ωκd / c).

[0183] The estimated parameters n, d, and κ are frequency-dependent.

[0184] Although the estimated d is frequency-dependent, a weighted average of the values of the frequency-dependent d can be taken and used as the estimated value of d. For example, d(ω) can be averaged over the values of ω when the measured data is of high quality (e.g., low noise).

[0185] By capturing the frequency dependencies of n, d, and κ, more accurate initial estimates can be obtained.

[0186] Complex refractive index

Number

[0187] Furthermore, optionally, the frequency dependencies of n, d, and κ can indicate which type of model to use in subsequent refinement steps. For example, the complex refractive index

Number

[0188] Additionally and optionally, the first spectrum and / or the second spectrum are filtered prior to any derivation of parameters therefrom. For example, a filter (e.g., a low-pass or band-pass) may be applied to the first and / or second spectrum to reduce noise. The filtered spectrum can then be transformed back to the time domain using an inverse FFT. The noise in the time domain signal can be reduced. The filtered time domain signal (or the filtered spectrum) is then used in the initial estimation of the parameters.

[0189] Additionally and optionally, a filter may be applied to emphasize the frequencies most sensitive to the measured parameters and attenuate other frequencies. An example of such a filter is a band-pass filter. The filtered spectrum is transformed back to the time domain. The time domain band-pass signal is then used in step S105. The parameters can be derived with higher accuracy.

[0190] In step S105, the entire waveform is used in the fitting / optimization stage. The purpose of separating the waveform into R0 and R1 is to enable the analytical solution of the initial estimated values. In step S105, the sample spectrum / reference spectrum is considered. While floating the fitted refractive index parameters and thickness, the expected reflection is modeled. This is optimized when (synthetic signal - sample waveform) is minimized. The optimization can be performed in the frequency space or the time domain.

[0191] When fitting to the frequency space, the error between the composite spectrum (the composite spectrum refers to the frequency domain form of the composite signal) and the sample spectrum (the conversion of the sample waveform to the frequency domain) is reduced. The spectrum can be optimized as it saves processing time.

[0192] Additionally and optionally, when fitting in the frequency space, multiple fittings with different frequency ranges may allow for improved parameter optimization. When fitting to the spectrum, the error function is (composite spectrum - sample spectrum) and is a function of frequency. This error function may be weighted to define which part of the spectrum is most important. For example, parts of the spectrum where the signal contains noise and is given a low weighting.

[0193] Furthermore, by way of non-limiting example, assuming that a part of the spectrum is sensitive to thickness but not to other parameters, initial fitting can be performed by using only this part of the spectrum and varying the thickness while keeping other parameters fixed. In other words, the error spectrum is weighted and at least one of the thickness and the complex refractive index is varied so as to reduce the weighted error spectrum. The advantage is that the error can be reduced more effectively (e.g., faster, more accurately).

[0194] Note that a part of the time domain signal may be selected by time gating to remove the influence of additional reflections not included in the model.

[0195] Figures 6(c) to 6(i) will be described. These figures are also relevant to obtaining initial estimates of the complex refractive index and thickness. The figures show how the method can be implemented. The above description regarding the processing of reflections in the frequency domain also applies to Figures 6(c) to (i).

[0196] Figure 6(c) shows a schematic diagram for estimating the real part of the refractive index.

[0197] In step S61, a reference waveform is acquired. In S63, the reference waveform is time-gated to select the peak shown as S in Fig. 6(a). The peak may be called the zero-order reflection. In S63b, the time-gated reference waveform is converted to the frequency domain to generate a reference spectrum (Ref). R In step S62, a sample waveform is acquired. In S64, the sample waveform is time-gated to select the peak shown as S in Fig. 6(a). The peak may be called the zero-order reflection. In S64b, the time-gated reference waveform is converted to the frequency domain to generate a sample spectrum (R0).

[0198] In step S66, R0 / Ref is obtained. R0 / Ref corresponds to the first ratio described above. R0 / Ref may be called r(ω). R0 / Ref is equal to r L (or r

[0199] (or r al (or r 12 ). The spectrum may be called the first spectrum.

[0200] In step S68, an estimated value of the real part of the refractive index n is obtained. n is frequency-dependent n(ω). The relationship between the refractive index and the ratio is obtained as r(ω) = (1 - n(ω)) / (1 + n(ω)) from Fresnel reflection.

[0201] Fig. 6(d) shows a schematic diagram for estimating the real part of the refractive index.

[0202] In step S71, a reference waveform is acquired. In step S73, the reference waveform is converted to the frequency domain by performing an FFT to generate a reference spectrum.

[0203] In step S72, a sample waveform is acquired. In S74, the sample waveform is converted to the frequency domain by performing an FFT to generate a sample spectrum.

[0204] In step S76, the sample is deconvolved from the reference. The sample spectrum is divided by the reference spectrum. In step S77, by performing an inverse FFT, (sample spectrum / reference spectrum) is converted to the time domain to generate a deconvolved waveform. An example of the deconvolved waveform is shown in FIG. 6(e). FIG. 6(e) shows the output of step S77. Note that the trace in FIG. 6(e) appears upside down with respect to the trace in FIG. 6(a) because the reflection between the low refractive index and the high refractive index is negative. In FIG. 6(e), the baseline is flatter compared to FIG. 6(a), and three separated peaks can be seen.

[0205] Returning to FIG. 6(d), in step S79, the deconvolved waveform is time-gated to select the zero-order reflection R0. Referring to FIG. 6(e), the zero-order reflection corresponds to the first peak and corresponds to the reflection from the surface of the layer.

[0206] In step S81, an FFT is performed to obtain a spectrum. The spectrum of S81 corresponds to the first spectrum and is related to the zero-order reflection R0 from the layer.

[0207] In step S83, the refractive index n(ω) is estimated from the spectrum of S81 using the relationship r(ω) = (1 - n(ω)) / (1 + n(ω)). Note that the spectrum of S81 here corresponds to r(ω). It should be noted that this relationship is a simplification used for normal incidence. The complete formula is the Fresnel reflection formula.

[0208] FIG. 6(f) shows a schematic diagram of estimating the reflection spectrum. In particular, the reflection spectrum is related to the reflection from the layer-substrate interface (peak S in FIG. 6(a)) S )

[0209] In step S90, a deconvolved waveform is obtained. The deconvolved waveform may be, for example, the waveform shown in FIG. 6(e).

[0210] In step S92, the waveform is time-gated to select the first reflection R1. R1 corresponds to the second peak shown in FIG. 6(e). R1 corresponds to the peak S shown in FIG. 6(a). S corresponds to.

[0211] In step S94, the spectrum of the time-gated waveform is acquired. This spectrum may be referred to as the R1 reflection spectrum. The said spectrum may also be referred to as the second spectrum.

[0212] FIG. 6(g) shows a schematic diagram for estimating the reflection spectrum. In particular, the reflection spectrum is related to the reflection from the layer substrate interface (peak S in FIG. 6(a)). S ) is related to.

[0213] In S400, an estimated value of the real part of the refractive index is obtained. The estimated value can be acquired, for example, using the method of FIG. 6(c) or (d).

[0214] In S402, a reference waveform is acquired, and in S405, a reference spectrum (Ref) is acquired from the reference waveform.

[0215] In S407, the zero-order reflection spectrum Ref is obtained using the reference spectrum and the estimated value of the real part of the refractive index. This calculation is based on the Fresnel relationship. For example, using the above formula r(ω)=(1 - n(ω)) / (1 + n(ω)), the reference zero-order reflection spectrum can be obtained. From the real part of the refractive index, the reflectance from the air layer interface can be calculated as r 12 =(1 - n) / (1 + n). The reference signal Ref is multiplied by r 12By multiplying, R0 is obtained. This is from R0 / Ref = r12 → R0 = Ref * r12. This is the calculated version of the zero - order reflection from the layer (i.e., the reflection from the surface of the layer). This is independent of time gating and thus applies to all times (unlike the time - gated measured R0 described in relation to FIG. 6(b)). Therefore, by subtracting the said R0 (converted to the time domain as in S408) from the sample waveform, the "slow" component of R0 that appears simultaneously with R1 is removed.

[0216] In step S408, an inverse FFT is performed to obtain a time - domain signal.

[0217] In step S404, a sample waveform is acquired. The sample waveform includes S S (first - order) reflection and S L (zero - order) reflection. In step S410, the signal from S408 is subtracted from the sample waveform.

[0218] In step S412, the signal from S410 is time - gated to select the first - order reflection R1 corresponding to the peak S S . In step S414, an FFT is performed to obtain the spectrum of R1.

[0219] In step S416, the ratio of R1 to the zero - order reference spectrum (Ref) is taken. The resulting ratio is also called the reflection spectrum of R1 and is represented as R1 / Ref.

[0220] FIG. 6(h) shows a schematic diagram for estimating the thickness of the layer.

[0221] In step S500, an R1 reflection spectrum is acquired. The R1 reflection spectrum can be derived, for example, using the method of FIG. 6(f) or FIG. 6(g). The R1 reflection spectrum means the ratio of R1 / Ref.

[0222] In step S502, the layer - surface reflection S LCorrect the reflection spectrum. In step S504, the correction of the layer substrate reflection S S is applied. The correction is as follows. ·R1 / Ref = t 12 t 21 r ls exp() is measured. · Since it is desirable that I / I0 = exp(), in step S502, the correction of the surface reflection is R1 / Ref * (1 / t 12 / t 21 ) is applied. Here, I / I0 refers to the ratio of (actual signal / signal without absorption). · Next, in step S504, the correction for the substrate is applied by multiplying by 1 / r ls (×1 / r ls ), where r ls is the reflection at the layer - substrate interface. · Therefore, I / I0 is obtained and the absorption can be calculated. · Transmittance t 12 = 1 - r 12 and t 21 =(r 12 - 1). Since r 12 is measured, r 12 is known. · Note that r ls is obtained as described in this specification.

[0223] In step S506, the real and imaginary parts of the corrected spectrum are taken.

[0224] Propagation through a layer of thickness d is

Equation

[0225] In S509, the absorption and the imaginary part of the refractive index κ are obtained from the real part S508.

[0226] In S512, the phase shift is calculated from the imaginary part S510, and the thickness is obtained from the phase shift and the estimated value S511 of the real part of the refractive index. The thickness is frequency-dependent.

[0227] In S513, a weighted average of the frequency-dependent thicknesses is taken to produce a single thickness. For example, d(ω) may be averaged over the values of ω when the measured data is of high quality (e.g., low noise).

[0228] FIG. 6(i) shows a schematic diagram of obtaining a layer model from the estimated complex refractive index. The estimated values of the real and imaginary parts of the complex refractive index can be determined as described above. The estimated values of the complex refractive index model are then fitted to the model. The model means the refractive index model of the layer. The refractive index model is used in subsequent optimization steps.

[0229] Complex refractive index

Number

[0230] Referring to FIG. 4, steps S101 and S103 relate to obtaining an initial estimated value of the layer parameters. Step S105 relates to optimizing the value of the estimated value. Step S106 relates to outputting the characteristics of the layer based on the refined value of the parameter.

[0231] FIG. 7 shows a schematic diagram of a method for analyzing a sample according to an embodiment. FIG. 7 shows step S105 of FIG. 4 in more detail. Step 105 is an iterative procedure executed to obtain a refined estimate of the parameter.

[0232] For example, step 105 is a least squares fitting procedure.

[0233] In step S106, using the initial estimated values of the parameters (n, κ, d) from S103, a reflection waveform is synthesized. To synthesize the reflection waveform, a model of the sample is assumed. For example, the model is formulated for a single layer on a metallic (reflective) substrate.

[0234] For example, to obtain the initial estimated values, the model is formulated for a single layer on a metallic (reflective) substrate. In the case of a metallic substrate, there is no penetration into the substrate.

[0235] The layer may be assumed to have a spatially uniform refractive index and thickness. For such a layer on a metallic substrate, the reflection may be calculated using the matrix form of the Fresnel equations.

[0236] The parameterized form of the complex refractive index is used in the iterative fitting procedure. For example, at least one of n and κ is a varying parameter. By using the parameterized form of the complex refractive index, the number of degrees of freedom is reduced and the fitting is simplified.

[0237] Optionally, the layer thickness d is another parameter that varies during the fitting procedure.

[0238] Optionally, when the complex refractive index estimated in S103 is frequency-dependent, the model is also frequency-dependent and the frequency-dependent form of the complex refractive index is used. Additionally, and optionally, the frequency-dependence of the complex refractive index is identified in S103 and used to select the refractive index model used in the fitting procedure.

[0239] In step S107, the synthetic signal is compared with the reflection received from the sample to generate an error. The error may be, for example, the mean squared error. The error represents the difference between the synthetic signal and the measured signal. Note that the comparison may be performed in the time domain or the frequency domain. When comparing in the frequency domain, the measured signal (sample waveform) may be converted to the frequency domain to generate a sample spectrum, and the synthetic signal is a frequency domain signal.

[0240] Optionally, the waveforms can be temporally aligned so that the reflection features (peaks and troughs) occur at the same time points, avoiding blurring of the reflection features. Temporal alignment is performed before fitting. For example, temporal alignment is performed before generating the error.

[0241] In step S109, it is determined whether a predetermined condition is satisfied. If the condition is satisfied, the iterative procedure ends and the method proceeds to step S113. The values of the parameters are provided to the output step S113.

[0242] If the condition is not satisfied, the method proceeds to step S111, the estimated values of the parameters (n, κ, d) are corrected, and the method returns to step S106. The range of values over which the parameters are swept is predetermined. For example, the range may be determined by performing a pre-measurement using a known sample.

[0243] In step S113, the output is generated from refined parameters (layer thickness and / or complex refractive index). The output includes at least one of layer thickness, conductivity, or density. The thickness, conductivity, or density is obtained from the layer parameters described herein.

[0244] Regarding steps S106 and S107, it should be noted that the iterative fitting procedure is performed in the time domain (i.e., using the received time domain reflection waveform and the synthesized time domain reflection waveform). Time domain fitting provides improved accuracy when the layer thickness is important. This is because the layer thickness appears in the separation of the positions of the first peak (S L ) and the second peak (S S ).

[0245] Alternatively, instead of the time domain formulation, the frequency response of the received reflection may be used. For example, an FFT may be performed on the reflection waveform received in S101 to obtain the frequency response. In step S106, the frequency response may be calculated using matrix form. In step S107, the difference between the frequency responses is obtained.

[0246] Figure 8(a) shows a plot of the reflection waveform. The plot shows the reflection of the substrate divided by a reference. Here, the substrate corresponds to an uncoated substrate. The plot does not show features other than surface reflection.

[0247] It should be noted that this is different from the reference trace of FIG. 6(a) which was the raw waveform.

[0248] Figure 8(b) shows a plot of the reflection waveform obtained from the cathode. The cathode includes a metal substrate and a coating. Figure 8(b) shows two traces, one for a thin coating (red, solid line) and one for a thick coating (blue, dashed line). Each trace has two reflection features, one for surface reflection S L for use, and one for coated substrate reflection S SThis is the case. In the case of a thin coating, the reflection features (peaks) are separated by a smaller delay than in the case of a thick coating.

[0249] Figure 8(c) shows a plot of the reflection waveforms obtained from the anode. The anode includes a metal substrate and a coating. Figure 8(c) shows two traces, one with a thin coating (black, solid line) and one with a thick coating (blue, dashed line). In the trace with the thin coating, the peak of the surface reflection S L is evident. The feature corresponding to the anode - metal interface S S is also visible (marked by the arrow). For the thick coating, the peak of the surface reflection S L is again evident, while the feature corresponding to the anode - metal interface S S is present (marked by the arrow), but is, for example, more subtle than in Figure 8(b). Further, in the thick anode, a further third structure (shown by the dashed circle) is distinguishable. The further structure is thought to indicate an additional layer or other interface.

[0250] Measuring a layer that is opaque due to strong absorption or other effects is difficult because less signal penetrates the film, is reflected, and returns to the detector.

[0251] The inventors of the present invention configured the sensor 3 of the system 1 to have an appropriate depth of focus that enables detection of reflections and to enable estimation of the layer parameters. The depth of focus is configured such that the beam reflected from the coated substrate reflection S S is detectable.

[0252] In a film having a high refractive index (e.g., due to high conductivity or other causes), the change in beam divergence is affected away from the focus of the beam by an amount different from that expected for a low refractive index material (n≈1).

[0253] Figure 9(a) shows a schematic of a beam focused on the surface of the substrate when no sample is present.

[0254] Figure 9(b) shows a schematic diagram of the change in the focus of the beam in Figure 9(a) when a sample is introduced. The beam without the sample is indicated by the gray dashed line (-·-). The beam with the introduced layer is shown by the solid line in Figure 9(b). The layer has a high refractive index, and the beam (solid line) shows that the focus has moved laterally away from the outer (upper) surface of the layer.

[0255] Materials with a refractive index > 1 shift the focus position. This is particularly serious in materials with high conductivity (e.g., conductive coatings used for battery electrodes). For example, the anode has a terahertz refractive index > about 6. A higher refractive index moves the focus away from the front surface of the layer being inspected and laterally from the nominal focus position. This has the effect of moving the focus and effectively taking the system out of focus and alignment. The higher the refractive index, the greater the focus shift and the greater the misalignment.

[0256] Figure 9(c) shows a schematic diagram of the beam width w(z) as a function of the distance z. Figure 9(c) shows an example of a Gaussian beam. The beam waist w0 is the beam size at its focus (i.e., z = 0).

[0257] The beam width changes as w(z)=w0[1+(z / z R ) 2 1 / 2 where z R is the Rayleigh range, and z R =πw0 2 n / λ, where n is the refractive index and λ is the wavelength of light. b is the confocal parameter, and b = 2z R . The confocal parameter and the Rayleigh range are related to the depth of focus. Points within the confocal parameter can be considered to be in focus. Points within the confocal parameter can be resolved by the optical element.

[0258] ​The beam waist w0 is related to the lateral resolution. A small spot size (small w0) is used to increase the lateral resolution (in the horizontal plane). A small spot size means a corresponding decrease in the depth of focus z R is meant.

[0259] Conventionally, in terahertz imaging, a small w0 (and thus a small depth of focus) has been used.

[0260] The inventors recognized that by using a longer depth of focus, it should be possible to focus on two interfaces of a layer on a substrate. A lens with a longer focal length for the same aperture has a larger depth of focus. Such an arrangement has proven to be the key for maintaining focus on the sample interface and detecting reflections from both interfaces in high refractive index layers such as the coatings used on the cathodes and anodes of lithium ion and other types of batteries.

[0261] In high-resolution imaging, a lens with a short focal length lens having a large aperture is used. This gives a small spot size but a small depth of focus ~2.44fλ / a.

[0262] Here, f is the focal length and a is the limiting aperture of the optical system. The ratio f / a is known as the f-number. For a fixed aperture a, as the focal length f increases, the f-number becomes longer and the depth of focus becomes larger.

[0263] To enable focusing on two interfaces of a layer on a substrate, the inventors configured the optical system to have an f-number such that the focus remains on the two interfaces. In other words, the f-number is adapted such that the confocal parameter is greater than or equal to the thickness d of the layer (l).

[0264] In one embodiment, the f-number is greater than 3.

[0265] In another embodiment, the f-number is greater than 4. In another embodiment, the f-number is greater than 5. In another embodiment, the f-number is greater than 6. In another embodiment, the f-number is greater than 7. In another embodiment, the f-number is greater than 8. In another embodiment, the f-number is greater than 9. In another embodiment, the f-number is greater than 10.

[0266] Figure 10(a) shows a schematic diagram using an optical system with a large f-number. Here, the f-number is 10 (sometimes denoted as f / 10). The sample is the anode described in FIG. 8(c). When using an f / 10 lens, the reflection from the surface of the coating and the reflection from the coating-substrate interface are resolved and appear as characteristics of the reflection waveform in the traces for both thick and thin samples. Figure 10(a) shows the reflection waveform obtained using a slower optical system (f-number = 10), and the depth of focus is larger along the optical axis (z). The reflection SS can be identified, and the film thickness can be estimated.

[0267] Figure 10(b) shows a schematic diagram using an optical system with a small f-number. Here, the f-number is 3 (sometimes denoted as f / 3). The sample is the same as in Figure 10(a). When using an f / 3 lens, the reflection profiles of the thin sample and the thin sample look similar. Both show the reflection from the surface of the coating. However, the characteristics from the reflection at the coating-substrate interface are not resolved and do not appear in the reflection waveform.

[0268] In Figure 10(b), the optical system includes a faster optical component (f-number = 3) and correspondingly a smaller depth of focus on the optical axis, and the reflection at the coating-substrate interface is not resolved.

[0269] FIG. 11(a) shows a schematic diagram of the internal configuration of the terahertz sensor 3 for measuring the sample 51. The sample 51 corresponds, for example, to the sample 200 described in this specification. The sample is provided at the focus of the terahertz sensor. The terahertz sensor includes a unit 53 that emits and detects terahertz radiation. The radiation when exiting the terahertz unit is focused by the optical element 54 onto the focal plane 57 near the sample 51. The optical element 54 includes a focusing element 55 and an aperture 56. The focusing element 55 may be a mirror or a lens. The focusing element 55 has a focal length f. The aperture 56 is also provided. The aperture 56 has an aperture size a, and both the focusing element 55 and the aperture 56 define the f-number of the optical element. The focusing element 55 and the aperture 56 are collectively referred to as the optical element 54.

[0270] In one embodiment, the f-number of the optical element is adjusted by changing the focusing element 55 to one having a different focal length. The aperture 56 is fixed.

[0271] Additionally or alternatively, the f-number of the optical element is adjusted by changing the aperture 56 to change the aperture size a.

[0272] By changing the focal length of the focusing element rather than the aperture, a reduction in power is avoided. However, the aperture can still be limited.

[0273] As described above, the device response can be determined. The device response is determined in a separate measurement, enabling a plane mirror (not shown) to be provided at the focus 57 and the terahertz beam to be reflected from the focus to the detector within the unit 53. Alternatively, an uncoated substrate may be used instead of the plane mirror.

[0274] Additionally and optionally, the arrangement of FIG. 11(a) includes a polarizer. The polarizer may be arranged, for example, between the terahertz unit 53 and the lens 55. Alternatively, the polarizer may be arranged at other positions within the beam path. The emitter and detector of the terahertz unit 53 may be highly polarized and may emit in one polarization. The addition of the polarizer enables consideration of the relevant polarization and improves the accuracy of the detected signal.

[0275] As an alternative to including a polarizer, the sample may be irradiated using a P-polarized beam and then measured at the Brewster angle for P-polarization. Such a measurement gives a direct measurement of the imaginary part of the complex refractive index.

[0276] FIG. 11(b) shows a schematic diagram of the internal configuration of the terahertz sensor 3-b. The terahertz sensor 3-b includes a terahertz unit 53-b that emits and detects terahertz radiation. For example, the terahertz unit 53-b includes an emitter and a detector.

[0277] The focusing element 55-b is a lens. For example, the focusing element 55-b is a silicon lens. The terahertz sensor 3-b further includes a mirror 59-b. The mirror 59-b may also be referred to as an internal reference mirror, an internal mirror, or a roof mirror.

[0278] The terahertz radiation can be emitted in the form of pulses. The pulses can include a plurality of frequencies in the range of 0.01 THz to 10 THz.

[0279] The lens 55-b, the unit 53-b, and the mirror 59-b are configured such that the radiation emitted by the unit 53-b is incident on the lens 55-b. A portion of the light incident on the lens 55-b passes through the lens 55-b and reaches point 57-b. Point 57-b is the focal point of the lens. A portion of the radiation incident on the lens 55-b is reflected towards the mirror 59-b.

[0280] In use, the sample is provided at the focus 57-b. The radiation incident on the sample is reflected back by the lens and directed towards the terahertz unit 53-b, where it is detected. The path along which the radiation travels may be referred to as the transmission path. The radiation passing through the transmission path depends on the sample. The sample can be any of the samples described herein.

[0281] A part of the radiation incident on the lens 55-b is reflected towards the mirror 59-b. At the mirror 59-b, the reflected radiation is directed towards the lens 55-b. The reflected radiation is further directed by the lens 55-b towards the terahertz unit 53-b, where it is detected. The path along which the radiation travels may be referred to as the reflection path. The radiation passing through the reflection path is independent of the sample. The said radiation provides an internal reference.

[0282] The mirror 59-b may be a roof mirror. The roof mirror 59-b is also called a roof reflector, a roof mirror reflector, or a prism.

[0283] Figure 11(c) shows a schematic view of the lens 55-b. The lens 55-b comprises a rear surface 551-b and a front surface 552-b. The radiation from the terahertz unit 53-b is incident on the rear surface 551-b of the lens 55-b. The rear surface 551-b includes a plane. A part of the incident radiation is reflected, and a part passes through the lens and exits from the front surface. The incident, reflected, and transmitted radiations are shown by dashed lines. The reflected radiation is transmitted towards the mirror 59-b and follows the above-described reflection path. The transmitted radiation is transmitted towards the sample in use. The said radiation follows the above-described transmission path.

[0284] The back surface 551-b of the lens 55-b is cut at a wedge angle with respect to the optical axis. This results in the following optical arrangement. · The angle between the optical axis of the lens 55-b and the back surface 551-b of the lens is θ w which is the wedge angle. · The angle between the incident (and reflected) beam and the surface normal is θi It is. · Let the refractive index of the lens be n Si Then, when the relationship of sin(θ i ) = n Si sin(θ t ) holds, the transmitted light passes through the optical axis. · The incident beam and the transmitted beam are each at an angle of θ i + θ w and θ i - θ w with respect to the optical axis.

[0285] FIG. 11(d) shows a schematic diagram of the lens 55-b and the mirror 59-b.

[0286] In the figure, the transmitted beam path is indicated by a dashed line (--). The reflected beam path is indicated by a two-dot chain line (-··-).

[0287] In the transmitted beam path, the incident radiation hits the rear surface 551-b of the lens, propagates through the lens 55-b, exits from the front surface 552-b of the lens, and is focused at the point 57-b. The front surface 552-b of the lens includes a convex surface. The convex surface is adapted to focus the beam at the focus 57-b.

[0288] In use, a sample is provided at the focal plane 57-b. The radiation is reflected towards the front surface 552-b of the lens 55-b, propagates through the lens 55-b, exits from the rear surface 551-b of the lens, and is directed towards the terahertz unit 53-b, where it is detected.

[0289] In the reflected beam path, the incident radiation hits the back surface 551-b of the lens, where it is reflected towards the mirror 59-b. At the mirror 59-b, the radiation is directed back towards the rear surface 551-b of the lens, where it is then directed towards the terahertz unit 53-b for detection.

[0290] The x-axis separation of the beams may be set so as to enable the THz emitter and detector devices to be conveniently arranged side by side.

[0291] The position of the roof mirror along the x-axis determines the x-axis separation of the beam (FIG. 11-d) and is selected such that when the beam reaches the THz unit 53-b, the x-axis offset of the reflected beam coincides with the x-axis offset of the transmitted beam, so that the two beams reach the terahertz unit 53-b at the same position.

[0292] The reflected light reflected by the back surface 551-b of the lens 55-b is indicated by a two-dot chain line (-··-) in FIG. 11(d).

[0293] The roof mirror is configured such that the beam reflected by the mirror is parallel to the incident beam.

[0294] The beam between the THz unit 53-b and the lens 55-b defines a first axis, axis A.

[0295] The beam between the back surface 551-b of the lens 55-b and the roof prism 59-b defines a second axis, axis B.

[0296] As shown in FIG. 11(c), the plane normal of the lens defines a third axis, axis C. Axis C does not coincide with the optical axis of the lens.

[0297] FIG. 11(c) also shows a fourth axis which is the optical axis.

[0298] The angle between axis A and axis C is equal to the angle between axis B and axis C (axis A∠axis C = axis B∠axis C), and A and B are located on the opposite side of the surface normal C. That is, B is the axis of specular reflection of the beam incident along axis A by the surface having the normal C.

[0299] The path length of the reflected beam is adjusted by moving the roof mirror 59-b along its axis B, and the total path of the reflected beam is somewhat shorter than the total path of the transmitted beam.

[0300] The purpose of the shorter reflection path is for the reference signal (i.e., the signal taking the reflection path) to arrive before the sample signal (i.e., the signal taking the transmission path). This enables the detection of the reference signal independently of the sample signal using a single transmitter and receiver (detector).

[0301] Accordingly, the reference signal is acquired together with the sample signal. Being acquired together means that the reference signal is measured as part of the same waveform as the sample signal. However, the measurement has a finite duration. The reference signal provides an internal reference and enables the correction of short-term system signal variations.

[0302] Optionally, note that for calibrated measurements, the internal reference is corrected with an external reference. The external reference is obtained by placing a gold mirror at the sample position and measuring the reflected signal. The external reference can be used to correct the internal reference.

[0303] The correction can be done as follows. · The signal from the external gold mirror reference is represented by R E and the signal from the internal reference is represented by R i and the signal from the sample is represented by S. t0 represents the time when the external mirror is measured. t1 represents the time when the sample is measured. · Using the external gold mirror reference R E (t0) and the internal reference R i (t0), as well as the sample measurement S(t1) measured at t1 and the internal reference R i (t1), the correction of the measurement values can be performed. · The sample relative to the mirror reference is S / R E = S(t1) / R E (t0) × R i (t0) / R i (t1). · Similarly, the sample measurement value can be corrected as follows. S = S(t1) × R i (t0) / R i(t1)

[0304] The configuration of FIG. 11(d) enables improved signal strength stability. This configuration also enables the optical delay of the sample path to be measured independent of the optical delay drift of the system. Thereby, signal amplitude fluctuations due to focus variations can be corrected.

[0305] The sensor 3-b described in connection with FIG. 11(b), as well as the components described in connection with FIGS. 11(c) and (d), enable the signal amplitude to be measured with improved accuracy.

[0306] For example, long-term system drift can be removed by a standard reference, but the standard reference involves interrupting the sample measurement while another reference is being made. The configurations of FIGS. 11(b) to 11(d) are self-referencing. This configuration maintains the measurement accuracy while avoiding interruption of the sample measurement. The sensor 3-b provides an arrangement that can measure the sample (transmission path) and the reference (reflection path) together without interrupting the sample measurement.

[0307] For example, the position of the sample relative to the focus of the lens also affects the signal amplitude. The configuration of the sensor 3-b enables this to be measured and calibrated using the peak position (optical delay), but only when other optical delay variations are removed. The sensor 3-b can perform this because the delay between the internal reference pulse and the sample pulse is independent of the optical delay variations of the system.

[0308] Therefore, the sensor 3-b described in connection with FIGS. 11(b) to 11(d) enables the signal amplitude to be accurately measured. Also, since there is no need to interrupt the sample measurement to obtain a reference measurement value, the sensor can operate continuously while maintaining accuracy.

[0309] Next, this enables the accurate measurement of the real number n and the thickness of the layer. This is because the measurement of the real number n and the thickness requires the accurate measurement of the signal amplitude. This becomes more important for samples with a high refractive index (since dr / dn decreases at high n). Here, r represents the Fresnel reflection at the interface of the layer.

[0310] Sensor 3-b can be combined with any of the systems and methods described herein. For example, sensor 3-b may be used in place of sensor 3 in the system of FIG. 3 or FIG. 11(a).

[0311] When sensor 3-b is used in the arrangement of FIG. 11(a), the focal length of lens 55-b can be varied by adapting the front face 552-b of lens 55-b. For example, the radius of curvature of the convex surface may be adapted.

[0312] When sensor 3-b is used in the arrangement of FIG. 11(a), the focal length and aperture of lens 55-b set the f-number.

[0313] FIG. 12 shows a schematic diagram of the analysis unit.

[0314] Analysis unit 451 may correspond to analysis unit 5 shown in the system of FIG. 3. Analysis unit 451 can implement any of the methods described herein.

[0315] Analysis unit 451 can determine the thickness of the layer in real time, or the data can be stored by the analysis unit and processed later. Analysis unit 451, which can be implemented on a standard computer, comprises a memory 453, a processor 455 that executes a program 457, and further includes an input module 459 and an output module 461.

[0316] In one embodiment, the input module 459 receives data from the sensor 401, and the input is in the form of a time-domain terahertz trace. This data is then passed to a processor 455 that executes a program 457. During the determination of the device response, the data passed to the input module 459 is processed by the processor 455 and stored in a memory 453. When analyzing data from a sample, the processor 455 calls the device response from the memory 453 to derive the sample response. The output is provided by an output module 461. In a further embodiment, the processor 455 is a multi-core processor. This enables much faster processing by using multiple cores of a PC to calculate thicknesses in parallel.

[0317] Battery

[0318] The terahertz technology described herein can be used to measure quantities such as the thickness, weight, density, and conductivity of coatings used on electrodes in the development and manufacture of lithium-ion batteries. Lithium-ion batteries are currently used to power most of the world's portable electronic devices such as smartphones, laptops, and tablets, and are increasingly being used in hybrid and electric vehicles (EVs), as well as in national power grid storage for renewable energy.

[0319] An important issue in lithium-ion battery manufacturing is to optimize the manufacturing process of electrode coatings (cathodes and anodes) in order to improve and optimize capacity while reducing and controlling manufacturing costs. Parameters to optimize during coating manufacturing include changing the coating gap, line speed, etc. Other performance metrics that determine electrode performance include coating density, coating thickness, and conductivity.

[0320] The methods and systems described herein enable at least one of coating thickness, coating density, and conductivity to be measured accurately and rapidly. Further, the methods and systems described herein enable these quantities to be measured simultaneously using a single sensor.

[0321] Electrode manufacturing process

[0322] An important issue in battery manufacturing is the optimization of the manufacturing process to reduce manufacturing costs while improving long-term cycle performance and capacity life. An important step in manufacturing that determines the final quality of the battery pack is the electrode manufacturing process, where the quality and consistency of the coatings used for both the cathode and anode are emphasized.

[0323] The manufacture of the electrode can include three steps: coating, drying, and calendaring.

[0324] FIG. 15 shows a schematic diagram of a method S1500 for manufacturing an electrode according to one embodiment. FIG. 15 illustrates important stages and parameters in the manufacture of the electrode coating, as well as feedback control of the manufacture using coating thickness and density measured by a terahertz sensor at different parts of the process.

[0325] There are several stages in the production processes of both the cathode and anode where terahertz sensors can play a role in process control and waste reduction (see FIG. 15). The in-line configuration can measure the thickness and density of the coating and feedback these parameters to control the coating speed and gap. It is further valuable to make these measurements both before and after drying. By comparing the wet thickness with the applied gap, information about the elasticity in the process can be obtained. The dried coat density and thickness determine the capacity of the coating and are thus the ultimate factors for optimization by varying the gap and coating production speed.

[0326] Step S1501 represents the coating process. In the coating process, a layer (coating) is deposited on the substrate. The substrate functions as a current collector. In the coating process, typically, current collectors such as aluminum for the cathode and copper for the anode are coated. The coating is a slurry mixture consisting of an active material, lithium nickel manganese cobalt oxide LiNi-MnCo (NMC), or lithium nickel cobalt aluminum oxide (NCA) or lithium iron phosphate (LFP) or lithium cobalt oxide (LCO) or lithium manganese oxide (LMO), and can be used for typical lithium-ion cathodes and anodes made of graphite. The coating may also include conductive carbon nanoparticles (e.g., carbon black), a polymer binder, and a solvent. It should be recognized that the methods and inventions described herein can be applied to any type of active material used for coatings for cathodes and anodes and are not limited to the examples shown herein.

[0327] Step S1503 represents the drying process. In the drying process, the mixture is then dried by exposure to an air stream, heat, or other processes. Another example of this method is described in F. Duffner, L. Mauler, M. Wentker, J. Leker, M. Winter, 2021, "Large-scale automotive battery cell manufacturing: Analysis of strategic and operational impacts on manufacturing costs", International Journal of Production Economics, 232, p. 107982.

[0328] Step S1505 represents the calendaring process. In the calendaring process, the dried coating is compressed, increasing the cell's energy density through a reduction in porosity while leaving sufficient porosity for lithium transport and other forms of conduction (see Journal of Power Sources 393 (2018) 177 - 185).

[0329] In step S1507, the performance indicators of the electrodes are measured. The performance indicators are measured using terahertz radiation using the systems and methods described herein.

[0330] In step S1509, the measurement value(s) from S1507 are fed back and the manufacturing process is adjusted and controlled. Each step of the manufacturing process is controlled by process conditions. In the case of S1501, the process conditions are coating speed, coating gap, web tension, and temperature. Any one of the process conditions such as coating speed, coating gap, web tension, and temperature in step S1501 may be adjusted. In the case of S1503, the process conditions are drying time, temperature, and air flow. Similarly, any one of the drying time, temperature, and air flow in step S1503 may be adjusted. In the case of S1505, the process conditions are gap, pressure, speed, and temperature. Any one of the gap, pressure, speed, and temperature in S1505 may be adjusted.

[0331] FIG. 15 shows that S1507 is performed after each of S1501 (coating), S1503 (drying), and S1505 (calendering), but it will be understood that the measurement of S1507 may be performed after any one or any two of the steps of coating, drying, and calendering.

[0332] Similarly, FIG. 15 shows that the feedback S1509 is applied to the three steps of coating, drying, and calendering, but it will be understood that the feedback may be applied to any one or any two of the steps of coating, drying, and calendering.

[0333] The performance indicators in the electrode manufacturing process that enable continuous optimization, monitoring, and maintenance of the uniformity, quality, and performance of the cathode and anode coatings include the following. · Coating density - important for maximizing the energy density of the battery cell, but leaving sufficient porosity for lithium transport and other conduction mechanisms. The mass of the active material per unit area determines the final capacity of the electrode, and a higher coating weight is desirable to increase the energy density, but typically also decreases the power density. Therefore, there must be a compromise between the two to give the maximum energy density while meeting the power requirements necessary for the application. For this reason, specific control of the coating density is highly desirable. Thus, density is an important variable in determining the electrochemical properties of the coating. · Coating thickness - Ensure the uniformity of the coating thickness in the manufacturing process and the final product. Thicker electrodes contain a greater amount of active material and increase the energy density, but have longer diffusion distances, reduce the power output, potentially cause non-uniform responses across the electrode, and result in faster degradation. Therefore, there is an optimal thickness to balance these effects, and control of the thickness is important (see Materials & Design 209(2021)109971). · Coating conductivity - High conductivity improves the ability at higher discharge rates, and more energy can be extracted from the battery in a given time (see https: / / undergraduateresearch.virginia.edu / investigating-conductivity-lithium-ion-batteries-across-porous-thin-films-through-manipulation-0).

[0334] In step 1507, any one or more of these performance metrics can be measured.

[0335] The measurement of the above performance metrics may be carried out at the following points in the manufacturing process. · When applying paint to the current collector metal substrate, it is performed before or after the coating drying process (S1503). It is currently reported that only the coating and drying processes constitute 22% of the total cost of electrode manufacturing (Materials & Design 209(2021)109971). · When the dried coating is compressed, it is before or after the calendaring process (S1505).

[0336] In step S1507, the performance indicators of the electrode are measured using terahertz radiation using the systems and methods described herein.

[0337] The advantage of using the systems and methods described herein is that the above important performance indicators can be quickly monitored in the manufacture of electrode coatings, and real-time feedback can be provided for process control.

[0338] For example, improved process control can eliminate waste costs and ensure supply to the market without production stoppages. For example, losses in the manufacture of lithium-ion batteries due to scrap of out-of-specification electrodes can reach 2 - 5% of output (Journal of Minerals, Materials and Metals 69(2017)1484 - 1496, and Int.J.Prod.Econ. 232(2021)107982), and higher numbers are possible in the increase of new production runs. This waste contributes to the overall battery cost and becomes very important as production scales up in gigafactories.

[0339] Another advantage of using the systems and methods described herein is that the above important performance indicators can be measured simultaneously.

[0340] In the manufacturing process of lithium-ion electrodes, various techniques have been used to monitor the above performance indicators. Attention to these techniques provides all of the important performance indicators described above. These techniques are often used in isolation and tend to be used offline as a quality control means. Offline techniques such as sampling and weighing of coated electrodes against uncoated substrates are also used. However, in large-scale production, the offline trial-and-error approach can result in further waste and equipment downtime.

[0341] None of these techniques can directly and simultaneously measure coating density, thickness, and conductivity in-line. For example, thickness can be estimated using laser triangulation or a laser caliper at near-infrared wavelengths, but it is difficult to perform on an opaque coating and often requires calibration against an uncoated substrate and maintenance of accurate alignment in the production environment, which leads to inaccuracies. Sensors are available for measuring coating weight. X-rays, beta, and gamma radiation can be used in transmission and reflection geometries, but suffer from both safety concerns and the need to integrate over a long time scale to collect an accurate signal (in the case of beta sensors). In the case of the high coating speeds used in the industry, this can result in a significant area of the coating being lost. Ultrasonics can also be used to measure the weight of the coated material, but depends on a stable and accurate calibration. Further, in all of the above, the coating weight is measured and the coating density, which is the desired performance indicator, is not measured. Coating density can be calculated and thus indirectly estimated, but depends on thickness measurements from yet another set of sensors, introducing further error.

[0342] The systems and methods described herein enable direct and simultaneous measurements.

[0343] An additional substantial advantage of terahertz over other techniques is that the refractive index of the coating n = n oThe real part (n o ) and the imaginary part (k) of +ik simultaneously. This unique ability of terahertz pulses provides important information regarding important performance metrics for coatings. · Coating thickness: The coating thickness d can be calculated using the formula d = cDt / 2n, where c is the speed of light, n is the refractive index, and Dt is the time delay of the terahertz pulse from the coating interface (see Figure 1). · Coating density: The real part of the refractive index n of the material in terahertz o is proportional to the bulk density of the material (Journal of Pharmaceutical Sciences On-line DOI 10.1002 / jps.23560(2013)) and can be used to directly measure the density using calibration. Figure 14(a) shows a plot of the change in refractive index with bulk density. · Coating conductivity: The imaginary part of the refractive index k of the material in terahertz is related to its high-frequency conductivity σ using the formula σ(ω) = 2n o ke o w, where ω is the terahertz frequency, enabling the monitoring and adjustment of the electrical properties of the film during the manufacturing process or for offline use in process modeling and prediction.

[0344] The systems and methods described herein also enable an inline approach. Inline techniques often examine more coatings than taking sample points and provide improved detection of coating defects through a wider range of inspections. Rapid inline measurements also provide opportunities for real-time process control, along with cost reduction and supply assurance benefits, as described above.

[0345] In one embodiment, a single sensor is used for both in-line thickness measurement and coating density measurement. This sensor enables in-line control, optimizes the coating, and maintains quality by providing real-time feedback of the coating thickness and density to the coating deposition control process (e.g., by modifying the production line speed, the gap used in the deposition system, etc.). Heretofore, this in-line process has not been proven possible with the measurement techniques described above.

[0346] FIG. 13 shows terahertz measurements of cathodes based on NMC (lithium-nickel-manganese-cobalt-oxide (LiNiMnCoO2)) on aluminum. To show a wide range of terahertz measurements, both thin (15 μm) and thick (70 μm) coatings were investigated. Twin peaks of the terahertz waveform are shown to demonstrate the ease with which the interfaces of the front and back surfaces of the coating are resolved and the coating thickness is accurately measured. However, in a production system, a simple value of the coating thickness is automatically returned to the operator or the plant data management system. Scanning the terahertz sensor across the width of the cathode reveals the surface and coating profile, as well as a linear plot of the coating thickness itself across the cathode. FIG. 13(a) shows terahertz measurements of an NMC coating on aluminum with a thin coating. FIG. 13(b) shows terahertz measurements of an NMC coating on aluminum with a thick coating. FIG. 13(c) shows a cross-sectional image of the coating thickness across the cathode, demonstrating the ability of terahertz to measure and map the thickness across the cathode used in a lithium-ion battery.

[0347] Figure 14(b) shows the terahertz measurement of an anode coating based on graphite mixed with carbon black and binder on copper. To show again a wide range of terahertz measurements, measurements of both thin (130 μm) and thick (150 μm) coatings are shown. Although graphite is more absorptive in terahertz, two peaks in the terahertz waveform that identify the front and back interfaces of the coating and accurately return the coating thickness are still resolved.

[0348] Another example of a sensor 1603 having a self-referencing (SR) mechanism in a linear configuration will be described below with reference to FIGS. 16(a), 16(b), and 17. The sensor 1603 can be combined with any of the systems and methods described herein. For example, the sensor 1603 may be used in place of the system of FIG. 3 or the sensor 3 of FIG. 11(a) described above.

[0349] A reference waveform is measured to remove the instrument response and recover the deconvolved waveform from the measured sample waveform. However, the characteristics of the terahertz emitter and receiver can change over time, which can involve moving the sensor to data points during a measurement campaign and can lead to data loss, so taking a regular reference waveform in an industrial environment can be inconvenient.

[0350] However, in this configuration, a classical Michelson interferometer configuration can be used. 1. The beam splitter device is difficult to manufacture for the terahertz region. 2. The beam splitter is relatively thin (5 mm of high-resistivity silicon), but can result in a configuration that is not sufficiently compact. 3. Using high-resistivity silicon can result in high insertion loss for the beam splitter alone. 4. Multiple reflections from the silicon-air interface can also occur. 5. In an industrial environment, the orthogonal arrangement of the emitter and receiver can cause problems for high-speed scanning on the production line.

[0351] The sensor 1603 described in this specification can overcome at least some of these problems by combining a beam splitter and a focusing element into a linear body. This configuration can be beneficial for attachment to conveyor belt type applications.

[0352] Figures 16(a) and 16(b) respectively show a schematic top view and a schematic side view of the self - reference sensor 1603. The sensor 1603 includes a lens 1655, an emitter 1660, a receiver 1661, and a retro - reflector 1659. The lens 1655 is used to reflect a portion of the beam to the retro - reflector 1659 and back to the terahertz receiver 1661. The retro - reflector 1659 may be used as an internal reference mirror such as the internal reference mirror 59 - b of Figure 11(b).

[0353] In the sensor 1603, the lens 1655 has a wedge as shown in Figure 16(b). The wedge is arranged such that the normal vector of the plane on the back surface of the lens 1655 forms an angle with the optical axis defined by the convex front surface of the lens 1655. In this configuration, the reflected beam from the plane of the lens 1655 is deflected away from the optical axis and reflected by the retro - reflector 1659 to the receiver 1661. The reflected beam can be aligned to be collinear with the beam coming from the sample 1602 onto the receiver 1661. By arranging the position of the retro - reflector 1659 to compensate for the path difference to the sample 1602, a prepulse on the measurement waveform can be achieved.

[0354] In use, the sample 1602 is placed at the focus of the lens 1655. The emitter 1660 emits a pulse of terahertz radiation that irradiates the sample 1602. The radiation reflected from the sample 1602 is collected by the lens 1655 onto the receiver 1661. A portion of the emitted radiation is reflected from the back surface of the lens 1655 through the retro - reflector 1659 to the receiver 1661, providing an internal reference pulse within the sample waveform.

[0355] The deconvolved waveform (also referred to herein as the sample response) can be used for data analysis. For example, the deconvolved waveform can be used to determine the refractive index of a sample and the thickness of a layer. Deconvolution is given by the following equation.

Equation

[0356] Here, b(t) is the baseline pulse that can be determined at the start of measurement, f(t) is the apodization function, r c (t) is the reference pulse, and s c (t) is the sample pulse. The subscript “c” indicates a corrected signal corrected using an internal reference signal, as will be described in more detail below. The apodization function f(t) is a type of frequency filter that can be used to remove or suppress edge effects and thereby improve the SNR. For example, f(t) can be a Tukey apodization function set to, for example, 10% of the length of the optical time delay. The reference pulse r(t) can be obtained from an external gold mirror as described above. The baseline pulse b(t), also called the background waveform, can be obtained by performing a measurement with no sample / obstacle in the path of the terahertz beam. The baseline pulse b(t), the reference pulse r(t), and the sample pulse s(t) all include an internal reference pulse, which can be removed from the data before the final analysis.

[0357] The external reference pulse (e.g., from a gold mirror) can be labeled as r e (t), and the internal reference pulse (from a retroreflector) can be labeled as r i (t). Both the initial references (internal and external) are measured at the first time t0. The sample measurement value at the second (later) time t1

Equation

[0358] Internal reference pulse r i (t) has no change over time, [Number] .

[0359] The corrected signal is given by the following equation. [Number]

[0360] The reference can be corrected in a similar way.

[0361] Figure 17 shows different waveforms used for data analysis. The background waveform b(t) contains one peak from the internal reference mirror. The reference waveform r(t) is obtained using a gold mirror placed at the focal position of the lens and contains two peaks, S R Peak [Number] and the internal reference mirror peak [Number] . The sample is placed at the focal position to give an uncorrected sample pulse S UC . The uncorrected sample waveform has a peak from the internal reference mirror at time t1 [Number] and includes sample information. The sample waveform is corrected for changes in signal amplitude, and the internal reference waveform is filtered out to provide a corrected sample response, which can be used to calculate the thickness and refractive index of the sample.

[0362] To test the sensor, an 18.5 mm focal length lens was used in a terahertz scanning system as described herein. This produced a frequency-dependent focus on the order of 1 mm. As a proof of concept, the lithium iron phosphate (LFP) cathode on an aluminum current collector was measured 100 times, and both the thickness and the terahertz refractive index were calculated. The determined thickness shown in FIG. 18 was 89.92 μm, the standard deviation was 0.44 μm, and the coefficient of variation was 0.495%. The determined terahertz refractive index shown in FIG. 19 was 2.22, with a standard deviation of 0.0096 and a coefficient of variation of 0.43%.

[0363] Although several embodiments of the present invention have been described, these embodiments have been presented by way of example and are not intended to limit the scope of the invention. These novel embodiments can be implemented in various other forms, and various omissions, replacements, and changes can be made without departing from the gist of the invention.

Claims

1. A method for analyzing a sample including a layer having a first interface and a second interface, comprising the step of irradiating the sample with a pulse of terahertz radiation, the pulse including a plurality of frequencies in the range of 0.01 THz to 10 THz; detecting the radiation reflected from the sample to generate a sample waveform; obtaining a first reflected waveform from the sample waveform, the first reflected waveform corresponding to the reflection from the first interface; obtaining a second reflected waveform from the sample waveform, the second reflected waveform corresponding to the reflection from the second interface; comparing the first reflected waveform with the second reflected waveform to generate an estimated value of the thickness and complex refractive index of the layer; generating a composite signal using the estimated values of the thickness and complex refractive index; varying at least one of the thickness and complex refractive index to reduce the error between the sample waveform and the composite signal; and outputting the thickness of the layer. A method comprising.

2. The method according to claim 1, comprising the step of outputting at least one of the density and conductivity of the layer, the density and conductivity being determined from the thickness and / or complex refractive index.

3. The method according to claim 1, comprising the step of obtaining a reference waveform, the reference waveform being obtained by irradiating a reference sample with a pulse of terahertz radiation including a plurality of frequencies in the range of 0.01 THz to 10 THz and detecting the radiation reflected from the reference sample to generate the reference waveform.

4. The method according to claim 1, wherein the step of obtaining the first reflected waveform and the second reflected waveform from the sample waveform includes the step of using time gating.

5. The method according to claim 4, wherein the first reflected waveform is converted into the frequency domain to obtain a first spectrum, and the second reflected waveform is converted into the frequency domain to obtain a second spectrum.

6. The step of deconvolving the sample waveform with the reference waveform to generate a deconvolved waveform; The step of time-gating the deconvolved waveform to obtain the first reflected waveform and the second reflected waveform; The step of converting the first reflected waveform into the frequency domain to obtain a first spectrum; The step of converting the second reflected waveform into the frequency domain to obtain a second spectrum, the method according to claim 3 including the steps.

7. The method according to claim 6, including the step of determining estimated values of the thickness and the complex refractive index from the first spectrum and / or the second spectrum, wherein the complex refractive index is frequency-dependent.

8. The step of converting the reference waveform into the frequency domain to obtain a reference spectrum; The method according to claim 7, including the step of determining an estimated value of the real part of the complex refractive index from the reference spectrum and the first spectrum.

9. The step of obtaining a second reflected spectrum; The step of correcting the second reflected spectrum; The step of determining the imaginary part of the complex refractive index from the corrected second reflected spectrum; The method according to claim 7, including the step of determining the thickness from the corrected second reflected spectrum.

10. The method according to claim 7, comprising the step of fitting the estimated value of the complex refractive index to a physical model to generate a model of the layer. **Claim 11** The step of varying the complex refractive index to reduce the error between the sample waveform and the synthesized signal is a step of varying the parameters of the model, wherein the parameters of the model are related to the complex refractive index, the method according to claim 10. **Claim 12** Determining the magnitude of the first reflected waveform; Generating a first ratio by comparing the magnitude of the first reflected waveform with the magnitude of the reference waveform; Estimating the real part of the complex refractive index using the first ratio, the method according to claim 3. **Claim 13** Determining the magnitude of the second reflected waveform; Generating a second ratio by comparing the magnitude of the first reflected waveform with the magnitude of the second reflected waveform; Estimating the imaginary part of the complex refractive index using the second ratio, the method according to claim 12. **Claim 14** Comparing the first reflected waveform and the second reflected waveform to obtain a time delay; Estimating the thickness using the time delay or the time delay combined with refractive index information, the method according to claim 12. **Claim 15** A system for analyzing a sample including a layer having a first interface and a second interface, A sensor comprising a pulse source of terahertz radiation adapted to irradiate the sample with a pulse of terahertz radiation including a plurality of frequencies in the range of 0.01 THz to 10 THz, and a detector for detecting the reflected radiation and generating a sample waveform derived from the reflected radiation. An analysis unit comprising a processor and a memory, wherein the processor is Obtaining a first reflected waveform from the sample waveform, wherein the first reflected waveform corresponds to the reflection from the first interface; Obtaining a second reflected waveform from the sample waveform, wherein the second reflected waveform corresponds to the reflection from the second interface; Comparing the first reflected waveform with the second reflected waveform to generate estimated values of the thickness and complex refractive index of the layer; Generating a synthetic signal using the estimated values of the thickness and complex refractive index; Changing at least one of the thickness and complex refractive index to reduce the error between the sample waveform and the synthetic signal; Outputting the thickness of the layer; An analysis unit adapted to perform the steps, and a system including the analysis unit.

16. A system for analyzing a sample including a layer having a first interface and a second interface, A sensor including a terahertz radiation pulse source adapted to irradiate the sample with a pulse of terahertz radiation including a plurality of frequencies in the range of 0.01 THz to 10 THz, and a detector for detecting the reflected radiation and generating a sample waveform derived from the reflected radiation, the sensor comprising an optical element adapted to resolve the reflected radiation from both the first interface and the second interface.

17. The system according to claim 16, wherein the optical element has an f-number of 3 or more.

18. The system according to claim 17, wherein the optical element has an f-number of 10 or more.

19. A method of adapting a system for analyzing a sample comprising a layer having a first interface and a second interface, said system comprising a sensor, said sensor comprising a pulsed terahertz radiation source adapted to irradiate the sample with a pulse of terahertz radiation comprising a plurality of frequencies in the range of 0.01 THz to 10 THz, a detector for detecting the reflected radiation, and an optical element, said method comprising: obtaining an estimated value of the refractive index of the layer; obtaining an estimated value of the thickness of the layer; determining the f-number of the optical element such that a confocal parameter scaled by the estimated value of the refractive index is greater than the estimated value of the thickness of the layer. **Claim 20** A process for manufacturing an electrode for a battery, comprising: coating a substrate with a layer; drying the coated layer; calendering the dried layer. The process further comprises: analyzing the layer using the method according to any one of claims 1 to 14, wherein the layer is analyzed at any one or more of before drying the layer, after drying the layer, before calendering the dried layer, and after calendering the dried layer. preparing process conditions for any one or more of the steps of coating a substrate with a layer, drying the layer, and calendering the dried layer. **Claim 21** A sensor for analyzing a sample comprising a layer having a first interface and a second interface, comprising: a pulsed terahertz radiation source adapted to generate a pulse of terahertz radiation, said pulse comprising a plurality of frequencies in the range of 0.01 THz to 10 THz; Direct the generated terahertz radiation pulses towards the sample using a first path, and a focusing element configured to direct the terahertz radiation pulses towards an internal mirror using a second path, A detector for detecting the reflected radiation to generate a sample waveform, wherein the sample waveform includes radiation reflected from the sample via the first path and radiation reflected from the internal mirror via the second path, and a detector, a sensor.

22. The sensor according to claim 21, wherein the focusing element and the internal mirror are configured such that the second path is shorter than the first path.

23. The sensor according to claim 22, wherein the focusing element and the internal mirror are movable relative to each other such that the length of the second path is adjustable.

24. The focusing element includes a front surface and a back surface, the front surface includes a convex surface, and the back surface includes a flat surface, In use, the front surface faces towards the sample, and the back surface faces away from the sample, the sensor according to any one of claims 21 to 23.

25. The sensor according to claim 24, wherein the normal vector of the flat surface of the back surface forms an angle with the optical axis defined by the convex surface of the front surface.

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