Apparatus for varying the path length of a beam of radiation

Abstract

A system for varying a path length of a beam of radiation. The system comprises a device comprising a centre portion rotatably mounted about a rotation axis, the centre portion being configured to allow the passage of radiation there through; and a plurality of element pairs. Each element pair comprises a first and second element protruding from the centre portion, wherein each of the first and second elements of the plurality of element pairs comprise a non-reflective face and a reflective face. When the beam of radiation is transmitted through a pre-determined area of the non-reflective face of one of the first elements, in a first direction, a path is defined as the beam of radiation is reflected by the reflective face of the first element towards the reflective face of the corresponding second element, and further reflected by the reflective face of the second element towards the non-reflective face of the second element. The system further comprises a motor configured to rotate the device about the rotation axis at a pre-determined rotational speed.

Description

[0001]Apparatus for Varying the Path length of a Beam of Radiation FIELD OF INVENTION The invention relates to a system, device and method system for varying a path length of a beam of radiation. BACKGROUND Terahertz time-domain spectroscopy is used to characterize materials based on their interaction with terahertz (THz) radiation. This offers a non-destructive and real- time characterization of materials. As such, an advantage of THz time-domain spectroscopy is that it can characterize materials whilst they are in use. Photoconductive switching can be used in order to generate the THz radiation. In particular, pulses of THz radiation are usually generated. The generated pulses of THz radiation can then be focused onto a sample being studied. As a THz pulse interacts with the sample, the THz pulse can be absorbed, reflected, scattered and / or transmitted through the sample. After the THz pulse has interacted with the sample, the resulting THz pulse can be detected (e.g. using a photoconductive detector) and converted into an electrical signal which can be analysed. In particular, analysis of the THz pulse can provide information of the sample such as thickness, refractive index etc. In one implementation of THz time-domain spectroscopy, the phase of theresulting THz pulse is used to determine information on the sample (e.g., the refractive index of the sample). The phase of the THz pulse can be determined by, for example,using a probe beam to gate the detector sensitivity relative to the time of arrival of theTHz pulse. The probe beam can be obtained by, for example, splitting a laser beam used to generate the THz pulse into a pump beam and a probe beam. A delay line canbe introduced to provide a known, controlled, delay to the probe beam.The probe beam enables the measurement of the phase of the THz pulse because it enables the waveform to be measured, rather than just the amplitude. By delaying the time of arrival of the probe beam, the whole THz waveform can be measured, including the shape of the decay after the main pulse. This is, in essence, a type of pump-probe time resolved spectroscopy. The purpose of the delay line is to 'sweep' the THz waveforms quickly in real time. However, there is a need for an improved delay line which can provide relatively high acquisition rates whilst reducing measurement errors. 14630892-1 SUMMARY Herein is provided a system for varying a path length of a beam of radiation, the system comprising: a device comprising: a centre portion rotatably mounted about a rotation axis, the centre portion being configured to allow the passage of radiation there through; and a plurality of element pairs, wherein each element pair comprises a first and second element protruding from the centre portion; wherein each of the first and second elements of the plurality of element pairs comprise a non-reflective face and a reflective face, such that, when the beam of radiation is transmitted through a pre- determined area of the non-reflective face of one of the first elements, in a first direction, a path is defined as the beam of radiation is reflected by the reflective face of the first element towards the reflective face of the corresponding second element, and further reflected by the reflective face of the second element towards the non-reflective face of the second element; and a motor configured to rotate the device about the rotation axis at a pre- determined rotational speed. Scanning delay lines are used to delay the path of a beam of radiation by a well-defined amount. For example, in terahertz, THz, imaging systems, two beams ofradiation are used – a pump beam which travels from an emitter to a detector via asample, and a probe beam which travels from the emitter to the detector via the delay line. The phase and amplitude of the pump beam will depend on the sample itself. As such, by determining the phase and amplitude of the waveform of the pump beam, information of the sample (e.g., thickness and refractive index) can be obtained. Here, the device can be used as a delay line for the probe beam, where the probe beam (i.e., the beam of radiation) reflects twice within the device to increase the path length of the beam. As the device is rotated, the beam of radiation will pass through different element pairs and output discrete beams with a well-defined delay. The acquisition rate of an imaging system will thus depend on the rate at which said delayed beams are output by the delay line. When using more than one element pairs, the number of delayed beams outputby the device per rotation will increase based on the number of element pairs. As such, 14630892-1 a higher acquisition rate can be achieved, or a lower rotational speed can be used toachieve the same acquisition rate.The device may be made from one or more optical glass materials. The refractive index may be 1.5 to 2 with respect to the relevant wavelength of the beam of radiation. The beam of radiation may be an optical laser beam. Preferably, the beam of radiation is transmitted out of second element, through the non-reflective face of the second element, in a second direction parallel to the first direction. The first and second elements of each of the element pairs may form a parallelepiped shape. In other words, the first and second elements of each of the element pairs, and a part of the centre portion between the first and second elements, may for a parallelepiped. The first and second elements of each of the element pairs may protrude from opposite sides of the rotation axis. When the beam of radiation is transmitted through the non-reflective face of one of the first elements, it is transmitted through the material of the device towards the corresponding second element. The pre-determined area of the non-reflective face of first element may be based on an area of at which an incoming beam of radiation is capable of reflecting from the reflective face of the first element and towards the reflective face of the second element. A reflective face is a surface for which a beam of radiation within the corresponding element is at least partially reflected, preferably fully reflected. A non- reflective face is a surface for which a beam of radiation is allowed to pass through. The device may be made at least from a material which allows the beam of radiation to pass through the device. When the beam of radiation is reflected from the reflective face of the first element to the reflective face of the second element, the beam passes through the material of the device. For each element pair in the plurality of element pairs, the first element may be a first triangular prism and the second element is a second triangular prism. For each element pair in the plurality of element pairs, the second element mayprotrude from a side of the centre portion opposite the first element. 14630892-1 Each of the plurality of element pairs may be rotated relative to neighbouring element pairs by a constant angle such that the order of rotational symmetry of the device is equal to twice a number of element pairs. Each of the plurality of element pairs, and the part of the centre portion between the corresponding first and second elements, may form a rhomboid prism shape. By forming a rhomboid prism shape for the element pairs, this guarantees that the direction of the beam of radiation exiting the device will be in a direction parallel to the direction at which the beam entered the device. The system may comprise three element pairs, wherein each of the plurality of element pairs is rotated at 60 degrees relative to the other two element pairs. The pre-determined rotational speed may be at least 3000 revolutions per minute, rpm. A rotational speed of 10000 rpm is equivalent to about 167 Hz. When using three element pairs, the device outputs six delayed beams per rotation (two per element pair). Thus, an imaging system using the delay system described would be able to achieve an acquisition rate of 1 kHz. Thus, the pre-determined rotational speed may be 10000 rpm. The pre-determined rotational speed may be at most 30000 rpm.The system may further comprise a reflective member adapted to reflect a beam of radiation exiting the non-reflective face of the second element back into the non-reflective face of the second element, such that, when the beam of radiation is reflected back into the non-reflective face of the second element, the path is further defined as the beam of radiation isreflected by the reflective face of the second element towards the reflective face of thecorresponding first element, and further reflected by the reflective face of the first element towards the non-reflective face of the first element to exit the first element in a second direction parallel to, and opposite, the first direction. The reflective member increases the path of the beam of radiation as it has to travel through the device twice. As such, the delay on the output beam also increases. Additionally, the output beam exits the device in a direction opposite to and parallel to the direction at which it initially entered. Herein is further provided a device for varying a path length of a beam of radiation, the device comprising: a centre portion configured to allow the passage of radiation there through; and 14630892-1 a plurality of element pairs, wherein each element pair comprises a first and second element protruding from the centre portion; wherein each of the first and second elements of the plurality of element pairs comprise a non-reflective face and a reflective face, such that, when the beam of radiation is transmitted through a pre-determined area of the non-reflective face of one of the first elements, in a first direction, a path is defined as the beam of radiation is reflected by the reflective face of the first element towards the reflective face of the corresponding second element, and further reflected by the reflective face of the second element towards the non-reflective face of the second element. For each element pair in the plurality of element pairs, the first element may be a first triangular prism and the second element may be a second triangular prism. For each element pair in the plurality of element pairs, the second element may protrude from a side of the centre portion opposite the first element. Each of the plurality of element pairs may be rotated relative to neighbouring element pairs by a constant angle such that the order of rotational symmetry of the device is equal to a number of element pairs. Each of the plurality of element pairs, and the part of the centre portion betweenthe corresponding first and second elements, may form a rhomboid prism shape. The device may comprise three element pairs, wherein each element pair is rotated at 60 degrees relative to the other two element pairs. In an example, for each element pair, the reflective face of the first element andthe reflective face of the corresponding second element are parallel. In one example,the reflective face of the first element and the reflective face of the correspondingsecond element are parallel to within ±60 arcseconds or less. In other words, there is amanufacturing tolerance of ±60 arc seconds or less between the angles of oppositereflective faces. In one example, the reflective face of the first element and the reflective face of the corresponding second element are parallel to within ±30arcseconds or less. In other words, there is a manufacturing tolerance of ±30 arcseconds or less between the angles of opposite reflective faces. In one example, thereflective face of the first element and the reflective face of the corresponding secondelement are parallel to within ±10 arcseconds or less. In other words, there is amanufacturing tolerance of ±10 arc seconds or less between the angles of oppositereflective faces. In one example, the reflective face of the first element and thereflective face of the corresponding second element are parallel to within ±5 14630892-1arcseconds or less. In other words, there is a manufacturing tolerance of ±5 arcseconds or less between the angles of opposite reflective faces. Providing a toleranceof ±5 arc seconds or less improves the stability of the imaging system. In one example,the reflective face of the first element and the reflective face of the correspondingsecond element are parallel to within ±2 arcseconds or less. In other words, there is amanufacturing tolerance of ±2 arc seconds or less between the angles of oppositereflective faces. Providing a tolerance of ±2 arc seconds or less further improves thestability of the imaging system. Herein is further provided a method for varying a path length of a beam of radiation, the method comprising: rotating the afore-mentioned device at a pre-determined rotational speed; and transmitting the beam of radiation through the non-reflective faces of the plurality of first elements of the plurality of element pairs. Herein is provided a method for determining a porosity, or density, of a coating formed on an electrode using terahertz radiation, the method comprising: controlling a terahertz radiation emitter to irradiate the electrode with a pulse of terahertz radiation; determining a measured real part of a refractive index, in terahertz, of the coating on the electrode based on the terahertz radiation reflected from the electrode; determining the porosity, or density, of the coating on the electrode based on the measured real part of the refractive index and a pre-determined relationship between porosity, or density, of coatings of the same material as the coating on the electrode, and the real part of the refractive index of said coatings. The predetermined relationship between porosity, or density, of coatings of the same material as the coating on the electrode, and the real part of the refractive index of said coatings may be a linear relationship such that the porosity, or density, of the coating is equal to the measured real part of the refractive index multiplied by a calibration factor. Herein is provided a computer program comprising code which, when executed on a processing system, causes the processing system to perform all of the steps of the afore-mentioned method Reference to terahertz radiation herein may generally refer to radiation having a frequency in the range of 0.01 to 100 THz. 14630892-1 Thickness Correction Herein is provided a method for determining a thickness and / or density of asample using terahertz radiation, the method comprising: controlling a terahertz radiation emitter to irradiate the sample with terahertz radiation; determining a measured real part of a refractive index and a thickness of the sample based on a waveform indicative of the terahertz radiation transmitted through, and reflected from, the sample; determining a correction value based on the measured real part of the refractiveindex and a first pre-determined relationship between the measured real part of the refractive index and the correction value; and correcting the measured thickness based on a second pre-determined relationship between measured thickness, a real thickness of the sample and the correction value. THz imaging can be used to determine a thickness and refractive index of a sample based on the THz radiation transmitted through, and reflected from, the sample. This is because the phase of the THz radiation transmitted through the sample will change depending on the thickness and refractive index of the sample. The density of the sample could be calculated using the refractive index andmaterial-dependant predetermined factors C^^ and C^^. The material-dependant pre-determined factors can be obtained by comparing the refractive indices of a material to real densities to obtain a relationship between the refractive index n^^^and the density.For battery electrodes, it has been found that said relationship is linear - D = n^^^ ×C^^ + C^^.It has been realized that, in some cases, the real part of the refractive index measured during THz imaging is overestimated whilst the thickness is underestimated. The degree of error seems to correlate to the porosity / density of the sample. In particular, the lower the density (higher porosity), the higher the degree of error. This seems to be caused by non-uniformity in the porosity distribution in the samples. The density / porosity can be found using the real part of the refractive index. Thus, it has been realized that a correction value for the thickness can be found by using the measured refractive index. The correction value may be material dependent. In order to find the first pre-determined relationship between the measured real part of the refractive index and the correction value, the real thickness of a plurality of 14630892-1 samples of the same material can be determined. Measured values of the thickness and refractive index can then be estimated using THz radiation. The correction factor may be, for example, the ratio of the real thickness to the measured thickness, and this can be plotted against the refractive index. As such, the first pre-determined relationship is determined by, for example, fitting a curve to the plot of the correction factor and the refractive index. The second pre-determined relationship may be ^^^^ = ^^^^^, where ^^^^ is thecorrected thickness, ^^^^ is the measured thickness and ^ is the correction value.The method may further comprise correcting the real part of the refractive indexbased on the correction value and a third pre-determined relationship. The third pre-determined relationship may be ^^^^ = ^^^^ / ^ , where ^^^^ is thecorrected real part of the refractive index, ^^^^is the measured real part of therefractive index and ^ is the correction value.A computer program is provided comprising code which, when executed on aprocessing system, causes the processing system to perform all of the steps of the afore-mentioned method. Reference to terahertz radiation herein may generally refer to radiation having a frequency in the range of 0.01 to 100 THz.Drift and Noise CorrectionHerein is provided a method for generating a drift-corrected sample signal, themethod comprising: controlling a terahertz radiation emitter to irradiate a spectroscopy system when it does not contain a sample with terahertz radiation; obtaining a baseline waveform from the terahertz radiation reflected from the spectroscopy system; controlling the terahertz radiation emitter to irradiate the spectroscopy system containing a sample with terahertz radiation; obtaining a sample waveform from the terahertz radiation reflected from the spectroscopy system; determining a ratio signal between a Fourier transform of the baselinewaveform and a Fourier transform of the sample waveform; anddetermining a corrected sample signal based on the Fourier transform of the sample waveform and the modulus of the ratio signal. 14630892-1The baseline waveform may be a first baseline waveform obtained at a firsttime, and the method may further comprise:controlling the terahertz radiation emitter to irradiate the spectroscopy system when it does not contain a sample with terahertz radiation at one or more second times; and obtaining one or more second baseline waveforms from the terahertz radiation reflected from the spectroscopy system at the one or more second times; and determining an average baseline waveform based on the first baseline waveform and the one or more second baseline waveforms, or determining an average baseline signal based on a Fourier transform of the first baseline waveform and a Fourier transform of the one or more second baseline waveforms, wherein the ratio signal is determined based on a Fourier transform of the average baseline waveform or the average baseline signal. By using multiple baseline waveforms, the noise in the ratio signal can be reduced whilst maintaining any drift information of the system. The sample waveform may be a first sample waveform obtained at a third time,and the method may further comprise: controlling the terahertz radiation emitter to irradiate the spectroscopy system containing the sample with terahertz radiation at one or more fourth times; obtaining one or more second sample waveforms from the terahertz radiation reflected from the sample at the one or more fourth times; and determining an average sample waveform based on the first sample waveform and the one or more second sample waveforms, or determining an average sample signal based on a Fourier transform of the first sample waveform and a Fourier transform of the one or more second sample waveforms, wherein the ratio signal is determined based on a Fourier transform of the average sample waveform or the average sample signal. The method may further comprise determining a characteristic of the samplebased on the corrected sample signal. The one or more characteristics of the sample may include one or more of:a thickness of the sample; a density of the sample; a porosity of the sample; 14630892-1 a sheet density of the sample; a refractive index of the sample; an absorption coefficient of the sample; and an optical conductivity of the sample. Acomputer program is provided comprising code which, when executed on aprocessing system, causes the processing system to perform all of the steps of the afore-mentioned method. Reference to terahertz radiation herein may generally refer to radiation having a frequency in the range of 0.01 to 100 THz. Pulse Offset Herein is provided a method for determining a characteristic of the sampleusing terahertz, THz, radiation, the method comprising: controlling a terahertz radiation emitter to irradiate the sample with THz radiation; determining a measured complex refractive index of the sample based on a sample waveform indicative of the terahertz radiation transmitted through, and reflected from, the sample; modelling a reflection pulse for the sample using the measured complex refractive index; determining a phase offset between the reflection pulse and the sample waveform; determining a corrected sample waveform by adding the pulse offset to a phase of the sample waveform; and determining one or more characteristics of the sample based on the corrected sample waveform. Determining the corrected sample waveform may further comprise iteratively: determining a corrected complex refractive index based on the corrected sample waveform; re-modelling the reflection pulse for the sample based on the corrected complex refractive index; re-determining the phase offset between the reflection pulse and the corrected sample waveform; and updating the corrected sample waveform by adding the pulse offset to the corrected sample waveform. 14630892-1The one or more characteristics of the sample may include one or more of:a thickness of the sample; a density of the sample; a porosity of the sample; a sheet density of the sample; a refractive index of the sample; an absorption coefficient of the sample; and an optical conductivity of the sample. Acomputer program is provided comprising code which, when executed on aprocessing system, causes the processing system to perform all of the steps of the afore-mentioned method. Reference to terahertz radiation herein may generally refer to radiation having a frequency in the range of 0.01 to 100 THz. BRIEF DESCRIPTION OF FIGURES Embodiments of the invention will now be described, by way of example only, with reference to the accompanying drawings, in which: Figure 1 shows the layout of an exemplary rapid scan delay line; Figure 2 illustrates the geometry of the rhomboid prism used in the delay line of Figure 1; Figure 3 shows a tri-prism rhomboid as defined herein; Figure 4 shows the path of a laser beam at a minimum and maximum rotation per waveform for a tri-prism rhomboid; Figure 5 shows an exemplary tri-prism rhomboid; Figure 6 illustrates the acquisition rate for waveforms when using an oscillating rhombus prism, a rotating rhombus prism and a rotating tri-prism rhombus; Figure 7 shows an exemplary imaging system using the tri-prism rhombus described herein; Figure 8 shows a plot of a sample waveform, a baseline waveform and a baseline corrected sample waveform; Figure 9 shows an example of a deconvolved waveform;Figure 10 shows an example of a deconvolved waveform and an example of a zeroed deconvolved waveform; Figure 11 shows a plot of an exemplary deconvolved waveform overlaid with the corresponding window functions; 14630892-1 Figure 12 shows a plot of exemplary 0thand 1storder reflection waveforms; Figure 13 illustrates a THz beam being reflected from, and transmitted through, a coating layer on a substrate; Figure 14 shows an exemplary 20% Tukey apodisation function; Figure 15 shows a plot of the correlation between density and THz refractiveindex for battery electrodes; Figure 16 shows a plot of the correlation between the THz porosity and THz refractive index for six electrodes; and Figure 17 shows a plot where the measured density of a coating correlates with the correction factor. DETAILED DESCRIPTION A system for varying a path length of a beam of radiation. The system comprises a device comprising a centre portion rotatably mounted about a rotationaxis, the centre portion being configured to allow the passage of radiation therethrough; and a plurality of element pairs. Each element pair comprises a first and second element protruding from the centre portion, wherein each of the first and second elements of the plurality of element pairs comprise a non-reflective face and a reflective face. When the beam of radiation is transmitted through a pre-determined area of the non-reflective face of one of the first elements, in a first direction, a path is defined as the beam of radiation is reflected by the reflective face of the first element towards the reflective face of the corresponding second element, and further reflectedby the reflective face of the second element towards the non-reflective face of thesecond element. The system further comprises a motor configured to rotate the device about the rotation axis at a pre-determined rotational speed. Achieving a relatively high (e.g., 1 kHz) terahertz waveform acquisition rate using time-domain photoconductive technology requires modulating the laser optical path delay at this frequency. There are many approaches in the literature using rotating prism(s) and / or rotating mirrors (i.e., involute spirals). However, sensitivity to alignment and manufacturing complexity of delay lines used can affect the accuracy of said modulation. Figure 1 shows the layout of an exemplary rapid scan delay line. The delay line uses a parallelogram prism 102 (parallelepiped, rhomboid prism) which has theproperty that the input beam and the output beam are always parallel, making the14630892-1 beam of radiation (e.g., an optical beam) more optically stable and insensitive to alignment. The input beam and output beam are shown. Note that four beams, 106, 108, 110 and 112, are shown between the rhomboid prism 102 and a reflective prism. These beams are shown to illustrate that the path for the beam of radiation changes height with respect to the reflective prism as the rhomboid prism is oscillated. Beams 106 and 108 correspond to a minimum extent of rotation and beams 110 and 112 correspond to a maximum extent of rotation. A galvanometer is used to cause the rhomboid to oscillate at a frequency of typically up to 50 Hz. This modulates the time-delay because the laser beam experiences a different path length through the rhomboid prism 102 at different points in its rotation. In this delay line, the maximum frequency is limited by the inertia of the rhomboid prism 102. Figure 2 illustrates the geometry of the rhomboid prism 102 used in the delay line of Figure 1. On the left, the rhomboid prism 102 is shown with minimum optical delay at anticlockwise deflection and, on the right, the rhomboid prism 102 is shown with maximum optical delay at clockwise deflection. For this example, the minimum rotation angle due to loss of total internal reflection is about -5.3 degrees and the maximum rotation angle due to clipping is about 20.0 degrees. Additionally, Figure 2 shows the path 202 of the laser beam through the rhomboid prism 102 towards the reflective prism 204. The laser input beam is incident on the bottom of the rhomboid prism 102 and is reflected off the top and bottom surfaces by total internal reflection, exiting at the same angle as the incident beam. Thelaser beam is then reflected off the reflective prism 204, and passes back through therhomboid prism 102, following the same path, but offset in the negative x-direction. Herein is proposed a method of achieving the time delay by continuously rotating the rhomboid prism 102, instead of oscillating backwards and forwards. This allows much higher rotational speeds. However, with this scheme, the duty cycle would be very low, because much of the 360° angular rotation does not correspond to any useful beam path through the rhomboid prism 102. By using a tri-prism rhomboid instead of a regular rhomboid prism, the duty cycle can be increased threefold, and the corresponding motor speed required for a 1 kHz waveform rate can be reduced threefold. The tri-prism rhomboid consists of three intersecting parallelepipeds, set at 60° apart. 14630892-1 Figure 3 shows a tri-prism rhomboid 302 as defined herein. The ‘tri-prism rhomboid’ is defined as three intersecting parallelepipeds. Figure 3 also shows the path 304 of the laser beam through tri-prism rhomboid 304. Here, the tri-prism rhomboid 302 has six elements 306 protruding from a centre portion. At six waveforms per revolution, a motor speed of only 10,000 rpm is required to yield a waveform rate of 1 kHz. For example, an easily available and inexpensive brushless DC motor can be used to rotate the tri-prism rhomboid 302. Of course, it will be appreciated that other acquisition rates can be achieved by using other rotational speeds (e.g., at least 1000 rpm, or between 3000 rpm and 30000 rpm). Figure 4 shows the path 304 of the laser beam at a minimum and maximumrotation per waveform. In particular, Figure 4 shows the path taken through the tri-prism rhomboid 302 for the first data point of the waveform (left) and last data point for the waveform. As the tri-prism rhomboid 302 spins in a clockwise direction, the path length is monotonically modulated from short to long. The next waveform begins when the next element of the tri-prism rhomboid 302 moves into position. The optical path length difference between the two extreme positions (found by summing the product of refractive index and propagation distance for all the individual paths) is unchanged from the single rhomboid to the tri-prism rhomboid. Figure 5 shows an exemplary tri-prism rhomboid. As can be seen in Figure 5, the tri-prism rhomboid has, in essence, six triangular prisms, each with a coated face and a non-coated face. The coating may be an anti-reflective coating applied to the non-reflective faces of the triangular prisms, such as silicon nitride. Of course, the particular coating and thickness of the coating will depend on the wavelength used for the beam of radiation. Figure 6 illustrates the acquisition rate for waveforms when using an oscillating rhombus prism, a rotating rhombus prism and a rotating tri-prism rhombus. When using an oscillating rhombus prism, there is no dead zones (i.e., time between acquisitions) between acquisitions. However, the time per acquisition is limited by how quick the rhombus can be oscillated. Thus, generally, the time per acquisition is lower when using an oscillating rhombus prism when compared to using rotating prisms. When using a continuously rotating rhombus prism, the dead zone is much larger than when using the oscillating rhombus prism as the beam can only be correctly reflected on two faces of the rotating rhombus prism. 14630892-1 When using a continuously rotating tri-prism rhombus, the dead zone between acquisitions is much smaller as, here, the beam can be correctly reflected six times perrotation. Thus, the number of acquisitions per rotation can be increased by increasingthe number of rhombus shapes superimposed on one another to form a prism. Additionally, a prism can be continuously rotated much faster than it can be oscillated. As such, the acquisitions per unit time can be further increased by using higher rotation speeds. The tri-prism rhomboid may be made from a glass material (e.g. BK7, N-SF11 or H-ZF13) having a refractive index of about 1.5 to 2 for the relevant wavelength of the beam of radiation. A variety of manufacturing techniques can be used to manufacture the tri-prism rhomboid. For example, individual glass prisms can be glued together. Alternatively, the tri-prism rhomboid can be machined from a single block of material. The tri-prism rhomboid described above is shaped from three superimposed rhombus prisms. However, it will be appreciated that the prism can be shaped from two or more superimposed prisms. This creates a compound shape for the prisms formed from the superimposed rhomboid prism shapes. When the compound shape is achieved using rhomboid prism shapes at equal rotations relative to each other, the compound shape will have an order of rotational symmetry equal to twice the number of rhomboid prism shapes used. More generally, the shape of the prism can be described as having a plurality of “element pairs” and a centre portion, where the element pairs have two elements protruding from the centre portion. The two elements in an element pair may protrude from opposite sides of the centre portion. For example, the element pairs may correspond to the extremities of the rhombus prism shapes and may have a triangular prism shape. The purpose of each element pair is to receive a beam of radiation and reflect it through the centre portion and out of the prism. The beam of radiation reflects from a reflective face of a first element, in an element pair, and is transmitted through the centre portion of the prism towards a second element, in the element pair. The beam of radiation is then further reflected by a reflective face of the second element and transmitted out of the prism. When the element pairs are shaped to form a rhombus shape, the beam of radiation output by the second element travels in a direction parallel to the beam of radiation input into the first element. 14630892-1 Figure 7 shows an exemplary imaging system using the tri-prism rhombus described herein. In this imaging system, a laser 702 is used to output a beam of radiation. The beam of radiation is divided by a beam splitter into a pump beam 704 and a probe beam 706. The radiation emitted by the laser 702 may be in the mid-infrared range. Inorder to produce THz radiation, the mid-infrared pump beam 704 impinges on a THzemitter 708. THz emitter 708 may comprise a photoconductive substrate and two electrodes provided on said substrate. Terahertz beam 710 is emitted from the emitter 708 and can be guided via mirrors onto a sample 712. The sample 712 can be provided on a motorised stage, which allows it to move in the X and Y directions, where the X and Y directions are orthogonal to one another such that a large area of the sample 712 can be scanned. The reflected radiation is then directed via more mirrors onto a detector 714. In this type of imaging system, a probe beam 706 is also provided to the detector 714. In order to determine the phase shifts introduced by the sample 712, the phase of probe beam 706 is varied over a small range in order to continually vary the phase of the probe beam reaching detector 714. To achieve this, the pulse beam is passed through the delay line 716 which, in this example, is the tri-prism rhombus as described above.In a further example, there is provided a device comprising a centre portion rotatablymounted about a rotation axis, the centre portion being configured to allow the passage of radiation there through and a plurality of element pairs, wherein each element pair comprises a first and second element protruding from the centre portion. Each of the first and second elements of the plurality of element pairs comprise a non-reflective face and a reflective face, such that, when the beam of radiation is transmitted through a pre-determined area of the non-reflective face of one of the first elements, in a first direction, a path is defined as the beam of radiation is reflected by the reflective face of the first element towards the reflective face of the corresponding second element, and further reflected by the reflective face of the second element towards the non-reflective face of the second element.The device may be a device as described in relation to one or more of Figures 3 to 6above for example. In one example, the device may be the device as shown in Figure5. In one example, the device may have a length of 14mm, a width of 10mm and a14630892-1thickness of 7mm. The thickness of 7 mm represents the thickness of the material(e.g., glass) from which the shape depicted in Fig 5 is manufactured - i.e. the directionin or out of the page as drawn. In geometry, this is the altitude of a prism. The length ofthe device (e.g., 14 mm) corresponds to the length of the longer face of the initialparallelopiped prism. The width of the device (e.g., 10 mm) corresponds to the lengthof the shorter face of the initial parallelopiped prism. These are related to the cartesiancoordinates shown in Fig 5 by (since the long and short faces are at 45 degrees toeach other): 21.07 = 14 + 10 sin(45°) and 7.07 = 10 cos (45°). The device may be usedin the imaging system as described in relation to Figure 7 above for example.For each element pair, the reflective face of the first element and the reflective face of the corresponding second element are parallel. In one example, the reflective face of the first element and the reflective face of the corresponding second element areparallel to within ±60 arcseconds or less. In other words, there is a manufacturingtolerance of ±60 arc seconds or less between the angles of opposite reflective faces.In one example, the reflective face of the first element and the reflective face of thecorresponding second element are parallel to within ±30 arcseconds or less. In otherwords, there is a manufacturing tolerance of ±30 arc seconds or less between theangles of opposite reflective faces. In one example, the reflective face of the firstelement and the reflective face of the corresponding second element are parallel towithin ±10 arcseconds or less. In other words, there is a manufacturing tolerance of±10 arc seconds or less between the angles of opposite reflective faces. In oneexample, the reflective face of the first element and the reflective face of thecorresponding second element are parallel to within ±5 arcseconds or less. In otherwords, there is a manufacturing tolerance of ±5 arc seconds or less between theangles of opposite reflective faces. Providing a tolerance of ±5 arc seconds or lessimproves the stability of the imaging system. In one example, the reflective face of the first element and the reflective face of the corresponding second element are parallel towithin ±2 arcseconds or less. In other words, there is a manufacturing tolerance of ±2arc seconds or less between the angles of opposite reflective faces. Providing atolerance of ±2 arc seconds or less further improves the stability of the imaging system.Thickness Herein is provided an exemplary method for analysing an electrode (e.g., in a battery) using terahertz time-domain spectroscopy. 14630892-1 Electrode analysis is designed to use a hybrid analytical / fit result to quickly measure thickness and complex refractive index from a reflection spectrum (assumes single layer on metal substrate). The measured optical parameters are calibrated to give density, conductivity and loading as well as the sample thickness. Three waveforms are used in the analysis (assume a self-reference head is used). A baseline waveform is used, which is a measure of THz radiation with nothing in front of the sensor (i.e., no sample) and is a measure of the internal reference signal. A reference waveform is also used, which is a measurement with a gold mirror (100% reflection) in front of the sensor. A sample waveform is also used, which is a measurement taken with a sample in front of the sensor. In this exemplary method, baseline and reference waveform measurements are only required to be taken intermittently, so will be stored for use in processing sample waveforms. The baseline waveform can be stored as a raw waveform and the reference waveform can be stored after internal reference correction. Internal reference correction is a function used to correct a sample or reference waveform for drift in the system. It can do this because there are two paths in the measurement head, an internal path (internal reference) and an external path (for sample measurement). A baseline waveform is typically collected prior to the sample waveform, where the baseline waveform is taken without a sample in the external beam. The baseline waveform therefore only has signal from the internal reference. The system is setup so the internal reference path is shorter than the external path, such that the internal reference appears before the sample signal in the waveform. This separation in arrival time can be used to process the internal and external signals separately. An internal monitor signal is measured for both the baseline waveform ^(^) andthe sample waveform ^(^). Note that the same processing can be performed for the reference waveform ^(^). A system configuration parameter ^^^^^^^^is defined as the time from the start of the waveform that is reserved for the internal monitor signal (typically 5-10ps). Additionally, a width ^^^^^^^defines the width of the internal monitor signal pulse (typically 2-4 ps). First, the time ^^^^^^ of an internal reference pulse maximum is found. In otherwords, the time at which a maximum signal amplitude occurs in the range ^ = ^^^^ to^ = ^^^^ + ^^^^^^^^, where ^^^^ is the time at the start of the scan.For ^^ = ^^ − ^^^^^^^, a window function ^(^) is defined as:14630892-1^^^: ^^ < ^^^^^^^ ^(^) = 1An internal reference baseline spectra ^^^(^) and an internal reference samplespectra ^^^(^), as well as the full sample spectra ^(^) and ^(^), are calculated using aFast Fourier Transform (FFT): ^^^(^) = ^^^(^(^) ∗ ^(^)) Corrections for the full spectra can be calculated using the internal reference spectra. First, the ratio of the sample and baseline internal reference spectra is calculated: Two corrections are generated, one for magnitude ^^^^and one for phase ^^^^^^. The correction for magnitude ^^^^is based on the modulus of the ratio ^^^^^(^): ^^^^ = |^^^^^(^)|An alternative to this correction for the magnitude ^^^^could be to calculate the average value in a particular frequency range. In this example, the frequency range is0.5 to 1.5 THz. However, the modulus of the ratio ^^^^^(^) allows for frequencydependent variations in the spectra. The correction for phase ^^^^^^is based on the average of the unwrappedphase gradient in the range 0.5 - 1.5 THz: To unwrap the phase, steps of 2π are removed. An alternative correction to the phase could be to take the gradient of a linear fit to the phase. A full correction can then be generated from the corrections to the magnitude and phase: 14630892-1The full sample spectra ^(^) can then be corrected into a baseline correctedsample spectra ^^^(^) using the full correction : Acorrected sample waveform ^^^(^) can then be obtained by performing aninverse Fourier transform on the corrected sample spectra ^^^(^). As discussed above, exactly the same processing can be performed on thereference waveform ^(^) to generate a baseline corrected reference waveform ^^^(^).Figure 8 shows a plot of a sample waveform ^(^), a baseline waveform ^(^)and a baseline corrected sample waveform ^^^(^) for illustrator purposes only. On theleft of the plot, the internal monitor region ^^^^^^^^is shown. On the right of the plot, the sample signal region is shown. For production systems where data is taken quickly, although the internal reference corrects for signal drift (occurs on longer time scales and affects both internal and external signal paths), it does not correct for noise (affects internal and external data independently). In fact, as there may be noise on the correction, this could increase the noise in the corrected sample data (although it would still remove drift). To account for this, rather than apply a correction from an individual measurement, thecorrection can be filtered / averaged using a range of data points from recentmeasurements. This means that the correction continues to correct drift without causing an increase in noise. The next step is to calculate a deconvolved waveform ^(^). In order to do this,a second ratio of the corrected sample spectra ^(^) divided by the corrected referencespectra ^(^) is taken: Here, the corrected reference spectra ^^^ (^) is multiplied by -1 as themeasured reference waveform ^(^) has been reflected by a gold mirror. This obeys theactual sign of reflection so that the reflection at an interface between layers 1 and 2 with refractive index n1 and n2, where n2 > n1, will be negative. This is done here because it means that standard reflection equations (e.g. Fresnel reflection equations) can be used without changing the signs. However, it will be appreciated that it is not necessary to use the negative sign here. Next, a filter ^(^) is generated. Here, the filter ^(^) is a low pass gaussian filterwith a width of F1 (e.g., 2 THz). The filter ^(^) also has a sharp cut-off at a higher14630892-1 frequency F2 (e.g., 4 THz). Preferably, F2 should be around twice F1 to prevent distortion to the Gaussian shape of the pulse. The deconvolved waveform ^(^) can then be obtained by multiplying the ratio^^^^^^(^) with the filter ^(^) and applying an inverse Fourier transform (IFFT). Note that if the pulses in the reference and sample waveforms occur at thesame time, the pulse in the deconvolved waveform ^(^) will be split between the startand end of the deconvolved waveform ^(^). This makes some of the further processing steps more onerous. In these cases, the array (i.e., the points of data in the deconvolved waveform) can be re-arranged by moving a number of elements from the start to the end. To be consistent, the reference pulse index can be used to do this. Ifthe index of the reference pulse maximum is ^^^^, and ^^^ is the number of points,then the array order can be re-arranged such that: ^(0 … ^^^) → ^(^^^^ … ^^^, 0 … ^^^^^^)Figure 9 shows a deconvolved waveform ^(^). Note that the deconvolvedwaveform ^(^) is shown with a positive peak for illustrator purposes only. Thedeconvolved waveform ^(^) is filtered based on the order of reflections. First, thestrongest pulse R0 in the deconvolved wave ^(^) is identified. The optical delay time atwhich the maximum pulse magnitude (positive or negative) occurs is labelled ^^.Using a parameter for the pulse width ^^^^^^^^^^(e.g., 0.3 ps for THz pulses)indicative of a half width of the 0th order reflection pulse, a mask function ^(^) iscreated of the form:^^^: |^ − ^^| > ^^^^^^^^^^ ^(^) = 1^^^: |^ − ^^| ≤ ^^^^^^^^^^ ^(^) = 0Thus, a new deconvolved waveform ^′(^) which does not include the pulsehaving the highest pulse magnitude can be generated: ^^(^) = ^(^) ∗ ^(^)Using the new deconvolved waveform ^′(^), a pulse time of second strongest pulse can be found based on the optical delay time ^^of the maximum pulse magnitudeof ^′(^). If ^^ < ^^, ^^ is swapped with ^^.14630892-1A delay time ^^ and a pulse separation ∆^ can then be found using the time atthe start of the optical delay axis ^^^^: ∆^ = ^^ − ^^A zeroed deconvolved waveform ^^^^^^^(^) can then be calculated by using thephase shift associated with the delay time ^^and the magnitude ^^^^and phase ^^^^^^of the deconvolved spectra ^(^): ^ℎ^^^ ^ℎ^^^ = −2^^^^ Figure 10 shows an example of a deconvolved waveform ^(^) in plot 1002 andan example of a zeroed deconvolved waveform ^^^^^^^ (^) in plot 1004.A window function ^^(^) is generated for the 0thorder reflection, where ^^= ^^^: ^ < ^^^^^^^^^^ ^^(^) = 1Similarly, for the 1st reflection order, a window function ^^(^) can be generatedwhere, in this case, ^^ = |^ − ^^^^ − ^^| and ^^^^ℎ = ^^ − ^^^^^^^^^^: ^^^: ^ < ^^^^ℎ ^^(^) = 1Figure 11 shows a plot of an exemplary deconvolved waveform overlaid with the corresponding window functions. It will be appreciated that, for the purpose of illustration, the deconvolved waveform and the windows shown in Figure 11 are not zeroed. 14630892-1Waveforms for the 0th and 1st order reflection signals, ^^(^) and ^^(^), can begenerated by multiplying the zeroed deconvolved waveform ^^^^^^^(^) by the 0thand1st order window functions, ^^(^) and ^^(^), respectively: Figure 12 shows a plot of exemplary 0thand 1storder reflection waveforms. The ^^(^) signal (from the 1storder reflection) contains information on both the propagation through the film layer, the surface reflection and the substrate reflection. Figure 13 illustrates a THz beam being reflected from, and transmitted through, a coating layer (l) on a substrate (s). The THz beam travels through air (a) and is partially reflected by the coating layer (l) and partially transmitted through the coating layer (l). The 0thorder reflection R0 is based on the reflectivity between the air (a) and the layer ^^^. The 1storder reflection R1 is based on the transmittance, ^^^and ^^^, of the THz beam at the boundary between the coating layer (l) and air (a), as well as the reflectivity ^^^of the THz beam at the boundary between the coating layer (l) and the substrate (s). However, for an analytical solution of the first order reflection, it is preferable to remove some of this information on the reflections. For example, it can be assumedthat the substrate reflection is from a perfect metal such that ^^^ = −1. The 0thordersurface reflection is known from ^^(^) so, this can be used to calculate the part of ^^(^)that is dependent on transmission through the coating layer (l): Where ^(^) is the low pass Gaussian filter previously used to filter the secondratio ^^^^^^(^). Thus, in essence, ^^,^^(^) is an unfiltered version of the 0thorder signalspectra. A corrected 1st order reflection signal ^^,^^^(^) can be determined as such: The corrected 1st order reflection signal ^^,^^^(^) can be used to calculate thephase of ^^(^). This is further discussed below. The rest of the analysis can be performed using a section of the spectrum between a minimum frequency ^^^^and a maximum frequency ^^^^. The ^^^^and ^^^^values will generally be pre-determined for different materials. For example, 14630892-1 sampled with a high absorption will often give low bandwidth, whereas relatively thin samples will often give poor low frequency data where pulses overlap. The signals corresponding to said section of thespectrum can be defined as: The magnitude and phase of are then determined because the refractiveindex ^(^′) and the absorption coefficient ^(^′) depend on magnitude and phase. Themagnitude can be calculated as ^^,^^^ = |^^(^^)|. The phase ^^^,^^^^^(^) isunwrapped such that the phase shift is not bound by ±^. A routine can be written to generate the phase ^^,^^^^^(^^) when the data has a pulse offset greater than zero as will be discussed below. ^^,^^^^^(^^) = The refractive index ^(^′) can be estimated from the signal ^^(^^) as it isindicative of the reflection from surface of the coating layer (l), as illustrated in Figure13. In particular, ^(^′) can be estimated as: where the angle ^^is the angle of incidence (in radians) for the incoming THz beam. This is a parameter which can be set to match the optics in use. The average thickness T of the coating layer (l) can be estimated from the phase ^^^,^^^^^(^) and a transmission angle ^^. The transmission angle ^^ can becalculated using Snell’s law: where the average thickness T is the average of ^ℎ^^^^^^^(^′), and where c isthe speed of light. Based on the estimated average thickness T, the refractive index ^(^′) can bere-estimated as: 14630892-1Additionally, the absorption coefficient ^(^′) can be estimated as: The calculations mentioned above thus provide initial estimates of characteristics (e.g., thickness, refractive index and absorption coefficient) of the sample in an analytical manner. This reduces the time taken for the THz waveforms to be analysed. Initially, the pulse alignment (determining the zeroed deconvolved waveform ^^^^^^^(^)) assumes a perfect Gaussian pulse shape, which in turn assumes that therefractive index is real, or at least that ^(^^) ≫ . However, for a complex refractiveindex, the pulse maximum may not be at ^^^^and will be offset instead. For example, measurements performed on graphite anodes have resulted in relatively high values of , thereby causing misalignment of the 0thorder reflection pulse R0 when compared to models for these measurements. This offset can be estimated by calculating the pulse position of an R0 reflectionwaveform, ^0^^^^(^), using the initial estimates for the refractive index ^(^^) and theabsorption coefficient ^(^^). Here, the refractive index and the absorption coefficient can be assumed to be constant over the spectrum between the frequency limits ^^^^and ^^^^, and theaverage values, ^ and ^, of ^(^^) and ^(^^) are used to calculate ^0^^^^(^):^^ = ^ + ^^(co ( ) ( ))^0^^^^(^) = s ^^ − ^^ × cos ^^(cos(^^) + ^^ × cos(^^)) where ^^ is the complex refractive index.The offset in time is given by the time between the start of the waveform and position of the pulse. For THz beams, the offset typically has a magnitude of around 10fs. The calculated offset can be converted into an offset phase ^ℎ^^^^^^^^^and ^(^), and ^^,^^^^^(^ ) can be corrected by adding the offset phase^ℎ^^^^^^^^^. Adding the offset phase can be performed by multiplying a spectrum by a complex number of magnitude one and phase ^ℎ^^^^^^^^^. For example: ^^^ℎ^^^ + ^ℎ^^^^^^^^^^ = ^(^ℎ^^^) ∗ 1^^^^^^ℎ^^^^^^^^^^ + ^ sin^^ℎ^^^^^^^^^^^14630892-1 Alternatively, the phase of the spectrum can be found and the offset phase can be added to said phase. Such an alignment step can be performed iteratively. This is because changingthe pulse alignment also changes the calculated refractive index ^(^^) and thethickness T, which in turn changes the offset phase ^ℎ^^^^^^^^^. This should converge quickly as the pulse alignment change only has a relatively small effect on these values compared to the magnitude of the peak separation. The initial estimate of the refractive index, the absorption coefficient and the thickness is taken using relatively minimal information from the deconvolved waveform. In addition to the improved alignment, the value of the thickness can be further improved by minimising an error function based on the difference between a modelled reflection spectra and the detected reflection spectra. The modelled reflection spectra could be obtained by modelling a number of reflection spectra for different orders of reflections and summing said reflection spectra. Preferably, the modelled reflection spectra is modelled simultaneously for all orders of reflection using a series summation. For example, a reflection spectra ^^from a single coating layer with refractive index ^^and thickness T on a substrate withcomplex refractive index ^^(^) can be calculated as: A root-mean-squared error function can then be used to calculate the error based on the difference between the model reflection spectra and the deconvolved spectra: This error can thus be minimized to obtain an improved value for the thickness T. Once an improved value for the thickness T has been obtained, a final calculation ofthe improved refractive index ^(^^) and absorption coefficient ^(^^) can be performed.14630892-1The refractive index ^(^^) can be used to determine the density (g / cm3) and / orporosity of the coating using, for example, a pre-determined polynomial calibration. Further details for determining the density and / or porosity are provided below. With overlapping pulses, there can be a small error in the calculated thickness. This error can be corrected using a calibration. Thus, the thickness can optionally be calibrated thickness using another pre-determined polynomial calibration. The loading of the coating layer (l) can be determined by multiplying the density by the thickness T. The dielectric function ε(^) and the optical conductivity ^(^) of the coating layer(l) can be determined by using the complex refractive index ^^: Windowing functions have been used in the calculations above. More generally, the window functions could be based on a Tukey apodisation function. As with most window functions, Tukey apodisation functions can be used to set ends of a waveformto zero. Tukey apodisation has one adjustable parameter ^ which is the fraction of thefull waveform length where the apodisation has a value of less than one. The middle section of the function has amplitude 1 so it does not affect the data of the waveform.The parameter ^ has a range between 0 and 1.Assuming the waveform for which the Tukey apodisation function is beinggenerated starts at ^ = 0 and ends at ^ = ^^^^, the form of Tukey apodisation function^^^(^) can be expressed as: Figure 14 shows an exemplary 20% Tukey apodisation function. In thisexample, the function is 20% Tukey apodisation because the parameter ^ = 0.2. Plot1402 shows the full 20% Tukey apodisation function, the sample waveform and the windowed sampled waveform (Sample x Apodisation). Due to the sample waveform 14630892-1 not having much variation at the ends, the difference between the sample waveform and the windowed sample waveform is not very noticeable at these scales. Plot 1404 shows a close up of a second sample waveform and a 20% Tukey apodisation function. Here, the difference between the sample waveform and the windows sample waveform noticeably changes as the 20% Tukey apodisation increases. Returning to the phase shift ^^,^^^^^(^^) calculation above, if only the phase ofthe complex number is taken, the phase will be limited to + / - π radians. In practice, thephase shift ^^^,^^^^^(^) used is a (frequency dependent) time delay, where phase andtime are linked by: ^ℎ^^^ = −^^^^ × ^ × 2^Such a phase can have values which are not limited to + / - π. The full phase inthe frequency calculation of phase can be determined by starting at a relatively low frequency and each time the phase steps by ~2π, add (or subtract, depending on sign of step) 2π. This method is often referred to as phase wrapping and phase unwrapping. A problem with doing phase unwrapping is identifying the steps as, for example, sharp absorption peaks (e.g., caused by water) also cause large changes in the phase. Preferably, the phase is calculated by starting with the time domain plot (i.e., the waveforms) and determining the peak position in time for the 1streflection order. For ^^,^^^(^), this is the peak separation ∆^. The phase shift for this time can be calculated to give ^^^(^). The waveform data can then be shifted such that the maximum is at the start of the axis, as performed above for ^^^^^^^(^). The zeroed peak can then be Fouriertransformed to give a complex spectrum. The phase ^^^(^) of each point on thespectrum is then calculated and phase unwrapped. This spectrum will typically have minimal steps. The final phase shift (e.g., ^^,^^^^^(^^)) can thus be calculated as: Final Phase = ^^^(^) + ^^^(^)The methods discussed above can be used to estimate the characteristics of a coating layer on an electrode, such as thickness, density, loading, refractive index, absorption coefficient etc. In general, the methods discussed above attempt to analytically solve for the initial estimates of the characteristics of a sample and, when models are used to improve said estimates, these models tend to have a low number of parameters. This means that the characteristics can be obtained faster than, for example, using multi- parameter models. 14630892-1 Porosity significantly influences lithium-ion battery performance, impacting cell capacity, voltage, and specific power. It has been realized that Terahertz technology enables non-destructive, remote porosity assessment, addressing limitations in current measurement methods. Lithium-ion batteries have become crucial components in the automotive industry, large-scale utility storage, and various advanced technologies due to their beneficial traits. They boast higher energy and power densities, along with remarkable cycle durability, rendering them highly desirable for diverse applications. Additionally, their reduced environmental impact and enhanced safety unquestionably demonstrate their superiority as alternatives to conventional fossil fuels. Within this context, understanding the characteristics of battery electrodes (e.g., porosity, thickness, density, and conductivity) is of paramount importance. Ensuring the homogeneity of coating thickness prevents uneven responses across electrodes and reduces degradation rates. The coating density must strike a balance between energy density and the necessary power requirements for the intended application. Moreover, coating conductivity enhances capacity at high discharge rates, which is essential for rapid energy release. The coating porosity directly influences the efficiency, performance, and longevity of lithium-ion batteries. Traditional methods for measuring these quantities often involve destructive techniques, limiting their applicability, especially in understanding real-time performance or in-operando behaviour. Porosity assessment traditionally involves destructive methods like Mercury Intrusion, Gas Adsorption, and Liquid Extrusion. X-Ray scanning, while effective, raises safety concerns due to its use of radiation. Additionally, electrochemical impedance spectroscopy offers indirect porosity measurement, but its complexity may limit its application. As discussed above, Terahertz spectroscopy is a technique which enables the direct, simultaneous in-line measurement of coating density, thickness, and conductivity. For instance, while laser triangulation can estimate thickness, its implementation with opaque coatings is challenging and frequently requires calibration. Other sensors exist for measuring coating weight, but techniques such as X-Ray, beta, and gamma radiation raise safety concerns and, in the case of beta sensors, necessitate long integration times for accurate signals. Ultrasound methods can gauge the weight of coated material, but relies on stable and precise calibrations. 14630892-1 It has been realized that Terahertz radiation can be used to assess battery porosity. In particular, Terahertz spectroscopy provides information on the electrode structures without altering them or compromising the electrode integrity, offering detailed insights into their internal architecture. Figure 15 shows a plot of the correlation between density and THz refractive index for battery electrodes. Here, (the real component of) the average refractive index (over a particular frequency range) was determined for six calendared electrodes C1 – C6 with known, varying, densities. The thicknesses of the electrodes ranged from 150- 180 µm. The plot of Figure 15 reveals a correlation between the refractive index, measured using a terahertz signal as discussed above, and the density of the coating on the electrodes. The application of stronger calendaring for lower thicknesses (e.g., electrode C1) resulted in higher density. The plot of Figure 15 demonstrates that an increase in refractive index aligns with higher density. Density and porosity are inversely correlated, meaning higher density implies a lower level of porosity. Thus, an empirical calibration factor can be derived to assist in establishing a relationship between THz refractive index and porosity, based on the coating properties such as density. This relationship allows direct density measurement with calibrated terahertz technology. As such, interpreting the refractive index measurement can provide insights into density and consequently to porosity. Figure 16 shows a plot of the correlation between the THz porosity and THzrefractive index for the electrodes C1 – D6. As expected, electrode C1 electrode withthe highest density has the lowest level of porosity. Figure 16 seems to show a linear relationship between the porosity and the THz refractive index. As such, the empirical calibration factor can be estimated as the gradient of the plot. Thus, an unknown porosity for an electrode coating can be determined by applying said empirical calibration factor to an estimated THz refractive index (e.g., as demonstrated above). Of course, a similar relationship can be pre-determined for the density as shown inFigure 15.It will be appreciated that the empirical calibration factor will depend on the material used for the coating layer on the electrodes. In summary, Figures 15 and 16 demonstrate that an increase in coating porosity corresponds to a decrease in refractive index, indicating a decrease in density. The changes of THz refractive index with respect to density and porosity seem to be linear and, thus, an empirical calibration factor can be determined, using known values 14630892-1 of density and / or porosity, to estimate density and / or porosity for new electrode coatings from a measured THz refractive index. This highlights that terahertz sensors can remotely measure porosity alongside conductivity, coating density, and thickness, showcasing the expanding versatility of terahertz technology in non-invasive characterization of battery electrodes. Returning to the thickness estimation for the samples, most of the estimated thickness values seem to be in good agreement with the actual thickness. This seems to be true over a range of porosities (densities) for electrode coatings. However, for a number of samples, it has been found that the real part of the refractive index is overestimated and the thickness is underestimated. The degree of this error has been found to be dependent on the sample porosity. A cause for this is believed to be a non-uniformity of the porosity distribution in the samples. For low porosities, the samples seem to generate an initial thickness ^^^^. As discussed above, the density ^^^^can be estimated using the refractive index ^^^^with a calibration factor ^^: ^^^^ = ^^^^ ∗ ^^For these samples, it has been realized that a single correction factor ^ can bepre-determined which allows for the correction of density ^^^^, loading ^^^^(i.e., sheet density) and thickness, where: The correction factor ^ can be predicted based on a relationship between themeasured and actual thickness for samples with a range of porosity. Additionally, it hasbeen realized that the correction factor ^ correlates with the density of the sample.Figure 17 shows a plot where the measured density of a coating correlates with the correction factor. The actual density was measured for calendared and uncalendared coating thicknesses. In this case, the relationship between the correction factor and the measured density ^^^^was found to be: 1+ ^ = 0.1331^ ^^^^ − 1.014^^^^ + 2.9755Therefore, as an extension to the electrode coating analysis, the characteristics of high porosity samples can be measured by: measuring the samples and determiningthe thickness ^^^^ and the density ^^^^, calculating the correction factor ^ based onthe density ^^^^and re-calculating a corrected thickness ^^^^and a corrected density ^^^^using the correction factor ^. A corrected loading ^^^^can thus also be calculated. 14630892-1 Of course, in practice, the relationship between the correction factor and the measured density may depend on the particular material used for the sample. Additionally, it may be the case that, during development, the real values ofrefractive index are known but the real values for density are not known. In this case, arelationship between the refractive index and the correction factor can be pre- determined in a similar manner as for the density, given the relationship between refractive index and density discussed above. In this case, the correction factor ^could be defined as ^^^^ = ^^^^ ∗ ^ and a pre-determined relationship between thecorrection factor ^ and real values for the refractive index can be determinedaccordingly. Similarly, the refractive index could also be corrected using the correctionfactor - ^^^^ = ^^^^ / ^.In general, the term “refractive index” as used herein refers to the real component of a complex refractive index unless otherwise stated. Similarly, the term “absorption coefficient” as used herein refers to the imaginary component of a complex refractive index. Reference to terahertz radiation herein may generally refer to radiation having a frequency in the range of 0.01 to 100 THz. Further embodiments are provided in the following clauses: Clause 1. A system for varying a path length of a beam of radiation, thesystem comprising: a device comprising: a centre portion rotatably mounted about a rotation axis, the centre portion being configured to allow the passage of radiation there through; and a plurality of element pairs, wherein each element pair comprises a first and second element protruding from the centre portion; wherein each of the first and second elements of the plurality of element pairs comprise a non-reflective face and a reflective face, such that, when the beam of radiation is transmitted through a pre- determined area of the non-reflective face of one of the first elements, in a first direction, a path is defined as the beam of radiation is reflected by the reflective face of the first element towards the reflective face of the corresponding second element, andfurther reflected by the reflective face of the second element towards the non-reflectiveface of the second element; and 14630892-1 a motor configured to rotate the device about the rotation axis at a pre- determined rotational speed. Clause 2. The system of clause 1, wherein, for each element pair in theplurality of element pairs, the first element is a first triangular prism and the second element is a second triangular prism. Clause 3. The system of clauses 1 or 2, wherein, for each element pair inthe plurality of element pairs, the second element protrudes from a side of the centre portion opposite the first element. Clause 4. The system of any of clauses 1 to 3, wherein each of the pluralityof element pairs are rotated relative to neighbouring element pairs by a constant angle such that the order of rotational symmetry of the device is equal to twice a number of element pairs. Clause 5. The system of any of clauses 1 to 4, wherein each of the pluralityof element pairs, and the part of the centre portion between the corresponding first and second elements, form a rhomboid prism shape. Clause 6. The system of clause 5, comprising three element pairs, whereineach of the plurality of element pairs is rotated at 60 degrees relative to the other two element pairs. Clause 7. The system of any of clauses 1 to 6, wherein the pre-determinedrotational speed is at least 3000 revolutions per minute, rpm. Clause 8. The system of any of clauses 1 to 7, further comprising areflective member adapted to reflect a beam of radiation exiting the non-reflective faceof the second element back into the non-reflective face of the second element, such that, when the beam of radiation is reflected back into the non-reflective face of the second element, the path is further defined as the beam of radiation is reflected by the reflective face of the second element towards the reflective face of the corresponding first element, and further reflected by the reflective face of the first 14630892-1 element towards the non-reflective face of the first element to exit the first element in a second direction parallel to, and opposite, the first direction. Clause 9. A device for varying a path length of a beam of radiation, thedevice comprising: acentre portion configured to allow the passage of radiation there through; anda plurality of element pairs, wherein each element pair comprises a first and second element protruding from the centre portion; wherein each of the first and second elements of the plurality of element pairs comprise a non-reflective face and a reflective face, such that, when the beam of radiation is transmitted through a pre-determined area of the non-reflective face of one of the first elements, in a first direction, a path is defined as the beam of radiation is reflected by the reflective face of the first element towards the reflective face of the corresponding second element, and further reflected by the reflective face of the second element towards the non-reflective face of the second element. Clause 10. The device of clause 9, wherein, for each element pair in theplurality of element pairs, the first element is a first triangular prism and the second element is a second triangular prism. Clause 11. The device of clauses 9 or 10, wherein, for each element pair inthe plurality of element pairs, the second element protrudes from a side of the centre portion opposite the first element. Clause 12. The device of any of clauses 9 to 11, wherein each of theplurality of element pairs are rotated relative to neighbouring element pairs by aconstant angle such that the order of rotational symmetry of the device is equal to a number of element pairs. Clause 13. The device of any of clauses 1 to 3, wherein each of the pluralityof element pairs, and the part of the centre portion between the corresponding first and second elements, form a rhomboid prism shape. 14630892-1Clause 14. The device of clauses 13, comprising three element pairs,wherein each element pair is rotated at 60 degrees relative to the other two element pairs. Clause 15. A method for varying a path length of a beam of radiation, themethod comprising: rotating the device of any of clauses 9 to 14 at a pre-determined rotational speed; and transmitting the beam of radiation through the non-reflective faces of the plurality of first elements of the plurality of element pairs. Clause 16. The system of any of clauses 1 to 8, wherein, for each of theplurality of element pairs, the reflective face of the first element and the reflective faceof the corresponding second element are parallel to within ±5 arcseconds or less. Clause 17. The device of any of clauses 9 to 14, wherein, for each of theplurality of element pairs, the reflective face of the first element and the reflective faceof the corresponding second element are parallel to within ±5 arcseconds or less. Clause 18. A system comprising:a laser configured to output a beam of radiation; a beam splitter configured to split the beam of radiation into a first beam and a second beam; a device comprising: a centre portion rotatably mounted about a rotation axis, the centre portion being configured to allow the passage of radiation there through; and a plurality of element pairs, wherein each element pair comprises a first and second element protruding from the centre portion, wherein each of the first and second elements of the plurality of element pairs comprise a non-reflective face and areflective face, wherein the device is configured to:receive the second beam through a pre-determined area of the non-reflective face of one of the first elements, in a first direction; reflect, by the reflective face of the first element, the secondbeam towards the reflective face of the corresponding second element; and14630892-1reflect, by the reflective face of the second element, the secondbeam towards the non-reflective face of the second element; and amotor configured to rotate the device about the rotation axis at a pre-determined rotational speed. Clause 19. The system of clause 18, wherein the system is a terahertz, THz,time-domain spectroscopy system. Clause 20. The system of clause 19, further comprising:a THz emitter configured to receive the first beam and, in response, emit a THz beam onto a sample; and a detector configured to receive reflected radiation from the sample and a beam of radiation output by the device. While certain embodiments have been described, these embodiments have been presented by way of example only, and are not intended to limit the scope of theinventions. Indeed the novel methods and apparatus described herein may beembodied in a variety of other forms; furthermore, various omissions, substitutions and changes in the form of methods and apparatus described herein may be made. 14630892-1

Claims

CLAIMS:

1. A system for varying a path length of a beam of radiation, the systemcomprising: a device comprising: a centre portion rotatably mounted about a rotation axis, the centre portion being configured to allow the passage of radiation there through; and a plurality of element pairs, wherein each element pair comprises a firstand second element protruding from the centre portion; wherein each of the first andsecond elements of the plurality of element pairs comprise a non-reflective face and a reflective face, such that, when the beam of radiation is transmitted through a pre- determined area of the non-reflective face of one of the first elements, in a first direction, a path is defined as the beam of radiation is reflected by the reflective face of the first element towards the reflective face of the corresponding second element, and further reflected by the reflective face of the second element towards the non-reflectiveface of the second element; anda motor configured to rotate the device about the rotation axis at a pre- determined rotational speed.

2. The system of claim 1, wherein, for each element pair in the plurality of elementpairs, the first element is a first triangular prism and the second element is a secondtriangular prism.

3. The system of claims 1 or 2, wherein, for each element pair in the plurality ofelement pairs, the second element protrudes from a side of the centre portion opposite the first element.

4. The system of any of claims 1 to 3, wherein each of the plurality of elementpairs are rotated relative to neighbouring element pairs by a constant angle such that the order of rotational symmetry of the device is equal to twice a number of element pairs. 14630892-15. The system of any of claims 1 to 4, wherein each of the plurality of elementpairs, and the part of the centre portion between the corresponding first and second elements, form a rhomboid prism shape.

6. The system of claim 5, comprising three element pairs, wherein each of theplurality of element pairs is rotated at 60 degrees relative to the other two element pairs.

7. The system of any of claims 1 to 6, wherein the pre-determined rotational speedis at least 3000 revolutions per minute, rpm.

8. The system of any of claims 1 to 7, further comprising a reflective memberadapted to reflect a beam of radiation exiting the non-reflective face of the second element back into the non-reflective face of the second element, such that, when the beam of radiation is reflected back into the non-reflective face of the second element, the path is further defined as the beam of radiation is reflected by the reflective face of the second element towards the reflective face of thecorresponding first element, and further reflected by the reflective face of the firstelement towards the non-reflective face of the first element to exit the first element in a second direction parallel to, and opposite, the first direction.

9. The system of any of claims 1 to 8, wherein the non-reflective face of each ofthe first and second elements of the plurality of element pairs comprises an anti-reflective coating.

10. A method for varying a path length of a beam of radiation in a system accordingto any of claims 1 to 10, the method comprising: rotating the device at a pre-determined rotational speed; and transmitting the beam of radiation through the non-reflective faces of the plurality of first elements of the plurality of element pairs.

11. A system comprising:a laser (702) configured to output a beam of radiation; a beam splitter configured to split the beam of radiation into a first beam (704) and a second beam (706); 14630892-1a device (716) comprising: a centre portion rotatably mounted about a rotation axis, the centre portion being configured to allow the passage of radiation there through; and a plurality of element pairs, wherein each element pair comprises a first and second element protruding from the centre portion, wherein each of the first and second elements of the plurality of element pairs comprise a non-reflective face and areflective face, wherein the device (716) is configured to:receive the second beam (706) through a pre-determined area of the non-reflective face of one of the first elements, in a first direction; reflect, by the reflective face of the first element, the secondbeam (706) towards the reflective face of the corresponding second element; andreflect, by the reflective face of the second element, the secondbeam (706) towards the non-reflective face of the second element; and amotor configured to rotate the device (716) about the rotation axis at a pre-determined rotational speed.

12. The system of claim 11, wherein the system is a terahertz, THz, time-domainspectroscopy system.

13. The system of claim 12, further comprising:a THz emitter (708) configured to receive the first beam (704) and, in response, emit a THz beam (710) onto a sample; and a detector (714) configured to receive reflected radiation from the sample and a beam of radiation output by the device (716). 14630892-1

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