System and method for surface profile estimation via optical coherence tomography

Abstract

An optical coherence tomography (OCT) system comprises an interferometer configured to split incident light into a reference beam and an inspection beam and to interfere the inspection beam reflected from a sample with the reference beam reflected from a reference mirror to generate an interference pattern. The OCT system also comprises a spectrometer configured to analyze spectral components of the interference pattern at non-uniformly sampled frequencies. The computer-readable memory of the OCT system is configured to store a measurement model having elements weighted with weights derived from the power spectral density (PSD) of the incident light for corresponding frequencies, connecting different depth values to different non-uniformly sampled frequencies. The OCT system further comprises a processor configured to obtain a profilometry measurement of the sample as a maximum likelihood estimate of the sample surface depth by back-projecting the measured intensity in the measurement model.

Description

Technical Field

[0001] The present disclosure generally relates to imaging, and more particularly, to optical coherence tomography (OCT) systems and methods for generating profilometry measurements of a sample.

Background Art

[0002] Profilometry is a technique used to extract topographic data from a surface. This can be a single point, a line scan, or even a complete three-dimensional scan. The purpose of profilometry is to obtain surface topography, step height, and surface roughness. In many applications, electromagnetic sensing is used to obtain information about the surface or subsurface of a particular sample for profilometry measurements. One such technique is tomography. Tomography can be used in various applications, such as radiology, biology, materials science, manufacturing, quality assurance, quality control, etc. Some types of tomography include, for example, optical coherence tomography (OCT), X-ray tomography, positron emission tomography, light projection tomography, etc.

[0003] OCT is a technique used to perform high-resolution cross-sectional imaging. This is often applied to image living tissue structures, such as the human eye, in real time at, for example, the microscopic scale. Light waves are reflected from an object or sample, and a computer uses information about how the light waves change upon reflection to generate an image of a cross-section or three-dimensional volume rendering of the sample.

[0004] OCT is an interferometer-based imaging technique that coherently mixes an optical signal from a target with a reference signal. OCT provides non-invasive, non-contact, non-label imaging of a sample with micron-scale resolution in three dimensions. Due to the ability of OCT to achieve micron-scale resolution, OCT is used across a variety of technical fields, including factory automation processes for checking the integrity of assembly or manufacturing operations, as well as in a variety of medical specialties, including ophthalmology and cardiology.

[0005] OCT can be performed based on time domain processing (time domain OCT or TD-OCT) or Fourier domain processing (Fourier domain OCT or FD-OCT). In time domain OCT (TD-OCT), the optical path length difference between the light returning from the sample and the reference light is temporally longitudinally converted to recover depth information within the sample. In frequency domain or Fourier domain OCT (FD-OCT), broadband interference between the reflected sample light and the reference light is acquired in the frequency domain, and Fourier transform is used to recover depth information.

Summary of the Invention

Problems to be Solved by the Invention

[0006] The advantages of the sensitivity of FD-OCT over TD-OCT are well established. However, FD-OCT still suffers from measurement noise and may require additional calculations and subsampling methods to improve the accuracy of profilometry estimation. See, for example, U.S. Patent 10502544.

Means for Solving the Problems

[0007] The object of some embodiments is to provide an optical coherence tomography (OCT) system and method for generating profilometry measurements of a sample. Additionally, or alternatively, the object of some embodiments is to provide a system and method for Fourier domain OCT (FD-OCT) with an improved signal-to-noise ratio (SNR) of the recovered depth information. Additionally, or alternatively, the object of some embodiments is to overcome the aforementioned drawbacks of the FD-OCT method.

[0008] OCT uses the interference of two light beams to measure the difference in optical path length. The beat frequency of the interfering light is much lower than the oscillation frequency of the light, enabling OCT to achieve fine depth resolution without the need for high-bandwidth electronics. FD-OCT profilometry utilizes fast Fourier transform (FFT)-based processing over the values of the wave number of the interfering signal. Applying a Fourier transform to an interfered signal that is uniformly sampled in wave number should result in sharp peaks in the depth domain. However, OCT systems typically sample the interfered light at a uniform wavelength λ, which means that the samples are non-uniformly spaced at the wave number k = 2π / λ. The processor of an FD-OCT system can interpolate the data and resample it uniformly at the wave number k so that it can use the inverse fast Fourier transform (IFFT) to process the measurements. However, the interpolation process also propagates noise to the non-sampled wave numbers, which reduces the robustness against noise, especially for the higher-frequency interference patterns corresponding to the deepest features of the sample.

[0009] Some embodiments are based on the recognition that depth can be recovered from the backprojection of measurements, rather than using an FFT to recover depth for a single reflector. Backprojection reverses the mapping from the depth domain to the measurement domain through a model of the measurement system. Since backprojection is not usually equivalent to reversing this mapping, it is not suitable for recovering the depths of multiple reflectors. Therefore, the FFT with interpolated wave numbers is usually advantageous for calculating the approximate inverse, compared to backprojection. Thus, it is not surprising that backprojection has not been used for profilometry measurements, to the extent of the available knowledge. However, some embodiments are based on the recognition that under certain conditions, backprojection can be adapted to perform better than the FFT.

[0010] Various embodiments adapt backprojection by modifying the measurement matrix and the structure of the restored data. Specifically, in some embodiments, the backprojection f = M *y generates the vector f from the measurement value y using the measurement matrix M. In the case of opaque surface measurement, the maximum element of the vector f can be obtained as an approximate maximum likelihood estimate for a single surface depth, avoiding interpolation of the input of the backprojection.

[0011] In addition, some embodiments are based on an understanding of the nature of profilometry measurements using an interferometer. The interferometer generates an interference pattern of a beat signal, which is analyzed to measure the intensity of wavelengths uniformly sampled in the interference pattern. This uniform sampling of wavelengths is due to the nature of the physical properties of diffraction. However, there is a non - linear relationship k n = 2π / λ n between the wave number and the wavelength, and the intensity of the uniformly sampled wavelengths corresponds to non - uniformly sampled wave numbers.

[0012] In contrast to the FFT which requires uniformly sampled wave numbers, the measurement matrix can be directly defined for non - uniformly sampled wave numbers corresponding to uniformly sampled wavelengths of the interference pattern. Further, for the depth range of the object, it is possible to obtain a measurement model having elements that connect different depth values to different non - uniformly sampled wave numbers corresponding to uniformly sampled wavelengths. In this way, interpolation inside the backprojection can also be avoided.

[0013] Furthermore, in contrast to the FFT, the measurement matrix of some embodiments includes not only depth and wave number, but also the power spectral density (PSD) S(k n ) required for different wave numbers k n . This is equivalent to the amplitude envelope that multiplies the measurement values.

[0014] The PSD in the measurement matrix accounts for the weights that each element of the data should receive in the backprojection. In this way, the PSD increases the robustness of the backprojection by depending more strongly on samples with higher envelope amplitudes.

[0015] Furthermore, some embodiments explicitly define a measurement matrix M for a set (m = 0, ..., M-1) of possible depths z. These possible depths can be selected with any coarse or fine resolution as desired, and over any range of depths, regardless of what that range is. For example, OCT measurements are typically made with respect to a reference depth z = 0. The sample is kept entirely above or below the reference depth; otherwise, ambiguity arises. Thus, it is possible to reconstruct only positive (or only negative) depth values. m

[0016] Some exemplary embodiments may be implemented for process monitoring in manufacturing. For example, without limitation, some exemplary embodiments may be included in computer numerical control (CNC) machines such as fabrication machines, electrical discharge machining (EDM) machines, wire EDM machines, and the like.

[0017] To achieve the foregoing objects and advantages, some exemplary embodiments provide a system, method, and program for profilometry measurements of a sample.

[0018] For example, some exemplary embodiments provide an OCT system for profilometry measurements of a sample. The OCT system includes an interferometer configured to split incident light into a reference beam and an interrogation beam and to interfere the interrogation beam reflected from the sample with the reference beam reflected from a reference mirror to generate an interference pattern. The OCT system also includes a spectrometer configured to analyze spectral components of the interference pattern at non-uniformly sampled frequencies. The computer-readable memory of the OCT system is configured to store a measurement model having elements weighted by weights derived from the power spectral density (PSD) of the incident light for corresponding frequencies, connecting different depth values to different non-uniformly sampled frequencies. The OCT system further includes a processor configured to determine a profilometry measurement of the sample as the maximum likelihood estimate of the sample surface depth by back-projecting the measured intensity in the measurement model.

[0019] ​ Some exemplary embodiments also provide a method for profile measurements of a sample in an OCT system. The method includes the interferometer splitting the incident light into a reference beam and an interrogation beam, and interfering the interrogation beam reflected from the sample with the reference beam reflected from the reference mirror to generate an interference pattern. The method further includes the spectrometer analyzing spectral components of the interference pattern at unevenly sampled frequencies. The computer-readable memory of the OCT system stores a measurement model having elements weighted by weights derived from the power spectral density (PSD) of the incident light for corresponding frequencies, connecting different depth values to different unevenly sampled frequencies. The method further includes obtaining a profile measurement of the sample as a maximum likelihood estimate of the surface depth of the sample by back-projecting the measured intensity in the measurement model.

[0020] Some exemplary embodiments also provide a non-transitory computer-readable medium storing computer-executable instructions that, when executed by a computer, cause the computer to perform a method for profile measurements of a sample in an OCT system. The method includes the interferometer splitting the incident light into a reference beam and an interrogation beam, and interfering the interrogation beam reflected from the sample with the reference beam reflected from the reference mirror to generate an interference pattern. The method further includes the spectrometer analyzing spectral components of the interference pattern at unevenly sampled frequencies. The computer-readable memory of the OCT system stores a measurement model having elements weighted by weights derived from the power spectral density (PSD) of the incident light for corresponding frequencies, connecting different depth values to different unevenly sampled frequencies. The method further includes obtaining a profile measurement of the sample as a maximum likelihood estimate of the surface depth of the sample by back-projecting the measured intensity in the measurement model.

[0021] According to some exemplary embodiments, the depth values are uniformly sampled from the depth measurement range at the resolution of the OCT system. The depth values may be relative values with respect to a reference depth selected outside the depth measurement range.

[0022] As part of this method, each profilometry measurement may be estimated by performing a maximum likelihood estimator (MLE) to generate an argument of the maximum likelihood estimate of the non-zero elements in the corresponding reflectivity vector. Further, each argument of the reflectivity vector corresponds to one of the depth values in the measurement model. Further, the MLE may be an approximate MLE, and performing the approximate MLE includes back-projecting a data vector through a measurement matrix. The MLE may be the depth value corresponding to the element of the largest magnitude in the back-projection.

[0023] According to some exemplary embodiments, the MLE may be an exact MLE, and performing the exact MLE includes refining the approximate MLE by using a gradient-free optimization method to maximize the maximum likelihood objective function.

[0024] Embodiments of the present disclosure will be described below with reference to the following drawings. The drawings shown are not necessarily to scale; instead, emphasis is generally placed on explaining the principles of the embodiments of the present disclosure. BRIEF DESCRIPTION OF THE DRAWINGS

[0025]

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Best Mode for Carrying Out the Invention

[0026] The drawings identified above describe embodiments of the present disclosure, but as discussed, other embodiments are also contemplated. The present disclosure is presented as representative and not as limiting exemplary embodiments. Those skilled in the art can devise numerous other modifications and embodiments that fall within the scope and spirit of the principles of the embodiments of the present disclosure. [Description of Embodiments]

[0027] The following description provides only exemplary embodiments and is not intended to limit the scope, applicability, or configuration of the present disclosure. Rather, the following description of exemplary embodiments provides those skilled in the art with a feasible description for implementing one or more exemplary embodiments. Various changes can be made in the functions and configurations of elements without departing from the spirit and scope of the disclosed subject matter as recited in the claims.

[0028] In the following description, specific details are provided for a complete understanding of the embodiments. However, those skilled in the art will be able to understand that the embodiments can be implemented without these specific details. For example, systems, processes, and other elements in the disclosed subject matter may be shown as components in the form of block diagrams so as not to obscure the embodiments with unnecessary details. In other instances, well-known processes, structures, and techniques may be shown without unnecessary details in order to avoid obscuring the embodiments. Further, like reference numbers and names in the various drawings indicate like elements.

[0029] Also, individual embodiments may be described as a process shown as a flowchart, a flow diagram, a data flow diagram, a structural diagram, or a block diagram. A flowchart can describe the operations as a sequential process, but many of the operations can be executed in parallel or simultaneously. Additionally, the order of the operations may be rearranged. The process may end when its operations are completed, but may have additional steps not discussed or included in the figure. Further, not all operations in any particular process described will occur in all embodiments. A process may correspond to a method, a function, a procedure, a subroutine, a subprogram, etc. When a process corresponds to a function, the end of the function can correspond to a return to the calling function or the main function of that function.

[0030] Furthermore, embodiments of the disclosed subject matter may be realized, at least in part, either manually or automatically. Manual or automatic realizations may be executed or at least assisted through the use of a machine, hardware, software, firmware, middleware, microcode, a hardware description language, or any combination thereof. When realized in software, firmware, middleware, or microcode, the program code or code segments for performing the necessary tasks may be stored on a machine-readable medium. The necessary tasks may be executed by a processor.

[0031] To measure the surface profile of a material surface, a quantified measurement value of the material surface is required. This can be done by profilometry where a mechanical (contact) or optical (non-contact) probe passes across the surface. The probe traces the contour at each point on the surface, the height of the probe at each point is recorded, and the resulting 1D scan or 2D map is analyzed. To quantify roughness, parameters such as the arithmetic mean (Ra) of the absolute values of all points of the profile, the root mean square value (Rq) of the total height of the mean perimeter, etc. are often used. The profilometer generates an image of the surface height. The size of the measured area and the size of the probe set upper and lower limits on the size of the features that can be characterized. The nature of the probe limits the range of surfaces that can be investigated by these techniques. In this regard, optical techniques are more suitable for relatively soft materials.

[0032] Optical profilometry is a more recent and modern technique that has been developed to improve accuracy. Briefly, a light source is used to scan the sample surface, and the light beam diffracted by the surface roughness is collected on a mirror. The generated image is the displacement of the light beam on the mirror. With this technique, theoretically, it is possible to evaluate roughness as low as on the order of nanometers.

[0033] Optical profilometry is a rapid, non-destructive, and non-contact surface measurement technique. An optical profiler is a type of microscope in which light from a lamp is split into two paths by a beam splitter. One path directs the light towards the surface under inspection, and the other path directs the light towards a reference mirror. The reflections from the two surfaces are recombined and projected onto an array detector. Interference can occur if the path difference between the recombined beams is on the order of light of a few wavelengths or less. This interference contains information about the surface profile of the inspected surface. The vertical resolution can be on the order of a few angstroms, but the lateral resolution depends on the purpose and is typically in the range of a few microns.

[0034] In many applications, electromagnetic sensing is used in profilometry measurements to obtain information about the surface or subsurface of a particular sample. One such technique is tomography. Some types of tomography include, for example, optical coherence tomography (OCT), x-ray tomography, positron emission tomography, optical projection tomography, and the like. OCT is a technique used to perform high-resolution cross-sectional imaging. This is often applied to image biological tissue structures, such as the human eye, in real time at, for example, the microscopic scale. Light waves are reflected from an object or sample, and a computer uses information about how the light waves change upon reflection to generate an image of a cross-section or three-dimensional volume rendering of the sample.

[0035] OCT uses the interference of two light beams to measure the difference in optical path lengths. The beat frequency of the interfering light is much lower than the oscillation frequency of the light, which reduces the need for high-bandwidth electronics. FD-OCT profilometry utilizes fast Fourier transform (FFT)-based processing over the values of the wave number of the interfering signal. Applying a Fourier transform to an interfering signal that is uniformly sampled in wave number should result in sharp peaks in the depth domain. However, OCT systems typically sample the interfering light at a uniform wavelength λ, which means that the sample is non-uniformly spaced at the wave number k = 2π / λ. The processor of an FD-OCT system can interpolate the data and resample it uniformly at the wave number k so that it can process the measurements using an inverse fast Fourier transform (IFFT). However, the interpolation process also propagates noise to the non-sampled wave numbers, which reduces the robustness to noise, particularly for the higher frequency interference patterns corresponding to the deepest features of the sample.

[0036] FIG. 1A shows a method 100A for profilometry measurements of a sample in an OCT system according to some exemplary embodiments. Method 100A may be performed by some or all of the components of the OCT system, which will be described in detail later with reference to FIG. 1B. The profilometry measurement method 100A includes a step 3 of splitting an incident light beam into a reference beam and an inspection beam. This can be performed by an interferometer 104 of the OCT system. According to some exemplary embodiments, a beam splitter may be utilized in this regard. In step 5, method 100A includes interfering the inspection beam reflected from the sample with the reference beam reflected from the reference mirror to generate an interference pattern.

[0037] In step 7, the method includes analyzing spectral components of the interference pattern at non-uniformly sampled wave numbers. In step 9, a processor 110 of the OCT system utilizes a computer-readable memory 112 to obtain a profilometry measurement of the sample as the maximum likelihood estimate of the sample surface depth by back-projecting the measured intensity in a measurement model. The computer-readable memory 112 of the OCT system stores a measurement model having elements weighted by weights derived from the power spectral density (PSD) of the incident light for corresponding wave numbers, connecting different depth values to different non-uniformly sampled wave numbers.

[0038] According to some exemplary embodiments, the depth values are uniformly sampled at the resolution of the OCT system from the depth measurement range. In some exemplary embodiments, the depth values are relative values with respect to a reference depth selected outside the depth measurement range. The profilometry measurement obtained in this way by the processor 110 may be output via an interface 120 of the OCT system (11).

[0039] One or more components, such as interferometer 104, spectrometer 106, interface 120, and / or memory 112, may be communicatively coupled to processor 110. Processor 110 may be additionally coupled to one or more additional processing circuits for performing additional processing. Processor 110 may execute one or more operations, such as communication operations, read / write operations, and / or control operations of the one or more components described above. The profilometry measurement method includes several modules that will be described in detail below. First, an overview of the OCT system is provided with reference to FIGS. 1B and 1C to understand the components and elements utilized to implement the OCT system.

[0040] FIG. 1B shows a schematic diagram of an OCT system 100B that generates a depth estimate 116 of a sample 118 from measurements 108 made at a non-uniformly sampled wave number 114. OCT system 100B includes a light source 102 (also referred to as an illumination source), an interferometer 104, a spectrometer 106, a processor 110, and a memory 112. In some exemplary embodiments, sample 118 may be opaque and may have a single visible surface.

[0041] Light source 102 may comprise any suitable illumination source that supplies a light beam or electromagnetic beam for investigating the sample. The choice of illumination source may depend on the target sample and / or the intended use of the OCT system. For example, without limitation, light source 102 may comprise one or more of a wavelength-variable laser, an LED array, an incandescent light source, a rare gas-based lamp, a radiation source such as an X-ray generator, a photon emitter, a positron emitter, etc. According to some exemplary embodiments, light source 102 includes one or a combination of a laser, a superluminescent diode (SLD), or a light-emitting diode (LED).

[0042] In some exemplary embodiments, the light source 102 may be configured to utilize a planar shape, a fan beam shape, a point illumination, or any combination thereof. The point illumination may be provided by any beam steering mirror-like device such as electromechanical, optoelectronic, acousto-optic, all optical system technologies, liquid crystal mirrors, and any other such devices.

[0043] The beam emerging from the light source 102 may include light having coaxial, orthogonally polarized and / or different optical frequencies. The beam is split by a beam splitter of the interferometer 104. In some exemplary embodiments, the interferometer 104 may be a Michelson interferometer. In some exemplary embodiments, the interferometer 104 may be a Linnik interferometer. According to some exemplary embodiments, the beam splitter may be a partially reflective mirror. In some exemplary embodiments, the beam splitter may be a non-polarizing beam splitter. The beam splitter can split the beam into a reference illumination transmitted to the reference mirror and a sample illumination transmitted to the sample 118.

[0044] According to some exemplary embodiments, the beam splitter may optionally comprise a series of beam splitters and / or polarizers. The sample illumination is incident on the sample 118, and all or part of the sample illumination may be reflected from the sample towards the beam splitter. The reflected signal from the sample 118 may be split by the beam splitter, and at least a portion thereof is combined with the reflected reference illumination and directed towards the detector array of the spectrometer 106 for further analysis and detection. The detector array of the spectrometer 106 may comprise a suitable imaging device such as a charge-coupled device camera. The detector array can provide one or more detection signals corresponding to the recombination of the reflected signal and the reference signal.

[0045] The sample illumination may include an electromagnetic two-dimensional (2D) field directed by the interferometer 104 to form an axial scan of the sample 118 such that the measured intensity of the interference pattern includes measurements corresponding to a succession of points on the line of the sample 118. In some exemplary embodiments, the OCT system 100B may also include one or more actuators for directing the incident light to another line parallel to the line of the previous scan.

[0046] The processor 110 can extract a succession of intensities corresponding to a succession of points on the line of the sample 118. Additionally, the processor 110 can process the intensities of different points simultaneously to generate profilometry measurements for the succession of points. In some exemplary embodiments, the OCT system 100B may comprise or be additionally coupled with one or more processing circuits for generating profilometry measurements for at least some of the points in the succession of points in parallel. The one or more processing circuits can comprise suitable processing means such as a processor and a memory.

[0047] According to some exemplary embodiments, the OCT system 100B may additionally include a line field generator that includes an extended light source with an angular size larger than the lateral resolution across the profilometry measurements, a lens disposed on the path of the light emitted by the extended light source to focus the light into an extended line field light with a width larger than the lateral resolution, and a filter disposed in the focal plane of the lens to spatially filter the extended line field light into a line field with a width equal to the lateral resolution of the incident light.

[0048]

Number

[0049]

Number

[0050]

Number

[0051] The steps of preprocessing and depth estimation may be performed by the processor 110 of the computer system 150. The DC component is removed from the measurement value 108. In some exemplary embodiments, the DC component is removed by subtracting the scaled PSD from the raw measurement as follows.

Number

[0052] In some exemplary embodiments, the DC component is removed by applying a high-pass filter to the raw measurement. The resulting interference data vector y has values given as follows for element n,

Number

[0053] As described above, the spectrometer 106 includes a diffraction grating 140 and a detector array 142. The detector array 142 may have detector elements arranged at different diffraction angles to measure the intensities of different beams corresponding to the intensities of wavelengths uniformly sampled in the interference pattern. The detector elements of the detector array 142 are calibrated to map each index of the detector elements within the detector array to the corresponding wavelength.

[0054] FIG. 2A shows a process 200 for obtaining a maximum likelihood estimator (MLE) 214 for the depth of the sample surface 118 given a data vector 202, a wavenumber calibration 204, a power spectral density calibration 206, and a set 208 of candidate depths. Assuming the noise is Gaussian noise, the likelihood of observing the data y is as follows.

Number

[0055] [Mathematics] Since the negative log-likelihood is highly multimodal, it is advantageous to perform the minimization in the two-step procedure of the coarse estimate 210 and the fine estimate 212. In some exemplary embodiments, the coarse estimate step 210 may be sufficient for the depth estimate 214, and in such cases, the depth fine estimate step 212 may be optional.

[0056] FIG. 2B shows a detailed process leading to the coarse estimate step 210 of FIG. 2A according to some exemplary embodiments. For the coarse step, some embodiments recognize that a slowly varying PSD is approximated by the value of z that maximizes the MLE. [Mathematics] The advantage of this approximation is that it can be efficiently evaluated via matrix-vector multiplication in a discrete set of candidate depths.

[0057]

[0058] [Mathematics] FIG. 2C shows an exemplary structure of a measurement matrix M220 according to some exemplary embodiments. As the measurement matrix M220, it is conceivable to select a matrix that satisfies the conditions described above for the elements.

[0059] [Mathematics] In many cases, the approximate coarse estimate is accurate enough. However, it is recognized in some embodiments that further accuracy can be achieved by directly maximizing F(z) when the measurements have a sufficiently high signal-to-noise ratio (SNR).

[0060]

Mathematics

[0061] The correct depth MLE minimizes the negative log-likelihood and is the value of z that maximizes F(z), i.e.,

Mathematics

[0062]

Mathematics

[0063] According to some embodiments, the processors 110 of FIGS. 1B and 1C may estimate each profilometry measurement by performing a maximum likelihood estimator (MLE) to generate an argument of the maximum likelihood estimate of the non-zero elements in the corresponding reflectivity vector. Each argument of the reflectivity vector corresponds to one of the depth values in the measurement model. The MLE may be an approximate MLE or an exact MLE. In some exemplary embodiments where the MLE is an approximate MLE, performing the approximate MLE includes back-projecting the data vector through the measurement matrix, and the MLE is the depth value corresponding to the element of the largest magnitude in the back-projection. In some exemplary embodiments where the MLE is an exact MLE, performing the exact MLE includes refining the approximate MLE by maximizing the maximum likelihood objective function using a gradient-free optimization method such as Brent's minimization method or the golden section search. Advantages of ML Depth Estimation

[0064] A common approach for OCT-based surface estimation is to compute the Fourier transform of the data and find the peaks. However, the fast Fourier transform (FFT) algorithm cannot be directly applied to the data vector y because the FFT requires that the samples of y be uniformly spaced in frequency. In a spectral domain OCT (SDOCT) system, a dispersive element such as diffraction grating 140 causes a nearly linear change in angle as a function of wavelength. As a result, referring to FIG. 3, detector array 142 uniformly samples the interference signal at wavelength 300, resulting in non-uniform samples at frequency 304.

[0065] In a swept source OCT (SSOCT) system, existing methods for achieving samples with uniform frequency require additional complex hardware. These involve either seeking a non-linear sweep of the drive current that would result in temporally uniform time samples, or using an arbitrary drive current and an additional reference (e.g., a k-clock based on an etalon or a Michelson interferometer) to determine when to sample non-uniformly in time corresponding to uniform frequency samples. Instead of using these complex methods, a uniform frequency interval is typically achieved via software post-processing. The measurements are made at non-uniform frequency samples, and the signal is interpolated and resampled such that the sample interval is uniform (i.e., uniform k-interval) at frequency 302 as shown in FIG. 3.

[0066] A second approach for OCT-based surface estimation attempts to invert the measurement y using M and sparse recovery methods that apply the assumption of a small number of surfaces. However, sparse solvers are too general for typical scenarios, which assume that a single surface may exist, and thus these solvers are much slower than the FFT method.

[0067] The ML method has the following advantages compared to the FFT method. First, the ML method avoids interpolation: the measurement matrix M measures the frequency k regardless of the distribution of the samples. nIt is explicitly defined in []. On the other hand, in the fast Fourier transform (FFT), it is necessary to uniformly sample the measured values at the frequencies. FIG. 3 shows how a sample that is uniform at a wavelength of 300 reaches a sample that is non-uniform at a frequency of 304. The measured values taken at the non-uniform frequency 310 must be interpolated and resampled at a uniform frequency 312 before the FFT can be applied, and the interpolation is undesirable because it also interpolates the noise.

[0068] Second, the ML method identifies the useful measurement range: The measurement matrix M is explicitly defined for a set of possible depths z from m = 0,..., M-1 m These z m can be selected at that resolution, regardless of what coarse or fine resolution is desired, and over that range of depths, regardless of what range of depths is relevant. For example, OCT measurements are typically made with respect to a reference depth z = 0. According to some exemplary embodiments, as shown in FIG. 4, the sample surface 404 is held entirely on one side of the reference plane 400 so that there is no ambiguity. That is, no calculations are required for negative depths 402. Therefore, it is easy to reconstruct only positive depth values. On the other hand, the FFT automatically calculates depth profiles for both positive and negative depths, which adds unnecessary calculations. Furthermore, the FFT resolution is inversely proportional to the length of the data vector. For finer depth resolution, the measured values are usually zero-padded to increase its length, which also increases the calculation time.

[0069] Third, the ML method includes all available information: The measurement matrix includes not only depth and frequency, but also the power spectral density (PSD) S(k n ). This is equivalent to the amplitude envelope that multiplies the measured values. Including the PSD in M appropriately accounts for the weights that each element of the data should receive in the backprojection. The FFT does not include the PSD. As a result, the depth region is convolved with the Fourier transform of the PSD, the peaks broaden, and it becomes more difficult to identify the true peaks.

[0070] Compared to sparse recovery methods, ML estimation has the additional advantage that the ML method enables fast implementation: backprojection multiplies the data by the adjoint measurement matrix M* for which the calculations (transpose and conjugate) are trivial. More general sparse reconstruction methods require regularized least-squares solutions which are iterative and much slower. Modified example of the SS-OCT system

[0071] In one SS-OCT configuration, the illumination source sweeps one wavelength at a time. Since the wavelengths are separated in time, the spectrometer is a single-pixel detector that measures the intensity of the combined light at the time samples n = 0, ..., N-1 that cover the wavelength sweep of the light source. Some implementations use a balanced detector to remove the DC component of the measurements in hardware. Calibration procedure

[0072] The measurements made by detector array 142 can be given as I D [n], but the data required for the estimation may require a conversion of the measurements to the form given by Equation 4(a). The detector measurements have a linear index n, but to accurately recover the absolute depth, k nThe actual value is required. FIG. 5 shows a schematic diagram of wavelength calibration for determining the wavelength associated with each detector pixel of the detector array 142 according to some exemplary embodiments. The illumination source 102, interferometer 104, spectrometer 106, and computer system 150 may be the same as those described with reference to FIG. 1C and may operate in the same manner as described with reference to FIG. 1C. As shown in FIG. 5, in some exemplary embodiments, a "reference only" measurement 504 is performed on the detector array 142 by blocking the sample arm 506. For example, the sample may be masked by a non-reflective surface to block the sample arm. In this case, the intensity at the detector 142 is due only to the reference arm reflected from the reference mirror 132, and thus it includes the power spectral density but does not include any of the interference terms. In some embodiments, the spectrometer 106 has a diffraction grating 140 that causes linear dispersion as a function of wavelength. In such a scenario, the only light reaching the detector array 142 is from the reference arm. In that case, the intensity at the detector is as follows.

Number

[0073] According to some exemplary embodiments, the OCT system 100B may further include a PSD calibrator for blocking the sample arm of the interferometer 104, and the measurement value includes only the light from the reference beam propagating within the reference arm of the interferometer 104. The measured intensity of the interference pattern is a function of the PSD of the incident light for the corresponding wavenumber scaled by the responsiveness of the spectrometer 106 and the reflectivity of the reference arm. During the execution of the PSD calibrator, the processor 110 is configured to calibrate the PSD of the incident light such that the wavenumber corresponding to each pixel of the spectrometer 106 is estimated.

[0074] The reference measurement has an index n and an associated wavenumber value k nis unknown. These associated wavenumber values are required to accurately recover the absolute depth. The method for determining the associated wavenumber values is based on the wavelength calibration procedure described below.

[0075] The wavenumber calibration procedure, shown in FIGS. 6A - 6E, involves aligning two measurements of the PSD, one made with a standard inspection device and one made with the OCT system 50. The light source spectrum is measured by an optical spectrum analyzer (OSA) 500 that measures the illumination intensity I GT (λ) as a function of wavelength. In that case, I GT (λ) is rescaled and aligned with respect to a reference measurement I C [n] to find a fit between the measurements that allows the pixel index to be directly mapped to the wavelength.

[0076]

Number

[0077] FIG. 6A shows a wavenumber calibration method according to some exemplary embodiments. FIG. 6A is described in relation to FIG. 5. The optical spectrum analyzer 500 of FIG. 5 provides a calibration measurement 502, and the detector array 142 provides a reference measurement 504 in the manner described above.

Number

[0078] Referring back to FIG. 6A, the calibration measurement values are interpolated and resampled (606) based on the spectral bandwidth of the reference measurement values. In particular, as shown in FIG. 6D, the LED source spectrum 650A is interpolated and resampled such that the uniform wavelength sample intervals

Number

[0079] The resampled calibration measurement values 650C and the reference mirror spectrum (i.e., the reference measurement value 650B) are then cross-correlated (608) to find a shift that maximizes the overlap between spectra 650C and 650B. The cross-correlation 655 between the LED spectrum and the reference mirror spectrum is shown in FIG. 6E. Wavelength calibration is ultimately achieved by aligning the two spectra 650C and 650B (610) and generating a direct mapping 657 between the wavelengths from the calibration wavelength and the reference measurement indices, which is used to assign detector indices along with their true wavelengths (612).

[0080] FIG. 7 shows a schematic comparison of the experimental results of applying the maximum likelihood estimator to simulated data of OCT surface measurement values. Note that the comparison shown in FIG. 7 is non-limiting and for illustrative purposes only, and it is contemplated that experimental values of various parameters may be set to different sets of values. The illumination source may be set to have a Gaussian spectrum with a central wavelength of about 550 nm and an FWHM bandwidth of 100 nm. The sample shown at 700 is a one-dimensional linear ramp spanning a depth of approximately 10 - 23 μm. The reflectivity at each pixel may be set to a constant value of 0.05, and a phase shift is added uniformly and randomly over [0, 2π). The measurement 702 is performed at 500 wavelengths, and reconstruction is calculated for a depth resolution of 25 nm and a maximum depth of 25 μm. Additive white Gaussian noise is added such that the measurement has a signal-to-noise ratio (SNR) of -10 dB.

[0081] In 704, a conventional method (IFFT) is applied that uses linear interpolation of the measured values 702 to obtain uniform wavenumber samples and is inverted via the FFT algorithm. The error between the FFT estimate 704 and the ground truth 700 is 712, which indicates a significant error in surface depth estimation. In 706, another conventional method (IDFT) is applied that uses linear interpolation of the measured values 702 to obtain uniform wavenumber samples, but the inversion is instead performed by explicitly specifying a partial inverse discrete Fourier transform matrix for a small range of only positive depth values. The error between the FFT estimate 706 and the ground truth 700 is 714, which is the same as 712 and indicates a significant error in surface depth estimation.

[0082] In 708, a coarse step (backprojection) of the proposed maximum likelihood estimator (ML grid: depth MLE on a discrete grid) is applied directly to the measured values 702 without interpolation. The surface estimation error between the ML coarse estimate 708 and the ground truth 700 is 716, which is significantly smaller than 712 and 714. In 710, a fine ML estimation step (ML-iter: depth MLE with iterative refinement) is applied directly to the measured values 702 using the result from the coarse step 708 as an initialization. The error between the ML fine estimate 710 and the ground truth 700 is 718, which is significantly smaller than 712. Thus, an exemplary embodiment based on the ML fine estimation method provides several advantages over conventional available solutions.

[0083] Figure 8 shows a performance comparison between surface depth estimation methods. Note that the comparison shown in Figure 7 is non-limiting and for illustrative purposes only, and it is contemplated that experimental values of various parameters may be set to different sets of values. At 800, the depth estimation root mean square error (RMSE) is plotted against the SNR averaged over 10 trials. The RMSE is compared to the square root of the Cramer-Rao lower bound (CRLB), which gives a lower bound on the range accuracy for unbiased estimators. The expected RMSE limits for discrete estimators are also plotted. The backprojection method, FFT method, and DFT method are limited to discrete grids with grid spacing δ z so that, assuming a uniformly distributed depth, the root mean square error (RMSE) is limited to

Equation

[0084] The ML coarse estimator (backprojection) is faster than the inverse DFT matrix because it avoids the wavenumber interpolation step. Both the ML coarse estimator method and the inverse DFT method using explicitly defined matrices are faster than the inverse FFT algorithm, which performs unnecessary calculations for negative and out-of-range depth values. The ML fine estimator method requires exactly twice the execution time of conventional FFT-based methods.

[0085] FIG. 9 shows a block diagram of a system for implementing OCT according to an embodiment of the present disclosure. Computer 911 includes a processor 940, a computer-readable memory 912, a storage 958, and a user interface 949 with a display 952 and a keyboard 951, which are connected via a bus 956. For example, when the user interface 949 that communicates with the processor 940 and the computer-readable memory 912 receives an input from the surface of the user interface 957 or the keyboard 953 by the user, it acquires image data and stores it in the computer-readable memory 912.

[0086] Computer 911 can include a power supply 954 according to the application, and the power supply 954 may optionally be disposed outside the computer 911. Via the bus 956, a user input interface 957 adapted to connect to a display device 948 can be linked, and the display device 948 can include, among other things, a computer monitor, a camera, a television, a projector, or a mobile device. A network interface controller (NIC) 934 is adapted to connect to a network 936 via the bus 956 and can, among other things, render image data or other data on a third-party display device, a third-party imaging device, and / or a third-party printing device external to the computer 911.

[0087] Referring further to FIG. 9, among other things, image data or other data may be transmitted via the communication channels of network 936 and / or stored within storage system 958 for storage and / or further processing. Further, time series data or other data may be received wirelessly or wired from receiver 946 (or external receiver 938), or transmitted wirelessly or wired via transmitter 947 (or external transmitter 939), and both receiver 946 and transmitter 947 are connected via bus 956. Computer 911 may be connected to external sensing device 944 and external input / output device 941 via input interface 908. For example, external sensing device 904 may include sensors that collect data before, during, and after the time series data collected by the machine. Computer 911 may be connected to other external computers 942. Output interface 909 may be used to output the data processed by processor 940. It should be noted that user interface 949, which communicates with processor 940 and non-transitory computer-readable storage medium 912, acquires region data and stores it in non-transitory computer-readable storage medium 912 when receiving an input from the surface of user interface 949 by the user.

[0088] The foregoing description provides only exemplary embodiments and is not intended to limit the scope, applicability, or configuration of the present disclosure. Rather, the following description of the exemplary embodiments provides a feasible description for implementing one or more exemplary embodiments to those skilled in the art. Various changes may be contemplated in the functions and arrangements of the elements without departing from the spirit and scope of the subject matter disclosed as set forth in the claims.

[0089] In the following description, specific details are given for a thorough understanding of the embodiments. However, one skilled in the art will be able to understand that the embodiments can be implemented without these specific details. For example, systems, processes, and other elements in the disclosed subject matter may be shown as components in the form of block diagrams so as not to obscure the embodiments with unnecessary details. In other instances, well-known processes, structures, and techniques may be shown without unnecessary detail to avoid obscuring the embodiments. Additionally, like reference numbers and names in the various drawings indicate like elements. Also, individual embodiments may be described as a process shown as a flowchart, a flow diagram, a data flow diagram, a structural diagram, or a block diagram. A flowchart may describe operations as a sequential process, but many of the operations can be performed in parallel or simultaneously. In addition, the order of the operations can be rearranged. The process may end when its operations are completed, but may have additional steps not discussed or included in the figure. Furthermore, not all operations in any of the processes specifically described may occur in all embodiments. A process may correspond to a method, a function, a procedure, a subroutine, a subprogram, etc. When a process corresponds to a function, the end of the function can correspond to a return to the calling function or the main function of that function.

[0090] Furthermore, embodiments of the disclosed subject matter may be implemented, at least in part, either manually or automatically. Manual or automatic implementations may be carried out through, or at least assisted by, the use of a machine, hardware, software, firmware, middleware, microcode, hardware description language, or any combination thereof. When implemented in software, firmware, middleware, or microcode, the program code or code segments for performing the required tasks may be stored on a machine-readable medium. The required tasks may be executable by a processor. The various methods or processes outlined herein may be coded as software executable on one or more processors using any one of a variety of operating systems or platforms. Additionally, such software may be written using any of several suitable programming languages and / or programming or scripting tools, and may be compiled as executable machine language code or intermediate code that is executed on a framework or virtual machine. Typically, the functionality of program modules may be combined or distributed as desired in various embodiments.

[0091] Embodiments of the present disclosure may be embodied as a method for which examples are provided. The acts performed as part of the method can be ordered in any suitable way. Thus, embodiments can be constructed in which some acts shown as consecutive acts in an exemplary embodiment are performed simultaneously, with the acts being performed in a different order than the example. Further, the use of ordinal language such as "first," "second," etc. to modify the elements of a claim does not, in itself, mean that the priority, precedence, or order of one element of a claim exceeds that of another element of a claim, or the chronological order in which the acts of the method are performed, but rather is used merely as a label to distinguish one claim element having a certain name from another element having the same name (but for which ordinal language is used) to distinguish those claim elements. The present disclosure has been described with reference to certain preferred embodiments, but it should be understood that various other adaptations and modifications can be made within the spirit and scope of the present disclosure. Accordingly, the claims are intended to cover all such variations and modifications that fall within the true spirit and scope of the present disclosure.

Claims

1. An optical coherence tomography (OCT) system for profilometry measurements of a sample, comprising: An interferometer that splits incident light into a reference beam and an inspection beam, and interferes the inspection beam reflected from the sample and the reference beam reflected from a reference mirror to generate an interference pattern; A spectrometer configured to analyze spectral components of the interference pattern at non-uniformly sampled frequencies; A computer-readable memory configured to store a measurement model having elements weighted by weights derived from the power spectral density (PSD) of the incident light for corresponding frequencies, connecting different depth values to different non-uniformly sampled frequencies; A processor configured to obtain a profilometry measurement of the sample as a maximum likelihood estimate of the surface depth of the sample by backprojecting the measured intensity in the measurement model. An OCT system comprising:

2. The OCT system according to claim 1, wherein the depth values are uniformly sampled at the resolution of the OCT system from a depth measurement range, and the depth values are relative values with respect to a reference depth selected outside the depth measurement range.

3. The OCT system according to claim 1, wherein the processor is configured to estimate each profilometry measurement by performing a maximum likelihood estimator (MLE) to generate an argument of a maximum likelihood estimate of non-zero elements in a corresponding reflectivity vector, and each argument of the reflectivity vector corresponds to one of the depth values in the measurement model.

4. The OCT system according to claim 3, wherein the MLE is an approximate MLE, and performing the approximate MLE includes backprojecting a data vector through a measurement matrix, and the MLE is a depth value corresponding to an element having the largest magnitude in the backprojection.

5. The OCT system according to claim 3, wherein the MLE is an exact MLE, and performing the exact MLE includes refining the approximate MLE by maximizing a maximum likelihood objective function using a gradient-free optimization method.

6. The spectrometer comprises: A diffraction grating configured to diffract different beams of different wavelengths forming the interference pattern at different diffraction angles; The OCT system according to claim 1, further comprising a detector array having detector elements arranged at different diffraction angles for measuring the intensities of different beams corresponding to the intensities of uniformly sampled wavelengths in the interference pattern.

7. The OCT system according to claim 6, wherein the detector elements of the detector array are calibrated to map each index of the detector elements in the detector array to a corresponding wavelength.

8. The incident light includes an electromagnetic two-dimensional (2D) field directed by the interferometer to form an axial scan of the sample such that the measured intensity of the interference pattern includes measurements corresponding to a succession of points on the line of the sample, and the processor is further configured to extract a succession of intensities corresponding to the succession of points, simultaneously process the intensities of different points respectively to generate profilometry measurements of the succession of points. The OCT system according to claim 1.

9. The OCT system according to claim 8, further comprising a plurality of processing circuits for generating the profilometry measurements in parallel for at least some of the points in the succession of points, including the processor.

10. The OCT system according to claim 8, further comprising an actuator for directing the incident light to another line parallel to the line of the previous scan.

11. The OCT system according to claim 1, further comprising an illumination source for generating the incident light, the illumination source including one or a combination of a laser, a superluminescent diode (SLD), or a light emitting diode (LED).

12. The OCT system according to claim 1, further comprising a line field generator including an extended light source having an angular size larger than the lateral resolution over the profilometry measurements, a lens disposed on the path of the light emitted by the extended light source for focusing the light into an extended line field light having a width larger than the lateral resolution, and a filter disposed on the focal plane of the lens for spatially filtering the extended line field light into a line field having a width equal to the lateral resolution of the incident light.

13. The OCT system according to claim 1, wherein the interferometer is a Michelson interferometer or a Linnik interferometer.

14. The OCT system according to claim 1, further comprising a PSD calibrator configured to block a sample arm of the interferometer, wherein the measured value includes light only from the reference beam propagating within the reference arm of the interferometer, and the measured intensity of the interference pattern is a function of the PSD of the incident light for corresponding wave numbers scaled by the responsivity of the spectrometer and the reflectivity of the reference arm, and during execution of the PSD calibrator, the processor is configured to calibrate the PSD of the incident light such that the wave number corresponding to each pixel of the spectrometer is estimated.

15. A method for profilometry measurements of a sample in an optical coherence tomography (OCT) system, comprising: an interferometer splitting incident light into a reference beam and an interrogation beam, and interfering the interrogation beam reflected from the sample with the reference beam reflected from a reference mirror to generate an interference pattern; a spectrometer analyzing spectral components of the interference pattern at non-uniformly sampled wave numbers; a computer-readable memory of the OCT system storing a measurement model having elements weighted by weights derived from the power spectral density (PSD) of the incident light for corresponding wave numbers, connecting different depth values to different non-uniformly sampled wave numbers, and the method further comprising: obtaining profilometry measurements of the sample as maximum likelihood estimates of the surface depth of the sample by backprojecting the measured intensity in the measurement model.

16. The method according to claim 15, wherein the depth values are uniformly sampled at the resolution of the OCT system from a depth measurement range, and the depth values are relative values with respect to a reference depth selected outside the depth measurement range.

17. The method according to claim 15, further comprising estimating each profilometry measurement by performing a maximum likelihood estimator (MLE) to generate an argument of a maximum likelihood estimate of non-zero elements in a corresponding reflectivity vector, wherein each argument of the reflectivity vector corresponds to one of the depth values in the measurement model.

18. The method according to claim 17, wherein the MLE is an approximate MLE, and execution of the approximate MLE includes backprojecting a data vector through a measurement matrix, and the MLE is the depth value corresponding to the element of the largest magnitude in the backprojection.

19. The MLE is an exact MLE, and the execution of the exact MLE includes refining an approximate MLE by maximizing a log-likelihood objective function using a gradient-free optimization method, the method according to claim 17. **Claim 20** A non-transitory computer-readable medium storing computer-executable instructions that, when executed by a computer, cause the computer to execute a method for profilometry measurements of a sample in an optical coherence tomography (OCT) system, the method comprising: an interferometer splitting incident light into a reference beam and an inspection beam, and interfering the inspection beam reflected from the sample with the reference beam reflected from a reference mirror to generate an interference pattern; a spectrometer analyzing spectral components of the interference pattern at non-uniformly sampled frequencies; a computer-readable memory of the OCT system storing a measurement model having elements weighted by weights derived from a power spectral density (PSD) of the incident light for corresponding frequencies, connecting different depth values to different non-uniformly sampled frequencies, and the method further comprising: obtaining profilometry measurements of the sample as maximum likelihood estimates of the surface depth of the sample by back-projecting the measured intensities in the measurement model.

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