HK1147124A - System and method to measure the transit time position(s) of pulses in a time domain data - Google Patents

Description

System and method for measuring the transit time position of a pulse in time domain data

Background

The invention relates to a method of measuring the transit time of a pulse in time domain waveform data. An example is given using time-domain terahertz data to determine sample properties. Terahertz electromagnetic radiation is potentially useful in many industrial measurement applications. At TD-THz, substantially monocycle pulses of radiation (approximately 1ps width, fig. 1) are generated and detected simultaneously. This synchronization method results in a high fidelity measurement of the electric field strength radiated over the waveform time window. The width of such a window may vary in a wide range depending on the instrumentation used. Since THz radiation pulses are very short in time, they will contain a very wide frequency band (10GHz up to 50 THz).

Once the TD-THz pulse interacts with the sample, some useful measurements can be extracted from the acquired time domain data. Possible measurements include, but are not limited to: sample mass, thickness, density, refractive index, density and surface variation, and spectroscopy (e.g., moisture content, polymorph identification).

Fig. 2 shows a terahertz transmitter 10 and a terahertz receiver 12. TD-THz, the change in THz pulse after it interacts with the material is recorded in the time domain waveform. For example, as a pulse passes through the sample 14, the arrival of the pulse at the receiver will be attenuated and delayed compared to the transmission of the same pulse through the air path (fig. 2). The amount of pulse delay is determined by the group refractive index value of the material and the value of the mass in the sample beam. The attenuation of the pulse also depends on the refractive index of the material (Fresnel reflection loss), the scattering of the radiation by the sample, and the attenuation of the frequency of the pulse by the material.

In the uppermost schematic, the THz pulse passes through the air with minimal time of flight and no amplitude loss. The addition of substantially transparent solid material (e.g., plastic, paper, and cloth) in the THz beam path (lower left corner) will result in longer pulse flight times. The increased time of flight is proportional to the mass and refractive index of the material. In the lower right diagram, in addition to creating a time-of-flight pulse delay, a scattering or absorbing medium (such as a cloth loaded with foam or water) reduces the pulse amplitude.

Many measurements can be made using the reflected TD-THz pulses from the sample (fig. 3). Figure 3 shows a set of possible interactions and properties of the sample that can be measured thereby. A consistent requirement for all measurements is an accurate determination of the time-of-flight value of the TD-THz pulse.

One exemplary measurement is a sample thickness measurement. Such measurements may be made in terms of transmission or reflection optical geometry. In transmission, the delay of the THz pulse caused by the sample 14 can be used to measure the thickness (fig. 4). In fig. 4, line 16 represents no sample. Line 18 represents a thin sample. Line 20 represents a thick sample. This approach requires determining the time position of the peak (i.e. the sample in and out of the beam) from two time domain waveforms acquired at two different times. This approach can lead to offset or calibration errors if the position of any peak shifts due to instrumentation or environmental conditions.

On the other hand, THz pulses will reflect some energy at any interface (e.g., Fresnel reflections). Referring to fig. 5, a multi-pass sample cell 21 is shown. As shown in fig. 6, reflections from the front and back surfaces of the specimen using mirrors 22 and 24 can be observed. The time delay between these two reflection peaks is determined by the mass and refractive index of the material. Thus, the mass of the sample, the thickness of the sample, and/or the density may be measured from a single time-domain waveform. Measurements made in this manner will exhibit reduced offset or time slope error.

Passing the THz pulse multiple times through the sample will increase the observed time delay without changing the inaccuracy of the pulse time measurement (as long as a sufficient signal-to-noise ratio is maintained). This concept is shown in fig. 4. This approach will increase the overall sample thickness measurement accuracy.

An interesting aspect of the reflected waveform is the polarity of the waveform pulse. TD-THz measures the direct electric field, so the polarity of the pulse indicates the electric field polarity. In transmission measurements, the presence of the sample does not affect the pulse polarity. However, for reflectance measurements, the pulse will reverse polarity when reflected from a low to high refractive index or metal interface. This is why the first pulse (air to sample) in the reflected waveform of fig. 6 reverses polarity. The intensity of the reflection depends inter alia on the refractive index difference between the two materials. This information can be used to determine the difference in refractive index change of the two materials across the interface, including the sign of the difference.

Drawings

FIG. 1 shows a time-domain terahertz (TD-THz) waveform;

FIG. 2 shows the interaction of THz pulses with a material;

FIG. 3 illustrates exemplary reflection interactions of TD-THz pulses;

FIG. 4 shows transmission measurements of air and samples of various thicknesses;

FIG. 5 shows a multi-pass sample cell;

FIG. 6 shows a reflected TD-THz waveform of a sample;

FIG. 7 shows simulation examples of TD-THz pulses reflected from various material thicknesses;

FIG. 8 shows a graph of an edge midpoint algorithm applied to various sample thicknesses;

FIG. 9 shows a plot of peak-to-peak reflected waveform amplitude versus sample thickness;

FIG. 10 shows pulse edges for midpoint determination;

FIG. 11 shows a linear fit of selected edge midpoints;

FIG. 12 is a flow chart illustrating midpoint determination;

FIG. 13 shows two criteria for a multi-peak search;

FIG. 14 shows two non-standard "bipolar" cases for multimodal search;

FIG. 15 shows the case where the first peak is labeled as bipolar and the second peak is labeled as standard type;

FIG. 16 shows a model waveform;

FIG. 17 shows a sample waveform in which the first two peaks are plastic pads and the last two peaks are silicon reference etalons;

FIG. 18 shows an enlarged view of the peaks of the model waveform;

FIG. 19 shows an enlarged view of a sample waveform peak;

FIG. 20 shows the starting point of a trial fit of a sample waveform using a model waveform of 256 points;

FIG. 21 shows the final result after method optimization;

FIG. 22 shows plastic shim thickness measurements;

FIG. 23 shows sample data for a deconvolution method;

FIG. 24 shows a Fourier transform of the data in FIG. 23;

FIG. 25 shows the frequency domain result of dividing a sample by a reference;

FIG. 26 shows that applying a Tikhonov filter to the results of FIG. 25 results in the reassignment of weights to data for frequencies having greater signal-to-noise ratios;

FIG. 27 shows an inverse Fourier transform of the data of FIG. 26, which is a completed deconvolution of the data of FIG. 23;

FIG. 28 shows model fitting applied to the deconvolution results;

FIG. 29 shows the result of convolving FIG. 27 with a Gaussian function;

FIG. 30 shows the model fitting of FIG. 29;

FIG. 31 shows the model fit of FIG. 30 including the reflection;

FIG. 32 shows a sensor with an internal calibration etalon;

FIG. 33 shows the reflected waveforms from the internal calibration standard sensor and single layer sample;

FIG. 34 shows an ICE/back reflector configuration, where the leftmost diagram is an empty configuration and the rightmost diagram is a configuration with a sample;

FIG. 35 shows a TD-THz waveform of a sample in an internal calibration standard/back reflector configuration;

FIG. 36 shows a sample multi-pass etalon;

FIG. 37 shows a multi-pass sample etalon chamber with simultaneous reference air paths;

FIG. 38 is a block diagram of a general purpose computer to which the principles of the present invention may be applied.

Detailed Description

In order to make some different sample property measurements, the time of flight of the TD-THz pulse needs to be accurately determined. This can be achieved in a number of ways. Three algorithms are provided for fast, high precision pulse time values using TD-THz pulses as example data: edge midpoint methods, model fitting methods, and deconvolution methods using fitting. Guidance in determining the best algorithm is described below.

The choice of algorithm depends on a number of factors, which is illustrated in fig. 7. Waveform 26 simulates a 10ps delay between the front and back reflections, equivalent to a 3mm material with n-1.5. Waveform 28 is 2ps (0.6mm), waveform 30 is 0.02ps (60 microns), and waveform 32 is 0.005ps (15 microns). The edge midpoint algorithm is generally a faster calculation method, but it does not generally work well for very thin samples (fig. 8). The peak-to-peak amplitude method is also fast, but only works for very thin samples (fig. 9). The model fitting and deconvolution algorithms generally work on all samples and provide higher accuracy (table 1), but are computationally slow.

In the edge midpoint method, the midpoints of the edges of the pulses are determined and shifted so that they are at 0V. Some points around the midpoint of 0V were selected for analysis (fig. 10). A linear fit of the data is then performed for points around the midpoint (fig. 10). The obtained linear fit equation is solved for its intercept and the value is the time assigned to the peak. This method is computationally simple and provides a high accuracy peak time position on the order of 1/75 of the waveform measurement point interval, which is a significant improvement in time accuracy.

The linear fit of the selected points provides the most computationally simple and therefore fastest method for accurately determining the midpoint of the edge. However, higher order fits can also be employed and further time accuracy improvements can be achieved. A third order polynomial has demonstrated improved fitting accuracy. Other non-linear or higher order fits may also be used.

A flow chart detailing the method 34 of midpoint determination is shown in fig. 12. Exemplary peak shapes are highlighted in fig. 13, where two standard cases of multi-peak finding are shown. If the ratio Vmin/Vmax is too large, the peak is considered to be a bipolar case as given below. If the ratio Vmin/Vmin2 (or Vmax/Vmax2) is too large, the peak is labeled as irregular (type 3).

Figure 14 provides two non-standard "bipolar" cases for the multi-peak search. The above results confirm that we can solve and quantify multiple peaks and the edges of each peak using an edge midpoint detection algorithm.

Referring to fig. 12, method 34 first acquires a waveform as shown in step 36. In step 38, it is determined whether the maximum voltage is greater than the absolute value of the minimum value of the waveform. If step 38 is true, then it is determined in step 40 whether the minimum occurs before the maximum. Thereafter, in step 42, the minimum value is found by backing off from the maximum value. Method 34 then proceeds to step 48, which step 48 is described in more detail in the following paragraphs.

If step 38 is false, it is determined in step 44 whether the maximum occurs before the minimum. If step 44 is false, the method proceeds to step 48, which will be described in more detail later. Otherwise, the method continues to step 46 where the maximum value is determined by backing off from the minimum value in step 46.

In step 48, the linear fit is accomplished by using a linear regression of the subset of data points between the minimum and maximum values of the waveform. Finally, in step 50, the pulse time is determined based on when the linear regression line is at the midpoint of the maximum voltage and the minimum voltage.

This model fitting method uses the edge midpoint method as a starting point for fitting the model waveform. The model waveform is a single THz pulse (fig. 16, 18) collected using the same conditions as the sample waveform (fig. 17, 19) as much as possible. The edge midpoints are used to generate a starting point for model fitting, which then evaluates a series of trial waveforms with varying times and amplitudes using a simplex optimization method and converges to optimal solution parameters.

For initial testing, two copies of the model waveform were used before and after fitting a 0.02 "thick plastic shim. In fig. 20, the waveform 52 is a sample waveform and the waveform 54 is a trial waveform, wherein fig. 20 shows a starting point of trial fitting of a model waveform to the sample waveform using 256 points. Fig. 21 shows the final result after method optimization, where waveform 56 is the sample waveform and waveform 58 is the trial waveform. Optimization changes the time of the two copies and the overall scale of the waveform. The time Δ T is changed by changing the phase of the Fourier transform:

this is because the frequency domain method gives accurate results without interpolation. Generating a test waveform from a model waveform in the following manner

Wherein, W_trialAnd W_refAre the time domain trial waveform and the reference waveform, C is the scale factor,is the phase shift from the model peak to the first sample peak, and δ (v) is the phase shift from the first sample peak to the second sample peak. Optimization attempts to minimize the RMS value of the difference between the test waveform and the sample waveform

The initial result is that this method is always slightly better than the zero crossing method, even when the model waveform is not optimal:

method of producing a composite material	Standard deviation (average 8 to 8.15ps)	Required time
			Zero crossing method	1.62fs	0.046 second
Peak fitting, 256 points	0.91fs	0.21 second
			Peak fitting, 512 points	0.90fs	0.33 second

Method of producing a composite material	Standard deviation (average 8 to 8.15ps)	When requiredWorkshop
			Peak fitting, entire waveform	0.91fs	2.80 seconds

Table 1-peak fitting results for 100 waveforms collected using a rotator. Each waveform is itself 99 averages. Time is measured on a slow personal computer.

For the deconvolution method, the THz measurement can be viewed as a convolution of the intrinsic instrument response (due to photoconductive material properties, laser pulse shape, transmitter and receiver antenna geometry, etc.) with the sample surface. If the meter response can be determined separately, for example by using a reference surface, deconvolution can be performed on the THz measurements to extract sample surface data separately. In the past, deconvolution has been reported for THz 3D reconstruction. The only element here is the deconvolution applied to the thickness measurement. Deconvolution prior to model fitting improves the accuracy of the results.

The convolution of two functions is equivalent to their Fourier transform multiplication:

for THz signals, the acquired terahertz waveform y (t) is the convolution of the reference waveform or instrument response h (t) with the Fresnel reflection x (t) of the actual object. The object may have multiple layers with different refractive indices, absorbances, and thicknesses.Andrepresenting Fourier transform and inverse Fourier transform, respectively.

Deconvolution is the inverse operation of reconstructing an unknown object given the acquired THz waveform and the THz reference waveform. The simplest method of deconvolution uses division in the Fourier domain:

this division produces very noisy results because the THz measurements are oversampled to prevent aliasing. The oversampled measurement has zero or minimum SNR over a portion of the frequency range where the division results in amplification of noise. The solution is to filter the measurement results. One approach is a simple band pass filter. However, this requires adjustments to the bandwidth of each THz system architecture and may create impulse response interference. Another alternative filter R (ω) is the Tikhonov filter, a simple, progressive filter that eliminates the low SNR part of the spectrum:

this method is proposed in the case of imaging deconvolution. Performing filtering and deconvolution simultaneously reduces the number of required transformations:

fig. 23 shows a sample THz measurement result and a reference measurement result. Line 58 is the THz measurement reflected from the panel. Line 60 is the individual measured meter response. Fig. 24 shows Fourier transform of the measurement results. Fig. 25 and 26 show the results of division in Fourier domain. In fig. 25, the signal has been band-pass processed to limit it to 2.2 THz. However, the increased variability of the signal at higher frequencies (e.g., vs. around 0.5 THz. around 2 THz) is a result of the SNR variation of the broadband THz pulses. In fig. 26, the results of the following operations are shown: the result of applying the Tikhonov filter to fig. 25 results in the re-assignment of weights for data for frequencies with greater SNR. Line 62 is the result of dividing the reference pulse by itself and applying the same filtering.

Fig. 27 shows the completed deconvolution. The peaks are narrower and symmetrical, which makes them easier to distinguish than the original waveform. Line 64 is the inverse Fourier transform of the data in fig. 26, which is the completed deconvolution of the data in fig. 23. Line 66 is a deconvolution of the reference pulse that can be fitted to the sample pulse to increase accuracy. The deconvolution method is followed by model fitting. As described above, the time and amplitude of the fitting function are varied using simplex multidimensional search techniques.

In fig. 28, model fitting is applied to the deconvolution results. Line 68 is the result of fitting 3 copies of the reference pulse to the sample pulse. Line 70 is the residual.

Another improvement is to convolve the deconvolved result with a balanced, compact function, such as a Gaussian function, as shown in fig. 29 and 30. Fig. 29 is a result of convolving fig. 27 with a Gaussian function. Although the FWHM of the peak is slightly increased, the Gaussian function has a relatively compact body and minimizes fluctuations. FIG. 30 is a model fit of FIG. 29.

Another improvement is to calculate the internal reflection within the sample using the known or assumed refractive index values of the layer materials, as shown in fig. 31. The dots indicate the amplitude and time of the primary reflections and the asterisks indicate multiple bounces of the terahertz pulse between the paint layers. The dashed lines again indicate residuals.

The main advantages of the deconvolution/fitting method are: the fitted expected linear behavior of the incremental time of flight between the reflections of the front and back surfaces of the sample relative to the thickness of the sample extends to thinner samples (fig. 22).

As discussed previously, the primary measured characteristic of a time-domain terahertz waveform pulse is the position in time and amplitude of the pulse. It should be appreciated that instrumentation or environmental conditions (e.g., noise, drift) affect the measurement, in which case the accuracy of the peak time position or amplitude measurement is reduced. The system internal reference can confirm proper system operation and can provide data for correction (e.g., amplitude scaling or time calibration) of the sample waveform results, if necessary.

As previously mentioned, any refractive index interface produces a reflection of the THz pulse. As shown in fig. 32, the invention and embodiments presented herein mount an internal calibration standard 72 in a sensor head 74 that provides reflected signals for system and measurement calibration. It is noted that for a sensor with such an internal calibration standard and the single film type sample shown, there are four interfaces (front and back faces of the etalon and front and back faces of the sample). Thus, four reflection peaks (represented by reference numerals 74, 76, 78 and 80 in fig. 33) are expected. The TD-THz waveform of this experimental result is shown in fig. 22. The expected 4 reflection peaks were observed.

Furthermore, it should be noted that this concept is not limited to 4 interfaces, but can be extended to any number of interfaces, such as multilayer or laminate samples.

The etalon is ideally fabricated from a stable material with a low coefficient of thermal expansion, a low refractive index at THz frequencies and very low absorption. High Density Polyethylene (HDPE) is a suitable target material. Low resistivity silicon or fused silica may also be selected.

The purpose of such calibration standards is to be able to measure both the time and amplitude of the etalon and the sample simultaneously for each waveform acquisition. Changes in the instrument or environmental conditions will be reflected as changes in the calibration standard peak. The etalon is selected to provide a stable signal whereby variations in the etalon measurement can be used to adjust the sample measurement. Exemplary adjustments include scaling the sample delta time measurements or using calibration signal amplitude information to improve the model fitting algorithm. It is important to note again that such calibration/reference information will be contained within each TD-THz reflection waveform. The TD-THz waveform time window must be long enough to ensure that both the calibration etalon and the sample reflection occur within this window. The pulse temporal position method and algorithm discussed previously is important to provide sufficient accuracy to make such calibration standards useful. In addition, the amplitude of the reflected pulse can be used to help form a reflected sample waveform pulse, especially for peak-to-peak amplitudes and model fitting methods.

The thickness of the etalon can be varied to provide the best alignment accuracy. Ideally, the etalon reflects only a small portion of the THz pulse, while advancing most of the pulse energy to the sample. If the etalon is relatively thick, two clearly separated reflection peaks can be obtained (as shown in fig. 6 and 22). Then, measuring the delta time between the two calibration standard peaks would be the preferred time analysis method. If the etalon is thin enough, it will reflect less THz energy, which is preferred. In this case, however, the interfaces are not time resolved and would require the previously discussed model fitting using amplitude fitting. Which method and algorithm provides better sample measurement accuracy will depend on the sample and experimental conditions (e.g., measurement rate).

As shown in fig. 34, the use of a back reflector 82 with an internal calibration standard (ICE)84 makes it possible to improve the accuracy of measurement of other sample properties. One example is the absolute thickness of the sample. The usual method of calculating the thickness of a sample requires the value of the incremental pulse transit times of the TD-THz pulse reflections of the front and back surfaces of the sample and knowledge of the refractive index of the sample. By using the ICE and the back reflector, higher accuracy thickness measurements can be made without knowing the refractive index of the sample material.

For this method, incremental pulse transit time values for empty structures need to be measured and recorded. This value is used to calculate the absolute sample thickness.

Once the sample was inserted into the structure, at least four pulses were observed (fig. 35). Using the high-precision method described above, it is necessary to find the transit times (T) of all the pulses_pk#)。

From these high precision values, the absolute thickness of the sample can be calculated according to the following formula:

thickness ═ T_Ref-T_Pk1-T_Pk3+T_Pk2)×c

All T_PkTime is relative to T_Pk0The time of day. Measuring T with empty ICE/retroreflector configuration_RefThe value is obtained. c is the value of the known speed of light. This calculation provides a high accuracy result of the sample thickness regardless of the composition of the sample material.

As shown in fig. 4, passing the THz pulse through the sample multiple times in the transmission measurement will increase the time-of-flight delay between the observed air reference and the sample scan without increasing the inaccuracy of the time measurement. This is a satisfactory situation.

However, an undesirable aspect of transmission measurements is that it is often not possible to distinguish between changes in the air reference scan (e.g., drift) or changes in THz transmitter/receiver spacing (e.g., mechanical motion) and changes in the sample. The following systems and methods address this issue.

Referring to fig. 36, a first multi-pass sample chamber 86 with sample 14 and a second multi-pass sample chamber 88 without sample are shown. First multi-pass sample cell 86 and second multi-pass sample cell 88 each include total reflection mirrors 90 and 92 and partial reflection mirrors 94 and 96. In addition, sample waveforms 98 and 100 obtained for multi-pass sample chambers 86 and 88, respectively, are shown. If one side of multi-pass sample chambers 86 and 88 is selected for only partial reflection, the transmitted THz pulse can be acquired multiple times within the same time domain waveform.

The thickness of the sample may be determined from the increase in the interval between transmission pulses of the sample compared to air. Therefore, a pulse increment time value of air is required. However, this value is relatively easy to measure, since the incremental time can be determined from the sample multi-pass etalon spacing. This spacing may be set at any suitable distance to allow for a clear separation between the transmitted pulses. This minimum separation is maintained for all sample thicknesses, that is, the two pulses do not convolve with each other and interfere with each other as shown in reflectance measurements of thin samples. This aspect is very beneficial for transmission measurements. In addition, the advantage of multiple transmission passes that increase the measured time-of-flight delay (as a multiple of the number of passes through the sample) still exists. That is, the delay between the first transmission pulse and the third transmission pulse will be four times the delay of the air/single transmission pass. This will result in a significant improvement in the accuracy of the incremental time measurement.

Another improvement can be achieved because the mechanical stability impact on the measurement will depend only on the etalon separation and not the transmitter to receiver distance. It will be easier to select materials and construction methods for etalons that result in greater thermal and mechanical stability. An increase in the stability of the sample cell etalon will directly result in an increase in the accuracy of the measurement.

FIG. 37 shows another embodiment of multi-pass sample cell 86, showing a first transmission pulse 102, a second transmission pulse 104, and a third transmission pulse 106. If the etalon could be added so that the set pulse etalon reflection would occur without the sample in the THz beam, the resulting waveform would provide increased information of both the etalon and the time of flight of the transmitted pulse through the sample. Fig. 37 also shows the obtained waveform 108.

Referring to FIG. 38, an illustrative embodiment of a computer system in general is shown and designated 110. The computer system 110 may include a set of instructions executable to cause the computer system 110 to perform any one or more of the methods or computer-based functions disclosed herein. The computer system 110 may function as a standalone device or may be connected to other computer systems or peripheral devices, for example, through the use of a network.

In a networked configuration, the computer system may operate in the capacity of a server or a client user computer in a server-client user network environment, or as a peer computer system in a peer-to-peer (or distributed) network environment. The computer system 110 may also be implemented as or included in various devices, such as: a Personal Computer (PC), a tablet PC, a set-top box (STB), a Personal Digital Assistant (PDA), a mobile device, a palmtop computer, a laptop computer, a desktop computer, a communicator, a wireless telephone, a landline telephone, a control system, a camera, a scanner, a facsimile machine, a printer, a pager, a personal security device, a network appliance, a network router, switch or bridge, or any other machine capable of executing a set of instructions (sequential or otherwise) that specify actions to be taken by that machine. In particular embodiments, computer system 110 can be implemented using electronic devices that provide voice, video, or data communication. Additionally, while a single computer system 80 is shown, the term "system" may also include any collection of systems or subsystems that individually or jointly execute a set or multiple sets of instructions for performing one or more computer functions.

As shown in fig. 38, computer system 110 may include a processor 112, such as a Central Processing Unit (CPU), a Graphics Processing Unit (GPU), or both. Further, the computer system 110 can include a main memory 114 and a static memory 116, the main memory 114 and the static memory 116 being capable of communicating with each other via a bus 118. As shown in the figure, the computer system 110 may further include a video display unit 120, such as a Liquid Crystal Display (LCD), an organic light emitting diode (0LED), a flat panel display, a solid state display, or a Cathode Ray Tube (CRT). In addition, the computer system 110 may include an input device 122 (such as a keyboard) and a cursor control device 124 (such as a mouse). The computer system 110 may also include a disk drive unit 126, a signal generation device 128 (such as a speaker or remote control), and a network interface device 130.

In certain embodiments, as shown in FIG. 28, the disk drive unit 126 may include a computer-readable medium 132, where one or more sets of instructions 134, such as software, may be embedded in the computer-readable medium 132. Additionally, the instructions 134 may implement one or more of the methods or logic described herein. In particular embodiments, the instructions 134 may reside, completely or at least partially, within the main memory 114, static memory 116, and/or within the processor 112 during execution thereof by the computer system 110. The main memory 114 and the processor 82 may also include computer readable media.

In another embodiment, dedicated hardware implementations (such as application specific integrated circuits, programmable logic arrays and other hardware devices) can be constructed to implement one or more of the methods described herein. Applications that may include the apparatus and systems of various embodiments may broadly include a variety of electronic and computer systems. One or more embodiments described herein may implement functions using two or more specific interconnected hardware modules or devices according to related control and data signals capable of communication between and through the modules, or as portions of an application-specific integrated circuit. Accordingly, the present invention encompasses software, firmware, and hardware implementations.

According to various embodiments of the present disclosure, the methods described herein may be implemented by software programs executable by a computer system. Additionally, in the illustrative non-limiting embodiment, implementations can include distributed processing, component/target distributed processing, and parallel processing. In another aspect, a virtual computer system process may be constructed to implement one or more of the methods or functions described herein.

The present disclosure includes a computer-readable medium that includes instructions 1345 or receives and executes instructions 134 in response to a propagated signal so that a device connected to the network 136 can communicate voice, video, or data over the network 136. Additionally, the instructions 134 may be transmitted or received over a network 136 via the network interface device 130.

While the computer-readable medium is shown to be a single medium, the term "computer-readable medium" includes a single medium or multiple media, such as a centralized or distributed database, and/or associated caches and servers that store one or more sets of instructions. The term "computer-readable medium" shall also be taken to include any medium that is capable of storing, encoding or carrying a set of instructions for execution by the processor or that cause the computer system to perform any one or more of the methodologies or operations disclosed herein.

In certain non-limiting, exemplary embodiments, the computer-readable medium can include solid-state memory, such as a memory card or other package containing one or more non-volatile read-only memories. Additionally, the computer readable medium may be random access memory or other volatile rewritable memory. Additionally, the computer readable medium may include a magneto-optical medium or an optical medium, such as a disk or tape or other storage device for capturing a carrier wave signal (such as a signal transmitted over a transmission medium). A digital file attachment to E-mail or other information file or set of files contained therein may be viewed as a distribution medium equivalent to a tangible storage medium. Accordingly, the disclosure is considered to include any one or more of a computer-readable medium or a distribution medium and other equivalents and successor media, in which data or instructions may be stored.

Although this specification describes components and functions that may be implemented in particular embodiments with reference to particular standards and protocols, the present invention is not limited to these standards and protocols. For example, standards for Internet and other packet-switched network transmissions (e.g., TCP/IP, UDP/IP, HTML, HTTP) represent examples of the prior art. These standards are periodically replaced by faster or more efficient equivalents having substantially the same function. Accordingly, replacement standards and protocols having the same or similar functions as the standards and protocols disclosed herein are considered equivalents thereof.

The illustrations of the embodiments described herein are intended to provide a general understanding of the structure of the various embodiments. The illustrations are not intended to serve as a complete description of all of the elements and features of apparatus and systems that utilize the structures or methods described herein. Many other embodiments will be apparent to those of skill in the art upon reading this disclosure. Other embodiments may be utilized and derived from the disclosure, such that structural and logical substitutions and changes may be made without departing from the scope of the disclosure. Additionally, these figures are merely representative and may not be drawn to scale. Some portions of these figures may be exaggerated while other portions may be minimized. The present disclosure and figures are, therefore, to be regarded as illustrative rather than restrictive.

The term "invention" may refer, individually and/or collectively, to one or more embodiments of the present disclosure for convenience only and is not intended to limit the scope of this application to any particular invention or inventive concept. Moreover, although specific embodiments have been illustrated and described herein, it should be appreciated that any subsequent arrangement designed to achieve the same or similar purpose may be substituted for the specific embodiments shown. This disclosure is intended to cover any and all subsequent adaptations or variations of various embodiments.

In "detailed description," various features may be grouped together or described in a single embodiment for the purpose of streamlining the disclosure. This disclosure is not to be interpreted as reflecting an intention that: embodiments require more features than are directly recited in each claim. Rather, as the following claims reflect, inventive subject matter may be directed to less than all of the features of any of the disclosed embodiments. Thus, the claims are included in the detailed description, and each claim independently defines separately claimed subject matter.

The above-disclosed subject matter is to be considered illustrative, and not restrictive, and the appended claims are intended to cover all such modifications, enhancements, and other embodiments, which fall within the true spirit and scope of the present invention. Thus, to the maximum extent allowed by law, the scope of the present invention is to be determined by the broadest permissible interpretation of the following claims and their equivalents, and shall not be restricted or limited by the foregoing detailed description.

Claims (4)

1. A method of determining peak time values in time data of a time-domain terahertz waveform, the method comprising the steps of:

receiving a time-domain terahertz waveform;

determining a midpoint of an edge of the waveform;

performing a linear fit of the waveform for points near the midpoint;

an intercept value is determined, wherein the intercept value is a peak time value in the time data.

2. The method of claim 1, further comprising the steps of: the waveform is offset so that the edges of the waveform are set to zero.

3. The method of claim 1, wherein the step of performing a linear fit is accomplished by using a linear regression of a subset of data points between a minimum and a maximum of the waveform.

4. The method of claim 1, wherein the intercept value is a time when a linear regression line of the waveform is at a midpoint of a minimum and maximum of the waveform.