Systematic phase errors in Fourier transform spectroscopy can severely degrade the calculated spectra. Compensation of these errors is typically accomplished using post-processing techniques, such as Fourier deconvolution, linear unmixing, or iterative solvers. This results in increased computational complexity when reconstructing and calibrating many parallel interference patterns. In this paper, we describe a new method of calibrating a Fourier transform spectrometer based on the use of artificial neural networks (ANNs). In this way, it is demonstrated that a simpler and more straightforward reconstruction process can be achieved at the cost of additional calibration equipment. To this end, we provide a theoretical model for general systematic phase errors in a polarization birefringent interferometer. This is followed by a discussion of our experimental setup and a demonstration of our technique, as applied to data with and without phase error. The technique’s utility is then supported by comparison to alternative reconstruction techniques using fast Fourier transforms (FFTs) and linear unmixing.
© 2016 Optical Society of America
Phase errors have been described by Mertz  and Forman , and these two technique represent the two most popular methods of correcting phase errors in the field of Fourier transform spectroscopy. In the Mertz method, phase correction is performed in the frequency domain, while the Forman method applies the phase correction in the spatial domain. A combination of the two methods, using an iterative FFT shift theorem, has also been described . In a typical interferometer, phase errors can contain both systematic and random contributions. Systematic phase errors are generally reproducible and happen predictably, e.g., phase errors caused by dispersion within a Michelson interferometer . Meanwhile, random phase errors are manifested by noise and can include random variations in sampling error in, for example, a step-scan Fourier transform infrared spectrometer (FT-IR) . Failures on applying proper phase correction algorithms on the asymmetric interferogram lead to increased error in the reconstructed spectra. For spatial heterodyne spectrometers without a moving mirror or scanning requirements , systematic phase errors are caused by refractive index dispersion or lens distortion. Correcting these effects can be accomplished by measuring a monochromatic interferogram, which can be used to correct later measured polychromatic interferograms, assuming that the phase error is frequency independent . For frequency dependent phase error, the phase correction procedure is applied in the spectral domain by multiplying the Fourier transform interferogram with pre-calculated phase errors obtained from a multiline source . However, application of these phase correction techniques creates additional computational overhead. For instance, the Mertz phase correction technique often requires several forward and inverse fast Fourier transforms (FFTs), in addition to interpolation. Performing this procedure on each interferogram from a high resolution imaging Fourier transform spectrometer can be computationally overwhelming for high throughput measurements .
In this paper, we present a method that has greater speed to process many parallel interferograms while simultaneously accounting for phase errors. Neural networks (NNs) are well known for their ability to identify statistical significance as applied to pattern recognition . However, an additional and less studied application of NNs are their use in providing a method to empirically calibrate various sensors and systems to overcome systematic errors [11–13]. In the case of sensor calibration, where the sampling rate is fixed and the phase errors are systematic and reproducible, NNs can offer great advantages for realizing real-time operation. Such is the case with birefringent interferometers. In section 2, we describe the theoretical model and the nature of the phase errors. Section 3 contains various calibration methods that were studied to reconstruct measured data. Section 4 demonstrates our proof of concept experimental configuration. Lastly, section 5 shows a comparison of the results, garnered from the proposed NN-based spectral calibration, as compared to both linear and Fourier-transform based approaches.
2. Theoretical model of systematic phase errors
Random phase errors, caused by variations in scanning and sampling, have been described in previous work . However, in a birefringent interferometer, phase errors of this nature are uncommon since the optical path difference is generally fixed with respect to the detector (i.e., there is no scanning, nor are there any moving parts). In our previously described birefringent interferometer, it is often the case that many interferogram segments are collected by multiple apertures or across multiple regions of a focal plane array (FPA) [9, 15]. In the case of our SHIFT spectrometer , the phase error is step-like and occurs primarily when transitioning from column to column. In such a system, a discrete superposition of monochromatic interference patterns can be described by16] asFig. 1(a)-1(c), respectively.
The Root Mean Square (RMS) error is calculated between the reconstructed spectra with and without phase error asFigure 2(b)-2(d) illustrates the reconstructed spectra extracted from the RMS error plot in Fig. 2(a) when is 0π, π/2, and π, respectively. Due the nature of the cosine term in the interferogram per Eq. (1), when is 0π and 2π, ideal recovery of the contaminated spectra is obtained. Conversely, the worst error occurs when is approximately .
3. Experimental configuration
While these phase errors are common in the 2-dimensional (2D) SHIFT spectrometer, to simplify the presented studies, a representative 1-dimensional (1D) experiment was configured. A schematic of this 1D Birefringent Fourier Transform Spectrometer (BFTS) is depicted in Fig. 3 . It consists of a generating polarizer (G) which linearly polarizes incident light at 135 degrees relative to the x axis. A quartz Wollaston prism (WP), which has a wedge angle of 6.2 degrees, splits the light into two orthogonal components that then transmit through an achromatic quarter wave plate (QWP) oriented with its fast axis at 45 degrees. This converts the two orthogonal linear polarizations, exiting the Wollaston prism, into two circular components. A relay lens (L4) then re-localizes the prism’s interference onto a phase mask (PM). This mask, which is created as a louvered waveplate , enables the experimental simulation of piecewise linear phase errors across the WP’s interference pattern. After transmission through the phase mask, the light is then analyzed by a linear polarizer (A) with a transmission axis of 0 degrees with respect to the x axis. Finally, the light is relayed from the phase mask onto a focal plane array (FPA) by relay lens L5.
3.1 Phase mask
The polarization phase mask (PM) consists of a single mm glass substrate, which was coated with a polymerized liquid crystal-based half wave plate (HWP) optimized for a 600 nm wavelength. As illustrated in Fig. 4(a), the phase mask was divided into two regions, A and B. For region A, the HWP’s fast axis orientation varied periodically between 0° and 45°, while for region B, its fast axis was varied between 0° and 22.5° along the x axis. The period (Λ) for both regions is 1 mm. This periodicity creates a geometric phase shift, along the x axis, that acts like a step function when the Wollaston prism’s interference fringes are imaged onto it, enabling us to experimentally model the systematic phase error. The phase, which has been modulated into a simulated monochromatic interferogram, is shown in Fig. 4(b). We will refer to these two phase patterns as “phase A” and “phase B” for the remainder of this manuscript.
BFTS interferograms, without the PM, were simulated at a wavelength of 633 nm, by using the Mueller matrix formalism. The BFTS’s Mueller matrix, without the PM, can be expressed asEq. (5) and extracting the component of . Hence, a general mathematical form can be yielded asEq. (7), the interferogram with phase contamination can be expressed asEq. (6) and Eq. (8), are illustrated in Fig. 4(c) and 4(d), respectively.
3.2 Spectral sources for calibration
Calibration spectra were generated using a spectrum generator, based on a Digital Light Processing (DLP) projector module [19, 20]. A schematic of the BFTS, integrated into the spectral calibration setup, is depicted in Fig. 5. The BFTS’s field of view is filled by a 100 mm diameter integrating sphere. This sphere is illuminated by either (1) the DLP-based spectrum generator; or (2) a monochromator. The DLP-based spectrum generator consists of a 75 W xenon arc lamp that is reimaged onto a slit (S) by lens L0. Between the slit and L0 is located a longpass filter (F), which is used to block ultraviolet (UV) light, with wavelengths less than 400 nm, from the lamp. Light from the slit is then collimated by lens L1 through a polarization grating (PG) . This PG takes the relatively unpolarized light from the lamp and diffracts it, with high efficiency, into a + 1st and −1st diffraction order that are orthogonally polarized. The + 1st order diffracted beam is then imaged onto the DLP chip directly by lens L2, and a mirror (M) is used to capture and redirect the −1st order onto the DLP chip. A final lens L3 is used to collect both beams after reflection from the DLP, and the beams are diverted into the integrating sphere’s entrance port. Ultimately, this configuration maps light of different colors onto the DLP. Based on the image that is loaded onto the DLP, the DLP’s micro-mirrors redirect the desired wavelengths into the integrating sphere and reject undesired wavelengths, creating an arbitrary spectrum once the light is homogenized inside the integrating sphere. The interferometer can thus be illuminated with arbitrary spectra up to the maximum spectral resolution of the DLP illuminator, which is approximately 20 nm.
While the DLP enables the generation of continuous spectra, its relatively low spectral resolution can be a limitation for calibrating the system using H-Matrix (or measurement matrix) based techniques. Thus, a Horiba Micro-HR monochromator, illuminated by a Tungsten-halogen light source, was used to create variable narrow spectral band illumination. Light from this monochromator was guided into the sphere through a fiber light guide. It should be mentioned that either the DLP-based light source or the monochromator-based light source were used to illuminate the sphere, and that both sources were never used simultaneously to generate spectra. A central processing unit (CPU) is used to synchronize the DLP or the monochromator to the FPA’s measurements. Finally, “truth” or target spectra were recorded by a fiber-based USB 4000 Ocean Optics Spectrometer. The targets’ spectra were used for the neural network training in the output layer and for calculating the RMS error between the various calibration techniques.
4. Sensor calibration methods
Implementing neural networks for the BFTS’s calibration provides several advantages. First, NNs can potentially minimize the post-processing that is required to calculate a spectrum from the measured interferogram. For example, in Fourier-based reconstruction techniques, compensating for phase errors requires several forward and inverse FFTs for deconvolution, interpolation, upsampling, etc. These calculations have to be performed for each interferogram, increasing the computational burden. The number of operations (NOA) required for the NN, FFT, and linear-unmixing (H-Matrix) techniques were estimated to be 19379, 454877, and 8446, respectively, for our data. While the NOA required for the H-Matrix reconstruction technique is two times smaller than the NN, the NN’s performance, as will soon be demonstrated, exceeds that of the H-Matrix. Meanwhile, the NOA of the NN reconstruction method is 23 times smaller than the FFT’s, primarily due to interpolation steps that were required for phase correction in the FFT technique. Conversely, provided appropriate training, an NN can be established to perform these operations directly. Additionally, once the neural network’s architecture is determined, the parallel processing capability, often associated with FFTs, is preserved. However, advantages of using NNs come at the cost of increased complexity associated with the calibration equipment or procedures needed for NN training, which can be more costly than conventional methods of calibration. Furthermore, one must be careful to avoid over-fitting the solution . Still, the possibility of transferring some of the post-processing burden away from the field and into the laboratory is an enticing aspect for real-time performance.
Three kinds of calibration approaches were applied to calibrate the spectrometer: (1) conventional Fourier transforms with Mertz phase correction; (2) a linear systems model, or “H-Matrix”; and (3) an artificial Neural Network (NN). In this section, we will briefly overview the theory and procedures of performing the conventional Fourier and H-Matrix calibrations. Furthermore, the NN’s architecture, training data preparation, and the calibration steps will also be detailed.
4.1 Fourier transform
Given an interferogram without periodic phase error, we could perform Mertz phase correction. However, when the artificial phase error is introduced into the interferogram, additional phase compensation and post processing are required to reconstruct the true spectra. For instance, the phase mask does not have a perfectly immediate transition, in the HWP’s fast axis orientation, between adjacent periods. To avoid sampling this transition, the 1D interferogram is downsampled, using a sparser uniform sampling grid, which avoids this transition. This maintains the Nyquist sampling rate while avoiding spurious phase measurements caused by samples lying within this region. To phase correct the periodic phase error, phase corrections are performed in the frequency domain to compensate the phase change within each period of the phase mask . Thus, numerical phase correction is applied only on the artificial phase contaminated interferogram, as illustrated in Fig. 6(a). Assuming monochromatic light with a wavenumber , the interferogram, with periodic phase modulation, can be expressed using Eq. (1). Fourier transformation of one period () of this function yieldsFig. 4(b).
The resulting interferogram, after inverse Fourier transforming the numerically phase-corrected spectra, is what we are referring to as a phase compensated (but not phase corrected) interferogram, as illustrated in Fig. 6(b). Since the interferogram was asymmetric due to dispersion within the prism, phase correction was incorporated via the Mertz phase correction algorithm . The procedure for applying Mertz phase correction is summarized in Fig. 6(c). The post processed 1D interferogram is first upsampled by a factor of 8. A ramp function (Apo2) was then applied on the upsampled interferogram. This function was then centered via zero-padding and apodized using the triangular function, illustrated as Apo3. Finally, the interferogram’s center burst is isolated and apodized by Apo1 to extract the low frequency phase errors. The corrected spectra are obtained by multiplying the uncorrected spectra with the phase angle obtained from the short double-sided interferogram [1, 24]. Inverse Fourier transformation of this spectra yields the symmetric double-sided interferogram, illustrated in Fig. 6(d).
The H-Matrix, used in our calibration procedure, is constructed by 20 post-processed 1D interferograms, acquired using a monochromator. Interferograms were sampled uniformly in wavenumber from 2500 to 13889 cm−1 (400 nm to 720 nm). Thus, an arbitrary interferogram can be represented as a weighted sum of the interferograms contained within the H-Matrix as 
4.3 Neural network
A supervised multilayer feed-forward back-propagation neural network, also known as a multilayer perceptron (MLP) , was used in our NN-based calibration approach. The NN training algorithm is operated by the MATLAB NN toolbox  and involved two steps. First, to enable convergence towards an optimal neural network, the training data set must be representative of many different classes of spectra. Second, once the training data set is created, the optimal NN architecture must be determined to enable spectral calculation without over-fitting.
4.3.1 Training data preparation
Identifying the target characteristics to include into the training data is important to ensure proper NN training. Since we are using a NN to model an experimental Fourier transform process, input data are 1D interferograms (i.e., intensity versus optical path difference) and output, or target, data are spectra (i.e., intensity versus wavelength). The training set comprised of monochromatic, dichromatic, and random spectra. A representative spectrum from each of these data sets are presented Fig. 7.
To ensure that the training data set was statistically significant, and to ensure that it included both spectral and intensity variations, we created the data set using the following procedure. First, 50 monochromatic spectra were generated using the monochromator, sampled linearly in wavenumber from 13,889 to 25,000 cm−1 (or 400-720 nm). These spectra were represented in the training data set to present the NN with high spectral resolution (10.3 nm) monochromatic inputs. Next, three sets of these monochromatic spectra (or 150 total spectra) were acquired at different intensity levels: the monochromator’s maximum light output, 73% brightness, and 49% brightness. This enables the NN to properly identify signals with differing illumination levels. Similarly, 56 monochromatic spectra (spectral bandwidth of 20 nm), were generated by the DLP for wavelengths spanning 400-720 nm. Similar to the monochromator data, three different intensity levels (or 168 spectra total) were generated corresponding to the DLP’s maximum light output, 77% brightness, and 58% brightness. Finally, the total number of dichromatic and random continuous spectra were 332 and 511, respectively. In these data sets, both the brightness of any particular wavelength and the wavelength’s spectral position were randomized. Thus, there were a total of 1161 training pairs, consisting of BFTS interferograms and Ocean Optics spectra. Of these data, 1011 pairs were used for NN training, and 150 pairs were used for validation.
4.3.2 NN architecture determination
The size of the input and output layers, as well as the number of hidden nodes and layers, influences the NN’s training speed and the risk of over-fitting . Generally, the size of the input and output layers should be kept small to accelerate the training time. Furthermore, as described later in Sections 4.4.1 and 4.4.2, some post-processing was performed on the input interferogram and target spectra to improve fitting performance. Ultimately, the BFTS’s post-processed interferogram and the OO’s post-processed spectrum contained 136 and 70 data points, respectively. The number of samples contained within the OO’s spectrum was determined by matching the OO’s spectral resolution to that of the BFTS, while the number of samples representing the BFTS’s interferogram was determined using the Nyquist limit.
NN training was performed using the Levenberg-Marquardt algorithm, and a mean least square error algorithm (cost function) was used to determine the training direction. The cost function can be written as27]
The selected NN architecture was a feed-forward design which contained 136 input nodes, 20 nodes within a hidden layer, and 70 nodes in the output layer. Figure 8 illustrates the Neural Network’s topology in which the aforementioned post-processed interferograms serve as inputs to be matched to their corresponding OO spectra.
4.4 Data acquisition and post-processing common to all methods
4.4.1 Interferogram preparation for calibration
The 2D interferograms were first captured by an 8-bit FPA with a sensor size of pixels. All 2D interferogram frames were first divided by a measured flatfield to remove illumination nonuniformity across the camera . Dark frames were also taken to remove dark noise. After manually choosing the highest quality row of pixels (the intensity patterns only varied on the horizontal axis) on the 2D interferogram, each point was averaged between the vertical pixels to create a 1D interferogram. Finally, the mean intensity was removed from the 1D interferogram. The outcome after applying the procedures mentioned above is shown in Fig. 9(a).
In order to make the interferogram symmetric around ZOPD, the interferogram was zero-padded. It should be mentioned that the ZOPD position was set to the pixel containing the interferogram’s maximum intensity value. Finally, the interferogram was apodized using the triangular function illustrated in Fig. 9(b), which produced the final post-processed interferogram per Fig. 9(c).
4.4.2 Spectra preparation for calibration
Spectra, taken from the OO spectrometer, were also processed before using them for validation and calibration testing. Dark spectra were acquired and subtracted from each measurement to remove dark noise and the OO spectra were filtered to match the BFTS’s spectral resolution. First, the BFTS’s spectral resolution, with a triangular apodization filter, can be calculated by
The results of the three calibration techniques will be discussed in detail here. These techniques, described previously in section 4.0, were applied to the post-processed interferograms contained within the calibration and validation data sets. Due to numerical scaling differences between the three techniques, comparison was performed by calculating the RMS error between transmission measurements.
5.1 Transmission measurements
In order to enable comparison between the three different techniques, the RMS error was calculated on transmission measurements that were calculated, from the DLP data, by
5.2 Reconstructed spectra
The results of the transmission measurements and the calculated absolute error, after applying the NN training, FFT, and H-Matrix calibration procedures, are shown in the following sections, as compared alongside the OO (truth) transmission spectra. All of the reconstructed spectra were interpolated onto the same wavelength axis for direct comparison. A more detailed discussion of the RMS error is reserved for section 6.0.
5.2.1 Case I: without phase error
When no artificial phase error was introduced into the system, the reconstructed spectra obtained by the three techniques performed well. Representative spectra from this configuration are provided in Fig. 10, illustrating the outcome from the three algorithms given a monochromatic, dichromatic, and random input spectrum. Meanwhile, RMS error, calculated using all 150 validation spectra, yielded the results depicted in Fig. 11.
5.2.2 Case II: phase B
With the 90 degree (Phase B) phase mask inserted into the interferometer, most of the reconstructed spectra obtained from the FFT had lower accuracy. Representative spectra for this Phase B error case are illustrated in Fig. 12 while the RMS error is presented in Fig. 13 for monochromatic, dichromatic, and random input spectra.
5.2.3 Case III: phase A
Finally, the 180 degree (Phase A) phase mask was inserted into the interferometer. Similar to previous sections, representative spectra for the Phase A error case are illustrated in Fig. 14, while the RMS error is presented in Fig. 15 for monochromatic, dichromatic, and random input spectra.
Generally, the three calibration methods succeed in all three cases. For cases I and II, RMS error of the H-Matrix method was highest, followed by the RMS error of the FFT-based calibration and the NN providing the best performance of all three. For the H-Matrix technique, the calibration process was intuitive and easy to configure; however, the mean RMS error for case I (without phase error) was approximately 2.3 times greater than it was for the Fourier transform technique. Since the H-Matrix was constructed by using the monochromatic spectra, which were linearly sampled in wavenumber, the performance on the monochromatic spectra yielded lower RMS error. However, when calibrating broadband or continuous spectra, the RMS error increased by nearly a factor of 2. This is likely caused by cross-talk in the matrix during data reduction. Meanwhile, for the FFT technique, the RMS error increased when going from case II to case III. With reference to Fig. 15, the FFT’s RMS error in case III was approximately 2.6 times larger than case II and 3.3 times greater than case I. Additionally, for case III, the H-Matrix performed better than the FFT, which is converse to cases I and II in which the FFT performed better than the H-matrix technique. Finally, the Neural Network-based calibration provided low RMS error for all three cases, and was relatively independent of the phase mask’s presence in the system. For case II, the NN’s RMS error was over 6 and 3.6 times smaller than that of the H-Matrix and FFT, respectively. Furthermore, the differences between RMS errors were greatest in case III, demonstrating a six and nine fold decrease in the RMS error between the H-Matrix and FFT, respectively, when using the NN to calibrate the system.
We have successfully demonstrated the first experimental application of NNs to the calibration of a birefringent Fourier transform spectrometer with systematic phase errors. With the described technique, we were able to experimentally achieve an RMS error of approximately 0.0046 when artificial systematic phase error was present in the system; a 4.76 and 10 fold improvement over the FFT-based reconstruction method and a 6 and 11.7 fold improvement over the H-Matrix approach for periodic π/2 (phase B) and π (phase A) phase errors, respectively. Also, with our promising results on using NNs to calibrate our proposed system, there is a great possibility of calibrating more complex systems such as hyperspectral imagers or heterodyne spectrometers with multiple aperture arrays. Future work will be focused on optimizing the NN’s parameters and applying them to imaging architectures.
The authors would like to thank Matthew Miskiewicz and Michael J. Escuti for providing the patterned phase mask that was used in this experimental work.
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