1. Introduction

Optical spectroscopy is an important technique that can examine the properties of a material by analyzing its interaction with light at a range of wavelengths [1]. An array detector, e.g., charge coupled device and complementary metal-oxide-semiconductor, in combination with a dispersive device such as a grating or prism are the most widely used configuration in optical spectroscopy including both bench-top and miniaturized instruments [2], where the input light at each wavelength is dispersed and focused onto a different spatial location thus suffering from low signal-to-noise ratio (SNR). In contrast, Fourier transform spectroscopy [3] based on interferometry takes advantage of wavelength multiplexing measurements thus possessing the “Fellgett advantage” [4] in terms of SNR due to the simultaneous detection of light at multiple wavelengths. However, the moving parts in Michelson interferometer imply low measurement throughput [5] and the array detector in stationary interferometers [6,7] shows lower sensitivity compared to single pixel detectors such as photomultiplier tube (PMT). Apart from interferometric methods, wavelength multiplexing measurements can be also achieved by combining the spatial modulation of dispersed light and a single-pixel detector in techniques such as Hadamard transform spectroscopy [8–10] and compressive sensing [11]. Due to the use of a single-pixel detector and Hadamard masks for spatial coding of light, akin to single pixel imaging [12], it is justified to include Hadamard transform spectroscopy in the category of single pixel spectroscopy [13]. Due to high SNR, single pixel spectroscopy holds potential for recording dynamic events such as gas leakage [14]. Furthermore, single pixel spectroscopy offers an alternative to spectroscopic applications in the IR region, where silicon-based array detectors are inefficient and expensive [15].

Single pixel spectroscopy based on Hadamard transform (SPS-HT) using various one-dimensional (1D) Hadamard masks has been demonstrated successfully from the visible to the near infrared spectral range [16–21]. Decker et al. experimentally demonstrated a 19-slot SPS-HT for the first time [16] by physically moving a multi-slit mask and then a 255-slot SPS-HT to realize a higher SNR [17]. Tilotta et al. proposed a Raman SPS-HT in the visible range based on a liquid-crystal spatial light modulator to eliminate the mechanical movement of the multi-slit mask in the previous systems [18]. Hanley thoroughly discussed the effect of mask error in SPS-HT systems [19]. Researchers have also successfully demonstrated the near infrared SPS-HT [20–22]. Recently, the compressive single pixel spectroscopy (CSPS) via 1D sequency-ordered Hadamard masks in the spatial domain is proposed to accelerate spectral reconstruction [9].

The common disadvantage of all the past SPS-HT is that coding is conducted for one Hadamard coefficient at a time along the wavelength dimension (i.e., the lower graph of Fig. 2(a)) and multiple Hadamard coefficients have to be measured sequentially using direct current (DC) measurements, which is subject to the influence of noise and signal drift thus prolonging measurements. In this paper, we propose sequency encoding single pixel spectroscopy (SESPS) employing two-dimensional (2D) masks, in which each Hadamard coefficient is encoded with a sequency value along one column of the 2D masks thus enabling the concurrent coding of all Hadamard coefficients (i.e., the upper graph of Fig. 2(a)). As a result, all Hadamard coefficients can be reconstructed simultaneously each with a pre-allocated sequency from the alternating current (AC) measurements of the time-domain signal. Note that sequency can be viewed as a general term for frequency and refers to the average number of zero-crossings per unit time interval [23]. This scheme is analogous to the frequency-division multiplexing scheme in telecommunications in that desired information is encoded in a range of frequencies and can be extracted from AC measurements. Compared to the DC measurements, the AC measurements of Hadamard coefficients can speed up the data acquisition because of lower 1/f noise at higher frequencies associated with detectors in general [24]. As a proof-of-principle demonstration, the spectral measurements of white light sources and fluorescence particles by the SESPS with 32 spectral channels are about 14 times and 70 times faster than those using a commercial spectrometer, respectively, when the relative root mean square error (RMSE) is around 3% or smaller. The data acquisition speed can be increased by an extra 4 times when only eight spectral channels are used to achieve a compression ratio (CR) of 4:1 with a slight increase in relative RMSE.

2. Experimental setup

Figure 1(a) schematically shows the experimental setup of the SESPS. The incoming beam from a light source coupled through a 600-µm multimode fiber (M29L05, Thorlabs, Newton, NJ, USA), after passing a 100-µm pinhole (P100D, Thorlabs, Newton, NJ, USA) and a collimating lens (AC254-030-A, Thorlabs, Newton, NJ, USA) with a focal length of 30 mm, is vertically dispersed by a volume phase holographic grating (VPHG, WP-1200/600-25 × 25 mm, Wasatch Photonics, Logan, UT, USA). Note that the grating is tilted to ensure that the beam of the first diffraction order is parallel to the optical table, which is favorable for optical alignment. After that, the dispersed light beams at all wavelengths impinge upon a cylindrical lens (ACY254-100-A, Thorlabs, Newton, NJ, USA) with a focal length of 100 mm and are focused onto different sets of micromirrors on a digital micro-mirror device (DMD) (DLP 9500, Texas Instruments, Dallas, TX, USA) with a resolution of 1920×1080 pixels, a micromirror pitch of 10.8 µm and a maximum switching rate of 17.8 kHz placed at the focal plane of the cylindrical lens. The 2D mask for sequency encoding is displayed on the DMD under the control of a computer. Because every micromirror hinge of the DMD is aligned along the diagonal of the micromirror, the micromirrors rotate about an axis that is 45° to the DMD edges. Therefore, the edge of the DMD is oriented 45° relative to the λ-direction as illustrated in the inset of Fig. 1, in which the λ-direction represents the dimension of the light dispersion. The light reflected from the DMD passes a collection lens, consisted of a lens with a focal length of 75 mm (AC508-075-A, Thorlabs, Newton, NJ, USA), a lens with a focal length of 30 mm (AC254-030-A, Thorlabs, Newton, NJ, USA), and an objective lens (RMS40x, Thorlabs, Newton, NJ, USA), then is focused into a lightguide with a core diameter of 5.1 mm (77637, Newport, Irvine, CA, USA) that leads to a PMT (11461-03-Y003, Hamamatsu, Shizuoka, Japan). An oscilloscope (DSO7014B Agilent digital oscilloscope, ValueTronics International Inc., Elgin, IL, USA) is used to read the PMT output, the maximum sampling rate of which is 2 GSa/s. Figure 1(b) is a photo of the experimental setup, where the light source and oscilloscope are not shown.

 

Fig. 1. Experimental setup of the SESPS. (a) Schematic of the experiment setup. VPHG: volume phase holographic grating; DMD: digital micromirror device; PMT: photomultiplier tube. The inset on the top right corner shows the orientation of the DMD. (b) Photo of the experimental setup.

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3. Theoretical analysis

Traditionally, SPS-HT is implemented by placing a set of Hadamard masks at the focal plane, i.e., the DMD plane in Fig. 1, of a grating-based spectrometer. In order to get N spectral elements x_j(λ) (j = 1,2, …, N) of a spectrum X(λ), X(λ)=$\sum _{j = 1}^N$ x_j(λ), each corresponding to a different channel with a unique central wavelength, a grating-based spectrometer needs N consecutive measurements, each measurement detecting one spectral element x_j(λ). However, such a measurement suffers from a poor SNR due to the small bandwidth in each channel. As an alternative, SPS-HT takes N measurements, where each measurement is the combination of at least half of the spectral elements determined by the Hadamard masks. The N measurements in SPS-HT are described by N linearly independent equations as [8]

Figure 2 schematically illustrates the 2D masks and Hadamard coefficients coding scheme of the SESPS by exemplifying the number of channels N = 4. Here we employed the sequency-ordered Hadamard matrix (SOHM), which facilitates compressed measurements because of the energy compaction characteristic of SOHM [9]. The 2D coding scheme is schematically displayed in the upper graph of Fig. 2(a), where the coding is simultaneously executed along both the wavelength dimension and Hadamard coefficient dimension. In contrast, the traditional SPS-HT only implements 1D coding along the wavelength dimension (the lower graph of Fig. 2(a)). Figure 2(b) shows a series of 2D sequency coding masks in the SESPS and corresponding SOHM, where the entries ‘‒1’ in the original SOHM are changed to ‘0’. Since each mask in Fig. 2(b) does not represent a complete Hadamard matrix, it is therefore called 2D mask instead of 2D Hadamard mask, which can be divided into N independent columns. Each column (λ-direction), corresponding to one Hadamard mask s_i in Eq. (1) as illustrated by different colors in the matrices superimposed on masks in Fig. 2(b), of the 2D mask is supposed to produce one Hadamard coefficient y_i upon incident light illumination. The incident light is dispersed along or parallel to the λ-direction (wavelength dimension), while each Hadamard coefficient is generated by the linear combination of the signals reflected by DMD micromirrors in a column perpendicular to the Coeff.-direction (coefficient dimension). When the number of channels (λ-direction), i.e., colors, increases, the channel width decreases since the total usable number of pixels along the λ-direction is fixed, which yields a better spectral resolution. The number of channels in each row (Coeff.-direction) of the 2D mask is identical to the number of Hadamard coefficients in use. Note that an increasing number of coefficients yields more details in the reconstructed spectrum. As a result, the coefficient corresponding to each column of the 2D mask is encoded by a different sequency as shown in Fig. 2(c), where a range of sequency values (i.e., numbers 1, 3, 5 and 7 enclosed in gray rectangles to the right) in one cycle designed for encoding four Hadamard coefficients is illustrated. In general, encoding N Hadamard coefficients needs all even rows of a 2N×2N SOHM with the sequency values of 2i–1 (i = 1, 2, …, N), which will facilitate the separation of Hadamard coefficients in the sequency domain after FWHT.

 

Fig. 2. Illustration of sequency encoding 2D masks in case of N = 4. (a) The upper graph denotes the proposed 2D encoding (i.e., along the wavelength and Hadamard coefficient dimensions) scheme in SESPS, the lower graph denotes the 1D encoding (i.e., along the wavelength dimension) scheme in the traditional SPS-HT. (b) 2D masks for sequency encoding that follow the layout of axes in (a) and corresponding matrices at eight different time points (t_i, i = 1, 2, …, 8). (c) Illustration of sequency encoding along the time dimension in one period of time, in which the sequency value corresponding to each coefficient is listed to the right.

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Based on the above description, the output for a 2D mask is the sum of spectral elements of the incoming light weighted by respective Hadamard coefficients, which yields y_i in Eq. (1), and each y_i is encoded by a different sequency in the time domain. The resulting output can be mathematically modeled as

If a spectrum as described by Eq. (1) is detected by a single-channel detector, i.e., a PMT in this work, the output of the PMT can be modeled by integrating both sides of Eq. (1) with respect to the wavelength λ, which yields

Figure 3 illustrates the simulation reconstruction procedure of a spectrum X(λ) from f(t), in which the spectrum has four spectral elements $\hat{x}_j$ as shown by the blue dots in Fig. 3(d). Note that the simulation does not take any noise into account. Figure 3(a) shows the simulated time domain signal f(t) in two periods. After performing a fast Walsh Hadamard transform (FWHT) on f(t), it is supposed to get the coefficient of w_2i(t) (i.e., ŷ_i), which is encoded by the corresponding sequency value as shown in Fig. 2. It should be stressed that FWHT requires the length of digitized f(t) to be the integer power of two, i.e., 2ⁿ. One can always pad the time-domain signal f(t) with zeros at the end to fulfill this condition. Similar to the Fourier transform [25], the sequency-domain representation of f(t) is depicted in Fig. 3(b). The magnitude of all sequency components lying on the abscissa equals half the magnitude of ŷ_i, the location of each ŷ_i is determined by the products between periods of f(t) and 2i. Based on that, all Hadamard coefficients ŷ_i are selected and plotted in Fig. 3(c). After multiplying the inverse of the 4 × 4 SOHM, the estimated spectral elements $\hat{x}_j$ are plotted by red dots in Fig. 3(d), which agree perfectly with the reference.

 

Fig. 3. Simulation procedure for estimating a spectrum with four channels (without compression). (a) Time-domain signal in two periods. (b) Sequency-domain representation obtained after conducting FWHT on the data in (a). (c) Hadamard coefficients extracted from peak values of various sequency components in (b). (d) Spectrum estimated by multiplying the Hadamard coefficients in (c) by the inverse of the SOHM as well as the reference spectrum for comparison. The dashed gray curves in all figures are visual guides.

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In practice, the spatially adopted complementary scheme [9] is effective in this experiment to minimize the influence of noise. In the SESPS, the complementary pattern is integrated into 2D mask by encoding each Hadamard coefficient into the sequence [s_i s_i ^* s_i s_i^* …] instead of [s_i 0 s_i 0 …], where s_i* is obtained by swapping ‘1’ and ‘0’ in s_i. Figure 4(e) shows the 2D masks incorporated with the complementary scheme. Similarly, the simulation results with this integrated complementary scheme are shown in Fig. 4. In fact, it is noted that this integrated complementary scheme is equivalent to implementing Hadamard encoding with the ‘H’ matrix rather than the ‘S’ matrix, which has already been demonstrated that the SNR using the ‘H’ matrix is 2N/(N+1) times higher than that using the ‘S’ matrix [25].

 

Fig. 4. Simulation procedure for estimating a spectrum with four channels (without compression) using the complementary scheme. (a) Time-domain signal in two periods. (b) Sequency-domain representation obtained after conducting FWHT on the data in (a). (c) Hadamard coefficients extracted from the peak values of various sequency components in (b). (d) Spectrum estimated by multiplying the Hadamard coefficients in (c) by the inverse of the SOHM as well as the reference spectrum for comparison. The dashed gray lines in (a)-(d) are visual guides. (e) Illustration of 2D masks for sequency encoding incorporated with the complementary scheme and corresponding matrices in case of N = 4.

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To code all coefficients simultaneously in practice, the beam needs to be spread out along the coefficient dimension as shown in Fig. 5(a) and then each coefficient, i.e., each column, is encoded separately. This process can be described by revising Eq. (1) slightly, i.e.,

Then one practical issue arises, which is that the incoming light spectrum may not be uniformly identical along the coefficient dimension of the DMD as illustrated in Fig. 5(a) due to imperfect alignment and physical constraints. In this case, each column is described mathematically as X_i(λ)=$\sum _{j = 1}^N$ x_ij(λ) varies with i (i = 1, 2, ···, N). The following paragraphs will give details about how to estimate X₁(λ) for an unknown input spectrum.

 

Fig. 5. Spectral intensity calibration. (a) Distribution of spectral elements of an incoming light illuminating a 2D mask with 4 channels (N = 4). The horizontal and vertical direction refer to the coefficient dimension and the wavelength dimension, respectively. (b) Representative spectra of all channels in the j-th column in (a) when assuming a uniform spectral shape but variable intensity along the coefficient dimension.

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It is found reasonable to assume that the spectral shape is uniform along the coefficient dimension while the spectral intensity varies, as shown in Fig. 5(b), which makes the following coefficient calibration step effective. As shown in Fig. 5(a) for the case of N = 4, x_ij(λ), in which both i and j vary among 1, 2, 3 and 4, represent the light distribution along the coefficient and wavelength dimensions, respectively. According to the assumption of the same spectral shape but variable intensity along the coefficient dimension, the spectrum for any one of three columns of grids in Fig. 5(a) can be expressed as the remaining one column of grids with a scaling factor, e.g., x_ij(λ)=a_ijx_1j(λ), where x_ij (λ) is the spectral element of the j-th channel along the wavelength dimension and the i-th channel along the coefficient dimension, a_ij is the corresponding scaling factor and x_1j(λ) is the spectral element from the j-th spectral channel in the first column. Such changes could be incorporated into Eq. (5) and yield

For given Hadamard coefficients y_i, x_1j(λ) could be calculated by solving the system of linearly independent equations in Eq. (6) in the wavelength-wise manner, which can then be summed to yield the corresponding X­₁(λ).

When a single-channel detector is used to detect the spectrum, the output can be obtained by incorporating the above scaling factor a_ij into Eq. (3), i.e.,

The calibration procedure includes two steps to obtain spectral elements x_1j(λ) and the corresponding output ${r_j}{\hat x_{1j}}$ of a single pixel detector from a broadband light source. In the first step, the configuration of the system is the same as that in Fig. 1 except that the PMT is replaced by a commercial spectrometer. The spectrometer is used to measure y_i(λ) first as modeled in Eq. (6). By solving Eq. (6) with known y_i(λ), x_1j (λ) can be obtained. In the second step, the light guide is connected to the PMT as shown in Fig. 1, and the 2D masks are loaded onto the DMD. The output of the PMT is the incoming spectrum X(λ) modified by the same 2D masks except that the light at all wavelengths is integrated by the PMT. This step can be described by Eq. (7), from which ŷ_i can be obtained by performing FWHT on the sequency encoded time series f (t) as modeled by Eq. (4). For any given set of ŷ_i, the corresponding ${r_j}{\hat x_{1j}}$ can be calculated by solving Eq. (7).

For the measurement of an unknown spectrum X′(λ), one just needs to obtain the output of the PMT, which is the integration of the incoming spectrum X′(λ) modified by the 2D masks with respect to wavelength. As Eq. (4) indicates, such a process is described as

Combined with the calibration datasets x_1j(λ) and ${r_j}{\hat x_{1j}}$ determined in the calibration step, any unknown spectrum, X₁′(λ) = $\sum\nolimits_{j = 1}^N x _{1j}^{\prime}$ (λ), can be estimated from the outputs of the PMT for the same set of 2D masks using the following equation

To enable compressive measurements, it is important to be aware that Hadamard coefficients ŷ_i (and ŷ_i′) as modeled in Eqs. (7) and (9) can be viewed as linear combinations of different sequency components in the spectrum weighted by ${r_j}{\hat x_{1j}}$ (and ${r_j}{\hat x_{1j}}'$). The Hadamard coefficients corresponding to low-sequency components contain information about the outline of the measured spectrum, and those corresponding to high-sequency components contain the information about the fine details of the spectrum. Therefore, only the low-sequency components ŷ_i′ are needed to retrieve the major information of the spectrum X′(λ) for the purpose of high-speed measurements when the accuracy requirement is moderate. As a result, we only need to encode the first M Hadamard coefficients of ŷ_i′ (M<<N) to estimate an arbitrary spectrum X′(λ) with a large CR when necessary [9].

The above scaling factors a_ij are measured as follows. According to the assumption x_ij(λ)=a_ijx_1j(λ), the spectral intensity x_ij(λ) of each grid in Fig. 5(a) needs to be measured first by the PMT in the calibration process from a broadband light source. Then, a_ij could be estimated by taking the ratio of x_ij(λ) to x_1j(λ). In this study, all calibration data are obtained using a broadband light source (UHP-LED-White, Prizmatix, Holon, Israel) and a commercial spectrograph (HoloSpec, Andor Technology, Belfast, UK) equipped with a camera (DU94OP-BV, Andor Technology, Belfast, UK).

4. Experimental results

Figure 6(a) shows the spectral resolution characterization results of the SESPS system for a range of channel widths in terms of number of pixels per channel on the DMD. When the channel width is varied from 1 to 16, 32 and 64 pixels per channel, which corresponds to the numbers of channels N = 1024, 128, 32 and 16, the spectral resolution changes from 2.3 nm to 2.4 nm, 2.6 nm and 4.1 nm, respectively. As the number of channels increases from N = 32 to N = 128 and N = 1024, the resolution improves by only 0.2 nm and 0.3 nm, respectively. To achieve a tradeoff between spectral resolution and data acquisition speed, the number of channels is set to N = 32 subsequently. Figure 6(b) shows the spectra of all individual channels, where the light source is an LED (UHP-LED-White, Prizmatix, Holon, Israel). It is obvious that all channels’ spectra share the nearly identical FWHM as expected. Figure 6(c) shows the center wavelength of each channel in the SESPS system, where the dark blue squares and red curve denote the measured and fitted results, respectively. It shows a clearly linear relation between the center wavelength of a channel and the channel’s index number, which agrees with the prediction by a grating’s linear dispersion in the first diffraction order.

 

Fig. 6. SESPS system characterization. (a) Spectral resolution for a range of channel widths. (b) Spectra of individual channels in the case of N = 32 when the light source is an LED. (c) Center wavelength as a function of the channel’s index number. The dark blue squares and red line stand for the measured center wavelength of each channel and the fitted results, respectively.

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Figure 7 shows the spectra of a halogen light source (OSL1-EC, Thorlabs, Newton, NJ, USA) measured by the Andor spectrometer (serving as the reference spectrum) and the SESPS system encoding separately with the SOHM and the naturally-ordered Hadamard matrix (NOHM) when N = 32. Moreover, all spectra shown in Figs. 7 through 9 have been denoised using the wavelet signal denoising function “wdenoise” with “BlockJS” options in MATLAB (R2020a, MathWorks, Natick, MA, USA) to improve the visualization. Similar to our previous work [9], we adopt the relative RMSE and CR to quantitatively evaluate the performance of the SESPS system. In Fig. 7(a) where the SOHM is used, the agreement between the reference spectrum and the spectrum measured by the SESPS system (CR = 1:1) is excellent, in which the relative RMSE is 1.62%. When the CR values are 2:1 and 4:1, the relative RMSEs are 1.62% and 1.94%, respectively. As the CR value is increased to 4:1, the estimated spectrum starts to show noticeable differences from the reference spectrum. In Fig. 7(b) where the NOHM is used, the agreement between the reference spectrum measured by the Andor spectrometer and the spectrum measured by the SESPS system (CR = 1:1) is still quite well, in which the relative RMSE is 1.88%. However, when the CR values are 2:1 and 4:1, the estimated spectra look totally different from the reference spectrum. In this case, the compressed measurements are also implemented according to the sequency value of the NOHM. Comparing the spectra estimated by the SESPS system using the SOHM and NOHM, the compressed measurements enabled by the SOHM is quite clear.

 

Fig. 7. White light source measurements. Spectra of a halogen light source measured by an Andor spectrometer (labeled as “Reference”) and the SESPS system using the (a) SOHM and (b) NOHM.

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Figure 8(a) shows the spectra of a Xenon light source (Gemini 180, HORIBA Scientific, Sunnyvale, CA, USA) measured by the Andor spectrometer (serving as the reference spectrum) and the SESPS system when N = 32. Note that the spectrum of the Xenon light source contains three peaks each with a FWHM of 4 nm in the tested spectral range, which is different from the previously tested halogen light source. The agreement between the reference spectrum and the spectrum measured by the SESPS system (CR = 1:1) is excellent, in which the relative RMSE is 1.87%. When the CR values are 2:1 and 4:1, the relative RMSE are ∼1.92% and 2.03%, respectively. Figure 8(b) shows the spectra of a mercury-argon light source (Photon control Inc., Camarillo, CA, USA) measured by the Andor spectrometer (serving as the reference spectrum) and the SESPS system when N = 32. Note that the mercury argon light source contains three peaks each with a FWHM of 0.8 nm, which is beyond the spectral resolution of our current system. Moreover, this source is pulsed with a repetition rate of around 63 kHz, which creates difficulty in estimating the amplitude of the light because of the principle of high-frequency AC measurements in the SESPS system. To facilitate the comparison in line shape, the spectra given in Fig. 8(b) have been normalized, which is performed by dividing all data in a spectrum by the greatest value of all. When all Hadamard coefficients are measured, i.e., CR = 1:1, the line shape is well recovered although the left peak appears broader than it should be and the two right peaks merge to one due to the inadequate resolution. When the CR value is increased to 2:1, the peaks become noticeably distorted in addition to being broadened further.

 

Fig. 8. Measurements of spectra with sharp peaks. (a) Spectra of a Xenon light source measured by an Andor spectrometer (labeled as “Reference”) and the SESPS system. (b) Spectra of a Mercury-Argon light source measured by an Andor spectrometer (labeled as “Reference”) and the SESPS system.

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Figure 9 depicts the time domain signal in two periods and the spectra of Nile Red fluorescence particles (FH-2056-2, Spherotech, Chicago, IL, USA) excited by a 532 nm laser (L1C-532, Oxxius, Lannion, France) with a power of 1.7 mW on the sample surface, which is measured by the Andor spectrometer (serving as the reference) and the SESPS system when N = 32. The relative RMSE between the reference spectrum and the spectrum measured by the SESPS system (CR = 1:1) is ∼3.34%, which proves good agreement. When the CR values are 2:1 and 4:1, the relative RMSEs are ∼3.36% and 3.63%, respectively. Note that the sharp cutoff wavelength around 540 nm and 610 nm are determined by the boundary of the 2D mask. It also shows that our current system is applicable to the visible spectral range limited by the spectral response of DMD.

 

Fig. 9. Fluorescence measurements. (a) Time-domain signal in two periods of Nile Red particles measured by the SESPS system. The dashed gray curve is a visual guide. (b) Fluorescence spectra of Nile Red particles measured by an Andor spectrometer (labeled as “Reference”) and the SESPS system (labeled with different CR).

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Table 1 compares data acquisition specifications including relative RMSE and data acquisition time for a single spectrum between the earlier CSPS [9] and SESPS (this study), among them the intensities of the fluorescence are almost the same and the intensities of the white light are much lower for SESPS than that of CSPS. The relative RMSEs of fluorescence measurements are always greater than those of white light measurements because fluorescence is weaker. While the relative RMSEs measured by the two methods are comparable, the data acquisition time using the SESPS is about three orders of magnitude shorter than that using the CSPS. Note that the data acquisition time is defined as the least display time of Hadamard patterns on DMD required for obtaining data to reconstruct spectra accurately. For comparison, the data acquisition times for measuring the reference spectra of white light in Fig. 7 and fluorescence in Fig. 9 with the Andor spectrometer are 100 ms and 500 ms, respectively. The SESPS is around 14 times and 70 times faster in white light and fluorescence measurements when all 32 spectral channels are used. The acceleration factors can be boosted by an extra 4 times when only eight channels are used to achieve a CR of 4:1, in which the relative RMSEs change only marginally.

Table 1. Comparison in data acquisition specifications for a single spectrum between the earlier CSPS [9] and SESPS (this study)

View Table

5. Discussion

Sequency coding that requires FWHT is used in this paper instead of a more commonly used frequency coding that involves Fourier transform for the following reason. Because a DMD micromirror can only simulate a binary number, i.e., 0 or 1, the resulting time-domain signal resembles more closely to a square wave than to a sinusoidal wave, which makes FWHT more applicable. In fact, the Fourier transform of a square wave with a single fundamental frequency yields accompanying odd harmonics, which would result in significant overlaps in the frequency domain among different frequency components. This problem can be prevented by using the FWHT as shown in Figs. 3(b) and 4(b), whereby no overlap among different sequency components occurs in the sequency domain. In addition, the FWHT requires a relative ease of computation because it involves only additions and subtractions while the FT requires multiplications as well. Moreover, Fourier transform based methods are subject to phase error [26].

As demonstrated in Table 1, this SESPS based on AC measurements is three orders of magnitude faster than the CSPS method based on DC measurements. The AC measurement brings the following advantages compared to the DC measurement. The first advantage is that 1/f noise is effectively suppressed because this noise decreases with an increasing frequency resulting in a higher SNR and a shorter data acquisition time. The second advantage is that AC measurements allow the multiplexing measurements of all Hadamard coefficients from a single measurement of the time-domain signal, which saves time for measuring every Hadamard coefficient sequentially in DC measurements. Because signal drift with time in general affects low-frequency components, such slow signal drift will have minimal influence on encoded coefficients if the encoding frequencies are chosen wisely. In contrast, DC measurements of multiple coefficients are sequential, which consequently leads to signal drift over time. The theoretical maximum of measurement throughput using the SESPS is mainly limited by the switching rate of the DMD and the required CR. If a faster spatial light modulator is available [27] or spinning masks are employed [28,29] and a larger RMSE associated with a greater CR is acceptable, the measurement throughput can be improved further.

One common advantage of both the SESPS and the CSPS is that one has a choice to make a tradeoff between the RMSE/resolution and the data acquisition time under low-noise regime due to the energy compaction trait of the SOHM. From a practical point of view, one can always measure an unknown sample with a CR of 1 to achieve the best resolution. After that one can make a tradeoff according to the need by setting a greater CR to achieve a faster acquisition with an acceptable resolution. The resolution of the system is shown in Fig. 6(a), which can be further improved using a small pinhole, a collimation lens with a longer focal length or a focusing cylindrical lens with a shorter focal length. The efficiency of the current SESPS system is characterized using a 532 nm laser, which is defined as the ratio of light power after the collimation lens to the light power after the light guide. The value is found to be about 51%, which is comparable to that of a grating-based commercial spectrometer [2].

6. Conclusion

In conclusion, we propose the SESPS for the simultaneous AC measurements of Hadamard coefficients to speed up the data acquisition, in which 2D masks are designed to code wavelength and Hadamard coefficients concurrently, in which each Hadamard coefficient is coded with a different sequency value. It is demonstrated that the SESPS with 32 spectral channels can accelerate spectral measurements from white light sources and fluorescence particles by around 14 times and 70 times, respectively, compared to measurements using a commercial spectrometer when the relative RMSE is less than 3%. Furthermore, we can improve the measurement speed by an extra 4 times when only eight spectral channels are used to achieve a CR of 4:1 with a tiny change in relative RMSE. This technique is expected to be useful for monitoring dynamic processes such as the photochemical degradation of scintillator liquid, reaction of ions and biomedical sample measurements in real time.