1. Introduction

Infrared (IR) spectroscopy is widely used in the field of medical diagnostics [1], pharmaceutics [2,3] and environmental monitoring [4–6] due to its capability in material characterization. While the availability of different IR spectroscopic devices has been increasing over the years, Fourier-transform infrared (FTIR) spectrometers [7–10] are most commonly employed due to their throughput and multiplexing advantages which provides higher signal-to-noise ratio (SNR). Furthermore, they offer a broad sensing wavelength range and high spectral resolution, thus making them capable for use across a multitude of applications. However, FTIR spectrometers are generally expensive and lack robustness due to the presence of moving optical components, hence limiting their potential in field use.

To overcome these problems, dispersive spectroscopy can be considered as an alternative approach to low-cost, robust, and compact devices for use outside of the laboratory environment. Dispersive IR spectrometers are commonly implemented with expensive array detectors. However, recent advancements in single-pixel imaging (SPI) technology [11–13] have allowed expensive array detectors to be replaced with a much cheaper alternative. SPI-based spectrometers use multiplexing schemes to encode the spectrum and capture the intensities of multiple wavelengths simultaneously with a single-pixel detector [14]. Multiplexing schemes like Hadamard transform [15–17], compressed sensing (CS) [18–20], and Fourier transform [21,22] have been demonstrated to provide higher achievable SNR. However, most spectrometers are still implemented with a narrow entrance slit (typically 100 µm or less) to maintain their spectral resolutions which comes with a tradeoff in optical throughput.

Single-pixel Hadamard transform spectrometers with double encoding schemes have been demonstrated in literature to increase optical throughput [23,24]. In devices that utilize a static encoding mask in conjunction with a moving mask, the spectral resolution is still coupled with their throughput. Devices that make use of dual moving masks remove this coupling effect, but the driving mechanism required to move and synchronize two encoding masks is complex. It is therefore still difficult to efficiently increase the throughput of Hadamard transform spectrometers. In this paper, we demonstrate a single-pixel spectrometer with a novel encoding mechanism exhibiting both high-throughput and multiplexing advantages. The proposed spectrometer involves a two-stage encoding process enabled by a dynamic encoding mask at the entrance aperture and a static mask at the exit aperture. The novel encoding scheme provides a workaround to previously known designs by decoupling the spectral resolution from its throughput to enhance the SNR of the system. Additionally, spatial information of the incident light is included in the spectral reconstruction process to remove the necessity for uniform illumination. These open up opportunities for low-cost and portable dispersive IR spectrometers with high SNR and high spectral resolution, and it is suitable for use in a wide range of field applications.

2. Working principle

2.1 Harwit’s doubly-encoded spectrometer

We first look at the work presented by Harwit et al. [24] to better understand the working principles of a doubly-encoded spectrometer and the design changes made in our proposed system. A schematic of Harwit’s spectrometer is shown in Fig. 1(a). A spectral filter is placed in front to permit only wavelengths of interest into the dispersive spectrometer. The entrance aperture is encoded with a mask comprising multiple slits, with some slits opened to transmit light and some closed to block light as illustrated in Fig. 1(b). A window restricts the field-of-view of the spectrometer so that light goes through only a controlled number of encoding slits. The spectrometer disperses the light and forms spectral images of the entrance aperture on the static encoding mask at the exit aperture. The light collected by the single-pixel detector is thus encoded twice.

 

Fig. 1. Schematic of (a) Harwit’s doubly-encoded Hadamard transform spectrometer, and (b) light passing through the dynamic encoding mask at the entrance aperture forming spectral images on the static encoding mask at the exit aperture.

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To understand the encoding process, we consider a system with N entrance slits and 2N−1 exit slits as seen in Fig. 1(b) where N is a number of the S-matrix order [15]. N is selected based on the number of spectral components to be sampled and the encoding scheme selection will be elaborated in a later section. The encoding masks each consist of a total of 2N−1 encoding patterns and the physical implementation of the masks includes printing binary patterns according to the sequence of the cyclic S-matrix, where slits that transmit light represent ‘1’ and slits that block light represent ‘0’.

The first N slits are printed according to the encoding pattern given by the S-matrix, and the next N−1 slits are simply a repeat of the first N−1 slits. The entrance encoding mask is dynamic and is stepped across the entrance aperture. In each step of the encoding mask, an encoding pattern corresponding to one row of the cyclic S-matrix is generated, imaged onto the static encoding mask, and the single-pixel detector records a new measurement. On the j^th encoding pattern (j = 1∼N), the measured signal on the detector, y_j, is given by:

We see in Harwit’s design that both the A and B matrices used are the same N × N cyclic S-matrix, and that the spectral image of each spectral component for a particular encoding pattern on the entrance mask that corresponds to one row of A should overlay an encoding pattern on the static exit mask that corresponds to a column of B. By the properties of S-matrix, the multiplication of A and B yields a diagonal matrix of size N × N where the values along its diagonal are double that of the rest of the matrix. Physically, it means that in each encoding step, one spectral component is selectively given double the weightage of the other spectral components. This allows us to recover the distinct information of each spectral component using just a single-pixel detector, and also simplifies the mathematical recovery of the above linear equation to obtain the spectrum vector X directly from the measurement vector Y.

While the above implementation simplifies the recovery process, there are drawbacks that can affect the performance of the spectrometer. One drawback is the need to ensure uniform illumination at the entrance aperture for accurate spectral recovery. This is necessary as the encoding matrices A and B used in Eq. (2) are binary, meaning that light passing through the entrance encoding mask must be uniform across all pixels for the model to be valid. A non-uniformly illuminated aperture will result in each spectral image formed on the static encoding mask to also be non-uniform as illustrated in Fig. 2(a). When that happens, the light intensity collected by the single-pixel detector in each encoding step does not accurately reflect the information of the spectral components. This error is brought into the measurement vector Y and the recovery of the incident radiation spectrum vector X can potentially be inaccurate.

 

Fig. 2. Schematic of (a) the difference in light intensity reaching the static mask as a result of non-uniform illumination at the entrance aperture and (b) the coupling of the spectrometer throughput with its spectral resolution.

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Another drawback is the coupling of the system throughput to its spectral resolution. The one-dimensional encoding scheme is based on the assumption that the light passing through the i^th entrance slit and leaving the j^th exit slit has the same wavelength as the light entering the (i + N−1)^th entrance slit and exiting the (j + N−1)^th exit slit as shown in Fig. 2(b). The wavelength components of light λ₁, λ₂, …, λ_N going through the first entrance slit should exit the 1^st, 2^nd, …, N^th exit slit, respectively. This means that for a system with N slits as its entrance aperture and N spectral components, the image of the N^th slit of the first spectral component must overlap with the image of the first slit of the N^th spectral component as shown in the figure. In other words, the total spatial separation of wavelength components on the static encoding mask is directly proportional to the width of the entrance aperture. The consequence is that in order to achieve higher spectral resolution, the size of the entrance aperture has to be larger as well. An enlarged entrance aperture results in more optical aberrations and distortions which, more often than not, requires the size of the system to be larger to correct. This can in turn limit the portability of the system. In the following sections, we show that the aforementioned drawbacks can be overcome by sampling the spatial information of the entrance aperture and using a redesigned encoding pattern to decouple the throughput of the system from its spectral resolution.

2.2 System description of the proposed spectrometer

A schematic of our proposed spectrometer is shown in Fig. 3. A set of fore-optics is used to image incoming light from the source (can be an extended source) onto the entrance aperture. At the entrance aperture, instead of a single slit, a digital micromirror device (DMD) is used to encode the incoming light and control the amount of light that enters the spectrometer. The spectrometer is based on the Czerny−Turner configuration and a diffraction grating within it separates the light into its individual wavelength components.

 

Fig. 3. Schematic of the proposed spectrometer with a DMD serving as the dynamic encoded entrance aperture, along with two single-pixel detectors to separately capture the spatial and spectral information of the incident radiation.

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The DMD serves as a multi-slit aperture with the slits arranged along the direction of dispersion and spatially encoded along a direction substantially transverse to the dispersion direction. Depending on the choice of encoding scheme, different patterns can be generated on the DMD which in turn affects the light pattern that reaches the dispersion optics. The grating diffracts light into several orders. The zeroth-order light is simply reflected off the grating and traditionally does not serve any purpose in a conventional dispersive spectrometer. In our system, however, the zeroth-order light contains spatially encoded information due to the first stage encoding by the DMD. The otherwise wasted light is thus directed to a single-pixel detector (detector 1) where intensity information of the entrance aperture is collected for spatial sampling. Concurrently, a selected non-zeroth order diffracted wave—typically +1^st or −1^st order—is collected by a set of imaging optics and focused onto the exit aperture where dispersed spectral images of the encoded entrance aperture are formed.

Like the entrance aperture, the exit aperture also comprises slits that are spatially encoded along the direction that is transverse to the direction of light dispersion. Unlike the DMD, the encoder is a fixed mask that is designed in accordance with the encoding scheme used at the entrance aperture. Hence, light that passes through the exit aperture is encoded twice and collected by another single-pixel detector (detector 2) for spectral recovery. The entire sampling process is completed once the DMD has generated all the encoding patterns and the detectors have recorded the corresponding measurement signals. In the following sections, we will describe how our method of encoding decouples the spectral resolution of our spectrometer from its throughput, and how the inclusion of the spatial information at the entrance aperture improves the performance of spectral reconstruction.

2.3 Spatial sampling of the entrance aperture

To describe both the spatial and spectral sampling processes, we can typically consider that the incident radiation consists of N spectral components from ${\lambda _1},{\lambda _2}, \ldots \; $ to ${\lambda _N}$ within the operating band of the spectrometer. The spectral shape of the radiation is represented as a column vector:

The illumination of light on the entrance aperture can be affected by several factors including the light source uniformity, optical alignment and focusing conditions, and sample homogeneity. As such, ensuring uniform illumination can be a difficult process and failure to achieve it can potentially affect the performance of spectral reconstruction. Our spectrometer overcomes this by capturing the zeroth-order signal from the grating using an additional single-pixel detector as shown in Fig. 4(a). This allow us to obtain information of the spatial intensity distribution at the entrance aperture which can be included in the spectral recovery process to correct for the non-uniformity at the aperture and improve the overall spectrometer performance. We first consider light passing through the j^th slit encoded along its height by K pixels as shown in Fig. 4(a). The zeroth-order light from the grating is then collected by a single-pixel detector.

 

Fig. 4. Schematic of (a) the spatial encoding process for the j^th column of the entrance mask comprising M columns of pixels on the DMD and (b) the two-stage spectral encoding process completed with the DMD and a fixed encoding mask for the j^th entrance slit.

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For spectroscopy applications, the light spectrum within each encoding pixel on the entrance aperture is typically the same. Hence, ${I^{\prime}_{jk}}\left( {\mathop \sum \nolimits_{i = 1}^N x^{\prime}({{\lambda_i}} )} \right)$ represents the total radiation intensity incident on the k^th encoding pixel of the j^th entrance slit, where ${I^{\prime}_{jk}}$ is the scaling factor that describes the non-uniform illumination on the pixel. As the encoding pixels either block or transmit light, $w_{jk}^m$ denotes the weightage of the k^th encoding pixel of the j^th slit in the m^th encoding pattern, where $w_{jk}^m{\; } = 1$ represents an ‘open’ pixel and $w_{jk}^m{\; } = 0$ represents a ‘closed’ pixel depending on the driven state of the DMD micromirrors. Thus, the light intensity on a pixel is encoded as $w_{jk}^m{I^{\prime}_{jk}}\left( {\mathop \sum \nolimits_{i = 1}^N x^{\prime}({{\lambda_i}} )} \right)$ and the signal measured by the zeroth-order single-pixel detector due to the contribution of the j^th slit is expressed as:

Subsequently, the overall zeroth-order single-pixel detector output in the m^th encoding pattern is:

With Eq. (6) established, we can also define a new scaling factor as:

With these definitions and a complete encoding process of M measurements (m = 1∼M), the spatial sampling process can be simplified into the following linear equation:

From Eq. (8), we can see that as we increase the number of slits J used on the entrance aperture, the size of the encoding matrix W increases column-wise. This means that in order to solve the above linear equation by direct inversion, the number of measurements M also has to increase which results in a longer sampling time. To shorten the sampling time with a large number of slits, CS theory is introduced in the reconstruction process. This theory exploits the principles of sparsity to represent a signal with fewer samples. In this case, CS compresses the spatial intensity during the sampling process, and the intensity information can be recovered with much fewer measurements by taking advantage of its sparsity in some bases. To do this, the intensity scaling vector I is represented by a basis $\boldsymbol{\mathrm{\Phi}}$ and a sparse vector $\boldsymbol{\mathrm{\theta}}$ as ${\textbf I} = \boldsymbol{\mathrm{\Phi}}\boldsymbol{\mathrm{\theta}}$. The total number of measurements can therefore be set at a value much smaller than the dimension of the signal ($M \ll JK$) [18]. By substituting the new representation into Eq. (8), the sparse vector can be reconstructed by solving:

It is worthwhile to note that the spectrum X that we have defined in Eq. (3) is normalized by $\mathop \sum \nolimits_{i = 1}^N x({{\lambda_i}} ){\eta _i} = 1$, and the overall light intensity incident onto the entrance aperture is ${I_{\textrm{total}}} = \left( {\mathop \sum \nolimits_{k = 1}^K \mathop \sum \nolimits_{j = 1}^J I{^{\prime}_{jk}}} \right)\left( {\mathop \sum \nolimits_{i = 1}^N x^{\prime}({{\lambda_i}} )} \right) = \left( {\mathop \sum \nolimits_{k = 1}^K \mathop \sum \nolimits_{j = 1}^J {I_{jk}}} \right)\left( {\mathop \sum \nolimits_{i = 1}^N x({{\lambda_i}} )} \right).$ Hence, the recovered spectrum X can be scaled by a factor of $\mathop \sum \nolimits_{k = 1}^K \mathop \sum \nolimits_{j = 1}^J {I_{jk}}$ to reflect the actual amount of light illuminating the entrance, which is necessary in some spectroscopy applications where it is required to compare absorption spectra of samples.

2.4 Spectral recovery

The spectral sampling is a two-stage encoding process involving a dynamic mask enabled by a DMD as the entrance aperture and a fixed mask at the exit aperture. Light that passes through the entrance aperture gets encoded and then dispersed into its individual wavelength components by a diffraction grating. A set of overlapped spectral images of the entrance aperture is thus formed on the exit aperture. To distinguish these wavelengths, the encoded light must be encoded a second time before it is collected by another single-pixel detector for the recovery of the spectrum.

Figure 4(b) illustrates the second stage encoding process. For the m^th measurement ($m = 1\sim M)$, the light intensity that reaches the k^th encoding pixel of the j^th slit of the entrance aperture is ${I_{jk}}\left( {\mathop \sum \nolimits_{i = 1}^N x({{\lambda_i}} )} \right)$ and the encoding weight of that pixel is denoted as $a_{jk}^m$. It is noted that $a_{jk}^m$ here can be different from $w_{jk}^m$ of the spatial sampling process. This encoded light is dispersed and then encoded again by a secondary encoder, where ${b_{jk}}({{\lambda_i}} )$ represents the weightage of the k^th encoding pixel on the fixed encoder along the j^th column of the i^th wavelength component, where ${b_{jk}}({{\lambda_i}} )= 1$ represents a transparent pixel and ${b_{jk}}({{\lambda_i}} )= 0$ represents an opaque pixel depending on the designed pattern on the fixed mask. Thus, light from this pixel is encoded as $a_{jk}^m{I_{jk}}x({{\lambda_i}} ){b_{jk}}({{\lambda_i}} ){\xi _i}$, where ${\xi _i}$ represents the detection efficiency of the i^th wavelength component. As the image of different spectral components fall on different parts of the second fixed mask, intensities of the encoded spectrum from ${\lambda _1},\; {\lambda _2}, \ldots $ to ${\lambda _N}$ reaches the first-order detector. Hence, the overall intensity detected by the first-order single-pixel detector in the m^th measurement is described by the following equation:

In the above equation, the matrices A and B are known matrices based on the encoding design used on the DMD and fixed mask, respectively, and the matrix O is derived from the zeroth-order measurement by detector 1 (see Fig. 3). Thus, we can solve for Z when sufficient measurements Y are made, that is, when the full set of spatial and spectral information are collected by the zeroth- and first-order detectors, respectively. Based on Eq. (12), the spectrum Z can easily be found by matrix inversion as the matrix AOB can be efficiently inversed with full rank. Hence, the spectral information can be recovered by:

Once vector Z is solved, the radiation spectrum X can be obtained using $x({{\lambda_i}} )= {z_i}/{\xi _i}$.

2.5 Encoding scheme selection

This section covers the encoding pattern that is used on the masks of our proposed spectrometer. As the encoding mask on the entrance aperture is dynamic, the encoding scheme used can be replaced by uploading different sets of patterns onto the DMD. The mask at the exit aperture, however, is static and its patterns fixed after printing them on a glass substrate. As seen in Fig. 3, the zeroth-order encoded light that reflects from the diffraction grating is not dispersed and gets collected by detector 1. On the other hand, the first-order light is dispersed, and the individual spectral components are imaged onto the static encoding mask and is then collected by detector 2. Hence, it is only for the spectral sampling process that the relationship between the encoding patterns used on the static and dynamic masks require careful consideration.

For the spatial sampling process, the encoding patterns used are a set of random sparse matrices to meet the CS requirement. The encoding matrices are formed by randomly selecting P% of elements from each column to be ‘1’ while the rest of the elements are assigned to be ‘0’, where P is the compression ratio—the ratio of collected data to original data—used for the CS recovery process. A high compression ratio results in a better recovery of the spatial distribution on the entrance aperture, but at the expense of sampling time. As the number of pixels JK on the DMD can be relatively large, P has to be carefully selected to leverage the reduction in sampling time enabled by CS while still achieving a sufficiently high fidelity of the intensity distribution on the entrance aperture.

For the spectral sampling process, we use Hadamard matrix to design the encoding patterns on both the DMD and the fixed mask. It is known that in a multiplexing optical instrument, the use of Hadamard and S-matrices can provide an SNR enhancement to the system [15]. We use an S-matrix encoding scheme as it can be designed in a cyclic fashion, thus allowing us to effectively decode the overlapping images of the different wavelengths on the fixed encoding mask. Furthermore, as the Hadamard matrices require entries that are +1 and –1, additional optics and detectors are required to complete the measurements. To keep our system portable and low-cost, S-matrices are used in the design of both the dynamic and static masks to encode light during the spectral sampling process.

The entrance aperture consists of J slits which can be varied depending on the desired throughput. Each slit has K pixels. To maximize the rank of AOB, K should be equal to or greater than N, which corresponds to the total number of wavelength components in the spectral band of interest. This can be understood by using a single slit as an example, i.e., setting J = 1 in Eq. (11). In this paper, we use K = N. Hence, the overlapping dispersed spectral images of the entrance aperture covers a total of J + N−1 columns on the fixed mask. The first N columns are designed with an N^th order cyclic S-matrix, while the remaining columns are generated by cyclically repeating the first J−1 columns of the S-matrix. This way, each dispersed spectral image of the entrance aperture overlays J columns of encoding pattern on the fixed mask. Such a design ensures that high correlation of encoding information is avoided for each spectral image due to the orthonormal properties of any two columns in the S-matrix, thus allowing the information of each spectral component to be easily distinguished. The j^th column vector of B is formed by expanding the J × N encoding pattern on the second fixed mask for the j^th wavelength. Hence, the second encoding matrix can be expressed as:

Compared to the 1-D encoding pattern design by Harwit, the proposed scheme further encodes each slit spatially along a direction that is substantially transverse to the direction of dispersion. Such a scheme provides the freedom to select the number of entrance slits used, thus decoupling the spectral resolution of the system from its optical throughput. This is highly desirable in the design of compact and portable optical systems.

2.6 2-D to 1-D sampling transformation

We described, in earlier sections, the spectral reconstruction process using a combination of spatial and spectral sampling, whereby the former uses the zeroth-order signal and CS to obtain the 2-D spatial information of the entrance aperture, while the latter combines the spatial information of the entrance aperture with the spectrally-encoded first-order diffraction to reconstruct the spectrum. This is a general model that can be used for any encoding pattern designs. We also note that the sampling process can potentially take a long time as the number of slits on the entrance aperture J increases due to the involvement of CS and convex optimization. In this section, we prove that the use of a cyclic S-matrix encoding scheme can greatly simplify the spatial sampling process and thus reduce the overall sampling time.

We recall Eq. (11), where A and B are the encoding patterns used on the DMD and the fixed mask, respectively. For simplicity and without loss of generality, we investigate the case where light passes through two slits, S₁ and S₂, and assume the total number of measurements M and number of pixels K in each slit are both equal to the number of spectral components N. Equation (11) then becomes:

On the DMD, we can choose our dynamic encoding patterns such that:

On the second fixed mask, we can design the cyclic encoding patterns such that:

With the properties of Eq. (17) and (18), Eq. (17) can be further simplified into:

Referring to Fig. 5, we see that the new intensity matrix in Eq. (19) is simply the sum of the intensities through the pixels along slits S₁ and S₂ grouped diagonally. We can thus define new scaling factors ${T_1} = {I_1} + {I_{2N}}$, ${T_2} = {I_2} + {I_{N + 1}}$, …, ${T_\textrm{N}} = {I_N} + {I_{2N - 1}}$, i.e. the diagonally aggregated intensities, and the equation for spectrum reconstruction now becomes:

 

Fig. 5. Schematic of light passing through two columns of encoding pattern to form spectral images on the fixed mask, and the diagonal aggregation of pixels to form superpixels.

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3. Experiments and results

3.1 Experimental setup

A construct of the proposed spectrometer is shown in Fig. 6(a). The setup demonstrates the ability of our encoding scheme in enhancing the throughput and SNR of a dispersive spectrometer. Even though the setup demonstrated in this paper is designed to work in the 1500 nm to 1600 nm wavelength range, the encoding scheme can be easily applied in other regions by reconfiguring the system with the relevant optics, such as with an encoding device, single-pixel detector, and other optical elements that work in the detection wavelength of interest.

 

Fig. 6. (a) Experimental setup of the proposed high-throughput spectrometer; schematic illustrating (b) one encoding pixel on the DMD consisting of eleven micromirrors colored in green, and (c) the static encoding mask pattern at the exit aperture.

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The spectrometer is fabricated by precision machining and blackened to reduce the effect of stray light. The optical components are purchased off-the-shelf and mounted precisely onto the baseboard with dowel pins and screws. Fore-optics is used to image the source input onto the DMD. The DMD (Texas Instruments DLP7000) is made up of 1024 × 768 micromirrors, each with a diagonal pitch of 19.3 µm and a flip angle of ±12°. Micromirrors that form the ‘1’ encoding direct light into the spectrometer while the other micromirrors dump the unwanted light. As discussed in the earlier sections, S-matrix patterns are used to encode the light that enters the system. One encoding pixel is made up of 11 micromirrors arranged in three rows in the slit height direction and seven columns in the slit width direction as illustrated in Fig. 6(b). The dimensions of each encoding pixel are hence roughly 77.2 µm in width and 38.6 µm in height. As the S-matrix selected is of the 83^rd order, the entrance aperture has a height of approximately 3.2 mm, while the aperture width can be varied according to the number of slits used. Light directed by the DMD into the spectrometer is then dispersed by a diffraction grating to form spectral images on the static encoding mask at the exit aperture. The static encoding mask is fabricated by depositing a 100 nm chromium layer on a glass substrate as shown in Fig. 6(c). Finally, two InGaAs single-pixel detectors (Hamamatsu G12180-030A) are used to collect information of the zeroth- and first-order light to recover the input light spectrum. Overall, the size of our prototype is 220 mm × 180 mm × 130 mm, thus demonstrating its potential for portability.

3.2 Effects of spatial intensity distribution on spectral recovery

The first experiment characterizes the relationship between the spatial intensity distribution of the light source on the entrance aperture and its effect on the spectral reconstruction performance of the spectrometer. A polarization maintaining (PM) fiber with an output diameter of 8.6 µm is used to feed the tunable laser (Santec TSL-710) at a fixed wavelength into the system. The laser spot on the DMD is intentionally defocused to illuminate a larger area on the entrance aperture. The laser position is shifted between measurements to vary the spatial distribution on the entrance aperture. This allows us to compare the spectrums reconstructed with a poorly illuminated entrance aperture and with a properly illuminated entrance aperture.

For visualization, the 2-D intensity distribution of the laser at the entrance aperture is ascertained by recovering its spatial information using CS as described in section 2.3. A set of random sparse matrices is loaded onto the DMD, and the undispersed zeroth-order light is collected by detector 1 (see Fig. 6(a)). In our experiment, the discrete cosine transform is used with a compression ratio of approximately 30%. Once the distribution pattern is established, the position of the PM fiber can be finely adjusted in the horizontal and vertical direction using linear stages to generate different types of illumination conditions on the entrance aperture.

Figure 7 shows two different illumination conditions for a 1550.24 nm laser input, where Fig. 7(a) has a less uniform spatial distribution than Fig. 7(b). Figure 7(c) and 7(d) show their respective 1-D spatial distribution representations which is considered during the spectral recovery process using Eq. (21). Figure 7(e) shows the recoveries of both without considering the spatial information on the entrance aperture.

 

Fig. 7. CS-reconstructed normalized 2-D spatial distribution on the entrance aperture with (a) poor uniformity and (b) good uniformity of the incident light radiation; reduced 1-D representation of the spatial distribution in (c) Fig. 7(a) and (d) 7(b) using aggregated intensity distribution of diagonally disposed encoding superpixels; (e) reconstructed spectra for the 1550.24 nm laser under the illumination conditions shown in Fig. 7(a) and 7(b) without considering the spatial information of the entrance aperture; (f) reconstructed spectra for the 1550.24 nm laser with and without considering the spatial information of the entrance aperture.

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From the plots, it is observed that the SNR of the reconstructed spectra are affected by the illumination conditions, whereby poorer illumination uniformity on the DMD results in a noisier reconstruction compared to one with better uniformity. When the laser spot is directed to the central region of the aperture, more pixels are well-illuminated by the incident light and hence more efficiently involved in the encoding process, resulting in a more accurate recovery of the laser spectrum.

To remove the need for uniformity at the entrance aperture, we consider the spatial intensity distribution on the aperture in the spectral recovery process. This can be seen in Fig. 7(f) for comparison, where the two spectral plots are recovered under the same poor-uniformity illumination condition at the entrance aperture as shown in Fig. 7(a). As shown in Fig. 7(f), the red curve is the recovered spectrum without considering its spatial intensity distribution, that is, the spectrum is directly obtained by multiplying the measurement vector Y with the inverse matrix of AB. This is equivalent to assuming that the spatial intensity on the entrance aperture is uniformly distributed. As seen, it is still possible to distinguish the laser peak at the correct wavelength but with considerably more noise in the reconstructed spectrum. The green curve is the spectrum recovered by Eq. (21) with its 1-D diagonally aggregated intensity distribution of the superpixels shown in Fig. 7(c) taken into consideration. Clearly, the noise in the reconstructed spectrum is significantly reduced, with the SNR improving from 15.1 dB to 17.8 dB and noise reduction by a factor of around 2.42 (2.3 dB to 0.95 dB), thus proving the effectiveness of our proposed method.

3.3 SNR improvement with increase in width of entrance aperture

In our second experiment, we demonstrate the throughput advantage and SNR enhancement of the proposed spectrometer. We do this by changing the width of the entrance aperture to vary the amount of light that enters the spectrometer. The light source was defocused to illuminate a sufficiently large area on the DMD and fixed in position, and 3, 5, 11, 13, 19 and 21 encoding slits on the DMD are used in the experiment. The tunable laser is employed to input a source with fixed power, and the wavelengths chosen are 1514.75 nm, 1523.34 nm, 1549.02 nm, 1557.55 nm, 1583.06 nm, and 1591.54 nm to better visualize the recovery of the individual laser peaks.

Figure 8(a) presents the relative intensity of the recovered spectra. From the results, it is evident that the throughput of our spectrometer is significantly higher with a larger entrance aperture (i.e. increasing the number of slits used at the entrance aperture) without any degradation in its spectral resolution. It is noted that the width of the entrance is about 1.6 mm when 21 slits are opened on the DMD. Furthermore, the peaks of the reconstructed spectra are located at the correct wavelengths. Figure 8(c) shows the calculated full width at half maximum (FWHM) for different number of slits used when the input is a 1591.54 nm laser at 3 mW power. From the plot, we can see that the calculated FWHM remains relatively constant despite more slits being opened, and that the throughput is indeed decoupled from its spectral resolution. The SNR of the spectra are quantified by using the root mean square (rms) of the reconstructed spectra ${\textbf z}$ and the real signal $\hat{{\textbf z}}$ to estimate the noise level of the spectral reconstruction. In this case, the real signal is assumed to be a single-element column vector based on the experimental conditions. We calculate the rms using:

Consequently, the SNR (in dB) for each of the recovered spectrum can be estimated with its highest intensity ${z_{max}}$ as follows:

 

Fig. 8. (a) Reconstructed spectra of different wavelength peaks with different number of entrance slits used; (b) reconstructed spectra of a SLD source with different number of entrance slits used; calculated (c) FWHM and (d) SNR under the same light source condition of 3 mW and 1591.54 nm laser input with increasing number of entrance slits.

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We calculate that the SNR with 3 slits to be 14.7 dB, while the SNR with 21 slits is calculated to be 20.3 dB. The graph plotted in Fig. 8(d) further shows that the SNR of the recovered laser spectrum increases with number of slits used at the entrance aperture.

Next, we replace the light source with a superluminescent diode (SLD) (Thorlabs SLD1550P-A40) to reconstruct its spectrum using different number of slits at the entrance aperture. The results presented in Fig. 8(b) show that with fewer slits, the recovered spectra has a high noise profile. As we increase the number of slits to 21, we can see that the noise profile of the spectrum decreases and the shape of the reconstruction gets closer to that of the reference spectrum. The slight difference between the recovered and reference SLD spectra is a result of errors in the optical system caused by the manual alignment of the optical components within our setup. As this is a lab prototype, some tolerance and misalignment issues are inevitable which may affect the shape of the spectrum. Furthermore, the absence of equipment to properly account for the optical efficiencies of the components can result in distortion of the reconstructed spectra. However, the reduction in noise of the recovered spectra with increasing number of slits proves that the proposed encoding scheme is effective in increasing system throughput to enhance the SNR of the reconstructed spectra.

4. Conclusion

In this paper, we demonstrate a high-throughput single-pixel dispersive spectrometer that uses a double encoding scheme. We show that the width of the entrance aperture can be increased to allow higher throughput into our system to boost the SNR of the reconstructed spectra without degrading the spectral resolution. This proves that the proposed encoding mechanism is effective in decoupling the throughput of the system from its spectral resolution and exhibits desirable throughput and multiplexing advantages. The robustness of a dispersive spectrometer setup, coupled with the absence of mechanical moving parts, give the proposed spectrometer huge potential in field sensing applications. Furthermore, the proposed system removes the requirement of uniform incident radiation by taking into consideration the spatial information at the entrance aperture, thus circumventing the potential cost and space constraints that may come with methods to ensure uniformity of the incident radiation. These improvements allow our spectrometer to be extremely useful as portable, low-cost, and field-applicable high-throughput sensors and detectors.