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Computational spectrometer based on a broadband diffractive optic was demonstrated with high spectral resolution over large bandwidth and high photon utilization efficiency. In this paper, we analyze such a spectrometer using singular value decomposition and propose a faster spectrum reconstruction algorithm with excellent accuracy by regularization. A new definition of spectral resolution based upon the Fourier analysis of singular vectors is described as well.
© 2014 Optical Society of America
1. Introduction
The capability of analyzing optical spectrum enables applications in a wide range of disciplines including materials science, electrical engineering, biology, chemistry, astrophysics, etc [1–5]. Traditional spectrometers work by detecting the incoming spectra dispersed by either gratings or prisms [1–3]. In contrast, computational spectrometers utilize dispersive media such that the unknown spectra can be reconstructed by measuring the dispersed image, knowing the dispersion response and applying proper numerical algorithms [6–14]. As a result, one can build compact, simpler spectrometers with enhanced resolution and bandwidths. Coded aperture [6,7], multi-mode fiber [8,9], random scattering media [10,11], disordered photonic crystal [12] and photonic resonance [13,14] have all been utilized as the “dispersive media”. However, the coded aperture has to sacrifice photon throughput due to magnitude modulation [6,7]. The other approaches suffer from limited bandwidths [8–14]. The multi-mode fibers require specific polarization state [8,9]. Besides, random nanophotonic scattering (as used in the integrated spectrometer), introduces undesired power loss due to out-of-plane scattering and poor coupling efficiency into the waveguides from free-space [10]. Also, photonic resonators are very sensitive to fabrication errors [13,14].
Recently, we demonstrated an alternative free-space approach by exploiting far-field diffraction [15]. A broadband diffractive optic that we call a polychromat was used to disperse collimated incident light onto a sensor array. The spectral response of the polychromat was well characterized via a prior calibration step. Then, iterative numerical techniques were applied to reconstruct the unknown incident spectra. Spectral resolution of ~1nm over a bandwidth of ~450nm was experimentally demonstrated [15]. The bandwidth is essentially constraint by the quantum efficiency of the sensor array. In addition, it is able to offer unprecedented bandwidth-to-resolution ratio and high photon throughput, since the polychromat is fully transparent. The polychromat is readily fabricated using a single step gray-scale lithography process. In this paper, we apply singular value decomposition of the spatial-spectral point-spread-function (SS-PSF) of the polychromat and regularization in order to achieve spectral reconstruction that is at least an order of magnitude faster. Furthermore, the new approach allows one to gain physical insight into the relationship between spectral resolution and the design of the spectrometer. All the calculations and analyses in this paper were conducted on the platform of a MATLAB regularization tool [16,17].
2. Methods
2.1 Computational spectrometer
The basic mechanism of the diffraction-based computational spectrometer is schematically illustrated in Fig. 1. For simplicity, we utilize a polychromat that is patterned only in 1 dimension, i.e., it is pixelated along the X direction and uniform along the Y direction. The height of each groove is quantized into multiple discrete levels, bounded by a maximum height of H = 1.2μm. Here, 6 levels are used, making a unit height of Δh = 0.24μm. The grooves have the same width Δx = 3μm, dictated by the resolution of our patterning process. One polychromat is comprised of 1600 grooves and periodic boundary condition is implied along X. It disperses the incident collimated light. After propagating a distance of d along the Z axis, each wavelength forms a unique image on the sensor array (the X’-Y’ plane). The final image is a linear combination of the images formed by each wavelength in the incident spectrum.
Fig. 1 Schematic explaining the principle of the diffraction-based computational spectrometer. The polychromat is a pixelated micro-optic with discretized heights. Each wavelength generates a specific intensity pattern after propagation through the polychromat. The image plane is at distance d away from the polychromat.
Like other computational spectroscopy techniques [6–14], the imaging process can be expressed by an integral equation:
In Eq. (1), f(λ) and g(x) represent the input spectrum and the output signal, respectively, and K(x,λ) is the kernel, describing the spatial-spectral response of the system. This is usually referred to as the Fredholm integral equation of the first kind [16]. In practice, this integral has to take the discrete form:where A is a model matrix. Here we only consider A∈Rm × m. Therefore, x is the m × 1 vector of unknown spectrum information and b is the m × 1 vector of measured data. The key point of all computational spectrometers is to derive x from A and b. Regularization seems to be a good tool in stably solving the inverse problem x = A−1b [16]. For this particular study on diffraction-based computational spectrometer, Eq. (2) is transformed into:In Eq. (3), PSF is the SS-PSF matrix of the polychromat, Ψ(λ) is the compound photon flux spectrum, which is an element-wise multiplication of the unknown photon flux spectrum Φ(λ) and the sensor’s quantum efficiency QE(λ), and S(x) is the captured image.2.2 Matrix analysis
The SS-PSF fully determines the computational spectroscopy system. It is basically a two-dimensional matrix that describes both spatial and spectral responses of a polychromat at the image plane [15]. Singular value decomposition (SVD) of this matrix aids the understanding of the regularization method that is developed in the following section. The SVD decomposes a complicated matrix into a representation of a sequence of singular values and a set of left and right singular vectors [18]:
In Eq. (4), Σ is a diagonal matrix with the singular values as its diagonal elements, arranged in a descending manner (σ1≥σ2≥σ3≥…≥σm). The columns of U and V matrices (u1, u2, u3…um, and v1, v2, v3…vm) contain the m × 1 left and right singular vectors, respectively. In this work, where A = PSF, vectors ui form orthogonal bases in the spatial domain, while vectors vi form orthogonal bases in the spectral domain. The SVD by Eq. (4) is equivalent to a Fourier expansion of the model matrix. The vectors ui and vi are the frequency components in the spatial and spectral domains, respectively, and σi are the coefficients of the components (i = 1,2,3…m). We emphasize the spectral domain analysis (vi) to investigate the achievable spectral resolution in section 3.2.3 Solving inverse problem
To solve Eq. (2) by direct matrix inversion is problematic, since the matrix A is ill-conditioned with extremely large condition number (cond(A) = σ1/σm). In order to obtain reasonable solution for these types of problems that do not satisfy the Hadamard requirements, regularization methods were developed, and are particularly applied in biomedical imaging, geophysics and image deblurring [16]. Note that in [15], we used a modified version of the direct-binary-search (DBS) algorithm [19] to reconstruct the unknown spectra. Previously, we have also used the DBS algorithm to design solar spectrum-splitters/concentrators [20], dispersive optics for fluorescence microscopy [21], phase masks for three-dimensional lithography [22] and nanophotonic resonators for ultra-thin-film solar cells [23,24]. However, since this algorithm is iterative, it is very slow. Therefore, here we are motivated by the need for faster spectrum reconstruction and also the potential to gain physical insights into the inverse problem.
From the SVD point-of-view, any forward problem, generalized by Eq. (2), diminishes the high-frequency components in ui and vi. However, the inverse process attempts to magnify those high-frequency parts [16]. The regularization technique stabilizes the problem by minimizing both the residual norm ||Ax-b||2 and the solution norm ||x||2. Usually, a regularization parameter ω is selected to balance these two terms. Here, we focus on a widely-used method called Tikhonov regularization. Its goal is stated as:
It can be equally formulated as applying filter factors to solution vectors, from the computational perspective:in which the filter factor is defined by:In Eq. (6), we call xω the Tikhonov regularization solution with parameter ω. The effects of ω are discussed in section 4.3. Singular value decomposition
3.1 SS-PSF characterization
The polychromat of the computational spectrometer is prepared on clean glass substrate and patterned by standard gray-scale lithography in a positive photoresist. More details can be found in [20,21]. A 3D view of the atomic-force-microscopy (AFM) measurement of a small segment of the fabricated polychromat is given in Fig. 2(a), clearly showing the details of the multi-level grooves. Figure 2(b) depicts the setup for characterizing the spatial-spectral point-spread-function of the polychromat (PSF matrix in Eq. (3)). The output from a super-continuum source (NKT Photonics) is first collimated, expanded and is made normally incident on the polychromat. A fiber-tip detector, controlled by a motorized stage, scans across the polychromat’s image plane with a step size of 3μm and feeds the signal to a conventional Ocean Optics Jaz spectrometer. The PSF matrix is then synthesized by combining the spectral responses (compared to blank reference without the polychromat) at various locations.
Fig. 2 (a) AFM measurement of a segment (70μm × 30μm) of the fabricated polychromat with maximum height of 1.2μm and groove width of 3μm. (b) Schematic of the setup for characterizing the SS-PSF of the polychromat. The beam from super-continuum source is collimated and expanded. The fiber-tip detector, connected to a conventional spectrometer, is scanned at the image plane.
As mentioned in [15], the propagation distance d between the polychromat and the image plane is critical in deciding the spectrometer’s performance. Therefore, we characterized the SS-PSF at different distances. Two examples at d = 49mm [Fig. 3(b)] and d = 450mm [Fig. 3(c)] are plotted. They are both 1600 × 1600 matrices (m = 1600). Figure 3(a) gives the vectors of singular values σi based on singular value decomposition. Generally speaking, σi remains high for low-frequency components (when i is small). Then it drops quickly to extremely small values and rests on a plateau (when i is large). Interestingly, the ‘turning point’ right before σi experiences rapid decrease shifts to larger index i with increased propagation distance d. This intuitively indicates that the SS-PSF at greater distance will include the higher-frequency components, and consequently better spectral resolution. Theoretically, this occurs before the diffraction pattern at the image plane fully develops in the far-field zone of the polychromat, therefore different wavelengths still experience strong correlation, especially in Fig. 3(b).
Fig. 3 Singular value decomposition analysis of the experimental data. (a) Singular values σi of the measured SS-PSFs at different propagation distances. (inset: the spectral resolution estimated by Fourier analysis of singular vectors versus propagation distance, and the spectral resolution predicted by a correlation function) (b) Measured SS-PSF of the polychromat at d = 49mm. (c) Measured SS-PSF of the polychromat at d = 450mm. (d) Each column i represents the Fourier transform of the singular vector vi of the SS-PSF at d = 450mm. The low- and high-frequency regimes are separated by i = 420. (e) The Fourier transform of the singular vector v420. Peak frequency f0 = 0.53nm−1 indicates a spectral resolution of 0.95nm.
3.2 Fourier analysis of singular vectors
Singular vectors ui and vi with larger index i represents the higher-frequency components. This is further shown in Fig. 3(d), the ith column of which is the Fourier transform of the right singular vector vi in the spectral domain. The corresponding SS-PSF is the one at d = 450mm (see Fig. 3(c)). The maximum frequency is:
For a bandwidth of BW = 400nm and m = 1600, it gives fmax = 2nm−1.In Fig. 3(d), a clear transition between low- and high-frequency regimes is observed, separated at exactly the ‘turning-point’ of the singular value plot in Fig. 3(a) for d = 450mm. The low-frequency components have high singular values such that they dominate the PSF matrix, while the high-frequency components have trivial contributions to PSF (σi<10−5). As a result, the latter ones are numerically impossible to recover. For direct matrix inversion without any regularization, the outcome tends to amplify these high-frequency parts, since the singular values appear as denominators in Eq. (6). This again proves the necessity of regularization. Note that the plateau of singular values at i>900 for d = 450mm is primarily due to rounding errors of the computer (see also the FFT of vi in Fig. 3(d)) [16].
3.3 Spectral resolution
Since the ‘turning-point’ of the singular value plot separates the resolvable and non-resolvable spectral frequency regions, we can readily define the achievable spectral resolution by pinpointing the peak frequency of the Fourier transform of the singular vector at the ‘turning-point’. This transition happens at i = 420 for d = 450mm in Fig. 3(d). The Fast Fourier Transform (FFT) of the v420 vector is given in Fig. 3(e). Its peak frequency is about f0 = 0.53nm−1, which indicates a spectral resolution of ~0.95nm. The rule is:
The appendix explains in detail how to derive this spectral resolution from the signal frequency of the singular vector based on correlation function analysis.According to the above description, it is straightforward to calculate the spectral resolution at different distances, as plotted as the inset of Fig. 3(a). The spectral resolution estimated by a correlation function [8–10,15] is plotted as well for comparison. Both methods agree very well. Intriguingly, this definition of spectral resolution is derived directly from the generalized Eq. (2), meaning that it can potentially serve as an alternative way to predict resolution for other computational spectroscopy techniques [6–14]. Additionally, the spectral resolution is inversely proportional to distance d, which is consistent with the results from previously reported multi-mode fiber spectrometer [8,9].
4. Tikhonov regularization
4.1 Choosing regularization parameter
In this part, we apply Tikhonov regularization to reconstruct an unknown spectrum in our diffraction-based computational spectrometer. A yellow LED is taken as an example source. Note that the results presented here are extracted from actual experimental data. Conventionally, L-curve is used as a criterion in determining the optimal regularization parameter ω [16]. It shows the solution norm versus residual norm at different parameter values, and is plotted in Fig. 4. The corner of the L-shaped curve often gives the optimal parameter [16], where both norms are properly balanced.
Fig. 4 L-curve for selecting the regularization parameter ω to reconstruct the unknown spectrum of a yellow LED by the Tikhonov regularization method. Spectrum reconstruction results by using ω = 0.001 (top left inset), ω = 0.1 (top right inset), ω = 1.857 (middle right inset), ω = 10 (bottom left inset) and ω = 1000 (bottom right inset). The best result is obtained by ω = 1.857. The reconstructed plots are in red and the reference measurements are in black. They are normalized. The captured grey-scale camera image is shown as the middle inset.
The image recorded by the sensor array is shown as the middle inset of Fig. 4. The other insets illustrate how ω affects the reconstruction results. Black curves are reference measurement using the conventional spectrometer (Ocean Optics Jaz) and red curves are the reconstructed data. All plots are normalized. When ω is too small, the residual norm is emphasized more than the solution norm (see Eq. (5)), leading to noise-dominated results (see top insets). In this situation, the high-frequency components in vi are not sufficiently weakened and thus overwhelm the useful information. On the other hand, too large of an ω leads to over-damped results with distorted spectrum (see bottom insets). This is because only components of very low frequency are taken into account to reconstruct the unknown spectrum, while other useful components of slightly higher frequency are discarded. When ω = 1.857, near the corner of L-curve, the best spectrum reconstruction is obtained, as shown in the middle right inset. The profile of spectrum is well reserved with strongly suppressed noise. This is a typical phenomenon when working on discrete inverse problems by regularization [16]. Again the importance of choosing optimal ω to balance the two terms in Eq. (5) is evident since the effort of minimizing either solely residual norm or solely solution norm ends up with noisy or over-damped reconstruction results, respectively. Other popular criteria such as discrepancy principle and generalized cross-validation may be applied to select ω as well [16].
4.2 Spectrum reconstruction results
Figure 5 summarizes another two spectrum reconstruction examples: a 532nm solid-state laser [Fig. 5(a)] and a super-continuum source [Fig. 5(b)]. The L-curves are shown as top right insets. Although the L-curve corners cannot be easily observed (which is common in practice), we can still predict the unknown spectra with reasonable accuracy by letting ω = 0.938 and ω = 4.004 for these two cases. The errors present in all three spectrum reconstruction experiments [Figs. 4 and 5] are ascribed to errors in measurement (camera image), inaccuracy in characterizing the SS-PSF of the polychromat, and numerical errors in computation. The regularization method is able to find the laser peak wavelength within 0.15nm (λ0 = 531.10nm and λ0 = 531.25nm by reference measurement and numerical reconstruction, respectively).
Fig. 5 Spectrum reconstruction results of a solid-state green laser with 532nm central wavelength (a) and a super-continuum source (b) by the Tikhonov regularization. The optimal parameters are ω = 0.938 and ω = 4.004, respectively. Red curves are the reconstructed results and black ones are the reference measurements. (Top insets: L-curves for selecting the best parameters; Bottom insets: photographs of the images projected on white paper at d = 450mm).
In this current work, we utilized the Tikhonov regularization as an example to reconstruct unknown spectra, because it allows for better smoothing performances (continuously-changing filter factors ) compared to other regularization techniques, such as truncated singular value decomposition (TSVD) and selective singular value decomposition (SSVD) [16]. Nevertheless, for large-scale inverse problems, iterative methods are preferred, including conjugate gradient for least square (CGLS), Krylov-subspace iteration, etc [16]. Although slow by nature, they are advantageous in their capability of solving large matrices stably. These methods could be promising candidates for ultra-high-resolution and ultra-broadband computational spectrometers.
5. Conclusion
In conclusion, we described a new approach for reconstructing unknown spectra in a diffraction-based computational spectrometer [15] by singular value decomposition. The SVD of the system matrix, the so-called SS-PSF of the polychromat, also predicts the achievable spectral resolution. To solve this ill-conditioned inverse problem more efficiently, we applied the Tikhonov regularization and successfully reconstructed the unknown spectra from an LED, a laser and a broadband supercontinuum source. The L-curve was shown to be extremely useful in choosing the proper regularization parameter. The technique proposed here is equally applicable to other previously reported computational spectroscopy techniques [6–14] and in general to any problem that can be fully modelled using Eq. (1) or Eq. (2).
Appendix: Derivation of spectral resolution
In order to approximate the real singular vectors derived from the SVD of the PSF matrix, we consider a perfect harmonic signal with period of Λ=50nm for simplicity in mathematical analysis. It is plotted in Fig. 6(a). Its peak frequency in the Fourier domain is f0=1/Λ=0.02nm−1. To determine the achievable spectral resolution of this frequency component, we introduce a correlation function analysis [8–10,15], which is expressed by:
Fig. 6 Explanation of the achievable spectral resolution. (a) A perfect harmonic signal in the spectral domain with period of Λ = 50nm from 500nm to 600nm. (b) Normalized correlation function of the harmonic signal, indicating a minimum at δλ = 25nm, which is half the period.
Here, vi is the ith singular vector in the spectral domain. In this simplified example, vi is replaced by the harmonic signal (Fig. 6(a)). The correlation basically describes the degree of similarity between any two signal segments separated by δλ. The larger the C(δλ), the harder it is to distinguish these two signal segments (vi(λ) and vi(λ+δλ)). Smaller C(δλ) means that these two segments are less correlated, and thus easier to identify. The normalized correlation function of the example signal is plotted in Fig. 6(b), which indicates a minimum at δλ=25nm. Due to the periodic nature of the original signal, this represents the minimum spectral spacing which is least correlated and consequently easiest to distinguish. We define this as the achievable spectral resolution of the oscillatory signal in Fig. 6(a). Therefore we can draw a conclusion that for any frequency component with peak frequency of f0, the spectral resolution is δλ=Λ/2=1/(2f0).
Acknowledgments
We would like to thank Prof. Christopher Johnson in the Scientific Computing and Imaging Institute (SCI) at the University of Utah for assistance on numerical analysis. The project was funded from a DOE Sunshot Grant, EE0005959 and a NASA Early Stage Innovations Grant, NNX14AB13G. R.M was partially funded by the Utah Science Technology and Research (USTAR) Initiative.
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