Photonic crystal spectrometers possess significant size and cost advantages over traditional grating-based spectrometers. In a previous work [Pervez, et al, Opt. Express 18, 8277 (2010)] we demonstrated a proof of this concept by implementing a 9-element array photonic crystal spectrometer with a resolution of 20nm. Here we demonstrate a photonic crystal spectrometer with improved performance. The dependence of the spectral recovery resolution on the number of photonic crystal arrays and the width of the response function from each photonic crystal is investigated. A mathematical treatment, regularization based on known information of the spectrum, is utilized in order to stabilize the spectral estimation inverse problem and achieve improved spectral recovery. Colorimetry applications, the measurement of CIE 1931 chromaticities and the color rendering index, are demonstrated with the improved spectrometer.
© 2013 Optical Society of America
Color measurements have wide application in informative display, printing , textile [2, 3], light source evaluation, anti-counterfeiting, process control, machine vision, dentistry , dermatology and pharmaceutical tests . Current approaches for color measurement include tris-timulus colorimetry based on three filters that resemble the CIE color matching functions, multichannel colorimetry with more than six filters , and continuous spectrum measurement using compact grating spectrometers.
Although inexpensive, tristimulus colorimeters are associated with large color errors because the colorimeter’s spectral response is usually not linearly related to the CIE color matching functions (Luther-Ives condition) due to manufacturing difficulties . Increasing the number of measurement channels can improve the precision, and multi-channel colorimetry with more filters provides such capability. Filters based on organic dyes are often of broad response and limited in choice; distributed Bragg reflector (DBR) based filters are expensive to manufacture and require precise multilayer vacuum depositions ; the recent development of continuously tunable filters based on LCDs is quite expensive . Another choice for color measurement is to integrate compact grating spectrometers to measure a continuous spectrum of emitted or reflected light, and the tristimulus coordinate of the light input is then calculated with high precision. However, the cost of this approach is prohibitive for certain portable applications, with the grating spectrometers costing hundreds of dollars.
Recently several interesting concepts for chip-size spectrometers have been proposed, and may be useful for inexpensive, high-resolution color measurement applications. A linearly variable filter based on a graded Fabry-Pérot cavity sandwiched between two homogeneous DBRs has been demonstrated. Such a filter transmits different wavelengths at different spatial locations and can be directly mounted to a linear photodetector array without a gap . A photonic crystal superprism and a detector array have been integrated on a chip to realize high-resolution spectrum recording [10, 11]. Band-pass filtering of visible light has also been observed in nanoslit structures because of their plasmonic interaction with light . A reflection-based spectrometer built with colloidal photonic crystals of different bandgaps has also been proposed . These self-assembled colloidal crystals reflect different wavelengths of light based on the matching of the wavelength to the Bragg diffraction condition.
We have demonstrated a new spectrometer based on leaky-mode photonic crystal arrays, and its application for recovering a white LEDs spectrum . This photonic crystal spectrometer is simpler to manufacture and potentially cheaper compared to the aforementioned chip-size spectrometers. It only consists of a waveguide slab and a layer of photonic crystal arrays formed on the surface of the waveguide by a single lithographic or imprinting step (Fig. 1(a)). The photonic crystal arrays extract different wavelengths of light from the waveguide based on the matching of the crystal periods and the wavelengths, and spatially distribute the intensities. Figure 1(b) is a microscopic photo of the 17-channel photonic crystal spectrometer when illuminated with a white LED. Each square channel is 30 μm by 30 μm. This spatially resolved information can be directly monitored by a 2D imager such as a CCD or CMOS sensor. The resolution is no longer dependent on the length of the light path as in a grating spectrometer, and thus the photonic crystal spectrometer can be significantly smaller, as small as 1 cm3 in an integrated format. The cost can be aggressively brought down from thousands of dollars to below ten dollars. Another advantage is the flexibility in customizing the photonic crystal spectrometers to address different applications with minimum cost. As mentioned previously, continuous spectrum recording is often overkill for applications where limited-channel representation is adequate, as in many colorimetry applications. The photonic crystal spectrometer can be customized to have varied resolution at different ranges of wavelengths.
The initial demonstration had only nine photonic crystal channels and wide response functions, limiting the quality and the resolution of the spectra recovered. In this paper we consider spectral estimation using a spectrometer with additional arrays (though still quite limited in number) and consider schemes for regularizing the associated ill-posed inverse problem in the presence of noise. We also show how to use this limited-channel photonic crystal spectrometer for colorimetry applications as simple as estimating CIE 1931 colorimetry parameters and as complex as color rendering index (CRI) values, without the intermediate step of estimating the spectrum itself. The very inexpensive build and extremely small footprint make the photonic crystal spectrometers an attractive solution to colorimetry applications where color filter technology is inadequate and compact grating spectrometers are too bulky or expensive.
2. Spectral estimation inverse problem
As described in , the photonic crystal spectrometer’s input-output response is linear. Suppose the spectrometer has m output channels and let bi denote the output intensity of the ith channel, 1 ≤ i ≤ m. The spectrometer’s output can be modeled as
Let x ∈ ℝn encode the spectral intensity of the input light at n equispaced wavelengths λ1,...,λn over some appropriate range. For actual computation we discretize Eq. (1) as
3. Regularizing the inverse problem
Although the linear system Ax = b is underdetermined, we can obtain a unique solution by imposing additional conditions on x. These conditions should use a priori information about the true spectrum, for example, smoothness conditions, concentration in a certain range of wavelengths, nonnegativity, etc. Because the response functions Ai peak at different locations, the matrix A has full row rank, and so we might expect the Moore-Penrose pseudoinverse to do a reasonable job of estimating the spectrum x from the data b, as the minimum norm solution to Ax = b. However, the data here is sufficiently noisy that some additional regularization is necessary, especially as the number of channels increases.
Let bmeas denote the measured (noisy) channel data. For the present work we regularize by seeking a vector xrec that satisfiesEq. (3) is unique, for if we differentiate with respect to x we see that any minimizer must satisfy the normal equation Eq. (4) has a unique solution and Eq. (3) has a unique minimizer.
In estimating the spectrum x we may choose Γ to produce smooth solutions to Eq. (3). Some typical choices areEq. (3), which encourages solutions that have smaller norms, but without a premium on smoothness.
The regularization parameter λ > 0 is chosen to strike a balance between fidelity to the original inverse problem (λ ∼ 0) and desirable qualities of the estimate xrec, especially in the presence of noise. We use the simple “discrepancy principle” [16, 17] for the selection of λ, illustrated in Section 4.18] and can be efficiently solved using a QR or singular value decomposition. Moreover, for estimating x the incorporation of nonnegativity constraints on xi ≥ 0 for the components of x is also desirable. Specific examples illustrating the regularization process are shown in the next section.
4. Estimating the spectrum
The estimation of the spectrum can become quite sensitive to noise in bmeas, even with a relatively small number of channels. In this section we illustrate this, and show how the regularization scheme outlined above can be used to provide stable spectral estimates.
Our photonic crystal spectrometer is constructed using 36 arrays (or channels) of photonic crystals with varied periodicities. Of the 36 channels, 17 are responsive to visible light (450 nm – 700nm); the other 19 channels are responsive to ultraviolet or infrared light, which cannot be imaged by the color camera we have. The response matrix A was characterized using a Xe arc light source, a monochromator, and a calibrated Si photodiode.
The value of the regularization parameter λ is determined using the discrepancy principle [17, 16], as follows: Let bmeas = b* + e, where b* is the true (noiseless) channel intensities and e is a noise vector with estimated magnitude ‖e‖2 = δ. We select the regularization parameter λ so that ‖Axrec − bmeas‖2 = δ, where xrec is the solution to Eq. (3). Using the techniques in section 4.7 of  one can easily show that this value of λ is unique, provided δ < ‖b‖2.
4.1. Perfect data example
We now give a simple example to show the performance of the procedure under ideal measurement conditions. We measured a white LEDs spectrum using the Ocean Optics USB 4000 commercial spectrometer. We will use this spectrum x* as our “ground-truth.” We then use x* to compute the “true” channel intensities b* = Ax*. We then use the regularized reconstruction procedure from Section 3 to attempt to recover x* from b*.
Since in this case the data b* is exact we use estimated noise level δ = 1.0×10−8 and choose λ to obtain ‖Axrec − bmeas‖2 = δ (though any choice δ < 1.0 × 10−3 makes little difference). Reconstructions using each of the regularization matrices in Eq. (5) are shown in Fig. 2, as well as that obtained from Γ = I. Second derivative regularization and first derivative regularization have almost identical recovery quality. Such quantification is done by calculating ‖xrec − x*‖2 for first derivative regularization (0.125), second derivative regularization (0.121) and identity matrix regularization (0.205).
4.2. Real data examples
For this example we use the actual 17-channel data to reconstruct the spectrum for the white LED. Reconstructions using each of the regularization matrices in Eq. (5) are shown in Fig. 3, as well as that obtained from Γ = I. The noise level in the data is estimated as ‖bmeas − b*‖2, and is equal to 12 percent of the magnitude of b*; that is, δ = 0.12‖b*‖2 in the discrepancy principle, and the regularization parameter λ is chosen accordingly. Spectral recovery quality is quantified by calculating ‖xrec − x*‖2 for first derivative regularization (0.211), second derivative regularization (0.208) and identity matrix regularization (0.290).
In Fig. 4 we show the reconstruction/regularization procedure applied to various LEDs of different colors using 17-channel data again with estimated noise level δ = 0.12‖b*‖2 and discrete first derivative regularization.
We believe there are two major sources of measurement noise. The first is intrinsic machine noise due to incomplete elimination of dark current, noisy imaging and processing. The other is due to the shift of response functions from the time that the response functions were taken to the time that these LED spectra were taken. The input fiber used in this system has a numerical aperture (NA) of 0.47 and the imaging optics has an NA of 0.096. The width (FWHM) of the response functions are affected by both NAs (lower NAs produce narrower response functions) but mostly constrained by the imaging optics. Significant measurement noise is introduced if the imaging NA or the fiber-waveguide coupling angle change, and the response functions are not retaken accordingly at the time of measurement.
4.3. Spectrum estimation resolution
Both the number of photonic crystal channels and the widths of response functions affect the resolution of spectrum recovery. With fixed widths for the response functions, more channels lead to a better resolution; however, after exceeding certain threshold (e.g., ∼ 50 channels for the wavelength range 450 – 700 nm) the number of channels does not affect the resolution. With sufficient channels, narrower response functions generate a better resolution.
We conduct a computational experiment in which the matrix of response functions A is known and represented by multiple Gaussian peaks. We consider a test spectrum xtest (λ) consisting of two Dirac delta spikes at wavelengths λ* and λ* + Δλ, so xtest (λ) = δ(λ − λ*) + δ(λ − λ* − Δλ). Similar to the approach in section IV (perfect data example), channel intensities can be calculated using this ground-truth xtest and known A, and these channel responses then used to recover an estimates of xtest. The distance Δλ between the two delta functions is varied until the recovered spectrum does not resolve the two delta peaks. The two peaks are resolved when the valley between the two peaks is lower than half of the peak value. In Fig. 5 the resolution of spectrum recovery varies with the number of photonic crystal channels. With three different widths of response functions, the resolution all hit a plateau when there are sufficient number of channels (15 for width of 50nm, 30 for width of 20nm and 45 for width of 10nm).
5. Colorimetric applications
5.1. Estimation of tristimulus parameters
Many real-world applications rely on colorimetry measurements, such as color comparison in paint and textile, light source qualification, media display and printing, anti-counterfeiting, packaging automation and production quality control. Current solutions rely on inconvenient filter-based photometry and expensive full-spectra photospectrometry technologies. Here we demonstrate a couple of applications with the highly customizable and very cost-effective photonic crystal spectrometer solutions.
The tristimulus parameters X, Y, and Z are computed from the spectral density function x(λ) asEq. (7) (or a discrete version thereof) with the appropriate weighting function. However, we can estimate these values directly from the spectrometer output data vector b without the intermediate step of directly estimating the spectrum itself, and regularize in a manner adapted to this setting.
Let w be a column-vector in ℝn that appropriately approximates the function w(λ) on the interval [λmin, λmax], and let x similarly be a vector approximating x(λ). The discrete version of Eq. (7) isEq. (3) or Eq. (4); let us also drop the non-negativity assumption on x for now (since we aren’t estimating x anyway). If A has singular value decomposition A = USVT with singular values σ1 ≥ σ2 ≥ ⋯ ≥ σ1 ≥ 0 then when λ > 0 the solution to Eq. (3) can be expressed as 16], for example. When λ = 0 has diagonal entries for σi > 0 and otherwise (so is the Moore-Penrose inverse for A). From Eq. (8) with x = xrec we can estimate
In the case that λ = 0 the vector c0 may also be realized as the minimizer of , since the minimum norm least squares solution to ATc − w is given byEq. (11) with λ = 0. That is, ATc0 is the best approximation to w that can be constructed from the rows of A. We might then consider as the best linear estimate of W in Eq. (8) that can be made with noiseless data b*. The vector cλ itself may be realized as the minimizer of , a regularized version of c0. In the presence of noise this regularization can be helpful.
To determine the optimal λ, let where bmeas = b* + e for some noise vector e. We choose that value of λ that minimizes the expected error E((Wλ − W*)2), under certain assumptions about e. We haveEq. (13)/Eq. (14) Eq. (15) and taking expected values yields Eq. (16) can be written Eq. (17), with a slight correction. Under the i.i.d. assumption for the ek one can compute that Eq. (17) with the measured values , corrected according to Eq. (18). This yields
Other error models are certainly possible, e.g., one in which the error is given by where the ek are independent with zero mean but with ; that is, the errors in the kth channel has standard deviation proportional to the magnitude |bk|. In this case and Eq. (19) becomes
Using the 17 channel data from Subsection 4.2 we estimate the tristimulus parameters X, Y, and Z using Eq. (10)/Eq. (11) with the regularization procedure of Section 5.1 and the same estimated noise level (‖e‖2 ≈ 0.12‖b*‖2) as in Subsection 4.2. We then use these values to compute the CIE 1931 chromaticity coordinates for each LED spectrum. We should note that for the case λ = 0 the best approximations w̃X, w̃Y, w̃Z for each of the tristimulus weighting vectors wX, wY, and wZ can be obtained using Eq. (12) and yield relative errors ‖w̃X − wX‖2/‖wX‖2 = 0.0747, ‖w̃Y − wY‖2/‖wY‖2 = 0.0649, and ‖w̃Z −wZ‖2/‖wZ‖2 = 0.0750, thus we should expect at least comparable error in our tristimulus or chromaticity estimates, even with perfect noise-free data.
The CIE 1931 chromaticities are computed as x = X/(X +Y +Z), y = Y/(X +Y +Z) and are shown in Table 1. The “Actual channel Data” column contains the estimates using the measured 17-channel data; the “Noiseless channel Data” shows the estimates using perfect data computed as b* = Ax*, where x* is measured using the commercial spectrometer and b = b* is used in Eq. (10). Finally, the column “True Values” contains the estimates obtained from Eq. (8) with x = x* and appropriate w. Figure 6 plots these results on the CIE 1931 color space. The triangles are the true CIE 1931 chromaticities of the color LEDs (near the periphery of the space) and the superwhite LED (in the center of the space) as measured by Ocean Optics spectrometer (USB 4000, calibrated). The cross near the center of the space is the CIE (x,y) for the superwhite LED estimated based on measurement by photonic crystal spectrometer. The circles adjacent to the periphery of the color space are the CIE 1931 chromaticities of the response functions of the photonic crystal spectrometer. In other words, if these circles are connected by a line, the enclosed space represents the color space range that the photonic crystal spectrometer can measure.
We can also use the term on the right in Eq. (14) to estimate the variance in the regularized X, Y, Z estimates due to noise, and the corresponding variation induced in the chromaticity coordinates. Under the i.i.d. assumption for the error ek with common variance σ2 we find
5.3. Color rendering index
An object looks drastically different under incandescent light and fluorescent light. The reason for this difference is that the incandescent and fluorescent light sources have different (correlated) color temperature and color rendering indices. The color rendering index (CRI) is an important quality factor for a light source. It evaluates the whiteness of a light source by comparing the color appearance of many given samples under a standard light source and the tested light source. The quantification of CRI has wide applications in cosmetics, illumination design, and graphics.
The calculation of CRI is based on the precise measurement of the color distance in the CIE 1976 color space between the chromatic coordinates of the same samples under the two light sources . First a standard light source is selected based on the correlated color temperature (CCT) of the tested light source (e.g., CIE illuminant D65 if the CCT is 6500K). Eight samples of different colors are selected and their chromatic coordinates under the standard light source and the tested light source are calculated. Both the correlated color temperature and the chromatic coordinates can be derived from a full spectrum (e.g., 350–700nm with 5nm step) measurement of the light source.
To calculate CRI under CIE 1976 amendment, the coordinates of the eight color samples under the reference light source and under the tested light source are plotted in CIE 1976 L*, u*, v* space in Fig. 7(a). The solid squares represent the chromaticities of the eight samples under the reference light source with CCT calculated from true spectrum of the white LED measured by an Ocean Optics spectrometer. The chromaticities of the eight samples under the reference light sources generated from the noiseless recovery (as in Fig. 2(b)) and noisy recovery of the LED spectra (as in Fig. 3(b)) from measurements by photonic crystal spectrometer are overlapping with the solid squares and not shown in the plot. This is because all the three spectra (true spectrum, noiesy recovery and noiseless recovery) have very similar CCTs (Fig. 7(b)).
The open squares, open circles and open triangles are the same 8 samples under the white LED’s true spectrum (dotted line in Fig. 2(b)), spectrum recovered in noiseless condition (solid line in Fig. 2(b)), and spectrum recovered in the noisy condition (solid line in Fig. 3(b)), respectively. The arithmetic mean of the distance between the solid and open symbols is (negatively) proportional to CRI. Figure 7(b) compares the CIE 1931 coordinates, CCTs, and CRIs calculated from measurements by Ocean Optics spectrometer and photonic crystal spectrometer. The difference in CCTs or reference light sources plays a minor role to produce the difference in CRI. This difference is predominantly due to the broadened peaks in the recovered spectrum in Fig. 3(b), from 450nm to 510nm, and from 600nm to 700nm. By decreasing the measurement noise (as indicated by ”noiseless recovery”) and the extent of regularization, the error in CRI measurement can be further minimized. Methods of minimizing experimental error such as using monochrome sensor (instead of sensors with color filters) and a more reliable coupling between light source and the spectrometer, are currently under investigation.
This work describes the use of a photonic crystal spectrometer, a novel and very inexpensive technology, in colorimetry applications. The size and cost of this technology provides unique advantages for color measurements than incumbent technologies such as tristimulus color filter approaches and grating spectrometers. It can be integrated into many handheld and portable devices for color measurement and calibration. We demonstrated chromaticity measurement directly from sensor readings instead of through the fully recovered spectrum, and CRI measurement with the improved photonic crystal spectrometer. The results are compared with measurements by a commercial grating spectrometer. Chromaticity measurements show < 10% error and CRI measurements show ∼ 19% error. We can further improve the performance of the photonic crystal spectrometer by minimizing the measurement noise, increasing the number of arrays of photonic crystals, decreasing the width of the response peaks, and introducing an appropriate regularization scheme in solving the estimation inverse problem. Based on the estimation from noiseless simulation (Fig. 5), in order to achieve 10 nm resolution, 50 channels with each channel having response function of ∼ 5 nm width are required.
This material is based upon work supported by the National Science Foundation under Award No. IIP-1152707. The authors would like to thank Prof. Xun Jia (University of California, San Diego) for helpful discussions.
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