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Abstract
To investigate the influence of phosphor coating on an UV-responsive coated enhancement CCD spectrometer, firstly, a phosphor coating mathematical model was established and its feasibility has been proved through experiments. Secondly, a spectral resolution improvement algorithm based on the MAP technique was introduced by employing the simplified model. Finally, the resolution improvement experiments of the spectra obtained from different coated CCDs with one CCD-based spectrometer were carried out, of which the results verify the effectiveness of the algorithm. The proposed phosphor coating mathematical model and spectral resolution improvement algorithm could be adequately applied to the field of UV-responsive coated enhancement.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Spectrometer is the basic equipment to measure, analyze and process the structure and composition of materials by using optical principles. Compared with the traditional spectrometer, the micro-spectrometer has the advantages of small size, high accuracy, wide measurement range, and high processing speed [1–4]. It was widely used in aerospace, food safety, drug testing, scientific research and other fields [5–7]. In some special research fields, it is not only necessary to analyze the spectrum of visible light, but also the ultraviolet. However, the response of ordinary charge-coupled devices (CCD) to ultraviolet spectrum is not high enough, which is due to the considerably small penetration depth (less than 2 nm) of ultraviolet light in polysilicon. To solve this problem, one of the feasible methods is to coat the CCD with a phosphor down-conversion coating [8–10] with enhanced ultraviolet response, so that the ultraviolet light can be converted into visible light and then be remarkably responded by the detector. This solution will undoubtedly have positive prospects for the development of complex and high cost back illuminated CCD [11].
Phosphor coating should meet the following requirements: high conversion efficiency, wide excitation spectrum, luminescence of phosphor body which is not limited by excitation wavelength. The emission spectrum should match the spectral response of the detector employed. Phosphor materials can be divided into organic materials [12,13] (such as coronene, Lumogen) and inorganic materials [14] (such as Y2O3: Eu). Compared with inorganic materials, organic materials have poorer photostability but higher fluorescence conversion efficiency with more uniform coating, so they are more commonly used to make phosphor coating. Which the solubility of organic materials is low, induce the coating made by spin coating method [15] is not uniform, evaporation method [13] is generally adopted.
However, the coating will cause the decline of the spectral resolution of the instrument, especially when the glass cannot be removed, the decline of the spectral resolution is particularly serious. Spectral resolution improvement is a classic problem in spectrometers. Several approaches for resolution enhancement have been proposed, such as Fourier self-deconvolution (FSD) [16], Stearns and Stearns (S-S) method [17], improved S-S method [18], maximum entropy deconvolution [19], high-order statistic method [20] and differential operator (DO) method [21]. Nevertheless, the problem itself is ill-posed. As a consequence, small deviations such as noise in the spectrum are amplified. When dealing with such problems, these algorithms often lack stability and uniqueness. In the process of spectral correction, the noise level increases with the increase of iteration process. As the noise reaches a certain level, the algorithm has to be stopped. The common method is to increase the interval between the data to reduce the spectral sensitivity to noise, but this will lead to the loss of spectral information. In order to solve the ill-posed problem of the inverse problem, an alternative approach based on the Richardson–Lucy (R-L) method [22] and Tikhonov regularization [23] was extended to the spectroscopy field by Eichstädt [24]. Because of the introduction of regularization terms, the method has a superior ability to suppress noise. In previous work, we proposed a spectral deconvolution method [25] based on Levenberg-Marquart (LM) algorithm [26,27] for LED spectral correction. By optimizing the parameters of the He-Zheng LED [28] model, the noise of the LED spectrum was eliminated, and the ill-posed problem was avoided. The maximum a posteriori (MAP) technique [29–31] is a common method for estimating the true spectra from a given measured spectra. This technique maximizes the probability of estimating the true spectrum from given measurement spectrum and convolution kernel function. Based on the conditional probability formula of Bayes, MAP can be transformed into a likelihood probability multiplied by a priori probability. The setting of a prior probability item has a certain inhibitory effect on noise.
In this paper, firstly, the influence of phosphor coating on ultraviolet light was explored. A mathematical model was established and the feasibility of it was analyzed through experiments. Then, considering that the effect of phosphor coating on ultraviolet light can be attributed to response enhancement and spectral broadening, the proposed mathematical model of the phosphor coating was simplified, and the response enhancement function and broadening function of the simplified model were obtained. Afterwards, in order to eliminate the influence of the broadening function and retain the influence of the response enhancement function on the spectrum, a spectral resolution improvement algorithm based on MAP technique was introduced. Finally, the resolution enhancement experiments of the fluorescence spectra obtained from different coated CCDs were carried out. This algorithm can effectively enhance the resolution of spectra.
2. Phosphor coating analysis and modeling
After absorbing ultraviolet light, the phosphor coating emits visible light, which then arrives at the pixel and is responded by CCD, as shown in Fig. 1. The whole process can be divided into two steps which is shown below. Firstly, the ultraviolet light is absorbed by the coating and excites visible light. After that, the visible light at the bottom of the coating reaches the pixels. The simulation are several functions that can help to show the trends involved with changing various aspects of the coating, and how they affect the properties of the coated CCD. Right from the start, the first step of the simulation process is as follows.
The incident light passing through the collimating system of the spectrometer is collimated. The coating is divided into N slices. The collimated incident light is absorbed partially while passing through each slice of the coating, and the absorption effect can be presented by Lambert-Beer law.
where I is the intensity of the incident light, ${I_{abs}}$ is the intensity of the light passing through the coating, ${\alpha _{abs}}$ is the absorption coefficient of ultraviolet photons in the coating, and t is the thickness of the slice.While one part absorbed, another part of the light is scattered, and the intensity of the scattered light is shown in Eq. (2). This is formally the same as Eq. (1), except that the parameter ${\alpha _{sca}}$ represents the scattering coefficient of ultraviolet photons in the coating.
It is assumed that any photon generated by the scattering in the coating will be lost without being received by CCD. Since most photons are scattered many times when passing through the coating, it is impossible to simulate the whole process completely. Therefore, the scattering coefficient here correspond to a measure of the loss caused by scattering, which is a small part of the actual scattering.The energy of ultraviolet photons is absorbed by phosphor which then re-emit visible photons. The percentage of visible photons re-emitted is determined by phosphor efficiency. Unlike the incident photons, these photons are isotropic. Therefore, in order to calculate the number of photons eventually reaching the position of each pixel at the bottom of the coating, it is necessary to calculate the ratio of the photons obtained at a given pixel position to the total photons by using the solid angle revision. The revision of solid angle is extremely complex. Since the sensor is a linear CCD, the structure as shown in Fig. 2. is used for approximate simulation. The formula for each angle is as follows.
In summary, if the incident spectrum is S, the output spectrum, ${S_{output1}}$, after passing through the coating, can be expressed as
- 1. One end of the fiber was connected with an ultraviolet lamp, and the other end was incident vertically to the CCD uncoated and the CCD coated on the glass surface, respectively, and read the spectral response.
- 2. Take the spectrum collected by CCD uncoated as the input spectrum, and calculate the output spectrum by employing the model as the simulation result.
- 3. Compare the difference between measured spectrum and the simulated spectrum.
Fig. 3. (a) The measured spectra by the CCD coated on the glass and the CCD uncoated, (b) The measured spectrum by the CCD coated on the glass, the simulated spectrum and the error between them.
3. MAP based resolution improvement method
Although the appealed model can effectively simulate the process of that ultraviolet light is absorbed and scattered with visible light emitted by the phosphor coating and then arriving at each pixel of the CCD, the process is slightly complex. The effect of phosphor coating on ultraviolet light mainly consists of two parts: the enhancement of ultraviolet response and the increase of spectral bandwidth. Hence, the following simplified models are established to simulate the process, as shown in Eq. (11).
where A and $Fb$ indicate the enhancement and broadening effects of coating on ultraviolet light, respectively. Using $(1,1,1\ldots \ldots 1,1)$ as the initial input spectrum, the output spectrum ${S_{output1}}$ obtained can be approximated as response enhancement function A. Utilizing $(0,\ldots 0,{1_{center}},0,\ldots 0)$ as the initial input spectrum and the area normalization is performed to the output spectrum ${S_{output2}}$ so as to gain broadening function $Fb$. Considering that all the measured spectra contain noise, Eq. (11) can be expressed as where $f = A \cdot S$, represents the spectrum affected by the response enhancement function, $Ne$ indicates the noise of measured spectrum. As the pixels of the detector are discrete, the convolution operation also can be expressed as $f(\lambda ) \otimes Fb(\lambda )\mbox{ = }\sum\limits_{}^{} {f({\lambda _i})} Fb(\lambda - {\lambda _i})$. Based on the measured spectrum and the proposed model, with the help of the available knowledge of the statistical characteristics of random noise, the deconvolution algorithm can be used to obtain the original spectrum, so as to improve the resolution. Here, a MAP framework is introduced for fluorescence spectrum to realize the MAP estimation of fluorescence spectrum which are unaffected by the broadening function. The mathematical expressions can be: Using Bayes formula, Eq. (13) can be transformed into Eq. (14). Since the fluorescence spectrum has been obtained, $p(M)$ is a constant. So, Eq. (14) can be expressed as Using logarithmic function to describe Eq. (15), it can be written as In Eq. (16), $p(M|f)$ denotes the probability that the measured spectrum behaves as $M(\lambda )$ when enhanced spectrum f and the broadening function $Fb$ are known. According to Eq. (12), $p(M|f)$ depends entirely on the distribution of the noise which can be expressed as $p(Ne)$. The noise level of each pixel in the spectrometer is independent of each other and obeys the Gaussian distribution with the mean value of 0, thus $p(M|f)$ can be expressed asThe last item $p(f)$ in Eq. (16) represents a priori of the enhanced spectrum, which is spectral space constraint item. In the spectral deconvolution problem, the introduction of this term is used to solve the ill-posed problem caused by errors like noise in the measurement spectrum, and make the spectra obtained tend to be smooth. The commonly form of Markov prior is expressed as
where ${C_2}$ is the constant coefficient. When $\rho (t) = {t^2}$, the prior model is Gauss-Markov prior. Nevertheless, Gauss-Markov prior will overly suppress high-frequency signals, cause the missing of partial details. Therefore, in this paper, Huber-Markov prior is utilized instead. Unlike Gauss-Markov prior, Huber-Markov prior has a threshold parameter in its potential function $\rho ( \cdot )$. The Huber function is expressed asThe Euler-Lagrange equation is a kind of linear local differential equation, which is a very important equation in functional, and also a very classical method of minimizing energy functional. The Euler-Lagrange equation is widely used in physics and computer fields. The objective function of f is minimized by employing the Euler-Lagrange equation with Neumann boundary conditions and the derivation process is shown below:
The corrected fluorescence spectrum can be obtained by using Eq. (26), Eq. (27) and Eq. (28). The algorithm involves the following steps:
- 1. Select the value of $\alpha$, $\mu$, $Th$ and $\varepsilon $, let ${f_0} = M$, $n = 0$ and calculate ${f_1}$;
- 2. While ${{||{{f_{x + 1}} - {f_x}} ||}/ {||{{f_x}} ||}} > \varepsilon $ and $x < Th$, fix ${f_{x + 1}} = {f_x}$;
- 3. All negative numbers in ${f_x}$ are converted to 0, output the corrected fluorescence spectrum ${f_x}$.
4. Experiments and discussion
An independently designed UV-VIS micro-spectrometer and six different types of coated CCDs were used in experiments, which as shown in Fig. 4 for the verification of the effectiveness of the MAP based resolution improvement method. Some parameters of the spectrometer and coated CCDs are shown in Table 1 and Table 2, respectively.
Table 1. Parameters of the independently designed UV-VIS micro-spectrometer
Table 2. Parameters of six different types of coated CCDs
The MAP based resolution improvement method mentioned above is essentially a spectral deconvolution algorithm. The requirement for accurate estimation of convolution kernel is necessary. As seen from Fig. 3 in Chapter 2, the simulated spectrum is in good agreement with the measured results, except for small errors near the peak. Therefore, the accuracy of the fluorescence broadening function (convolution kernel) can be satisfied. Since there is no spectrum that is only affected by the enhancement effect without broadening effect, the spectrum of Hg-Ar lamp obtained from the CCD 1# is used as the criterion to evaluate the correction effect of the algorithm. We also employ R-L algorithm, which is exceedingly classical in the field of spectral correction, to correct the fluorescence spectrum as a comparison of the MAP algorithm.
Firstly, the spectrometer was equipped with CCD 2# to collect the spectrum of Hg-Ar lamp. The broadening function at $H = 0.08mm$ obtained by using the fluorescence model mentioned above and MAP algorithm and R-L algorithm were respectively utilized to correct the measured spectrum. The original spectrum, measured spectrum, corrected spectra and broadening function are shown in Fig. 5. Comparing the measured spectrum with original spectrum, it can be seen that the spectral resolution and spectral response of the measured spectrum decrease to some extent due to the influence of the broadening function. Compared with the original spectrum, the corrected spectra are basically the same as the original spectrum, which shows that both of the two algorithms are of good correction effect. Specifically, two peaks in the original spectrum (296.728 nm, 302.150 nm) degenerated into weak response in the measured spectrum near 300 nm wavelength. Both algorithms can restore them to two peaks, and the restoring effect of MAP algorithm is better than that of R-L algorithm. It can be seen from the revised spectrum of MAP algorithm that the Huber-Markov priori method is adopted in the MAP algorithm, which not only has obvious noise suppression effect, but also retains the local details of the original spectrum, which proves the effectiveness and superiority of the algorithm.
Fig. 5. The original spectrum, measured spectrum, corrected spectra and broadening function when H = 0.08 mm.
Then, the spectrometer was equipped with CCD 2#, 3# and 4# respectively, and the spectra of Hg-Ar lamp were acquired respectively. The fluorescence model was used to simulate the broadening effect functions when $H = 0.12\, mm\, H\mbox{ = }0.18\, mm$ and $H = 0.23mm$. The span of the broadening effect function increased with the increase of the distance between coating and pixels. The MAP algorithm and R-L algorithm were employed to correct several measured spectra, and the corrected spectra are shown in Fig. 6, Fig. 7 and Fig. 8. Compared with the measured spectra, the resolution of all the corrected spectra is greatly improved. Nevertheless, with the increase of H, the correction effect of the two algorithms were also decreasing. Firstly, the resolution of the corrected spectra was decreasing. When $H = 0.12mm$ and $H\mbox{ = }0.18mm$, the two peaks near 300 nm can be clearly distinguished by the corrected spectra using the MAP algorithm, but can’t when $H\mbox{ = }0.23mm$, while the R-L algorithm can’t distinguish the two peaks when H is large than 0.12 mm. Secondly, the ringing effect appeared at the edges of the corrected linear spectra, which will lead to a large gap between the corrected spectral and the original spectrum. Because the negative value produced by ringing effect has no specific physical meaning, the general algorithm will choose to change the negative value to zero. Moreover, with the increase of H, the corrected spectra of MAP and R-L algorithms will gradually deviate from the original spectrum. The spectrum corrected by MAP algorithm is undercorrected, that is, the intensity of the corrected spectrum is less than that of the real spectrum, and the broadening of the corrected spectrum is larger than that of the original spectrum, as if affected by a smaller bandwidth function. To the contrary, the spectrum corrected by R-L algorithm is overcorrected. From this point alone, it is unlikely to judge which algorithm works better in this situation, which needs to be considered comprehensively according to the shape of the measured spectrum and its use. However, considering the accuracy and spectral resolution (the ability to distinguish adjacent peaks) of corrected spectra, the MAP algorithm probably has better performance in dealing with such problems. Hence, under the condition of this experiment (the selected phosphor material was Lumogen Yellow S 0790, the thickness of coating was $0.42\mu m$ and the pixel length is $8\mu m$), when $H \le 0.18mm$, the resolution of corrected spectra was high enough and the ringing effect was weak. It can be considered that the MAP algorithm can basically eliminate the influence of the broadening function, while filtering out the noise with a strong correction effect. This implies that when $H \le 0.18mm$, the MAP algorithm can make the spectra obtained by the CCD coated on glass achieve the same resolution as the CCD coated on pixels.
Fig. 6. The original spectrum, measured spectrum, corrected spectra and broadening function when H = 0.12 mm.
Fig. 7. The original spectrum, measured spectrum, corrected spectra and broadening function when H = 0.18 mm.
Fig. 8. The original spectrum, measured spectrum, corrected spectra and broadening function when H = 0.23 mm.
Finally, in order to explore the correction effect of the MAP algorithm on the spectrum affected by broadening function with very large span (the broadening function contains more than 2000 pixels, while the measured spectrum contains only 662 pixels), the spectrometer was equipped with CCD 6# to collect the spectrum of Hg-Ar light. It should be aware that the influence of this broadening function was too large, which resulted in serious degradation of the measured spectrum, so it is necessary to increase the integration time to enhance the spectral response. The fluorescence model was used to simulate the broadening function when $H = 0.88mm$ and the two algorithms were employed to correct the measured spectrum. The normalized original spectrum, measured spectrum and corrected spectra are shown in Fig. 9. It can be seen that the overall trend of the measured spectrum is too flat, with the considerably low resolution and seriously lost information. Although they were still far from the original spectrum and the spectrum corrected by R-L algorithm is also undercorrected due to the excessive broadening function, compared with the measured spectrum, the spectral resolution of the corrected spectra by both two algorithms were greatly improved and the corrected spectra can identify some spectral peaks more clearly. Moreover, the spectrum corrected by the MAP algorithm has higher resolution than that corrected by the R-L algorithm in this situation, which proves that the MAP algorithm still has an effect even for the severely degraded spectrum.
Fig. 9. The normalized original spectrum, measured spectrum, corrected spectra and broadening function when H = 0.88 mm.
5. Conclusions
Phosphor material coating is one of feasible and effective methods to enhance the detector's UV response. In this paper, firstly, a phosphor coating mathematical model was established and the feasibility of it have been proved. Then, the complex mathematical model was simplified into a response enhancement function and a broadening function. A MAP based algorithm was introduced to eliminate the influence of the broadening effect. Experimental results suggest that the algorithm can improve spectral resolution as well as suppress noise effectively. In particular, since the proposed phosphor coating mathematical model can accurately estimate the broadening function, the ill-posed problem caused by convolution kernel can be avoided. Moreover, the MAP algorithm can make the spectra obtained by CCD coated on glass achieve the same resolution as the spectra obtained by the CCD coated on pixels when the distance between the coating and the pixels is less than 0.18 mm. We hope the proposed phosphor coating mathematical model and the spectral resolution enhancement algorithm could be of promising prospect in the field of UV coating.
Funding
Key Research and Development Program of Anhui Province (No.1804d08020310); Chinese Academy of Sciences (CAS) (JZ2016QTXM1135).
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