# A Correction Method for Measuring Spectral Irradiance of Light Sources Based on Differential Quadrature Method

Authors: Yanfei Wang; Caihong Dai; Zhifeng Wu; Binhua Chen
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

A novel correction method was demonstrated for measuring spectral irradiance of light sources with a narrow
bandwidth. Using the correction method based on differential quadrature method, an estimate of the true value
was achieved with measured values of seven adjacent points. The formula of this correction method was derived.
Numerical simulations and experimental validation of this correction method were also performed, respectively.
This correction method could be used in radiometry, photometry, colorimetry, and other spectrometry fields,
especially in the spectrum measurement of LED lamp.


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## Body

## 1. Introduction

Differential quadrature method plays a key role in engineering science and has received intense attention [1–3]. In the area of radiometry, measurement of spectrum with a narrow bandwidth such as light-emitting diode (LED) is a challenging topic. We have obtained ultranarrow spectra with a bandwidth approaching the natural linewidth [4, 5]. Spectral irradiance is measured by spectroradiometers with a finite bandwidth. The finite bandwidth of spectroradiometers could cause significant errors when the measured light source has a narrow bandwidth compared to that of spectroradiometers [6]. Therefore, it is necessary to apply correction methods to measured data and obtain an estimate of the true value.In order to satisfy the metrological and industrial demand on bandwidth correction, the International Commission on Illumination (CIE) has set up a technical committee (TC2-60) who is responsible for creating guidelines on bandwidth correction [7, 8]. In 1988, E. I. Stearns and R. E. Stearns demonstrated S–S method [9]. This method is applicable to monochromators with a triangular bandpass function and it requires that the measured wavelength step is equal to the bandwidth. To solve this problem, the differential quadrature approach has been demonstrated, which is applicable to spectrometers with an arbitrary bandpass function and at an arbitrary wavelength step [7]. Several other correction methods have been proposed [10–12].In the present paper, we present a novel correction method for measuring spectral irradiance of light sources with a narrow bandwidth. The correction method is based on differential quadrature method. Unlike differential quadrature method with three-point formula and five-point formula, we derive seven-point formula theoretically. Using a sine function and a Gaussian function, we validate this correction method through numerical simulations. We also validate seven-point formula experimentally. This correction method could be used in radiometry, photometry, colorimetry, and other spectrometry fields, especially in the spectrum measurement of LED lamp [9, 11, 12].


## 2. Theoretical Calculation

We present the derivation of seven-point formula in this part. The theoretical calculation is based on differential quadrature method [13]. The true spectrum and the measured spectrum are represented by S(λ) and M(λ), respectively. The bandpass function of a spectroradiometer is b(λ) and the relation between S(λ) and M(λ) could be written as(1)M(λ0)=∫Sλbλ-λ0dλ,where M(λ0) represents measured value at the λ=λ0 point. For monochromators, the bandpass function has a triangular shape in general case, which is shown in Figure 1. The bandwidth is 10 nm and the bandpass function is normalized. In this paper, we only consider monochromators with a triangular bandpass function.Figure 1
A typical bandpass function for monochromators.The true valueS(λ) could be written in the form of Taylor series expansion around the measured point at the wavelength λ=λ0(2)S(λ)=S(λ0)+∑n=1n=∞‍1n!λ-λ0nSnλ0,in which S(n)(λ) means the nth order derivative of S(λ). Using the notation(3)In=∫λnbλdλ,the measured value could be written as(4)Mλ0=I0Sλ0+I1S′λ0+12I2S′′λ0+⋯.From (4), we obtain the true value S(λ0) in the form of measured value and its derivative(5)Sλ0=A0M(λ0)+A1M′λ0+A2M′′λ0+A3M′′′λ0+A4Miv(λ0)+A5Mvλ0+A6Mviλ0+⋯,where the coefficients A are expressed as follows:(6)A0=1,A1=-I1,A2=I12-I22,A3=-I13+I2I1-I36,A4=I14-3I12I22+I3I13+I224-I424,A5=-I15+2I13I2-I3I122-3I1I224+I4I112+I3I26-I5120,A6=I16-5I14I22+2I13I33+3I12I222-I4I128-I1I2I32+I5I160-I238+I4I224+I3236-I6720.In order to obtain the true value, we should calculate each order derivative of the measured spectrumM(λ), which is shown as (5). The differentiation theory for an ordinary function is introduced in [13]. A function f(x) is tabulated at equal intervals h of the independent variable x. Further notational simplifications are obtained by writing(7)fx=fx0+ph=fxp=fp.The symbolp represents an integer and could be expressed as p=0,±1,±2,±3,…. f0(n) is used to denote the nth derivative of f0. One has(8)hf0′=μδ0-16μδ03+130μδ05-⋯h2f0′′=δ02-112δ04+190δ06-⋯h3f0′′′=μδ03-14μδ05+7120μδ07-⋯h4f0iv=δ04-16δ06+7240δ08-⋯h5f0v=μδ05-13μδ07+13144μδ09-⋯h6f0vi=δ06-14δ08+13240δ010-⋯,where μ is averaging operator and δ0(n) is nth order difference of f0.The displacement operatorE is defined as Efp=fp+1, using the displacement operator and the following formula:(9)δ2n=E-nE-12n2μδ2n+1=E-n-1E2-1E-12n.Equation (8) could be written as(10)hf0′=12f1-f-1-16μδ3f0+⋯=112-f2+8f1-8f-1+f-2+130μδ5f0-⋯=160f3-9f2+45f1-45f-1+9f-2-f-3-1140μδ7f0+⋯,h2f0′′=f1-2f0+f-1-112δ4f0+⋯=112-f2+16f1-30f0+16f-1-f-2+190δ6f0-⋯=11802f3-27f2+270f1-490f0hhhhhh+270f-1-27f-2+2f-3-1560δ8f0+⋯,h3f0′′′=12(f2-2f1+2f-1-f-2)-14μδ05+⋯=18-f3+8f2-13f1+13f-1-8f-2+f-3+7120μδ07-⋯,h4f0iv=f2-4f1+6f0-4f-1+f-2-16δ06+⋯=16-f3+12f2-39f1+56f0hhhhh-39f-1+12f-2-f-3+7240δ08-⋯,h5f0v=12f3-4f2+5f1-5f-1+4f-2-f-3-13μδ07+⋯,h6f0vi=f3-6f2+15f1-20f0+15f-1-6f-2+f-3-14δ08+⋯.Neglecting the high order differences, each order derivative of the measured spectrum could be written in the form of Mp. One has(11)M′=160h(M3-9M2+45M1-45M-1+9M-2-M-3),M′′=1180h22M3-27M2+270M1-490M0hhhhhhhh+270M-1-27M-2+2M-3,M′′′=18h3-M3+8M2-13M1hhhhhh+13M-1-8M-2+M-3,Miv=16h4-M3+12M2-39M1+56M0-39M-1hhhhhh+12M-2-M-3,Mv=12h5M3-4M2+5M1-5M-1+4M-2-M-3,Mvi=1h6M3-6M2+15M1-20M0hhhhh+15M-1-6M-2+M-3.We obtain seven-point formula by neglecting terms after the seventh term in (5). The three-point formula and five-point formula could also be obtained by neglecting terms after the third term and the fifth term in (5). For three-point formula, we obtain(12)S3=a-1M-1+a0M0+a1M1,a-1=-A12h+A2h2,a0=A0-2A2h2,a1=A12h+A2h2,where Sn represents n-point correction value of the measured point and an is the coefficient of Mn.For five-point formula, we obtain(13)S5=b-2M-2+b-1M-1+b0M0+b1M1+b2M2,b-2=A112h-A212h2-A32h3+A4h4,b-1=-2A13h+4A23h2+A3h3-4A4h4,b0=A0-5A22h2+6A4h4,b1=2A13h+4A23h2-A3h3-4A4h4,b2=-A112h-A212h2+A32h3+A4h4,where bn is the coefficient of Mn in five-point formula.Finally, we obtain seven-point formula by inserting (6) and (11) into (5). One has(14)S7=c-3M-3+c-2M-2+c-1M-1+c0M0+c1M1+c2M2+c3M3,c-3=-A160h+A290h2+A38h3-A46h4-A52h5+A6h6,c-2=3A120h-3A220h2-A3h3+2A4h4+2A5h5-6A6h6,c-1=-3A14h+3A22h2+13A38h3-13A42h4-5A52h5+15A6h6,c0=A0-49A218h2+28A43h4-20A6h6,c1=3A14h+3A22h2-13A38h3-13A42h4+5A52h5+15A6h6,c2=-3A120h-3A220h2+A3h3+2A4h4-2A5h5-6A6h6,c3=A160h+A290h2-A38h3-A46h4+A52h5+A6h6,where cn is the coefficient of Mn in seven-point formula.


## 3. Numerical Simulations

In order to validate seven-point formula, we use a sine function and a Gaussian function to simulate the correction method in this section. We simulate the true spectrum through a function and then obtain the measured spectrum using (1). At last, we apply seven-point formula to measured data. For simplicity, we neglect noise effect in simulation process and experimental validation.


### 3.1. Sine Function

Firstly, we use a sine function to validate seven-point formula. The expression of the chosen sine function is(15)y=12sinπ10λ-π2+12,where the wavelength λ is the independent variable and y is the amplitude of the spectrum. The period of this function is 20.Figure2 shows the simulated measurement and correction of the sine function expressed as (15). For the purpose of comparing the three formulae, bandwidth of the bandpass function is chosen as B=12 and the wavelength step h=0.1. The true spectrum (solid blue line) is obtained from (15) and the measured spectrum (solid green line) is obtained from (1). We apply three-point formula which is shown as (12) to the measured spectrum and obtain the corrected spectrum (solid black line). Similarly, we could obtain the corrected spectrum using five-point formula (solid yellow line) which is shown as (13) and seven-point formula (solid red line) which is shown as (14). From Figure 2, we note that the measured spectrum is severely distorted compared with the true spectrum. In this particular condition, the measured value at the centre wavelength is 62.7% of the true value. The percentages for three-point method, five-point method, and seven-point method reach 77.8%, 88.5%, and 94.6%, respectively. Therefore, seven-point method does have an obvious correction effect.Figure 2
The simulated measurement and correction of the sine function. The bandwidth of the bandpass function isB=12.The bandwidth determines how effective the correction method would be. We change the bandwidth toB=8 and keep other conditions unchanged. The simulated measurement of the sine function and the corrected spectrum with seven-point formula are shown in Figure 3. Similar to that described above, the true spectrum (solid blue line) is obtained from (15) and the measured spectrum (solid green line) is obtained from (1). We apply seven-point formula to the measured spectrum and obtain the corrected spectrum (dotted red line). The measured value at the centre wavelength is 78.6% of the true value. After the correction using seven-point method, the percentage reaches 99.7%. The corrected spectrum and the true one almost coincide. With the bandwidth decreasing, the corrected spectrum becomes closer to the true spectrum.Figure 3
The simulated measurement and correction of the sine function. The bandwidth of the bandpass function isB=8.


### 3.2.  Gaussian Function

We use a Gaussian function to validate seven-point formula again. The expression of the Gaussian function is(16)y=e-(λ-μ)2/2σ2,where μ represents the centre wavelength and μ=500 nm. The symbol σ is σ=FWHM/8ln⁡2. FWHM (Full Width Half Maximum) of this function and the bandwidth are chosen as FWHM=15 nm and B=10 nm.The simulated measurement of the Gaussian function and the corrected spectrum with seven-point formula are shown in Figure4. The true spectrum (solid blue line) is obtained from (16) and the measured spectrum (solid green line) is calculated from (1). Seven-point formula is applied to the measured spectrum and corrected spectrum (dotted red line) is obtained. The measured value at the centre wavelength is 83.6% of the true value. After the correction using seven-point method, the percentage reaches 99.3%. Therefore, we validate seven-point formula once again.Figure 4
The simulated measurement of the Gaussian function and the corrected spectrum with seven-point formula.


## 3.1. Sine Function

Firstly, we use a sine function to validate seven-point formula. The expression of the chosen sine function is(15)y=12sinπ10λ-π2+12,where the wavelength λ is the independent variable and y is the amplitude of the spectrum. The period of this function is 20.Figure2 shows the simulated measurement and correction of the sine function expressed as (15). For the purpose of comparing the three formulae, bandwidth of the bandpass function is chosen as B=12 and the wavelength step h=0.1. The true spectrum (solid blue line) is obtained from (15) and the measured spectrum (solid green line) is obtained from (1). We apply three-point formula which is shown as (12) to the measured spectrum and obtain the corrected spectrum (solid black line). Similarly, we could obtain the corrected spectrum using five-point formula (solid yellow line) which is shown as (13) and seven-point formula (solid red line) which is shown as (14). From Figure 2, we note that the measured spectrum is severely distorted compared with the true spectrum. In this particular condition, the measured value at the centre wavelength is 62.7% of the true value. The percentages for three-point method, five-point method, and seven-point method reach 77.8%, 88.5%, and 94.6%, respectively. Therefore, seven-point method does have an obvious correction effect.Figure 2
The simulated measurement and correction of the sine function. The bandwidth of the bandpass function isB=12.The bandwidth determines how effective the correction method would be. We change the bandwidth toB=8 and keep other conditions unchanged. The simulated measurement of the sine function and the corrected spectrum with seven-point formula are shown in Figure 3. Similar to that described above, the true spectrum (solid blue line) is obtained from (15) and the measured spectrum (solid green line) is obtained from (1). We apply seven-point formula to the measured spectrum and obtain the corrected spectrum (dotted red line). The measured value at the centre wavelength is 78.6% of the true value. After the correction using seven-point method, the percentage reaches 99.7%. The corrected spectrum and the true one almost coincide. With the bandwidth decreasing, the corrected spectrum becomes closer to the true spectrum.Figure 3
The simulated measurement and correction of the sine function. The bandwidth of the bandpass function isB=8.


## 3.2.  Gaussian Function

We use a Gaussian function to validate seven-point formula again. The expression of the Gaussian function is(16)y=e-(λ-μ)2/2σ2,where μ represents the centre wavelength and μ=500 nm. The symbol σ is σ=FWHM/8ln⁡2. FWHM (Full Width Half Maximum) of this function and the bandwidth are chosen as FWHM=15 nm and B=10 nm.The simulated measurement of the Gaussian function and the corrected spectrum with seven-point formula are shown in Figure4. The true spectrum (solid blue line) is obtained from (16) and the measured spectrum (solid green line) is calculated from (1). Seven-point formula is applied to the measured spectrum and corrected spectrum (dotted red line) is obtained. The measured value at the centre wavelength is 83.6% of the true value. After the correction using seven-point method, the percentage reaches 99.3%. Therefore, we validate seven-point formula once again.Figure 4
The simulated measurement of the Gaussian function and the corrected spectrum with seven-point formula.


## 4. Experimental Validation

In this section, experimental validation of seven-point method is investigated. We measure spectral irradiance of a LED lamp whose centre wavelength is 365 nm using a double grating spectroradiometer (OL 750D). The bandwidth of this lamp is about 10 nm and the measured wavelength step is 1 nm. We change the bandwidth of the spectroradiometer by changing width of entrance slit and exit slit. We select proper width of entrance slit and exit slit so that the bandwidth of the spectroradiometer isB=5 nm or B=1 nm. In each case, the spectroradiometer is calibrated and traced to the national primary standard of spectral irradiance, which is based on a high temperature blackbody (HTBB) BB3500M [14]. And then we measure the spectral irradiance when the bandwidth is 5 nm and 1 nm. The spectroradiometer bandwidth B=1 nm is narrow enough compared to the LED bandwidth. Therefore, we consider the measured spectral irradiance at B=1 nm as the true spectral irradiance. After that, we apply seven-point formula to the measured spectral irradiance at B=5 nm and obtain the corrected spectral irradiance. Finally, we compare the corrected spectral irradiance with the measured one at B=1 nm. If they are close to each other, it means that the seven-point method is effective.The experimental results are shown in Figure5. The peak value of the measured spectral irradiance at B=5 nm (solid green line) is 16.7 μW/cm2 and the peak value at B=1 nm (solid blue line) is 19.4 μW/cm2. The ratio between the two values is 86.1%. The peak value of the corrected spectral irradiance for the measured spectral irradiance at B=5 nm (solid red line) is 18.9 μW/cm2. The ratio between the peak value of the corrected spectral irradiance and that of the measured spectral irradiance at B=1 nm is 97.4%. Thus we validate seven-point formula experimentally.Figure 5
The measured spectral irradiance (B=5 nm and B=1 nm) and corrected spectral irradiance (B=5 nm) with seven-point formula.


## 5. Conclusion

In conclusion, we have presented a novel correction method for measuring spectra distribution of light sources with a narrow bandwidth. Seven-point formula is derived based on differential quadrature method. We validate this correction method through numerical simulations using a sine function and a Gaussian function. By selecting proper parameters, the corrected value at the centre wavelength could reach above 99% of the true value. We also validate this correction method experimentally. We measure the spectral irradiance of a LED lamp atB=5 nm and B=1 nm. After that, we apply seven-point formula to the measured spectral irradiance at B=5 nm and obtain the corrected spectrum. The ratio between the peak value of the corrected spectral irradiance and that of the measured spectral irradiance at B=1 nm is 97.4%. Therefore, we validate the correction method both theoretically and experimentally.



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