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

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.


Introduction
Differential quadrature method plays a key role in engineering science and has received intense attention [1][2][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 - 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][11][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].

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 () and (), respectively.The bandpass function of a spectroradiometer is () and the relation between () and () could be written as where ( 0 ) represents measured value at the  =  0 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.
The true value () could be written in the form of Taylor series expansion around the measured point at the wavelength  =  0 in which  () () means the th order derivative of ().Using the notation the measured value could be written as From ( 4), we obtain the true value ( 0 ) in the form of measured value and its derivative where the coefficients  are expressed as follows: In order to obtain the true value, we should calculate each order derivative of the measured spectrum (), which is shown as (5).The differentiation theory for an ordinary function is introduced in [13].A function () is tabulated at equal intervals ℎ of the independent variable .Further notational simplifications are obtained by writing The symbol  represents an integer and could be expressed as  = 0, ±1, ±2, ±3, . ...  ()  0 is used to denote the th derivative of  0 .One has where  is averaging operator and  () 0 is th order difference of  0 .
The displacement operator  is defined as   =  +1 , using the displacement operator and the following formula: Equation ( 8) could be written as Neglecting the high order differences, each order derivative of the measured spectrum could be written in the form of   .One has We obtain seven-point formula by neglecting terms after the seventh term in (5).The three-point formula and fivepoint formula could also be obtained by neglecting terms after the third term and the fifth term in (5).For three-point formula, we obtain where   represents -point correction value of the measured point and   is the coefficient of   .
For five-point formula, we obtain where   is the coefficient of   in five-point formula.
Finally, we obtain seven-point formula by inserting ( 6) and ( 11) into (5).One has where   is the coefficient of   in seven-point formula.

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.

Sine Function.
Firstly, we use a sine function to validate seven-point formula.The expression of the chosen sine function is where the wavelength  is the independent variable and  is the amplitude of the spectrum.The period of this function is 20.
Figure 2 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  = 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.
The bandwidth determines how effective the correction method would be.We change the bandwidth to  = 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.

Gaussian Function.
We use a Gaussian function to validate seven-point formula again.The expression of the Gaussian function is where  represents the centre wavelength and  = 500 nm.The symbol  is  = FWHM/ √ 8 ln 2. FWHM (Full Width Half Maximum) of this function and the bandwidth are chosen as FWHM = 15 nm and  = 10 nm.The simulated measurement of the Gaussian function and the corrected spectrum with seven-point formula are shown in Figure 4.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 sevenpoint method, the percentage reaches 99.3%.Therefore, we validate seven-point formula once again.

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 is  = 5 nm or  = 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  = 1 nm is narrow enough compared to the LED bandwidth.Therefore, we consider the measured spectral irradiance at  = 1 nm as the true spectral irradiance.After that, we apply seven-point formula to the measured spectral irradiance at  = 5nm and obtain the corrected spectral irradiance.Finally, we compare the corrected spectral irradiance with the measured one at  = 1 nm.If they are close to each other, it means that the seven-point method is effective.The experimental results are shown in Figure 5.The peak value of the measured spectral irradiance at  = 5 nm (solid green line) is 16.7 W/cm 2 and the peak value at  = 1 nm (solid blue line) is 19.4 W/cm 2 .The ratio between the two values is 86.1%.The peak value of the corrected spectral irradiance for the measured spectral irradiance at  = 5 nm (solid red line) is 18.9 W/cm 2 .The ratio between the peak value of the corrected spectral irradiance and that of the measured spectral irradiance at  = 1 nm is 97.4%.Thus we validate seven-point formula experimentally.

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 at  = 5nm and  = 1nm.After that, we apply seven-point formula to the measured spectral irradiance at  = 5nm and obtain the corrected spectrum.The ratio between the peak value of the corrected spectral irradiance and that of the measured spectral irradiance at  = 1 nm is 97.4%.Therefore, we validate the correction method both theoretically and experimentally.

Figure 2 :Figure 3 :
Figure 2: The simulated measurement and correction of the sine function.The bandwidth of the bandpass function is  = 12.

Figure 4 :
Figure 4: The simulated measurement of the Gaussian function and the corrected spectrum with seven-point formula.