Correcting for bandwidth in monochromator measurements

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Everett Mathews
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1 Correcting for bandwidth in monochromator measurements Emma R. Woolliams, Maurice G. Cox, Peter M. Harris Presentation to: ORM Club 8 th June 6

2 Yep, maths again ( ) ( ) I( λ) = I ( λ) Δ λβi ( λ) + Δλ β γ I ( λ) +K m m m Sl () sl () = N 1 δλ S r= 1 N r β = Δλ N N 1 r= 1 N rs 1 Δλ γ = + 1N r, N 1 3 rsr N r= 1 N. I( λ δλ) I( λ) + I( λ + δλ) I% ( λ) % % % ( δλ)

3 Bandwidth where does it come from? - Monochromatic Throughput Rotation of grating

4 Wavelength Bandwidth where does it come from? - Polychromatic Throughput

5 Bandwidth what effect can it have?.8.7 original measured.6 Spectral output wavelength / nm

6 Correcting for triangular bandpass functions S(λ, λ) Δλ Useful to see if you have a problem and a reasonable first correction λ Δλ λ λ + Δλ iv V( λ) = V% ( λ) ( Δ λ) V% ( λ) + ( Δ λ) V% ( λ) + L. 1 4 V( λ δλ) V( λ) + V( λ + δλ) V% ( λ) % % % ( ) 4 ( ) 6 ( ) 4 ( ) ( ) V% ( ) V λ δλ V λ δλ + V λ V λ + δλ + V λ + δλ iv λ % % % % % 4 ( δλ) ( δλ) 1 ( Δλ ) 1 * = - ( - + ) ( δλ)

7 What does the equation show us? iv V( λ) = V% ( λ) ( Δ λ) V% ( λ) + ( Δ λ) V% ( λ) + L. 1 4 No correction for linearly changing signals Fourth derivative term much smaller correction (usually) original measured corrected to nd derivative corrected to 4th derivative Spectral output wavelength / nm

8 And for real experimental data 7 Bandwidth, slits and equivalent triangle Radiance of the UHTBB Wavelength UHTBB Deuterium Lamp 1.4E-3 1.E-3 1.E-3 8.E-4 6.E-4 4.E-4.E-4.E+ Irradiance of the Deuterium Lamp Signal on PMT detector / V Signal on PMT detector / V Measured Equivalent triangle Wavelength / nm Δ λ = 4.6 Bandwidth, slits and equivalent triangle Intensity Equivalent triangle Wavelength / nm Δ λ = 1.46

9 The correction applied V( λ) = V% ( λ) ( Δ λ) V% ( λ) + ( Δ λ) V% iv ( λ) % Bandwidth correction with increasing terms Slits Correction to 1 Correction 1 to % Correction to next term % % -4% -6% -8% -1% Wavelength / nm

10 Experimental results 1% 8% 6% Uncertainty, k=1 Uncertainty, k=1 Uncertainty, k= Uncertainty, k= Uncorrected difference Uncorrected difference corrected difference Signal on Si detector / V 4% % % -% -4% -6% -8% -1% Wavelength / nm

11 What about asymmetrical/arbitrary bandpass functions? Spectral output original measured corrected wavelength / nm

12 What is the correction For triangular bandpass iv V( λ) = V% ( λ) ( Δ λ) V% ( λ) + ( Δ λ) V% ( λ) + L. 1 4 For arbitrary bandpass ( ) ( )( ) V( λ ) = V% ( λ ) β Δ λ V% ( λ ) + β γ Δ λ V% ( λ ) + L. First derivative: Affects linearly changing signals too

13 Determining β and γ ( ) ( )( ) V( λ ) = V% ( λ ) β Δ λ V% ( λ ) + β γ Δ λ V% ( λ ) + L. Function of bandpass only need to determine once for your monochromator β = Δλ N N 1 r= 1 N 1 γ = + 1N rs 3 r Δλ N N 1 r= 1 N rsr s r r N

14 Doing it for real ( ) ( )( ) V( λ ) = V% ( λ ) β Δ λ V% ( λ ) + β γ Δ λ V% ( λ ) + L. Determine β and γ for monochromator Apply correction at each wavelength of MEASUREMENT in turn It introduces correlation between neighbouring points and therefore affects the covariance matrix

15 Doing it for real with matrices a1 b1 c1 a b c a b c A = O O O a b c a b c am bm c M M = AM % T U = AUA %

16 Warning Function nm Spectrum Measurement Corrected spectrum Wavelength / nm

17 Warning Function Spectrum Measurement Corrected spectrum nm Wavelength / nm

Joint Industry Project RESULT 1: RAW DATA Monochromator slit function Measurements of test lamp + associated uncertainties Measurements of reference lamp + associated uncertainties Reference lamp

irradiance + associated covariance matrix RESULT 3: BANDWIDTH CORRECTED Bandwidth corrected measurements of test lamp + associated covariance matrix Bandwidth corrected measurements of reference lamp

18 Joint Industry Project RESULT 1: RAW DATA Monochromator slit function Measurements of test lamp + associated uncertainties Measurements of reference lamp + associated uncertainties Reference lamp irradiance + associated uncertainties RESULT : WITH COVARIANCE MATRICES Measurements of test lamp + associated covariance matrix Measurements of reference lamp + associated covariance Reference lamp irradiance + associated covariance matrix RESULT 3: BANDWIDTH CORRECTED Bandwidth corrected measurements of test lamp + associated covariance matrix Bandwidth corrected measurements of reference lamp + associated covariance matrix RESULT 4: TEST LAMP SPECTRAL IRRADIANCE Test lamp spectral irradiance + associated uncertainties/covariance matrix OUTPUT: Spectral irradiance, bandwidth corrected Multiplicative function, e.g. V(λ) RESULT 5: LUMINOUS INTENSITY, COLOUR ETC Test lamp luminous intensity / colour + associated uncertainty OUTPUT: Integrated quantities

Summary The maths isn t that bad No problem with bandwidth if + vs You can easily correct for triangular functions, and check scale of problem 1 1 4 iv V( λ) = V% ( λ) ( Δ λ) V% ( λ) + ( Δ λ) V% ( λ)

19 Summary The maths isn t that bad No problem with bandwidth if + vs You can easily correct for triangular functions, and check scale of problem iv V( λ) = V% ( λ) ( Δ λ) V% ( λ) + ( Δ λ) V% ( λ) + L. 1 4 It s not that much more difficult to correct for arbitrary bandpass functions V( λ ) = V% ( λ ) β Δ λ V% ( λ ) + β γ Δ λ V% ( λ ) + L. ( ) ( )( ) Nor to consider covariances properly M = AM % T U = AUA % But no method is a miracle cure for very fine changes

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