Conversion to wavelength
  • Function to convert pixel values to wavelength uses the equation:
 \begin{equation} \lambda(m,p)= C_0+C_1(p-p_0)-D_1(m-m_0)-E_1(p-p_0)(m-m_0)+C_2(p-p_0)^2+D_2(m-m_0)^2\end{equation}
  • Calibration completed for Helium and Cadmium so far (Below are some examples of the values used for the function of lambda):
Filename Micrometer setting Peaklocation Wavelength
Calib_He_0100 1.00 145.582, 690.276 728.1, 778.3
Calib_He_0150 1.50 174.077, 403.700 706.5, 728.1
Calib_He_0200 2.00 33.349, 433.886, 668.112 668.7, 706.5, 728.1
Calib_He_0250 2.50 289.445, 699.538 668.7, 706.5
Calib_Cd_0550 5.50 0.0299, 110.860, 380.483 467.945, 480.125, 508.724
Calib_Cd_0600 6.00 262.870, 377.610, 651.768 467.945,480.125,508.724
Calib_Cd_0650 6.50 530.697, 646.576 467.945,480.125
  • The cut off value (wrt the second derivative) for a fluctuation to be considered a peak was: -0.05.
  • $x(i+\Delta x) - x(i-\Delta x)=1$
  • If the x range was greater than 2 than it was considered a peak.
Parameter Initial guess values Output values
$C_0$ $446.6$ $5.87656692e+02$
$C_1$ $0.1$ $1.05911606e-01$
$C_2$ $3.8 \times 10^{-4}$ $-1.83590351e-05$
$D_1$ $0.2$ $5.42576832e+01$
$D_2$ $2.0 \times 10^{-3}$ $-2.97124355e-03$
$E_1$ $0.8$ $-7.79129936e-01$
  • Relation between wavelength and micrometer position for pixel postion of 3.50mm:
    Micro_waverelation.png

  • Plot of pixel to wavelength conversion. Lower micrometer settings the output wavelength values would be: $\Delta\lambda\leq 3nm$ than the actual wavelengths.
  • Higher micrometer settings the output wavelengths would be greater than the actual wavelength by: $\Delta\lambda\geq 5nm$

  • Test Helium plot for micrometer setting of 6.50mm:
    Helium_0650.png
Fitting to observation data

  • Gaussian equation used:
 \begin{equation}  f(x;a,\mu, \sigma, b) = \frac{a}{\sqrt{2\pi}} \: \rm{exp} \bigg[ \frac{-(x - \mu)^2}{2\sigma ^2} \bigg] + \textit{b}  \end{equation}
  • where a is amplitude, μ is the mean of the distribution, σ is the standard deviation of the curve and b is the background

  • Cauchy - Lorentz equation used:
 \begin{equation}  f(x;a, \mu, \gamma, b) = \frac{a}{\pi \gamma} \bigg[ \frac{\gamma ^2}{(x - \mu)^2 + \gamma ^2} \bigg] + b  \end{equation}
  • where a is amplitude, μ is peak position, γ is the HWHM and b is the background

  • For a pixel range of plus minus 100 pixels of the estimated μ and with the same guess parameters passed to each function, the following fits were obtained
  • Gaussian fit to the peak in Vega, centered at 6.00mm:
    Vega_0600_180_gauss.png
  • Chi squared of ~338345

  • Lorentzian fit to the peak in Vega, centered at 6.00mm:
    Vega_0600_180_lorentz.png
  • Chi squared of ~278812
-- WillBurrows - 24 Jan 2017
Topic attachments
I Attachment Action Size Date Who Comment
PNGpng Helium_0650.png manage 78.3 K 25 Jan 2017 - 09:31 JamesAngthopo Test Helium plot for micrometer setting of 6.50
PNGpng Helium_0700.png manage 67.0 K 25 Jan 2017 - 09:43 JamesAngthopo Test Helium plot for micrometer setting of 7.00
PNGpng Micro_waverelation.png manage 36.1 K 25 Jan 2017 - 09:30 JamesAngthopo Relation between wavelength and micrometer position for pixel postion of 350
PNGpng Vega_0600_180_gauss.png manage 57.4 K 24 Jan 2017 - 20:21 WilliamBurrows Gaussian fit to the peak in Vega, centered at 6.00mm
PNGpng Vega_0600_180_lorentz.png manage 48.3 K 24 Jan 2017 - 20:20 WilliamBurrows Lorentzian fit to the peak in Vega, centered at 6.00mm

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Topic revision: r4 - 25 Jan 2017 - JamesAngthopo

 
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