Conversion from pixel to wavelength

  • Same equation used:
    CodeCogsEqn.gif

  • cut off value = -0.05
  • $\frac{\Delta y}{2j}= y(i+j)-y(i-j), j=1$
  • $\Delta x= 2$ for it to be greater than 2 for it to be considered a peak.
  • Each of the coefficient found using the function to convert the pixel to wavelength:
Parameters Output coefficient
C_0 564.36
C_1 1.02e-01
C_2 -5.06e-06
D_1 54.13
D_2 -5.05e-06
E_1 -2.75e-03
  • p0=354 and m0=4.5
  • Plot showing the relation between micrometer and the wavelength for different pixel values:
    microm_diffpixpos.png

  • Plot showing the relation between pixel and the wavelength for different micrometer setting:
    pixelplot_diffmsettings.png
  • Plot showing the relation between pixel and the wavelength for different micrometer setting:
    pixelplot_0.5invtervals.png
  • Was expecting the second graph (pixel and wavelength relation) to be a quadratic graph due to $(p-p0)^2$, however the coefficient infront is small therefore the quadratic term doesn't have any impact on the curve.

  • Calibrated plot of Helium at the micrometer setting of 6 with where the peaks should be:
    ExpObtainedwave.png
Issues with Cadmium data
  • For some unknown reason, the use of cadmium data would cause an error in the calibration.
  • Looked for causes of this:
  • Could be due to peak finder algorithm :- ruled out by checking if the outputted peak location matched the ones on the graph. Also peak finding algorithm works for all other lamps so couldn't be due to this.
  • Looked at peak location(in wavelength) and the line indicating where the peak should be (red line) for different micrometer setting using the coefficients found without using cadmium data. In each case the red line was shifted left to that where the peak was (LOOK AT EXAMPLE PLOT THAT IS ATTACHED)
  • New Theory : - Could be due to misidentifying the cadmium lines or could be due to micrometer not being homed properly.
  • To test this hypothesis rearranged the equation for converting to pixel to wavelength so that it take in pixel and wavelength and give micrometer setting.
 \begin{equation} m(\lambda,pixel)=\frac{-b\pm\sqrt{(b)^2-4ac}}{2a} \end{equation}

  • where the coefficients are $a= D_2$
  • $b=-2D_2m_0-D_1-E_1(p-p_0)$
  • $c=-(\lambda-C_0-C_1(p-p0)-C_2(p-p0)^2-E_1m_0(p_p_0)-D_1m_0-D_2m_0^2$
  • Done 2 different calibration: one with just cadmium lamp, and another with the rest of the lamp.
  • For the 2 calibration recieved different coefficient.
Parameters Cadmium_calib(coefficient) Other lamps(coefficient)
C_0 348.96 593.63
C_1 1.14e-01 1.01e-01
C_2 -6.37e-06 -4.84e-06
D_1 61.13 53.11
D_2 -9.35e-06 -8.83e-01
E_1 -3.13e-03 -2.70e-03
  • Calculated the wavelength at 6 micrometer using these two coefficients. Then using the equation above, converting pixel and wavelength to micrometer, re-obtained the micrometer position. But this time only used the coefficients from the "Other lamps". - Converted the wavelengths obtained from cadmium lamps to micrometer position using the other lamp's coefficients.
  • If due to misidentifying spectral lines then expected the micrometer position to not be consistent. If due to the shift in micrometer position then the micrometer position given back should be similar in value.
  • Plots for micrometer setting for the two hypothesis:
    diffplot.png
  • On the left is the plot showing the variation in the obtained micrometer values when using the hypothesis that the spectral lines have been misidentified. This gave the range in the variation of the micrometer position of 1.6911$\mu m$. On the right it is the same plot but under the assumption that the spectral lines identified are correct and there is shift to the right (increase in the micrometer setting). The range in the variation for this is 0.007. Much smaller than the misidentified one. In conclusion unless there are some other reason for the issue with cadmium data the error from the calibration is most likely from some form of shift in the micrometer.
  • Adding 0.16 to the micrometer setting to all the cadmium micrometer setting, which is the calculated mean in the right hand graph, seems to make the callibration more accurate.

Vega line in wavlength

  • This shows the vega spectrum at micrometer setting of 6 with a red line which indicates the wavelength of the hydrogen line for vega at that location:
    Vega.png

Background fitting

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

  • Background fit is a polynomial:
 \begin{equation}  f(x) = \sum_{i = 0}^{n} c_i x^i  \end{equation}

  • Total fit is the sum of the Lorentzian fit, and the (second order) background fit:
 \begin{equation}  f_{\rm{L}, \rm{BG}}(x; a, \mu, \gamma, A, B, C) = f_{\rm{L}}(x; a, \mu, \gamma) + f_{\rm{BG}}(x; A, B, C)  \end{equation}

  • Lorentzian + background fit to the peak in Vega, centered at 6.00mm,:
    Vega_Lorenztian_bg_6.png
  • Chi squared of 1579606

-- WillBurrows - 31 Jan 2017
Latex rendering error!! dvi file was not created.

Topic attachments
I Attachment Action Size Date Who Comment
GIFgif CodeCogsEqn.gif manage 2.3 K 01 Feb 2017 - 01:03 JamesAngthopo  
PNGpng ExpObtainedwave.png manage 135.5 K 31 Jan 2017 - 23:36 JamesAngthopo Calibrated plot of Helium at the micrometer setting of 6 with where the peaks should be
PNGpng Vega.png manage 98.2 K 01 Feb 2017 - 01:32 JamesAngthopo This shows the vega spectrum at micrometer setting of 6 with a red line which indicates the wavelength of the hydrogen line for vega at that location
PNGpng Vega_Lorenztian_bg_6.png manage 59.9 K 01 Feb 2017 - 10:32 WillBurrows Lorentzian + background fit to the peak in Vega, centered at 6.00mm,
PNGpng diffplot.png manage 170.6 K 01 Feb 2017 - 01:13 JamesAngthopo Plots for micrometer setting for the two hypothesis
PNGpng exampleplot.png manage 105.7 K 01 Feb 2017 - 00:03 JamesAngthopo example plot to show the effect of cadmium lamp on calibration
PNGpng microm_diffpixpos.png manage 166.5 K 31 Jan 2017 - 23:27 JamesAngthopo Plot showing the relation between micrometer and the wavelength for different pixel values
PNGpng pixelplot_0.5invtervals.png manage 172.6 K 01 Feb 2017 - 10:30 JamesAngthopo  
PNGpng pixelplot_diffmsettings.png manage 101.2 K 31 Jan 2017 - 23:27 JamesAngthopo Plot showing the relation between pixel and the wavelength for different micrometer setting

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Topic revision: r9 - 01 Feb 2017 - WillBurrows

 
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