Correction of Extracted Spectra

The functional form of the relation between the intra-pixel position and quantum efficiency depends on detector properties, the point spread function and the size of the imaged object. For point sources, the function can be determined empirically. It is easy to formulate the problem in terms of a correction factor f_l to the extracted spectrum, where the true spectrum $F_{l,\rm prf}$ relates to the raw extracted spectrum F_l as

$$F_{l,\rm prf}= f_l \cdot F_l$$ (23)

where l labels the different wavelength bins. For simplicity, the dispersion relation here is assumed to be linear,

$$\lambda_l= a_0 + a_1x_l$$ (24)

where x_l is the deflection of the lth wavelength bin in pixels relative to the position of the object on the direct image. A simple model for the correction factor was fitted to the calibration data of the first NICMOS campaign,  

$$f_l(\lambda)= 1 + A \cos [p (\lambda-a_0) + \psi]$$ (25)

The period p is given by

$$p= 2\pi\tan(\theta)/a_1$$ (26)

where a₁ is the dispersion in $\mu$/pixel, and the phase $\psi$ by

$$\psi= 2\pi (y_{\rm obj}+ y_{\rm offset})$$ (27)

where $y_{\rm obj}$ is the position of the object on the direct image, and $y_{\rm obj}+ y_{\rm offset}$ is the y coordinate of the spectrum at the reference wavelength a₀. For high signal-to-noise spectra, $\theta$ and $y_{\rm obj}+ y_{\rm offset}$ can easily be determined. The only free parameter to be directly determined from the extracted calibration spectra is the amplitude A. Figure 3.16 shows the fit to the calibration data taken during the first NICMOS campaign. The amplitudes determined from NICMOS campaigns (when NIC3 was in focus) is about 0.1, whereas outside the campaigns the amplitudes are about 0.05. Significant deviations from these average values have been detected, and it is currently not known whether these changes in amplitude correspond to changes in the PSF or are due to the calibration data taken at different positions on the detectors. Because of these variations in amplitude, we estimate that correction for PRF will on average reduce the amplitude of the wave like pattern in spectra by a factor of 2 to 5.

 

Figure 3.16: Fit of the PRF correction factor. The data are the same as in figure 3.14 and 3.15. With the amplitude A as the only free parameter, the behavior of both grisms can be fitted with the model. Note in particular that the amplitude seems to be almost independent of wavelength.