1.4.1 Sytematic Deformations: Defocus, Coma, Astigmatism

There are three basic systematic surface/wavefront deformations (occasionally associated with pointing errors) with which the observer may be confronted, i.e. defocus, coma, and astigmatism (a transient feature on the IRAM 30-m telescope).

1.

The most important systematic wavefront/beam error is due to a defocus of the telescope. This error is easily detected, measured, and corrected from the observation of a strong source at a number of focus settings. Figure 1.7 shows, as example, the beam pattern measured on Jupiter with the telescope being gradually defocused. Evidently, the peak power in the main beam decreases, the power in the side lobes increases, until finally the beam pattern has completely collapsed. To be on the safe side for observations, the defocus of the telescope should not exceed $\sim 1/10 \lambda$. A defocus does not introduce a pointing error.

  

Figure 1.7: Effect on the beam pattern (scans across Jupiter) introduced by defocusing the IRAM 15-m telescope (shifts of the subreflector in steps of a quarter of the wavelength (3 mm)).

2.

A telescope may have a comatic wavefront/beam error due to a misaligned subreflector, shifted perpendicular off the main reflector axis. Figure 8 shows, as example, a cross scan through a comatic beam of the IRAM 15-m telescope, especially produced by displacement of the subreflector. A comatic beam pattern introduces a pointing error. It may be useful for the observer to recognize this error, in particular if unexplained pointing errors occur in an observations. [The IRAM telescopes are regularly checked for misalignments, and correspondingly corrected.]

  

Figure 1.8: Illustration of a comatic beam (scanned in the direction of the coma) especially produced on the IRAM 15-m telescope. The shift of the subreflector is indicated by S. The beam pattern is perfect at S = 0. Note the shift of the beam (pointing error) when the subreflector is shifted.

3.

A telescope may have an astigmatic wavefront/beam error, usually introduced by complicated mechanical and/or thermal deformations (a transient feature on the IRAM 30-m telescope). While this beam deformation is easily recognized by the observer from the difference in beam widths measured from in-and-out-of-focus cross scans, the improvement of the telescope usually is difficult, and out of reach of the observer. A focused astigmatic beam does not introduce a pointing error. Figure 1.9 shows the focused beam pattern measured on a telescope which has a strong astigmatic main reflector (amplitude of the astigmatism $\sim$0.5mm).

  

Figure 1.9: Illustration of an astigmatic beam pattern; well focused.

The beam deformation of systematic wavefront deformations occurs close to the main beam, and the exact analysis should be based on diffraction calculations. A convenient description of systematic deformations uses Zernike polynomials of order (n,m) [Born and Wolf 1975]. Without going into details, the Zernike-type surface deformation $\delta_{\rm n,m} = \alpha_{\rm n,m}\,R_{\rm n}(\rho)\,\cos(m\theta)$[with ( $\rho,\theta$) normalized coordinates of the aperture, and R special polynomial functions] with amplitude $\alpha_{\rm n,m}$ has a quasi rms-value $\sigma$ = $\alpha_{\rm n,m}/\sqrt{\rm n+1}$ and introduces a loss in main beam intensity of

$${\epsilon}_{\rm sys}/{\epsilon}_{o}\ {\approx}\ {\rm exp[-(4{\pi}{\alpha}/{\lambda})^{2}/(n+1)]}$$ (1.16)

For primary coma n = 1, for primary astigmatism n = 2. Although the beam deformation may be very noticeable and severe, the associated loss in main beam intensity may still be low because of the reduction by the factor (n+1).