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The result for the signal modulation is identical to the result we obtained in Eq (1451) This should not be surprising since the modulation of the signal is independent of the starting value of the reference phase ( p/2 vs p/4) Similarly, the difference between analyzing the three intensity frames with Eq (1450) instead of Eq (1454) is a constant wavefront phase offset of 3p=4 These later expressions are useful as they are analytically less complex, and can be used for any set of three measurements with p=2 phase shifts regardless of the starting phase An unimportant phase offset or piston term will appear in the result of the calculation The three step algorithm requires the minimum amount of data and is the simplest to use However, as we will see later, this algorithm is also the most sensitive to errors in the phase shift between frames.

143-150 161-174 131-142 1-22

1472 Least-Squares Algorithms As we have noted several times, the measured intensity at a given location in the interferogram varies as a sinusoidal function of the reference phase with a known period and three unknowns: the average signal, the data modulation and the unknown reference phase It should not be surprising that the wavefront phase can be determined through a least-squares t of the measured intensities to a sinusoidal function The general least-squares solution for a series of N interferograms recorded at phase shifts di was rst discussed by Bruning, Herriott, Gallagher, Rosenfeld, White and Brangaccio (1974) and more rigorously by Greivenkamp (1984) Eq (14.

Figure 27-10: The Insert Object dialog box. 3. If you want to choose an existing file, choose Create from File. Click Browse to find the file. Click Open. You return to the Insert Object file, where you can choose Link to link the data (described in the next section of this chapter). Choose OK. The file appears at the top-left corner of your screen with handles that you can use to move and/or resize the object. Here s the second way: 1. Open the source application, select the data, and copy it to the Clipboard. (Click Copy on the Standard toolbar or choose Edit Copy.) Leave the source application open. 2. If your drawing is open, switch to it by choosing its button on the task bar. Otherwise, open it. 3. In your drawing, choose Edit Paste Special. 4. In the Paste Special dialog box, choose the first option, which lets you embed the object as an object of the source application. Click OK. 5. You can now close the other application. The third way to insert data is to use drag-and-drop: 1. If your drawing is not open, open it. Open the drawing where you want to embed the data. 2. Open the source application and select the data. 3. Press Ctrl and click the selected data again, holding down the mouse button.

37) is rst rewritten as Ii x; y I 0 x; y I 00 x; y; cos f x; y cos di I 00 x; y; sin f x; y sin di or Ii x; y a0 x; y a1 x; y cos di a2 x; y sin di where a0 x; y I 0 x; y a1 x; y I 00 x; y; cos f x; y 14:57 14:58 14:56.

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When the third derivative term disappears, this expression becomes identical to Eq. (9.3) (with j 0 ), which was the basis of the geometrical theory of the Ronchi test. Therefore, in such cases the physical and geometrical patterns coincide. This result was derived by Toraldo di Francia (1947, 1954) who pointed out that the third derivative cancels out when the focus shifts and third-order astigmatism and coma cancel out when no power of x higher than the second appears [see Eq. (9.4)]. With regard to spherical aberration, we nd substituting the expression for spherical aberration with defocusing from Eq. (9.4) in Eq. (9.65), 2 2 lr md 4A x2 y2 x1 2 A 2 D x1 1 1 r 2D 9:66

The Polar Coordinates filter is a great way to edit gradations. After drawing a linear gradation with the gradient tool (as discussed in 6), try applying Filter Distort Polar Coordinates with the Polar to Rectangular option selected. (Rectangular to Polar just turns it into a radial gradation, sometimes with undesirable results.) You get a redrawn gradation with highlights at the bottom of the selection. Press Ctrl+F to reapply the filter to achieve another effect. You can keep repeating this technique until jagged edges start to appear. Then press Ctrl+Z to go back to the last smooth effect.