At the FEL technical review, the committee suggested that we should use the best BPMs available for the two BPMs around the wiggler. As a result of this, three sets of SEE BPM electronics were ordered. At the April 30 Commissioning meeting the question arose as to whether one could really specify the electron beam position and angle in the wiggler sufficiently accurately with only two BPMs in the two positions now available. The rms optical mode size and divergence at a 3 µm laser wavelength are 310 µm and 773 µrad respectively. If the electron beam average angle and position are off from their optimal values by one tenth of these numbers, the laser should lase with almost no reduction in power or gain. If they are off by one third this value, the laser will lase but may need some reoptimization to get maximum power. In order to determine the effectiveness of these two BPMs in re-establishing the beam position and angle in the wiggler, I needed the M_12 and M_34 matrix elements from the first BPM to the wiggler center and from the first to the second BPM. Dave Douglas provided me with these numbers:

	                                       BPM1 to wiggler	        BPM1 to BPM2
        M12	                                 1.0 m	                     -2.0 m
        M34	                                 1.0 m	                       1.2 m

The angular tolerance at BPM1 is determined by the relative resolution of the BPMs divided by effective distance between them. Since the resolution of each BPM is 50 microns the relative resolution between them is the square root of two times the resolution or 71 microns. The minimum angle which one can resolve and therefore correct is therefore 36 microradians in the horizontal plane and 59 microradians in the vertical plane. The beam movement in the wiggler center under these conditions will be 36 microns horizontal and 59 microns in the vertical. The position at other points in the wiggler might be slightly larger than this judging from the betatron tunes between the BPMs of 227 degrees horizontally and 140 degrees vertically, but the mean position will only be off by these values. From the discussion above about the reproducibility requirements, one can see that the net rms reproducibility in position should be better than one third of the rms mode size, so the laser should lase if the angle in the wiggler is also reproduced.

One can set an upper limit on the angle at the wiggler center by using the ratio of the beta functions and assuming the worst possible phase advance. The beta functions at the wiggler center are 50 cm in both planes while the beta functions at the first BPM are 2.8 m horizontal and 1.9 m vertical. Using this argument, the angle in the wiggler might be as large as 85 microradians in the horizontal plane. The vertical error should be comparable to or smaller than the original error of 59 microradians since the last quad before the wiggler focuses the beam and therefore reduces any angular error at the first BPM. Both of these angular errors are one tenth or less of the rms angular divergence and so will contribute minimally to laser performance degradation.

When the energy is lowered or if the wiggler strength is increased, the horizontal focusing and tune will increase. This will lead to a larger M_12 between the BPMs and so the angle at the first BPM will even be closer to the original value. The optical mode divergence in that case will also increase so the 3 micron laser operation is actually a worst case. In conclusion, the two SEE BPMs should be sufficient to reproduce the beam position and angle in the wiggler to within the requirements needed for lasing. A further optimization of the position and angle might be necessary to fully optimize the laser power output but the power gain from this final optimization will probably be quite small. A program to optimize laser performance versus steering and focusing via computer has been written and tested at Duke if it is necessary to automate the process.