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Another variation (Hariharan, 1975) of the basic parallel plate interferometer uses two plates whose separation may be adjusted to expand the range of the possible shear. In Figure 4.35, one of the plates is mounted on a movable platform so that the air gap is variable and, hence, variable shear is obtained without plate rotation. The inside surfaces of these plates are uncoated so that about 4% re ection occurs there. The outer surfaces are treated with high quality antire ection coatings to minimize re ections. The plate mounting may be equipped with additional degree of freedom so that it can be rotated orthogonally to introduce tilt. The physical separation of two plates makes this system sensitive to disturbances and fragile in handling. The stability of the fringe system depends on the quality of mechanical mounts, their design, fabrication, and environmental conditions. The lateral shearing interferograms displayed in Figures 4.4 4.11 were generated employing a laser shearing interferometer laid out in Figure 4.32. An uncoated glass plate with a laser source results in a suf ciently intense fringe pattern, projected on a ground glass in a dimly illuminated room. An exposure time of only a fraction of a second is suf cient to photograph the fringes. With the currently available CCD detector arrays, the experimental work is even simpler.

Hence, one can imagine a beam of protons, each particle having suffered various random deflections due to scattering from gas molecules, intrabeam scattering, perturbations in magnetic fields, power supply noise, mechanical vibrations, and so on. So long as the various events are random and uncorrelated among the various particles, the distribution function describing the position of a particle which has been subjected to all of these random processes will be Gaussian, to good approximation. At the beginning of this chapter it was mentioned that discussion of electron beam emittance would be left to 8. There we will see that the dominant source of emittance fluctuation is the emission of photons due to synchrotron radiation, an inherently random process. Since the ensuing damping times are, in general, short, one would expect that even correlated processes such as injection errors might be swamped by the synchrotron radiation effects. Thus, electron beams in circular accelerators are certainly expected to have Gaussian distributions, and that is what is observed. The fact that proton beams in large synchrotrons also typically appear Gaussian suggests that the central limit theorem is at work here as well.

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12.3 12.4

23 O L Antipov and A S Kuzhelev, Self-pumped phase conjugation of laser beams in a nematic liquid-crystal layer with nonreciprocal feedback, Quantum Electron 25(1), 49 52 (1995) 24 OL Antipov, S I Belyaev, and A S Kuzhelev, Orientational self-pumped phase conjugation in a layer of the nematic liquid crystal with a nonrecoprocal feedback, Izv Vuzov: Radiophysics, XXXVIII(3 4), 304 311 (1995) (in Russian) 25 O L Antipov, A S Kuzhelev, and V V Turygin, Optimization of phase-conjugate mirrors made of nematic liquid crystals in a two-pass ampli er, Quantum Electron 24(5), 411 415 (1994) 26 Yu A Arutyunov, and A I Khizhniak, Generation conditions of phase conjugated waves in ring laser with local nonstationary nonlinearity, Sov Quantum Electron 16(4), 789 792 (1989) 27 A P Mazur, A D Novikov, S G Odulov, and M S Soskin, Parametric generation in media with local response in incompleted resonator, JETP Lett.

51(10), 503 506 (1990) 28 A V Mamaev and V V Shkunov, Instability of near-contrunning waves in medium with local nonlinearity, Sov Quantum Electron 19(3), 560 566 (1991) 29 O L Antipov, Parametric oscillation by joint stimulated scattering of near-counterruning laser beams, PhD Thesis, Nizhny Novgorod State University (1992) 30 O L Antipov, N A Dvorjaninov, and V Sheshkauskas, Parametric generation and selfpumped phase conjugation of intersecting laser beams in NLC layer doped by dye, JETP Lett 53, 610 612 (1991) 31 Iam-Choon Khoo, Hong Li, and Yu Liang, Self-starting optical phase conjugation in dyed nematic liquid crystals with stimulated thermal-scattering effect, Opt Lett 18, 1490 1492 (1993) 32 P Meindl, R Macdonald, H J Eichler, and O L Antipov, Low threshold self-pumped phase conjugation of an Ar -laser beam in dye-doped nematic liquid crystals, Mol Cryst Liq Cryst 282, 429 435 (1996) 33 M A.

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