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The molecular iodine is rapidly dissociated to atomic iodine by the presence of the singlet oxygen in a complex, multistep mechanism so that the flow is fully dissociated as it reaches the exit plane of the nozzle and before entering the gain region. The iodine dissociation process is poorly understood and provides the greatest difficulty in modeling the gas-phase kinetics of the COIL device. The energetics of the dissociation process are shown in Fig. 8. The second electronically excited HO–2 → H20 + HO–2 + K– K+ + OH– + H202 ← state of oxygen, O2(b 1⌺), is produced from the energy-pooling reaction: O2 (a 1 ) + O2 (a 1 ) → O2 (b 1 ) + O2 (X 3 ) (22) and O2(b 1⌺) is sufficiently energetic to dissociate iodine.
Note that these transverse modes, which satisfy the boundary conditions of the waveguide FEL, are spatially coherent. The degree of mixing of the transverse modes is a direct measurement of the transverse spatial coherence of the radiation generated in the FEL. The transverse wave equation, for cylindrical geometry, is − 1 2 1 ∂ Ar − 2 (Ar + 2∂θ Aθ ) = µ0 jr c2 t r (33) 546 COHERENCE It is easily seen that in the case of a spatially extended charge distribution propagating in a helical wiggler, the transverse electric (TE) modes couple to the wiggler-induced motion, while the transverse magnetic (TM) modes are driven by the uniform motion of the space-charge distribution in the cylindrical waveguide.
The transition from coherent to incoherent radiation is modeled by considering the ratio of the electron bunch length to the radiation wavelength. The spatial coherence corresponds to the excitation of transverse modes in the system, and phase noise can be analyzed by considering the dispersion characteristics of the structure. For the sake of illustration, a fairly specific example is considered: coherent synchrotron radiation in a cylindrical waveguide 545 FEL structure. The ideas presented here are easily generalized to other free-electron devices.