The "phase matching" FEL

Both a theoretical treatment and a design of a device, intended to systematically investigate the CSE as a function of the longitudinal phase space distribution, have been carried out.

Coherent spontaneous emission (CSE) occurs in an FEL when the electron bunch length is comparable to the wavelength of the radiation that is generated in the undulator. Under such conditions all electrons in the bunch have a similar phase and emit coherently. The emission is proportional to N², where N is the number of electrons in the single bunch (~10⁸), while the incoherent emission is proportional to N. Moreover if the electron current is generated in a RF accelerator, it carries a modulation at harmonics of the RF w_RF with a relative amplitude that depends on the electron bunch shape [1].

Utilising the physical model described in [1] it is possible to calculate the field amplitude for the CSE by integrating the usual product, that represent the energy exchange rate between the electron beam current and the radiation field, over the interaction volume. Following the analysis reported in [2] it is possible to treat each electron bunch at the undulator entrance as an ensemble of N_e particles distributed in the phase space (y,g), each having energy g_j and phase y_j=w_RFt_j with respect to a reference charge injected at t=0 with velocity b_z0. The total radiated power can be calculated as a sum of the single electron contributions:

where (1)

where is the usual definition of the phase shift parameter describing the FEL resonance condition in a waveguide, Z₀=377 W is the free space impedance, b_gl is the normalized group velocity of the waveguide mode at the frequency w_l with wave vector k_0,n , F is a form factor describing the overlapping between the e-beam transverse distribution and the waveguide mode, K is the undulator parameter , L is the undulator length, a and b are the transverse waveguide dimensions and q is the charge of each particle. The factor in (1) shows the dependence of the phase of the radiated field on the electron drift velocity, which is implicit in , and on the phase of the electron at the undulator entrance. The emitted power P_l,0,n can be maximised when the single electron contributions in the sum of the expansion coefficient A_l,0,n interfere constructively with each other. This happens when the electrons are distributed in the longitudinal phase space as close as possible to the "phase-matching" curve:

(2)

The effect of emittance can be taken into account, considering that an angular spread s ’ will imply a spread in the electron drift velocity, and thus a spread in phase around the phase matching curve. Calculations lead to the condition , in agreement with the usual requirement that the emittance should be smaller than the emission wavelength.

The phase-matching curve (2) has been plotted in Fig.2 for the mm-wave FEL parameters discussed later in the text, and the effect of phase matching has been calculated.

Fig. 2: Phase Matching Curve for g = 4.56, K = 1, N = 16 and l_u = 2.5 cm.

Results reported in [2] show that a given temporal profile of the electron bunch can correspond to different electron distributions in the longitudinal phase space. The emission from a "correlated" distribution of electrons in the (y,g) plane, satisfying the phase-matching condition, can be up to a factor ten higher than that from an "uncorrelated" distribution. More recently, further simulations have been performed keeping the longitudinal emittance constant when manipulating the longitudinal phase space. Calculations on a bunch composed by 1250 macro-particles, each one with a 0.1 pC charge, are shown in Fig. 3 for two different distributions in the longitudinal phase space. The "horizontal" distribution is totally uncorrelated, while the other one fulfils the phase matching condition. Also in this case a considerable increase of the output power is calculated for the correlated distribution as can be seen from the spectra in the figure.

Fig. 3 : Longitudinal phase space distributions and corresponding power spectra.

It has been thus designed a new device able to "manipulate" the electron beam longitudinal phase space. With such a device, together with a systematic investigation of the CSE, a high output power can be extracted from the e-beam, without the use of an optical resonator.

The experimental setup consists of a linear accelerator composed by an electron gun, a short transport line and two accelerating sections followed by a permanent magnet undulator located at a distance of 30 cm, as sketched in Fig.4.

Fig. 4 : Experiment design

A triplet of quadrupoles provides focusing at the undulator entrance. The undulator is 40 cm long, consists of 16 periods and operates with a magnetic field of 6000 Gauss corresponding to K_w=1. A rectangular waveguide, with cross section axb=0.668 x 4.318 mm², is placed inside the undulator.

The electron beam (1 A - 13 kV) is produced by a pulsed triode gun equipped with a 7.7 mm diameter osmium treated dispenser cathode. A suitable optical magnetic lens system matches the input admittance of the accelerator, which is composed by two modules:

a) a b -graded on axis S band (2998 MHz) linear accelerator (LINAC), with three full cells and two half end cells, operating in the p /2 mode. This section accelerates an electron macro bunch current of 0.40 A, to an energy of 1.8 MeV; the coupling coefficient to the waveguide b is 3.2.

b) a phase matching section (PMS) placed 4 cm downstream of the linac output. The drift space and the phase shift of the PMS are set to have the reference electron passing through the centre of the PMS with a phase close to zero. The coupling coefficient to the waveguide b is 1.1. The PMS is composed by three on axis coupled cavities (one and two halves) operating in the p /2 mode tuned at the same frequency of the linac. Two motorised plungers inserted in the end cavities adjust the frequency at the exact design value. In addition a cooling system keeps both LINAC and PMS at the fixed temperature of 30°C ± 0.05 °C.

The distribution of the bunched electrons in the phase space at the PMS output can be changed by varying the phase and the amplitude of the RF field driving the PMS with respect to the linac. The total RF power required is about 2 MW. In order to control phase and amplitude independently, a suitable RF system has been designed. It makes use of a high stability low power cavity controlled oscillator, an amplifying chain and a 10 MW klystron equipped with a 3- dB power splitter: one arm feeds and controls the RF accelerating field amplitude to the linac, the other is equipped with a high power variable attenuator and a variable high power phase shifter (0°- 360°) enabling power and phase control of the PMS module.

Fig. 5: Longitudinal phase space distribution of the e-beam at the linac output (a), at the PMS

output (b), at the undulator input (c) and at the center of the undulator (d)

Numerical simulations of the beam dynamics were performed using the PARMELA code [3]. The calculation takes into account the space charge effects and uses 3000 particles leaving the gun with a charge per particle of 0.11 pC corresponding to a beam current of 1 A with an energy of 13 KeV and a unnormalized emittance of 17 p mm mrad. It is interesting to notice that a phase space distribution satisfying the matching condition (2) at the undulator entrance results in the shortest bunch length at the undulator center [4]. 

A net increase in the CSE is expected for a phase space distribution at the PMS output with optimum setting with respect to the distribution at the linac output, as shown in Fig. 6. In the case shown, the electric field in the PMS is set to 50% of the electric field in the linac (E_linac= 25 MV/m) and the relative phase is adjusted in order to have the proper correlation at the undulator input.

It has been be observed that a dephasing of 7 degrees between linac and PMS produces a band power variation of about 50% that can be easily traced experimentally. In this way it is possible not only to control the total emitted power, but also the power emitted on a single harmonic, adjusting the phase shift between linac and PMS to optimise the emission at a selected wavelength.

In 1999 we will be able to experimentally test the whole apparatus: the linac is being built by Hitesys, while the PMS is being built by "Busato e Satta", both according to the ENEA design and we are waiting for the delivery of the necessary RF components and of the e-gun. An experimental area was set up, in the former "20 MeV microtron" area and work are in progress to deliver the RF power inside the selected area. Linac will be delivered in february 1999, RF components in the beginning of 1999, while the RF gun (Thomson) and PMS will be delivered before june 1999.

We will thus be able to test the complete apparatus in autumn 1999.