### The Physics of Free Electron Lasers

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Both spontaneous and stimulated emission occur, allowing for optical amplification in a certain wavelength range.

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Compared with other synchrotron radiation sources pure undulators and wigglers , FELs can generate an output with a much higher spectral brightness and coherence. This is very useful for a number of applications, including fields such as atomic and molecular physics, ultrafast X-ray science, advanced material studies, ultrafast chemical dynamics, biology and medicine.

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The big drawback of FELs is that their setups are very large and expensive, so that they can be used only at relatively few large facilities in the world. The most ambitious free electron laser project is currently pursued in Hamburg XFEL, within the TESLA project with the goal of obtaining hard X-ray output with unprecedented performance features: wavelengths down to 0.

So far, wavelengths down to 6. Suggest additional literature! See also: ultraviolet light , ultraviolet lasers , X-ray lasers and other articles in the category lasers. If you like this article, share it with your friends and colleagues, e. Virtual Library.

## Free Electron Lasers

Sponsoring this encyclopedia:. Sorry, we don't have an article for that keyword! The periodically varying magnetic field forces the electron beam blue on a slightly oscillatory path, which leads to emission of radiation. The greatest attractions of free electron lasers are: their ability to be operated in very wide wavelength regions the large wavelength tuning range possible with a single device the spectacular performance in extreme wavelength regions, not reachable with any other light source Compared with other synchrotron radiation sources pure undulators and wigglers , FELs can generate an output with a much higher spectral brightness and coherence.

Bibliography [1] D.

Deacon et al. Kim and A. Shorter pulses naturally have broader bandwidths, so in order to achieve the narrowest harmonic bunching spectrum, a balance must be found between the transform limited bandwidth of the short laser and the broadening from the nonlinear phase contributions. Here, using an extension of the formalism in Hemsing [ 1 ], we examine this more practically relevant case where the seed laser pulse can vary in duration, and also carry nonlinear temporal phase structures.

This allows analysis of the bunching spectrum in the presence of nonlinearity in the electron beam and laser phase.

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From a general expression for the bandwidth, we show that laser pulses optimized in duration can produce much narrower bunching spectra once the e-beam energy distortions reach a threshold value. We also define the requirements that allow the laser phase to precisely compensate the e-beam distortions to obtain transform limited pulses. The paper is arranged as follows. We first present an extension to the theory in Hemsing [ 1 ] that includes a finite seed laser pulse that produces a modulation with a longitudinal profile given by the function g z.

Since the spectral-temporal properties in EEHG are inherited primarily from the second laser, we assume the first laser is infinite. Assuming that the EEHG bunching amplitude is optimized, we then derive an analytic expression for g z with a Gaussian seed laser. Arbitrary phase distortions are then included as a Taylor series, and their impacts to each order on the total bandwidth and time-bandwidth product TBP are then calculated analytically. We then derive the conditions on the laser pulse length and e-beam length to minimize the bandwidth, and study a few examples.

Notation closely follows that of Xiang [ 11 ]. Consider an EEHG electron beam transformation of the form,. The first laser, A 1 , is assumed to be ideal and infinite in length. The beam is assumed to have an uncorrelated Gaussian initial energy distribution. It is also assumed that the system is far from the minimum pulse duration limit [ 21 ]. The z -dependence of the second laser envelope A 2 z is still within the argument of the J m Bessel function, which complicates simple analytic solutions.

We therefore search for an ansatz of the form,. Now the instantaneous spatial bunching frequency is just the z -derivative of the full longitudinal phase,.

This gives the bandwidth due to the combination of the laser modulation g z with the electron beam distribution function f z. It is dominated by the shorter of the two if they coincide longitudinally. We consider the effect of a Gaussian laser pulse in the second echo modulator on the longitudinal profile of the bunching. Slippage in the modulator is ignored. Small relative changes in the bunching due to variations from the optimum modulation are given by [ 20 ].

We posit then, that the functional dependance of g z may be modeled as a super-Gaussian distribution that has the same series expansion to lowest order:. This form for the z -dependent bunching envelope induced by the laser closely matches exact solutions, as shown in Figure 1.

## The Physics of Free Electron Lasers | Evgeny Saldin | Springer

The rms width of g z 2 is, to a good approximation,. Figure 1. The approximate super-Gaussian g z is slightly narrower than the exact solution.

Bottom Corresponding frequency spectra. These scalings are plotted against exact values in Figure 2. Figure 2. Normalized RMS and FWHM lengths of the bunching envelope from exact numerical calculations solid lines and from the super-Gaussian approximation dashed lines. Harmonic compression is the result of the high harmonics being increasingly more sensitive to the optimal modulation amplitude to produce bunching, so the longitudinal region of the Gaussian modulation that matches this condition becomes narrowed.

The flattened, super-Gaussian form of g z is thus characteristic of the optimized bunching envelope for a Gaussian second seed laser. We note that it slightly underestimates the exact longitudinal width of the bunching envelope, as shown in Figures 1 , 2. With an analytic form for g z in hand, the transform-limited bandwidth in Equation 9 can be calculated for a specified electron beam distribution f z.

The rms transform-limited bandwidth of the bunching spectrum then has simple analytic solution that is approximately see Supplementary Material ,. Two limiting regimes can be identified. This is the regime studied in Hemsing [ 1 ]. Figure 3. Because of harmonic compression, the laser pulse length needs to increase slightly with increasing harmonic number to maintain a fixed bandwidth. Consider the example of an electron beam with a quadratic chirp, as shown in Figure 4.

On one hand, a narrowband laser with a longer pulse length could work, but it may extend over or add more of the undesirable nonlinearities that add bandwidth during the harmonic up-conversion. Conversely, a short laser pulse is intrinsically broadband, but may be less impacted by the presence of nonlinearities. As we will see, the optimization is straightforward once the phase is known, and the answer depends on the amplitude and source of the non-linearity.

Figure 4. Example electron beam phase space distribution from FERMI [Allaria, personal communication] with dominant quadratic structure and higher order structure near the head and tail. The optimized laser pulse length to produce a minimum bandwidth depends on the amplitude of the non-linearities in the beam.

Without regard to the origin of the nonlinear phase structure i.

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We will then use the results to derive conditions for obtaining the minimum bandwidth. The second term is the excess bandwidth from the phase nonlinearity. The analytic expression for the numerical coefficient G N is given in the Supplementary Material , and the lowest order numerical values are given in Table 1.