**6. High-gain single-pass FELs**

There are several reasons why it is more difficult to build free-electron lasers for X-ray region thatis forlarger wavelengths. First of all, for small wavelengths we need high-energy electrons, but high electron energy also increases the gain length *LG* ∝ *γ*, and we must keep the gain length short, as required for an X-FEL. Then, the undulator parameters *H*<sup>0</sup> and *λ<sup>u</sup>* must be maximized, keeping in mind, however, that *λ<sup>u</sup>* also determines the radiation wavelength: *λ* ∝ *λu*/*γ*<sup>2</sup> . Then, the electron beam current *i* must be high and its transverse cross section *σ* small. However, the γ-factor cannot be freely decreased if we want to obtain X-ray wave‐ lengths. These challenges can be betterfaced in single-pass FELs with typical high-gain regime.

In the high-gain regime the radiation power increases exponentially as the electron beam and radiation co-propagate along the FEL undulator, and this happens on a single pass of the radiation along the FEL. These kinds of FELs are sometime called amplifiers, since every‐ thing starts from an initially small source, which may originate as noise, and it can be amplified by many orders of magnitude before the process saturates. In such FEL, there are no mirrors to form an oscillator cavity and this is particularly good for X-ray region, where mirrors are the most compromised element of the cavity-based FEL. Such FEL essentially works as an amplifier, in which the radiation forms on the single pass through a very long undulator, reaching peak pulse power ~ 1010 *W* for few dozens of femto-seconds. Overwhelming majority of current X-ray FELs are based on this type of design, which has been made possible due to the twenty-first century advances in magnet technology, accelerator constructions, and electron beam production.

**Figure 10.** Radiation mechanism in a SASE FEL (amplifier).

where *j* is the current density, *ν* is the detuning parameter (4). **Figure 9** demonstrates the FEL

**Figure 9** in the rear vertical plane, where *β* = 0. The constant magnetic field *H*d = *κH*0, where

*<sup>H</sup>*<sup>0</sup> is the amplitude ofthe undulator periodic field, produces the bending angle *<sup>θ</sup><sup>H</sup>* <sup>=</sup> <sup>2</sup>

<sup>∂</sup> *<sup>ν</sup><sup>n</sup>* , which describes FEL gain in an undulator in external field.

There are several reasons why it is more difficult to build free-electron lasers for X-ray region thatis forlarger wavelengths. First of all, for small wavelengths we need high-energy electrons, but high electron energy also increases the gain length *LG* ∝ *γ*, and we must keep the gain length short, as required for an X-FEL. Then, the undulator parameters *H*<sup>0</sup> and *λ<sup>u</sup>* must be maximized, keeping in mind, however, that *λ<sup>u</sup>* also determines the radiation wavelength:

small. However, the γ-factor cannot be freely decreased if we want to obtain X-ray wave‐ lengths. These challenges can be betterfaced in single-pass FELs with typical high-gain regime. In the high-gain regime the radiation power increases exponentially as the electron beam and radiation co-propagate along the FEL undulator, and this happens on a single pass of the radiation along the FEL. These kinds of FELs are sometime called amplifiers, since every‐ thing starts from an initially small source, which may originate as noise, and it can be amplified

. Then, the electron beam current *i* must be high and its transverse cross section *σ*

*<sup>ω</sup>*<sup>0</sup> <sup>=</sup> <sup>1</sup>

/∂*νn*. The commonly known shape − ∂(*sinc*<sup>2</sup>

<sup>2</sup>*<sup>N</sup>* . In the presence of constant magnetic

0 1 *dτe*

<sup>1</sup> <sup>+</sup> (*<sup>k</sup>* <sup>2</sup> / 2) <sup>+</sup> (*γθ<sup>H</sup>* )2 [35, 37, 39] and shifts the maximum of the

*<sup>i</sup>*(*νnτ*+*<sup>β</sup> <sup>τ</sup>* 3) and the

*νn*)/∂*ν<sup>n</sup>* is seen in

3 *k <sup>γ</sup> π N κ*1,

gain; the homogeneous bandwidth is given by *Δω*

resulting in nonzero values of *<sup>β</sup>* <sup>=</sup> (2*πnN* <sup>+</sup> *<sup>ν</sup>n*)(*γθ<sup>H</sup>* )2

curve to lower frequencies (see **Figure 9**).

(*νn*))

**6. High-gain single-pass FELs**

derivative modifies into − ∂*S*<sup>2</sup>

214 216High Energy and Short Pulse Lasers

**Figure 9.** Function <sup>−</sup> <sup>∂</sup> (*<sup>S</sup>* <sup>2</sup>

*λ* ∝ *λu*/*γ*<sup>2</sup>

field, the line shape is described by the Airy-type function *S*(*νn*, *β*)≡*∫*

The principle of the high-gain FEL interaction is based on the positive feedback process. The electrons emit undulator radiation, which corrects their position in space and their phase with respect to the electromagnetic wave; this groups the electrons on the radiation wavelength scale and thus the more and more coherent radiation is emitted along the undulator. First, the electrons in the bunch have random phases and produce incoherent emission. Already the first waves, emitted by these electrons, trigger formation of microbunches as discussed above (see **Figures 4**, **6** and **10**). Contrary to non-micro-bunched electrons, which emit incoherent waves, the emission of electrons, collected in micro-bunches, which are separated from each other by one wavelength, is correlated (see **Figures 4** and **10**). This causes an exponential intensity increase with the distance that continues until saturation is reached (see **Figure 10**). The schematic drawing in **Figure 10** represents modeling of the performance of the Linac Coherent Light Source—LCLS, with the parameters of the undulator: *L* = 100 m, *λu* = 3 cm, *LG* = 3.3 m, *λ* = 1.5Å = 0.15 nm. The whole installation has the length of *L* = 1 km, current *I* = 3 kA, *E* = 13.6 GeV, Linac and bunch compression: *γεx,y* = 0.4 μm (slice), *σE*/*E* = 0.01% (slice).

Simulation of the electron density in a bunch of the electron beam as it develops from the entrance toward the exit of the undulator is demonstrated in **Figure 11**. Left picture simu‐ lates the electron density in the bunch at the beginning of the undulator, middle picture represents the simulation in the middle of the undulator length, and the right picture demon‐ strates the electron density in the bunch at the end of the undulator.

**Figure 11.** Simulation of the density modulation of the electron beam along the undulator: undulator beginning—left picture, undulator middle—middle picture, undulator end—right picture.

The transverse structure of the electron bunch is much larger than its longitudinal substructure. Note the length between the slices can be of the order of nm, and there can be up to hundreds of thousands of slices in a bunch.
