**2. Synchrotron radiation, undulator radiation, and free-electron lasers**

Charged particle radiates energy in the form of electromagnetic radiation when it acceler‐ ates. Following relativistic Lorentz transforms tan*<sup>θ</sup>* <sup>=</sup> sin *<sup>θ</sup>* ' *<sup>γ</sup>*(*<sup>β</sup>* <sup>+</sup> cos *<sup>θ</sup>* ') , it is easy to conclude that the radiation emission of the relativistic electron is focused in the narrow angle *<sup>θ</sup>* <sup>≈</sup> <sup>1</sup> *<sup>γ</sup>* (see, for example, [46]). While the charge is moving on a circular orbit ofradius*R*, it emits SRin a narrow cone of emission, which illuminates the receiver for a very short period of time, while passing from the point A to the point B. Lorentz transforms is applied to the period of time in the reference frames, related to the electron and to the observer and yield the time of the SR pulse <sup>Δ</sup>*<sup>t</sup>* <sup>=</sup> *<sup>m</sup>* <sup>2</sup>*eB<sup>γ</sup>* <sup>2</sup> <sup>=</sup> *<sup>R</sup>* <sup>2</sup>*c<sup>γ</sup>* <sup>3</sup> , where *<sup>R</sup>* <sup>≅</sup> *<sup>γ</sup>mc eB* . For the UR, it is not so, since it is gathered all along the undulator and the characteristic length then equals that of the undulator, by far exceeding the arc, from which the SR, reaching the user, is gathered. Denoting the unit vector *n* <sup>→</sup> =*R* <sup>→</sup> / *<sup>R</sup>* and *n* <sup>→</sup> ≅(*ψ*cos*ϕ*, *ψ*sin*ϕ*, 1−*ψ*<sup>2</sup> / 2), it is easy to obtain the wavelength of the radiation, emitted off the axis in the angle *θ*

$$\mathcal{A}\_{\mathbf{s}} = \frac{\mathcal{A}\_{\mathbf{u}}}{2n\gamma^2} \left( 1 + \gamma^2 \left\langle \theta^2 \right\rangle + \gamma^2 \psi^2 \right) \tag{1}$$

from the simple condition of positive interference of the wavelengths, emitted on each magnetic poles of the undulator. Since the SR from a relativistic charge is emitted in a narrow cone, which includes the undulator axis all the time, the electron drifts in the undulator at relativistic speed, if the electrons transversal oscillations are small. At the exit of the undula‐ tor the intense radiation appears (see **Figure 1**).

The spectral range of SR and UR extends up to Röntgen band. However, due to the fact that the SR is perceived as a very short pulse, its spectral range is very broad, starting from the synchrotron frequency *ω*0 = *v*/*R*, while the UR has few harmonics and in some cases, such as that of a spiral undulator, it can contain a single harmonic.

**Figure 1.** Schematic drawing of a planar undulator.

For spontaneous radiation, emitted by an electron, the undulator selects resonant UR wave‐ lengths. The idea about it can be given by the simple consideration, demonstrated in **Figure 2**, where electric fields for two resonant wavelengths *λ<sup>n</sup>* at the fundamental (*n* = 1; red) and at the third harmonics (*n* = 3; blue) are shown. A non-resonant electric field for the second harmon‐ ic is shown in green. The fundamental and the third harmonics are phase-matched with the electron after one undulator period as highlighted in **Figure 2**. The electron trajectory is drawn in gray. Such a phase matching of the radiation proceeds on each next undulator period. Thus, the radiation from one electron constructively interferes over many periods, and in this sense we obtain coherent radiation from one electron along the whole length of the undulator.

**Figure 2.** Undulator selects resonant frequencies for the emitted SR.

The following conditions are common in modern undulators: the electrons are ultrarelativis‐ tic, which is natural in contemporary accelerators, they have small transverse momentum, and the electric field is absent:

Undulators for Short Pulse X-Ray Self-Amplified Spontaneous Emission-Free Electron Lasers http://dx.doi.org/10.5772/64439 203 205

$$\forall \gamma \gg 1, \nexists \beta\_{\perp} << 1, H\_{\parallel} << H\_{\perp'} \vec{E} = 0 \tag{2}$$

The UR from <sup>a</sup> planar undulator with *<sup>N</sup>* periods of *<sup>λ</sup>*<sup>0</sup> with the undulator parameter *<sup>k</sup>* <sup>=</sup> *<sup>e</sup> mc* <sup>2</sup> *H*<sup>0</sup> *kλ* , where *kλ* = 2*π*/*λu*, *λ<sup>u</sup>* is the undulator period, *Hy* = *H*0*sin*(*kλz*) is the sinusoidal magnetic field, has the following peak frequencies:

$$\alpha o\_{\boldsymbol{n}} = n o\_{\boldsymbol{n}} = \frac{2n\alpha\_0 \boldsymbol{\gamma}^2}{1 + \frac{k^2}{2} + (\boldsymbol{\gamma}\boldsymbol{\nu})^2}, \; o\_{\boldsymbol{n}0} = \frac{2\alpha\_0 \boldsymbol{\gamma}^2}{1 + k^2/2}, \; o\_{\boldsymbol{n}0} = n o\_{\boldsymbol{n}0}, \tag{3}$$

where *ω*<sup>0</sup> =*kλβ<sup>z</sup>* 0 *c*, *β<sup>z</sup>* <sup>0</sup> =1<sup>−</sup> <sup>1</sup> <sup>2</sup>*<sup>γ</sup>* <sup>2</sup> (1 <sup>+</sup> *<sup>k</sup>* <sup>2</sup> <sup>2</sup> ) is the average drift speed of the electrons along the undulator axis, *k* ≈ *H*0*λ*0 [*T cm*] and *ψ* is off the undulator axis angle. The shape of the UR emission line is described by the *sin*(*ν<sup>n</sup>* / 2) *<sup>ν</sup><sup>n</sup>* / <sup>2</sup> =*sinc νn* <sup>2</sup> function, dependent on the detuning param‐ eter

$$\nu\_n = 2\pi N n \left(\frac{\alpha}{\alpha\_n} - 1\right). \tag{4}$$

The homogeneous bandwidth, sometimes called half-width of UR spectrum line at its halfheight or simply half-width, is <sup>1</sup> <sup>2</sup>*nN* and the half-width equals

$$\frac{\Delta\alpha o}{\alpha o\_{n0}} = \frac{\alpha - \alpha o\_{n0}}{\alpha o\_{n0}} = \frac{1}{nN}.\tag{5}$$

In real devices <sup>1</sup> *nN* < <1. The emitted wavelength *λn* = 2*π*/*kn* can be expressed through the speed of the electrons in the undulator as follows:

$$
\hat{\lambda}\_u = \frac{\lambda\_1}{n} = \frac{\lambda\_u}{n} \left( \frac{1 - u/c}{u/c} \right) \equiv \frac{\lambda\_u}{2n\gamma^2} \left( 1 + \tilde{k}^2 \right), \tag{6}
$$

where *λu* is the undulator period, *u* is the electron speed

$$
\mu = c \left( 1 - \left( 1 + \tilde{k}^2 \right) \Big/ 2\gamma^2 \right),
\tag{7}
$$

The undulator parameter is *k* or *k* ˜:

**Figure 1.** Schematic drawing of a planar undulator.

202 204High Energy and Short Pulse Lasers

**Figure 2.** Undulator selects resonant frequencies for the emitted SR.

the electric field is absent:

For spontaneous radiation, emitted by an electron, the undulator selects resonant UR wave‐ lengths. The idea about it can be given by the simple consideration, demonstrated in **Figure 2**, where electric fields for two resonant wavelengths *λ<sup>n</sup>* at the fundamental (*n* = 1; red) and at the third harmonics (*n* = 3; blue) are shown. A non-resonant electric field for the second harmon‐ ic is shown in green. The fundamental and the third harmonics are phase-matched with the electron after one undulator period as highlighted in **Figure 2**. The electron trajectory is drawn in gray. Such a phase matching of the radiation proceeds on each next undulator period. Thus, the radiation from one electron constructively interferes over many periods, and in this sense we obtain coherent radiation from one electron along the whole length of the undulator.

The following conditions are common in modern undulators: the electrons are ultrarelativis‐ tic, which is natural in contemporary accelerators, they have small transverse momentum, and

$$
\tilde{k}^2 \equiv \frac{k^2}{2} = \gamma^2 \mathcal{J}\_\perp^2 = \gamma^2 \left\{ \Theta^2 \right\},
\\
k = eH\_0 \mathcal{J}\_u \Big/ 2\pi mc^2,\tag{8}
$$

*θ* is the off-axis angle, which in essence indicates how much the electron deviates from the axis in its motion along the undulator due to transversal oscillations, caused by the periodic magnetic field. In real conditions for a weak undulator *k* ~ 1, while for wiggler or strong undulator *k* ~ 10. Thus, for a weak undulator the radiation is essentially directed all along the undulator axis, while for the wiggler, it is in much wider angle. Considering *n* = 1, i.e., the fundamental harmonic, we write *u* = *ck*1/(*k*1 + *ku*), *k*1 = 2*π*/*λ*1. Both SR and spontaneous UR are incoherent. In the full spectrum of the radiation, the components, whose wave length is longer than the bunch length, are coherent, while the radiation of the length, significantly exceed‐ ing the size of the bunch, is approximately coherent. The region of coherent radiation will enlarge as the bunch gets shorter. The difference between the radiations, emitted by various SR sources, is demonstrated in **Figure 3**. It shows how the radiation intensity and the degree of coherency of the emitted radiation varies from one device scheme to the other; *Nf* is the number of emitted photons, *Ne* is the number of the electrons in the beam, *N* is the number of undulator periods. For SR from a bending magnet, the intensity is roughly proportional to the number of electrons emitting photons in a bending magnet. Wiggler, or strong undulator, is essentially a number of bending magnets, where electrons deviate significantly from the axis; the character of the radiation remains that from the bending magnets, but the intensity is *N* times higher. The radiation is incoherent. In a weak undulator, where the electrons slightly oscillate in transversal to the axis plane, so that the cone of the emission of the radiation always includes the undulator axis, the radiation of one single electron is coherent along the undula‐ tor axis, but the radiation of the bunch of electrons is incoherent in between them. The intensity of the radiation is proportional to *N*<sup>2</sup> . In free-electron laser, the electrons within a bunch emit largely coherent radiation, which transforms into a significant increase of the intensity *Nf* ~ (*Ne N*) 2 .

New type of radiation source—X-ray free-electron laser (X-FEL)—provides a combination of sub-picosecond pulse duration of a conventional laser with the X-ray wavelength of a synchrotron radiation source. Practical research potential of short wave radiation in nature andin technology can hardly be overestimated. Indeed, wavelength about 200 nm allows study of viruses, the scale of an atomic corral is ~14 nm, and the wavelength of 1–2 nm gives the potential to resolve DNA helix width and carbon nanotubes, while shorter wavelengths could visualize the small molecules, such as water molecule, and atoms. On the other hand, studies of processes at mircoscale often require very short time resolution. Indeed, the processes of a water molecule dissociation takes as short time as 10 fs, Bohr period of valence electron is ~1 fs, shock wave propagates by 1 atom in 100 fs, and electron spin processes in the magnet‐ ic field of 1 T within 10 ps. Thus, it is of great practical importance to have a device, which produces radiation with a combination of very short wavelength and short duration. One of the most prominent devices of such a kind is a free-electron laser.

Undulators for Short Pulse X-Ray Self-Amplified Spontaneous Emission-Free Electron Lasers http://dx.doi.org/10.5772/64439 205 207

**Figure 3.** Different sources of SR: synchrotron, wiggler, undulator, and free-electron laser.

## **3. Keynotes on free-electron lasers**

2

gb

*<sup>k</sup> <sup>k</sup>*

204 206High Energy and Short Pulse Lasers

of the radiation is proportional to *N*<sup>2</sup>

*N*) 2 .

2 22 2 2 2 <sup>0</sup> ,= 2 , <sup>2</sup> º= = ^ % *<sup>u</sup>*

*θ* is the off-axis angle, which in essence indicates how much the electron deviates from the axis in its motion along the undulator due to transversal oscillations, caused by the periodic magnetic field. In real conditions for a weak undulator *k* ~ 1, while for wiggler or strong undulator *k* ~ 10. Thus, for a weak undulator the radiation is essentially directed all along the undulator axis, while for the wiggler, it is in much wider angle. Considering *n* = 1, i.e., the fundamental harmonic, we write *u* = *ck*1/(*k*1 + *ku*), *k*1 = 2*π*/*λ*1. Both SR and spontaneous UR are incoherent. In the full spectrum of the radiation, the components, whose wave length is longer than the bunch length, are coherent, while the radiation of the length, significantly exceed‐ ing the size of the bunch, is approximately coherent. The region of coherent radiation will enlarge as the bunch gets shorter. The difference between the radiations, emitted by various SR sources, is demonstrated in **Figure 3**. It shows how the radiation intensity and the degree of coherency of the emitted radiation varies from one device scheme to the other; *Nf* is the number of emitted photons, *Ne* is the number of the electrons in the beam, *N* is the number of undulator periods. For SR from a bending magnet, the intensity is roughly proportional to the number of electrons emitting photons in a bending magnet. Wiggler, or strong undulator, is essentially a number of bending magnets, where electrons deviate significantly from the axis; the character of the radiation remains that from the bending magnets, but the intensity is *N* times higher. The radiation is incoherent. In a weak undulator, where the electrons slightly oscillate in transversal to the axis plane, so that the cone of the emission of the radiation always includes the undulator axis, the radiation of one single electron is coherent along the undula‐ tor axis, but the radiation of the bunch of electrons is incoherent in between them. The intensity

largely coherent radiation, which transforms into a significant increase of the intensity *Nf*

the most prominent devices of such a kind is a free-electron laser.

New type of radiation source—X-ray free-electron laser (X-FEL)—provides a combination of sub-picosecond pulse duration of a conventional laser with the X-ray wavelength of a synchrotron radiation source. Practical research potential of short wave radiation in nature andin technology can hardly be overestimated. Indeed, wavelength about 200 nm allows study of viruses, the scale of an atomic corral is ~14 nm, and the wavelength of 1–2 nm gives the potential to resolve DNA helix width and carbon nanotubes, while shorter wavelengths could visualize the small molecules, such as water molecule, and atoms. On the other hand, studies of processes at mircoscale often require very short time resolution. Indeed, the processes of a water molecule dissociation takes as short time as 10 fs, Bohr period of valence electron is ~1 fs, shock wave propagates by 1 atom in 100 fs, and electron spin processes in the magnet‐ ic field of 1 T within 10 ps. Thus, it is of great practical importance to have a device, which produces radiation with a combination of very short wavelength and short duration. One of

 l p

*k eH mc* (8)

. In free-electron laser, the electrons within a bunch emit

~ (*Ne*

 g q

> In 1971, Madey published a theory of the FEL [47], where he described a small gain process in a system: relativistic electron beam/undulator. In his study he hypothesized that it could generate coherent X-ray radiation. Few years later the first demonstration of FEL amplifica‐ tion and lasing was performed in a low-gain infrared oscillator FEL at Stanford. At approxi‐ mately the same time, Colson and Hopf described classically the FEL interaction, which had god quantum description by Madey before. Since 1970s, extensive classical description of the high-gain regime of FEL operation has been developed. Since the first X-rays were discov‐ ered by Wilhelm Röntgen in Würzburg in 1895, the peak "brilliance" of X-ray sources has increased by ~16 orders of magnitude. The so-called first-generation synchrotron sources were in fact particle accelerators, designed for experiments in high-energy physics. The secondgeneration labs were custom-built facilities, while the third-generation sources at labs such as the European Synchrotron Radiation Facility (ESRF) feature undulators, emitting light in a very narrow cone. The fourth generation of synchrotron-radiation sources includes freeelectron lasers.

> To account for the processes, which occur in a free-electron laser, we need to understand the fundamental difference between the coherent and non-coherent radiation, emitted by many electrons, which pass through undulator at the same time. The power, emitted by electrons

$$P \propto \left| \sum\_{j=1}^{N\_\epsilon} E\_j e^{i\phi\_j} \right|^2 = \sum\_{j=1}^{N\_\epsilon} E\_j^2 + \left| \sum\_{j=1, j \neq k}^{N\_\epsilon} \sum\_{k=1}^{N\_\epsilon} E\_j E\_k e^{i\left(\phi\_j + \phi\_k\right)} \right|^2,\tag{9}$$

where *φ<sup>j</sup>* are the relative phases of the emitted radiation electric fields *Ej* in a system of big number of electrons *N* ≫ 1, includes two terms. For a system with uncorrelated phases, the terms in the second double sum, which is of the order of ~*N*<sup>2</sup> , tend to destructively interfere. This happens in incoherent spontaneous UR sources. Total power emitted then approximate‐ ly equals to the sum of the powers from the *N* independent scattering electrons, which originates from the first term in Eq. (9). For correlated phases of the electric fields, we have *φ<sup>j</sup>* ≈ *φ<sup>k</sup>* for all the electrons and then the coherent second term ~*N*<sup>2</sup> contributes. Forit to happen, the electron sources must be periodically bunched at the resonant radiation wavelength. **Figure 4** demonstrates how incoherent radiation from a bunch of electrons in an undulator becomes coherent toward the undulator's end. Indeed, at the beginning, the electrons in the bunch enter the undulator with initially random phases, which ensures that mostly incoher‐ ent radiation is emitted at the resonant radiation wavelength. Because of the electrons interact collectively with the radiation they emit, small coherent fluctuations in the radiation field grow along the undulator length and simultaneously begin to bunch the electrons at the resonant wavelength. This collective process continues until the electrons are strongly bunched toward the end of the undulator, where the process saturates and the electrons begin to de-bunch.

**Figure 4.** From incoherent to coherent emission along the undulator length.

Behind this phenomenon stands a simple physical mechanism, which is based on the fact that the speed of electrons, while being close to that of the radiation, is still smaller, and, there‐ fore, the electrons appear behind its radiation, propagating in the undulator. It can be best illustrated in **Figure 5**.

**Figure 5.** Slippage of the radiation and the electron bunch in FEL.

( ) 2 2

j j+

, tend to destructively interfere.

µ =+ å å åå (9)

, *<sup>e</sup> <sup>e</sup> e e j j k*

where *φ<sup>j</sup>* are the relative phases of the emitted radiation electric fields *Ej* in a system of big number of electrons *N* ≫ 1, includes two terms. For a system with uncorrelated phases, the

This happens in incoherent spontaneous UR sources. Total power emitted then approximate‐ ly equals to the sum of the powers from the *N* independent scattering electrons, which originates from the first term in Eq. (9). For correlated phases of the electric fields, we have *φ<sup>j</sup>* ≈ *φ<sup>k</sup>* for all the electrons and then the coherent second term ~*N*<sup>2</sup> contributes. Forit to happen, the electron sources must be periodically bunched at the resonant radiation wavelength. **Figure 4** demonstrates how incoherent radiation from a bunch of electrons in an undulator becomes coherent toward the undulator's end. Indeed, at the beginning, the electrons in the bunch enter the undulator with initially random phases, which ensures that mostly incoher‐ ent radiation is emitted at the resonant radiation wavelength. Because of the electrons interact collectively with the radiation they emit, small coherent fluctuations in the radiation field grow along the undulator length and simultaneously begin to bunch the electrons at the resonant wavelength. This collective process continues until the electrons are strongly bunched toward the end of the undulator, where the process saturates and the electrons begin to de-bunch.

2 1 1 1, 1

*j j j j kk P Ee E EEe*

= = =¹ =

j

206 208High Energy and Short Pulse Lasers

terms in the second double sum, which is of the order of ~*N*<sup>2</sup>

**Figure 4.** From incoherent to coherent emission along the undulator length.

illustrated in **Figure 5**.

Behind this phenomenon stands a simple physical mechanism, which is based on the fact that the speed of electrons, while being close to that of the radiation, is still smaller, and, there‐ fore, the electrons appear behind its radiation, propagating in the undulator. It can be best

*<sup>N</sup> N NN <sup>i</sup> <sup>i</sup> j j j k*

> Indeed, the photons move at the speed of light *vγ* = *c*, while the electrons move at the speed *u* < *c*, *β*|| ≈ 1, i.e., slower than photons. Therefore a full slip between the electron bunch and the photon pulse accumulates along the length of the undulator and reads as follows:

$$
\Delta = (1 - \beta\_{\parallel}) N \mathcal{X}\_{\boldsymbol{\upiota}} \equiv N \mathcal{X}, \tag{10}
$$

where *<sup>λ</sup>* is the emitted wavelength *<sup>λ</sup>* <sup>=</sup> *<sup>λ</sup><sup>u</sup>* <sup>2</sup> *<sup>γ</sup>* <sup>2</sup> (1 <sup>+</sup> *<sup>k</sup>* <sup>2</sup> <sup>2</sup> ) and the parallel component of the *β* is *<sup>β</sup>*||≅1<sup>−</sup> <sup>1</sup> <sup>2</sup> *<sup>γ</sup>* <sup>2</sup> (1 <sup>+</sup> *<sup>k</sup>* <sup>2</sup> <sup>2</sup> ). Thus the slip on the whole undulator length is *N* times the emitted wavelength. Respectively, the slip on a distance, equal half an undulator period equals half of the emitted wavelength:

$$
\Delta\_{\mathbb{A}\_4 \uparrow 2} = N \lambda \big| \big( 2N \big) = \lambda \big| \big2 . \tag{11}
$$

In other words, as the radiation wave travels over a distance *λ*/2 in a time *λ*/(2*c*), the electron travels over a smaller distance *uλ*/(2c), and on one undulator period the electrons slip, respectively, to the photons by one emitted wavelength. The whole photon packet, having a higher velocity that the electrons, slips over the electron packet and the wave emitted by the electrons on a certain undulator period comes in phase with the wave, emitted on the next undulator period. To illustrate the formation of microbunches, which lead to coherent radiation of the electrons within a bunch, we present the following explanation in **Figure 6** (see [48]).

Suppose we have the magnetic field *Bw* of the already existing electromagnetic wave. Then its interaction with the electron transverse velocity *vt* creates Lorentz force *F*bunch, which we denote as *f* for brevity (see **Figure 6**). This force pushes the electron toward a wave node as seen in **Figure 6**. After the electron travels over one-half undulator period, its transverse velocity becomes reversed. In the meantime the electromagnetic wave travels ahead of the electron by one-half wavelength as follows from (10) and (11). Its field *Bw* is now also reversed, so that the

Lorentz force keeps its direction and the microbunching continues. For the charge, which is ahead with respect to the microbunch, which is in the wave node (see **Figure 6**), the Lorentz force *F*bunch or *f* pushes the charge back to the node, thus grouping electrons in a bunch at the wave node. So it proceeds on other periods, because while the electrons move through a full oscillation period *λu*, the electromagnetic wave propagates by *λ<sup>u</sup>* plus one wavelength *λ*. Consequently, the transverse movement of each electron has a constant phase with respect to the electromagnetic field.

**Figure 6.** Interaction of the radiation with the electrons and formation of microbunches.

## **4. Correlated amplification in FEL**

Amplification in FEL occurs because of the energy is transferred from the electrons to the previously emitted waves. This effect is due to the negative work of the force, produced by the transverse electric field of the wave, since the magnetic field of the wave does not work. The time rate of the energy transfer for a single electron is proportional to the product *EWvt* , where *EW* is the electric field of the radiation wave and *vt* is the electron transverse velocity. Then *EW* ∝*I* 1 2 , where *I* is the wave intensity and *dI dt* ∝*I* 1 2 *vt*. Note that uncorrelated combi‐ nation of the effects of individual electrons would not lead to an exponential increase of the intensity with the distance (or time), but to a quadratic law: *I*(*z*) ∝ *z*<sup>2</sup> . Transverse velocity and the field *B* produce longitudinal Lorentz force *F*bunch =*vtBW* , which pushes the electrons and forms microbunches. This force is proportional to the transverse electron velocity, and it is also orthogonal to the wave field *B* of the strength *BW*. Since *BW* ∝*I* 1 2 , the microbunching force is also proportional to *I* 1 2 : *B*bunch∝*I* 1 2 . We now assume that this force enhances the correlated emission by a factor, proportional to the microbunching force. Multiplied by the energy transfer rate for each electron, this factor gives *dI dt* = *AI*, *A* = constant. Proceeding on the supposition that *A* = *u*/*LG*, where *LG* is the gain length, we obtain the exponential growth:

Lorentz force keeps its direction and the microbunching continues. For the charge, which is ahead with respect to the microbunch, which is in the wave node (see **Figure 6**), the Lorentz force *F*bunch or *f* pushes the charge back to the node, thus grouping electrons in a bunch at the wave node. So it proceeds on other periods, because while the electrons move through a full oscillation period *λu*, the electromagnetic wave propagates by *λ<sup>u</sup>* plus one wavelength *λ*. Consequently, the transverse movement of each electron has a constant phase with respect to

**Figure 6.** Interaction of the radiation with the electrons and formation of microbunches.

, where *I* is the wave intensity and *dI*

intensity with the distance (or time), but to a quadratic law: *I*(*z*) ∝ *z*<sup>2</sup>

1 2

orthogonal to the wave field *B* of the strength *BW*. Since *BW* ∝*I*

: *B*bunch∝*I*

1 2

Amplification in FEL occurs because of the energy is transferred from the electrons to the previously emitted waves. This effect is due to the negative work of the force, produced by the transverse electric field of the wave, since the magnetic field of the wave does not work. The time rate of the energy transfer for a single electron is proportional to the product *EWvt*

where *EW* is the electric field of the radiation wave and *vt* is the electron transverse velocity.

nation of the effects of individual electrons would not lead to an exponential increase of the

the field *B* produce longitudinal Lorentz force *F*bunch =*vtBW* , which pushes the electrons and forms microbunches. This force is proportional to the transverse electron velocity, and it is also

emission by a factor, proportional to the microbunching force. Multiplied by the energy

*dt* ∝*I* 1 2

> 1 2

. We now assume that this force enhances the correlated

,

*vt*. Note that uncorrelated combi‐

. Transverse velocity and

, the microbunching force is

**4. Correlated amplification in FEL**

Then *EW* ∝*I*

1 2

also proportional to *I*

the electromagnetic field.

208 210High Energy and Short Pulse Lasers

$$I = I\_0 \exp\left(\frac{ut}{L\_G}\right) = I\_0 \exp\left(\frac{z}{L\_G}\right). \tag{12}$$

The exponential gain only occurs after bunching is established. This process continues until the saturation is reached. Provided at the beginning, the initial position of the electron, respectively, the existing wave, is favorable for the energy transfer from the electron to the wave and the direction of the electron transverse velocity, respectively, to the wave electric field *E* result in negative work, the electron energy is successfully transferred to the wave. This results in the decrease in the electron energy and of the longitudinal speed of the electron *u*, which becomes *u* − Δ*u.* This decrease in longitudinal speed in its turn changes the above conditions, making them less favorable for the energy transfer from the electron to wave. At a certain point, as this process continues and the electron speed reduction Δ*u* becomes more and more significant, the electrons can give no more energy to the wave and the wave starts giving its energy to the electrons of the beam. It increases the electron speed *u* and restores the favorable forthe energy transferfrom the electrons to the wave conditions. Thus, in this regime the exponential wave energy growth is over and instead the energy oscillates between the wave and the electrons.

To find the amplification value, we must evaluate *vt* and the degree of bunching. The trans‐ verse speed of the electron reads as follows:

$$\mathbf{v}\_{\iota} = \left(\frac{euH\_{\iota}}{\gamma m\_{0}}\right) \left(\frac{\lambda\_{u}}{2\pi u}\right) \cos\frac{2\pi ut}{\lambda\_{u}}.\tag{13}$$

The energy transfer rate by a single electron to preexisting wave is proportional to *I* 1 2 *H*0*λ<sup>u</sup>* / *γ*; the field *BW* of the electromagnetic wave is proportional to the square root of the wave intensity *BW* ∝*I*<sup>0</sup> 1/2 exp(*ut* / <sup>2</sup>*<sup>L</sup> <sup>G</sup>*). Then the bunching force, which is the longitudinal force, can be written as follows:

$$F\_{\text{bunch}} = \text{const} \times \left(\frac{H\_0 \mathcal{A}\_u}{\mathcal{I}}\right) I\_0^{\text{J}^2} \exp\left(\frac{\omega t}{2L\_G}\right),\tag{14}$$

which, in turn, yields the following equation of motion for longitudinal mass *γ*<sup>3</sup> *m*:

$$\gamma^3 m \frac{d^2 \Delta \mathbf{x}}{dt^2} = F\_{\text{bunch}} = \text{const} \times \left(\frac{H\_0 \lambda\_u}{\mathcal{I}}\right) I\_0^{\text{fl}} \exp\left(\frac{\mu t}{2L\_c}\right). \tag{15}$$

The bunching force *F*bunch induces a small longitudinal electron displacement Δ*x*:

$$\Delta \mathbf{v} = \text{const} \times \frac{1}{\mathcal{V}^3} \frac{H\_0 \lambda\_u}{\mathcal{V}} L\_G^2 I\_0^{1/2} \exp\left(\frac{u t}{2L\_G}\right) = \frac{H\_0 \lambda\_u L\_G^2}{\mathcal{V}^4} I^{1/2}. \tag{16}$$

As discussed above, the electrons are concentrated in narrow slabs, separated from each other by a distance of the wavelength *λ*. The degree of microbunching, corresponding to the fraction of electrons that emit in a correlated way, can be assumed to be proportional to (Δ*x*/*λ*). The corresponding number of electrons is proportional to *Ne*(Δ*x*/*λ*). Their contribution to the wave intensity reads as follows:

$$j\frac{\Delta\mathbf{x}}{\lambda} \propto j\frac{I^{\text{l}2}H\_0\lambda\_u L\_G^2 / \gamma^4}{\lambda\_u / \gamma^2} = jI^{\text{l}2}H\_0L\_G^2 / \gamma^2, \quad j \equiv \frac{i}{\Sigma} = \frac{\text{current}}{\text{cross-section}}.\tag{17}$$

Microbunching effects correspond to a factor, proportional to the longitudinal microbunch‐ ing force *F*bunch and, therefore, proportional to ~ *I*1/2, where *I* is the electromagnetic wave intensity. Multiplying this factor by the energy transfer rate for electrons, we obtain the total transfer rate as follows:

$$\frac{d\mathbf{I}}{dt} = \text{const} \times j \frac{H\_0 L\_G^2}{\gamma^2} I^{1/2} \left( I^{1/2} \frac{H\_0 \lambda\_u}{\gamma} \right) = \text{const} \times j \frac{H\_0^2 \lambda\_u L\_G^2}{\gamma^3} I. \tag{18}$$

The solution of this equation reads *I* = *I*<sup>0</sup> exp(*z*/*LG*) and for ultrarelativistic electrons *u* / *L <sup>G</sup>* ≅*c* / *L <sup>G</sup>* ∝ *jHo* 2 *λuL <sup>G</sup>* <sup>2</sup> / *γ* <sup>3</sup> . Thus we obtain the FEL gain length

$$L\_G = \text{const} \times j^{-\text{J3}} H\_0^{-2\text{J3}} \lambda\_\mu^{-\text{J3}} \gamma. \tag{19}$$

Gain length is related to the Pierce parameter as follows:

$$\rho = \frac{\lambda\_u}{4\pi\sqrt{3}L\_\odot} \propto j^{4\dagger 3} H\_0^{2\dagger 3} \lambda\_u^{4\dagger 3} \gamma^{-1} \tag{20}$$

which can be also written as *<sup>ρ</sup>* <sup>=</sup> <sup>1</sup> <sup>2</sup>*<sup>γ</sup>* ( *Ie IA* ( *<sup>λ</sup>uk*˜ *<sup>f</sup> <sup>B</sup>* 2*πσ<sup>r</sup>* ) 2 ) 1/3 , where *IA* ≅ 17 kA is the Alfven current, *Ie* is the electron beam current, *σ<sup>r</sup>* is the beam radius, *fB*(*Jn*) is the bunching coefficient, and *Jn* are the Bessel functions. The arguments and the varieties of the Bessel functions, in charge of the description of the harmonics of the UR, will be explored in what follows.

### **5. Cavity-based FELs**

2

g

210 212High Energy and Short Pulse Lasers

intensity reads as follows:

transfer rate as follows:

*u* / *L <sup>G</sup>* ≅*c* / *L <sup>G</sup>* ∝ *jHo*

2 *λuL <sup>G</sup>* <sup>2</sup> / *γ* <sup>3</sup>

Gain length is related to the Pierce parameter as follows:

r 4 3 *u*

p

l

*G j H <sup>L</sup>*

l

3 0 1 2

The bunching force *F*bunch induces a small longitudinal electron displacement Δ*x*:

*G*

æ ö = ´ ç ÷ <sup>=</sup>

0 12 2 2 2 0

gg

*x IH L <sup>i</sup> j j jI H L j*

 g

<sup>Δ</sup> / current / , . / cross-section

*G*

Microbunching effects correspond to a factor, proportional to the longitudinal microbunch‐ ing force *F*bunch and, therefore, proportional to ~ *I*1/2, where *I* is the electromagnetic wave intensity. Multiplying this factor by the energy transfer rate for electrons, we obtain the total

The solution of this equation reads *I* = *I*<sup>0</sup> exp(*z*/*LG*) and for ultrarelativistic electrons

. Thus we obtain the FEL gain length

13 23 13

l g

13 23 43 1 0

l g

*u*

<sup>0</sup> const . *L jH G u*

2 2 2 00 0 1/2 1/2 2 3 const const . *<sup>G</sup> <sup>u</sup> u G dI H L H H L j II j I dt* l

 l

 g



æ ö = ´ ç ÷ = ´ è ø (18)

g

<sup>µ</sup> <sup>=</sup> º = <sup>S</sup> (17)

l

g g

12 2 4

l

*u*

 lg

*u G*

*<sup>H</sup> ut H L <sup>x</sup> L I <sup>I</sup>*

*d x <sup>H</sup> ut m F <sup>I</sup> dt L*

<sup>2</sup> bunch <sup>0</sup> const exp . <sup>2</sup> *u*

3 4 0 <sup>1</sup> Δ const exp . <sup>2</sup>

As discussed above, the electrons are concentrated in narrow slabs, separated from each other by a distance of the wavelength *λ*. The degree of microbunching, corresponding to the fraction of electrons that emit in a correlated way, can be assumed to be proportional to (Δ*x*/*λ*). The corresponding number of electrons is proportional to *Ne*(Δ*x*/*λ*). Their contribution to the wave

l

0 0 2 1/2 1/2

*L*

*G*

*u u G*

g

*G*

2

è ø (16)

 l

g

<sup>D</sup> æ ö æ ö ==´ ç ÷ ç ÷ è ø è ø (15)

There are essentially two main types of design for FEL: cavity-based FELs and single-pass FELs. Cavity-based FELs exploit many passes of the UR in the undulator and use mirrors. Relativistic electron beam passes through periodic magnetic field of an undulator and the mirrors feed spontaneous emission back onto the beam. Consequently, the spontaneous emission is enhanced by the stimulated emission.

**Figure 7.** FEL mirror limitations.

Mirror design imposes limitations on the wavelength due to the mirror material. Common mirror materials limit the wavelength to IR-UV (see **Figure 7**). Recent improvements in materials and technology result in new multilayer X-ray reflective mirrors, which have the efficiency up to 70%, such as MoRu/Be mirrors have 69.3% reflectivity maximums at 11.43 nm, Mo/Si mirrors have 67.2% reflectivity peak at 13.51 nm.

**Figure 8.** Longitudinal mode structure in mirror-based FEL design.

The mirror design has significant advantage: mirrors naturally select longitudinal modes, which are determined by the integer number of half-periods of the wavelength in between them. The longitudinal modes in optical cavity are equally spaced in frequency and time (see the middle plot in **Figure 8**):

$$
\Delta\nu = \frac{c}{2L}, \Delta\tau = \frac{1}{\Delta\nu}.\tag{21}
$$

The interval between them depends on the cavity length *L* (see the middle plot in **Figure 8**).

Taking into account the laser gain bandwidth

$$
\Delta\nu\_{\text{Caln}} \approx \frac{1}{\tau\_{\text{Pulso}}} = \frac{c}{2\Delta} \tag{22}
$$

where *Δ* = *Nλ* (see (10), see the top plot in **Figure 8**) and, imposing it on the longitudinal picture of the middle plot in **Figure 8**, we obtain the laser spectrum as shown in the bottom plot of **Figure 8**. Inside the gain bandwidth, the longitudinal modes of the laser will periodically and constructively interfere with each other when in phase, producing an intense burst of light. These light bursts are separated by the period of time *τ* = 2*L*/*c*, necessary for the light to make

exactly one round trip of the laser cavity; the comb of equally spaced modes in the output spectrum with spacing

$$
\Delta\nu\_{\text{RoundTip}} = \Delta\nu = \frac{1}{\tau\_{\text{RoundTip}}},
\text{ where}
\tau\_{\text{RoundTip}} = \frac{2L}{c} \tag{23}
$$

and *L* is the cavity size. In other words, only phase-matched wavelengths will constructively interfere and form the modes of the radiation field to create a comb of equally spaced modes in the output frequency spectrum. Such a laser is said to be mode-locked or phase-locked. Mode locking modifies the temporal envelope of the output field from a continuous wave to a series of short, periodically spaced pulses. The homogeneous gain bandwidth of a FEL is determined by the slippage length *Δ* = *Nλ* (10). This last plays central role in FEL physics. Indeed, it is easy to see that only finite number of longitudinal modes with positive gain *N*Gain exist: *Δν*Gain <sup>≈</sup> <sup>1</sup> *<sup>τ</sup>*Pulse <sup>=</sup> *<sup>c</sup>* <sup>2</sup>*<sup>Δ</sup>* (see (21)–(23)), where *Δ* = *Nλ*, *λ* is the radiated wavelength, and *N* is the number of the undulator periods. Therefore the number of gained modes reads as follows:

$$N\_{\rm Gain} = \frac{\Delta \nu\_{\rm Gain}}{\Delta \nu\_{\rm BoundTrip}} \approx \frac{L}{\Delta},\tag{24}$$

and the duration of the pulse is evidently

**Figure 8.** Longitudinal mode structure in mirror-based FEL design.

(see the middle plot in **Figure 8**):

212 214High Energy and Short Pulse Lasers

Taking into account the laser gain bandwidth

The mirror design has significant advantage: mirrors naturally select longitudinal modes, which are determined by the integer number of half-periods of the wavelength in between them. The longitudinal modes in optical cavity are equally spaced in frequency and time

> <sup>1</sup> Δ ,Δ . <sup>2</sup> *c L*

 t

The interval between them depends on the cavity length *L* (see the middle plot in **Figure 8**).

Pulse 1

where *Δ* = *Nλ* (see (10), see the top plot in **Figure 8**) and, imposing it on the longitudinal picture of the middle plot in **Figure 8**, we obtain the laser spectrum as shown in the bottom plot of **Figure 8**. Inside the gain bandwidth, the longitudinal modes of the laser will periodically and constructively interfere with each other when in phase, producing an intense burst of light. These light bursts are separated by the period of time *τ* = 2*L*/*c*, necessary for the light to make

t

D» =

2 *c*

D

n

= = <sup>D</sup> (21)

(22)

n

Gain

n

$$
\pi\_{\text{Pulse}} \approx \frac{1}{\Delta \nu\_{\text{Gain}}} = \frac{\pi\_{\text{RoundTrip}}}{N\_{\text{Gain}}}.\tag{25}
$$

For a Gaussian-shaped pulse, we have *τ*Gaussian Pulse≅ 0.44 *Δν*Gain . Thus, a laser macro pulse of, to say, microsecond duration consist of a train of short micro pulses, which are picoseconds or less in length. Such a short duration of the pulses is useful for studies of ultrashort physical and chemical processes. It is used for ultrafast spectroscopy and in femto-chemistry. The macro pulses repeat at a repetition ratio, limited by the accelerator usually at 10–100 Hz. The micro pulse repetition rate can vary from several MHz to even THz.

We omit in this work the field gain equations, which are well known and can be found elsewhere, and without derivation we just state the formulation of the Madey's theorem [47], which claims that the gain in the so-called Compton or low-gain regime is proportional to the slope of the spontaneous UR spectrum *f*(*ν*): *G*(*ν*)<sup>∝</sup> <sup>∂</sup> *<sup>f</sup>* (*ν*) <sup>∂</sup> *<sup>ν</sup>* , *<sup>f</sup>* (*ν*)=*sinc* 2(*ν*), and, therefore,

$$G(\nu) = -\frac{j}{2} \partial\_{\nu} \left[ \sin \frac{\nu}{2} \sqrt{\frac{\nu}{2}} \right]^2,\tag{26}$$

where *j* is the current density, *ν* is the detuning parameter (4). **Figure 9** demonstrates the FEL gain; the homogeneous bandwidth is given by *Δω <sup>ω</sup>*<sup>0</sup> <sup>=</sup> <sup>1</sup> <sup>2</sup>*<sup>N</sup>* . In the presence of constant magnetic field, the line shape is described by the Airy-type function *S*(*νn*, *β*)≡*∫* 0 1 *dτe <sup>i</sup>*(*νnτ*+*<sup>β</sup> <sup>τ</sup>* 3) and the derivative modifies into − ∂*S*<sup>2</sup> /∂*νn*. The commonly known shape − ∂(*sinc*<sup>2</sup> *νn*)/∂*ν<sup>n</sup>* is seen in **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> 3 *k <sup>γ</sup> π N κ*1,

resulting in nonzero values of *<sup>β</sup>* <sup>=</sup> (2*πnN* <sup>+</sup> *<sup>ν</sup>n*)(*γθ<sup>H</sup>* )2 <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 curve to lower frequencies (see **Figure 9**).

**Figure 9.** Function <sup>−</sup> <sup>∂</sup> (*<sup>S</sup>* <sup>2</sup> (*νn*)) <sup>∂</sup> *<sup>ν</sup><sup>n</sup>* , which describes FEL gain in an undulator in external field.
