Free-Electron Laser

Abstract

In a free-electron laser (FEL), free-electrons of a velocity near the speed of light are passing through a periodic transverse magnetic field.

Problems

19.1

Acceleration energies. Given is wiggler ($\lambda _\mathrm{w} = 2.4$ cm and $K = 1$). [Hint: make use of data of Table 19.1 to solve this and the following problems.]

• (a) Determine the electron energy necessary to drive a terahertz FEL at 1 THz and determine the change of energy necessary to change the frequency by 1%

• (b) Determine the electron energy necessary to drive an X-ray at a wavelength of 10 nm and determine the change of energy necessary to change the frequency by 1%

19.2

Frequency tuning. Relate a small change of energy relative to the energy E to the relative change of frequency and to the relative change of wavelength of a free-electron laser.

19.3

Show that the inhomogeneous broadening of the gain profile of a free-electron laser due to energy smearing is negligibly small if the condition $\varDelta \gamma \ll \gamma /(2N_\mathrm{w})$ is fulfilled.

19.4

Refractive index of a free-electron laser medium.

• (a) Estimate the frequency shift of a mode of a free-electron laser resonator that occurs when a free-electron laser is switched on.

• (b) Determine the speed of light in a free-electron medium.

• (c) Estimate the difference of the time it takes light and the time it takes an electron to propagate through the wiggler.

19.5

Determine the absolute number of electrons present in an active medium of a free-electron laser.

19.6

Estimate the time of onset of laser oscillation in a free-electron laser.

19.7

Determine characteristic quantities of the spectrum of spontaneously emitted radiation on the angular frequency scale: resonance frequency, halfwidth, frequency of the first minima, distance between this frequency and the resonance frequency.

19.8

Show that, on the time average, an electromagnetic field cannot exchange energy with an electron in free space.

19.9

Continuous wave free-electron laser. How is it possible to operate a continuous wave free-electron laser at the frequency of maximum gain although the initial gain at the frequency of maximum gain at steady state oscillation is negligibly small? [Hint: Consider the onset of oscillation.]

19.10

Energy spread of the electrons in an electron beam that enters a free-electron laser. (a) Estimate the inhomogeneous broadening of the frequency distribution of the radiation emitted by a mode locked free-electron laser due to a finite energy distribution of the electrons in the electron beam. (b) Estimate the broadening of the pulse duration. (c) If you would plan a mode locked free-electron laser for a wavelength optimized at a wavelength of 3 $\upmu$m, what tolerance would you allow for the energy spread of the electrons? [Hint: the accelerator is the most expensive part of a free-electron laser; an answer like energy spread should be small compared with a value that you find by analyzing laser operation is not sufficient.]

19.11

X-ray SASE FEL. Show that an electron pulse (duration 100 fs) and a pulse of radiation do not separate in an X-ray SASE FEL (wavelength 0.1 nm) at a wiggler length of 100 m.

19.12

Relativistic electron in a periodic magnetic field. We describe, in the laboratory frame, the motion of an electron moving at a relativistic velocity (along the z axis).

• (a) Which is the Lorentz force, assuming that an electric field is absent? Answer: The Lorentz force is equal to $\mathbf{F}=q\mathbf{v} \times \mathbf{B},$ where $q (= -e)$ is the electron charge.

• (b) Determine the equation of motion of an electron (mass $m_0$) moving at a relativistic velocity in a periodic magnetic field; assume that an optical field is absent and that loss of energy due to spontaneous emission of radiation is negligibly small. Answer:

  $$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}(\gamma m_0 \mathbf{v})=q\mathbf{v}\times \mathbf{B}. \end{aligned}$$

  (19.138)

  Since the electron does not lose energy, the Lorentz factor is a constant and we can write $\gamma \mathrm{d}{} \mathbf{v} /\mathrm{d}t=q\mathbf{v}\times \mathbf{B}$.

• (c) Write the equation of motion for the x component and the z component of the velocity for the case that the magnetic field is oriented along the y direction, namely $\mathbf{B}=(0, B, 0)$ and $B=B_\mathrm{w}\sin k_\mathrm {w} z$, where $k_\mathrm{w}=2\pi / \lambda _\mathrm{w}$ and $\lambda _\mathrm{w}$ is the period of the wiggler. Answer: $\gamma m_0 \mathrm{d}\mathrm {v}_x /\mathrm{d}t=q \mathrm {v}_z B$ and $\gamma m_0 \mathrm{dv}_z /\mathrm{d}t=q \mathrm {v}_x B$.

• (d) Replace in the equations of motion the term $k_\mathrm{w}z$ by the corresponding time dependent term. Answer. We suppose that $\mathrm {v}_x<< \mathrm {v}_z$ so that $z\approx \mathrm {v}_z t$ and define $\varOmega _0 =k_\mathrm{w} \mathrm {v}_z$. We find $\gamma m_0 \mathrm{d}\mathrm {v}_x /\mathrm{d}t=q\mathrm {v}_z B_\mathrm{w} \sin \varOmega _0 t$ and $\gamma m_0 \mathrm{d}\mathrm {v}_z /\mathrm{d}t = q\mathrm {v}_x B_\mathrm{w} \sin \varOmega _0 t$.

• (e) Determine the solutions. Answer:

  $$\begin{aligned} \mathrm {v}_x =\frac{qcB_\mathrm{w} \lambda _\mathrm{w}}{2\pi \gamma \mathrm{m}_{0}}\cos (\varOmega _0 t)=-\frac{K_\mathrm{w} c}{\gamma }\cos (\varOmega _0 t). \end{aligned}$$

  (19.139)

  Using this expression, we find the differential equation

  $$\begin{aligned} \frac{\mathrm{d}\mathrm {v}_z}{\mathrm{d}t} = -\frac{qK_\mathrm{w} cB_\mathrm{w}}{\gamma ^{2}m_0}\sin \varOmega _0 t\cos \varOmega _0 t=-\frac{qK_\mathrm{w} cB_\mathrm{w}}{2\gamma ^{2}m_0 }\sin 2\varOmega _0 t. \end{aligned}$$

  (19.140)

  The solution is

  $$\begin{aligned} \mathrm {v}_z =\bar{{\mathrm {v}}}_z -\frac{K_\mathrm{w}^{2} c}{4\gamma ^{2}}\cos 2 \varOmega _0 t, \end{aligned}$$

  (19.141)

  where $\bar{{\mathrm {v}}}_z =c\left( {1-\frac{1+K_\mathrm{w}^{2} /2}{\gamma ^{2}}} \right)$ is an average velocity for propagation along z and where $K_\mathrm{w} =\displaystyle \frac{e\lambda _\mathrm{w} B_\mathrm{w} }{2\pi m_{0} c}$ is the wiggler parameter. Instead of using the differential equation for $\mathrm {v}_z$, we can make use of the Lorentz factor, $\gamma =\displaystyle \frac{1}{\sqrt{1-(\mathrm {v}_x^2 +\mathrm {v}_z^2 )}}$. From this relation we find the same expression for $\mathrm {v}_z$.

• (f) Determine the effective Lorentz factor (that is the Lorentz factor related to $\bar{{\mathrm {v}}}_z )$ for $K_\mathrm{w} =1$ and $\gamma =100$.

• (g) Show that the wiggler parameter can be written as the ratio of two energies. One energy term is equal to $ec\lambda _\mathrm{w} B_\mathrm{w} /2\pi$ and the other is the rest energy $m_0 c^{2}$ of the electron; determine the value of $\lambda _\mathrm{w} B_\mathrm{w}$ for $K_\mathrm{w} =1$.

• (h) Determine the orbit of the electron. Answer. Integration of the velocity components leads to

  $$\begin{aligned} x(t)=-\frac{K_\mathrm{w} \lambda _\mathrm{w} }{2\pi \gamma }\sin \varOmega _0 t, \end{aligned}$$

  (19.142)
  $$\begin{aligned} z(t)=\bar{{z}}(t)-\frac{K_\mathrm{w}^{2} \lambda _\mathrm{w}}{8\pi \gamma ^{2}}\sin 2\varOmega _0 t, \end{aligned}$$

  (19.143)

  with $\bar{{z}}(t)=\bar{\mathrm{v}}z$. The vector $(z-\bar{{z}}, x)$ describes the form of an “eight” in the z, x plane (see Fig. 19.7). The x component oscillatees with the frequency $\varOmega _0$ and the y component with $2\varOmega _0$. The frequency $\varOmega _0 \approx k_\mathrm{w} c$ is determined by the wiggler period. It is independent of $\gamma$ and thus of the kinetic energy of the electron; $\varOmega _0 =2\times 10^{7}\mathrm{s}^{-1}$ for $\lambda _\mathrm{w} =1\mathrm{cm}$.

19.13

Lorentz factor. Relate the relativistic momentum and the relativistic energy with the Lorentz factor.

Answer. The Lorentz factor is given by

$$\begin{aligned} \gamma =\frac{1}{\sqrt{1-\mathbf{v}^{2}/c^{2}}}. \end{aligned}$$

(19.144)

The relativistic momentum is equal to

$$\begin{aligned} \mathbf{p}=\gamma m_0 \mathbf{v} \end{aligned}$$

(19.145)

and the relativistic energy

$$\begin{aligned} E=\sqrt{m_0^2 c^{2}+c^{2}\mathrm{p}^{2}}. \end{aligned}$$

(19.146)

19.14

Relativistic electron submit to both a periodic magnetic field and a high frequency electric field. We describe, in the laboratory frame, the motion of an electron that propagates at a relativistic velocity through a periodic static magnetic field and is submit to a high frequency electric field. Which is the Lorentz force?

Answer: The Lorentz force is equal to

$$\begin{aligned} \mathbf{F}=q(\mathbf{E}+\mathbf{v}\times \mathbf{B}). \end{aligned}$$

(19.147)

The electric field, which is oriented along the x direction, is given by

$$\begin{aligned} E=A\cos \omega t \end{aligned}$$

(19.148)

and the magnetic field, which is oriented along y, is equal to

$$\begin{aligned} B=B_\mathrm{w} \cos \varOmega _0 t. \end{aligned}$$

(19.149)

The Lorentz force is oriented along x and is approximately given by

$$\begin{aligned} F=q(E+cB). \end{aligned}$$

(19.150)

The frequency $\omega$ is much larger than $\varOmega _0$.

19.15

Critical field—a speculation. We speculate that a distortion of the electron orbit can occur when the amplitude of the electric force is equal to the amplitude of the magnetic force. We denote the corresponding electric field amplitude as critical field amplitude $A^{*}$.

(a) Determine the critical field amplitude $A^{*}$.

Answer.

$$\begin{aligned} A^{*}=cB_\mathrm{w} . \end{aligned}$$

(19.151)

The critical field amplitude is determined by the amplitude of the wiggler field.

In the case that $B_\mathrm{w} =1\,\mathrm{T}$, the critical field amplitude is equal to $A^{*}=3\times 10^{8}\,\mathrm{V}\,\mathrm{m}^{-1}$.

19.16

Modulation model: an estimate of the distortion energy.

We assume that the critical field amplitude $A^{*}$ plays a role for distortion of an electron orbit that we described in connection with the modulation model.

Characteristic of the modulation model is the distortion energy $E^{*}$. We now attribute the distortion energy a critical frequency $\omega ^{*}$ by the relationship

$$\begin{aligned} \hbar \omega ^{*}=E^{*}. \end{aligned}$$

(19.152)

If an electron performs the work $E^{*}$, then, stimulated emission of radiation can result in a strong distortion of the electron orbit. We will differ between two cases:

• $\omega _0 \le \omega ^{*};$ the electron oscillation frequency $\omega _0 (\approx \omega )$ is smaller or equal to the critical frequency.

• $\omega _0>>\omega ^{*};$ the electron oscillation frequency is large compared with the critical frequency (see Problem 19.17).

Here, we consider the case that the electron oscillation frequency is equal or smaller than the critical frequency, $\omega _0 \le \omega ^{*}$. We ask for the work performed by an electron submitted to the critical field using the relation

$$\begin{aligned} q\int _0^T {E\mathrm {v}_\mathrm{x} } \mathrm{d}t=-\frac{eA^{*}cK_\mathrm{w}}{\gamma ^{*}\omega ^{*}}=-E^{*}, \end{aligned}$$

(19.153)

where $\gamma ^{*}$ is the Lorentz factor that corresponds to the energy of the electrons driving a free-electron laser at the frequency $\omega ^{*}$. We find, with $A^{*}=cB_\mathrm{w}$, the distortion energy

$$\begin{aligned} E^{*}=\frac{ec^{2}B_\mathrm{w} K_\mathrm{w} }{\gamma ^{*} \omega ^{*}}. \end{aligned}$$

(19.154)

We find, from (19.152) and (19.154), using the relation between frequency and Lorentz factor,

$$\begin{aligned} \omega _0^{*} =\frac{1}{1+K_\mathrm{w}^{2} /2} \frac{4\pi c (\gamma ^{*})^{2}}{\lambda _\mathrm{w}}, \end{aligned}$$

(19.155)

the critical frequency

$$\begin{aligned} \omega ^{*}=(4\pi )^{1/5}(e/\hbar )^{2/5}cB_\mathrm{w}^{2/5} \lambda _\mathrm{w}^{-1/5} K_\mathrm{w}^{-1/5} (1+K_\mathrm{w}/2)^{1/5}. \end{aligned}$$

(19.156)

The distortion energy is equal to

$$\begin{aligned} E^{*}=\left[ {4\pi \hbar ^{3}e^{2}c^{5}\frac{B_\mathrm{w}^{2} (1+K_\mathrm{w} /2)}{\lambda _\mathrm{w} K_\mathrm{w} }} \right] ^{1/5}, \end{aligned}$$

(19.157)

where $K_\mathrm{w} =\displaystyle \frac{ecB_\mathrm{w} \lambda _\mathrm{w} }{4\pi m_0 c^{2}}$. Accordingly, the distortion energy depends on the strength of the wiggler field $B_\mathrm{w}$ and the wiggler wavelength $\lambda _\mathrm{w}$.

• (a) Determine $E^{*}$ and $\omega ^{*}$ for $B_\mathrm{w} = 1\,\mathrm{T}$ and $\lambda _\mathrm{w} =1\,\mathrm{cm};\ K_\mathrm{w} \approx 1$.

  Answer.

  $E^{*}=1\mathrm{eV}$. The calculated value of the distortion energy is comparable with the value (2 eV) that we extracted, using the modulation model, from experimental data of infrared and far infrared free-electron lasers. The critical frequency is equal to $\omega ^{*}/2\pi =2.5\times 10^{14}$ Hz.

• (b) Determine the distortion energy and the critical frequency for the limits $K_\mathrm{w}<<1$ and $K_\mathrm{w}>>1$.

• (c) Determine the coupling strength $\kappa$ that describes, in the modulation model, the coupling between the electron oscillation and the high frequency electric field. Answer. The coupling strength is equal to

  $$\begin{aligned} \kappa =\frac{\pi ^{4/5}e^{3/5}}{4^{1/5}\hbar ^{3/5}} \frac{\lambda ^{1/5}K_\mathrm{w}^{6/5} }{B_\mathrm{w}^{2/5} (1+K_\mathrm{w}/2)^{1/5}}. \end{aligned}$$

  (19.158)

  It depends on the strength of the wiggler field and on the wiggler wavelength.

• (d) Determine the distortion energy, the critical frequency, and the coupling strength in the limits $K_\mathrm{w}<<1$ and $K_\mathrm{w}>>1$.

• (e) What is the reason that many expressions imply complicated dependences on the parameters; see, for instance the dependence of the distortion energy on the wiggler field and the wiggler wavelength. Answer. Complicated dependences stem mainly from the dependence of the laser frequency on the Lorentz factor, $\omega _0 (\gamma )$.

• (f) What is the process that leads, according to the modulation model, to strong distortion of the electron orbit at saturation of the laser field?

  Answer. At saturated laser field, the work done by an electron during one period of the electron oscillation is equal to the distortion energy. During a period of the electron oscillation, one photon is generated if the electron oscillation frequency is about equal to the critical frequency, $\omega _0 \approx \omega ^{*}$. Many photons are generated by a cascade process if the electron oscillation frequency is small compared with the critical frequency. Then, the number n of photons generated in a cascade process is given by the relationship $n\omega _0 \approx \omega ^{*}$. At saturated laser field, an absorption process in a period of the electron oscillation follows a stimulated emission process occurring in the preceding period.

  Thus, the condition of saturation implies that stimulated emission and absorption processes follow each other at a period that is twice the period of the electron oscillation (= period of the laser field).

19.17

Saturation of the X-ray SASE free-electron laser. The laser frequency and thus the electron oscillation frequency of an X-ray SASE free-electron laser is much larger than the critical frequency, $\omega _0>>\omega ^{*}$. Stimulated emission of a photon is joined with a large distortion of the electron orbit. However, this distortion is repaired as long as an absorption process does not occur. Saturation does occur if the amplitude of the laser field is so strong that stimulated emission and absorption processes follow each other at a period that is twice the period of the electron oscillation.

• (a) Determine the saturation field amplitude $A_\mathrm{{sat}}$ for an X-ray SASE free-electron laser. Show that $A_\mathrm{{sat}}$ shows an $\omega$ dependence; the amplitude increases, for frequencies $\omega>>\omega ^{*},$ less strong than assumed in Sect. 19.11, equation (19.97), showing approximately a $\omega ^{3/2}$ dependence.

• (b) Which is the gain coefficient for radiation in an X-ray SASE free-electron laser? This is a yet open question. It may decrease, with increasing frequency, less strongly than discussed in Sect. 19.11, equation (19.96). If we assume that the product of the saturation field amplitude and the gain coefficient $\alpha$ is independent of frequency as at small frequencies, we would expect that $\alpha$ is proportional to $\omega ^{-1}$ rather than proportional to $\omega ^{-3/2}$.

19.18

Discuss the general equation of motion. The generals equation of motion, (19.146), takes into account that the Lorentz factor depends on time. Due to stimulated emission of radiation by an electron, the Lorentz factor decreases. We can write

$$\begin{aligned} m_0 \mathbf{v}\frac{\mathrm{d}\gamma }{\mathrm{d}t}+\gamma m_0 \frac{\mathrm{d}{} \mathbf{v}}{\mathrm{d}t}=q(\mathbf{E}+\mathbf{v}\times \mathbf{B}). \end{aligned}$$

(19.159)

This differential equation is treated in many books and is used to derive the pendulum model of the electron motion that leads to the expression (19.127) for the small-signal gain; see, for instance, [5, 166, 312].

19.19

One-electron FEL. We consider a hypothetical one-electron free-electron laser. The FEL is driven by electron pulses, each pulse containing one electron. The pulse repetition rate is equal to the round trip transit rate of the optical pulses in the laser resonator. We assume that the laser resonator shows no loss. We choose the following data. Laser wavelength $1.6\,{\mathrm \upmu \!{m}}$; $\nu = 188$ THz, $E_\mathrm{{el}} = 115$ MeV, $\gamma = 230$, $N_\mathrm {w}=100$, $L_\mathrm {w}=1\,\mathrm {m}$, $\lambda _\mathrm {w} = 1\,$cm, $B_\mathrm {w} = 1\mathrm {T}$, $K_\mathrm {w} = 1$, $E^{*} = 2\,\mathrm {eV}$, $L_{\text {res}} = 30\,\mathrm {m}$, mode radius of the near concentric resonator $r = 2\,$mm. Describe the dynamics by using the modulation model; instead of Gaussian distributions of the optical pulse we describe the pulse by a rectangular distributions of the high frequency field perpendicular to the beam and along the beam.

• (a) Estimate the power of a radiation pulse in the laser resonator.

• (b) Determine the oscillation onset time of the laser.

• (c) Discuss the dynamics of the FEL during the oscillation onset and at steady state oscillation.

• (d) Make clear that the one-electron FEL shows main features of a free-electron laser.