live BOOKLET FOR FEL DESIGN

A collection of practical formulae with interactive graphs

Marcello Artioli (1), Giuseppe Dattoli (2) and Simonetta Pagnutti (1)

(1) ENEA - Centro Ricerche Energia Bologna, Via Martiri di Monte Sole 4, Bologna, Italy
(2) ENEA - Centro Ricerche Energia Frascati, Via Enrico Fermi 45, Frascati, Rome, Italy

Dedicated to Pierluigi "Gigi" Ottaviani

Generalities on FEL and definitions

Physical constants used in the practical formulae

1-D equations and parameters

The dynamic of the system of an e-beam travelling in an undulator, along the longitudinal z-direction (see Fig. 1.1 for axes orientation), interacting with a plane e.m. wave is specified by the Lorentz’s equations of motion (Gaussian system)

Axes orientation

where:

for linearly polarized undulator:

for helically polarized undulator:

with

Halbach  configuration  undulator  parameters:

FEL equations

where

Radiation features

An approximation of the line shape useful to evaluate inhomogeneous broadening effects is (see Appendix A)

Correspondence between Gaussian, MKS and practical units

Betatron motion in an undulator focusing in both vertical directions

e–beam energy distribution

e–beam phase space distribution

No coupling between x,y planes

Matched beam

Bazarov-Sinclair formula for optimized DC-gun photo-injectors

e–beam transverse normalized rms emittance vs bunch charge q and rms bunch length

e–beam optimized longitudinal rms emittance

Pendulum equations and Colson’s dimensionless variables

where

e–beam power and saturation intensity

The FEL–saturation intensity is defined as the field intensity halving the small signal gain and is implicitly contained in the definition of Colson’s dimensionless intensity.

In Panel 1.1 FEL characterstics for linear or helical undulators (on the right side) can be calculated as a function of electron beam, seed radiation and undulators parameters (on the left side).

Design parameters

FEL gain and saturation intensity

We denote as the maximum gain and

Homogeneous broadening case

where

for ν → 0 :

A particularly useful approximation for integrals computation is:

with

relative error < 1.7% in the range  |ν|≤10

In Panel 2.1 FEL gain, saturation intensity and detuning coefficient can be calculated as a function of the small signal gain coefficient.

Gain, saturation and detuning, with no broadening effects

Energy spread corrections

For

This quantity represents the intrinsic FEL efficiency which remains, within large limits, not affected by inhomogeneous broadening effects.

In Panel 2.2 FEL maximum gain, saturation intensity and detuning coefficient can be calculated as a function of the small signal gain coefficient, taking into account the energy spread as well.

Maximum gain, saturation and detuning vs. inhomogeneous energy distribution  parameter

Emittance corrections (matched and round e-beam case)

For

In low gain regime the use of integral representation for the field amplitude allows to derive the following expression for the gain including emittance corrections:

From above equations the maximum gain as function of can be derived.

For we get the following formulas

and

The FEL efficiency remains nearly independent for values of the μ parameters below 0.4.

In Panel 2.3 FEL maximum gain, saturation intensity and detuning coefficient can be calculated as a function of the small signal gain coefficient and emittance.

Maximum gain, saturation and detuning vs. emittance parameter

Combined effects of energy spread and emittance

For and

In Panel 2.4 FEL maximum gain, saturation intensity and detuning coefficient can be calculated as a function of the small signal gain coefficient, inhomogenous energy spread parameter and emittance.

Maximum gain, saturation and detuning vs. inhomogeneous energy spread parameter

Assigning μ

Gain saturation formula

with

where

Gain versus X example

High gain amplifier and non-linear harmonic generation

Colson’s equations including harmonic generation

where  
n   =  harmonic number (odd ones for linear undulator, n = 1 only for helical undulator)
   =  dimensionless amplitude
   =  field amplitude phase
   =  dimensionless current
  corresponds to a,  φ, j of previous sections.

Fundamental harmonic

In the case of high gain amplifier devices having

a quick evaluation of the maximum gain and of the final intensity yields

Logistic function power growth (fast growing root only) without lethargy contribution (see Appendix C)

Power growth including lethargy

FEL induced relative energy spread

The total energy spread is

In Panel 3.1 the evolution of the main harmonic power, without and with lethargy correction, and energy spread can be evaluated as a function of seed power and undulator period.

Main harmonic power and energy spread evolution

Inhomogeneous broadening effects

Inhomogeneous broadening parameters

Saturated power and saturation length, eq. (3.5) and (3.6) are replaced by

If , are negligible and reduces to .The high gain power growth is reproduced by logistic type equations, upon replacing

Diffraction correction

Previously defined quantities maintain the same expressions provided that
obtaining, e.g., a function in place of χ.The saturated power results

This equation yields a slightly larger value than the one predicted by the Xie formula.A closer agreement is given by

Bunching coefficients

The bunching generated by main harmonic in the exponential power growth part is characterized by

For n ≤ 5:

As an example, in Panel 3.2, the main harmonic power and the first bunching coefficient evolution is shown as a function of seed power and undulator period.

In Panel 3.3, the main harmonic power and the first five bunching coefficients evolution is shown as a function of seed power and undulator period, with optional lethargy correction.

Main harmonic power and evolution

Main harmonic power and bunching coefficients

It can be seen that the behaviour of in the lethargic region is not well reproduced by eq. (3.23); a perfect agreement with simulation results in all region is given by the following formula:

Higher harmonics generation (linear undulator)

Higher harmonic power evolution in a high gain FEL amplifier is shown.
The first part of the evolution, indicated as , is the linear contribution, the second one, denoted by , specifies the non linear contribution induced by the bunching due to the fundamental.

where

with

The bunching coefficient evolution, in presence of subharmonic generation, is given by

In Panel 3.4, the power growth of the first three harmonics is shown as a function of the harmonics seed and the undulator period, with optional lethargy correction.

In Panel 3.5, the bunching coefficients evolution, in presence of subharmonic generation is shown as a function of the harmonics seed and the undulator period.

Power growth of the first three harmonics in a high gain FEL amplifier

Evolution of bunching coefficients in a high gain FEL amplifier

Segmented undulators

FEL operating with segmented (linear or helical) undulators in the SASE mode employ two (or more) undulators sections, tuned at the same wavelength or arranged in such a sequence that each undulator is tuned at a sub - harmonic of the preceding one.We consider negligible seeding in these subsequent sections so that the field growth is initially dominated by the electron bunching.We first deal with sections with same undulator parameters .The power growth evolution of the main harmonic, in sections after the first one, is well reproduced by

where z refers to each single section and stands for .

The subharmonics growth is reproduced by

It can be deduced that, at the beginning of each section, a length is required in order that the main harmonic regains the previous power.

In Panel 3.6, the power growth of the first three harmonics in a two-section undulator is shown as a function of the harmonics seed and the undulator period. Dispersive zones between sections are not represented.

Power growth of the first three harmonics in a two-section undulator for a high gain FEL amplifier

The behaviour of the bunching coefficient in sections after the first is described by

Let us now consider a second section tuned at a subharmonic of the preceding one.In order to optimize such a device, determined cutting points, , has to be chosen quite before saturation in the previous section. The first section parameters are

The parameters of the second sections are

The cutting point in the first section is fixed by demanding that the induced energy spread satisfies the condition . Accordingly, from eq.(3.11), it can be obtained

The growth of the field in the second section is reproduced by

where F(z) and B(z) are the functions of eq. (3.33) and (3.24) respectively with the gain length of the second section and n is the order of the harmonic in the first section amplified in the second section.
The behaviour of the bunching coefficient in second section is reproduced by

Exotic undulators

In order to enhance harmonic generation in high gain FEL lasers, non conventional undulator schemes are considered in which the on - axis field oscillates in both transverse directions or in the same direction with different periods.These types of undulators are called bi - harmonic.

Generalized bessel functions

We introduce the generalized Bessel functions which will be used in the FEL equations for bi - harmonic undulator.

1 Bi - harmonic undulator

We consider here the more interesting case of linear orthogonal polarization with

The FEL equations maintain the same structure as eq. (3.1) where now the summation index n runs over both the sets of harmonics with x and y polarization with harmonic numbers and respectively.

The Bessel function factors become

with

When d=h=3 one obtains the more simple expressions

Pulse propagation: high gain linearly polarized amplifier

Definitions and parameters

In the following we will deal with main harmonic.A gaussian form is assumed for the e - bunch longitudinal profile

with

For inhomogeneous broadening parameter χ≅1 (Sections 3.3 and 3.4) and are related to and of the continuous e-beam case by

Fundamental harmonic optical packet profile

We derive the optical packet profile from a generalization of the logistic function (3.9)

The e-bunch is modelled as a set of "slices" of equal length.The core of the SASE pulse dynamics is contained in the integral Ι(z,t) which takes into account the number of slices at position t in the e-bunch coupled by the FEL interaction after that the electrons have travelled for a distance z in the undulator.

here  Δt'   is the slice length, is the "effective" number of coupled slices,

FEL oscillators: the continuous beam case

In the following we will consider linearly polarized undulator.We will denote the cavity round trip number as r and the cavity total losses as η. Intensity refers to the undulator entry.

FEL oscillator rate equation

where is the saturation intensity accounting for high gain and inhomogeneous broadening effects (see Section 2)

Net gain

Theoretical expressions for are

and

where is the maximum gain (see Section 2).

The intra - cavity FEL induced relative energy spread is reproduced by

The equilibrium intra-cavity intensity, obtained from the condition

using eq. (6.2), results

Round trips number necessary to reach 10% of the equilibrium intracavity intensity (from eq. (6.7))

Intracavity power growth and logistic functions

Main harmonic logistic function

which results from rate equation for constant G.

Higher harmonics logistic functions: no intracavity power accumulation

FEL oscillators: pulsed regime

In the following we will consider linearly polarized undulator and negligible inhomogeneous broadening effects.

Parameters

Low gain (from supermode theory)

Maximum gain and corresponding optimum cavity detuning parameter

In the following the definition and notations of chap .6 are used.

Pulsed FEL gain

The saturation intensity is denoted by because it contains pulse propagation corrections.

Intra cavity equilibrium normalized intensity ()

Intra cavity power logistic function

Optical Pulse width before saturation at optimum cavity detuning

High gain correction

Small signal gain including cavity detuning effects

with maximum gain

and optimum cavity detuning parameter

Gain saturation formula

Intra cavity equilibrium normalized intensity

The comparison between theoretical formulas and numerical results, shown below, is performed assuming for numerical the average power of the optical pulse.It can be deduced that

For small the low gain formula (7.5) well reproduces the saturated power numerical results, while the related gain formula (7.4) does not satisfactory fit the gain numerical results which instead are in good agreement with the general formula (7.8).

Panel 7.1 shows the shape of the equilibrium intensity and net gain vs. cavity mismatch for different values of the longitudinal mode coupling parameter, small signal gain coefficient, cavity loss and number of oscillator periods.

Dimensionless equilibrium intensity and net gain vs. cavity mismatch

The optical klystron

The optical klystron (O.K.) is a FEL device characterized by two undulators with the same characteristics (identical polarization, period length and on axis field) separated by a drift section. In the following the small signal gain and the saturation intensity refer to the single undulator section.

Parameters

Small signal gain (no inhomogeneous broadening effects)

Optical klystron gain line shape (Panel 8.1)

Maximum gain

Saturation intensity (halving the maximum small signal gain)

High gain corrections (no inhomogeneous broadening effects)

Detuning corresponding to the maximum gain

where is from eq. (2.6). The range of validity of the above formulas is

In Panel 8.1 the gain-line shape (G, vs. detuning parameter), the small signal gain (), the high gain correction () and the maximum gain () calculated as a function of dispersive section parameter δ and .

Panel 8.2 shows detuning and saturation intensity vs. dispersive section parameter as a function of dispersive section parameter δ and .

Gain line-shape, maximum gain and small signal gain

Detuning and saturation intensity vs. dispersive section parameter

Gain saturation

Panel 8.3 shows the optical klystron gain saturation vs. X for different values of the small signal coefficient and the dispersive section parameter.

Optical klystron gain saturation (no inhomogeneous broadening effects)

Energy spread corrections

Inhomogeneous broadening parameters

Gain reduction due to energy spread

Optimum gain dispersive parameter

In Panel 8.4 the optical klystron maximum gain is reported vs. the broadening parameter and the dispersive section parameter for different values of the small signal gain coefficient with optional energy sèread correction.

Energy spread corrections - small signal gain

Appendix A

Convolution and inhomogeneous broadening

The effect of the inhomogeneous broadening either on the spontaneous emission or gain can be evaluated by convolving the gain or spontaneous emission line shapes on the energy and/or phase space distribution of the e-beam, thus finding e.g., in the case of the energy distribution Φ(ε) 

where the subscript i stands for inhomogeneous.The advantage of using approximations of the spontaneous line shape as given in Section 1.4 or of the gain as given in eq. (2.4) is that the integral can be derived in an explicit form

The effect of the energy spread on the spontaneous emission line shape is indeed provided by

Appendix B

The small signal limit of the Colson equation

a) Integro-differential form

b) Operatorial form

The operator is the negative derivative of order defined as

c) Third order differential equation form

The solution of the eq. (B.4) can be obtained using standard means, the derivation of the growth of the small signal FEL amplitude is rather cumbersome, because it involves the solution of a cubic equation, without any simplifying assumption.The explicit form writes

which can be exploited to derive the small signal gain and other quantities of practical interest.It is worth stressing that in the case of a pre - bunched e-beam the problem can be reduced to the same eq. (B.4) with only difference in the initial conditions, which for a bunched e-beam operation writes

Where is the bunching coefficient defined as

The high gain FEL small signal equation with the inclusion of inhomogeneous broadening terms can be derived from the integral eq. (B.1) by performing a convolution on the e-beam energy and phase space distribution (matched beam).The integral equation modifies as it follows

A useful operatorial form is

which can be exploited to treat the problem using a perturbative expansion.

Appendix C

Logistic equations

The logistic equation used in Section 3.2 to model the power growth, satisfies the following differential equations.

a=0

or

which imply a saturation mechanism of the gain depending on the quadratic power of the laser.

a≠0

The presence of a non vanishing a does not imply that we are considering a different saturation mechanism (the above form does not guarantee any saturation at large z); this factor has just been added in a phenomenological way to better reproduce the numerical data before that final saturation occurs.

Differential equation for power growth including lethargy (the prime denotes derivative with respect to z)

where A(z) is given in eq.(3.10)

Differential equation for power growth from an initially bunched beam

The previous logistic equations take into account the fast growing root only; more in general the FEL power density evolution can be written as

where a(τ) is the small signal FEL complex amplitude given in eq. (B.5), and is the dimensionless field saturated intensity defined as

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