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U.S. Particle Accelerator School January 25 – February 19, 2021

VUV and X-ray Free-Electron Lasers Introduction, Electron Motions in an
Undulator, Undulator Radiation & FEL

Dinh C. Nguyen,1 Petr Anisimov,2 Nicole Neveu1

1 SLAC National Accelerator Laboratory
2 Los Alamos National Laboratory

LA-UR-21-20610

Monday (Jan 25) Lecture Outline

Time
• VUV and X-ray FELs in the World • Properties of electromagnetic radiation • Break
10:00 – 10:30 10:30 – 10:50 10:50 – 11:00 11:00 – 11:20

11:20 – 11:40

11:40 – Noon
• Electron motions in an undulator

• Undulator radiation

• Introduction to FELs

2

VUV and X-ray FELs in the World

3

World Map of VUV and X-ray FELs

FLASH
FERMI

European XFEL

Italy
Germany

SPARC

Italy

LCLS
SwissFEL

POLFEL

Poland

LCLS-II & LCLS-II-HE

USA
Switzerland

PAL XFEL

S. Korea

SACLA

Japan

SDUV-SXFEL

SHINE

China

Blue=VUV to Soft X-ray

Purple=Soft to Hard X-ray

4

Sub-systems of an RF Linac Driven X-ray FEL

An RF-linac driven XFEL has the following sub-systems in order to produce

Low-emittance

• PHOTOINJECTOR to generate low-emittance electrons in ps bunches

electron beams

• RF LINAC to accelerate the electron beams to GeV energy • BUNCH COMPRESSORS to shorten the bunches and produce kA current

High peak current

• LASER HEATER to reduce the microbunching instabilities

• BEAM OPTICS to transport the electron beams to the undulators

Single-pass, high-

• UNDULATORS to generate and amplify the radiation in a single pass

gain X-ray FEL

• DIAGNOSTICS to characterize the electron & FEL beams

Laser

  • Photoinjector heater
  • BC2

L2

L3
Undulators
Optics
L1
L0
X-rays

electrons

5

Layout of the sub-systems of the LCLS first X-ray FEL

Linac Coherent Light Source

LCLS-II
LCLS-II-HE

Photoinjector

The last 1 km of the SLAC linac

accelerates electrons up to 15 GeV

Starting at Sector 20, electron bunches are injected at 135 MeV

Linac-to-Undulator Transport (340m)

140-m long undulator hall with 2 side-by-side undulators, Hard X-ray (HXR) and Soft X-ray (SXR)

Near Experimental Hall

Far Experimental Hall

6

RF-linac Driven FEL Pulse Format

Low-repetition-rate Mode (e.g., LCLS CuRF)

~8 ms (1/120 Hz)
~10 fs

Burst Mode (e.g., Eu-XFEL)

  • ~1 ms macropulse
  • ~1 ms macropulse

~100 ms (1/10 Hz)

~200ns

micropulses

Continuous-Wave Mode (e.g., LCLS-II/HE, SHINE)

~1 ms (1/MHz)

7

LCLS-II-HE Soft X-ray Undulator

Planar Undulators

strongback

magnetic field electron beam

magnets

푢

X-ray polarization (same direction as electron oscillations)

Undulator magnetic field varies sinusoidally with z and points in the y direction

ෝ

B = 퐵0푠푖푛 푘푢푧 풚

Planar undulators produce linearly (plane) polarized radiation at the fundamental frequency and also at harmonic frequencies.

8

Helical Superconducting Undulators

Superconducting coils are wound around

the electron beam pipe in a helical pattern.

Undulator magnetic field varies sinusoidally

with z and points in both x and y directions in a helical fashion.

  • ෝ
  • ෝ

B = 퐵0 푐표푠 푘푢푧 풙 + 푠푖푛 푘푢푧 풚
2휋

Undulator wavenumber

푘푢 =

푢

Helical undulators produce circularly

polarized radiation at the fundamental

frequency. Helical undulators do not produce undulator radiation at the harmonic frequencies.
Snapshots of the helical undulator magnetic field at different locations in one undulator period.

9

Delta Undulators (Variable Polarization)

Varying the linear positions of the magnet jaws changes radiation polarization from linear to circular

Delta undulator cross-section

10

X-ray FEL Wavelength for a Planar Undulator

Undulator period
FEL resonant wavelength

푢

2훾2

퐾2

2

Undulator parameter

 =

1 +

ꢃ퐵0

ꢃ푢퐵0

K =

=

Lorentz relativistic factor

(Dimensionless beam energy)

  • 푘푢ꢂ푒푐
  • 2휋ꢂ푒푐

Dimensionless K parameter is a measure

of how much the electron beam is

deflected transversely in the undulator

퐸푡ꢁ푡푎푙

훾 =

K = 0.934 퐵0푢

ꢂ푒푐2

퐾

휃푚푎ꢀ

=

푥

훾

B0 in tesla 푢in cm

u + 

radiation wave

휃푚푎ꢀ

푧

e- trajectory

u

11

Electron Beam Kinematics

Dimensionless electron beam energy
Electron velocity relative to the speed of light

푣
훽 =
푐

퐸푡ꢁ푡푎푙

훾 =
ꢂ푒푐2

Using the g-b relation, we can calculate b

퐸푡ꢁ푡푎푙 = 퐸ꢄ + ꢂ푒푐2

1

1

훾2

훾 =
훽 = 1 −
1 − 훽2

Rest mass energy
Kinetic energy
0.511 MeV

Approximate b for

GeV electrons

1

Approximate g for GeV electrons

훽 ≈ 1 −
2훾2

퐸푡ꢁ푡푎푙 ≈ 퐸ꢄ = 퐸푏

We shall see later the longitudinal component of the velocity, ʋz is reduced when the electrons undergo transverse oscillations in the undulator.

훾 ≈ 1957 퐸푏 퐺ꢃ푉

12

X-ray FEL Wavelength Tuning

The FEL x-ray wavelength can be tuned by one of the following methods 1. varying the electron beam energy, Eb and thus the beam g 2. varying the gap by moving the magnet jaws symmetrically in and out, thus changing the K value

2

The on-axis magnetic field amplitude depends

on the gap-to-period ratio. The remanence

  • 푔
  • 푔

퐵0 푔, 푢 = 3.13퐵푟 ꢃ푥푝 −5.08
+ 1.54

  • 푢
  • 푢

퐵

magnetic field, of NdFeB is about 1.2 tesla.

푟

Larger gap – Smaller K – Shorter 
Small gap – Large K – Long 

gap

gap e- beam e- beam

magnets poles

13

period

LCLS-II-HE Variable-Gap Soft X-ray Undulator

K

SXR K parameter vs gap (mm)

10

This example illustrates how we change the LCLS X-ray FEL

9

wavelength by varying the magnet gap of the Soft X-ray (SXR) undulator which has an undulator period of 56 mm.

87

6

5

K

The undulator K parameter decreases and photon energy

4

increases as we increase the gap (right figures), using the LCLS-II-HE 8-GeV electron beams as an example.

321

However, opening the gap to increase the photon energy also reduces the FEL output pulse energy (below).

0

7.2 9.2 11.2 13.2 15P.2h1o7t.2o1n9.2en21e.2rg23y.2 25.2 27.2 29.2 31.2
5000

3.5

4500

Photon energy (eV) vs gap (mm)

3
2.5
2

4000

3500 3000 2500 2000 1500 1000
500

Photon energy (eV)
Output pulse energy (mJ)

1.5

1

0.5
0
0
7.2 9.2 11.2 13.2 15.2 17.2 19.2 21.2 23.2 25.2 27.2 29.2 31.2

14

Photon energy (eV) gap (mm)

Properties of Electromagnetic Radiation

15

EM Radiation in Free Space

ℎ = 4.1357 ∙ 10−15 ꢃV ∙ 푠

푐 = 2.9979 ∙ 108 ꢂ/푠

  • Low frequency
  • Long wavelength

= 1.23984 ∙ 10−6ꢃV−m
 ∙ ℎ휈 = ℎ푐
1239.84 ꢃ푉 ℎ휈 =

Photon energy

λ 푛ꢂ

푥

푐

푦
푧

  • High frequency
  • Short wavelength

16

Accelerated charged particles emit EM radiation

Radiation from circular motions

Bremsstrahlung Synchrotron radiation

Radiation from oscillatory motions

17

Generations of Beam-based Radiation Sources

• X-ray Tubes

– X-ray tubes emit Bremsstrahlung and characteristic peaks

– Characteristic peaks are narrow-line atomic transitions

• Synchrotron Radiation (1st and 2nd Generation Light Sources)

– Electrons going around bends produce synchrotron radiation

– First generation SR operated as parasitic radiation devices

– Second generation SR are dedicated to radiation production

• Synchrotron Radiation (3rd Generation Light Sources)

– 3rd Gen SR have low-emittance lattice with straight sections

– Bending magnet and wiggler radiation is broadband – Undulator radiation has narrow spectral lines

• X-ray Free-Electron Lasers (4th Generation Light Sources)

– XFEL produce coherent, tunable, narrow-band x-rays – X-ray pulses are typically a few femtoseconds long – FEL peak brilliance is typically 10 orders of magnitude brighter

than third-generation SR.

18

Electric Field of a Gaussian Wave-Packet

2

Complex electric

field of a Gaussian

wave-packet

ꢈ−ꢈ

0

−

2

  • 퐸(푧, ꢅ) = 퐸0ꢃꢆ ꢄꢇ−휔푡+휓
  • ꢃ

+ c.c.

2휎
ꢈ

Phase
Amplitude

Gaussian envelope rms temporal width
Wavenumber

Angular frequency

2휋
푘 =

2휋푐
ꢉ =

Radiation intensity

1

2

  • 퐼 =
  • 퐸 푧, ꢅ

2푍0

Normalized

intensity envelope

Fast carrier waves at w

1

2

퐼0 =

퐸0

2푍0

Gaussian envelope

Impedance of free space = 120p 

19

Fourier Transform a Gaussian Pulse

2

Consider only the time-dependent part of a

푡

−

퐸0

න ꢃ−ꢆ 휔−휔 푡

  • ꢃ
  • 2 푑ꢅ

Gaussian wave-packet. Its Fourier Transform

2ꢋ

ꢊ

ℰ(ꢉ) =

ꢈ

2휋
ꢉ

is a Gaussian spectrum centered at ±

푟

2
2

휔−휔
푡

ꢊ

ꢃ−

−

퐸0

2
2

2ꢋ

  • 퐸(ꢅ) = 퐸0ꢃ−ꢆ휔 푡
  • ꢃ

2ꢋ

  • ꢍ
  • ꢊ

ℰ(ꢉ) =

ꢈ

2ꢌ휔

Fourier Transform limit

of time-bandwidth

2ꢌ푡

product (rms widths)

2

ℰ ꢉ

ꢌ휔ꢌ푡 = ½

2ꢌ휔

In general, TBW is

larger than 1/2

ꢉ푟

Time domain

Frequency domain

20

Radiation Pulse & Time-Bandwidth Product

Linear frequency
Full-width-at-half-maximum (FWHM) in
FWHM

ꢉ

  • 훿ꢅ
  • 훿휈

time and linear frequency domain

휈 =

2휋

• Time-bandwidth product for a Gaussian pulse

2

퐸 ꢅ

dꢅ

4ꢎ푛2

  • 훿휈 ∙ 훿ꢅ =
  • ꢌ휔ꢌ푡 = 0.44

훿ꢅ = 2 2ꢎ푛2 ꢌ푡

휋

• Multiply both sides by the Planck’s constant in eV-s

  • -4.0
  • -3.0
  • -2.0
  • -1.0
  • 0.0
  • 1.0
  • 2.0
  • 3.0
  • 4.0

ꢅ − ꢅ0

ℎ = 4.136 ∙ 10−15 ꢃV-s

ꢌ휔

2휋
훿휈 = 2 2ꢎ푛2 ℎ훿휈 ∙ 훿ꢅ = 1.82 ꢃ푉 ∙ 푓푠

훿휈

Energy (eV) – time FWHM product

2

ℰ 휈

훿휀 ∙ 훿ꢅ ≥ 1.82 ꢃ푉 ∙ 푓푠

  • -4.0
  • -3.0
  • -2.0
  • -1.0
  • 0.0
  • 1.0
  • 2.0
  • 3.0
  • 4.0

휈 − 휈푟

21

Wave Equation & Helmholtz Equation

1 휕2

푐2 휕ꢅ2

• Wave equation

2 −

ꢏ 풓, ꢅ = 0

ꢏ 풓, ꢅ = Re  풓 ꢃ−ꢆ휔 푡

ꢊ

• Solution to the wave equation

Time-independent wave amplitude
Time-dependent oscillatory term

• Helmholtz equation for the time-independent wave amplitude

2 + 푘2  풓 = 0
 풓 = 퐴 풓 ꢃꢆ풌·풓

22

Paraxial Approximation

Paraxial wave equation for a wave

휕

ꢒ푇 − 2푖푘 퐴 푥, 푦, 푧 = 0

propagating in the z direction

휕푧

Gaussian beam transverse amplitude (beam propagating in the z direction)

T denotes transverse spreading due to optical diffraction

  • 휕ꢒ
  • 휕ꢒ

ꢒ푇 =

+

휕푥ꢒ 휕푦ꢒ and 푘 = 푘ꢀꢒ + 푘ꢐꢒ + 푘ꢇꢒ

푘ꢀꢒ + 푘ꢐꢒ ≪ 푘ꢇꢒ

Paraxial approximation

푦

푘 = 푘

For axisymmetric Gaussian beams,

  • ꢀ
  • ꢐ

2
2

  • 푟
  • 푟

ꢃ−

ꢃꢆ ꢄꢇ−휔푡+휓 ꢇ ꢃꢆ ꢄ

ꢑ

0

2

  • ꢑ ꢇ
  • ꢒ푅 ꢇ

퐸(ꢓ, 푧) = 퐸0

ꢑ ꢇ

푥

Gouy phase shift

푤0 is the radius where the field

Radius of curvature of the Gaussian beam wavefront

decays to 1/e of E0 at the beam waist

푤 푧 is the 1/e radius at location z

23

Gaussian Beam Intensity & Diffraction

퐼0

1

• Optical intensity

ꢒ

  • 퐼 ꢓ, 푧 =
  • 퐸 ꢓ, 푧

2푍0

퐼 ꢓ

2

ꢒ푟
ꢑ
ꢒ

• Gaussian beam

ꢃ−

ꢑ

0
2

퐼(ꢓ, 푧) = 퐼0

푤0
퐼0

ꢑ

ꢃꢒ

푧ꢒ 푧Rꢒ

  • -4.0
  • -3.0
  • -2.0
  • -1.0
  • 0.0
  • 1.0
  • 2.0
  • 3.0
  • 4.0

푤 = 푤0 1 +

Beam profile at beam waist

• Gaussian beam diffracting from the beam waist

Far-field divergence half-angle

푤0

휆
휃 = =

푧푅 휋푤0

휆
휃푤0 =
휋

푧푅

24

Radiation Beam FWHM, Radius and Emittance

Gaussian beam radial FWHM

퐼0

( 푤0 = 1/e2 radius)

훿ꢓ퐹푊퐻푀 = 2ꢎ푛2 푤0

Gaussian beam angular divergence FWHM

ꢌ푟

ꢌ푟′

rms 퐼 = .606퐼0

(휃 = 1/e2 half-angle)

퐼 ꢓ

훿ꢓꢔ퐹푊퐻푀 = 2ꢎ푛2 휃

FHWM

퐼 ꢓ′

푤0

ꢌ푟 =

rms beam radius

1

ꢃꢒ

2

푤0

휃
퐼 = .135퐼0

휃

2

rms angular divergence ꢌ푟′ =

  • -4.0
  • -3.0
  • -2.0
  • -1.0
  • 0.0
  • 1.0
  • 2.0
  • 3.0
  • 4.0

휖푟 = ꢌ푟ꢌ푟′

Radial dimension (or angle)

Gaussian beam emittance

Photon beam emittance for

transversely incoherent (not

diffraction limited) radiation

휆
휆

ꢕꢒ > 1

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  • The Free Electron Laser Klystron Amplifier Concept

    The Free Electron Laser Klystron Amplifier Concept

    E.L. Saldin et al. / Proceedings of the 2004 FEL Conference, 143-146 143 THE FREE ELECTRON LASER KLYSTRON AMPLIFIER CONCEPT E.L. Saldin, E.A. Schneidmiller, and M.V. Yurkov Deutsches Elektronen-Synchrotron (DESY), Hamburg, Germany Abstract as short as 10-100 µm have never been measured and are challenging to predict. We consider optical klystron with a high gain per cas- The situation is quite different for klystron amplifier cade pass. In order to achieve high gain at short wave- scheme described in our paper. A distinguishing feature of lengths, conventional FEL amplifiers require electron beam the klystron amplifier is the absence of apparent limitations peak current of a few kA. This is achieved by applying which would prevent operation without bunch compression longitudinal compression using a magnetic chicane. In in the injector linac. As we will see, the gain per cas- the case of klystron things are quite different and gain of cade pass is proportional to the peak current and inversely klystron does not depend on the bunch compression in the proportional to the energy spread of the beam. Since the injector linac. A distinguishing feature of the klystron am- bunch length and energy spread are related to each other plifier is that maximum of gain per cascade pass at high through Liouville’s theorem, the peak current and energy beam peak current is the same as at low beam peak cur- spread cannot vary independently of each other in the injec- rent without compression. Second important feature of the tor linac. To extent that local energy spead is proportional klystron configuration is that there are no requirements on to the peak current, which is usually the case for bunch the alignment of the cascade undulators and dispersion sec- compression, the gain will be independent of the actual tions.
  • Optical Klystrons P

    Optical Klystrons P

    OPTICAL KLYSTRONS P. Elleaume To cite this version: P. Elleaume. OPTICAL KLYSTRONS. Journal de Physique Colloques, 1983, 44 (C1), pp.C1- 333-C1-352. <10.1051/jphyscol:1983127>. <jpa-00222556> HAL Id: jpa-00222556 https://hal.archives-ouvertes.fr/jpa-00222556 Submitted on 1 Jan 1983 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL,est archive for the deposit and dissemination of sci- destin´ee au d´epˆotet `ala di↵usion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publi´es ou non, lished or not. The documents may come from ´emanant des ´etablissements d’enseignement et de teaching and research institutions in France or recherche fran¸cais ou ´etrangers, des laboratoires abroad, or from public or private research centers. publics ou priv´es. JOURNAL DE PHYSIQUE Colloque Cl, supplément au n°8, Tome 44, février 1983 page Cl-333 OPTICAL KLYSTRONS P. Elleaume Département de Physico-Chimie, CEN Saolay, 91191 Gif-sur-Yvette, France Résumé : Le klystron optique est une version modifiée de l'onduleur utilisé pour améliorer le gain du Laser à Electrons Libres. L'émission spontanée et le gain sont calculés en fonction de l'énergie des électrons et de la longueur d'onde. Quelques effets limitant l'augmentation du gain sont étudiés : d i s p e r s i o n en énergie, dispersion angulaire, tailles transverses du faisceau d'électrons. Je discute brièvement l'utilisation de la modulation en densité introduite par le klystron optique sur le faisceau d'électrons pour générer du rayonnement synchrotron cohérent.
  • A New PLS-II In-Vacuum Undulator and Characterization Of

    A New PLS-II In-Vacuum Undulator and Characterization Of

    A New PLS-II In-Vacuum Undulator and Characterization of Undulator Radiation D-E. Kim, H-H. Lee, K-H. Park, H-S. Seo, T. Ha, Y-G. Jeong, H-S. Han, W.W. Lee, J-Y. Huang, S. Nam, K-R. Kim and S. Shin Pohang Accelerator Laboratory, POSTECH, Pohang, 790-784, KOREA This paper describes the result of overall studies from development to characterization of undulator radiation. After three years of upgrading, PLS-II [1, 2] has been operating successfully since 21st March 2012. During the upgrade, we developed and installed an in-vacuum undulator (IVU) that generates brilliant X-ray beam. The IVU with 3 GeV electron beam generates undulator radiation up to ~ 21 keV using 11th higher harmonic. The characterizations of the undulator radiation at an X-ray beam line in PLS-II agreed well with the simulation. Based on this performance demonstration, the in-vacuum undulator is successfully operating at PLS-II. PACS number: 07-85-Qe Keywords: Synchrotron radiation, Undulator, X-ray beamline Email: [email protected] Fax: +82-54-279-1799 1 I. INTRODUCTION Newer generation light sources with medium electron energy around 3 GeV as well as SASE-FEL are adopting in-vacuum undulator (IVU) to obtain short-wavelength and high-brilliance X-rays from undulator with short magnet period and required magnetic field. Pohang Accelerator Laboratory (PAL) [3] had developed IVU during PLS-II project. With operating experience of in-vacuum revolver undulator from SPring-8 [4] and conventional IVU from ADC [5] at the PLS [6], the new IVU with 20 mm magnetic period and 1.8 m long undulator length was developed and have been operated in PLS-II.
  • Optical Klystron Enhancement to Self Amplified Spontaneous Emission At

    Optical Klystron Enhancement to Self Amplified Spontaneous Emission At

    hv photonics Article Optical Klystron Enhancement to Self Amplified Spontaneous Emission at FERMI Giuseppe Penco 1,*, Enrico Allaria 1, Giovanni De Ninno 1,2, Eugenio Ferrari 1,3 , Luca Giannessi 1,4, Eléonore Roussel 1 and Simone Spampinati 1 1 Elettra-Sincrotrone Trieste S.C.p.A., Strada Statale 14-km 163.5 in AREA Science Park, Basovizza, 34149 Trieste, Italy; [email protected] (E.A.); [email protected] (G.D.N.); [email protected] (E.F.); [email protected] (L.G.); [email protected] (E.R.); [email protected] (S.S.) 2 Laboratory of Quantum Optics, University of Nova Gorica, 5001 Nova Gorica, Slovenia 3 Paul Scherrer Institut, 5232 Villigen, Switzerland 4 Enea, via Enrico Fermi 45, Frascati, 00044 Roma, Italy * Correspondence: [email protected] Received: 1 February 2017; Accepted: 23 February 2017; Published: 1 March 2017 Abstract: The optical klystron enhancement to a self-amplified spontaneous emission free electron laser has been studied in theory and in simulations and has been experimentally demonstrated on a single-pass high-gain free electron laser, the FERMI FEL-1, in 2014. The main concept consists of two undulators separated by a dispersive section that converts the energy modulation induced in the first undulator in density modulation, enhancing the coherent harmonic generation in the first part of the second undulator. This scheme could be replicated in a multi-stage: the bunching is enhanced after each dispersive section, consistently reducing the saturation length. We have applied the multi-stage optical klystron (OK) scheme on the FEL-2 line at FERMI, whose layout includes three dispersive sections.
  • Jørgen S. Nielsen Institute for Storage Ring Facilities (ISA) Aarhus University Denmark

    Jørgen S. Nielsen Institute for Storage Ring Facilities (ISA) Aarhus University Denmark

    Jørgen S. Nielsen Institute for Storage Ring Facilities (ISA) Aarhus University Denmark ESLS-RF 13 (30/9-1/10 2009), ASTRID2 and its RF system 1 ASTRID2 is the new synchrotron light source to be built in Århus, Denmark Dec 2008: Awarded 5.0 M€ for ◦ Construction of the synchrotron ◦ Transfer of beamlines from ASTRID1 to ASTRID2 ◦ New multipole wiggler ◦ We did apply for 5.5 M€ Cut away an undulator for a new beamline Has to be financed together with a new beamline Saved some money by changing the multipole wiggler to better match our need ESLS-RF 13 (30/9-1/10 2009), ASTRID2 and its RF system 2 ◦ Electron energy: 580 MeV ◦ Emittance: 12 nm ◦ Beam Current: 200 mA ◦ Circumference: 45.7 m ◦ 6-fold symmetry lattice: DBA with 12 combined function dipole magnets Integrated quadrupole gradient ◦ 4 straight sections for insertion devices ◦ Will use ASTRID as booster (full energy injection) Allows top-up operation ESLS-RF 13 (30/9-1/10 2009), ASTRID2 and its RF system 3 ESLS-RF 13 (30/9-1/10 2009), ASTRID2 and its RF system 4 ESLS-RF 13 (30/9-1/10 2009), ASTRID2 and its RF system 5 General parameters ASTRID2 ASTRID Energy E [GeV] 0.58 0.58 Dipole field B [T] 1.192 1.6 Circumference L [m] 45.704 40.00 Current I [mA] 200 200 Revolution time T [ns] 152.45 133.43 Length straight sections [m] ~3 Number of insertion devices 4 1 Lattice parameters Straight section dispersion [m] 0 2.7 Horizontal tune Qx 5.23 2.29 Vertical tune Qy 2.14 2.69 Horizontal chromaticity dQ x/d( ∆p/p) -6.4 -4.0 Vertical chromaticity dQ y/d( ∆p/p) -11.2 -7.1 Momentum
  • Small Signal Bi-Period Harmonic Undulator Free Electron Laser

    Small Signal Bi-Period Harmonic Undulator Free Electron Laser

    Progress In Electromagnetics Research C, Vol. 105, 217–227, 2020 Small Signal Bi-Period Harmonic Undulator Free Electron Laser Ganeswar Mishra*, Avani Sharma, and Saif M. Khan Abstract—In this paper, we discuss the spectral property of radiation of an electron moving in a bi- period harmonic undulator field with a phase between the primary undulator field and the harmonic field component. We derive the expression for the photons per second per mrad2 per 0.1% BW of the radiation. A small signal gain analysis also is discussed highlighting this feature of the radiation. A bi-period index parameter, i.e., Λ is introduced in the calculation. According to the value of the index parameter, the scheme can operate as one period or bi-period undulator. It is shown that when Λ = π, the device operates at the fundamental and the third harmonic. However, when Λ = π/2, it is possible to eliminate the third harmonic. 1. INTRODUCTION Insertion devices are the key components of the synchrotron radiation and free electron laser sources. There are interests and technology upgrades of undulator technology for synchrotron radiation source (SRS) and free electron laser (FEL) for producing tunable electromagnetic radiation over the desired electromagnetic spectrum [1–12]. In the synchrotron radiation source, the relativistic electron beam propagates in the magnetic undulator field and emits spontaneous radiation. In the free electron laser scheme, the relativistic electron beam exchanges the kinetic energy to a co-propagating radiation field at the resonant frequency. Most insertion devices are undulators designed either as pure permanent magnet or hybrid permanent magnet according to Halbach field configuration.