Light Sources I
Takashi TANAKARIKEN SPring-8 Center

Outline• Introduction• Fundamentals of Light and SR• Overview of SR Light Source• Characteristics of SR (1)• Characteristics of SR (2)• Practical Knowledge on SR

Lecture I?

Lecture II?

Outline• Introduction• Fundamentals of Light and SR• Overview of SR Light Source• Characteristics of SR (1)• Characteristics of SR (2)• Practical Knowledge on SR

Overview of SR Facility
SR Light Source(Undulator)

Accelerators

ExperimentalHall4

What’s the Advantage of SR?
Brilliance vs. wavelengthfor various light sourcesLight
Total PowerSource(W)Sun1026
X-ray Tube 103
Light Bulb 102
SR103~104
Radiation power
Comprehensive understandingofforvarious light sources
SR (and light source) is required
for efficient experiments.

5Topics in This Lecture (1)• Fundamentals of Light and SR– General description of light– Why we need SR?
– Physical quantity of light– Uncertainty of light: Fourier anddiffraction limits– SR: Light from a moving electron• Overview of SR Light Source– Types of light sources– Magnet configuration

6

Topics in This Lecture (2)• Characteristics of SR– Radiation from bending magnets– Electron Trajectory in insertion devices– Radiation from insertion devices• Practical Knowledge on SR– Finite emittance and energy spread– Heat load and photon flux
– Evaluation of optical properties of SR– Definition of undulators and wigglers– Numerical examples
7

Outline• Introduction• Fundamentals of Light and SR– General description of light– Why we need SR?
– Physical quantity of light– Uncertainty of light: Fourier anddiffraction limits– SR: Light from a moving electron

What is Light?
What is light? It is a kind of wave, but…
Surf
Water

VariousWaves

RippleWater

String Phone
StringSound
Air

What propagates the light wave?
Electric and Magnetic Field

9Electric Field+++++++e--

The E-field is generated byelectric charges, and gives aforce on a charged particle.
10Magnetic Fielde-

The M-field is generated by moving
electric charges, and give a force ona moving charged particle.

11

Electro- and Magnetostatic Phenomena
Thunder(charge in cloud)Van de Graaff
Generator

Motions of electrons (charges)
Electrostaticare stationary or steady.
We do not observe any
Electromagnet
radiation in such a steady state.
PermanentMagnet

Magnetostatic 12

Thought Experiment3x108me-Electric Field Linee+e+e+

Does e+ moves simultaneously? If so,
the information on e- propagateswith an infinite speed!
13Thought Experiment3x108me-e+e+e+

Information on e- (Efield) propagates withthe speed of light!
14

E-Field Line Is Not “Rigid”

e-e+e+e+

Electric field line can
propagate as a wave
Source of Light

15

Light as an Electromagnetic Wave

𝑬𝑯

𝑯 (magnetic vector)

𝑬 (electric vector)𝜆 (wavelength)

Electric and magnetic
fields can propagate
as a transverse wave

𝑯(𝒓, 𝑡) = 𝒛� × 𝑬(𝒓, 𝑡)/𝜇0 𝑐
Analysis of 𝑬(𝒓, 𝑡) gives all
the information on light.

𝒛 (axis of propagation)

16

Light as a Photon
Light is not only an electromagnetic
wave but also a particle, or a photon.
EM wave with awavelength of 0.1nm
𝜆 =0.1nm (1Å)

𝐸𝑝𝑝𝑝𝑝𝑝𝑝 = ℏ𝜔 = ℎ𝑐/𝜆Photons with anenergy of 12.4 keV

17

Wave? Photon?
EM Wave

PhotonPhotoelectric EffectInterference

𝐸𝑝𝑝𝑝𝑝𝑝𝑝 = ℎ𝑐/𝜆Evaluation of SR
characteristics(brilliance, flux, …)

Polarization

Formulation of SR withclassical electrodynamics(field amplitude)18

Outline• Introduction• Fundamentals of Light and SR– General description of light– Why we need SR?
– Physical quantity of light– Uncertainty of light: Fourier anddiffraction limits– SR: Light from a moving electron

Observation with Light
Light Source(Bulb)

ComputerSR

Analyzer(Brain)

High-quality light source foran accurate observation!
Reflection

Diffraction

Object(flower)

Detector(eye)Protein Crystal

ＣＣＤ Camera

20Which Quality is Better?
Specs of SPring-8- E = 8GeV- I = 100mA- L = 1500m

1mW Laser (pointer)

100W Bulb

Lighting equipment in a room: Bulb
Pointer during a presentation: Laser

Depends on the Object!
21

How to Define the Quality of Light?(1)ObjectLight Source Quality:
Light Source
How efficiently these
conditions are satisfied?
Detector

Illumination& Interaction

Detection &Analysis
22How to Define the Quality of Light?(2)Important Features of the Light Source
Item

ObjectFlower Protein

Radiation Power◎○Source Size×△△◎◎◎Directivity
Monochromaticity

Why?
# Emitted
Photons
IlluminatedArea
Accuracy ofAnalysisBrilliance23

What is Brilliance ?
Brilliance(photons/sec/mm2/mrad2/0.1%B.W.)
Total Power~Source Size x Angular Divergence x Band Width• Brilliance specifies the quality of lightfor observation of microscopic objects.
• The brilliance of a light source with ahigh total power is not necessarily
high.
24

Example of Brilliance Estimation
Item

Bulb

Laser Pointer

Total Power (W)

100

10-3

Angular Div. (mrad2) 4πx106 1
Source Size: (mm2)1021Bandwidth: (%)

1000.01

Brilliance(photons/sec/....)~108~1016

Laser is the best light source toobserve the microscopic object!
25

X ray as a Probe• Definition (not unique)– Electromagnetic wave (= light) with l of
10 nm(10-8 m) ~ 0.1Å(10-11 m)• Properties– High Energy/Photon– High Penetration (Roentgen etc..)• Application to Microscopic Objects– X-ray Diffraction– Fluorescent X-ray Analysis• No Practical Lasers!! (until recently)

Synchrotron Radation (SR)26

Outline• Introduction• Fundamentals of Light and SR– General description of light– Why we need SR?
– Physical quantity of light– Uncertainty of light: Fourier anddiffraction limits– SR: Light from a moving electron

Phase Space Representation
-x'1

z

x'1

x'x'1

x1
x'1x2x-x'1
Real Spacex1x2x-x'1
Phase Space28

Photon Propagation in Phase Spacezx'
Beam
Envelopexx

Liouville’s Theorem:
Area of the phase ellipse(beam emittance) is constant.
29

zx'

Beam Size

Beam
Envelope
Source Size

Beam Sizex

AngularDivergence

Photon Propagation in Phase Spacex

Beam Size
Beam Size

Source Size
30

Energy-Time Phase Space
E=ℏ𝜔

E=ℏ𝜔

Dispersivemedium

Pulse Lengthtt

Band
Width

Pulse compressionscheme (optical laser)

31

6D Phase Space & Brilliancex’,y’,ω

Motion of each photon can be
described in a space spanned by
the 6 coordinates (phase space)
x,y,t
Photon distribution function in
the 6D phase space gives full
information on light source′′

Typical form of the photon distribution

Central Freq.

𝑛 𝑥, 𝑥 , 𝑦, 𝑦 , 𝑡, 𝜔 =𝑥2
𝑦2
𝑥𝑥2
𝑦𝑦2𝑡2
(𝜔 − 𝜔𝑐 )2
𝐵0 exp −−−−−−
222222𝜎𝑥2𝜎𝑦2𝜎𝑥𝑥
2𝜎𝑦𝑦2𝜎𝑡2𝜎𝜔 2Brilliance

Source Size

AngularDivergence

Pulse
LengthBandWidth32

Photon Flux & Radiation Power𝑛 𝑥, 𝑥 ′ , 𝑦, 𝑦 ′ , 𝑡, 𝜔
Photon Density in 6D Phase Space
Integrateover (𝑥′, 𝑦𝑦)

𝐹 𝑡, 𝜔
Total Flux

Integrate over (𝑥, 𝑦)

𝑓 𝑥 ′ , 𝑦 ′ , 𝑡, 𝜔Angular Flux Density

Integrateover 𝜔

Integrateover 𝜔

𝑝 𝑥 ′ , 𝑦𝑦, 𝑡Angular Power Density
𝑃 𝑡
Total Power

Integrateover (𝑥′, 𝑦𝑦)

33

To Be More Specific…

34

To Be More Specific…
AngularAngular
BrilliancePower Density
Flux Density

y' n x=y=0,y' n ω=ωcy' nx'x,yx,yx'
Spectrum ofFlux (Density)SlitExperimental Setup forLight Source Measurementx'

Profilex’,y’
Monitorωx’,y’
Monochromator
35

Outline• Introduction• Fundamentals of Light and SR– General description of light– Why we need SR?
– Physical quantity of light– Uncertainty of light: Fourier anddiffraction limits– SR: Light from a moving electron

Uncertainty of Lightx’,y’,ω

(x’2,y’2,ω2)(x1,y1,t1)(x2,y2,t2)

Light as a photon can be
fully described by the 6
coordinate variables.
x,y,t∆x’, ∆y’, ∆ω(x’1,y’1,ω1)∆x,∆y,∆t

Because of the wave nature,they have some uncertainty
characterized by the so-called
Fourier transform.

37

Monochromatic & Plane Wave
λMonochromatic & Plane Wave
 Purely sinusoidal
 Constant amplitude

x'Electric-field wave

38

In Reality…
• A monochromatic and plane wave isan (ideal) form of a wave.
• In practice, a wave is composed ofmany ideal waves having different λand x’.
• Fourier Transform is a mathematicaloperation to decompose an arbitrary
wave into a number ofmonochromatic and plane waves.
39Composition of Monochromatic Waves
E(t)

ω=0

ω=ω0ω=2ω0

ω=3ω0

ω=4ω0tE(t)t40

ご連絡ありがとうございました。それでは人事公募の内容が確定次第 原稿を作成し、お送りします。遅くとも月末までには準備できると思 います。

Time & Frequency Domains
1.00.5

Fourier
Transform
0.5E(t)Amplitude

1.00.0Inv. Fourier
Transform

-0.5

0.0
-0.4

-0.2

0.0

t/T00 1 2 3 4 5 6 7 8 9 100.4
FourierFrequency/ω0Transform Pair

0.2

Time Domain: waveform asa function of t

Frequency Domain: wave
amplitude as a function of ω 41

How Does the Waveform Changes?
1.01.011ω7ω5ω3ω00 19ω
00040ω0∆ω-0.5When ∆ω increases, ∆t decreases,
and vice versa
0.0E(t)0.0

Amplitude

0.5∆t-0.4-0.2

0.0

t/T00.20.4

0.501020304050

Frequency/ω042

Uncertainty Relation• ∆t and ∆ω are regarded as uncertainty
of a photon– ∆t: longitudinal position (pulse width)– ∆ω: photon energy (bandwidth)
1.0

1.0

E(t)

Amplitude

0.5

0.0

∆ω∆t Uncertainty relation:∆𝑡 × ∆𝜔 = const. ~10.5

0.0

-0.5
-0.4

-0.2

0.0

t/T0

0.2

0.4

0102030Frequency/ω040

5043

Composition of Plane Waves
x𝐸 𝑥 = 𝐸0f0E(x)xx'𝜆/𝑥𝑥𝐸 𝑥 = 𝐸0 cos 𝑘𝑥 𝑥
2𝜋𝑥 ′=E(x)𝜆Wavenumber
Vector44

Composition of Plane Waves
xx
E(x)

E(x)

kx=0
kx=k0
kx=2k0kx=3k0kx=4k045

Spatial & Angular Domains
1.00.5

Fourier
Transform
0.5E(x)Amplitude

1.00.0Inv. Fourier
Transform

-0.5

0.0

-0.4

-0.2

0.0x

Fourier
0 1 2 3 4 5 6 7 8 9 10
0.4
k/k0
Transform Pair0.2

Spatial Domain: waveform
as a function of x

Angular Domain: wave
amplitude as a function of k 46

Fourier Transform Examples
𝐹(𝑡) Fourier Transform𝐹� 𝜔 = � 𝐹(𝑡)𝑒 𝑖𝜔𝑡 𝑑𝑑

Boxcar Function
Product of “widths”“Sinc” Function
sin 𝜔Δ𝑡/2≡ sinc(𝜔Δ𝑡/2)
transform
1⁄∆𝑡 of; Fourier𝑡 ≤ ∆𝑡/2
𝜔Δ𝑡/2
0; 𝑡 > ∆𝑡/2
�pair (𝑭 and 𝑭) is aconstant∆tGaussian Function
1𝑡2
exp −2𝜎𝑡 22𝜋𝜎𝑡σt∆ω=2π/∆t

Gaussian Function
𝜔2 𝜎𝑡 2exp −2σω=1/σt47

Coherent Gaussian Beam

2.35σtIn an ideal case, 𝝈𝒕 𝝈𝝎 and 𝝈𝒙 𝝈𝒙𝒙reach minimum values defined
Diffraction
by the Fourier transform theory.
Coherent Gaussian Beam

2.35σx’

2.35σxSpectralDecomposition
2.35σω

48

Fourier and Diffraction LimitsFourierLimitConditiona) temporal
1𝜎𝑡 𝜎𝜔 ≥2Diffraction b) spatial
𝜆𝜎≥𝜎Limit𝑥 𝑥𝑥
4𝜋𝜆𝜎𝑦 𝜎𝑦𝑦 ≥ 4𝜋

Extreme Cases
𝜎𝑡 = 0White (or Pulse)𝜎𝜔 = ∞ Light𝜎𝑡 = ∞ Monochromatic𝜎𝜔 = 0 Light
𝜎𝑥 = ∞ Parallel Light
𝜎𝑥𝑥 = 0𝜎𝑥 ~ 𝜆Minimum Focal
𝜎𝑥𝑥 ~ 1 Size• If equality a) holds, the light is:– Temporally Coherent or Fourier Limited• If equality b) holds, the light is:– Spatially Coherent or Diffraction Limited 49

Outline• Introduction• Fundamentals of Light and SR– General description of light– Why we need SR?
– Physical quantity of light– Uncertainty of light: Fourier anddiffraction limits– SR: Light from a moving electron

SR: Light from a Moving Electron• Unlike the ordinary light source (sun,light bulb,... ), the light emitter of SR(electron) is ultra-relativistic.
• The characteristics of SR is thus quite
different because of relativistic effects.
• What we have to take care is:
1.Speed-of-light limit
2.Squeezing of light pulse
3.Conversion of the emission angles

51

Speed-of-Light Limit
Within the framework of relativity, the velocityof any object never exceeds the speed of light.
Energy
1MeV
10MeV
100MeV
8GeV

β0.941
0.99880.999987
0.999999998:Lorentz Factor(relative electron energy, mc2=0.511MeV)
52

Squeezing of Light Pulse Duration
e-𝒄𝜟𝒕𝒗𝜟𝒕(𝒄 − 𝒗)𝜟𝒕e-(𝑐−𝑣)Δ𝑡
1Pulse Duration Ratio ==1−β ~ 2𝑐Δ𝑡2𝛾

53More Generally…

P1P2Time Squeezing

γ>>1,θ~054

Photons Emitted by a Moving Electrone-Rest Electron

e-Moving Electron (𝑣~𝑐)55

Conversion of Emission Angles
rest frame
θ’how to convert?

lab. frame
θβIsotropic emission
from a rest electron

γ-1

Emission from anelectron with β=0.95

Light emitted from a moving object(β~1) concentrates within γ-156

SR from a High-Energy ElectronHigh Energy Electron

Large Time
Squeezing

Radiation in theForward Direction

Short Wavelength

High Directivity
57