Beams, Beamlines and Beam DeliveryBeams, Beamlines and Beam Delivery

Jay Flanz, Ph.D.PTCOG, Educational Workshop

May, 2007

Beam and RayBeam and Ray• Beam is a Collection of Particles generated by a source.

• Trajectory of an individual particle in that beam is called a Ray.

• The collection of motion of the rays in a beam produce a beam thathas an overall beam size which is modified by a beam transportsystem.

• However the centroid of the beam behaves like a ray.

• Louiville’s Theorem - Conservation of phase space area in theabsence of external forces. (Extra Credit)

RayBeam Centroid

Ray Optics - Light Analogy

Beam Optics - Drift SpaceBeam Optics - Drift Space• Below is a representation of the trajectory of a ray which begins with coordinates (xo, θo)

• After traveling a along a drift length L, winds up with coordinates (xf, θf).

•The drift length is a region which applies no external forces to the beam, therefore one canwrite the equation which relates the final coordinates to the initial coordinates.

• In this case the transverse displacement of the ray will change according to the angle it’strajectory makes with the reference axis.

•Since there are no outside forces acting on the particle, the transverse momentum of the raycannot change and it’s angle remains constant. A matrix representation of that equation canalso be written

xo

xf

L

θo

=

o

o

f

f xLxϑϑ 10

1xf = xo + Lθo

θf = θo

Beam Optics - FocusingBeam Optics - FocusingThin Focusing Element

•Below is a representation of the trajectory of a ray which begins with coordinates (xo, θo)

• After traveling a through a focusing element with focal length f, winds up with coordinates (xf, θf).

• Note that the drift lengths before and after the focusing element are not considered here.

• In this case, if we consider the coordinates of the ray immediately before entering theinfinitesimally thin lens compared to the coordinates immediately after we know that the effect of thefocusing force was to change the angle of the beam.

• Since no drift length was traversed there has been no opportunity for the transverse offset to change.

• One can now write the equation which relates the final coordinates to the initial coordinate.

xo xfθf

−

=

o

xff

x of

ϑϑ 1/101xf=xo

θf = -(1/f) xo + θo

Beam Optics - MomentumBeam Optics - Momentum• Thus far we have considered a 2x2 matrix, or offset and transverse momentum of aparticle.

• It is also important to consider the “longitudinal” momentum of a particle and understandit’s effect a particle.

• We define δ=∆p/p, as the fractional deviation of a particle’s momentum (p) from areference momentum.

• Such a reference momentum can be the magnetic field set in a dipole that is required tobend a particle of momentum p a certain desired angle.

• Rays of different momentum will bend different amounts. The transfer matrix for awedge (Lengthless) dipole can be written as:

−=

100

sin1sin001

)/()/()/()/()/()/()/()/()/(

ϑρϑ

δδϑδδδϑϑϑϑδϑ

xx

xxxx

(x/δ) = Dispersion and (θ/δ) = Angular Dispersion

Ray Beam Optics – Building BlocksRay Beam Optics – Building Blocks

L1 L2In this case the final transverse offset x is independent of the initial angleθo. In this case (x/θ)=0. For the particular case above this implies that

L1+L2-(L1L2/f)=0, or f = L1L2/(L1+L2).

In this case the final angle θf is independent of the initial angle θo. Inthis case (θ/θ)=0. For the particular case above this implies that

1-(L1/f)=0, or f = L1.

Point-to-parallel focusingPoint-to-parallel focusing

Point-to-point focusingPoint-to-point focusing

In this case the transverse offset xf is independent of the initial transverseoffset xo. In this case (x/x)=0. For the particular case above this impliesthat 1-(L2/f)=0, or f = L2.

Parallel-to-point focusingParallel-to-point focusing

e.g. Drift + Focusing + Drift

MagneticsMagnetics: Quadrupoles: Quadrupoles

NS

SN

B

x

F = q v x B

v

B

F

MagneticsMagnetics: Quadrupoles: Quadrupoles

NS

SN

B

x

F = q v x B

v

B

F

Achromaticity

(x/δ) = (θ/δ) = 0

(x/δ)≠0

A Dipole behaves like a prism. Higher energy beams arebent less, therefore after a dipole, there is a correlationbetween the position and/or angle with the Ray Momentum.

Achromaticity

(x/δ) = (θ/δ) = 0

Slit

(x/δ)≠0

A Dipole behaves like a prism. Higher energy beams arebent less, therefore after a dipole, there is a correlationbetween the position and/or angle with the Ray Momentum.

Gaussian Folding in EnergyAnalysis ????

Beam Size = √ xo2 + (x/δ)2* dP/P

The animations above graphically illustrate the convolution of two rectangle functions (left)and two Gaussians (right). In the plots, the green curve shows the convolution of the blueand red curves as a function of , the position indicated by the vertical green line. Thegray region indicates the product as a function of , so its area as a function of isprecisely the convolution. mathworld.wolfram.com/Convolution.html

BeamBeam• A beam is a collection of many particles all of whose longitudinal andtransverse momenta are relatively close enough to be transported througha beam transport system and remain more or less close to each other in allcoordinates.

• One characterizes the transverse properties of a beam by plotting thephase space diagram below in which the transverse particle position andtransverse momentum of each particle in the beam is plotted.

x

θ

Distribution of Particles inDistribution of Particles in“Phase Space”“Phase Space”

x

θ

x

θ

•f(x) = 1/[ (2π)σ] * exp(-½[(x-µ)/σ]2)

UniformGaussian

Phase Space RepresentationPhase Space Representation

γxx2 + 2αxxx’ + βxx’2 = εx,full

εx,rms = (⟨x2⟩⟨x’2⟩ − ⟨xx’⟩2)½

Phase space Area = πε (mm•mrad)

Due to the mechanism by which beams are produced,usually the particle distribution can be represented as agaussian distribution. Therefore, in two dimensions,the particle number density can be represented as:

)( 22

),( βθγρϑρ +−= xoex

It can be seen from this equation that the locus of constantparticle distribution is = Constant. Note thatthis is the equation of an ellipse, and this is the form normallyused for the outline of the phase space. Actually this is theoutline inside which is contained 1/e (~65% of the particles).The area of this ellipse is παβ, otherwise written as πε where εis the ‘emittance’ of the beam. Note that when an emittance isquoted, it is important to ask what fraction of particles areincluded within the ellipse, it is not always 1/e. This isespecially important when aperture restrictions are an issue.

22 βθγ +x

a

b Area = πab

Evolution of Beam Phase SpaceEvolution of Beam Phase Space

x

θ

x

θ

Drift

Upright Ellipse = Waist(no correlations)

Cyclotron Extracted Beam Profiles (x)X Profiles and Fits

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-25 -20 -15 -10 -5 0 5 10 15 20 25

X [mm]

1 #REF! 2 #REF! 3 #REF! 4 #REF! 5 #REF! 6 #REF!

0.000

10.000

20.000

30.000

40.000

50.000

60.000

0.700 0.800 0.900 1.000 1.100 1.200 1.300 1.400 1.5001 / focal lenght

Squa

re o

f be

am s

ize

(mm2

)

Y Square of Size Y Fit

Waist fit plot

0

2

4

6

8

10

12

14

16

18

20

22

0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6Focal lenght

Squa

re o

f si

ze

X beam squared X beam fit

Beam Size Plotted against Quad Strength

What happens to the beam fromWhat happens to the beam fromthe last magnet to Isocenter?the last magnet to Isocenter?

Gantry/Beamline Dipole

Isocenter

Beam Size, Beam Size, 3 m drift,3 m drift, From Gantry to From Gantry to IsocenterIsocenterEmit = 18mm mrad, sigma = 3mm Emit = 18, sigma = 9mm

Emit = 5, sigma = 3mm Emit = 5, sigma = 9mm

ISOC

ENTER

ISOC

ENTER

ISOC

ENTER

ISOC

ENTER

Typical Cyclotron Degraded Beam

Typical Synchrotron Beam

X2DIP ~ X2

ISO + L2*θ2ISO

Gantry Dipole Power/Weight Including Scanning

0

2

4

6

8

10

12

14

0 2 4 6 8 10 12

1 sigma Isocenter Beam Size

Dip

ole

Pow

erEffect of Beam Size on Gantry Dipole and PowerEffect of Beam Size on Gantry Dipole and Power

Gantry Dipole Power / Weight

0

2

4

6

8

10

12

0 5 10 151 s igm a Isocenter Beam Size

Dip

ole

Pow

er/W

eigh

t

PS Current ~ Gap (2.1 sigma)

Magnet weight ~ gap 2

Power ~ Current 2 ~ gap 2

Smaller Emittance

Larger Emittance

For Larger Emittance Beams, thepower and weight requirementsincrease a factor of 3 when reducingthe beam size from 6mm to 3mm

Reducing emittance means reducingthe current intensity.

Even smaller optical beam size isnecessary when considering scatter.

Multiple ScatteringMultiple Scattering

θf = √ θi2 + θMS

2

θ

x

θ

x

θfθiScatterer

Beamline Example:Beamline Example:

Beamline Tuning / SteeringBeamline Tuning / Steering

ESS

VH

H

V

V

H H

V

T8BS

IS

ECU/BTCU

TCU

NS

SN

Adjust or Predictable ?

Beam Delivery: Double ScatteringBeam Delivery: Double ScatteringFrom a Beam Point of view:From a Beam Point of view:

Scatterer Scatterer

BeamPost

θ

x

θ

x

θfθi

θ

x

θ

x?

Step 1

Step 2Step 3

Step 1

Step 2 Step 3

Step 1 - First ScattererX - X’ X - Y

Step 2 - Drift and PostX - X’ X - Y

Step 3 - Second Scatterer and Drift

Beam Delivery: ScanningBeam Delivery: Scanning• Simpler ?

– No beam modifying devices ???• More Difficult ?

– Control the beam - no safety net– Many issues to consider

•• Beam ShapeBeam Shape•• Beam SizeBeam Size•• Beam TrajectoryBeam Trajectory•• Materials in the beam pathMaterials in the beam path Scanning relies on the

SUPERPOSITION ofUnscattered beams

Beam Shape:Beam Shape: Superimposing Gaussians

• Gaussians have a nice property when adding themtogether in a Voxel type manner.

Beam Shape:Beam Shape: Monte Carlo Studies ofAberration Effects and Correction

Don’t assumeDon’t assumebeam is alwaysbeam is always

GaussianGaussian??

Beam Shape: Beam Shape: How to IMPLEMENT aHow to IMPLEMENT asmall small penumbrapenumbra at the edge and conform to at the edge and conform to

the desired dose distribution inside the target?the desired dose distribution inside the target?Use of Beam Edge in Scanning?Use of Beam Edge in Scanning?

• There are so many disadvantages to a small beam– Tolerances (e.g. 1mm/40cm = .25%)

– Time for Scanning– Safety/Dosimetry– Hard to Make

• What are the Requirements?– Small Beam ?– Or sharp edge ?

• Is a sharp edge the same as a small beam?

A Sharp Edged non-A Sharp Edged non-gaussiangaussian beam beam

• This is a sharp Edged beam.• Disadvantages in a type of scanning.

– Tighter Tolerances when adding them together in a Voxel type scanningScheme Of in the line direction of continuous scanning

This has less sensitivity to periodic addition,but is still sharper

Phase space representationPhase space representation of an Aperture

θθ

x

NoParticlesGetThroughHere

NoParticlesGetThroughHere

θθ

x

NoParticlesGetThroughHere

NoParticlesGetThroughHere

One way to make a beamedge sharper is by use ofan aperture.

Beam Shape:Beam Shape: Apertures

θ

x

θ

x

θi

Hard to keep an edgeHard to keep an edgeInitial Rectangular Beam 2.5 m Drift in Vacuum

4.5 cm of Water 18cm of Water

DriftDrift

Multiple ScatteringMultiple Scattering

Penumbra

Beam Size/Shape:Beam Size/Shape:How can one achieve a sharp edge beamHow can one achieve a sharp edge beamWithout a collimator right at the patient?Without a collimator right at the patient?

• With a selectable effective driftselectable effective drift one can alsocontrol the ‘penumbra’ which could be useful inmatching.

Upstream CollimatorGaussianBeam

Focussing Elements

Rectangular Beam

PSI Gantry 2PSI Gantry 2• The Ultimate IMPT Device?

– Or The Ultimate IMPT R&DDevice?

– Upstream Instrumentationminimized (1)

– Vacuum System (1 & 2)– Beam size ~ 3mm (1 & 2)– Moderate Field Size

• 20x PPS (1), 20cmx12cm (2)– Infinite SAD (1 &2)

• (Adjustable?)– EDGE CONTROL !!! (2)

Beam Size:Beam Size:Materials in the Nozzle Beam PathMaterials in the Nozzle Beam Path

• Instrumentation (Two points make a straightline; (x, px, y, py) - One point implies you haveto be confident that the angle is right !(Commisioning/QA)

• Gas (Air, Vacuum (+ Vacuum Window), Helium)• Windows

– Gantry Dipole– Helium Chamber Windows– End of one Vacuum System

• Range Shifter/Ridge Filter

Field Size vs. Beam SizeField Size vs. Beam Sizedue to Window Thickness

0

5

10

15

20

25

30

35

40

45

50

0 5 10 15 20 25 30 35 40 45

1/ Angle ~ Better

Field Size ~ Better

Windowthicknessdepends uponminimum foildimension.

Foildimension isdetermined byfield size.

AB

B

CD

egree of Betterness

Min Field Size Dimension

CME Credits !

Homework:Homework:•• What are the requirements for the beamWhat are the requirements for the beam

size/shape for scanning? (e.g. Tradeoffs)size/shape for scanning? (e.g. Tradeoffs)•• What time structure are which instrumentsWhat time structure are which instruments

capable of monitoring?capable of monitoring?•• What will it take for ‘industry’, to implementWhat will it take for ‘industry’, to implement

the ‘innovations’ that we have been discussing?the ‘innovations’ that we have been discussing?•• Now that you’ve heard this discussion, is itNow that you’ve heard this discussion, is it

harder or easier to make a beam, than youharder or easier to make a beam, than youthought it was?thought it was?

End

PTCOG 46

Beams

Flanz