lecture 3 - uspas.fnal.govuspas.fnal.gov/materials/17niu/3_chromatic_optics.pdf · lecture 3:...

Lecture 3:

Chromatic Optics

Yunhai Cai SLAC National Accelerator Laboratory

June 12, 2017

USPAS June 2017, Lisle, IL, USA

Periodic Cell: FODO

5/30/17 Yunhai Cai, SLAC 2

QF B Q

D

B

QF

QD

How to compute the Courant-Synder parameters and dispersions? For simplicity, we can use thin-lens approximation for quadrupoles, and small angle approximation for dipoles, and no gaps between any magnets. What’s the problem if we use these FODO cells to build entire ring? Why do we need to introduce sextupole magnets? How they work? Can we do better?

φ/2 φ/2

-f, -κd

2f, κf/2 2f, κf/2

Concept of Transfer Map


s1 s2

element

L

direction of propagating

segment plane

M1→2

1s

y

x

1 pypx

sz

⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

=

ℓδ

)( )( 2

s

y

x

szpypx

2

=

⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

ℓδ

A set (six) of functions of canonical coordinates. It’s called symplectic if its Jacob is symplectic.

z(s2 ) = M1→2 (z(s1)).

abbreviated map notation

Property of Symplectic Maps


⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

−

−

−

=

0110

0110

0110

0000

0000

0000

0000

0000

0000

JJ(M ) =

∂M1

∂x∂M1

∂px∂M2

∂x∂M2

∂px

∂M1

∂y∂M1

∂py∂M2

∂y∂M2

∂py

∂M1

∂δ∂M1

∂ℓ∂M2

∂δ∂M2

∂ℓ

∂M3

∂x∂M3

∂px∂M4

∂x∂M4

∂px

∂M3

∂y∂M3

∂py∂M4

∂y∂M4

∂py

∂M3

∂δ∂M3

∂ℓ∂M4

∂δ∂M4

∂ℓ

∂M5

∂x∂M5

∂px∂M6

∂x∂M6

∂px

∂M5

∂y∂M5

∂py∂M6

∂y∂M6

∂py

∂M5

∂δ∂M5

∂ℓ∂M6

∂δ∂M6

∂ℓ

⎛

⎝

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟

!J(M ) ⋅ J ⋅ J(M ) = J

Jacobian of a map: constant J matrix:

Symplectic condition:

Specifically, R-matrix is given by J(M)|x=px=y=py=d=l=0. So it is sympletic as well.

Transfer Map for Sector Bend with Small Bending Angle

M1 = x +L1+δ

(px +θδ2),

M2 = px +θδ,

M3 = y+L1+δ

py,

M4 = py,M5 = δ

M6 = ℓ+θx +L

2(1+δ)2[px

2 + py2 +θ(1+ 2δ)(px +

θδ3)],

The transfer map is given by:

where L is the length of the magnet and θ bending angle. Note that it becomes map of drift if θ=0.


Transfer Map for Thin Quadrupole, Sextupole

M1 = x,

M2 = px −xf−κ2(x2 − y2 ),

M3 = y,

M4 = py +yf+κ xy,

M5 = δ

M6 = ℓ,

Transfer map is given by a kick:

where f is the focusing (in horizontal plane) length of quadrupole and κ, is the integrated strengths of sextupole.


Concatenation of Maps


s2

Element 1

L1 s1 s3 L2

Element 2 propagating direction

z(s2 ) = M1→2 (z(s1)),z(s3 ) = M2→3 (z(s2 )).

If we have the transfer map for each individual elements:

z(s3 ) = M1→2 !M2→3 (z(s1)) ≡ M2→3 (M1→2z(s1)),

Then the transfer map for the combined elements is given by

nested functions M1→3

Courant-Snyder Parameters

⎟⎟⎠

⎞⎜⎜⎝

⎛

−−

+=

µαµµγ

µβµαµ

sincossinsinsincos

M⎟⎟⎠

⎞⎜⎜⎝

⎛

−=

µµ

µµ

cossinsincos

R

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−=

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

=−

ββαβ

ββαβ 1

0,

01

1 AA

Matrix of periodic system: Rotation matrix:

where A-1 is a transformations from physical to normalized coordinates:

We have: 1ARAM −=

A is an “ascript” and is not unique. Since two-dimensional rotational group is commutative AR(θ) is also an “ascript”. Courant and Synder choose to have A12=0.

8 Yunhai Cai, SLAC 5/30/17

Linear Optics


βx =L(1+ sin µ

2)

sinµ,βy =

L(1− sin µ2)

sinµ,

,42

sinfL

=µ

Using the transfer map of the cell and the R-matrix, we find that the betatron phase advances in both planes are the same µx=µy=µ and given by,

where L is the cell length. The beta functions at the beginning:

and the periodical dispersion:

η0 =Lϕ(1+ 1

2sin µ2)

4sin2 µ2

.

No surprises. They agree with the well-known results.

To the first-order of δ


Mη0 = Aη0 !Mcell ! A-1η0,

Make a similar transformation to obtain the feed-down effects from the dispersive orbit,

A1 = x +η0δ,A2 = px,A3 = y,A4 = py,A5 = δ,A6 = ℓ−η0p x ,

where the dispersive map is given by,

Rη0 (δ) = J[Mη0 ]≡ J(Mη0 ) |x=px=y=py=l=0Introducing a Jacobian operator, we find the matrix with dependence of δ:

Like the R-matrix, it is symplectic.

Linear Chromaticity


µx (δ) = µ − tanµ2[2− 1

4sin µ2

(12+

1

sin2 µ2

)(κ f −κd ) fLϕ −3

8sin2 µ2

(κ f +κd ) fLϕ ]δ,

µy (δ) = µ − tanµ2[2− 1

4sin µ2

(12−

1

sin2 µ2

)(κ f −κd ) fLϕ −1

8sin2 µ2

(κ f +κd ) fLϕ ]δ,

Betatron phase advances up to the first-order of δ:

where κf,κd are the integrated strengths of the sextupoles. We can set their values:

κ f =4sin2 µ

2fLϕ(1+ 1

2sin µ2),κd =

4sin2 µ2

fLϕ(1− 12sin µ2),

to cancel the linear chromaticities in both planes. The settings are expected for the local compensation to the chromatic errors by quadrupoles.

To the second-order of δ


Mη1 = Aη1 ! Aη0 !Mcell ! A-1η0 ! A

-1η1,

Make a similar transformation to obtain the feed-down effects from the dispersive orbit,

A1 = x +η1δ 2

2,

A 2 = px,A 3 = y,A 4 = py,A 5 =,A 6 = ℓ−η1p xδ,

where the new dispersive map is given by,

Rη1(δ) = J[Mη1]Using the Jacobian operator, we find the matrix with dependence of δ:

Like the R-matrix, it is symplectic.

.21φ

ηf

−=

where the first-order dispersion is found in the same way as the zeroth-order one, we have

Second-Order Beta Beating

•  half integer resonance seen

•  not good if µ >1350


βx (δ) = βx[1−δ +10−13sin2 µ

2− sin3 µ

2+3sin4 µ

22(4− 5sin2 µ

2+ sin4 µ

2)

δ 2 ],

βy (δ) = βy[1−δ +3(2−3sin2 µ

2− sin3 µ

2+ sin4 µ

2)

2(4− 5sin2 µ2+ sin4 µ

2)

δ 2 ].

The beta functions at the beginning of the cell, up to the second-order of δ:

Second-Order Chromatic Effects


.)2

sin411(2

)2

sin211(

2tan

)(,)2

sin411(2

)2

sin211(

2tan

)( 2

2

2

2

2

2

δµ

µµ

µδµδµ

µµ

µδµ−

++=

−

−−= yx

The betatron phase advances up to the second-order of δ:

Comparison to a numerical simulation in LEGO in a ring that consists of 101 900 cells.

horizontal

vertical

Transfer Map for Thin Quadrupole, Sextupole,Octupole, and Decapole

M1 = x,

M2 = px −xf−κ2(x2 − y2 )− o

6x(x2 −3y2 )− ξ

24(x4 − 6x2y2 + y4 ),

M3 = y,

M4 = py +yf+κ xy+ o

6y(3x2 − y2 )+ ξ

6xy(x2 − y2 ),

M5 = δ

M6 = ℓ,

Transfer map is given by a kick:

where f is the focusing (in horizontal plane) length of quadrupole and κ, ο, ξ are the integrated strengths of sextupole, octupole, and decapole respectively.


Cell with Nonlinear Elements


QF B Q

D

B

QF

QD

φ/2 φ/2

-f, -κd,-οd,-ξd

2f, κf/2, οf/2, ξf/2 2f, κf/2, οf/2, ξf/2

,)2

sin211(

)2

sin2

sin24(2

sin16,

)2

sin211(

)2

sin2

sin24(2

sin16

,)2

sin211(2

sin8,

)2

sin211(2

sin8

,)2

sin211(2

sin4,)2

sin211(2

sin4

533

26

533

26

322

5

322

5

22

µφ

µµµ

ξµ

φ

µµµ

ξ

µφ

µ

µφ

µ

µφ

µ

κµ

φ

µ

κ

−

++=

+

+−=

−−=

+=

−=

+=

fLfL

fLo

fLo

fLfL

df

df

df Sextupoles

Octupoles

Decapoles

Four-Order Chromatic Effects


µx (δ) = µ +tan µ

2(−352+312sin2 µ

2+ 60sin4 µ

2+ sin6 µ

2)

12(4− sin2 µ2)3

δ 4,

µy (δ) = µ +tan µ

2(992+840sin2 µ

2+84sin4 µ

2+ sin6 µ

2)

12(4− sin2 µ2)3

δ 4.

The betatron phase advances in the cell up to the fourth-order of δ:

Comparison to a numerical simulation in LEGO in a ring that consists of 101 900 cells.

horizontal

vertical

Beta Functions and Phase Advances within cell


ψx (δ, s) = tan−1[tan µ

2(1− sin µ

2) 2sL

1− 2sLsin µ2

],

ψy (δ, s) = tan−1[tan µ

2(1+ sin µ

2) 2sL

1+ 2sLsin µ2

].

The betatron phase advances up to the third-order of δ:

βx (δ, s) = βx[1−4sLsin µ2(1−

2ssin µ2

L(1+ sin µ2))](1−δ +δ 2 +δ 3),

βy (δ, s) = βy[1+4sLsin µ2(1+

2ssin µ2

L(1− sin µ2))](1−δ +δ 2 +δ 3).

The beta functions in region 0<s<L/2, up to the third-order of δ:

no δ dependence !

Mismatch Parameter


)2(21

000 γβααβγξ +−=

x

px

Machine

Beam

( , Sands )

Meaning of Mismatch Parameters


Ao−1ΣAo

−1T = ε!β − !α− !α !γ

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

Transformations to normalized coordinates:

Σ = εβ −α

−α γ

⎛

⎝⎜⎜

⎞

⎠⎟⎟

Beam sigma matrix:

Transform it to the normalized space:

Ao−1 =

1β0

0

α0

β0β0

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

⎟⎟⎠

⎞⎜⎜⎝

⎛

1001

εMatched

Mismatch parameters: ξ =

12( !β + !γ ) M = ξ + ξ 2 −1

Mismatch Parameter and Montague Functions


)]()(2)([21)( 000 δγβδααδβγδξ +−=

Applying to a chromatic mismatch, we introduce:

as a function of momentum deviation δ. Then we have its Taylor expansion:

ξ (δ) =1+ 12[(β 'β0)2 + (α '−α0

β 'β0)2 ]δ 2 +O(δ 3)

M (δ) =1+ (β 'β0)2 + (α '−α0

β 'β0)2δ +O(δ 2 )

Here we see how it related to Montague functions

A = β '

β0,B =α '−α0

β 'β0,W = A2 +B2

Higher Order Chromatic Mismatch


ξ (δ) =1+ 128[(β ''β0)2 + (α ''−α0

β ''β0)2 ]δ 2 +O(δ 3)

Higher order terms are much more complicated. But if we assume that The first order deviations β’ and α’ have been zeroed out then we have

An =β (n)

β0,Bn =α

(n) −α0β (n)

β0,Wn = An

2 +Bn2

We found that these functions are very useful to describe the high order chromatic mismatch.

Summary

•  Hamiltonian and symplectic maps are fundamental for the beam dynamics in storage rings, including the linear and chromatic optics.

•  A perfect chromatic optics can be achieved order-by-order using high-order harmonics, such as sextupoles, octupoles, and decapoles in FODO cell.


References 1)  E.D. Courant and H.S. Synder, “Theory of the Alternating-

Gradient Synchrotron,” Annals of Physics: 3 1-148 (1958). 2)  R.D. Ruth, “Single-particle dynamics in circular

accelerators,” AIP proceedings, volume 153, pages 150-235, 1987.

3)  K.L. Brown and R.V. Servranckx, “First- and second-order charged particle optics,” SLAC-PUB-3381, July 1984.

4)  Yunhai Cai, “Symplectic Maps and Chromatic Optics in Particle Accelerators,” Nucl. Instr. Meth. A 797 (2015) 172-182.

5)  M. Sands, “A Beta Mismatch Parameter,” SLAC-AP-85, April, 1991.


Acknowledgements

I would like to thank Johan Bengtsson, Alexander Chao, Etienne Forest, John Iwrin, and Yiton Yan for many stimulating and helpful conversations.

25 Yunhai Cai, SLAC 5/30/17

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