Velocity Compliant Bunching Scheme

with Amplitude Modulation

Yoshihisa IwashitaAdvanced Research Center for Beam

Science, Institute for Chemical Research, Kyoto

University, Gokanosho, Uji, Kyoto 611-0011, JAPAN

iwashita@kyticr.kuicr.kyoto-u.ac.jphttp://wwwal.kuicr.kyoto-u.ac.jp

— Neat Bunching & ø-E Rotator Based on Fast Amplitude Modulation

by use of Beat —

(t=0) 350 400 450 ns500

40

60

80

100

120

140

160MeV

100m downstream5ns time slicing (200MHz)

Drift-1

at Production Target(Full width = 3ns)

(t=0) 350 400 450 ns500

40

60

80

100

120

140

160MeV

100m downstream5ns time slicing (200MHz)

Drift-1

at Production Target(Full width = 3ns)

Drift-2

200m downstream10ns split (100MHz)(offsetted by -320ns)

We need this distribution!

Drift-2

200m downstream10ns split (100MHz)(offsetted by -320ns)

We need this distribution!Flipping phase space distribution for each5ns time slice allows neat bunching. But how?Flipping phase space distribution for each5ns time slice allows neat bunching. But how?

Slope changesrapidlySlope changesrapidly

•Neat Bunching (Velocity Compliant)

•Local phase space

a t100m

kic k

A t produc tion ta rge ta t200m

a t100m

kic k

A t produc tion ta rge ta t200m

τµτbτµτb

Uses ToF information. τb >2τµτb >2τµ

•3D Simulation from the Target in Solenoid

100m after the production target

0

5

10

15

20

-1 0 1 2 3 4 5 6 7 8 9 10 11

Bz [T]

Z [m]

Target: W ø10xL150Bz=16T max

Bz=15T @ target

Bz=1T after 9m

0

5

10

15

20

-1 0 1 2 3 4 5 6 7 8 9 10 11

Bz [T]

Z [m]

Target: W ø10xL150Bz=16T max

Bz=15T @ target

Bz=1T after 9m

Bz=1T, Ø30cm

Target: W ø10 x L150

100m

50GeV 100k protons

•Time Spread at 100m (PARMSOL)

99MeV<E<101MeV

~10ns

0

100

200

300

400

500

350 400 450 500 550

Mu-Mu+

ToF [ns]

t-HIST T100m 50-150MeV

0

100

200

300

400

500

350 400 450 500 550

Mu-Mu+

ToF [ns]

t-HIST T100m 50-150MeV

0

10

20

30

40

50

370 380 390 400 410 420

Mu-Mu+

ToF [ns]

t-HIST T100m 99-101MeV

0

10

20

30

40

50

370 380 390 400 410 420

Mu-Mu+

ToF [ns]

t-HIST T100m 99-101MeV

0 50 100 150 200 250 30050

100

150

Mu-Mu+

µ/2MeV

MeV

0 50 100 150 200 250 30050

100

150

Mu-Mu+

µ/2MeV

MeV

50GeV 100k protons

VelocityCompliantBuncher

100m

Drift Drift

ProductionTarget

100m

Phase Rotator

VelocityCompliantBuncher

100m

Drift Drift

ProductionTarget

100m

Phase Rotator

•1D simulation — System

Buncher Frequency ??

•1D simulation — Before the ø-E Rotator

ωbuncher = 2ωø-E

= 40MHz

τμ=10ns

buncher@100mprofile @200msingle harmonic(not sawtooth)

~30%

30

40

50

60

70

80

90

0 20 40 60 80 100 120 140

W50% yield 3nsW30% yield 3nsW50% yield 10nsW30% yield 10ns

Yield [%]

ø-E frequency [MHz]

30

40

50

60

70

80

90

0 20 40 60 80 100 120 140

W50% yield 3nsW30% yield 3nsW50% yield 10nsW30% yield 10ns

Yield [%]

ø-E frequency [MHz]

-8

-6

-4

-2

0

2

4

6

8

0.36 0.38 0.4 0.42 0.44 0.46

2/ω /dW dt2 (Sin ω )/t ω /dW dt

µs

MV

-8

-6

-4

-2

0

2

4

6

8

0.36 0.38 0.4 0.42 0.44 0.46

2/ω /dW dt2 (Sin ω )/t ω /dW dt

µs

MV

•Required Waveform for Velocity Compliant Bunching

Sawtooth is ideal.Sine wave OK.Sawtooth is ideal.Sine wave OK.

Amplitude changes rapidly.(Envelope)

V t( ) = 2sin ω

bt( )

ωb

dWμ

t( )

dt,W

μt( ) = m

μ

ct

c 2 t 2 − L 2− 1

⎛

⎝⎜

⎞

⎠⎟,

Amplitude Modulation

Amplitude changes rapidly.(Envelope)

V t( ) = 2sin ω

bt( )

ωb

dWμ

t( )

dt,W

μt( ) = m

μ

ct

c 2 t 2 − L 2− 1

⎛

⎝⎜

⎞

⎠⎟,

Amplitude Modulation

Two frequency componentscan fit the envelope.

The variation is faster thanan exponential fn.

Fitting:

Ve n v e lo p

t( ) = 2 0 sin ωet + ϕ

1( ) + A sin 2ω

et + ϕ

2( )( ) [ M V ]

Two frequency componentscan fit the envelope.

The variation is faster thanan exponential fn.

Fitting:

Ve n v e lo p

t( ) = 2 0 sin ωet + ϕ

1( ) + A sin 2ω

et + ϕ

2( )( ) [ M V ]

€

Vbuncher =Venvelope ×Vsine or

=Venvelope ×Vsawtooth

€

Vbuncher =Venvelope ×Vsine or

=Venvelope ×Vsawtooth

•Trigonometric Reduction (some math.)

V t( ) = 1 0 c o s ωe

− ωb

( ) t + ϕ1

( ) − c o s ωe

+ ωb

( ) t + ϕ1

( ){ }

+ 1 0 A c o s 2 ωe

− ωb

( ) t + ϕ2

( ) − c o s 2 ωe

+ ωb

( ) t + ϕ2

( ){ } .

.Broken into four components

V t( ) = 1 0 c o s ωe

− ωb

( ) t + ϕ1

( ) − c o s ωe

+ ωb

( ) t + ϕ1

( ){ }

+ 1 0 A c o s 2 ωe

− ωb

( ) t + ϕ2

( ) − c o s 2 ωe

+ ωb

( ) t + ϕ2

( ){ } .

.Broken into four components

V t( ) = 3 0 s i n ωb

t( ) s i n ωe

t + ϕ1

( ) + A s i n 2 ωe

t + ϕ2

( )( )

Buncher :waveform to be synthesized

EnvelopeBuncher

Trigonometric reduction makes...

V t( ) = 3 0 s i n ωb

t( ) s i n ωe

t + ϕ1

( ) + A s i n 2 ωe

t + ϕ2

( )( )

Buncher :waveform to be synthesized

EnvelopeBuncher

Trigonometric reduction makes...

• Layout (Single station @100m)

VelocityCompliantBuncher

100m

Drift Drift

ProductionTarget

100m

Phase Rotator

VelocityCompliantBuncher

100m

Drift Drift

ProductionTarget

100m

Phase Rotator

Four Frequencies, length:3m each.Cavities:

Voltage Frequency#1 15 MV 37.5 MHz#2 15 MV 45.5 MHz#3 8.7 MV 35.0 MHz#4 8.7 MV 45.0 MHz

fbuncher = 40MHz

fø-E = 20MHz

No 3D sim. result yet...But suppose the following:

-2

-1

0

1

2

740 760 780 800 820 840 860 880 900ns-2

-1

0

1

2

740 760 780 800 820 840 860 880 900ns

0

20

40

60

80

100

120

740 760 780 800 820 840 860 880 900ns0

20

40

60

80

100

120

740 760 780 800 820 840 860 880 900ns

€

Vø−E =Venvelope ×Vsquarewave

€

Vø−E =Venvelope ×Vsquarewave✕

€

Venvelope = A1Sin(ωe (t − 740ns))

+ A2Sin(2ωe (t − 740ns))

€

Venvelope = A1Sin(ωe (t − 740ns))

+ A2Sin(2ωe (t − 740ns))

€

Vsquare = B1Sin(ωe (t))

+ B2Sin(3ωe (t))

€

Vsquare = B1Sin(ωe (t))

+ B2Sin(3ωe (t))

€

Vφ−E = CiSin(ωit +ϕ i)i=1

8

∑

€

Vφ−E = CiSin(ωit +ϕ i)i=1

8

∑

• ø-E Rotaor

Before Phase Rotaor

-60-40-200204060

760 780 800 820 840 860 880 ns

200m

After Phase Rotaor

This exampleshows rough ideahow it works!

~71% @∆E/E<10%

Before Phase Rotaor

-60-40-200204060

760 780 800 820 840 860 880 ns

200m

After Phase Rotaor

This exampleshows rough ideahow it works!

~71% @∆E/E<10%

• Simple simulation results: after the Phase Rotator (CPEC)