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Quantitative single shot and spatially resolved plasma wakefield - - PowerPoint PPT Presentation

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Quantitative single shot and spatially resolved plasma wakefield diagnostics Muhammad Firmansyah Kasim University of Oxford, UK PhD Supervisors: Professor Peter Norreys & Professor Philip Burrows University of Oxford, UK JAI-Fest,

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Muhammad Firmansyah Kasim University of Oxford, UK PhD Supervisors: Professor Peter Norreys & Professor Philip Burrows University of Oxford, UK JAI-Fest, Oxford, 10 December 2015

Quantitative single shot and spatially resolved plasma wakefield diagnostics

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JAI-Fest, 10 November 2015 2 University of Oxford, RAL, UCL

Acknowledgement

University of Oxford:

  • Peter Norreys
  • Philip Burrows
  • James Holloway
  • Matthew Levy
  • Naren Ratan
  • Luke Ceurvorst
  • James Sadler

AWAKE Collaboration, including:

  • Allen Caldwell
  • Matthew Wing
  • Patric Muggli
  • Edda Gschwendtner

Rutherford Appleton Laboratory:

  • Robert Bingham
  • Raoul Trines
  • Rajeev Pattathil
  • Dan Symes
  • Pete Brummitt
  • Chris Gregory
  • Steve Hawkes

OSIRIS Consortium, including:

  • Luis O. Silva
  • Jorge Vieira

Indonesian Endowment Fund for Education

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JAI-Fest, 10 November 2015 3 University of Oxford, RAL, UCL

Introduction

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JAI-Fest, 10 November 2015 4 University of Oxford, RAL, UCL

Introduction

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JAI-Fest, 10 November 2015 5 University of Oxford, RAL, UCL

Introduction

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JAI-Fest, 10 November 2015 6 University of Oxford, RAL, UCL

Introduction

𝑕 𝑧, 𝑨 =

𝑄

𝑔 𝐬, 𝑨 𝑒𝐬

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JAI-Fest, 10 November 2015 7 University of Oxford, RAL, UCL

Introduction

𝑕 𝑧, 𝑨 =

𝑄

𝑔 𝐬, 𝑨 𝑒𝐬

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JAI-Fest, 10 November 2015 8 University of Oxford, RAL, UCL

Introduction

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JAI-Fest, 10 November 2015 9 University of Oxford, RAL, UCL

Introduction

Abel transform

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JAI-Fest, 10 November 2015 10 University of Oxford, RAL, UCL

Introduction

Abel transform

Abel transform: 𝑕 𝑧, 𝑨 = 2

𝑧 ∞

𝑔(𝑠, 𝑨) 𝑠 𝑒𝑠 𝑠2 βˆ’ 𝑧2

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JAI-Fest, 10 November 2015 11 University of Oxford, RAL, UCL

Introduction

Abel transform

Abel transform: 𝑕 𝑧, 𝑨 = 2

𝑧 ∞

𝑔(𝑠, 𝑨) 𝑠 𝑒𝑠 𝑠2 βˆ’ 𝑧2 Abel inversion: 𝑔 𝑠, 𝑨 = βˆ’ 1 𝜌

𝑠 ∞ πœ–π‘•(𝑧, 𝑨)

πœ–π‘§ 𝑒𝑧 𝑧2 βˆ’ 𝑠2

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JAI-Fest, 10 November 2015 12 University of Oxford, RAL, UCL

Moving case

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JAI-Fest, 10 November 2015 13 University of Oxford, RAL, UCL

Moving case

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JAI-Fest, 10 November 2015 14 University of Oxford, RAL, UCL

Moving case

𝐝𝐩𝐭 𝜾 = π’˜/𝒗

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JAI-Fest, 10 November 2015 15 University of Oxford, RAL, UCL

Moving case

𝐝𝐩𝐭 𝜾 = π’˜/𝒗

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JAI-Fest, 10 November 2015 16 University of Oxford, RAL, UCL

Moving case

𝐝𝐩𝐭 𝜾 = π’˜/𝒗 Abel transform still works!

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JAI-Fest, 10 November 2015 17 University of Oxford, RAL, UCL

Moving case

𝐝𝐩𝐭 𝜾 = π’˜/𝒗 Abel transform still works!

How if π’˜ > 𝒗?

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JAI-Fest, 10 November 2015 18 University of Oxford, RAL, UCL

Moving case

The probe goes through different longitudinal position, 𝑨, during the interaction

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JAI-Fest, 10 November 2015 19 University of Oxford, RAL, UCL

Moving case

The probe goes through different longitudinal position, 𝑨, during the interaction Normal Abel transform does not work.

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JAI-Fest, 10 November 2015 20 University of Oxford, RAL, UCL

  • Normal Abel transformation

𝒉 𝒛, π’œ = πŸ‘

𝒛 ∞ π’ˆ 𝒔, π’œ 𝒔

π’”πŸ‘ βˆ’ π’›πŸ‘ 𝒆𝒔

Forward transform

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JAI-Fest, 10 November 2015 21 University of Oxford, RAL, UCL

  • Normal Abel transformation

𝒉 𝒛, π’œ = πŸ‘

𝒛 ∞ π’ˆ 𝒔, π’œ 𝒔

π’”πŸ‘ βˆ’ π’›πŸ‘ 𝒆𝒔

  • Modified Abel transformation

𝒉 𝒛, 𝒍 = πŸ‘

𝒛 ∞

𝐝𝐩𝐭 𝒍𝒃 π’”πŸ‘ βˆ’ π’›πŸ‘ π’ˆ 𝒔, 𝒍 𝒔 π’”πŸ‘ βˆ’ π’›πŸ‘ 𝒆𝒔

Forward transform

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JAI-Fest, 10 November 2015 22 University of Oxford, RAL, UCL

  • Normal Abel transformation

𝒉 𝒛, π’œ = πŸ‘

𝒛 ∞ π’ˆ 𝒔, π’œ 𝒔

π’”πŸ‘ βˆ’ π’›πŸ‘ 𝒆𝒔

  • Modified Abel transformation

𝒉 𝒛, 𝒍 = πŸ‘

𝒛 ∞

𝐝𝐩𝐭 𝒍𝒃 π’”πŸ‘ βˆ’ π’›πŸ‘ π’ˆ 𝒔, 𝒍 𝒔 π’”πŸ‘ βˆ’ π’›πŸ‘ 𝒆𝒔 where 𝒃 = 𝐝𝐩𝐭 𝜾 βˆ’ π’˜/𝒗 /𝐭𝐣𝐨 𝜾,

Forward transform

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JAI-Fest, 10 November 2015 23 University of Oxford, RAL, UCL

  • Normal Abel transformation

𝒉 𝒛, π’œ = πŸ‘

𝒛 ∞ π’ˆ 𝒔, π’œ 𝒔

π’”πŸ‘ βˆ’ π’›πŸ‘ 𝒆𝒔

  • Modified Abel transformation

𝒉 𝒛, 𝒍 = πŸ‘

𝒛 ∞

𝐝𝐩𝐭 𝒍𝒃 π’”πŸ‘ βˆ’ π’›πŸ‘ π’ˆ 𝒔, 𝒍 𝒔 π’”πŸ‘ βˆ’ π’›πŸ‘ 𝒆𝒔 where 𝒃 = 𝐝𝐩𝐭 𝜾 βˆ’ π’˜/𝒗 /𝐭𝐣𝐨 𝜾, 𝒉(𝒛, 𝒍) and π’ˆ(𝒔, 𝒍) are Fourier Transform of 𝒉 𝒛, π’œ and π’ˆ(𝒔, π’œ) in longitudinal direction, respectively

Forward transform

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JAI-Fest, 10 November 2015 24 University of Oxford, RAL, UCL

  • Normal Abel inversion

π’ˆ 𝒔, π’œ = βˆ’ 𝟐 𝝆

𝒔 ∞ 𝝐𝒉 𝒛, π’œ

𝝐𝒛 𝒆𝒛 π’›πŸ‘ βˆ’ π’”πŸ‘

Inverse transform

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JAI-Fest, 10 November 2015 25 University of Oxford, RAL, UCL

  • Normal Abel inversion

π’ˆ 𝒔, π’œ = βˆ’ 𝟐 𝝆

𝒔 ∞ 𝝐𝒉 𝒛, π’œ

𝝐𝒛 𝒆𝒛 π’›πŸ‘ βˆ’ π’”πŸ‘

  • Modified Abel inversion

π’ˆ 𝒔, 𝒍 = βˆ’ 𝟐 𝝆

𝒔 ∞

𝐝𝐩𝐭𝐒 𝒍𝒃 π’›πŸ‘ βˆ’ π’”πŸ‘ 𝝐 𝒉 𝒛, 𝒍 𝝐𝒛 𝒆𝒛 π’›πŸ‘ βˆ’ π’”πŸ‘

Inverse transform

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JAI-Fest, 10 November 2015 26 University of Oxford, RAL, UCL

  • Objective: diagnose electron density profile, 𝒐(𝒔, 𝜼), in the

wakefield

  • Can be done by sending the laser probe with oblique angle
  • f incidence relative to the wakefield

Application on plasma accelerators

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JAI-Fest, 10 November 2015 27 University of Oxford, RAL, UCL

  • Objective: diagnose electron density profile, 𝒐(𝒔, 𝜼), in the

wakefield

  • Can be done by sending the laser probe with oblique angle
  • f incidence relative to the wakefield
  • What can we detect?

Application on plasma accelerators

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JAI-Fest, 10 November 2015 28 University of Oxford, RAL, UCL

  • By photon ray theory in plasma wake (Wilks,

1989) 𝚬𝝏 𝝏𝟏 β‰ˆ βˆ’ 𝝏𝒒

πŸ‘

πŸ‘ππŸ

πŸ‘

𝒅 π’πŸ 𝝐𝒐 𝝐𝜼 πžπ’–

Theory of photon acceleration

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JAI-Fest, 10 November 2015 29 University of Oxford, RAL, UCL

  • By photon ray theory in plasma wake (Wilks,

1989) 𝚬𝝏 𝝏𝟏 β‰ˆ βˆ’ 𝝏𝒒

πŸ‘

πŸ‘ππŸ

πŸ‘

𝒅 π’πŸ 𝝐𝒐 𝝐𝜼 πžπ’–

  • 𝚬𝝏 and 𝝏𝟏: change in frequency and the central

frequency of the EM wave

Theory of photon acceleration

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JAI-Fest, 10 November 2015 30 University of Oxford, RAL, UCL

  • By photon ray theory in plasma wake (Wilks,

1989) 𝚬𝝏 𝝏𝟏 β‰ˆ βˆ’ 𝝏𝒒

πŸ‘

πŸ‘ππŸ

πŸ‘

𝒅 π’πŸ 𝝐𝒐 𝝐𝜼 πžπ’–

  • 𝚬𝝏 and 𝝏𝟏: change in frequency and the central

frequency of the EM wave

  • π’πŸ and 𝒐: the unperturbed and perturbed electron

density

Theory of photon acceleration

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JAI-Fest, 10 November 2015 31 University of Oxford, RAL, UCL

  • By photon ray theory in plasma wake (Wilks,

1989) 𝚬𝝏 𝝏𝟏 β‰ˆ βˆ’ 𝝏𝒒

πŸ‘

πŸ‘ππŸ

πŸ‘

𝒅 π’πŸ 𝝐𝒐 𝝐𝜼 πžπ’–

  • 𝚬𝝏 and 𝝏𝟏: change in frequency and the central

frequency of the EM wave

  • π’πŸ and 𝒐: the unperturbed and perturbed electron

density

  • 𝜼: the distance in the wakefield’s frame

Theory of photon acceleration

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JAI-Fest, 10 November 2015 32 University of Oxford, RAL, UCL

  • From photon ray theory, we get

𝒉 𝒛, 𝒍 = πŸ‘

𝒛 ∞

𝐝𝐩𝐭 𝒍𝒃 π’”πŸ‘ βˆ’ π’›πŸ‘ π’ˆ 𝒔, 𝒍 𝒔 π’”πŸ‘ βˆ’ π’›πŸ‘ 𝒆𝒔

Theory of photon acceleration

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JAI-Fest, 10 November 2015 33 University of Oxford, RAL, UCL

  • From photon ray theory, we get

𝒉 𝒛, 𝒍 = πŸ‘

𝒛 ∞

𝐝𝐩𝐭 𝒍𝒃 π’”πŸ‘ βˆ’ π’›πŸ‘ π’ˆ 𝒔, 𝒍 𝒔 π’”πŸ‘ βˆ’ π’›πŸ‘ 𝒆𝒔

  • Where

𝒉 𝒛, 𝜼 = 𝚬𝝏 𝒛, 𝜼 𝝏𝟏 π’ˆ 𝒔, 𝜼 = βˆ’ 𝟐 π’πŸ 𝝐𝒐 𝒔, 𝜼 𝝐𝜼 𝝏𝒒

πŸ‘

πŸ‘ππŸ

πŸ‘

π’˜ 𝒗 𝐭𝐣𝐨𝜾

Theory of photon acceleration

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JAI-Fest, 10 November 2015 34 University of Oxford, RAL, UCL

  • From photon ray theory, we get

𝒉 𝒛, 𝒍 = πŸ‘

𝒛 ∞

𝐝𝐩𝐭 𝒍𝒃 π’”πŸ‘ βˆ’ π’›πŸ‘ π’ˆ 𝒔, 𝒍 𝒔 π’”πŸ‘ βˆ’ π’›πŸ‘ 𝒆𝒔

  • Where

𝒉 𝒛, 𝜼 = 𝚬𝝏 𝒛, 𝜼 𝝏𝟏 π’ˆ 𝒔, 𝜼 = βˆ’ 𝟐 π’πŸ 𝝐𝒐 𝒔, 𝜼 𝝐𝜼 𝝏𝒒

πŸ‘

πŸ‘ππŸ

πŸ‘

π’˜ 𝒗 𝐭𝐣𝐨𝜾

  • 𝚬𝝏 𝒛, 𝜼 can be detected using SPIDER, S3I, FDH, or other

interferometry method.

Theory of photon acceleration

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JAI-Fest, 10 November 2015 35 University of Oxford, RAL, UCL

  • From photon ray theory, we get

𝒉 𝒛, 𝒍 = πŸ‘

𝒛 ∞

𝐝𝐩𝐭 𝒍𝒃 π’”πŸ‘ βˆ’ π’›πŸ‘ π’ˆ 𝒔, 𝒍 𝒔 π’”πŸ‘ βˆ’ π’›πŸ‘ 𝒆𝒔

  • Where

𝒉 𝒛, 𝜼 = 𝚬𝝏 𝒛, 𝜼 𝝏𝟏 π’ˆ 𝒔, 𝜼 = βˆ’ 𝟐 π’πŸ 𝝐𝒐 𝒔, 𝜼 𝝐𝜼 𝝏𝒒

πŸ‘

πŸ‘ππŸ

πŸ‘

π’˜ 𝒗 𝐭𝐣𝐨𝜾

  • 𝒐 𝒔, 𝜼 can be obtained by the modified Abel inversion.

Theory of photon acceleration

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JAI-Fest, 10 November 2015 36 University of Oxford, RAL, UCL

  • Plasma and probe parameters

– 𝒐 = 𝟐. πŸ— Γ— πŸπŸπŸπŸ˜ππ§βˆ’πŸ’ – 𝝁𝒒𝒔𝒑𝒄𝒇 = πŸ—πŸπŸ 𝐨𝐧 (plane wave) – π‰πŠ = πŸ”πŸ”. πŸ• 𝐠𝐭

  • Driver parameters (electron beam)

– π’π’π’ƒπ’š = πŸ”. πŸ“ Γ— πŸπŸπŸ‘πŸππ§βˆ’πŸ’ – 𝝉 = πŸ“. πŸ“ π›Žπ§ – 𝑭 = πŸ‘πŸ’ π‡πŸπ– – 𝜾 = πŸ‘πŸπ©

  • Repetitive boundaries

3D simulation

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JAI-Fest, 10 November 2015 37 University of Oxford, RAL, UCL

3D simulation

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JAI-Fest, 10 November 2015 38 University of Oxford, RAL, UCL

  • Transform the density profile to the frequency-

shift profile

Results from density profile

Probe’s frequency profile

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JAI-Fest, 10 November 2015 39 University of Oxford, RAL, UCL

  • Transform the density profile to the frequency-

shift profile

π’ˆ 𝒔, 𝒍 = βˆ’ 𝟐 𝝆

𝒔 ∞

𝐝𝐩𝐭𝐒 𝒍𝒃 π’›πŸ‘ βˆ’ π’”πŸ‘ 𝝐 𝒉 𝒛, 𝒍 𝝐𝒛 𝒆𝒛 π’›πŸ‘ βˆ’ π’”πŸ‘

Results from density profile

Probe’s frequency profile

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JAI-Fest, 10 November 2015 40 University of Oxford, RAL, UCL

  • Transform the density profile to the frequency-

shift profile

π’ˆ 𝒔, 𝒍 = βˆ’ 𝟐 𝝆

𝒔 ∞

𝐝𝐩𝐭𝐒 𝒍𝒃 π’›πŸ‘ βˆ’ π’”πŸ‘ 𝝐 𝒉 𝒛, 𝒍 𝝐𝒛 𝒆𝒛 π’›πŸ‘ βˆ’ π’”πŸ‘

Results from density profile

Inverse Modified Abel transform

Probe’s frequency profile Retrieved density profile

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JAI-Fest, 10 November 2015 41 University of Oxford, RAL, UCL

  • Qualitative comparison with the retrieved density profile

from the probe pulse

Results

Obtained from the electron density profile Retrieved from the probe pulse’s frequency profile

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JAI-Fest, 10 November 2015 42 University of Oxford, RAL, UCL

  • Quantitative comparison between the density data and the

measured data.

Results

Less than 15% relative error

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JAI-Fest, 10 November 2015 43 University of Oxford, RAL, UCL

  • Quantitative comparison between the density data and the

measured data.

Results

Less than 15% relative error

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JAI-Fest, 10 November 2015 44 University of Oxford, RAL, UCL

  • Quantitative comparison between the density data and the

measured data.

Results

Less than 15% relative error

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JAI-Fest, 10 November 2015 45 University of Oxford, RAL, UCL

  • Quantitative comparison between the density data and the

measured data.

Results

Less than 15% relative error diffraction

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JAI-Fest, 10 November 2015 46 University of Oxford, RAL, UCL

  • Frequency-shift profile after interacting with the

wakefield

Limitations

Just after the interaction with the wakefield

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JAI-Fest, 10 November 2015 47 University of Oxford, RAL, UCL

  • Frequency-shift profile after interacting with the

wakefield

Limitations

Propagates 12 ΞΌm

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JAI-Fest, 10 November 2015 48 University of Oxford, RAL, UCL

  • Frequency-shift profile after interacting with the

wakefield

Limitations

Propagates 30 ΞΌm

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JAI-Fest, 10 November 2015 49 University of Oxford, RAL, UCL

  • Frequency-shift profile after interacting with the

wakefield

Limitations

Propagates 30 ΞΌm Initial profile

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JAI-Fest, 10 November 2015 50 University of Oxford, RAL, UCL

  • Frequency-shift profile after interacting with the

wakefield

Limitations

Propagates 30 ΞΌm Initial profile

Lower value

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JAI-Fest, 10 November 2015 51 University of Oxford, RAL, UCL

  • As 𝜾 decreases, interaction length increases.

Limitations

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JAI-Fest, 10 November 2015 52 University of Oxford, RAL, UCL

  • As 𝜾 decreases, interaction length increases.
  • If the interaction length β‰₯ diffraction length, it

reads lower value.

Limitations

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JAI-Fest, 10 November 2015 53 University of Oxford, RAL, UCL

  • As 𝜾 decreases, interaction length increases.
  • If the interaction length β‰₯ diffraction length, it

reads lower value.

  • Diffraction limit:

𝐭𝐣𝐨 𝜾 β‰₯ 𝝁𝟏 𝝆𝒔𝒒

  • 𝝁𝟏 is the probe’s wavelength
  • 𝒔𝒒 is the wakefield’s radius

Limitations

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JAI-Fest, 10 November 2015 54 University of Oxford, RAL, UCL

  • We have developed a mathematical model for diagnostics of cylindrically

symmetric & relativistic moving objects.

  • Using photon acceleration method, we could diagnose the plasma

wakefield density quantitatively.

  • More information:
  • We are planning to try the diagnostic on laser plasma wakefield

accelerator at Rutherford Appleton Lab and on AWAKE at CERN.

Conclusions

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JAI-Fest, 10 November 2015 55 University of Oxford, RAL, UCL

THANK YOU!

Image credit to: http://sf.co.ua/13/07/wallpaper-2937024.jpg

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JAI-Fest, 10 November 2015 56 University of Oxford, RAL, UCL

  • Frequency and phase shift relation:

πš¬π” = 𝚬𝝏 πžπ’–

  • Phase modulation

πš¬π” β‰ˆ 𝝏𝟏 𝝏𝒒 𝚬𝝏 𝝏𝟏

  • Detecting πš¬π” can be done using Spectral Interferometry

method

  • Then 𝚬𝝏 can be obtained by 𝚬𝝏 = ππš¬π”/𝝐𝒖

Detecting πš¬π” and 𝚬𝝏

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JAI-Fest, 10 November 2015 57 University of Oxford, RAL, UCL

  • Uses 2 chirped pulses: (1) reference and (2) probe pulses.
  • Separated by time separation 𝝊.
  • The reference isn’t modulated, but the probe is.
  • The reference can be obtained from unmodulated part of a

single probe (self-referenced).

  • Successfully implemented in some experiments:

– Single-shot supercontinuum spectral interferometry (Kim, et al., 2002) – Frequency domain holography (Matlis, et al., 2006) – Frequency domain streak camera (Li, et al., 2010) – etc.

Spectral interferometry

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JAI-Fest, 10 November 2015 58 University of Oxford, RAL, UCL

  • If the reference electric field has the profile: 𝑭𝒔(𝒖)
  • And the probe has the profile:

𝑭𝒒 𝒖 = 𝑭𝒔 𝒖 βˆ’ 𝝊 π’‡π’‹πš¬π”(π’–βˆ’πŠ)

  • where:

– πš¬π” 𝒖 is the probe’s phase modulation

  • The spectrum of the pulses:

𝑱 𝝏 = πŸ‘ | 𝑭𝒔 𝝏 |πŸ‘ 𝟐 + 𝐝𝐩𝐭 𝚬 𝝔 𝝏 + 𝝏𝝊

  • Using FFT, we can extract 𝚬

𝝔 𝝏 , hence πš¬π”(𝒖)

Spectral interferometry

  • scillating part
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JAI-Fest, 10 November 2015 59 University of Oxford, RAL, UCL

  • The spectrum of the pulses:

𝑱 𝝏 = πŸ‘ | 𝑭𝒔 𝝏 |πŸ‘ 𝟐 + 𝐝𝐩𝐭 𝚬 𝝔 𝝏 + 𝝏𝝊

Spectral interferometry

FFT HPF IFFT

Phase

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JAI-Fest, 10 November 2015 60 University of Oxford, RAL, UCL

  • If the modulation too large β†’ the sideband of the

sideband will overlap the central element and/or the sideband gets out of range.

  • If the modulation too small β†’ the noise will spoil

the modulation information.

Spectral interferometry limitations

𝜐 πœπ‘ž

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JAI-Fest, 10 November 2015 61 University of Oxford, RAL, UCL

  • If the modulation too large β†’ adjust the delay

between two pulses, 𝝊, and/or get spectrometer with higher resolution.

  • If the modulation too small β†’ reduce the noise

and/or increase the probe’s energy.

Spectral interferometry limitations

𝜐 πœπ‘ž