Evaluation of Transfer Matrix of Plasma Ramp with Squared Cosine Shape
Analyzing the transfer matrix of a plasma ramp with a squared cosine shape using an approximate solution of Mathieu differential equation. The evaluation includes optimization of emittance preservation, theoretical plasma focusing strength, and analytical and numerical solutions. Various methods like Hill's equation are utilized to assess the transfer matrix.
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Evaluation Evaluation of the transfer transfer matrix of a of aplasma ramp plasma ramp with squared cosine squared cosine shape via via an an approximate solution approximate solution of of Mathieu Mathieu differential differential equation equation Stefano Romeo Stefano Romeo stefano.romeo@lnf.infn.it stefano.romeo@lnf.infn.it Evaluation Evaluation of of of matrix matrix the the the transfer transfer matrix of a of aplasma ramp plasma ramp with squared cosine squared cosine shape via via an an approximate solution approximate solution of of Mathieu differential Mathieu differential equation equation with with shape shape Island of Elba 2023-09-18 6th European Advanced Accelerator Concepts Workshop
Outline: Outline: Plasma transport line optimization Emittance preservation inside plasma is difficult, since in order to balance the extreme focusing field inside plasma the beam needs to be focused up to few micrometers/sub micrometer scale 4 2 ?? ?? Guide ?0= Vacuum ? Capillary 2? ?? ?0= High sensitivity to transverse instability! Plasma channel Electrode A PLASMA RAMP CAN HELP TO RELAX MATCHING CONDITIONS!!! Courtesy of A. Biagioni Evaluation of ramp transfer matrix Analytical solution: Find a general rule for a suitable shape At the edge of discharge capillaries, regions with a gradually decreasing plasma density form, connecting the plasma plateau to the vacuum Which plasma ramp? Numerical solution: For any working point perform several simulation scans These regions are commonly referred to as plasma ramps Stefano Romeo stefano.romeo@lnf.infn.it 6th European Advanced Accelerator Concepts Workshop Island of Elba 2023-09-18
Theory: Theory: Plasma focusing strength +[? ? ]2 2?(?) 2 2 ? (?) + 2???? ? ? ? = Envelope equation ? ? Neglecting constants, the focusing force inside a plasma ramp depends only from the plasma density, with the energy taken as a parameter Ion column model (only the particles inside the bubble) 2(?) 2? ? =?? 2 ???? Plasma focusing strength? ?2 ??(?) ? 2 ???? ? = = ?(?;?) 2?0???2 2? ?(?) = ?? Matching conditions on the plateau ? ? = 0 ? ? = ? ? = 0 1 ? ? Stefano Romeo stefano.romeo@lnf.infn.it 6th European Advanced Accelerator Concepts Workshop Island of Elba 2023-09-18
Method: Method: Hill s Equation ? ? + ? ? ? ? = 0 Hill s Equation ?20 ?0+?20 ? 20 ?0+? 20 ? 0 ? 0 ?0 ? 0 ? 0 ? 0 = 0 Liouville s Theorem = ?0 ?0 1 = ?(?) is a continuous and derivable function Negligible deceleration ?0 ?0 2 ???? ? = ? ? ??(?) = 0 ?0 ?0 0 1 ?0 The computation is performed over the inverse matrix in order to have simplified matching conditions ?(?) ?(?) ?(?) ?2 ?? ? 2 ?2 2?? ?? + ? ? 2? ? = ?? ? 2 Stefano Romeo stefano.romeo@lnf.infn.it 6th European Advanced Accelerator Concepts Workshop Island of Elba 2023-09-18
Framework: Framework: Squared Cosine plasma ramp If ? ? is even Hill s equation becomes a symmetric differential equation Solutions are continuous and derivable One solution is even, the other is odd THE CHOICE ? ?? ?cos2?? ? ? = 2? Realistic ramp No discontinuity at any derivative order Partially concave, partially convex Never adiabatic Ramps with an even functional form guarantee the possibility to respect matching conditions 1 ??(?) ?? ? = 4?3/2(?) We can normalize the solutions such as ? ? = ? and ? ? = ? Stefano Romeo stefano.romeo@lnf.infn.it 6th European Advanced Accelerator Concepts Workshop Island of Elba 2023-09-18
Expansion: Expansion: Mathieu Equation Mathieu Equation Hill s Equation 2?2 ?2?0 1 ?0 2cos2?? ? ? + ? ? + ?(?) = 0 21 + cos(2?) ?(?) = 0 2? Even and odd normalized Mathieu Functions 2?2 ?2?0 2?2 ?2?0 ?2 2?2 ?2?0 2?2 ?2?0 ?2 2,?? 2? 2,?? 2? ??? 2, ??? 2, ?2?0 ?2 ?2?0 ?2?0 ?(?) =2? ?(?) = ? ?2 ??? 2, 2,0 ??? 2, 2,0 ?2?0 Stefano Romeo stefano.romeo@lnf.infn.it 6th European Advanced Accelerator Concepts Workshop Island of Elba 2023-09-18
Simplify: Simplify: Normalized transfer matrix Normalization Extraction ? Geometric factor ? = ??0 ?2 ?2 ?1/?0 ?1 ?1?0 ?2/?0 ?2 ?2?0 2?? ?? + ? ? 2? ? ??? 2?2, ?2,? ???2?2, ?2,0 ??? 2?2, ?2,? ??? 2?2, ?2,0 ??? 2?2, ?2,? ???2?2, ?2,0 ??? 2?2, ?2,? ??? 2?2, ?2,0 = ?? ? 2 ?? ? 2 ? 2 2 ??= ? = ?? Normalized transfer matrices 2 2 ??= = 2? Injection ?0 ? ?= ? ? 1 ?2/?0 ?2 ?2?0 ?1/?0 ?1 ?1?0 ? 2 ? ? ? 2 ?2 ?? ?2 2 2?? ?? + ? ? 2?? ?0= 2 2? = ? 2 2 ? ?= ? = Stefano Romeo stefano.romeo@lnf.infn.it 6th European Advanced Accelerator Concepts Workshop Island of Elba 2023-09-18
Approximation: Approximation: Behavior of Mathieu functions Sinusoidal behavior Phase delay between Mathieu functions and their first derivative The product ?? ? ? Transfer matrix is unimodular (Liouville s theorem) ?+ ? ? ? ?+ ?? ? saturates for ? ? ?? ? ??cos 2? + ? ?? ? ??sin 2? + ? 1 cos2? sin2?? ?sin 2? ? 1 cos2? sin2?? ?cos 2? ? ? ? 1 ? ? 1 ? ? Stefano Romeo stefano.romeo@lnf.infn.it 6th European Advanced Accelerator Concepts Workshop Island of Elba 2023-09-18
Alignment: Alignment: Evaluation of the constants 1 4cos 2? +? ?? 3 ? 2 ? =? 8 1 2+ ? ? 2 2+ ?? 2 ? ? 1 4sin 2? +? ?? = ? ? cos2? sin2? ?? 3 ? 8 2 3 ? 1 4sin 2? ? ? ? 8 1 2 2+1 2 2+ ?? 2= ??? ?? 2 3 ? 1 4cos 2? ? 2+ ? ? 2 ? ? ? ? 8 ? = ? =? ? ? Stefano Romeo stefano.romeo@lnf.infn.it 6th European Advanced Accelerator Concepts Workshop Island of Elba 2023-09-18
Check: Check: Numerical vs. Analytical Approximation Stefano Romeo stefano.romeo@lnf.infn.it 6th European Advanced Accelerator Concepts Workshop Island of Elba 2023-09-18
Application: Application: Stabilizing effect of the ramp Transfer matrix ????/?0 ???? ?????0 ? ? ? = ? Mehrling-Floettmann equation for emittance growth 1 + ?2 ? ??,???=??,??? + ? 2 Mehrling, T., et al. "Transverse emittance growth in staged laser-wakefield acceleration." Physical Review Special Topics-Accelerators and Beams 15.11 (2012): 111303. Stefano Romeo stefano.romeo@lnf.infn.it 6th European Advanced Accelerator Concepts Workshop Island of Elba 2023-09-18
Perspective: Perspective: Experimental ramps Plasma ramps with squared cosine shape were actually realized at the plasma lab of SPARC_LAB facility by means of tapered capillaries The measured shapes were replied in numerical simulations by means of segmented lines and tested in comparison with analytical squared cosine shape The results are consistent with the expectations Stefano Romeo stefano.romeo@lnf.infn.it 6th European Advanced Accelerator Concepts Workshop Island of Elba 2023-09-18
Conclusion: Conclusion: Summary We opted for a purely analytical approach trying to evaluate the transfer matrix of a plasma ramp We opted for a purely analytical approach trying to evaluate the transfer matrix of a plasma ramp We concluded that a ramp with a symmetric form could simplify the problem We concluded that a ramp with a symmetric form could simplify the problem We opted for a ramp with a squared cosine shape We opted for a ramp with a squared cosine shape We evaluated that this ramp is never adiabatic and its application lead to Mathieu Equation We evaluated that this ramp is never adiabatic and its application lead to Mathieu Equation We compared the solution of the Mathieu Equation with actual envelope evolution We compared the solution of the Mathieu Equation with actual envelope evolution We deduced that the ramp transfer matrix can be normalized to a geometric parameter We deduced that the ramp transfer matrix can be normalized to a geometric parameter We found an analytical approximation that allows to write the transfer matrix in a simple way We found an analytical approximation that allows to write the transfer matrix in a simple way Tons of applications on their way! Tons of applications on their way! Stefano Romeo stefano.romeo@lnf.infn.it 6th European Advanced Accelerator Concepts Workshop Island of Elba 2023-09-18
Thank you for for Thank you for for the attention! the attention! the attention! the attention! Stefano Romeo Stefano Romeo stefano.romeo@lnf.infn.it stefano.romeo@lnf.infn.it Island of Elba 2023-09-18 6th European Advanced Accelerator Concepts Workshop
Very special thanks to those who collaborated with me in the realization of this work realization of this work Very special thanks to those who collaborated with me in the Angelo Biagioni Lucio Crincoli Alessio Del Dotto Massimo Ferrario Anna Giribono Gianmarco Parise Andrea Renato Rossi Gilles Jacopo Silvi Cristina Vaccarezza Cristina Vaccarezza Angelo Biagioni Lucio Crincoli Alessio Del Dotto Massimo Ferrario Anna Giribono Gianmarco Parise Andrea Renato Rossi Gilles Jacopo Silvi Stefano Romeo Stefano Romeo stefano.romeo@lnf.infn.it stefano.romeo@lnf.infn.it Island of Elba 2023-09-18 6th European Advanced Accelerator Concepts Workshop
