Undulator Field Integral Measurements Page: 2 of 9
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measurements lab, but typically with larger signals. Extra care must be taken in the present system
design to make these low level measurements.
We begin the note with a discussion of the field integrals and their importance. We then discuss
how to measure the field integrals with a coil. The measurement system is described. Estimates
are made of the signal levels and noise. The required number of turns on the coil is determined. A
second coil system for use in the tunnel is discussed.
2 Discussion Of The Field Integrals
The field integrals determine the overall effect of the undulator on the electron beam. Let x be the
horizontal position of an electron, y the vertical position, and z the position along the undulator.
The equations of motion for the electron can be expressed as follows4
" = - q By (1)
y = q Bx (2)
The prime indicates a derivative with respect to z. In these equations, q is the electron charge, 'y is
the Lorentz factor, m is the electron rest mass, vz is the velocity along the undulator, and Bx and
By are the horizontal and vertical magnetic field components, respectively.
The horizontal and vertical slopes of the trajectories are found by integrating these equations
along z. At the initial point z = zo, we take the initial slopes x'(zo) and y'(zo) to be zero. The
slopes for arbitrary z are given as follows.
'(z) = zx"(zi) dz = - q jBy(zi) dzi (3)
Jz 7'mvz Jz
y'(z) = ]y"(zi) dz = qmv Bx(zi) dzi (4)
Note that the slope depends on the first integral of the magnetic field. The exit slope from the
undulator is found by integrating through the undulator. If the region of interest, which includes
the undulator, has length L, the exit slope is given by
=' 'Zn="(zi1) dz1 = - J By (z1) dz1 (5)
z -Imvz z
zoa+L q zo-rL
yi y"i(zi) dzi = qBx(z1) dz1 (6)
These equations explain why the first field integrals should be small, so that the electrons receive a
minimal transverse kick as they go through the undulator.
To find the horizontal and vertical positions of the electrons, the slopes are integrated again.
The initial positions x(zo) and y(zo) are taken to be zero.
x(z) = j '(z2)lz2=- B (zi)dz dz2 (7)
y(z) = j y'(Z2) dz2 = q 1:1:2 B x(zi) dz dz2 (8)
Jz W nmuz Jzo Jzo
4Z. Wolf, "Introduction To LCLS Undulator Tuning", LCLS-TN-04-7, June 2004.
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Wolf, Zachary. Undulator Field Integral Measurements, report, December 7, 2010; United States. (digital.library.unt.edu/ark:/67531/metadc836025/m1/2/: accessed April 27, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.