Fully Coherent X-ray Pulses from a Regenerative Amplifier Free Electron Laser Page: 4 of 11
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the frequency yields
En+1(t) = dt'rummeC wr(t-t')_e6U(t-t/)2g(t')En(t'). (4)
Since there is no initial seed signal, E1(t) = 0, and
E2 (t) = dt,ru-m e (t-t/), a2(t-t/) oE1(t') (5)
is the spectrally filtered SASE from the first pass that seeds the second pass.
For n 1, we look for an exponentially growing solution
E,(t) = A-A(t)e- rt. (6)
Eq. (4) is then transformed to an integral equation:
AA(t) = dt'K(t, t')A(t'), (7)
with the kernel
K(t, t') = rume,_2t t_)g(t') .(8)
Since both r and g(t') may be complex, K(t, t') is in general not a hermitian kernel.
We expect that a Gaussian fundamental mode will have the largest gain A0l, i.e.,
Ao(t) = exp - . (9)
Here uo- is the rms pulse duration of the returning filtered radiation. Inserting Eq. (9) into
Eq. (7), we obtain
2 22ma (2nt2
Ao exp - 4u2 ) gor exp - ,t (10)
4uxo j1 + 42 22 1 + 4n ga
where Oxa = _o(ow/ 2uo + oT is the rms x-ray pulse duration at the undulator end (see
Eq. (14)). The self-consistent solution of Eq. (10) is
2 _1+ 4ao2(,a 1+ 8(2 UT+ 1
oz= 4uI2 8u1,
2 _ 1 +8O2 _
Oxa= 8u2 - (11)
A gm 4 a
Ao -gor1 + 4o2Oxa
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Huang, Zhirong & Ruth, Ronald D. Fully Coherent X-ray Pulses from a Regenerative Amplifier Free Electron Laser, report, February 17, 2006; [Menlo Park, California]. (digital.library.unt.edu/ark:/67531/metadc876282/m1/4/: accessed December 15, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.