Formulation of energy-dependent pathlength variation for a two Double Bend Achromat dogleg

Recently, the generation of ultrashort x-ray free-electron laser (XFEL) pulses of less than 1 fs has been reported using additional electron bunch compression in front of undulators. In this bunch compression, a self-modulated energy chirp provided by space charge force is used. In the present paper, we discuss an optimal bunch compression scheme using a two-Double Bend Achromat (DBA) dogleg and an electron bunch having the energy chirp which is reversed with respect to that used for a conventional magnetic chicane bunch compressor. After giving analytical expressions of <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mrow> <a:msub> <a:mi>R</a:mi> <a:mn>56</a:mn> </a:msub> </a:mrow> </a:math> and <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mrow> <c:msub> <c:mi>T</c:mi> <c:mn>566</c:mn> </c:msub> </c:mrow> </c:math> of an ordinary DBA section, a method for evaluating <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mrow> <e:msub> <e:mi>R</e:mi> <e:mn>56</e:mn> </e:msub> </e:mrow> </e:math> and <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:mrow> <g:msub> <g:mi>T</g:mi> <g:mn>566</g:mn> </g:msub> </g:mrow> </g:math> is presented for varied bending angle cases to achieve additional bunch compression. When changing the bending angles for controlling <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:mrow> <i:msub> <i:mi>R</i:mi> <i:mn>56</i:mn> </i:msub> </i:mrow> </i:math> and <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:mrow> <k:msub> <k:mi>T</k:mi> <k:mn>566</k:mn> </k:msub> </k:mrow> </k:math> , the electron beam no longer passes the magnetic center of focusing quadrupole magnets of the DBA section, but the electron beam trajectory and transverse envelope remain the same downstream of the two-DBA dogleg, including the undulators. The analytical framework proposed in this paper would be useful to design and finely tune the bunch compression in front of the undulators to generate ultrashort XFEL pulses.