Extreme radiation emission regime for electron beams in strong focusing ion channels and undulators

A fundamental comparison between a magnetic undulator and an ion channel, or betatron, radiation from relativistic electrons is presented. While conventional theories nominally range from the undulator (<a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>K</a:mi><a:mo>&lt;</a:mo><a:mn>1</a:mn></a:math>) to the wiggler (<c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>K</c:mi><c:mo>&gt;</c:mo><c:mn>1</c:mn></c:math>) regime, they are only applicable for sufficiently large Lorentz factors (<e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:msub><e:mi>γ</e:mi><e:mn>0</e:mn></e:msub><e:mo>≫</e:mo><e:mi>K</e:mi></e:math>). They therefore do not account for high <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>K</g:mi><g:mo>/</g:mo><g:msub><g:mi>γ</g:mi><g:mn>0</g:mn></g:msub></g:math> cases, for which we show that particle trajectories and radiation characteristics strongly deviate from the linear predictions in both magnetic undulators and ion channels. This problem arises from the fundamental differences between a magnetostatically and electrostatically induced oscillation. A reformulation of both the ion channel betatron wavelength and amplitude, as well as the same parameters in a magnetic undulator, permits us to compare cases with equivalent oscillation period and amplitude in the two different scenarios. The notable differences in spectral features of the two radiation mechanisms can then be addressed via numerical simulations of single particle as well as full beam dynamics. Additionally, we identify and quantify a novel transverse orbit precession effect in ion channels for particles with initial angular momentum relative to the device axis. This effect, which is significant in cases of strong transverse kinetic energy oscillations, alters both the radiation divergence and the beam emittance. In this paper, we present this new theoretical framework and compare its results with numerical simulation applied to realizable experimental tests of such radiation sources.