Fluorescence imaging of nonlinear x-ray propagation and lasing

We introduce a general approach for probing nonlinear x-ray propagation by imaging secondary fluorescence emitted transverse to the driving field. When a short, intense x-ray pulse excites a deep <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mrow> <a:mn>1</a:mn> <a:mi>s</a:mi> </a:mrow> </a:math> core orbital, subsequent <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mrow> <b:mi mathvariant="normal">K</b:mi> <b:mi>α</b:mi> </b:mrow> </b:math> emission from spin-orbit-split <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"> <d:mrow> <d:mn>2</d:mn> <d:mi>p</d:mi> </d:mrow> </d:math> states can undergo stimulated amplification. This nonlinear process reshapes the relative populations of the <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"> <e:mrow> <e:mn>2</e:mn> <e:msub> <e:mi>p</e:mi> <e:mrow> <e:mn>1</e:mn> <e:mo>/</e:mo> <e:mn>2</e:mn> </e:mrow> </e:msub> </e:mrow> </e:math> and <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"> <f:mrow> <f:mn>2</f:mn> <f:msub> <f:mi>p</f:mi> <f:mrow> <f:mn>3</f:mn> <f:mo>/</f:mo> <f:mn>2</f:mn> </f:mrow> </f:msub> </f:mrow> </f:math> levels along the propagation path, leaving distinct signatures in the delayed L-edge fluorescence. By solving the coupled density-matrix and Maxwell equations, we show that these fluorescence signals provide a direct and experimentally accessible probe of x-ray amplification dynamics. We demonstrate the concept for argon atoms and extend it to molecular systems containing third-row elements, where competing effects of lifetimes, transition intensities, and nonresonant absorption determine the efficiency of stimulated emission. Our results establish L-edge fluorescence as a broadly applicable diagnostic of nonlinear x-ray phenomena, opening opportunities for studying light-matter interactions in regimes where direct detection of amplified x-ray signals is technically challenging.