[Paper Review] Nonlinearity in the sequential absorption of multiple photons

Christian Buth|arXiv (Cornell University)|Dec 21, 2016

Advanced X-ray Imaging Techniques39 references3 citations

TL;DR

This paper classifies multiphoton absorption into separable, linked, and simultaneous processes, using analytical solutions of rate equations and Volterra integral equations to model sequential x-ray absorption in nitrogen atoms under intense free-electron laser pulses. It demonstrates that standard rate equations and short-pulse approximations fail to capture nonlinearity when decay processes are present, but a successive approximation of the Volterra integral equation accurately reproduces numerically exact solutions.

ABSTRACT

I classify multiphoton absorption into separable, linked, and simultaneous processes. The first and second types can be distinguished when the rate-equation approximation is valid whereas the third type refers to the case when the full description of multiphoton absorption is essential. For this purpose, rate equations are solved analytically without decay processes which shows that even if many photons are absorbed the interaction with the light field is linear and one has the case of separable multiphoton absorption. Next a short-pulse approximation is investigated in which I first solve the rate equations without decay processes and then solve only rate equations for the ensuing decay. Finally, the full rate equations are examined and a successive approximation of the underlying Volterra integral equation of the second kind is derived leading to linked multiphoton absorption by the involved decay widths. The three methods are applied to a nitrogen atom in intense and ultrafast x rays from free-electron lasers (FELs). The linearity theorem barely approximates the results in the presence of decay processes which is also not satisfactorily corrected for by the short-pulse approximation. The successive approximation gives excellent agreement with the numerically-exact solution of the rate equations.

Motivation & Objective

• To classify multiphoton absorption into separable, linked, and simultaneous processes based on the validity of the rate-equation approximation.
• To investigate the breakdown of linearity in sequential multiphoton absorption when decay processes are included.
• To develop and validate a successive approximation method for solving the underlying Volterra integral equation of the second kind.
• To compare the accuracy of rate equations, short-pulse approximations, and the successive approximation against numerically exact solutions.
• To assess the applicability of these methods to intense, ultrafast x-ray interactions in atoms such as nitrogen using free-electron lasers.

Proposed method

• Solves rate equations analytically without decay to demonstrate linearity in the absence of decay, establishing the separable multiphoton absorption regime.
• Applies a short-pulse approximation by solving rate equations without decay first, then solving only the decay equations afterward.
• Derives a successive approximation for the Volterra integral equation of the second kind to model linked multiphoton absorption via decay widths.
• Uses the full system of rate equations as the reference for numerical comparison.
• Applies all three methods to a nitrogen atom under intense, ultrafast x-ray pulses from free-electron lasers.
• Validates results by comparing the successive approximation against the numerically exact solution of the rate equations.

Experimental results

Research questions

• RQ1 How do separable, linked, and simultaneous multiphoton absorption processes differ in the context of intense x-ray pulses?
• RQ2 To what extent does the rate-equation approximation fail to capture nonlinearity when decay processes are present?
• RQ3 Can the short-pulse approximation adequately correct for decay-induced nonlinearity in multiphoton absorption?
• RQ4 Does a successive approximation of the Volterra integral equation accurately reproduce the numerically exact solution of the rate equations?
• RQ5 What is the role of decay widths in enabling linked multiphoton absorption in the presence of intense x-ray fields?

Key findings

• The rate-equation approximation alone fails to capture nonlinearity when decay processes are included, as the linearity theorem breaks down under such conditions.
• The short-pulse approximation does not satisfactorily correct for decay-induced nonlinearity and shows poor agreement with the numerically exact solution.
• The successive approximation of the Volterra integral equation of the second kind yields excellent agreement with the numerically exact solution of the rate equations.
• Even with many photons absorbed, the interaction remains linear in the absence of decay, confirming the separable multiphoton absorption regime.
• The method is generalizable to include resonance-enhanced x-ray multiple ionization (REXMI) and simultaneous two-photon absorption by extending the decay term.
• The approach can be extended to fine-structure-resolved states and heavy atoms by using matrix-vector products for sparse matrices, improving computational feasibility.