David Shekhtman, Muhammad Ammar Mustafa, and Nicholaus Joseph Parziale, "Two-photon cross-section calculations for krypton in the 190–220 nm range," Appl. Opt. 59, 10826-10837 (2020)
This paper presents multi-path, two-photon excitation cross-section calculations for krypton, using first-order perturbation theory. For evaluation of the two-photon-transition matrix element, this paper formulates the two-photon cross-section calculation as a matrix mechanics problem. From a finite basis of states, consisting of $4\!p$, $5s$, $6s$, $7s$, $5\!p$, $6\!p$, $4d$, $5d$, and $6d$ orbitals, electric dipole matrix elements are constructed, and a Green’s function is expressed as a truncated, spectral expansion of solutions, satisfying the Schrödinger equation. Electric dipole matrix elements are evaluated via tabulated oscillator strengths, and where those are unavailable, quantum-defect theory is used. The relative magnitudes of two-photon cross-sections for eight krypton lines in the 190–220 nm range are compared to experimental excitation spectra with good agreement. This work provides fundamental physical understanding of the Kr atom, which adds to experimental observations of relative fluorescence intensity. This is valuable when comparing excitation schemes in different environments for krypton fluorescence experiments. We conclude that two-photon excitation at 212.556 nm is optimal for single-laser, krypton tagging velocimetry or krypton planar laser-induced fluorescence.
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${-}{1^{- {J_j} - {M_i} + 1}}$${J_j} = 0,1$ corresponds to two-photon transitions, and ${J_j} = 0,1,2$ corresponds to three-photon transitions.
The term ${-}{1^{- {J_j} - {M_i} + 1}}$ does not contribute to the transition matrix element summation because it is consistently the same value for each stage of a multiphoton transition for all possible pathways.
Table 2.
Addition of the Angular Momentum of Two Electrons and: ( for both electrons)
State
0
0
1
1
Table 3.
Input Parameters for Quantum-Defect Radial Wave Functionsa,b,c
This table also provides the basis of states used to calculate two-photon transition matrix element. Data were obtained from NIST [36].
States $|5\rangle$, $|6\rangle$, $|9\rangle$, $|11\rangle$, $|12\rangle$, $|15\rangle$, $|16\rangle$, and $|17\rangle$ are of critical interest for the laser excitation lines considered in this paper.
The ${\lambda _L}$ column lists the laser excitation wavelength required for two-photon excitation, as measured in vacuum.
Two notations were used. (1) For the Kr ground state, Russell–Saunders $^{2S + 1}{L_J}$ notation is used ($\textit{LS}$ coupling). (2) For excited Kr states, Racah $({^{2{S_1} + 1}P_{{J_1}}^o})\;n{l^{(2{S_1} + 1)}}{[K]_J^o}$ notation is used ($L{S_1}$ coupling). $\vec J = \vec K + \vec s$ and $\vec K = \vec L + {\vec S_1}$ [36]. ${S_1}$ is the total electron spin of the ion, $s$ is the spin of the excited electron, and $L$ is the total orbital angular momentum. $\vec S = {\vec S_1} + \vec s$.
Table 4.
Calculation of Einstein Coefficients Using Quantum-Defect Functions and Comparison with NIST Experimental Dataa [36]
${-}{1^{- {J_j} - {M_i} + 1}}$${J_j} = 0,1$ corresponds to two-photon transitions, and ${J_j} = 0,1,2$ corresponds to three-photon transitions.
The term ${-}{1^{- {J_j} - {M_i} + 1}}$ does not contribute to the transition matrix element summation because it is consistently the same value for each stage of a multiphoton transition for all possible pathways.
Table 2.
Addition of the Angular Momentum of Two Electrons and: ( for both electrons)
State
0
0
1
1
Table 3.
Input Parameters for Quantum-Defect Radial Wave Functionsa,b,c
This table also provides the basis of states used to calculate two-photon transition matrix element. Data were obtained from NIST [36].
States $|5\rangle$, $|6\rangle$, $|9\rangle$, $|11\rangle$, $|12\rangle$, $|15\rangle$, $|16\rangle$, and $|17\rangle$ are of critical interest for the laser excitation lines considered in this paper.
The ${\lambda _L}$ column lists the laser excitation wavelength required for two-photon excitation, as measured in vacuum.
Two notations were used. (1) For the Kr ground state, Russell–Saunders $^{2S + 1}{L_J}$ notation is used ($\textit{LS}$ coupling). (2) For excited Kr states, Racah $({^{2{S_1} + 1}P_{{J_1}}^o})\;n{l^{(2{S_1} + 1)}}{[K]_J^o}$ notation is used ($L{S_1}$ coupling). $\vec J = \vec K + \vec s$ and $\vec K = \vec L + {\vec S_1}$ [36]. ${S_1}$ is the total electron spin of the ion, $s$ is the spin of the excited electron, and $L$ is the total orbital angular momentum. $\vec S = {\vec S_1} + \vec s$.
Table 4.
Calculation of Einstein Coefficients Using Quantum-Defect Functions and Comparison with NIST Experimental Dataa [36]