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Topical Meeting on Laser Techniques in the Extreme Ultraviolet

Sum Frequency Generation Spectroscopy of Rydberg States

A. Schnitzer, K. Hollenberg, Wolfgang Behmenburg

Published Jan 1, 1984• 0 citations
Publisher

Abstract

Consider sum frequency generation (SFG) ω3 = 2 ω1 + ω2 in a phase matched metal-noble gas mixture, using two lasers ω1 and ω2, where 2 ω1 is tuned to a two photon transition Ωg2 and ω2 is shifted across Rydberg transitions Ω2n (g groundstate, 2 excited state, n Rydberg state of the metal atom). Then the intensity of the SFG spectrum I3 (ω3)~ |χ(3) (ω3)|2 displays resonances and related antiresonances, separated by (1) which are caused respectively by constructive and destructive interference of resonant and nonresonant parts of the 3. order nonlinear susceptibility χ(3) (ω3 ≃ Ωng) = χR(ω3)+ χNR (3) The products μ2nμng:= xn of the dipole transition matrix elements may then be obtained from a simple measurement of the frequency spacings Δωn at a large number of transitions Ωgn and solving iteratively the coupled equation system (1),(3); neither absolute radiation intensity nor vapour density deter mination is needed.

DOI
10.1364/euv.1984.tud5

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Consider sum frequency generation (SFG) ω3 = 2 ω1 + ω2 in a phase matched metal-noble gas mixture, using two lasers ω1 and ω2, where 2 ω1 is tuned to a two photon transition Ωg2 and ω2 is shifted across Rydberg transitions Ω2n (g groundstate, 2 excited state, n Rydberg state of the metal atom). Then the intensity of the SFG spectrum I3 (ω3)~ |χ(3) (ω3)|2 displays resonances and related antiresonances, separated by (1) which are caused respectively by constructive and destructive interference of resonant and nonresonant parts of the 3. order nonlinear susceptibility χ(3) (ω3 ≃ Ωng) = χR(ω3)+ χNR (3) The products μ2nμng:= xn of the dipole transition matrix elements may then be obtained from a simple measurement of the frequency spacings Δωn at a large number of transitions Ωgn and solving iteratively the coupled equation system (1),(3); neither absolute radiation intensity nor vapour density deter mination is needed.
It has recently been shown1 that extremely large values of the second-order nonlinear suceptibility can be obtained through use of Rydberg atomic states perturbed by an external electric field.2 In these experiments a value of χ(2) = 10-9 esu was obtained using a sodium vapor of number density ~1014 atoms cm-3 and a dc electric field of magnitude 1 kV/cm. One laser was tuned close to the sodium D2 resonance line while a second laser completed the two-photon resonance with a Rydberg state. A strong coherent UV beam at the sum frequency was generated in the forward direction. The splitting of the highly excited atomic levels induced by the external electric field created a broad region of overlapping levels, leading to continuous tunability. This technique can be applied to any system displaying a Rydberg series, and should be capable of producing coherent, tunable radiation in the extreme ultraviolet.
We calculate the sum- and difference-frequency-generation (SFG and DFG) susceptibilities, ${\mathrm{\ensuremath{\chi}}}^{(2)}$(-(${\mathrm{\ensuremath{\omega}}}_{1}$+${\mathrm{\ensuremath{\omega}}}_{2}$);${\mathrm{\ensuremath{\omega}}}_{1}$,${\mathrm{\ensuremath{\omega}}}_{2}$) and ${\mathrm{\ensuremath{\chi}}}^{(2)}$(-(${\mathrm{\ensuremath{\omega}}}_{1}$-${\mathrm{\ensuremath{\omega}}}_{2}$);${\mathrm{\ensuremath{\omega}}}_{1}$,-${\mathrm{\ensuremath{\omega}}}_{2}$), of a homogeneously broadened two-level system with permanent dipole moments in the presence of pump fields at frequencies ${\mathrm{\ensuremath{\omega}}}_{1}$ and ${\mathrm{\ensuremath{\omega}}}_{2}$. Both SFG and DFG susceptibilities have two maxima at frequencies corresponding to the one- and two-photon resonances of the two-level system. Knowledge of the magnitude of one of the maxima of either SFG or DFG is sufficient for determining the magnitude of the second peak of that process and also the two peaks of the other process. In both processes the peak at the two-photon resonance has dominant contributions arising from off-diagonal elements of the density matrix. The interference between these contributions is constructive in SFG and destructive in DFG. Therefore, SFG is more efficient at the two-photon resonance than DFG. An expression for the ratio between these peaks is obtained. The peak of the spectrum of each process corresponding to the one-photon resonance has two main contributions, one of which is diagonal and the other off diagonal. In DFG the interference between the two contributions is constructive and in SFG the interference is destructive. Hence, DFG is more efficient in the vicinity of the one-photon resonance. However, since the off-diagonal contributions to SFG and DFG on the one-photon resonance are equal in magnitude (and differ by a phase difference of \ensuremath{\sim}\ensuremath{\pi}) the efficiencies of the two processes are equal when the duration of the pulses is comparable to or shorter than T1{\mathit{T}}_{1}T1​.
The measurements of microwave (μw) and radio-frequency (RF) radiation quantitative parameters may be based on the quantum–optical approach to determine the spectral characteristics of radiation transitions between the Rydberg states of atoms. Frequencies and matrix elements are calculated for dipole transitions between opposite-parity Rydberg states nL 1L and n′L±1 1L±1 (where n′= n,n±1,n±2) of the singlet series in the alkaline–earth–metal-like atoms of group IIb (Zn, Cd, Hg) and Yb. The matrix elements determine the shifts of Rydberg-state energy levels in the field of resonance μw or RF radiation, splitting the resonance of electromagnetically induced transparency (EIT) for intensely absorbed probe radiation. Numerical computations based on the single-electron quantum defect method (QDM) and the Fues’ model potential (FMP) approach with the use of the most reliable data from the current literature on quantum defect values are performed for frequencies and matrix elements of transitions between singlet Rydberg states of 1S0-, 1P1-, 1D2-, and 1F3-series in Zn, Cd, Hg, and Yb atoms. The calculated data are approximated by polynomials in the powers of the principal quantum numbers. The polynomial coefficients are determined with the use of a standard curve-fitting interpolation polynomial procedure for numerically calculated functions. These approximation expressions provide new possibilities for accurately evaluating the frequencies and matrix elements of dipole transitions between Rydberg states over a wide range of quantum numbers n >> 1, accompanied by the emission and absorption of μw and RF photons.