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Physical Review A

Narrowing of Level-Crossing Signals by Radiative Transfer in Diffuse Reflection. I. Theory

B. Dodsworth, A. Omont

Published Jun 1, 1971• 5 citations
Publisher

Abstract

The multiple diffusion theory of Barrat has been extended by expanding the density matrix in irreducible tensors to calculate level-crossing signals arising from observation of the back-scattered light. In the framework of an approximation equivalent to the coherent scattering of radiative-transfer theory, we first calculate the different Hanle-effect signals, using, in particular, the results of Chandrasekhar which give light intensities in zero magnetic field. We then roughly evaluate the effect of the redistribution in frequency of the radiation and the variation of absorption with frequency. The narrowing predicted for the Hanle-effect curves is much less important than for lateral diffusion.

DOI
10.1103/physreva.3.1901

Related papers

Narrowing of Level-Crossing Signals by Radiative Transfer in Diffuse Reflection. II. Experiments on the 6P13 Level of Mercury

B. Dodsworth, J. C. Gay, A. Omont

We present experimental verification of a certain number of theoretical predictions given in the preceding paper. The narrowing of the different Hanle-effect signals by multiple diffusion is clearly weaker if observed using the light scattered in the backward direction than if observed using the laterally scattered light. This narrowing can become very small if the vapor is illuminated with light whose frequency is in the wings of the Doppler absorption width of the vapor. The values of the ratios of the widths and intensities of the different Hanle-effect signals that were measured are in good agreement with the values predicted by theory.

Physical Review A•Jun 1, 1971•8 citations

The forward scattering of resonance radiation, with special reference to double resonance and level-crossing experiments

Alan Corney, B P Kibble, George William Series

Abstract A theoretical and experimental study has been made of the intensity of forward scattered light in conditions of radio-frequency double resonance and level-crossing. The main conclusions are (i) that the magnetic resonance and level-crossing effects which are familiar in the laterally scattered light are to be found also in the forward scattered light, but (ii) that the resonance curves are Doppler-broadened, and the strength of the oscillatory field necessary to generate them is correspondingly increased, and (iii) that coherence narrowing is much more pronounced in forward than in lateral scattering. The differences derive from the coherence between the radiation from different atoms which is present in forward, but not in lateral scattering, and which is implicitly recognized in the classical theory of the propagation of light in polarizable media. The equations of electromagnetism form the basis of the present analysis. In the presence of the oscillatory magnetic field the polarization tensor is complex and time-dependent. The time dependence leads to the result that the number of eigenwaves necessary to describe propagated waves in the medium is four, as against the usual two. An arbitrary wave is resolved into a superposition of the four eigenwaves, all of which have different propagation constants. The analysis is very general, and is applicable to experiments with lasers. The components of the polarization tensor are derived by finding first the polarizability of individual atoms from a semi-classical theory of their interaction with light and magnetic fields. The macroscopic polarization, obtained by summing over individual atoms, is then given by folding the individual polarizabilities with the Doppler distribution of resonance frequencies. In the particular case studied in detail (the inter-combination resonance line, 2537 A, in mercury), the Doppler width is much greater than the radiation width, and the imaginary (absorptive) part of the polarizability is Gaussian. Expressions are derived in a ‘weak scattering’ approximation for the intensity of the forward scattered light. They are magnetic resonance functions which are found to be the Doppler-broadened equivalents of the familiar functions, based on Lorentzians, characteristic of scattering by a single atom. In the case of double resonance the forward scattered light is strongly modulated, as is the laterally scattered light. To investigate coherence narrowing, the zero-field level-crossing has been studied with the restriction of ‘weak scattering’ removed. The result is obtained that the width of levelcrossing curves is inversely proportional to the vapour density for magnetic fields such that the Zeeman splitting is small in relation to the Doppler width. This analytical result has been verified experimentally.

Proceedings of the Royal Society of London A Mathematical and Physical Sciences•Jul 5, 1966•87 citations

Optical Hanle effect

V.Paul Kaftandjian, Christian Delsart, J. Keller

The optical Hanle effect (light-shift-induced zero-field level crossing) has been investigated theoretically and experimentally. In the case of a $J=0\ensuremath{\rightarrow}1\ensuremath{\rightarrow}0$ three-level system and of a $J=0\ensuremath{\rightarrow}1$ two-level system, the resonance line shape has been obtained using a nonlinear-response-function method. Approximate expressions, valid under the conditions for the observation of light shifts, are obtained and compared with exact calculations as well as with the corresponding expression for the magnetic Hanle effect. As concerns the weak laser beam which produces the coherent excitation of atomic substates, we have calculated the linear response in the limiting case of monochromatic and of broad-band excitation. The experimental investigation has been performed for the Ba resonance line using two cw dye lasers at $\ensuremath{\lambda}=5535$ \AA{} and an atomic beam. The absorption- and dispersion-shaped level crossing resonances have been observed, for narrow- and broad-band excitation, by varying the power of the strong nonresonant laser beam. We have checked that the line shapes are only dependent upon the reduced variable $\frac{{P}_{L}}{\ensuremath{\delta}}$ (${P}_{L}$; beam power; $\ensuremath{\delta}$: frequency detuning). Using the theoretical expressions, we have tried to fit the experimental curves. Provided that the Gaussian distribution of the beams is taken into account, we have obtained a good agreement between theory and experiment for the whole set of experimental data.

Physical Review A•Mar 1, 1981•13 citations

Selective reflection from an atomic vapour in double-resonance and level-crossing experiments

George William Series

The intensity of polarized light reflected from an absorbing vapour has been calculated for optical radio-frequency double resonance and level-crossing. The reflection coefficient was obtained from the refractive index of the vapour, using the equations of classical electromagnetism. The result obtained by integration over a range of frequencies corresponding to broad-band illumination shows that all the familiar magnetic resonance effects are to be expected in the reflected light, except that the resonance curves should be Doppler- and pressure-broadened. A study of the zero-field level-crossing curve, as a function of vapour pressure, should enable one to follow the gradual transition from diffuse to coherent scattering, and to test whether the simple theory of selective reflection is adequate.

Proceedings of the Physical Society•Jun 1, 1967•12 citations

Hanle effect in spectrally broadened light

K. Arnett, Stuart James Smith, Robert E. Ryan +4 more

We have carried out experimental and theoretical investigations of the Hanle effect in the $^{1}\mathrm{S}_{0}$${\ensuremath{\rightarrow}}^{3}$${\mathrm{P}}_{1}$ transition at \ensuremath{\lambda}=555.6 nm in $^{174}\mathrm{Yb}$, emphasizing the dependence of the Hanle signal on the intensity and on the statistical parameters of a well-characterized phase-diffusing optical field. The phase and frequency fluctuations imposed on the laser beam by a system of modulators are described by two parameters corresponding to the bandwidth \ensuremath{\beta} and the spectral density b of the noise spectrum. The shape of the laser power spectrum is continuously variable between nearly Gaussian (b/\ensuremath{\beta}\ensuremath{\gg}1) and nearly Lorentzian (b/\ensuremath{\beta}\ensuremath{\ll}1). Measurements are made with a well-collimated Yb atomic beam crossed at right angles by a linearly polarized laser beam. In accord with theoretical predictions, the full width at half maximum (FWHM) of the Hanle effect coherence dip is found to increase as the square root of the laser intensity for monochromatic excitation. For a given intensity, the FWHM decreases as the laser bandwidth increases. With the same laser intensity and spectral width, the FWHM of the Hanle coherence dip increases as the laser spectral profile approaches the Gaussian limit, particularly when \ensuremath{\beta}/2\ensuremath{\pi} is within an order of magnitude of the natural width of the atomic line. The Hanle signal shapes agree well with theoretical calculations using a Brownian motion phase-diffusion model of the laser field. For large \ensuremath{\beta}, it suffices to apply substitution rules for simple phase diffusion to the density-matrix equations.

Physical Review A•Mar 1, 1990•26 citations

Coherent Forward Scattering Spectroscopy (Cfs): Present Status and Future Perspectives

Gerd M. Hermann, Kay Niemax

I. INTRODUCTION A. Level-Crossing Phenomena and the Effect of Coherent Forward Scattering Coherent forward.scattering (CFS—used for the physical effect as well as for the approach of coherent forward scattering spectroscopy) has been investigated first in connection with atomic resonance radiation phenomena like double resonance and level-crossing effects.1 The very early level crossing has been measured in lateral fluorescence in a weak magnetic field.2 This “zero-field level crossing” in lateral detected resonance fluorescence is called “Hanle effect”3 and can be classically described by the model of an atomic oscillator, emitting during its Larmor precession with respect to the applied magnetic field (Figure 1).

C R C Critical Reviews in Analytical Chemistry•Jan 1, 1988•6 citations
The multiple diffusion theory of Barrat has been extended by expanding the density matrix in irreducible tensors to calculate level-crossing signals arising from observation of the back-scattered light. In the framework of an approximation equivalent to the coherent scattering of radiative-transfer theory, we first calculate the different Hanle-effect signals, using, in particular, the results of Chandrasekhar which give light intensities in zero magnetic field. We then roughly evaluate the effect of the redistribution in frequency of the radiation and the variation of absorption with frequency. The narrowing predicted for the Hanle-effect curves is much less important than for lateral diffusion.
We present experimental verification of a certain number of theoretical predictions given in the preceding paper. The narrowing of the different Hanle-effect signals by multiple diffusion is clearly weaker if observed using the light scattered in the backward direction than if observed using the laterally scattered light. This narrowing can become very small if the vapor is illuminated with light whose frequency is in the wings of the Doppler absorption width of the vapor. The values of the ratios of the widths and intensities of the different Hanle-effect signals that were measured are in good agreement with the values predicted by theory.
Abstract A theoretical and experimental study has been made of the intensity of forward scattered light in conditions of radio-frequency double resonance and level-crossing. The main conclusions are (i) that the magnetic resonance and level-crossing effects which are familiar in the laterally scattered light are to be found also in the forward scattered light, but (ii) that the resonance curves are Doppler-broadened, and the strength of the oscillatory field necessary to generate them is correspondingly increased, and (iii) that coherence narrowing is much more pronounced in forward than in lateral scattering. The differences derive from the coherence between the radiation from different atoms which is present in forward, but not in lateral scattering, and which is implicitly recognized in the classical theory of the propagation of light in polarizable media. The equations of electromagnetism form the basis of the present analysis. In the presence of the oscillatory magnetic field the polarization tensor is complex and time-dependent. The time dependence leads to the result that the number of eigenwaves necessary to describe propagated waves in the medium is four, as against the usual two. An arbitrary wave is resolved into a superposition of the four eigenwaves, all of which have different propagation constants. The analysis is very general, and is applicable to experiments with lasers. The components of the polarization tensor are derived by finding first the polarizability of individual atoms from a semi-classical theory of their interaction with light and magnetic fields. The macroscopic polarization, obtained by summing over individual atoms, is then given by folding the individual polarizabilities with the Doppler distribution of resonance frequencies. In the particular case studied in detail (the inter-combination resonance line, 2537 A, in mercury), the Doppler width is much greater than the radiation width, and the imaginary (absorptive) part of the polarizability is Gaussian. Expressions are derived in a ‘weak scattering’ approximation for the intensity of the forward scattered light. They are magnetic resonance functions which are found to be the Doppler-broadened equivalents of the familiar functions, based on Lorentzians, characteristic of scattering by a single atom. In the case of double resonance the forward scattered light is strongly modulated, as is the laterally scattered light. To investigate coherence narrowing, the zero-field level-crossing has been studied with the restriction of ‘weak scattering’ removed. The result is obtained that the width of levelcrossing curves is inversely proportional to the vapour density for magnetic fields such that the Zeeman splitting is small in relation to the Doppler width. This analytical result has been verified experimentally.
The optical Hanle effect (light-shift-induced zero-field level crossing) has been investigated theoretically and experimentally. In the case of a $J=0\ensuremath{\rightarrow}1\ensuremath{\rightarrow}0$ three-level system and of a $J=0\ensuremath{\rightarrow}1$ two-level system, the resonance line shape has been obtained using a nonlinear-response-function method. Approximate expressions, valid under the conditions for the observation of light shifts, are obtained and compared with exact calculations as well as with the corresponding expression for the magnetic Hanle effect. As concerns the weak laser beam which produces the coherent excitation of atomic substates, we have calculated the linear response in the limiting case of monochromatic and of broad-band excitation. The experimental investigation has been performed for the Ba resonance line using two cw dye lasers at $\ensuremath{\lambda}=5535$ \AA{} and an atomic beam. The absorption- and dispersion-shaped level crossing resonances have been observed, for narrow- and broad-band excitation, by varying the power of the strong nonresonant laser beam. We have checked that the line shapes are only dependent upon the reduced variable $\frac{{P}_{L}}{\ensuremath{\delta}}$ (PL{P}_{L}PL​; beam power; $\ensuremath{\delta}$: frequency detuning). Using the theoretical expressions, we have tried to fit the experimental curves. Provided that the Gaussian distribution of the beams is taken into account, we have obtained a good agreement between theory and experiment for the whole set of experimental data.
The intensity of polarized light reflected from an absorbing vapour has been calculated for optical radio-frequency double resonance and level-crossing. The reflection coefficient was obtained from the refractive index of the vapour, using the equations of classical electromagnetism. The result obtained by integration over a range of frequencies corresponding to broad-band illumination shows that all the familiar magnetic resonance effects are to be expected in the reflected light, except that the resonance curves should be Doppler- and pressure-broadened. A study of the zero-field level-crossing curve, as a function of vapour pressure, should enable one to follow the gradual transition from diffuse to coherent scattering, and to test whether the simple theory of selective reflection is adequate.
We have carried out experimental and theoretical investigations of the Hanle effect in the 1S0^{1}\mathrm{S}_{0}1S0​${\ensuremath{\rightarrow}}^{3}$P1{\mathrm{P}}_{1}P1​ transition at \ensuremath{\lambda}=555.6 nm in 174Yb^{174}\mathrm{Yb}174Yb, emphasizing the dependence of the Hanle signal on the intensity and on the statistical parameters of a well-characterized phase-diffusing optical field. The phase and frequency fluctuations imposed on the laser beam by a system of modulators are described by two parameters corresponding to the bandwidth \ensuremath{\beta} and the spectral density b of the noise spectrum. The shape of the laser power spectrum is continuously variable between nearly Gaussian (b/\ensuremath{\beta}\ensuremath{\gg}1) and nearly Lorentzian (b/\ensuremath{\beta}\ensuremath{\ll}1). Measurements are made with a well-collimated Yb atomic beam crossed at right angles by a linearly polarized laser beam. In accord with theoretical predictions, the full width at half maximum (FWHM) of the Hanle effect coherence dip is found to increase as the square root of the laser intensity for monochromatic excitation. For a given intensity, the FWHM decreases as the laser bandwidth increases. With the same laser intensity and spectral width, the FWHM of the Hanle coherence dip increases as the laser spectral profile approaches the Gaussian limit, particularly when \ensuremath{\beta}/2\ensuremath{\pi} is within an order of magnitude of the natural width of the atomic line. The Hanle signal shapes agree well with theoretical calculations using a Brownian motion phase-diffusion model of the laser field. For large \ensuremath{\beta}, it suffices to apply substitution rules for simple phase diffusion to the density-matrix equations.
I. INTRODUCTION A. Level-Crossing Phenomena and the Effect of Coherent Forward Scattering Coherent forward.scattering (CFS—used for the physical effect as well as for the approach of coherent forward scattering spectroscopy) has been investigated first in connection with atomic resonance radiation phenomena like double resonance and level-crossing effects.1 The very early level crossing has been measured in lateral fluorescence in a weak magnetic field.2 This “zero-field level crossing” in lateral detected resonance fluorescence is called “Hanle effect”3 and can be classically described by the model of an atomic oscillator, emitting during its Larmor precession with respect to the applied magnetic field (Figure 1).