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The Journal of Chemical Physics

Line shape theory for a Doppler-broadened signal in presence of an infrared pump

Sharmistha Ghoshal, Pradip Narayan Ghosh

Published Oct 15, 1985• 10 citations
Publisher

Abstract

A theory of line shape for a Doppler broadened probe in the presence of a strong infrared pump is presented. The optical analog of Bloch equations has been developed; the T1, T2 relaxations are introduced phenomenologically in these equations. Doppler broadening of both the pump and signal transitions are taken into account to obtain the absorption coefficient for the signal. This leads to a Doppler broadened Gaussian function weighted by a pump induced Lorentzian function whose position and linewidth are controlled by the detuning and power respectively of the pump radiation. The absorption coefficient is further modulated by a Lorentzian function containing the fourth power of the radiation. The off-resonance pumping leads to two symmetrically located ‘‘Lamb dips’’ or ‘‘peaks’’ around the Gaussian pedestal for co- and counter propagating laser radiations. A calculation of the line shape of the central sub-Doppler feature for the on-resonance pumping is in good agreement with the observed variation of linewidth with pressure. Numerical results are presented graphically to demonstrate the pump and signal power dependence of the central hole or peak for the bent and cascade-type double resonances. The results obtained confirm the expected behavior based on level populations.

DOI
10.1063/1.449116

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Physical Review A•Jun 1, 1993•18 citations
A theory of line shape for a Doppler broadened probe in the presence of a strong infrared pump is presented. The optical analog of Bloch equations has been developed; the T1, T2 relaxations are introduced phenomenologically in these equations. Doppler broadening of both the pump and signal transitions are taken into account to obtain the absorption coefficient for the signal. This leads to a Doppler broadened Gaussian function weighted by a pump induced Lorentzian function whose position and linewidth are controlled by the detuning and power respectively of the pump radiation. The absorption coefficient is further modulated by a Lorentzian function containing the fourth power of the radiation. The off-resonance pumping leads to two symmetrically located ‘‘Lamb dips’’ or ‘‘peaks’’ around the Gaussian pedestal for co- and counter propagating laser radiations. A calculation of the line shape of the central sub-Doppler feature for the on-resonance pumping is in good agreement with the observed variation of linewidth with pressure. Numerical results are presented graphically to demonstrate the pump and signal power dependence of the central hole or peak for the bent and cascade-type double resonances. The results obtained confirm the expected behavior based on level populations.
This paper presents a theory of lineshapes for molecular Optical-Optical Double Resonance (DR) Spectroscopy with arbitrary strength for both pump and probe fields using the steady-state solutions for the 3-level density matrix. When Doppler broadening can be neglected, the results are analytical, and the probe spectrum is a pair of Lorentzian lines that display Autler-Townes splitting, each with an angular frequency half-width half maximum equal to the relaxation rates, which are assumed to be equal. When Doppler broadening is introduced, one must resort to numerical integration except for the limit of weak pump and probe fields. When the Doppler width is assumed to be much larger than the pump and probe Rabi frequencies and the probe frequency is higher than that of the pump, the calculated DR lineshapes are found to be Lorentzian with a strong pump field limit width that is proportional to the pump Rabi frequency, which is commonly known as power broadening. However, the width does not equal that Rabi frequency and is different for co- and counter-propagating pump and probe fields. Furthermore, that broadening is largely inhomogeneous, despite the Lorentzian shape. The saturation power is found to be about 4 times higher than for the bare probe transition with the same relaxation rate, dramatically lower than that expected if the width is interpreted as homogeneous.
The line shape of a pair of Doppler-broadened closely spaced transitions sharing a common lower level and under the effect of a standing-wave monochromatic radiation field is studied. The eight steady-state optical Bloch equations are solved analytically to obtain the polarization by retaining terms up to the third power in the electric-field amplitude. The absorption coefficient derived for the standing-wave case consists of two Lamb dips for the two allowed transitions at ${\mathrm{\ensuremath{\omega}}}_{1}$ and ${\mathrm{\ensuremath{\omega}}}_{2}$ and a crossover resonance dip at the intermediate frequency ${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$=(${\mathrm{\ensuremath{\omega}}}_{1}$+${\mathrm{\ensuremath{\omega}}}_{2}$)/2. The Lamp-dip peak frequencies show intensity-dependent shifts. Numerical computations are presented for different values of Rabi frequencies and relaxation linewidth parameters, which are small compared to the frequency difference \ensuremath{\Delta}=${\mathrm{\ensuremath{\omega}}}_{2}$-${\mathrm{\ensuremath{\omega}}}_{1}$. The computed line shape and frequency shift are compared with the observed results in the Balmer ${\mathrm{H}}_{\mathrm{\ensuremath{\alpha}}}$ line of hydrogen and the infrared Zeeman spectra of methane.
In the domain of semiclassical formulation and for the Doppler-free atom–field interaction, we construct the optical Bloch equations involving an M-type six-level system coupled to two pump fields and a probe field. The response of the system is probed for different pump-induced transitions in double and triple-resonance situations. In order to obtain the coherent lineshapes (absorptive and dispersive), we use the usual perturbation method for obtaining the approximate analytical solutions to these coupled optical Bloch equations for the density matrix elements. The interferences between the probability amplitudes for different energy levels (dipole allowed and dipole forbidden) are taken care of. For off-resonance pump positions, the linewidths of the three probe transitions are insensitive to the pump Rabi frequencies. On the other hand, the shifts of the three resonance peaks are extremely sensitive to the pump Rabi frequencies. However, for on-resonance pump conditions, the sensitivities of pump Rabi frequencies on the linewidths of the resonance peaks and on the shifts of the resonance peak positions are opposite to those of their off-resonance counterparts. In particular, we have shown the asymmetric and symmetric Rabi splittings under different physical conditions, for non-zero and near-zero probe detuning, respectively. The Rabi splitting under triple-resonance conditions, significantly, modifies the dispersive lineshape at the centre of the absorption line. The two- and three-photon absorptions are also reported for different off-resonant pump positions.
We calculate the steady-state probe absorption line-shape function for a strongly driven, Zeeman-degenerate molecular system. The probe laser is treated to lowest order while the pump laser is dealt with to all orders. We obtain the probe line shape for the cases of parallel and perpendicular linear polarization of the two lasers. As expected, the effects of MMM degeneracy, as well as differences due to the relative laser polarizations, are most pronounced when Doppler broadening is not important. However, even in the presence of large Doppler broadening we find a narrowing of the population hole by including the Zeeman degeneracy and a further narrowing if perpendicular laser polarizations are used.
We calculate the two-level pump-probe absorption coefficient including both upper-to-lower-level decay and level decays to a still lower lying reservoir level. We carried out calculations of probe absorption profile for arbitrary ratios of natural width and detuning to the Doppler width associated with a given transition. This theory accounts for contributions from all velocity subclasses of atoms, not only those which are Doppler shifted into resonance with the laser field (Doppler limit).1 We observe and explain new line shape features based on dissimilar decays.
The interaction of a four-level atomic system with two electromagnetic radiations has been studied. The two intermediate levels are closely spaced such that a single radiation frequency can induce transitions from the ground state to both of them. In a cascade-type double resonance, the pump electric-field amplitude is much stronger than the amplitude of the signal field, which can induce transitions from both of the intermediate levels to the upper state. The 15 optical Bloch equations obtained from the Liouville equations are solved analytically to obtain the Doppler-free signal line shape. The computed signals, exhibiting two Lorentzian line profiles, are presented for different pump frequencies. In addition to these one-photon transitions, we obtain a weak two-photon resonance peak. The two-photon resonance becomes weaker when the detuning of the pump frequency from a resonance frequency is increased. When the pump frequency ${\mathrm{\ensuremath{\Omega}}}_{\mathit{p}}$ is intermediate between the pump resonance frequencies ${\mathrm{\ensuremath{\omega}}}_{1}$ and ${\mathrm{\ensuremath{\omega}}}_{2}$, the two-photon resonance is very weak and almost vanishes when ${\mathrm{\ensuremath{\Omega}}}_{\mathit{p}}$=(${\mathrm{\ensuremath{\omega}}}_{1}$+${\mathrm{\ensuremath{\omega}}}_{2}$)/2. When the pump frequency is on resonance with one of the transition frequencies, the Rabi splitting of the signal shows asymmetry because of the presence of a close-lying transition which is pumped off resonance. The intensity-dependent shifts of the transitions are such that they repel each other more and more as the intensity of the pump field is increased. These shifts arise from the nonresonant interaction between the atom and the field.