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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Saturation effects and the dip in single-mode laser radiation curve

T. V. Radina

Published Jun 15, 2007• 0 citations
Publisher

Abstract

We show that the dip in the intensity line shape of a single-mode laser can appear at definite experimental conditions in cases both of nonhomogeneously broadened line as well as homogeneously broadened line of the atomic transition. In both cases the dip is explained by the nonlinear character of the phenomenon. The saturation of the population difference of the involved atomic energy levels has its maximum in the vicinity of the line center. Therefore the saturation of the amplification is at maximum near the atomic line center, and, as a consequence, the lasing power is reduced.

DOI
10.1117/12.752239

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We show that the dip in the intensity line shape of a single-mode laser can appear at definite experimental conditions in cases both of nonhomogeneously broadened line as well as homogeneously broadened line of the atomic transition. In both cases the dip is explained by the nonlinear character of the phenomenon. The saturation of the population difference of the involved atomic energy levels has its maximum in the vicinity of the line center. Therefore the saturation of the amplification is at maximum near the atomic line center, and, as a consequence, the lasing power is reduced.
Previous theoretical results on the influence of a laser on the line shape of a coupled transition-laser-induced line narrowing-have been restricted to the case where the laser is detuned from the center of its atomic gain profile or is in the form of a traveling wave. This paper extends those results to include the case where the laser is an intense standing-wave field tunable to the center of its atomic gain profile (conditions for Lamb dip). The effect may be observed either in transmission, by probing the coupled transition with a weak traveling-wave field coaxial with the laser field, or else in spontaneous emission from the coupled transition viewed along the axis of the laser field. It is now well known that for a laser detuned from the center of its atomic gain profile, two narrow Lorentzian resonances of different widths appear superimposed upon the broad background signal at frequencies symmetrically located about the corresponding line center. When the laser is tuned to the center of its gain profile, however, additional fine structure develops. This structure, which is particularly significant when the laser field is intense, may have important applications in highresolution spectroscopy and laser-frequency stabilization. In this paper the laser frequency may be smaller or larger than the frequency of the coupled transition. In the latter case an intense laser introduces additional splitting effects, even when the laser is detuned. Splitting effects due to weakly saturating laser fields are also discussed. The problem is formulated by expanding elements of the ensemble-averaged density matrix in an infinite series of spatial Fourier components. A perturbation technique is employed, valid for a weak probe field and a standing-wave field of arbitrary intensity. One obtains an expression for emission induced by the probe field due to atoms moving with one velocity, written in terms of continued fractions in the general case and with Bessel functions in an important special case. This expression is integrated over the atomic velocity distribution by means of a computer to obtain the total emission due to atoms moving with all velocities. In some cases the integrated expressions may be written in closed algebraic form. A detailed discussions of line shapes and of the physical processes involved is included.
An analysis is made of the operation of a laser with a saturable absorber and dominant inhomogeneous line broadening. A four-level model is used to determine the characteristics of mode intensities and conditions for the appearance of an optical bistability. By analogy with a two-level model, the results obtained demonstrate that in single-mode operation this bistability appears only if the relaxation coefficients of the upper levels of atomic or molecular systems are different in the amplifying and absorbing parts of a laser. Moreover, a change in the inhomogeneous broadening alters greatly the optical bistability interval. The calculation method used in the present study can be readily extended to analyze a multimode laser.
It is very important to understand the characteristics of gain saturation for homogeneously broadened laser medium. Assuming a Lorentzian lineshape function with a linewidth of ΔνH, we have derived the analytical functions of the gain coefficient under three different conditions: 1) a small and 2) a large signal incident on the medium respectively, and 3) a small signal accompanied by a large signal simultaneously incident on the medium. We have found that the bandwidth Δν of the gain coefficient is equal to the linewidth ΔνH for condition 1) and 3) while it is Δ ν = 1 + I ν 1 / I s Δ ν H for condition 2). Here, I v 1 and Is are the intensity of the large signal and the saturation intensity at the center frequency Vo, respectively. The reasons are also presented for such results. For condition 2), gain saturation effect is strongly dependent on the frequency of the large signal: the more the frequency deviates from the center frequency, the weaker the gain saturation effect. This results an increase of the bandwidth. For condition 3), the intensity of the large signal only changes the distribution of populations on the upper and lower energy levels. Hence, the shape of the gain coefficient does not change with the gain reduction of the small signal, thus, its bandwidth remains the same.
It is very important to understand the characteristics of gain saturation for homogeneously broadened laser medium. Assuming a Lorentzian lineshape function with a linewidth of ΔνH, we have derived the analytical functions of the gain coefficient under three different conditions: 1) a small and 2) a large signal incident on the medium respectively, and 3) a small signal accompanied by a large signal simultaneously incident on the medium. We have found that the bandwidth Δν of the gain coefficient is equal to the linewidth ΔνH for condition 1) and 3) while it is Δν = √1+Iν1 / IsΔνH for condition 2). Here, Iν1 and Is are the intensity of the large signal and the saturation intensity at the center frequency ν0, respectively. The reasons are also presented for such results. For condition 2), gain saturation effect is strongly dependent on the frequency of the large signal: the more the frequency deviates from the center frequency, the weaker the gain saturation effect. This results an increase of the bandwidth. For condition 3), the intensity of the large signal only changes the distribution of populations on the upper and lower energy levels. Hence, the shape of the gain coefficient does not change with the gain reduction of the small signal, thus, its bandwidth remains the same.
The effects of different laser line shapes on the intensity and spectrum of resonance fluorescence from a two-level atom are investigated. The laser model considered is that of an ideal single-mode laser operating high above threshold, with constant amplitude and undergoing phase-frequency fluctuations analogous to Brownian motion. As the correlation time of the frequency fluctuations increases from zero to infinity, the laser line shape changes from Lorentzian to Gaussian in a continuous way. It is shown that for intermediate and strong fields, the average intensity of fluorescence in the case of a resonant broadband Lorentzian line shape is higher than that in the case of a Gaussian line shape with the same bandwidth and total power. This is in contrast to the weak-field case where the higher peak power of the Gaussian line shape makes it more effective than the Lorentzian line shape. It is also shown that in the case of a nonzero frequency correlation time (non-Lorentzian line shape) the intensity of fluorescence undergoes non-Markovian fluctuations. In relation to the spectrum of resonance fluorescence, it is shown that as the line shape is varied from Lorentzian to Gaussian the following changes take place: In the case of off-resonance excitation, the asymmetry of the spectrum decreases. In the case of resonant excitation, the center-peak to side-peak height ratio for the triplet structure increases. Moreover, the recently predicted center-line dip, which develops in the spectrum in the case of broadband excitation when the Rabi frequency and the bandwidth are nearly equal, becomes increasingly deeper.