1 Introduction

Hyperfine structure studies provide information on the magnetic and higher-order electric interactions between electrons and the nucleus. In turn, measurements of the Zeeman structure provide information about the mixing of electronic configurations. All these experimental data are needed for the correct designation of electron levels.

The methods of laser absorption spectroscopy enable precise studies of the atomic structure. The problem arises in situations where the spectrum is very complex and it is impossible to separate the individual components of the structure. The use of computer programs to simulate complex spectra becomes helpful here. These programs require the use of line profile models and theoretically predictable intensity ratios of structure components.

An additional difficulty in interpreting the spectra of complex atoms is the saturation effect that occurs at the high power density of the exciting laser light. Sometimes it is possible to reduce the laser power to a level where the observed intensities are as predicted, but in many cases the saturation effect cannot be avoided, since lowering the light intensity usually increases the experimental noise. Due to the saturation the stronger components (with larger transition probabilities) have relatively lower intensities compared to the weaker components.

The effect of saturation was described in the well-known monograph [1]. There is a large number of papers related to the effect of saturation and experiments with optical pumping (see, e.g. [2,3,4,5]). These work concerns studies of the population of states in optical pumping and the observation of effects related to the polarization of radiation. The effect of saturation also has a huge impact on the observation of reaction dynamics in studies using the LIF (laser-induced fluorescence) technique. This was described in an extensive review [3]. There are also papers (see [6, 7]) describing the effect of saturation on the registration the Zeeman and hyperfine structure spectra by the use of laser absorption spectroscopy. Our current work falls within this line of research. In the paper [6], a difference was noted in the hfs (hyperfine structure) spectra recorded using Fourier transform emission and laser optogalvanic spectroscopies. The differences in the spectra were reduced by introducing the saturation parameter into the intensity formula for hfs components. The paper [7] describes the effect of saturation on the LIF spectrum of the Zeeman structure of xenon and an attempt to take this effect into account in the analysis.

Based on the theoretical description of the saturation effect, which can be found in the monograph [1], we present a method that allows to take this effect into account in computer simulations.

2 The saturation effect in the Zeeman spectra

Let us consider a two-level atomic system in the state of thermodynamic equilibrium. In such a system, absorption transitions balanced by emission transitions (spontaneous and stimulated emission) may occur between the atomic levels. The system of kinetic equations describing changes in the population of states associated with these transitions leads to the following relation [1]

$$\begin{aligned} \Delta N=\frac{\Delta N_0}{1+S}. \end{aligned}$$
(1)

Here \( \Delta N = N_1 - N_2 \) is the population density difference for levels 1 and 2, while \(\Delta N_0\) is the difference in population density in thermal equilibrium (without radiation field) and S is the saturation parameter. When to neglect stimulated emission, i.e. assume that spontaneous emission is the only relaxation mechanism from the upper “2” level we get

$$\begin{aligned} S=\frac{2B_{12} \rho (\omega _{12}) }{A_{12}}. \end{aligned}$$
(2)

Here \(B_{12} \rho (\omega _{12})\) is the absorption probability and \(A_{12}\) is the spontaneous decay rate.

Changes in the population of states can be taken into account by modifying the values of transition probabilities. According to Eq. (1) we can define the saturation absorption coefficient as

$$\begin{aligned} \alpha ^\textrm{sat}=\frac{\alpha _{0}}{1+S}, \end{aligned}$$
(3)

where \(\alpha _0\) is the unsaturated absorption coefficient without pumping.

In order to adapt the theory to the Zeeman transitions, let us denote S as the ratio

$$\begin{aligned} S=A_{MM'}/A_S \end{aligned}$$
(4)

Here \(A_{MM'}\) is the unsaturated transition rate relating to the transition \(MM'\) directly proportional to the intensity of radiation and \(A_S\) is the saturation rate (see also [8]). The \(A_{MM'}\) rate for a given direction of observation and polarization can be calculated from appropriate theoretical formulas (see e.g. Ref. [18]).

From Eq. (3) we get the following proportionality:

$$\begin{aligned} A^\textrm{sat} _{MM'} \sim \frac{A_{MM'}}{1+A_{MM'}/A_{S}} \end{aligned}$$
(5)

Equation (5) shows that the influence of the saturation effect on the intensity of the Zeeman component depends on the ratio \(A_{MM'}/A_S\). Because the \(A_S\) parameter has the same constant value for all components of the structure the stronger \(A_{MM'}\) components have a relatively lower intensity in relation to the weaker components.

Formula (5) was used by us in computer programs simulating the shape of the Zeeman spectra deformed by the saturation effect. Transition rates \(A_{MM'}\) were calculated from the appropriate theoretical formulas while the coefficient \(A_S\) was treated as a free fitting parameter.

Fig. 1
figure 1

Experimental LIF spectra of the Zeeman effect (\(\Delta M = 0\)- components) of the 620.4306 nm line of praseodymium at magnetic field of 834 G. \(I=5/2\), \(J^\textrm{up}=15/2\), \(J^\textrm{low}=17/2\), \(A^\textrm{up}=490(2)\) MHz [9, 10], \(A^\textrm{low}=245(1)\) MHz \(B^\textrm{low}=-20(15)\) MHz [11], \(g_J^\textrm{up}= 1.11189(33) \) [12], \(g_J^\textrm{low}=\) 1.19883(17) [12], \(B^\textrm{up}\) factor was assumed to be zero. The bottom figure shows the computer generated best fit that does not take into account the saturation effect. The modified Lorentzian shape (with FWHM 680 MHz) was used. The top figure shows the best fit considering the saturation effect. The red vertical lines represent the individual components of the structure

Full size image

The presented method of considering the saturation effect in the analysis of the Zeeman spectra was previously tested by us in the case of spectra of vanadium and niobium [13,14,15,16]. One more example of the method we use is presented in Fig. 1. It presents the LIF spectra of the Zeeman effect of the 620.4306 nm line of praseodymium at magnetic field of about 834 G. The top figure shows the best fit when the effect of saturation is not taken into account. The best fit in the bottom figure takes into account the effect of saturation. It is evident that the agreement between the simulation and the experiment is much higher when the saturation effect is taken into account.

3 Influence of the saturation effect on hyperfine structure spectra

The saturation effect should also be taken into account when analyzing hyperfine structure spectra (when the magnetic fields \(B=0\)). The pump laser light is always at least partially linearly polarized. This implies that the pumping process produces a nonuniform population of M sublevels [1].

When analyzing hyperfine structure spectra, the following intensity formula is usually used [17, 18]:

$$\begin{aligned} A^{FF'} \sim \left\{ \begin{array}{ccc} J &{} I &{} F \\ F' &{} 1 &{} J' \end{array} \right\} ^2 (2F+1)(2F'+1)S_{\gamma J, \gamma ' J'}. \end{aligned}$$
(6)

Here the expression in curly braces stands for the 6J symbol and \(S_{\gamma J, \gamma ' J'} \equiv |<\gamma J||P^{(1)}|| \gamma ' J ' >| ^2\) is the electric-dipole line strength.

Expression (6) is the result of summing up all the \(M, M'\) contributions over polarization states of the emitted or absorbed radiation. The sum rule for 3J symbols included in the expression (6) tacitly assumes that the populations of the respective sublevels are the same. However the saturation effect changes this situation.

Fig. 2
figure 2

An exemplary computer-generated spectrum of a hyperfine structure pattern with the following parameters: \( I = 9/2 \), \(J^\textrm{up} = 5/2 \), \( J^\textrm{down} = 5/2 \), \( A^\textrm{up} = 100 \) MHz, \( B^\textrm{up} = 0 \), \( A^\textrm{down} = 100\) MHz, \( B^\textrm{down} = 0\) and the same value of the saturation rate \(A_S=50\). The top figure shows the contour being the sum of the \( \Delta M = \pm 1 \) components, and the bottom one the sum of the \( \Delta M = \) 0 components. The ratios for the Zeeman components of the highest intensity \( A_{\nu } ^\textrm{max}\) to the saturation coefficient \(A_s\) equals 4.57 and 14.28. The modified Lorentzian shape (with FWHM 600 MHz) was used. The red vertical lines represent the individual components of the structure. The boxes at the bottom of each subfigure show the deviations between the experimental and simulated contours

Full size image
Fig. 3
figure 3

Experimental hfs spectrum (at zero magnetic field) of the 586.889 nm niobium line (\(I=9/2\), \(J^\textrm{up}=7/2\), \(J^\textrm{down}=5/2\), \(A^\textrm{up}=593.9(10)\) MHz, \(B^\textrm{up}=-17(15)\) MHz [19], \(A^\textrm{low}=834.1(10) \) MHz, \(B^\textrm{low}=-108(30)\)MHz [19]) recorded using the LIF technique with linearly polarized exciting light. The top a shows the experimental spectrum the best computer fit. The b shows the experimental trace together with simulations taking into account the saturation effect for all \(\Delta M\) components. c shows the best computer fit which consists of only the \(\Delta M = \pm 1\) components. Te bottom d shows the best result which was obtained when only the \(\Delta M = 0\) components were taken into account. The modified Lorentzian shape (with FWHM 1110 MHz) was used. The red vertical lines represent the individual components of the structure. The boxes at the bottom of each subfigure show the deviations between the experimental and simulated contours

Full size image

The simulations shown in Fig. 2 present a computer generated hfs spectrum in two cases: /1/ assuming that only the \(\Delta M =0\) or /2/ \(\Delta M = \pm 1\) components are observed.

If there was no saturation effect, both spectra should be the same. The differences in the shape of the spectra due to the saturation effect are clearly visible. Each computer calculated hfs component was found as a sum of contributions from all M, \(M'\) components.

Figure 3 presents experimental hfs spectrum of the 586.889 nm niobium line recorded using the LIF technique with linearly polarized exciting light. The top figure (a) shows the experimental trace together with computer generated best fit in the case when the saturation effect is not taken into account. All three figures below show best computer fits that take into account the effect of saturation, but are obtained as a sum of selected \(\Delta M\) components. The best computer fit was obtained in the case (d) when the simulated spectrum is the sum of contributions only from transitions \(\Delta M= 0\) (the light from the pump laser was completely linearly polarized probably in the direction of the residual magnetic field in which the atomic system was located).

The presented example shows that hyperfine structure measurements should be carried out in conditions where the polarization direction of the excited laser light is known, what should be reflected in simulations matching the shape of the experimental spectrum to the theory.

4 Conclusion

In the paper, we present a method of analysis of the spectra of the Zeeman and hyperfine structures, taking into account the effect of saturation, which deforms the intensity ratios between the individual components of the structure. Computer simulations, as well as examples of the analysis of experimental spectra affected by the saturation effect were presented.