1 Introduction

The diode-pumped rare gas laser (DPRGL) has been recently demonstrated as a viable high-energy laser (HEL) system with three-level laser cycles analogous to the diode-pumped alkali laser (DPAL). Since its inception in 2012, the DPRGL has developed rapidly with pulsed lasing demonstrations conducted in large volume, pulsed discharges [1,2,3] as well as CW demonstrations occurring in small-volume atmospheric pressure discharges [4, 5]. Recently, a larger volume CW demonstration was conducted in a high-rep rate pulsed discharge providing 2.6 W of output power at an estimated optical-to-optical efficiency of 30% [6]. A DPRGL uses helium (He) partnered with another rare gas (Rg), usually neon, argon, krypton, or xenon (Ne, Ar, Kr, and Xe) in a weakly ionized discharge to excite the \(s_5 \left( \left[ 3/2\right] _2\right) \) level which is the lower laser level in a three- or four-level cycle. To efficiently operate the DPRGL, there must be sufficient spectral broadening to approximately match the diode-pump laser and the atomic absorption line shape, making measuring the pressure broadening and shift rates particularly important [7, 8]. For a three-level system using Ar-He and pumped by volume Bragg grating (VBG) narrowed diode bars at 811.7542 nm, this matching generally requires pressures on the order of an atmosphere. At this pressure, the atomic transitions are homogeneously broadened and the absorption line shape is expected to be Lorentzian. To achieve high operational powers, the gain volume needs to be pumped at intensities of more than 10 kW/cm\(^2\) making the behavior of these line shapes under high-intensity fields important. Furthermore, in the low-pressure regime, hole burning of inhomogeneously broadened transitions is commonly used for frequency stabilization using Lamb-dip cells [9,10,11,12]. Studying the behavior of the rare gas gain medium in the presence of strong pump lasers provides insight into the complex kinetics that will inevitably influence the operation and efficiency of the DPRGL. The Ar-He system has been the subject of several laser models that attempt to characterize the overall efficiency of a potential laser system [13,14,15,16]

Saturation spectroscopy of optically thin samples is a commonly used technique to remove the influence of the Doppler component of the line shape allowing for a high-resolution study of the Lorentzian components [17,18,19,20]. The earliest work focused on alkali [21,22,23,24,25,26,27,28,29,30,31] and excited rare gas [32,33,34,35,36,37,38] transitions but has shifted to molecular–atomic interactions [39,40,41]. The work described has focused on determining the influence of velocity-changing collisions on the line shape and to investigate the saturation broadening rate of the Lorentzian core. There is a significant research thread involved in deriving the collisional kernels for a wide variety of atomic and molecular collision partners [26, 42,43,44,45,46]. Saturation spectroscopy in optically thick samples has been studied less [25, 47,48,49], but several models for the line shapes under specific conditions have been suggested [49, 50] with a recent effort to develop analytic line shapes in highly saturated systems [51,52,53,54,54]. Most of the prior studies have focused on a limited set of measurements, particularly low pressure (less than 10 Torr) with pump intensities kept below the saturation intensities of the transitions. The models have dealt with, at most, radially averaged pump and probe profiles. The influence of velocity-changing collisions on a highly saturated and extremely optically thick (opaque) medium does not appear to have been considered in the literature either experimentally or theoretically.

Smith and Hänsch were the first to consider the influence of velocity-changing collisions (vcc) on the Lamb-dip spectra of atomic species. They observed the influence of vcc on neon–neon and neon–helium collisions, and developed a two-level model in the weak pump, low-pressure limit, where the Doppler width of the lineshape is much greater than the Lorentzian contribution. They applied a strong-collision model which assumes that every collision results in a Maxwellian velocity distribution, independent of the initial velocity [38]. Building on their work, Tenebaum et al. developed a model for both strong and weak vcc applying the strong-collision model of Smith and Hänsch combined with the Fokker–Planck model for weak collisions which includes small changes in velocity as well as diffusion. The resulting line shape was applied to the spectrum of uranium perturbed by helium and xenon [20]. This derivation, like the previous one, assumed that the pump intensity and pressure were sufficiently low. In contrast to these previous works, Pappas et al. developed a model for a three-level system in which the upper pump state is both radiatively and collisionally coupled to two lower states. The model was developed without the influence of vcc, but did consider the influence of the pump and probe polarizations in the pumping scheme.

In this work, we use saturation spectroscopy techniques with phase-sensitive detection to measure the Doppler-free line shape at Ar pressures from 1.3 to 50 Torr, pump intensities from 1.8 to 17.7 W/cm\(^2\), probe intensities from 0.1 to 1.8 W/cm\(^2\), and metastable densities, for which RF power is a surrogate varied from 1 to 40 W. The highly saturated, Doppler-limited line shapes are measured simultaneously. The unsaturated line shapes of each transition were also studied, for nearly all the same pressures and RF powers, along with longitudinally resolved inter-modulated fluorescence spectra to corroborate that optically thick conditions are the main driver of the observed line shapes. An absorption coefficient is derived using a rate equation formalism for an equivalent two-level system, motivated by a similar derivation in barium conducted by Pappas et al. [19], but considering velocity-changing collisions in the strong limit [38]. In addition, the model developed does not limit the pressure or intensity of the pump and probe fields. The signal is modeled by propagating the pump and probe beams through the absorbing media, numerically averaging the integrated pump and probe intensities and examining the difference spectra that results in the sub-Doppler line shape. This approach allows us to account fully for the spatial and spectral variation of both the pump and probe laser, and can include any known spatial variation of the absorbing medium. In addition, modification of the model to employ a collision kernel that differs from the hard-collision kernel chosen in this work is straight forward.

2 Experiment

There are several different configurations used in our analysis of the sub-Doppler spectra, but all are modifications of the standard saturation spectroscopy (SS) technique. This system is shown in Fig. 1. The laser source is a Sirah Matisse-TX Titanium:Sapphire (Ti:S) ring laser with maximum output power of 3.5 W, a line width of \(\approx \) 100 kHz, a horizontal polarization, and an \(M^2\), or beam quality factor, of 1.14. The spot size of the laser at the source is 1.4 mm with a beam divergence angle of less than 2 mrad which corresponds to a spot size of less than 3.4 mm after 1 m of propagation. A small fraction of the beam is sampled and fiber-coupled into a WS7 High-Finesse wavemeter for absolute wavelength calibration with error of 60 MHz at 632.8 nm. The laser is tuned over a range of 20 GHz (4.4 Å) centered at the 811.7542 nm \(s_5 (4s(^2P_{3/2})[3/2]_2) \rightarrow p_9 (4p(^2P_{3/2})[5/2]_3)\) transition in Ar. The beam is propagated through an achromatic half-wave plate (Thorlabs AHWP05M-980) and then a polarizing beam splitter (PBS, Thorlabs PBS252) to split the beam into a pump and probe of varying intensities. The pump is sent through an amplitude modulating chopper wheel (SR540) at 1.8 kHz and is propagated through another PBS, where a small fraction is picked off and directed onto a fast photodiode (Thorlabs PDA10A) to track incident pump intensity. The remaining optical power is transmitted through the discharge cell after which it is propagated through another PBS and an ND filter to a second photodiode to track the transmitted pump intensity. Before transmission through the discharge cell, the pump laser spot size was measured to be approximately 4 mm. Tracking these two intensities allows us to measure the highly saturated absorbance of the medium. The probe beam is propagated around the discharge tube using silver mirrors (Thorlabs PF10-03-P01) and is reflected off a PBS into the discharge tube, where it is counter-propagating with the pump beam and overlaps the same volume. After exiting the cell, the probe beam is reflected off the PBS and sent to a third photodiode, where the transmitted probe intensity is tracked. All three photodiodes have a 25 mm focal length lens mounted on a 1\(^{\prime \prime }\) long tube. They are connected to three SR850 lock-in amplifiers for phase-sensitive detection at the chopper frequency. The lock-ins are referenced to the chopping frequency of the pump beam. The signal on the probe detector is limited to atoms that interact with both the pump and probe beam, yielding a sub-Doppler spectral feature for the zero velocity group. The lock-ins are connected to a National Instruments USB-6251 BNC DAQ unit and controlled via an Agilent model 82357B GPIB-to-USB module. Matlab is used to collect data with the DAQ as well as to control the instruments through the USB-to-GPIB interface.

Fig. 1
figure 1

Experimental diagram for the saturation absorption spectroscopy experiment. The lock-in amplifiers and computer are not shown

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The second configuration is a Doppler-limited laser absorption spectroscopy (LAS) experiment which only slightly modifies the above apparatus. The half-wave plate is rotated from 90\(^{\circ }\) to 0\(^{\circ }\) to send most of the laser power down the probe channel which is then incident on a graphite beam block and not used. The remaining beam, with only \(\approx \)40 mW of power, an intensity of 0.35 W/cm\(^2\), is sent through the original pump beam path with both the incident and transmitted intensities tracked by the fast photodiodes. The lock-in amplifiers are referenced to the chopping frequency of the probe beam.

The third configuration is the inter-modulated fluorescence spectroscopy (IFS) experiment which removes the photodiodes from the original configuration and replaces them all with beam blocks. The photodiodes are then placed along the glass tube to observe the fluorescence from the discharge at various positions. The 25 mm lenses now serve to collimate the view of the photodiode which are placed at 2.5, 7.5, and 17.5 cm relative to the probe propagation direction. This is annotated in Fig. 1 by the three circles along the discharge cell. The chopper wheel is also shifted, so that both the pump and probe beam are chopped at different frequencies (1.8 and 1.5 kHz). In this case, the SR540 chopper controller outputs the sum frequency to which the lock-in amplifiers are referenced. For the fluorescence to be at the sum frequency, the emitting atoms must be interacting with both beams yielding a sub-Doppler feature. This allows the absorption of the pump and probe beam along the path to be investigated.

The Ar metastable state is formed in a radio-frequency dielectric barrier discharge. The gas is Matheson UHP (99.999%) Ar with flow rate set by a Sierra instruments flow controller (C50L) to 25 or 50 sccm. After the controller, Ar is filtered using an SAES gas purifier (MC200-904FV). The gas then flows to the cell which consists of a 1/4\(^{\prime \prime }\) O.D. (0.15\(^{\prime \prime }\) I.D.) Pyrex tube and out to vacuum at the exit. On the outside of the tube, there are two copper electrodes that were cut by wire electrical discharge machining (EDM) to fit the O.D. of the Pyrex tube. The electrodes were designed, so that there would be a 1/8\(^{\prime \prime }\) gap between the edges. These electrodes are held together using a custom Macor mount and the air gaps between them are filled with Macor bars milled to the width of the gaps. Power is fed to the electrodes using an Advanced Energy Cesar 1310 radio-frequency power generator which can output between 1 W and 1.25 kW at 13.56 MHz. The plasma load is impedance matched to the source load using a Vario matching network (VM1000). This system is capable of generating Ar plasmas at pressures as high as 1000 Torr with sustaining powers in the tens of watts. Figure 2 shows the images of the discharge that were collected at 50 Torr and RF powers of 5, 10, and 25 W. All images were collected using the same exposure time. The images show that as RF power increases, the effective path length and fluorescence intensity increase. These two visual parameters are surrogates for the total absorbance, the longer path lengths increase the total absorption path length, and the fluorescence is correlated with the number density of absorbers. The path length is shown to increase from the 2\(^{\prime \prime }\) electrode length to the full 8.4\(^{\prime \prime }\) tube length as the RF power increases from 1 to 25 W. The flow velocities are 2100–112 cm/s for 1.3–50 Torr corresponding resonance times of 1.2–45 ms, where the resonance time is defined as the time of travel between the electrodes. The circles designated by 1–3 are representative of the photodiode locations discussed in the previous paragraph and in Fig. 1.

Fig. 2
figure 2

Images of the discharge at 50 Torr of Ar for RF powers of 5 (a), 10 (b), and 25 (c) W

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3 Results

The original goal of this work was to investigate the velocity-changing collision rate of Ar–Ar collisions in an Ar plasma. Based on surveys of the literature, the line shape was expected to be well represented by a Lorentzian core on top of a Doppler pedestal [30, 38]. Towards this end, we set up the saturation spectroscopy apparatus and collected data at several different Ar pressures (1.3–50 Torr) and RF powers (1–40 W). The expectations were that as pressure increases so does the velocity-changing collision rate and, therefore, the Doppler pedestal should increase in area with a corresponding decrease in the pressure-broadened Lorentzian core. Figure 3 reveals that the measured line shapes did not fulfill these expectations as a multi-peak structure was observed. Focusing first on the 1 W case at 5 Torr, the spectra is well-represented by a Voigt profile for frequencies greater than 1 GHz from line center. At the core, the feature is a Lorentzian profile which is inset. As the RF power is increased, this dip deepens significantly. In the high-pressure case, we see a similar behavior except the core only becomes inset at higher RF powers. These line shapes are similar to self-reversed lines observed in optically thick fluorescence experiments [55]. The effect of self-reversal leads the core of the line shape to be removed because of absorption of the pump and probe laser along the path. This is consistent with the observed trend in both pressure and RF power, since increasing pressure broadens the original Voigt line shape, lowering the absorption cross section, which leads to greater transmission. Increasing the RF power has the opposite effect, whereby both the absorption path length and number density of absorbers are increased. As expected, higher RF powers create a significantly more “self-reversed” line.

Fig. 3
figure 3

(SS) Probe transmission spectra are shown for 5 (a) and 50 (b) Torr, where the different RF powers are 1 W (), 5 W (), 10 W (), 25 W (), and 40 W (). The 5 Torr case shows a significant transmission only in the wings and in the Lorentzian core. The 50 Torr case shows absorption in the core about the Lorentzian center only at higher (>25 W) RF Powers. The Lorentzian core in the 50 Torr case shallower and broader than the 5 Torr case, as expected from the pressure broadening of the line

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With absorption of the probe beam (at lower intensity) being a significant contributor to the sub-Doppler line shape, an LAS experiment was conducted to characterize this absorption. Figure 4 shows the absorbance of a 40 mW (0.35 W/cm\(^2\)) probe beam through the cell for various RF powers at 5 and 50 Torr. The absorbance is defined as follows:

$$\begin{aligned} Abs(\nu ) = -\text {ln}\left( \frac{I}{I_0}\right) = \sigma (\nu ) n l, \end{aligned}$$
(1)

where \(\sigma (\nu )\) is the absorption cross section, n is the number density of absorbers, and l is the absorption path length. The absorption cross section is as follows:

$$\begin{aligned} \sigma (\nu ) = \frac{g_2}{g_1}\frac{A \lambda ^2}{8\pi }g(\nu ), \end{aligned}$$
(2)

where \(g_2=7\) is the degeneracy of the upper state, \(g_1 = 5\) is the degeneracy of the lower state, \(A = 3.31\times 10^7\) s\(^{-1}\) is the Einstein A coefficient, and \(\lambda \) is the wavelength of the transition. The line shape, \(g(\nu )\), contained in the absorption cross section is area normalized, so that the integral over the absorbance is proportional to number density. For this experiment, the detection limit was absorbance \(\approx \) 3 corresponding to number densities of \(7.6\times 10^{10}\) and \(1.1\times 10^{11}\) cm\(^{-3}\) at 5 and 50 Torr, respectively. It is immediately apparent at 5 Torr that the medium is just barely opaque to the probe, even at the lowest RF power of 1 W. In the higher pressure case, the media becomes opaque to the probe at powers above 25 W. This is consistent with the previous observation in Fig. 3 that the “self-reversed” structure is only apparent at RF powers greater than 25 W in the high-pressure case and is a significant influence to the 5 Torr line shape even at a power of 1 W.

Fig. 4
figure 4

(LAS) Absorption spectra are shown for the 5 (a) and 50 (b) Torr cases with the RF powers as 1 W (), 5 W (), 10 W (), 25 W (), and 40 W (). Higher RF powers correspond to the highest absorbances. The maximum measurable absorbance for the experiment is at \(\approx \) 3, and so, all data above absorbance 3 is excluded from the fitting routine (as shown in the residuals)

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The intensity of 0.35 W/cm\(^2\) was chosen to minimize the influence of saturation on the line shape. The saturation intensity is defined as follows:

$$\begin{aligned} I_{\mathrm{sat}} = \frac{h \nu }{\sigma (\nu _0)}A, \end{aligned}$$
(3)

where h is Planck’s constant and \(\nu \) is the frequency of the transition. The cross section is evaluated at line center which is its maximum value. For a Doppler-limited profile at 300 K, the saturation intensity is 1 W/cm\(^2\) rising to 1.5 W/cm\(^2\) as the pressure rises to 50 Torr and the lineshape transitions from a Doppler-limited Gaussian to a Voigt. The saturation coefficient, defined \(S = I/I_{\mathrm{sat}}\), provides the metric for how strongly influenced a transition is by saturation with \(S \ll 1\) being entirely unsaturated and \(S \gg 1\) implying highly saturated conditions. For the initial SS experiment, the intensity of the pump was 17.7 W/cm\(^2\) which corresponds to \(S = 17.7\) and 11.8 at 5 and 50 Torr, respectively. For the LAS experiment, the intensity of 0.35 W/cm\(^2\) corresponds to unsaturated conditions, \(S = 0.35\) and 0.23.

Recorded line shapes are a line-of-sight average of the local line shape along the cross section and length of the cell. To observe this longitudinal dependence, the experiment was modified to configuration three, where emission was observed at the sum frequency of the pump and probe. This setup allows us to investigate the contribution of opacity as a function of probe propagation distance through the cell, further characterizing the influence of opacity on the observed line shapes. Figure 2 shows the positioning of the three photodiodes overlaid on images of the discharge cell with RF powers of 5, 10, and 25 W at a pressure of 50 Torr. At 5 W, (a) the plasma does not extend to zone 3 and just barely half fills zone 2, whereas at 10 W, (b) the plasma fills zones 1 and 2 but does not yet extend to zone 3. At 25 W, (c) all three zones are filled with plasma. These three powers offer a good characterization of the absorption as a function of number density and path length. The IFS experiment resulted in the spectra, as shown in Fig. 5. If we first look at the 5 W (a) case, we see that there is no plasma in zone 3 and very little in zone 2. In fact, the spectra in the middle panel appears to be of the expected form with no obvious opacity effects in the core. By zone 1, the plasma has fully absorbed the laser in the core. In the 10 W (b) case, we see similar trends except the measured intensity has been reduced, likely due to the absorption of the pump beam before reaching zone 2. By the time we reach 25 W, (c) there is significant “self-reversal” in the core, even at the shortest path length, with near full absorption of the probe by the time it propagates to zone 1.

Fig. 5
figure 5

(IFS) Inter-modulated fluorescence spectra for the 50 Torr case at positions 1 (top), 2 (mid), and 3 (bottom), as shown in Fig. 2. The RF powers are given by 1 W (), 5 W (), 10 W (), 25 W (), and 40 W ()

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With the cause of the line shapes revealed, we chose to investigate the influence of various parameters. The probe and pump intensity were both varied and an optic was introduced only into the pump path and symmetrically into both paths to raise the pump intensity to 15 kW/cm\(^2\) at the focus, corresponding to \(S = \) 15,000 and 10,000 at 5 and 50 Torr, respectively. The pump beam was focused down to a calculated spot size of 52 \(\upmu \)m corresponding to a reduction in th Rayleigh range from 871 to 1 m. It also changes the spatial intensity distribution of the pump and probe. By only focusing the pump laser, the volume overlap between the pump and probe was greatly reduced from nearly 100 to 0.5%. This led to significant absorption on the probe beam at radii greater than that of the focused pump and a weaker overall signal in the core of the sub-Doppler feature. Similarly, by placing an optic on both beams, so that they share a common focal point, the probe beam was found to be more attenuated in the core. Increasing the intensity of the probe beam, when the pump was focused, did not significantly change the observed line shape as the absorption in the unsaturated volume was too high. An increase in the intensity of the probe beam by nearly a factor of 10 leads only to a factor of 5 increase in the core transmission.

As these tests did not provide any illuminating information, we instead focus on the influence of the laser intensity in the absence of an optic, on the observed line shape. Figure 6a shows that as laser intensity is increased, the opacity in the medium is reduced causing the peak probe transmission to increase from 0.75 to 4 and for the Lorentzian core to increase from absent to an amplitude of 1. This increased transmission is directly due to the pump laser saturating a larger volume of the medium. Furthermore, the Lorentzian core is not significantly broadened by the increased saturation. This may be due to the strong influence of velocity-changing collisions in the larger volume. At the higher pressure of 50 Torr, in Fig. 6b, we see that increasing the intensity of the pump leads to a factor of two increase in probe transmission, for the same reasons as the 5 Torr case, as well as a factor of two increase in width. Due to the pressure-broadened Lorentzian core, it is difficult to see if the resulting broadening is because of saturation in the core or increased width of the pedestal.

Fig. 6
figure 6

(SS) Probe transmission spectra are shown for 5 (a) and 50 (b) Torr and for incident laser intensities of 17.7 (), 14.1 (), 10.6 (), 7.1 (), 3.5 (), and 1.8 () W/cm\(^2\). As the pump intensity increases, the opacity to the probe decreases due to the wings of the atomic feature becoming more saturated. At 5 Torr, the increasing pump intensity leads to a taller core feature and less absorbed core. For the 50 Torr case, the probe transmission increases and the line shape broadens slightly with intensity

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At pressures less than 5 Torr and at the low pump intensity of 17.7 W/cm\(^2\), a significant structure is observed in the line shape that cannot be accounted for by considering only the spatial or spectral variation of the pump or probe lasers. This structure appears in the form of a secondary line shape that is blue-shifted and broadened relative to the main feature. The relative amplitude of the two features as well as the width and shift of the secondary peak shows a dependence on the pressure, RF power, and pump laser intensity. Figure 7 shows the RF power and laser intensity dependence. Both the pump absorbance and probe transmission are displayed and separated by a residual that is the result of fitting a sum of Voigt profiles to the measured pump absorbance. Figure 7a, b corresponds to the 1.3 Torr case at a pump intensity of 17.7 W/cm\(^2\) and Fig. 7c, d uses the 15 kW/cm\(^2\) pump intensity for the same pressure. The probe transmission shows no asymmetry for the 1 W case, but even at 5 W, there is a clear asymmetry, in both the wings and core feature, that increases with RF power. This behavior is distinct for the low-intensity and high-intensity cases, and this structure appears to be strongly correlated to the absorption of the pump laser with a larger absorption leading to a more observable influence on the line shape.

Fig. 7
figure 7

(SS) Pump absorbance (a), (c) and probe transmission spectra (b), (d) for the 1.3 Torr case with a pump intensity of 17.7 W/cm\(^2\) (left) and 15 kW/cm\(^2\) (right) are shown. The RF powers are given by 1 W (), 5 W (), 10 W (), 25 W (), and 40 W (). The pump absorbance shows a clear secondary feature in the blue wing, even at 5 W. The residuals are shown for a dual-Voigt fit to the line shape

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There are several potential causes of the observed structure. First, we note that there are no other Ar or Ar\(^+\) lines in the vicinity of this transition, so this spectral feature must arise from the influences on the internal state of the atom. The least likely candidate for this structure lies in the inhomogeneity of the plasma itself. Spatial images show that the peak Ar(\(s_5\)) number densities exist in the sheath and are significantly diminished towards the center of the plasma. Furthermore, the electron density is increasing in the sheath and is maximized in the central column. The consideration of electron density is important as they significantly broadened and blue-shift spectral features. This Stark broadening and shift has been the subject of several reviews [56]. In fact, at the high intensity, 40 W RF Power, the secondary Voigt profile is blue-shifted by 0.7 GHz. The measured Lorentzian width is saturation broadened, so it is not useful for this analysis. Stark shifts are notoriously hard to measure, but if we assume that the shift is caused purely by electrons and that the published rates are accurate, then the required density would be approximately \(2\times 10^{15}\) cm\(^{-3}\) which is somewhat higher than expected densities below 10\(^{14}\) cm\(^{-3}\) from preliminary modeling. It is worth noting, however, that these spatial inhomogeneities can cause significant variations in the line shape of a single transition [57]. This is an unlikely candidate because it does not account for the dependence on the pump laser intensity. It also seems unlikely that the change in RF power from 1–40 W would be able to account for the required increase in electron number density as the observed RMS current only increases by about a factor of 3.

The most likely candidate for this structure appears to be coupling between the plasma and the propagating pump laser field, where the plasma becomes driven by the field generated in the laser spot and not the RF discharge field. At 5 Torr, the 17.7 and 15 kW/cm\(^2\), laser intensities correspond to local reduced electric fields of 72 and 2100 Td which are fields of sufficient strength to rapidly accelerate electrons and ions within the laser spot and could reasonably increase the local ionization rates. The observation of this effect would depend both on the radial intensity of the propagating field and the longitudinal intensity which is absorption, and, therefore, RF power dependent. In essence, the laser creates a significant radial gradient in the electron density with peak densities occurring in the spatial core of the pump laser and decreasing along the path as the pump laser is absorbed. The probe laser then interrogates the complete plasma volume and these gradients would be observable if their influence on the production of the Ar(\(s_5\)) level and the local electron density was significant enough. Both these sources of structure amount to speculation at this point and more work is required to be able to more definitively state the cause.

4 Discussion

A model was developed to give physical insight into the observed optically thick line shapes. We consider the propagation of a laser of arbitrary intensity through an absorbing media. The laser propagation in free-space is well described by a Gaussian propagation:

$$\begin{aligned} I(r,z)&= I_0 \left( \frac{w_{0}}{w(z)}\right) ^{2} \mathrm{Exp}\left( -\frac{2r^{2}}{w(z)^{2}}\right) \end{aligned}$$
(4)
$$\begin{aligned} w(z)&= w_0\left( 1+\left( z\frac{M^{2}}{z_{R}}\right) ^2\right) ^{1/2}, \end{aligned}$$
(5)

where \(I_{0}\) is the incident laser intensity, \(w_0\) is the beam waist, \(M^{2}\) is the laser beam quality factor, \(z_R = \frac{\pi w_{0}^{2}}{\lambda }\) is the Rayleigh range of the laser, and \(\lambda \) is the laser wavelength. When considering an absorbing media, we must consider the longitudinal (z) variation of the laser intensity. This involves the solution to a differential equation:

$$\begin{aligned} \frac{\mathrm{d}I(r,z)}{\mathrm{d}z} = -\alpha (\nu ,I(r,z))I(r,z) + \frac{2}{w(z)}\frac{\mathrm{d}w(z)}{\mathrm{d}z}\left( \frac{2r^2}{w(z)^2}-1\right) I(r,z), \end{aligned}$$
(6)

where \(\alpha (\nu )\) is the frequency and intensity-dependent absorption coefficient, and I(r, z) is the local intensity of the laser. In the absence of absorption, the solution to Eq. (6) is Eq. (4). The absorption coefficient is further defined in terms of atomic parameters as follows:

$$\begin{aligned} \alpha (\nu ) = -\int _{-\infty}^{\infty}{\sigma (\nu ,\pm v_z) \, \Delta n(v_z,I)\,\mathrm{d}v_z}, \end{aligned}$$
(7)

where \(\Delta n(v_z) = n_2(v_z)-\frac{g_2}{g_1}n_1(v_z)\) is the population inversion of a particular velocity group, which is dependent on the laser intensity due to saturation effects, and the cross section is defined as follows:

$$\begin{aligned} \sigma (\nu ,\pm v_z)&= \frac{g_2}{g_1} \frac{A \lambda ^2}{8 \pi n^2} \frac{\gamma /(2\pi )}{(\nu -\nu _0{\mp } k v_z)^2+(\gamma /2)^2} \nonumber \\&= \sigma _0 \frac{\gamma /(2\pi )}{(\nu -\nu _0{\mp } k v_z)^2+(\gamma /2)^2}, \end{aligned}$$
(8)

where A is the Einstein A coefficient for the transition, \(\lambda \) is the vacuum wavelength, n is the non-resonant index of refraction (taken to be 1), \(\gamma \) is the homogeneous width, \(\nu _0\) is the central frequency, \(k = \nu _0/c\) is the wave vector, and \(v_z\) is the axial velocity. The negative on the RHS of Eq. (7) arises due to the fact that the population inversion is negative, but the absorption coefficient is provided as a positive number. We note that ± on the LHS of Eq. (8) corresponds to \({\mp }\) on the RHS and refers to the co-propagating and counter-propagating beams. The velocity- and intensity-dependent population inversion are derived from the rate equations for an equivalent two-level system. Figure 8 shows the energy levels and considered mechanisms.

Fig. 8
figure 8

Energy level diagram and corresponding mechanisms, defined in Eqs. (9) and (10), are shown. The velocity-changing collisions are internal to each level population and are not displayed

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We use the rate equation formalism despite the high pump intensities as the elevated pressures will destroy any coherence effects that would make the use of a density matrix formalism necessary. Henceforth, we make the substitution \(v_{z} \rightarrow {v}\) with the understanding that v refers to the axial velocity. The rate equations for the system are then as follows:

$$\begin{aligned} {\dot{n}}_1(v)&= C_1(v)-\Gamma _1 n_1(v)+\frac{I \sigma (\nu ,v)}{h\nu }(n_2(v)-\frac{g_2}{g_1}n_1(v))-n_1(v)G_1(v)+F_1(v) \end{aligned}$$
(9)
$$\begin{aligned} \dot{n}_2(v)&= C_2(v)-\Gamma _2 n_2(v)-\frac{I \sigma (\nu ,v)}{h\nu }(n_2(v)-\frac{g_2}{g_1}n_1(v))-n_2(v)G_2(v)+F_2(v), \end{aligned}$$
(10)

where \(C_i(v) = C_i f(v)\) is the production rate of the \(i{\text {th}}\) level, \(\Gamma _i\) is the total loss rate of the i \({\text {th}}\) level, \(g_2\) and \(g_1\) are the degeneracies of the upper and lower levels, respectively, and \(G_i(v)\) and \(F_i(v)\) are the production and destruction of a specific velocity group given by the following equations:

$$\begin{aligned} G_i(v) = \int _{-\infty}^{\infty}{W_i(v\rightarrow v') \, \mathrm{d}v'} \end{aligned}$$
(11)
$$\begin{aligned} F_i(v) = \int _{-\infty}^{\infty}{n_i(v') W_i(v'\rightarrow v)\, \mathrm{d}v'}, \end{aligned}$$
(12)

where \(W_i(v'\rightarrow v)\) is the velocity-changing collision kernel for a change from the initial velocity \(v'\) to final velocity v. The selection of this collisional kernel has been a source of significant research over the past 3 decades [26, 42,43,44,45,46], but the simple strong-collision kernel derived by Rautian [58] will be used. The equations are expressed in general terms of the collision kernel until the end of the derivation to allow for a straight forward substitution of another kernel. In the absence of the pump laser, Eqs. (9) and (10) yield the initial population inversion:

$$\begin{aligned} \Delta n_0(v)&= n_2^0(v) - \frac{g_2}{g_1}n_1^0(v) \nonumber \\&= \frac{C_2(v)+F_2(v)}{\Gamma _2 + G_2(v)} -\frac{g_2}{g_1}\frac{C_1(v)+F_1(v)}{\Gamma _1 + G_1(v)}. \end{aligned}$$
(13)

With a pump laser of arbitrary intensity, the inversion from solving Eqs. (9) and (10) is as follows:

$$\begin{aligned} \Delta n(v) = \frac{\Delta n_0(v)}{1+I/I_{\mathrm{sat}}(\nu ,v)}, \end{aligned}$$
(14)

where \(I_{\mathrm{sat}}(\nu ,v) = \frac{h \nu }{\sigma (\nu ,v)}\Gamma \text {*}\) and \(\Gamma \text {*}\) is defined as follows:

$$\begin{aligned} \Gamma \text {*} = \frac{(\Gamma _1+G_1(v))(\Gamma _2+G_2(v))}{(\Gamma _1+G_1(v)+\frac{g_2}{g_1}(\Gamma _2+G_2(v))}. \end{aligned}$$
(15)

In the limit of strong collisions, which should be satisfied for Ar*–Ar collisions due to the equivalent mass of the emitter and perturber, we use the collisional kernel defined by Hänsch and Smith [38] as follows:

$$\begin{aligned} W(v'\rightarrow v) = \Gamma ^c_i f(v), \end{aligned}$$
(16)

where \(\Gamma ^c_i\) is the velocity-changing collision rate of the \(i{\text {th}}\) level and f(v) is the Maxwell–Boltzmann velocity distribution in a single dimension:

$$\begin{aligned} f(v) = \left( \frac{m_{\mathrm{Ar}}}{2\pi k_b T}\right) ^{3/2}\text { Exp}\left( -m_{\mathrm{Ar}}v^2/2k_bT\right) , \end{aligned}$$
(17)

where \(m_{\mathrm{Ar}}\) is the mass of an Argon atom, \(k_b\) is Boltzmann’s constant, and T is the gas temperature. This collision kernel is equivalent to the statement that all collisions are fully thermalizing collisions and are, therefore, independent of the initial velocity. Using this kernel, we find that \(G_i(v) = \Gamma ^c_i\) and \(F_i(v) = N_i \Gamma ^c_i f(v)\), where \(N_i = \int _{-\infty}^{\infty}{n_i(v')\mathrm{d}v'}\) is the total population in the \(i^{\text {th}}\) level. In the absence of velocity-changing collisions, \(n_i(v) = N_i f(v)\). This collisional kernel will result in a zeroth-order population inversion and \(\Gamma \)* defined as follows:

$$\begin{aligned} \Delta n_0(v)&= \frac{C_2(v)+F_2(v)}{\Gamma _2 + G_2(v)} -\frac{g_2}{g_1}\frac{C_1(v)+F_1(v)}{\Gamma _1 + G_1(v)} \nonumber \\&= f(v)\left( \frac{C_2+N_2 \Gamma ^c_2}{\Gamma _2 + \Gamma ^c_2} -\frac{g_2}{g_1}\frac{C_1+N_1 \Gamma ^c_1}{\Gamma _1 + \Gamma ^c_1}\right) \nonumber \\&= f(v)\Delta n_0 \end{aligned}$$
(18)
$$\begin{aligned} \nonumber \\ \Gamma \text {*}&= \frac{(\Gamma _1+\Gamma ^c_1)(\Gamma _2+\Gamma ^c_2)}{(\Gamma _1+\Gamma ^c_1+\frac{g_2}{g_1}(\Gamma _2+\Gamma ^c_2)}, \end{aligned}$$
(19)

where we have defined the velocity-independent inversion as \(\Delta n_0\). It is clear from Eqs. (18) and (19) that the main influence of strong collisions is to increase the saturation intensity due to increased destruction of a given velocity group. Substituting Eqs. (8) and (18) into (7) for the two cases of a co-propagating and counter-propagating probe beam yields:

$$\begin{aligned} \alpha _{\mathrm{co}}(\nu ,I)&= \frac{g_2}{g_1} \frac{\sigma _0\Delta n_0}{(1+S_0)^{1/2}\sqrt{\pi }} \frac{2 \sqrt{\text {ln}(2)}}{\Delta \nu _D} \int _{-\infty}^{\infty}{\frac{\text {Exp}\left[ -(a_1 y-u(\nu ))^2\right] }{y^2+1} \, \mathrm{d}y} \end{aligned}$$
(20)
$$\begin{aligned} \alpha _{\mathrm{counter}}(\nu ,I)&= \frac{g_2}{g_1} \frac{\sigma _0 \Delta n_0}{\sqrt{\pi }} \frac{2 \sqrt{\text {ln}(2)}}{\Delta \nu _D} \int _{-\infty}^{\infty}{\frac{\text {Exp}\left[ -(a_0 y-u(\nu ))^2\right] (y^2+1)}{((y-2 u(\nu )/a_0)^2+1)(y^2+1+S_0)} \, \mathrm{d}y}, \end{aligned}$$
(21)

where \(\Delta \nu _D\) is the inhomogeneous (Doppler) width; the line-center saturation parameter is as follows:

$$\begin{aligned} S_0 = I/I_{\mathrm{sat}}(\nu _0)=I \sigma (\nu =\nu _0\pm kv_z)/(h\nu \Gamma \text {*}), \end{aligned}$$
(22)

and the dimensionless parameters in the integrals are defined as follows:

$$\begin{aligned} a_0&= \sqrt{\text {ln}(2)}\frac{\gamma }{\Delta \nu _D} \end{aligned}$$
(23)
$$\begin{aligned} a_1&= \sqrt{\text {ln}(2)}\frac{\gamma _s}{\Delta \nu _D} \end{aligned}$$
(24)
$$\begin{aligned} u(\nu )&= 2\sqrt{\text {ln}(2)}\frac{(\nu _0-\nu )}{\Delta \nu _D} \end{aligned}$$
(25)

and \(\gamma _s = \gamma (1+S_0)^{1/2}\) is the saturation broadened homogeneous width.

The derivation of the sub-Doppler line shape will proceed in three steps, as displayed in Fig. 9. First, Eq. (6) is solved for the pump beam with the boundary condition that at \(z=0\), the intensity is equal to the measured incident pump intensity and the absorption coefficient defined by Eq. (20), as shown in Fig. 9a. We then solve Eq. (6) for the probe beam in the absence of the pump. Since the experimental observations are a line-of-sight average, the observed probe signal will be an integrated average of the simulated probe signal. The solution, \(I_{\mathrm{probe}}(r,z)\), is then radially averaged as follows:

$$\begin{aligned} I_{\mathrm{probe}}(z=L) = \frac{2\pi }{\pi r_{\max }^2} = \int _0^{r_{\max }}{r I_{\mathrm{probe}}(r,L) \mathrm{d}r}, \end{aligned}$$
(26)

where \(r_{\max } = 2 w(L)\) is the point at which the probe beam intensity is down by a factor of \(10^4\) and is no longer contributing. This result is shown in Fig. 9b.

Fig. 9
figure 9

Results of the line shape model are shown for the 5 Torr case and RF powers of 1 W (), 5 W (), 10 W (), 25 W (), and 40 W (). a Pump transmission intensity at the end of the cell, b spatially averaged probe transmission in the absence of the pump, and c spatially averaged probe transmission in the presence of the pump

Full size image

Finally, after solving Eq. (6) for the spatial and spectral dependencies of the pump and probe, the observation of counter-propagating probe beam in the presence of the derived pump intensity is considered. In this case, the absorption coefficient is defined by Eq. (21). I in \(\alpha (\nu ,I)\) will now be given by the solution to the pump intensity, \(I_{\mathrm{pump}}(r,L-z)\), and not be defined by the local intensity of the propagating probe. The pump intensity has been flipped along the axial direction to account for the counter-propagating pump, \(z\rightarrow L-z\). Equation (21) is substituted into (6) and solved for the intensity of the counter-propagating probe beam in the presence of the pump. This intensity must also be numerically averaged at the final probe position \(z=L\) as in (26) and shown in Fig. 9c. The sub-Doppler spectra are given by the difference between the probe beam in the absence of the pump and the probe in the presence of the pump.

With this line shape derivation, the influence of different parameters on the calculation can be directly compared to the data. In Fig. 3, it was observed that as the RF power increased, the depth of the core increased. This was posited and experimentally verified as being caused by the opacity of the media to the probe laser. It was also observed that the system only became opaque for RF powers \(>\,25\) W for the 50 Torr case. Figure 10 verifies these trends. In Fig. 10a, it is clear that as the opacity increases, the core depth feature of the line shape does increase. In addition, we see in Fig. 10b that the 50 Torr case does only become opaque to the probe in the core for RF powers greater than 25 W. To calculate these, and the following line shapes, the homogeneous and inhomogeneous widths, \(\gamma \) and \(\Delta \nu _D\), the shifted line centers, \(\nu _0\), and the initial inversion, \(\Delta n_0\), were obtained from Voigt fits to the LAS data for the corresponding pressures and RF powers. While the model agrees qualitatively with the observations, there are several deviations from the observations. For the 5 Torr case (top), the Lorentzian feature at the core of the line is emphasized in the model output as compared to the observations. In addition, the opacity of the model is greater at the higher RF powers (25, 40 W) which corresponds to the Lorentzian core being absent from the lineshape; this is not observed in the data where the Lorentzian core persists at the highest RF powers. The 50 Torr case contains a reversed trend as a function of RF power from 1 to 10 W where an increase in probe transmission is observed, in contrast to the model which predicts a decreasing probe transmission. The 50 Torr prediction also displays the same exaggerated influence of opacity at the highest RF powers.

Fig. 10
figure 10

Output of the line shape model is shown for 5 (a) and 50 (b) Torr and for RF powers of 1 W (), 5 W (), 10 W (), 25 W (), and 40 W (). Increasing the RF power leads to an increased opacity and a line shape that is reduced significantly in the core. Both the 5 and 50 Torr predictions are red-shifted due to pressure effects with a stronger influence of the 50 Torr case, as expected

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Figure 6 shows the pump laser intensity dependence on the line shape. The model results for the same parameters are shown in Fig. 11. It is clear that as the laser intensity increases from 1.8 to 17.7 W/cm\(^2\), the line shape becomes less opaque in the core. The reduced opacity is a direct result of the increased saturation in the wings of the laser profile. As the pump intensity increases, the saturation will increase for the full spatial profile and will lead to a larger volume being saturated. Furthermore, both the model and experimental data show that the width of the Lorentzian core changes minimally with pump intensity, likely due to the redistribution of population by velocity-changing collisions. Similar deviations between the model and observations are observed for the laser intensity predictions. Specifically, at 5 Torr, the predicted trend appears to emphasize the Lorentzian core and overemphasize the opacity at line-center at the highest RF powers. At 50 Torr, the model shows a factor of eight reduction in the probe transmission in contrast to the factor of four reduction in the observation as the laser intensity decreases 17.7–1.8 W/cm\(^2\).

Fig. 11
figure 11

Output of the line shape model is shown for 5 (a) and 50 (b) Torr and for incident laser intensities of 17.7 (), 14.1 (), 10.6 (), 7.1 (), 3.5 (), and 1.8 () W/cm\(^2\). Increasing the laser intensity leads to a more saturated line shape resulting in less absorption of the probe beam in the core. Both the 5 and 50 Torr predictions are red-shifted due to pressure effects with a stronger influence of the 50 Torr case, as expected

Full size image

A model prediction of the dual-Voigt profile shown in Fig. 7 could be treated in several ways. Considering the line shape to likely be the result of spatial variation in the Ar (\(s_5\)) number density as well as the electron density, spatially dependent number densities and absorption cross sections could be substituted in place of those given above. This would further modify the absorption coefficient both radially and longitudinally. The spatial dependence of both the Ar (\(s_5\)) and electron densities would have to be analyzed; however, using spatially resolved laser absorption spectroscopy. This would require experimentation not conducted in this work and is left to a future analysis.

5 Conclusion

Saturation spectroscopy was conducted in an optically thick Ar plasma. The expected line shape consisted of a single, narrow Lorentzian centered on a Doppler pedestal, consistent with a sub-Doppler spectrum in the presence of velocity-changing collisions. The observed structure was that of a multi-peak line shape where the Lorentzian core appeared to be centered on a recessed pedestal. The structure was discovered to be due to the plasma opacity relative to the transmitted probe intensity and was characterized using laser absorption spectroscopy and inter-modulated fluorescence spectroscopy. The line shape was observed for Ar pressures of 1.3–50 Torr, pump laser intensities of 1.8–17.7 W/cm\(^2\), probe intensities of 0.1–1.8 W/cm\(^2\), and RF powers of 1–40 W. Increased laser power was shown to correlate with increased probe transmission, especially about the core of the line shape, reducing the influence of opacity due to more saturation of the pump volume. The opaque line shape was observed at all powers for the low-pressure tests, but only resulted at RF powers greater than 25 W for the high-pressure cases, consistent with the level of opacity measured by the LAS experiment. At the lowest measured pressures (1.3 and 3.6 Torr), the Doppler-limited line shape was shown to be non-Voigt-like and well described by a dual-Voigt profile with a second shifted and broadened peak. The source of this structure may be from the spatial inhomogeneities of the plasma or from coupling of the high-intensity laser field and the ions and free electrons in the plasma.

A line shape model was developed that used the laser propagation equation combined with an absorption coefficient for co- and counter-propagating beams derived from the rate equations for an equivalent two-level system. The resulting line shape was shown to have good qualitative agreement with the observations and can be extended in the future to consider more complex plasma features, such as spatial inhomogeneity.

The high absorbance measured in these spectra is a positive result for the DPRGL. Optically thick conditions are caused directly by the number density of the ground state which will need to be greater than \(10^{13}\) cm\(^{-3}\) for efficient operation. While this work does not reach this order of magnitude, it does display a moderate increasing trend of the number density as a function of the RF power. Most of the complex structure associated with the sub-Doppler spectra is expected to be resolved by the shift away from a Voigt profile to a simple Lorentzian at pressures above 50 Torr. Spatial inhomogeneities in the plasma would result in spatial variations of the output laser intensity leading to a reduced beam quality for a diode-pumped rare gas laser. Coupling between the electromagnetic field of the laser and the plasma could either be a boon, if it was to cause increased production of the metastable state within the pump beam, or a detractor if increased ionization along the laser path causes laser-induced arcing. In the latter case, this arcing threshold would set an upper bound for the maximum pump intensity of the DPRGL. More research must be conducted on this subject to fully understand this phenomena and to discover how it will influence the rare gas laser system.