On identifying critical parameters in an amplification without inversion setup in mercury

Lasing without inversion is a compelling method based on the generation of coherences between atomic levels in order to produce UV radiation. While the proof-of-principle of this scheme has been shown for several decades, so far no radiation at a significant shorter wavelength than the necessary drive fields has been observed. In a recent publication Rein et al (2022 Phys. Rev. A 105 023722) have made experimental progress towards this goal. In this paper, we investigate the necessary improvements to their setup and discuss the experimental steps taken to achieve those goals. Specifically, we report on the improvement with respect to the laser sources.


Introduction
Despite considerable progress in the realm of laser development, continuous-wave (cw) lasers in the ultra-violet (UV) or even vacuum ultra-violet (VUV) wavelength region remain a * This paper is dedicated to the memory of Bruce W. Shore, a great friend and superb scientist. We will always remember the many discussions and conversations on physics, science in general as well as life. Rarely, one encounters such widely interested individuals such as Bruce. During his many visits to Darmstadt, he was always keen to learn about the progress of the experiments described in this paper. One of us (T.W.) is deeply grateful for the hospitality Bruce and his wife Randi offered him throughout the years. We miss him and he will always be part of our lives. * * Author to whom any correspondence should be addressed.
Original Content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. challenge. The pump power required for a population inversion on the laser transition scales at least with the fourth power of the laser frequency ω 4 [1]. Thus, most often nonlinear processes are used to generate cw laser radiation in the UV or even VUV [2,3]. However, these techniques are limited by the availability of suitable materials. Furthermore, to generate radiation at an increasingly shorter wavelength, the fundamental or coupling lasers have to operate at an increasingly shorter wavelength as well.
In the late 1980s, a different concept, i.e. lasing without inversion (LWI), was introduced in [4][5][6][7] to overcome this limitation. The key idea of this concept is the suppression of the coherent absorption on the lasing transition by employing quantum interference effects such as coherent population trapping or electromagnetically induced transparency [8][9][10][11].
Since then, the feasibility of cw LWI has been demonstrated in three-level schemes for rubidium [12] and sodium [13,14], but until now there has been no demonstration of LWI where the laser transition had a significantly shorter wavelength than the coupling lasers. In three-level systems, the Doppler effect reduces the gain once the wavelength difference between the coupling laser and the lasing transition becomes large [15]. However, a four-level scheme in mercury first proposed in reference [16] makes it possible to overcome this limitation, making LWI at 253.7 nm feasible while the shortest wavelength of the coupling lasers is at 435.8 nm.
For a detailed theoretical description on the underlying mechanisms of the aforementioned LWI scheme in mercury, we refer to [16][17][18]. The four-level coupling scheme of mercury is depicted in figure 1. It utilizes both a strong driving field at 435.8 nm and a weak driving field at 546.1 nm as well as a weak probe beam at 253.7 nm. If those laser beams are superimposed inside a mercury filled absorption cell employing a particular geometric arrangement, the Doppler effect can be canceled, greatly simplifying the setup as no atomic beam apparatus is necessary. An additional incoherent repump laser at 404.7 nm is necessary to prevent population trapping in the 6 3 P 0 level.
Recently, our group observed a reduction of absorption in this three-photon coherence when the two driving fields were applied and the 6 1 S 0 − 6 3 P 1 -transition was incoherently pumped [18]. However, the available incoherent pump was not sufficient for the actual observation of amplification without inversion (AWI). If that was the case, placing the cell inside a cavity would lead to the observation of LWI provided that the gain is larger than the losses in the resonator.
In [17], a detailed theoretical model of this LWI scheme was derived. By obtaining a steady state solution of the radiation damped optical Bloch equations of the full 13-level system caused by Zeeman splitting, the imaginary part of the linear susceptibility χ ′′ can be calculated. Depending on the sign, χ ′′ directly represents absorption or gain in mercury for the weak probe beam. In [18], this model was further refined by including the finite interaction time of the light and the mercury atoms due to the thermal velocity and by modeling the incoherent pump as a laser beam with a specified (broad) spectral width. The experiment showed excellent agreement with the model predictions. Assuming the knowledge of all relevant parameters, this allows to calculate real transmission/ absorption values of a weak probe beam through a mercury cell.
As stated above, the observed experimental results in [18] did not warrant the observation of AWI. As a critical parameter, the power and spectral width of the incoherent pump was identified. In this paper, we use these models of [17,18] to thoroughly study the transmission of the probe beam in relation to several other parameters in order to identify the most critical ones to achieve AWI. These include the power and spectral width of the incoherent pump as well as the power levels and the linewidths of both driving fields resulting in a six-dimensional parameter space. Since the solution of the optical Bloch equations in the stationary state has to be calculated numerically, we rasterize the parameter space with the parameters given in table 1. This leads to 6 6 = 46656 separate calculations. On a computer with a modern six core processor with twelve threads employing parallelization techniques, this resulted in a total calculation time of several days. Based on the results of these simulations, the experimental setup was specifically improved in order to best match the critical parameters.
In this paper, we discuss the simulations and describe which steps have been taken in order to satisfy these requirements. In section 2 we will show an increase in the incoherent pump power by utilizing an elliptical focusing enhancement cavity. The spectral width of the incoherent pump will be artificially broadened by modulating white noise onto the pump beam via an acousto-optical modulator (AOM) driven by a home built AOM driver. Section 3 shows an increase in the power of the strong driving field by resonant frequency doubling employing periodically poled potassium titanyle phosphate (PPKTP). Additionally, a theory is derived for resonant second harmonic generation taking self heating effects into account. In section 4, we discuss the importance of the linewidth and frequency stability of the driving fields. Different frequency stabilization schemes employing a mercury bulb and modulation transfer techniques are evaluated.

Incoherent pump at 253.7 nm
Although the parameter space is six-dimensional, we show a two-dimensional excerpt of the calculated simulation data in figure 2 to exemplary discuss the influence of the incoherent pump beam on the achievable probe beam transmission. The power of the strong driving field is set to 170 mW, the power of the weak driving field to 3.95 mW. The linewidths of both driving fields are kept at 60 kHz. These are the same parameters as used for the calculations in section 5 of [18]. In the experiment, a maximum pump power of 40 mW with a spectral width of approximately 0.5 MHz was used. For a better readability, a two dimensional spline fit with the arguments of pump power and spectral width was used to interpolate the data between individual points of the simulation. A higher spectral width is hugely beneficial, and so is an increase in pump power. Of course, in order to be able to observe AWI, a transmission larger than 1 is required.

Increase in power
The incoherent pump beam was derived from the same laser system that was used to generate the weak probe beam. It is based on an external cavity diode laser (ECDL) at 1014.8 nm which is amplified in an ytterbium-doped fiber. A resonant frequency doubling stage employing lithium triborate (LBO) generates radiation at 507.4 nm, which is then transported to the optical table of the LWI experiment by an optical fiber to an additional frequency doubling stage employing barium borate (BBO) in a standard bow-tie configuration cavity. As reported in [18], in the original setup a stable output of 40 mW at 253.7 nm was generated, limited by severe degradation effects of the BBO crystal [19][20][21]. As mentioned above, this was not enough to achieve AWI.
After improving the source of the fundamental radiation at 1014.8 nm and the first doubling stage, the 507.4 nm radiation is now transferred by a 7 m long photonic crystal fiber suitable for higher optical power levels in order to avoid detrimental effects by stimulated Brillouin scattering. The fiber (LMA-PM-15) has a large mode field diameter of 12.2 ± 1.5 µm. With a measured transport efficiency of up to 70%, a maximum of 2 W is available for frequency conversion into the ultra violet.
The degradation in BBO can be overcome by an elliptical focusing cavity as reported in [22]. Elliptical focusing inside the crystal leads to much smaller peak intensities while still maintaining a high conversion efficiency. By using a design process employing evolutionary algorithms described in [23], a novel enhancement cavity was designed that allowed maintaining high UV powers over hours of operation [22]. We were able to utilize the very same design in our experiment. Therefore, we refer to previous publications [22,23] for further details about the enhancement cavity.
Overall, we succeeded in generating up to 420 mW at a wavelength of 253.7 nm, representing an increase by a factor of ten in available UV power compared to [18].

Spectral width of the incoherent pump
The results of the simulation show that a further broadening of the incoherent pump's spectral width is the most crucial parameter to enhance the gain and achieve AWI. In [18], an acousto optical modulator (AOM) was employed for the separation of the probe and pump beam. The AOM was modulated by a commercial AOM driver with a frequency of 200 MHz, leading to a deflection of the pump beam into the first diffraction order and a frequency shift of 200 MHz. By modulating the driving signal with white noise, a broadening of the spectral width of the pump is feasible. However, employing commercial drivers did not yield satisfactory results [18].
Therefore, based on the ideas of [24], we designed an analog AOM driver from scratch using discrete elements. A schematic sketch of our driver is shown in figure 3. The core of this setup is a voltage controlled oscillator (VCO) transferring the amplitude modulated white noise of a frequency generator into the frequency space. Since the generated frequency noise is centered around 1174 MHz, we employ a second local oscillator VCO. Employing a frequency mixer, we can generate the non-linear differential frequency of both VCOs. By controlling the voltage of the local oscillator, we can shift the differential frequency to be located at 200 MHz.
The frequency mixed signal is filtered by a low-pass filter with a cut-off frequency of 505 MHz to suppress harmonic and sum frequencies. Since our AOM needs an rf-power of up to 2 W to achieve the highest diffraction efficiency, we amplify the generated signal in two consecutive amplifiers. Since both of the amplifiers have a fixed gain, we employ a variable attenuator before the second amplifier to control the overall output power of the whole driver. Note that the low-pass filter, the attenuator and the first amplifier are in principle interchangeable. However, the chosen configuration (cf. figure 3) was found to produce the highest measured side mode suppression ratio.
With this setup, we were able to generate the driving signal shown in figure 4 measured with an electrical spectrum analyzer (ESA). The signal was taken before the second amplifier in order not to exceed the damage threshold of the ESA.  When driving the AOM with our generated signal (with the second amplifier), the diffraction efficiency is identical to driving with the commercial driver.

Additional broadening techniques.
For the further course of the experiment, there are some possibilities to further broaden the spectral width. A double pass configuration of the AOM can be employed to modulate the pump beam twice. Since this will result in a convolution of two Gaussian distributions, the FWHM will be enhanced by a factor of √ 2. A second possibility is the use of an electro optical modulator (EOM) after the AOM. If the modulation index of the EOM is sufficiently high, the first few orders of sidebands will be of similar magnitude. The resulting spectrum of the light passing through AOM and EOM will approximately become an overlay between the Gaussian distributions with a FWHM determined by the AOM located at the sideband frequencies of the EOM. If the sideband spacing and therefore the modulation frequency of the EOM is chosen to be smaller than the spectral width of the beam after the AOM, this will result in a further broadening of the spectrum.
Additionally, a particular pump spectral width can be linked to a certain range of atom velocities via the Maxwell-Boltzmann distribution, addressing approximately all mercury atoms, whose velocity component in beam direction lies inside this range. If the spectral width is significantly smaller than the Doppler broadened linewidth of the transition, the pump radiation could be separated, so that two perpendicular pump beams overlap in the cell. Hence all atoms with one velocity component in any of both beam propagation directions inside the velocity range could be addressed.

Power levels at the driving fields
In figure 5 the dependence of the transmission on the power levels of the driving fields are shown. The power of the incoherent pump beam was set to 200 mW with a spectral width of 50 MHz. The linewidth of the driving fields was set to the experimental values. While the power of the weak driving field at 546.1 nm has a negligible impact on the transmission, a strong dependence on the power of the strong driving field at 435.8 nm exists. Even for a power above 1 W a significant gain in transmission can still be achieved indicating that the power of the strong driving field is one of the most crucial parameters for scaling the transmission in order to overcome optical losses in an LWI setup. In this configuration AWI becomes feasible if the power of the strong driving field is doubled with respect to the previous setup, i.e. to 340 mW.

Resonant frequency doubling employing PPKTP
Previously the strong driving field was generated by a system consisting of an ECDL at 871.6 nm, a tapered amplifier and an enhancement cavity containing a potassium niobate (KNbO 3 ) crystal. For a detailed description of this system, we refer to [25]. Overall, a frequency stabilized power of 224 mW at 435.8 nm was generated mainly limited by blue light induced infrared absorption (BLIIRA) followed by the generation of a thermal lens inside the KNbO 3 crystal as also reported in [26,27].
To overcome the limitations of BLIIRA, we replaced the KNbO 3 crystal by a periodically poled potassium titanyle phosphate (PPKTP) crystal. In first-order quasiphasematching, PPKTP has a lower effective nonlinear coefficient of 9.5 pm V −1 in comparison to 13.8 pm V −1 of KNbO 3 [28], but is shown to be less prone to BLIIRA [27]. To still achieve a high conversion efficiency, we employed a non-waveguided crystal with a poling period of 4.45 µm and dimensions of 2 mm × 1 mm × 15 mm inside an enhancement cavity. Despite the bigger length compared to 10 mm of the KNbO 3 and a different refractive index of 1.74 compared to 2.28 at the fundamental wavelength, the preexisting enhancement cavity was calculated to still be stable and was reused.  [18]. Note that in contrast to the parameters given in table 1 we additionally calculated with even higher powers for the strong drive in this figure. The beam waist inside the crystal is 52.6 µm and is approximately twice as large as the optimum waist according to Boyd and Kleinman's theory [29] to avoid thermal lenses in the crystal but is still small enough to keep the conversion efficiency high.
Over the span of six hours, an average power of 416 mW with a peak-to-peak-stability of 20 mW could be generated (cf figure 6) at 435.8 nm. A slight adjustment of the tapered amplifier led to an increased power level at the fundamental compared to the measurements in figure 7. Also, the crystal temperature has been reduced to 106.8 ± 0.1 • C. The laser had to be frequency stabilized, as a frequency drift of the ECDL corresponded to a decline in SH power over time. A remaining slow drift of approximately 1.8 mW h −1 , depicted as an orange line in figure 6, can most likely be attributed to thermal effects. All in all, the power of the strong driving field in combination with the enhancements made in section 2 should allow the observation of AWI of the 235.7 nm probe beam. Figure 7 shows the generated SHG power (left) and the corresponding conversion efficiency (right) versus the fundamental power before the cavity. For the diamond shaped data points, the temperature of the crystal was optimized for each separate measurement point to compensate for self heating, ranging from 110.5 ± 0.1 • C for the lowest fundamental power to 108.8 ± 0.1 • C for the highest. In this measurement, a maximum power of 352 ± 18 mW at a conversion efficiency of 40.7% has been achieved. Note that in contrast to [25], we calculate the conversion efficiency based on the fundamental power before the cavity, not based on the power that is coupled in. This allows for a better comparability between different experimental setups. Taking into account our measured coupling efficiency of 72%, which is strongly limited by the beam quality due to the tapered amplifier, a conversion efficiency of 56.6% is achieved. In principle, this could be further increased by spatially filtering the incident fundamental beam, but since the enhancement cavity itself acts as a spatial filter for the fundamental beam, this has no benefit for the overall experiment.
Clearly, it is important to optimize the temperature for each specific power level of the fundamental. However, as observed before [27], it is equally important to carefully adjust the position of the crystal in the direction of beam propagation in regard to the maximum SH power to minimize the effects of thermal lensing.
As shown in figure 5 the transmission of the probe beam through the mercury cell increases even with a strong drive power far greater than 400 mW. In order to achieve even higher power, a careful redesign of the enhancement cavity could be performed. For simplicity, we used the identical cavity for the 15 mm PPKTP and the previous 10 mm KNbO 3 crystals. Specifically, taking into account the findings of [27], the generation of a thermal lens causing a mode-mismatch between the cavity eigenmode and the incident optical field could be included during the design process. This will lead to a higher mode matching efficiency and therefore to a greater conversion efficiency of the setup. An alternative possibility would be the use of a Ti:sapphire laser employing intracavity frequency doubling. Setups like these demonstrated output powers of 1 W in the blue spectral range [30], although here thermal lenses are the limiting factor, too.

Depleted pump resonant SHG.
In the following, we derive a theoretical model in order to fit the data shown in figure 7. A theory on resonant SHG is given in [31]. Here, a formalism is developed to calculate the circulating fundamental power P c inside a cavity while taking into account nonlinear frequency conversion of the fundamental characterized by the parameter γ SHG . This parameter has the physical unit of 1/W and can be calculated by the formalism published by Boyd and Kleinman [29]. The circulating power is dependent on the transmission t 1 of the coupling mirror, the coupled fundamental power P 1 and the ABCD cavityreflectance parameter r m . Note that we choose which differs from [31] due to the fact that we employ a ring resonator as also used in [32] in contrast to a linear cavity. Also, it is convenient to combine the mirror reflectivities and The generated SH power can then be calculated as To achieve a better agreement with our experimental data, we expand on the formalism in [31] in two steps. First, we substitute the coupled fundamental power in equation (1) with the product of the incident power P inc right before the cavity multiplied by a factor ϵ mode representing the mode matching efficiency. This allows for an easier comparison between different setups. Second, we replace the product of the Boyd-Kleinman parameter and the circulating power in equations (2) and (3). This term reflects the decrease of the single pass conversion efficiency due to depletion of the fundamental power along the crystal caused by frequency conversion. This becomes relevant for high-efficiency setups [33] and ensures that the calculated conversion efficiency does not exceed 100%.
This model is fitted to our experimental data and is depicted by the red lines in figure 7. The mode matching efficiency was measured to be ϵ mode = 0.72 and the transmission of the coupling mirror is t 1 = 0.035. The linear losses were estimated by a measurement of the cavity finesse and fine tuned with the experimental data to L = 0.0275. In summary, a near perfect agreement between theory and measurement for optimized temperature is found.
The temperature difference of 1.7 K between the lowest and highest measured power of figure 7(red squares) is comparable to the temperature acceptance bandwidth (full width half maximum) of 1.4 K, measured in single pass configuration. This shows that optimizing the crystal temperature in a given setup is crucial while varying the input power. This is further corroborated by the data (blue circles) in figure 7, which shows the SH power and conversion efficiency without separately optimizing the crystal temperature. Here, the crystal temperature was only optimized once at the highest fundamental power. Especially for low powers, this has a huge impact on the conversion efficiency.
To reflect this, we expand the above resonant SHG theory by taking into account the self heating of the crystal by absorbed fundamental radiation. If the crystal temperature changes, a deviation from the optimal phase matching condition comes into play making a modification of the Boyd-Kleinman parameter necessary: This substitution has to be executed after the former replacements 4 and 5. Here, T 0 represents the low power phase matching temperature without taking any additional heating processes into account, T set denotes the set temperature of the crystal oven and ∆T the temperature difference in the middle of the crystal due to self heating. The factor of 2.05 was found numerically such that the expression 6 has a FWHM equal to our measured temperature bandwidth of 1.4 K. To calculate ∆T, we use a simplistic approach employing Fouriers law of heat conductivity. In a steady state, the energyQ de posit deposited per second by fundamental radiation inside the crystal can be approximated bẏ where l denotes the crystal length and α denotes the linear absorption coefficient for the fundamental, enhanced due to BLIIRA by the term ∆α BLIIRA P 2 in dependence of the SH power P 2 . If we now assume that this energy deposit causes an increased temperature by ∆T along the beam axis in the middle of the crystal compared to the temperature of the copper block used to stabilize the crystal's temperature, we can employ Fouriers law of heat conductivity in a simplified approach. The heat transferQ trans to the copper block is given by a rough estimation oḟ where l 1 and l 2 denote the perpendicular dimensions of the crystal and k its thermal conductivity. Note that this simplified model will in general overestimate the heat transferQ trans . Therefore, we added a correction factor c smaller than one. The copper block is treated as an infinite reservoir with a constant temperature. To get a steady state for ∆T, we can equate equations (7) and (8). This allows us to calculate the generated SH power for non-optimized crystal temperatures. Calculated results of this model can be seen as the blue lines in figure 7 accompanying the circular markers. The thermal conductivity k = 3.3 Wm −1 K −1 [27,34] as well as the fundamental absorption coefficient α = 0.01 cm −1 are taken from the literature [27]. T 0 was estimated by the optimized phase matching temperature of 110.5 K for the lowest input power and therefore for minimized absorption effects. For a better agreement between theory and experiment, we chose a slightly lower temperature of T 0 = 110.45 K. T set is the set temperature of 108.8 K of the crystal oven. The unknown parameters were found to be c = 0.4 and ∆α BLIIRA = 6.0 W −1 . Even with a very simplistic approach to crystal self heating, the overall qualitative form can be reproduced. This could be further refined by taking into account more complex heating simulation techniques such as finite element methods.

Frequency stability and linewidth of the driving fields
The current laser systems at 435.8 nm and 546.1 nm have approximately a linewidth of 60 kHz, as this results in the best fit of the simulations to the experimental data [18]. In the simulations described in this work we also varied the linewidths of both driving lasers. Exemplary results can be seen in figure 8. For this figure, the power of the pump beam was set to 200 mW with a spectral width of 50 MHz. The power levels were 340 mW for the strong drive at 435.8 nm and 3.95 mW for the weak drive at 546.1 nm, respectively. It can be clearly seen that a decrease in linewidth benefits the transmission. However, for linewidths smaller than 10 kHz no significant improvement is found. Therefore, special care should be taken when setting up the frequency stabilization schemes of both driving fields as the latter have an enormous influence on the linewidth of the radiation.
Although the linewidths of our lasers are near the optimum, in the following we describe two alternative approaches for their laser stabilization. On the one hand we replaced the rf discharge cells by ordinary mercury bulbs. On the other hand we investigated modulation transfer spectroscopy (MTS) instead of polarization spectroscopy (PS).

Comparison of mercury bulb and rf discharge cell
In the experimental setup used in [18], both driving laser systems were frequency stabilized on the corresponding transition of the 202 Hg isotope by Doppler-free PS [35,36]. Since the lower states of those transitions are not populated at room temperature, a radio frequency (rf) discharge was employed to excite atoms into the lower states of the respective transitions [25]. However, some caveats with this setup were encountered. The windows of the mercury absorption cell show a discoloration due to condensation and consecutive diffusion of mercury into the glass leading to increased absorption and a significantly reduced error signal. Furthermore, the ignition of the discharge plasma is highly sensitive to the position of the coil used to couple the rf field into the cell. Those effects were observed on both driving field laser systems and significantly hamper the handling of the experiment.
To overcome those difficulties, we investigated a possible substitution of the discharge cell with a mercury bulb intended for disinfection readily available at a fraction of the cost of absorption cells produced specifically for scientific applications. For characterization and comparison with the discharge cell, a parallel setup of two standard PS experiments [36] was implemented with the laser system at 546.1 nm and two homebuilt differential photo diodes. The pump fields were circularly polarized whereas linear polarized probe beams were used since this configuration yielded better slopes of the error signal than the swapped polarization configuration. The parallel setup allows simultaneous measurements of the PS signals of both the cell and the bulb and individual optimization of the error signals due to separate sets of polarization optics. A suitable lens must be inserted in the bulb setup to compensate for the beam divergence due to the cylindrical shape of the bulb.
The optimized difference signals for the transition of the 202 Hg isotope are shown in figure 9(left). The influence of the 199B Hg-transition with a frequency difference of +31.7 MHz relative to the 202 Hg transition [37] is visible in the error signal of the discharge cell. Despite the slight distortion of the error signal, frequency stabilization was possible. This influence is not visible in the error signal with the bulb since it shows a significant broadening of the signal width. Using the frequency difference between the minimum and maximum of the dispersive features of the isotopes 198 Hg, 200 Hg and 204 Hg, the effective linewidth of the bulb was estimated to be 90 ± 10 MHz in comparison to the width of the discharge cell which was estimated to be 40 ± 5 MHz. Since parameters like the internal pressure of the bulb are not documented, the exact reasons for this broadening are not directly known but effects like pressure broadening appear to be reasonable. Figure 9. PS signals of a rf discharge cell (P probe =0.28 mW, Ppump =1.12 mW) and a mercury bulb (P probe = 0.18 mW, Ppump = 0.98 mW) for the 6 3 P 2 -7 3 S 1 -transition of the isotope 202 Hg (left) and 198 Hg (right). The calculation of the frequency shift is schematically shown on the right. The zero point of the frequency axis was manually set to be located on the gradient of the 202 Hg transition of the discharge cell. Since both data sets were acquired simultaneously, this is sufficient to determine frequency shifts.
A shift of the transition frequency of the bulb towards lower frequencies is clearly visible in figure 9. Using the differences between the frequencies halfway between maximum and minimum of the dispersive signal features of the same isotopes as above, a frequency shift of −17.6 ± 4.6 MHz was obtained.
In conjunction with a smaller signal amplitude, the signal broadening of the PS signal obtained with the mercury bulb leads to a smaller signal gradient and therefore a decreased frequency discrimination. Yet, the laser system can still be stabilized to the same frequency as the discharge cell despite the frequency shift due to the broader stabilization range by applying an electrical offset to the signal. In both setups, a maximized gradient is obtained with high probe powers as long as the pump power is sufficiently high to saturate the pumping process of the atoms into one of the maximum |m F | ground states. This ratio is different to the ratios typically used for PS where small probe powers are desirable to prevent saturation effects from the probe beam. Such an effect is not relevant for our application since we are mainly interested in a maximum signal slope.
We conclude from these results that a commercially available mercury bulb is a very cost efficient alternative to more expensive and elaborate absorption cells for applications not requiring the smallest linewidths. Furthermore, a population of the excited states is more straightforward and allows for easier handling in the lab.

MTS on the 6 3 P 2 -7 3 S 1 -transition in mercury
As shown in [18], not only the linewidth of the lasers is important, but also the overall frequency stability. In a PS setup, the polarization optics and the medium can be sensitive to power or temperature fluctuations, leading to a change of the zero point of the error signal and consequently a frequency drift of the laser frequency.
A possible approach to minimize these influences are methods that include modulation of the laser field, e.g. frequency modulation spectroscopy [38] or MTS [39]. In the MTS scheme, the pump beam is frequency modulated and the modulation is transferred to the initially unmodulated probe beam. The spectroscopic signal is extracted by phase-sensitive detection of the probe signal. Due to the non-linearity of the process, a narrow signal with a flat zero-baseband is created. The zero crossing of the signal is independent of fluctuations in laser power and temperature [40] as long as the effects of residual amplitude modulation [41] are minimized. We therefore performed initial measurements to examine the possibility of using MTS for the frequency stabilization on the 6 3 P 2 -7 3 S 1transition in Mercury.
The rf discharge cell was used for the population of the 6 3 P 2 -state since an electro-optical modulator with a suitable range of modulation frequencies of 14.4 MHz to 22.2 MHz was already available in the laboratory. We used a standard MTS setup [40,42] with a modulation depth of β ≈ 0.9 mostly limited by the rf amplifier used to drive the EOM. The signal component at the modulation frequency of the probe beam signal from a fast, low-noise photodiode is extracted by a bias-tee, amplified and demodulated with a STEMLab 125-14. A comparison between an optimized PS signal and the MTS signal at a modulation frequency of 17.5 MHz is shown in figure 10. The influence of the 199B-transition is also visible in the MTS signal and its asymmetry, but the influence on the stabilization gradient can be eliminated by a suitable choice of phase and the smaller signal width. Despite the higher probe power, the signal-to-noise-ratio (SNR) as well as the overall amplitude of the MTS signal are lower compared to the PS signal. Possible reasons are the high absorption of laser power in the cell windows leading to low powers for the nonlinear modulation transfer process and the limited modulation depth of the EOM. The advantages of MTS for frequency stabilization, mainly the flat baseband and a smaller signal width can still be observed. A stabilization on the transition of the isotope 202 Hg was successful, but the lower SNR lead to a decreased frequency stability compared to PS. This can be improved by a new discharge cell and higher modulation index.
In summary, the MTS setup is a viable alternative to PS for applications where power or temperature fluctuations are the prevalent limiting factors.

Conclusion
In conclusion, we have performed extended simulations of the transmission of the probe beam at 253.7 nm through a mercury vapor cell in dependence of several parameters. Based on these results, we were able to identify the power and spectral width of the incoherent pump as well as the power of the strong driving field at 435.8 nm as the most influential parameters. For the power of the incoherent pump, we employed a frequency doubling stage containing BBO inside an elliptical focusing enhancement cavity, which resolves the degradation problems usually found with BBO. Combined with the use of a photonic crystal fiber for transport of the green laser light, we achieved a UV output of over 400 mW, representing an increase by a factor of ten compared to the setup in [18]. To further broaden the spectral width of the incoherent pump, we built an analogue AOM driver to superimpose white noise onto the laser light. A signal with a FWHM of 39.3 MHz was generated.
For the strong driving field, we replaced the KNbO 3 with a PPKTP crystal. We achieved a stable output above 400 mW, which represents a doubling of the power used in [18]. Additionally, thermal effects and the impact on SHG caused by self heating of the crystal were discussed. By modeling a simplified heating process employing Fouriers law of heat conductivity, the measured SH data can be retraced in excellent agreement.
Also, we discussed the importance of the linewidth of both driving fields onto the transmission. Special care has to be taken to stabilize both laser systems onto the particular mercury line. Additionally to the rf discharge cell used in [18], a commercially available mercury bulb was tested in a Doppler free PS setup. The bulb is found to have a frequency shift of −17.6 ± 4.6 MHz compared to the cell. Nevertheless, the capture range allows for effective locking onto the same frequency as the discharge cell. We conclude that the bulb is a good alternative for a cost effective stabilization. It is much easier in handling, but this comes at the cost of a decreased frequency discrimination while stabilizing. Subsequently, MTS was successfully shown in the rf discharge cell and its performance is compared to PS.
Overall, the calculated simulation data allowed us to modify the experiment in a way that AWI should in principle be achievable. In the near future, measurements in AWI configuration will be performed and compared to the theoretical model at hand. For LWI, a cavity around the mercury cell has to be built. Since the setup employs an uni-directional gain due to the Doppler free configuration, a ring cavity rather than a linear one has to be employed. This will introduce at least three sources of losses by reflective optics in addition to the losses at the windows of the mercury cell. Therefore, the transmission has to be further increased to overcome these losses in order to get an effective gain.
With the improvements introduced in this paper, our simulation show that the spectral width of the incoherent pump still remains the most crucial parameter in scaling the transmission to higher values, followed by the power of the strong driving field. In order to get a better understanding in terms of the scalability, table 2 exemplary shows calculated transmissions for specific selected laser parameters. For the spectral width of the pump further broadening possibilities such as a double pass configuration through the AOM, an additional EOM or splitting of the pump beam were discussed. In light of generating more power at 435.8 nm, the design of a new enhancement cavity for the PPKTP taking thermal lensing effects into account will lead to better mode matching and therefore a higher conversion efficiency. Also, a Ti:sapphire laser with intracavity doubling could potentially lead to an even higher output power.
In view of possible applications of LWI, the frequency quadrupled incoherent pump laser at 253.7 nm has to be replaced to get an effective gain in laser wavelength. An incoherent light source can be found in mercury vapor lamps. Although they can emit sufficient overall power, their isotropic spatial emission profile limits the achievable power density. Since the power has to be focused into the interaction region of the driving lasers, the construction of a reflecting chamber has to be considered allowing for concentrating the radiation of the vapor lamp into the mercury cell. Here, an integrating cavity design based on quartz powder could also be very helpful [43,44].

Data availability statement
The data that support the findings of this study are available upon reasonable request from the authors.