Investigation of spatiotemporal output beam profile instabilities from differentially pumped capillaries
Martin Gebhardt, Emmanuel B. Amuah, Robert Klas, Henning Stark, Joachim Buldt, Albrecht Steinkopff, Jens Limpert
RResearch Article
Vol. 29, No. 5 / 1 March 2021 /
Optics Express
Investigation of spatiotemporal output beamprofile instabilities from differentially pumpedcapillaries M ARTIN G EBHARDT , E MMANUEL
B. A
MUAH , R OBERT K LAS , H ENNING S TARK , J OACHIM B ULDT , A LBRECHT S TEINKOPFF , AND J ENS L IMPERT Institute of Applied Physics, Abbe Center of Photonics, Friedrich-Schiller-Universität Jena,Albert-Einstein-Str. 15, 07745 Jena, Germany Helmholtz-Institute Jena, Fröbelstieg 3, 07743 Jena, Germany Current address: Department of Physics and Astronomy, Aarhus University, Ny Munkegade 120, DK-8000Aarhus, Denmark Fraunhofer Institute for Applied Optics and Precision Engineering, Albert-Einstein-Str. 7, 07745 Jena,Germany * [email protected] Abstract:
Differentially pumped capillaries, i.e., capillaries operated in a pressure gradientenvironment, are widely used for nonlinear pulse compression. In this work, we show that strongpressure gradients and high gas throughputs can cause spatiotemporal instabilities of the outputbeam profile. The instabilities occur with a sudden onset as the flow evolves from laminar toturbulent. Based on the experimental and numerical results, we derive guidelines to predict theonset of those instabilities and discuss possible applications in the context of nonlinear flowdynamics. © 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The use of gas-filled, hollow-core waveguides has revolutionized fundamental research andapplications in optics and photonics. Nowadays, they are an enabling component for nonlinearfrequency conversion to the ultraviolet [1,2] or soft X-ray [3] spectral regions, they are usedfor enhanced gas spectroscopy [4] as well as for nonlinear pulse compression [5]. With theever-growing performance of table-top, solid-state laser technology [6–10], the generation ofextremely short and intense temporal waveforms through external pulse shortening is in highdemand within the community [11]. Because of the broadband waveguiding capabilities, thestraightforward experimental handling and the possibility to realize a broad range of corediameters on the order of 0.1 mm – 1 mm, glass capillaries have been widely used in thespectral broadening step of nonlinear pulse compression. This is especially true when the goalwas to push the pulse durations of high-energy, femtosecond laser systems to the few-cycleregime [12–17]. In fact, it has been demonstrated recently that this approach is also scalableto hundreds of watts in average power [13,18]. Operating a hollow-core fiber compressor in adifferential pumping scheme, i.e. implementing low pressure or vacuum at the optical input ofthe fiber and high pressure at its optical output, was established as a very helpful measure toachieve well-defined input coupling conditions without perturbations due to ionization or thespatial Kerr effect [12–14,16,17,19]. Furthermore, it can be shown that the gas particle densitygradient along the fiber can be employed to wash out the phase-matching of unwanted parametricgeneration via position-dependent changes in the dispersion landscape. Another successfulstrategy to avoid detrimental ionization effects [20] associated with too high laser intensitiesis to use capillaries with very large core diameters (i.e. several hundred times the operation esearch Article
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Optics Express wavelength [16]). Clearly, this also leads to a low fundamental guidance loss for short capillarylengths and can be exploited to build very efficient fiber-based compressors [21]. However, atreasonable lengths of the straight fibers, which are certainly <
10 m for standard laboratories, thegas throughput for such differentially pumped, large-core capillaries can become considerableat a certain pressure gradient. Because the strength of the nonlinear phase accumulation isclosely related to the gas particle density, it is typically desired to increase the pressure at theoptical output of the capillary as much as possible (ideally up to the self-focusing limit), whilethe region around the optical input is evacuated for the above described reasons. This schemeleads to the aforementioned strong pressure gradients and high throughputs. Interestingly, ithas been reported that certain instabilities (of the optical output) occur, if the pressure gradientfor a differentially pumped capillary with 350 µm core diameter and 1 m length exceeds 3 bar(nitrogen) [22], but the authors neither investigated this effect nor explained its origin. This raisesthe question of whether and to which extent a high gas throughput can have a detrimental impacton the output laser characteristics. Answering this question could be especially interesting for thewavelength (up-)scaling of nonlinear pulse compression using differentially pumped capillaries.This is because the higher possible gas pressures (due to the wavelength dependence of theself-focusing limit [23]) and the increased fiber dimensions (to allow for reasonable transmissionand spectral broadening [20]) are not necessarily compatible with a low gas throughput.In this work, we show that extensive gas volume flow through capillary waveguides can lead todynamic changes of the output beam’s spatial power distribution. In the experiments describedherein, we found that the spatiotemporal fluctuations occur mostly in time scales in the 1 ms– 10 ms region. Furthermore, we have identified that the instability onset is closely linked tothe Reynolds number, which is a benchmark for different volume flow conditions [24]. Basedon the detailed observations in this work, including experiments with multiple geometries andgas species, we derive guidelines that will help to predict the onset of the observed instabilities,which are clearly harmful to laser applications. At the same time, this work opens up a potentialroute toward easy and straightforward experimental characterization of volume flow throughcapillaries. Such optical measurements will be a valuable tool for fundamental and appliedresearch in the field of complex nonlinear flow dynamics [24–26].
2. Experimental setup and results
The principal outline of the experimental setup can be seen in Fig. 1(a). For the experimentspresented herein, we used a mode-locked oscillator, that provided a femtosecond pulse train ata repetition rate of 40 MHz. The central wavelength of the pulse spectrum was around 1.03µm. In the initial experiment, we launched 100 mW of average power to the capillary under test(500 µm inner diameter, 5 m length) and optimized the input coupling for maximum transmissionand fundamental mode operation. In fact, the capillary dimensions were chosen such that theyrepresent reasonable experimental conditions for high-energy nonlinear pulse compression. Thetransmitted power was 70 mW, which is about 80% of the expected power transmission of thewaveguide’s fundamental mode (the deviation from the theoretical transmission is attributed tocoupling losses). In the experiments, the capillary was set up such that it connected two sealedvacuum/pressure chambers. The pressure in the chamber containing its optical input was held at p =
10 mbar using a vacuum pump (XDS35i, Edwards, nominal pumping speed 35 m /h),while the upstream pressure p at its optical output was varied.We characterized the diverging output beam with a high-speed camera (HSC (V611, Phantom))and with a fast photodiode (PD (PDA36A(-EC), Thorlabs)). For the photodiode measurements,only a fraction ( ∼
20% of the total power) of the beam was selected with a stationary rectangularaperture, while most of the power was blocked (see Fig. 1(b)). This way, fluctuations in the modeprofile translate to a temporally varying photodiode signal, detected with a fast oscilloscope(HDO6104, Teledyne Lecroy), from which comparable values such as a standard deviation esearch Article
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Optics Express
HSCp p to vacuum pump from gas supply PD a) b) PD Fig. 1. a) Experimental setup consisting of the capillary under test, an evacuated, opticalinput vacuum chamber held at a pressure p and an optical output pressure chamber heldat a pressure p . The laser light was coupled to the fundamental mode of the differentiallypumped capillary and the spatial output was characterized with a fast photodiode (PD) or ahigh-speed camera (HSC). b) Typical output beam profile and approximate position of therectangular aperture (dashed line) employed in the photodiode measurements. The color barshows the normalized counts. or a frequency spectrum can be derived. Note that this procedure has been established tocharacterize the dynamics of transverse mode instabilities in high-power fiber amplifiers [27].For the photodiode measurements presented herein, the maximum bandwidth limited by theelectronic signal amplification was 1.6 MHz. As it will become clear in short, this is sufficientfor the investigation of the temporal instabilities observed herein.In the initial experiment, nitrogen was used to create the pressure gradient and volume flowthrough the capillary under test. At first, the time domain photodiode signal was analyzed for anabsolute pressure p = p at the optical output side was increased, leadingto increasing gas throughput, while the photodiode signal was carefully monitored. Figure 2presents the behavior of the standard deviation as the upstream pressure is increased. At anupstream pressure p = Fig. 2.
Standard deviation of the photodiode signal for different upstream pressures. Thedata points are connected by the dashed line to guide the eye.
Subsequently, we have performed a more detailed evaluation of the measurements for p = p = <
10 kHz when the pressure p was set to 4.2 bar. In this case,the integration of the power spectral density starting from the high frequency side (from 2 MHzdown to 2 Hz) yields a relative intensity noise (RIN) of about 10% (Fig. 3(f)). In contrast to this, esearch Article Vol. 29, No. 5 / 1 March 2021 /
Optics Express the spatial RIN for p = Fig. 3.
Measured photodiode signals for a) p = = It appears that most instabilities manifest themselves at frequencies between 0.1 kHz and1 kHz. This enabled us to monitor the suspected spatiotemporal changes in the output beamprofile using a high-speed camera with a 21 kHz frame rate and 5 µs integration time. Figure 4presents an evaluation of the frames, captured at p = p = σ -method) in both of these axes. The comparison between Fig. 4(a) esearch Article Vol. 29, No. 5 / 1 March 2021 /
Optics Express b) p = 4.2 bara) p = 2.8 bar Frame Frame
241 1824 3139 241 1824 3139
Fig. 4.
Evaluation of high-speed camera frames (time between capturing two consecutiveframes is 47.6 µs) for a) 2.8 bar at the optical output and b) 4.2 bar at the optical output.The variable x represents the horizontal direction and the variable y represents the verticaldirection. Insets: representative beam profiles at frame numbers 241, 1824 and 3139normalized to the maximum counts of frame 3139, respectively. The insets show the full512 ×
512 pixels of the recorded frames (pixel pitch: 20 µm). For the full high-speed videos,see Visualization 1 and Visualization 2, respectively. The real-time duration of the fullhigh-speed videos is 0.164 seconds (2.8 bar at the optical output) and 0.389 seconds (4.2 barat the optical output). The visualizations are rendered in slow-motion (350x). and 4(b) shows that above the threshold pressure, there is a significant temporal variation inthe output spatial beam profile (e.g. compare frames 1824 and 3139, respectively). Generallyspeaking, it is found that much of the output power is still spatially confined in a central lobe, ascould be expected from the fundamental mode of the capillary without strong pressure gradient.However, it can be observed that the beam area, center of mass and peak intensity undergotemporal fluctuations. Clearly, a laser output with such temporal and spatial properties is hardlyusable, especially if the laser system operates at a repetition rate in the 0.1 kHz – 10 kHz range.In order to better understand this behavior and to potentially predict the instability threshold fordifferent experimental conditions, it is important to investigate the origin of the above describedobservations.
3. Discussion
Because the observations described so far are clearly linked to the gas volume flow through thecapillary, we have implemented an iterative numerical formalism to calculate the throughputand to estimate the Reynolds number Re . The theoretical basis for this is described in detail inRef. [28]. Herein, we assume that the transition region between laminar and turbulent flow islocated between Re = Re = Q , while elsewhere, it is alsopossible to use the following explicit equations [26,29,30]: Q lam = π · η · d L · ( p − p ) , (1) Q tur = d · (︄ π · d ( p − p ) L )︄ / · (︃ RTM mol )︃ / · (︃ πη )︃ / . (2)Here, Q lam is the throughput assuming laminar flow conditions, η is the dynamic viscosityof the fluid, d is the capillary inner diameter, L is the capillary length. Q tur is the throughput esearch Article Vol. 29, No. 5 / 1 March 2021 /
Optics Express assuming turbulent flow conditions, R is the universal gas constant, T is the temperature of theinflowing gas, M mol is its molar mass and the pressures p and p are defined above. In Ref. [28],the Reynolds number is defined as Re = M mol QRTB η , (3)where B = π d is the perimeter of the circular waveguide channel. With this formalism, it ispossible to calculate the throughput and the Reynolds number for a differentially pumped capillary(nitrogen) with 5 m length and 500 µm inner diameter. The results of this calculation are shownin Fig. 5. We note that for p > p =
10 mbar,because the flow is chocked.
Fig. 5.
Comparison of volume flow and Reynolds number based on analytic solutions forlaminar flow (Eq. (1)), turbulent flow (Eq. (2)) and for the numerical model as described inRef. [28] depending on the nitrogen pressure p . The capillary under consideration is 5 mlong with 500 µm inner diameter. It can be derived from the Reynolds number, that the above described measurements at around p ∼ Re < p = esearch Article Vol. 29, No. 5 / 1 March 2021 /
Optics Express
Reynolds number a) d = 200µm,L = 1 m,N2e) d = 350µm,L = 1 m,N2 f) d = 350µm,L = 1 m,Arc) d = 500µm,L = 5 m,N2 d) d = 500µm,L = 5 m,Ar h) d = 500µm,L = 0.35 m,Arg) d = 500µm,L = 0.35 m,N b) d = 200µm,L = 1 m,Ari) d = 500µm,L = 0.35 m,He Fig. 6.
Compilation of measured RIN values for 9 different experimental configurationsas specified in a) to i). The length to diameter ratio ranges from 700 (g-i) to 10000 (c,d) with gas species argon, nitrogen and helium. The top x-axis represents the Reynoldsnumber boundaries for the transition region between laminar and turbulent flow as well asthe Reynolds number for maximum RIN. The colored background is to guide the eye withregard to the flow regimes. esearch Article
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Optics Express output facets. The measurements are compared to the respective Reynolds numbers (calculatedusing the above-mentioned numerical methods) in Fig. 6.The RIN measurements presented in Fig. 6 show, without exception, a significant increasein spatiotemporal instability of the output beam profile, if the flow is in the transition regionbetween laminar and turbulent. For a variety of different geometries ranging from L = d to L = d , and for different gas species, we found that the peak of our (instability-) measurementvariable is around a mean Reynolds number of 2400. Interestingly, the measured RIN drops forflow conditions in the fully turbulent regime at Re > ∼ > Re > Re >
4. Conclusion and outlook
In this work, we have observed that high throughputs can cause spatiotemporal instabilities in theoutput beam profile from differentially pumped capillaries. Resulting from numerical modellingand the experimental investigations presented herein, it is found that the onset of those stronginstabilities is related to the Reynolds number Re = Re = p , the downstream pressure p , the fiber dimensionsand the gas species in a way, such that the Reynolds number (Eq. (3)) does not exceed a value of2000. The necessary throughput calculations can be performed using Eq. (1), since laminar flowis desired. This gives, for the special case that the pressure at the optical input of the capillary isnegligible as compared to the upstream pressure p : p < · (︃ RT η M mol )︃ · (︃ Ld )︃ · √ p < esearch Article Vol. 29, No. 5 / 1 March 2021 /
Optics Express compression system operating at the beginning of the transition region, a reduction in core size orincrease in fiber length could suffice to reduce the throughput such that laminar flow is ensured.However, this comes at the cost of increased intensity and increased transmission losses, which isnot desired either. Hence, such analytic formulas represent another important building blockfor the challenging design of high-energy, differentially pumped hollow fiber compressors andthe formulas discussed herein are also applicable to a nonzero pressure at the optical input [13].Furthermore, it is worth noting that the iterative numerical flow calculations allow obtaining thedownstream pressure for chocked flow conditions, which is not necessarily equal to the ambientpressure in the evacuated recipient. Hence, these methods can be used to check the desired “zero”pressure condition at the optical input of a differentially pumped capillary.In addition to the discussed implications for direct laser applications, we believe that theanalysis of the optical output properties can be used as a novel tool for the investigation ofvolume flow through capillaries. By analyzing the light that is guided within the gas-filledhollow core fiber, we have not only confirmed the commonly known boundary between laminarand turbulent flow, but we have also characterized the frequency spectrum of the turbulencesfor our experimental case. Provided that our interpretations of this observational work can beconfirmed by further studies, such measurements could be an interesting complement to theinvestigation of gas flow through capillaries in the context of mass spectrometry [26]. Themethods discussed herein can be readily expanded to investigate the influence of externallytriggered disturbances, to include the flow of liquids through hollow fibers [32] and to probe ahollow-core fiber perpendicular to the flow direction [33]. Because of the fast light propagation,optical probing of turbulences could be especially interesting for pump-probe measurements.
Funding.
Fraunhofer-Gesellschaft (CAPS); H2020 European Research Council (835306).
Acknowledgements.
The authors would like to thank Dr. César Jáuregui and Dr. Walter Wißdorf for fruitfuldiscussions. This work was supported by the European Research Council (ERC) under the European Union’s Horizon2020 research and innovation program (grant 835306, SALT) and the Fraunhofer Cluster of Excellence Advanced PhotonSources (CAPS).
Disclosures.
The authors declare no conflict of interest.
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