Invited Paper
`
`Laser Resonators, Microresonators, and Beam Control XV, edited by Alexis V. Kudryashov,
`Alan H. Paxton, Vladimir S. Ilchenko, Lutz Aschke, Kunihiko Washio, Proc. of SPIE
`Vol. 8600, 860002 · © 2013 SPIE · CCC code: 0277-786X/13/$18 · doi: 10.1117/12.2003658
`
`Proc. of SPIE Vol. 8600 860002-1
`
`Bistable behavior of a continuous optical discharge as a laser beam
`propagation effect
`
`V.P. Zimakov, V.A. Kuznetsov, A.N. Shemyakin, N.G. Solovyov, A.O. Shilov, M.Yu.Yakimov*
`
`A.Ishlinsky Institute for Problems in Mechanics,
`101, bld. 1, Vernadskogo prsp., Moscow, 119526, Russia
`
`
`
`ABSTRACT
`
`Two stable configurations of a continuous optical discharge (COD) were observed in experiments with plasma sustained
`continuously in xenon at high pressure by radiation of a medium power CW ytterbium fiber laser.
`
`One is the plasmoid of relatively small length with one temperature maximum and laser beam absorption of 10-30%. The
`other one is the plasma formation stretched along the laser beam with two or three local temperature maxima. The laser
`beam absorption in the second plasma configuration is increased dramatically up to 70-80% due to increased plasma
`length.
`
`Both plasma shapes were obtained under close conditions, so that oscillations between the two states were possible and
`also have being observed.
`
`The effect was studied and explained on the base of simplified consideration of the laser beam propagation through lens-
`like plasma medium surrounded by refractive near-spherical bounds between cold and hot gas.
`
`Other experimental results on the sustaining conditions of COD and plasma properties are also presented.
`
`Keywords: laser sustained plasma, continuous optical discharge, COD, laser beam refraction, ytterbium fiber laser,
`plasma bistability
`
`
`1. INTRODUCTION
`
`
`Continuous optical discharge (COD) in which dense plasma is sustained due to absorption of CW laser radiation is now
`one of a few and maybe most effective and convenient method to produce stationary plasma with temperature 20-25 kK
`under atmospheric and elevated pressure in the laboratory. The phenomenon of COD was theoretically predicted and
`first obtained in the experiment in A.Ishlinsky Institute for Problems in Mechanics in 1969-19701,2.
`
`The review of the main results of theoretical and experimental studies of COD carried out since that time may be found
`in3,4. For years as the possibilities of producing COD have being further developed with the improvement of high power
`CW lasers, mainly CO2-lasers, general COD characteristics and sustaining conditions were found experimentally, COD
`plasma diagnostics were developed, theoretical models for calculation and prediction COD properties were created.
`Nevertheless in spite of the success in COD studies and understanding, its industrials and technological applications are
`greatly limited by high maintenance and operation costs together with relatively low efficiency of high power CO2 lasers,
`almost exclusive used for COD production and investigation.
`
`Now we can observe a tremendous upgrowth of the industrial solid state lasers technologies. The combination of the
`characteristics of modern industrial fiber or disc solid-state CW lasers, such as output laser power, laser efficiency and
`high beam quality, just of mass produced models, are close or above that of gas lasers applied in industry for years. Good
`choice as candidate laser for COD sustaining at 1.07 um wavelength may be modern industrial solid state ytterbium fiber
`
` yakimov@lantanlaser.ru; phone: 7 495 433 8218; fax: 7 495 434 1556; ipmnet.ru
`
` *
`
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`

`
`Proc. of SPIE Vol. 8600 860002-2
`
`laser6. Now multikilowatt CW ytterbium lasers of up to tens kilowatts output power are available as well as perfect beam
`quality very high brightness CW lasers of up to 6 kW output power. Total efficiency of high power fiber lasers may
`achieve from 20% up to 30% depending on beam quality.
`
`Till now authors know one or two correspondences that may be treated as containing some scientific information on the
`practical realization of COD with lasers emitted radiation around 1.07-1.09 um5,16, and there are a big lack of detailed
`papers reported on systematic scientific studies of the properties of COD plasmas and its sustaining conditions by
`radiation around 1 um in more or less wide range of parameters. Nevertheless the results of studies like that if they were
`carried out would have decisive impact on the real prospects of the near-IR lasers in the promising field of dense plasma
`sustaining and generation.
`
`The studies are required because laser beam absorption mechanisms around 1 um laser wavelength are different than that
`of around 10 um, where COD is studied relatively well. For instance, the quantum energy of CO2 laser radiation is h =
`0.117 eV is too low for photoionization (bound-free) absorption mechanism to be effective, so that radiation of CO2 laser
`is absorbed mainly through inverse bremsstrahlung mechanism (free-free transitions). The quantum energy of the
`ytterbium laser is h = 1.16 which may also cause photoionization from the upper levels of the electronically excited
`atoms – they only should already be thermally excited – or photoelectronic transitions between the upper levels. If one
`takes into account that absorption through free-free transition is proportional to 2, he will recognize that bound-free and
`bound-bound transitions are the only possibility for the continuous optical discharge to be realized around 1 um.
`Different absorption mechanisms should lead to different properties of COD plasmas produced by CO2 or Yb laser
`radiation.
`
`In this paper we have summarized our first results obtained on the sustaining of COD with ytterbium fiber lasers at  =
`1.07 um in Xe under high pressure from 10 up to 22 bar. The lasers we have used in the experiments operated at low
`order mixed transverse mode (M2 = 5) and close to the lowest order transverse single mode (M2 < 1.1). Continuous
`optical discharge obtained with these two different lasers also demonstrated distinctly different behavior, which may be
`also attributed to the shorter wavelength. The most interesting effects observed with single mode laser – two locally
`stable plasma states and transition between that states – should be treated as the effect of the shorter wave beam
`propagation in plasmas. The data obtained in the paper may be already used for theoretical generalization and
`estimations of the effectiveness of plasma generation with lasers around 1 um in various gases at different pressures.
`
`2. LASER POWER REQUIRED TO SUSTAIN COD (PRELIMINARY ESTIMATIONS)
`
`Preliminary estimations of the laser power required to sustain COD were done for xenon at elevated pressure as the most
`favorable gas to sustain COD because of its low thermal conductivity and low ground state ionization energy (I = 12,12
`eV). Let us estimate minimum laser power required to sustain COD (so called threshold of sustaining) by laser radiation
`at  = 1.07 um in xenon under high pressure p = 15 bar. The estimation may be done on the base of simplified energy
`balance considerations as it was done in 7. Focused laser beam at  = 1.07 um has relatively small diameter from several
`to several tens microns depending on the focusing lens and beam quality 6. The estimation is further simplified by the
`fact that near the threshold plasma is localized near the focus where plasma dimensions are small and plasma thermal
`radiation losses proportional to the plasma volume are small. Also near the threshold the absorption of the laser beam is
`also small due to small plasma length along the beam pass, so that one can treat laser beam as not attenuating in plasma
`(low absorption approximation). So laser power deposited and removed by thermal conductivity, which are equal in
`stationary case, may be written as 7:
`
`
`λ(T)P = AΘ(T), (1)
`
`where Θ – heat flux potential (Θ(T) = T
`, where C(t) – heat conductivity depending on temperature), P – laser
`dt
`tC0
`)(
`beam power, A – geometry factor, depending on radial temperature profile (suppose A ≈ 2 7), λ(T) – laser beam
`absorption coefficient. Θ and  are sharply increased with temperature at 10-15 кК, while their ratio defining P 
`Θ(T)/λ(T) have minimum, which magnitude (Θ/λ)min may be used for the threshold laser power estimations,
`transforming formula (1) to the expression for a threshold power Pt:
`
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`

`
`4 3
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`
`Proc. of SPIE Vol. 8600 860002-3
`
` (2)
`
`
`
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`
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`
`
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`Dependence Θ(T) can be found from known temperature dependence C(t) for equilibrium xenon plasma when electron
`component of thermal conductivity dominates (see 8 for instance). Dependence λ(T) can be found by Kramers-Unsoeld
`formula 9 that describes free-free and bound free absorption processes and does not take into account absorption due to
`possible bound-bound transition. Plots Θ(T), λ(T), and Θ(T)/λ(T), for Xe at p = 15 бар, are presented at Fig.1.
`
`Threshold laser beam power for λ = 1.07 um for sustaining COD in xenon under the pressure p = 15 бар, calculated from
`data presented in Fig. 1 by formula (2), is about Pt ≈ 180 W. Thus one can found that in the pressure range p = 10-20 bar
`the estimation for Pt is varied correspondingly from 310 to 125 W decreasing with pressure.
`
`If one turn to more detailed computer based calculations for the spectral absorption continuum λ (also without taking to
`the account bound-bound processes)10-12 for the pressure p = 15 bar and wavelength λ = 1.07 um, our estimation for COD
`sustaining threshold power should be further increased up to Pt ≈ 550 W because of the decreased value of the calculated
`continuum absorption coefficient of λ ≈ 0.5 см-1 – considerably lower than used in our rough estimation on Fig. 1.
`
`To estimate possible bound-bound transitions component in the laser beam absorption coefficient, that is not so easily
`calculated, we can use experimental data for radiation absorption in the center of the spectral emission line of the cathode
`region of high pressure xenon arc discharge, presented in 13. At measured pressure p  16 bar, temperature 11 kK and
`absorbing plasma thickness d = 0.15 cm, the absorption in the center of spectral line λ = 1.053 um was measure to be
`28.5%, which correspond to absorption coefficient value λ ≈ 2 cm-1 at T ≈ 11 kK. Based on the data from Fig. 1, where
`both Θ(T) and λ(T) are growing with T, but their ratio Θ(T)/λ(T) weakly depends on T and have minimum in the
`temperature region of interest, let us use value λ ≈ 2 cm-1 for COD threshold estimation to obtain Pt ≈ 50 W for the laser
`radiation at wavelength λ = 1.053 um. But at λ = 1.07 um situation may be different because this laser wavelength does
`not coincide with stronger emission/absorption lines λ = 1.053 um and λ = 1.089 um in this spectral region.
`
`Thus different estimations based both on the rough analytical and more precise computer calculations, as well as on the
`experimental data on the absorption coefficient of the radiation with wavelength around λ = 1.07 um in plasma under p =
`15-16 bar lead to the COD sustaining threshold values which differ more the by the order of magnitude Pt ≈ 50-550 W.
`Nevertheless this estimations could help us to define required laser power which can guarantee COD sustaining in the
`experiment – above 1 kW.
`
`
`3. SUSTAINING COD IN XENON WITH LOW ORDER MIXED MODE Yb LASER
`
`We have used ytterbium fiber laser YLS-16 granted by NTO “IRE-Polus” (Russian branch of IPG Photonics, Inc.)
`especially for our experiment. The experimental layout is presented at Figure 2.
`
`
`
`Figure1. Temperature dependences of a heat flux potential Θ(T), laser radiation absorption coefficient λ(T), and threshold laser
`power factor Θ(T)/λ(T),for xenon under p = 15 bar.
`
`
`
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`

`
`X6 91
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`
`Proc. of SPIE Vol. 8600 860002-4
`
`
`
`
`Figure 2. Experimental set up for sustaining COD and measuring sustaining beam characteristics.
`
`
`Laser maximum power was 1 kW CW, beam quality corresponded to mixed low order mode of up to 3-rd order with M2
`= 4.9, laser emitted radiation in spectral band centered at λ = 1.07 um with width changed from 2 to 5 nm as the power
`was increased. Focused beam spot diameter was about 50 um with F/7.2 beam focusing number.
`
`Stable xenon plasma sustained by absorption of a part of incident laser radiation was obtained at p = 14-18 bar inside the
`spherical quartz bulb of a standard high pressure xenon arc lamp. The lamp was filled with p = 11 bar of xenon at room
`temperature, and during the experiment temperature of the bulb has being changed in the limits of 100-2000C depending
`on the incident and absorbed laser power. Incident laser power was varied in the limits P = 50-200 W.
`
`Laser sustained plasma has being initiated from an electric arc column when laser power was increased above the
`threshold value. COD threshold value in Xe at p  15-16 bar was found to be Pt ≈ 50 W. Upper power level of 200 W in
`the experiment was limited by safety reasons to protect 200W arc lamp from overheating.
`
`Plasma was located in the convergent part of the focused laser beam, with rear front displaced by 0.3-1 mm from focal
`point toward the focusing lens (Figure 3). The plasma ball shape was corresponded to the conical shape of the sustaining
`beam, so that the diameter of the glowing part of the plasma ball defined through the plasma image luminosity isolines
`was close to the beam diameter but slightly higher.
`
`The dimensions of plasma were measured as dimensions of plasma image luminosity isoline on the level of 20% of
`maximum luminosity. When incident power has being set slowly growing from 55 to 185W, plasma length was
`increased correspondingly from 0.6 to 1.2 mm and its diameter from 0.23 to 0.42. Beam diameter corresponded to the
`plasma front edge (closest to the lens) was then changed from 0.1 to 0.3 mm.
`
`Incident laser power fraction absorbed in plasma was increased with incident power tending to saturation at 30%. Mean
`absorption coefficient defined from Beer-Lambert law was decreased from 3.5-3.6 cm-1 at lowest COD supporting
`incident power 55 W to 2.7-2.8 cm-1 at upper power limit. Absorption at the lowest power limit is higher because in this
`case plasma diameter is twice higher than beam diameter, so that beam is passed through the most heated central parts of
`the plasma. At the upper power limit plasma diameter is closer to the beam one while a periphery of the beam cross
`section is less absorbed in the outer cold plasma layers.
`
`Deposited power density in the volume of beam-plasma interaction is achieved 3 MW/cm3. Plasma energy balance has
`being studied by absorbed power measurements and absolute plasma radiation balance measurements, the last has being
`compared to that of standard electric arc xenon lamp of equal power. It was found that radiated power fraction in spectral
`region from 200 to 1100 nm is on the level of 50-60% of the absorbed laser power. Residual power is removed out of the
`plasma volume by heat conductivity and convection.
`
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`

`
`il
`
`ME
`MIT
`
`126 W
`
`146 W
`
`165 W
`
`185 W
`
`87 W
`
`96 W
`
`106 W
`
`Proc. of SPIE Vol. 8600 860002-5
`
`
`
`Figure 3. Plasma appearance, position and dimensions of COD sustained by CW Yb fiber laser YLS-1. Electrodes and laser beam
`arrangement are depicted by solid white lines. Photos were taken with light attenuation by 105 times.
`
`When compared to thermal radiation of an electric arc lamp, COD plasma luminosity is about or higher than most
`luminous part of the arc column near the arc cathode. Also COD thermal radiation spectra demonstrate intensive
`continuum in the violet and ultraviolet parts of spectrum, giving the evidence of high plasma temperature.
`
`3. ON THE MECHANISMS OF HIGH LASER RADIATION ABSORPTION IN PLASMA
`
`It was treated for a long time since COD was obtained for the first time with CO2 laser ( = 9.4-10.6 um) that near
`infrared lasers (  1 um) can not be used for efficient sustaining of COD because of low absorption coefficients of laser
`radiation and correspondingly high sustaining threshold7,9. This thesis has being confirmed by experimental data
`obtained for so called laser combustion waves sustained by radiation of powerful long pulsed Nd-glass lasers ( = 1.06
`um, pulse energy about kilojoules, pulse length – milliseconds) in the atmospheric pressure air14. In this case bound-
`bound transitions apparently did not contribute to the laser beam absorption, because absorption coefficient at  = 1.06
`um was 200 times lower than that for  = 10.6 um, as it is predicted by Kramers-Unsoeld formula9.
`
`In the case of high pressure xenon high absorption coefficients observed in our experiments could not be explained only
`by free-free and bound-free transitions. As can be seen from COD power threshold estimations in the above chapter,
`absorption is being mainly determined by interatomic bound-bound transitions between energy levels of the upper
`excited states of xenon. Spectral band of YLS-1 laser radiation centered at  = 1070 nm is 2-5 nm wide. Laser band is
`close to the lines of the emission spectrum of xenon plasma centered at  = 1053, 1071, 1076, 1084, 1090 nm. The only
`Xe line within the band of the laser ( = 1071 nm) is the weakest line. The strongest lines in close vicinity of the laser
`band ( = 1053 nm and  = 1090 nm) may participate in the absorption only with their wings. The last presumption is
`confirmed by our observation that laser radiation spectrum is subjected no change after passing through absorbing
`plasma. As the pressure and temperature increased, emission and absorption lines are broadened by collisional and Stark
`broadening, so that line participation in absorption becomes more and more significant with pressure.
`
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`

`
`p =10 bar
`-p =15 bar
`p =20 bar
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`Proc. of SPIE Vol. 8600 860002-6
`
`4. COD SUSTAINING THRESHOLD POWER
`
`More detailed investigation of COD sustaining at  = 1.07 um have being performed with a single mode CW fiber
`ytterbium laser YLS-200-SM also granted by NTO “IRE-Polus”6 for our experiments. YLS-200-SM has output power
`continuously adjusted from 0 to 200 W, output beam is almost pure Gaussian (M2 < 1.1) with collimated beam diameter
`d = 5.0 mm just after built in collimator output. Gaussian beam propagation in optical systems could be predicted in a
`simple way to make easy experimental data handling and interpretation. Laser radiation spectrum distribution was
`centered at 1.07 um, the distribution FWHM was increased from 2 nm to 5 nm with the increase of output power from
`minimum to maximum value.
`
`Continuous optical discharge was initiated in high pressure xenon inside sealed off quartz bulb of a standard short arc
`xenon lamp, as shown on Fig. 2. The beam was focused inside the lamp by optical systems with different F = f/d, where f
`is equivalent focal length and d is equivalent diameter of the beam. The F value define the full angle α of the cone
`carrying of 86% power of the convergent focused beam α = arctan(1/F) which is important characteristic strongly
`affected COD properties. The F value range used in the experience was from F = 3 to F = 15, so that α  1/F with less
`than 3 % deviation.
`
`The other important parameter for COD is gas pressure p. Two types of arc lamps were used in the experiments. One
`was 200 W nominal power lamp filled with 11 bar of Xe at room temperature 200C. The second was 1000 W lamp filled
`with 8 bar of Xe. In the experiments the pressure was changed accordingly to the quartz bulb average temperature which
`has being measured by thermocouple probes. The actual pressure was than recalculated accordingly to the temperature
`readings.
`
`Power threshold of sustaining COD dependence on F and p was first to be determined in our experiments. In a single
`experiment quartz bulb was heated by arc discharge. Then after COD initiation arc was switched off and then COD was
`quenched or by smooth decreasing of laser power at almost constant pressure, or by gradually decreasing pressure with
`bulb cooling at constant power. After COD quenching the actual laser power was measured in two ways – by
`recalculating output laser power taking into account all reflections in optical system between laser output and plasma,
`and also by recalculating measured value of passed through beam also passing through reflecting surfaces before coming
`to a measuring probe.
`
`The results of the threshold measurements are presented on the Figure 4 in the form of the curves: the parameters
`possible for sustaining COD are above the curves.
`
`
`The most characteristic features of the threshold power Pth are the dependences on xenon pressure p (Pth is decreasing as
`p is increased) and on focusing number F (Pth is decreasing as F increased). The dependences on p and F are similar to
`each other – stronger growing at lower argument value and the existence of the upper threshold value after which stable
`
`
`
`
`Figure 4. COD threshold laser radiation power Pth depending on Gaussian beam focusing number F and gas pressure p obtained in
`xenon.
`
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`

`
`3
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`Proc. of SPIE Vol. 8600 860002-7
`
`
`
`
`
`
`Figure 5. Typical layout and main characteristics of a stable plasma shape.
`
`
`COD does not exist if sustaining beam axis is arranged horizontally. The upper thresholds are not shown on the plot. The
`only upper threshold detected in the experiments was that for F=15 stable COD does not exist at pressures above p = 16
`bar in horizontal position. Under pressure above p = 16 bar with F = 15 or higher one could observe plasma motion along
`the beam periodically starting from arc column plasma. The laser plasma in the last case will immediately blow out just
`when the arc will be switched off.
`
`One can see that actual threshold power at  = 1.07 um is definitely below our theoretical estimation presented above, as
`10.6  40 W) under the pressure of xenon p > 15 bar.
`well as below that of obtained with CO2 lasers at  = 10.6 um (Pth
`
`5. PLASMA APPEARANCE AND BISTABILITY
`
`Plasmoid shape when sustained by Gaussian beam resembles the surface of equal intensity in converging Gaussian beam
`that can be easily calculated. The intensity value is generally in the limits of 105-106 W/cm2 decreasing as laser power
`and dimensions of plasma are increased. Typical layout and main characteristics of a stable plasma shape are shown on
`Figure 5. Under stable conditions plasma is continuously growing with increasing laser power, absorbed part of incident
`laser power is also growing monotonously achieving high values of 75% or even 80%.
`
`Nevertheless, conditions of stability were violated with growing pressure or focusing number F. In experiments pressure
`was actually changing all the time due to sealed quartz bulb heating or cooling. Generally xenon pressure was in the
`limits 15-20 bar due to the heating by COD plasma and laser beam. Also the pressure was growing when laser power
`was increased. So the unstable or bistable plasma behavior was observed easily when laser power was high enough or the
`bulb was preheated by an arc discharge. Figure 6 shows generalized parameters of COD plasma presented in dependence
`on focusing number F. The region that exhibits bistability is presented with shaded areas. At F<6 and F>10 we have
`different plasma states of “long” and “short” plasma respectively. Shaded region 6-7.5<F<10 represents the region of
`bistability. In this region plasma may exist both in “short” or “long” states, passing from one to another if the pressure or
`temperature correspondingly increased or decreased, or may artificially be switched from one state to another by
`manipulation with power up and down or moving the focal point of a beam.
`
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`

`
`0.8
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`
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`P = 146 W
`
`P = 146 W
`
`Proc. of SPIE Vol. 8600 860002-8
`
`
`
`Figure 6. Maximum plasma absorption μ, mean absorbed power density, plasma length for incident power P = 150W and absorbed
`power to incident power ratio (Pin-Pout)/Pin are presented as functions of laser beam focusing number F. At F<6 and F>10 we have
`different plasma states of “long” and “short” plasma respectively. Shaded regions 6-7.5<F<10 represent the regions of bistability,
`which exhibit theirselves in sharp plasma parameters changes together with hysteresis-like behaviour. Data have being obtained for
`COD plasma sustained by single mode Yb laser of up to 200 W output power in xenon under p = 15-20 bar.
`
`
`
`
`Figure 7. Isolines of COD plasma luminosity corresponded to the temperature range 10-16 kK (equivalent to lines of equal plasma
`absorption) in the plane of the optical axis with corresponded laser beam 1/e2 intensity lines without plasma (straight lines). Two
`stable plasma configurations in “long” and “short” states stabilized in physically vertical beam (shown as horizontal) may be observed
`under the same conditions. Laser beam is Gaussian TEM00 mode, focusing parameter F = 10.
`
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`Energetiq Ex. 2029, page 8 - IPR2015-01277
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`

`
`F =10
`p = 17 -19 bar
`
`g 0.5
`
`m 0.4
`
`o s
`
` 0.2
`
`oác
`
`o 0.1
`
`0
`
`100
`150
`50
`Incident laser power, W
`
`150
`100
`50
`incident laser power, W
`
`-B A. Lf, mm
`-0$ Lr,mm
`t LP, mm
`F = 7.5
`p = 16 -20 bar
`
`50
`
`100
`incident laser power, W
`
`150
`
`50
`
`100
`incident laser power, W
`
`150
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`Proc. of SPIE Vol. 8600 860002-9
`
`
`
`Figure 8. Plasma length Lp and plasma front Lf and rear Lr bound positions in vertically stabilized bistable configuration with F = 10
`corresponding to the data presented on Figure 8. Plasma absorbed to incident laser beam power ratio against incident beam power
`dependence both for “long” (highly absorbing) and “short” (low-absorbing) are also presented. Bistability regions are shadowed.
`
`Regular or irregular oscillations between two states followed by corresponding changes of pressure and temperature
`(“long” plasma corresponds to high absorption and higher temperature and pressure than a “short” one) were also
`observed. In bound bistability state F = 10 “long” plasma state can be stabilized if the beam was transmitted to the
`plasma upward along vertical optical axis. Convection flow coaxial with the laser beam stabilized “long” plasma state.
`On the contrary, when laser beam axis was horizontal, transversely directed convection flow prevented plasma from
`being in the “long” state and only “short” state has being observed with F=10 at horizontal position. The appearance of
`the plasma and its characteristics both in “long” and “short” states are presented on Figures 7 and 8. Figure 9 shows
`bistable behavior parameters for F = 7.5 in horizontal position.
`
`6. LASER BEAM PROPAGATION IN PLASMA AND PLASMA BISTABILITY
`
`The reason why plasma exhibits bistability when focused laser beam F number or gas pressure exceed certain values may
`be laser beam refraction in plasma, the effect which have being extensively studied in the case of COD supported by CO2
`laser radiation ( = 10.6 um)17-22. Now we deal with COD in high pressure xenon at  = 1.07 um and we will use earlier
`developed and approved methods22 to understand our new experimental results.
`
`To estimate the lensing effect in a model plasma region with dimensions corresponded to that measured for plasma
`length Lp = Lf – Lr, and diameter dp, that also was determined from data like shown on Figure 3, 5, 7, let us follow the
`procedure elaborated in22. Plasma dimensions measured in our experiments correspond to the length on which free
`
`
`
`
`
`
`Figure 9. Plasma length Lp and plasma front Lf and rear Lr bound positions in horizontal bistable configuration with F = 7.5. Plasma
`absorbed laser power to incident laser power ratio against incident beam power dependences both for “long” (highly absorbing) and
`“short” (low-absorbing) are also presented. Bistability regions are shadowed.
`
`
`
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`

`
`beam F7.5
`- - - -
`beam F7.5,
`- - beam F7.5,
`
`no plasma
`long plasma -
`short plasma
`
`5
`
`- béam no plasma
`- -
`long plasma calc
`- short plasma calc
`O short plasma da
`O O long plasmad.
`
`2.5
`
`EE Wä
`
`0.3
`
`0.2
`
`0.1
`
`EE
`
`m
`
`Short plasma,
`P = 150W
`
`Long plasma,
`P = 130W
`
`71
`
`73
`distance from lens, mm
`
`0
`
`75
`
`100
`
`200
`150
`distance from lens, mm
`
`250
`
`Proc. of SPIE Vol. 8600 860002-10
`
`electron density drops from 2 to e times from the center of plasma ball to the bound. Assume this drop to be equal to Ne
` 21018 см-3 corresponded to a half value for equilibrium xenon plasma at p = 16 bar that may be found from well
`known Saha equation9. Refraction coefficient gradients in plasma are the result of free electron density gradients and of
`gas density gradients. Electronic part of refraction index at  = 1.07 um may be expressed through the electron density as
`
`
`n = 1 – 5.110-22Ne
`
`(3)
`
`
`or for n we have n = – 5.110-22Ne  –10-3. Gas density component in refraction may contribute about the same value
`in n under p = 16 bar with the same sign as electron component, because in spite of a gas density refraction index is
`more than 1, neutral gas density considerably drops from periphery to the center of the plasma. For the following
`simplified model calculations let us suppose that refraction index is increased from center to the bound accordingly to
`parabolic law
`
`
`n(r) = n0 (1+ 1/2(n2r)2)
`
`(4)
`
`
`(5)
`
`
`
`where n0, n(r)  1, and n2 = 4-5, the last value correspond to n(<dp>/2) = 210-3, where <dp>is estimated average plasma
`diameter equals to 0.25-0.3 mm. Radiation transport in the media with parabola like refraction index profile may be
`described in terms of matrix optics, the matrix for negative parabolic lens will be the following23:
`
`
`
`
` 
`
`nLch
`(
`
`
`p
`2
`nLshn
`(
`
`p
`2
`
`
`
`)
`
`)
`
`2
`
`
`
`1
`n
`
`
`nLsh
`(
`
`p
`2
`2
`nLch
`)
`(
`
`p
`
`2
`
`)
`
` 
`
`
`Figure 10 depicts optical scheme of the beam transport calculation region where modeling medium with parabolic
`refraction index profile Lp = (Lf, Lr) is presented in half tone. The refraction on nearly spherical bound between hot gas
`bubble (temperature 1-10 kK) surrounding plasma and cold gas was also taken into account and was predomina