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Simulations of recombination lasing in Ar7+ driven by optical field ionization in a capillary discharge waveguide
Optics Communications 249 (2005) 501–513 www.elsevier.com/locate/optcom
Simulations of recombination lasing in Ar7+ driven by optical field ionization in a capillary discharge waveguide D.J. Spence, S.M. Hooker
*
Department of Physics, Clarendon Laboratory, University of Oxford, Parks Road, Oxford OX1 3PU, UK Received 3 August 2004; received in revised form 19 January 2005; accepted 21 January 2005
Abstract We present calculations of the small-signal gain coefficient, gain length, and output energy of a recombination laser in Ar7+ driven by optical field ionization. Simulations are presented for both 400 and 800 nm pump radiation, and for two targets containing mixtures of argon and hydrogen: a gas cell, and a gas-filled capillary discharge waveguide. Extremely high values for the small-signal gain coefficient are calculated for the 4s–3p transition at 23.2 nm using a pump wavelength of 400 nm for both the gas cell and waveguide. Operation in the waveguide is predicted to greatly increase the XUV laser output owing to a large increase of the gain length. The calculations also show that use of the waveguide allows significant single-pass gain to be achieved even with pump radiation of 800 nm wavelength. 2005 Elsevier B.V. All rights reserved. PACS: 42.55.Vc; 52.38.Hb; 42.60.By
1. Introduction As has been known for some years, plasmas generated by optical field ionization (OFI) with linearly polarized radiation are promising candidates for recombination XUV lasers because, in principle, extremely low electron temperatures
* Corresponding author. Tel.: +44 1865 282209; fax: +44 1865 282296. E-mail address: [email protected] (S.M. Hooker).
can be achieved in relation to the ion stages produced [1]. The first report of gain arising from recombination in an OFI plasma was by Nagata et al. [2], who employed radiation from a nanosecond-pulse KrF laser to form a singly ionized Li plasma from a solid target, which was then field ionized by radiation from a picosecond KrF laser. A gain of 20 cm1 on the n = 2 ! 1 transition of H-like Li was reported for plasma lengths of up to 2.0 mm. In more recent work, the length of the gain region was increased to 5 mm by Korobkin et al. [3] who employed a laser-ablated LiF capillary to guide the pump beam. By this means
0030-4018/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2005.01.031
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the single-pass, small-signal gain was increased to approximately exp(5.5). Further progress in driving recombination lasers in OFI plasmas has been hampered by the difficulty in practice of achieving simultaneously a highly ionized, yet cold plasma. One potential solution to this problem is to dilute the lasant gas in a gas of lower ionization energy so as to reduce the average electron temperature produced by OFI of the mixture. For example, Grout et al. [4] have compared the gain generated on recombination laser transitions in nitrogen and argon ions produced by OFI of the pure gases and dilutions of those gases in hydrogen. In that work small-signal gain coefficients on a 24.7 nm transition in N4+ and a 23.2 nm transition in Ar7+ of a = 25 and 10 cm1 respectively were predicted for ionization of the pure gases by KrF laser radiation at 250 nm. Diluting the lasant gas with hydrogen was calculated to increase the gain significantly: the small-signal gain coefficients were predicted to increase to 65 and 360 cm1 respectively for nitrogen–hydrogen and argon–hydrogen gas mixtures. In addition to difficulties in achieving a low electron temperature, it is also difficult to generate the precursor to the lasant ion stage over a sufficiently long length that saturation of the shortwavelength laser transition is achieved. In fact, saturation of an OFI-driven recombination laser has yet to be reported. These difficulties are caused by the fact that longitudinal pumping is necessary to achieve efficient travelling-wave excitation of the short-lived gain medium, and for this pumping geometry diffraction and ionization-induced refraction severely restrict the gain length that may be achieved. In order to overcome this limitation it is desirable to operate the recombination laser in a waveguide. In this paper we address both of these problems, and analyze pumping the argon-doped hydrogen OFI-driven recombination laser within the plasma channel of a gas-filled capillary discharge waveguide. This waveguide [5] offers two distinct advantages for driving recombination lasers of this type. First, a plasma channel is formed that is able to guide the pump laser pulses over long lengths. Second, for channels formed in
mixtures of hydrogen and the lasant gas, the capillary discharge creates a dense pool of cold electrons by discharge ionization of the hydrogen. The cold discharge electrons can greatly enhance the recombination rate into the upper laser level following optical field ionization of the lasant species. These effects combine to enhance the small-signal gain coefficient, and the intensity and brightness of the output XUV pulse. The paper is organized as follows. In Section 2, we analyze the gain produced by pumping neutral Ar and Ar/H gas targets using driving pulses with wavelengths of 400 and 800 nm. Very high gain coefficients are predicted for 400 nm pumping of Ar/H mixtures such that the population inversion would be extremely heavily saturated. We argue that in this case the greatest output would be achieved for plasma conditions that are very different than those which yield the greatest small-signal gain, and show that the output intensity can be orders of magnitude greater than the saturation intensity Isat. In Section 3, we present the results of simulations of the propagation of the driving laser pulses through a passive gas cell containing that gas mixture which is predicted to yield the greatest XUV output. We find that defocusing limits the gain length to 20 mm for a pump wavelength of 400 nm, leading to an estimated output pulse energy of approximately 50 lJ. In Sections 4 and 5 we consider the operation of OFI recombination lasers driven within the plasma channel of a gas-filled capillary discharge waveguide. In Section 4 we analyze the effects on the kinetics of the laser system of pre-ionization of the lasant mixture by the discharge. It is found that for 400 nm pump radiation pre-ionization has little effect on the maximum intensity and output energy that can be obtained. For 800 nm pumping, however, pre-ionization is extremely beneficial, increasing the small-signal gain coefficient by an order of magnitude. In Section 5, we present the results of calculations of the gain length that may be achieved within the waveguide. It is shown that extended propagation of both 400 and 800 nm pump pulses is feasible, leading to calculated output pulse energies of several hundred microjoules.
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2. Laser kinetics for argon–hydrogen mixtures in a gas cell The argon-doped hydrogen recombination laser has been investigated by Grout et al. [4]. Grout considered pumping with a 100-fs, 250-nm KrF laser pulse, and calculated a small-signal gain coefficient of 360 cm1 and a saturation intensity of 2 · 107 W cm2 for a gas mixture comprising 2 · 1018 cm3 of H and 1 · 1017 cm3 of Ar. This can be compared to a calculated gain coefficient of less than 10 cm1 for pure argon gas targets. The calculations presented in the present paper use the same computational model used by Grout et al. [6] The model includes plasma ionization, heating and expansion, and collisional-radiative modelling of the excitation of selected ion stages. The model assumes that after the passage of the laser pulse, the electron energy distribution relaxes instantly to a Maxwellian distribution: this will lead to an underestimate of the gain for recombination lasers since the excess of cold electrons in the real distribution will enhance the recombination rate, as modelled recently by Avitzour et al. [7]. The argon-doped hydrogen recombination laser operates as follows. For a gas cell, the gas mixture is rapidly ionized by OFI to H+ and Ar8+ by a linearly polarized pump pulse. The electrons are heated by above-threshold ionization (ATI) and inverse bremsstrahlung (IB), reaching a final temperature of a few eV after the passage of the laser pulse. Since the ATI heating of the electrons originating from ionization of hydrogen is far lower than the average ATI heating of the eight electrons stripped from argon, hydrogen has the effect of decreasing the average electron temperature compared to OFI of pure argon gas. The cold electrons recombine rapidly with Ar8+ ions, leading to population inversions on several n = 4 ! 3 transitions in Ar7+. The gain is transient, with a full width at half maximum duration of around 10 ps. In this section, results are presented for argondoped hydrogen recombination lasers pumped using 400 and 800 nm pump pulses that could be obtained from a Ti:Sapphire laser. We note that terawatt Ti:Sapphire lasers are compact and convenient, and are much more widely available than terawatt KrF laser systems.
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2.1. Pumping at 400 nm We first consider pumping with frequencydoubled Ti:Sapphire laser radiation at 400 nm. Simulations were performed for both zero dimensions (a single point in space) and one dimension (as a function of radius, assuming cylindrical symmetry). Plasma expansion and heat flow in the 1D simulations were found to have very little effect on the population inversion density on axis, and so the results presented in this paper are from zerodimensional simulations. Simulations were performed for a laser pulse with a sech-squared temporal profile of 30-fs-fullwidth at half maximum duration focused to a peak intensity of 1 · 1017 W cm2. The gain coefficient on several n = 4 ! 3 transitions was calculated for a range of total ion densities and for a range of Ar densities. Fig. 1 shows three calculated parameters plotted as a function of the total ion density, for the range of fixed argon densities shown in the legend; moving to the right along each curve therefore corresponds to an increasing hydrogen density. The calculated parameters are the average electron temperature Te immediately after the passage of the pump pulse, the peak small-signal gain coefficient a of the 4s1/2–3p3/2 transition in Ar7+ at 23.2 nm, and the maximum output energy of the XUV laser per unit volume of gain (discussed below). The same parameters are shown for pure argon and pure hydrogen gas. The results for pure argon in Fig. 1 show that for low total densities, Te is independent of density, and for high densities it is proportional to density. The reason for this is that at low density ATI heating is dominant – ATI heating is determined only by the ratio of Ar to H and is independent of the total density. However, at high total densities IB heating begins to dominate since the IB heating rate is proportional to the ion density. The results for pure hydrogen illustrate the fact that ATI heating of H is smaller than that of argon. The IB heating asymptote is also moved to higher densities for pure hydrogen since the electron–ion collision frequency increases rapidly with the ionic charge. The remainder of the Te curves on Fig. 1 demonstrate the effect of adding hydrogen to a fixed
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heating by the hydrogen ions increases the plasma temperature. For argon densities greater than approximately 1019 cm3, a minimum in Te is no longer formed as hydrogen is added. In this regime the dominant heating mechanism for all electrons is IB heating by the argon ions; the hydrogen electrons are also heated by the argon ions and no reduction in plasma temperature occurs. Fig. 1 also shows the calculated small-signal gain coefficient of the 4s1/2–3p3/2 transition in Ar7+ as a function of the total ion density for the same Ar/H mixtures as above. Extremely high gain coefficients are predicted; gains of over 200 cm1 are calculated for a wide range of plasma conditions, the largest being more than 430 cm1 for an argon density of 1 · 1017 cm3 and a hydrogen density of 8 · 1018 cm3. We note that small-signal gain coefficients of between 10 and 400 cm1 are also calculated for eight other transitions in Ar7+. These are, in order of decreasing gain, 4s1/2– 3p1/2, 4f5/2–3d3/2, 4p3/2–3d5/2, 4f7/2–3d5/2, 4p1/2–3d3/2, 4p3/2–3s1/2, 4d5/2–3p3/2, and 4p3/2–3d3/2. 2.2. Estimate of output energy
Fig. 1. Calculated electron temperature Te, small-signal gain coefficient a, and the XUV output pulse energy per unit volume of gain Eex for argon–hydrogen mixtures as a function of total ion density for a range of fixed argon densities. Curves are also shown for pure hydrogen, and pure argon. The parameters are calculated for pump pulses of 400 nm wavelength and 30 fs duration focused to an intensity of 1 · 1017 W cm2.
density of argon. For argon densities lower than approximately 3 · 1018 cm3, a pronounced minimum in temperature occurs as hydrogen is added. For this range of argon densities, IB heating by the argon ions is still small, and cooler electrons from the additional hydrogen, heated dominantly by ATI, decrease the average electron temperature. As the hydrogen density is increased further, IB
It is clear that the single-pass small-signal gain in a system of this type can reach extremely high values, especially, as we consider below, if the pump laser pulse is guided over long lengths. It is important to be clear about the significance of the gain coefficient a and saturation intensity Isat in this regime. The intensity of the output XUV laser pulse I is characterized by exponential growth with propagation distance for I Isat but growth which is linear, or sub-linear, for I Isat. For ASE lasers it is often estimated that the output will be saturated for a single-pass small-signal gain of order exp[alsat] = e15, which may therefore be taken to define a saturation length lsat. Using this estimate it is clear that transitions with a gain coefficient of order 150 cm1 will saturate for gain lengths as short as 1 mm, and hence for gain lengths much longer than this the gain will be very strongly saturated. Calculation of the output of a pulsed ASE laser operating under conditions of saturation is complex. However, some insight may be obtained by first considering a homogeneously broadened laser
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amplifier operating under steady-state conditions. The growth in intensity of the beam in such an amplifier is described by, dI a0 j0 ¼ I; ð1Þ dz 1 þ I=I sat 1 þ I=I 0sat
we will assume a0 j0 and hence that the amplification is strongly saturated for much of the length of the laser. Within this region of strongly saturated amplification the intensity will grow linearly with propagation distance according to
where a0 is the small-signal gain coefficient, j0 the unsaturated absorption coefficient, I 0sat the saturation intensity for the absorption, and z is the distance along the axis of propagation. Clearly at low intensities such that I I sat ; I 0sat the intensity will grow exponentially as I(z) = I(0)exp([a0j0]z). If I 0sat I sat the absorption saturates before the gain, and hence the beam intensity grows essentially exponentially until the gain also saturates, whereupon the intensity continues to grow according to,
ð4Þ
dI a0 I sat j0 I 0sat ; dz
ð2Þ
i.e. the intensity of the beam grows linearly with distance at a rate slower than a0Isat by the term j0 I 0sat . The situation will be rather different, however, if I 0sat > ða0 =j0 ÞI sat . In this case the gain saturates faster than the loss such that the right-hand side of Eq. (1) decreases to zero when the intensity of the beam reaches a maximum value of, I max ¼
a0 =j0 1 I sat : 1 ja00 II sat 0
ð3Þ
sat
We note that this can be orders of magnitude greater than Isat. From this simple analysis we see that the maximum intensity to which the beam may be amplified is determined by losses for which I 0sat > I sat , i.e. losses which saturate slowly relative to the gain, or those which do not saturate at all. For the present case the dominant slowly saturating loss is likely to be photoionization of excited states of the lasant ion. The cross-section for this process will be several orders of magnitude smaller than the optical gain cross-section of the laser transition owing to the greater spectral width. As a consequence it is likely that a0 j0. Detailed calculation of the absorption arising from excited states is beyond the scope of the present paper, and so
dI a0 I sat ; dz
until the intensity reaches values of order Imax. However, dI/dz is equal to the power per unit volume extracted by the beam, and is a maximum under conditions of extreme saturated amplification. Hence we may define the maximum possible power extraction per unit volume as, J ex ¼ a0 I sat ¼
N hx ; sR
ð5Þ
where sR is the recovery time, N* is the population inversion density that would be achieved in the absence of saturation, and hx is the energy of the laser transition. We note that Jex is independent of the saturation intensity. For pulsed ASE lasers also, the output pulse energy is not necessarily limited by Isat provided the single-pass small-signal gain is sufficiently high. This is particularly important for OFI lasers for which the optical gain cross-section is usually large as a consequence of the small laser line width. This causes the saturation intensity to be low, but the small-signal gain coefficient to be high. The high gain coefficient that may, in principle, be achieved in OFI lasers means that conditions of extreme saturation may be reached for relatively short gain lengths, in which case the laser output can be orders of magnitude greater than Isat. Eq. (5) may be used to deduce the power extracted from a pulsed amplifier provided two conditions are met. First the amplifier must operate in a quasi CW regime in which the upper and lower laser level pump rates vary only slowly on the timescale of the gain recovery time sR. Second, the upper and lower level pump rates must be independent of the populations in the upper and lower laser levels. For the recombination scheme under investigation here the first condition is met, but the second condition is not and consequently a modified method must be employed to estimate the output energy of the laser transition.
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In the limit of strong saturation the peak intensity of the ASE laser pulse greatly exceeds the saturation intensity over most of the gain region. Under such conditions the rate of stimulated emission will be the maximum possible, that is at a rate which just reduces the population inversion density to zero throughout the pump pulse. Hence we may calculate Eex within the radiative-collisional code by forcing additional transitions between the upper and lower laser levels at a sufficient rate to keep the population inversion equal to zero. These additional transitions can be counted, and correspond to stimulated emission into the short-wavelength laser pulse. The result is an extracted power per unit volume Jex that can be integrated over the duration of the gain to give the extracted energy per unit volume Eex. We recognize that it will only be possible to extract Eex Joules per unit volume for regions of the gain medium which are strongly saturated. Nevertheless, this provides a useful upper limit, and we note that once the intensity of the ASE pulse becomes so high that absorption arising from unsaturated losses dominates, the ASE intensity can always, in principle, be reduced by allowing the beam to expand, or otherwise coupling some fraction of it out of the gain region. Further, we note that the plasma conditions which optimize Eex are found to be quite different from those which maximize the unsaturated gain coefficient; the former conditions may well be closer to those that optimize the output of a real ASE laser than the latter. 2.3. Results Fig. 1 shows the calculated value of Eex for the range of argon and hydrogen mixtures under consideration. For the conditions corresponding to the peak small-signal gain coefficient of 430 cm1, Eex is 0.45 J cm3. However, since the gain is so high for a wide range of plasma conditions, we can consider operating at conditions away from those which maximize the peak gain coefficient, provided the gain is still sufficient to ensure that extreme saturation would be achieved. It is clear from Fig. 1 that the plasma conditions which optimize Eex lie at higher argon and hydrogen density: for an Ar density of 3 · 1017 cm3 and
an H density of 4 · 1019 cm3, we have Eex = 1.2 J cm3. For these conditions, the predicted small-signal gain is still 150 cm1, sufficient to ensure extreme saturation for gain lengths of order 10 mm. It then remains, as presented in Section 3, to estimate the gain length and the gain volume that may be achieved so that the peak output intensity and pulse energy may be estimated. 2.4. Pumping at 800 nm It would be preferable to pump recombination lasers using 800 nm laser pulses rather than 400 nm owing to the additional complexity as well as energy loss incurred by frequency doubling the output of a Ti:Sapphire laser system. The change of wavelength from 400 to 800 nm has two main effects on the interaction of the laser and the plasma [1,8]: ATI heating increases with wavelength approximately as k2 owing to the increased quiver energy of electrons in the lower frequency field; at the same time, the electron-ion collision rate is decreased, and for a given ion density the IB heating rate varies approximately as k1/2. Since ATI heating determines the baseline temperature for all plasma densities, this leads in general to higher plasma temperatures for longer pump wavelengths. The collisional-radiative code was used to simulate the plasma properties and small-signal gain achieved with 800 nm pumping. Fig. 2 shows the electron temperature Te and gain coefficient a for the case of pumping with a 30 fs, 800 nm laser pulse focused to an intensity of 1 · 1017 W cm2. The curves and their interpretation are the same as in Fig. 1. First consider the electron temperature of pure argon and pure hydrogen plasmas. At low densities, where ATI heating dominates, the calculations predict that a pure argon plasma is heated to over 40 eV, compared to 10 eV for 400 nm pumping. However, IB heating is decreased sufficiently that IB heating of pure argon is not significant for argon ion densities less than approximately 1018 cm3. ATI heating of pure hydrogen increases to 10 eV compared to 2.4 eV for 400 nm pumping. For argon–hydrogen mixtures, there is still a pronounced temperature minimum as hydrogen is added to a fixed amount of argon; however, the plasma temperature is con-
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Fig. 2. Calculated electron temperature and small-signal gain coefficient as in Fig. 1 but for a pump laser of wavelength 800 nm.
strained to be greater than 10 eV owing to the ATI heating of the hydrogen electrons. Fig. 2 shows the gain calculated for various plasma conditions. Optimal conditions are shifted to higher ion densities compared to 400 nm pumping owing to the decrease in IB heating. However, because of the increased plasma temperature, and the consequently much smaller recombination rate, the gain is drastically reduced, to the point that 800 nm pumping of this laser system seems impractical. A peak gain coefficient of only 6 cm1 means that the laser output will not saturate for inversion lengths of the order of 1 cm, and so energy will not be extracted efficiently from the laser medium.
3. Pump laser propagation for a neutral gas target The gain length that can be achieved by longitudinal pumping of partially ionized plasmas is lim-
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ited by refraction and by ionization-induced defocusing. This has been discussed previously in relation to recombination lasing in Ar/H mixtures pumped using 250 nm KrF laser pulses [9], where it was concluded that it would be possible to create Ar8+ along the full length of a 4 mm gas jet. In this section, we discuss the maximum gain length that can be achieved in an argon-doped hydrogen gas mixture for 400 nm pump pulses. An estimate of the gain length allows us to confirm that the single-pass gain is sufficient to ensure extreme saturation, and to estimate the output intensity and energy of the XUV laser pulse. Numerical simulations were performed using a propagation code [10] that simulates propagation of an intense pulse propagating through a plasma undergoing optical field ionization. The nonrelativistic code accounts for absorption of the pulse energy owing to ionization, but not due to ATI and IB heating – these will later be shown to be negligible. Fig. 3 shows, as a function of radial position and distance z from the cell entrance, the calculated argon ion stage produced by the passage of an intense laser pulse through a gas cell containing a mixture of argon and hydrogen gas. The gas cell was taken to contain Ar at a density of 3 · 1017 cm3 and H at a density of 4 · 1019 cm3, both initially un-ionized. The 400nm-wavelength, 30-fs-duration sech-squared laser
Fig. 3. Contour plot showing the argon ion stage produced by the passage of a 30 fs, 235 mJ laser pulse of wavelength 400 nm through a gas cell containing 3 · 1017 cm3 Ar and 4 · 1019 cm3 H. In vacuum, the beam would focus 5 mm into the gas cell with a spot size of 20 lm, and a peak intensity of 1.2 · 1018 W cm2.
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pulse was taken to be focused into the gas cell so that in vacuum it would reach a 20 lm focal spot radius at a position 5 mm into the gas cell, with a peak intensity of 1.25 ·1018 W cm2. The position and size of the focus were chosen heuristically. The simulation shows that the laser pulse is defocused by the ionizing plasma, and reaches a peak intensity of 1 · 1018 W cm2 just 4 mm into the gas cell. The desired species, Ar8+, is generated over a length of 20 mm, and out to a radius of over 30 lm at the cell entrance. Absorption due to ATI and IB are estimated to be just 4% of the pulse energy, and so are safely neglected by the simulation. The peak intensity reached during the simulation was 1 · 1018 W cm2; at this intensity the normalized laser potential a0 = 0.4, and so relativistic effects may also be neglected. The gain region calculated above may be used to predict the output characteristics of the XUV laser pulse. For these conditions we calculate a = 150 cm1 and Eex = 1.2 J cm3 (see Fig. 1). It is clear that for the majority of the gain length, the XUV signal will be sufficiently intense to extract the maximum possible energy from the population inversion. While different regions of the gain volume experience different peak pump intensities, since Eex and Jex are only slow functions of pump intensity we may apply the values calculated for the intermediate pump intensity of 1 · 1017 W cm2 to the whole gain volume. The gain volume is 4.5 · 105 cm3, and so we estimate the output energy to be 50 lJ. The calculated peak value of Jex is 1.2 · 1011 W cm3, which together with the calculated 2 cm gain length leads to an estimated peak output intensity of 2.4 · 1011 W cm2. We note that the calculated value of Isat for these plasma conditions is just 8.7 · 109 W cm2, more than an order of magnitude smaller than the peak output intensity estimated above. We also note that Isat is usually calculated for the conditions which maximize the small-signal gain coefficient; for the present case, this saturation intensity is only 4.4 · 107 W cm2, almost four orders of magnitude below the peak output intensity calculated for conditions that optimize Eex.
4. Laser kinetics for argon–hydrogen mixtures in a plasma waveguide The gas-filled capillary discharge waveguide would seem to be ideal for increasing the gain length of OFI recombination lasers since the guiding channel comprises an essentially fully ionized plasma of low temperature [11,12]. Guiding of pulses with intensities in excess of 1017 W cm2 has been demonstrated for hydrogen-filled capillaries up to 50 mm long [5]. By doping the hydrogen with a small fraction of the lasant gas, the properties of the waveguide will be changed only slightly, and the gain length can be extended beyond that which can be achieved in a gas cell. Indeed we note that recently this approach was used to demonstrate lasing on the collisionally excited 5d–5p transition at 41.8 nm in Xe8+ in a Xe/ H-filled capillary discharge waveguide [13]. The hydrogen-filled capillary discharge waveguide operates as follows. A discharge is constrained in a long capillary initially filled with unionized hydrogen; the energy dissipated by the discharge current heats and ionizes the hydrogen to create an essentially fully ionized plasma. Thermal conduction to the capillary wall causes the plasma temperature to be greatest on axis, and, since pressure is rapidly equilibrated across the capillary diameter, the plasma density is a minimum on axis. Magnetohydrodynamic simulations of the capillary discharge [14] and measurements [11] have shown that the plasma channel formed is approximately parabolic. In the absence of further ionization, and neglecting relativistic effects, a parabolic plasma channel will guide a Gaussian laser pulse with a constant spot size W provided that W is equal to the ‘‘matched spot size’’ of the channel W M ¼ ðr2ch =pre DN e Þ1=4 , in which re is the classical electron radius, and DNe is the increase in electron density at radius rch. Simulations have shown that a capillary discharge in pure hydrogen can form parabolic plasma channels with plasma temperatures as low as 3 eV, with more than 99% of the hydrogen atoms ionized [14]. As we demonstrate in Section 5, a plasma channel of this type will allow the pump laser to be guided over extended distances, creating a large volume of Ar8+. However, preheating of the plas-
ma by the discharge affects the kinetics of the XUV laser and must be considered in detail. To this end, the collisional-radiative excitation code was used to simulate the plasma conditions produced by the passage of pump laser pulses through plasma targets preheated to 3 eV, as expected to be found in the guiding channel of the waveguide prior to the arrival of the pump laser pulse. The code calculates the steady-state populations of each ion stage and its energy levels at this initial temperature produced by the discharge, before simulating the excitation of that plasma by an intense laser pulse. Prior to the arrival of the laser pulse, the discharge is calculated to ionize the hydrogen fully, and the argon to between Ar+ and Ar2+.
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4.1. Pumping at 800 nm Fig. 4 shows the electron temperature, peak small-signal gain coefficient, and Eex after the passage of an 800 nm, 30 fs laser pulse with a peak intensity of 1 · 1017 W cm2 through plasmas preheated to 3 eV by a discharge. The key difference from Section 2 is that here the hydrogen atoms are ionized thermally by the discharge rather than by optical field ionization. This means that there is no ATI heating of the hydrogen electrons. Accordingly, at low ion densities where IB heating is negligible, the predicted temperature of a pure hydrogen plasma is now only 3 eV, compared to 10 eV for a neutral hydrogen target. Similarly, the pre-ionization of the argon ions marginally reduces the temperature of a low density argon plasma to 43 eV, compared to 44 eV for a neutral argon target. The result of the discharge pre-ionization is that there is an enhanced cooling effect as hydrogen is added to a fixed density of argon. Peak smallsignal gain coefficients of over 30 cm1 are predicted for a range of plasma conditions, with a peak gain coefficient of 62 cm1 for an Ar density of 1 · 1017 cm3 and an H density of 2 · 1019 cm3. This gain coefficient is approximately an order of magnitude greater than that predicted for OFI at this wavelength of Ar/H gas mixtures contained in a passive gas cell. Operating at the conditions for peak gain, extreme saturation of the laser medium will occur for gain lengths of order one centi-
Fig. 4. Calculated electron temperature, small-signal gain coefficient, and XUV pulse energy per unit volume for preheated plasmas. An 800 nm pump laser pulse was assumed with a duration of 30 fs, and an intensity of 1 · 1017 W cm2. The plasma was taken to be pre-heated to 3 eV causing ionization to Ar1.2+ and H+ before the arrival of the pump pulse.
metre, allowing efficient extraction of the energy stored in the population inversion. For these conditions, we calculate Eex = 0.3 J cm3 and Jex = 1.6 · 1010 W cm3. 4.2. Pumping at 400 nm Fig. 5 shows the calculated final temperature, gain coefficient, and Eex after the passage of a 400 nm, 30 fs laser pulse with a peak intensity of
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1 · 1017 W cm2 through plasmas preheated to 3 eV by a discharge. For this pump wavelength, the calculated plasma temperatures are similar to those calculated for neutral gas targets, as presented in Fig. 1. The electrons from the hydrogen ions are heated to 3 eV by the discharge in the present case, and to 2.5 eV by ATI heating in the unionized gas mixture. The calculated gain coefficients are decreased by just over a factor of two for the preheated plasma. This reduction is almost entirely caused by the higher ion temperature
(0.6 eV for the unionized plasma and 3.2 eV for the preheated plasma), and hence larger line width, which decreases the optical gain cross-section in the pre-ionized plasma. However, Isat is corresponding increased, and thus the calculated value for Eex is not strongly affected, reflecting the fact that the population inversion density is similar for the two cases. This is a good example of a situation in which increasing the saturation intensity by increasing the line width of the laser transition does not lead to an increase in XUV output. A peak value for Eex of 1.0 J cm3 for an Ar density of 3 · 1017 cm3 and an H density of 4 · 1019 cm3 is calculated, which is similar to the value of 1.2 J cm3 in the absence of preheating.
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5. Pump laser propagation in a plasma waveguide
Fig. 5. Calculated electron temperature, small-signal gain coefficient, and XUV pulse energy per unit volume for preheated plasmas as for Fig. 4 but for a pump laser wavelength of 400 nm.
We now consider the propagation of the pump laser pulse through Ar/H-filled capillary discharge waveguides. For waveguides formed in pure hydrogen, the plasma is essentially fully ionized before the arrival of the laser pulse and proper matching to the plasma profile will allow the pulse to propagate with a constant spot size. The matched spot size of gas-filled capillary discharge waveguides is typically of the order 20–40 lm, and may be controlled by varying the initial gas density and the capillary radius. Perfect matched guiding cannot be achieved when driving OFI lasers within the plasma channel owing to the requirement that the lasant gas must be further ionized by the pump pulse. These additional electrons will be created predominantly in the axial region of the waveguide axis where the laser intensity is greatest, perturbing the parabolic electron density profile and defocusing the laser pulse. However, whilst perfect matched guiding is not possible, laser energy that is defocused away from the axis will be collected and refocused by the waveguide, greatly increasing the length over which the peak intensity of the laser pulse remains high. Simulations show that variations in the axial intensity of the guided laser pulse are smallest when the matched spot-size of the waveguide is smaller than the input spot size of the laser pulse; the additional focusing power of the waveguide is
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better able to collect and refocus the energy refracted away from the axis. Fig. 6 shows, as a function of radial position and distance z from the waveguide entrance, the calculated argon ion stage produced by the passage of an intense, 400 nm laser pulse through a plasma waveguide. The plasma channel was taken to comprise a mixture of Ar1.2+ (as determined by the collisional-radiative code for this plasma at 3 eV) at a density of 3 · 1017 cm3 and H+ at a density of 4 · 1019 cm3, corresponding to the mixture calculated in Section 4 to yield the highest value of Eex. A parabolic initial electron density profile with a matched spot size of 22 lm was found heuristically to provide the best results – this guiding profile would be formed [14] in a capillary with a diameter of approximately 350 lm. The 400 nm pump laser pulse was assumed to be of 30 fs duration focused to a peak intensity of 5 · 1017 W cm2 and a 30 lm spot at the entrance to the waveguide. This corresponds to a laser energy of 235 mJ, and peak laser power of 8 TW. The simulations show that the laser spot size oscillates as the pulse propagates, with the peak axial intensity varying between 5 · 1016 and 1 · 1018 W cm2. Owing to its stable closed shell configuration Ar8+ is produced for almost all of
Fig. 6. Contour plot showing the argon ion stage in an argonhydrogen waveguide after the passage of a 30 fs, 235 mJ laser pulse of wavelength 400 nm. The beam is focused at the entrance of the waveguide with a spot size of 30 lm, and a peak intensity of 5 · 1017 W cm2. The matched spot size of the waveguide was 22 lm. The gas mixture was 3 · 1017 cm3 Ar and 4 · 1019 cm3 H.
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this range of intensities. Hence, the desired species is generated over a length of more than 30 cm, and out to a radius of approximately 30 lm. From these results we can estimate the XUV output intensity that might be achieved. In Section 4 it was determined that Eex = 1.0 J cm3 and Jex = 9.5 · 1010 W cm3 for these plasma conditions. Hence, for a 30 cm gain length the output intensity is predicted to be 2.9 · 1012 W cm2 on the waveguide axis. Note that this is more than three orders of magnitude greater than Isat = 9.3 · 108 W cm2. The output energy may be estimated by multiplying Eex by the volume of Ar8+ generated by the pump pulse, leading to an output energy of 850 lJ. This would be a very high output energy, and compares very favourably with the highest pulse energy yet achieved from a tabletop short-wavelength laser: 880 lJ at 46.9 nm obtained from a in a z-pinch capillary discharge laser [15]. It is also much greater than can be achieved by high harmonic generation. For example, high harmonic output energies of 7 lJ at 72.7 nm and 0.33 lJ at 29.6 nm have been reported [16,17]. We also note that the Ar/H recombination could be operated at multi-Hertz pulse repetition rates, thereby providing mean output powers of several milliwatts. We acknowledge that the longest gas-filled capillary discharge waveguide which has been demonstrated to date is only 50 mm long. However, increasing the length of the waveguide to tens of centimetres is expected to be relatively straightforward, in which case the calculations presented here suggest that short-wavelength outputs with pulse energies of hundreds of microjoules should be possible. Simulations of the propagation of 800 nm, 30 fs laser pulses through the optimum gas mixture calculated in Section 4 (argon density of 3 · 1017 cm3 and a hydrogen density of 2 · 1019 cm3) show that Ar8+ could be generated over a length of 15 cm, and out to a radius of 30 lm. From the calculated values of Eex = 0.3 J cm3 and Jex = 1.6 · 1010 W cm3, we deduce an output pulse energy of 125 lJ, and a peak output intensity of 2.4 · 1011 W cm2 would be obtained.
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6. Summary In summary we have presented the results of calculations which show that a gas-filled capillary discharge waveguide could be used to increase the output of OFI-driven recombination lasers in two ways. First, the presence of a high density of relatively cold electrons from discharge ionization of hydrogen reduces the mean electron temperature of the plasma after OFI of the lasant gas, leading to greatly increased rates of recombination. Second, the formation of a plasma channel allows gain to be achieved over very much longer lengths than is possible in a gas cell. We have also argued that for very high values of the single-pass small-signal gain, such as those calculated for the Ar–H recombination laser, the laser will operate under conditions of extreme saturation such that the output will be significantly greater than deduced from the saturation intensity. Further, the output energy will be greatest under conditions which maximize Eex and Jex, rather than those which yield the largest single-pass unsaturated gain. These conditions can be markedly different. For the Ar–H recombination laser considered here, the peak output intensity was estimated to be as much as three orders of magnitude greater than the saturation intensity. To test these ideas we have calculated the electron temperature and small-signal gain coefficient for recombination lasing on the 4s1/2–3p3/2 transition at 23.2 nm in Ar7+ for both neutral gas cell targets and pre-ionized waveguide targets pumped by Ti:Sapphire laser radiation at 400 and 800 nm. For 400 nm pump radiation, peak small-signal gain coefficients of 440 and 200 cm1 were calculated for the gas cell and waveguide respectively. The XUV output was found to be maximized at higher hydrogen and argon densities, for which the extracted power densities Jex were 1.2 and 0.95 · 1011 W cm3 respectively. For 400 nm pump laser radiation the plasma channel did not greatly affect the kinetics of the laser, but did allow an increase in the gain length by more than an order of magnitude. For an 8 TW pump laser at 400 nm, a gain length of 2.0 cm was calculated for a neutral gas cell target, giving an output intensity of 2.4 · 1011 W cm2, and an output energy of
50 lJ. It was calculated that use of a plasma waveguide could increase the length over which gain could be generated to more than 30 cm, raising the potential output peak intensity and pulse energy by more than an order of magnitude to 2.9 · 1012 W cm2 and 850 lJ respectively. We note that these predicted energies may be reduced by non-saturating losses not considered in this current model. A more accurate prediction will require a detailed analysis of such losses. For 800 nm pumping, a peak gain coefficient of only 6 cm1 was calculated for neutral gas targets owing to increased ATI heating at the longer pump wavelength. However, for the plasma channels, small-signal gain coefficients of over 60 cm1 were predicted. At conditions yielding the largest peak small-gain coefficient, which in this case is required to ensure extreme saturation, Eex = 0.3 J cm3 and Jex = 1.6 · 1010 W cm3. Simulations of the propagation of 800 nm driving laser pulses through a plasma channel with optimum argon and hydrogen densities show that a gain length of over 15 cm could be achieved, resulting in an output intensity of 2.4 · 1011 W cm2 and an output energy of 125 lJ. We emphasize that the use of a plasma waveguide is essential if the laser is to be pumped with 800 nm radiation; the preionization of the gas mixture is crucial to achieving a high gain coefficient, as well as increasing the gain length which may be achieved by guiding the pump laser pulse. Finally we add that it seems likely the gas-filled capillary discharge waveguide will prove beneficial to a range of other recombination-pumped as well as collisionally pumped XUV lasers.
Acknowledgements The authors would like to thank Prof. G.J. Pert for supplying us with the collisional-radiative code used in the calculations, for advice on its use, and for many helpful discussions on the subject of this paper. SMH would like to thank the Royal Society for the support of a University Research Fellowship.
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References [1] S.C. Rae, K. Burnett, Physical Review A 46 (4) (1992) 2077. [2] Y. Nagata, et al., Physical Review Letters 71 (23) (1993) 3774. [3] D.V. Korobkin, et al., Physical Review Letters 77 (26) (1996) 5206. [4] M.J. Grout, et al., Optics Communications 141 (3–4) (1997) 213. [5] A. Butler, D.J. Spence, S.M. Hooker, Physical Review Letters 89 (18) (2002), p. art. no.-185003. [6] K.A. Janulewicz, M.J. Grout, G.J. Pert, Journal of Physics B – Atomic Molecular and Optical Physics 29 (4) (1996) 901. [7] Y. Avitzour, S. Suckewer, E. Valeo, Physical Review E 69 (4) (2004), p. art. no.-046409. [8] G.J. Pert, Physical Review E 51 (5) (1995) 4778. [9] M.J. Grout, G.J. Pert, A. Djaoui, Journal of Physics B – Atomic Molecular and Optical Physics 31 (1) (1998) 197.
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[10] D.J. Spence, S.M. Hooker, Journal of the Optical Society of America B – Optical Physics 17 (9) (2000) 1565. [11] D.J. Spence, S.M. Hooker, Physical Review E 6302 (2) (2001), p. art. no.-015401. [12] D.J. Spence, A. Butler, S.M. Hooker, Journal of Physics B – Atomic Molecular and Optical Physics 34 (21) (2001) 4103. [13] A. Butler, et al., Physics Review Letters 91 (18) (2003) 205001. [14] N.A. Bobrova, et al., Physical Review E 6501 (1) (2002), p. art. no.-016407. [15] J.J. Rocca, et al., Nuclear Instruments and Methods in Physics Research Section A – Accelerators Spectrometers Detectors and Associated Equipment 507 (1–2) (2003) 515. [16] E. Takahashi, Y. Nabekawa, K. Midorikawa, Optics Letters 27 (21) (2002) 1920. [17] E. Takahashi, et al., Physical Review A 66 (2) (2002), p. art. no.-021802.
Simulations of recombination lasing in Ar7+ driven by optical field ionization in a capillary discharge waveguide D.J. Spence, S.M. Hooker
*
Department of Physics, Clarendon Laboratory, University of Oxford, Parks Road, Oxford OX1 3PU, UK Received 3 August 2004; received in revised form 19 January 2005; accepted 21 January 2005
Abstract We present calculations of the small-signal gain coefficient, gain length, and output energy of a recombination laser in Ar7+ driven by optical field ionization. Simulations are presented for both 400 and 800 nm pump radiation, and for two targets containing mixtures of argon and hydrogen: a gas cell, and a gas-filled capillary discharge waveguide. Extremely high values for the small-signal gain coefficient are calculated for the 4s–3p transition at 23.2 nm using a pump wavelength of 400 nm for both the gas cell and waveguide. Operation in the waveguide is predicted to greatly increase the XUV laser output owing to a large increase of the gain length. The calculations also show that use of the waveguide allows significant single-pass gain to be achieved even with pump radiation of 800 nm wavelength. 2005 Elsevier B.V. All rights reserved. PACS: 42.55.Vc; 52.38.Hb; 42.60.By
1. Introduction As has been known for some years, plasmas generated by optical field ionization (OFI) with linearly polarized radiation are promising candidates for recombination XUV lasers because, in principle, extremely low electron temperatures
* Corresponding author. Tel.: +44 1865 282209; fax: +44 1865 282296. E-mail address: [email protected] (S.M. Hooker).
can be achieved in relation to the ion stages produced [1]. The first report of gain arising from recombination in an OFI plasma was by Nagata et al. [2], who employed radiation from a nanosecond-pulse KrF laser to form a singly ionized Li plasma from a solid target, which was then field ionized by radiation from a picosecond KrF laser. A gain of 20 cm1 on the n = 2 ! 1 transition of H-like Li was reported for plasma lengths of up to 2.0 mm. In more recent work, the length of the gain region was increased to 5 mm by Korobkin et al. [3] who employed a laser-ablated LiF capillary to guide the pump beam. By this means
0030-4018/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2005.01.031
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the single-pass, small-signal gain was increased to approximately exp(5.5). Further progress in driving recombination lasers in OFI plasmas has been hampered by the difficulty in practice of achieving simultaneously a highly ionized, yet cold plasma. One potential solution to this problem is to dilute the lasant gas in a gas of lower ionization energy so as to reduce the average electron temperature produced by OFI of the mixture. For example, Grout et al. [4] have compared the gain generated on recombination laser transitions in nitrogen and argon ions produced by OFI of the pure gases and dilutions of those gases in hydrogen. In that work small-signal gain coefficients on a 24.7 nm transition in N4+ and a 23.2 nm transition in Ar7+ of a = 25 and 10 cm1 respectively were predicted for ionization of the pure gases by KrF laser radiation at 250 nm. Diluting the lasant gas with hydrogen was calculated to increase the gain significantly: the small-signal gain coefficients were predicted to increase to 65 and 360 cm1 respectively for nitrogen–hydrogen and argon–hydrogen gas mixtures. In addition to difficulties in achieving a low electron temperature, it is also difficult to generate the precursor to the lasant ion stage over a sufficiently long length that saturation of the shortwavelength laser transition is achieved. In fact, saturation of an OFI-driven recombination laser has yet to be reported. These difficulties are caused by the fact that longitudinal pumping is necessary to achieve efficient travelling-wave excitation of the short-lived gain medium, and for this pumping geometry diffraction and ionization-induced refraction severely restrict the gain length that may be achieved. In order to overcome this limitation it is desirable to operate the recombination laser in a waveguide. In this paper we address both of these problems, and analyze pumping the argon-doped hydrogen OFI-driven recombination laser within the plasma channel of a gas-filled capillary discharge waveguide. This waveguide [5] offers two distinct advantages for driving recombination lasers of this type. First, a plasma channel is formed that is able to guide the pump laser pulses over long lengths. Second, for channels formed in
mixtures of hydrogen and the lasant gas, the capillary discharge creates a dense pool of cold electrons by discharge ionization of the hydrogen. The cold discharge electrons can greatly enhance the recombination rate into the upper laser level following optical field ionization of the lasant species. These effects combine to enhance the small-signal gain coefficient, and the intensity and brightness of the output XUV pulse. The paper is organized as follows. In Section 2, we analyze the gain produced by pumping neutral Ar and Ar/H gas targets using driving pulses with wavelengths of 400 and 800 nm. Very high gain coefficients are predicted for 400 nm pumping of Ar/H mixtures such that the population inversion would be extremely heavily saturated. We argue that in this case the greatest output would be achieved for plasma conditions that are very different than those which yield the greatest small-signal gain, and show that the output intensity can be orders of magnitude greater than the saturation intensity Isat. In Section 3, we present the results of simulations of the propagation of the driving laser pulses through a passive gas cell containing that gas mixture which is predicted to yield the greatest XUV output. We find that defocusing limits the gain length to 20 mm for a pump wavelength of 400 nm, leading to an estimated output pulse energy of approximately 50 lJ. In Sections 4 and 5 we consider the operation of OFI recombination lasers driven within the plasma channel of a gas-filled capillary discharge waveguide. In Section 4 we analyze the effects on the kinetics of the laser system of pre-ionization of the lasant mixture by the discharge. It is found that for 400 nm pump radiation pre-ionization has little effect on the maximum intensity and output energy that can be obtained. For 800 nm pumping, however, pre-ionization is extremely beneficial, increasing the small-signal gain coefficient by an order of magnitude. In Section 5, we present the results of calculations of the gain length that may be achieved within the waveguide. It is shown that extended propagation of both 400 and 800 nm pump pulses is feasible, leading to calculated output pulse energies of several hundred microjoules.
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2. Laser kinetics for argon–hydrogen mixtures in a gas cell The argon-doped hydrogen recombination laser has been investigated by Grout et al. [4]. Grout considered pumping with a 100-fs, 250-nm KrF laser pulse, and calculated a small-signal gain coefficient of 360 cm1 and a saturation intensity of 2 · 107 W cm2 for a gas mixture comprising 2 · 1018 cm3 of H and 1 · 1017 cm3 of Ar. This can be compared to a calculated gain coefficient of less than 10 cm1 for pure argon gas targets. The calculations presented in the present paper use the same computational model used by Grout et al. [6] The model includes plasma ionization, heating and expansion, and collisional-radiative modelling of the excitation of selected ion stages. The model assumes that after the passage of the laser pulse, the electron energy distribution relaxes instantly to a Maxwellian distribution: this will lead to an underestimate of the gain for recombination lasers since the excess of cold electrons in the real distribution will enhance the recombination rate, as modelled recently by Avitzour et al. [7]. The argon-doped hydrogen recombination laser operates as follows. For a gas cell, the gas mixture is rapidly ionized by OFI to H+ and Ar8+ by a linearly polarized pump pulse. The electrons are heated by above-threshold ionization (ATI) and inverse bremsstrahlung (IB), reaching a final temperature of a few eV after the passage of the laser pulse. Since the ATI heating of the electrons originating from ionization of hydrogen is far lower than the average ATI heating of the eight electrons stripped from argon, hydrogen has the effect of decreasing the average electron temperature compared to OFI of pure argon gas. The cold electrons recombine rapidly with Ar8+ ions, leading to population inversions on several n = 4 ! 3 transitions in Ar7+. The gain is transient, with a full width at half maximum duration of around 10 ps. In this section, results are presented for argondoped hydrogen recombination lasers pumped using 400 and 800 nm pump pulses that could be obtained from a Ti:Sapphire laser. We note that terawatt Ti:Sapphire lasers are compact and convenient, and are much more widely available than terawatt KrF laser systems.
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2.1. Pumping at 400 nm We first consider pumping with frequencydoubled Ti:Sapphire laser radiation at 400 nm. Simulations were performed for both zero dimensions (a single point in space) and one dimension (as a function of radius, assuming cylindrical symmetry). Plasma expansion and heat flow in the 1D simulations were found to have very little effect on the population inversion density on axis, and so the results presented in this paper are from zerodimensional simulations. Simulations were performed for a laser pulse with a sech-squared temporal profile of 30-fs-fullwidth at half maximum duration focused to a peak intensity of 1 · 1017 W cm2. The gain coefficient on several n = 4 ! 3 transitions was calculated for a range of total ion densities and for a range of Ar densities. Fig. 1 shows three calculated parameters plotted as a function of the total ion density, for the range of fixed argon densities shown in the legend; moving to the right along each curve therefore corresponds to an increasing hydrogen density. The calculated parameters are the average electron temperature Te immediately after the passage of the pump pulse, the peak small-signal gain coefficient a of the 4s1/2–3p3/2 transition in Ar7+ at 23.2 nm, and the maximum output energy of the XUV laser per unit volume of gain (discussed below). The same parameters are shown for pure argon and pure hydrogen gas. The results for pure argon in Fig. 1 show that for low total densities, Te is independent of density, and for high densities it is proportional to density. The reason for this is that at low density ATI heating is dominant – ATI heating is determined only by the ratio of Ar to H and is independent of the total density. However, at high total densities IB heating begins to dominate since the IB heating rate is proportional to the ion density. The results for pure hydrogen illustrate the fact that ATI heating of H is smaller than that of argon. The IB heating asymptote is also moved to higher densities for pure hydrogen since the electron–ion collision frequency increases rapidly with the ionic charge. The remainder of the Te curves on Fig. 1 demonstrate the effect of adding hydrogen to a fixed
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heating by the hydrogen ions increases the plasma temperature. For argon densities greater than approximately 1019 cm3, a minimum in Te is no longer formed as hydrogen is added. In this regime the dominant heating mechanism for all electrons is IB heating by the argon ions; the hydrogen electrons are also heated by the argon ions and no reduction in plasma temperature occurs. Fig. 1 also shows the calculated small-signal gain coefficient of the 4s1/2–3p3/2 transition in Ar7+ as a function of the total ion density for the same Ar/H mixtures as above. Extremely high gain coefficients are predicted; gains of over 200 cm1 are calculated for a wide range of plasma conditions, the largest being more than 430 cm1 for an argon density of 1 · 1017 cm3 and a hydrogen density of 8 · 1018 cm3. We note that small-signal gain coefficients of between 10 and 400 cm1 are also calculated for eight other transitions in Ar7+. These are, in order of decreasing gain, 4s1/2– 3p1/2, 4f5/2–3d3/2, 4p3/2–3d5/2, 4f7/2–3d5/2, 4p1/2–3d3/2, 4p3/2–3s1/2, 4d5/2–3p3/2, and 4p3/2–3d3/2. 2.2. Estimate of output energy
Fig. 1. Calculated electron temperature Te, small-signal gain coefficient a, and the XUV output pulse energy per unit volume of gain Eex for argon–hydrogen mixtures as a function of total ion density for a range of fixed argon densities. Curves are also shown for pure hydrogen, and pure argon. The parameters are calculated for pump pulses of 400 nm wavelength and 30 fs duration focused to an intensity of 1 · 1017 W cm2.
density of argon. For argon densities lower than approximately 3 · 1018 cm3, a pronounced minimum in temperature occurs as hydrogen is added. For this range of argon densities, IB heating by the argon ions is still small, and cooler electrons from the additional hydrogen, heated dominantly by ATI, decrease the average electron temperature. As the hydrogen density is increased further, IB
It is clear that the single-pass small-signal gain in a system of this type can reach extremely high values, especially, as we consider below, if the pump laser pulse is guided over long lengths. It is important to be clear about the significance of the gain coefficient a and saturation intensity Isat in this regime. The intensity of the output XUV laser pulse I is characterized by exponential growth with propagation distance for I Isat but growth which is linear, or sub-linear, for I Isat. For ASE lasers it is often estimated that the output will be saturated for a single-pass small-signal gain of order exp[alsat] = e15, which may therefore be taken to define a saturation length lsat. Using this estimate it is clear that transitions with a gain coefficient of order 150 cm1 will saturate for gain lengths as short as 1 mm, and hence for gain lengths much longer than this the gain will be very strongly saturated. Calculation of the output of a pulsed ASE laser operating under conditions of saturation is complex. However, some insight may be obtained by first considering a homogeneously broadened laser
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amplifier operating under steady-state conditions. The growth in intensity of the beam in such an amplifier is described by, dI a0 j0 ¼ I; ð1Þ dz 1 þ I=I sat 1 þ I=I 0sat
we will assume a0 j0 and hence that the amplification is strongly saturated for much of the length of the laser. Within this region of strongly saturated amplification the intensity will grow linearly with propagation distance according to
where a0 is the small-signal gain coefficient, j0 the unsaturated absorption coefficient, I 0sat the saturation intensity for the absorption, and z is the distance along the axis of propagation. Clearly at low intensities such that I I sat ; I 0sat the intensity will grow exponentially as I(z) = I(0)exp([a0j0]z). If I 0sat I sat the absorption saturates before the gain, and hence the beam intensity grows essentially exponentially until the gain also saturates, whereupon the intensity continues to grow according to,
ð4Þ
dI a0 I sat j0 I 0sat ; dz
ð2Þ
i.e. the intensity of the beam grows linearly with distance at a rate slower than a0Isat by the term j0 I 0sat . The situation will be rather different, however, if I 0sat > ða0 =j0 ÞI sat . In this case the gain saturates faster than the loss such that the right-hand side of Eq. (1) decreases to zero when the intensity of the beam reaches a maximum value of, I max ¼
a0 =j0 1 I sat : 1 ja00 II sat 0
ð3Þ
sat
We note that this can be orders of magnitude greater than Isat. From this simple analysis we see that the maximum intensity to which the beam may be amplified is determined by losses for which I 0sat > I sat , i.e. losses which saturate slowly relative to the gain, or those which do not saturate at all. For the present case the dominant slowly saturating loss is likely to be photoionization of excited states of the lasant ion. The cross-section for this process will be several orders of magnitude smaller than the optical gain cross-section of the laser transition owing to the greater spectral width. As a consequence it is likely that a0 j0. Detailed calculation of the absorption arising from excited states is beyond the scope of the present paper, and so
dI a0 I sat ; dz
until the intensity reaches values of order Imax. However, dI/dz is equal to the power per unit volume extracted by the beam, and is a maximum under conditions of extreme saturated amplification. Hence we may define the maximum possible power extraction per unit volume as, J ex ¼ a0 I sat ¼
N hx ; sR
ð5Þ
where sR is the recovery time, N* is the population inversion density that would be achieved in the absence of saturation, and hx is the energy of the laser transition. We note that Jex is independent of the saturation intensity. For pulsed ASE lasers also, the output pulse energy is not necessarily limited by Isat provided the single-pass small-signal gain is sufficiently high. This is particularly important for OFI lasers for which the optical gain cross-section is usually large as a consequence of the small laser line width. This causes the saturation intensity to be low, but the small-signal gain coefficient to be high. The high gain coefficient that may, in principle, be achieved in OFI lasers means that conditions of extreme saturation may be reached for relatively short gain lengths, in which case the laser output can be orders of magnitude greater than Isat. Eq. (5) may be used to deduce the power extracted from a pulsed amplifier provided two conditions are met. First the amplifier must operate in a quasi CW regime in which the upper and lower laser level pump rates vary only slowly on the timescale of the gain recovery time sR. Second, the upper and lower level pump rates must be independent of the populations in the upper and lower laser levels. For the recombination scheme under investigation here the first condition is met, but the second condition is not and consequently a modified method must be employed to estimate the output energy of the laser transition.
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In the limit of strong saturation the peak intensity of the ASE laser pulse greatly exceeds the saturation intensity over most of the gain region. Under such conditions the rate of stimulated emission will be the maximum possible, that is at a rate which just reduces the population inversion density to zero throughout the pump pulse. Hence we may calculate Eex within the radiative-collisional code by forcing additional transitions between the upper and lower laser levels at a sufficient rate to keep the population inversion equal to zero. These additional transitions can be counted, and correspond to stimulated emission into the short-wavelength laser pulse. The result is an extracted power per unit volume Jex that can be integrated over the duration of the gain to give the extracted energy per unit volume Eex. We recognize that it will only be possible to extract Eex Joules per unit volume for regions of the gain medium which are strongly saturated. Nevertheless, this provides a useful upper limit, and we note that once the intensity of the ASE pulse becomes so high that absorption arising from unsaturated losses dominates, the ASE intensity can always, in principle, be reduced by allowing the beam to expand, or otherwise coupling some fraction of it out of the gain region. Further, we note that the plasma conditions which optimize Eex are found to be quite different from those which maximize the unsaturated gain coefficient; the former conditions may well be closer to those that optimize the output of a real ASE laser than the latter. 2.3. Results Fig. 1 shows the calculated value of Eex for the range of argon and hydrogen mixtures under consideration. For the conditions corresponding to the peak small-signal gain coefficient of 430 cm1, Eex is 0.45 J cm3. However, since the gain is so high for a wide range of plasma conditions, we can consider operating at conditions away from those which maximize the peak gain coefficient, provided the gain is still sufficient to ensure that extreme saturation would be achieved. It is clear from Fig. 1 that the plasma conditions which optimize Eex lie at higher argon and hydrogen density: for an Ar density of 3 · 1017 cm3 and
an H density of 4 · 1019 cm3, we have Eex = 1.2 J cm3. For these conditions, the predicted small-signal gain is still 150 cm1, sufficient to ensure extreme saturation for gain lengths of order 10 mm. It then remains, as presented in Section 3, to estimate the gain length and the gain volume that may be achieved so that the peak output intensity and pulse energy may be estimated. 2.4. Pumping at 800 nm It would be preferable to pump recombination lasers using 800 nm laser pulses rather than 400 nm owing to the additional complexity as well as energy loss incurred by frequency doubling the output of a Ti:Sapphire laser system. The change of wavelength from 400 to 800 nm has two main effects on the interaction of the laser and the plasma [1,8]: ATI heating increases with wavelength approximately as k2 owing to the increased quiver energy of electrons in the lower frequency field; at the same time, the electron-ion collision rate is decreased, and for a given ion density the IB heating rate varies approximately as k1/2. Since ATI heating determines the baseline temperature for all plasma densities, this leads in general to higher plasma temperatures for longer pump wavelengths. The collisional-radiative code was used to simulate the plasma properties and small-signal gain achieved with 800 nm pumping. Fig. 2 shows the electron temperature Te and gain coefficient a for the case of pumping with a 30 fs, 800 nm laser pulse focused to an intensity of 1 · 1017 W cm2. The curves and their interpretation are the same as in Fig. 1. First consider the electron temperature of pure argon and pure hydrogen plasmas. At low densities, where ATI heating dominates, the calculations predict that a pure argon plasma is heated to over 40 eV, compared to 10 eV for 400 nm pumping. However, IB heating is decreased sufficiently that IB heating of pure argon is not significant for argon ion densities less than approximately 1018 cm3. ATI heating of pure hydrogen increases to 10 eV compared to 2.4 eV for 400 nm pumping. For argon–hydrogen mixtures, there is still a pronounced temperature minimum as hydrogen is added to a fixed amount of argon; however, the plasma temperature is con-
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Fig. 2. Calculated electron temperature and small-signal gain coefficient as in Fig. 1 but for a pump laser of wavelength 800 nm.
strained to be greater than 10 eV owing to the ATI heating of the hydrogen electrons. Fig. 2 shows the gain calculated for various plasma conditions. Optimal conditions are shifted to higher ion densities compared to 400 nm pumping owing to the decrease in IB heating. However, because of the increased plasma temperature, and the consequently much smaller recombination rate, the gain is drastically reduced, to the point that 800 nm pumping of this laser system seems impractical. A peak gain coefficient of only 6 cm1 means that the laser output will not saturate for inversion lengths of the order of 1 cm, and so energy will not be extracted efficiently from the laser medium.
3. Pump laser propagation for a neutral gas target The gain length that can be achieved by longitudinal pumping of partially ionized plasmas is lim-
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ited by refraction and by ionization-induced defocusing. This has been discussed previously in relation to recombination lasing in Ar/H mixtures pumped using 250 nm KrF laser pulses [9], where it was concluded that it would be possible to create Ar8+ along the full length of a 4 mm gas jet. In this section, we discuss the maximum gain length that can be achieved in an argon-doped hydrogen gas mixture for 400 nm pump pulses. An estimate of the gain length allows us to confirm that the single-pass gain is sufficient to ensure extreme saturation, and to estimate the output intensity and energy of the XUV laser pulse. Numerical simulations were performed using a propagation code [10] that simulates propagation of an intense pulse propagating through a plasma undergoing optical field ionization. The nonrelativistic code accounts for absorption of the pulse energy owing to ionization, but not due to ATI and IB heating – these will later be shown to be negligible. Fig. 3 shows, as a function of radial position and distance z from the cell entrance, the calculated argon ion stage produced by the passage of an intense laser pulse through a gas cell containing a mixture of argon and hydrogen gas. The gas cell was taken to contain Ar at a density of 3 · 1017 cm3 and H at a density of 4 · 1019 cm3, both initially un-ionized. The 400nm-wavelength, 30-fs-duration sech-squared laser
Fig. 3. Contour plot showing the argon ion stage produced by the passage of a 30 fs, 235 mJ laser pulse of wavelength 400 nm through a gas cell containing 3 · 1017 cm3 Ar and 4 · 1019 cm3 H. In vacuum, the beam would focus 5 mm into the gas cell with a spot size of 20 lm, and a peak intensity of 1.2 · 1018 W cm2.
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pulse was taken to be focused into the gas cell so that in vacuum it would reach a 20 lm focal spot radius at a position 5 mm into the gas cell, with a peak intensity of 1.25 ·1018 W cm2. The position and size of the focus were chosen heuristically. The simulation shows that the laser pulse is defocused by the ionizing plasma, and reaches a peak intensity of 1 · 1018 W cm2 just 4 mm into the gas cell. The desired species, Ar8+, is generated over a length of 20 mm, and out to a radius of over 30 lm at the cell entrance. Absorption due to ATI and IB are estimated to be just 4% of the pulse energy, and so are safely neglected by the simulation. The peak intensity reached during the simulation was 1 · 1018 W cm2; at this intensity the normalized laser potential a0 = 0.4, and so relativistic effects may also be neglected. The gain region calculated above may be used to predict the output characteristics of the XUV laser pulse. For these conditions we calculate a = 150 cm1 and Eex = 1.2 J cm3 (see Fig. 1). It is clear that for the majority of the gain length, the XUV signal will be sufficiently intense to extract the maximum possible energy from the population inversion. While different regions of the gain volume experience different peak pump intensities, since Eex and Jex are only slow functions of pump intensity we may apply the values calculated for the intermediate pump intensity of 1 · 1017 W cm2 to the whole gain volume. The gain volume is 4.5 · 105 cm3, and so we estimate the output energy to be 50 lJ. The calculated peak value of Jex is 1.2 · 1011 W cm3, which together with the calculated 2 cm gain length leads to an estimated peak output intensity of 2.4 · 1011 W cm2. We note that the calculated value of Isat for these plasma conditions is just 8.7 · 109 W cm2, more than an order of magnitude smaller than the peak output intensity estimated above. We also note that Isat is usually calculated for the conditions which maximize the small-signal gain coefficient; for the present case, this saturation intensity is only 4.4 · 107 W cm2, almost four orders of magnitude below the peak output intensity calculated for conditions that optimize Eex.
4. Laser kinetics for argon–hydrogen mixtures in a plasma waveguide The gas-filled capillary discharge waveguide would seem to be ideal for increasing the gain length of OFI recombination lasers since the guiding channel comprises an essentially fully ionized plasma of low temperature [11,12]. Guiding of pulses with intensities in excess of 1017 W cm2 has been demonstrated for hydrogen-filled capillaries up to 50 mm long [5]. By doping the hydrogen with a small fraction of the lasant gas, the properties of the waveguide will be changed only slightly, and the gain length can be extended beyond that which can be achieved in a gas cell. Indeed we note that recently this approach was used to demonstrate lasing on the collisionally excited 5d–5p transition at 41.8 nm in Xe8+ in a Xe/ H-filled capillary discharge waveguide [13]. The hydrogen-filled capillary discharge waveguide operates as follows. A discharge is constrained in a long capillary initially filled with unionized hydrogen; the energy dissipated by the discharge current heats and ionizes the hydrogen to create an essentially fully ionized plasma. Thermal conduction to the capillary wall causes the plasma temperature to be greatest on axis, and, since pressure is rapidly equilibrated across the capillary diameter, the plasma density is a minimum on axis. Magnetohydrodynamic simulations of the capillary discharge [14] and measurements [11] have shown that the plasma channel formed is approximately parabolic. In the absence of further ionization, and neglecting relativistic effects, a parabolic plasma channel will guide a Gaussian laser pulse with a constant spot size W provided that W is equal to the ‘‘matched spot size’’ of the channel W M ¼ ðr2ch =pre DN e Þ1=4 , in which re is the classical electron radius, and DNe is the increase in electron density at radius rch. Simulations have shown that a capillary discharge in pure hydrogen can form parabolic plasma channels with plasma temperatures as low as 3 eV, with more than 99% of the hydrogen atoms ionized [14]. As we demonstrate in Section 5, a plasma channel of this type will allow the pump laser to be guided over extended distances, creating a large volume of Ar8+. However, preheating of the plas-
ma by the discharge affects the kinetics of the XUV laser and must be considered in detail. To this end, the collisional-radiative excitation code was used to simulate the plasma conditions produced by the passage of pump laser pulses through plasma targets preheated to 3 eV, as expected to be found in the guiding channel of the waveguide prior to the arrival of the pump laser pulse. The code calculates the steady-state populations of each ion stage and its energy levels at this initial temperature produced by the discharge, before simulating the excitation of that plasma by an intense laser pulse. Prior to the arrival of the laser pulse, the discharge is calculated to ionize the hydrogen fully, and the argon to between Ar+ and Ar2+.
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4.1. Pumping at 800 nm Fig. 4 shows the electron temperature, peak small-signal gain coefficient, and Eex after the passage of an 800 nm, 30 fs laser pulse with a peak intensity of 1 · 1017 W cm2 through plasmas preheated to 3 eV by a discharge. The key difference from Section 2 is that here the hydrogen atoms are ionized thermally by the discharge rather than by optical field ionization. This means that there is no ATI heating of the hydrogen electrons. Accordingly, at low ion densities where IB heating is negligible, the predicted temperature of a pure hydrogen plasma is now only 3 eV, compared to 10 eV for a neutral hydrogen target. Similarly, the pre-ionization of the argon ions marginally reduces the temperature of a low density argon plasma to 43 eV, compared to 44 eV for a neutral argon target. The result of the discharge pre-ionization is that there is an enhanced cooling effect as hydrogen is added to a fixed density of argon. Peak smallsignal gain coefficients of over 30 cm1 are predicted for a range of plasma conditions, with a peak gain coefficient of 62 cm1 for an Ar density of 1 · 1017 cm3 and an H density of 2 · 1019 cm3. This gain coefficient is approximately an order of magnitude greater than that predicted for OFI at this wavelength of Ar/H gas mixtures contained in a passive gas cell. Operating at the conditions for peak gain, extreme saturation of the laser medium will occur for gain lengths of order one centi-
Fig. 4. Calculated electron temperature, small-signal gain coefficient, and XUV pulse energy per unit volume for preheated plasmas. An 800 nm pump laser pulse was assumed with a duration of 30 fs, and an intensity of 1 · 1017 W cm2. The plasma was taken to be pre-heated to 3 eV causing ionization to Ar1.2+ and H+ before the arrival of the pump pulse.
metre, allowing efficient extraction of the energy stored in the population inversion. For these conditions, we calculate Eex = 0.3 J cm3 and Jex = 1.6 · 1010 W cm3. 4.2. Pumping at 400 nm Fig. 5 shows the calculated final temperature, gain coefficient, and Eex after the passage of a 400 nm, 30 fs laser pulse with a peak intensity of
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1 · 1017 W cm2 through plasmas preheated to 3 eV by a discharge. For this pump wavelength, the calculated plasma temperatures are similar to those calculated for neutral gas targets, as presented in Fig. 1. The electrons from the hydrogen ions are heated to 3 eV by the discharge in the present case, and to 2.5 eV by ATI heating in the unionized gas mixture. The calculated gain coefficients are decreased by just over a factor of two for the preheated plasma. This reduction is almost entirely caused by the higher ion temperature
(0.6 eV for the unionized plasma and 3.2 eV for the preheated plasma), and hence larger line width, which decreases the optical gain cross-section in the pre-ionized plasma. However, Isat is corresponding increased, and thus the calculated value for Eex is not strongly affected, reflecting the fact that the population inversion density is similar for the two cases. This is a good example of a situation in which increasing the saturation intensity by increasing the line width of the laser transition does not lead to an increase in XUV output. A peak value for Eex of 1.0 J cm3 for an Ar density of 3 · 1017 cm3 and an H density of 4 · 1019 cm3 is calculated, which is similar to the value of 1.2 J cm3 in the absence of preheating.
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5. Pump laser propagation in a plasma waveguide
Fig. 5. Calculated electron temperature, small-signal gain coefficient, and XUV pulse energy per unit volume for preheated plasmas as for Fig. 4 but for a pump laser wavelength of 400 nm.
We now consider the propagation of the pump laser pulse through Ar/H-filled capillary discharge waveguides. For waveguides formed in pure hydrogen, the plasma is essentially fully ionized before the arrival of the laser pulse and proper matching to the plasma profile will allow the pulse to propagate with a constant spot size. The matched spot size of gas-filled capillary discharge waveguides is typically of the order 20–40 lm, and may be controlled by varying the initial gas density and the capillary radius. Perfect matched guiding cannot be achieved when driving OFI lasers within the plasma channel owing to the requirement that the lasant gas must be further ionized by the pump pulse. These additional electrons will be created predominantly in the axial region of the waveguide axis where the laser intensity is greatest, perturbing the parabolic electron density profile and defocusing the laser pulse. However, whilst perfect matched guiding is not possible, laser energy that is defocused away from the axis will be collected and refocused by the waveguide, greatly increasing the length over which the peak intensity of the laser pulse remains high. Simulations show that variations in the axial intensity of the guided laser pulse are smallest when the matched spot-size of the waveguide is smaller than the input spot size of the laser pulse; the additional focusing power of the waveguide is
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better able to collect and refocus the energy refracted away from the axis. Fig. 6 shows, as a function of radial position and distance z from the waveguide entrance, the calculated argon ion stage produced by the passage of an intense, 400 nm laser pulse through a plasma waveguide. The plasma channel was taken to comprise a mixture of Ar1.2+ (as determined by the collisional-radiative code for this plasma at 3 eV) at a density of 3 · 1017 cm3 and H+ at a density of 4 · 1019 cm3, corresponding to the mixture calculated in Section 4 to yield the highest value of Eex. A parabolic initial electron density profile with a matched spot size of 22 lm was found heuristically to provide the best results – this guiding profile would be formed [14] in a capillary with a diameter of approximately 350 lm. The 400 nm pump laser pulse was assumed to be of 30 fs duration focused to a peak intensity of 5 · 1017 W cm2 and a 30 lm spot at the entrance to the waveguide. This corresponds to a laser energy of 235 mJ, and peak laser power of 8 TW. The simulations show that the laser spot size oscillates as the pulse propagates, with the peak axial intensity varying between 5 · 1016 and 1 · 1018 W cm2. Owing to its stable closed shell configuration Ar8+ is produced for almost all of
Fig. 6. Contour plot showing the argon ion stage in an argonhydrogen waveguide after the passage of a 30 fs, 235 mJ laser pulse of wavelength 400 nm. The beam is focused at the entrance of the waveguide with a spot size of 30 lm, and a peak intensity of 5 · 1017 W cm2. The matched spot size of the waveguide was 22 lm. The gas mixture was 3 · 1017 cm3 Ar and 4 · 1019 cm3 H.
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this range of intensities. Hence, the desired species is generated over a length of more than 30 cm, and out to a radius of approximately 30 lm. From these results we can estimate the XUV output intensity that might be achieved. In Section 4 it was determined that Eex = 1.0 J cm3 and Jex = 9.5 · 1010 W cm3 for these plasma conditions. Hence, for a 30 cm gain length the output intensity is predicted to be 2.9 · 1012 W cm2 on the waveguide axis. Note that this is more than three orders of magnitude greater than Isat = 9.3 · 108 W cm2. The output energy may be estimated by multiplying Eex by the volume of Ar8+ generated by the pump pulse, leading to an output energy of 850 lJ. This would be a very high output energy, and compares very favourably with the highest pulse energy yet achieved from a tabletop short-wavelength laser: 880 lJ at 46.9 nm obtained from a in a z-pinch capillary discharge laser [15]. It is also much greater than can be achieved by high harmonic generation. For example, high harmonic output energies of 7 lJ at 72.7 nm and 0.33 lJ at 29.6 nm have been reported [16,17]. We also note that the Ar/H recombination could be operated at multi-Hertz pulse repetition rates, thereby providing mean output powers of several milliwatts. We acknowledge that the longest gas-filled capillary discharge waveguide which has been demonstrated to date is only 50 mm long. However, increasing the length of the waveguide to tens of centimetres is expected to be relatively straightforward, in which case the calculations presented here suggest that short-wavelength outputs with pulse energies of hundreds of microjoules should be possible. Simulations of the propagation of 800 nm, 30 fs laser pulses through the optimum gas mixture calculated in Section 4 (argon density of 3 · 1017 cm3 and a hydrogen density of 2 · 1019 cm3) show that Ar8+ could be generated over a length of 15 cm, and out to a radius of 30 lm. From the calculated values of Eex = 0.3 J cm3 and Jex = 1.6 · 1010 W cm3, we deduce an output pulse energy of 125 lJ, and a peak output intensity of 2.4 · 1011 W cm2 would be obtained.
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6. Summary In summary we have presented the results of calculations which show that a gas-filled capillary discharge waveguide could be used to increase the output of OFI-driven recombination lasers in two ways. First, the presence of a high density of relatively cold electrons from discharge ionization of hydrogen reduces the mean electron temperature of the plasma after OFI of the lasant gas, leading to greatly increased rates of recombination. Second, the formation of a plasma channel allows gain to be achieved over very much longer lengths than is possible in a gas cell. We have also argued that for very high values of the single-pass small-signal gain, such as those calculated for the Ar–H recombination laser, the laser will operate under conditions of extreme saturation such that the output will be significantly greater than deduced from the saturation intensity. Further, the output energy will be greatest under conditions which maximize Eex and Jex, rather than those which yield the largest single-pass unsaturated gain. These conditions can be markedly different. For the Ar–H recombination laser considered here, the peak output intensity was estimated to be as much as three orders of magnitude greater than the saturation intensity. To test these ideas we have calculated the electron temperature and small-signal gain coefficient for recombination lasing on the 4s1/2–3p3/2 transition at 23.2 nm in Ar7+ for both neutral gas cell targets and pre-ionized waveguide targets pumped by Ti:Sapphire laser radiation at 400 and 800 nm. For 400 nm pump radiation, peak small-signal gain coefficients of 440 and 200 cm1 were calculated for the gas cell and waveguide respectively. The XUV output was found to be maximized at higher hydrogen and argon densities, for which the extracted power densities Jex were 1.2 and 0.95 · 1011 W cm3 respectively. For 400 nm pump laser radiation the plasma channel did not greatly affect the kinetics of the laser, but did allow an increase in the gain length by more than an order of magnitude. For an 8 TW pump laser at 400 nm, a gain length of 2.0 cm was calculated for a neutral gas cell target, giving an output intensity of 2.4 · 1011 W cm2, and an output energy of
50 lJ. It was calculated that use of a plasma waveguide could increase the length over which gain could be generated to more than 30 cm, raising the potential output peak intensity and pulse energy by more than an order of magnitude to 2.9 · 1012 W cm2 and 850 lJ respectively. We note that these predicted energies may be reduced by non-saturating losses not considered in this current model. A more accurate prediction will require a detailed analysis of such losses. For 800 nm pumping, a peak gain coefficient of only 6 cm1 was calculated for neutral gas targets owing to increased ATI heating at the longer pump wavelength. However, for the plasma channels, small-signal gain coefficients of over 60 cm1 were predicted. At conditions yielding the largest peak small-gain coefficient, which in this case is required to ensure extreme saturation, Eex = 0.3 J cm3 and Jex = 1.6 · 1010 W cm3. Simulations of the propagation of 800 nm driving laser pulses through a plasma channel with optimum argon and hydrogen densities show that a gain length of over 15 cm could be achieved, resulting in an output intensity of 2.4 · 1011 W cm2 and an output energy of 125 lJ. We emphasize that the use of a plasma waveguide is essential if the laser is to be pumped with 800 nm radiation; the preionization of the gas mixture is crucial to achieving a high gain coefficient, as well as increasing the gain length which may be achieved by guiding the pump laser pulse. Finally we add that it seems likely the gas-filled capillary discharge waveguide will prove beneficial to a range of other recombination-pumped as well as collisionally pumped XUV lasers.
Acknowledgements The authors would like to thank Prof. G.J. Pert for supplying us with the collisional-radiative code used in the calculations, for advice on its use, and for many helpful discussions on the subject of this paper. SMH would like to thank the Royal Society for the support of a University Research Fellowship.
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References [1] S.C. Rae, K. Burnett, Physical Review A 46 (4) (1992) 2077. [2] Y. Nagata, et al., Physical Review Letters 71 (23) (1993) 3774. [3] D.V. Korobkin, et al., Physical Review Letters 77 (26) (1996) 5206. [4] M.J. Grout, et al., Optics Communications 141 (3–4) (1997) 213. [5] A. Butler, D.J. Spence, S.M. Hooker, Physical Review Letters 89 (18) (2002), p. art. no.-185003. [6] K.A. Janulewicz, M.J. Grout, G.J. Pert, Journal of Physics B – Atomic Molecular and Optical Physics 29 (4) (1996) 901. [7] Y. Avitzour, S. Suckewer, E. Valeo, Physical Review E 69 (4) (2004), p. art. no.-046409. [8] G.J. Pert, Physical Review E 51 (5) (1995) 4778. [9] M.J. Grout, G.J. Pert, A. Djaoui, Journal of Physics B – Atomic Molecular and Optical Physics 31 (1) (1998) 197.
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[10] D.J. Spence, S.M. Hooker, Journal of the Optical Society of America B – Optical Physics 17 (9) (2000) 1565. [11] D.J. Spence, S.M. Hooker, Physical Review E 6302 (2) (2001), p. art. no.-015401. [12] D.J. Spence, A. Butler, S.M. Hooker, Journal of Physics B – Atomic Molecular and Optical Physics 34 (21) (2001) 4103. [13] A. Butler, et al., Physics Review Letters 91 (18) (2003) 205001. [14] N.A. Bobrova, et al., Physical Review E 6501 (1) (2002), p. art. no.-016407. [15] J.J. Rocca, et al., Nuclear Instruments and Methods in Physics Research Section A – Accelerators Spectrometers Detectors and Associated Equipment 507 (1–2) (2003) 515. [16] E. Takahashi, Y. Nabekawa, K. Midorikawa, Optics Letters 27 (21) (2002) 1920. [17] E. Takahashi, et al., Physical Review A 66 (2) (2002), p. art. no.-021802.