Spatial coherence of short wavelength high-order harmonics

Appl. Phys. B 65, 313 328 (1997) C Springer-Verlag 1997 Invited paper Spatial coherence of short wavelength high-order harmonics T. Ditmire 1, J.W.G. Tisch 1, E.T. Gumbrell 1,R.A.Smith 1, D.D. Meyerhofer 2, M.H.R. Hutchinson 1 1 Blackett Laboratory, Imperial College of Science Technology and Medicine, London SW7 2BZ, United Kingdom 2 Department of Mechanical Engineering, University of Rochester, Rochester, New York 14627, USA Received: 12 May 1997 Abstract. Nonperturbative high-order harmonic conversion of high-intensity, ultrafast laser pulses in gases represents a unique means to generate high-brightness, coherent XUV, and soft X-ray radiation. We review the physics controlling the spatial qualities of harmonic radiation and recent experimental measurements of high-order harmonic spatial profiles. We also examine the factors controlling the spatial coherence of the harmonics. A detailed series of Young s two-slit experiments that measure the spatial coherence of the harmonics is presented. These measurements indicate that the harmonics exhibit good fringe visibility and high spatial coherence. PACS: 42.65.Ky; 42.25.Kb Since the first observation of nonperturbative high-order harmonic generation of laser light in a gas by McPherson et al. in 1985 [1], high-order harmonics have received considerable attention as a potential high brightness, coherent source of X-rays [2]. High-order harmonics are produced when a high intensity laser is focused into a gas, and the nonlinear oscillations of the atoms in the medium re-emit photons at odd harmonics of the incident laser light. Though low-order harmonics, such as those with harmonic order less then the seventh or ninth, have been studied for many years [3], the experiment of [1] and many others that followed soon after illustrated that at high intensities (> 10 13 W/cm 2 ) the exponential drop in yield with increasing harmonic order usually associated with a lowest-order perturbation theory treatment does not continue with higher harmonic orders [4 7]. Instead, above a low order the harmonic spectrum exhibits a plateau of many harmonics with roughly equal yield, which can reach to very high orders. For example, harmonics as high as the 155th order have been observed from the harmonic conversion of 800 nm light in helium [8]. These very high-order harmonics have wavelengths that stretch into the XUV and soft X-ray region. The high-order harmonics may represent a very attractive source of short wavelength radiation. The extensive research of high-order harmonic generation has been motivated by the potential of using these harmonics as a source of highbrightness, coherent, soft X-ray radiation. A number of recent experiments have explored the possibility of using the harmonics in various applications requiring short pulses of XUV or soft X-ray radiation. Haight et al. have used harmonics in time resolved pump probe photoelectron spectroscopy of GaAs [9, 10]. Balcou et al. utilized harmonics in photoionization spectroscopy of noble gases with photon energy as high as 100 ev [11] and Larsson et al. exploited the short pulse nature of the harmonics to measure the lifetime of the 1s2p 1 P state in helium [12]. XUV harmonics have also been used to probe the dynamics of laser plasmas [13]. Aside from these applications, the properties of these harmonics have been very carefully studied by a number of groups around the world in recent years [4, 5, 14, 15]. Probing the shortest wavelengths attainable with harmonic generation has been a topic of extensive research. In addition to the 155th harmonic of a Ti:sapphire laser, the 141st harmonic from a 1ps, 1054 nm laser has been observed [16] and the 37th harmonic of a 248 nm KrF laser has been reported [17]. The harmonic spectra that result in most experiments typically have a number of important features in common. The harmonic yield drops exponentially over the first few orders (up to the 7th or 9th). This drop is followed by a long plateau of roughly constant yield between harmonics, which is in turn followed by an abrupt cutoff. Theoretical and experimental studies of this behavior indicate that the location of the harmonic cutoff, and hence the shortest wavelengths achievable, is usually given by a universal scaling law [18 20]: hω cut I p + 3.2U p, (1) where I p is the atom s ionization potential, U p is the ponderomotive field of the laser, and hω cut is the cutoff harmonic photon energy. This law has been confirmed in many experiments though some deviations from this scaling have been observed [21, 22]. Other properties of the high-order harmonics have been studied as well. The energy yields and conversion efficiency of the harmonics have been measured. Yields as high as 60 nj in the 200 Å wavelength range have been reported [23]. The

314 spectral bandwidth properties have also been examined, indicating that the harmonics exhibit linewidths comparable to or larger than perturbation theory estimates for the transform limited bandwidth [21, 24]. Recently techniques have also been developed to measure the harmonic pulse width as well [25 28]. These studies have confirmed that the harmonic pulse width is close to that of the driving laser and therefore using femtosecond laser pulses, soft X-ray pulses of under 100 fs can be generated. Glover et al. measured the harmonic pulse width by cross correlating the harmonic with the laser pulse by using photoionization [26]. This measurement indicated that the spatially integrated harmonic pulse width from a 100 fs laser varied from 80 to 300 fs, depending upon the conditions. Similar results have been obtained using ionization gated absorption [27]. Tisch et al. have also measuredthe harmonicpulsewidthsfrom1pslaser pulses by using chirped pulse temporal spectroscopy and have found that if the pulses are not spatially integrated, the harmonic pulse can be substantially shorter then the drive laser pulse [28]. In addition to these studies of harmonics produced in gases of single atoms, the properties of harmonics produced in molecules [29, 30] and clusters [31] have also been examined in experiments. Recently harmonics produced from laser interactions with solid target plasmas are beginning to receive attention as well [32]. Aside from these experiments, there is particular interest in the spatial properties of the high-order harmonics. Knowledge of the spatial divergence of the harmonics is important if the harmonics are to be used in applications. It also serves further to illuminate the nature of the physics controlling the harmonic generation. In addition, characterization of the spatial coherence is of particular importance if harmonics are to be used in interferometric applications. The coherence of high-order harmonics is expected to be high since the radiation is created by the parametric conversion of coherent, single spatial mode, laser radiation. Measurements of the harmonics far-field profiles have indicated that the harmonics largely preserve the low divergence, Gaussian character of the laser radiation [33 35]. However, a more fundamental question is whether the high-order harmonics preserve the high spatial coherence of the fundamental laser radiation. Until recently, the spatial coherence of the harmonics had not been measured [36]. While a number of previous experimental studies have characterized the far-field profiles of the high-order harmonics, knowledge of the far-field profile alone does not indicate the actual transverse spatial coherence of the radiation. Measuring the transverse coherence requires performing some manner of interference experiment such as a Young s two-slit experiment. Previously, such techniques have been applied to measure the spatial coherence of short wavelength XUV [37] and soft X-ray lasers [38 40] as well as laser-plasma X-ray sources [41] and low-order harmonics from solid target plasmas [42]. In this article, we briefly review the equations governing the propagation of the harmonic field in a nonlinear medium, and discuss the physics important in determining the harmonics far-field profiles. This is followed by a brief review of the experiments that have examined the harmonics spatial profiles. We then discuss the physics of spatial coherence in the context of harmonic generation. Finally, we present a detailed experimental study of the harmonics spatial coherence. In particular, we have conducted a series of Young s two-slit experiments to measure the spatial coherence of soft X-ray, high-order harmonics in the 270 Åto480 Åwavelength range. In general, we find that these harmonics exhibit good fringe visibility and high spatial coherence. 1 Harmonic production in a nonlinear, ionizing medium To investigate the physical mechanisms affecting the spatial properties of the high-order harmonic production, it is useful to examine the propagation equations governing the build-up of the harmonic field in a nonlinear medium driven by the intense laser pulse. Because the efficiency of the harmonic process is generally low, we can usually ignore any pump depletion on the fundamental laser beam. We also consider laser pulses that have pulses widths much greater then the time scale of a single wavelength. This fact allows us to use the slowly varying envelope approximation in deriving the wave equation governing the harmonic s growth [2]. In the slowly varying envelope approximation, the wave equation for the qth harmonic s field oscillating at a frequency ω q is 2 A q + n2 q ω2 q c 2 A q = 4πω2 q c 2 P q, (2) where A q is the field strength of the qth harmonic, n q is the spatially and temporally varying refractive index of the media at ω q,andp q is the polarization induced by the laser field at a frequency ω q. Equation (2) ignores the group velocity dispersion of the harmonic pulse as well as the group velocity walk-off of the harmonic pulse with respect to the laser pulse. Both of these effects are negligible for pulses of 100 fs or longer in a low density ( 10 19 atoms/cm 3 ) gas medium. To further simplify this equation, we can introduce the slowly varying spatial envelopes for the harmonic field and the polarization into (2). These are given by [ z a q (x, t) = A q (x, t) exp i [ z p q (x, t) = P q (x, t) exp i [ qk0 (x, t) + k(x, t) ] dz ], (3) qk 0 (x, t)dz ], (4) where k 0 is the wave number of the fundamental laser field, and the phase mismatch of the harmonic with the laser field is defined as k k q qk 0. Finally, using the fact that 2 a q z 2 k a q q z, (5) we arrive at the paraxial wave equation for the qth harmonic: 2 a a q q + 2ik q z 2k q ka q + ik q Nσ abs a q = 4πω2 q c 2 p q. (6) To derive this we have explicitly separated the real and imaginary part of the harmonic wave number, where k q is the

315 real part of the harmonic wave number, σ abs is the absorption cross section for harmonic photons, and N is the gas density. In deriving (6) we have ignored all terms that vary as k q ; this is equivalent to saying that n q / x = n q / y 0. In doing this we have ignored any refraction of the harmonic field by a spatially varying refractive index arising from the plasma formation. This is a very good approximation since short wavelength radiation will be very resistant to refraction by plasmas because of the large critical density associated with soft X-ray radiation. The refractive index of a low density plasma is n e n q 1 1, (7) 2 n crit where n crit is the critical density (the density at which the plasma frequency equals the laser frequency) and n e is the electron density. For harmonics with wavelength < 1000 Å, n crit > 10 23 cm 3, while the average gas density is typically < 10 19 cm 3 in most harmonic experiments. Thus n q 1and the radiation is very insensitive to radial variations in electron density; n crit for the laser, however, is 10 21 cm 3. Thus the radially varying phase imparted to the laser light may not be negligible and refraction may have an effect on the spatial profile of the laser [43]. Because the phase of the laser is then imparted to the harmonic through p q, the far-field profile and divergence of the harmonics can be affected. The nonlinear polarization, p q, is the term that drives the harmonic field. The amplitude of the nonlinear polarization is proportional to the nonlinear dipole moment of the atoms in the medium oscillating at the harmonic frequency times the density of the atoms in the medium. At low intensity, the phase of the polarization will generally be q times the phase of the laser. At higher intensity, the phase of the polarization will be intensity dependent. If we explicitly separate the intensity dependence of the polarization from the traditional geometric phase, however, we can write the following for the nonlinear polarization: p q (x, t) = 2N(x, t) dq [I(x, t)] { exp iϕdip [I(x, t)] } exp { iq[ϕ Gaus (x) + ϕ ran (x, t)] }, (8) where d q [I(x, t)] is the amplitude of the intensity dependent dipole moment oscillating at a frequency q. This dipole has been calculated numerically from the Schrödinger equation by a number of groups [18, 44]. In a perturbative treatment it is simply proportional to the qth power of the laser intensity, but, in strong laser fields its dependence with laser intensity is usually slower than this [2]. In (8), we have explicitly written the three different possible contributions to the phase of the nonlinear polarization, as it is these terms that have the greatest effects on the harmonics profiles and coherence. ϕ dip [I(x, t)] is the intensity dependence of the phase of the dipole, d q [I(x, t)]. This term has been shown to be very important in shaping the harmonics far-field profiles and may be important in a determination of their coherence [45, 46]; ϕ Gaus (x) is the phase associated with a focused laser beam. For a focused Gaussian laser beam with an intensity distribution given as [ ] 1 2(x 2 + y 2 ) I(x, t) = I 0 1 + 4z 2 /k0 2 exp w4 0 w 2 0 (1 + 4z2 /k0 2w4 0 ) exp ( 4ln2t 2 /τfwhm) 2, (9) (where w 0 is the 1/e 2 radius of the laser at focus), this phase is ϕ Gaus (x) = tan 1( 2z/k 0 w 2 ) 2k 0 (x 2 + y 2 )z 0 k0 2w4 0 +. (10) 4z2 Finally we have included a third contribution to the phase, ϕ ran (x, t), which accounts for randomly varying temporal phase initially on the laser beam. It is this rapidly varying phase that accounts for imperfect coherence of the laser beam and may subsequently degrade the coherence of the harmonic. In the tight focusing limit, namely, that case when the laser confocal parameter is smaller then the length of the medium, the geometric phase is very important. It is usually the limit on the conversion efficiency. The origin of this can be seen in (10) because the phase of the harmonics produced on one side of the focus have a phase nearly 180 different from those produced on the other side of the focus (due to the tan 1 term) and consequently destructively interfere. This phase can give rise to rings in the harmonics far-field profiles from geometric phase interferences. However, because of the much greater conversion efficiency when the laser is weakly focused (kw 2 0 l), most experiments are conducted in a regime where geometric phase effects are minimized and it is the intensity dependent phase and the production of free electrons that are most important in controlling the harmonics spatial properties. In addition to the phase contributions outlined above, examination of (6) indicates that the value of the phase mismatch k contributes to the harmonics phase. Though there is some phase mismatch associated with the refractive index of the neutral medium, a much greater contribution to the phase mismatch is the production of free electrons by the laser field during the harmonic generation. Using (7) for the refractive index of the fields in a plasma, this phase mismatch can be written (when q 1) k (πq/λ 0 )(n e /n crit ), (11) where n crit is the plasma critical density at the laser wavelength. The importance of this term can be simply seen if we find the solution of (6) in the plane wave limit. (This limit is a good approximation if the laser is weakly focused through the harmonic generating medium.) In this limit, ignoring the geometric phase mismatch and the absorption cross section, the qth harmonic field exiting a uniform medium of length l is A q (t) = 2iπk q n 0 l dq [A 0 (x, t)] e iϕ d sinc( kl/2)e i kl/2. (12) From this expression it is clear that the production of free electrons not only limits the conversion efficiency through the sinc term but also imparts phase on the harmonics. This phase

316 can alter the far-field profile [47]. Furthermore, if the time history of the production of free electrons is not uniform across the beam, a degradation of spatial coherence can result. 2 Measurements of high-harmonic far field spatial profiles A number of experiments have been conducted to investigate the spatial profiles of the harmonics [33 35]. In general these experiments produced harmonics by focusing a laser into the output of a gas jet or a flowing gas medium. Harmonics produced in the medium are spectrally dispersed in one dimension with a reflective or transmission grating and the spatial profiles were measured in the direction perpendicular to the spectral dispersion. This arrangement permits simultaneous measurement of the spatial profiles on a range of harmonics. Tisch et al. reported a measurement of the profiles from harmonics produced by a 1054 nm, 1ps, Nd:glass laser focused into low-density helium at intensities up to 3 10 14 W/cm 2 [33]. This experiment examined harmonics as high as the 119th order (88 Å). Raw data from that experiment are shown in Fig. 1a. Harmonics of increasing order are produced toward the right side of these data. In general this experiment found that the divergence of the harmonic radiation decreased with increasing harmonic order. The overall divergence of the harmonics tended to be larger then the perturbation theory prediction for harmonics produced in an unionized medium [i.e., when I q (x) I 0 (x) q ]. In addition, the harmonics in the plateau exhibited substantial structure. Figure 1b shows the lineout of the 71st harmonic. The harmonic exhibits a smooth central feature that is surrounded by sharp spikes. This behavior was attributed to the phase imparted to the harmonic beam by the production of free electrons. As the gas is ionized by the high intensity pulse, n e increases resulting in a commensurate increase in k. Because the ionization history varies radially in the focus, the exp{i kl/2} term imparts a rapidly varying radially phase that translates to structure in the far field. Experiments of Peatross et al. also observed structure in the lower-order harmonics produced in gases of very low density [35, 45]. These experiments were conducted using a 1ps Nd:glass laser focused into a low-density tubular target containing gas. The experiments were conducted in a density regime in which free electron phase matching effects were expected to be negligible. Figure 2 shows the measured far-field pattern of the 11th through 17th harmonics generated in 0.3torrof Xe at an intensity of 5 10 13 W/cm 2 [45]. The smooth, solid line shows the angular pattern of the f/70 incident 1 µm laser. These profiles are typical of those generated in that experiment, with a narrow central feature and broad wings or shoulders. The width of the central feature is consistent with that predicted from a perturbative calculation where the order of the harmonic production is of order 5. This is typical of calculations of high-order harmonic generation [2]. Because of the low gas density, good focal quality, and large f number (thin target) used in these experiments, the only apparent explanation of the wing structure observed was the effect of the phase of the atomic dipole response to the driving field. For a thin target, the emission can be considered to come from a single plane at the focus. At any time there is a radial intensity profile, which means that the harmonics are emitted with different phases at different radial locations. The interference of the different radial locations causes the harmonic emission to be spread over a larger angular region than predicted by perturbation theory. a b Fig. 1. a Spatial profiles from harmonics produced by a 1054 nm, 1ps, Nd:glass laser focused into low density helium at an intensity of 3 10 14 W/cm 2 (taken from [33]). b Lineout of the 71st harmonic from (a). The dashed line is the prediction of lowest-order perturbation theory

317 a Fig. 2. Spatial profiles from harmonics produced by a 1054 nm, 1ps, Nd:glass laser focused into 0.3torrof xenon at an intensity of 5 10 13 W/ cm 2 (taken from [35]) The spatial profiles from a 100 fs laser have been studied by Salières et al. [34, 48]. These experiments utilized a 825 nm, 140 fs Cr:LiSrAlF 6 laser focused with an f/25 lens into an argon or neon gas jet. These studies were conducted in gas at higher density then the studies of [33] and [35]. These experiments found that at low intensity, namely, that below the ionization saturation intensity (I sat ), the harmonics exhibited very smooth, single mode Gaussian profiles. (The saturation intensity is the peak intensity at which the ionization rate integrated over the laser pulse is approximately two, i.e., when nearly 90% ofthe atoms have been ionized by the end of the laser pulse.) Figure 3a shows data taken in Ne under these conditions. Once again, the harmonics are spectrally dispersed in one axis and spatially resolved in the other. This data illustrates that all the harmonics exhibit reasonably smooth profiles and that the divergence decreases with increasing order. Figure 3b shows lineouts of selected harmonics. These harmonics exhibit a divergence of roughly 10 20 mrad. This is approximately the divergence predicted by perturbation theory. At higher density and peak laser intensity above the ionization threshold of the gas, the smooth single mode spatial profiles of the harmonics is no longer preserved [49]. This is largely due to the refraction of the fundamental beam. This refraction imparts phase distortion on the fundamental, which is transferred to the harmonic through (8). This effect is illustrated in Fig. 4 in which the profiles of harmonics produced in argon at intensities corresponding to I sat (Fig. 4a) and an intensity 2 I sat (Fig. 4b) are compared. In these images, the spectra are dispersed horizontally and the spatial distribution is in the vertical direction. The images are of the 19th through the 29th harmonic and were taken 75 cm from b Fig. 3. a Spatial profiles from harmonics produced by a 825 nm, 140 fs, Cr:LiSrAlF 6 laser focused into a Ne gas jet at an intensity of 5 10 14 W/cm 2 (taken from [34]). b Lineouts of the 29th, 35th, and 41st harmonics from (a) the laser focus. At the lower intensity the profiles are very smooth and nearly Gaussian. However, at the higher intensity, the profiles exhibit some breakup from the refraction of the laser. Even more dramatic is a consideration of the twodimensional images of the harmonics from a refracted laser beam. Figure 5 shows the images of harmonics produced from a frequency doubled Nd:glass (526 nm) laser in a helium jet focused with an f/50 lens. These images were taken on the laser axis with an X-ray CCD detector. The detector was located 250 cm from the laser focus. A 7500 Å thick aluminum filter blocked the laser light and passed the soft X-ray radiation. The images are essentially a sum of the harmonic profiles of the harmonics that fall between the roll-over of the CCD detector on the long wavelength side at 350 Å and the cutoff of the aluminum filter at 173 Å on the short wavelength side (i.e., the 15th through the 31st harmonics.) The first image is of harmonics produced at 2 10 15 W/cm 2 (i.e., I sat ) the second and third images are at intensities of 5 10 15 W/cm 2 and 8 10 15 W/cm 2, respectively (i.e., > I sat ). The harmonics exhibit a clean, single mode Gaussian

318 3 Spatial coherence of harmonic radiation a Though the spatial profile studies presented in the previous section have been very important in unraveling much of the physics controlling the harmonic generation process, as mentioned earlier, they do not contain all pertinent information about the spatial properties. Of equal importance is the spatial coherence of the harmonic beam, or in other words how well two parts of the harmonic profile interfere with each other. The smooth, single mode character of the harmonics implies that the good coherence of the laser beam is probably preserved; however, an interferometric based measurement is required to quantify it. Because of the more difficult nature of such experiments, until recently no data existed to help quantify the spatial coherence of the harmonics in the XUV and soft X-ray region [36]. The simplest interferometric measurement possible is the well-known Young s two slit experiment. Before we discuss the physics affecting the coherence of the harmonics, it is useful to briefly review the mathematics of spatial coherence and place that in the context of a two-slit experiment. 3.1 Mathematical treatment of coherence b Fig. 4. a Spatial profiles from harmonics produced by a 825 nm, 140 fs, Cr:LiSrAlF 6 laser focused into an Ar gas jet of density 5 10 18 cm 3 at an intensity of 4 10 14 W/cm 2. b Spatial profiles of harmonics produced in Ar when the peak intensity is increased to 1 10 15 W/cm 2 a b c Fig. 5a c. Images of harmonics in the 173 Åto350 Å range produced from a frequency doubled Nd:glass (526 nm) laser in a helium jet focused with an f/50 lens at an intensity of (a) 2 10 15 W/cm 2,(b)4 10 15 W/cm 2, and (c) 8 10 15 W/cm 2 shape at the lower intensity but exhibit severe distortion at higher intensities. It is this refraction that sets an upper limit on the useful conversion efficiency that is achievable with high-order harmonic radiation [49]. Though the harmonic yield scales as the square of the gas density, implying that increasing the density is advantageous in achieving high yield, increasing the density and the laser intensity begins to cause a degradation in the harmonics profiles. This may be of concern for some applications. Equation (7) suggests that higher electron densities can be tolerated for a short wavelength laser. Therefore, the refraction effects will be more severe for the fundamental of a near IR laser at 800 or 1000 nm than for their second harmonic. This plasma induced phase can also cause a degradation of the harmonics coherence. The coherence of any light field can usually be quantified by the mutual coherence function [50]. This function is defined as Γ 12 = a(p 1, t + τ)a (P 2, t), (13) where a is the complex electric field amplitude of the field at points P 1 and P 2. The brackets denote an ensemble average. For all light fields we are concerned with, the ensemble average can be taken to mean a time average over some appropriate time interval. For an ultrashort pulse, such as a harmonic, this time interval will almost always be the entire pulse width [51]. The mutual coherence function is essentially the correlation function between two points in the field of the light source delayed with respect to each other by a time t. Note that Γ ii (0) is the intensity of the light source at point P i. It is usually convenient to work with the normalized quantity usually called the complex degree of coherence, which is γ 12 (τ) = Γ 12 (τ) Γ11 (0)Γ 22 (0), (14) where γ satisfies the relation 0 γ 12 (τ) 1. (15) If γ is one, the field exhibits complete correlation between the points 1 and 2 at time delay of t and is said to be fully coherent. We are most concerned with the coherence of the field with no delay between the two points. We will consequently refer to the complex coherence factor (CCF) µ 12 = γ 12 (0). (16) Next, we need to relate these quantities to the experimental setup, namely, the two-slit interference. The experimental configuration is generically shown in Fig. 6. We will assume that the slits are very narrow compared to the harmonic beam

319 Fig. 6. Generic experimental configuration of the Young s two-slit experiment to measure spatial coherence so that effectively only one point on the beam is sampled. If a(p) is the harmonic field at the slit mask, then the harmonic field at the detector plane, Q, will be a(q, t) = K 1 a(p 1, t r 1 /c) + K 2 a(p 2, t r 2 /c), (17) where the K i are constant amplitude factors, which depend on the diffraction of the field from each slit. We note that the harmonic intensity detected if light passes through only one slit will be I j (Q) = K j 2 a(pj,t r j /c) 2. (18) Consequently, the intensity detected when both slits are illuminated is [ ( )] I(Q) = I (1) (Q) + I (2) r2 r 1 (Q) + 2K 1 K 2 Re Γ 12. c (19) Examining (14), we note that the complex degree of coherence can always be written as γ 12 (τ) = γ 12 (τ) exp[ iω q τ iϕ(τ)]. Thus we see that the intensity observed, given by (19), of course oscillates as r 1 r 2 varies. If the detector is far from the slits, then the fringes have a spatial period, L,of L= λr x. (20) If we then define the fringe visibility on the basis of the maximum and the minimum intensities of this intensity pattern as V = I max I min (21) I max + I min (where I max and I min the maximum and minimum intensities of the fringe pattern), then the fringe visibility at the center of the pattern is simply V = 2 I (1) I (2) I (1) + I (2) γ 12(0). (22) We can therefore obtain a direct measurement of the complex coherence factor by a measurement of the fringe visibility that results from the introduction of a slit pair in the harmonics beam. From a practical standpoint, no slits used in any experiment are infinitely thin. They must simply be thin enough to assure that the diffraction from each slit is sufficient to assure complete overlap of both diffracted patterns at the detector. In practice this requires that the xz slit width, D,isD λz/ x. For partially coherent light, it will be useful to compare the measured dependence of the complex coherence factor with a known case. It is well known that an extended source of light, emitting incoherently over its surface, will exhibit some level of partial spatial coherence when the coherence is measured a distance z from the source. The complex coherence factor of an incoherent source can be quite easily calculated using the well-known van Cittert Zernike theorem [51]. This theorem states that the CCF of an incoherent source with an intensity function I(x) is given, far from the source as [ I(ξ, η) exp i 2π µ(x 1,y 1 ; x 2,y 2 ) = λz ( x ξ + y η) ] dξ dη, I(ξ, η)dξ dη (23) where ξ and η are the spatial coordinates at the source. This result, very similar to the result for Fraunhofer diffraction, indicates that the CCF is simply the Fourier transform of the source function. For a uniform, circular disk with radius a, the CCF measured between two points separated by x and y is µ( x, y) =2 ( ) J 2πa 1 λz x 2 + y 2, (24) x 2 + y 2 2πa λz where J 1 (ζ) is the Bessel function of the first kind, order 1. We can further quantify the coherence of a light source through the definition of its coherence area. The coherence area is defined as [51] A c = µ( x, y) d x d y. (25) An incoherent disk of diameter d will have a coherence area of A c = 4λ 2 z 2 /πd 2 measured at a distance z from the source. 3.2 Factors affecting the coherence of the harmonics With these mathematical preliminaries in mind, we are in a position to consider the factors that can degrade the coherence of the harmonics. We point out initially that if the fundamental laser is beam is completely coherent, in the absence of any phase variations imparted on the harmonic, the harmonics will also exhibit full coherence. Their phase is simply q times the phase of the laser; so, if the laser phase is completely correlated over all space, the harmonic s phase will also exhibit complete correlation. However, as we have seen there are a number of factors that alter the phase of the harmonics during their generation. If these factors impart a time-varying phase that differs from one point on the laser beam to the next, then the normalized time averaged correlation of (14) is less than one. There are a couple of factors that can do this.

320 First, from (8), we see that the harmonic has a phase associated with the intensity dependent dipole. If the intensity at two points on the laser beam are equal, then the time history of the dipole phase will be the same at these two points. If the CCF is then calculated between these two points the phases will cancel and the coherence is not affected. On the other hand, if the peak intensities differ slightly, the phase histories at the two points on the beam in question will not be exactly the same. This will give rise to interference in the time integral of (13) and may cause a decrease in coherence. By this reasoning, we note that if the two-slit experiment is performed on the harmonic beam with the slits placed symmetrically about the beam center and the laser beam intensity is radially symmetric, we expect that the fringe visibility will not be degraded. On the other hand, if the dipole phase does exhibit significant variation with intensity, as conjectured in [46] and [45], the CCF measured on two points not radially symmetric will exhibit a decrease in the CCF from unity as a result of the differing time histories of the phase. Another contribution to the phase of the harmonics is the k term as manifested in (12). As already discussed, this term is largely due to the production of free electrons by ionization when the laser intensity is sufficiently high. Thus, if there are any density fluctuations or fluctuations in the ionization time history between two points, the time history of n e, and consequently k will differ. This will also result in a decrease in the CCF upon time integration in (13). This physics is illustrated schematically in Fig. 7. Consider two points on the beam generated by equal peak intensities. If there is a slight density difference between the two points, the time history of the electron density will differ slightly, as shown in Fig. 7. This electron density time history translates into a phase time history. The phases between the two points do not cancel and will affect the time average correlation integral. We can further quantify this effect by making a estimation for the harmonic pulse shape. We can derive a simple scaling for the complex coherence factor if we assume that the harmonic pulse is square in time and that the electron density ramps up linearly over the harmonic pulse. In this case, we approximate the harmonic pulse as A q (x i, t) = A q0 exp{ i kl/2} { = A q0 exp i πqni e (t) } l, 0 t τ p, (26) 2λ 0 n crit If the linearly increasing electron density between points x 1 and x 2 differs by the amount δn e at the end of the pulse, the electron density can be written { (ne0 n i e (t) = + δn e ) τ t p, i = 1 t (27) n e0 τ p, i = 2 and the complex coherence factor is [ ( )] πql t µ 12 = exp i δn e. (28) 2λ 0 n crit τ p Time integration yields for the absolute value of the CCF: [ ] µ 12 = πql sinc δn e. (29) 4λ 0 n crit The maximum electron density at which harmonics will be produced is that at which kl 2π, implying a maximum electron density of approximately 1 5 10 17 cm 3 for the harmonics in the 500 100 Å range. It is reasonable, therefore, to assume that the electron density can fluctuate by an amount that is comparable to this value. Equation (29) implies that electron density variations of this magnitude will degrade the coherence to µ 0.9 0.6, a significant decrease in harmonic coherence. Equation (29) also implies that the coherence will be further degraded as q increases. This coherence degradation by the creation of free electrons is not due to the phase fluctuations imparted directly on the harmonic by refractive index changes, fluctuations that are very small for short wavelength harmonic light in low density plasma, but is due to the phase fluctuations placed on the laser beam, which are then transferred to the harmonic. Because the phase of the harmonic is q times that of the laser s, for high orders, it is possible for a very small degradation in the laser s coherence to result in a sizable degradation on the harmonic. This is a general statement, independent of the actual mechanism by which the laser s coherence is originally altered. This fact can be simply illustrated if we consider two points on a laser beam with small phase fluctuations between them. The laser field at the two points 1 and 2 can be written: A 1,2 = E 0 (t) exp [ iωt ikx +iϕ 1,2 (t) ], (30) which means that the CCF of the laser is µ 1,2 (ω) = A 1 A 2 A1 2 A 2 2 = exp [ i(ϕ 1 ϕ 2 ) ], (31) = τ 1 p τ p 0 exp [ i(ϕ 1 ϕ 2 ) ] dt, (32) for a square pulse, where we have simply averaged over the laser pulse. (τ p is the laser pulse width). Since we assume that the phase fluctuations are small, we can say that the absolute value of the CCF can be approximated as Fig. 7. General explanation of how a variation in density across the laser beam can affect the harmonic s spatial coherence. If the density between two points in the beam is slightly different, as the medium is ionized, the electron density will follow the thick solid and dashed curves µ12 (ω) 1 1 2 τp 1 τ p 0 ϕ dt. (33)

321 The harmonic field generated by this laser beam at points 1 and 2 will be approximately A 1,2 (q) = E q (t) exp [ iqωt iqkx + iqϕ 1,2 (t) ], (34) which means, by the same reasoning, that the magnitude of the harmonic s CCF is µ 12 (qω) 1 q 2 τ 1 p τ p 0 ϕ dt. (35) (now assuming that q ϕ is small, which is equivalent to saying that the laser s CCF is very close to one). So we see that the CCF of the harmonic and the laser are related by µ 12 (qω) 1 q ( 1 µ 12 (ω) ). (36) Thus the deviation of the harmonics CCF from unity will be q times larger than that of the laser. This reasoning implies that from the standpoint of preserving the inherent high coherence of the laser in harmonic generation, it may be advantageous to use short wavelength laser light to reach short wavelength harmonics since q can be kept small. 4 Measurement of high-order harmonic spatial coherence 4.1 Experimental Details To measure the coherence of XUV radiation produced by high-order harmonic generation, we executed a Young s twoslit experiment. The experimental configuration is shown in Fig. 8. The majority of the data described in the sections that follow were of harmonics generated in helium gas. This gas was chosen because the harmonics produced reach to higher order and shorter wavelength then the other gases, a feature essentially due to the high ionization potential of helium. Experiments were also performed in other gases with similar results. In our experiment, harmonics were generated with a Nd: glass laser based on the well-known technique of chirped pulse amplification. In brief, the laser consists of a diode pumped, additive pulse mode-locked Nd:YLF oscillator, whose 1053 nm, 2ps pulses are stretched to 500 ps with a dispersive grating/lens stretcher. These pulses are amplified in a Nd:glass regenerative amplifier and three additional amplifiers to an energy of 1J. A grating pair recompresses the pulses to their original pulse width of 2ps. The resulting pulse energy produced is up to 0.5J. The beam profile after recompression is near Gaussian with a 1/e 2 diameter of 35 mm. The laser pulses were then frequency doubled in a 1cm thick KDP crystal to a wavelength of 526 nm with a conversion efficiency of 45%. The high-order harmonics were produced with 526 nm light instead of 1053 nm light because the shorter wavelength drive has been shown to yield higher conversion efficiency into harmonics in the 200 500 Å range [23]. As discussed above, it is also expected to be less susceptible to free electron phase fluctuations. Furthermore, the shorter wavelength laser light is also less susceptible to ionization induced refraction effects within the gas medium. The Gaussian laser beam spatial profile was apertured to a diameter of 1.5cm immediately prior to the focusing lens to produce a uniform, near flattop profile. This beam was then focused with a 75 cm focal length, plano-convex lens. The focal spot of the laser light was measured to be a diameter of 70 µm, indicating that it is very near the diffraction limit. The laser pulses were focused into a gas plume produced by a pulsed gas jet that could be backed with up to 50 bar of pressure (yielding an estimated gas density of 5 10 18 cm 3 [52]). The high-order harmonics produced in the 200 Åto500 Å range were detected with a flat-field soft X-ray spectrometer located on the laser axis. This spectrometer utilized a horizontal 50 µm entrance slit. The harmonic radiation was dispersed with a flat field, grazing incidence, gold-coated 1200 line/mm grating. The soft X-rays were then detected with a Cs:I coated, two-stage microchannel plate that was coupled to a phosphor readout. The signal was then collected with a lens and a CCD camera. The total distance from harmonic source (laser focus) to the entrance slit was 130 cm and the distance from source to detector was 180 cm. Because the grating focuses only in the dispersing direction and does not alter the divergence of the radiation in the transverse direction, this configuration permits simultaneous measure of spectral information on one axis, with a measure of the spatial profile of each harmonic on the transverse axis. 4.2 Raw data A typical spectrum of harmonic radiation collected with this configuration is shown in Fig. 9. This spectrum shows the spatially integrated spectrum of high-order harmonics produced with 526 nm light in He at an intensity of 4 10 15 W/cm 2. The spectrum exhibits the classic highorder harmonic behavior, the production of a plateau of Fig. 8. Experimental configuration used to measure the high-order harmonics spatial coherence

322 slits in 20 µm thick Ti foil. Each slit had a width of 8 ± 1 µm and slit pairs with spacings of 28, 50, 75 and 100 µm were used for the measurements. Because of the proximity of the slit pairs to the laser focus (the laser was roughly 1mmin diameter at the slits), the slits began to show laser damage after 10 to 20 full-power laser shots. To monitor this degradation, the slits were backlit at a slight angle with a HeNe laser and the image of the slits was imaged outside the vacuum cham- Fig. 9. Spatially integrated spectrum of high-order harmonics produced with 526 nm light in He at an intensity of 4 10 15 W/cm 2 Fig. 11. Measured profile of the 13th harmonic at the location of the double slits (4cm from focus) generated at an intensity of 4 10 15 W/cm 2. This was measured by scanning a single slit across the harmonic profile. The solid line is a Gaussian fit to the data Fig. 10. Angularly resolved spectrum of high-order harmonics produced with 526 nm light in He at an intensity of 4 10 15 W/cm 2 harmonics to a high order with a sharp cutoff below a certain wavelength (in this case occurring at the 23rd harmonic λ 229 Å). The harmonics produced under these conditions exhibit smooth, low divergence spatial profiles. A twodimensional raw image of harmonics produced under the same conditions as Fig. 9 is shown in Fig. 10. The harmonics are dispersed in the vertical direction; their far-field spatial profile is manifested in the horizontal profile. These harmonics clearly exhibit smooth, single mode Gaussian far-field profiles, similar to the results reported in [34]. To measure the spatial coherence, we introduced slit pairs into the harmonic s beam before the spectrometer (see Fig. 8) to generate interference fringe patterns on each harmonic. Because of the short wavelength of the harmonics, it is necessary to use open slits to produce the interference fringe pattern. These slits need to be sufficiently narrow to ensure that only a small portion of the harmonic beam is sampled by each slit. Furthermore, we require that each slit be narrow enough to assure that the diffraction of the beam from each slit is fast enough to yield complete overlap of the radiation passing both slits at the detector. This implies that the slits need to be no wider than 10 µm. On the other hand, the slits must be wide enough to pass enough radiation for the fringe pattern to be detectable on each laser shot. With these considerations, we found that the optimal configuration was to use slit pairs of roughly 8 µm width placed 4cm from the gas jet plume and laser focus. The slit pairs used in the experiment were produced by laser drilling the Fig. 12. CCD image of fringes generated on the 11th to the 19th harmonic in helium with a peak intensity of 6 10 15 W/cm 2

323 ber with magnification. During the experiment when the slit exhibited some damage, it was replaced by an unused slit foil in the position of the original slits. The slit pairs were placed perpendicular to the 50 µm entrance slit of the spectrometer. The total distance from the slit pair to the detector was 176 cm. This distance assured that the harmonic fringe spacing (> 600 1000 µm) was larger than the estimated 100 µm spatial resolution of the detector. The profile of the harmonics at the position of the slit pairs was measured by placing a single 8 µm slit at the same position as the slit pairs and scanning the slit across the beam. The measured profile of the 13th harmonic generated at an intensity of 4 10 15 W/cm 2 is shown in Fig. 11. This measurement is characteristic of all of the harmonics, which, in general, have a diameter of roughly 350 µm. A typical raw image of a fringe pattern generated in the configuration of Fig. 8 by a slit pair with a 50 µm spacing placed in a beam of harmonics is shown in Fig. 12. This is data of harmonics generated at an intensity of 6 10 15 W/cm 2 in helium. In this image, fringes on all the harmonics (the 11th to the 19th) are clearly visible. All the harmonics exhibit some coherence under these conditions. From an analysis of such images it is possible to determine the harmonic fringe visibility under a variety of conditions. It is also important to note that the coherence of the laser used to generate these harmonics is very good. Fig. 13 shows lineouts of fringes generated by the laser itself. These fringe Fig. 14. Comparison of fringes produced by the laser with a 50 µm slit pair when the laser propagates only through vacuum at its focus and when the laser propagates through the ionizing gas jet at focus. Both fringe patterns were produced when the laser peak intensity at focus was 5 10 15 W/cm 2. (The gas jet backing pressure was 50 bar) patterns were generated with the same slits used for the harmonics; however the laser beam was reflected out of the chamber after the slit pairs and detected with a CCD camera. Figure 13 shows the laser fringes in vacuum (no gas jet) with three slit spacings. The fringe visibility is > 0.9 for all three slit separation. We also note that propagation of the laser through the gas jet does not significantly degrade the coherence of the laser. Figure 14 compares the fringe pattern of the laser generated with 50 µm slit separation when the laser propagates only in vacuum inside the chamber and the fringe pattern at the same laser intensity with the gas jet pulsed, so that the laser has propagated through the harmonic generating medium. As can be seen in this figure, the laser s fringes do not change dramatically upon propagation through the gas jet. 4.3 Coherence measurement results Fig. 13. Lineout of fringes generated by the 526 nm laser beam with three different slit separations. The fringes were generated with the same slit pairs used for the harmonics in vacuum, 4cmfrom the laser focus with the detector located 200 cm from the slit pairs To derive information about the harmonics mutual coherence function and how it changes with changing parameters, we have examined the fringe visibility of the harmonics as a function of a number of parameters. Lineouts of typical interference patterns obtained by using slits with a 50 µm spacing centered on the harmonic beam are shown in Fig. 15. Here fringe patterns on the 11th to the 19th harmonic are shown (covering the wavelength range of 277 to 479 Å). These data were taken with a peak laser intensity of 4 10 15 W/cm 2, an intensity above the ionization saturation intensity in helium ( 1 10 15 W/cm 2 ). As shown above in Fig. 12, all the harmonics exhibit interference fringes. However, the fringe visibility is < 1.0, indicating that all the harmonics exhibit some degree of partial coherence over a separation of 50 µm. In fact, the fringe visibility decreases with increasing harmonic order. Figure 16 shows the measured visibility with a 50 µm slit spacing as a function of harmonic order from the 11th to the 19th harmonic for two peak intensities: 1 10 15 W/cm 2 and 4 10 15 W/cm 2. Each point in this figure represents the average of 6 laser shots within a ±10% energy bin. The error bars in the measurement were determined by the extent of the shot to shot variation of the fringe visibility. At the lowest intensity, the fringe visibility is virtually constant over the harmonic order considered here, though

324 Fig. 17. Images and the resulting lineouts of fringes from the 11th harmonic (λ = 479 Å) generated with slits of 50 um separation at two different peak intensities Fig. 15. Lineouts of typical interference patterns obtained using slits with a 50 µm spacing centered on the harmonic beam produced with an intensity of 4 10 15 W/cm 2 and a gas jet backing pressure of 50 bar Fig. 18. Fringe visibility of the 15th harmonic as a function of laser peak intensity Fig. 16. Fringe visibility as a function of harmonic order for two different peak intensities there is a slight decrease in visibility as the order increases: the visibility is 0.6 for the 11th harmonic while it drops to 0.45 for the 19th harmonic. At the higher intensity, however, the variation of the fringe visibility with increasing harmonic order is more pronounced, falling faster with increasing order than the lower intensity harmonics. Here the visibility on the 11th harmonic is 0.45 and the visibility drops to 0.18 on the 19th harmonic. Figure 16 also indicates that there is a variation in the harmonic fringe visibility with laser peak intensity. This effect is illustrated in Fig. 17 where the images and the resulting lineouts of the fringes from the 11th harmonic (λ = 479 Å) generated with slits of 50 µm separation are shown at two different peak intensities. In both cases the slits were centered on the beam. The first image shows the fringes generated with a peak intensity of 8 10 14 W/cm 2, below the onset of significant ionization in the helium gas. Here the fringes are sharp and well defined with a corresponding visibility of 0.8. In contrast, at the higher intensity of 5 10 15 W/cm 2, an intensity at which significant ionization has occurred in the helium [53], the fringes are broadened, with a drop in visibility to 0.45. This trend is reflected over the intensity range from the mid 10 14 W/cm 2 to the mid 10 15 W/cm 2, the range over which helium ionization becomes important. Figure 18 illustrates the visibility for fringes of the 15th harmonic (λ = 351 Å) with 50 µm spaced slits as a function of intensity. Below the ionization saturation intensity the harmonic exhibits good fringe visibility of between 0.7 and 0.8. This visibility falls and levels off at about 0.4 at the highest intensity. The drop in fringe visibility illustrated here corresponds closely to the onset of significant ionization [53]. This data also indicates that at low intensity, the coherence of the harmonics can be quite high, nearly that of the laser itself with the same slit separation. One interesting trend of the fringe visibility is illustrated in Fig. 19. Here is shown the fringe visibility of the 13th harmonic with a 50 µm slit spacing as a function of the position of the center of the slit pair across the harmonic beam. The peak laser intensity is 5 10 15 W/cm 2, an intensity above ionization saturation and a regime where the fringe visibility is degraded. In the center of the beam, the fringe visibility is 0.4. However, as the slit pair is moved out to larger radius, the visibility increases. At a radius of 200 µm, near the edge of the harmonic beam (see Fig. 11), the fringe visibility has increased to nearly 0.7. This trend is reproduced in all of the harmonics studied and is also evident when larger slit spacings are used. Furthermore, we find that, at the high backing pressures used in our experiments, the fringe visibility does not significantly change with changing backing pressure. Figure 20