Abstract:
Phase-matched high-order harmonic generation of soft and hard X-rays is accomplished using infrared driving lasers in a high-pressure non-linear medium. The pressure of the non-linear medium is increased to multi-atmospheres and a mid-IR (or higher) laser device provides the driving pulse. Based on this scaling, also a general method for global optimization of the flux of phase-matched high-order harmonic generation at a desired wavelength is designed.

Description:
PRIORITY 
     This application claims benefit of U.S. Provisional Patent Applications No. 61/171,783 filed Apr. 22, 2009, and 61/172,686, filed Apr. 24, 2009, and 61/327,065 filed Apr. 22, 2010. 
     U.S. Pat. No. 6,151,155 is incorporated herein by reference. 
    
    
     GOVERNMENT SUPPORT 
     This invention was made with government support under grant number DE-FG02-04ER15592 awarded by the Department of Energy and grant number EEC 0310717 awarded by the National Science Foundation. The government has certain rights in the invention. 
    
    
     BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The present invention relates to phase-matched high-order harmonic generation of soft and hard X-rays using infrared driving lasers in a high-pressure non-linear medium. In particular, the present invention relates to efficient generation of coherent x-ray radiation by coherent upconversion of light from an intense mid-infrared pulsed laser in a high pressure gas nonlinear medium. The invention further relates to a general method for global optimization of the flux of coherent light of desired wavelength by selecting the optimal wavelength of the driving laser and its parameters, in combination with the optimal nonlinear medium and its parameters. 
     2. Background of the Invention 
     High-order harmonic generation (HHG) is a unique source of femtosecond-to-attosecond duration soft x-ray beams that has opened up new studies of atoms, molecules, and materials, as well as enabling new high-resolution coherent imaging using a table-top light source. To date, however, most applications of HHG radiation employ extreme ultraviolet wavelengths (photon energy ˜20-100 electron volts), because the efficiency of the HHG process decreases rapidly at higher photon energies. This decrease is not fundamental to the HHG process, but rather results from the large phase mismatch between the generated HHG field and the driving laser field at 800 nm, which to date is used in nearly all HHG experiments because of the availability of high-power ultrashort pulse lasers generating light at this wavelength. The obstacle in phase-matching HHG upconversion to very short wavelengths is the higher required laser intensity, which results in high levels of ionization and thus large free electron dispersion. This dominant plasma dispersion limits phase matching of HHG to relatively low levels of ionization, where neutral atom dispersion can balance the anomalous free-electron plasma dispersion. For a 0.8 μm driving laser, the “critical” ionization levels above which true phase matching is not possible are ≈5% for argon and ≈0.5% for helium. As a result, the highest photon energies that can be phase matched in Ar and He are ˜50 eV and ˜130 eV respectively. 
     Another important limit in HHG is the highest photon energy that can be generated by the laser regardless of phase matching—the so-called cutoff energy. This cutoff is given by hν max =I p +3.2U p , where I p  is the ionization potential of the gas and U p ∝I L λ L   2  is the quiver energy of the liberated electron, λ L  is the wavelength of the laser driving the process, and I L  is its intensity. The favorable λ L   2  scaling has motivated studies of HHG with mid-infrared driving pulses with wavelength longer than 800 nm. Significant extension of the cutoff energy hν max  to higher energy was demonstrated in several experiments. However, it was recently found theoretically that the actual EUV or x-ray yield of an atom radiating HHG light scales as λ L   −5.5±0.5 . The use of a longer wavelength driver, although it increases the energy of the individual HHG photons, greatly reduces the total conversion efficiency and thus the total energy in the burst of HHG photons [6]. Thus, increasing the HHG yield by finding new methods of phase-matching the conversion process is critical to obtain a usable flux at shorter wavelengths. 
     It is an object of the present invention to generate high-order harmonic light in the soft and hard X-ray regions of the spectrum in a more-efficient manner that optimizes phase matching of the light. This can be accomplished by using a mid-infrared driving laser in combination with a very high-pressure non-linear medium. This method of optimizing efficiency of high-harmonic generation conversion to short wavelengths has not heretofore been recognized. Past teaching in the area of high-order harmonic generation mostly employed sub-atmosphere target pressures, with the use of a very short-wavelength driving laser to maximize high-harmonic flux. 
     SUMMARY OF THE INVENTION 
     An object of the present invention is to generate phase-matched high-order harmonic generation of soft and hard X-rays using infrared driving lasers in a high-pressure non-linear medium. 
     For example, driving lasers having a wavelength of 1.3 μm and 2.0 μm generate HHG light in the water window region of the spectrum where the HHG is macroscopically phase-matched over centimeter distances. The optimal phase matching pressures of the non-linear medium are multi-atmosphere and are preferably combined with very moderate ionization levels of the medium (≈10 1 -10 −3 %). In this regime, the driving laser beam experiences minimal nonlinear distortion, resulting in an excellent spatial coherence of the HHG beam even when conversion is happening in a high pressure gas, well above one atmosphere. 
     To phase-match a nonlinear conversion process, the driving pulse phase velocity is matched to the phase velocity of the generated x-rays. The phase mismatch comprises the pressure-dependent neutral atom and free electron dispersions and pressure-independent geometric dispersions. These factors can cancel each other out within certain ranges. Therefore, the pressure and the ionization (and other factors) may be adjusted to minimize the phase mismatch. 
     Apparatus according to the present invention provides significant conversion efficiency of laser light into the x-ray spectrum. Soft x-rays allow coherent diffractive imaging/sensing (lensless imaging or holography) of biological specimens with resolution &lt;&lt;100 nm, using a table-top microscope. Hard x-ray applications include x-ray crystallography, diffraction imaging, and x-ray medical and biomedical imaging and treatment. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIGS. 1A-1D  (Prior Art) are schematic diagrams illustrating high-harmonic emission generation (HHG) without phase matching. 
         FIGS. 2A-2D  are schematic diagrams illustrating how the output x-ray intensity increases with phase matching within the medium, as well as the parameters of the HHG scheme that depend on the driving laser wavelength. 
         FIG. 3  is a diagram illustrating HHG with phase matching according to prior art A and according to the present invention B in the spectral domain. 
         FIGS. 4A  (Prior Art) and  4 B are schematic diagrams illustrating the resulting macroscopic HHG flux in conventional systems and in the present invention. 
         FIG. 5  is a diagram illustrating the driving lasers pulses and the HHG pulses with phase matching in conventional systems and in the present invention in the time domain. 
         FIGS. 6A-6D  are schematic diagrams illustrating input pulse shapes and output pulse shapes according to the present invention. 
         FIG. 7A  is a block diagram illustrating a general system for phase-matched HHG of x-rays according to the present invention.  FIG. 7B  is a block diagram illustrating a guided beam geometry embodiment of the present invention.  FIG. 7C  is a block diagram illustrating a loose-focusing geometry embodiment of the present invention. 
         FIG. 8  is a flow diagram illustrating steps in the process of global optimization of HHG with phase matching according to the present invention, and steps in selecting the optimal parameters for most efficient HHG at a desired HHG wavelength. 
         FIG. 9  is a plot of laser energy versus wavelength. 
         FIG. 10A  is a plot of ionization level versus wavelength.  FIG. 10B  is a plot of laser intensity versus wavelength. 
         FIG. 11A  is a plot of pressure energy and wavelength.  FIG. 11B  is a plot of medium length, energy and wavelength. 
         FIG. 12A  is a plot of harmonic intensity, energy, and wavelength.  FIG. 12B  is a plot of intensity, energy, and wavelength. 
         FIGS. 13A  and B are plots photon energy versus laser wavelength. 
     
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT 
       FIGS. 1A-1D  (Prior Art) are schematic diagrams illustrating high-harmonic emission generation (HHG) without phase matching.  FIG. 1A  shows the interactions within waveguide  104  in medium  120 , wherein harmonics are generated, but are added incoherently. In  FIG. 1B , driving pulse  102  comprises, for example, a femtosecond laser pulse, which enters the HHG medium  120 . The diagram in  FIG. 1C  indicates the regions within the HHG medium in which constructive and destructive interference of HHG emissions will occur, due to the short coherence length of the HHG beam. The areas indicated by a plus (+) are areas of constructive interference, while the areas indicated by a minus (−) are areas of destructive interference. The length of one positively interfering area (+) in the absence of a modulating pulse is one coherence length. Note that the lengths of the (+) and (−) varies over the interaction length, which will make the task of phase matching more complicates. In general, the non-phase-matched HHG beams are of higher relative divergence. 
     Briefly, an ultrashort light pulse  102  is focused into a medium  120  (for example a noble gas) to generate high-order harmonics  106  in the x-ray regions of the spectrum (generally termed “x-ray HHG” herein). However, particularly for conversion to very short wavelengths, the high-harmonic process is not well phase-matched, for a variety of reasons. The most significant is usually the presence of a plasma, generated either through pre-ionization of the medium or through the intense laser-matter interaction itself, that affects the speed of propagation of the driving laser pulse  102 . The “polarization” of the medium (i.e. where the high harmonics are generated) follows the propagation of the driving laser pulse, while the generated signal  106  travels at the (different) speed of light of the signal in the medium. 
     Note that  FIG. 1D  graphs the intensity of the x-ray HHG signal as it propagates. Thus, the x-axis for this curve corresponds to propagation distance through the medium, while the y-axis corresponds to the intensity of the signal. 
     Hence, as indicated by output HHG signal  106  in  FIG. 1D , at first the HHG signal increases, but as the coherence length is reached destructive interference causes the signal to decrease. This process is repeated with a period of two coherence lengths, with the HHG beam being amplified in portions  108  of the waveform and the HHG beam being attenuated in portions  110  of the waveform. Hence, output HHG beam  106  will contain little HHG signal. The coherence length may vary in space and in time L coh (z,t). The largest HHG signal would be obtained if the waveguide length corresponded to an odd number multiple of coherence lengths, but the HHG signal never gets very large. 
       FIG. 2A  shows the interactions within waveguide  104  in medium  120 , wherein harmonics are generated, and are added coherently.  FIG. 2B  is a schematic diagram illustrating how the output x-ray intensity increases with phase matching within the medium. Driving pulse  102  phase velocity is matched to the phase velocity of the generated x-rays  206 . Thus, the coherence length of the phase matching is extended well beyond the optimal medium length L medium  in  FIG. 2C  (ideally L coh →∞)—in other words, the entire interaction region is now equivalent to a (+)  208  region similar to the (+)  108  region in  FIG. 1C . No significant destructive interference occurs. In the present art, the coherence length L coh (λ L , z,t) for any HHG wavelength can be maintained always relatively long by properly selecting the laser wavelength λ L  and the denoted parameters in  FIG. 2  that depend on λ L .  FIG. 2D  shows the resulting intensity of the output x-rays over the interaction region. It increases monotonically with interaction length, until non-resonant absorption of the generated HHG or evolution of the laser pulse in the gas limits further buildup of HHG signal. Under phase matching conditions, there is coherent radiation build-up in a very specific direction that reduces the HHG beam divergence, as shown in  FIG. 2B  compared to  FIG. 1B . 
     Methods of phase matching the generation of high harmonic radiation in a waveguide are known. See for example Rundquist et al., Science 280, 1412 (1998); Durfee et al., Phys. Rev. Lett. 83, 2187 (1999), and U.S. Pat. No. 6,151,155 (incorporated herein by reference).  FIGS. 3A-C  and  4 A-B as well as Table 1 illustrate the differences between the present invention and these known methods. Typically, in the old technique a 0.8 μm wavelength Ti:Sapphire laser drives the high harmonic generation and full phase matching of the process can extend only into the generated soft x-ray wavelengths to about 10 nm (E=hν˜130 eV). Therefore, in practice the x-ray flux produced is significant enough to enable applications only down to about 10 nm. In addition, the prior art does not teach determining what the optimal laser wavelength and/or nonlinear medium are for obtaining the highest possible conversion efficiency in the limited photon energy region that is accessible. In this 0.8 μm driving wavelength scheme, the generation of soft x-rays of shorter wavelengths (&lt;10 nm) would require increasing the driving laser intensity. Other factors such as ionization prevent extending phase matching with this process to shorter x-ray wavelengths (see typical values in Table 1, Regime I). Thus, the x-ray emission falls within a region of the nonlinear medium of short length, low density of emitters, and large number of free electrons, dramatically decreasing the x-ray yield to a level not useful for most applications. All experiments in this phase-matched geometry, until the present invention, thus taught that the efficiency of HHG upconversion inevitably decreased as one attempted to generate high-energy photons. 
     Table I shows a comparison of the practical parameters when attempting to scale the HHG process to achieve shorter X-ray wavelengths, using 0.8 μm driving laser (conventional Regime I), and longer-wavelength driving lasers (current invention Regime II). Note the contrasting parameters. 
     
       
         
               
               
               
             
               
               
               
             
           
               
                 TABLE 1 
               
               
                   
               
               
                   
                 Regime I (conventional) 
                 Regime II (invention) 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                   
                 (non-phase-matched) 
                 (phase-matched) 
               
               
                   
                 λ L  = 0.8 μm = const &amp; 
                 λ L  increases &amp; 
               
               
                   
                 I L  (Laser Intensity) increases 
                 I L  decreases (slightly) 
               
               
                 Peak Laser Intensity (I L ) 
                 Increases from 10 13 -10 14  to &gt;10 16 . 
                 Decreases from 10 14 -10 15  to 
               
               
                 [W/cm 2 ] 
                   
                 10 14  under phase matching 
               
               
                   
                   
                 conditions. 
               
               
                 Ionization Level of the  
                 Increases from ~10% to &gt;&gt;200% 
                 Low ionization initially and 
               
               
                 Medium 
                 Highly ionized or multi- 
                 decreases from ~10% to  
               
               
                   
                 ionized nonlinear medium. 
                 ~0.001%. 
               
               
                   
                   
                 Moderately ionized 
               
               
                   
                   
                 nonlinear medium. 
               
               
                 Nonlinear Medium 
                 Decreases from ~10-500 torr to 
                 Increases from ~10-500 torr 
               
               
                 Density 
                 ~1 torr. 
                 to ~10000 torr. 
               
               
                   
                 Range: fraction of atmosphere. 
                 Range: multi-atmosphere. 
               
               
                 Useful Nonlinear Medium 
                 Decreases from mm-cm to μm. 
                 Increases from mm-cm to m. 
               
               
                 Length (Coherence length) 
                   
                   
               
               
                 Density of the Ionized 
                 Increases from 10 17  to 10 18  cm −3 . 
                 Constant: ~10 17  cm −3 . 
               
               
                 Electrons 
                   
                   
               
               
                 X-ray Flux 
                 Non-phase matched addition 
                 Phase-matched addition of 
               
               
                   
                 of X-ray fields: rapidly 
                 X-ray fields: flux does not 
               
               
                   
                 decreasing at shorter X-ray 
                 vary strongly with 
               
               
                   
                 wavelengths. 
                 wavelength. 
               
               
                   
               
             
          
         
       
     
       FIG. 3  compares the accessible HHG spectrum  306 B within spectrum  318  with phase matching according to the present invention compared to results achieved with the phase matching technique of the prior art (Regime I). Arrow  302 A shows the phase-matched frequency upconversion achievable in the prior art. The broad driving laser spectrum  304 A is of 0.8 μm central wavelength. The phase-matched output HHG spectrum range is indicated by white bar  306 A. At best it extends into the UV, VUV and EUV regions. 
     Arrow  302 B indicates the results available employing the method of the current invention (Regime  2 ). First, the central wavelength of the driving laser spectrum is tuned. The spectral bandwidths of the ultrafast lasers can be relatively broader  304 B when moving to the mid-IR spectral region. Correspondingly, the HHG output  306 B can be tuned from a few hundred nanometers (few eV in photon energy) up to a few Å in wavelength (multi-keV in photon energy). It is well known for a person skilled in the art, that a single emitter can radiate in these soft (SXR) and hard X-ray (HXR) spectral ranges. In a macroscopic picture, ionization induced effects do not allow prior phase matching techniques to access these regions. In practice, Regime I reaches these spectral ranges with very poor efficiency—the HHG flux is below the threshold required by applications. The present invention enables access to these spectral regions under phase matching conditions. Counter-intuitively to a person skilled in low harmonic generation, the essence of the new method is to increase the laser wavelength (from VIS-IR to mid-IR) in order to extend phase matching of HHG towards shorter wavelengths, from VUV-EUV to SXR and HXR. 
       FIGS. 4A  (Regime I) and  4 B (Regime II) are schematic diagrams illustrating macroscopic HHG flux in conventional systems and in the present invention. Regime I ( FIG. 4A ) generates photons in the SXR and HXR regions by using the same laser wavelength  406  for which the HHG flux from a single emitter does not vary significantly. Optimization here requires increasing the laser intensity  408  which reduces the usable density-length product  410  (lower number of potential emitters) due to ionization induced effects. The resulting HHG flux  412  drops rapidly at short wavelengths. 
     The present invention relies on simultaneous increase of the laser wavelength  426  and slight decrease in laser intensity  428 , which maintain phase matching conditions. Thus HHG flux from a single emitter scales even faster than the predicted λ L   −5.5±0.5 ). However, phase matching conditions at shorter HHG wavelengths using longer driving wavelengths favors large optimal or phase matching density-length products (large number of potential emitters). As a result, the macroscopic phase-matched HHG flux  432  varies slightly with wavelength. 
       FIG. 5  is a diagram illustrating HHG with phase matching according to the present invention (Regime  2 ), compared to results achieved with phase matching using prior art techniques in the time domain. Again, Arrow  502 A indicates the results achievable in the prior art—bright femtosecond-to-attosecond HHG pulses. A smaller range of HHG output pulse durations  506 A is achievable due to the narrower phase-matched HHG bandwidth  306 A available. Arrow  502 B indicates the results achievable with the present invention (Regime  2 )—femtosecond-to-attosecond-to-zeptosecond HHG pulses. With devices according to the present invention, the available HHG pulse duration range  506 B is larger due to the larger phase-matched HHG bandwidth  306 B. Time scale  518  shows log time. 
       FIGS. 6A-D  are diagrams illustrating the structure/shape of the driving laser pulses and the generated HHG pulses  FIG. 6A  shows a single driving ultrashort pulse input.  FIG. 6B  indicates that a more complex series of pulses (for example, FEL pulses, consisting of several ultrashort micro pulses within a long (macro) pulse) could also be used.  FIG. 6C  shows that a series of pulses may be generated by the driving pulse of either  6 A or  6 B.  FIG. 6D  shows that a single output pulse may be achieved. 
       FIG. 7A  is a general block diagram illustrating the invention. The embodiment of  FIG. 7A  might utilize a driving laser source  702  providing, for example, a 1.3 μm pulse or 2.0 μm pulse  102  with a peak intensity of about 10 14  W/cm 2 -10 15  W/cm 2 . For example source  702  might be a laser, a laser in conjunction with an optical parametric amplifier for converting the laser light to the desired driving pulse  704  wavelength, or an optical parametric chirped pulse amplifier, or for example a VIS or IR free electron laser (FEL). The nonlinear medium  120  is moderately ionized, for example from about 10% down to 0.001%. The pressure of the medium can be multi-atmosphere, for example 10,000 torr. The density of the ionized electrons can be about 10 17  cm −3 . This results in a useful nonlinear medium length L medium  of multi-centimeters to a meter or more. The phase-matched addition of the x-ray fields means that flux does not vary strongly with wavelength. 
     The laser source  702  energy, wavelength, and pulse  704  duration are selected to maintain phase matched HHG  206  generation. Driving source  702  may produce ultrashort driving pulses at any repetition rate or long “macroscopic” pulses at any repetition rate with multiple driving pulses under the envelope. 
     Medium  730  or  732  might comprise atomic gases (for example, noble gases: helium, neon, argon, etc.), mixtures  730 ,  734 ,  732 , of molecular gasses, and mixtures of atomic and molecular gases. In mixtures, phase matching relies on the presence of a target that is less or non-ionized compared to the other species targets for a given peak laser intensity. Therefore, a mixture of targets with different ionization potentials is desirable. Mixtures allow the less ionized medium to contribute to the neutral index of refraction. Therefore, higher laser intensity can be employed. As a result higher photon energies may be phase matched with further increase in phase-matched HHG flux. Since the mid-IR driving laser require higher density medium to phase match the HHG process, the nonlinear medium may be liquid, or mixture of liquids, or for example solid state He, Ne, etc. 
       FIG. 7B  is a block diagram illustrating a guided beam geometry embodiment of the present invention. A driving laser beam  708  originating from, for example, a wavelength tunable ultrafast laser-optical parametric amplifier system, is focused into a gas-filled  120  hollow waveguide  104  to facilitate near plane-wave propagation. For perfect phase matching of HHG, the driving laser phase velocity must equal that of the generated x-rays. In any HHG geometry, the phase mismatch Δk is a sum of contributions from the pressure-dependent neutral atom and free electron dispersion, as well as from pressure-independent geometric dispersion. Phase matching, i.e. Δk→0, can be achieved by varying the gas pressure inside the waveguide since the sign of the neutral atom contribution to the dispersion is opposite that of the generated free-electron plasma. This dispersion balance mechanism has been directly verified through in situ measurements of the coherence length of the HHG process as the phase matching conditions were varied. 
     The embodiment may require differential pumping to vacuum. Differential pumping to vacuum may be required on both sides of the geometry containing the nonlinear medium. For soft x-ray generation, the geometry containing the nonlinear medium is preceded and followed by a vacuum chamber. When x-rays of shorter wavelengths are generated, a solid window may be used at the entrance and/or exit of this geometry to confine the high density medium, and to obtain vacuum outside of this region. 
     Phase matching is possible only if the ionization is less than a critical ionization level, η CR (λ L ). Values for η CR  are on the order of a few percent in the near-IR region, e.g. approximately 1.5% for Ar, 0.4% for Ne, and 0.2% for He at 1.3 μm driving laser wavelengths. This critical ionization level monotonically decreases as the driving laser wavelength increases from VIS-IR into the mid-IR and higher. 
     Under the illumination conditions of this embodiment (laser intensities of 10 14 -10 15  W/cm 2  and 8-cycle laser pulses), ionization of an atom by an intense laser pulse is well-described by the Ammosov-Delone-Krainov (ADK) tunneling ionization model. Using the ADK model, the laser intensity for which ionization in the medium approaches η CR . This phase matching cutoff hν PM  corresponds to the maximum photon energy that can be generated from a macroscopic medium with near-optimum conversion efficiency (full phase matching).  FIG. 9  plots the phase matching cutoff hν PM (λ L ) for values of λ L  up to 10 μm, assuming a hyperbolic-secant laser pulse with 8 optical cycles FWHM (35 fs at λ L =1.3 μm). This plot shows that phase matching of HHG can extend to 1 keV for driving laser wavelengths approaching 3 μm, and extends even to the multi-keV x-ray region when longer mid-IR laser wavelengths are used. Use of a shorter 3-cycle pulse (FWHM) can increase these phase matching cutoffs by an additional 15%, due to decreased ionization levels for shorter laser pulses. Finally, phase matching cutoffs may increase by an additional few percent due to non-adiabatic effects, which also lower the ionization level and which are not captured by the quasi-static ADK approximation. Using other pulse shapes, for example FEL pulses with rectangular envelopes, may require re-evaluating the optimal pulse intensity based on the accumulated ionization level under such illuminating conditions. 
     In order to experimentally verify the predicted scaling of the HHG phase matching cutoffs with driving laser wavelength, we generated driving laser beams either from an optical parametric amplifier, tuned to λ L =1.3 μm (signal) and λ L =2.0 μm (idler), with energy of up to 5.5 mJ and 3.5 mJ, respectively, and with a pulse duration under 35 fs. The driving laser was focused into a hollow capillary filled with Ar, Ne or He gas. Harmonics generated using (prior art) 0.8 μm driving beams serve as a reference. At this reference wavelength, the phase matching cutoff extends in the EUV region of the spectrum to ˜50 eV, 90 eV, and 130 eV in Ar, Ne, and He, respectively. Equivalent pressure-tuned phase matching spectra using longer 1.3 μm and 2.0 μm driving lasers resulted in phase matching cutoffs that extend from the EUV into the water window of the soft X-ray region of the spectrum: to ˜100 eV and ˜165 eV for Ar, and ˜200 eV and ˜395 eV for Ne, while for He phase matching extends to ˜330 eV and ˜520 eV for the two laser wavelengths respectively (see  FIGS. 13A and 13B  respectively). These phase matching cutoff values are all well beyond what can be achieved using a reference 0.8 μm driving laser. As predicted, full phase-matched harmonic emission was achieved at high gas pressures (&gt;&gt;atm) over centimeter distances. 
     Finally, because of the very large bandwidths that are simultaneously phase matched, these data also show great promise for generating bright, attosecond pulses at much higher photon energies than have been possible to date. A Fourier Transform of the HHG spectra from He in the water window indicates the potential for generating an 11±1 attosecond duration pulse. Past theoretical work has shown that since phase matching is confined to only a few half-cycles of the laser—even when using relatively long driving laser pulses (15 fs at 0.8 μm or 45 fs at 2 μm)—the harmonic emission can emerge as a single attosecond burst. Moreover, this prediction has been confirmed experimentally using 15 fs 0.8 m pulses, where pulses as short as 200 attoseconds were generated even without carrier envelope phase (CEP) stabilization. 
       FIG. 7C  is a block diagram illustrating a loose focusing geometry embodiment of the present invention. The predicted scaling of the critical ionization level and driving laser intensity is also applicable to this HHG scheme. In this embodiment, a near-plane-wave propagation is achieved by using a loosely focused  1026  laser beam  708 . A shorter focusing optic  742  is used compared to focusing optic  740 . However, a larger laser pulse energy may be required in this case compared to HHG in a wave guide, because the laser beam cross-section increases. For a laser confocal parameter 2z R  significantly longer than the interaction region L medium , any geometric contribution to the phase mismatch can be neglected. To achieve phase matching, the ionization level must be close to the critical level η CR . Thus, in contrast with a waveguide geometry embodiment ( FIG. 7B ), where there is optimal phase-matching pressure, here, density of the medium is decoupled from the phase-matching process (this is also equivalent to a waveguide with a large inner diameter). In both cases (and also for tight focusing embodiment, i.e. using confocal parameter on the scale of L medium ), the density-length product of the nonlinear medium that optimizes the HHG emission near the phase-matching cutoff is set by the absorption cross-section of the generated X-rays. Therefore, the optimal pressure-length product and phase-matched HHG intensity are the same as the predicted in  FIGS. 11A and 6B . 
       FIG. 8  is a flow diagram illustrating steps in the process of flux optimization of HHG with phase matching according to the present invention. In step  802 , λ L  of the driving pulse  704  is chosen. In step  804 , nonlinear medium  120  is selected. In step  806 , the index of refraction of medium  120  for a range of λ L  is computed. In steps  808 ,  810 , and  812 , the geometry of the device is chosen. In step  814 , the phase matching limits as a function of XL are evaluated (see  FIGS. 9 and 10A ) based on the index of refraction of the driving laser light, and that of the generated light, for a specific nonlinear medium and a specific laser beam geometry. In step  816 , the optimal laser pulse  704  parameters are determined (see  FIG. 10B ). In step  820 , the HHG flux from a single emitter as a function of λ L  is determined. The generated HHG light in the nonlinear medium  120  can be absorbed by the medium  120 . Thus the monotonic growth of the HHG signal may saturate which sets an optimal density-length product  822  (or number of potential emitters) that can be used. The density-length product  822  is determined by the properties of nonlinear medium and in general increases as λ L  under the phase matching conditions of the present art (Regime II). See  FIGS. 11A and 11B . For geometries where phase matching is pressure dependent (for example, waveguide geometries), first optimal pressure  824  is determined and then optimal length  826 , set by the same density-length product  822 . Step  828  combines the microscopic, single emitter HHG flux  820 , and the optimization of macroscopic nonlinear medium parameters and laser parameters to get the macroscopic phase-matched HHG flux as a function of laser wavelength. Flux  828  can give relative brightness of phase-matched HHG of a desired wavelength for various nonlinear medium, laser wavelengths, etc.  FIGS. 12A and 12B  illustrate such an optimization for photon energies close to the phase matching limit as a function of laser wavelength  802 . Determining the maximum flux  830  at a desired HHG wavelength allows proper selection of optimal laser wavelength  836 , optimal nonlinear medium  838 , optimal laser pulse parameters  834 , optimal density length product  822 , or  824  and  826 . 
     While the exemplary preferred embodiments of the present invention are described herein with particularity, those skilled in the art will appreciate various changes, additions, and applications other than those specifically mentioned, which are within the spirit of this invention. For example, the following may further optimize the HHG geometry:
         A. A waveguide of optimized length, and diameter, as required for maximum x-ray signal. The laser may be guided in the lowest waveguide mode. Nonlinear medium with density gradient may be used.   B. A tapered waveguide of optimized length, input and output diameter/slab, as required for maximum x-ray signal.   C. A converging laser beam may be used in a loosely focused geometry to compensate for laser energy losses.   D. Shaped driving pulses can be used to mitigate group velocity mismatch between the fundamental and the generated x-ray light. More specifically, a temporal intensity ramp in the envelope of the driving pulse would be useful to keep the high energy part of the driving pulse propagating at the same group velocity as the generated light. This or similar temporal pulse shaping may be also combined with spatial shaping (propagation in a tapered waveguide, converging laser beam, etc.) to mitigate group velocity walk-off.