H igh H armonic Generation - The Budker Groupbudker.berkeley.edu/Physics250_Spring0607/HHG.pdf ·...

High Harmonic GenerationBasic Techniques & Challenges

C. Michael R. Greaves

Regimes of Nonlinear Opticslomb field. As a consequence, the laser field only slightlyperturbs the atomic quantum states under nonresonantexcitation conditions. The energy levels suffer only afaint shift proportional to Ea

2 , which is referred to as theac Stark shift (Landau and Lifshitz, 1977). The atomsremain, with a high probability, in their ground state andthe extension of the wave function of the ground stateremains on the order of the Bohr radius aB . Nonlinearinteractions taking place under these conditions can bewell described by a perturbative approach, and hence werefer to this parameter range as the regime of perturba-tive nonlinear optics.

If the electric-field strength becomes comparable to(or higher than) the binding atomic Coulomb field expe-rienced by the outer-shell electrons, an electron can es-cape with a substantial probability from its bound state(via tunneling or above-barrier detachment) before thelaser electric field reverses its sign. The electron wavepacket liberated by optical-field ionization subsequentlywiggles in the linearly polarized electric field. The am-plitude of the wiggle exceeds the Bohr radius by severalorders of magnitude and the cycle-averaged kinetic en-ergy of the electron exceeds the binding energy Wb .The parameter range giving rise to these processes isreferred to as the strong-field regime of nonlinear optics.The atomic polarization response is dominated by theionization process and the contribution from the boundelectrons is negligible.

Nonlinear polarization induced by optical-field ioniza-tion emerges only as long as the electron remains inclose proximity to its parent ion. Once the electron is setfree, its trajectory is governed by the Newtonian equa-tions of motion, resulting in a linear response (with asmall remaining nonlinearity originating from the parention’s reduced polarizibility). A strong nonlinearity arisesonly at intensities order(s) of magnitude higher due tooptical-field ionization stripping the next electron and/orto the wiggle energy of free electrons becoming compa-rable to their rest energy mc2, indicating the onset ofrelativistic nonlinear optics.

Figure 24 assigns the relevant intensity regions to theabove regimes of nonlinear optics for visible and near-infrared radiation. Here we focus on nonrelativisticlight-matter interactions. We shall (i) review how exploi-tation of processes in the perturbative regime allowsgeneration of intense light pulses in the few-cycle regimeand (ii) demonstrate that these pulses are capable ofsignificantly extending the frontiers of strong-field non-linear optics. Whereas in the perturbative regime theintensity envelope governs the evolution of nonlinearoptical processes, in the strong-field regime the electric(and at relativistic intensities also the magnetic) fieldstake control. Because the generation of ultrashort lightpulses relies on perturbative processes, the phase !0 [seeEq. (14)] is unknown, and the electric field of thesepulses is indefinite. The light fields are expected to be-come accessible when strong-field processes with few-cycle wave packets are induced (see Sec. VIII). In Secs.IV. A-IV.C, we analyze the polarization response of anatomic medium irradiated with strong laser fields and

introduce field-strength scale parameters that roughlydefine the borders of the above regimes. The relation ofthe induced medium polarization to the incident fields isindispensable for describing the propagation of intenseradiation in atomic media, i.e., for the interaction of in-tense light with matter in a macroscopic volume.

A. Perturbative nonlinear optics

At low and moderate intensities the polarization, P"As/m2# (where As represents Ampere seconds), of anatomic ensemble can be expanded into a Taylor serieswith respect to the electric field and can be given as thesuperposition of the linear and nonlinear responses P!$0%(1)E"Pnl , where

Pnl!$0%(2)E2"$0%(3)E3"$0%(4)E4"••• , (19)

$0!8.85#10$12 As/V m is the vacuum permeability, and%(k) "(m/V)k$1# is the kth-order susceptibility (Bloem-bergen, 1965). It is implicit in Eq. (19) that the atomicpolarization instantly follows the change of the field,which is usually a good approximation even on a timescale of a few femtoseconds. This is because the inducedatomic dipole moment is of purely electronic origin witha response time on the order of 1/& , where &!!' ik$'0!, ' ik represents the transition frequency from theinitial (usually ground) quantum state i into some ex-cited state k for which !' ik$'! is the minimum and thedipole transition matrix element ( ik is nonzero, and '0is the laser carrier frequency [see Eq. (14)]. Since thetypical transition frequency from the atomic groundstate to the lowest excited state significantly exceeds thelaser frequency in the visible and near-infrared range,1/& is typically less than 1 fs. In molecules and con-densed matter, nuclear motion may also provide a sig-nificant contribution to the induced dipole moment. Thiscontribution has a response time of hundreds of femto-

FIG. 24. Regimes of nonlinear optics. Only nonresonant inter-actions have been considered by assuming the detuning &) tobe larger than the photon energy (*1 eV). The boundariesbetween different regimes are not sharply defined. In thestrong-field regime, the intensity scale applies to the visibleand near-infrared spectral range.

564 T. Brabec and F. Krausz: Intense few-cycle laser fields

Rev. Mod. Phys., Vol. 72, No. 2, April 2000

Outline

• Optical Field Atomic Ionization

• High Harmonic Generation (HHG)

• Microscopic Description

• Macroscopic Description

• Techniques for Phase Matched Generation of Coherent Soft X-rays

• References

Optical Field Atomic Ionization

• Harmonics are generated by the tunneling, transport and recombination of electrons

• Atomic ionization proceeds by perturbation of the Coulomb potential in the presence of the electric field of the driving laser pulse

• Ionization rate by the Quasistatic Approximation

energy loss suffered by the laser pulse according to Eq.(24) is in good agreement with experimental observa-tions (Fig. 29).

Equations (31) and (23) take proper account of non-linear interactions in the strong-field regime where theuse of intense few-cycle pulses pushes the nonlinear re-sponse of (ionizing) matter to unprecedented extremes.As a result, coherently generated harmonics of the driv-ing laser radiation now exceed the 300th order. Model-ing even predicts significant further improvement (interms of both photon energy and conversion efficiency)in previously unaccessed parameter ranges and providesrelevant guidelines for the experimenters (Sec. VII).This same theoretical framework also accounts for thefascinating temporal behavior of x-ray harmonic radia-tion (Sec. VII) as well as the dependence of strong-fieldprocesses on the carrier phase in few-cycle wave packets(Sec. VIII).

VI. OPTICAL-FIELD IONIZATION OF ATOMS

When matter is exposed to intense laser fields, awealth of exciting phenomena can be observed, includ-ing high harmonic generation (Corkum, 1993), above-threshold ionization (Corkum et al., 1989), atomic stabi-lization (Pont and Gavrila, 1990), x-ray lasing (Lemoffet al., 1995; Hooker et al., 1995), laser-induced damageof dielectrics (Du et al., 1994; Stuart et al., 1995; Lenzneret al., 1998; Tien et al., 1999) and molecular dissociation(Seidemann et al., 1995; Chelkowski et al., 1996). Thekey process triggering all of these strong-field phenom-ena is ionization. The theoretical investigation of theseprocesses is significantly simplified by the analytic de-scription of ionization that is available in the limitingcases characterized by !!1 and !"1 [see Eq. (21)]. Thetwo limiting cases of multiphoton ionization and optical-field ionization are illustrated schematically in Figs.30(a)–(c), respectively.

In what follows we shall restrict our discussion to thestrong-field limit !"#1, for which significant ionizationtakes place. In this parameter range the quasistatic ap-proximation is valid, which relies on the assumption thatthe perturbed electron wave function reaches a quasi-static state before the electric field changes significantly(Shakeshaft et al., 1990). Then the fraction of electronsionized in the laser field E(t) as a function of time t maybe calculated by Eq. (23). At the threshold for the de-tachment of the second (or higher) electrons, Eq. (23)must be generalized by calculating a sum over the indi-vidual ionization processes, (1/na)# inei(t), determinedby the ionization rates wi for the ith electron. Further-more, possible nonsequential ionization channels mustbe taken into account (Fittinghoff et al., 1992; Corkum,1993; Walker et al., 1994; Augst et al., 1995; Lablanquieet al., 1995; Brabec et al., 1996; Dorner et al., 1998;Rosen et al., 1999).

The power of the quasistatic approximation rests onthe fact that ionization in time-varying laser fields maybe calculated by using the static field ionization rate w .There exist two approaches for analytic calculation of

static field ionization rates, namely, the Keldysh theory(Keldysh, 1965; Reiss, 1980) and the Ammosov-Delone-Krainov theory (Oppenheimer, 1928; Perelomov et al.,1996; Ammosov et al., 1986). In Fig. 31, the Ammosov-Delone-Krainov and Keldysh ionization rates for hydro-gen and helium atoms exposed to a static electric fieldare compared with the rates obtained from an exact nu-merical solution of the time-independent Schrodinger

FIG. 30. Regimes of atomic ionization. Exposing an atom toan intense laser field will result in a modified potential (solidcurve) composed of the Coulomb potential (dashed curve) andthe time-dependent effective potential of the optical pulse. (a)At moderate intensities the resulting potential is close to theunperturbed Coulomb potential and an electron can be liber-ated only upon simultaneous absorption of N photons, result-ing in multiphoton ionization. The multiphoton ionization ratescales with the Nth power of the intensity of the optical pulse.(b) At sufficiently high field strengths the Coulomb barrierbecomes narrow, allowing optical tunneling ionization to takeover and resulting in a tunneling current that follows adiabati-cally the variation of the resultant potential. (c) At very highfield strengths, the electric field amplitude reaches values suf-ficient to suppress the Coulomb barrier below the energy levelof the ground state, opening the way to above-barrier ioniza-tion.

FIG. 31. Static field ionization rates in hydrogen and heliumatoms vs the electric-field strength in atomic units: solid curve,numerical result; dashed curve, Ammosov-Delone-Krainovformula; dash-dotted curve, Keldysh theory for hydrogen. Thedotted line denotes the barrier suppression field strength for Hand He.

570 T. Brabec and F. Krausz: Intense few-cycle laser fields

Rev. Mod. Phys., Vol. 72, No. 2, April 2000

Strong Field Regime

• Measure of field strength is given by Keldysh scale parameter:

• defines the strong field regime

• Coulomb potential is suppressed

• Electron can tunnel through the potential barrier

• Corkum model: Classical description of the electron evolution after ionization

1!

=eEa

"0!

2mWb1!

> 1

Outline






• References

HHG Microscopic Analysis

• HHG photon radiation dependent on the acceleration of the electron

• Two simplifying assumptions:1. Electron is a free particle in continuum states2. Only ground state interactions are considered

• Under these assumptions:

!2

!t2< "e|r|"e >

dn(!) =! 1!

iaion(!b)apropogation(!b, !)arecombination(!)

!b

Microscopic II

• Model valid for linearly polarized light

• Elliptically polarized light is less efficient

• time is on the order of the oscillation period of the electric field

• Photon emission occurs upon recombination with the ground state

apropogation

Microscopic III

τbE-field potentialsuppresses Coulombfield

τpropogationfree electron

τrecombinationhigh KE electronrecombination yieldsharmonic photon

t

E(t)

Microscopic IV

• Periodic process yields odd harmonics of the laser field

(!!)max = Ip + 3.2Up

Up =2!ca0IL

"2

Outline






• References

Macroscopic Analysis (Propagation)

• Macroscopic effects reduce HHG efficiency

1. Absorption - excitation of core atomic states by harmonic photon

2. Dephasing - difference in propagating velocities of laser and harmonic photon

3. Defocusing - distribution in electron density profile due to laser focus

Outline






• References

Phase Matching

• Goal: Increase the coherence length of the harmonic beam and the driving laser pulse for a given harmonic order

• Phase match condition:

• Use fiber to increase path length of laser relative to harmonic beam

!k = qklaser ! kx!ray = 0

klaser !2!

"+

2!Na#(")"

"Nere""u2

nm"

4!a2

Phase Matching II

Phase-Matched Generation of CoherentSoft X-rays

Andy Rundquist, Charles G. Durfee III, Zenghu Chang,Catherine Herne, Sterling Backus, Margaret M. Murnane,*

Henry C. Kapteyn

Phase-matched harmonic conversion of visible laser light into soft x-rays was demon-strated. The recently developed technique of guided-wave frequency conversion wasused to upshift light from 800 nanometers to the range from 17 to 32 nanometers. Thisprocess increased the coherent x-ray output by factors of 102 to 103 compared to thenon–phase-matched case. This source uses a small-scale (sub-millijoule) high repetition-rate laser and will enable a wide variety of new experimental investigations in linear andnonlinear x-ray science.

Nonlinear optical techniques for frequen-cy conversion have played a pivotal role inthe development of efficient coherent lightsources (1), second in significance only tothe invention of the laser itself. In recentyears, nonlinear optics has led to the devel-opment of efficient visible-wavelength lasersources and sources with broad tunabilitysuch as optical parametric oscillators (2)and amplifiers (3). Most recently, the use ofstructured materials has made possible anew class of efficient nonlinear optical de-vices based on quasi–phase-matching (4).

However, nonlinear frequency-conver-sion techniques have almost exclusively re-lied on crystalline solids as nonlinear media.For efficient conversion, the electromag-netic waves corresponding to the drivingand the generated signal must be phase-matched—both colors of light must havethe same phase velocity as they travelthrough the generating medium. In thiscase, the nonlinear polarization adds coher-ently as the waves co-propagate, resulting ina rapid increase in signal intensity. To date,phase-matching has generally beenachieved by relying on a combination ofanisotropic materials and differing sourceand signal polarizations, because unless con-ditions are chosen carefully, light of differ-ent colors travel at different speeds througha material.

This reliance on solid materials has se-verely limited the application of nonlinearoptical techniques to very short wave-lengths. The best solid nonlinear materialssuch as lanthanum boron oxide (LBO) andbarium boron oxide (BBO) can be used togenerate light at wavelengths as short as200 nm (5). However, few solids are trans-parent at wavelengths shorter than this, and

none are transparent in the extreme ultra-violet (XUV) and soft x-ray regions of thespectrum. In contrast, many gases are trans-parent well into the vacuum ultraviolet(VUV), and even shorter wavelengths canpropagate with moderate absorptionthrough low-pressure gases. However, be-cause gases are isotropic, established phase-matching techniques are not applicable,and thus, phase-matched frequency conver-sion techniques have not been developedfor the XUV and soft x-ray regions of thespectrum.

Nevertheless, in recent years the tech-nique of high-harmonic generation (HHG)has proven to be a useful source of XUVand soft x-ray light (6, 7). In this process,atoms in a dilute gas radiate harmonics ofthe incident light in the process of beingionized at the focus of an intense ultrashort-pulse laser. The highly nonlinear nature ofthe ionization process makes it possible togenerate harmonics up to order 299 andhigher at wavelengths below 3 nm, usingvery short laser pulses (8, 9). Despite thelow efficiency of this process (!10–8 con-version of laser light into one harmonicpeak), the photon flux generated is stillsufficient to be useful for some applicationsin ultrafast x-ray spectroscopy. The HHGtechnique is also of considerable theoreticalinterest as a possible method for generatingattosecond-duration light pulses (10–12).

However, many more applications would bepossible if the process were made more ef-ficient by phase-matching. Unfortunately,the rapidly changing free-electron densityassociated with ionization of the atoms ledto the generally held opinion that phase-matching would result in, at most, modestimprovements in efficiency (13–15), be-cause the length over which the harmonicphase slips by " from the laser (the coher-ence length) is # 50 $m for a fully ionizedgas. Phase-matching would compensate forthis phase slippage (Fig. 1, inset) and allowthe x-ray emission from much longerlengths to add coherently.

We report a phase-matching techniquethat circumvents the limitations of crystal-line nonlinear optics by making phase-matched frequency conversion of ultra-short-pulse laser light into the XUV possi-ble. Phase-matched frequency conversionin gases has been achieved in the UV spec-trum in past experiments by exploiting thenegative dispersion that exists for light atwavelengths just shorter than a resonantabsorption line (16, 17). However, the res-onant requirement limits the phase-match-ing bandwidth, making it difficult to usethis technique for applications that requirebroad tunability or ultrashort pulses, andalso limits the conversion efficiency. More-over, most experiments in HHG to datehave been done in the tight focusing geom-etry, where the phase shift due to the laserfocus dominates and limits phase-matching(18). Although proper choice of focusingconditions can somewhat increase the co-herence length, the useful interactionlength is still fundamentally limited by thelaser beam divergence. In contrast, by prop-agating a light pulse through a wave guide,its phase velocity can be carefully con-trolled and the beam divergence effectseliminated. Previous work has proposed theuse of a transient wave guide created by alaser-produced plasma for phase-matchedfrequency conversion using high-order fre-quency mixing (19). However, as we show,a hollow capillary tube allows phase-match-ing in neutral gas with single frequencyinput, while employing a much simpler ex-perimental configuration. The phase veloc-

Center for Ultrafast Optical Science, University of Michi-gan, Ann Arbor, MI 48109–2099, USA.

*To whom correspondence should be addressed. E-mail:[email protected]

iris

lensfilter

fibers

gas detectorx-ray

spectrometer

20fs laser

Fig. 1. Experimental set-up forphase-matching of soft x-rays in acapillary wave guide. The insetshows the growth of the x-ray wavewhen phase-matched. Note thatthe laser and x-ray wavelengths arenot to scale.

REPORTS

SCIENCE ! VOL. 280 ! 29 MAY 1998 ! www.sciencemag.org1412

ic generation in a pulsed-gas nozzle or free-flowing cell, which have been used in thepast for HHG experiments.

If phase-matching of conversion of laserlight into x-rays is achieved, we shouldobserve a marked increase in x-ray output atthe optimum pressure for a given harmonicorder. Propagation in the wave guide isbasically plane-wave, and losses of the laserover the propagation length in the fiber (afew centimeters in these experiments) areminimal. Thus, for any given position tem-porally within the pulse and copropagatingwith it, the level of ionization is constant,and an optimum pressure and intensity canbe found in the constant-density middlesection. Typical data for the 29th and 31stharmonic of 800 nm at 28 and 26 nm,respectively, generated in a 150-!m capil-lary filled with argon, are shown in Fig. 2.The theoretically predicted output is shownin Fig. 3. At pressures of 35 torr, phase-matching is predicted and observed to occurand caused a dramatic increase in harmonic

output. We checked that for pressures up to200 torr, no change in the transmission ofthe laser light occurred because of ioniza-tion effects. Therefore, the decrease in sig-nal at high pressures is due to the loss ofphase-matching because of excess gas dis-persion, and not due to ionization-induceddefocusing. A further confirmation of thisexplanation is that the optimum phase-matched pressure shifts to higher pressuresif the laser intensity is increased. The in-creased intensity adds a negative contribu-tion to the refractive index because of in-creased ionization, which must be balancedby a higher gas pressure. At pressures of 30to 50 torr, a total of five harmonic peaks(orders 23 through 31) are phase-matched.However, the optimum phase-matchingpressure for each harmonic is slightly dif-ferent, because the higher harmonics aregenerated at higher values of ionization.No signal was observed for harmonic or-ders below the 23rd because of strong ab-sorption in the argon gas. This is a strongindication that the observed harmoniclight is generated primarily within thecentral fiber segment. Also, at the peakintensities used in these experiments(" 2 # 1014 W cm–2), we only expect toobserve harmonics up to order 31. Thehigher harmonic orders (orders 33 through45) are phase-matched by increasing thegas pressure and laser intensity.

Figure 3 shows the theoretically predict-ed output for the 29th harmonic, with andwithout the presence of absorption and ion-ization. The presence of absorption and ion-ization tends to dampen and broaden theexpected oscillations of the output signalwith pressure, both because the absorptiondepth (8 mm) is shorter than the coherencelength and because the harmonics are gen-erated over a range of ionization levels,each optimizing at a slightly different pres-sure. The presence of ionization (" 2%when the harmonic is being generated)tends to shift the optimum phase-matchedpressure to higher values (Fig. 3, B and D)because of the need to balance the refrac-tive index contribution introduced by ion-ization to maintain phase-matching. Ab-sorption in the gas reduces the expectedoutput by an order of magnitude (Fig. 3).This absorption explains why our output isenhanced only by factors of 102 to 103 (seebelow), instead of the maximum expectedenhancements of 104 due to the increasein coherence length from 50 !m to aninteraction length of a few centimeters.The coherence length is infinite whenphase-matched.

The conversion efficiency of laser lightinto a single harmonic peak was measuredin two independent ways. First, a vacuumphotodiode (XSI) was inserted before thex-ray spectrometer, and the current wasrecorded with a picoammeter (Keithley485; Keithley, Cleveland, OH). A metalfilter was placed between the source anddetector to block any low-order harmonicsthat might have been present. In practice,this procedure was not necessary, becausethe low-order harmonics are not phase-matched at this pressure and also are strong-ly absorbed in argon, and therefore, theirsignal levels are orders of magnitude belowthat of the five phase-matched harmonics.Signal levels of $100 pA were observed.Given our repetition rate of 1 kHz, a detec-tor quantum efficiency of at best 0.08, ameasured filter transmission of 10%, and aninput laser energy of 150 !J per pulse, weestimate that % 2 # 107 photons per har-monic peak per pulse were emitted. Thisvalue corresponds to an energy of % 0.2 nJper harmonic peak per pulse, or a conver-sion efficiency of 10–6 to 10&5 into asingle harmonic peak—an increase of 100times over previously measured efficien-cies (6). We also estimated the enhance-ment factors by comparing our outputfrom the fiber with that from a gas jet (7).We estimate an increase in efficiency of102 to 103 due to phase-matching. Wenote that the use of the vacuum photo-diode x-ray detector significantly im-proved the accuracy of our conversion ef-ficiency measurements, compared with

Fig. 2. Measured signals of the (A) 29th and (B)31st harmonics as a function of gas pressure.

Fig. 3. (A) Calculated output of the 29th harmonic,(B) in the presence of constant 2% ionization, (C)in the presence of absorption, and (D) net outputin the presence of absorption and varying levels ofionization around 2%.

A

B

Fig. 4. Soft x-ray output beam for harmonic or-ders 23 through 31: (A) unphase-matched, detec-tor set to high gain, and (B) phase-matched, de-tector set to low gain.


ic generation in a pulsed-gas nozzle or free-flowing cell, which have been used in thepast for HHG experiments.

If phase-matching of conversion of laserlight into x-rays is achieved, we shouldobserve a marked increase in x-ray output atthe optimum pressure for a given harmonicorder. Propagation in the wave guide isbasically plane-wave, and losses of the laserover the propagation length in the fiber (afew centimeters in these experiments) areminimal. Thus, for any given position tem-porally within the pulse and copropagatingwith it, the level of ionization is constant,and an optimum pressure and intensity canbe found in the constant-density middlesection. Typical data for the 29th and 31stharmonic of 800 nm at 28 and 26 nm,respectively, generated in a 150-!m capil-lary filled with argon, are shown in Fig. 2.The theoretically predicted output is shownin Fig. 3. At pressures of 35 torr, phase-matching is predicted and observed to occurand caused a dramatic increase in harmonic

output. We checked that for pressures up to200 torr, no change in the transmission ofthe laser light occurred because of ioniza-tion effects. Therefore, the decrease in sig-nal at high pressures is due to the loss ofphase-matching because of excess gas dis-persion, and not due to ionization-induceddefocusing. A further confirmation of thisexplanation is that the optimum phase-matched pressure shifts to higher pressuresif the laser intensity is increased. The in-creased intensity adds a negative contribu-tion to the refractive index because of in-creased ionization, which must be balancedby a higher gas pressure. At pressures of 30to 50 torr, a total of five harmonic peaks(orders 23 through 31) are phase-matched.However, the optimum phase-matchingpressure for each harmonic is slightly dif-ferent, because the higher harmonics aregenerated at higher values of ionization.No signal was observed for harmonic or-ders below the 23rd because of strong ab-sorption in the argon gas. This is a strongindication that the observed harmoniclight is generated primarily within thecentral fiber segment. Also, at the peakintensities used in these experiments(" 2 # 1014 W cm–2), we only expect toobserve harmonics up to order 31. Thehigher harmonic orders (orders 33 through45) are phase-matched by increasing thegas pressure and laser intensity.

Figure 3 shows the theoretically predict-ed output for the 29th harmonic, with andwithout the presence of absorption and ion-ization. The presence of absorption and ion-ization tends to dampen and broaden theexpected oscillations of the output signalwith pressure, both because the absorptiondepth (8 mm) is shorter than the coherencelength and because the harmonics are gen-erated over a range of ionization levels,each optimizing at a slightly different pres-sure. The presence of ionization (" 2%when the harmonic is being generated)tends to shift the optimum phase-matchedpressure to higher values (Fig. 3, B and D)because of the need to balance the refrac-tive index contribution introduced by ion-ization to maintain phase-matching. Ab-sorption in the gas reduces the expectedoutput by an order of magnitude (Fig. 3).This absorption explains why our output isenhanced only by factors of 102 to 103 (seebelow), instead of the maximum expectedenhancements of 104 due to the increasein coherence length from 50 !m to aninteraction length of a few centimeters.The coherence length is infinite whenphase-matched.

The conversion efficiency of laser lightinto a single harmonic peak was measuredin two independent ways. First, a vacuumphotodiode (XSI) was inserted before thex-ray spectrometer, and the current wasrecorded with a picoammeter (Keithley485; Keithley, Cleveland, OH). A metalfilter was placed between the source anddetector to block any low-order harmonicsthat might have been present. In practice,this procedure was not necessary, becausethe low-order harmonics are not phase-matched at this pressure and also are strong-ly absorbed in argon, and therefore, theirsignal levels are orders of magnitude belowthat of the five phase-matched harmonics.Signal levels of $100 pA were observed.Given our repetition rate of 1 kHz, a detec-tor quantum efficiency of at best 0.08, ameasured filter transmission of 10%, and aninput laser energy of 150 !J per pulse, weestimate that % 2 # 107 photons per har-monic peak per pulse were emitted. Thisvalue corresponds to an energy of % 0.2 nJper harmonic peak per pulse, or a conver-sion efficiency of 10–6 to 10&5 into asingle harmonic peak—an increase of 100times over previously measured efficien-cies (6). We also estimated the enhance-ment factors by comparing our outputfrom the fiber with that from a gas jet (7).We estimate an increase in efficiency of102 to 103 due to phase-matching. Wenote that the use of the vacuum photo-diode x-ray detector significantly im-proved the accuracy of our conversion ef-ficiency measurements, compared with

Fig. 2. Measured signals of the (A) 29th and (B)31st harmonics as a function of gas pressure.

Fig. 3. (A) Calculated output of the 29th harmonic,(B) in the presence of constant 2% ionization, (C)in the presence of absorption, and (D) net outputin the presence of absorption and varying levels ofionization around 2%.

A

B

Fig. 4. Soft x-ray output beam for harmonic or-ders 23 through 31: (A) unphase-matched, detec-tor set to high gain, and (B) phase-matched, de-tector set to low gain.


References

T. Brabec and F. Krausz, “Intense few-cycle laser fields: Frontiers of nonlinear optics”, Rev Mod Phys. 72, 545 (2002)

X. Zhang, et al. “Phase matching, quasi-phase matching and pulse compression in a single waveguide for enhanced high-harmonic generation”, Optics Letters, 30, 1971 (2005)

Rundquist, et al. “Phase-Matched Generation of Coherent Soft X-rays”, Science, 280, 1412 (1998)

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