Includes bibliographical references (page 348).Transient birefringence can be induced in a gas of anisotropic molecules by an intense polarized laser pulse. We propose to use this birefringence to phase match nonlinear optical frequency-conversion processes. The conditions for anisotropic phase matching are derived, and experimental conditions required for phase-matched third-harmonic generation in a gas-filled hollow-core fiber are presented. We show that these conditions are experimentally feasible over a significant parameter range, making possible a new type of nonlinear optics

Baertschy, M. D.

Bartels, R. A.

Kapteyn, H. C.

Murnane, M. M.

Wagner, N. L.

Wyss, J.

English

Transient birefringence can be induced in a gas of anisotropic molecules by an intense polarized laser pulse. We propose to use this birefringence to phase match nonlinear optical frequency-conversion processes. The conditions for anisotropic phase matching are derived, and experimental conditions required for phase-matched third-harmonic generation in a gas-filled hollow-core fiber are presented. We show that these conditions are experimentally feasible over a significant parameter range, making possible a new type of nonlinear optics

Kapteyn, H. C., author

Murnane, M. M., author

Wyss, J., author

Baertschy, M. D., author

Wagner, N. L., author

Bartels, R. A., author

Optical Society of America, publisher

Mountain Scholar (Digital Collections of Colorado and Wyoming)

346 OPTICS LETTERS / Vol. 28, No. 5 / March 1, 2003Phase-matching conditions for nonlinear frequency conversionby use of aligned molecular gasesR. A. Bartels, N. L. Wagner, M. D. Baertschy, J. Wyss, M. M. Murnane, and H. C. KapteynJILA and the Department of Physics, University of Colorado, Boulder, Colorado 80309-0440Received August 15, 2002Transient birefringence can be induced in a gas of anisotropic molecules by an intense polarized laser pulse.We propose to use this birefringence to phase match nonlinear optical frequency-conversion processes. Theconditions for anisotropic phase matching are derived, and experimental conditions required for phase-matchedthird-harmonic generation in a gas-filled hollow-core fiber are presented. We show that these conditions areexperimentally feasible over a significant parameter range, making possible a new type of nonlinear optics.© 2003 Optical Society of AmericaOCIS codes: 190.0190, 190.2620.The development of nonlinear optics1,2 has dramati-cally expanded the utility of laser sources, making itpossible to coherently convert radiation from visible ornear-infrared lasers into the terahertz,3 visible,4 andextreme-ultraviolet regions of the spectrum.5 –7 A keyissue in nonlinear optics is phase matching of the con-version process, i.e., to create a situation in which thepropagation velocity of the pump and signal fields isthe same, allowing the signal to accumulate construc-tively over extended propagation.The most common implementation of phase-matchedfrequency conversion is birefringent frequency dou-bling, in which an oriented anisotropic crystal allowsthe pump and signal waves to propagate with differ-ent polarizations but the same phase velocity. Sincegases and noncrystalline materials normally exhibitneither birefringence nor second-order nonlineari-ties, this process has been demonstrated only usinganisotropic crystals. However, gases are transparentover a much broader range of frequencies than solids,making them an attractive candidate for vacuumultraviolet frequency conversion. Existing gas-phasefrequency conversion techniques rely on either reso-nances8,9 or waveguide propagation,7,10 but each haslimitations. New techniques are needed to generatelight over a broader range of frequencies and tosupport shorter-duration pulses.Here, we propose a new phase-matched frequency-conversion technique for gaseous media: transientbirefringent phase matching. In a molecular gas theintrinsic anisotropy in polarizability of the individualmolecules normally is not observed, since the moleculesare randomly oriented. However, a transient align-ment can be imposed on the gas by use of an intenselight field that induces a torque on any molecule thatis not oriented either parallel or perpendicular to thelaser field.11 For pulses shorter than the rotationalperiod of the molecule, the induced torque is impulsiveand results in a change in the rotational angularmomentum and an excited distribution of rotationalenergy levels. The ensemble of molecules experiencesa periodic realignment determined by the excited an-gular momentum states and the molecular rotationalenergy level structure. These alignments modify the0146-9592/03/050346-03$15.00/0macroscopic polarizability of the ensemble of moleculesand induce a transient birefringence. Recently, wedemonstrated that this time-dependent index can beused to spectrally broaden an ultrafast optical pulsewith a negative chirp that is time compressed with atransparent window.12In this Letter we calculate the conditions underwhich the birefringence created by an ensemble ofaligned molecules can be used to phase match non-linear frequency conversion. In particular, wecalculate the conditions required to phase matchthird-harmonic generation in a hollow-core f iber, butthese results can also be generalized to other nonlinearprocesses and interaction geometries.The polarizability tensor am of an isolated linear,axially symmetric molecule (e.g., CO2, N2) isa diagonal second-rank tensor with elementsax0x0 ,ay 0y 0 ,az0z0  a,a,ak, where k is the directionalong the long axis of the molecule,  is perpendicularto the long axis, and the primed coordinates are alongthe principal directions of the molecule. If all thesemolecules are aligned along the same direction, themacroscopic polarizability of the ensemble of moleculeswill be dictated by the polarizability properties of asingle molecule, and the gas behaves as a positiveuniaxial crystal. More generally, the molecules arenot completely aligned. To calculate the macroscopicpolarizability of the entire ensemble, we transformthe linear polarizability tensor through the rotationaldegrees of freedom, am  am:f,u, c . The macroscopicpolarizability, a, is given by the average of the en-semble of molecules over the rotational degrees offreedom, a  am:f,u,c fuc . The transformationused is a standard Euler transform, where f and care rotational degrees of freedom that are unaffectedby the alignment f ield and u is the rotation angle fromthe alignment polarization direction. The averageover the unaligned rotational degrees of freedom isdefined as am:f, u,c f,c  1/2pR2p0 am:f,u, cf,cdf, dc,and the average over u is determined by the degree ofalignment imposed by the alignment pulse.The aligned linear polarizability tensor is diago-nal in the coordinates defined by the polarizationdirection of the alignment pulse. The polarization© 2003 Optical Society of AmericaMarch 1, 2003 / Vol. 28, No. 5 / OPTICS LETTERS 347of the pump pulse defines the extraordinary axis ofthe aligned quasi-crystal, with a polarizability givenby azz  a 1 Dacos2u. The polarizability of thedirections perpendicular to this axis corresponds toaxx  ayy  1/2a 1 ak 2 Dacos2u and defines theordinary index of the quasi-crystal; the inset of Fig. 1shows the geometry. Here, Da  ak 2 a is themolecular polarizability anisotropy.We can measure the components of the linearpolarizability tensor of the aligned ensemble byrecording the frequency shift imposed on a probepulse by the temporal index of refraction varia-tion of a revival.12 This frequency shift of Dvt 212pdw̃tdt	, where t is the delay between thepump and probe pulses, is integrated to yield therevival phase and thus the instantaneous index ofboth polarizations. Figure 1 shows a measurementof the index as a function of time for the ordinaryand extraordinary polarization for the f irst partialrevival in CO2 gas and verifies the predicted ordinaryand extraordinary indices imposed by the molecularalignment.This calculation and measurement demonstrates theuniaxial nature of the aligned-molecule quasi-crystal.Moreover, we see that the gas is a positive uniaxialcrystal at times when the molecules exhibit a net align-ment cos2u . 13 and a negative uniaxial in thecase of a net antialignment cos2u , 13. This po-larizability can be translated into an index of refrac-tion by use of the approximate expression nij 2 1 aij pNTorr2e0  pdnij, where p is the gas pressurein Torr, NTorr is the number density of molecules at1-Torr pressure, e0 is the dielectric permittivity, anddnij is a convenient grouping of constants.Given the induced birefringence, it is possible todetermine phase-matching conditions in an alignedmolecular gas. However, it is also necessary toconsider the properties of the nonlinear susceptibilityin the aligned frame. Here, we consider the simplestcase of third-harmonic generation (THG). Thus,we need to consider the properties of the ensemble-averaged third-order nonlinear tensor and to deter-mine which tensor elements are nonzero and canconnect the fundamental and harmonic polarizationsto make THG possible. This calculation follows theprocedure outlined for the linear susceptibility; how-ever, the third-order polarizability [x 3] is a fourth-rank tensor and transforms with the relation x 3ijkl riarjbrkcrjdx3abcd, where rnm are direction cosines de-fined by the transformation from the aligned referenceframe.After evaluating the x 3 tensor components of thealigned ensemble of molecules, we find that the ten-sor elements depend on the alignment of the molecu-lar gas and the nonzero tensor components allow fortwo types of anisotropic phase matching. The firstphase-matching condition requires the third harmonicto be polarized along the extraordinary (z) polarizationdirection (3v0 ! z) and the fundamental along boththe ordinary ( y) and the extraordinary (z) axes (v0 !y,y, z), whereas the other nonzero tensor elements al-low phase matching with the polarizations (3v0 ! y)and (v0 ! y, z, z). We define these phase-matchingconditions as type IIa and type IIb, respectively.Because the hollow-core f iber geometry allows foran extended interaction length in a simplif ied, effec-tive one-dimensional geometry, this is the case thatwe consider here. Since the optic axis of the alignedquasi-crystal is coincident with the aligning pumppulse polarization, a probe pulse that is travelingcollinearly with the aligning pump pulse propagatesorthogonally to the optic axis, resulting in the so-callednoncritical phase-matching geometry.The k vector of a light wave propagating in a hollow-core fiber filled with an anisotropic gas can be writtenas kij  2p/l1 1 pdnijxy 2 1/2unml2pa2	, where lis the free-space wavelength, unm is a mode constant,p is the gas pressure, a is the hollow-fiber radius,dnijxy Rdnijx, yI x, ydxdyRI x, ydxdy is themodal averaged index, and I x, y is the spatial modein the hollow-core fiber.Under conditions of negligible pump deple-tion, the f ield of the qth-harmonic signal gener-ated in a nonlinear medium can be written asEqv ~RL0 EqvexpiDkzdz, where the phase mis-match is defined as Dk  kqv 2Pqj1 kiv. Foroptimal signal conversion, the phase mismatch iszero, corresponding to an equalization of the phasevelocity of the fundamental and harmonic f ields.When Dk  0, we can relate the optimum pres-sure, popt, to the required alignment cosine cos2uby the expression P0  poptcos2u 2 K, wherethe waveguide pressure, P0, and the critical align-ment, K, are determined by the characteristics ofthe molecular gas, the waveguide, and the type ofphase matching. For type IIa phase matching withall f ields propagating in the fundamental mode,we find that P0  249dDn3v lu112pa2 andK  dnvk 1 2dnv 2 3dn3v 3dDn3v, whereas fortype IIb phase matching P0  89dDn3v 1 dDnv	 3lu112pa2 and K  dn3v 1 dn3vk 2 5/3dn3v 21/3dnvk dDn3v 1 dDnv. We note that K is thedegree of alignment required by the molecular gas forphase matching of the nonlinear interaction in freespace when the alignment, fundamental, and har-monic pulses propagate collinearly. At high optimalFig. 1. Measured phase of the f irst rotational revival inCO2 for a probe polarized parallel (solid curve) and perpen-dicularly (dashed curve) to the alignment pulse polariza-tion direction. The inset shows the alignment geometry.348 OPTICS LETTERS / Vol. 28, No. 5 / March 1, 2003Fig. 2. Phase-matching curves for type IIa (solid curves)and type IIb (dashed curves) THG processes for the mo-lecular gases listed in Table 1. (a) Free-space alignmentrequired for THG in molecular gases across a broad fre-quency range. (b) Gas pressure needed for THG with an800-nm fundamental beam in a 175-mm hollow fiber.pressures, the required alignment approaches that offree space because the dispersion of the waveguidebecomes negligible compared with that of the gas.Published dispersion curves13,14 can be used tocalculate the alignment required to phase match theTHG process in an aligned molecular gas. Figure 2(a)shows the free-space alignment required for THG inmolecular gases across a broad wavelength range; thedispersion of the critical alignment and the dispersionof the hollow-fiber balance make it possible to obtainextremely broad phase-matching bandwidths thatcan support few-cycle (i.e., ø10-fs) pulse durationsin the vacuum ultraviolet. The lowest requiredalignment occurs for gases with a combination oflow spectral dispersion between the fundamental andthe harmonic fields and a relatively high intrinsicanisotropy. Both CO2 and N2O exhibit relativelylow required alignment levels; N2O is easier to phasematch due to its higher anisotropy. The alignmentbandwidth, defined by the FWHM of the THG yield,is 60.015 for a 10-cm-long fiber with 100 Torr ofN2O. We also evaluated phase-matching conditionsfor phase matching at 800 nm for a number of linearmolecules contained in a 175-mm-diameter hollow-corefiber. The results are summarized in Table 1 andplotted in Fig. 2(b). The minimum alignment pulsetemporal duration required for suff icient alignment ofthe molecular gas to phase match THG frequency con-version in a hollow-core fiber is shown in Table 1 fora range of peak pulse intensity values. We estimatethe conversion efficiency of the THG process to rangefrom 1% to 20% for THG pulse durations of the orderof 1.8–8 fs.In summary, we have demonstrated that phase-matched third-harmonic generation is possible in anensemble of aligned linear molecules over a very broadwavelength range. One may also effectively combinethis technique with time-dependent phase modula-tion12 to obtain spectral broadening and subsequentcompression, or it may be used with other nonlinearinteractions. For example, a combination of transientbirefringent and resonant phase matching may makeTable 1. Conditions Required for Phase MatchingThird-Harmonic Frequency Conversionof 800-nm Light with Aligned Linear Moleculesin a Hollow-Core Fiber for PeakIntensities (in femtoseconds) of 3 3 1013 (ta),6 3 1013 (tb), and 9 3 1013 (tc) Wcm2Gas Ka Kb ta tb tcCO2 0.2277 0.4420 162 72 48N2O 0.2257 0.4447 122 59 36O2 0.1995 0.4782 – – 82it possible to tune resonant phase-matched frequenciesover a larger range than previously possible. Second-harmonic generation in gases may also be possible inthe case of polar molecules aligned with a half-cycleoptical pulse. Finally, by control of the degree ofalignment along the propagation direction, the non-linear coefficient of the material can be periodicallystructured, allowing quasi-phase matching.This work was supported by the Chemical Sciences,Geosciences and Biosciences Division of the Off ice ofBasic Energy Sciences, U.S. Department of Energy,and by the National Science Foundation. We thankChris H. Greene for useful discussions. R. A. Bartels’se-mail address is bartels@colorado.edu.References1. N. Bloembergen, Nonlinear Optics (World Scientif ic,Teaneck, N.J., 1996).2. P. A. Franken, A. E. Hill, C. W. Peters, and G. Wein-reich, Phys. Rev. Lett. 7, 118 (1961).3. N. Katzenellenbogen and D. Grischkowsky, Appl. Phys.Lett. 58, 222 (1991).4. M. Nisoli, S. D. Silvestri, V. Magni, and O. Svelto, Opt.Lett. 19, 1973 (1994).5. A. McPherson, G. Gibson, H. Jara, U. Johann, T. S.Luk, I. A. McIntyre, K. Boyer, and C. K. Rhodes,J. Opt. Soc. Am. B 4, 595 (1987).6. M. Ferray, A. L’Huillier, X. F. Li, L. A. Lompré, G.Mainfray, and C. Manus, J. Phys. B 21, L31 (1987).7. A. Rundquist, C. G. Durfee III, S. Bakus, C. Herne, Z.Chang, M. M. Murnane, and H. C. Kapteyn, Science280, 1412 (1998).8. A. J. Merriam, S. J. Sharpe, M. Shverdin, D.Manuszak, G. Y. Yin, and S. E. Harris, Phys. Rev.Lett. 84, 5308 (2000).9. A. H. Kung, J. F. Young, and S. E. Harris, Appl. Phys.Lett. 22, 301 (1973).10. C. G. Durfee, S. Backus, M. M. Murnane, and H. C.Kapteyn, Opt. Lett. 22, 1565 (1997).11. B. Friedrich and D. Herschbach, Phys. Rev. Lett. 74,4623 (1995).12. R. A. Bartels, T. C. Weinacht, N. Wagner, M. Baertchy,C. Greene, M. M. Murnane, and H. C. Kapteyn, Phys.Rev. Lett. 88, 019303 (2002).13. U. Hohm, Chem. Phys. 179, 533 (1994).14. D. Nikogasyan, Properties of Optical and Laser-RelatedMaterials (Wiley, New York, 1997).

Phase-matching conditions for nonlinear frequency conversion by use of aligned molecular gases

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Phase-matching conditions for nonlinear frequency conversion by use of aligned molecular gases

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