1750 J. Opt. Soc. Am. B/Vol. 20, No. 8 /August 2003 Schwarz et al.

Stimulated anti-Stokes Raman scattering withBessel beams in hydrogen gas

Ulrich Theodor Schwarz, Jurgen Zeitler, Jurgen Baier, and Max Maier

Naturwissenschaftliche Fakultat II Physik, Universitat Regensburg, D-93040 Regensburg, Germany

Suren Sogomonian

Naturwissenschaftliche Falultat II Physik, Universitat Regensburg, D-93040 Regensburg, Germany,and Epygi Labs AM, Arshaguniants Avenue 41, 375026 Yerevan, Armenia

Received December 22, 2002; revised manuscript received April 11, 2003

Stimulated anti-Stokes Raman scattering experiments were performed with a Bessel pump beam in hydrogengas in the pressure range from p 5 6 3 105 to 5 3 106 Pa. The interrelation between two conical Stokesmodes and several anti-Stokes far-field rings could be explained if we take into account planar transversewave-vector matching only. Planar and nonplanar transverse and longitudinal wave-vector matching are dis-cussed in detail. The experiments are compared with numerical solutions of the coupled equations for Stokesand anti-Stokes fields. © 2003 Optical Society of America

OCIS codes: 190.4420, 190.4380, 190.5650, 190.5890.

1. INTRODUCTIONBessel beams are quasi nondiffracting beams1,2 that havebeen used for the investigation of stimulated Ramanscattering.3–11 It has been demonstrated that, dependingon the experimental conditions, the Raman Stokes light isemitted either as a gain-guided conical mode6,7 or as anaxial spotlike mode.3,4

Bessel beams can be considered as a superposition ofplane waves, the wave vectors of which lie upon the sur-face of a cone with a half-angle qP . This allows the gen-eration of anti-Stokes light in stimulated Raman scatter-ing under broad experimental conditions. Matching thelongitudinal and transverse components of the wave vec-tors of conical pump, Stokes, and anti-Stokes waves hasbeen treated in detail theoretically.3,8–10 However, in theexperiments only the anti-Stokes component, which isgenerated by the axial spotlike Stokes emission, has beenobserved in liquids3,9 and hydrogen gas8 and the corre-sponding cone angles have been measured. In the case ofliquids the results have been discussed in terms of longi-tudinal and transverse phase matching. In hydrogen gassome experiments were explained with exact nonplanarphase matching, where the wave vectors of the pump,Stokes, and anti-Stokes components do not lie in a plane.8

The phase-matching conditions for generation of the sec-ond Stokes and second anti-Stokes components were in-vestigated in Ref. 10. The corresponding experimentalresults in liquid acetone have been explained with a spe-cial case of exact planar phase matching.

In this paper we present a detailed investigation of theeffects of longitudinal and transverse planar and nonpla-nar wave-vector matching on the generation of anti-Stokes light by stimulated Raman scattering with aBessel pump beam. The experiments were carried outwith a J0 Bessel beam with a pump cone angle of 2.7mrad in hydrogen gas in the pressure range between 6

0740-3224/2003/081750-08$15.00 ©

and 50 bars (6 3 105 and 5 3 106 Pa). At high pressureboth Stokes and anti-Stokes emission is conical and bothfar fields are sharp rings. The conical Stokes emission isthe consequence of gain guiding in the propagation-invariant central maximum of a Bessel pump beam.6

With decreasing pressure a second, smaller and partiallyfilled ring appears in the Stokes far field, which aftersome transition region becomes the only Stokes emissionat low pressure. The anti-Stokes emission shows severalrings in the far field and is of a particularly rich structurein the transition region where two Stokes modes are vis-ible.

Here we show that most of the angles of the conicalanti-Stokes emission, even in the transition region ofpressure, where the angular spectrum of the anti-Stokeslight is quite complex, can be explained just by taking intoaccount planar transverse wave-vector matching. We ex-plain this dominance of the planar scattering geometry byintroducing a density-of-states-like quantity for the anti-Stokes far-field angles with respect to the contributingazimuthal angles of the pump and Stokes beam. In theexperiment, in addition to the anti-Stokes emission onmultiple cones, we observed a continuous anti-Stokesbackground, which is taken as evidence for nonplanarscattering processes.

We not only calculated the anti-Stokes angles fromwave-vector matching considerations, but we also solvednumerically the coupled paraxial wave equations for theStokes and anti-Stokes light. The calculated correspond-ing intensity distributions were found to be in good agree-ment with the measured ones.

2. THEORYA. Longitudinal Wave-Vector MatchingWe treat the case of generation of anti-Stokes light byfour-photon processes of the type kP 1 kP8 5 kS 1 kA ,

2003 Optical Society of America

Schwarz et al. Vol. 20, No. 8 /August 2003 /J. Opt. Soc. Am. B 1751

where wave vectors kP , kP8 , kS , and kA of the pump,Stokes, and anti-Stokes light lie on the surfaces of coneswith different angles but do not necessarily lie in oneplane, i.e., we consider the most general case of nonplanarphase matching [see Fig. 1(a)]. To pronounce that eithertransverse or longitudinal components of the wave vec-tors can be matched while the respective other compo-nents are mismatched, we prefer the term transverse (orlongitudinal) wave-vector matching. Actually for none ofthe prominent anti-Stokes rings discussed in this articleis exact phase matching achieved, i.e., simultaneouslytransverse and longitudinal wave-vector matching.

The condition for matching the longitudinal compo-nents kii of the wave vectors can be written as kSi 1 kAi

5 2kPi , where kii 5 ki cos(qi) ' ki(1 212q i

2). In the fol-lowing discussion we assume that cone angles q i of thelight beams are small, i.e., q i ! 1 (i 5 P, S, A). As ameasure of the dispersion of the medium we introduce thequantity

DK 5 2kP 2 kS 2 kA , (1)

which in hydrogen gas is approximately given by

DK 5 2Cp, (2)

with C 5 0.130 cm21/bar in the pressure range from 6 to50 bars, using the linear refractive indices of hydrogengas obtained from Ref. 12.

The following equation for longitudinal wave-vectormatching is derived3,9,10:

kAqA2 1 kSqS

2 5 2kPqP2 2 2DK. (3)

Introducing the normalized Stokes and anti-Stokesangles cS 5 qSkS /(qPkP) and cA 5 qAkA /(qPkP) we getthe following relation from Eq. (3):

Fig. 1. (a) Nonplanar phase-matching geometry for conicalbeams. The configuration is to scale in the x –y plane for p5 10 bars, q 5 2.7 mrad, and cS 5 1. This nonplanar phase-matching configuration with f 5 86° and w 5 30° (see text) cor-responds to exact phase matching and is marked with an aster-isk in Fig. 2. (b) Transverse wave-vector matching geometry forthe same configuration as in (a).

cA2

RA2

1cS

2

RS2

5 1. (4)

Equation (4) represents an ellipse in the cS 2 cA planewith the axial intersections10

Ri 5 ki1/2F 2

kPS 1 2

DK

kPqP2 D G 1/2

~i 5 S, A !, (5)

which depend on the pump cone angle qP and pressure p[see Eq. (2)]. Ellipses are drawn in Fig. 2 as thin curvesfor pump angle qP 5 2.7 mrad and for four different pres-sures.

B. Transverse Wave-Vector MatchingNext we discuss transverse wave-vector matching for thenonplanar case, where the wave vectors of the light wavesdo not lie in a plane. The transverse components of thewave vectors are shown in Fig. 1(b). w and f are theangles between the transverse components kP' and kP'8of the pump wave vectors kP and kP8 and between the vec-tor sum kV of these components and transverse compo-nent kS' of the Stokes wave vector, respectively. Thetransverse wave-vector matching condition can be writtenas kS' 1 kA' 5 kV . Introducing angles w and f be-tween the transverse components, we derived the follow-ing equation for transverse wave-vector matching:

Fig. 2. Central plot spans the cS 2 cA plane of the normalizedStokes and anti-Stokes angles. Thick straight lines mark pla-nar transverse wave-vector matching; the corresponding planarscattering configurations are sketched in the small boxes. Theposition of these lines is independent of pump beam angle qP andhydrogen pressure. Within the shaded area nonplanar exacttransverse wave-vector matching is possible. Thin curves corre-spond to exact longitudinal wave-vector matching [Eq. (4)], de-pending on qP (here qP 5 2.7 mrad) and pressure. Dashed ver-tical lines mark the normalized Stokes angles cS 5 0.3 and 1.0.Intersections with thick lines of planar transverse wave-vectormatching are marked by greek letters for association with peaksin the anti-Stokes spectra of Figs. 5 and 7.

1752 J. Opt. Soc. Am. B/Vol. 20, No. 8 /August 2003 Schwarz et al.

kA2 qA

2 5 kS2 qS

2 2 4kPkSqPqS cosw

2cos f

1 4kP2 qP

2 cos2w

2. (6)

Using normalized angles this equation reads

cA2 5 cS

2 2 4cS cosw

2cos f 1 4 cos2

w

2. (7)

The anti-Stokes cone angle qA for transverse wave-vectormatching depends on angles w and f, which are charac-teristic for nonplanar wave-vector matching. When wand f are equal to 0° and/or 180°, all wave vectors lie in aplane. This is the case of planar wave-vector matching,which is especially important to explain the experimentalresults.

As an example we refer to angle w of 0° between thetransverse components kP' and kP'8 of the pump wavevectors and angle f of 180° between the vector sum ofthese components and the transverse component of theStokes wave vector kS' . The relation between anti-Stokes angle cA and Stokes angle cS for transversematching is calculated from Eq. (7) and yields

cA 5 cS 1 2. (8)

This equation represents a straight line in the cS 2 cAplane, which is shown in Fig. 2 as a thick solid line. Thecorresponding wave-vector configuration for w 5 0° andf 5 180° is shown in one of the small boxes. Using theother possible 0° and 180° combinations of w and f, threefurther equations for straight lines in the cS 2 cA planewere obtained, which are also shown in Fig. 2 togetherwith the corresponding wave-vector configurations.These relations between cA and cS have already been ob-tained by Gadonas et al.,9 who calculated the overlap in-tegrals over the electric field distributions of pump,Stokes, and anti-Stokes conical beams.

For nonplanar matching of the transverse componentsof the wave vectors the relation between the anti-Stokesangle qA (normalized angle cA) and the Stokes angleqS ( cS) can be calculated from Eq. (6) for different valuesof angles w and f. When these angles are varied in therange of 0° , w, f , 180°, the corresponding values ofcA and cS cover the area between straight linescA 5 cS 1 2 and cA 5 cS 2 2 (shaded area in Fig. 2),i.e., in this range nonplanar transverse wave-vectormatching is possible.

There is an interesting special case that is importantfor the experiments discussed below. When the Stokeslight is emitted as a conical M00 mode,6 the normalizedStokes angle is cS

00 5 1, meaning that the transversewave vectors of pump and Stokes light are equal. Thisvalue of cS

00 5 1 is marked by one of the dashed verticallines in Fig. 2, from which many different possibilities forwave-vector matching of the M00 Stokes mode, the anti-Stokes mode, and the pump light can be derived. Planartransverse wave-vector matching (without longitudinalwave-vector matching) occurs for anti-Stokes angles cA5 1 and 3, where the vertical broken line (at cS

00 5 1) in-tersects the thick solid lines at points a and b in Fig. 2.

Exact planar phase matching for the M00 Stokes mode,i.e., simultaneous transverse and longitudinal matching,occurs at the intersection of the dashed vertical line atcS

00 5 1, a thick solid straight line for transverse wave-vector matching, and a thin ellipse for longitudinal wave-vector matching. This is the case for a pressure of23 bars (2.3 3 106 Pa) at point b in Fig. 2, giving a nor-malized phase-matched anti-Stokes angle cA 5 3. Exactphase matching is not possible for pressures larger than23 bars but can be realized for all pressures below 23 barsby use of suitable nonplanar wave-vector configurations(shaded area in Fig. 2). For example, for a pressure of10 bars (1 3 106 Pa) exact nonplanar phase matching oc-curs at the intersection of the vertical dashed line (atcS

00 5 1) with the corresponding ellipse (marked by a starin Fig. 2).

C. Numerical SimulationsThe wave-vector matching calculations presented aboveare useful for calculation of the angles of the conical anti-Stokes emission but not for determining the intensities.We calculated the intensity distributions of the Stokesand anti-Stokes light by solving the following coupledparaxial wave equations for field amplitudes AS and AA :

]

]zAS 5

i

2kS¹'

2 AS

11

2gS@ uAPu2AS 1 AP

2 AA* exp~iDKz !#, (9)

]

]zAA 5

i

2kA¹'

2 AA

21

2gA@ uAPu2AA 1 AP

2 AS* exp~iDKz !#. (10)

Here, g i (i 5 S, A) is the gain factor for stimulated Ra-man scattering and AP is the amplitude of the pump field.We solved these equations numerically using the Crank–Nicholson scheme.13 The far field was obtained by Fou-rier transformation of the light field at a length corre-sponding to the end of the gas cell.

3. EXPERIMENTAL RESULTS ANDDISCUSSIONIn the experiments we generated the J0 Bessel pumpbeam by illuminating a diffractive axicon with the lin-early polarized expanded beam of a frequency-doubled,injection-seeded, Q-switched Nd:YAG laser that produced13-ns-long pulses at 532 nm with a repetition rate of 10Hz and a pulse energy of 60 mJ. The experimental setupis shown in Fig. 3. After attenuation by a l/2 plate andpolarizer P the beam is expanded by telescope T1 to a di-ameter of approximately 4 cm (full width at half-maximum) and collimated. The central part of the ex-panded beam illuminates off-axis grating G with adiameter of dPP 5 20 mm to form the Bessel pump beam.We used a four-level off-axis phase grating with a theoret-ical efficiency of 81%. A second telescope, T2 , controlsthe cone angle of the Bessel beam. The desired order of

Schwarz et al. Vol. 20, No. 8 /August 2003 /J. Opt. Soc. Am. B 1753

the diffracted light beam is separated by pinhole B. Forthis experiment the cone angle is qP 5 2.7 mrad, corre-sponding to a diameter of d 5 76 mm of the first ring ofzero intensity of the Bessel beam. The propagationrange of the Bessel beam before its collapse is zmax' dPP /(2qP) 5 3.7 m. The Raman medium is hydrogengas (Raman shift of 4155 cm21) in a 150-cm-long cell.The peak intensity of the central spot of the Bessel beamat the entrance of the Raman cell is of the order of 100MW/cm2. The pulse-to-pulse fluctuations of the power ofthe Bessel beam are approximately 5% for laser pulsesthat generate Stokes pulses just above the detectionthreshold. For the pump far field the ratio of ring thick-ness (FWHM) to ring diameter is 1:50. The backgroundof the pump beam far field lies below the camera detectionthreshold and is thus at least 3 orders of magnitudesmaller than the far-field peak intensity. The far-field in-tensity distribution (angular spectrum) of the generatedStokes and anti-Stokes beams at 683 and 436 nm, respec-tively, was measured simultaneously at one laser pulse inthe focal plane of lens L with two CCD cameras. Themeasurements were carried out just above the detectionthreshold of the CCD cameras. Thus the Raman conver-sion efficiency for the anti-Stokes light was kept below1023.

A. Stokes and Anti-Stokes Far-Field AnglesTypical experimental results for a J0 Bessel pump beamwith a cone angle of qP 5 2.7 mrad and a hydrogen pres-sure of 30 bars are shown in Fig. 4. Stokes [Fig. 4(a)] andanti-Stokes [Fig. 4(b)] far-field distributions were takenfor an individual single pump pulse, as in all other corre-sponding pairs of Stokes and anti-Stokes angular spectrathroughout this article. Two Stokes rings can be ob-served in Fig. 4(a), the inner one being partially filled inthe center. The azimuthal intensity distribution of theStokes light in Fig. 4(a) is constant, i.e., the angular mo-mentum is zero (l 5 0). The anti-Stokes emission inFig. 4(b) shows two rings corresponding to a conical emis-sion and a continuous distribution up to a cutoff angle ofqA 5 6.5 mrad. The distribution of the anti-Stokes light

Fig. 3. Experimental setup: the intensity of the frequency-doubled (lP 5 532-nm) Nd:YAG laser is adjusted by a l/2 waveplate and polarizer P, telescope T1 for beam expansion and colli-mation, Bessel beam generation by off-axis diffractive grating G,second telescope T2 to vary Bessel beam cone angle qP and toseparate the desired order of the off-axis grating with pinhole B,mirrors M, dumps D, hydrogen gas cell H2 , dichroic mirror S toseparate the pump beam from Stokes and anti-Stokes beams.The far fields of the Stokes and anti-Stokes light are measured inthe focal plane of lens L with cameras CCD1 and CCD2. FiltersF1 and F2 transmit Stokes and anti-Stokes wavelengths lS5 683 nm and lA 5 436 nm, respectively.

is azimuthally symmetric as well, although less regular.In contrast to experiments in which either a pumpbeam7,11 or a Stokes seed beam11 of higher angular mo-mentum was used, we observed neither a Stokes nor ananti-Stokes beam with nonzero angular momentum.

To reveal more details about angular distributions andto facilitate comparison with calculated spectra, we azi-muthally averaged the measured distributions and we de-termined the radial intensity distribution along a linethrough the center as shown in Fig. 5 for a Bessel pumpbeam with a cone angle of qP 5 2.7 mrad and for a hydro-gen pressure of 28 bars. The dominating Stokes mode inFig. 5(a) can be shown to be the M00 Stokes mode ob-tained from the theory of gain-guided modes.6 The ob-served Stokes angle of qS 5 3.4 mrad or cS 5 0.98 is ingood agreement with that obtained from the relation qS5 qPkP /kS or cS 5 1 from the mode theory.6 Figure5(c) shows an example of a different pump pulse at thesame pressure at which a Stokes mode with qS5 1.1 mrad corresponding to cS 5 0.3 dominates. Weobserved these fluctuations of the Stokes field betweentwo modes in a transition region from approximately 25 to35 bars for qP 5 2.7 mrad. It is not uncommon to ob-serve both modes simultaneously, as seen in Fig. 5(c).The relative intensity of both modes fluctuates from pulse

Fig. 4. Measured (a) Stokes and (b) anti-Stokes far-field distri-bution. Hydrogen pressure was p 5 30 bars and the pump coneangle was qP 5 2.7 mrad.

Fig. 5. Measured (a), (c) Stokes and (b), (d) anti-Stokes far-fieldspectra. The corresponding pairs (a), (b) and (c),(d) were mea-sured simultaneously with one pump pulse, respectively. Hy-drogen pressure was p 5 28 bars and the pump cone angle wasqP 5 2.7 mrad.

1754 J. Opt. Soc. Am. B/Vol. 20, No. 8 /August 2003 Schwarz et al.

to pulse, suggesting that Stokes emission occurs throughtwo independent guided modes.

Both Stokes modes with cS00 5 1.0 and cS 5 0.3 are

marked in Fig. 2 as vertical dashed lines. At a pressureof 28 bars exact phase matching is not possible for eitherof these two modes, because the intersections of thesedashed lines with the ellipse for longitudinal wave-vectormatching for 28 bars lies outside the shaded area of non-planar transverse wave-vector matching in Fig. 2. Wenow show that the structure of the anti-Stokes far fieldcan be explained by transverse wave-vector matching,only, under the assumption that planar scattering pro-cesses dominate. In Fig. 2 the intersection of the dashedline at the normalized Stokes angle cS

00 5 1 and the solidstraight lines corresponding to planar transverse wave-vector matching are at normalized anti-Stokes angles cA5 1(a) and 3(b), corresponding to qA 5 2.2 and 6.6mrad. These angles are in good agreement with the mea-sured peaks marked a (qA 5 2.1 mrad) and b (qA5 6.4 mrad) in Fig. 5(b).

The narrow Stokes ring of Fig. 5(a) produces a weakcontinuous background of anti-Stokes angles [Fig. 5(b)]from qA 5 0 –6.6 mrad ( cA 5 0 –3). This background isdue to nonplanar transverse wave-vector matching in theregion of the cS 2 cA plane (Fig. 2), where the dashedvertical line at cS

00 5 1 traverses the shaded area. It isseen in Fig. 2 that these processes are effective betweencA 5 0 and 3, in agreement with the experimental result.The important observation is that the planar processesdominate, producing the multiple sharp far-field rings ofthe anti-Stokes field. The origin of this dominance is dis-cussed in Subsection 3.B.

For the normalized Stokes angle cS 5 0.3 correspond-ing to Fig. 5(c) the anti-Stokes angles determined fromthe intersections g, d, and e in Fig. 2 are cA 5 0.3, 1.7,and 2.3 corresponding to qA 5 0.7, 3.8, and 5.1 mrad ingood agreement with the experimental measured posi-tions marked g (qA 5 0.7 mrad), d (qA 5 3.5 mrad), ande (qA 5 4.9 mrad) in the anti-Stokes far field in Fig. 5(d).This figure shows additional weak lines a and b corre-sponding to the weak M00 Stokes mode in Fig. 5(c) ( cS5 0.98). For none of these anti-Stokes peaks a to e isthe longitudinal wave-vector mismatch zero. Thereforethey are not exactly phase matched.

Additional small peaks can be observed in Figs. 5(b)and 5(d) for anti-Stokes angles in the 7 mrad , qA, 8 mrad range. Their position depends on the normal-ized Stokes angle cS and on hydrogen pressure. We sup-pose that they originate either from longitudinal wave-vector matching of one of the Stokes modes with theconical pump beam or from exact phase matching of oneof the Stokes modes with the background of the pumplight angular distribution.

B. Discussion of Planar and Nonplanar TransverseWave-Vector Matching ProcessesThe good coincidence in the comparison between simula-tions and experiments in Subsection 3.C demonstratesthat the physics of the four-wave mixing process nearanti-Stokes threshold can be described by the two coupledEqs. (9) and (10) for Stokes and anti-Stokes fields. Yetthis knowledge does not provide an answer to the ques-

tion of why the planar scattering processes are preferredcompared with nonplanar processes. In the originalarticle6 on gain guiding with Bessel beams only theStokes Eq. (9) with AA 5 0 was solved. A comparisonwith the solution of the coupled Eqs. (9) and (10) showsthat the shape of the Stokes mode is dictated by themechanism of gain guiding with slight corrections be-cause of the presence of the anti-Stokes field.

In addition it has been shown experimentally thatblocking sections of the azimuthal distribution of thepump beam in nonplanar Raman scattering causes adepletion of corresponding sections of the azimuthal dis-tribution of the anti-Stokes emission.8 This demon-strates that combinations of azimuthal sections of theBessel pump beam and of the Stokes beam contribute in-dependently of other combinations to the anti-Stokesbeam.

Keeping both facts in mind—the little effect of the anti-Stokes emission on the gain-guided Stokes field and theindependent contribution of the azimuthal sections ofpump and Stokes field to the anti-Stokes field—we intro-duce a new concept for a semiquantitative measure of thecontributions of planar and nonplanar transverse wave-vector matching processes to the anti-Stokes intensity.In analogy to the concept of the density of states in solid-state physics we introduce a density Nwf of combinationsof angles w and f contributing in the scattering process toa peculiar anti-Stokes angle cA . This quantity will behigh when both derivatives, ]cA /]w and ]cA /]f, calcu-lated from Eq. (7) are zero, which is true for planar scat-tering configurations. So in analogy to the increaseddensity of states at the Brillouin zone boundaries we ex-pect a dominance of the planar scattering processes.

One can calculate this density Nwf by counting thenumber of combinations of angles w and f for which val-ues of the anti-Stokes angle between cA and cA 1 DcAare calculated from Eq. (7) and by dividing this numberby DcA . For this calculation the continuous distribu-tions of angles w and f were replaced by discrete andequally spaced distributions. We plotted Nwf (normal-ized to a maximal value of one) in Figs. 6(a) and 6(c) as afunction of the normalized anti-Stokes angle cA for twovalues of the normalized Stokes angle cS 5 1 and 0.3, re-spectively, corresponding to the dashed vertical lines inFig. 2. Although the height and width of the peaks ofNwf depend on the chosen discretization steps, the overallstructure, the positions of the maxima, and the cutoff areindependent of these steps. For the computations in Fig.6 we used a step size Dw and Df of 10 mrad for w and fand of DcA 5 0.01 for cA .

There is one distinct peak a at cA 5 1 in Fig. 6(a) andthere are two distinct peaks g and d at cA 5 0.3 and 1.7in Fig. 6(c). All three maxima correspond to planar scat-tering processes, i.e., intersections of the dashed verticallines with straight solid lines for planar transverse wave-vector matching in Fig. 2. There are two more planarcombinations, b in Fig. 6(a) and e in Fig. 6(c), which donot show a maximum but an abrupt cutoff, because thetransverse wave vectors cannot be matched beyond b ande, respectively.

The maxima in Figs. 6(a) and 6(c) sit on a continuousbackground that is due to nonplanar transverse wave-

Schwarz et al. Vol. 20, No. 8 /August 2003 /J. Opt. Soc. Am. B 1755

vector matching processes (shaded area in Fig. 2). Incontrast to the overlap integral approach of Ref. 9, our ap-proach does not show a maximum for cA 5 cS 5 0. Allthe basic features of the anti-Stokes spectra can be de-rived from the density Nwf of anti-Stokes far-field angleswith respect to angles w and f that contribute to a conicalanti-Stokes emission with normalized angle cA .

To illuminate the role of phase matching in the scatter-ing process in Figs. 6(b) and 6(d) we plotted the absolutevalue of phase mismatch uDku 5 ukP 1 kP8 2 kS 2 kAu forthe same values of cS 5 0.3 and 1 as in Figs. 6(a) and6(c), respectively, and for pressure p 5 28 bars. Thisvector sum of transverse and longitudinal wave-vectormismatch uDku is not to be confused with the scalar quan-tity DK in Eq. (1), the latter being a material propertythat contains no information on the geometry of the scat-tering process.

For cS 5 1 [Fig. 6(b)] and cA , 3 the transverse wave-vector mismatch can always be chosen to be zero, becausethese combinations lie within the shaded area in Fig. 2.A small contribution of the longitudinal wave-vector mis-match (,4 cm21) is left, which shows only a weak depen-dency on cA . It has a minimum of uDku 5 0.7 cm21 atcA 5 3. For cA . 3 the transverse wave-vector mis-match increases rapidly with cA . Exact phase matchingis not possible for these configurations. Figure 6(d)shows a similar picture for cS 5 0.3. The distinct de-crease in uDku toward cA 5 3 in Fig. 6(b) is the cause forpeak b in Fig. 5(b). The corresponding minimum of uDku

Fig. 6. (a), (c) Plot of the density-of-states-like quantity Nwf fortwo different values of normalized Stokes angle (a) cS 5 1 and(c) cS 5 0.3. (b), (d) Absolute values of the total phase mis-match uDku 5 ukP 1 kP8 2 kS 2 kAu for the same values of (b)cS 5 1 and (d) cS 5 0.3, and a pressure p 5 28 bars.

at cA 5 2.3 in Fig. 6(d) is less pronunced, in accordancewith the observation of a shoulder only at e in Fig. 5(d).

As a result we state that the phase-matching mecha-nism suppresses anti-Stokes generation outside theshaded area in Fig. 2 because of the large phase mis-match, which is mostly due to transverse wave-vectormismatch. On the other hand the phase-matchingmechanism is not responsible for the predominance ofplanar scattering geometries, as seen in the distinctivepeaks in the anti-Stokes far field, because the phase mis-match uDku is only smoothly varying over the range of cAallowed by transverse wave-vector matching [Fig. 6(b)].Only one of the anti-Stokes rings [b in Fig. 5(b)] is causedby a minimum in the phase mismatch, whereas all otherrings are due to the dominance of the planar scatteringprocesses caused by the higher number of combinationsNwf of azimuthal pump and Stokes angles that contributeto a specific value of normalized anti-Stokes cone anglecA .

C. Comparison of Measured Angular IntensityDistributions with Numerical CalculationsFor comparison between the experiments and the numeri-cal calculations typical results of the Stokes and anti-Stokes far-field distributions for a Bessel pump beamwith the same cone angle of qP 5 2.7 mrad but differenthydrogen pressures are shown in Fig. 7. The solid curvescorrespond to the experimental results, the dashed curvesrepresent the numerical solution of Eqs. (9) and (10),where the only free parameter is amplitude AP of thepump beam, which is related to the pump intensity. At apressure of p 5 47 bars [Figs. 7(a) and 7(b)] only the coni-cal Stokes emission with qA 5 3.3 mrad ( cS 5 0.96) as-sociated with the M00 mode is observed, together with thecorresponding anti-Stokes peak a and cutoff angle b.There is good agreement between the measured far-fielddistributions and the calculated curves.

Figure 7(c) for p 5 33 bars shows an example for thetransition range of pressure 25 bars , p , 35 bars forqP 5 2.7 mrad at which both Stokes modes are of compa-rable intensity. The corresponding anti-Stokes far fieldin Fig. 7(d) has a particularly rich structure. Five peaksor shoulders a–e corresponding to planar scattering pro-cesses in Fig. 2 can be resolved in this experiment.These peaks have been discussed above for a pressure of28 bars (Fig. 5). In Fig. 7(c) the agreement between themeasured and the calculated Stokes far field is worse(solid and dashed curves, respectively) compared with thegood agreement in Fig. 7(a). From gain-guiding theory itis known that the Stokes far-field ring broadens with in-creased pump amplitudes AP (or parameter v as used inmode theory6). But in the numerical simulations of thecoupled equations both Stokes modes are observed onlysimultaneously in a narrow range of not-too-large pumpamplitudes AP , yielding the narrow rings shown in Fig.7(c). The agreement between measured and calculatedanti-Stokes far field is better than for the Stokes far field,because nonplanar transverse wave-vector matching pro-cesses fill the gaps for peaks from a to e in Fig. 7(d).

At low pressures p , 25 bars the inner Stokes modedominates. An example for p 5 10 bars is shown in Fig.7(e). The Stokes cone angle qS 5 0.8 mrad or cS

1756 J. Opt. Soc. Am. B/Vol. 20, No. 8 /August 2003 Schwarz et al.

5 0.23 at this low pressure is smaller compared with thevalue cS 5 0.3 and 0.34 of the same mode in Figs. 5(c)and 7(c), respectively. The mode is still conical, but al-ready filled in the center to a large extent. In Fig. 7(f)there is good agreement between the measured and thecalculated anti-Stokes far field except for outermost ringe. For this ring the anti-Stokes angles of transverse andlongitudinal wave-vector matching are close to each other(see Fig. 2 near point e). In this range of close to exactphase matching the refractive indices must be known tohigh accuracy. Even slight deviations cause differencesbetween measured and calculated anti-Stokes angles andtheir respective intensities as observed in Fig. 7(f). Foreven lower pressures down to p 5 6 bars the outer ring ofthe anti-Stokes far field becomes more intense than theinner part, with little change in the shape of the Stokesmode.

4. CONCLUSIONSThe far field of the anti-Stokes light generated by stimu-lated Raman scattering with a J0 Bessel pump beam witha cone angle of 2.7 mrad was investigated in hydrogen gasin the pressure range from 6 to 50 bars. We demon-strated experimentally that for Raman scattering in hy-drogen gas with a Bessel pump beam conical Stokes andconical anti-Stokes emission can be observed over thiswide pressure range. Each Stokes far-field ring cancause one or several anti-Stokes far-field rings. The coni-cal Stokes emission itself is caused by gain guiding.

We have discussed the wave-vector matching condi-tions for the generation of anti-Stokes light by stimulated

Fig. 7. Solid lines represent Stokes and anti-Stokes far fields asin Fig. 5 but for three different pressures: (a), (b) p 5 47 bars;(c), (d) p 5 30 bars; (e), (f ) p 5 10 bars. The dashed lines rep-resent results from numerical integration of coupled Eqs. (9) and(10) of Stokes and anti-Stokes fields.

Raman scattering with a Bessel pump beam. Particularemphasis was given to the treatment of nonplanar andplanar transverse wave-vector matching. Exact phasematching applied to none of the observed anti-Stokesrings. On the other hand we could explain most peaks inthe anti-Stokes far-field spectra by considering transversewave-vector matching only, by assuming that planar scat-tering processes, in which all four wave vectors lie withinone plane, predominate. This allowed us to explain eventhe complex anti-Stokes structure that consisted of threerings and two different cutoff angles in the transition re-gion from approximately 25 to 35 bars. The continuousanti-Stokes background distribution between these dis-tinct features shows that nonplanar scattering processesalso take place.

For a physical interpretation of the predominance ofplanar scattering processes in the absence of exact phasematching we introduced a density-of-states-like quantityfor the anti-Stokes far-field angles with respect to the azi-muthal angles of pump and Stokes light contributing toan anti-Stokes emission with a certain cone angle. Themaxima of this quantity correspond to planar configura-tions and peaks observed in the anti-Stokes far-field spec-tra. The sharp cutoff of this quantity at a maximal anti-Stokes far-field angle for a given Stokes far-field angle isseen as a shoulder in the experimentally observed anti-Stokes emission.

This detailed study of phase matching in anti-StokesRaman scattering is made possible by the singular trans-verse wave-vector distribution of Bessel beams. Simula-tions starting from a set of two coupled paraxial waveequations that describe the evolution of the Stokes andanti-Stokes fields were in good agreement with the ex-periment. Deviations were largest for parameters closeto exact phase matching, because of the critical influenceof errors in the pressure- and temperature-dependent in-dices of refraction of hydrogen.

The e-mail address for U. T. Schwarz is uli.schwarz@physik.uni-regensburg.de.

REFERENCES1. J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, ‘‘Diffraction-

free beams,’’ Phys. Rev. Lett. 58, 1499–1501 (1987).2. A. Vasara, J. Turunen, and A. T. Friberg, ‘‘Realization of

general nondiffracting beams with computer-generated ho-lograms,’’ J. Opt. Soc. Am. A 6, 1748–1754 (1989).

3. I. Golub, ‘‘Superluminal-source-induced emission,’’ Opt.Lett. 20, 1847–1849 (1995).

4. S. Klewitz, P. Leiderer, S. Sogomonian, and S. Herming-haus, ‘‘Tunable stimulated Raman scattering by pumpingwith Bessel beams,’’ Opt. Lett. 21, 248–250 (1996).

5. L. Niggl and M. Maier, ‘‘Efficient conical emission of stimu-lated Raman Stokes light generated by a Bessel pumpbeam,’’ Opt. Lett. 22, 910–912 (1997).

6. L. Niggl and M. Maier, ‘‘Gain-guided modes in stimulatedscattering processes with diffraction-free pump beams,’’Opt. Commun. 154, 65–69 (1998).

7. U. T. Schwarz, L. Niggl, and M. Maier, ‘‘Gain guiding instimulated scattering processes with hollow Bessel pumpbeams,’’ Opt. Commun. 181, 413–423 (2000).

8. S. Sogomonian, L. Niggl, and M. Maier, ‘‘Nonplanar phase-matching of stimulated anti-Stokes Raman scatteringpumped by a Bessel beam,’’ Opt. Commun. 162, 261–266(1999).

Schwarz et al. Vol. 20, No. 8 /August 2003 /J. Opt. Soc. Am. B 1757

9. R. Gadonas, V. Jarutis, A. Marcinkevicius, V. Smilgevicius,A. Stabinis, and V. Vaicaitis, ‘‘Transverse phase-matchingin stimulated Raman scattering by a Bessel beam,’’ Opt.Commun. 169, 189–197 (1999).

10. V. Vaicaitis, A. Stabinis, A. Marcinkevicius, and V. Jarutis,‘‘Second-order Stokes and anti-Stokes coupling in stimu-lated Raman scattering by Bessel beam,’’ Opt. Commun.178, 461–467 (2000).

11. S. Sogomonian, U. T. Schwarz, and M. Maier, ‘‘Phase-front

transformation of a first-order Bessel beam in Raman-resonant four-wave mixing,’’ J. Opt. Soc. Am. B 18, 497–504(2001).

12. Landolt-Bornstein, Zahlenwerte und Funktionen: II.Band, 8. Teil, Optische Konstanten (Springer-Verlag, Ber-lin, 1962), Chap. 286, Table 5, pp. 6–885.

13. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P.Flannery, Numerical Recipes in C (Cambridge University,Cambridge, England, 1992).