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Sogomonian et al.

Vol. 18, No. 4 / April 2001 / J. Opt. Soc. Am. B

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Phase-front transformation of a first-order Bessel beam in Raman-resonant four-wave mixing S. Sogomonian Institute for Physical Research of the Armenian National Academy of Sciences, 378410 Ashtarak-2, Republic of Armenia

U. T. Schwarz and Max Maier Naturwissenschaftliche Fakulta¨t II—Physik, Universita¨t Regensburg, D-93040 Regensburg, Germany Received May 24, 2000 We have studied the evolution of the phase front of an apertured first-order Bessel beam (J 1 ) generated by a diffractive axicon during free-space propagation and its transformation in a Raman-resonant four-wave mixing process in hydrogen gas. It is demonstrated that the order of the phase singularity (vortex) of a Bessel beam can easily be detected by interferometry at the boundaries of the beam propagation range. With a J 1 beam and a Gaussian beam at the Stokes and the pump frequencies and vice versa, anti-Stokes light is generated as a singular, conical beam. In both cases the observed phase front agrees with theoretical predictions. In particular, the transformation of the topological charge is consistent with the conservation of angular momentum. © 2001 Optical Society of America OCIS codes: 350.5030, 190.4380, 290.5910.

1. INTRODUCTION The optics of laser beams that possess phase singularities1 has become an active research topic in recent years.1–9 Screw dislocations of the wave front, or optical vortices, are a common type of phase singularity. The dislocations appear as helical phase variations of 2 ␲ l about a point where the phase of the wave is undetermined (singular) and thus the wave’s amplitude vanishes, resulting in a dark spot. Integer l is known as the order of the screw dislocation, or the topological charge of the optical vortex. Much interest stems from the fact that such a beam with azimuthal phase dependence exp(il␾) has an orbital angular momentum of lប per photon associated with the helical structure of the wave front.3 Consequently, transformations of these beams are constrained by the conservation of angular momentum. Frequency conversion of laser beams that contain optical vortices has been investigated by several authors for the case of three-wave interaction in a quadratic nonlinear medium.4–9 Transformation of the topological charge of an optical vortex has been demonstrated in secondharmonic generation,4–6 sum-frequency mixing,7 parametric amplification,8 and parametric downconversion.9 Most of these studies concentrated on the use of the Laguerre–Gaussian mode family; beams in this family are shape-invariant fields, that is, they retain their transverse intensity structure, but not their scale, on propagation in free space. Another class of light beams that carry phase singularities is the family of propagationinvariant higher-order Bessel beams that maintain a constant transverse intensity distribution during 0740-3224/2001/040497-08$15.00

propagation.10–14 Owing to their nondiffractive nature and narrow zero-intensity spot in the center, the size of which can be of the order of the wavelength of light, these beams have potential applications in precision alignment and for collimation and trapping of particles. The lth-order Bessel beam has a screw wave-front dislocation of charge l at its center, which is stable on propagation in a linear medium. We are concerned here with the experimental and theoretical study of the spatial evolution of the helical phase front of a first-order Bessel beam during propagation in free space and the transformation of the phase front in Raman-resonant four-wave mixing in hydrogen gas. The paper is organized as follows: In Section 2 we study the peculiarities in the detection of the phase singularity of a finite-aperture Bessel beam. We present numerical calculations of the dependence of the radial phase distribution on the propagation coordinate of a finite-aperture first-order Bessel beam and experimental results on the interferometric detection of the beam’s phase singularity at different propagation distances. In Section 3 we study the transformation of the phase singularity of a Bessel beam in the generation of anti-Stokes light by means of four-wave mixing, ␻ A ⫽ 2 ␻ P ⫺ ␻ S . We demonstrate in Section 4 that Stokes and anti-Stokes beams are propagation invariant and calculate their near-field and far-field distributions. In Section 5 these results are compared with experimental results obtained by four-wave mixing in hydrogen gas. We observe a reversal of the charge of the singularity or its doubling by use of different combinations of Gaussian and Bessel beams at the laser and Stokes frequencies. © 2001 Optical Society of America

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2. PHASE FRONT OF A FINITE-APERTURE BESSEL BEAM The field of an lth-order Bessel beam propagating in direction z is given by E 共 r, ␾ , z 兲 ⬀ J l 共 ␣ r 兲 exp关 i 共 ␤ z ⫺ ␻ t 兲兴 exp共 il ␾ 兲 ,

(1)

where J l is an lth-order Bessel function of the first kind, r is the radial coordinate, and ␾ is the azimuthal angle. The transverse intensity distribution of a higher-order Bessel beam consists of a dark spot of radius r ⬇ l/ ␣ surrounded by the characteristic ring structure. Bessel beams of any order can be considered a superposition of plane waves with k vectors that form a cone with angle ␽ about the propagation axis. Interference of these component waves yields a propagation-invariant transverse intensity profile. The lth-order Bessel beam has a zerointensity minimum in the center that is associated with a screw wave-front dislocation of charge l described by the azimuthal phase term exp(il␾) in relation (1). Parameters ␣ and ␤ are the transverse and longitudinal components, respectively, of the wave vector k ⫽ n ␻ /c. From the geometrical relation ␤ 2 ⫹ ␣ 2 ⫽ k 2 they can be expressed through cone angle ␽, which is a characteristic parameter of the beam:

␣ ⫽ k sin ␽ , ␤ ⫽ k cos ␽ .

(2)

A versatile technique for determining the charge of the beam singularity is to measure the phase structure of the beam by means of its interference with a reference wave.15 The use of this technique for the analysis of the phase front of a Bessel beam has some peculiarities, as is described below. First we analyze the phase front of the field produced by a diffractive axicon placed at plane z ⫽ 0 with the phase transmission function T 共 r, ␾ 兲 ⫽



exp关 ⫺2 ␲ i 共 r/ ␳ 兲兴 exp共 il ␾ 兲

r⭐R

0

r⬎R

,

angle ␾ the phase is independent of radial coordinate r, except for ␲ phase shifts between successive rings of Bessel function (the central dashed curve in Fig. 1b shows the phase dependence for an ideal Bessel beam). In any other plane within the interference region the Bessel profile is confined to a smaller transverse region about the axis. The phase is independent of radial coordinate r within this region, whereas it depends linearly on r outside this region. Outside the interference region (for z ⭓ z max) the equal phase surface formed by the diverging components of a conical wave is kz ⫹ 2 ␲ r/ ␳ ⫹ l ␾ ⫽ constant. Here again, for a given ␾, the phase depends approximately linearly on r (upper-most dashed curve in Fig. 1b), but the sign of proportionality constant is opposite that for z ⫽ 0. These physical-optics predictions for the phase behavior of the field produced by a finite-aperture axicon may also be obtained by direct numerical calculation. It is to be noted that evaluation of a Fresnel integral with the stationary phase method does not provide a correct description of the phase function; it gives an artificial phase term with quadratic radial dependence, which is not detected in the experiment.14 We do not use this approximation but rather evaluate the Fresnel diffraction integral directly by numerical integration. The solid curves in Fig. 1b show the calculated radial dependence of the phase at different propagation distances z. In the initial part, z ⫽ z max/10, of the propagation range the phase shows the plateaus of constant phase within the Bessel beam rings inside the overlap re-

(3)

where ␳ is the period of the grating and R is the radius of the aperture of the axicon. When it is illuminated by a plane wave, such an axicon produces a good approximation to an lth-order Bessel beam with cone angle sin ␽ ⫽ ␭/␳. Because of the finite value of R, the propagation range of the Bessel beam produced is limited to a distance z max ⫽ R/tan ␽ (see Fig. 1a). A defining characteristic of the field formed immediately behind the axicon is a conical wave with an lth-order screw dislocation of the wave front. The equal phase surface of this wave is a spiral conical sheet given by kz ⫺ 2 ␲ r/ ␳ ⫹ l ␾ ⫽ constant, and for a given azimuthal angle ␾ the phase depends linearly on the radius (lowest dashed curve in Fig. 1b). In the range 0 ⬍ z ⬍ z max the converging and diverging components of the conical wave interfere, producing the Bessel beam (shaded area in Fig. 1a). The transverse size of the interference region depends on longitudinal coordinate z: It takes the largest value equal to radius R of the beam aperture in the middle of the propagation range z ⫽ z max/2 (Fig. 1a). In this plane for a given azimuthal

Fig. 1. Wave fronts of the field produced by a finite-aperture axicon: (a) Schematic illustration of the spatial evolution of the wave front. (b) Calculated dependence of the phase on normalized radial coordinate r/R for a given azimuthal angle at several distances z from the axicon.

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Fig. 2. Interferometry of the field produced by a diffractive axicon. Top, optical setup: G, diffractive axicon; T, telescope. A reference plane wave propagates along the axis of the Bessel beam with cone angle ␽ P ⫽ 1.6 mrad. Interference patterns at different propagation distances: a, b, at the beginning and the end of the propagation range, respectively; c, d, within the propagation range at distances separated by 11.5 cm. The white lines mark the 90° rotation of the axis of symmetry.

gion r/R ⬍ 1/10, and the phase depends linearly on the radial coordinate r outside this region. In the middle of the propagation range, z ⫽ z max/2, the phase front is independent of radial coordinate r over the whole aperture of the beam (between the ␲ phase shifts the curve is horizontal). At the end of the propagation range of the Bessel beam, z ⫽ z max , the dependence of the phase on r is again approximately linear. The different signs of the slopes for the curves in Fig. 1(b) correspond to converging and diverging components of the conical wave. We experimentally studied this evolution of the phase front of the field produced by a diffractive axicon by interfering this field with a reference plane wave. We produced a J 1 beam by illuminating the axicon with collimated light from a He–Ne laser and subsequently imaged the beam with a Kepler telescope, T, consisting of two positive lenses (Fig. 2, top). We used a 2R ⫽ 20-mmdiameter axicon with grating period ␳ of 400 ␮m. Imaging of the beam provides access to the beginning of the propagation range of the beam, which otherwise is in the plane of the axicon (see Fig. 2). The evolution of the interference pattern with variation of propagation distance is depicted in Fig. 2. As can be seen, the patterns of Figs. 2a and 2b recorded at the beginning and at the end of the propagation range of the beam show spirals of opposite handedness. Whereas the sign of the phase singularity determined by the azimuthal term exp(il␾) with l ⫽ 1 re-

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mains unchanged during propagation, the opposite handedness of the two spiral patterns is due to the converging and diverging components of the conical wave at the beginning and the end of the propagation range, respectively. Note that, immediately behind the axicon, that is, in the beginning of the propagation range, the r term of the phase factor in Eq. (3) is negative (the waves are converging toward the axis). Thus the screw dislocation of charge l ⫽ 1 results in a left-handed spiral at the beginning (Fig. 2a) and in a right-handed spiral at the end (Fig. 2b) of the propagation range. Now we discuss the behavior of interference within the propagation range of the beam. Figures 2c and 2d show interference patterns obtained at two propagation distances close to z max/2. Detailed examination of the fringe patterns shows that at a given distance z the intensity of the rings depends on azimuthal angle ␾; the intensity is going through a single maximum as ␾ varies from 0 to 2␲. Because of the ␲ phase shift between successive rings of the Bessel beam the azimuthal positions of maximum intensity for odd and even rings are reversed. For example, in Fig. 2c the first and the third rings have maxima approximately at the bottom, whereas the second and fourth rings have maxima at the top. Figures 2c and 2d show that the interference pattern rotates approximately 90° (indicated in Fig. 2d by a white arrow) as z is increased by 11.5 cm. The full rotation by 360° is thus achieved as the propagation distance varies by ⌬z ⫽ 46 cm. To understand the physical origin of this effect, we consider the interference of a Bessel wave and a copropagating plane wave of unit amplitude and note that the interfering waves have different propagation constants ␤ ⫽ k cos ␽ and k, respectively; i.e., the phase velocity of the Bessel wave is larger. The interference pattern is given by I 共 r, ␾ , z 兲 ⫽ 1 ⫹ J l 2 共 ␣ r 兲 ⫹ 2J l 共 ␣ r 兲 cos关共 ␤ ⫺ k 兲 z ⫹ l ␾ 兴 . (4) The last term in Eq. (4) indicates the rotation of the interference pattern during propagation. For a given radial distance r, the azimuthal position ␾ m of maximum intensity is found from the condition ( ␤ ⫺ k)z ⫹ l ␾ m ⫽ 0, which gives

⳵␾m ⳵z



k⫺␤ l



k l

共 1 ⫺ cos ␽ 兲 .

(5)

The rate and the direction of the rotation thus depend on the value and the sign of charge l of the phase singularity and on the difference of the propagation constants k ⫺ ␤ , which is a function of the cone angle ␽ [see Eq. (2)]. Note that the determination of the sign of the phase singularity from the direction of rotation of the interference pattern requires recording the pattern at least for two propagation distances. As the field distribution of a beam with a screw wavefront dislocation of charge l has an azimuthal symmetry of order l, the rotation of the interference pattern by ⌬ ␾ m ⫽ 2 ␲ /l about the axis of the dislocation gives indistinguishable patterns. Therefore the spatial period ⌬z of rotation of the interference pattern defined as the distance between two successive identical patterns is independent of l:

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⌬z ⫽ ⌬ ␾ m

冉 冊 ⳵␾m ⳵z

⫺1



2␲ k⫺␤



Sogomonian et al.

2␭



2

.

(6)

For our experimental conditions (␭ ⫽ 632.8 nm, ␽ ⫽ 1.6 ⫾ 0.2 mrad) the expected period of rotation is ⌬z ⬇ 49 cm, which is in good agreement with the observed value of 46 cm. The rotation of the interference pattern observed in the earlier study14 has been interpreted as being due to an additional axially varying phase ⌿(z) ⫽ ␲ ␭z/ ␳ 2 term in the phase function of the field formed at a distance z from the axicon. It is to be noted that in the paraxial approximation we get ␤ ⬇ k 关 1 ⫺ (1/2) ␽ 2 兴 and ␽ ⬇ sin ␽ ⫽ ␭/␳. Therefore the additional phase term ⌿(z) considered in Ref. 14 is simply the phase shift that arises from the difference in propagation constants (k ⫺ ␤ )z ⬇ ␲␽ 2 z/␭ ⫽ ␲ z␭/ ␳ 2 . A conclusion from these observations is that the phase singularity of a Bessel beam can easily be detected by the interferometry at the boundaries of the beam propagation range, where the interference pattern is much more distinct and easier to interpret than that displayed within the propagation range. We utilize this feature to study the phase singularity of an anti-Stokes beam generated by four-wave mixing, as described below.

3. TRANSFORMATION OF THE PHASE SINGULARITY For a description of the transformation of the phase singularity from the singular pump or Stokes beam to the anti-Stokes beam, we need consistent treatment of the coherent process. The existence of propagation-invariant modes has been demonstrated for stimulated Raman scattering with Bessel pump beams.16,17 In the Stokes amplification process, diffraction can be compensated for by Raman gain, which leads to the existence of a gainguided, propagation-invariant Stokes mode. With a similar ansatz we demonstrate the existence of propagation-invariant modes in the process of four-wave mixing with Bessel beams. The resultant predictions for the radial field distribution of the Stokes and anti-Stokes fields and for their azimuthal phase functions that correspond to the angular momentum of the beams is compared in Section 5 with the experiment. The frequencies of the four-wave mixing process are related as

␻A ⫽ 2␻P ⫺ ␻S .

(7)

Our pumping scheme with two input waves is thus a coherent anti-Stokes Raman spectroscopic process, with ␻ 1 ⫽ ␻ 3 ⫽ ␻ P , ␻ 2 ⫽ ␻ S , and ␻ 4 ⫽ ␻ A . We treat two different combinations of pump and Stokes beams. First, we have a plane pump wave E P ˆ exp(ik z) and a J Bessel Stokes beam with a screw ⫽ A P P 1 dislocation of the order l ⫽ 1. Second, we treat a J 1 Bessel pump beam and a plane Stokes wave. These two cases are not symmetric, as the factor-of-2 difference and the different sign in Eq. (7) have already shown. To employ the paraxial wave approximation we write the electrical fields as E i (r, ␾ , z) ⫽ A i (r, ␾ , z)exp(iki z), where i ⫽ P,S,A. As the experiment is performed near

the threshold for Stokes and anti-Stokes detection, the depletion of the pump beam can be neglected. We also neglect the real parts of the nonlinear susceptibility. This leaves two coupled paraxial wave equations for the Stokes and anti-Stokes fields:

⳵ ⳵z

AS ⫽

ic 2n S ␻ S

ⵜ⬜ 2 A S ⫹

1 2

˜g S 关 兩 A P 兩 2 A S

⫹ A P 2 A A * exp共 i⌬ ␬ z 兲兴 ,

⳵ ⳵z

AA ⫽

ic 2n A ␻ A

ⵜ⬜ 2 A A ⫺

1 2

(8)

˜g A 关 兩 A P 兩 2 A A

⫹ A P 2 A S * exp共 i⌬ ␬ z 兲兴 . (9) 2 ˆ 兩 for For gain factor ˜g i the definition 1/2 g i Iˆ P ⫽ 1/2 ˜g i 兩 A P i ⫽ A, S is used, where the peak intensity of the pump ˆ 兩 2 . The Raman gain factor g beam is Iˆ P ⫽ 1/2 n P ⑀ 0 c 兩 A P i ⫽ g S ␻ i / ␻ S is proportional to the frequency, and we use the Stokes gain factor g S ⫽ 2.6 ⫻ 10⫺3 cm/MW for hydrogen gas at a pump wavelength ␭ P ⫽ 532 nm. In Eqs. (8) and (9) we introduced ⌬␬ ⫽ 2

n P␻ P c



n S␻ S c



n A␻ A c

.

(10)

This collinear phase mismatch in the linear medium is a material property that is due to the dispersion of the refractive index. For hydrogen, ⌬ ␬ ⬇ ⫺p ⫻ 1.27 ⫻ 10⫺6 cm⫺1/Pa is a good approximation for the pressure range up to p ⫽ 107 Pa. 17 One has to be aware that the propagation constant of the wave at ␻ i generated in the nonlinear medium does not generally obey the relation k i ⫽ n i ␻ i /n i , with index of refraction n i in the linear medium. A z-independent pump beam profile is a prerequisite for the existence of propagation-invariant Stokes and anti-Stokes modes. An ansatz for both fields with arbitrary radial distribution u i (r) for i ⫽ A, S, paraxial propagation constant offset ␤ i p ⫽ ␤ i ⫺ k i , and exponential amplification with gain coefficient ␥ i in direction z is ˆ exp关 i 共 ␤ p z ⫹ l ␾ 兲兴 u 共 r 兲 exp共 ␥ z 兲 . Ai ⫽ A i i i i i

(11)

For a plane-wave pump beam we show that Bessel Stokes and anti-Stokes beams u i (r) ⫽ J l i ( ␣ i r) are analytical solutions that fulfill Eqs. (8) and (9). For a higher-order Bessel pump beam numerical integration will demonstrate the possibility of a propagation-invariant scattering process. Insertion of the ansatz Eq. (11) into paraxial wave ˆ u (r) lead to equation (9) and subsequent division by A A A

␥ A ⫹ i ␤ Ap ⫺ ⫹

ic



1



2n A ␻ A u A r ⳵ r



r

⳵ ⳵r

冊 册

uA ⫺

l A2 r2

1 ˜g 兩 A 兩 2 2 A P

ˆ *u * A 1 S S ⫽ ⫺ ˜g A A P 2 exp关 i 共 2l P ⫺ l S ⫺ l A 兲 ␾ ˆ u 2 A A A ⫹ i 共 ⌬ ␬ ⫹ 2 ␤ P p ⫺ ␤ S p ⫺ ␤ A p 兲 z ⫹ 共 ␥ S ⫺ ␥ A 兲 z].

(12)

A similar equation holds for the Stokes field. Because the left-hand side of Eq. (12) is independent of ␾ and z,

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the same must be true for the right-hand side, and thus the exponential terms have to be constant. This stricture leads to the following conditions for the gain coefficients ␥ i , paraxial propagation constant offsets ␤ i p , and angular momenta l:

␥ ⫹ i ␤ Ap ⫹ i

␣ A2c 2n A ␻ A

⫹ ␥ A pw ⫽ ⫺␥ A pw R

J ⫺l 共 ␣ S r 兲 J l共 ␣ Ar 兲

501

. (17)

(13)

Using the relation J l ( ␣ r) ⫽ J ⫺l ( ␣ r) for Bessel functions of real arguments, we can fulfill both equations for all r if the radial parameters ␣ A and ␣ S are equal:

⌬ ␬ ⫹ 2 ␤ P p ⫺ ␤ S p ⫺ ␤ A p ⫽ 0,

(14)

␣A ⫽ ␣S .

2l P ⫺ l S ⫺ l A ⫽ 0.

(15)

Satisfying Eq. (18) reduces Eqs. (16) and (17) to a simple set of two coupled linear equations for gain coefficient ␥ and amplitude ratio R of the Stokes and anti-Stokes fields. The input parameters are the plane-wave gain coefficients ␥ i pw , which are varied by pump beam intensity Iˆ P , and the radial parameter ␣ S , derived from cone angle ␽ S of the Stokes Bessel beam by Eqs. (2). The properties of the anti-Stokes beam are then fixed by Eqs. (13)–(15) and (18). From the last relation, ␣ A ⫽ ␣ S , and Eq. (2) we get the ratio ␽ A / ␽ S ⬇ ␭ A /␭ S for the cone angles of the anti-Stokes and Stokes beams. At exact phase-matching condition the gain ␥ is zero (gain suppression). For all other input values of Stokes cone angle ␽ S and planewave gain ␥ S pw a solution of propagation-invariant Stokes and anti-Stokes modes with positive gain exists. Next we treat the situation of an incident J 1 Bessel pump beam and a plane Stokes beam with angular momentum l S ⫽ 0, resulting in an anti-Stokes beam with angular momentum l A ⫽ 2l P ⫽ 2. Because of the radial dependence of the gain in Eq. (12), which is proportional to J 1 2 ( ␣ P r), it is not possible to find analytical solutions for radial field distributions u A and u S . One way to find solutions is to integrate Eq. (12) and the corresponding equation for the Stokes field numerically, which constitutes a boundary value problem. We have chosen a different approach, integrating paraxial wave equations (8) and (9) numerically. These equations are formally equal to coupled nonlinear Schro¨dinger equations and describe the evolution of Stokes and anti-Stokes beams in direction z. Input parameters are the Bessel pump beam parameters Iˆ P , ␽ P , and l P ⫽ 1 and the Stokes and antiStokes angular momenta l S ⫽ 0 and l A ⫽ 2. For a given angular momentum the azimuthal part of the Laplace operator gives a term ⫺l 2 /r 2 [see Eq. (12)] that can be interpreted as a centrifugal potential. Then only the onedimensional problem of the evolution of the radial distribution has to be integrated for a given angular momentum. Starting from a plane Stokes wave and a small antiStokes field with arbitrary radial distribution, propagation-invariant solutions for Stokes and antiStokes fields are found. In the range of stationary amplification they have identical gain, in accordance with Eq. (13). The far fields consist of a sharp central spot for the Stokes beam and a sharp ring for the anti-Stokes beam. For comparison with the experiment, ⌬ ␬ ⫽ ⫺1.52 cm⫺1 is used, corresponding to a hydrogen pressure p ⫽ 1.2 ⫻ 10 6 Pa, 17 a pump beam cone angle ␽ P ⫽ 3.5 mrad, and a pump wavelength of 532 nm. The result is a calculated ratio of the transverse wave-vector components ␣ A / ␣ P ⫽ 2.1 and a ratio of cone angles ␽ A / ␽ P ⬇ ( ␣ A ␭ A )/( ␣ P ␭ P ) ⫽ 1.7 [see Eqs. (2)]. In the limit of small gain, the radial distribution of the anti-

␥A ⫽ ␥S ,

Following Eq. (13), Stokes and anti-Stokes field are amplified exponentially with the same mode gain coefficient ␥ ⫽ ␥ A ⫽ ␥ S . Thus the amplitude ratio of both fields ˆ * /A ˆ is constant and independent of z. Equation R⫽A S A (14) constitutes the phase-matching condition for the longitudinal components of the wave vectors in the paraxial approximation. In the experiment the propagation constants and paraxial offsets of the input Stokes and pump beam are known. Thus we can calculate propagation constant ␤ A of the generated anti-Stokes beam through Eqs. (10) and (14). Condition (15) states the conservation of the total angular momentum in the four-wave mixing process. This condition originates from the last terms of Eqs. (8) and (9), relating the complex amplitudes of the pump, Stokes, and anti-Stokes fields. For the first case, a plane pump wave (l P ⫽ 0) and a J 1 Stokes beam with l S ⫽ ⫹1, an anti-Stokes beam with angular momentum l A ⫽ ⫺1 is obtained; i.e., in the scattering process, pairs of photons at ␻ S and ␻ A with opposite angular momentum are generated. In the second case, that of a J 1 pump beam with l P ⫽ ⫹1 and a plane Stokes wave (l S ⫽ 0), we expect an angular momentum l A ⫽ ⫹2 for the anti-Stokes beam, representing the transfer of the angular momentum of two photons at ␻ P to one photon at ␻ A . The angular momentum is independent of the frequency, and thus the total angular momentum is conserved.

4. RADIAL DISTRIBUTION OF STOKES AND ANTI-STOKES FIELDS For a plane pump beam with ␤ P p ⫽ 0, l P ⫽ 0, and u P ⫽ 1 it is possible to calculate the radial field distribution of the propagation-invariant Stokes and anti-Stokes modes analytically. With a proper choice of origin of the ˆ 兩 2 ⫽ 兩 A 兩 2 ⫽ A 2 , we define plane-wave gain z axis, 兩 A P P P ˆ 兩 2 and insert Bessel functions coefficient ␥ i pw ⫽ 1/2 ˜g i 兩 A P u i (r) ⫽ J l i ( ␣ i r), with i ⫽ S, A, as trial solutions into Eq. (12) and the equivalent equation for the Stokes field. Using l A ⫽ ⫺l S ⫽ l from condition (15) and the general relation 1 ⳵ r ⳵r



r

⳵ ⳵r

册 冉

J l 共 ␣ r 兲 ⫽ ⫺␣ 2 ⫹

l2 r2



J l共 ␣ r 兲 ,

we get

␥ ⫹ i ␤ Sp ⫹ i

␣ S2c 2n S ␻ S

⫺ ␥ S pw ⫽ ⫹␥ S pw

1 J l共 ␣ Ar 兲 R * J ⫺l 共 ␣ S r 兲

, (16)

(18)

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Stokes field has approximately the radial shape of the squared pump field intensity, leading to a radial intensity distribution that is proportional to J 1 4 ( ␣ A r).

5. EXPERIMENT Experiments with Raman-resonant four-wave mixing were performed with a frequency-doubled, injectionseeded, Q-switched Nd:YAG laser that produced 12-nslong pulses at 532 nm with a repetition rate of 10 Hz. We studied two schemes for the generation of anti-Stokes light by four-wave mixing ␻ A ⫽ 2 ␻ P ⫺ ␻ S by pumping the Raman medium with a first-order Bessel (J 1 ) beam at Stokes frequency ␻ S and a Gaussian beam at pump frequency ␻ P or with a J 1 beam at ␻ P and a Gaussian beam at ␻ S . The optical setup is shown schematically in Fig. 3. A fraction of the linearly polarized laser beam at 532 nm was focused with an f ⫽ 1-m lens, L1 , in a 155-cm-long Raman cell, RC1 , filled with hydrogen gas. The Stokes beam (683 nm) of backward stimulated Raman scattering from cell RC1 and a fraction of the laser beam were used to generate an anti-Stokes beam in a 60-cm-long second Raman cell, RC2 , by Raman-resonant parametric fourwave mixing. Each of the two beams at ␻ P and ␻ S was expanded and collimated by a telescope (not shown in Fig. 3) to 10-mm diameter and 0.2–0.4-mrad divergence angle. The intensities of the two beams in the second cell were kept below the threshold of stimulated Raman scattering. The Bessel beam was produced either at the Stokes frequency (first scheme) or at the pump frequency (second scheme) by diffractive axicon G placed in the Stokes or the pump channel, respectively. The cone angle of the Bessel beam produced by an axicon with a grating period of ␳ ⫽ 150 ␮ m was ␽ P ⫽ 3.5 mrad at the pump frequency and ␽ S ⫽ 4.6 mrad at the Stokes frequency. By variation of gas pressure near p ⫽ 1.2 ⫻ 106 Pa, which was kept nearly equal in both Raman cells, we were able to study the generation of the anti-Stokes wave close to phase matching for the four-wave mixing process. With typical intensities of pump and Stokes beams of 50–100 MW/cm2 and 5–20 MW/cm2, respectively, in cell RC2 , the

Fig. 3. Optical setup for the generation of anti-Stokes emission by means of Raman-resonant four-wave mixing: G’s, diffractive axicons; F1 , F2 , colored glass filters to select Stokes and antiStokes light, respectively; L1 , focusing lens; L2 , L3 , expanding telescope; RC1 , RC2 , Raman cells. Inset, schematic illustration of the position of the generated anti-Stokes beam at the output window of RC2 .

Fig. 4. Properties of an anti-Stokes beam generated with a J 1 Stokes beam at 683 nm and a Gaussian pump beam at 532 nm: a, k-vector diagram; b, far-field pattern of pump, Stokes, and anti-Stokes beams; c, near-field pattern of Stokes and antiStokes beams; d, interference pattern of the anti-Stokes beam.

energy conversion efficiency from the pump to the antiStokes beam was of the order of 0.1%. Forward anti-Stokes emission (436 nm) from the first Raman cell, RC1 , was used as a reference beam for the interferometric study of the phase structure of the antiStokes beam generated in the second cell, RC2 . A fraction of the anti-Stokes beam from RC1 selected by an aperture was expanded and collimated by a telescope consisting of lenses L2 and L3 . The two beams at 436 nm were combined by a beam splitter to propagate collinearly and produced an interference pattern, which was recorded by a CCD camera. The far field pattern of the radiation at the output of RC2 was recorded at the focal plane of an f ⫽ 500-mm lens. We recorded the near-field pattern of the antiStokes beam generated in RC2 by projecting the intensity distribution at the output window of the cell to the camera. First we discuss the experimental results obtained with a Stokes J 1 beam at 683 nm and a Gaussian pump beam at 532 nm. The wave vector diagram appears in Fig. 4a. Two collinear on-axis pump photons with the resultant wave vector 2kP and a Stokes photon with kS on the surface of the cone with angle ␽ S of the Stokes Bessel beam generate an anti-Stokes photon with kA on the surface of the cone with angle ␽ A . The far-field intensity distribution of the output of RC2 (Fig. 4b) shows the central spot (overexposed) of the Gaussian pump beam, the outer ring of the input J 1 Stokes beam with cone angle ␽ S ⫽ (4.6 ⫾ 0.2) mrad, and the inner ring of the generated antiStokes beam with cone angle ␽ A ⫽ (2.9 ⫾ 0.2) mrad. The ratio of cone angles is thus ␽ A / ␽ S ⫽ 0.63, in good agreement with the expected value of ␽ A / ␽ S ⫽ ␭ A /␭ S ⫽ 0.64. A pattern of the near-field intensity distribution of Stokes and anti-Stokes beams is shown in Fig. 4c. It can be seen that the anti-Stokes beam has a dark spot in

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the center and a characteristic ring structure identical to that of the J 1 Stokes beam; that is, it is also a Bessel beam, and the transverse components of k vectors of Stokes and anti-Stokes beams are equal: ␣ A ⫽ ␣ S [Eq. (18)]. The phase structure of the anti-Stokes beam from RC2 displayed by interference with the reference beam is shown in Fig. 4d. The interference pattern is recorded at the end of the propagation range of the anti-Stokes beam with the camera placed ⬃15 cm away from the output window of the Raman cell. The inset in Fig. 3 illustrates schematically that the anti-Stokes beam has its diverging part at the ouput window of the cell, and only a fraction (typically 10–15 cm) of the beam range close to its end is outside the cell. It is important to note that the handedness of the spiral pattern of the anti-Stokes beam recorded at the end of the beam propagation range is opposite that of the Stokes beam (not shown in Fig. 4) recorded under the same conditions. The spiral pattern of the Stokes beam is similar to that shown in Fig. 2b. We thus conclude that anti-Stokes emission generated with this scheme is a first-order Bessel beam with reversed charge of the phase singularity with respect to the J 1 Stokes beam, as is expected from Eq. (15). Now we discuss the properties of the anti-Stokes beam generated with the second pumping scheme. With this pumping geometry two types of phase matching exist: the usual planar phase matching involving two collinear photons at the pump frequency, and nonplanar phase matching involving two noncollinear pump photons with wave vectors on the surface of the cone of the Bessel pump beam.18 By choosing suitable conditions in our experiment ( ␽ P ⫽ 3.5 mrad and p ⫽ 1.2 ⫻ 106 Pa) we minimize the contribution of nonplanar phase matching and get planar interaction. Two collinear pump photons with the resultant wave vector 2kP on the surface of the cone of the Bessel pump beam and an on-axis Stokes photon with kS generate an anti-Stokes photon with kA on the surface of the cone with angle ␽ A (Fig. 5a). The far-field pattern in Fig. 5b shows the central spot of the Gaussian Stokes beam, the inner ring of the input J 1 pump beam with cone angle ␽ P ⫽ (3.5 ⫾ 0.2) mrad, and the outer ring of the generated anti-Stokes beam with cone angle ␽ A ⫽ (5.9 ⫾ 0.2) mrad. The ratio of cone angles is thus ␽ A / ␽ P ⫽ 1.68, in good agreement with the expected value of 1.7. The near-field intensity distribution of the anti-Stokes beam at the output window of the cell (Fig. 5c) has a ring structure, the intensity of which decreases rapidly with radius, and a zero-intensity minimum in the center. Although the anti-Stokes beam generated with this pumping scheme has a simple far-field pattern in the form of single sharp ring [shown in Fig. 5(b)], it is no longer a Bessel beam. Its near-field radial intensity distribution observed in the experiment is in a qualitative agreement with the J 1 4 ( ␣ A r) profile expected from the theoretical model in the limit of small gain (see Section 4). The phase structure of the anti-Stokes beam shows a double-start spiral (Fig. 5d) with the same handedness as the single-start spiral of the pump Bessel beam at ␻ P , which is similar to that shown in Fig. 2b. We thus conclude that the anti-Stokes light is a conical beam with

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Fig. 5. Properties of an anti-Stokes beam generated with a J 1 pump beam at 532 nm and a Gaussian Stokes beam at 683 nm: a, k-vector diagram; b, far-field pattern of pump, Stokes, and anti-Stokes beams; c, d, near field pattern and interference pattern, respectively, of the anti-Stokes beam.

phase singularity of charge l ⫽ 2, in agreement with the conservation of angular momentum [Eq. (15)]. In the field of the Stokes beam carrying zero angular momentum, two photons at ␻ P with total angular momentum 2ប are transformed into Stokes and anti-Stokes photons with angular momenta of 0 and 2ប, respectively.

6. CONCLUSIONS It has been demonstrated that the phase singularity of an apertured lth-order Bessel beam can be detected by interference with a reference plane wave at the boundaries of the beam propagation range, where a simple spiral-type interference pattern gives both the value and the sign of the singularity. The phase singularity may also be detected by interferometry within the beam propagation range. In this case, however, the interference pattern must be recorded for at least two different propagation distances. The Raman-resonant four-wave mixing process ␻ A ⫽ 2 ␻ P ⫺ ␻ S involving a J 1 Bessel beam with a helical wave front either at the pump ␻ P or at the Stokes ␻ S frequency was studied. In both cases propagation-invariant modes for the Stokes and anti-Stokes fields with positive gain exist. For a plane pump beam it was shown analytically and experimentally that Stokes and anti-Stokes modes are Bessel beams with identical transverse components of the wave vectors. For a J 1 Bessel pump beam, numerical integration of the paraxial wave equation gives radial field distributions that are in good agreement with experiment. It has been demonstrated that the angular momenta of the input beams associated with the helical structure of their wave fronts are transferred to the generated anti-Stokes beam. The observed reversal or dou-

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bling of the charge of the phase singularity in the antiStokes beam is consistent with the conservation of angular momentum.

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ACKNOWLEDGMENTS The authors thank L. Niggl for helpful discussions and gratefully acknowledge financial support by the Deutsche Forschungsgemeinschaft. U. T. Schwarz’s e-mail @physik.uni-regensburg.de.

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