Holographic generation and orbital angular momentum of high ...

SChávez-Cerda et alA 1Hf 1L 1A 2-2-1f 20L 2+1f 1+2f 2H′zCFigure 3. Schematic of the experiment for the holographicgeneration of higher-order Mathieu beams. The phase hologram His illuminated by a beam truncated by the circular aperture A 1 .Inthe focal plane of lens L 1 the circular aperture A 2 picks out thebright ring of the first diffraction order, which is then turned into adiffraction-free beam by the lens L 2 . The CCD array C allows theintensity cross sections of the resulting beam to be recorded atvarious distances z behind the image of the hologram, H ′ . The focallengths used in our experiment were f 1 = 600 mm andf 2 = 250 mm.(a)(b)-2 -1 0 +1 +2Figure 4. Experimentally recorded intensity cross sections in thefocal plane of a lens behind Mathieu-beam phase holograms undertop-hat illumination for k r = 3 × 10 4 m −1 (recorded behind m = 5and q = 7 hologram, see figure 2) (a) andfork r = 4 × 10 4 m −1(recorded behind m = 5andq = 20 hologram) (b). The diffractionorders are identified at the bottom.z max =rtan θ . (5) allowing the zeroth-order diffracted beam to pass through theaperture A 2 . Comparison with figure 2 confirms that the beamsFigure 5. Intensity cross sections of different experimentallygenerated higher-order Mathieu beams at different distances zbehind the plane H ′ (see figure 3). The three different beamscorrespond to the three holograms shown in figure 2. The arearepresented by each image is approximately 0.5 mm by 0.4 mm.Here r is the radius of the beam in the plane of the aperture (thebase of the cone) and θ is the inclination angle of the beam’splane-wave components with respect to the optical axis. Therelevant aperture in our experiment is the image of A 1 in theplane H ′ , which has a radius of r ≈ 1.6 mm (the radius of A 1is about 3.75 mm, and its image in the plane H ′ is magnifiedby a factor of M =−f 2 /f 1 ≈−0.42). The angle θ can becalculated from the equation tan θ ≈ sin θ = k r /k, whichinour experiment evaluated to θ ≈ 0.55 ◦ and θ ≈ 0.42 ◦ forholograms with k r = 4 × 10 4 m −1 and k r = 3 × 10 4 m −1 ,respectively. (Note that imaging the beam from the plane Hinto the plane H ′ with magnification M changes k r by afactor 1/|M|.) These values lead to respective diffractionfreedistances of r max ≈ 162 mm (for θ ≈ 0.55 ◦ )andr max ≈ 215 mm (θ ≈ 0.42 ◦ ), which are in good agreementwith the distances after which the centre of the beams in ourexperiment became much fainter (figure 5).Figure 6 shows interference patterns between the Mathieubeams and an inclined plane wave, illustrating the phasestructure of the beams. The plane wave was obtained by alsoperfect circle. As this plane also happens to be the front focalplane of lens L 2 , our set-up is essentially of the type shownin figure 1; the first part of the set-up, including illumination,hologram, lens L 1 , and aperture A 2 , is therefore simply a wayof creating, in the front focal plane of lens L 2 , a ring-shapedlight field with the azimuthal field dependence A(φ) that leadsto a Mathieu beam behind the lens.Figure 5 shows intensity cross sections through theMathieu beams produced with the holograms shown in figure 2.It can clearly be seen that the beams are diffraction free onlyover a finite distance, z max , the so-called propagation-invariantregion, behind which the centre of the beam rapidly becomesfainter. This is a well known effect due to the finite size ofthe beam. The slight rotation about the optical axis withinthe propagation-invariant region, which can also be seen infigure 5, has been predicted theoretically very recently [11]and is also due to the beam’s finite size. The distance z maxcan be estimated geometrically to be the length of the conein which all the plane-wave components in the beam intersect(see figure 1), which is given by [1]S54