The results of a theoretical and experimental investigation of the Gouy effect in Bessel beams are presented. We point out that the peculiar feature of the Bessel beams of being nondiffracting is related to the accumulation of an extra axial phase shift (i.e., the Gouy phase shift) linearly dependent on the propagation distance. The constant spatial rate of variation of the Gouy phase shift is independent of the order of the Bessel beam, while it is a growing function of the transverse component of the angular spectrum wave-vectors, originated by the transverse confinement of the beam. A free-space Mach-Zehnder interferometer has been set-up for measuring the transverse intensity distribution of the interference between holographically-produced finite-aperture Bessel beams of order from zero up to three and a reference Gaussian beam, at a wavelength of 633 nm. The interference patterns have been registered for different propagation distances and show a spatial periodicity, in agreement with the expected period due to the linear increase of the Gouy phase shift of the realized Bessel beams.
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More than one hundred years ago Louis George Gouy discovered that a spherical converging waves experiences an abrupt phase change of π when it passes through the focal region . This phase anomaly explains the imaginary factor i in the Huygens’ integral, corresponding to a phase shift of π/2 acquired by the secondary wavelets diverging from each point of the primary incident wave-front . In general the Gouy effect manifests itself as an additional axial phase shift acquired by any diffracting optical beam (hence characterized by a transverse spatial confinement) with respect to a reference on-axis plane wave. It was been also recognized that the Gouy effect can be seen as another example of topological or Berry’s phase [3–5].
It is well known that Hermite- and Laguerre-Gaussian beams, which are the modes of curved mirrors resonators with respective indexes (m,n) and (p,l), show a Gouy phase shift equal to and , respectively, where z R is the Rayleigh range [2,6,7]. This axial phase shift adds to the usual phase term −kz, corresponding to an on-axis plane wave, and therefore the Gouy effect plays a fundamental role in determining the resonant frequencies of the laser cavities [6,8]. For the fundamental Gaussian mode the accumulated additional phase shift propagating from −∞ to + ∞ is π, and the maximum variation rate is at the beam waist. For higher-order modes the phase shift is higher, due to the moltiplicative factor depending on the mode indexes. Explicit measurements of the Gouy phase shift for Hermite- and Laguerre-Gaussian beams have been only recently reported [9,10], confirming the theoretical previsions.
In this article we investigate the Gouy effect in nondiffracting Bessel beams. These beams were recognized by Durnin  as solutions of the scalar wave equation having a transverse intensity distribution independent of the propagation distance. Actually the Durnin’s solutions are exact in infinite free-space, but they are not square integrable (i.e., the energy is infinite) and thus are not physically realizable. However, finite-aperture approximated Bessel beams was experimentally generated, possessing a very small diffraction spreading over significant distances [12–15]. This remarkable feature is attractive in applications requiring a large depth of field, such as long-range optical alignement, optical interconnections, optical manipulation and guiding of microscopic particles [16–18].
The Bessel beams can be seen as a set of free-space propagation modes, hence characterized by a propagation constant β. These beams, although nondiffracting, present a transverse spatial confinement inducing a reduction of β, which is just coincident to the axial component of the angular spectrum wave-vectors, with respect to k. The Gouy extra axial phase shift is then and hence it is directly proportional to the propagation distance z. We report here the first experimental verification of this linear z-dependence of the Gouy phase shift in Bessel beams up to third order, by using an interferometric technique.
A monochromatic free-space scalar optical field, which is an eigenfunction of the translation operator along the propagation axis z, obeys the following general expression7]Eq. (1) represents free-space propagation modes. In this condition the action of the propagation along z is simply the multiplication of the field by the phase term , where β represents the propagation constant of the mode, therefore the intensity is not dependent on z.
A well-known example of free-space propagation modes is given by the monochromatic plane wavesEq. (2). The plane waves are infinite-energy nondiffracting beams with constant intensity. In this case the reduction of β with respect to k is simply due to the inclination of the wave-vector k with respect to the axis z. A change of reference framework, which aligns k to the axis z, makes β coincident with k.
In the following we point out that the Bessel beams are another set of nondiffracting free-space propagation modes, which show a transverse spatial confinement differently from plane waves. It is suitable to express the spatial dependence on the scalar field of Eq. (1) in cylindrical coordinates:Eq. (2), i.e. the transverse Helmoltz equation, becomes
This equation has rotational symmetry and we can search for modes that are eigenfunction of the rotation operator around the axis z, hence with a transverse field described byEq. (7) in Eq. (6) yields the following differential equation for the radial part of the field:19,20]. The continuous and bounded solutions of Eq. (8), for any , are , that is the Bessel functions of first kind of order l with argument , where
Therefore the Bessel beams are described by the scalar field
These beams propagate along z without any diffraction, although they present a transverse spatial confinement, which is tighter as α increases. For instance, the radius of the central spot of the zero-order Bessel beam is .
Now we can determine the Gouy phase shift for the Bessel beams, by evaluating the extra axial phase shift with respect to a reference plane wave propagating along z. We obtain from Eqs. (9) and (10), by means of straigthforward algebra:
The Gouy phase shift for the nondiffracting Bessel beams accumulates proportionally to the propagation distance z and it is not explicitly dependent on the order of the beam, differently from diffracting Hermite- and Laguerre-Gaussian beams. Moreover the Gouy shift is a growing function of the parameter α, which measures the spatial confinement.
A meaningful explanation of the physical origin of the Gouy effect is given in . The transverse spatial confinement, through the uncertainty principle, causes a transverse spreading of the angular spectrum wave-vectors and a reduction of the average axial component with respect to k, hence giving the Gouy phase shift. The Bessel beams can be obtained as superposition of plane waves with wave-vectors describing a cone around the axis z [22,23], as depicted in Fig. 1 . All these vectors have the same axial component and also the same transverse componentEq. (9). In conclusion the parameter α, ruling the transverse field shaping and consequently the Gouy phase shift, is exactly the transverse component of the wave-vectors generating the Bessel beams.
3. Holographic generation of finite-aperture Bessel beams
The ideal Bessel beams have infinite energy, since the square modulus of the transverse field, that is , is not integrable over a plane orthogonal to the propagation axis. In our experiments we have realized approximated l th-order Bessel beams with finite transverse extent, by using suitable computer generated holograms (CGHs).Eq. (10).
For simplifiying the realization of the hologram, it is possible to obtain good reconstructions also by means of binary-amplitude coding, since the main effect of the binarization is the redistribution of the energy among the various hologram diffraction orders. In this case is replaced by the binarized transmission function
In this work we have fabricated four binary-amplitude off-axis CGHs with transmission functions , shown in Fig. 2 , to reproduce Bessel beams of order l respectively from 0 to 3. In particular, each CGH has been realized by plotting the corresponding bidimensional transmission pattern with a laser printer and photoreducing it on Ilford Pan film. All these holograms have the same parameters mm−1, mm and mm.
Figure 3 reports some measured intensity distributions of the finite-aperture Bessel beams obtained by illuminating the binary CGHs with an expanded He-Ne laser beam (wavelength nm). The distributions have been acquired by a CCD camera collecting the first diffraction order at a distance of 60 cm from each hologram. This distance is sufficient for spatially separating the desired first diffraction order from the order zero and is less than the maximum nondiffracting distance , which in our experimental conditions is about 2 m.
4. Experimental verification of the Gouy phase shift in Bessel beams
For observing the Gouy phase shift of the Bessel beams, we have set-up the free-space Mach-Zehnder interferometer schematically shown in Fig. 4 and we have acquired the interference patterns between the holographically-produced l th-order Bessel beams, with l varying from zero to three, and a reference Gaussian beam. More in detail, a Gaussian beam with wavelength of 633 nm and spot size of about 0.3 mm is emitted by a He-Ne laser source and then split by a 50/50 cube beam splitter (BS) in two beams propagating respectively in the two different arms of the interferometer. The beam in the first arm is just attenuated, to equalize the optical powers of the interfering beams, and remains Gaussian. The beam in the second arm is expanded of a factor 8 by a two-lens system and goes through the proper CGH for producing the required l th-order Bessel beam. The unwanted diffraction order of the CGH is blocked by an absorber, while the desired order (that is the Bessel beam) is aligned so as to interfere, after the output 50/50 BS, with the reference Gaussian beam of the other arm. Finally the transverse intensity distribution of the interference between the Bessel and Gauss beams has been acquired by a CCD camera, placed over a motorized stage movable along the propagation direction z.
In the following we give an analytical expression of the expected interference intensity as a function of the cylindrical coordinates . The finite-aperture l th-order Bessel beam reproduced by the CGH is an approximation of the ideal nondiffracting beam characterized by the fieldEqs. (10) and (14). The propagation constant β, from Eq. (9), is given by
Since and hence , we can express β as
In this case the Gouy phase shift of Eq. (11) can be approximated byEq. (17), considering the expression of β given by Eq. (19), and the Gaussian beam of Eq. (21) is then
This phase is the sum of five contributions. The first two contributions are respectively the angular phase term and the Gouy phase shift of the Bessel beam. The next two contributions are respectively the Gouy phase shift and the wave-front curvature term of the Gaussian beam, both changed of sign. The last contribution is a constant phase offset. The variation of the transverse intensity distribution of the interference as a function of the propagation distance z is mainly due to the Gouy phase shift of the Bessel beam, as we will verify later.
Considering an ideal plane wave as reference beam, that is neglecting the two contributions to ζ related to the Gaussian beam in Eq. (23), the phase difference ζ depends linearly on z and hence gives a periodicity of the interference intensity pattern in the z-propagation. The spatial period Λ is equal to 2π divided by the spatial variation rate of the Gouy phase shift, that is, according to Eq. (20):
This period is independent of the order l of the Bessel beam, but depends just on the wavelength λ and on the transverse component α of the angular spectrum wave-vectors, which is simply related to the size parameter by Eq. (14). In our experimental conditions mm and nm, hence the expected value of Λ is 19.75 cm.
Actually the reference beam used in the interferometric measure is not a plane wave, but a Gaussian beam, which is diffracting and perturbs the periodicity of the interference pattern. For evaluating the entity of such a perturbation we calculate the variation of ζ due to the two Gaussian phase contributions in Eq. (23), for a variation of the propagation distance z equal to Λ. The initial distance is m, corresponding to the length of the propagation path of the reference Gaussian beam between its waist (with spot size mm) and the initial position of the CCD camera. At the final distance the variation of the Gouy phase shift of the Gaussian beam is 2.1°, yielding a relative error of 0.58% in the evaluation of ζ, since the corresponding variation of the Gouy phase shift of the Bessel beam is 360°. Moreover, the contribution to ζ due to the wave-front curvature of the Gaussian beam is dependent on the radial coordinate ρ. If we evaluate the perturbation of induced by the two Gaussian phase contributions, in the radial position of the intensity peak of the Bessel beams with order l from 0 to 3 (resulting in , 0.293, 0.486 and 0.669, respectively), then we obtain an overall relative error of 0.58%, 0.60%, 0.66% and 0.73%, respectively. So we have shown that the expected perturbation of the ideally periodical interference pattern, due to the diffracting Gaussian reference beam, is very small in the considered experimental conditions.
In Fig. 5 we report the transverse intensity distributions of the interference between the l th-order Bessel beams and the reference Gaussian beam, calculated at different propagation distances by using Eqs. (22) and (23), considering the actual experimental values of the parameters. The interference patterns shown in the first column are calculated at the initial position . Then the position is increased by a quantity d, obtaining the patterns shown in the second and third column, respectively after the increments and . We can see that the patterns remains practically unchanged after the ideally calculated period Λ for all Bessel beams considered, having orders from 0 to 3.
In the case of zero-order Bessel beam (i.e., the first raw of Fig. 5) the interference patterns are circularly symmetric and show a series of bright concentric rings alternated by dark concentric rings. After a semi-period Λ/2 the Gouy phase shift of the Bessel beam increments of π and consequently the bright and dark rings of the interference pattern mutually exchange. On the other hand, after a period Λ the Gouy phase shift increment is 2π and the resulting interference pattern is practically identical to the initial pattern.
In the case of higher-order Bessel beams the interference patterns are invariant for rotations being integer multiples of , due to the angular phase term in Eq. (23). The intensity maxima are located on a series of concentric rings: each one of these rings contains exactly l maxima, which are equally spaced and alternated by l minima. The Gouy phase shift of the l th-order Bessel beams varies in the propagation along z with constant rate, causing a uniform rotation of the intensity with angular rate . After a semi-period Λ/2 the interference pattern rotates of and the intensity maxima and minima mutually exchange, while after a period Λ the pattern rotates of and returns coincident to the initial pattern, apart from a slight perturbation of the pattern periodicity due to the Gaussian reference beam. The only relevant effect of employing a Gaussian reference beam is the growing shift of the angular position of the maxima (minima) in the outer rings, giving spiral-like interference patterns. In fact, considering an ideal plane wave as reference beam, the maxima and the minima should be alternated in radial direction, since the consecutive intensity maxima rings of the Bessel beams have a phase difference of π. Actually the phase term due to the wave-front curvature of the Gaussian beam gives a contribution to ζ proportional to the square of the radial distance and consequently the maxima (minima) of the interference pattern acquires an additional angular shift proportional to ρ 2.
We point out that the interference patterns of Fig. 5 have been calculated considering ideal Bessel beams. Actually in the experiments we have used finite-aperture Bessel beams obtained by means of CGHs. However, the Bessel beams propagate from the respective CGHs to the CCD camera for a maximum distance of about 1.1 m, at the final position , that is less than the maximum nondiffracting length . Therefore the CGH-realized Bessel beams are a good approximation of the ideal Bessel beams whose field is described by Eq. (17).
The interference patterns measured at different distances, incrementing the initial position z 0 of the CCD camera by the quantity d, confirms the expected periodical behavior due to the linear accumulation of the Gouy phase shift of the CGH-produced Bessel beams. As shown in Fig. 6 , the registered patterns are in good agreement with the simulated ones, apart from a phase offset, and recur after Λ. In particular, the measured pattern appears circularly symmetric and pulsates cyclically for , inverting maxima with minima after Λ/2, while it shows l bright spots in the inner ring and rotates uniformly for l from 1 to 3, as z increases.
We have investigated both theoretically and experimentally the Gouy effect in Bessel. Such beams possess the very remarkable feature of being nondiffracting: in this case the Gouy effect manifests itself as a reduction of the propagation constant β with respect to the length k of the wave-vector of a plane wave with same direction and wavelength. Therefore the Gouy extra axial phase shift accumulates linearly as the propagation distance increases, with a constant rate equal to the difference between k and β. This rate is independent of the order of the Bessel beam and depends only on the wavelength and on the transverse component α of the angular spectrum wave-vectors, which describe a cone around the propagation axis. In particular the Gouy phase shift is a growing function of α, according to the physical interpretation of the Gouy effect as originated by the transverse confinement of the beam. In fact the angular spectrum spreading, induced by the spatial confinement, causes an increase of α and hence a reduction of β, which is exactly the axial component of the angular spectrum wave-vectors.
For obtaining an experimental characterization of the Gouy effect in Bessel beams, we have realized finite-aperture Bessel beams of order from zero up to three, by means of off-axis binary-amplitude CGHs. We have measured, at different propagation distances, the transverse intensity distribution of the interference between each CGH-produced Bessel beam and a reference Gaussian beam. The measured interference patterns are in good agreement with the expected patterns. The linear dependence of the Gouy phase shift on the propagation distance, which is a remarkable property of the nondiffracting Bessel beams (differently from diffracting beams, as the well known Hermite- and Laguerre-Gauss beams), implies that the calculated interference patterns must have a spatial periodicity depending on the wavelength and on the parameter α. Our experiments show that the patterns actually recur after the expected period, therefore providing a direct observation of the Gouy effect in nondiffracting Bessel beams.
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