Due to their immunity to diffraction, Bessel light modes potentially offer advantages in various applications. However, they do exhibit significant intensity variations along their axial propagation length which hampers their applicability. In this paper we present a technique to generate Bessel beams with a tunable axial intensity within the accessible range of spatial frequencies. The beam may be engineered to have a constant intensity along its propagation length. Finally, we demonstrate how one can form a Bessel beam with a varying propagation constant along its axial extent which results in a tunable scaling of its lateral cross-section.
© 2009 Optical Society of America
The Bessel beam of the zeroth order is representative of the special class of the “non-diffractive” optical fields that were introduced by Durnin et al. [1, 2]. These beams are special solutions of the Helmholtz equation. Their intensity profile remains unchanged whilst the beam propagates in free space. As the name implies, the lateral cross-section follows a Bessel function of the zeroth order. Bessel beams as well as the “non-difrracting” Mathieu beams propagate in the free space along a straight axis, however, other types of “non-diffractive” fields are known namely Airy or parabolic beams [3, 4, 5] that propagate along a curved trajectory. The zeroth-order Bessel beam has a lateral profile which features a high intensity central core surrounded by numerous concentric rings that in case of the ideal Bessel beam would extend indefinitely in the radial direction. This would dictate the total energy of the beam to be equal to infinity and therefore such idealisations of the beam cannot exist. Nevertheless, a spatially limited approximation known as a quasi-Bessel beam can be implemented. Such light modes have found increasing utilizations in a variety of multi-disciplinary applications including optical trapping [6, 7], optical binding , cell transfection , optical coherence tomography , nonlinear physics  and imaging . Such beams are attractive not only because of their immunity to diffraction but as they also possess another remarkable feature, the ability to reconstruct themselves after encountering a disturbing obstacle [7, 13].
The spatial spectrum (k-space) of an ideal Bessel beam lies on a ring which is a single off-axis delta function. The simplest way to generate such a beam is to expose an annular aperture placed in the back focal plane of a lens to a collimated input beam that then generates the Bessel mode in the Fourier plane , see Fig 1. This approach, however, is very inefficient so the more popular choice is the use of a conical lens, termed an axicon. As will be shown in the next section both of these options form a Bessel beam with a non-uniform axial intensity, peaking at a certain point along the optical axis. Furthermore, it was also shown by Brzobohaty et al. that any real experimentally achievable Bessel beam field generated by an axicon usually results in undesired axial intensity oscillations due to the imperfect, oblate axicon tip . Here the unwanted oscillations were eliminated by spatial filtration. Such a technique was adopted in a Bessel beam based biophotonics workstation  where the spatial filtration was provided by an adaptive optical element, a spatial light modulator (SLM). Nevertheless in many applications such as optical trapping, binding, imaging and coherence tomography it would be significantly advantageous if the axial intensity of the beam was near constant as presently one can only use a small fraction of the beam’s “diffraction-free” axial extent. In this paper we address this key issue. We show how one can engineer the axial intensity profile of a Bessel mode at will by direct generation of a pre-designed spatial spectrum on the SLM rather than the method of spatial fitration. In particular, we show how one can engineer Bessel beams with a highly uniform axial intensity as well as a variety of other interesting and useful forms of axial intensity variations for a quasi-Bessel beam. In addition we discuss and experimentally verify a technique that introduces variations in the Bessel beam lateral cross-section scale whilst the beam propagates: the central core of such beams shrinks or extends along the axis. The paper is organised as follows: we first describe the theoretical considerations based on a Fourier analysis for Bessel light modes. The subsequent section describes the typical axial intensity variations of these modes generated using either an annulus or axicon. We then progress to describing how we engineer the axial intensity by a modified Gechberg-Saxton (G-S) algorithm, show experimental demonstrations and then draw conclusions from our studies.
2. Azimuthally symmetric fields
Within a scalar and paraxial description, the two-dimensional Fourier transform of a field U(x,y) defined in one lateral plane can be used to decompose any optical field into its plane wave components. Each plane wave component is defined by a wave-vector k=[kx,ky,kz] in the orthogonal system of coordinates [x,y,z] and if we assign a zero value to the axial position of the selected plane we can express this decomposition as:
This plane-wave representation is known as the spectrum of spatial frequencies, k-space or spatial spectrum. For azimuthally independent optical fields such as the Fourier transform can be written in the form of a zero-order Hankel transform:
where and J 0 is the zero-order Bessel function. The inverse transform to (2) is given by:
This has a very interesting physical interpretation: any azimuthally independent field may be considered as a composition of ideal Bessel beams of the zeroth order. Each of these ideal Bessel beam components then propagates in a free space as: , where kz is the axial wave-vector component: usually termed as the propagation constant. For non-evanescent modes the kz wave vector component lies in the interval 〈0,k〉. Each of these Bessel beam components then has a specific spatial frequency along the optical axis which contribute to the resulting on-axis field which as we know is a consequence of the interference of the individual modes. Using the inverse transform (3) we obtain for the on-axis field the following:
This formula shows that for any azimuthally independent field its on-axis evolution is linked with its spatial spectrum by a one-dimensional Fourier transform. In turn this gives us the insight and ability to generate an optical field with an arbitrary on-axis field dependence within the range of possible spatial frequencies kz∊〈0,k〉.
3. Axial intensity of frequently used quasi Bessel beams
In this section we discuss the axial intensity evolution of the two most common methods for generating quasi-Bessel beams: the Bessel beam generated by an annular aperture and an axicon generated Bessel beam.
3.1. Quasi-Bessel beam generated by an annular aperture
A Bessel beam generated by an annular aperture placed in the back focal plane of a lens is a clear demonstration of the above described principles. Since such a Bessel beam is in fact an isolated Bessel beam mode from the Gaussian beam that would appear behind the lens without the annular aperture present. Hard aperturing of the collimated beam in the back focal lens is identical with a binary amplitude filtration of the spatial spectrum since the fields in the back and the front focal planes are related by a Fourier transform. Using equation (4) one can show that the on-axis field is given by:
where a and b are given by:
R is the radius of the annular aperture, d is the annulus thickness, f is the focal length of the lens as shown in Fig. 1.
3.2. Axicon-generated Bessel beam
The axicon-generated beam is probably the most common representation of the quasi-Bessel beams. It is formed behind an axicon illuminated by a Gaussian beam as shown in Fig. 2. It was shown by Jarutis et al.  that its on-axis field can be approximately described by:
where w is the half-width of the Gaussian beam and α is the angle under which the light is refracted by the axicon. This, however, is the case where an optimal axicon is used. In reality axicons suffer a number of imperfections, with the oblate tip being the most important one. Such an imperfection causes a strong on-axis oscillations that destroys the overall smooth profile of the propagating Bessel beam . An example of such a field is shown in Fig. 3.
4. Shaping of the axial intensity profile of a quasi-Bessel beam
In the previous section we saw that for the commonly generated quasi-Bessel beams the on-axis axial intensity features a peak-like dependence. One may thus operate only in a very limited axial region where the intensity might be considered as uniform unless this undesirable property is taken into account. In this section we show how to engineer several types of the quasi Bessel beams each with a pre-shaped axial intensity profile along the propagation length, primarily one with its axial intensity uniform along almost the whole range of the beam existence. In addition we show how we may achieve tailored size variations of the quasi-Bessel beam lateral cross-section (central core size) along the optical axis.
The spatial spectrum for any quasi-Bessel beam forms a ring with a thickness inversely proportional to the beam axial extent. The ideal beam spectrum is then an off-axis delta function kr=δ(k r0). Let us define a quasi-Bessel beam with an arbitary axial envelope A(z) defined on the optical axis as:
where . Since the spatial spectrum is related with the on-axis field by the Fourier transform (4), we can use this relation to obtain the spatial spectrum of our model Bessel beam (7):
In the general case this spectrum will be defined by spatial frequencies outside the range of accessible values kz∊〈0,k〉. This limits the possibility of the axial beam shaping and so we have to circumvent these spectral values in the optical field design.
4.1. Bessel beam with an uniform axial intensity
There have been several attempts to address the non-uniformity of the quasi-Bessel beams in the past decade. Since illuminating the axicon by a plane wave brings linearly growing axial intensity profile, Honkanen and Turnen suggested a technique how to compensate for this by introducing a 1/r illumination of the axicon in their theoretical study . They achieved a significant improvement of the axial intensity uniformity but since this idea was based on a geometrical optics approach their wave-propagation simulations of the resulting field indicated some residual oscillations. This paragraph shows that all the axial oscillations can be suppressed in an unprecedented way. Let the arbitrary function A(z) from the relation (8) be a step function defined by:
The spatial spectrum can be for this case found in an analytical form:
An example of such a spatial spectrum is shown in Fig 4. This function is defined by the spatial frequencies covering the whole interval 〈-∞,∞〉, however, the spectrum peaks in the selected value of k r0. In reality we cannot even use the whole range of possible spatial frequencies kr∊〈0,k〉 because of the limited aperture size of the optics. Equation (4) however reveals which on-axis intensity profile one can reach using the spectrum with such a limited range of the spatial frequencies. Numerical evaluation of (4) for the spectrum presented in the Fig. 4 and for the range of the shown frequencies is seen in Fig. 5. Here we see some on-axis oscillations that are present because of the hard-aperturing of the spatial spectrum. To avoid this we can introduce a Gaussian amplitude envelope around the spatial spectrum centered at the value of k r0. The resulting on-axis intensity is then shown in Fig. 6. These types of fields have their spectrum concentrated in the vicinity of a ring with the radius of k r0 and have their on-axis intensity uniform over a limited range. From this perspective this would represent the best achievable approximation to the ideal Bessel light mode. It is important to say that the selection of the Gaussian envelope width directly influences the magnitude of the persisting oscillations. The narrower envelope used the smaller are the residual oscillations. The price for this is the reduction of the uniform intensity extent because of rounding the sharp corners of the selected step function (9). This way one has to trade-off these two influences for a particular application. The magnitude of the persisting oscillations gets always smaller towards the central part of the Bessel beam region (see Fig. 5). In our case presented in the Fig. 6 the magnitude of the oscillations in the central region covering 50% of the beam axial extent the ratio between the oscillation and the mean intensity does not exceed the level of 10-7. If this region is extended to 80% of the beam existence the ratio increases to the value of 10-4 and for 90% we still get less than 10-2. We believe that these values are more than satisfactory for the majority of applications but it is worth remarking that this evaluation was done for the spatial spectrum enveloped by the Gaussian function and that a different enveloping function might bring a future improvement.
4.2. Bessel beams with shaped on-axis intensity and the propagation constant
Similarly to the previous case we can adjust the model function A(z) to yield an arbitrary shape. In several cases the spatial spectrum can be found in an analytical form. For example, this includes the generation of a Bessel beam with either a uniformly increasing or decreasing on-axis intensity or even the axicon-generated Bessel beam with the on-axis intensity described by the approximative formula (6). For other on-axis axial intensity profiles one can find the spatial spectrum by numerical evaluation of the formula (8). For brevity we do not describe all of the possible cases but concentrate upon the example of a Bessel mode with an uniformly growing axial intensity. We present the relevant spatial spectrum in the Fig. 7 and the resulting intensity after the spectrum enveloped by the Gaussian function as in the previous sub-section (Fig. 8).
As mentioned earlier, such axial shaping is not restricted to the cases of a fixed propagation constant as the beam propagates. Variation in the propagation constant along the beam extent thus results in a modulation of the Bessel beam’s lateral profile (central core size). Changing the model function (7) to:
one can arbitrarily select both functions A(z) and k z0(z) and calculate the corresponding spatial spectrum evaluating the equation (8). Several examples of this will be shown in the next section together with their experimentally realized counterparts.
5. Experimental realization of the novel quasi-Bessel beam types
To generate the described axially tunable quasi-Bessel beams, one has to be able to shape both the amplitude and the phase of the field in a well controllable manner. For this purpose we used a computer-controlled diffractive optical element, a spatial light modulator (SLM, Hamamatsu LCOS X10468-04). The most direct manner of beam generation is to form the beam’s spatial spectrum directly on the SLM plane. If the SLM is uniformly illuminated (or illuminated by a large Gaussian mode overfilling the SLM chip) this approach has a very low power efficiency, but we shall describe how to overcome this drawback later.
5.1. The hologram design
There are a number of ways by which a complex modulation may be applied to a phase-only spatial light modulator. It is based on reducing the diffraction efficiency in the first diffractive order of light coming from the specific place on the SLM. Since our technique is extremely sensitive to the precision of the amplitude modulation we developed an iterative algorithm based on the Pasienski and DeMarco modification  of the well known G-S method . The standard G-S method iteratively calculates the field in the image plane from the field in the spectral plane and vice versa using two-dimensional Fourier transform. In each iteration the calculated phase is kept, but the intensity is replaced in the spectral plane with the given one and in the image plane with the required one. The modification of  relates to the step of replacing the desired field amplitude in the image plane. It is not the whole field amplitude what is replaced, rather than a spatially limited area of interest. Outside this area no amplitude modifications are made and a freedom is given to the algorithm. This then results in reassembling of the light outside the selected area by the most convenient way to match the required field amplitude inside the selected area with the desired pattern and it brings a significant improvement of the quality of the obtained structure comparing to the standard G-S algorithm. We have modified this technique by replacing not only the field amplitude but rather the complex field (both, phase and amplitude) in the selected area and we found that this algorithm converges quickly and generates the desired field giving us a much higher quality when compared to other methods. We also found that experimentally much better results are obtained if the initial phase at the spectral plane is set to a linear modulation that results in diffracting the light off the zero order in the image plane. A more detailed evaluation of this method will be a subject of our future studies. Figure 9 shows a hologram for the uniform intensity quasi-Bessel beam generation and a simulation of its image field obtained by this procedure.
5.2. Experimental procedure
To generate the designed quasi-Bessel modes we used laser light (Coherent, Verdi V5) operating at 532nm with an output power of 10mW. The output beam was expanded by a telescope to slightly overfill the SLM chip. The intensity overlay on the SLM chip was measured and used in the hologram design to achieve the best agreement between the simulations and the experiment. The diffracted light was then collected by a lens (f=400mm) placed at the distance of one focal length from the SLM. In the front focal plane we obtained the pseudo-Bessel beam that was separated from the unused light by a pinhole. The size of the pinhole was set so it corresponded to the restricted area in the hologram generating algorithm. This beam was then demagnified by a telescope consisting of f=250mm and f=50mm lenses to reduce the quasi-Bessel beam to the operating range of our motorized actuator (Newport CMA-12CC). The resulting field was imaged onto a CCD camera chip (Basler piA640-210gm) by a micro-scope objective (20x NA=0.4). The imaging system of the camera and the objective was placed on a positioning stage controlled by the motorized actuator. The quasi-Bessel beam intensity was then captured at 1200 axial positions along the axial distance of 10mm. The experimental results are summarized in the following section.
6. Experimental results
To compare the obtained results with the theoretical predictions, we analyzed each of the captured frames by the following procedure. We fitted the recorded transverse Bessel beam intensity profile with the cross-section of the ideal Bessel beam to find its central position on the SLM chip. The beam was than azimuthally averaged to its 1-dimensional lateral cross-section. This could not be applied for the parts of the record outside the axial range of the Bessel beam existence (where no high intensity central core is formed), so we linearly extrapolated the beam center drift over the camera into these regions. This gave us the axial beam-cross-section that is compared with its theoretical prediction. The quasi-Bessel beams generated with varying axial intensity are shown in Fig. 10. The quasi-Bessel beams with the uniform intensity and varying lateral cross-section (central core size) are seen in Fig. 11. One can see a good agreement between the experimental results and the simulations with deviations between them no more than 5%. The most significant source of the intensity variations seen on the resulting cross-sections is caused by the beam center traveling across the CCD chip as the record is made. We have attempted to align the beam so the central spot lies on the same place on the camera as the record is taken but we found that it is the stage itself that does not move along an exact straight line. The measured intensity then varies due to differing sensitivity of individual camera pixels and due to an interference effect when the coherent light passes through the layer protecting the CCD chip. To demonstrate this issue we made a record of the beam at a fixed axial plane but in this instance moved the stage laterally so the same field travels across the camera chip. As shown in the Fig. 12 the signal suffers from a periodical dependence from light interference due to multiple reflections from the camera cover glass. Avoiding this problem might be a complicated task that we have not undertaken in these demonstration studies, but we note that future improvement is possible in this area.
As already mentioned the power efficiency of this approach is very low. For the quasi-Bessel beams with the fixed central core size and the varying intensities the total power carried by the resulting field is around 1.5% of the initial laser power sent onto the SLM. Slightly better power balance of about 7% was achieved for the case of quasi-Bessel beams with the varying central core size. This is due to the fact that the spatial spectrum for these cases is not so highly concentrated within the narrow annular region as in the first case.
In this paper we have shown how one can tune the axial intensity of azimuthally symmetric optical fields. We discussed the limitation of the traditionally used quasi-Bessel beams in terms of their varying axial intensities. We have shown how to achieve various types of quasi-Bessel beams namely the one featuring an uniform axial intensity along its propagation length opening up new applications in a variety of interdisciplinary fields. We have presented a G-S algorithm modification to achieve an optimal amplitude and phase modulation in the process of the diffraction element design and we used these holograms to experimentally generate the novel quasi-Bessel beams using a spatial light modulator. The obtained results showed a very good agreement between the experimental data and the simulations. In the process we generated the spatial spectrum of the required beams at the SLM chip plane illuminated by a Gaussian laser mode. This is the most direct way to demonstrate the described phenomena since the modulation is performed in a single plane, however, this approach is very lossy in terms of power balance. We note that this is not the only possibility and one can find a number of ways to obviate this issue using a tandem system similarly as in . For example one can use a spatial spectrum of an axicon-generated Bessel beam to illuminate the SLM chip . Naturally one has to include this illumination field in the procedure of the hologram design. For the uniform intensity quasi-Bessel beam demonstrated in this study and for the optimal set of parameters of such an illumination our initial numerical simulations indicate that the power efficiency would reach a value of 45% in this case. A very efficient way might be deploying the phase contrast method to directly generate the desired amplitude modulation using another SLM or a phase element thus perfectly matching the desired amplitude pattern on the SLM chip .
We thank Michael Mazilu and Martin Ploschner for valuable discussions and the UK Engineering and Physical Sciences Research Council for funding. KD is a Royal Society-Wolfson Merit Award Holder.
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