1. Introduction

Bessel beams are well-known propagation invariant beams, which can be produced, e.g., using axicons [1–4]. They consist of a self-interfering conical wavefront with an intensity profile that can be described by a Bessel function [2]. Using the extended focal zone and the self-reconstruction of this beam, there are many different applications, where Bessel beams are used in optical imaging, or material processing [5–8]. General analytical studies of the propagation behaviour of Bessel beams compared to Gaussian beams with different apertures including circular apertures were already discussed in literature [9,10].

While the propagation invariance of Bessel beams is usually considered a virtue, it can also be a disadvantage. There are various applications where segmented Bessel beams with a limited focal zone could be of great interest, e.g., depth-selective material processing or for structured illumination in optical imaging. In this paper, we investigate segmented Bessel beams that are created by placing a ring aperture into the initial wavefront [Fig. 1(A)]. We assume that self-reconstruction is guaranteed since the wavefront in the illuminated region is still conical and an obstacle sufficiently before and after the segmented region will not effect the beam. The main advantage of the segmented Bessel beams compared to the change of e.g. the axicon angle is that the length and position of the Bessel beam can be varied by the ring aperture and ’scanning’ the Bessel beam can be achieved with an adjustable ring aperture as illustrated in Fig. 1(B). Furthermore, the core radius remains constant, which would not be the case if the axicon angle was changed.

 

Fig. 1 A: Schematic drawing of the segmented Bessel beam region Δz using a ring aperture with aperture size d and an axicon with apex angle γ. The ring aperture directly in front of the axicon limits the Bessel beam region such that we get a smaller, segmented Bessel beam region Δz at position z_start until z_end. B: Simulated transverse intensity profiles of segmented Bessel beams with axicon angle γ = 170° and wavelength λ = 632.8 nm. The intensity is scaled logarithmically to illustrated the side lobes. The optimal apertures d_min that we used at different radial positions r are: (i) r = 1 mm, d_min = 197.85 μm, (ii) r = 2 mm, d_min = 281.58 μm and (iii) r = 3 mm, d_min = 344.90 μm.

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In the classical regime of wide apertures, we expect the length of the segmented beam to be proportional to the aperture, and for narrow apertures the length should increase with decreasing aperture width due to diffraction. Hence, we expect to find the optimal aperture to be between the Fraunhofer and classical limit, which can only be solved numerically. This makes the solution of the inverse problem, to find the optimal aperture generating the shortest segmented Bessel beam, difficult, in particular since the optimal aperture depends on its radius, the conical angle of the axicon and the wavelength. We intend to derive a scale invariant analytical estimate in an easy to use expression by finding the intersection of the Faunhofer and classical limits and verify it using simulations and measurements.

Such an analytical estimate could be used, e.g., to design and optimize devices with different small actuated apertures to generate segmented Bessel beams as described in [11].

This paper is organized as follows: we discuss our analytical approximations in section 2 and the experimental setup with simulations in section 3. In section 4 we present our results and conclude our findings in section 5.

2. Analytic approximation

In general we can relate the transverse z-position of a Bessel beam and the radius r of the incident beam as illustrated in Fig. 1(A). Using the conical wavefront angle α = (n_axicon −n_air)((π − γ)/2), where γ is the axicon apex angle and n the corresponding refractive index, we can relate the incident beam radius r to the corresponding z-position:

To obtain simple analytic parametric expressions, we approximate the ring aperture by a slit with width d that is much narrower than its radius r, d ≪ r. In the far field, the well known diffraction integral can be approximated by the Fraunhofer diffraction equation [12]. The intensity with opening angle θ is then:

 

Fig. 2 Left: Normalized segmented Bessel beam width Δz̃ as a function of the aperture sizes d̃. The intersection of the Fraunhofer and classical regime is marked with d̃_min, where the smallest segmented Bessel beam width Δz̃_min is expected. Right: Comparison of the simulation to the analytical estimate as a function of the apertures sizes d̃, at a fixed scale invariant radius r̃ ≈ 79200, different axicon apex angles (γ = 178° and γ = 170°), wavelength (632.8 nm and 473 nm).

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3. Experimental setup and simulations

As our approximations are only exact in the far limits and we want to determine how well the estimate holds near the optimal aperture width, we numerically simulated the beam propagation using Fast Fourier transformations ℱ (·), as described in previous publications [13]. The initial wave with amplitude Ψ₀ is a conical wavefront,

To verify the analytic estimate and the simulations, we measured the intensity profile using a setup which consists of an axicon, a ring aperture that is placed directly behind the axicon tip [Fig. 1(A)]. We recorded the beam profile with a movable CMOS camera with 5 megapixels and a pixel pitch of 2.2 μm that is mounted on the axis which moves along the propagation direction. To verify the dependence of the minimal segmented Bessel beam region on the wavelength and the axicon angle, we used two different light sources, a 473 nm diode pumped solid state laser and a 632 nm helium-neon laser and two different commercially available axicons with apex angle of γ = 178° ± 0.5° and γ = 170° ± 0.1° made from UV fused silica with anti-reflection coating. The initial beam radius r_initial = 10 mm is generated using an f = 100 mm collimation lens. We chose an aperture with size of r_initial to obtain an approximate plane wave. For the ring aperture, we used high resolution photo plots (25 000 dpi) on a polyethylene terephthalate (PET) film with a thickness of 180 μm and an accuracy of ±1 μm. We measured the Bessel beam region with different apertures sizes, this will be discussed in more detail in section 4. The radius of the apertures is defined in the middle of each aperture, neglecting the small deviation coming from the conical wavefront that changes the effective width of the aperture by a factor of cos(α).

To compare the simulations and measurements we analyzed the data by reconstructing the image plane along the optical axis taking the center of gravity as central intensity of each plane to avoid possible transverse motion due to camera movements in the measurement [see Fig. 3]. Then, the core intensity of the segmented beam at each position is normalized by the corresponding initial core intensity measured without a ring aperture. Additionally, we added a factor of 1.098, which appears from surface reflections (on both sides) of the ring aperture substrate. The same procedure is used for the evaluation of the simulation. Afterwards, we found the segmented Bessel beam region Δz, where the intensity drops first to 1/e², starting from the center of the expected segmented Bessel beam region. We varied the aperture sizes, the axicon angle (γ = 178° and γ = 170°) and the wavelength (632.8 nm and 473 nm) with constant radius r̃ as illustrated in Fig. 2 (right). The chosen axicon angles and wavelengths from blue to red cover a wide range of parameters to verify our approximation to be generally valid.

 

Fig. 3 Comparison of the normalized core intensity around the segmented Bessel beam region at a radius r = 2mm in (i) the Fraunhofer limit d = 100 μm, (ii) near the minimal aperture d = 200 μm and (iii) in the classical limit d = 800 μm for simulations and measurements with wavelength λ = 632.8 nm and an axicon apex angle of γ = 170°. The shaded areas indicate the analytical estimate for the segmented Bessel beam region. The corresponding transverse simulated intensity profile along the optical axis is shown at the bottom of each graph. To illustrate the side lobes the intensities were scaled logarithmically. The actual aperture width in the measurement are: 101.9 ± 0.7 μm, 199.6 ± 1.1 μm and 801.0 ± 0.6 μm.

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4. Results

First, let us compare the analytic approximation with the corresponding simulations for four different configurations as illustrated in Fig. 2. The correction factor of Eq. (8) that accounts for the 1/e² intensity drop is ζ = 0.70. The estimate for the optimal ring aperture d̃_min, which results in a minimal Bessel beam region Δz̃_min, is indicated in Fig. 2 (left) at the intersection of the predictions for the classical limit and the Fraunhofer diffraction limit. In Fig. 2 (right), we see that the simulations are in good agreement with the analytical approximation and that our scaling is suitable as all simulations predict approximately the same beam length in the scaled coordinates. The deviations in the first point at the Fraunhofer limit appear at d̃ = 100 due to the violation of the condition in Eq. (13).

To compare the simulation with measurements, let us first qualitatively compare the normalized central maximum of the Bessel beam using the axicon apex angle 170°, radius r = 2 mm and the wavelength λ = 632 nm, as illustrated in Fig. 3 in the Fraunhofer limit d = 100 μm, near the minimal aperture d = 200 μm and in the classical limit d = 800 μm. In the Fraunhofer regime, the intensity decreases approximately proportional to the ratio of the classical length to the actual beam length. We can observe that both the measurements and the simulations show the quadratic sinc function profile in the Fraunhofer limit (at d = 100 μm) which appears increasingly asymmetric with decreasing aperture width. We expect this to originate from the projection of the Fraunhofer intensity profile onto the optical axis with varying angle α′, where the linear approximation in the last step of Eq. (4) is not valid anymore when the diffraction angles are not small as assumed in Eq. (5). Another effect that contributes to this asymmetry is the circular geometry, which we ignored in our analysis and becomes also increasingly relevant for small radii and large diffraction angles. In this case, our approximation of a long slit instead of a circular aperture doesn’t hold true anymore, and the intensity inside and outside the ring, or before and after the illuminated region will not be the same [Fig. 3 (i)]. The further we move outside of this regime the more we can see the classical unit step intensity profile. We can observe oscillations of the intensity profile in all measurements and simulations. These arise presumably from the interference of the diffracted wave from both edges of one slit. We verified this by comparing the oscillation period of this pattern to the period of the interference of two waves coming from both edges of the aperture. As they modulate the intensity profile, we can have a mismatch in the 1/e² determination by at most half a period.

In Fig. 4, we quantitatively compare the analytical approximation with simulations and measurements for two different combinations: λ = 473 nm, γ = 178° and λ = 632 nm, γ = 170°. To estimate the uncertainty in the measurement and simulation we define the error to be half of an oscillation period [see Fig. 3]. The deviations of the ring apertures themselves are sufficiently small (±1 μm) such that we neglect this error in the comparison. The results are in good agreement within the uncertainties.

 

Fig. 4 Comparison of the analytical approximation and simulation of the beam length Δz̃ as a function of different aperture size d̃ with measurements for two different configurations of the axicon apex angles, radius and wavelength.

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Finally, let us look at the minimal segmented Bessel beam region Δz̃_min and the corresponding ring aperture size d̃_min as predicted in Eq. (14) and illustrated in Fig. 5. We found the minimum simulated Bessel beam length Δz̃_min by determining the optimal aperture d̃_min using simulations with finer iterations steps around the theoretical minimum [Fig. 4]. The obtained optimal aperture was then used to simulate the corresponding Bessel beam length Δz̃_min. We found that the optimal simulated aperture width d̃_min is larger than the estimated optimal aperture. We assume that this results from the fact that we considered only both limits separately, i.e. the beam length due to diffraction in the Fraunhofer limit or the geometric beam length in the classical limit, but not the combination of both of them. The combined effects increase the beam length near the minimum where both effects are significant. The oscillations then, however, reduce the minimum to near its estimated value, at a slightly larger width of the aperture. Still, the trend of d̃_min shows the same square root behavior, scaled by a factor of 1.155 from the analytical estimate to the simulation. The analytical estimate and simulation for the minimal Δz̃_min are, however, in good agreement. We can in fact see this effect in Fig. 4, where the minimum of Δz̃ is shifted towards slightly larger values d̃, while the minimum value of Δz̃ is approximately the one of the intersection of the classical and Fraunhofer limits.

 

Fig. 5 Comparison of the analytical considerations and simulation of d̃_min (red) and Δz̃_min (green) as a function of different radii r̃.

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5. Summary and conclusion

We have investigated the shortest possible segment of a quasi Bessel beam, which can be generated using ring apertures placed in a conical wavefront. We have developed a simple analytical estimate in the classical and Fraunhofer limits to determine this segmented Bessel beam, choosing appropriate dimensionless parameters, which are independent of the wavelength and the axicon angle. Assuming that the shortest Bessel region occurs near the intersection of these two limits, we derived an easy to use expression for the shortest possible beam segment and corresponding optimal aperture width. While it is, in principle, possible to solve the full diffraction integral, there seems to be no exact analytical solution for our case. In particular the inverse problem of finding the optimal aperture would not yield a nice, simple parametric expression.

For verification, we performed a simulation using fast Fourier techniques and measured beam profiles created by placing a ring aperture behind an axicon that is illuminated with an approximate plane wave. Comparing different wavelengths, axicon angles and aperture sizes at different radial positions, we were able to verify our analytical estimate to be generally valid. The remaining small deviations appear due to oscillations in the intensity profile, which we expect to result from the hard aperture and cause a small increase in the optimal aperture width by 15.5%. Nevertheless, the simulated shortest possible segmented Bessel beam agrees almost exactly with the analytical estimate. We hope that this general expression can be used for many different applications, e.g., when developing ring aperture designs for depth-control or structured illumination.