Optical Bessel beams are used in numerous applications like fluorescence microscopy, material processing and optical trapping. These applications require Bessel beams having a central core with defined full width at half maximum and a defined axial length. Often, the side lobes of Bessel beams, which are associated with their non-diffracting properties, can interfere with the experimental process. We theoretically describe and practically verify the performance of a new refractive optical system to generate zoomable annular ring intensities. The ability to zoom the output ring diameter allows for flexibly choosing the Bessel beam parameters. Secondly, we introduce the use of a Michelson interferometer for destructively interfering Bessel beam side lobes in one direction. If two Bessel beams of zeroth order and first kind are coherently superposed with a small shift with respect to each other, their side lobes are enhanced in one direction and cancelled in the other direction. We suggest that applications like light-sheet microscopy can exploit the axis of destructive interference to improve their contrast.
Jana Hüve and Jürgen Klingauf contributed equally to this work.
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1 Introduction
Optical Bessel beams (BBs) are non-diffracting beams which are found to be useful in numerous applications such as optical trapping, material processing, fluorescence microscopy and more (Duocastella and Arnold 2012; Gao et al 2014; Ayala et al 2016; Power and Huisken 2017; Khonina et al 2020). Analogous to plane waves, ideal BBs as derived from the electrodynamic theory of light (Durnin 1987) cannot be created in the experiment because finite energy and diffraction constrain the field such that the beam will also have Gaussian properties (Durnin et al 1987). Classical experimental methods to generate BBs are the use of annular slits (Durnin et al 1987), axicons (Herman and Wiggins 1991) and spatial light modulators (SLMs) (Chattrapiban et al 2003). While using SLMs has the advantage of getting multi-level phase profiles and dynamic control, SLMs often have low damage thresholds and low reflectivity going along with high costs (Khonina et al 2021). Additionally, due to the pixelation of an SLM, there will be undesirable zeroth-order light (George et al 2023). In comparison, standard axicons are not expensive and have got a high transmission, though missing the flexibility of SLMs. Examples of improvements of classical axicons include fluidic axicons for generating tuneable BBs (Milne et al 2008) and variable meta-axicons specifically designed for certain BB patterns (Zhu et al 2018). Furthermore, classical axicons have been combined with Pancharatnam-Berry phase elements (Gotovski et al 2021) and linear polarizers to achieve a more constant axial intensity distribution. These customized optical elements have got a high damage threshold compared to SLMs, which in previous studies also have been successfully used to generate a nearly constant axial intensity (Čižmár and Dholakia 2009; Orlov et al 2018).
Classical axicon lenses are still a good option for the generation of BBs, though. If a fast change in parameters or engineering of the axial intensity distribution are not needed to fulfill the experimental needs, classical axicons are often a better choice. For example, an appropriate combination with lenses leads to optical systems to create focused ring intensities (Rioux et al 1978) or zoom systems that can continuously adapt the cone angle (Vaičaitis V, Paulikas Š 2003) or the incident ring diameter (Dickey and Conner 2011). The last mentioned system consists of a lens followed by two axicons and is able to create a focused ring in the focal plane of the lens. By changing the second axicon’s axial position the ring diameter changes while the position of the focal plane is maintained.
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Here, we present that adding a third axicon and a second lens to this system can create a collimated ring intensity with adaptable diameter. The output ring is not focused any more, but the ring width is diminished by a factor of \(f_2/f_1\) where f denotes the corresponding lens’ focal length. The choice of the lenses thus determines the width of the output ring so that we will refer to the system as SWAN system (set ring width, adapt NA). We show that the properties of the SWAN system can be derived by considering ABCD matrices and demonstrate experimentally the claimed behaviour.
In a second step, we combine the SWAN system with a Michelson interferometer. The concept of introducing a Michelson interferometer for interferometric cancellation of BB side lobes can be used independently of the SWAN system. For instance, a single axicon behind an interferometer is sufficient to generate a BB with destructively interfered side lobes in one axis as discussed in Sect. 3 of the supplement. If both techniques are combined, the SWAN system introduces the flexibility to adapt the BB parameters for the experimental needs whereas the Michelson interferometer is able to cancel the BB side lobes in one axis.
In general, many BB applications exploit the central core of the zeroth order BB of first kind \(J_0\) (Khonina et al 2020). Consequently, the surrounding side lobes are often degrading the results of experiments by increasing the noise level so that numerous methods have been developed to either suppress the detection of the side lobes (Fahrbach and Rohrbach 2012) or directly adapt the interaction between beam and sample. This adaption includes nonlinear excitation (Planchon et al 2011; Olarte et al 2012), stimulated emission depletion of excitation caused by side lobes (Zhang et al 2014) and beam shaping. The latter has been reported in terms of interfering BBs with different wave vectors (Mori 2015) or phases (Zhang et al 2014; Yu et al 2016).
The idea to superpose two BBs having different wave vectors in order to intentionally suppress BB side lobes has first been realised with the help of diffractive slits by Mori (2015). Later, the resulting intensity distribution has been generated with SLMs and baptised "droplet beam" in the context of fluorescent (light-sheet) imaging because besides the suppression of side lobes, an oscillation of the axial intensity is introduced (Antonacci et al 2017; Di Domenico et al 2018). The periodicity p of this oscillation is given by
$$\begin{aligned} p = \frac{2\pi }{|k_{z_1}-k_{z_2}|} \end{aligned}$$
(1)
with \(k_{z_i}\) being the axial component of the wave vector (Chávez-Cerda et al 1998). Recent studies report that the suppression of side lobes is also possible by applying a phase function to the SLM rather than to mimic the diffractive slits (George et al 2023). Furthermore, the use of meta-optical elements for the generation of Bessel-like beams (George et al 2023) or of Bessel-like light-sheets with suppressed side lobes has been reported (Fan et al 2022).
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What the approaches mentioned above have in common is that the side lobes are suppressed in an isotropic manner. For an efficient minimisation of the power traveling in the side lobes, the ratio of \(k_{z_1}\) to \(k_{z_2}\) has to be optimised. According to Eq. (1), this directly affects the periodicity of the axial oscillation. The approach to use a metalens for side lobe suppression differs from this consideration, but still shows residual power with a side-to-main lobe ratio down to \(7.3\,\%\) (Fan et al 2022).
In this study, we report nearly perfect interferometric cancellation of BB side lobes in one axis by superposing two BBs having the same wave vector k. Theoretically, we calculate on the base of classical vectorial electrodynamic theory which interference pattern is expected if different phase modulations are applied to one beam. We derive that for the considered ideas, a simple lateral shift of one BB with respect to the other cancels the BB side lobes in one direction most efficiently. We propose that using a Michelson interferometer behind the SWAN system is an efficient method to experimentally realise the shift of two BBs with respect to each other and thus the axis of destructive interference.
Although other methods to generate BBs are expected to achieve the same results when combined with an interferometer, the SWAN system introduces the advantage of being cheap compared to methods that require SLMs or custom optics (Antonacci et al 2017; Di Domenico et al 2018; George et al 2023) because it only contains stock optics. For an incident Gaussian beam with radius r, the SWAN system reduces the output ring width by a magnification factor defined by the choice of the focal lengths of the lenses. This results in a larger ratio of inner to outer ring diameter and leads to enhanced BB properties. Furthermore, the SWAN system generates a collimated output beam which is beneficial for interference. Finally, it is purely refractive and has got a high transmission of more than \(83\,\%\) in our conditions.
2 Results and discussion
2.1 Collimating three-axicon zoom system (SWAN system)
Whereas the classical axicon zoom system consisting of one lens and two axicons is able to create a focused ring intensity with adaptable diameter (Rioux et al 1978; Dickey and Conner 2011), we demonstrate that the SWAN system as sketched in Fig. 1 maintains the possibility to zoom the ring’s diameter and creates a collimated output beam. In the following, we will first describe the system’s properties in terms of geometrical optics. Secondly, measured annular intensity profiles created by the SWAN system as well as BBs generated by applying a lens to the output ring intensity are shown to demonstrate the claimed behaviour. A detailed comparison between the classical axicon zoom system and the SWAN system is drawn in Sect. 2.2.
Fig. 1
Two lenses and three positive axicons forming the SWAN system. The distance from the first axicon Ax\(_1\) to the second one Ax\(_2\) has to be the focal length f\(_2\) of the second lens to collimate the output beam. Axially displacing the third axicon Ax\(_3\) results in a change in diameter of the collimated output ring intensity
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The SWAN system transforms an incident collimated beam with diameter 2r to a collimated ring intensity distribution where the ring width as compared to the incident radius r is decreased by a diminution factor \(f_2/f_1\) with \(f_{1,2}\) denoting the corresponding lens’ focal length (Fig. 1). Consequently, the ratio \(\epsilon\) of the inner diameter \(d_i\) to outer diameter \(d_o\) of the output ring is increased. This cannot be achieved by reducing a given ring intensity in size with a telescope.
For the system’s behaviour described above, the two lenses have to be positioned in a standard 4f configuration. Additionally, the three axicons must have the same aperture angle \(\alpha\) and the first two axicons Ax\(_1\) and Ax\(_2\) must be located the focal length \(f_2\) of the second lens away from each other in order to achieve a collimated output beam. The third axicon Ax\(_3\)’s axial position then defines the diameter of the output beam.
The claims in the paragraph above can be derived using ABCD matrices \(\underline{{\textbf{L}}}\) for lenses with focal length f and \(\underline{{\textbf{D}}}\) for propagation distances d (Paschotta 2007). The components of the \(2\times 2\) matrices are given by
The parameters \(\alpha\) and n are the apex angle and refractive index of the axicon, respectively. In Eq. (2), y denotes the beam’s axial position and \(\theta\) the angle with respect to the optical axis. Despite the use of five optical elements, the overall transfer matrix of the whole system simplifies significantly if the output beam shall be collimated. A more detailed description is provided in the supplement.
Fig. 2
a and b Annular intensity patterns exiting the SWAN system for different positions of the third axicon Ax\(_3\). c calculated lateral intensity distribution of an exemplary BB to be found in the focal plane of an f=\(300\,\text {mm}\) lens (L\(_5\) in Fig. 6). d experimental measured BB profile. The white graphs are the normalized intensity curves along the dashed lines
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To demonstrate the SWAN system’s zoom capability we present two exemplary ring intensity distributions generated by different axial positions of the third axicon Ax\(_3\) (Fig. 2a and b). The images have been acquired in the focal plane of a 4f-system consisting of achromatic lenses with focal lengths of 500 and \(100\,\text {mm}\), respectively (lenses L\(_3\) and L\(_4\) in Fig. 6), in order to avoid that the ring intensity distribution is larger than the camera chip. We calculated the expected lateral BB intensity for a uniform ring intensity with inner diameter of \(3.20\,\text {mm}\) and outer diameter of \(3.65\,\text {mm}\) being incident on an achromatic lens with a focal length of \(300\,\text {mm}\) (Fig. 2c). The corresponding lens in the experimental setup is L\(_5\) in Fig. 6. Although the paraxial approximation can be used for calculating the expected field given the parameters above, we decided to use the more general equations for strongly focused fields which were needed for calculating interfering fields with higher NAs. This has the advantage that the same calculation program could be used for all fields of interest without the need to distinguish between low and high NAs. For this reason, the corresponding equations are presented below in the context of interfering fields and in detail in the supplement.
The measured BB presented in Fig. 2d corresponds well to the calculated pattern in Fig. 2c. The white graph is an intensity plot along the dashed line. The measured BB has got a central core with Gaussian standard deviation of \((16.0\pm 0.9)\,\mu \text {m}\) which is in good agreement with the expected value of \((15.9\pm 0.6)\,\mu \text {m}\) drawn from Fig. 2c. Qualitatively, the measured intensity profile shows deviation from perfect rotational symmetry as well as faster decaying side lobes as compared to the calculated pattern. This is due to small asymmetries in the ring power distribution and a finite extension of the focusing optics.
2.2 Comparison of the SWAN system with a two-axicon system
The SWAN system presented here has complementary properties as compared to the classical axicon zoom system (Dickey and Conner 2011). We refer to the term classical axicon zoom system (here abbreviated as CLAZ) by assuming a lens followed by two positive axicons. This configuration is included in Fig. 1 in the first three elements with the adaption that in this case, the distance between Ax\(_1\) and Ax\(_2\) serves as a zoom parameter. We are going to compare the two optical systems with respect to three different aspects: the created annular ring pattern, combination with annular diffractive slits and collimation.
Both the SWAN and the CLAZ system create annular intensity distributions with adaptable diameter. The CLAZ system produces a focused ring whereas the SWAN system forms an annular pattern with a ring width diminished by a factor \(f_2/f_1\) compared to the incident beam radius. As ideal BBs are the Fourier transformation of an infinitesimally thin ring, the CLAZ system is more suitable for applications where BB properties like being non-diffracting and having a long axial length are needed. As a consequence, the SWAN system is useful for applications that benefit from beams with Bessel-Gaussian properties. If the application needs a beam with specific axial length and less strong side lobes as a nearly ideal BB, a broader ring width and thus the SWAN system may be a better choice than the CLAZ system.
Secondly, the SWAN system can be easily adapted to a configuration with similar properties as the CLAZ system by introducing an annular diffractive slit directly behind the SWAN system. If we describe the output ring intensity by an inner radius \({\tilde{r}}\) and an outer radius \({\tilde{r}}+\Delta r\), inserting any annular slit with inner and outer radii \(\in [{\tilde{r}},{\tilde{r}}+\Delta r]\) maintains still a lot of the incident power. The diffractive element destroys the beam’s collimation, but may be an excellent option for applications which require illumination with a thin ring. Adding an annular slit into the beam path can be done fast without the need of much readjustment. If the slits’ width is narrow, nearly ideal BBs can be created, though the power transmission will in general be lower than in the CLAZ system.
Finally, the SWAN system collimates the beam which can be useful for applications exploiting interference. As an application example, we present an experimental means to cancel the side lobes of BBs in one axis as described below.
2.3 Cancellation of side lobes in BBs using a Michelson interferometer
Developing the idea of using a Michelson interferometer for the cancellation of BB side lobes, we used the vectorial description of focused fields \(\vec {E}\) (Foreman and Török 2011; Novotny and Hecht 2012) to calculate interference patterns of BBs. From this theoretical approach, we drew expectations of how to manipulate the electromagnetic field to achieve destructive interference in one lateral direction. The introduction of a Michelson interferometer is a simple means to modify one beam without affecting the other one in order to experimentally realise the calculated intensity patterns.
To our knowledge, the theoretical description as presented below as well as the experimental verification of the cancellation of BB side lobes in one direction have not been reported before. Similar theoretical considerations have been presented in the context of paraxial scalar diffraction theory (Zhang et al 2014). In this case, it was suggested that a specialised phase plate could generate a coherent superposition of a \(J_0\) and a \(J_2\) BB. This leads to destructive interference in one direction. An experimental realisation has however not been reported so far which may be due to the need of manufacturing a custom phase plate.
Our approach to describe interferometric cancellation of BB side lobes improves the previous description in two manners: firstly, applications like fluorescence microscopy often make use of NAs for which the paraxial theory is not a good approximation any more. Hence, the vectorial description of focused fields \(\vec {E}\) defined by
can describe focal fields for more cases than the paraxial theory. In Eq. (3), \(\vec {e}(\Theta ,\Phi )\) is the projection of the incident field on a Gaussian reference sphere, as described in more detail in the supplement. The symbols k and f denote the wave number and the focusing lens’ focal length, respectively. Secondly, we pay attention to the fact that destructive interference in the side lobes is reached for a phase difference of \(\pi\) so that different periodic modulations can be used for this task.
For qualitative comparison between resulting interference patterns, we assumed that the incident field is non-vanishing only on an infinitesimally thin ring so that only one integral in Eq. (3) remains:
\({\tilde{E}}_x, {\tilde{E}}_y\) are the Jones vector components (Foreman and Török 2011).
Fig. 3
Calculated interference patterns for a \(J_0\) BB and a modulated beam. Beams have been calculated for an effective NA of 0.55 and phase modulations as written above. The discrete phase modulation contains jumps between 0 (black) and \(\pi\) (white)
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We here compare the interference patterns of a normal BB (\({\mathcal {E}}=1\)) with three distinct modulation options in order to choose the most efficient one in terms of disappearing side lobes while maintaining a small central core (Fig. 3). The \(2\Phi\) modulation corresponds to the interference of a \(J_0\) and a \(J_2\) beam. It is worth noting that in our calculations assuming an effective NA of 0.55, the central core is broadened and embraces two local minima close to the axis of destructive interference. In contrast to the paraxial case reported before (Zhang et al 2014), this means that the advantage of destructively interfering side lobes comes at the cost of a distorted central core for this NA.
As alternative periodic modulation functions, we tested a \(\pi \sin (\Phi )\) modulation as well as discrete jumps between 0 and \(\pi\). Both tries lead to destructive interference of side lobes, but in the case of discrete modulation, strong side maxima appear for the central axis which shall show minimal intensity. The \(\pi \sin (\Phi )\) modulation corresponds to a simple tilt of the modulated incident beam (Born and Wolf 1999). Practically, the theoretically calculated interference pattern can thus be generated by using a Michelson interferometer and by tilting one mirror of the interferometer arms. In the Fourier plane of BBs, this corresponds to a lateral shift with respect to the reference beam.
This interference pattern of two \(J_0\) BBs slightly displaced with respect to each other shows perfect destructive interference in the vertical axis (cf. Fig. 3). This comes at the cost of an elongated central core by a factor of about 2 in the axis of destructive interference compared to the BB generated by the unmodulated reference arm alone. In contrast to the coherent superposition of a \(J_0\) and a \(J_2\) beam, the approach to interfere two laterally displaced \(J_0\) beams does not introduce local minima into the central core. Therefore, for the options presented here, it leads clearly to the most favorable interference pattern for e.g. STED microscopy or light-sheet microscopy.
Fig. 4
Proof of principle measurements for interference patterns of two BBs shifted with respect to each other. The interference pattern as predicted in Fig. 3 appears both in the paraxial case as well as in the focal plane of the microscope objective. a–d BBs generated by the SWAN system only, e–h BBs generated by SWAN system and an additional annular slit. The effective NA is approximately 0.009 in the paraxial case and approximately 0.5 behind the microscope objective (MO). The ratio \(\epsilon\) between inner and outer diameter of the incident ring is: a–d\(\epsilon =0.82\) and e–h\(\epsilon =0.98\)
Fig. 5
Intensity along the white dashed lines as shown in Fig. 4g. Whereas the first side maximum is enhanced to about 90\(\,\%\) of the main peak in the direction of constructive interference, the lobes effectively disappear in the direction of destructive interference. A typical magnitude of the first side maximum of destructive interference is 2\(\,\%\) of the central peak
×
×
To realise the theoretical expectations, we finished the experimental setup by adjusting the beam into an air microscope objective (MO) with a numerical aperture (NA) of 0.6 in order to observe highly focused beams as used e.g. in microscopy. The focused beam is reflected back into the MO and separated from the incoming beam with the help of a beam splitter as sketched in Fig. 6 in order to image the focus. In Fig. 4, we present measured intensity profiles both for the paraxial case and the focal plane behind the MO (dotted lines behind L\(_7\) and MO in Fig. 6). Furthermore, we compare intensity distributions created by the SWAN system only and those created by the introduction of an additional annular slit. The patterns in Fig. 4a, c, e, g appear for vertically aligned BBs and those in Fig. 4b, d, f, h for horizontally aligned BBs. The axis of destructive interference is defined by the line through both BBs’ central cores and can be arbitrarily adjusted.
Inserting an annular slit introduces sharp edges and thus a defined inner ring diameter \(d_i\) and outer ring diameter \(d_o\). Choosing an annular slit with narrow width increases the ratio \(\epsilon\) between inner and outer diameter of the incident ring, here from \(\epsilon =0.82\) (SWAN system only) to \(\epsilon =0.98\) (additional slit). Accordingly, the profiles presented in Fig. 4a–d contain less side lobes as they are decreased by the Gaussian envelope. The high value of \(\epsilon\) in Fig. 4e–h leads to numerous side lobes which however are effectively suppressed by destructive interference.
The experimental results are in good agreement with the expectations drawn from the numerical calculation. Although a \(\delta\)-distribution has been assumed for calculating Eq. (3) which corresponds to a ratio of inner and outer ring diameter \(\epsilon =1\), we also observe the interferometric cancellation for the smaller value of \(\epsilon =0.82\). Quantitatively, the first side lobe is reduced to below \(2\,\%\) of the main peak’s intensity while the constructively enhanced side ring has got peak intensities of about \(90\,\%\) of the main peak. Figure 5 contains the plot of the intensities along the white dashed lines printed in Fig. 4g. The central core has got a FWHM of \((0.88\pm 0.04)\,\mu\)m along the axis of destructive interference. We measured the central core’s diameter of a corresponding single BB to be \((0.47\pm 0.03)\,\mu\)m. The broadening factor is thus about 2 while the central core along the axis of constructive interference has got a FWHM of \((0.32\pm 0.03)\,\mu\)m. The interferometric central core is decreased in this direction compared to the single BB.
Previous approaches to weaken the intensity in BB side lobes by interference (Antonacci et al 2017) have superposed two BBs with different wavevectors. This resulted in an isotropic weakening of the first side lobes while further side lobes were enhanced. Our approach to achieve destructive interference by superposing two identical BBs which are laterally shifted with respect to each other achieves perfect destructive interference at the cost of losing isotropy. The intensity pattern is not rotationally symmetric anymore but one direction undergoes constructive interference while the other one destructive interference. For applications like light-sheet microscopy isotropy is not necessary because one direction is chosen to be the detection axis. If this axis coincides with the axis of disappearing side lobes we assume that the contrast of BB light-sheet microscopy can be significantly enhanced. This has to be verified in further studies.
3 Methods
Fig. 6
CW Laser beams of two different wave lengths are combined with a DC and switched with an AOTF. The first lens pair broadens the laser focus to better fill the entrance aperture of the SWAN system. Subsequently, the output beam’s diameter is decreased with a second lens pair. Optionally, an annular slit mask (BM) can be added to better define the ratio between inner and outer ring diameter. Its plane is imaged on the mirror surfaces of the Michelson interferometer arms and then onto a mirror hold by a galvanometric scanner. The last lens pair generates a conjugate plane in the MO’s entrance pupil. The beam is focused onto a mirror and coupled back into the MO by reflection. The second BS separates the output beam from the input beam so it can be focused with a TL and detected by the camera
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3.1 Illumination path
Two monochromatic laser beams exiting from a \(491\,\text {nm}\) (Cobolt Calypso \(491\,\text {nm}\)\(200\,\text {mW}\), Hübner Photonics GmbH, Solna, Sweden) and a \(561\,\text {nm}\) solid-state laser (Cobolt Jive \(561\,\text {nm}\)\(200\,\text {mW}\), Hübner Photonics GmbH, Solna, Sweden) are superposed using a dichromatic mirror (DMSP550, Thorlabs GmbH, Bergkirchen, Germany; in the following abbreviated as Thorlabs GmbH) as sketched in Fig. 6. The beams have been expanded using an \(f=200\,\text {mm}\) and an \(f=400\,\text {mm}\) achromatic lens (AC254–200-A and AC254–400-A, Thorlabs GmbH) to better fill the aperture of an acousto optical tuneable filter (AOTFnC\(-\)400.650 with driver MDS4C-B66–22\(-\)74.158.RS, AA opto-electronic, Orsay, France).
The two different laser wavelengths, the AOTF as well as the galvanometric scanner GM (dynAXIS 3 M Galvo, SCANLAB GmbH, Puchheim, Germany) are implemented for later using the setup as a light-sheet microscope. In the context of the data presented here, all experiments qualitatively provided the same results for the different wavelengths. More precisely, the ring intensity patterns as shown in Fig. 2a and b have got the same diameter. The ring’s intensity is not uniform, but contains local minima and maxima which are probably caused by diffraction (Goodman 2017) and influences from the undefined conical tip of the axicons (cf. Sect. 3.2). The positions of these local minima and maxima are wavelength-dependent. The core of the BB is dependent on the wavelength and thus smaller for \(491\,\text {nm}\). Data and calculations shown from Figs. 2, 3 and 4 are consistently presented for the wavelength of \(561\,\text {nm}\).
Finally, the beams exiting from the AOTF are expanded by using an \(f=50\,\text {mm}\) and an \(f=250\,\text {mm}\) lens (AC254–050-A and AC254–250-A, Thorlabs GmbH).
3.2 SWAN system
The focal lengths of the two lenses in the SWAN system are \(f=400\,\text {mm}\) and \(f=100\,\text {mm}\), respectively (AC254–400-A and AC508–100-A, Thorlabs GmbH). The three axicons have got an apex angle of \(\alpha =10^\circ\) (XFL25–100-U-A, asphericon GbmH, Jena, Germany). Their central dead area where the elements significantly deviate from the cone shape is \(2.3\,\text {mm}\). In order to reduce the influence of these undefined regions, the incident beam is expanded to fill a large area of the first axicon as described above.
The tips of the axicons have to be carefully aligned in order to generate a symmetric ring distribution which leads to a \(J_0\)-BB if focused. Displacements with respect to the optical axis and astigmatism introduced by tilt easily distort the focal intensity distribution (Tanaka and Yamamoto 2000). We applied 5-axis kinematic mounts to each axicon (K5X1, Thorlabs GmbH) and a single-axis translation stage (MT1/M, Thorlabs GmbH) to the axicons Ax\(_2\) and Ax\(_3\) for reliable and stable adjustment. In addition, we built a custom protective housing in order to protect the system from ambient air flow (Medizinische Feinmechanische Werkstatt, University of Münster, Germany).
For imaging, a CMOS camera has been used (UI-3260CP-M-GL Rev.2, IDS, Obersulm, Germany). For the images shown in Fig. 2a and b, a telescope consisting of lenses L\(_3\) and L\(_4\) with focal lengths of \(f=500\,\text {mm}\) lens and an \(f=100\,\text {mm}\), respectively (AC254–500-A and AC254–100-A, Thorlabs GmbH), has been used in order to be able to image the large scale of possible ring diameters. Within the actual setup, the lens L\(_3\) is an \(f=300\,\text {mm}\) lens (AC254–300-A, Thorlabs GmbH). The lower diminution factor was better suited for the illumination of the interferometer mirrors.
3.3 Michelson interferometer
In the Michelson interferometer, the beam is split by a non-polarizing beam splitter cube (BS013, Thorlabs GmbH) and reflected by dielectric mirrors (BB1-E02, Thorlabs GmbH). In order to minimize relaxation after manual adjustment in the interferometer arm, we applied a low distortion optic mount (POLARIS-K1F1, Thorlabs GmbH) to one mirror. In addition, we use a custom protective housing (Medizinische Feinmechanische Werkstatt, University of Münster, Germany) to minimise the effects of exterior disturbances.
3.4 Detection path
Behind the Michelson interferometer, the beam is expanded with the lens pair L\(_7\) and L\(_8\) having focal lengths \(f=200\,\text {mm}\) and \(f=300\,\text {mm}\), respectively (AC254–200-A and AC254–300-A, Thorlabs GmbH), so that a higher effective NA is reached. Figure 6 simplifies the setup in that sense that we have got two identical GMs mounted 90\(^\circ\) with respect to each other so that two different dimensions can be scanned. This is not important, however, for the current study so that only one GM is sketched.
Finally a lens system having a total magnification of \(\text {M}=1\) images the plane of the GM onto the back focal plane of the MO (Obj. HC PL FLUOTAR L 40x/0.60 CORR, Leica Microsystems, Wetzlar, Germany) which focuses the beam onto a dielectric mirror. That mirror is mounted on a motorized stepper (8SMC5-USB, Standa, Vilnius, Lithuania) so that the beam can be reliably defocused. In that manner, not only the focal plane but a 3D volume can be measured.
The beam is reflected and coupled back into the MO. A plate beam splitter (BSW16, Thorlabs GmbH) separates the reflected from the incoming beam. The output beam is focused with a tube lens having a focal length \(f=250\,\text {mm}\) so that the combination of MO and TL has got a total magnification of \(\text {M}=50\). For detection, the CMOS mentioned above is used.
3.5 Numerical calculations
The intensity distributions presented in Fig. 2c and in Fig. 3 have been numerically calculated via implementing Eqs. (3) and (4) with Python 3.8.5. More details about Eqs. (3) and (4) are provided in Sect. 2 of the supplement.
For calculating the intensity distribution presented in Fig. 2c, we assumed a pixel size of \(5.86\,\mu \text {m}\), an inner NA\(_i\approx \,0.0053\) and an outer NA\(_o\approx \,0.0061\). This corresponds to the pixel size of the CMOS camera we used (UI-3260CP-M-GL Rev.2, IDS, Obersulm, Germany) and to the effective NA we achieved by illumination of an \(f=300\,\text {mm}\) lens with the experimentally observed ring intensity, respectively. Numerically, we assumed that the annular ring intensity is uniform. Furthermore, we assumed a wavelength of \(\lambda =561\,\text {nm}\) and linear polarization which corresponds to the parameters of the laser (Cobolt Jive \(561\,\text {nm}\)\(200\,\text {mW}\), Hübner Photonics GmbH, Solna, Sweden) for which we present the experimental data.
The parameters used for calculating the intensity distributions of Fig. 3 are chosen differently because higher NAs are of interest for the expected application of light-sheet microscopy. The pixel size has been set to \(100\,\text {nm}\) to obtain highly resolved intensity distributions. Each intensity distribution corresponds to a square space of \(8\,\mu \text {m}\times 8\,\mu \text {m}\). The parameters of the MO (Obj. HC PL FLUOTAR L 40x/0.60 CORR, Leica Microsystems, Wetzlar, Germany), i.e. an NA of 0.6 and a magnification of 40, have been used to calculate the NA from the incident ring diameter. The effective illumination NAs have been set to NA\(_i=\,0.54\) and NA\(_o=\,0.55\). The wavelength has been set to \(\lambda =561\,\text {nm}\) and a linearly polarized incident beam has been assumed.
In both cases, the refractive indices in front of the focusing lens \(n_1\) and behind the focusing lens \(n_2\) have been set to 1 for propagation in air. In summary, the parameters are chosen to closely match the experimental conditions.
4 Conclusion
In summary, we present an optical system for generating Bessel beams consisting of three axicons and two lenses (SWAN system) which is able to create collimated annular intensity patterns with adjustable diameter. We demonstrate that the system’s properties can be derived using ABCD matrices and that its flexibility allows for the manipulation of BB parameters according to the experimental need.
We use the SWAN system in combination with a Michelson interferometer in order to cancel the beam’s side lobes in one axis. Based on the vectorial description of the electric field we calculate that the superposition of two BBs with their central core slightly laterally shifted with respect to each other will result in nearly ideal destructive interference. Experimentally, we demonstrate that destructive interference occurs both for Bessel-Gaussian beams (\(\epsilon =0.82\)) and beams with emphasised Bessel properties (\(\epsilon =0.98\)). The interferometric cancellation reduces the first side lobe to a peak intensity \(\le \,2\,\%\) of the main peak in one axis.
As one application idea, we suggest that light-sheet microscopy could benefit from the presented intensity pattern used for excitation by exploiting the axis of destructive interference as detection axis.
Acknowledgements
We would like to thank Marvin Wärmeling and Yannik Vaas from the Medical Mechanical Workshop, University of Münster, Germany, for constructing a protective housing around the SWAN system and the interferometer, custom optic holders and helpful tools needed for this work. This work was supported by grants from the Deutsche Forschungsgemeinschaft (SFB 1348).
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