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Abstract
A pair of axicons with an adjustable separation between them is used to generate a variable diameter ring beam with high efficiency. This beam illuminates a lens to produce quasi-diffraction-free beams with a tunable spot size and depth of field. We studied the generated beam characteristics while changing either the ring diameter or its thickness. Such a scheme has applications in adjustable imaging, including nondiffracting beam microscopy, material processing with an irradiance above a certain threshold value, and particle trapping/manipulation.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. INTRODUCTION
In biomedical applications, such as imaging/optical microscopy, it is beneficial to tune the system’s optical resolution. One may track, for example, moving objects at low resolution and then continuously increase it to view the object’s (e.g., the cell’s) details. Similar, in laser micromachining/drilling it may be required to tune the resolution/spot aspect ratio, first to align the beam and then to decrease the spot size to achieve an irradiance above a threshold value. Due to diffraction, the increase in resolution/decrease of the focal spot size is accompanied by a decrease in the depth of field (DOF). This limitation can be overcome by the use of diffraction-free or Bessel beams [1]. These beams, in addition to possessing a DOF that can exceed the Rayleigh range by orders of magnitude, have a transverse spot size whose full width at half maximum (FWHM) is ${0.36}\lambda {\rm /NA}$, which is below Abbe’s diffraction limit or the Airy disk with a ${\rm FWHM} = {0.51}\lambda {\rm /NA}$ [2]. Here, $\lambda$ is the wavelength and NA is the numerical aperture. In many cases, it is advantageous to tune the DOF of Bessel beams to study tissues of different thicknesses and label densities as well as to reduce the sensitivity to axial motion of the sample. Due to the importance of diffraction-free beams in many applications, making these beams tunable (i.e., controlling their cone convergence angle) [3–6] has been a major priority. However, most of the proposed schemes have disadvantages. For example, spatial light modulators are expensive, have low efficiency, and a low optical damage threshold; devices using an acousto-optic effect are expensive and have low efficiency [5]; and digital micromirror devices have finite pixel sizes, which result in stepwise beam control and are inefficient when only a small sample must be illuminated [3,4,6]. Fluidic, liquid crystal, and other adjustable axicons [7–9] have limited tuning ranges and can produce aberrations due to a lack of control degrees. One of the original methods to produce a Bessel beam uses a thin ring aperture placed in the front focal plane of a lens [1]. This scheme, while extensively studied [10–12], suffers from very low efficiency. Consequently, several works used axicons and their combinations to produce a ring-shaped beam [13,14] that can generate quasi-diffraction-free beams when focused by a lens. This approach is supported by the wide range and diversity of available and studied axicons such as Fresnel axicons [15,16], reflective axicons [17,18], and solid immersion axicons [19]. In addition, axicon–lens combinations were exploited [20–24]. All of these schemes are either complex or have limitations that include: a polarizing beam splitter and several quarter-wave plates, resulting in polarization alignment sensitivity [18,23]; a limited spot size range due to limits on the separation between the mirrors [18]; generation of a single angular component only for small angle axicons [21]; the inclusion of several lenses and an annular aperture mask, while producing resolution variation by only a factor of ${\sim}{1.5}$ [22]; spot size changes along the optical axis [23]; or a setup that requires a large number of lenses [24]. In this study, we present a simple scheme consisting of a two-axicon beam expander producing a tunable diameter ring beam that illuminates a lens. Also, in contrast to some other studies (e.g., [18,23]), producing long-range Bessel beams with lengths of tens of centimeters with ${\sim}{100}\;\unicode{x00B5}{\rm m}$ spot size, we focus on generating beams for imaging applications that require a DOF of hundreds µm length and a few µm spot size. In addition, we demonstrate the transition between the quasi-Bessel beam and Gaussian one upon changing the thickness of the ring that illuminates the lens. The proposed scheme to generate the ring of light may be used in dark field illumination. We note that a variable diameter ring or hollow beam generated by two axicons to tune the lens resolution was proposed in [25]; however, the concomitant DOF variation was not considered and, to the best of our knowledge, no experimental work was published about the control of these parameters when using two-axicon-generated ring beam illumination.
2. THEORETICAL CONSIDERATIONS
In cylindrical coordinates, the intensity of a plane wave of wavelength $\lambda$ focused by a lens of focal length $f$ and radius $R$ is [26]
Here ${{J}_0}$, ${{ J}_1}$ are first kind Bessel functions of zero and first order, ${ k} = 2\pi /\lambda$, and $\rho$ is the distance from the axis.
Fig. 1. Experimental setup block diagram for tunable diameter ring beam generation and its focusing. The optional insert with a beam expander and an aperture is used in the second part of the experiments.
The focused spot is minimized when a narrow annular mask of width $\delta$ and radius ${R}$ is used to converge only the high spatial frequencies. Then, the spot becomes
The size of the bright spot, defined as the distance from the peak to the first zero of ${{J}_0}$, is ${\rho _{\rm bs}} = \frac{{2.4f}}{{\rm kR}} \approx \frac{{0.38\lambda}}{{\rm NA}}$. The corresponding FWHM is
The depth of the field for a quasi-diffraction-free beam generated by the ring illumination of a lens is given in [12] as
where ${{R}_{\rm av}}$ is the average ring radius and ${\Delta}\! {R}$ is the ring’s thickness. In deriving this relation, it was assumed that the intensity over the ring thickness is constant.3. EXPERIMENTAL SETUP
Figure 1 illustrates the experimental setup for the above-mentioned configuration. Two variations were used. In the first one, applied to the study of the lens’ resolution tuning, the beam of a 594 nm HeNe CW laser illuminated a pair of identical 5° axicons (manufactured by B-Con Engineering Inc. from Zeon E48R plastic material), and a 10 cm focal length diffraction-limited lens. The axicon telescope produces a ring beam with a diameter that can be adjusted by changing the distance between the axicons, as shown in Fig. 1. Note that this arrangement effectively uses the incoming beam energy versus the low efficiency achieved when using a narrow ring aperture [1,10–12]. Now, to study the influence of the thickness of the hollow beam/light ring on the DOF, one has to change the diameter of the incoming beam, which can be done using different telescopes. (Note that the ring width is about half the incoming beam diameter, as shown in Fig. 1.) However, it is a laborious process and may produce results that depend on alignment of different telescopes unless one uses, for example, flexible incoming laser beam diameter scaling employing modular/additive telescopes [27]. Consequently, we chose a simple scheme where the incoming beam is expanded by a ${10\times}$ telescope and then a circular variable aperture is used to define the beam diameter and thus the thickness of the ring beam produced by the axicon pair. In this case, as a source we used a 543.5 nm HeNe CW laser, a pair of 30° axicons (made from PMMA by asphericon GmbH), and an 8.8 cm focal length aspheric lens. The axicons and the lens were installed in an asphericon GmbH high-precision modular mounting system, significantly simplifying the individual optical elements alignment. In both arrangements the second axicon was positioned close to the front focal plane of the lens. The generated ring beam impinges on a diffraction-limited lens whose DOF and focused spot size/resolution were characterized by imaging it with a ${20\times}$ microscope objective onto a camera.
4. RESULTS AND DISCUSSION
A. Tuning the Resolution of a Lens
First, we demonstrate the tuning of the lens resolution procured by changing the diameter of the ring of light impinging on the lens. It was done by varying the distance between the two 5° axicons; the ring thickness was constant in this process and equal to about half the diameter of the unexpanded, 594 nm laser beam with a 0.37 mm waist. In all cases, the second axicon was positioned close to the front focal plane of the lens, as required for the generation of quasi-Bessel/diffraction-free beams [1,10–12]; however, we found this condition not to be critical. Figure 2 shows the generated quasi-Bessel-type beam with a scale for calibration. In all cases, the width of the ring aperture is more than an order of magnitude smaller than the ring diameter, a characteristic requirement for a quasi-diffraction-free/Bessel beam generation [10–12]. Figure 3 presents the FWHM of the focal spot as a function of the distance between the axicons. (Note that we characterize the beam by FWHM and not waist to be able to compare Bessel beams with Gaussian ones.) It closely follows the 1/NA dependence of a focal spot on the incoming beam diameter. In our experiment, the spot size is decreased by a factor of ${\sim}{3.5}$ as the outer ring diameter increased from 5.2 mm to 18.3 mm. In our setup, this factor was limited due to mechanical restrictions and the size of clear aperture of the lens; in principle, however, it could be larger. The size of the focal spot was constant over a long distance, exceeding the Rayleigh range/DOF of a lens illuminated by a Gaussian beam with a waist comparable to our ring radius. A detailed study of the DOF dependence will be presented in the next section. The FWHM of the smallest spot, obtained for a ring with 18.3 mm outer diameter (${\sim}{17.9}\;{\rm mm}$ average ring diameter) was measured to be 2.5 µm, while an ideal Bessel beam focal spot should have a ${\rm FWHM} = {0.36}\lambda {\rm /NA} = {2.38}\;\unicode{x00B5}{\rm m}$.
Fig. 3. FWHM of the focal spot versus distance between the axicons. The solid line is plotted using Eq. (3).
To compare our experimental results with the theory in Eq. (3), we relate the ${\rm NA} = {\sin}[{\rm arctan}({{R}_{\rm av}}/\!{f})]$ to the axicons parameters and the distance L between them, using ${{ R}_{\rm av}} = {L}$ tanβ. Here $\beta$ is the convergence angle after an axicon that has a base angle $\alpha$ and a refractive index ${ n}$, and it is given as $\beta = {\rm arcsin}({n \sin}\alpha ) - \alpha$. The results, using the parameters of our setup, are shown as a solid line in Fig. 3. They are in a good agreement with the experimental values.
In many experiments to increase the effective NA and the resolution, the focusing microscope objective is usually overfilled by the incoming beam (with a concomitant energy loss). Thus, it would be fair to compare our results with the theoretical ones for a plane incoming wave, which is a close approximation to an overfilling beam. For a plane wave illuminating a 17.9 mm aperture, the ${\rm FWHM} = {0.51}\lambda {\rm /NA} = {3.38}\;\unicode{x00B5}{\rm m}$, which is significantly larger than the 2.5 µm focal spot measured for the hollow beam. This demonstrates the advantage of using an axicon-pair-produced tunable size ring beam that, upon focusing, generates a quasi-Bessel-type beam with a controllable spot size.
Note that our scheme is very energy efficient because no annular aperture or similar means were used to obtain the ring beam.
B. Tuning the Depth of Field of a Lens
As mentioned, the influence of the thickness of the ring on the DOF of the focused spot requires either varying magnifications of the incoming beam by different telescopes, which requires elaborate alignment and complicates comparison, or employing an adjustable telescope. Thus, we chose to use a variable diameter aperture to change the beam diameter impinging on the axicon pair, as shown in Fig. 1 with the beam expander version. The resulting hollow beam/ring thickness is about half of the impinging beam diameter. Note that while changing the aperture size, the outer diameter of the ring is kept constant and can be matched to the NA/diameter of the focusing lens. In contrast, using a telescope to expand the ring changes all of its characteristics. Our arrangement allows smooth transfer from the case of a ring-generated Bessel beam to a lens-only generated Gaussian-type one, albeit introducing energy loss. Here, while the external ring radius is constant, the thickness of the ring is increased, which fills the aperture with light and gradually transforms the beam from doughnut-like to a one without an on-axis minima. In fact, the two positive axicons produce a ring beam with an inverted intensity distribution; in other words, the intensity increases, going from the internal radius to the outer one. This effect can be reversed (i.e., a “normal” intensity profile that grows with the radius can be achieved, for example, when replacing the first axicon with a negative or diverging axicon).
Fig. 5. Peak intensity of the focused beam and FWHM of its spot size versus distance from the lens focus for a 1 mm aperture. The continuous line is a numerical simulation of longitudinal intensity distribution.
Fig. 7. Peak intensity of the focused beam and FWHM of its spot size versus distance from the lens focus for a 3 mm aperture. The continuous line is numerical simulation of longitudinal intensity distribution.
Fig. 9. Peak intensity of the focused beam and FWHM of its spot size versus distance from the lens focus for a fully open aperture. The continuous line is numerical simulation of longitudinal intensity distribution.
Fig. 11. Peak intensity of the focused beam and FWHM of its spot size versus distance from the lens focus for a case with a lens only.
The axicon pair in this experiment produces a ring of about a 11.5 mm external diameter. The incoming beam waist is 4.2 mm. The experimental results are presented in Figs. 4–9 for aperture diameters of 1 mm, 3 mm, and a 25 mm/fully opened one that produces an almost “filled doughnut” beam. Figures 10 and 11 show similar characteristics when using an “equivalent” Gaussian beam (whose description is given below) to illuminate the same lens. For each aperture, the transverse intensity profile of the generated focal spot, its FWHM, and the intensity as a function of the distance from the lens focus are depicted in the corresponding graphs. The on-axis beam intensity distribution was measured along the optical axis until the beam started to become distorted/asymmetrical. The DOF of the quasi-Bessel beam, defined as the distance from the peak to half its value, is changed by a factor of ${\sim}{3}$, from 500 µm to 160 µm, as shown in Figs. 5, 7, and 9. The DOF of the “equivalent” Gaussian beam is ${\sim}{75}\;\unicode{x00B5}{\rm m}$, or half of the DOF of the case with the fully open aperture, while the spot sizes have about the same FWHM; in both cases, it is 4.6 µm, as shown in Figs. 9 and 11. The transition from a quasi-Bessel beam with multiple side lobes and a long DOF to a Gaussian-type beam with very few small lobes and a much smaller DOF is evident. Note that this transition is important not only for didactical purpose but also when recollecting that each lobe/ring of the Bessel beam carries the same amount of energy as the central lobe, rendering such a beam less efficient than the Gaussian one.
Fig. 12. Transverse intensity profiles of the beam at different positions along the optical axis ${z}$ for a 1 mm aperture case. (a) ${z} = {650}\;\unicode{x00B5}{\rm m}$, (b) ${ z} = {400}\;\unicode{x00B5}{\rm m}$, (c) ${z} = {0}$, (d) ${z} = - {400}\;\unicode{x00B5}{\rm m}$, and (e) ${z} = - {650}\;\unicode{x00B5}{\rm m}$.
Since our incident ring has a half-Gaussian intensity distribution and is not like the constant one used in [12] for derivation of Eq. (4), this equation is only a rough guide for DOF estimation. Still, the measured DOF values (Figs. 5, 7, and 9) follow Eq. (4) within a 30–50% discrepancy. We also performed a numerical simulation, based on Fourier optics, of the longitudinal intensity distribution for our optical setup, shown as continuous lines in Figs. 5, 7, and 9. When compared to the experimental results, there is a reasonable correspondence, within a 10–20% difference. Note that the change in the shape/symmetry of the DOF/axial point spread function when there is a change in the illumination ring parameters is consistent with results of [22].
It is not straightforward to compare the size of a focal spot produced by a hollow incoming beam with the one produced by a Gaussian beam due to the very different characteristics of these beams. However, one can assume that since 99% of a Gaussian beam energy is contained in a radius equal to ${\sim}{1.5}\omega$ (while only $\sim86\%$ is enclosed within the radius equal to the waist $\omega$), this value, ${1.5}\omega$, is “equivalent” to the thin ring radius for the comparison between the focusing properties of two beams. Thus, we can use our ring beam with ${{R}_{\rm ring}}\;\sim\;{1.5}{\omega _{\rm in}}$ to compare between focusing by a hollow beam and an “equivalent” Gaussian one. For a Gaussian incoming beam with a waist ${\omega _{\rm in}} = {4.2}\;{\rm mm}$, the focal spot ${\rm FWHM} = \omega {({2\rm ln2})^{1/2}} =(\lambda {\rm f}/\pi {\omega _{\rm in}}){({2\rm ln2})^{1/2\:}} = \;{4.3}\;\unicode{x00B5}{\rm m}$, which is close to the one measured for a lens only, at 4.6 µm, as shown in Fig. 11. The 11.5 mm diameter ring-generated spot was measured to be 3.3 µm (Fig. 5), while the theoretical Bessel beam ${\rm FWHM} = {0.36}\lambda {\rm /NA} = {3}\;\unicode{x00B5}{\rm m}$. (Note that the Gaussian beam Rayleigh range/${\rm DOF} = \pi {(\omega)^2}/\lambda = (\lambda /\pi {\rm }){x}{({f}/{\omega _{\rm in}})^2} = {76}\;\unicode{x00B5}{\rm m}$, while the measured DOF was 75 µm.) This demonstrates the advantage of the hollow illuminating beam compared to a Gaussian one.
Finally, Fig. 12 depicts the transverse intensity profiles of the beam at different positions along the optical axis for a 1 mm aperture, confirming the Bessel-type beam behavior of the focal spot.
5. CONCLUSION
In conclusion, we have shown that using an axicon-pair-generated tunable annular ring/hollow beam to illuminate a lens, one can control the lens resolution and its depth of field. Depending on the impinging ring parameters, either one or simultaneously both of these characteristics can exceed those produced by focusing a plane wave or a Gaussian incoming beam. The tunability of the focal spot size and DOF can be beneficial in areas such as imaging and material processing/nonlinear light–matter interaction.
Disclosures
The authors declare no conflicts of interest.
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