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Abstract
Optical aberrations can greatly distort the image created by an optical element. Several aberrations can affect the image simultaneously and discerning or visualizing specific aberrations can be difficult. By making use of an optically levitated droplet as a light source, we have visualized the spherical aberration and coma of a lens. The droplet approximates a point source in the ray optic regime but, at the same time, creates a diffraction pattern in the far-field region similar to that used in the Ronchi test. When focused by a lens, this micro double point source creates patterns that resemble comets, barreling, hyperbolic triangles, and, most strikingly, a spider. We show how all these patterns are a consequence of spherical aberration and coma. The Zernike polynomials were used to quantify the value of several individual aberrations by comparing them to patterns resulting from numerical simulations.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The study of optical aberrations is an important part of the manufacturing and design of optical instruments in astronomy, physics, optometry, and optical engineering [1]. In particular, spherical aberration can be experimentally tested by some simple techniques to verify both the failure of paraxial optics and the importance of aberrations in optical systems. A common testing technique is the use of resolution test targets, where a set of known features are observed through an optical element, and the faults in the image can be traced back to different aberrations.
There is another group of tests where an object is placed close to the focus point of the optical element. One of these is the Foucault test [1,2] where the object is a knife-edge. It was originally developed to assess the sphericity of concave mirrors for astronomical use. Another of these is the Ronchi test [1,3] where a converging beam coming from an optical element being tested crosses a Ronchi grating made of straight parallel lines. The beam is then projected on a surface, where the resulting image, called a Ronchigram, is analyzed [1]. The deviation of the Ronchigram from straight lines is a consequence of the aberrations in the tested optical element. Due to their simplicity, both the Foucault and Ronchi tests are still some of the most widely used methods to evaluate the aberrations of optical systems [4] and similar procedures have been used more recently in estimations of the phase wave-front aberrations [5].
Apart from the previous classic tests, there are many other ways to quantify optical aberrations. For example, Furlan et al. [6] propose the use of a low-cost optometric instrument. Another technique is the use of interferometry to measure the emitted wave-front [1,7]. The deviations from a perfect wave-front can then be traced back quantitatively to the individual aberrations by fitting it with Zernike polynomials [8], which are a sequence of orthogonal polynomials on the unitary disk. They are commonly used to simulate aberrations in circular optical elements [1] since most aberrations are related to a particular polynomial.
Another novel method consists of using a small particle in an optical tweezer trap as a light source to study optical aberrations. One of the main advantages of this technique is that a trapped micro-particle produces a tiny ($\mathrm {\mu m}$ to nm sized) and highly controllable point source. Abdelsalam et al. [9] observed the diffraction spot of a nanosphere through a microscope objective along the direction of the trapping laser and used the Nijboer-Zernike model to quantify the spherical aberration of the microscope objective.
Despite the large range of testing techniques, optical aberrations are still hard to visualize. The original wave-front cannot be seen directly, the aberration of the point spread function is very subtle and the Ronchigrams cannot show effects of the aberration before the focus point. Some attempts can be found in the literature that visualize spherical aberration. Escobar et al. [10] present an experiment for the visualization of the impact of spherical aberration in the point-spread function based on the analogy between point-spread function of spherical aberrated systems and the defocused pattern of a 1-D slit-like screen. Another proposal to visualize and characterize spherical aberration is given by Amir et al. [11] by using Fourier Transforms spectral interferometry in multi-photon microscopy.
In this paper, we use the $90^{\circ }$ scattered light of an optically levitated micro-metric droplet as a light source. Since the droplet scatters mostly from the bottom and the top, it creates two point sources micro-meters apart. This was first observed by Ashkin [12,13] in 1970 during the work on optical tweezers that awarded him the Nobel prize in 2018. When observed perpendicularly to the trapping laser, the particle creates a striped, double-point diffraction pattern similar to the Ronchi pattern. The striped structure of the light in the far-field produced by this micro double point source allows us to recognize different patterns that arise after passing through the optical element and to quantify its optical aberrations in a similar way to the Ronchi test. Unlike the Ronchi test, this method shows the whole focus and defocus of the light and allows us to see the patterns formed before and around the area where the beam converges. Furthermore, this method can also be used to test other aberrations than just the spherical aberration. We present here what we call the barreling, spider, and hyperbolic triangles patterns created solely by spherical aberration, as well as a comet pattern formed by coma. Numerical simulations were performed using the Zernike polynomials confirming the same patterns. These were compared to the experimental images to quantify the values of different aberrations. Moreover, it is shown how these patterns can help in the alignment of optical elements even without quantitatively measuring the optical aberrations.
2. Experimental method
Our experimental setup has already been used to study optically levitated droplets interacting with radiation [14], as well as for the analysis of harmonic oscillations in the motion of the droplets [15]. Consequently, we only provide a short description of the experimental setup as used for the optical aberration analysis.
A silicone oil droplet ($\simeq$ 30 $\mu$m in diameter) was optically levitated in air inside an aluminum vacuum chamber 60 mm high, 50 mm wide and 50 mm deep with windows on all sides. The trap was created using a 532 nm CW linearly polarized laser (Laser Quantum gem532) with a beam diameter of 0.9(1) mm directed upwards and weakly focused with a biconvex lens (f = 100 mm). A liquid micro-dispenser (GeSiM Bent Steel Capillary) was used to dispense droplets into the laser beam. The liquid in the dispenser consisted of a solution of 5 parts of isopropanol to 1 part of silicone oil. Once trapped, the isopropanol quickly evaporates and only the silicone oil remains. Measurements of the shape parameter of falling droplets performed by Malot et al. [16] found increasing sphericity as the diameter of the droplet was reduced from 1400 to 200 $\mathrm {\mu }$m. This stands to reason since the surface tension of droplets increases as the diameter decreases [17]. We therefore expect a very high surface tension in the droplets used to perform our experiments (diameter $\simeq 30$ $\mu$m). This was experimentally confirmed by studying the diffraction pattern formed by the laser light and by imaging the droplet using a high resolution camera.
A trapped droplet scatters mostly from the bottom and top as light enters and exits the droplet vertically from below. Perpendicularly to the trapping laser, these two shiny spots micro-meters apart produce an axially symmetric diffraction pattern similar to the double-slit diffraction pattern. In Fig. 1, we show a graphical sketch of the laser, the droplet, the two light sources, and a photograph of the diffraction pattern.
Fig. 1. A levitated droplet (left blue sphere) scatters light mostly from the top and bottom creating a diffraction pattern on the far-field seen on the right ($90^\circ$ scattered light). The droplet and pattern are not shown to scale.
The two points are so close together that, in a ray optics regime, they appear as a single point source, but are sufficiently apart to create a diffraction pattern in the far-field. If a smaller source is needed for further applications, such diffraction patterns still appear in particles as small as 5 $\mu$m [18].
Three different lenses were tested for aberrations: a 25 mm in diameter plano-convex lens (f = 100 mm), a 50 mm in diameter biconvex lens (f = 100 mm) and a 25 mm in diameter plano-convex diffraction-limited aspheric lens (f = 50 mm, ThorLabs AL2550G). To perform a test, a lens was placed with its principal axis perpendicular to the trapping laser and at the same height as the levitated droplet. The droplet was placed slightly outside of the focal distance of the lens to produce a large magnification.
3. Spherical aberration
3.1 Experimental results
Since the ideal shape of a focusing lens is a parabola, a spherical lens will always induce spherical aberration. This is shown schematically in Fig. 2, where the rays of light from a point source that pass through the edges of the lens are focused before the rays passing close to the center of the lens. Two vertical arrows mark the circle of least confusion [19], where the beam waist is the smallest and the beam seems to focus, although the true point where the unaberrated image would be formed is further away from the lens. The vertical dotted lines show the image planes of the outer and inner rays, which are at different positions due to the spherical aberration.
Fig. 2. The spherical aberration of a lens focusing a micro double point source creates a set of recognizable structures associated to specific positions on the focusing region of the lens. A schematic drawing (top) shows the paths of different rays emanating from a point source and being focused by a lens with spherical aberration. Insets (a) to (g) show hand-drawings (above) and real images (below) of the patterns visible at each marked distance from the lens.
Insets (a)-(g) in Fig. 2 show hand-drawings to highlight the features of the experimental results (above) and examples of real images created by spherical aberration (below), together with the names we have given them. Figure 2(a) shows the initial stripy pattern. After the lens, in Fig. 2(b), the image barrels slightly. Then, in Fig. 2(c), the outer rays visibly cross over the inner rays and create a new pattern on the edges of the former barreling pattern called the half spider. At the image plane of the outer rays, this pattern on the edge converges into the center, forming the optical aberration spider shown in Fig. 2(d). The spider then shrinks, giving way to the hyperbolic triangles pattern that grow behind it, as shown in Fig. 2(e). The spider pattern continues to shrink until it becomes a small dot at the image plane of the inner rays in Fig. 2(f). Here the dominating pattern is the hyperbolic triangles. Finally, in Fig. 2(g), the spider has completely disappeared and the hyperbolic triangles start to straighten out. At infinity the aberration becomes negligible and the hyperbolic triangles tend to horizontal lines again.
To verify that these patterns were caused by spherical aberration, a diaphragm was placed on the side of the lens facing the levitated droplet. Figure 3 shows the effects on the spider pattern as the diaphragm was progressively closed. The spider pattern disappears from inside out first (Figs. 3(I)–3(III)), revealing the horizontal stripes behind it. Then, the horizontal stripes disappear from outside in (Figs. 3(IV)–3(VI)). Therefore, the spider is caused by aberrations from the outer edge of the lens, strongly suggesting its origin arising from spherical aberration. Enlargements of Fig. 3(II) are shown to the right in Figs. 3(VII) and 3(VIII) to highlight an additional structure on top of the spider. This additional structure consists of two sets of concentric circles coming from two vertically separated shiny dots in the center, highlighted by the two arrows in Fig. 3(VIII). These circles eventually merge into a single set of concentric circles near the edges of the image.
Fig. 3. Effects on the spider pattern of Fig. 2(d) as a diaphragm reduces the aperture size of the lens (small left circle). First, the spider disappears from inside-out, showing the stripes behind it. Then, the stripes disappear from outside-in. Enlarging II shows the overlap of the spider with a pattern of concentric circles coming from two vertically separated dots in the center in pictures VII and VIII. The images were taken 1.46 m away from the lens.
We performed four tests on different lenses to compare their spherical aberration, shown in Fig. 4. Figure 4(a) shows the effects of a 50 mm biconvex lens (f = 100 mm), Fig. 4(b) of a 25 mm plano-convex lens (f = 100 mm) with its convex side towards the droplet, Fig. 4(c) the same lens now with its plano side towards the droplet, and, finally, Fig. 4(d) of a 25 mm plano-convex diffraction-limited aspheric lens (f = 50 mm, ThorLabs AL2550G) with its plano side towards the droplet too. Figure 4 shows how the spider pattern becomes consecutively smaller in relation to the barreling or the hyperbolic triangles as the spherical aberration decreases from Figs. 4(a) progressively to 4(d). In 4(d) the spherical aberration is negligible and only horizontal stripes are formed.
Fig. 4. Images resulting from four different lens configurations with decreasing spherical aberration. The tests were performed on: (a) a 50 mm biconvex lens (f = 100 mm) (last photo of (a) at 75.9% of original size); (b) a 25 mm plano-convex lens (f = 100 mm) with its convex side towards the droplet; (c) the same lens now with its plano side towards the droplet; and (d) a 25 mm plano-convex diffraction-limited aspheric lens (f = 50 mm, ThorLabs AL2550G) with its plano side towards the droplet. The distance of the image to the lens, in meters, is marked below the images. Below the position of the images the calculated distance to the true image plane l is stated (see Sec. 3.2).
3.2 Simulation
As a next step, we proceeded to simulate these previous patterns using the Zernike polynomials [1] to quantitatively characterize our findings. When a point source is focused by an optical element, the ideal wave-front would be a segment of a sphere with its center at a distance l from the lens, where the image of the point forms. Each ray of light follows a path perpendicular to the wave-front, and they all coincide in the center. Optical aberrations alter this initial wave-front, change the direction of propagation of each ray of light, and hence, alter the image.
Approximating our source to a point, we initially modeled our wave-front as a sphere. We used cylindrical coordinates where z follows the principal axis of the optical lens. Placing the lens on the plane perpendicular to the z axis at $z = 0$, the wave-front is the surface given by [20]
where $\rho$ is the radial distance from the center of the lens. Aberrations can be added to this wave-front by adding specific Zernike polynomials to the surface in Eq. (1). Table 1 shows some important Zernike polynomials [1,8,21] and their relation to specific aberrations. For example, adding Spherical and X-Coma aberrations results in
Table 1. Zernike polynomials and their relation to specific aberrations [1] in an unitary circle.
This surface can be written implicitly in the form $\mathbf {W}(\rho , \theta , z) = 0$ and the normal vector at any point is given by $\mathbf {n} = \nabla \mathbf {W} (\rho , \theta , z)$. In vector component notation, one gets
Since the Zernike polynomials are defined in the unitary disk, the simulations were performed with normalized radii, $\rho \in [0,1]$. The resulting length to the image plane l and the size of the pattern were then scaled up to their real size by multiplying by the lens’s radius. The Zernike coefficient was not scaled up, providing a comparison of the aberration for all the tested configurations independent of the lens’ radius.
The simulation’s known constants were the diameter of the lens and the density and placement of the lines in the circle of light entering the lens. The parameters that defined the resulting pattern were the coefficients of the Zernike polynomials $Z_3^1$ and $Z_4^0$ as well as the distance to the true image plane $l$. The structure and size of the simulations was compared to the structure and size of the observed patterns at known distances from the lens. The parameters were changed until the best comparison of size and pattern between simulation and image was found. The uncertainties were estimated by the sensitivity of the simulations when compared to the real images as the parameters were varied.
Figure 5 shows a comparison of the images and simulations for the plano-convex lens with its convex side towards the drop. In this case, the lens was not rotated around the y-axis and no coma was induced, i.e., $Z_3^1 = 0$. The top row shows how the images are qualitatively similar, while the bottom row shows a comparison of the obtained image diameters to scale. The spherical aberration, expressed in its Zernike polynomial coefficient, was found to be of $Z_4^0 = (2.2 \pm 0.1) \times 10^{-4}$ and the distance to the image was found to be $l = 2.44 \pm 0.13$ m.
Fig. 5. Comparison of the real spherical aberration (left half) to the simulated images (right half) during the test for the spherical plano-convex lens with the convex side towards the droplet. The top row shows the qualitative similarities between them. The bottom row compares the diameters of the image and the simulation. As expected, a minimum diameter is reached around the circle of least confusion where the spider pattern is formed. The distance to the lens in meters is marked below the images.
Similar simulations were performed to compare the images taken from the lens with its plano side facing the droplet and from the biconvex lens. Table 2 shows the resulting coefficients of the spherical aberration.
Table 2. Values of the spherical aberration obtained for three different lenses.
3.3 Discussion
The transition from large to negligible spherical aberration in Fig. 4 can be seen straight away from the decreasing size of the spider pattern. This could also be seen when measuring the circle of least confusion produced by focusing a simple point source, but using a micro double point source adds structure to the resulting images. This gives an important point of reference and allows us to determine the magnitude of different aberrations by comparing the experimental results with simulations. It also creates eye-catching patterns that serve to visualize the effects of spherical aberration.
We can state that these patterns are caused by spherical aberration since they disappear gradually as an aperture is closed, hence blocking the off-axis rays. This conclusion is supported by the good agreement with the simulation. Closing the diaphragm slightly also produced the most spectacular pattern that we have observed, shown enlarged in Fig. 3. The concentric circles coming from the two dots in the center are expected to be the point spread function around each shiny spot coming from the levitated droplet.
As expected, the spherical aberration is smaller when the plano side of a plano-convex lens faces the point source than when the convex side of the same lens faces it, as shown in Tab. 2. This can be seen when comparing Fig. 4(b) and Fig. 4(c).
4. Coma
4.1 Experimental results and simulation
Focusing the droplet with the aspheric lens results in an image of the near field of the micro double point source. This procedure defined the original light beam, thus marked with $0.0^{\circ }$ in Fig. 6(a). To induce coma, the lens was rotated around its vertical axis. As the lens is rotated, the direction of the beam changed. The angle between the direction of the original beam and the resulting beam is marked as $\theta$. As the lens was rotated in steps of $0.56^{\circ }$, the two points slowly expand into two comets, as shown in the lower row of Fig. 6.
Fig. 6. Comet pattern resulting from a distortion on the original unaberrated double point image created by the aspheric lens. Coma was induced by rotating the lens, as shown in the lower axis. The inset shows a simulation of the comet pattern when adding X-coma. Taken 2.41 m away from the lens.
The comet pattern can be simulated by keeping X-coma ($Z_3^1 \neq 0$) and removing the spherical aberration ($Z_4^0 = 0$) in Eq. (2). The result of the simulation is shown in the inset of Fig. 6. The value of the aberration was found to be $Z_3^1 = (-9.0 \pm 2.5) \times 10^{-7}$ at $\theta = 1.12^\circ$. $Z_3^1$ carries a larger uncertainty because of the smaller size of the patterns and the fact that only one pattern was being compared to the simulation, as opposed to the spherical aberrations where a sequence of patterns was compared. The rotation of the lens slightly altered the length from the lens to the screen. The change was at most 4.0 mm and was also included in the simulation.
In Fig. 7, we show the effects of inducing coma on an image with spherical aberration. The first row repeats Fig. 2(c) (half-spider), Fig. 2(d) (spider) and Fig. 2(f) (hyperbolic triangles). Each row displays the patterns obtained when the lens has been rotated in steps of $1.2.^{\circ }$
Fig. 7. Distortions on the original half spider, spider, and hyperbolic triangles patterns as the lens is rotated, thereby inducing coma. The distance to the lens in meters is marked above the images. See text for discussion.
4.2 Discussion
Figure 6 shows how even an aspheric lens can produce aberrations because of a slight misalignment. Note that visible effects already occur at half a degree of rotation from the original image.
In the first two columns of Fig. 7, we can see how coma affects, in different ways, parts of the image that are in the front or in the back. In the first column, the incipient spider is affected first, while the lines on the back only start disappearing in the last two rows. In the second column, the spider in the front survives longer, while the hyperbolic triangles behind it expand outwards. The spider only starts disappearing in the last three rows where the hyperbolic triangles have almost disappeared. In the third column, the right side of the image is the one that disappears, as opposed to the first two rows where the left side disappears. Since the third column has no overlap between inner and outer rays, it all disappears simultaneously.
5. Other aberrations
Spherical aberration and coma, which have been discussed above, are two of the five Seidel aberrations, which are the basic optical aberrations due to the geometry of lenses or mirrors in interaction with monochromatic light [22]. Our simulations can, however, include any term in the Zernike polynomial including defocus, tilt, astigmatism and higher orders. Two other Seidel aberrations, field curvature and distortion, affect the image created by different off-axis rays. Such rays can, for example, be the ones that come from different points outside of the lens’ principal axis when focusing a 2D image. By using a source of light that is to a good approximation a point source, we eliminate both of these aberrations. The remaining Seidel aberration is astigmatism.
A simulation of the effects of oblique astigmatism ($Z_2^{-2}$) after focusing the micro double point source with a lens is shown in Fig. 8. In this simulation both spherical aberration and coma were removed. This represents what we would expect to see if a correctly aligned aspheric lens were astigmatic. The value of the oblique astigmatism coefficient was chosen to be of the same order of magnitude as the smallest spherical aberration out of the three lenses we measured, that is $Z_2^{-2} = 1 \times 10^{-4}$. These aberrations were never observed in our experimental images. In Fig. 6(a), we were able to focus the points coming from the droplet without the appearance of any lateral elongations as shown in Fig. 8(e). The absence of astigmatism stand to reason, since optical lenses, as the ones used for our experiments, are produced with axial symmetry around the principal axis of the lens.
Fig. 8. Expected effects of oblique astigmatism ($Z_2^{-2} = 1 \times 10^{-4}$) on the focusing of the micro double point source. The top row shows the patterns formed and the bottom shows the size of each pattern to scale.
6. Conclusions
Today, optical manipulation is a common technique, but, to our knowledge, the $90^{\circ }$ light scattering of such a system has not been used before to assess optical aberrations of lenses. The use of this micro double point source adds structure to the focused light and has allowed us to visualize eye-catching patterns resulting from different optical aberrations.
Of the five Seidel aberrations: spherical aberration, coma, astigmatism, field curvature and distortion, we focused only on coma and spherical aberration in this work since these two were sufficient to reproduce all the patterns we could see or induce. Field curvature and distortion are removed by using a point source, and the expected effects of astigmatism were not observed.
The findings are useful in two separate ways. First, recognizing these patterns can help the user of an optical lab to have an intuition of the aberration and alignment of an optical element in situ and straightaway. The comet pattern shows how even diffraction-limited lenses can induce aberrations when badly aligned and the pattern is recognizable even with rotations as small as half a degree. On the other hand, the whole sequence from barreling to spider to hyperbolic triangles comprises a new way to isolate and visualize spherical aberration, an aberration which is commonly very hard to observe.
Second, the measurements of coma and spherical aberration presented in this work are a proof of principle of a new way to characterize different optical aberrations. The effects of the aberrations can be measured in real-time, as opposed to other methods (e.g. interferometry), and are much easier to see (e.g. fitting a diffraction spot). Moreover, in our method, the light carries its own structure, as opposed to the Ronchi method where it has to be introduced by placing an obstructing structure in the beam path. This freedom has allowed us to use this light source to check the alignment of an optical element and could be valuable for further applications. An image recognition system could be applied in conjunction with these results in the characterization or in real-time, in situ aberration correction of optical elements.
7. Contributions
J.T.M. performed the experimental and theoretical work. D.H. provided help with experiments and their interpretation. R.C.-T. provided help with the interpretation and the theoretical work. B.B. help in processing the results. All authors contributed to the final version of the manuscript.
Funding
Vetenskapsrådet (2019-02376); Linnaeus-Palme Foundation; Dirección General de Asuntos del Personal Académico, Universidad Nacional Autónoma de México (PAPIIT-IN-111-820).
Acknowledgments
Financial support from the Swedish Research Council (2019-02376) is acknowledged. J.T.M. acknowledges support from the Linneaus-Palme foundation. R.C.-T. acknowledges support from the Linneaus-Palme foundation, DGAPA-PASPA and PAPIIT-IN-111-820, as well as to the University of Heidelberg for its hospitality during his sabbatical.
Disclosures
The authors declare no conflicts of interest.
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