We explore the diffraction and propagation of Laguerre-Gaussian beams of varying azimuthal index past a circular obstacle both experimentally and numerically. When the beam and obstacle centers are aligned the famous spot of Arago, which arises for zero azimuthal index, is replaced for non-zero azimuthal indices by a dark spot of Arago, a simple consequence of the conserved phase singularity at the beam center. We explore how the dark spot of Arago behaves as the beam and obstacle centers are progressively misaligned, and find that the central dark spot may break into several dark spots of Arago for higher incident azimuthal index beams.
©2007 Optical Society of America
The spot of Arago has had a long and illustrious history in optics . It was originally put forward by Poisson as evidence against Fresnel’s 1819 wave theory of light, Poisson being an advocate of the corpuscular theory of light, since Fresnel’s theory “violated common sense” by predicting that a bright spot would appear in the shadow region of an illuminated circular obstruction. However, when the experiment was subsequently performed by Arago the bright spot appeared in the shadow region thereby vindicating Fresnel’s wave theory of light. For a collimated incident beam, for example a Gaussian beam, diffraction leads to the formation of a centered bright spot in the shadow region surrounded by a series of concentric rings, the pattern being described mathematically by a Bessel function . More recently the spot of Arago has been studied and utilized in various settings including imaging , the measurement of superluminal fringe velocity , as well as in combination with broadband radiation . Furthermore, it is known that aberrations in the incident wave-front modify the spatial distribution of the spot of Arago which in turn can provide direct and sensitive information on Seidel aberrations present on the incident field .
In this paper we consider diffraction past a circular obstacle for incident light fields containing phase singularities, which are points in the plane of the incident field at which the phase is indeterminate, all phases are effectively present, and destructive interference creates a dark spot. For a phase singularity of azimuthal index l (sometimes referred to as the topological charge or winding number) the phase of the field varies from zero to 2πl as one traverses a closed curve encircling the dark spot (a Gaussian beam has zero azimuthal index). The canonical examples of optical fields with phase-singularities, also termed optical vortices, are the Laguerre-Gaussian (LG) modes of free-space with non-zero azimuthal index, which have helical wave-fronts and carry orbital angular momentum. The motivation for our study was the realization that if the spot of Arago experiment was repeated using LG fields with non-zero azimuthal index, and the centers of the LG beam and circular obstacle were aligned, the bright spot alluded to by Poisson turns into a dark spot of Arago by virtue of the fact that the phase singularity is conserved at the beam center upon propagation past the obstacle. Simple though this observation is it also constitutes a quite spectacular validation of the wave theory of light in its own right: the beam center remains dark for all propagation distances in and beyond the shadow region, a result that also violates common sense from the perspective of the corpuscular theory of light. The goal of this paper is to explore the dark spots of Arago using this simple observation as a starting point. For example, we investigate if the bright spot of Arago can appear with an LG input beam of non-zero azimuthal index, and how the dark spot of Arago behaves if the centers of the LG beam and circular obstacle are displaced with respect to one another.
2. Propagation model
We model propagation of a LG beam along the z-axis past the obstacle using the paraxial wave equation for the slowly-varying scalar electric field envelope ε(r⃗) of a monochromatic field of frequency ω
where beam diffraction is described by the transverse Laplacian ∇2⊥=∂ 2/∂x 2+∂ 2/∂y 2. Here we have assumed that the field is linearly polarized so that a scalar wave analysis is sufficient. The effect of the circular obstacle of radius R on the incident LG beam is included in the initial condition
where εin characterizes the input field strength, l≥0 is the azimuthal index of the incident LG beam, and w0 is the incident Gaussian beam spot size. LG beams are characterized by two indices, the azimuthal index l, and a radial index p such that the number of intensity rings of the mode is (p+1). Here we have chosen p=0 so that the LG beams are single ringed with a intensity peak at radius rl=(l/2)1/2 wyay0. The spatially dependent field transmission function t(x,y,D)=Θ((x-D)2+y 2-R 2), where Θ(ζ) is the Heaviside function which is zero for negative argument ζ and unity for positive arguments, and D is the displacement of the center of the circular obstacle along the x-axis from the center of the incident LG beam. The Heaviside function in the field transmission function ensures that the field just past the circular obstacle vanishes for transverse positions (x,y) that intersect the obstacle.
We have numerically solved Eq. (1) with the initial condition (2) using standard Fourier transform methods  for a wavelength λ=632.8 nm, an obstacle of diameter 2R=2.38 mm, and a propagation distance L=645 mm past the obstacle. For each non-zero incident azimuthal index the Gaussian beam spot size w0 is adjusted so that the peak intensity of the single-ringed LG beam occurs at a radius of about 1.1 mm. Specifically, for l=1 we set w0=1.6 mm, and for l=3 we set w0=0.87 mm. In order to minimize numerical constraints due to edge diffraction effects in the simulations we have used a super-Gaussian approximation to the transmission function
with m a positive integer. Typically m=7-10 in the simulations, and this ensures that the results vary little with increasing m.
For the special case that the LG beam and obstacle centers are aligned, D=0, the field just past the circular obstacle has an annular shape, retains the same azimuthal index as the incident field, and hence has the same azimuthal phase structure. Then standard analysis, formally in terms of the exact Lommel function solutions for a circular obstacle [6, 7], shows that for an incident field of azimuthal index l the near-axis field beyond the obstacle is proportional to the Bessel function Jl(αr)exp(ilθ), with α=2πR/λL, and (r,θ) the radial position and azimuthal angle in cylindrical coordinates. More generally, for non-zero azimuthal index the intensity is zero on-axis for all propagation distances past the obstacle. This underpins the observation of the dark spot of Arago, which is a simple consequence of the fact the same phase winding and interference that enforces the vanishing at the core of the incident LG beam applies to the field propagated past the centered obstacle. When the obstacle is not centered the situation is more complex and we shall explore that case below using numerical solutions.
A generic experimental setup to observe the Arago spot is shown in Fig. 1. A standard linearly polarized Helium Neon laser (λ=632.8nm, average power =20 mW) was used for the experiment. The Laguerre-Gaussian (LG) beams were generated using a spatial light modulator (SLM). The SLM (Holoeye LC-R 2500) was used as a dynamic diffractive optical element using a computer generated hologram . We deliberately adjusted the ring diameter (maximum intensity point) of each annular LG beam used to be around 2 mm, which is slightly smaller than the diameter of the obstacle, 2R=2.38 mm, which was a ball bearing.
An imaging lens (f=25 mm) was placed such that the image projected on the camera originated from a plane at a distance of L=645 mm after the obstacle. The image projected on the camera (Watec WAT-250D, 752×582 pixels, chip size 4.78 mm×4.3 mm) was still clearly in the shadow region of the obstacle, where the Arago spot can be observed. The shadow region behind the obstacle was found to extend far beyond the optical bench. In order to obtain a controlled displacement of the obstacle with respect to the beam, the obstacle was placed on an x-y translation stage that permitted positioning to an accuracy of 2.5 micron and translation in steps of 50 microns.
4.1 Aligned case
In this section we consider the aligned case where the centers of the circular obstacle and LG beams coincide, and for azimuthal indices l=0 and l=1. Figure 2(a) shows the measured intensity beam profile past the obstacle for zero incident azimuthal index l=0, and the bright spot of Arago is evident at beam center. In contrast, a dark spot of Arago is evident at the center of the measured intensity beam profile in Fig. 2(b) which corresponds to an incident azimuthal index l=1. In this case the phase singularity at the center of the incident beam is maintained past the obstacle, and this is verified by the appearance of the characteristic fork at the center of the interferogram shown in Fig. 2(c), which is obtained by interference of the field of the dark Arago spot with a reference collimated field (a broad Gaussian beam).
Furthermore, Fig. 3 shows a comparison of the measured and calculated beam intensity cross sections for l=1 with good overall agreement. In particular, we see that the near-axis intensity exhibits radial oscillations in keeping with the expectation that the near-axis field should display a Bessel function profile J1(αr)exp(iθ), meaning the near-axis intensity profile should vary as |J1(αr)|2. Using the result that αr=3.83 for the first zero of the Bessel function J1(αr)=0, and α=2πR/λL, we find that the first zero should appear at a radius of 209 µm, in reasonable agreement with experiment. The Bessel function approximation clearly breaks down away from the axis where the intensity increases near the boundary of the obstacle.
4.2 Displaced case (l=1)
Having exhibited the basic dark spot of Arago in the last section we next explore a variant of the experiment where the centers of the circular obstacle and the LG beam are displaced a distance D along the x-axis that is transverse to the propagation direction z. Fig. 4 shows the measured (right) and calculated (left) intensity profiles for l=1 and a variety of displacements. For the centered case D=0 (top row) the dark spot of Arago appears, but for the non-zero displacement D=1 mm (second row), which is close to the obstacle radius R=1.19 mm, a combination of dark and bright spots is apparent close to the center of the obstacle, and the diffracted intensity profile is losing the cylindrical symmetry of the aligned case (the center of each 2D plot is the center of the incident LG beam). For the largest displacement of D=1.75 mm (bottom row) the dark Arago spot has been replaced by a bright Arago spot that appears at the center of the obstacle, and the diffracted intensity profile has developed a distinct angular asymmetry.
The physical interpretation of the results seen in Fig. 4 is quite straightforward: When the centers of the obstacle and LG beam are displaced, with respect to one another, the field diffracted past the obstacle is no longer cylindrically symmetric around the z-axis, which is chosen to pass through the center of the obstacle, meaning the measured intensity profile need no longer be cylindrically symmetric as seen in Fig. 4. More specifically, the phase singularity present at the center of the incident field need no longer be preserved, and the diffracted field past the obstacle must then be considered as a superposition of LG beams with differing azimuthal indices with respect to the center of the obstacle, including l=0. This is why the bright spot of Arago can reappear in the presence of misalignment.
To provide a more quantitative theoretical picture of these results we employ the statistical uncertainties for angular position θ on the range [-π,π), and angular momentum as described by Franke-Arnold et al. . In units for which ħ=1, the angular momentum operator L̂z=l̂=-i∂/∂θ acts also as a azimuthal index operator in that l̂ acting on an LG beam of azimuthal index m produces m times that LG beam. Averages of the angular position operator (Ô=θ) and azimuthal index operator (Ô=l̂) raised to the power m are evaluated using the definition
from which we evaluate the statistical uncertainties using ΔO=(<O2>-<O>2)1/2. We have numerically evaluated the statistical uncertainties in angular position Δθ and azimuthal index Δl using the field in Eq. (2) just past the obstacle. (The apodized approximation to the transmission t(x,y,D) in Eq. (3) was used to avoid numerical issues associated with performing numerical derivatives at a hard edge, but care was taken to ensure that further apodization did not significantly modify the results).
Figure 5(a) shows the variation of the angular position uncertainty Δθ (solid curve), mean azimuthal index <l> (dotted curve), and azimuthal index uncertainty Δl (dashed curve) versus displacement D. For the centered case D=0 we see that <l>=1 and Δl=0, meaning that the field after the obstacle retains the incident azimuthal index l and has zero dispersion in azimuthal index, and Δθ=π/31/2 which corresponds to a cylindrically symmetric intensity profile with maximal angular uncertainty . (We remark that for these calculations the origin is chosen as the center of the obstacle.) As the displacement increases the angular uncertainty steadily decreases to Δθ=0.4 at D=3 mm, which signals that the underlying field is no longer cylindrically symmetric, and for D=1.5 mm we find Δθ≈1. Furthermore, the range of azimuthal indices present in the transmitted beam is roughly between <l>-Δl→<l>+Δl, and we have plotted these boundaries as a function of displacement in Fig. 5(b). We see that for D>1.5 mm Δl takes values such that <l>-Δl<0, meaning that the diffracted field may have a significant component of l=0: this explains why the bright spot of Arago can make an appearance at the center of the obstacle in the bottom row of Fig. 4.
The statistical uncertainties for the angular position and angular momentum thus provide a means to understand the effects of displacing the obstacle. Furthermore, we see that a displacement D=1.5 mm larger than the obstacle radius R=1.19 mm is required to replace the dark Arago spot by the more familiar bright Arago spot at the center of the obstacle.
4.3 Displaced case (l=3)
As a final example we consider the effect of displacing the obstacle on an incident LG beam with higher azimuthal index l=3, as shown in Fig. 6 for a variety of displacements. For each row in Fig. 6 the right hand side profile is the result of the simulation for the marked displacement, and the left hand side is the corresponding experimental profile, with good overall agreement between the two.
Here we find that as the displacement between the incident beam and obstacle centers is increased the single dark spot of Arago present in the intensity profile for the centered l=3 LG beam shown in the top row of Fig. 6 breaks up under diffraction into a number of spatially separated dark spots as evidenced in rows three and four: We shall argue below that these dark Arago spots are associated with phase singularities that have been broken apart by the displacement. Furthermore, as the displacement between the obstacle and incident field centers is increased the intensity profiles are seen to develop distinct angular symmetries similar to the l=1 case studied in the last section.
To explore these results in Fig. 7 we plot (a) the statistical uncertainties in the angular position Δθ and azimuthal index Δl versus displacement D, and (b) the boundaries of the range of azimuthal indices present in the transmitted beam for an l=3. Once again we see that as the displacement D is increased the angular uncertainty Δθ decreases from π/31/2 in keeping with the development of angular asymmetry. In contrast to the case l=1 considered in the last section, a bright spot or Arago does not appear at the center of the obstacle in the present example even for the largest displacement D=1 mm, and this is consistent with Fig. 7(b) according to which the lower bound on the azimuthal index <l>-Δl becomes negative only for D>1.8 mm. Nonetheless, there is a statistical uncertainty Δl≈1 for the larger displacements, meaning that the output field should be considered as a superposition of a few LG modes with azimuthal indices between <l>±Δl, and centered around the obstacle. The effect of superposing LG beams of differing azimuthal index, centered of the same axis, is illustrated using the example of two collimated beams with l=0 and l=1,
where the first and second terms on the left hand side represent a Gaussian solution with l=0, and an LG solution with l=1, respectively, and η is a complex constant. Simple algebra shows that this superposition can be rearranged into the right hand side which represents a single l=1 LG field but now centered at x0=-Re(η), y0=-Im(η). In general superpositions of LG beams centered on the same axis leads to a field distribution with multiple phase singularities with their centers at differing positions over the transverse (x-y) plane.
Based on the general argumentation given above, we expect that when the beam and obstacle centers are displaced the transmitted field, viewed as a superposition of LG fields centered on the obstacle, will exhibit a number of phase singularities over the transverse plane. This is verified in Fig. 8 where we show the experimental interferograms taken near the center of the obstacle for (a) the centered case D=0, (b) D=0.25 mm, and (c) D=0.5 mm, all corresponding to the case l=3 in Fig. 6 (the dashed lines are included to aid the eye in viewing the interference patterns). For the centered case a three pronged structure appears in the interferogram which reflects the fact that a single l=3 phase singularity is present at the obstacle center. In contrast, for the displaced cases shown in Figs. 8(b) and 8(c), three individual single pronged structures with the same orientation are seen indicating that the incident l=3 phase singularity has broken up into three l=1 phase singularities that are centered at different positions.
We have verified the breakup of the phase singularities in our simulations and the results are shown in Figs. 9(a)–9(c) which show the spatial profile of the field phase for l=3 and displacements D=0, 0.25, 0.5 mm, respectively. These plots have the phase color coded with dark red representing π and dark blue representing -π as shown in the color bar accompanying each plot. Then if we take the centered case in Fig. 9(a) in one transit around the origin we see the phase going through three 2π cycles going clockwise, as expected for an l=3 phase singularity. (The complicated phase structure outside the central region is due to the phase curvature of the diffracted field.) If we now consider the displaced case in Fig. 9(c) for D=0.5 mm, then we see that in the central region the single phase singularity for the centered case has been replaced by three displaced phase singularities whose phase goes through a 2π cycle going clockwise. The distance between the three l=1 phase singularities increases with increasing displacement [compare Figs. 9(b) and 9(c)], as one might expect intuitively.
Our results show that the single l=3 dark Arago spot that arises for zero displacement breaks up into what is close to a row of three l=1 dark Arago spots in the presence of non-zero displacements, and our simulations show corresponding behavior for incident LG fields with larger azimuthal indices. The break up of a higher-order vortex into a row of vortices of unit azimuthal index has been predicted by Dennis for elliptical perturbations of the initial field . Although different from the present situation in that the effect of the displaced obstacle is neither elliptical nor kept perturbative, we contend that aspects of Dennis’s predictions are reflected in our experiments and simulations.
Finally we remark that for even larger displacements the phase structure of the fields exhibits further interesting structure, a numerical example of which is shown in Fig. 9(d) for a displacement of D=1 mm. In particular, the phase singularities in the central region are now accompanied by a series of quasi-aligned vortices of index unity with alternating helicity that appear in the upper left quadrant outlined by the dashed white lines in Fig. 9(d): The phase singularities higher (right) up in the quadrant go from blue-to-yellow-to-red in their color coding going in a clockwise direction around their origin, whereas those lower (left) in the quadrant go from blue-to-red-to yellow, indicating the upper and lower phase singularities have opposite helicity. Such arrays of quasi-aligned vortices of index unity with alternating helicity, termed vortex streets, have been predicted and observed within optical resonators operating on higher-order modes , and also nonlinear media due a combination of diffraction and Poynting vector walk-off [12, 13]. Vortex streets constitute quasi-aligned arrays of unity azimuthal index vortices and have interesting analogues within fluid mechanics.
In Fig. 10 we show an experimental interferogram corresponding to the upper left quadrant enclosed in white dashed lines in Fig. 9(d) for a displacement D=1 mm, and we see direct experimental evidence for the predicted vortex streets. We draw particular attention to the fact all the forks observed in the lower (left) portion of the interferogram have the same orientation which is opposite to the orientation of the forks in the upper (right) portion, indicating that the vortices in the upper and lower portions have opposite helicity as is the case for vortex streets.
In summary, we have re-visited the classic Arago spot experiment for incident Laguerre-Gaussian light fields which have phase singularities. In particular, we have shown that for non-zero incident azimuthal index the bright spot of Arago is replaced by a dark spot of Arago which persists in the presence of misalignments of the centers of the obstacle and LG beam which are somewhat smaller than the obstacle size. When the displacement approaches the obstacle size, and for unity azimuthal index, the dark spot may be replaced by a bright spot, a phenomenon we explained using the statistical uncertainties for the angular position and azimuthal index of the field. For a higher incident azimuthal index l the single dark Arago spot for the centered case is seen to break into l dark Arago spots of unit azimuthal index as the centers of the incident LG beam and obstacle are displaced, an effect analogous to the vortex rows predicted by Dennis for elliptic perturbations of LG and other beams . Furthermore, for displacements approaching the obstacle radius vortex streets in the form of arrays of quasi-aligned vortices of index unity with alternating helicity have been observed. In all cases studied the experimentally recorded intensity profiles agree well with numerical calculations.
The results presented in this paper might find application for accurate alignment control of optical systems involving LG beams, could potentially be used for probing the azimuthal phase front of a beam or even for generating multi-vortex traps for low index or other microscopic particles.
We would like to thank the UK Engineering and Physical Sciences Research Council for their support of this work. Casey Streuber is supported by the Biomedical Imaging Spectroscopy (BMIS) Fellowship Program, and Ewan Wright is funded in part by the Joint Services Optical Program (JSOP). The authors would like to thank Prof. Brian P. Anderson of the College of Optical Sciences at the University for helpful discussions.
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