Are Bessel beams resilient to aberrations and turbulence?

Nokwazi Mphuthi; Nokwazi Mphuthi; Roelf Botha; Andrew Forbes

doi:10.1364/JOSAA.35.001021

1. INTRODUCTION

Since the seminal theoretical and experimental work on Bessel beams [1,2], such beams have been extensively studied to date [3–10], having found a myriad of applications including optical trapping [11,12], communication [13–17], and quantum studies [18–22], and have even been produced as matter waves [23,24]. A particular property that has been exploited is the self-healing ability of such beams [19,25–27]. While it has been known for a long time that this property is not unique to Bessel beams [28–33], they have nevertheless become the de facto test bed for this optical phenomenon.

Traditionally the self-healing ability of such beams has been studied with opaque objects, and the reconstruction has been explained as the interference of unobstructed conical waves [34,35], as is depicted in Fig. 1. In this scenario the conical waves bypass the obstruction, which in turn produces a shadow region. After the shadow region the conical waves again overlap, and by interference reproduce the original Bessel beam, albeit it with less power and over a limited range in both the transverse and propagation planes. This reconstruction is possible only because the unobstructed conical waves are (for the most part) not influenced by the obstruction, that is, apart from some small diffraction effects, they appear to bypass the obstruction. When the obstruction is transparent, say in the form of an aberrated screen, then clearly this argument does not hold. Despite this, many studies have suggested that Bessel beams are not only resilient to all obstructions [36], but are also less affected by inhomogeneous scattering media when compared to the conventional Gaussian beam [37,38], while the literature has mixed conclusions (mostly theoretical) on the resilience of Bessel beams to turbulence [39–44].

Fig. 1. Self-healing principle of BG beams. Plane waves incident on an axicon of apex angle $α$ are refracted to form conical waves, which interfere to form a BG beam in the region $z_{\max}$ . An opaque obstruction placed at $\frac{1}{2} z_{\max}$ blocks the waves while allowing unblocked waves to interfere at a short distance behind the obstruction, i.e., after $z_{\min}$ . The inset depicts the theoretical profiles (top row) of the beam at different distances before and after the obstruction together with corresponding experimental images (bottom row). The plane $z_{0}$ is the region where the beam is expected to self-heal completely.

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Here we consider this hypothesis both theoretically and experimentally, showing that Bessel beams do not always self-heal from transparent obstructions, and never self-heal from turbulence where the entire beam is affected. Using digital holograms written to a spatial light modulator (SLM), we program controlled aberrations and study the influence on the self-healing. We find that the scattering always affects the Bessel beam but to a degree that varies with the severity of the aberration. Rather counterintuitively we show that the effect can be minimized even for strong aberrations. Finally, by combining aberrations to form a simulation of atmospheric turbulence, we show that when the entire beam passes through the screen, as it would in the real world, that the beam is always perturbed in a manner that is concomitant with the strength of the turbulence. Our results debunk the myth that Bessel beams always self-heal and provide a systematic approach to future studies in this field.

2. THEORY

In this section we outline the basic concepts for the benefit of the reader, beginning with an overview of Bessel beams, their propagation, and how to create aberrated phase screens to perturb them.

A. Bessel Beams

The ideal Bessel beam is an exact, propagation-invariant solution to the Helmholtz equation containing an infinite amount of power. This cannot be realized physically, so, often, approximations of these beams, known as Bessel–Gauss (BG) beams [45], are generated in the laboratory through techniques such as annular slits [46], axicons [47], holograms [48], interferometers [49], and inside laser resonators [50–55]. They are also easily detected using holograms and custom optics [9,10, 17]. The complex amplitude of a BG beam is given by

U (r, ϕ, z = 0) = J_{l} (k_{r} r) \exp (- \frac{r^{2}}{w_{0}^{2}}) \exp (i l ϕ),

where

J_{l}

denotes the

l

th order Bessel function of the first kind,

k_{r}

is the radial component of the wavevector with wavenumber

k = \frac{2 π}{λ}

, and

w_{0}

is the waist of the Gaussian beam. The radial and longitudinal wave vectors are related by

k^{2} = k_{r}^{2} + k_{z}^{2}

. BG beams are geometrically formed by a superposition of coherent plane waves lying on a cone. The principle behind the self-healing property is illustrated in Fig. 1, where the conical plane waves are seen to be blocked by an opaque obstruction placed at

\frac{1}{2} z_{max}

. The shadow region,

z_{\min}

, depends on the obstruction size. The self-healing occurs behind

z_{\min}

, where the unobstructed plane waves are seen to interfere. We can analytically determine the amplitude of the field transmitted at a particular distance

z

in free space by using the known solutions to the wave equation for this function [45], and from the obstruction by using the angular spectrum method or by using suitable geometric approximations [34,35]. In this work we apply the angular spectrum approach to our numerical simulations. An example of this is shown in the insets of Fig. 1 where we show a comparison of the experimental and theoretical self-healing.

B. Aberrations

To illustrate the effect of a transparent obstruction, we considered phase-changing aberrations in the form of Zernike polynomials. Zernike polynomials are a convenient way to represent aberrations since they form an orthogonal basis set over the unit circle and are relatively scaled to unit variance, thereby setting their magnitudes on equal basis for easy comparison. Zernike polynomials can be expressed in the form

Z_{n m} (ρ, θ) = {\begin{cases} U_{n m} (ρ, θ) & : m < 0; | m - n | = even \\ V_{n m} (ρ, θ) & : m \neq 0; | m - n | = odd \\ R_{n}^{0} (ρ) & : m = 0 \end{cases},

where

U_{n m} (ρ, θ) = R_{n}^{m} (ρ) \cos (m θ), V_{n m} (ρ, θ) = R_{n}^{m} (ρ) \sin (m θ),

and

R_{n}^{m} (ρ) = \sum_{s = 0}^{(n - m) / 2} \frac{{(- 1)}^{s} (n - s)!}{s (\frac{n + m}{2} - s)! (\frac{n - m}{2} - s)!} ρ^{n - 2 s} .

Here,

n

and

m

denote the radial and azimuthal indices, respectively. Figure 2 depicts phase functions of a few low-order Zernike polynomials, namely, tilt, defocus, astigmatism, coma, trefoil, and spherical aberrations. The orthogonality of the Zernike basis functions enables us to represent any phase function,

ψ (ρ, θ)

, as a sum of weighted Zernike polynomials, given by

ψ (ρ, θ) = \sum_{n = 0}^{\infty} \sum_{m = 0}^{n} a_{n m} U_{n m} (ρ, θ) + b_{n m} V_{n m} (ρ, θ),

with amplitudes

a_{n m}

and

b_{n m}

for the even and odd terms given by

a_{n m} = K (m) (\frac{n + 1}{π}) \int_{0}^{2 π} \int_{0}^{1} ψ (ρ, θ) U_{n m} (ρ, θ) ρ d ρ d θ,

b_{n m} = K (m) (\frac{n + 1}{π}) \int_{0}^{2 π} \int_{0}^{1} ψ (ρ, θ) V_{n m} (ρ, θ) ρ d ρ d θ .

K (m)

is given by

K (m) = {\begin{cases} 2 : & for m = 0, n \neq 0 \\ 1 : & otherwise \end{cases} .

Fig. 2. (a) Phase maps of Zernike aberrations representing tilt ( $Z_{1 - 1}$ ), defocus ( $Z_{20}$ ), astigmatism ( $Z_{22}$ ), coma ( $Z_{31}$ ), trefoil ( $Z_{33}$ ), and spherical ( $Z_{40}$ ). (b) An example turbulence phase mask generated from the summation of appropriately weighted Zernike polynomials [56,57].

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Here, the polynomials $Z_{n m}$ represent the type of aberration, while the coefficients $a_{n m}$ and $b_{n m}$ represent the magnitude of the aberration. One particular class of phase function is that produced by atmospheric turbulence. One can generate turbulence phase functions by summing aberrations with a particular weighting, found from Noll’s covariance method [56], and produce these as phase screens on a SLM [57]. In this approach, valid for Kolmogorov turbulence, the coefficients of each Zernike aberration $Z_{n m}$ mode are assumed to be normally distributed with mean zero and a variance determined by

σ_{n m}^{2} = I_{n m} {(D / r_{0})}^{5 / 3},

where

I_{n m} = \frac{0.15337 {(- 1)}^{n - m} (n + 1) Γ (14 / 3) Γ (n - 5 / 6))}{Γ^{2} (17 / 6) Γ (n + 23 / 6)}

and

Γ

is the usual Gamma function. Here,

D

represents the diameter of the optical aperture of the system (perhaps the laser beam or the delivery optics) and

r_{0}

is the Fried parameter that represents the turbulence coherence length. For Kolmogorov turbulence, through a path length

L

, this is given as

r_{0} = 1.68 {(C_{n}^{2} k^{2} L)}^{- 3 / 5},

where

C_{n}^{2}

is known as the refractive index structure constant and is a measure of the turbulence strength. The phase screens are then generated by making random drawings from a normal distribution with the appropriate variance [Eq. (8)] to find each coefficient

a_{n m}

and

b_{n m}

, and using these in the summation of Eq. (4) to build a phase function that is representative of turbulence. Each drawing from the random distribution will produce new coefficients and, therefore, a new phase screen. Running these phase screens one after the other then represents changing turbulence of the same strength and following the same Kolmogorov statistics. We illustrate one such phase screen in Fig. 2(b).

3. EXPERIMENT AND RESULTS

A. Experimental Implementation

To study the self-healing of BG beams after encountering a transparent obstruction, we used the experimental setup outlined in Fig. 3. An argon-ion laser operating at a wavelength of 514 nm was collimated to produce a Gaussian beam of radius $w_{0} = 0.6 mm$ , which was directed to the SLM (Holoeye PLUTO-2-VIS-056, with $1920 \times 1080$ pixels of pitch 8 μm). The SLM screen was equally divided into two parts, one for the generation of the desired beam and the other for digitally implementing the perturbation (obstruction).

Fig. 3. (a) Schematic of the experimental setup used to study the self-healing of BG beams after encountering a phase-changing obstruction. A collimated argon-ion laser source was used to generate a BG from an axicon encoded on one half of the SLM screen (labeled 1) while the second half (labeled 2) was encoded with the obstruction. The distance between SLM1 and SLM2 was $\frac{1}{2} z_{max}$ , thereby placing the obstruction in the middle of $z_{max}$ . (b) A conceptual illustration of the core optical planes, showing the position of the two phase screens of the SLM.

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A digital axicon was encoded on the first half of the SLM for the creation of the BG beam, while the obstruction side was encoded with aberration masks of a desired radius, or, finally, the turbulence phase mask. The obstruction was placed at an effective distance of $\frac{1}{2} z_{max}$ (by imaging the created beam to this plane) to ensure that the obstruction was at the center of the Bessel region. The diameter of the obstruction was chosen to be $\frac{2}{3} w_{0}$ , corresponding to the chosen Zernike unit circle. This was to ensure that the polynomials remain valid over the definition radius. Aperture A1 was placed at the focal plane of L2 ( $f = 200 mm$ ) to ensure that only the desired first order was imaged onto second part of the SLM. The resulting beam was allowed to propagate and the self-healing observed at different propagation distances after the obstruction by detection on a CCD camera (PointGrey), with most measurements to be reported at the distance $z = z_{0}$ , as shown in Fig. 3. A blazed grating was applied to all masks in order to separate the desired first diffraction order from the others, which were filtered by lenses L3 and L4 ( $f = 250 mm$ ) together with an aperture.

B. Aberration Results

Figure 4 depicts some examples of the evolution of the beam after encountering various primary aberrations, where the first two rows display the beam at the shadow region and the last two rows display the self-healing region, along with their corresponding theoretical predictions. Measurements were first taken for the reference case of an opaque obstruction and are shown in column 1. As expected, the beam is seen to self-heal at $z_{0}$ , where the beam regenerates a certain portion of its initial intensity profile. This is in agreement with previous analytical predictions [31].

Fig. 4. Theoretical and experimental images depicting the evolution of the BG beam after encountering different aberrations. Columns (a)–(g) represent different obstructions, while the rows represent different observation planes within $z_{max}$ . Column (a) depicts the reference case of an opaque obstruction, while columns (b), (c), (d), (e), (f), and (g) depict tilt, defocus, astigmatism, coma, trefoil, and spherical aberrations, respectively, with a magnitude coefficient of unity ( $a_{n, m} = 1$ ). Rows 1 (experimental) and 2 (theoretical) illustrate the influence of the obstruction at the shadow region ( $z_{\min}$ ), while row 3 (experimental) and 4 (theoretical) illustrate how the beam has evolved at the self-healing region ( $z_{0}$ ).

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To investigate phase perturbations, aberrations were encoded inside the obstruction, and the effects on the resulting self-healing observed at both the $z_{\min}$ and $z_{0}$ planes are shown in Fig. 4. In this case, the disturbance of the beam due to the aberrations is more evident at the shadow region for all aberrations types. For the case of tilt ( $Z_{1 - 1}$ ), the beam centroid is seen to be laterally displaced in the vertical direction and is still evident at the self-healing region. Not surprisingly, we find that the aberration type influences the manner of the self-healing and visually it appears that the asymmetric aberrations have a more pronounced effect on the ability to self-heal. For example, the triangular pattern due to trefoil ( $Z_{33}$ ) remains distinctive even at $z_{0}$ .

To investigate how the strength of a given aberration influences the self-healing property, we performed a correlation of the perturbed and unperturbed intensity profiles at the self-healing plane ( $z = z_{0}$ ) as a function of the aberration strength, with the results shown in Fig. 5 for astigmatism, trefoil, and spherical aberrations. These aberrations were chosen as representations of even, odd, and radial aberrations, respectively.

Fig. 5. Correlation of the perturbed beam against a reference unperturbed beam at the plane $z = z_{0}$ , plotted as a function of aberration strength, for odd and even aberrations as well as aberrations with no azimuthal dependence (i.e., $m = 0$ ).

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Intensity correlations serve as a measure of beam quality with values between 0 and 1, denoting poor and high beam quality, respectively. In this case, a high quality beam is expected for an undisturbed beam, while poor quality will characterize a beam so greatly disturbed that it no longer resembles the reference beam. Initially, with no aberrations programmed, the beam is unperturbed. As aberrations are introduced, the beam quality begins to deteriorate, as demonstrated by the declining correlation values in Fig. 5. This decline is a result of the aberration distorting the conical waves, thereby prohibiting full reconstruction. Intriguingly, after aberration strengths of 1 and above (by coefficient value), an improvement in the self-healing is noted, where the beam quality starts to increase, and thereafter converges for higher coefficients. One can understand this result by observing Fig. 6, which illustrates the relationship between peak energy and aberration strength at the plane $z = z_{0}$ . From this it can be seen that the peak energy decreases as the aberration strength increases. This decrease in energy can be attributed to diffraction spreading that occurs as the aberration strength is increased, similar to scattering. Diffraction redistributes the energy outwardly away from the beam centroid where conical reconstruction occurs. This effect is graphically illustrated in Fig. 7, where diffraction effects are seen to increase with trefoil aberration strength.

Fig. 6. Relationship between beam energy and aberration strength. The measurements are shown for astigmatism, trefoil, and spherical aberrations for the same aberration strengths as in Fig. 5. It is clear that energy decreases with an increase in aberration strength for all the displayed aberrations. The dotted lines are estimated based on a least-squared fit on the measurements.

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Fig. 7. Theoretical and experimental (insets) images of Bessel beams at the self-healing plane after encountering trefoil of magnitude, $a_{3,3}$ , of (a) 1, (b) 5, (c) 10, and (d) 20. The circles indicate the region of the beam visible to the detector. Self-healing is seen to improve with increasing aberration strength around the center of the beam.

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The diffraction effects outside the circles increase with aberration strength but the quality of the beam inside the circle is seen to improve. It is therefore evident that the greater the distortion, the more the beam seems to be diffracted. This is because large aberrations will cause significant deviations from a spherical wavefront, thereby causing the beam to spread, consequently weakening the detectable overlap with the original unperturbed conical waves. The result is the illusion of a weak aberration acting on the beam.

C. Turbulence Results

The turbulence phase screens were created by summing the weighted aberrations as described in Section 3.A. Rather than disturbing a small part of the beam, the turbulence was encoded to disturb the entire beam, as would be the case in a real atmospheric propagation. The strength of the turbulence is denoted as the normalized coefficient $D / r_{0}$ , with large values indicating strong turbulence. In Fig. 8 we show the results of the beam initially and at the self-healing plane for $D / r_{0}$ values of 3, 10, and 20, representing weak, moderate, and strong turbulence, respectively. In all these cases one finds that the beam is seen to deteriorate with propagation distance concomitant with the strength of the turbulence, i.e., no self-healing at all, which can be understood from the fact that all conical waves are disturbed, and so there is no by-passing of the “obstacle.”

Fig. 8. Experimental images of the BG beams after encountering turbulence. (a)–(c) show the BG beam immediately behind the turbulence screen while (d)–(f) show the BG beam at $z = z_{0}$ . In all plots the turbulence increases from left to right with $D / r_{0}$ values of 3, 10, and 20.

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4. DISCUSSION

It is undeniable that Bessel beams present certain advantages over other beam types used in free-space optical communications, in terms of power delivery [41,43] and individual channel efficiency [44]. However, in order to claim resilience through phase perturbations, the Bessel beam has to self-heal after encountering random phase errors. The self-healing aspect can be demonstrated in the form of intensity similarity checks of the beam before and after a disturbance has occurred. It is visually evident from Fig. 4 that obstructions in the form of aberrations do not lead to complete self-healing, as remnants of the disturbance are seen at the self-healing region. This result clearly contradicts [36], which claims that Bessel beams are able to self-heal after encountering phase disruptions. The discrepancies can be attributed to the form and size of the obstruction used. In [36] the obstruction was a combination of an amplitude and a phase obstacle of minute size ( $d = 4.8 / k_{r}$ ). In this case, the conical waves blocked by the amplitude part of the obstacle do not interact with the phase-changing obstacle. Instead, what is seen behind the obstacle is a hollow center as a result of the opaque disk, with no influence from the phase perturbation. The influence of the phase object becomes noticeable only after further propagation, where there is interference of the undisturbed plane waves with the phase obstacle. The self-healing therefore occurs as a result of the amplitude perturbation introduced before the phase object. However, in our study, where the size of the obstacle is significantly larger and the disturbance is only phase-changing, we see that self-healing of the BG beam is not always guaranteed.

The self-healing process has always been based on the intuition that undisturbed conical waves will interfere to reform the BG beam. This has previously been demonstrated for opaque obstructions of finite size, but here we show that this is not in general true for arbitrary phase obstructions. As a caveat we note that the influence of an aberration on the resulting beam is an interplay between the aberration strength and the distance at which the observation takes place, where under some conditions it is possible for the self-healing to again improve with an increase in the aberration. We have also shown that it is impossible for the beam to self-heal when the disturbance covers the entire beam, as is the case with turbulence. Here the beam is seen to deteriorate with propagation distance. It is important to note that this does not necessarily contradict previous works that have demonstrated the resilience of partially coherent and Hankel–Bessel beams [40,42]. The concept of resilience in those cases refers to the rate at which the beam quality factor degrades, as well as the beam’s ability to focus in turbulence channels. The aforementioned merits do not extend to the self-healing aspect when the beam is subjected to phase errors. Despite this realization, obstructed Bessel beams can still be efficiently utilized for optical communication and imaging applications, as the lost information can be recovered even after significant distortions. We have used the Zernike aberration weightings as our quantitative measure of aberration strength and the integrated path over turbulence [Eq. (10)] to take into account both the length ( $L$ ) and turbulence strength ( $C_{n}^{2}$ ). The point is that while the correlation of the complete optical field, obstructed to unobstructed, may degrade, it is possible that specific aspects of the field may remain useful, for example, the energy transport or the orbital angular momentum (of higher order beams) and so on. This was demonstrated for obstructed pulsed Bessel beams using adapted image-processing procedures to restore losses in contrast and spatial frequencies [58].

5. CONCLUSION

In this work, we have shown that the notion of Bessel beam self-healing under phase perturbations does not always hold true. This has been demonstrated by disturbing a BG beam with transparent obstructions in the form of Zernike aberrations. We have shown that in such a case, the influence of the perturbation is noticeable in both the shadow and self-healing regions; therefore, we cannot affirm complete self-healing as in the case with opaque obstructions. On the other hand, we can reduce the effects of the aberrations by scattering the aberrated energy to the outskirts of the beam in order to minimize its influence on the beam seen by the detector. Moreover, when the entire beam is disturbed, as in real-life atmospheric turbulence, no self-healing can occur. All impinging conical waves become disturbed, diminishing any chance of self-healing. Therefore, contrary to some suggestions, Bessel beams are not resilient to atmospheric turbulence. Finally, we point out that this study has been restricted to spatial self-reconstruction, yet there is significant interest in ultrafast pulsed Bessel pulses [59–61]. Future work could consider time–space coupling in the context of self-healing and incorporate effects from dispersive media, too.

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Are Bessel beams resilient to aberrations and turbulence?

Abstract

1. INTRODUCTION

2. THEORY

A. Bessel Beams

B. Aberrations

3. EXPERIMENT AND RESULTS

A. Experimental Implementation

B. Aberration Results

C. Turbulence Results

4. DISCUSSION

5. CONCLUSION

REFERENCES

Cited By

Figures (8)

Equations (11)

Journal of the Optical Society of America A