1. Introduction

Vortex beams carrying non-zero orbital angular momentum (OAM) have gathered great interest in the last years due to their potential applications in fields like quantum computing [1, 2], optical communication [3, 4], optical tweezers [5, 6], vortex coronagraphs [7] and optical characterization of optical systems [8]. The angular dependence of the phase of these beams is what provides them with non-zero orbital angular momentum. It is well-known that when vortex beams propagate through a medium which distorts its wave-front phase (be it, for example, an optical system or the atmosphere), the OAM content of the beam is usually modified. The resulting wave-front will be, in general, a superposition of several OAM states, based on the initial superposition of states and on the particular optical aberrations of the medium. As the detection optical systems themselves are never completely free of aberrations, a measurement of the OAM spectrum of an incoming beam will always be modified by the quality of the optical system that detects it. As we will point out later, the distortion introduced on the wave-front affects the OAM spectrum of the beam only if it has angular dependent (azimuthal) aberrations. We will also show that, although radial wave-front distortions do not affect OAM content, they can affect the efficiency, and even the accuracy, of the detection techniques traditionally used.

Some studies have been carried out about the effect of aberrations on helical beams, in which only statistical predictions are made [9, 10, 11], or calculations for the effect of just a few low order Zernike polynomials on OAM spectra [12], and mostly focused on the effect of the turbulent atmosphere but not on the detection systems. Some works [13] are concerned only with the performance of vortex coronagraphs in presence of aberrations, but those optical systems are not intended to measure OAM spectra. The more in-depth study is that on Ref. [14], but they focus exclusively on the point spread function (PSF) shape deformation produced by azimuthal aberrations, without taking into account the OAM spectrum deterioration. This is so because they use the sensitivity of the PSF to Zernike aberrations to correct them. Our objective is a different, broader one: in order to measure with great accuracy the OAM spectrum of light, or to produce a vortex beam with an OAM spectrum known to great accuracy, one has to take into account the effect on that spectrum of the optical instrumentation used to detect or create it. This is a parallel to the now routine calibration procedures of polarization optics: as the accuracy and precision of ellipsometers and polarimeters increased over time, new calibration techniques were developed to take into account the effect on the measurements of instrumental polarization [15, 16]. Although up to a certain accuracy degree it is possible to assume the generators and detectors as polarization free, beyond some point a careful calibration is needed, as correction is impractical or impossible. This is our proposal for the field of OAM: a framework to characterize the optical instrumentation that deals with vortex beams, whichever its concrete application [1, 2, 3, 4, 5, 7]. We stress that the correction of aberrations is not our objective, as correction is not always feasible. However, the route to high accuracy OAM spectrum measurements is not only through aberration correction, as sometimes the quantitative knowledge of their effect is sufficient to correct, a posteriori, the experimental data. We already applied this work in a new technique for the measurement of the OAM superposition state of light [17].

In this work we propose a new basis for helical beams which takes into account radial aberrations. With this new tool, it will be possible to completely describe the effect of the aberrations of any medium or system on the OAM states of light. This simplifies greatly the characterization of systems that create or detect helical beams, and bridges the gap between the use of traditional tools for optical characterization and the perturbation of the OAM spectrum of the generated or detected OAM carrying beams. We will also study the effect of the radial aberrations in some common detection systems, and find that although they do not change the OAM content of the beam, they can affect not only the efficiency but the accuracy of the measurement by introducing cross-talk between OAM modes.

2. Theoretical basis

To take into account not only the angular dependence of phase, but also the radial dependence in the propagation through any medium which interacts with the phase structure of the beam, it is necessary to define a new basis to simplify the analysis. Because the natural basis for optical aberrations is the Zernike polynomials [18], we will first study the relation between OAM beams and Zernike polynomials.

A common description for the electric field of vortex beams is the Laguerre-Gauss (LG) functions, in which the electric field in a plane in z is

where c _1,2 are constants and $L_p^{\left|l\right|}$ is the associated Laguerre polynomial. This equation describes the eigenmodes for the electric field of electromagnetic radiation inside a cylinder, and as can be seen, the electric field has a helical phase e^ilϕ.

In Fig. 1 we have the projection of the phase of a helical beam, lϕ, with topological charge + 1 on the Zernike basis. Although in mathematical expressions we use the Zernikes $Z_n^{\pm m}(\rho ,\varphi )$ defined in [19], for easier visualization we plot using the Zernike ordering defined by Wyant [18]. We also have chosen here the convention that ϕ spans [0, 2π]. If we had chosen [−π, π], the only difference would be a zero-valued piston term. Analytically, the projection of the phase of a beam with topological charge l on the Zernike basis is

where the arg function extracts the phase of its argument, and $a_n^{\pm m}$ is the normalization constant for the Zernike polynomials defined by

where κ = 2 for m = 0, and κ = 1 otherwise.

 

Fig. 1 Zernike projection of vortex +1. The piston term 0, not shown completely, is equal to π ^3/2.

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Defining $Z_n^{+m}(\rho ,\varphi )=R_n^m(\rho )cos(m\varphi )$ and $Z_n^{-m}(\rho ,\varphi )=R_n^m(\rho )sin(m\varphi )$ , we now have

where cs corresponds to cos (sin) if the sign is + (−), and $K_n^m$ and I ^{± m,l} correspond to each one of the integrals on the left. Clearly, this projection (the value of I ^{± m,l}) is not unique. We chose the one that reproduces the OAM phase modulo 2π (that is, wrapping the phase). If one chooses to unwrap the phase, then it follows trivially that I ^{± m,l} = l×I ^{± m, 1}. This is an indication of the different nature of the OAM functions and Zernike polynomials: the first is a basis for the angular part of complex functions in a circular domain, while the second is a basis for real functions in the unit circle, which is usually used to describe the pure imaginary phase of the generalized pupil of optical systems. Now, using $R_n^m(\rho )$ from [19], the spectrum depicted in Fig 1 corresponds to

where $K_n^m$ is independent of l. It's clear that the + coefficients are zero due to the odd nature of vortex +1 phase. In Fig. 2 we can see the OAM +1 phase reconstructed from the first (Wyant ordering) 527 Zernike terms. However, one should not infer from this result that even (+) Zernike coefficients do not affect OAM: if one chooses to wrap the phase, the even coefficients will be non-zero in general, which means that they are not orthogonal to the OAM basis functions, and therefore affects the OAM spectrum, a result well-known for the case of astigmatism [14]. As we see, an infinite amount of Zernikes are necessary to reproduce the phase of a vortex beam, which is an expected result, due to the discontinuity of the phase. However, we see clearly that the weights decrease, and the most important terms are the lower order Zernikes (which explains the strong dependence on tilt found in [12]). Now, the radial terms of the Zernikes are those with m = 0. We have from the integral definition of $K_n^m$ , taking into account that $R_0^0(\rho )=1$ ,

from the definition in [19] of the normalization constants a_n and the orthogonality of $R_n^m(\rho )$ . Therefore, there are no contributions from radial terms, independent from the wrapped or unwrapped phase convention. This means that the radial Zernikes aberrations will never change the topological charge of a beam.

 

Fig. 2 Reconstructed vortex +1 from 527 Zernike polynomials. In white, lines of equal phase from 0 to 2π rad each 0.5π rad.

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At this point we want to indicate that if we invert the projection in Eq.(2), thus projecting the Zernike basis onto the OAM basis, it is possible to find the OAM content of each Zernike aberration. What this projection tells us is how much a given aberration affects each OAM state, and this is useful to determine the most important aberrations to be corrected, for example by means of adaptive optics, in a creation/detection system to optimize the purity of the created or measured OAM spectra. It could be possible to focus on a certain range of aberrations to minimize the impact on a given group of OAM states. However, the correction issue is out of the scope of this work and will be published elsewhere, as to optimize the correction of aberrations it is important, first, to have proper way to measure them in the optical system at issue.

With all of the above information, we can define a new basis taking into account the orthogonality between OAM states and the radial Zernike polynomials. As all the OAM states and the non-radial Zernike terms span the same (angular) subspace, we will take the OAM states as a basis for that subspace, and complement them with the radial Zernike terms to form a new basis for the complete space (angular and radial), called the OAM-Zernike basis. Therefore, a general cylindrically symmetric light beam with any phase form can be expressed as

where $R_n(\rho )\equiv R_n^0(\rho )$ describes the radial part of the phase, with real weights b_n, which should take into account the normalization factor of the radial polynomials. The complex weights c_l,p define the azimuthal phase (the OAM spectrum); and r(ρ) is a real-valued function, which modulates the amplitude of the beam in the radial direction (such as the Laguerre polynomials of Eq. (1)). As we see, this is a more general description than that of Eq. (1), as this can describe, for example, diverging Laguerre-Gaussian beams, or LG beams with radial aberrations. We want to point out that the fact that Eq. (9) applies only to cylindrically symmetric beams is a limitation of all basis, and is proper to the very nature of a basis. When a basis defined in some domain (here, unit circle) needs to be used in a different one, a Gram-Schmidt orthogonalization process is performed to adapt the basis to the new domain [20].

3. Usage of the new basis in an optical system

As an example, we will consider an OAM based optical communication system, analogous to that of [17]. A beam with well defined OAM spectrum exits the emitter (the OAM state generator, OSG), travels through free space and enters an optical system used to measure OAM (OAM state analyzer, OSA). As an OSG a simple computer controlled spatial light modulator can be used to produce a forked hologram [21] of the proper topological charge. As an OSA, apart from some coupling optics, a common measurement system is that used for the first time in [21], and used many other times such as in [2, 4, 22]. It consists of a forked hologram used to subtract from the incoming light the particular OAM state intended to be measured, and which focuses the resulting light in a pinhole or a mono-mode fiber. If the beam has content of the same OAM state which subtracts the hologram, that portion of beam after the hologram will be a plane wave and can be tightly focused by the system into the pinhole, thus triggering the detector.

Now, the light produced by the OSG can be described by

After traveling through free space (we can consider it aberration-free for now) and upon arrival at the OSA, including coupling and focusing optics after the forked hologram, the beam can be described by

where Ψ_i ≠ Ψ_o in general, due to the non-ideal nature of the OSA. The weights of each vortex mode can be different d_l,p ≠ c_l,p due to azimuthal aberrations, such as astigmatism or coma, or non-symmetrical amplitude changes, such as non-circular apertures or misalignment. The functions r_o are in general different from r_i, and represent radial changes in intensity, such as diffraction at a circular aperture. Finally, $e^{i{\sum }_n{b}_n{R}_n(\rho )}$ introduces a radial dependent aberration common to all vortex modes, such as defocus and spherical. Therefore, the image onto the pinhole of the OSA will not be a perfect Airy pattern, but will be the point spread function corresponding to a beam described by Eq. (11). Using the common tools of Fourier optics, the propagation of a beam of light through the non-ideal OSA system can be described by a generalized pupil (GP) [23] of the form

which is the quantity that describes the interaction of the system with the incoming beam phase, and can be easily separated in the OAM spectrum modification via the properly normalized d_l coefficients, and radial aberrations via the b_n coefficients. We will concentrate now on the effect of the d_l coefficients, and postpone to the next section how the radial aberrations affect the accuracy of the detection.

The generalized pupil of the system expressed in the OAM-Zernike basis is highly useful. For example, the phase of a pure OAM m beam coming from the OSG will be modified by the OSA into

in which the second summation represents the new OAM spectrum of the incoming beam, which will be the one ultimately measured by the OSA. A similar simple multiplication is required for an incoming beam of any form. It is easy to see that the squared absolute value of the d_l coefficients, correspond to the OAM spectrum that an incident 0 OAM beam will have after passage through the system. For a perfect system with no aberrations, only d₀ ≠ 0 and therefore GP is a constant.

We want to emphasize that Eq. (4) is equivalent to calculating a change of basis. This permits, for example, the use of traditional optical characterization devices, such as interferometers or Shack-Hartmann wave-front detectors, to measure the optical aberrations of an optical system (the GP). Afterwards, the change of basis is made to have the OAM-Zernike projection of those aberrations, Eq. (12), and therefore a straightforward characterization of the system in terms of its effect on OAM beams. Using Eq. (13) allows us to calculate the relation (and subsequent correction) between the measured OAM spectrum and the original spectrum. In [17] we already applied this GP form to a classical phase-diversity technique for the characterization of an OSG-OSA system, and the measured GP provides the necessary correction for all the spectra that the system measures.

4. Effect of aberrations in measurements

As stated before, azimuthal aberrations affect directly the OAM spectrum of light, and as such have been studied before [14]. Taking the fork-hologram and pinhole detection scheme discussed above, it is easy to see the strong effect of azimuthal aberrations on OAM spectrum by looking at the PSF of the focusing system. Here, we consider scalar PSFs, which are calculated as the squared absolute value of the Fourier Transform of the Generalized Pupil, assuming homogeneous illumination of a circular aperture.

We now define the signal-to-noise ratio (SNR) for such a system, due to contamination in the measurement of one state from light from other OAM states, in absence of aberrations. Because the null intensity in a vortex beam occurs only on-axis (ρ = 0), we will always have unwanted non-zero intensity entering the pinhole if we do not have a pure OAM state, regardless of how small the pinhole is. This is a limitation of this particular measurement technique, and presents a trade-off between the amount of light arriving at the detector and the maximum SNR of the system. To measure the m OAM state, one uses a −m forked hologram, and therefore the original OAM spectrum […,m − 1,m,m+1,…] is transformed into […,−1,0,+1,…]. When focusing onto the pinhole, some unwanted light from the [m±1,m±2,…] states, now [±1, ±2,…], enters the pinhole. As the intensity of vortex beams goes as ρ^{| 2 l|} for ρ → 0, the ratio of the intended signal intensity to the unwanted intensity inside a circular pinhole of radius r_p is

where A_m is the weight of the m OAM state in the beam, and only contributions from neighboring states where considered. We see that the reduced SNR comes from a crosstalk between neighboring states, and therefore depend directly on the amplitude of A_{m ±1}. We stress now the importance of this kind of calculations to asses the accuracy of OAM spectrum measurements. Although this technique has been used in several works, a discussion about its accuracy, to our knowledge, has never been made.

4.1. Azimuthal aberrations

Although the effects of azimuthal aberrations are known [14], we present a short example that illustrates straightforwardly their effect on the SNR. If we have an aberration-free optical system, the image onto the fiber's core or pinhole of a plain wave (0 vortex) will be that of Fig. 3(a), and for a +1 vortex the one in Fig. 3(b). The PSFs when spherical and coma aberrations are present ( $0.5Z_2^0+0.25Z_3^1$ , the Zernike coefficients in waves) are depicted in Fig. 3(c), for the plain wave, and Fig. 3(d) for the +1 vortex. It is clear now why coma affects the OAM spectrum of the beam: light which should not be detected enters through the pinhole, because the zero-intensity-on-axis property of the +1 vortex disappears, as it is now a superposition of vortices, including 0 OAM. However, if the azimuthal aberrations of the optical system are measured, and represented in the OAM-Zernike basis, the simple calculations of Eqs.(13) and (12) provide the way to correct the measured spectra.

 

Fig. 3 The effect of non-radial aberrations. PSF for plane wave (a) and for +1 vortex (b) with no aberrations. PSF for spherical and coma-aberrated $(0.5Z_2^0+0.25Z_3^1)$ plane wave (c) and +1 vortex (d).

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4.2. Radial aberrations

We now move to the topic of radial aberrations, which has not been treated before. We depict the PSFs of a system with only radial aberrations in Fig. 4(a) and (b). As can be seen, a radial aberration induced by the medium can affect strongly the intensity profile of a vortex beam, even emulating the profile of an l vortex with a lower l′ < l vortex with some radial aberrations, but without changing the OAM content of the beam. This is the reason why, if only intensity measurements of OAM spectra are carried out, compensation of the radial aberrations is very important for a correct measurement. In Fig. 5 we have the case of PSF profiles with 0, $0.25Z_2^0$ and $0.5Z_2^0$ (defocus). Also the pinhole radius for SNR 100 and 10 are shown, as well as the Airy radius. We see that for $0.25Z_2^0$ the intensities of both 0 and +1 vortex diminish, but maintain similar proportionality near on-axis. Therefore, the efficiency of the system diminishes (the coupling of the light), but its accuracy is not severely compromised and Eq. (14) still holds. For $0.5Z_2^0$ the linearity is completely lost, and the vortex +1 has larger intensity inside both pinhole sizes than the plane wave. This implies that in this case not only A_{m ±1} of Eq. (14) affects SNR, but now the radial dependence of intensity is changed completely, so that ρ ^{2 l} for ρ → 0 is no longer satisfied. Therefore, an optical characterization of this type of system is mandatory, to asses the effect of the system in the measurement of the OAM spectrum of the incoming light.

 

Fig. 4 The effect of radial aberrations. PSF for vortex +1 (a) with defocus and spherical aberration $(0.4Z_2^0+0.07Z_4^0)$ and PSF for non aberrated vortex +3 (b). Peak intensity in (a) is 0.43 times of that in (b).

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Fig. 5 PSF profile for no aberration, $0.25Z_2^0$ and $0.5Z_2^0$ defocus. The vertical black bars correspond to pinhole radius for SNR 100 and 10, and the dashed vertical bar indicates Airy radius. At $0.5Z_2^0$ defocus, the fiber or pinhole detector would confuse l = 1 for l = 0.

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In the case of interferometric measurements, such as the well-known cascaded Mach-Zehnder interferometers [24], the OAM spectrum is not compromised by radial aberrations as long as the two interferometer arms do not have different aberrations. This occurs because, in this interferometric approach, any radial aberrations that the beam has prior to entering the system cancel out at interference. On the other hand, if both arms have different aberrations, results will be affected. More over, if differential radial aberrations are present, the interferometer exit ports will behave differently depending upon the distance ρ from the center of the beam, in the same way that fringes would appear in a normal interferometer. Although this would be evident in applications with high light throughput, not so in low-light applications, and this aberrations can become critical. Therefore, it is important also to characterize this type of measurement devices.

As indicated in the introduction, some techniques are being proposed for astronomical applications of OAM, including vortex coronagraphs [7] and direct OAM spectrum measurement of astronomical objects [25, 26], which could also benefit from using the OAM-Zernike basis to characterize the measurement device.

5. Conclusions

The OAM-Radial Zernike basis we propose is a useful tool for describing the aberrations introduced by a medium through which vortex beams propagate. With the recent increase in applications of vortex beams, high accuracy measurements will become necessary, and the instrumental OAM spectrum should be determined. This new basis, because it contains the same states intended to be measured, simplifies greatly the assessment of the quantitative impact on the OAM spectrum of generated/measured beams from the aberrations of the system. A useful procedure would be the optical characterization in terms of Zernike aberrations using traditional equipment, such as interferometers or Shack-Hartmann wave-front detectors, and then performing a change of basis to OAM-Zernike to determine the effect of the system on the OAM spectrum of the incoming light. Additionally, we found that the radial aberrations of OAM beams should be taken into account, especially when measuring via intensity profiles, as those aberrations impact the efficiency of the measurement, affects the accuracy, and could even invalidate results.