## Abstract

We present a method of photon orbital angular momentum selection at very low light levels using spatial interference between a strong local oscillator field and a weak beam. By using Fourier phase recovery techniques familiar in classical interferometry, we can experimentally obtain a quantum-limited ${Q}$ distribution with a standard deviation consistent with the quantum noise floor. Further, by projecting the complex Fourier peak on a Laguerre–Gauss basis, we can distinguish states of different orbital angular momentum with high fidelity for small numbers of counts per acquisition frame. The noise equivalent photoelectron count for this measurement is ${10^{- 5}}$ counts per pixel per frame.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. INTRODUCTION

The orbital angular momentum (OAM) of light has a rich scientific history [1–3]. It links such fundamental concepts as geometric phase, the existence and evolution of phase singularities, to the (potential) use of angular momentum in quantum information [4] and space-multiplexed communication systems [5]. For optical systems, OAM provides a degree of freedom in the light field that has shown promise for dark-field confocal microscopy [6], superresolution microscopy [7], new approaches to multimode fiber communications [8], secure communication schemes utilizing quantum key distribution [1], and free-space communication systems that are robust to atmospheric turbulence [9,10]. OAM mode discrimination is central to many of these applications, and there is still a need for robust quantum-limited detection methods that also preserve global phase and can discriminate the phase difference between any two modes in a coherent or partially coherent mixture. Efficient sorting of photons by angular momentum influences not only the detection efficiency of space-multiplexed communication signals but also enables quantum protocols designed around the so-called quantum random walk.

A number of methods have been developed over the years to measure the OAM content of a beam, including interference and diffraction [11–19], holograms [4,20–22], phase plates such as ${ q}$-plates and spiral phase plates [23–25], mapping OAM into transverse momentum [26,27], and quantum weak measurements [28].

Prior angular momentum sorting has generally emphasized the physical separation of modes combined with photon counting. One scenario makes use of a spatial light modulator (SLM)-based programmable forked grating that imparts an OAM proportional to the order of the diffraction component. Energy is distributed to a range of diffracted orders, but only one corresponds to a Gaussian shape that can efficiently couple into a single-mode fiber. This allows the output channel that matches the input OAM state to couple efficiently into the fiber and register on a photon counter [29] or, in the implementations of Jassemnejad [30] and Mirhosseini and coworkers [26], to project a spot array on a camera. Generalizations of this approach have also been published in order to perform a decomposition of spatially multimode fields. For applications in astronomy, multipoint interference/diffraction was used by Berkhout and coworkers [15] to deduce angular momentum components in light collected from distant objects.

These approaches have the important advantage of providing good discrimination, but they often lose the global phase information in the beam and give up overall quantum efficiency in the process. Beginning in 2000, Beck and coworkers [31–34] proposed and implemented an approach for the detection of multimode fields using spatial interference information acquired using a 1d detector array. By analyzing in the (spatial) Fourier domain, it is possible to measure multimode fields in a way that lends itself to quantum tomographic information in both space and time/frequency. The use of a strong local oscillator (analogous to a reference beam in conventional interferometry) assures a signal-to-noise ratio (SNR) that is limited by quantum fluctuations.

In this paper we introduce the use of spatial interferometry in two dimensions and show that, when applied using the principles of heterodyne detection with reference beam correction and phase tracking, it is possible to measure OAM at very low light levels in a way that provides good discrimination down to a few photons per camera frame and, at the same time, *gives relative phase information between the OAM modes.* When the weak OAM beam mixes with a strong local oscillator, it is possible to approach the quantum limit of heterodyne detection, but this is done spatially such that one can display the ${Q}$ distribution in a quadrature plot in real time and for several (potentially many) beams simultaneously with a single camera and a single local oscillator (for a good explanation on constructing the ${Q}$ distributution of modes, see [35]). Quadrature distributions for the $l = 0$ and $l = 1$ modes show the expected SNR of unity at one count per frame.

## 2. DESCRIPTION OF THE EXPERIMENT

The essence of the experiment is to interfere an attenuated beam (hereafter referred to as the weak beam) containing OAM information with a strong reference (local oscillator) beam at an angle to produce a carrier fringe pattern that also contains the OAM information. The arrangement is a familiar one for conventional interferometric measurements; however, interferometric metrology generally aims for a test and reference beam of approximately equal power (for good fringe visibility). Our scheme, while it follows the general principles of interferometry, bears a close resemblance to heterodyne detection in which a weak signal is interfered with a strong local oscillator at a power ratio of ${10^6}$ to ${10^8}$ The interference fringes are detected by an EMCCD camera (Rolera Thunder) set at a low gain, acquired using a MATLAB/Micro-Manager-based computer interface and numerically filtered to give both the amplitude and phase of the OAM components. We describe each of these in turn.

The experiment, illustrated in Fig. 1(a), employs a stabilized He–Ne laser (Thorlabs) as the primary source and local oscillator. A variable neutral density filter (ND) is followed by a half-wave plate and polarization beamsplitter to control the irradiance and separate the laser into local oscillator (LO) and signal beams. The ND filter permits the control of the LO power incident on the camera, while the wave plate controls the power ratio between the LO and the signal beam. The usual double-mirror arrangement provides the ability to precisely align the signal and LO beam at the camera.

For the measurement of beams with nonzero OAM, a stress-engineered optic (SEO) is employed [36,37]. The SEO is a nominally flat window with symmetric (in this case trigonal) stress that results in a space-variant birefringence. If the window has sufficient stress, there exist a circular region of half-wave retardance (the half-wave ring ) whose fast axis follows the symmetry of the force distribution on the window. The theory of stress engineering in systems like this is covered in detail in Refs. [36–41]; in our application it is sufficient to know that the geometric phase distribution of the half-wave ring provides a near-perfect scalar vortex of left- (right-)handed circular polarization if the window is illuminated with right- (left-)handed polarization and then filtered with a left- (right-)handed circular analyzer. A practical way to accomplish the same thing is to illuminate the window with circularly polarized light and filter the transmitted light with a quarter-wave plate (QWP) and linear polarizer. In such a case, the light leaving the linear polarizer (Fig. 1) has a phase vortex.

The QWP placed after the SEO transforms the circular polarized components into linear polarization. The angle of the QWP was set so that the nonvortex mode is in the horizontally polarized component (in the laboratory frame) and the vortex mode is in the vertical component. The linear polarizer in the reference beam path after the first mirror provides the capability of selecting the state of linear polarization that interferes with the weak beam. Setting this polarizer in between 0° and 90° results in an interference of a mixed state that contains both the $l = 0$ and $l = 1$ modes.

For any interference experiment (including heterodyne detection) it is important that the signal and LO beams be matched in polarization. However, the particular spatial heterodyne scheme used here does not require spatial mode matching in the same way that is required for fiber-based homodyne or heterodyne detection systems [42–44]. One advantage (sometimes called the heterodyne advantage) of approaches like this is that the LO power can be kept as high as possible without detector saturation, thereby guaranteeing that the quantum noise (shot noise if viewed in a semiclassical context) dominates the dark count, thermal noise, and camera read noise. When the signal beam is provided with significant power, one can readily view the fringe in the direct image [Fig. 1(a)], in which the phase vortex is marked by a forked fringe pattern.

The experiment is controlled through a MATLAB-based graphic user interface that allows both real-time processing as well as image archival. All data was taken with an exposure time of 10 ms and a gain setting of 4. In order to establish a signal power spectrum at the theoretical limit of quantum fluctuations, we carry out an acquisition and 2D Fourier transform of the reference beam. A root-mean-squared (power spectral density) estimate is generated by averaging the square magnitude of 100 frames. Figure 2(b) shows a logarithmic plot of power spectral density against spatial frequency, normalized to unity at its peak. When normalized in this way, and when analyzed in the semiclassical approximation, the noise floor is given by $1/\sqrt {{N_S}}$, where ${N_S}$ is the total number of counts per frame in the reference beam.

In an ideal arrangement, the reference beam would be represented by a perfectly Gaussian envelope so that, after Fourier transformation, the power spectrum of the bare reference beam is confined to a small number of data points near zero frequency. In ordinary experiments, the reference beam is modified by imperfections of optical components. To remove artefacts, we record a 500 frame average of the strong beam, deduce a best-fit Gaussian function, and compute the ratio between the actual beam and the best-fit Gaussian. That correction function is then used to multiply the acquired data. The correction function is normalized so as to preserve the total count in each frame.

## 3. STRESS-ENGINEERED OPTICS

The SEOs employed in this experiment are optical windows with stress applied symmetrically about the element. Their fabrication is described in Refs. [36,37]. When used in an optical system, the SEO is either apodized with a circular aperture, or the illumination (in our case, a Gaussian beam) is confined to the center region.

For the case of an SEO with trigonal stress, the Jones matrix can be written as

## 4. ANALYSIS

The energy in the two SEO modes in the electric field given in Eq. (3) will, in general, differ according to the value of $\beta$. Thus, we may use ${I_l}$ to denote the irradiance of the $l = 0$ and $l = 1$ modes. Then the electric fields can be written as

We begin by considering the case of a pure single mode of angular momentum $l$ in which the strong reference beam has a polarization that matches the OAM mode. In this case, the interference of two fields at a pixel location ${\boldsymbol x}$ may be described in the usual formalism describing the superposition of two complex amplitudes:

The recorded number of counts per pixel is a function of the detection quantum efficiency, amplifier gain (if used), and digital scaling of the A/D converter. For the purpose of this (semiclassical) analysis, we consider an ideal CCD detector that accumulates photoelectrons with some quantum efficiency $\eta$ yielding a signal and local oscillator count per pixel per frame of

and where $\int_{{A_j}}$ indicates the integrated photon flux over the area of pixel j. This yields a frame-to-frame interferogram (measured in photoelectron counts) ofIn the limit of large local oscillator power, the frame-to-frame count distribution will have the form

The strong and weak beams will have a slowly varying mode profile that is somewhat distorted by deterministic spatial variations due to weak diffraction effects from particles, edges, and so on. To correct for this, we capture a 500 frame average of the strong beam alone (blocking the weak beam), fit the result to a Gaussian envelope, and store a correction function that, for each acquired frame, removes the weak artifacts while preserving the total count.

We now carry out a 2D discrete Fourier transform of each frame, yielding two fringe peaks that are broadened by the envelope function. While the mathematics is straightforward, the numerical implementation includes filtering/postprocessing and calibration of the peak height to the incident photon flux. We discuss these in the following section.

## 5. ACQUISITION AND ANALYSIS

The experiment is configured so that the analysis can either be done in real time (for aid in alignment) or in postprocessing by saving an image stack. The alignment procedure is tedious but straightforward: the weak and strong beams are located on a common center pixel both by viewing the irradiance profile of each beam separately and by viewing the phase vortex through real-time Fourier processing.

#### A. Laguerre–Gaussian Mode Decomposition

Once an acquired image of the interferogram is corrected for the background, the analysis can be done in the Fourier domain. A $32 \times 32\; {\rm pixel}$ window is selected, centered on one of the Fourier peaks. The width of the window is arbitrary, but it should fully contain the peak without cropping. In this experiment setting, the Fourier peak was about 10 pixels wide. Let us denote the Fourier peak as $F({\boldsymbol u})$, where ${\boldsymbol u}$ is Fourier domain position vector relative to the peak center as illustrated in the inset of Fig. 2(a). Then the complex-valued projection ${p_l}$ can be defined as

where $LG({\boldsymbol u};l)$ are the Laguerre–Gaussian (LG) functions of radial index 0 and azimuthal index $l$. When the LG modes are properly normalized, the complex projection ${p_l}$ has a length equal to the square root of the equivalent photon number and a phase equal to the phase difference between the particular OAM mode and the LO. For a given OAM mode sampled over a sequence of camera frames, the collected values of ${p_l}$ then represent a ${Q}$ distribution when plotted on the complex plane. For each frame captured, this projection was computed for $l$ values ranging from ${-}2$ to ${+}2$. The projection values are then divided by the noise floor; in this way, the normalized peak amplitude corresponds to the square root of the number of counts per frame of the camera. In the limit of unit quantum efficiency and perfect mode overlap, ${p_l}$ is a single complex value that that represents the field incident on the EMCCD. When plotted on the complex plane, these projection values show a distribution centered about the mean and having a standard deviation of $1/\sqrt 2$, the expected result of a semiclassical shot-noise-limited measurement or a quantum mechanical measurement of a coherent state of the field.#### B. Numerical Test of OAM Mode Sorting

To test the complex Fourier domain LG mode decomposition, a numerical simulation of the experiment was carried out in the MATLAB platform. A sampled fringe image was defined numerically in which the strong beam distribution was Gaussian and the weak beam is defined by either an ($l = 0$) or ($l = 1$) field profile. Quantum noise was modeled using the unit-variance Gaussian random number generator scaled to the square root of the local photoelectron count and using a separate instance for each frame. The average strong beam power was fixed at ${10^9}$ total counts per frame (consistent with the experiments), while the weak beam varied from 1 to 1000 counts per frames. It should be emphasized that this mean count rate is simply used to determine the equivalent amplitude of the weak beam in the simulation and, in the spirit of a semiclassical approximation, we are simulating continuous, classical fields without any formal quantization of the electromagnetic (EM) field. This type of analysis successfully describes coherent optical receivers, even when close to the quantum limit, provided that the signals are well represented (in a quantum mechanical sense) by coherent states.

Figure 2 shows a centered 2D Fourier transform of the interferogram between two Gaussian beams. Also shown is a log-scale, one-dimensional slice across the Fourier transform and running through the Fourier peak. The trace is normalized to unity at the zero frequency (DC) peak. The power spectrum is generated by taking a root-mean-squared average (of the magnitude of the complex signal) averaged over 500 frames. The noise floor in Fig. 2(b) represents the quantum (shot) noise limit which, when normalized, is equal to ${({\sum\nolimits_j {N_j}})^{- \frac{1}{2}}}$. This noise floor is consistent with the range used in the experiments.

For interference between two plane waves, it is straightforward to show that the magnitude of the fringe peak in the Fourier spectrum is equal to $\sqrt {{N_W}/{N_S}}$; thus, in Fig. 2, for 200 counts per frame, the Fourier peak amplitude is a factor of 14.14 above the noise floor. This is the theoretical limit of heterodyne efficiency, and it yields a SNR of 1 for ${N_W} = 1$ count. We can use this to quantify the SNR penalty for the interference of two Gaussian modes and for the interference between a Gaussian and Laguerre–Gauss mode.

In addition to examining the power spectrum, one can also accumulate frame-by-frame samples of the complex-valued fringe amplitude and plot the ${Q}$ distribution in the usual quadrature plot (Fig. 3). The solid circle corresponds to a one-count radius, which is equal to $\sqrt 2$ times the standard deviation of the actual count distribution in the complex plane centered about the mean, and it provides a guide to the eye. In this way, we can associate the center of mass of the distribution with the square root of the mean number of counts in the weak field.

#### C. Optimizing the Mode Discrimination

To accurately analyze the experimental data, the center of the LG functions should be the exact center of the Fourier peak for the best discrimination of modes. This is done by shifting the LG coordinates by a fractional number of pixels. The amount of shift is found by setting the polarizer to 90° to select the pure $l = 1$ mode and using the projection method where the LG functions are shifted along the $x$ and $y$ coordinates a fractional number of pixels. The leaked energy into the modes other than $l = 1$ mode is computed and used as an error metric. The fractional shift that minimizes the energy leak is used in the proceeding analysis. Even though it is possible to interpolate the Fourier peak to obtain a finer sampling, it was possible to obtain similar results by the fractional shift method with less computational time. The fractional shifts found using the $l = 1$ mode also minimize the leaked energy when the $l = 0$ mode is used. The histograms corresponding to these pure modes are shown in Figs. 6(a) and 6(d).

#### D. Measurement of Pure Modes and Mode Mixtures

When the polarizer is set to either 0° or 90°, a pure mode is generated; these are used for system calibration, making an initial evaluation of the effective number of counts per frame present in each mode, and establishing the baseline fidelity of the measurement. The fidelity of the mode discrimination can be evaluated by comparing the distributions for the known mode with the distributions of the rejected modes. These can be evaluated either in the complex plane or by computing a histogram of the number of counts per frame in an image stack. In the complex plane, the fidelity of a pure mode measurement can be evaluated in a straightforward manner: If we consider the transmitted mode the “1” state in a digital system, its ${ Q}$ distribution is Gaussian, having a standard deviation of $1/\sqrt 2$ and a mean equal to $\sqrt {{N_W}}$; the phase reference can be rotated so that the distribution is centered on the real axis. All other OAM states are “0” states in this scheme, having a Gaussian distribution centered at the origin and a standard deviation of $1/\sqrt 2$. Establishing a decision threshold (familiar in optical communications theory [47]) $\gamma$ to discriminate between the 0 and 1 states yields a straightforward result for the error probability:

When the polarizer is set to an angle in between 0° and 90°, a mixed mode can be measured. As with the pure mode, the complex projection onto each mode can either be displayed in the complex plane or can be represented as an energy distribution by displaying a histogram of counts. Quantifying the fidelity of a mixed state is more complicated both conceptually and experimentally [48]; a proper quantum optical treatment of this is out of the scope of this paper. However, it is worth emphasizing that there is a strong connection between this approach to OAM discrimination and the use of quadrature amplitude modulation in optical communication systems [49].

#### E. Phase Error Corrections

During the acquisition of images, it was observed that the phase of the Fourier peak drifted from frame to frame, possibly due to air currents and system vibrations. Covering the setup and tightening up all the mounts helped to slightly improve the phase drift but was not capable of fully eliminating it. The residual drift was corrected using a postprocessing scheme, in which the phase of the Fourier peak center location of each frame was computed. The low-frequency phase drift (seen in Fig. 4) was removed using Fourier filtering, along with a 30 Hz vibration present in the phase data. When this scheme is applied to the data set, it results in a well-centered distribution of values on the ${Q}$ distribution, which fits well within a unit circle on the plane.

## 6. RESULTS AND DISCUSSION

Measurements were taken both on pure modes and on mode mixtures, determined according to the angle of the polarizer. Figure 5 illustrates the measurements of a stack of 1000 interferograms displayed in the complex plane (each acquired with a 10 ms exposure time). The distance from the origin corresponds to square root of the equivalent number of counts per frame (this would be equal to the number of photons in a system with unit quantum efficiency). Figure 5 clearly shows the ability to discriminate, at very low light levels, between angular momenta. Using the fidelity criteria outlined earlier, an average of four counts per mode per frame yields an expected error probability of $2 \times {10^{- 3}}$ for the coherent state used here. This could be improved by the use of suitably design squeezed states.

It is also well known from communication theory that heterodyne detection exhibits twice the noise of homodyne detection. In a spatial heterodyne experiment, this can be explained using semiclassical shot noise arguments; for nonclassical fields, it can be understood by the fact that there are two possible weak beam angles that can produce fringes of a given spatial frequency. If we think of our OAM beam as a kind of signal beam channel, there is also an idler channel that provides vacuum fluctuations. It should also be emphasized that heterodyne detection provides both amplitude and phase information for many parallel channels, each occupying a different component in the Fourier plane.

We now turn to the measurement of mode mixtures, illustrated in Figs. 6 and 7, with $\theta = {0^ \circ}$ and $\theta = {90^ \circ}$ providing the reference for the peak average count rate of each of the pure modes. The mean of each of the two pure-mode distributions was used to compute the theoretically expected value and is indicated with dashed lines in each figure. Each histogram is accompanied by a Gaussian having a variance equal to the square root of the mean number of counts in order to provide a visual comparison to the predicted quantum limit. There are several things of note: 1) The two nonzero modes follow Malus’s law as predicted, but with different peak efficiencies. 2) The projection includes modes that are known not to be present in the beam but can appear as nonzero components due to beam irregularities. However, when the projection is optimized, these modes are excluded with quite high fidelity, resulting in an efficient sorting process with accurate phase and amplitude discrimination of the modes with nonzero amplitudes. In summary, it is possible to see the discrimination of the two modes in the mixture with a very high fidelity.

We interpret the difference in peak count rate for the $l = 0$ and $l = 1$ modes in Fig. 6 to arise from the fact that the reference beam and the test beam overlap can be somewhat different for the lowest-order Gaussian compared to a $l \ne 0$ Laguerre–Gauss. This is due in part to the fact that $l \ne 0$ modes have an intensity null at the center. This reduces the energy transformed into the Fourier peaks, reducing the effective count compared to the $l = 0$ mode. Optimizing the overlap could, in principle, provide an equal count for the two modes, but with an overall penalty in effective quantum efficiency.

Finally, we note the striking fact that a beam with very few counts per frame can be detected and the OAM discriminated despite the fact that the weak beam is spread over the ${512} {\times} {512}\;{\rm pixel}$ sensor; this means that the noise-equivalent photoelectron count per pixel per frame is, at its peak, approximately ${10^{ -5}}$. While this may seem strange, the measurement itself is taking place on the (spatial) Fourier spectrum of the signal; it is more accurate, then, to state that the noise-equivalent power of the detection system is 0.5 counts per resolution element per frame in Fourier space. This opens up important questions about the information capacity of a measurement of this type; with many OAM modes carrying information in both amplitude and phase, and with many parallel channels in Fourier space, the potential for accessing and processing large volumes of quantum information with a high-frame-rate camera are intriguing.

## 7. CONCLUSION

We have successfully shown the discrimination of OAM modes of light beams at very low light levels using an interferometric method. It was possible to recover the complex-valued field of the test beam after correcting for the inhomogeneities of the test beam and the phase errors that occurred during the experiment. We were able to achieve a good fidelity, discriminating the modes where the count distributions were consistent with the $1/\sqrt 2$-count standard deviation arising from the quantum shot noise floor. This shows promise for either classical or quantum information systems in which space, amplitude, phase, and OAM can all be used as useful channels.

## Funding

National Science Foundation (PHY-1507278).

## Acknowledgment

The authors gratefully acknowledge helpful conversations and advice from Mark Beck (Reed College), Gerd Leuchs (Max Planck Institute, Erlangen), S. A. Wadood, and A. Nick Vamivakas (The Institute of Optics). We also acknowledge support from the National Science Foundation (PHY-1507278).

## Disclosures

The authors declare that there are no conflicts of interest related to this article.

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