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Modal analysis of structured light with spatial light modulators: a practical tutorial

Jonathan Pinnell, Isaac Nape, Bereneice Sephton, Mitchell A. Cox, Valeria Rodríguez-Fajardo, and Andrew Forbes

Open Access Open Access

Abstract

A quantitative analysis of optical fields is essential, particularly when the light is structured in some desired manner, or when there is perhaps an undesired structure that must be corrected for. A ubiquitous procedure in the optical community is that of optical mode projections—a modal analysis of light—for the unveiling of amplitude and phase information of a light field. When correctly performed, all the salient features of the field can be deduced with high fidelity, including its orbital angular momentum, vectorial properties, wavefront, and Poynting vector. Here, we present a practical tutorial on how to perform an efficient and effective optical modal decomposition, with emphasis on holographic approaches using spatial light modulators, highlighting the care required at each step of the process.

© 2020 Optical Society of America

1. INTRODUCTION

The emergence of spatial light modulators (SLMs) in the form of liquid crystal devices [1–3] and digital micromirror devices (DMDs) [4–7] for the on-demand creation of structured light fields has allowed unprecedented control over the amplitude, phase, and polarization structure of classical light [8] as well as quantum entangled states using the spatial modes of light as a basis [9,10]. Given the power of the creation toolkit, it is inevitable that one would like to have an equally powerful detection toolkit in order to verify the properties of the light. Even beyond the context of structured light, quantifying the properties of laser beams is an industry relevant task for both the producer and user of the light source.

There are many tools to achieve this, each with advantages and disadvantages. Statistical tools can be applied to scalar light, treating the intensity as a pseudo probability density function, from which generalized parameters such as beam size, divergence, radius of curvature, and beam quality factor can be deduced [11,12]. More recently, quantum tools have been applied to vector fields to define a vector beam quality factor [13,14]. The advantage in these two approaches is the reduction of the beam into a single parameter, particularly useful for industry standardization. Mode sorters are devices that deterministically direct the modal components of a field into different paths for identification [15–25]. Unfortunately, such devices exist for very specific component analysis, and sometimes provide only a qualitative measure. The benefits of a high-throughput, single-shot analysis of a certain subset of a field has driven their development. Weak measurements can be applied to both classical and quantum fields [26] and interference-based methods are an alternative way to measure modal content [27–29], with derivatives successfully used on orbital angular momentum (OAM) fields of low and partial coherence [30], while simple diffraction-based devices [31–33] yield more limited information.

A more versatile approach is to resolve a field into a coherent sum of modes, each with a particular amplitude weighting and phase: a so-called modal decomposition [2,6,34–36]. In this approach, a field is described as a superposition of basis functions, called modes, each weighted by a complex expansion coefficient. Determining these coefficients is the main task of modal decomposition. There are several commonly used modal bases. OAM modes have received significant attention in recent years, particularly for mode division multiplexed communications [37,38]. OAM modes are a subset of the Laguerre–Gauss (LG) modes that form an orthonormal basis in cylindrical coordinates [39], as do the Bessel–Gauss (BG) modes [40]. In addition, Hermite–Gauss (HG) and Ince–Gauss (IG) modes are orthonormal bases in Cartesian and elliptical coordinates, respectively [41–44].

Modal decomposition was first done with hard-coded diffractive optics [45–47] and later digitally on SLMs [48–52] as well as computationally enhanced versions [53].

Modal decomposition has been applied in a broad range of fields, including oceanography [54], acoustics [55], and plasmonics [56,57]. In this tutorial, we are interested in optical fields, for which modal decomposition yields all the physical quantities associated with them, such as intensity, phase, wavefront, beam quality factor, Poynting vector, OAM density, and OAM purity [48,58–71], as well as offering insights on dynamic effects in lasers [72–77] and fibers [36,50,78–80]. Crucially, despite requiring only point-like detectors [e.g., simple photodiodes (PDs)], it yields results with very high resolution, since the resolution is determined by the basis functions (stored in computer memory), which are not measured. This is a key advantage of the technique: while the precision and accuracy of the measurement outcome depends on the detector and the optical system, the spatial resolution of the reconstructed field is independent of this. A good example of this was demonstrated for the case of wavefront sensing, resulting in pixellated reconstructed wavefronts when using conventional tools but smooth reconstructed wavefronts using a modal approach [62].

 figure: Fig. 1.

Fig. 1. Modal decomposition concept. The field on the left-hand side, perhaps unknown in its properties, is resolved into a sum of modes from a known orthonormal basis, shown here as the ${\rm LG}_p^\ell$ modes. The task is to find the unknown coefficients ${c_{\ell ,p}}$. Summing the right-hand side then provides all the information on the left-hand side.

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Modal decomposition is based on the premise that any optical field can be expressed as a linear combination of functions constituting a complete basis. This is depicted in Fig. 1, where an unknown beam (left) is expanded into the LG basis, with each element being a particular LG mode (right). Each coefficient ${c_{\ell ,p}}$ weights how much the (unknown) field correlates to the corresponding basis mode (${\rm LG}_p^\ell$ in this case). Such coefficients are in general complex, with their amplitude representing the mode weightings. Their phase is an intermodal phase, usually taken with respect to some reference mode in the series (which would then have a phase of zero). The measurement of these coefficients is called modal decomposition, and is achieved by projecting the field onto the set of basis modes. Optically, this translates to measuring the inner-product between the field and appropriate correlation filters (one per basis element), which is proportional to the modal power content. Once the right-hand side of Fig. 1 has been determined, which is nothing more than a series of complex numbers, the left-hand side is immediately known in all its salient properties. Despite the simplicity of the approach, there are multiple factors that can lessen the technique’s performance. Here, we highlight in a tutorial fashion the key theoretical and experimental considerations to successfully implement modal decomposition as a laboratory tool. We start with the basic theory, followed by the essential experimental aspects, and finish with supporting fully worked examples. We expect this paper to be of interest to both new and experienced modal decomposition users from a broad range of communities.

2. BACK TO BASICS

The main purpose of any modal decomposition is to describe a (perhaps unknown) light field in terms of some chosen set of basis modes. An analogy can be drawn as to how one specifies a vector ${\textbf v} \in {{\mathbb R}^n}$ in Euclidean space. One first chooses a basis in which to expand the vector, such as the Cartesian unit vectors ${{\textbf e}_n}$. The completeness of this basis (it must span ${{\mathbb R}^n}$) ensures that we can expand the vector as

$${\textbf v} = \sum\limits_n {c_n}{{\textbf e}_n} ,$$
where ${c_n}$ are the expansion coefficients. Since the basis vectors are known (we chose them), all that is required to fully specify the vector is to determine the expansion coefficients. This is done by employing the orthogonality of the basis vectors, which means that they obey
$${{\textbf e}_m} \cdot {{\textbf e}_n} = \langle {{\textbf e}_m}|{{\textbf e}_n}\rangle = {\delta _{m,n}} ,$$
where ${\delta _{m,n}}$ is the Kronecker delta symbol, and the bra-ket notation $\langle \cdot | \cdot \rangle$ is used to denote an inner-product, also called a projection. Using this property, we can take the inner-product of both sides of Eq. (1) with ${{\textbf e}_m}$, which (after some elementary manipulations) gives
$${c_n} = \langle {{\textbf e}_n}|{\textbf v}\rangle .$$
Once all $n$ projections are computed and the set of ${c_n}$ coefficients is determined, we can reconstruct the vector ${\textbf v}$ by simply adding up the sum on the right-hand side of Eq. (1), whereupon we can describe its properties.

Optical mode projections are analogous, except the inner-product is defined differently for the complex valued functions that define an optical field instead of that for vectors in Euclidean space. We have a light field in the transverse plane and, after having chosen a basis, we want to measure (optically) the expansion coefficients to characterize the field. Importantly, the modes of said chosen basis must constitute a complete and orthogonal set satisfying the paraxial Helmholtz equation.

In a mathematical sense, given an optical field $U({\textbf x})$ that we want to describe/analyze (where ${\textbf x}$ are transverse spatial coordinates), we expand this field in a chosen basis ${\Phi _n}({\textbf x})$ as

$$U({\textbf x}) = \sum\limits_n {c_n}{\Phi _n}({\textbf x}) ,$$
where the subscripts $n$ are called mode indices. Note that $n$ may represent multiple indices, for example, in the LG basis the radial and azimuthal indices $(p,\ell)$, respectively. The basis functions satisfy a similar orthogonality relation as for the Cartesian basis vectors:
$$\int {\rm d^2}{\textbf x} \Phi _m^*({\textbf x}) {\Phi _n}({\textbf x}) = \langle {\Phi _m}|{\Phi _n}\rangle = {\delta _{m,n}} ,$$
where the integration is performed over the entire transverse plane; these are planes perpendicular to the propagation axis (chosen to be the $z$ axis). Similarly as before, using the orthogonality of the basis functions, we can invert Eq. (4) to find
$${c_n} = \langle {\Phi _n}|U\rangle = \int {\rm d^2}{\textbf x} \Phi _n^*({\textbf x}) U({\textbf x}) .$$
This type of integral is sometimes called an overlap integral, since it determines how much of the field $U(\cdot)$ “overlaps” with the basis function ${\Phi _n}(\cdot)$.

How can this overlap integral be performed optically? First, we let the field $U({\textbf x})$ interact with an optical element, such as a SLM, which has a transmission function $\Phi _n^*({\textbf x})$. Knowing the amplitude and phase of the basis functions, one can use complex amplitude modulation [81,82] to determine the correct hologram that should be displayed on the SLM [52]. Such a hologram is sometimes referred to as a match filter. Note that one can equivalently use a DMD with the only difference being how the holograms are generated [7].

Now, consider what happens as we pass the modulated field through a Fourier lens of focal length $f$. At the back focal (Fourier) plane of the lens, the new field is

$$\begin{split}U(\textbf{x})\;\xrightarrow{\text{SLM}{ +}\text{lens}}\;W(\textbf{k})&=C\int_{S}{{d}^{2}}\textbf{x}\Phi _{n}^{*}(\textbf{x})U(\textbf{x})\\ &\quad\times \exp \left({- i \frac{{2\pi}}{{\lambda f}} {\textbf k} \cdot {\textbf x}} \right) ,\end{split}$$
where $\lambda$ is the wavelength of the light, and ${\textbf k}$ are transverse spatial frequency coordinates. If we restrict our gaze to the optical axis (the region where ${\textbf k} = 0$), then the field there corresponds to
$$W({0}) = C\int_S {\rm d^2}{\textbf x} \Phi _n^*({\textbf x}) U({\textbf x}) \propto {c_n} .$$
Note that optical devices do not have infinite spatial extent; the integration domain $S$ is not the entire transverse plane. However, this equation is valid if the fields fall to zero sufficiently rapidly at the boundary of $S$ (we’ll assume this to be true for the remainder of the tutorial).

In summary, the expansion coefficients can be found optically by modulating the field with a device whose transmission function is $\Phi _n^*$, passing the field through a Fourier lens, and observing the field at the optical axis.

3. EXPERIMENTAL IMPLEMENTATION

The key and most laborious part in the experimental implementation of modal decomposition is the extraction of the complex-valued field at the optical axis of the Fourier plane, since we do not (yet) have instruments that can directly acquire this. Fortunately, it is possible to extract both the amplitude and phase of the expansion coefficients ${c_n} = {\rho _n}{e^{i{\phi _n}}}$ using only an intensity detector such as a camera or PD. An example of a typical setup is shown in Fig. 2. A simple design for a “Do It Yourself” Arduino-based PD is provided in Appendix A.

 figure: Fig. 2.

Fig. 2. Modal decomposition with an intensity detector. Lenses ${{\rm L}_1}$ and ${{\rm L}_2}$ image the input spatial mode $U({\textbf x})$ onto the SLM encoding $\Phi _n^*({\textbf x})$. The resulting mode is then Fourier transformed with lens ${\rm L_3}$ and detected at the origin of the Fourier plane with a camera or single pixel detector.

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In what follows, we concentrate on the detection step, assuming that the holograms have been appropriately encoded [52]. Specifically, taking the modulus squared of both sides of Eq. (8), we have

$$|W({0}{)|^2} \propto |{c_n}{|^2} = \rho _n^2 .$$
Thus, by extracting the square root of the on-axis intensity with a detector placed at the Fourier plane, the magnitude of the expansion coefficients ${\rho _n}$ can be measured.

Since such a detector has access only to the intensity of the field, it necessarily destroys the modal phase information. To obtain the inter-modal phases ${\phi _n}$ requires two further on-axis intensity measurements. Once ${\rho _n}$ is determined from the measurement in Eq. (9), one performs the following two additional measurements:

$${({\rho _n^{{\cos}}} )^2} = {\left| {\frac{1}{{\sqrt 2}}\langle {\Phi _0} + {\Phi _n}|U\rangle} \right|^2} ,$$
$${({\rho _n^{{\sin}}} )^2} = {\left| {\frac{1}{{\sqrt 2}}\langle {\Phi _0} + i {\Phi _n}|U\rangle} \right|^2} .$$
This is exactly the same kind of overlap integral as before, except that one encodes a superposition on the SLM instead of a single basis element, where ${\Phi _0}$ is used as the reference against which all other inter-modal phases will be measured. Using the inner-product relations in Eq. (5), these can be expanded to give
$${({\rho _n^{{\cos}}} )^2} = \frac{1}{2}({\rho _0^2 + \rho _n^2 + 2{\rho _0}{\rho _n}\cos {\phi _n}} ) ,$$
$${({\rho _n^{{\sin}}} )^2} = \frac{1}{2}({\rho _0^2 + \rho _n^2 + 2{\rho _0}{\rho _n}\sin {\phi _n}} ) .$$
Since $\rho _0^2$ and $\rho _n^2$ can be measured directly, the above equations can be solved for ${\phi _n}$ by dividing Eq. (13) by Eq. (12), which yields
$${\phi _n} = {\rm arctan} \left({\frac{{2{{({\rho _n^{{\sin}}} )}^2} - \rho _0^2 - \rho _n^2}}{{2{{({\rho _n^{{\cos}}} )}^2} - \rho _0^2 - \rho _n^2}}} \right) .$$
We see then that the intensity measurements $\rho _0^2$ and $\rho _n^2$, ${({\rho _n^{{\cos}}})^2}$, ${({\rho _n^{{\sin}}})^2}$ for each $n \ne 0$ are sufficient to determine the full complex valued expansion coefficients.
 figure: Fig. 3.

Fig. 3. ${\rm LG}_{p = 0}^\ell$ decomposition illustrating the effect of the chosen detection cell size for (a) $d = \frac{1}{4}{d_{{\rm min}}}$ (very sensitive to noise), (b) $d \approx {d_{{\rm min}}}$ (accurate), and (c) $d = 3{d_{{\rm min}}}$ (robust but with some crosstalk). Transverse intensity profiles on the diagonal for $\ell = 0$ and $\ell = 5$ are shown above with red disks indicating the detection cell area in each case.

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As straightforward as this procedure may seem, there are numerous experimental complexities that can significantly affect the accuracy of the results. The following sections describe and provide solutions to these potential pitfalls.

A. Choosing the Correct Camera Settings

When using a camera for optical mode projections, several settings of the camera must be carefully configured. The two most important settings are the exposure (gain and shutter time), and the bit depth of the camera.

If the gain and shutter time of the camera are set too high, the image will be saturated. In this case, the intensity of the on-axis region (which is often the brightest part of the image) will no longer be proportional to the expansion coefficient. Conversely, if these settings are set too low, the signal-to-noise ratio (SNR) will be too low to obtain accurate results. One solution is to determine beforehand which overlap will yield the highest on-axis intensity. This can be done numerically, but intuitively this is usually the overlap with the largest basis mode, since more light from the incoming laser is used. This overlap should then be implemented optically so that the camera settings can be optimized such that the on-axis intensity is just below the point of saturation. These settings must not be modified for subsequent measurements.

Note that it is always more favourable to have no gain (a gain of one or zero, depending on the camera brand) and a camera shutter time as large as possible while avoiding saturation. If the optimal camera settings are such that the shutter time is close to its minimum value, an optical filter (such as a neutral density filter) should be placed in front of the camera to reduce the power on the sensor. The shutter time can then be increased, which effectively averages the measurement over a longer time to reduce the impact of noise.

It is also beneficial to use the highest bit depth possible, which maximizes the dynamic range of the measurements; a higher bit depth improves the SNR of the measurement. In the best case, the ${\rm SNR} \approx 20\;\log {(2^k})$, where $k$ is the bit depth. This expression is approximate because background light and the inherent quantum noise of the sensor and readout electronics also affect the SNR.

B. Crosstalk Matrices: Characterizing the Detection System

At any stage, it is useful to characterize the detection system. Generating a so-called crosstalk matrix can be beneficial in this regard. Here, one sequentially generates a mode in the basis state, $| {{\Phi _n}} \rangle$, and performs a modal decomposition in the same basis, $\langle {{\Phi _m}} |$. In doing so, a matrix of measurement results is obtained, called the crosstalk matrix (since it reveals information about mode crosstalk in the optical system). Due to the orthogonality of the basis, this array of measurement results should ideally be proportional to the identity matrix ${\delta _{m,n}}$. This is illustrated in Fig. 3 where each row of the matrix indicates the generation of a single ${\rm LG}_{p = 0}^\ell$ mode and its decomposition in the same mode set. A clean diagonal with uniform weights between rows indicates that the system can accurately and precisely detect the desired basis modes and is not biased between different mode indices. From experience, it is unlikely that an identity matrix will be obtained on the first pass, often due to residual errors in the detection system such as misalignment. Fortunately, deviations from the ideal result serve as a great diagnostic tool, as will be elaborated on in the coming sections.

C. Decomposition Detection Cell

An accurate measurement of the intensity at the point ${\textbf k} = 0$ is critical for modal decomposition. One might assume that this point implies the “center pixel” of the detector at the Fourier plane of the decomposition lens. While this may lead to reasonable looking measurements results, it is susceptible to noise and also not entirely accurate.

Since a lens is used to perform the Fourier transform, the system’s angular resolution is finite, and hence the minimum resolvable distance at the focal plane of the lens is given by the Rayleigh criterion

$${d_{{\rm min}}} \approx \frac{{\lambda f}}{D} ,$$
where $\lambda$ is the wavelength of the laser, $f$ is the focal length of the Fourier lens, and $D$ is the maximum possible diameter of the beam before the lens. If the latter is not known in advance, this can be taken to be the width of the SLM’s smallest dimension, as that is the largest possible beam size that the system could support. Thus, the measurement of the “point” at which ${\textbf k} = 0$ is in fact the average intensity of the region around that point, with the diameter given by Eq. (15).

The size of the detection cell at the Fourier plane should not be below the angular resolution of the Fourier transforming lens placed after the SLM (or equivalent element). In this regard, it is important to appropriately select this lens’ focal length in relation to the limiting aperture of the physical optics being used in the system and the resolution of the detector. Accordingly, one should seek to choose a lens such that the detection cell area is larger than or equal to the resolution of the detector being used.

In Fig. 3, a simulation of three cases is shown, corresponding to detection cell sizes (a) below, (b) the same as, and (c) thrice that of the angular resolution of the system. Crosstalk matrices are displayed for the case of the generation and detection of ${\rm LG}_{p = 0}^\ell$ beams. Above these plots are transverse intensity profiles at the Fourier plane for the $\langle {{\rm LG}_{p = 0}^{\ell = 0}| {{\rm LG}_{p = 0}^{\ell = 0}}} \rangle$ and $\langle {{\rm LG}_{p = 0}^{\ell = 5}| {{\rm LG}_{p = 0}^{\ell = 5}}} \rangle$ cases. The red regions indicate the pixel array corresponding to the chosen detection cell area. It should be noted that while the decomposition for a detection cell that is below the angular resolution [see Fig. 3(a)] adheres to the criteria for a good detection system, in practice it is highly susceptible to noise and is also not an accurate depiction of the mathematical overlap integral. Particular attention may be drawn to Fig. 3(c) where the detection cell is thrice that of the minimum value. Here, it can be seen that crosstalk becomes an issue for higher OAM charges, since the large detection cell is sampling field intensities outside of the on-axis region. Additionally, the detected weights are no longer uniform, causing a bias towards higher-order OAM modes. In summary, the above illustrates the need for appropriately selecting or tuning the size of the detection cell, which should adhere to Eq. (15) in most scenarios.

D. Finding the Optical Axis

A fundamental step for an effective optical modal decomposition using a camera is to find the optical axis, which is the center of the detection cell. When placing a camera at the Fourier plane, it is very unlikely that the optical axis will coincide with the exact center of the camera, meaning that the central pixel will not be the optical axis. One should endeavor to manually centralize the camera as well as possible to minimize the probability that the camera edges will clip the beam, but a quantitative procedure should be used to precisely determine the optical axis. Luckily, there are several simple ways to do this, and so the experimenter can choose whichever is most convenient. Note that this step should be done only after the optical element that encodes $\Phi _n^*$ is properly aligned and centered on the input beam.

An example of the computation of the optical axis is shown in Fig. 4. Here, a uniform circular field of size $D$, corresponding to Eq. (15), is sent through the detection system. If using a SLM, one can simply encode a binary disk of diameter $D$ on it [see Fig. 4(a)] and illuminate it with an overfilling Gaussian beam. The resulting camera image at the Fourier plane is an Airy disk with the minimum possible spot size (the region where ${\textbf k} = 0$), the center of which is precisely the optical axis [see Fig. 4(d)]. One can then fit a Gaussian profile to each coordinate direction ($x$ and $y$), and the maximum of these profiles is then the camera coordinates of the optical axis [see Figs. 4(b) and 4(c)]. Crucially, this method has the advantage of finding, in addition to the optical axis, the size of the resolution cell: the Gaussian fit will also output their widths, from which the value of ${d_{{\rm min}}}$ can be approximated. Care should be taken to ensure that the wavefront of the uniform disk is flat; otherwise, the image of the Airy disk will be distorted. Unfortunately, most SLMs and DMDs introduce wavefront error because they are not perfectly flat. We will discuss how to correct for this in Section 3.I.

 figure: Fig. 4.

Fig. 4. Finding the optical axis: by allowing a large beam to propagate through the detection system, such as a uniform disk (a), a Gaussian fit (b), (c) can be performed at the detection plane (d) to obtain the pixel coordinates $({c_x},{c_y})$ of the optical axis. The size of the detection cell, denoted by the red circle in (d), is then found using Eq. (15) or approximately from the width of the fitted Gaussian.

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In the case where it is not possible to send a uniform disk of diameter $D$ through the detection system, a Gaussian beam of any size can be sent through, and the same procedure described above can be carried out to determine the pixel coordinates of the optical axis. Unfortunately, the fit will not yield the detection cell size, but this can simply be computed using Eq. (15).

E. Using Optical Fibers

As an alternative to collecting the light at the Fourier plane using an intensity detector, one can instead utilize a single-mode fiber (SMF) and couple the output to a power meter. Here, we consider a SMF with a step-index profile, whose fundamental mode is localized mainly in the fiber’s core and decays radially into the cladding, and that can be well approximated by a Gaussian mode.

A typical experimental schematic is shown in Fig. 5—as before but now the Fourier transforming lens is replaced by an imaging system (two lenses). The aforementioned theory is slightly altered: the probability of exciting the fundamental Gaussian mode in the SMF is given by

$${\tilde c_n} = \int {\rm d^2}{\textbf x} \Phi _n^*({\textbf x}) U({\textbf x}) G({\textbf x}) ,$$
where $G({\textbf x})$ corresponds to the fundamental Gaussian mode that is supported by the SMF, described by
$$G({\textbf x}) = \sqrt {\frac{2}{{\pi {w^2}}}} \exp \left({- \frac{{{r^2}}}{{{w^2}}}} \right),$$
where $w = M {w_{{\rm SMF}}}$ is the radius of the “virtual” Gaussian mode at the SLM plane, which is related to the actual SMF mode size ${w_{{\rm SMF}}}$ by the magnification of the imaging telescope $M$. Although the Gaussian term in Eq. (16) breaks the orthonormality of the basis functions, this decomposition method is nevertheless still useful for mode projections.
 figure: Fig. 5.

Fig. 5. Modal decomposition with a single-mode fiber. Lenses ${{\rm L}_1}$ and ${{\rm L}_2}$ image the incoming spatial mode $U({\textbf x})$ onto the SLM. The resulting mode is then imaged into the core of the SMF using lenses ${\rm L_3}$ and ${\rm L_4}$. The light from the fiber is coupled to a detector (such as a power meter) that registers a power reading.

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There are various ways to correct for the extra Gaussian factor due to the optical fiber, including dividing it out from the basis mode encoded on the SLM or by adjusting the ratio between the encoded mode size and the mode size of the optical fiber [83]. As the former method suggests, one encodes the modified basis function ${\tilde \Phi _n}({\textbf x}) = {\Phi _n}({\textbf x})/G({\textbf x})$ on the SLM, thus absorbing the Gaussian component due to the fiber, resulting in the desired inner-product. However, this method is impractical for most basis mode sets, since the resulting function diverges quickly for large values of ${\textbf x}$. The second method is more favorable, since it requires only a re-scaling of the basis modes and therefore preserves their propagation characteristics in the overlap. To achieve this, the modulated field after the SLM simply needs to be smaller than the Gaussian factor $G(x)$. As a consequence, the product $\Phi _n^*({\textbf x}) U({\textbf x})$ lies within the near-uniform part of the virtual Gaussian, thereby producing the desired inner-product.

Now that we have shown how the overlap can be performed with optical fibers, in practice, how do we know if we have the right fiber? To answer this, one first has to consider that, in general, step-index SMF guided modes are more accurately described by linearly polarized (LP) modes [48], and can support none or many of them depending on the wavelength. Step-index SMFs propagate the fundamental mode (the Gaussian-like ${{\rm LP}_{01}}$ mode) only at a specific wavelength, and will either not propagate any light above this wavelength or will act as a multimode fiber below it, therefore guiding higher-order LP modes. As such, the results of the decomposition can be negatively affected if the fiber is chosen incorrectly. For example, if the fiber supports modes other than the fundamental mode (approximated by a Gaussian mode), the resulting modal crosstalk can significantly reduce the fidelity of the reconstructed field.

To confirm whether the fiber is indeed a SMF, one can compute the number of guided modes, which is approximated by

$${N_{{\rm max}}} = \frac{{{V^2}}}{2} ,$$
with
$$V = \frac{{2\pi}}{\lambda}\;a\;{\rm NA} \\ = \frac{{2\pi}}{\lambda}\;a\;\sqrt {n_{{\rm core}}^2 - n_{{\rm clad}}^2} ,$$
where $a$, NA, ${n_{{\rm core}}}$, and ${n_{{\rm clad}}}$ are the fiber core radius, numerical aperture, and core and cladding refractive indices, respectively. When ${N_{{\max}}} \approx 2$ (due to the degeneracy in polarization) the fiber is single mode.

Once we have chosen the correct fiber and the detection system is well aligned, we can proceed with the decomposition. The field from the SMF is coupled to a PD detector that integrates the signal over time and registers a measured power reading. The measured optical power is proportional to

$${P_n} = \eta |{c_n}{|^2}\int \langle {\textbf S}({\textbf x})\rangle \cdot {\rm d}{\textbf x},$$
where ${\textbf S}({\textbf x})$ is the Poynting vector characterizing the flux energy per unit time per unit area of the fundamental mode exiting the fiber, and $\eta$ is the detection efficiency. The angled brackets $\langle \cdot \rangle$ denote time averaging. Assuming that the field has a uniform polarization state, it suffices that
$${P_n} = \eta |{c_n}{|^2}\int |G({\textbf x}{)|^2}\;{\rm d}x = \eta |{c_n}{|^2}.$$
Therefore, the measured power is proportional to the coupling efficiency at the fiber tip, which is proportional to the modulus squared of the expansion coefficient. Crucially, the coupling efficiency depends on the quality of the imaging system. In particular, the lens’ finite aperture sizes can cause diffraction effects, while aberrations can result in distortions of the transfer function of the system. Such effects can negatively affect the coupling into the fiber, especially if the fields under consideration are non-paraxial (have large beam angles).

Finally, to measure the output power, a power meter or a PD can be be used. In Appendix A, we show how one can easily and cheaply build a DIY PD detector. It converts the current output of a PD to a voltage, which can be measured accurately using a multi-meter, oscilloscope, or even an Arduino.

F. Correction Factors

When implementing an optical mode decomposition, it may be the case that the optical overlap integral is not exactly the same as the theoretical overlap integral, such as when using an optical fiber in the detection step, or due to the manner in which the basis functions are encoded [63,84]. This results in measured expansion coefficients ${\bar c_n}$ that do not correspond to the true values ${c_n}$. This artifact of the optical implementation can be corrected by applying correction factors ${\gamma _n}$, such that

$${c_n} = {\gamma _n} {\bar c_n} .$$
Figure 6 illustrates the need for such a correction through the optical decomposition of an OAM-containing field using a SLM into the LG basis as well as the OAM eigenstates (these two bases are sometimes interchanged but are distinct).
 figure: Fig. 6.

Fig. 6. Observe how the raw measured spectrum coefficients of a uniform ${\rm LG}_0^\ell$ superposition (a), (b) depend on the choice of basis. In the case of the full LG basis (c), (d), the SLM rescales the amplitudes. In the case of the OAM eigenstates (e), (f), these are not a full basis of the transverse plane.

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Often when encoding holograms (see [52] for full details or the provided MATLAB code [85]), the amplitude of the encoded field is rescaled to be in the range [0,1]. This is done to utilize the full bit depth of the SLM. However, when used for optical projections, this has the inadvertent effect of breaking the orthonormality of the basis modes [36]. In this case, the optical overlap that is being implemented is

$$W({0}) \propto \int_S {\rm d^2}{\textbf x} \frac{{\Phi _n^*({\textbf x})}}{{{M_n}}} U({\textbf x}) ,$$
$$= \frac{1}{{{M_n}}} {c_n} ,$$
where ${M_n}$ is the maximum amplitude of the basis mode ${\Phi _n}$. It is quite straightforward to identify the correction factor that needs to be applied here: ${\gamma _n} = {M_n}$.

There are two manners in which this type of correction can be applied: before measuring the expansion coefficients or after. In the case of the former, the orthonormality of the basis functions is reinstated by maintaining the relative amplitudes of the basis modes in the encoded holograms. Since the detectors we consider provide only relative intensity measurements, it is thus important only to relatively rescale the amplitudes of the encoded basis functions. This can be achieved by numerically identifying which basis state has the largest value of ${M_n}$ and simply dividing the amplitude of all other encoded basis functions by this value for each hologram. Note that this will not work for binary holograms such as those displayed by DMDs.

Alternatively, one can apply the correction factors to the measured raw spectrum post hoc. Specifically, one generates the holograms in the usual way, performs the modal decomposition procedure, and then numerically applies ${\gamma _n}$ to the resulting spectrum. One reason for doing it this way is to maximize the SNR, since the full bit depth of the SLM is utilized. A disadvantage is that it is more prone to errors, whereas in the previous approach, the application of the correction factors forms part of the hologram generation process and can be forgotten thereafter.

It is sometimes the case that one would select a basis that has no amplitude component, i.e., a phase-only function $|{\Phi _n}| = 1$. A good example of this is the OAM eigenstate ${\Phi _\ell}(\phi) = \exp (i\ell \phi)$, which can be used to determine the OAM content of an optical field. However, it should be noted that these are not a true basis of the transverse plane, and so correction factors will also need to be applied [86]. To see this, consider an OAM-containing field of the form

$$U(r,\phi) = \sum\limits_\ell {c_\ell} {A_\ell}(r)\exp (i\ell \phi) ,$$
where ${A_\ell}(r)$ is the OAM-dependent amplitude function, and ${c_\ell}$ are the coefficients we want to determine. Virtually all OAM-containing fields can be written in this way. In this case, the optical overlap integral in the basis $\exp (in\phi)$ is
$$\begin{split}\!\!\!\!W({0}) &\propto \sum\limits_\ell {c_\ell}\int_0^\infty r {\rm d}r {A_\ell}(r)\int_{- \pi}^\pi {\rm d}\phi \exp [i(\ell - n)\phi] , \!\!\!\!\\[-2pt] &\propto {c_n}\int_0^\infty r dr {A_n}(r) .\end{split}$$
The correction factor here can be read off as ${\gamma _n} = {[{\int_0^\infty r {\rm d}r {A_n}(r)}]^{- 1}}$, which can be easily computed if the amplitude profile is known. This breaks down if the radial integral evaluates to zero; however, the authors have yet to find a realistic example of when this is the case. Even though this decomposition is in terms of a nonorthonormal basis, we see that the optical overlap integral still reveals the modal content. Specifically, there is an on-axis signal if the helical phase of the basis function matches an OAM mode in the field $U(\cdot)$. If the radial amplitude profile is unknown, a workaround has been offered where one decomposes the field in terms of azimuthal annular slits [49]. However, this requires many measurements per ${c_n}$, and it would most likely be easier to use the full LG basis.

In summary, the general rule for whether and how to compute correction factors for a given implementation involves determining the actual overlap integral that the detection system is performing. A careful consideration of the encoding procedure and type of basis used is a good first port of call.

G. Alignment

As the decomposition process is dependent on the physical positioning of the light field with respect to the SLM match filter ($\Phi _n^*$), lenses, and the detection equipment, it follows that such a system requires precise alignment for an effective implementation. In this regard, careful attention should be given to aspects such as the plane in which the detector is placed, positioning of the match filter, and, in some cases, the encoded sizes of the match filter. Errors here can result in the incorrect determination of the mode weightings as well as the detection of erroneous modes (crosstalk).

As the decomposition technique with an intensity detector is dependent on obtaining the Fourier transform of the overlap performed at the SLM, it follows that the detector position along the longitudinal axis is of importance. For instance, if the detector is not positioned in the Fourier plane of the SLM, Eq. (8) is no longer true, and as such, what is detected is no longer the expansion coefficient for the mode displayed on the SLM. Figure 7 illustrates the effect of incorrect positioning of the detector relative to the Rayleigh length (${z_R}$) of the focused beam for the modes used in Fig. 3. Here, the crosstalk matrix is shown for the detector positioned (a) $\frac{1}{4}{z_R}$ before the Fourier plane, (b) in the correct plane, and (c) $\frac{1}{2}{z_R}$ after the Fourier plane. Images of the amplitude and phase distributions at each plane are given to the left for the decomposition from $\ell = - 1$ to 1 in each case. Away from the correct plane, as in (a) and (c), the phase profiles are seen to carry curvature (associated with the beam divergence), which causes a spiraling of the phase profiles. It may be noted that the results are symmetric about the Fourier plane, and so the spectra at ${\pm}\alpha {z_R}$ are the same, where $\alpha$ is some fraction. As the detector is moved out of the correct plane, the measured values are no longer directly proportional to the coefficients of the modes encoded on the SLM, as seen by the non-uniform diagonal entries of the crosstalk matrix. We note that if the detector is displaced sufficiently far away from the Fourier plane, crosstalk may also occur with erroneous modes being detected. The degree of crosstalk also depends on the robustness of the basis set into which the mode is being decomposed; the LG-OAM basis is more robust to longitudinal displacements than the LG-radial basis, for example.

 figure: Fig. 7.

Fig. 7. ${\rm LG}_0^\ell$ decomposition illustrating the consequences of detection (a) $\frac{1}{4}{z_R}$ before the Fourier plane, (b) in the Fourier plane, and (c) $\frac{1}{2}{z_R}$ after the Fourier plane. Transverse intensity profiles (left) at the corresponding plane for a subset of $\ell$ are shown with phase profile insets of each mode in the top-right corner.

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As the Fourier plane is formed at one focal length after a lens, it follows that this plane may be found by observing the Gaussian beam size and intensity on the detector as it is moved along the longitudinal axis. If the beam before the SLM is collimated, then the waist plane (where the beam size is smallest) is a good approximation to the focal plane. Another more quantitative method would be to encode the Fourier transform of a shape (such as a square) on the SLM and illuminate it with a Gaussian while observing the spatial distribution after the lens with a camera. When the encoded shape is observed and is sharply focused, it follows that both the decomposition lens is correctly positioned and the detector is situated in the Fourier plane.

The mathematical overlap being implemented relies on the match filter and input beam having the same central axis. Hence, one should ensure that this is adhered to physically in the setup. Accordingly, the match filter should be accurately aligned with the center of the input field. Figure 8 shows an example of how a displaced hologram encoding $\Phi _n^*$ affects the crosstalk matrix for LG-OAM modes when the filter is displaced by 20% of the beam size in the (a) negative horizontal direction and (b) positive diagonal direction ($x \text{ displacement} = y$ displacement). Bar graphs below show the detected weights along the diagonal.

 figure: Fig. 8.

Fig. 8. ${\rm LG}_0^\ell$ decomposition illustrating the consequence of displacement of the filter’s central axis from that of the input beam in the (a) negative horizontal and (b) positive diagonal directions for 20% of the beam waist. Bar graphs show the decomposition weights along the diagonals (${\ell _{{\rm input}}} = {\ell _{\rm{filter}}}$). Transverse intensity profiles are shown as insets for the subset $\ell = [- 1,1]$ (lower left corner) as well as $\langle \ell = - 10|\ell = - 10\rangle$ (left) and $\langle \ell = 10|\ell = 10\rangle$ (right) (top right corner). Red cross-hairs show the optical axis of the system.

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Observe that while the zeroth ($\ell = 0$) mode is largely unaffected, the higher-order modes become biased in the axis of displacement. Depending on the filter, either the central null (indicating no mode present) or central bright spot (indicating the mode is present) is itself displaced in accordance with the filter’s displacement. This is illustrated in the lower left insets for modes $\ell = - 1$ to $\ell = 1$. Furthermore, this effect becomes more pronounced the higher the mode order becomes: the insets in the top-right corner for $\ell = - 10$ and $\ell = 10$ show greater relative displacement of the intensity distribution from the central axis (cross-hairs) than in the $\ell = \pm 1$ case. As a result, there is both a decrease in the weightings of the correct modes (as the central spot is displaced) and erroneous detection of adjacent modes (as the central null is displaced) as the magnitude of the mode order is increased.

Interestingly, the direction of the field displacement depends on the handedness of the OAM charge, with opposite charges causing the displacement to be mirrored along the axis in which the match filter is displaced. Additionally, asymmetry in the axial displacement of the match filter results in asymmetry in the weights of the detected modes with a bias seen for $\ell \gt 0$ in (a). In contrast, this bias in the detected weightings disappears when the filter is equally displaced in the $x-y$ directions.

An approach to align the hologram may be to initially adjust the filter such that the on-axis null for a mismatched overlap (e.g., ${\ell _{\rm{filter}}} = {\ell _{\rm{input}}} \pm 1$) appears centered with respect to the beam. Thereafter, the filter may then be finely adjusted until there is no observed shift between the central intensity spots for $\langle \ell |\ell \rangle$ and $\langle - \ell | {-} \ell \rangle$. If the basis modes are encoded digitally on a SLM, it is significantly more precise to translate the displayed hologram than it is to move the SLM itself or the incident beam.

H. Choosing an Optimal Scale

The size of the basis modes relative to that of the input beam can have a significant effect on the decomposition [86,87]. Most basis modes have an inherent scale parameter, such as the beam waist ${w_0}$, which characterizes the size of the modes. A sub-optimal choice for this parameter may lessen the performance of the decomposition. For two different bases having a different size, we have that

$$U({\textbf x}) = \sum\limits_n {c_n}{\Phi _n}({\textbf x}) = \sum\limits_n {\bar c_n}{\bar \Phi _n}({\textbf x}) ,$$
with ${c_n} \ne {\bar c_n}$, in general. Even though both expansions are able to reconstruct the field, it may be the case that the basis ${\Phi _n}$ more effectively characterizes the field than the basis ${\bar \Phi _n}$. This is illustrated in Fig. 9 for the radial $p$ mode decomposition of $U = {\rm LG}_{p = 2}^{\ell = 1}$ in the range $p \in [0,4]$. Observe that when the match filter mode size equals that of the input beam [see Fig. 9(a)], the decomposition returns the “correct” weightings of the input mode ($p = 2$). However, in the case of a mismatch in size [see Fig. 9(b)], many modes of this size are required to reconstruct the input field. The downside here is that a larger number of projections is required for a complete decomposition as well as the fact that the decomposition does not effectively portray the input field.
 figure: Fig. 9.

Fig. 9. Radial mode decomposition of a ${\rm LG}_{p = 2}^{\ell = 1}$ beam in the ${\rm LG}_{\ell = 1}^p$ basis over $p = [0,4]$ for a match filter mode size ${w_M}$ (a) equal to and (b) at 60% of the input mode size ${w_0}$ with (i) transverse phase and intensity profiles, (ii) intensity distributions in the Fourier plane, and (iii) mode weights.

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The implications of not selecting an optimal beam size are related largely to the purpose of the decomposition. For example, if one were using the radial degree of freedom to encode information in each radial mode, a size-mismatched basis would provide erroneous information. However, if the preservation of the particular radial mode was not important, this may be viewed as a parameter that could be tuned to one’s advantage. For instance, one may adjust the encoded size to boost the amount of light seen in the zeroth order ($p = 0$) for phase-only input beams [88].

A method for determining the appropriate scale of the basis functions was proposed in [89]. Here, it is required to determine the second moment size of the beam $\omega$ and the beam quality factor ${M^2}$. The former can be computed from a single image of the beam. The latter can be determined in various ways, but a fast and simple all-digital approach was developed in [61] that measures the beam size at various propagation distances with no moving parts. The advantage of this approach in our context is that the required experimental setup is almost identical to the detection system in Fig. 2. With these two quantities in hand, the optimal scale parameter with which to decompose the beam is then determined using ${w_0} = \omega /M$.

 figure: Fig. 10.

Fig. 10. Setup to perform wavefront correction (a): this is the same as the detection setup in Fig. 2, but the beam should overfill the SLM screen, and the wavefronts should be approximately planar. By encoding only a helical phase (b) on the SLM and using the image of the distorted vortex at the Fourier plane (c), a phase retrieval algorithm can determine the phase map (d) that produces such a beam. The wavefront distortions (e) are isolated by subtracting the ideal phase (b) from the retrieved phase (d). The corrected beam (f) contains very few residual wavefront errors.

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Alternatively, one can select any size for the decomposition and mathematically adjust the basis later to find the optimal scale. Specifically, if one set of expansion coefficients ${c_n}$ is known in Eq. (27), then the coefficients in an alternative basis can be found to be

$${\bar c_n} = \sum\limits_m {c_m}\langle {\bar \Phi _n}|{\Phi _m}\rangle .$$
The above can be computed numerically, since all terms on the right-hand side are known. Practically, however, choosing a near optimal basis to start with is more favorable, since fewer measurements are required and the SNR of the measurements is higher.

I. Aberration Correction

Wavefront distortions (aberrations) within the beam can have a large negative impact on multiple parts of the optical mode projection process. One of these negative impacts was discussed earlier in relation to determining the camera coordinates of the optical axis. Another is when utilizing a SMF; if the wavefront is distorted, then the coupling efficiency is negatively impacted and performance will suffer. In the context of using SLMs, the screens are not optically flat due to imperfections in the manufacturing process. If left uncorrected, this can add even more unwanted aberrations to the beam.

Luckily, there is a simple experimental (although computationally expensive) protocol to simultaneously correct the wavefront distortions induced by the SLM and many of the aberrations induced by the optical system [90]. The procedure is outlined in Fig. 10 and has the advantage of requiring only a single image of a distorted vortex (OAM) beam at the Fourier plane. The method relies on a phase-retrieval algorithm (such as the Gerchberg–Saxton algorithm) to iteratively find the phase of the vortex beam, which will be a sum of the ideal helical phase and the wavefront distortions. Then, by subtracting the ideal phase, the resulting phase map is precisely the residual wavefront of the beam, which can be stored and subtracted from the phase of all subsequently generated holograms to compensate for the aberrations.

Note that the magnitude of the SLM-induced aberrations will scale with the beam size. This is because imperfections on the SLM screen are less noticeable when utilizing a subset of the screen area. Therefore, if the beam size is small relative to the SLM screen, it may not be necessary to implement wavefront correction (assuming the rest of the optical system is largely aberration free).

4. WORKED EXAMPLES

A. Back to the Beginning

We will walk through the decomposition of the field as shown in Fig. 1, which corresponds to the superposition

$$\begin{split}\!\!\!U &= \frac{1}{{\cal N}}\left(\frac{1}{2}{e^{\textit{i}\pi}} {\rm LG}_0^1 + \frac{1}{4}{e^{i\pi /4}} {\rm LG}_1^{- 1} + {e^{- i\pi /2}} {\rm LG}_2^1 + \frac{1}{2}{\rm LG}_0^0 \right.\!\!\!\!\\ &\quad+\left.\frac{1}{4}{e^{i\pi /4}} {\rm LG}_1^1 + {e^{- i\pi /2}} {\rm LG}_2^{- 1}\right) ,\\[-1.4pc]\end{split}$$
where ${\cal N}$ is a normalization constant, and ${\rm LG}_p^\ell (r,\phi)$ are the well-known LG modes. Before performing the modal decomposition, one should ensure that the detection system has been characterized, in accordance with the guidelines and practices outlined in the previous sections. A crosstalk matrix that is close to an identity matrix is a good indicator of a well-aligned and effective detection system.

The very first optical overlap that one implements is with respect to the reference mode. In principle, any mode can be chosen as the reference, but it is often favorable that such a mode forms part of the field, so that the SNR of ${\rho _0}$ is high. In this example, the Gaussian mode ${\rm LG}_0^0$ was chosen as the reference. Then, in addition to the measurement of $\rho _0^2$, for each basis mode ${\Phi _{p,\ell}} = {\rm LG}_p^\ell$ in a predetermined set, we perform three optical inner-products and extract the square root of the detector intensity within the detection cell, as shown in Fig. 11 for the single example where $(p,l) = (1,1)$. A total of nine expansion coefficients was measured for $p \in [0,2]$ and $\ell \in [- 1,1]$. In general, one would have to perform sufficiently many overlaps to be sure that no modes in the spectrum were left out.

The result of the full experimental decomposition is shown as bars in Fig. 12(a), where the height of each bar denotes the amplitude of ${c_{p,\ell}}$, and the color denotes the inter-modal phase relative to the reference mode. Since we used a SLM to encode the basis functions, we have to consider the fact that the hologram rescales the basis functions. In this example, we chose to apply correction factors before the spectrum was measured, by reinstating the relative amplitudes of the basis functions in the encoded hologram. The red lines in Fig. 12(a) indicate what the raw spectrum amplitudes would be if applying the correction factors ad hoc. Up to experimental error, there is good agreement between the measured spectrum and that corresponding to the generated field as specified by Eq. (29).

 figure: Fig. 11.

Fig. 11. Optical overlaps showing the reference mode and the three optical inner-products required to determine ${c_{{p\ell}}}$ in the case where $p = \ell = 1$. Here, ${\cal M}$ denotes the square root of the maximum measured intensity in the detection cell of all the optical overlaps performed.

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Once the expansion coefficients have been determined, the full complex field $U$ is known and can be used to subsequently compute any desired property of the field. For example, this was used to reconstruct the intensity and phase of the field, as shown in Figs. 12(b) and 12(c).

B. Quantum Measurements of OAM

The fiber decomposition technique is commonly used in quantum experiments with entangled photons [9]. In the typical example, two photons are entangled using a nonlinear crystal and each directed to its own modal decomposition setup, but with the detectors in each photon arm time correlated so that only coincident measurements are counted [10].

The probability amplitude for detecting such an event can be written as

$$\begin{split}\!\!\!{{c}_{\ell ;\gamma }}&={{\tilde{\mathcal{N}}}_{\ell }}(\gamma ,w,{{w}_{p}})\iint \Phi _{{{\ell }_{1}}}^{*}(r,\phi ,\gamma w)\Phi _{{{\ell }_{2}}}^{*}(r,\phi ;\gamma w)\!\!\!\\[-5pt]&\quad \times \exp \left({- \frac{{{r^2}}}{{w_p^2}}} \right)\exp \left({- \frac{{2{r^2}}}{{{w^2}}}} \right){\rm d^2}r.\end{split}$$
Here, ${\tilde {\cal N}_\ell}$ absorbs the normalization factors for the mode functions, and ${w_p}$ is the mode size of the pump photon at the crystal plane. The $\gamma$ parameter scales the size of the encoded mode on the SLM relative to the size of the virtual Gaussian mode ($w$) from the fiber.
 figure: Fig. 12.

Fig. 12. (a) Full set of experimentally determined expansion coefficients (bars) for each mode in the predetermined set. Red lines indicate the measured amplitude spectrum before correction. With the mode weightings in hand, one can reconstruct the (b) intensity and (c) phase of the field.

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In Fig. 13, we present the basic optical setup for biphoton coincidence measurements for OAM modes programmed on each SLM, $\Phi _{{\ell _1},{\ell _2}}^*$ [see Fig. 13(a)] as well as the coincidence measurement results [see Figs. 13(b) and 13(c)]. The strict diagonal indicates a correlation between the conjugate ${\pm}\ell$ modes and demonstrates orthogonality, since the rest of the entries are zero. Notably, the fiber decomposition can detect orthogonality on a complete basis, even though the modes as measured are not orthonormal (due to the extra Gaussian factor). Importantly, there is a disparity in the crosstalk matrices between a smaller and larger scaling factor. We further illustrate the dependence of the mode distribution by considering only the diagonal of the measured probabilities in Fig. 14. As shown, the mode distribution becomes narrower with increased $\gamma$, meaning that higher-order modes have lower probability of being detected while lower-order modes increase in detection probability. As a result, the relative weightings decrease until eventually only the $\ell = 0$ mode can be detected. This indicates that there needs to be a good balance between maintaining a wide spectrum and selecting the appropriate $\gamma$.

C. A Comment on Coherence

So far we have used coherent fields as the example. In the case when the incoming field is an incoherent mix of modes, or some degree of coherence exists, then it is more convenient to write the resulting intensity as a sum of the incoherent term (amplitudes only) and the interference term: $I = {U^*}U = {I_{{\rm incoh}}} + {\gamma _{{tc}}}{I_{{\rm inter}}}$, where ${\gamma _{{tc}}}$ is the complex degree of temporal coherence. When ${\gamma _{{tc}}} = 1$, we have complete coherence, and when ${\gamma _{{tc}}} = 0$, we have complete incoherence. We have already covered the former case. In the case of the latter, the same approach works, but no modal phases are needed to describe the field, and it reverts to a sum of modal intensities, $I({\textbf x}) = \sum\nolimits_n |{c_n}{|^2}|{\Phi _n}({\textbf x}{)|^2}$, rather than a sum of modal fields as given in Eq. (4). For more details on this, the reader is referred to [91].

 figure: Fig. 13.

Fig. 13. (a) Example of a typical quantum experiment where photons from an entanglement source (ES) are decomposed in the OAM eigenstates using a SLM with the projected photons coupled to single-mode fibers and avalanche photodiodes (APD). Detected correlations for different OAM modes when the encoded mode has a Gaussian argument with (b) $\gamma = 0.4$ and (c) $\gamma = 0.9$.

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5. CONCLUDING REMARKS

In this tutorial, we have outlined how to get started and avoid pitfalls with the modal decomposition of light. Because the tool has the ability to unravel any incoming field into its constituent parts, it is highly versatile and powerful, revealing the full physical nature of the field. For example, with the field known, the modal changes can be tracked dynamically, used to infer wavefront aberrations without a wavefront sensor, or simply used to infer otherwise impossible to measure quantities, such as the OAM density of light [64]. Finally, in order to support the reader, we have provided all the MATLAB code needed to reproduce the key results shown in this tutorial [85].

APPENDIX A: DIY PHOTO-DETECTOR

It is relatively easy to construct an accurate, low-speed (kHz) photo-detector using a PD and trans-impedance amplifier (TIA). The TIA converts the current output of the PD to a voltage, which can be measured accurately using a multi-meter, oscilloscope, or even an Arduino. Figure 15 shows a breadboard layout of the required electronics with an Arduino Nano—the circuit is standard, and there are numerous online resources describing its operation. This DIY circuit is significantly cheaper than purchasing a calibrated power meter or good camera, but since it is not calibrated, it is not suitable for absolute measurements. It is sufficient, however, for the relative measurements required for modal decomposition. In addition, since the presented design is Arduino based, it allows for easy laboratory automation.

 figure: Fig. 14.

Fig. 14. Theoretical ($T$) and experimental ($E$) mode distribution for detected modes using an optical fiber and a SLM as a detector. The $\gamma$ parameter controls the size of the mode on the SLM relative to the fiber mode size.

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 figure: Fig. 15.

Fig. 15. Breadboard connection diagram for a photodiode and transimpedance amplifier (with variable gain set by a potentiometer), connected to an Arduino Nano via the green wire. The opamp choice is flexible, such as a TL071A.

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Listing 1. Arduino code to continuously read from ADC pin A0 and write the value to the serial port. In addition, the built-in LED is dimmed proportional to the input reading for easy alignment.

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The microcontroller on the Arduino measures the output of the TIA and outputs the values to a serial port for input into software on a PC (such as MATLAB). The analog-to-digital converter has a 10 bit ADC accuracy (i.e., ${2^{10}} - 1 = 1023$ voltage levels from 0 V to 5 V). More precise measurements can be made by connecting an oscilloscope or multi-meter to pin 6 of the opamp, which is the output of the TIA. If an Arduino is not desired, an alternative 5 V power supply for the TIA should be provided.

The PD should be chosen according to your wavelength and package requirements. For instance, fiber-coupled PDs enable straightforward SMF measurements. For higher-speed measurements, the PD must be chosen carefully, and the circuit design will change. This is out of the scope of this basic guide; however, the provided design will typically be sufficient for measurements up to at least several kHz.

Due to limitations of the Arduino, the TIA, and the serial port, the sample rate is limited to about 9 kHz. In the code provided in Listing 1, the readout to the serial port is approximately 500 Hz. Listing 2 provides a simple MATLAB code to read out a number of values from the provided Arduino code.

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Listing 2. A MATLAB script to read 1000 values from the serial port, normalize them, and add them to a vector.

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Funding

Department of Science and Technology (South Africa); National Research Foundation of South Africa (121908).

Disclosures

The authors declare no conflicts of interest.

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Figures (15)
Fig. 1.
Fig. 1. Modal decomposition concept. The field on the left-hand side, perhaps unknown in its properties, is resolved into a sum of modes from a known orthonormal basis, shown here as the ${\rm LG}_p^\ell$ modes. The task is to find the unknown coefficients ${c_{\ell ,p}}$. Summing the right-hand side then provides all the information on the left-hand side.

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Fig. 2.
Fig. 2. Modal decomposition with an intensity detector. Lenses ${{\rm L}_1}$ and ${{\rm L}_2}$ image the input spatial mode $U({\textbf x})$ onto the SLM encoding $\Phi _n^*({\textbf x})$. The resulting mode is then Fourier transformed with lens ${\rm L_3}$ and detected at the origin of the Fourier plane with a camera or single pixel detector.

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Fig. 3.
Fig. 3. ${\rm LG}_{p = 0}^\ell$ decomposition illustrating the effect of the chosen detection cell size for (a) $d = \frac{1}{4}{d_{{\rm min}}}$ (very sensitive to noise), (b) $d \approx {d_{{\rm min}}}$ (accurate), and (c) $d = 3{d_{{\rm min}}}$ (robust but with some crosstalk). Transverse intensity profiles on the diagonal for $\ell = 0$ and $\ell = 5$ are shown above with red disks indicating the detection cell area in each case.

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Fig. 4.
Fig. 4. Finding the optical axis: by allowing a large beam to propagate through the detection system, such as a uniform disk (a), a Gaussian fit (b), (c) can be performed at the detection plane (d) to obtain the pixel coordinates $({c_x},{c_y})$ of the optical axis. The size of the detection cell, denoted by the red circle in (d), is then found using Eq. (15) or approximately from the width of the fitted Gaussian.

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Fig. 5.
Fig. 5. Modal decomposition with a single-mode fiber. Lenses ${{\rm L}_1}$ and ${{\rm L}_2}$ image the incoming spatial mode $U({\textbf x})$ onto the SLM. The resulting mode is then imaged into the core of the SMF using lenses ${\rm L_3}$ and ${\rm L_4}$. The light from the fiber is coupled to a detector (such as a power meter) that registers a power reading.

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Fig. 6.
Fig. 6. Observe how the raw measured spectrum coefficients of a uniform ${\rm LG}_0^\ell$ superposition (a), (b) depend on the choice of basis. In the case of the full LG basis (c), (d), the SLM rescales the amplitudes. In the case of the OAM eigenstates (e), (f), these are not a full basis of the transverse plane.

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Fig. 7.
Fig. 7. ${\rm LG}_0^\ell$ decomposition illustrating the consequences of detection (a) $\frac{1}{4}{z_R}$ before the Fourier plane, (b) in the Fourier plane, and (c) $\frac{1}{2}{z_R}$ after the Fourier plane. Transverse intensity profiles (left) at the corresponding plane for a subset of $\ell$ are shown with phase profile insets of each mode in the top-right corner.

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Fig. 8.
Fig. 8. ${\rm LG}_0^\ell$ decomposition illustrating the consequence of displacement of the filter’s central axis from that of the input beam in the (a) negative horizontal and (b) positive diagonal directions for 20% of the beam waist. Bar graphs show the decomposition weights along the diagonals (${\ell _{{\rm input}}} = {\ell _{\rm{filter}}}$). Transverse intensity profiles are shown as insets for the subset $\ell = [- 1,1]$ (lower left corner) as well as $\langle \ell = - 10|\ell = - 10\rangle$ (left) and $\langle \ell = 10|\ell = 10\rangle$ (right) (top right corner). Red cross-hairs show the optical axis of the system.

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Fig. 9.
Fig. 9. Radial mode decomposition of a ${\rm LG}_{p = 2}^{\ell = 1}$ beam in the ${\rm LG}_{\ell = 1}^p$ basis over $p = [0,4]$ for a match filter mode size ${w_M}$ (a) equal to and (b) at 60% of the input mode size ${w_0}$ with (i) transverse phase and intensity profiles, (ii) intensity distributions in the Fourier plane, and (iii) mode weights.

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Fig. 10.
Fig. 10. Setup to perform wavefront correction (a): this is the same as the detection setup in Fig. 2, but the beam should overfill the SLM screen, and the wavefronts should be approximately planar. By encoding only a helical phase (b) on the SLM and using the image of the distorted vortex at the Fourier plane (c), a phase retrieval algorithm can determine the phase map (d) that produces such a beam. The wavefront distortions (e) are isolated by subtracting the ideal phase (b) from the retrieved phase (d). The corrected beam (f) contains very few residual wavefront errors.

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Fig. 11.
Fig. 11. Optical overlaps showing the reference mode and the three optical inner-products required to determine ${c_{{p\ell}}}$ in the case where $p = \ell = 1$. Here, ${\cal M}$ denotes the square root of the maximum measured intensity in the detection cell of all the optical overlaps performed.

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Fig. 12.
Fig. 12. (a) Full set of experimentally determined expansion coefficients (bars) for each mode in the predetermined set. Red lines indicate the measured amplitude spectrum before correction. With the mode weightings in hand, one can reconstruct the (b) intensity and (c) phase of the field.

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Fig. 13.
Fig. 13. (a) Example of a typical quantum experiment where photons from an entanglement source (ES) are decomposed in the OAM eigenstates using a SLM with the projected photons coupled to single-mode fibers and avalanche photodiodes (APD). Detected correlations for different OAM modes when the encoded mode has a Gaussian argument with (b) $\gamma = 0.4$ and (c) $\gamma = 0.9$.

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Fig. 14.
Fig. 14. Theoretical ($T$) and experimental ($E$) mode distribution for detected modes using an optical fiber and a SLM as a detector. The $\gamma$ parameter controls the size of the mode on the SLM relative to the fiber mode size.

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Fig. 15.
Fig. 15. Breadboard connection diagram for a photodiode and transimpedance amplifier (with variable gain set by a potentiometer), connected to an Arduino Nano via the green wire. The opamp choice is flexible, such as a TL071A.

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Tables (2)

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Table 1 Listing 1. Arduino code to continuously read from ADC pin A0 and write the value to the serial port. In addition, the built-in LED is dimmed proportional to the input reading for easy alignment.

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Table 2 Listing 2. A MATLAB script to read 1000 values from the serial port, normalize them, and add them to a vector.

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Equations (30)

Equations on this page are rendered with MathJax. Learn more.

v = ∑ n c n e n ,
e m ⋅ e n = ⟨ e m | e n ⟩ = δ m , n ,
c n = ⟨ e n | v ⟩ .
U ( x ) = ∑ n c n Φ n ( x ) ,
∫ d 2 x Φ m ∗ ( x ) Φ n ( x ) = ⟨ Φ m | Φ n ⟩ = δ m , n ,
c n = ⟨ Φ n | U ⟩ = ∫ d 2 x Φ n ∗ ( x ) U ( x ) .
U ( x ) → SLM + lens W ( k ) = C ∫ S d 2 x Φ n ∗ ( x ) U ( x ) × exp ⁡ ( − i 2 π λ f k ⋅ x ) ,
W ( 0 ) = C ∫ S d 2 x Φ n ∗ ( x ) U ( x ) ∝ c n .
| W ( 0 ) | 2 ∝ | c n | 2 = ρ n 2 .
( ρ n cos ) 2 = | 1 2 ⟨ Φ 0 + Φ n | U ⟩ | 2 ,
( ρ n sin ) 2 = | 1 2 ⟨ Φ 0 + i Φ n | U ⟩ | 2 .
( ρ n cos ) 2 = 1 2 ( ρ 0 2 + ρ n 2 + 2 ρ 0 ρ n cos ⁡ ϕ n ) ,
( ρ n sin ) 2 = 1 2 ( ρ 0 2 + ρ n 2 + 2 ρ 0 ρ n sin ⁡ ϕ n ) .
ϕ n = a r c t a n ( 2 ( ρ n sin ) 2 − ρ 0 2 − ρ n 2 2 ( ρ n cos ) 2 − ρ 0 2 − ρ n 2 ) .
d m i n ≈ λ f D ,
c ~ n = ∫ d 2 x Φ n ∗ ( x ) U ( x ) G ( x ) ,
G ( x ) = 2 π w 2 exp ⁡ ( − r 2 w 2 ) ,
N m a x = V 2 2 ,
V = 2 π λ a N A = 2 π λ a n c o r e 2 − n c l a d 2 ,
P n = η | c n | 2 ∫ ⟨ S ( x ) ⟩ ⋅ d x ,
P n = η | c n | 2 ∫ | G ( x ) | 2 d x = η | c n | 2 .
c n = γ n c ¯ n .
W ( 0 ) ∝ ∫ S d 2 x Φ n ∗ ( x ) M n U ( x ) ,
= 1 M n c n ,
U ( r , ϕ ) = ∑ ℓ c ℓ A ℓ ( r ) exp ⁡ ( i ℓ ϕ ) ,
W ( 0 ) ∝ ∑ ℓ c ℓ ∫ 0 ∞ r d r A ℓ ( r ) ∫ − π π d ϕ exp ⁡ [ i ( ℓ − n ) ϕ ] , ∝ c n ∫ 0 ∞ r d r A n ( r ) .
U ( x ) = ∑ n c n Φ n ( x ) = ∑ n c ¯ n Φ ¯ n ( x ) ,
c ¯ n = ∑ m c m ⟨ Φ ¯ n | Φ m ⟩ .
U = 1 N ( 1 2 e i π L G 0 1 + 1 4 e i π / 4 L G 1 − 1 + e − i π / 2 L G 2 1 + 1 2 L G 0 0 + 1 4 e i π / 4 L G 1 1 + e − i π / 2 L G 2 − 1 ) ,
c ℓ ; γ = N ~ ℓ ( γ , w , w p ) ∬ Φ ℓ 1 ∗ ( r , ϕ , γ w ) Φ ℓ 2 ∗ ( r , ϕ ; γ w ) × exp ⁡ ( − r 2 w p 2 ) exp ⁡ ( − 2 r 2 w 2 ) d 2 r .
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