When a light beam passes through or is reflected off a physical object or medium, its intensity and phase can be uniquely affected [1]. The information about the object or medium could be resolved by investigating the beam’s intensity as captured by a camera or its phase as obtained by an interferometer [2]. Analogous to time signals, which can be decomposed into multiple orthogonal frequency functions, a light beam can also be decomposed into a set of spatial modes that are taken from an orthogonal basis [3]. Such decomposition can potentially provide a tool for spatial spectrum analysis, which may enable the extraction of object information that might not be readily achievable using traditional approaches.

An arbitrary beam (e.g., an object-truncated Gaussian beam) can also be described by its complex spatial spectrum. Such a complex spectrum is formed by the beam’s complex decomposition coefficients over a mutually orthogonal modal basis set [3,4], such as Laguerre–Gauss (${\mathrm{LG}}_{p,\ell }$ with $p=0,1,2,\dots$, as the radial order, and $\ell =0,\pm 1,\pm 2,\dots$ as the azimuthal order) modes [4–6]. Generally, the spectrum of a beam comprising a single pure mode peaks only at the value corresponding to the order of the mode, while the spectrum of an arbitrary beam could have complex non-zero values for many mode orders.

A subset of LG modes are orbital-angular-momentum (OAM) modes with zero radial index, e.g., ${\mathrm{LG}}_{0,\ell }$ [5]. An OAM mode has a phase front of $\exp (\mathrm{j}\ell \varphi )$, which “twists” in a helical fashion as it propagates [5], where $\ell$ is also referred to as the OAM order, and $\varphi$ is the azimuthal angle. OAM modes are incomplete in the radial coordinate, but they do form a complete orthogonal basis in the azimuthal coordinate. Therefore, when they are used as the orthogonal basis on which to characterize an object-truncated beam, the coefficient of various orders of OAM modes may represent the various azimuthal properties of the object [3,7–10].

The properties of OAM modes include the following: (1) The phase-change rate of an OAM mode is proportional to its order, meaning that a larger-order OAM beam has a smaller phase-change spatial periodicity [9,10]; (2) the complex OAM spectrum of an arbitrary beam could form a Fourier pair with its spatial-intensity distribution in the azimuthal direction [3,9,10]; (3) the intensity of an OAM mode is circularly symmetric, which means the intensity of a beam’s OAM spectrum is generally rotation insensitive [5]; and (4) an OAM mode is relatively stable in a homogeneous medium, which indicates that the amount of OAM of a beam could be constant during propagation regardless of the beam diffraction [1,2].

The use of structured beams has recently been investigated in imaging [4,11–25], sensing [4,11,26–36], communications [4,11,37,38], and other applications [39]. Such investigations include using an OAM basis for single-frequency imaging and sensing in the radio frequency domain [24] and the quantum domain [20]. It may be desirable to design a complex OAM spectrum analyzer in the classical optical domain and explore its potential to provide information that could not be readily obtained using the traditional approach.

In this Letter, we demonstrate the use of OAM-based complex spectral analysis in the classical optical domain to measure object parameters. Through measuring the complex OAM spectrum, we could identify the object’s relative shape information from the OAM intensity spectrum and its position information from the OAM phase spectrum. Specifically, we explore the potential of a complex OAM-spectrum-based system to detect the opening angle and orientation of a sector-shaped object, achieving a $>15\mathrm{dB}$ signal-to-noise ratio (SNR). Our results show the following: (1) The OAM intensity spectrum is dependent on the opening angle of the object but insensitive to its orientation; and (2) the OAM phase spectrum is dependent on the orientation of the object but insensitive to its opening angle.

Figure 1 shows the concept of using an OAM spectral analyzer to measure object parameters. To measure an object’s parameters, including shape, thickness, and temperature, a light beam (Gaussian beam, OAM beam, or other beams) can be shone onto an object, and, by investigating this beam after the object interaction, the parameter of interest could be retrieved. In this work, we use the measurement of the opening angle and the opening’s orientation of an object as an example. A camera-based system may suffer the following problems: (1) Due to the diffraction caused by object truncation and beam propagation, the image may become too blurry to quantify the opening angle of the object [Fig. 1(c)], especially when the object is not in focus on the camera; (2) if the object is rotating at a relatively high speed, the camera may not be able to capture a clear image; and (3) a camera is usually a multi-pixel device, which may cost more than a single-pixel device.

 

Fig. 1. Concept of using the OAM spectrum to measure an object’s parameters. The beam’s intensity profile and OAM spectrum (a) before object truncation, (b) right after object truncation, and (c) some distance after object truncation.

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However, the complex OAM-spectrum-based approach could potentially determine the opening angle and opening’s orientation of an object without suffering the abovementioned issues. This approach is based on the following facts: (1) Object truncation of the probe beam could change its OAM spectrum, and the OAM spectrum before and after the object truncation differ sufficiently that the “truncation” could be identified; (2) the complex OAM spectrum of a beam forms a Fourier pair with its spatial distribution in the azimuthal direction [7–10], in which an opening on the object would lead to a Sinc function in its OAM intensity spectrum; (3) the light, propagating onward from the truncated object, maintains the powers of its constituent OAM modes so that its OAM intensity spectrum is insensitive to the radially symmetric diffraction kernel [2]; and (4) the phase-change rate of an OAM is proportional to its order; therefore, rotating an object may cause different phase delays to different components on the OAM phase spectrum.

Figure 2(a) shows the experimental setup. A spatial light modulator (SLM-1) is used to generate the desired probe beam with a certain OAM order. We generally use a Gaussian beam ($\ell =0$) as the probe, and we also show the cases when various orders of OAM beams are used as the probes to measure object parameters. SLM-2 is used to emulate objects with various parameters (opening angles, orientations, and numbers of the opening slot). Figure 2(b) shows the properties of these objects. We made the following assumptions: (1) The object is larger than the probe beam; (2) the object has a wedge-shaped opening with angular extent $\theta$; (3) the orientation of the opening ($\delta$) is defined as the angle between the left edge of the slot opening and the y axis; and (4) due to the hardware limitation, we change the orientation of the opening manually without real-time rotation of the object.

 

Fig. 2. (a) Experimental setup. The SLM-3, L-1, L-2, collimator, and power monitor form the complex OAM spectrum analyzer. Col, collimator; HWP, half-wave plate; M, mirror; SLM, spatial light modulator; L, lens. (b) The shape and position of the object. $\theta$, opening angle of the object; $\delta$, orientation of the object relative to the y axis.

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The beam truncated by the “object” is then collected by a complex OAM spectrum analyzer, which is composed of SLM-3, a subsystem coupling light from free space to single-mode fiber (SMF), and a power monitor. To obtain a complex OAM spectrum, we measure the OAM intensity spectrum and the phase spectrum separately. To measure the OAM intensity spectrum, SLM-3 is loaded with various spiral phase patterns, so that the decomposition of incoming light onto various OAM orders is measured sequentially and together forms an OAM intensity spectrum. To measure the OAM phase spectrum, we measure the relative phase between OAM $\ell$ and OAM 0, sequentially. For each phase measurement, four different specially designed patterns are sequentially loaded on SLM-3, and together the four intensities measured by the power monitor enable calculation of the phase. In this experiment, the distance between the object (SLM-2) and the OAM spectrum analyzer (SLM-3) varies from 0 to 35, 52, and 104 cm. We note that, for zero distance, SLM-2 serves only as a mirror, and the “object” emulation pattern and OAM spectrum analyzer pattern are combined and loaded on SLM-3.

First, we use the complex OAM-spectrum-based approach to measure the opening angle of an object. As Fig. 3(a) shows, the sample object has an opening angle of $2\pi /3$ and is placed in the propagation path of a Gaussian probe beam. For comparison, we also measure the spatial distribution of the object-truncated light with a lensless camera. As Fig. 3(b) shows, it is difficult to determine the opening angle of the object due to diffraction without an imaging system in place. However, when we measure the OAM intensity spectrum, we observe several local minima (dips) in the OAM spectrum, as shown in the Fig. 3(c). The power dips appear on OAM orders $|\ell |=3N$ ($N$ is a non-zero integer), indicating that the object has an opening angle of $2\pi /3$. When an object, azimuthally opened with an angle of $\theta$, truncates an OAM probe with order ${\ell }_1$ (here ${\ell }_1=0$), the probe’s spectral OAM component of order ${\ell }_2$ vanishes when ${\ell }_1-{\ell }_2=2\pi N/\theta$ [7,8,9,13]. This is due to the “vanishing effect” of the overlap integral between ${\ell }_1$ and ${\ell }_2$ over an azimuthal angle of $\theta$, that is ${\int }_0^{\infty }{\int }_0^{\theta }u(r,\varphi ,{\ell }_1)u^{*}(r,\varphi ,{\ell }_2)\mathrm{d}\varphi \mathrm{d}r=0$, when $({\ell }_1-{\ell }_2)\theta =2N\pi$, where $u(·)$ is the electric field. The power difference between a dip and its neighbors has a difference of $>15\mathrm{dB}$, indicating that our approach has a high SNR.

 

Fig. 3. a1–a8: various orientations (states) of an object having an opening angle of $2\pi /3$. b1–b8: the image of the light beam truncated by objects a1–a8, respectively. (c) The OAM intensity spectrum measured for the light truncated by objects a1–a8. (d) The OAM intensity spectrum measured for the light truncated by objects having various opening angles. (e) The relationship between the opening angles and the first-dip position in the OAM intensity spectrum. (f) The OAM intensity spectrum measured for the light truncated by an object having an opening angle of $2\pi /3$ when the distance from the object to the OAM spectrum analyzer is 0, 35, 52, and 104 cm. The distance from the object to the OAM spectrum analyzer is zero in (c)–(e).

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When we rotate the object in Fig. 3(a) to various angles [0, $\pi /4$, $\pi /2$, $3\pi /4$, $\pi$, $5\pi /4$, $3\pi /2$, and $7\pi /4$, corresponding to the states 1–8 in Fig. 3(b)], the measured OAM spectrum is almost unchanged, as Fig. 3(c) shows. We believe that this is due to the circular symmetry of the intensity of an OAM beam. Therefore, our approach could potentially be used to determine the opening angle of the object even when the object is rotating.

Furthermore, we verify the relationship between an object’s opening angle and the OAM intensity spectrum of the beam truncated by that object. As Fig. 3(d) shows, when the object’s opening angle is modified to $2\pi /4$, minima are observed on the modes with $|\ell |=4N$. When the opening angle is $2\pi /5$, the minima are observed on the modes with $|\ell |=5N$. Our simulation [Fig. 3(e)] shows a more general case, and the position of the first dip is inversely proportional to the object’s opening angle.

As Fig. 3(f) shows, the OAM spectrum varies within a small range when the spectrum analyzer is placed at various distances (0, 35, 52, and 104 cm) from the object. This is because the kernel of the diffraction (the effect of propagating through free space) is proportional to the phase-changing direction of the OAM beam. However, we observe some power fluctuation at various distances. We believe this is because our system ignores LG modes with non-zero radial orders ($p\ne 0$). When the beam is truncated by an object and propagates in free space, higher-radial-order LG modes may arise, whose power is not collected by our OAM spectrum analyzer, since it contains a free space to SMF coupling subsystem.

In addition to the OAM intensity spectrum, the resultant OAM modes may also have various phases. Here, we sequentially measured the relative phase between OAM $\ell$ and OAM 0 for $\ell =-8,-7,\dots ,+7,+8$ to get an OAM phase spectrum. In general, to measure the phase difference between OAM ${\ell }_1$ and OAM ${\ell }_2$, we load phase masks ${\mathrm{T}}_0$, ${\mathrm{T}}_{45}$, ${\mathrm{T}}_{90}$, and ${\mathrm{T}}_{135}$ on SLM-3, where

We note that reports have shown the complex OAM spectrum measurement using an interferometer [34,35,41], but these systems are generally sensitive to environmental vibrations. However, the approach in this work is based on direct methods [29,36,40,42] and may provide more stable phase measurements.

We measure the OAM phase spectrum of the object-truncated beam when the object has an opening angle of $2\pi /3$ with various orientation angles $\delta$. As Figs. 4(a) and 4(b) show, the orientation angle is closely related to the slope of the OAM phase spectrum. When the orientation angle of the object is $\delta$, the slope of the measured phase is approximately $\delta$. We believe this is because different OAM modes have different phase-change rates. Rotating the object by a certain angle may cause different phase delays for different OAM modes, and such phase delays are proportional to the OAM orders [see Fig. 4(c)]. We also fix the orientation of the object to be $\pi /8$. When we change the object’s opening angle, the slope of the OAM spectrum is nearly constant [Fig. 4(d)], indicating that the OAM phase spectrum is insensitive to the object’s opening angle.

 

Fig. 4. (a), (b) OAM phase spectrum measured for the light truncated by an object having an opening angle of $2\pi /3$ but various negative/positive orientation angles. (c) The relationship between the orientation angle of the object and the slope of the OAM phase spectrum. (d) The OAM phase spectrum measured for the light truncated by an object having various opening angles and an orientation angle of $\pi /8$. In this figure, the distance from the object to the OAM spectrum analyzer is zero. In the measurement, the data is calculated by an arctan function, which is between $-\pi /2$ and $\pi /2$, and we may also add a $\pi$ or $-\pi$ phase shift to the measurement for the convenience of slope calculation.

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We note that other modal basis sets could also be applied for object parameter measurement according to their unique properties.