Measuring the OAM spectrum of a fractional helical beam in a single shot

Tushar Sarkar; Jiapeng Cai; Xiang Peng; Wenqi He

doi:10.1364/PRJ.538320

1. INTRODUCTION

A helical beam or vortex beam, known as an optical vortex, is a special type of electromagnetic beam characterized by a helical phase front, resulting from a helical rotation of the phase front along the direction of the optic axis [1 –3]. It is quantitatively described by a phase term $\exp (i m φ)$ , where $m$ and $φ$ refer to topological charge (TC) and azimuthal angle, respectively. This TC quantifies the number of twists in the helical phase front and helps to determine the amount of orbital angular momentum (OAM) $m ℏ$ per photon carried by the beam [2]. The beam with integer TC ( $m$ ) is termed an integer vortex beam (IVB). The helical phase of the IVB, represented by a $2 π m$ step, ensures that the TC is always an integer. An OAM carrying IVB reveals an annular “ring” intensity distribution at the center of the beam owing to the point singularity around the helical phase [3]. The IVBs with different OAM modes are intrinsically orthogonal to each other in Hilbert space, used as a promising degree of freedom for encoding optical information [4], and have shown tremendous potential applications in optical communication [5], particle trapping [6], super-resolution imaging [7], quantum information processing [8], and OAM holography [9], to name a few. Most of the studies on helical beams usually restrict the value of TC to an integer. Indeed, a non-integer value of TC ( $l$ ) is also possible, which shows the phase variation around the axis of the beam is $2 π l$ but no longer $2 π m$ . Such a type of beam with non-integer TC is referred to as a fractional vortex beam (FVB) [10,11]. The intensity distribution of the FVB is no longer circularly symmetric due to a single mixed screw edge dislocation phase structure. In contrast to IVB, where the phase singularity is concentrated at a single point, FVB contains multiple phase singularities, creating a more complex phase pattern [11]. It is demonstrated that an FVB can be explicit as a superposition state of the IVBs. In other words, an FVB could be decomposed into a Fourier series of IVBs with different intensity weights [12,13]. The characteristics of the FVB offer unique advantages in various prominent applications in contrast to IVB. For instance, FVB is utilized to sort cells and precisely control their orientation [14,15], encode the optical information and improve the transmission capacity of the optical channels [16], increase the number of available OAM modes [17], and enhance the anisotropic edge [18]. Apart from these, such beams are also useful in quantum digital spiral imaging [19], cryptography [20], quantum entanglement [21], etc. Subsequently, to meet these significant applications, how to characterize the properties and behavior of the FVB becomes a pivotal issue. This directly makes the accurate measurement of the TC and OAM spectrum be a hot subject.

Over the years, pioneering works on the detection of the TC and OAM spectrum of the FVB have been researched. Usually, the TC measurement methods relied on interferometry and diffraction. In the interferometry method, a modified Mach-Zender interferometer with a Dove prism was first used to measure the magnitude of TC, and the TC is determined by counting the number of petals from the interference pattern [22,23]. Recently, a spiral interferometer was designed to record spiral-shaped interference patterns between FVB and a spherical Gaussian beam to estimate the TC [24]. Besides these techniques, a method based on diffraction was reported using a cylindrical lens, and a mode conversion method was also demonstrated [25,26]. In addition, fractional TC measurements based on diffraction have also been reported with rotating phase plates and intersections of circular apertures [27,28]. Additionally, measurement of TC was carried out by using a dynamic annular double slit by recording the interference pattern at the far field. To investigate the TC, the angle between these two slits was continually varied at a specific interval [29]. Recently, an optical-correlator-based technique has been demonstrated to detect fractional TC by measuring the correlation between the interference pattern of FVB and its nearest integer TCs [30]. Machine learning was also utilized to effectively recognize TC using a convolutional neural network [31].

On the other hand, evaluating the OAM spectrum of the helical beam requires decomposing the beam into its constituent OAM modes. This is inherently more challenging than measuring a TC. The OAM spectrum essentially describes how the total OAM content of the beam is distributed among the different OAM modes. Typically, the FVB with fractional TC consists of multiple phase singularities, which breaks the orthogonality of OAM, making it difficult to measure the complicated OAM spectrum [32]. A few methods have been explored to measure the OAM spectrum of the FVB. A multifocal array was designed to measure the OAM spectrum of the FVB [32,33]. This array was used to transform the spot with an integer state $m$ into a Gaussian-like spot when the incident state of the OAM beam is $- m$ and spot intensity equals OAM spectrum weight. Nevertheless, the design of the multifocal array was employed using the pixel checkerboard method in these techniques. The generation of a multifocal array by a pixel checkerboard method needs some precautions such as pixelation artifacts, interference and crosstalk between adjacent vortices, spatial resolution, and phase discontinuity. If these parameters are not accurately designed then it can affect the measurement of the OAM spectrum [34,35]. Moreover, these techniques demand extensive computation, specialized masks, precise alignment, and calibration of the optical setup. Recently, an attractive technique has been reported to measure the OAM spectrum of the FVB in a single shot based on Mach-Zehnder interferometry combined with the Kramers-Kronig relation [36]. However, this technique is sensitive to external vibration and atmospheric turbulence owing to the use of Mach-Zehnder interferometry and also requires substantial computation. Although, the measurement of the OAM spectrum of the FVB is indeed a promising research area, it requires further explorations, involving the development of robust, simple, efficient, and cost-effective techniques.

In this paper, we propose an alternative technique to precisely measure the OAM spectrum of the FVB in a single shot. This is realized using a single-path interferometer configuration with polarization phase-shifting and a strategy of space division multiplexing. In this technique, multiple phase-shifted interferograms are captured, to compute the phase profile of the incident FVB, in a single shot by using the proposed parallel phase-shifting and space division multiplexing. This single-shot ability is thanks to the measuring of vectorial light fields coming from a single-path interferometer that contains a quarter-wave plate (QWP) and micro polarizer array encoded in the polarization camera. The OAM spectrum of the incident FVB is assessed using an orthogonal projection method where the measured phase is projected over the spiral harmonics [37]. The on-axis configuration of the proposed experimental setup provides high spatial and temporal stability in the measurement of the OAM spectrum against the external environment due to its single-shot nature. Moreover, the proposed technique does not invoke any computation or specialized mask to measure the OAM spectrum in contrast to Refs. [32,36]. Furthermore, a single-shot OAM spectrum retrieval for different FVBs is experimentally demonstrated to evaluate the effectiveness of the proposed technique. A detailed theoretical framework, simulation, and experimental results are presented below.

2. THEORETICAL FRAMEWORK

The proposed technique leverages a highly stable single-path interferometer to measure the OAM spectrum of the FVB. The on-axis configuration implies that the two orthogonally polarized beams propagate along the same optical axis where the FVB is encoded into the horizontal polarization basis ( $x$ ) while the vertical polarization basis ( $y$ ), i.e., plane wave, serves as a reference beam. The orthogonally polarized beam is encoded with fractional helical and non-helical beams that self-interfere with each other and generate an interference pattern at the observation plane. The single-shot OAM spectrum measurement is enabled by simultaneously recording multiple phase-shifted interference patterns using a strategy of space division multiplexing and polarization phase-shifting as shown in Fig. 1. The use of a single-path interferometer combined with space division multiplexing and polarization phase-shifting makes the technique simpler and compact and also free from the requirement of sophisticated optics, specialized masks, and computation. Consider a transversely polarized monochromatic light consisting of two orthogonal polarization bases i.e., $x$ and $y$ , coaxially travelling along the $z$ -axis. The complex amplitude at the $z = 0$ plane is expressed as $U (r) = A_{l} \exp [i (l φ - φ_{R})] \hat{x} + A_{R} (r) \hat{y},$ (1)where $\hat{x}$ , $\hat{y}$ denote horizontal and vertical polarization bases, respectively. $A_{l} (r)$ , $A_{R} (r)$ indicate the amplitude of the FVB and reference beam, respectively. $φ$ is the azimuthal coordinate while $φ_{R}$ signifies the constant phase shift. $l$ represents fractional TC and $r$ is the spatial coordinate at the transverse plane. The fractional TC ( $l$ ) breaks the orthogonality of the OAM; therefore FVB could be decomposed into the basis of integer OAM states. In other words, the FVB could be deemed as a multiplexing IVB with different weights and represented [38] as $\exp (i l φ) = \sum_{m = - \infty}^{\infty} C_{m} (l) \exp (i m φ),$ (2)where $C_{m}$ represents the Fourier coefficient that can be described as $C_{m} (l) = \frac{\exp (i l π) \sin (l π)}{π (l - m)},$ where $l = m + ε$ . Here $m$ is an integer number and $ε$ represents a small fraction.

Figure 1.Conceptual representation of the proposed technique. A coaxially propagating orthogonally polarized beam encoded with helical and non-helical beams travels through the QWP placed at 45° for space division multiplexing and polarization filters that capture the multiple phase-shifted interferograms in a single shot.

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The coaxially propagating orthogonally polarized beams represented in Eq. (1) travel through a wave plate retarder (QWP) that is placed at 45° with respect to coaxially propagating beams. The QWP converts the linearly polarized beams into mutually orthogonal circularly polarized beams, represented as $U_{p} (ρ) = \frac{1}{\sqrt{2}} (\begin{matrix} 1 & i \\ i & 1 \end{matrix}) U_{p} (r),$ (3)where $p = x, y$ , $U_{x} (r) = {(1 0)}^{T}$ denotes the object beam loaded with FVB, $T$ is the transpose, and $U_{y} (r) = {(0 1)}^{T}$ stands for the reference beam, i.e., a plane wave. The matrix in Eq. (3) represents the Jones matrix of the QWP placed at 45° with respect to the incident orthogonally polarized beams. The symbol $ρ$ signifies spatial coordinates at the observation plane.

Now, the coaxially propagating orthogonal circularly polarized beams pass through a linear polarizer, sitting at an angle $β$ to the object beam (horizontal direction). The linear polarizer orients orthogonal circularly polarized beams into a linearly polarized beam with the same polarization bases and it is represented as $U_{p}^{'} (ρ) = (\begin{matrix} \cos^{2} β & \sin β \cos β \\ \cos β \sin β & \sin^{2} β \end{matrix}) U_{p} (ρ),$ (4)where $U_{p}^{'} (ρ)$ represents the field after the linear polarizer. As the orthogonal circularly polarized beams travel through the linear polarizer, they acquire similar polarization directions. Owing to this, an interference pattern forms at the observation plane, and the intensity at the observation plane is expressed as $I_{β} (ρ) = U_{p}^{' *} (ρ) U_{p}^{'} (ρ) \propto [{| U_{x}^{'} (ρ) |}^{2} + {| U_{y}^{'} (ρ) |}^{2} + U_{x}^{' *} (ρ) U_{y}^{'} (ρ) + U_{x}^{'} (ρ) U_{y}^{' *} (ρ)] \propto {{| U_{x}^{'} (ρ) |}^{2} + {| U_{y}^{'} (ρ) |}^{2} + | U_{x}^{'} (ρ) | | U_{y}^{'} (ρ) | \cos [2 β - (φ - φ_{R})]} \propto [{| U_{x}^{'} (ρ) |}^{2} + {| U_{y}^{'} (ρ) |}^{2} + | U_{x}^{'} (ρ) | | U_{y}^{'} (ρ) | \cos (2 β - φ_{d})],$ (5)where $φ_{d} = φ - φ_{R}$ indicates the phase difference between an object and a reference beam and * denotes a complex conjugate. $U_{p}^{'} (ρ) = U_{x}^{'} (ρ) + U_{y}^{'} (ρ)$ depicts the resultant field at the observation plane. From Eq. (5), it is noted that an interferogram with different phase shifts can be recorded by orienting the angle $β$ of the polarizer.

On putting $U_{x}^{'} (ρ) = A_{l} \exp [i (l φ - φ_{R})]$ and $U_{y}^{'} (ρ) = A_{R}$ , Eq. (5) modifies to $I_{β} (ρ) \propto [A_{l}^{2} (ρ) + A_{R}^{2} (ρ) + 2 A_{l} (ρ) A_{R} (ρ) \cos (2 β - φ_{d})] .$ (6)

The interference of the FVB and reference beam, i.e., plane wave, generates a petal-like structure. In the proposed technique, to record the multiple phase-shifted interferograms simultaneously, a parallel phase-shifting approach is adopted using a polarized camera. This is realized by achieving space division multiplexing using the amalgamation of an orthogonal circularly polarized beam generated from the QWP and a polarized camera comprising the micro-polarized array [39]. The micro-polarized array consists of four different polarizing filters angled at 0°, 45°, 90°, and 135° placed in a systematic pattern across the camera, which enables the recording of four phase-shifted interferograms simultaneously. Single-shot intensity patterns captured by the camera are used to extract four phase-shifted interferograms. Therefore, the intensity distribution of four phase-shifted interferograms captured by the polarized camera is represented as $I_{0} (ρ) \propto [A_{l}^{2} (ρ) + A_{R}^{2} (ρ) + 2 A_{l} (ρ) A_{R} (ρ) \cos φ_{d}],$ (7) $I_{π / 2} (ρ) \propto [A_{l}^{2} (ρ) + A_{R}^{2} (ρ) + 2 A_{l} (ρ) A_{R} (ρ) \sin φ_{d}],$ (8) $I_{π} (ρ) \propto [A_{l}^{2} (ρ) + A_{R}^{2} (ρ) - 2 A_{l} (ρ) A_{R} (ρ) \cos φ_{d}],$ (9) $I_{3 π / 2} (ρ) \propto [A_{l}^{2} (ρ) + A_{R}^{2} (ρ) - 2 A_{l} (ρ) A_{R} (ρ) \sin φ_{d}] .$ (10)

A conventional four-phase-shifting technique is utilized to extract the phase distribution of the incident FVB from the single-shot recorded multiple-phase shifted interferograms using a polarized camera and it is expressed [40] as $φ (ρ) = \arctan [\frac{I_{4} (ρ) - I_{2} (ρ)}{I_{1} (ρ) - I_{3} (ρ)}],$ (11)where $I_{4} (ρ)$ , $I_{3} (ρ)$ , $I_{2} (ρ)$ , and $I_{1} (ρ)$ indicate phase-shifted interferograms of phase shifts of $π / 2, π, 3 π / 2$ , and 0, respectively. Using Eq. (11) the incident FVB is recovered and represented as $U (ρ)$ .

To assess the OAM spectrum of the FVB, an orthogonal projection method is employed [37,41]. This is implemented by projecting the experimentally measured complex field $U (ρ)$ over spiral harmonics $\exp (i m φ)$ , which helps to acquire the information of TC composition embedded into the complex optical field. This projection method makes it possible to investigate the OAM spectrum of the incident light beam. To evaluate the OAM spectrum of the incident light beam, an angular Fourier transform is applied over a complex field $U (ρ)$ to obtain the complex coefficient $C_{m} (ρ)$ . Each complex coefficient $C_{m} (ρ)$ contains a specific TC value depending on the radial coordinates. Furthermore, the OAM spectrum of the FVB is determined by integrating the modulus square of $C_{m} (ρ)$ with respect to the radial coordinates.

By applying the angular Fourier transform to the experimentally measured complex field, the complex coefficient $C_{m} (ρ)$ is obtained [41,42] as $C_{m} (ρ) = \int_{0}^{2 π} \exp (- i m φ) U (ρ) d φ .$ (12)

An integration along with the radial coordinates is performed over the square modulus of $C_{m} (ρ)$ to assess the OAM power spectrum of the incident FVB: $P (m) = \frac{1}{S} \int_{0}^{\infty} {| C_{m} (ρ) |}^{2} r d r,$ (13)where $P (m)$ and $S = \sum \int_{0}^{\infty} {| C_{m} (ρ) |}^{2} r d r$ signify the OAM power spectrum or weight of the OAM modes and beam power, respectively.

3. EXPERIMENTS AND DISCUSSION

We demonstrate the viability of the proposed technique by measuring the OAM spectrum of the different FVBs in a single shot. For this purpose, a simple and compact single-path interferometer experimental configuration is designed. The schematic of the experimental configuration is depicted in Fig. 2. A linearly vertically polarized beam with a wavelength of 532 nm illuminating from the diode laser is utilized to generate a uniform beam with a plane wavefront using a spatial filter assembly and a lens $L$ . A half-wave plate (HWP) placed at 22.5° orients a linearly polarized beam into an orthogonally polarized beam, i.e., $x$ - and $y$ -polarization bases. The orthogonally polarized beam coaxially propagates and transmits through the beam splitter (BS), illuminating a spatial light modulator (SLM) (LCOS-SLM from Hamamatsu Photonics). To generate FVB, the phase profile of the FVB is encoded into the $x$ -polarization basis using SLM. Here, a phase-only SLM is utilized that modulates only the $x$ -polarization basis while the $y$ -polarization basis remains unmodulated acting as a reference beam. Now, the orthogonally polarized beam after reflection from the BS travels through the quarter-wave plate (QWP) with its fast axis oriented at 45°. The QWP orients a linearly orthogonally polarized beam into a circularly orthogonally polarized beam and enables to achieve space division multiplexing. The transmitted beam from the QWP is captured using a polarized camera. The camera is equipped with a Sony IMX264MZR CMOS polarized sensor that has four different polarizing filters angled at 0°, 45°, 90°, and 135° (pixel resolution of $2448 \times 2048$ with pixel size 3.45 μm and 36 frames per second). A parallel phase-shifting approach to record multiple phase-shifted interferograms in a single shot is achieved by the amalgamation of a QWP and a polarized camera. The linearly orthogonally polarized beam after transmission through a QWP and a micro polarizer array generates a petal structure interference pattern at the camera plane. The phase of the incident FVB is extracted from the multiple phase-shifted interferograms using Eq. (11). Finally, the OAM spectrum is measured from the extracted phase using Eqs. (12) and (13).

Figure 2.Schematic of the experiment to measure the OAM spectrum of the FVB in a single shot. MO, microscopic objective; P, pinhole; L, lens; HWP, half-wave plate; BS, beam splitter; SLM, spatial light modulator; QWP, quarter-wave plate. In the inset, the arrangement of the polarization filters embedded into a polarized camera and respective orientation angles are shown.

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4. RESULTS

We measure the OAM spectrum of different FVBs. The polarized camera captured a single-shot interferogram of FVB and a reference beam and the recorded interferogram depicted in Fig. 3(a) exhibits a petal structure. Figures 3(b)–3(e) indicate the four phase-shifted interferograms extracted from the single-shot interferogram. The phase profile of the incident FVB extracted from the multiple phase-shifted interferograms is shown in Fig. 3(f). The OAM spectrum of the FVB represented in Fig. 3(g) is assessed from the measured phase using an orthogonal projection method. The simulation results in Figs. 4(a)–4(d) show the phase distribution of the incident FVBs with TC $l = 3.50$ , 3.75, 3.90, and 4.80, respectively. Corresponding experimental results are shown in Figs. 4(e)–4(h), directly providing the information of the fractional TC. Unlike an integer vortex beam, the FVB consists of a fractional TC, which reveals that the phase variation is not a complete integer of $2 π$ around the beam axis as shown in Figs. 4(a)–4(d) and 4(e)–4(h). This phase profile also reveals that the singularity is distributed over several points rather than being confined to a single central point and these multiple singularities interact with each other, generating a more complex phase structure. The proposed technique can also determine the sign of the FVB. Furthermore, to measure the OAM spectrum of the FVB from the experimentally measured phase, an orthogonal projection method as explained earlier is utilized and the OAM spectrum results are shown in Fig. 5. Figures 5(a)–5(d) show the OAM spectrum of the FVBs with TC $l = 3.50$ , 3.75, 3.90, and 4.80, respectively, which no longer occupy a single OAM, but also consist of other OAM constituents. It can be seen in Figs. 5(b) and 5(c) that the OAM spectrum of TC $m = 4$ dominates as $l$ approaches 4. Likewise, as $l$ approaches 5, the OAM spectrum of TC $m = 5$ dominates in Fig. 4(d).

Figure 3.Measuring the OAM spectrum from the experimentally recorded interferogram. (a) denotes single-shot recording interferogram and (b)–(e) show multiple phase-shifted interferograms corresponding to phase shifts of $0, π / 2, π$ , and $3 π / 2$ , respectively, extracted from the single-shot recording interferogram. (f) represents the measured phase profile of FVB and the corresponding OAM spectrum is shown in (g).

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Figure 4.(a)–(d) Simulation results of FVBs with TC $l = 3.50$ , 3.75, 3.90, and 4.80, respectively. (e)–(h) Corresponding experimental results.

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Figure 5.OAM spectra of FVBs with TC $l = 3.50$ , 3.75, 3.90, and 4.80.

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Furthermore, to demonstrate the applicability of the proposed technique, the OAM spectrum of the FVBs with higher-order fractional TC is measured. Figures 6(a) and 6(b) depict the OAM spectrum of the FVBs with TC $l = 15.87$ and 29.80, respectively. In Fig. 6(a) the predominant contribution is from the OAM spectrum of TC $m = 16$ , as $l$ is closest to 16, and a stronger contribution from the OAM mode corresponds to $m = 30$ , as $l$ approaches 30 as depicted in Fig. 6(b). The accuracy of the proposed technique is also checked by measuring the OAM spectrum of three different FVBs with TCs $l = 4.458$ , 4.498, and 4.537 where the true topological charge is $l = 4.50$ as represented in Fig. 7(a). The accuracy is confirmed by comparing the results in Fig. 7(a). It can be seen from Fig. 7(a) that the accuracy of the proposed technique is less than 0.1.

$OAM spectra of the FVBs with higher-order fractional TC l=15.87 and 29.80, and corresponding experimentally measured phases are shown in the inset.$

Figure 6.OAM spectra of the FVBs with higher-order fractional TC $l = 15.87$ and 29.80, and corresponding experimentally measured phases are shown in the inset.

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Figure 7.(a) OAM spectra of three different FVBs with TCs $l = 4.458$ , 4.498, and 4.537 and (b) accuracy verification.

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In addition, 10 experimental data are taken for TC $l = 4.750$ to calculate the standard deviation (orange error bars) to further verify the accuracy of the proposed technique and it is verified that the accuracy of the experimental results is less than 0.1. Figure 7(b) depicts the average value of 10 experimental data for TC $l = 4.750$ . The accuracy of the proposed technique is affected due to the size of the polarization optics used in the experiment. Any leakage in the orthogonal polarization bases due to polarization optics will affect the reconstruction quality and hence the accuracy. This issue can be addressed by using bigger-sized polarization optics.

5. CONCLUSIONS

In summary, we have developed a new compact and simpler technique to precisely measure the OAM spectrum of the FVB in a single shot. The OAM spectrum of the FVB is assessed by recording single-shot multiple phase-shifted interferograms using a single-path interferometer combined with space division multiplexing and polarization phase-shifting. The proposed technique offers high accuracy and efficiency, as evidenced by the good agreement between the simulation and experimental results. The viability of the proposed technique is experimentally demonstrated by measuring the OAM spectrum of the different FVBs. The single-shot nature of the proposed method may be useful for the real-time characterization of fractional OAM-based systems. Moreover, this technique may also be beneficial in characterizing the properties and behavior of the FVB.

Category: Physical Optics

Received: Aug. 1, 2024

Accepted: Sep. 14, 2024

Published Online: Nov. 1, 2024

The Author Email: Wenqi He (he.wenqi@qq.com)

DOI:10.1364/PRJ.538320