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Abstract
We propose the use of an intensity technique to decompose superpositions consisting of two, three, or four basis Laguerre-Gaussian (LG) modes, and measure the orbital angular momentum (OAM) of such superpositions. The mode generation and decomposition are both accomplished only on a 2f optical imaging system. We demonstrate numerically and experimentally that the squared amplitudes of superpositions can be determined by recording a single frame of the intensity distribution. This is accomplished by measuring the intensity along certain circles and solving a linear set of equations relating the sampled intensities to squared amplitudes. The accuracy of better than 98% for composite beams consisting of two, and about 90% for composite beams consisting of more than two basis modes are achieved. Finally, we report the value of the measured OAM of the superpositions with excellent accuracy regarding theoretical values, for small and large integer and non-integer OAM.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
In recent years, laser beams carrying orbital angular momentum (OAM) have been widely utilized in optical communications [1], optical tweezers [2], quantum information [3,4], micromanipulation [4], and single-photon level for realizing high-dimensional entanglement [5]. Such optical modes contain a phase factor of exp(imφ) where φ is the azimuthal angle and m is an integer named topological charge [6]. Coming up with a technique to measure the OAM of light has been a topic of intense study. For instance, diffracting the beam from apertures [7], interfering with a Gaussian Ref. [8,9], using cylindrical lenses [10], and modal decomposition methods [11] are different demonstrated techniques to measure light’s OAM. In modal decomposition, a superposition of the basis modes is decomposed into its components, so the expansion coefficients of each component can be determined [12]. Furthermore, modal decomposition has been used to obtain information about the amplitude and phase of the coefficients, which are important parameters in some researches areas, especially in optical communications [13]. In general, all physical parameters such as intensity, phase, wavefront, beam quality factor, and the value of OAM of an unknown field will be extracted, if the expansion coefficients are found [14]. There are several methods to carry out the modal decomposition. For example, it was performed with hard-coded diffractive optics based on the decomposition of the field at certain cross sections of the laser beam into a system of orthogonal functions [15]. Also, log-polar optical transformations were utilized in the form of converting the OAM beams with a helical phase into a beam with a transverse phase gradient [16]. Recently, optical correlation techniques based on the computer-generated holograms (CGHs) have been investigated to decompose multiplexed Laguerre-Gaussian (LG) beams with different radial or azimuthal orders [17]. Such techniques are performed for measurement of the beam quality factor [18], the wavefront and the phase of the light [19], and determination of the OAM of light as well [20]. However, optical correlation techniques need additional optical elements such as spatial light modulators (SLMs) to achieve the modal decomposition. Bayesian analysis, as a widely used technique in machine learning, has been proposed to determine the mode components of a free-space multimode laser beam under various circumstances [21]. Via this transverse mode analysis, the expansion coefficients were numerically determined with an error of about 5%. Digital mode sorting has been utilized to develop and implement the basic principles of digital sorting of the Laguerre-Gauss modes by radial numbers [22]. In such digital sorting, superpositions consisting of three radial LG modes were decomposed based on the high-order intensity moments. Furthermore, the stochastic parallel gradient descent algorithm as an optimization tool was implemented on near-field images to analyze modes and extract the modal coefficients with an accuracy of better than 0.03% [23]. High-order intensity moments method is the other technique that was employed to measure the spectrum of modes having the same signs of the topological charges of the vortices, by studying the properties of the related Wigner function distribution [11]. The measurement error for this technique did not exceed 4% and increased with increasing mode numbers in the superposition. An extended version of this method was implemented for the vortices with different signs of the topological charges [24]. Kotlyar and et. al, proposed a technique based on evaluating intensity moments to measure the squared modules of expansion coefficients [25]. They experimentally determined the fractional OAM of superpositions consisting of two LG modes, with an error of about 10%.
In this work, we demonstrate a modernized intensity moments technique to decompose the superposition of LG modes with fractional OAM and measure the squared modules of expansion coefficients (SMEC). Both modal decomposition and the measurement of the OAM-spectrum of light are numerically and experimentally carried out just by recording a single frame of the intensity distribution of the LG modes with integer and non-integer OAM. A superposition of LG modes is generated using a digital micromirror device (DMD) and recorded by a CCD camera on a 2f optical imaging system. As a new approach, we just consider specified parts of the intensity distribution of the beam. Consequently, the needed time duration to extract the modal coefficients is significantly decreased. Furthermore, the presented method is implemented on various superpositions consisting of two, three, and four basis LG modes with different topological charges. The OAM spectrum is measured with an accuracy of better than 99.5% for superpositions of two and 98.2% for superpositions of more than two basis modes. The presented model can pave the way for the online and high-speed measurement of the OAM spectrum with high accuracy which has important potential applications in optical communications.
2. Basic principle
The aim of the presented method is to measure the spectrum of squared amplitudes or the SMEC and the OAM of a laser beam. The method is based on mode intensity distribution processing. It is performed by a single frame recording of a LG beam intensity, considering the specified parts of the intensity in the form of concentric circles, and solving a system of linear equations.
An OAM beam can be presented by a superposition of orthogonal basis LG Modes in cylindrical coordinates (r, φ, z), and is expressed by [25]:
where n and m are the radial and azimuthal indices, respectively, N is the number of basis functions and Cmn is the complex expansion coefficient. The basis function of LG mode consists of a phase factor exp(imφ), where φ is the azimuthal angle, and the field amplitude profile is given by [26]:In order to implement and validate the proposed method for extracting the absolute value of the expansion coefficients Cmn and the OAM of the light, it is assumed that the basis modes of superposition are initially known, likewise what is performing in the optical telecommunication. Therefore, for a beam profile of the superposition, we know the included basis modes, but the weight of each basis mode or the expansion coefficients and the OAM of the beam are unknown and are determined by the presented method. To explain the method, at first, two superpositions of LG modes according to Eqs. (1) and (2) are produced. Accordingly, superpositions of C4,0 LG04 + C6,0 LG06 and C70,0 LG070 + C73,0 LG073, are assumed as the understudy beams and the expansion coefficients are selected C4,0 = (0.20)1/2, C6,0 = (0.80)1/2 and C70,0 = C73,0 = (0.50)1/2. The selected expansion coefficients are satisfying the condition of ∑m,n|Cmn|2 = 1 [18]. Figure 1 shows the intensity profiles of assumed LG beams.
Fig. 1. Intensity profile of superpositions (a) C4,0 LG04 + C6,0 LG06 and (b) C70,0 LG070 + C73,0 LG073. The circles are drawn to sample the intensity profile of superpositions. Circles with the numbers related to the LG modes included in superpositions are highlighted in red color.
To evaluate the expansion coefficients by using a recorded intensity profile of the supposed LG beams, the concentric circles to sample the intensity are drawn on two beams profile. Therefore, the circles contain a specified part of the beam intensity. The number of circles is determined by the azimuthal index of the basis mode which has the bigger azimuthal index in the superposition. Therefore, due to the basis modes LG06 and LG073 in mentioned superpositions, 6 and 73 circles are drawn in Fig. 1(a) and 1(b), respectively. The radii of the circles are determined based on the largest circle contains 98% of the total intensity of the beam. Considering the pixel size and the pixel numbers of the beam profile, the radius of the largest circle is calculated. The parameter Δr is defined as the radius of the largest circle divided by the azimuthal index of the basis mode, which has the larger azimuthal index in the superposition. Therefore, the radii of the other circles, rm are chosen in the form of m×Δr.
To calculate the SMEC, the intensities which are enclosed by circles are determined. The intensity is defined by squared modules of the field in Eq. (1) as I(r,φ,z)=|U(r,φ,z)|2 and via integration with respect to the angular variable, the intensity equation is given by [25]:
The field of the superposition is assumed at the beam waist plane z = 0 and, the LG modes with zero radial index n = 0 are investigated. Considering these assumptions, the intensity enclosed by a circle is written in the form:
where Ic is a part of the intensity of the superposition and | Ψm(rc, z = 0)|2 is a part of the intensity of the basis function which is enclosed by a sampling circle with radius rc . The parameter m is a positive integer equal to the azimuthal indices of LG modes included in the superposition. Substituting the Ic and |Ψm(rc, z = 0)|2 in Eq. (4) and solving the equation, by the inverse matrix method, for example, determine the squared modules of all coefficients of Eq. (4). For instance, for the superposition in Fig. 1(a), only the intensities related to the circles fourth and sixth (which are distinguished with red dashed circles) are calculated. Thus, via solving this linear set:The value of the normalized total OAM of the beam is determined by dividing Eq. (6) by Eq. (7) and substituting Eq. (1) into both equations:
3. Results and discussion
The experimental setup for the measurement of the SMEC is illustrated in Fig. 2. The He-Ne laser beam at λ = 632.8 nm wavelength is collimated and filtered via a telescope system (L1: f1 = 25.4 mm; L2: f2 = 750 mm) and a spatial filter SF, respectively. Mirrors M1 and M2 are used to direct the beam to illuminate the amplitude SLM from Texas Instrument (XR-325 DMD chip) with 800 × 600 resolution. An appropriate fork grating element is loaded on the SLM to generate the desired superposition of LG modes. The reflected beam is focused by a lens L3 (f3 = 400 mm). The positive first-order diffracted beam is recorded using a CCD camera (DMK 23U274) at the back focal plane of the lens L3. To avoid damaging and saturating the CCD, an attenuator filter is placed in front of the CCD. The Gaussian beam waist was measured w0 = 100µm and some key experimental considerations are applied to the imaging process to perform an effective and optimal modal decomposition. For example, the Gaussian laser beam is diverged via the telescope to cover the whole of the DMD area. Furthermore, the physical position of the laser beam is aligned with respect to the DMD, lenses, filters, and CCD, to record the more uniform and perfect intensity distributions of generated LG beams. To investigate only the effect of varying the coefficients on intensity distributions, the attenuator filter and all CCD properties such as gain, contrast and exposure are constantly considered. In addition, the focal plane of the lens L3 is placed on the CCD to record all intensity distributions. Examples of the intensity distributions of the generated various superpositions are shown in Fig. 3. Similar to the superpositions of Fig. 1, the presented method is implemented on the experimental intensity distributions to measure the SMEC and the OAM of the laser beam.
Fig. 2. Schematic experimental setup for generating superposition of LG modes and measuring the SMEC. Laser: He-Ne laser beam (λ = 632.8 nm); SF: spatial filter; L1, L2 and L3: spherical lens; M1,2: mirrors; SLM: spatial light modulator; Filter: attenuator filter; CCD: charge-coupled device.
Fig. 3. The simulation and experimental intensity profiles for two superpositions of UA = C4LG04 + C6LG06 and UB = C12LG012 + C15LG015 by variation of the expansion coefficients from 0 to 1.
As it is shown in Fig. 2, by using a 2f imaging setup, intensity distributions |U|2 of two superpositions UA = C4LG04 + C6LG06 and UB = C12 LG012 + C15 LG015 are recorded and shown in Fig. 3. A spectrum of the expansion coefficients from 0 to 1 with the step of 0.1 (i.e., 0, 0.1, …, 0.9 and 1) is selected and also the condition of ∑m|Cm|2 = 1 is satisfied for each superposition.
It is clear from Fig. 3 that, the experimentally recorded profiles are coinciding with simulations while the coefficients set is varying. However, for the single LG modes with the SMEC set (0:1) and (1:0), an incompatibility can be observed. The physical reason is the phase distortions of the DMD on the shaped beam wavefront [28]. To determine the measurement error and investigate the repeatability of the method, 45 images are recorded for each composition and via solving the Eq. (4), the SMEC are determined for all images. Furthermore, the measured SMEC are averaged to determine the mean value, and the standard deviation value is considered as the measurement error.
Labeled column plots are associated with error bars, present the means of the measured SMEC and the standard deviation, respectively, and are shown in Fig. 4.
Fig. 4. The mean of the measured SMEC on 45 recorded images of the beam intensity profile for superpositions (a) A (UA = C4LG04 + C6LG06) and (b) B (UB = C12LG012 + C15LG015). The error bar corresponds to the standard deviation of the measured SMEC.
The means of the measured SMEC are nicely matched with the real values. Considering the error bars, the standard deviation increases from the center mode with the coefficient set (0.5:0.5) to the single modes with the coefficient sets (0:1) and (1:0). The minimum and maximum of the standard deviation are obtained to be 0.05 and 0.81 for the LG beams of superposition A with the coefficient sets (0.5:0.5) and (0:1), respectively. In addition, for superposition B, the minimum and maximum of the standard deviation are obtained to be 0.04 and 0.18 for the LG beams with the coefficient sets (0.5:0.5) and (1:0), respectively. As mentioned before, the reason behind the high error of single mode measurement is that small OAM LG single modes are very sensitive to phase distortion [28].
Substituting the measured SMEC of each composition, in Eq. (8), yields the OAM of laser modes. To compare the measured and theoretical value of the OAM, Fig. 5(a) and 5(b) are plotted and labeled for superpositions A and B, respectively. In both plots, the standard deviation of the measured OAM is shown as error bars. Due to matching the measured SMEC with theoretical values, a good agreement between the measured and theoretical values of the OAM is achieved, except for the case of (0:1) of superposition A, in which a considerable difference is observed. Figures 5(c) and 5(d) show the relative uncertainly of the OAM measurement, which is defined as the standard deviation divided by the mean value of the measured OAM, for the superpositions A and B, respectively. The minimum and maximum of the relative uncertainly are obtained to be 0.02 and 0.20 for the LG beams of superposition A with the coefficient sets (0.5:0.5) and (0:1), respectively. Furthermore, for the superposition B, the minimum and maximum of the relative uncertainly are obtained to be 0.01 and 0.05 for the LG beams with the coefficient sets (0.5:0.5) and (1:0), respectively. It can be understood that in the measurement of both the SMEC and the OAM, the minimum of the standard deviation and the relative uncertainly are achieved for the composition with the coefficient set (0.5:0.5) that the weight of both basis modes of the composition are the same.
Fig. 5. Comparison of the theoretical and the measured OAM, for superpositions (a) A and (b) B. Also, the relative uncertainty is shown for the superpositions (c) A and (d) B.
To extend the research, various superpositions consisting of three or four LG basis modes are experimentally generated and based on the presented intensity method, the SMEC and the OAM of the laser mode are measured. Superpositions of three basis modes, similar to the superpositions of two basis modes, are investigated via solving the linear Eq. (4). However, the Eq. (4) which consist of the superposition of four LG basis modes, is considered as two relations, each containing two basis modes. Then, each relation is solved separately to determine the squared amplitudes. This technique reduces the measurement error caused by increasing the number of the basis modes in the superposition and also saves the computation time. To understand more, consider the superposition UT:
it is assumed that the intensity of UT is a correlation of two intensities of two superpositions which can be written in the form of two relations:Table 1 presents five selected superpositions as examples of the superpositions which included three or four LG basis modes.
Table 1. Superpositions of LG basis modes
Figure 6(a) shows the simulation and experimental intensities of the presented superpositions in Table 1. Using the presented intensity method, the SMEC, the corresponding OAM, and the standard deviation on 45 recorded images from each profile, are measured and summarized in Table 2.
Fig. 6. (a) The simulation and experimental intensity profiles, (b) the theoretical and measured OAM, and the standard deviation of five presented superpositions in Table 1.
Table 2. Comparison of the real and the mean of measured (Meas.) SMEC, and the standard deviation (Std) of presented superpositions (Sup.) of Table 1
It is observed that the mean of the measured SMEC nicely follows the real values. The minimum and maximum of the standard deviation of the SMEC measurement are achieved to be 0.01 and 0.08, respectively.
To determine the OAM of investigated superpositions of Table 1, the measured SMEC are substituted in Eq. (8). The result of the measured OAM is compared with theoretical values in the form of a column plot that is shown in Fig. 6(b). Considering the labeled columns of the plot, a high accordance is observed between the measured and theoretical values of the OAM for all superpositions.
In general, the presented intensity method can be nicely used to decompose the superpositions consisting of two or three LG basis modes. In addition, the method can be extended for the superpositions consisting of four or more basis modes, by considering the linear set in the form of some small equations of two or three modes, like what was applied to the presented superpositions of Table 1.
It is worth noting that the measurement of SMEC for a superposition of five modes with the described experimental setup is associated with an error about 10%, and the measurement error becomes significant as the number of modes increases. The physical reason for this error is the limited resolution of the used SLM. Increasing the number of modes contributing to the superposition increases the complexity of the generated fork patterns and to display such delicate patterns, a higher resolution SLM is required. The effects of DMD resolution on the beam quality and intensity profile of the generated LG modes have been studied and confirmed [28]. It has also been shown that SLM resolution has a strong influence on the modal content of the generated superposition of LG modes [9].
4. Conclusion
In this paper, an intensity moments method has been reported for the modal decomposition and measurement of the OAM of the laser beam. Various superpositions with different mode numbers equal to two, three, or four of LG basis modes with a spectrum of the expansion coefficients, have been generated. Using the intensity moments and solving the linear equations, the superpositions have been decomposed. Finally, for the superpositions of two basis modes, the SMEC and the OAM of light were obtained with better than 98% and 99.5% accuracy, respectively. Moreover, for the superpositions consisting of more than two basis modes, the SMEC and the OAM of light were measured with an accuracy of about 90% and better than 98.2%, respectively. Therefore, the proposed intensity method has been numerically and experimentally demonstrated to extract the information of the OAM beams which have been utilized in various optical systems, especially in optical communication.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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