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Abstract
In this paper, the amplitude properties of the Laguerre-Gaussian (LG) vortex beams are analyzed theoretically and demonstrated experimentally. Firstly, the ring width of the LG vortex beam is almost a constant, which can be considered to be independent of the topological charge (l) and only determined by the waist radius ω(z), and its general expression is derived via investigating Lambert W function. On this basis, concise expressions for inner and outer ring radii, ratio of inner and outer ring radius, ring area and beam divergence are also given out. Moreover, modification functions are added to obtain more accurate expressions. In the experiment, a spatial light modulator is employed to generate the LG vortex beams with l = 1∼20 at ω0 = 0.3mm and ω0=0.4mm, and the LG vortex beams with l=10 at different propagation distances. The experimental results are in good agreement with the theoretical values. This work may help researchers to better understand the LG vortex beam and provide a useful guideline for its further applications.
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1. Introduction
The angular momentum of a photon includes spin angular momentum and orbital angular momentum (OAM), where spin angular momentum has only two eigenvalues, while the eigenvalues of OAM can be taken as any integer. In 1992, Allen et al. demonstrated the Laguerre-Gaussian (LG) beam carries OAM of $l\hslash$ per photon, where l is the topological charge and $\hslash$ is the reduced Planck constant [1]. Later, Beijersbergen proved that the beam can carry OAM as long as the phase is helical [2]. Since then, vortex beams have been intensively researched and applied in various fields, especially in optical communications [3–5], where OAM as a new dimension of light, can increase channel capacity. Additionally, in the field of optical tweezers, it is possible to capture particles, and transfer the OAM to particles, then rotate them as an optical wrench [6]. Due to its unique intensity distribution, it can also be employed in optical processing [7]. In these applications, it is essential to investigate the amplitude properties of the vortex beam.
In practical applications, the single-ring LG vortex beams with p=0 (referred to as LG beam in the following) or the vortex beam obtained by utilizing a spiral phase plate (referred to as general vortex beam in the following) are usually employed, and researchers have also investigated the amplitude properties of these beams. In 1995, Allen et al. [8] analyzed the LG vortex beams and obtained the expression for the radius of maximum field amplitude, which is proportional to the square root of the l. Jennifer [9] investigated general vortex beam on the focal plane, and derived approximate expression of the radius of the principal maximum in intensity, which is proportional to the l. In our previous work, the general vortex beam on the focal plane was also investigated and mathematical expression of the outer ring radius was obtained [10]. As the research on the vortex beam is continuously deepened, the amplitude properties of the vortex beams that can not fully described via the radius of maximum field amplitude. Thus, in 2014, Reddy [11] defined inner and outer ring radii of the general vortex beam, which are the distance from 1/e2(13.5%) of the maximum intensity to the singularity, and analyzed area of the bright ring, which is proportional to the l. Furthermore, in 2015, Cleberson demonstrated that general fractional-order vortex beams are also consistent with this conclusion, and proposed that the l of the vortex beam can be obtained by calculating the area of the bright ring or the central dark region [12]. Moreover, Reddy presented approximate expressions that contain distance z for the inner and outer ring radii of the general vortex beam, and analyzed the beam divergence [13]. Padgett [14] also researched the divergence of the LG vortex beam, and proved that the vortex beam generated by a column lens has a different divergence than that generated by a forked grating. Additionally, the ratio of inner and outer ring radius of the LG vortex beam was investigated, which may be useful to couple of the OAM to fibers with an annular index profile [15].
However, the expressions reported above are complicated, and the amplitude properties of the LG vortex beam cannot be quantitatively described directly, and further numerical analysis is required. Therefore, A clear physical law and the intrinsic connection between LG vortex beam and l, beam waist, propagation distance are expected to be obtained, rather than just numerical simulation results. Moreover, the ring width of LG vortex beam has not been investigated deeply, which is an important parameter in optical processing based vortex beam.
In this paper, the amplitude properties of the LG vortex beam are investigated completely, and concise expressions of the parameters are obtained. The general expression of the ring width is obtained via further investigating the Lambert W function. Furthermore, concise expressions of the inner and outer ring radii are given out. On this basis, the concise expressions of the ring area, ratio of inner and outer ring radius and beam divergence are also derived. The LG vortex beams are clearly understood by these quantitative descriptions and improve for further applications.
2. Theoretical analysis
Here, the LG vortex beam with radial index p = 0 is researched and the expression in cylindrical coordinates is
Fig. 1. The intensity distribution, inner and outer ring radii and the ring width of a LG vortex beam with l=4.
Here, r0 is the radius of maximum field amplitude, and is given by [8]
The insertion Eq. (5) and Eq. (3) into Eq. (4) gives
This is a transcendental equation and it is difficult to solve. Therefore, the Lambert W function, the inverse function of y = x·exp(x), is employed to solve it, which is a special function with important applications in photovoltaic systems modeling [16] and solving renormalization group equations [17]. The Lambert W function has numerous branches, including two real branches W0(x) and W−1(x) [18], as shown in Fig. 2.
According to the properties of this function [18], the equation x·exp(x)=ξ, when ξ ∈ (-e−1, 0), has two real roots W0(ξ) and W−1(ξ). Therefore, the conversion of Eq. (6) yields the following
Here, $- \textrm{exp} ( - \frac{{\textrm{|}l\textrm{|+ 2}}}{{\textrm{|}l\textrm{|}}})$ ∈ (-e−1, -e−3], thus, the inner ring radius r1 and outer ring radius r2 can be solved.
The ring width is defined as the difference inner and outer ring radius, as shown in Fig. 1, and the expression is as follows
Equation (8) and Eq. (9) are still complicated and require further numerical analysis. In the simulation, the variations of inner and outer ring radius and ring width with the l are shown in Fig. 3, here, ω0=0.3mm, z=0. From Fig. 3, the inner and outer ring radii increases with the l, however, the ring width is almost constant. Nevertheless, these conclusions cannot be directly derived from Eq. (8) and Eq. (9). Consequently, the Lambert W function is further investigated. Although this function lacks an analytic expression, it has asymptotic expansions in certain positions [19].
In Eq. (9), $\mathop {\lim }\limits_{|l|\to \infty } - {\textrm{e}^{ - (|l|+ 2)/|l|}} ={-} {\textrm{e}^{ - 1}}$. Accordingly, substituting Eq. (10) into Eq. (9) and utilizing two infinite approximations, $\mathop {\lim }\limits_{x \to 0} {\textrm{e}^x} = x + 1$ and $\mathop {\lim }\limits_{x \to 0} {(1 + x)^\alpha } = \alpha x + 1$, then, the following results can be obtained.
Equation (11) is derived at l→∞ and is an asymptote of the Eq. (9). Figure 4 displays the ring width described by different expressions, ω0=0.3mm, z=0. In Fig. 4(a), the red circular dashed line indicates Eq. (9) that contains Lambert W function and the green diamond dashed line represents asymptote Eq. (11). As the l increases, the ring width described by Eq. (9) gradually approximates to Eq. (11), in contrast, as the l decreases, the difference between them increases. The relative error is the l dependent and inversely proportional to the l, therefore, an inverse proportional function is employed to modify Eq. (11), as follows.
Here, g(l) = 1 + 0.05625/|l|. Coincidentally, the modification function g(l) is a standard inverse proportional function and the inverse proportionality coefficient 0.05625 is the ratio of Eq. (9) and Eq. (11) at l=1 minus 1. Figure 4(b) displays the comparison of real value Eq. (9) and modified Eq. (12). It is noted that the two curves match up perfectly. As a result, for LG vortex beams with arbitrary l, Eq. (12) can be employed as a general expression for the ring width. The g(l) is a decreasing function on the domain l∈[1, ∞), which its range from 1 to 1.05625, almost equal to 1. Therefore, the ring width can be considered as a constant, especially in large l, when the waist ω(z) is unchanged. This discovery may be useful to better understand the spatial distribution of LG vortex beams with different l.
Equation (9) has been simplified, for Eq. (8), although it is an analytical solution, the expressions are complex and require further numerical analysis with specialized calculation software. As a result, the following analysis is performed. From Eq. (3), the intensity expression of the LG vortex beam contains a Gaussian term. Thus, the intensity distribution is Gaussian in the radial direction, that is, the intensity on the ring is symmetrical about the position of the maximum intensity. Therefore, utilizing the radius at the maximum intensity r0 and the ring width d, the inner and outer ring radii expressions are re-derived, as follows
Figure 5 displays the comparison of the simulation results of real value Eq. (8) and re-derived Eq. (13). The inner and outer ring radii described by both are basically the identical, except for the slight errors when the l is small. This error will be carefully explored in the discussion section. However, compared with Eq. (8), Eq. (13) is more concise. Previously, the amplitude characteristics of the LG vortex beam were analyzed by numerical simulation, and from Eq. (13), the relations between the inner and outer ring radii and the l and beam waist ω(z) can be directly obtained. Additionally, according to Eq. (13) and Eq. (12), the expression of the ring area, ratio of inner ring radius and outer ring radius and beam divergence can be derived, this can greatly help researchers analyze the LG vortex beams.
The ratio of inner ring radius and outer ring radius can be expressed as
From Eq. (14), it is obvious that the ratio of the inner and outer ring radius is independent of the waist ω(z), and is inversely proportional to the l, for g(l) is almost equal to 1, which is consistent with the conclusions obtained by numerical analysis in Ref. [15].
The ring area can be expressed as
The beam divergence can be characterized as the derivative of the inner and outer ring radii with respect to the propagation distance z, and can be expressed by
Equation (5) can only be utilized to analyze the divergence of the LG vortex beam at the maximum intensity. Since the ring width increases as the propagation distance increase according to Eq. (12), therefore, Eq. (16) can be employed to completely characterize the beam divergence. Generally, the propagation distance z>>zR, thus, Eq. (15) can be simplified as follows
From Eq. (17), it is noted that the beam divergence is already determined at source, independent of the propagation distance z. For LG vortex beams with different l, the larger the l, the more divergence, which can explain the concentric LG vortex beam transmitted in space without crosstalk. For the l is constant, the larger the beam waist ω0, the less divergence. Moreover, the beam divergence is also wavelength-dependent; the longer the wavelength, the more divergence. Furthermore, Eq. (17) can also be considered as the divergence angle of the LG vortex beam. Compared with the conclusions in Ref. [13], Eq. (17) is more concise and more intuitive.
As results of the above theoretical derivation, concise expressions for the parameters of the LG vortex beam are obtained, and like the Gaussian beam, the amplitude properties of the LG vortex beam can be analyzed intuitively without numerical simulations, which is friendly to researchers.
3. Experimental setup and results
Here, the complex amplitude modulation is utilized to generate the LG vortex beam with p=0 for experimental demonstration [20,21]. The experimental setup as shown in Fig. 6. In the experiment, a 532 nm laser with single-mode fiber (SMF) coupling output is employed. The beam emitted is collimated and expanded by employed a collimated beam expander (CBE). Afterward, the beam size is controlled by utilizing an adjustable diaphragm 1 to fit the active area of the spatial light modulator (SLM), and the polarization state of the beam is adjusted by employed a rotatable polarizer placed behind the diaphragm 1 to conform to the polarization state of the SLM. A phase-only SLM (1920×1200pixels, 8um per pixel) is used to generate the LG vortex beam, after the beam is reflected by a beam splitting prism (BS). The phase pattern loaded into the SLM is superimposed with a blazed grating to separate the target light field from the zero-level diffraction. For the convenience of observing the generated LG vortex beam, a 4f system is built, consisting of lens L1 and lens L2, with focal lengths of f1 = 300 mm and f2 = 150 mm respectively. An adjustable diaphragm 2 is placed at the focal plane of the lens L1 to remove other diffraction orders and obtain only first-order diffraction. Then, the generated LG vortex beam is recorded by CCD. Furthermore, the interference experiment of the LG vortex beam is realized by the following operations. Increasing the aperture of the adjustable diaphragm 2 to allow the LG vortex beam of the first-order diffraction and the zero-order diffracted light passing through, and move the position of the CCD backward, which can make them interfere and generate forked patterns.
Figure 7 displays the generated LG vortex beams with the different l, ω0=0.3mm, z=0. Figure 7(a1) and (a2) indicate the simulation and experimental results for the interference of LG vortex beam with l = 2 and plane wave. Both are forked patterns and branch out two fringes, corresponding to l = 2, proving that the generated annular optical field is a LG vortex beam with corresponding l. Figure 6(b1)∼(b11) display the generated LG vortex beams with l = 1∼20, from which it can be noticed that the ring radius increases as the l increases.
Fig. 7. The interference of a LG vortex beam with l=2 and plane waves, (a1) simulation results (a2) experimental results. The generated LG vortex beams, (b1)∼(b11) l=1∼20.
According to the above theoretical analysis, ω(z) is affected by the propagation distance z and the waist ω0, for convenience, the following experiments are set to z = 0, except for the demonstration of the beam divergence. Figure 8(a) illustrates the comparison of the derived Eq. (13) and the experimental results for the inner and outer ring radii of the generated LG vortex beam. The red square dashed line and red circular dashed line indicate the theoretical values of r2 and r1 from Eq. (13), respectively, and the inverted triangle and the diamond denote the experimental results of r2 and r1, respectively. Figure 8(a1) and (a2) indicate the LG vortex beams with ω0=0.3 mm and ω0=0.4 mm, respectively. From Fig. 8(a), the experimental results are basically consistent with the results of Eq. (13) at different waist ω0, except for slight errors at small l, which is consistent with the theoretical analysis in Section 2. Figure 8(b) displays the experimental results of the ring width and the theoretical results of Eq. (12). The red square dashed line and green positive triangle indicate ω0=0.3mm, and the red circular dashed line and green inverted triangle represent ω0=0.4mm. From Fig. 8(b), the experimental results and theoretical results are in almost agreement, and the ring width is nearly constant. Moreover, the ring width increases with the waist ω0 increases, which is consistent with the above theoretical analysis.
Fig. 8. (a1) and (a2) depict the inner and outer ring radii for ω0=0.3mm and ω0=0.4mm, respectively, (b) represent the ring width at ω0=0.3mm and ω0=0.4mm.
Figure 9(a) illustrates the ratio of the inner and outer ring radius of the LG vortex beams with different l. The red circular dashed line depicts the theoretical values of Eq. (14), and the inverted and positive triangles indicate the experimental results for ω0=0.3mm and ω0=0.4mm, respectively. From Fig. 9(a), when the l is small, the experimental results are slightly in error with the theoretical values of Eq. (14), this error will be carefully explored in the discussion section, and the experimental results are good consistent with the simulation results at large l. Moreover, the ratio of the inner and outer ring radius with the same l for different waist ω0 remains basically the same, which is in good agreement with the above theoretical analysis that the ratio of the inner and outer radius is independent of the beam waist and inversely proportional to the l. Figure 9(b) is the comparison of the theoretical and experimental results of the ring area of the LG vortex beams. The red circular dashed line and the red diamond dashed line indicate the theoretical results for ω0=0.3mm and ω0=0.4mm, respectively, and the green positive triangle and green inverted triangle represent the experimental results for ω0=0.3mm and ω0=0.4mm, respectively. The experimental results are in good agreement with the theories that the ring area increases with the l at the constant waist, and the ring area increases when the waist increases.
Fig. 9. (a) The experimental results and the theoretical results of the ratio of inner and outer ring radius. (b) The experimental results and the theoretical results of the ring area.
Figure 10 depicts the divergence of the LG vortex beams with l = 10 during the propagation. The horizontal coordinate unit in Fig. 10 is the Rayleigh distance zR, here, ω0=0.3mm, zR=0.531 m. The red circular dashed line and the red diamond dashed line indicate the theoretical values of r1 and r2, respectively, and the green positive triangle and the green inverted triangle display the experimental results of r1 and r2, respectively, the blue solid line and the green solid line represent the divergence angles of r1 and r2, respectively. From Fig. 10, the inner and outer ring radii of the LG vortex beam increases with the propagation distance, and the experimental results are consistent with the theoretical results. Moreover, the divergence of the beam is relatively small within the one-fold Rayleigh distance, and the divergence angle of the beam approaches the theoretical value as the distance increases.
4. Discussion
In the above theories and experiments, the experimental results deviate slightly from the derived theoretical value at small l, yet the overall evolution trend is uniform. These errors are relatively small, and since the inner and outer ring radii are defined as the position at 1/e2 (13.5%) of the maximum intensity, consequently, the inner and outer ring radii described by Eq. (13) are still within the range of the intensity distribution. Simultaneously, the experimental results remain in good agreement with derived theoretical values at larger l. Thus, the derived concise expressions can be employed to characterize LG vortex beams with arbitrary l.
To further investigate the causes of these errors, the normalized transverse intensity profiles of the LG vortex beams are investigated, as shown in Fig. 11. It is noted that the amplitude profile of the LG vortex beam with small l is not a standard Gaussian distribution, this is perhaps caused by intensity collapse, due to phase singularities at the center, such as in Fig. 11(a), where there is a clear distribution of intensity discontinuities near the singularity. This effect decreases as the l increases, as shown in Fig. 11(d) for the profile of the LG vortex beam with l = 10, the amplitude is Gaussian symmetrically in the radial direction. This phenomenon can clearly explain why the derived theories have slight errors at small l, and the theories and experiments are in good agreement at large l.
Fig. 11. the normalized transverse intensity profiles of the LG vortex beam, (a) l = 1, (b) l = 2, (c) l = 3, (d) l = 10.
If a precise expression is required, the modification functions related to the l needs to be added to the Eq. (13), as follows.
Here, ${h_1}(l) = 1.003 + \frac{{0.2186}}{{|l{|^{0.6017}}}}$, ${h_2}(l) = 1.001 + \frac{{0.1817}}{{|l{|^{0.5370}}}}$, obtained by numerical fitting. Figure 12 displays the theoretical results of Eq. (8) containing the Lambert W function and modified Eq. (18), it is obvious that the modified Eq. (18) is perfectly consistent with Eq. (8). Compared to Eq. (8), Eq. (18) is more concise. However, on the basis of Eq. (18), further analysis of the ratio of inner and outer radius, ring area, etc. is relatively complicated. Moreover, in the above results, at l > 10, the error is almost neglected and the modification function h1(l) and h2(l) are almost equal to 1. Therefore, in order to obtain more concise expressions, for the LG vortex beams with large l, these modification functions can be removed and the accuracy can be guaranteed. Furthermore, if accuracy is not required, the modification function can also be removed for arbitrary l, since overall evolution trends of the modified and unmodified expressions are uniform.
This work is useful for the application of vortex beams in various fields. For instance, generation of LG vortex beams in the cavity, the appropriate a spot-defect [22,23] or annular pump laser [24] or cavity structure [25,26] can be selected to output the LG mode utilizing our derived theories. In spatial optical communication, the LG vortex beam with a suitable beam waist is selected to minimize the beam divergence angle, and the receiver aperture size can also be calculated. Simultaneously, these parameters can be solved quickly with a pocket calculator, such as for the ring area, using Eq. (15), which can be calculated simply with the beam waist and l. Otherwise, according to Eq. (8), the inner and outer ring radii containing the Lambert W function requires to be calculated with specialized calculation software first, and then the ring area can be obtained. Furthermore, One can also make further investigation based on these theories, for example, according to Eq. (12), calculating the first-order derivative of the optical ring width about the propagation distance, which can obtain the variation rule of the ring width with the propagation distance, rather than stop at the numerical analysis. Moreover, this work may be helpful for expansion of Lambert W function.
5. Conclusion
In this paper, the amplitude properties of the LG vortex beam are investigated. The ring width of the LG vortex beam is almost a constant, which can be considered to be independent of the l and only determined by the waist radius. Then, the concise expressions of ring width, inner and outer ring radii, ratio of inner and outer ring radius, ring area and the beam divergence are derived, and accurate modified expressions are also obtained. Simultaneously, the amplitude properties in relation to the l and the waist ω(z) can be analyzed directly via these expressions without numerical analysis. Additionally, the experimental results are in a good agreement with the theoretical results. These conclusions can contribute to a better understanding for the LG vortex beams and are of good guidance for applications in various fields.
Funding
National Natural Science Foundation of China (61901449); Equipment Pre-research Key Laboratory Fund (6142A04190212); Frontier Research Fund of Institute of Optics and Electronics, China Academy of Sciences (C21K006).
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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