
==== Front
Heliyon
Heliyon
Heliyon
2405-8440
Elsevier

S2405-8440(24)13047-2
10.1016/j.heliyon.2024.e37016
e37016
Research Article
Generalized dark hollow sine-Gaussian beam and its propagation properties
Wang Taofen a
Su Qin b
Zhu Jie jiegit_16@163.com
b⁎
a School of Physics, Hunan University of Science and Technology, Xiangtan, 411201, China
b College of Science, Guizhou Institute of Technology, Guiyang, 550003, China
⁎ Corresponding author. jiegit_16@163.com
28 8 2024
15 9 2024
28 8 2024
10 17 e3701624 6 2024
20 8 2024
26 8 2024
© 2024 The Authors
2024
https://creativecommons.org/licenses/by-nc/4.0/ This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).
A model of the generalized dark hollow sine-Gaussian beam (GDHsGB) is proposed to uniformly describe both conventional dark hollow beams (DHBs) and anomalous dark hollow beams (ADHBs) with circular or elliptic geometrical patterns. Using the Collins formula, we derive the analytical expression for GDHsGBs propagating in ABCD paraxial optical systems. We analyze the evolution of the intensity pattern and beam width of circular ADHBs, as well as the ellipticity of elliptic ADHBs, providing mathematical expressions for these physical quantities. The results reveal various evolution forms based on beam parameters, with elliptic ADHBs exhibiting more intricate propagation behavior compared to circular ADHBs. By controlling parameters, the intensity pattern of elliptic ADHBs undergoes a transformation into a petal-like distribution in the near field, later reverting to its original elliptic configuration but rotated 90° from its initial orientation on the source plane in the far field.

Keywords

Dark hollow beams
Beam propagation
Elliptic and circular beam patterns
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pmc1 Introduction

Laser beams exhibit diverse configurations of light intensity patterns, each possessing unique properties and applications across various fields. The study of different types of laser beams and the exploration of their transmission characteristics in different optical systems or materials have been a longstanding and challenging pursuit in scientific research for decades [1,2]. To facilitate the study of beam propagation dynamics, numerous mathematical models have been devised. These models can generally be classified into two categories. The first category comprises pure eigenmodes or fundamental solutions derived directly from the Helmholtz equation or its corresponding paraxial approximation. Examples include Gaussian beams [1,3], Hermite–Gaussian beams [4], Laguerre–Gaussian beams [4,5], Bessel (Bessel-Gaussian) beams [6], Mathieu-Gaussian beams [7], Ince-Gauss (IG) beams [8,9], Weber-Gaussian beams [10], Airy beams [11], Pearcey beams [12], Tricomi (Tricomi-Gaussian) beams [13], Olver beams [14], and others [1]. The second category encompasses complex beams resulting from the combination of basic modes or basis modes with simple analytic functions representing optical elements or apparatuses, which yields a diverse range of beam types, including dark hollow beams [15], hollow elliptical Gaussian beams [16], anomalous dark hollow beams [17,18], hollow sinh-Gaussian beams [19], elliptical Bessel beams [20], and others [[14], [15], [16], [17], [18], [19], [20]]. Through analytical modeling, novel and practical dynamical properties of structured optical beams have been unveiled, laying a firm foundation for their expanded applications [21,22].

In recent decades, conventional dark hollow beams (DHB), characterized by their zero central intensity, have received significant attention and emerged as invaluable tools in various fields such as atomic physics, free space optical communications, binary optics, optical particle trapping, and medical sciences [23,24]. To facilitate analytical or numerical investigations into their dynamic evolution properties, several mathematical models have been developed [[23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41]]. In a seminal experiment in 2005, Wu et al. observed the first instance of an anomalous hollow electron beam exhibiting elliptical symmetry with a solid elliptical core [42]. This anomalous dark hollow beam (ADHB) serves as a distinctive model system for studying transverse instabilities and offering insights into linear and nonlinear particle dynamics within storage rings [42]. To date, only a few mathematical models have been formulated to describe ADHBs [17,18], with previous research primarily focusing on deriving propagation formulas for coherent and partially coherent ADHBs as they pass through paraxial ABCD optical systems [17,18,[43], [44], [45], [46], [47], [48], [49], [50], [51], [52], [53], [54]].

Expanding upon the prior research, this paper introduces the generalized dark hollow sine-Gaussian beam (GDHsGB) to comprehensively describe both conventional DHBs and ADHBs with circular or elliptical geometrical patterns. Using this flexible model, we investigate the propagation characteristics of circular and elliptical ADHBs in free space, revealing diverse propagation and transformation characteristics. The structure of this paper is as follows: Section 2 derives the propagation expression of GDHsGB through an ABCD optical system. Section 3 analyzes the evolution properties of the transverse intensity pattern, beam width, and ellipticity of GDHsGB in free space, both analytically and numerically. Finally, Section 4 provides a summary of the paper.

2 Propagation formula of GDHsGBs through ABCD optical system

By extending and combining previously introduced mathematical models for the ADHB [15,17] and the sin-Gaussian DHB [19,21], we define the GDHsGB at the initial incident plane z = 0 as follows:(1) En(x0,y0)=exp(−x02+ey02w02)sinn(ax02+by02w02−α)

where a, b, e and α are limited to be real constant parameters, n is a non-negative integer, and w0 denotes the beam width of the Gaussian beam at the input plane. Such an expression (1) with real parameters characterizes a beam with a plane wavefront at the source plane. Depending on the values chosen for n, a, b, e and α, this mathematical model can depict various types of beams. When α = a = b = 0, Eq. (1) describes circular or elliptic Gaussian beams by selecting e = 1 or not, respectively. When α ≠ 0, a ≠ 0 and b ≠ 0, Eq. (1) represents circular or elliptic ADHBs [43,44] depending on the setting of e, a and b. Note that the intensity pattern of circular or elliptic ADHBs respectively consists of a central intensity peak surrounded by a bright circular ring or an elliptic encirclement. When α = 0, a ≠ 0 and b ≠ 0, Eq. (1) reduces to circular or elliptic DHBs [41,42] by adjusting e, a and b. Since the last case has been widely explored in previous literature, it will not be discussed further here. Equation (1) clearly offers a unified description of both DHBs and ADHBs with circular or elliptic pattern configurations, thus leading to the term GDHsGB for this beam model. Additionally, it indicates that GDHsGB can be experimentally realized by passing a Gaussian beam through amplitude gratings with an appropriate transmittance function.

Based on Eq. (1), we plot in Fig. 1, Fig. 2 to show how the beam parameters n, a, b and α affect the initial intensity distribution configurations for circular ADHBs (e = 1). In Fig. 1, each row analyzes the effect of parameters n and α on the pattern configuration while keeping a = b constant within the same row. It is evident that, for identical a = b values, the bright ring exhibits a larger size and a finer edge as both n and α increase. Moreover, a comparison of corresponding columns across the two rows reveals that, with constant n, the bright ring sizes notably shrink when α and a = b increase. Fig. 2 further illustrates the influence of parameter α on the pattern configuration, revealing its ability to adjust the relative peak intensity between the central bright spot and the bright periphery. Specifically, for circular ADHBs with a parameter set of (n, a, b, e) = (5, 1, 1, 1), a perfect pattern configuration--where the central spot and the surrounding bright periphery have equal maximum brightness--can be achieved at approximately α = 0.70.Fig. 1 Intensity distribution configurations for circular ADHBs (e = 1) at the source plane with different parameter sets (n, a, b, α). The top row shows a = b = 1; the bottom row shows a = b = 2. For both rows, the parameter sets from left to right are (n, α) = (1, 0.25), (4, 0.65), and (7, 0.80) for the top row, and (1, 0.45), (4, 0.85), and (7, 1.0) for the bottom row.

Fig. 1

Fig. 2 Intensity distribution configurations for circular ADHBs at the source plane with parameters (n, a, b, e) = (5, 1, 1, 1), shown for different values of α: (a) α = 0.65, (b) α = 0.70, and (c) α = 0.75.

Fig. 2

We further use Eq. (1) to mimic elliptic ADHBs by adjusting e ≠ 1 along with other beam parameters. Fig. 3 shows elliptic ADHBs exhibiting an elliptic anomalous dark hollow configuration at the source plane, with varying values of n and α, while a = 1 and e = b = 1/3. Notably, a similar perfect pattern configuration as mentioned above can also be observed for n = 4 and α = 0.65, confirming the attainability of perfect elliptic ADHBs under suitable beam conditions.Fig. 3 Intensity distribution configurations for elliptic ADHBs at the source plane with parameters (a, b, e) = (1, 1/3, 1/3), shown for different values of n and α: (a) (n, α) = (1, 0.25), (b) (n, α) = (4, 0.65), and (c) (n, α) = (7, 0.80).

Fig. 3

Using [55].(2) sinnx=12nin(eix−e−ix)n=n!2n∑l=0n(−1)l(n−l)!l!exp[i(n−2l)x]

Eq. (1) can be rewritten as(3) En(x0,y0)=n!2neinα∑l=0nCn,lexp{−[1+i(n−2l)a]x02w02−[1+i(n−2l)b]y02w02}

with Cn,l=(−1)lexp(−2ilα)(n−l)!l!, which means a coaxial combination of fundamental Gaussian beams with the same waist width but different spherical wavefronts and weighted coefficients. The curvature radius of the spherical wavefront Rl=zR(n−2l)b (l = 0, 1 2, …, n) where zR=kw02/2 is the Rayleigh length corresponding to Gaussian beam and k=2π/λ is the wave number related to the wavelength.

We then proceed to study the propagation of GDHsGB along the z axis through an ABCD optical system. The field in the half space z > 0 can be expressed using the Collins integral [[15], [16], [17], [18], [19]].(4) E(x,y,z)=−iλB∫∫−∞∞dx0dy0En(x0,y0,0)exp[−ikA2Br02+ikB(xx0+yy0)−ikD2B(x2+y2)]

where A, B, C, and D are the transfer matrix elements of the paraxial optical system. Substituting Eq. (1) into Eq. (4) and performing some mathematical manipulations aided by Eq. (3), we obtain the following electric field expression for GDHsGB in any z plane:(5) E(x,y,z)=−iλBexp[−izRDB(xw2+yw2)]∫∫−∞∞dx0dy0sinn(ax02+by02w02−α)exp[−(1+ikw02A2B)x02w02−(e+ikw02A2B)y02w02+ikB(xx0+yy0)]=n!zR2nin+1Bexp[inα−izRDB(xw2+yw2)]∑l=0nCn,lΩxlΩylexp(−zR2xw2ΩxlB2−zR2yw2ΩylB2)

where(6) Ωxl=1+iAzRB−ia(n−2l),Ωyl=e+iAzRB−ib(n−2l)

and the scaled coordinates qw = q/w0 (q = x, y, r) are used.

Equation (5) serves as the general propagation formula for GDHsGBs through an ABCD optical system, offering a framework for conveniently analyzing their novel features and transformations during propagation. It should be noted that while the product form of the Gaussian function and nth-order sine function in Eq. (1) cannot be maintained during transmission, the superposition of Gaussian beams in Eq. (3) remains valid throughout transmission.

3 Evolution of field properties in free-space propagation of GDHsGBs

3.1 Intensity distribution evolution of GDHsGBs

By virtue of Eq. (5), the intensity distribution I(x, y, zi) = |E(x, y, zi)|2 of the GDHsGBs can be analyzed via numerical simulations as it propagates in free space, with the transfer matrix expressed as(7) (ABCD)=(1z01)

In the following investigations, without any loss of generality, intensity distributions will always be normalized to their maximum and scaled coordinates will be utilized. Additionally, for comparative analysis, intensity distributions at the original plane will also be provided.

Fig. 4 displays the intensity patterns at various propagation distances z/zR and the corresponding intensity variation in the (x-z) plane for circular ADHBs with a = b = e = 1 at different values of n and α. It is clearly shown that, for circular ADHBs propagating in free space, their initial perfect circular anomalous dark hollow configurations, characterized by a central bright spot and a surrounding bright ring with the same extreme intensity, degrade steadily and rapidly. As the propagation distance further increases, the pattern consistently evolves into one featuring a prominent central bright spot surrounded by much weaker bright rings in the far field.Fig. 4 Intensity distribution evolution I(x, y, z) for circular ADHBs with a = b = e = 1 and (n, α) = (1, 0.25) in (a), (4, 0.65) in (b), (7, 0.80) in (c), and (10, 0.90) in (d). Left panels show the contour intensity distribution in the transversal plane at various propagation distances, while right panels illustrate the contour intensity variation in the (x–z) plane.

Fig. 4

Fig. 5 illustrates the intensity patterns at various propagation distances z/zR and the corresponding intensity variation in the (x-z) plane for elliptic ADHBs with a = 1, e = b = 1/3 at different values of n and α. It is obvious that elliptic ADHBs exhibit similar degradation behavior during propagation, eventually transitioning into a pattern characterized by a stronger central brightness surrounded by weaker elliptical bright areas in the far field. However, during near-field propagation, elliptic ADHBs display more complex evolution behaviors than circular ADHBs. The initial elliptic anomalous dark hollow configurations gradually lose their spatial elliptic structures and transform into petal-like distribution patterns around z = zR. At certain distances in the far field, these patterns partially restore the elliptic configuration with a 90° rotation around the propagation axis relative to their initial configurations. These results clearly demonstrate that the proposed beams are not propagation invariant.Fig. 5 Same as Fig. 4 but for elliptic ADHBs with a = 1, b = e = 1/3.

Fig. 5

3.2 Evolution of beam widths

Now we discuss how beam width evolves when the GDHsGBs propagate through free space. The mean-squared beam widths versus propagation distance, which provides approximate insights into the evolution of beam spot size spread during propagation, is defined by [56].(8) wσ2=1Itot∫∫−∞∞dxdy(σ−σ‾)2|E(x,y,z)|2,σ=x,y

where σ‾ = 0 denotes the centroid position of laser beams, and Itot represents the total power of the GDHsGBs. Using the formula ∫−∞∞exp(−ax2)dx=πa, we can derive the total power of the GDHsGBs from Eq. (5):(9) Itot=∫−∞∞dxdy|E(x,y,z)|2=(n!zR2nB)2∑m,l=0nCn,lCn.m*ΩxlΩylΩxm*Ωym*×∫∫−∞∞exp[−(1Ωxl+1Ωxm*)zR2xw2B2−(1Ωyl+1Ωym*)zR2yw2B2]dxdy=(n!w02n)2πI0

with I0=∑m,l=0nCn,lCn,m*(Ωxl+Ωxm*)(Ωyl+Ωym*). Similarly, wxs2 (wys2) can be derived using the same method.

Based on the analytic findings mentioned above, the mean-squared beam width of the GDHsGBs in two transverse directions can be obtained as(10) wxs2=wx2w02=B22zR2I0∑m,l=0nCn,lCn,m*ΩxlΩxm*(Ωxl+Ωxm*)3(Ωyl+Ωym*)

(11) wys2=wy2w02=B22zR2I0∑m,l=0nCn,lCn,m*ΩylΩym*(Ωxl+Ωxm*)(Ωyl+Ωym*)3

We conduct numerical simulations using Eqs. (10), (11) to analyze the evolution process of mean-squared beam width for circular and elliptic ADHBs, respectively. Our findings reveal that in the case of circular ADHBs with e = 1 and a = b, the beam widths uniformly broaden in both the x and y directions due to diffraction effects, maintaining pattern symmetry throughout propagation regardless of variations in n and α. Fig. 6 illustrates some typical examples of the wxs2 (wys2) against the propagation distance. Notably, we observe that, when holding other parameters constant, beam patterns broaden more rapidly with smaller values of α and a = b.Fig. 6 Beam width wxs2 (wys2) of circular ADHBs (e = 1) as a function of propagation distance z/zR. Left: for n = 4 with (a) α = 0.65, a = b = 1, (b) α = 0.7, a = b = 1, and (c) α = 0.85, a = b = 2. Right: for n = 7 with (a) α = 0.8, a = b = 1, (b) α = 0.9, a = b = 1, and (c) α = 1, a = b = 2.

Fig. 6

Fig. 7 depicts the evolutions of wxs2 and wys2 with respect to propagation distance for elliptic ADHBs with a = 1 and e = b = 1/3 across different values of n. It is shown that, as the propagation distance increases, the beam width in the direction where the initial beam has a smaller width undergoes faster broadening compared to the direction with a larger width. This difference in broadening rates between the x and y directions can be attributed to the varying diffraction effects in the two transversal directions. Taking wx < wy for example, it implies that the diffraction effect in the x direction is stronger than in the y direction, resulting in a more rapid increase in beam width along the x direction. Note that this phenomenon leads to both widths becoming equal at a specified propagation distance z0. Beyond this point, the beam width in the x direction exceeds that in the y direction, leading to an interchange of the major and minor axes of the astigmatic beam. The position of point z0 can be determined by(12) ∑m,l=0nCn,lCn,m*ΩxlΩxm*(Ωxl+Ωxm*)3(Ωyl+Ωym*)=∑m,l=0nCn,lCn,m*ΩylΩym*(Ωxl+Ωxm*)(Ωyl+Ωym*)3

which can be solved numerically.Fig. 7 Beam width wxs2 and wys2 of elliptic ADHBs (e = 1/3, a = 1, b = 1/3) as a function of propagation distance z/zR. Left: for n = 1 with α = 0.25 (red lines) and α = 0.3 (black lines). Right: for n = 4 with α = 0.65 (red lines) and α = 0.7 (black lines).

Fig. 7

3.3 Ellipticity of the elliptic ADHBs

To gain a comprehensive understanding of the propagation properties of GDHsGBs, we further investigate the evolution law of ellipticity for elliptic ADHBs propagating in free space. Ellipticity, denoted by ε(z), plays a pivotal role in characterizing the geometric configuration of optical patterns and can be defined as [57].(13) ε(z)=wxs2(z)−wys2(z)wxs2(z)+wys2(z)

where wxs(z) and wys(z) are the z-dependent pattern half widths in the x and y directions, respectively. Thus, the ellipticity varies with z ranging from −1 < ε(z) < +1, providing flexibility for adjustment as needed.

Fig. 8 shows the evolution of the ellipticity of elliptic ADHBs with a = 1 and b = e = 1/3 for various values of n and α. We find that, no matter what the values of n and α are, a consistent trend emerges: the ellipticity of elliptic ADHBs experiences rapid growth from its initial negative value in the near field, and eventually stabilizes at specified positive values in the far field. This phenomenon reflects the interchange of the major and minor axes of the elliptic ADHB patterns after the point z0 during propagation, which is in accordance with the previously discussed results. Note that the specified positive value the ellipticity approaches in the far field can be adjusted by the beam parameters n and α.Fig. 8 Ellipticity variations of elliptic ADHBs with propagation distance z/zR for different values of n and α (as denoted in the figures) with a = 1 and b = e = 1/3.

Fig. 8

4 Conclusions

In conclusion, we introduce GDHsGBs, a unified model that combines sin-Gaussian DHBs, characterized by a dark central region surrounded by bright rings, and ADHBs, featuring a bright central spot surrounded by bright rings. Using the Collins integral, we derive the analytical propagation equation for GDHsGBs through paraxial ABCD optical systems. Numerical simulations allow us to analyze their evolution in free space, focusing on intensity distribution and beam width. Circular ADHBs display symmetrical pattern evolution with uniformly broadening beam widths due to diffraction effects. In contrast, elliptic ADHBs exhibit more complex behavior during propagation, including changes in beam configuration, width, and ellipticity. These characteristics can be adjusted via beam parameters n, a, and e, providing flexibility in beam manipulation. Notably, we observed rapid growth of ellipticity from its initial negative value in the near field, stabilizing at positive values in the far field, revealing a 90° rotation of the intensity pattern relative to its initial configuration on the input plane during propagation. Due to these interesting properties, particularly the inherent central hollow region, GDHsGBs serve as ideal models for DHBs and ADHBs with circular or elliptic geometrical patterns, with potential applications in atom guiding and trapping, among others.

Funding

This research is supported by 10.13039/501100018555 Science and Technology Program of Guizhou Province ([2020]1Y025 ) and 10.13039/501100001809 National Natural Science Foundation of China (62163008 ).

Data availability

Data will be made available on request.

CRediT authorship contribution statement

Taofen Wang: Writing – original draft, Software, Methodology, Investigation, Formal analysis. Qin Su: Writing – review & editing, Software. Jie Zhu: Funding acquisition, Data curation, Conceptualization.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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