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Abstract
As abruptly autofocusing beams, autofocusing Bessel beams (ABBs) have been proven to be a class solution for the Helmholtz equation [Opt. Express 31, 33228 (2023) [CrossRef] ]. In this paper, we use the Fresnel number as the basic parameter and accurately compare the focusing property and radiation force of ABBs versus focused Gaussian beams (FGBs) under the same Fresnel number. Unlike FGBs, ABBs can achieve autofocusing without the need for an initial focusing phase. Our analysis of the beam width defined by power in the bucket, revealed that FGBs exhibit uniform focusing along the straight line, whereas ABBs demonstrate accelerated focusing along the elliptic curve. At the same Fresnel number, FGBs exhibit a higher peak intensity in the focal plane, yet ABBs excel in gradient force on particles. In comparison to FGBs, ABBs exhibit smaller potential well widths, allowing for stable and precise trapping of high refractive index particles at the focal point. While FGBs are considered suitable for laser processing and ablation due to their high peak power density, ABBs possess significant advantages in optical manipulation due to their great gradient force. Furthermore, we conduct a comparative analysis between ABBs and circular Airy beams (CABs). The peak intensity and gradient force exhibited by CABs are slightly lesser than those of ABBs. CABs are appropriate for multi-point trapping along the axis, whereas ABBs are more suited for precise single-point trapping.
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1. Introduction
The focusing of light beams, as a dynamic topic, has unique applications in optical manipulation [1–3], laser medicine [4], and high-resolution imaging [5]. Focused Gaussian beams (FGBs) are the most common focused beams. Many studies have been conducted on the propagation characteristics of FGBs [6–9]. Recently, structured beams with abruptly autofocusing property have attracted extensive attention. The maximum intensity of abruptly autofocusing beams remains nearly constant during the propagation process but undergoes a sudden increase of several orders of magnitude near the focus. Circular Airy beams and circular Pearcey beams exhibit abruptly autofocusing property and can be generated using computer holography [10–13]. Circular Mathieu and Weber autofocusing beams were found to automatically focus along parabolic and ellipsoidal planes [14,15]. In our previous work, we found a new kind of abruptly autofocusing beams called autofocusing Bessel beams (ABBs). ABBs have been proven to be a class solution to the Helmholtz equation. The experiment observed the autofocusing of ABBs and the dual focus of ABBs passing through a focusing lens [16].
Ashkin first observed the trapping of dielectric particles by a single beam in 1986 [17]. The results confirmed the existence of negative optical pressure and indicated that the trapping well is generated by an axial gradient force. Optical tweezers typically employ focused Gaussian beams (FGBs) [18]. High refractive index particles will be trapped at the focus due to the gradient force of the FGBs [19,20]. Manipulation of micron-sized strongly absorbing particles in pure liquid glycerol by a single FGB has been reported [21]. In addition, the radiation force of abruptly autofocusing beams on Rayleigh particles has become a research hotspot [22–26]. The circular Airy beam can trap high refractive index particles on the propagation axis, and the trapping position can be adjusted by controlling the corresponding parameters of the circular Airy beam [27]. Optical manipulation using circular Pearcey vortex beams has achieved the rotational motion of micron-sized particles and the transport of light-absorbing particles in the air [28]. The trapping capability enhanced by autofocusing beams offers many advantages for biological samples that may be subjected to optical heating and damage. As focused beams, ABBs and FGBs are the solutions to the Helmholtz equation. Analyzing these two beams is necessary to clarify the distinctions in focusing behavior and applications between focused beams relying on the focusing phase and autofocusing beams.
We accurately compare the focusing property and radiation force of ABBs and FGBs at the same Fresnel number. The electric field expressions of ABBs and FGBs are derived based on the Fresnel diffraction theory, and the propagation dynamics are investigated. Through an analysis of the variations in maximum intensity and beam width during propagation, we found that FGBs exhibit higher peak intensity and smaller beam width than ABBs in the focal plane. ABBs have a great gradient force near the focus due to their accelerated focusing characteristic. Compared to FGBs, the potential well formed by ABBs has smaller potential well widths, enabling stable and precise trapping of high refractive index particles at the focal point. Overall, FGBs exhibit a great peak power density, yet ABBs have advantages in optical confinement force. As classic abruptly autofocusing beams, CABs have been widely studied in the field of optical tweezers. We compared ABBs with CABs to clarify their differences in the characteristics of trapping particles.
2. Propagation dynamics of ABBs and FGBs
The electric field distribution for the FGBs in the initial plane (z = 0) is given by
Under the finite energy limitation, the initial field of the ABBs is [16]
To accurately compare the focusing property of ABBs and FGBs, we require the amplitude factor of the ABBs. The beam power in the initial plane can be expressed as
Utilizing Eq. (3) and the following standard integral [29]
${\boldsymbol K}({\bullet} )$ is the complete elliptic integral of the first kind.
We use the Fresnel diffraction formula to find the electric field distribution at z
Using the following standard integral [29]
Utilizing the following standard integral [29]
Figures 1 and 2 represent the propagation dynamics of the FGB and ABB, respectively, A = 1 and Nf=30. In the initial plane, the FGB exhibits only one central bright spot, while the ABB extends outward as a series of increasingly tightly arranged concentric intensity rings [Fig. 1(a) and Fig. 2(a)]. Both the FGB and the ABB achieve focusing [Fig. 1(c) and Fig. 2(c)]. The FGB has the focusing phase in the initial plane [Fig. 1(b)]. However, the ABB can achieve autofocusing without the focusing phase [Fig. 2(b)]. In the focal plane, the FGB and the ABB exhibit no sidelobe [Fig. 1(d) and Fig. 2(d)], which is different from circular Airy beams and circular Mathieu beams.
Fig. 1. Propagation dynamics of the FGB. (a)(d) Intensity distribution; (b)(e) phase distribution; (c) variation of intensity with z.
Fig. 2. Propagation dynamics of the ABB. (a)(d) Intensity distribution; (b)(e) phase distribution; (c) variation of intensity with z.
3. Comparison of focusing property between ABBs and FGBs
3.1 Maximum intensity
In this section, we investigate the maximum intensity of ABBs and FGBs during the propagation. According to Eq. (8) and Eq. (10), the maximum intensity is
From Eqs. (11) and (12), Imax is determined by the amplitude factor and Fresnel number. The diffraction ability is closely related to the Fresnel number. To fairly compare the focusing property of FGBs and ABBs, we set them at the same Fresnel number and average power density. Figure 3 shows the variation of Imax with propagation distance, A = 1. The diffraction effect becomes significant when the Fresnel number is small, resulting in an obvious focus shift [Fig. 3(a)]. The ABB has a smaller focus shift than the FGB. Due to the abruptly autofocusing property of ABBs, their maximum intensity changes slowly during propagation and abruptly increases by orders of magnitude near the focus. Therefore, near the focus, Imax of the ABBs increases more abruptly than that of the FGBs [Figs. 3(b), (c) and (d)].
Substituting ${z_\varsigma } = 1$ into Eq. (11) and Eq. (12), we obtain the expression for the maximum intensity in the focal plane
Figure 4(a) represents Imax in the focal plane versus Fresnel number, A = 1. Imax in the focal plane increases as the Fresnel number increases. This implies that the focused beams with a larger wave number, a wider initial beam width, and a shorter focal length result in higher peak intensity in the focal plane. For Fresnel numbers below 30, both the FGBs and ABBs exhibit similar Imax in the focal plane. However, for Fresnel numbers exceeding 30, the FGBs surpass the ABBs in terms of Imax in the focal plane. The high peak intensity of FGBs renders them advantageous for laser processing and ablation.
Fig. 4. (a) Imax in the focal plane versus Fresnel number; (b) Imax under critical Fresnel number versus z.
Achieving focusing requires the Imax in the focal plane to surpass that in the initial plane. The critical Fresnel number is reached when Imax in the initial plane is equal to that in the focal plane. By setting Imax(z = 0)=Imax(z = f), we calculate the critical Fresnel numbers for FGBs and ABBs based on Eqs. (11) and (12). The critical Fresnel numbers of the FGB and the ABB are 2 and 0.5, respectively. The variation of Imax under the critical Fresnel number with propagation distance is shown in Fig. 4(b).
3.2 Power in the bucket and beam width
Power in the bucket (PIB) is defined as the power proportion within a given radius r0 of the bucket
Figure 5 represents PIB of the ABB and the FGB, A = 1, Nf=30. In the initial plane, the FGB and the ABB have the same PIB when ${{{r_0}} / w} = 0.9$ [Fig. 5(a)]. In the focal plane, PIB of the FGB is always greater than that of the ABB [Fig. 5(b)]. This indicates that, in the focal plane, the power distribution of the FGB is more concentrated compared to that of the ABB.
The beam widths defined by PIB can be utilized to evaluate the spot size. We define r0 at PIB = 86.5%, 63%, 48%, 25% as beam widths ${w_{86.5\%}}$, ${w_{63\%}}$, ${w_{48\%}}$, ${w_{25\%}}$, respectively. Figure 6 shows the variation of the beam widths with propagation distance. In Fig. 6(a), ${w_{86.5\%}}$ of the ABB increases with the propagation distance but abruptly decreases near the focus. For ${w_{63\%}}$, ${w_{48\%}}$, and ${w_{25\%}}$, the beam widths of the FGB are uniformly decreasing, while the beam widths of ABB are accelerating decreasing [Figs. 6(b), (c) and (d)]. Hence, the FGB exhibits uniform focusing, while the ABB demonstrates accelerated focusing. It proves that ABBs exhibit the abruptly autofocusing property.
Fig. 6. Variation of the beam widths with z. (a)${w_{86.5\%}}$; (b)${w_{63\%}}$; (c)${w_{48\%}}$; (d)${w_{25\%}}$.
According to $I \propto {P / {{w^2}(z )}}$ and Eq. (11), it can be approximated that the beam width of FGBs adheres to $w(z )\propto |{{z_\varsigma } - {F_{\textrm{FG}}}} |$, i.e., FGBs exhibit uniform focusing along the straight line [Figs. 6(a), (b), (c) and (d)]. However, the beam width of ABBs conforms to $w(z )\propto \sqrt {|{z_\varsigma^2 - F_{\textrm {AB}}^2} |}$, i.e., ABBs exhibit accelerated focusing along the elliptical curve [Figs. 6(b), (c) and (d)].
4. Comparison of radiation force between ABBs and FGBs
In this section, we calculate the radiation force on the Rayleigh particle in the focused beams and the potential well distribution. It is well known that Rayleigh particles can be regarded as point dipoles in a light field. Therefore, a Rayleigh particle is subjected to the combined effect of the scattering force Fs and the gradient force Fg [30]
where np and nm are the refractive indexes of the particle and the medium, respectively. a is the particle radius, and ${\varepsilon _0}$ is the dielectric constant in vacuum. Suppose a glass micro particle with np = 1.59 and a = 15 nm is placed in a medium (i.e., water) with nm=1.33. The axial gradient force Fgz and transverse gradient force Fgr can be calculated using Eq. (16). The scattering force Fs can be determined by applying Eq. (17). The scattering force points to the propagation direction. Since we consider the particle whose refractive index is higher than the surrounding medium, the axial gradient force is parallel to the propagation axis and points to the focal point, and the transverse gradient force points to the beam center along the radial direction. To ensure a fair comparison of the radiation forces of FGBs and ABBs, we adjust them at the same Fresnel number and average power density. The calculation parameters P = 1W, w = 0.1 mm, f = 2 mm, k = 6 × 106m-1, and Nf = 30 are chosen.Substituting the electric field expressions Eqs. (8) and (10) into Eqs. (16) and (17), we obtain the distributions of radiation force on the Rayleigh particle, as shown in Fig. 7. In the focal plane, the ABB exhibits the greater transverse gradient force than the FGB [Fig. 7(a)]. Due to the accelerated focusing characteristic of ABBs, the axial gradient force of the ABB near the focus is greater than that of FGB [Fig. 7(b)]. Compared to the FGB, the ABB exhibits a smaller scattering force [Fig. 7(c)]. Achieving optical trapping requires the axial gradient force to be large enough to overcome the scattering force. From Fig. 7(d), the ABB and the FGB overcome the influence of scattering force. Near the focus, the ABB has a stronger axial confinement force than the FGB [Fig. 7(d)]. Due to the advantage of ABBs in the transverse and axial confinement force, ABBs can trap the particle more stably than FGBs near the focus. The confinement force is proportional to the beam power. From another perspective, ABBs with lower power can also achieve the same value of confinement force as FGBs. The trapping capability enhanced by ABBs offers many advantages for biological samples that may be subjected to optical heating and damage.
The gradient force is dominant in trapping particles with a single laser beam. Since the gradient force is a conservative force, the potential energy corresponding to the gradient force is
Figure 8 represents the potential well corresponding to the gradient force. High refractive index particles will be trapped in the deep potential well region. In both the axial and transverse scales, the ABB exhibits smaller potential well widths than the FGB [Figs. 8(c) and (d)]. A smaller potential well width indicates a more precise trapping region. Compared to the FGB, the potential well formed by the ABB has a more precise trapping region [Figs. 8(a) and (b)].
Fig. 8. (a)(b) Potential well corresponding to the gradient force; (c) axial potential well, r = 0; (d) transverse potential well, z = f.
5. Comparison between ABBs and circular Airy Beams
As classic autofocusing beams, circular Airy Beams (CABs) have been widely studied in the field of optical manipulation. In this section, we compare ABBs with CABs to clarify their differences in focusing property and radiation force. The electric field distribution for the CABs in the initial plane (z = 0) is given by
The calculation parameters P = 1W, r0 = 0.1 mm, $s = 6.5\,{\mathrm{\mu} \mathrm{m}}, \alpha = 0.05,$ k = 6 × 106m-1. According to Eq. (21), the focal length of CABs is f = 2 mm. If r0 is considered as the initial beam width of CABs, then the Fresnel number of the CAB under the above parameters is Nf = 30. For a fair comparison with the CAB, we set the parameters of the ABB as P = 1W, w = 0.1 mm, f = 2 mm, k = 6 × 106m-1, and Nf = 30. These parameters are the same as in section 4. We simulate the electric field of the CAB at z by fast Fourier transform.
Figure 9 shows the intensity distribution of the CAB and the ABB. From Fig. 9, the peak intensity of the ABB at the focal point is greater than that of the CAB. In the transverse scale, the CAB has a sidelobe in the focal plane [Fig. 9(a)]. In the axial scale, the CAB exhibits multiple peak intensities beyond the focus [Fig. 9(b)]. However, the ABB near the focus displays only one peak intensity in both the transverse and axial scales. Utilizing ABBs for optical manipulation can circumvent interference from multiple peak intensities, giving the potential well generated by ABBs a precise trapping region [Fig. 8(b)].
Fig. 9. (a) Intensity distribution in the transverse scale, z = f; (b) intensity distribution in the axial scale, r = 0.
Figure 10 represents the radiation force and potential well of the CAB and the ABB. In the transverse scale, the ABB exhibits a greater gradient force than the CAB [Fig. 10(a)]. The transverse gradient force caused by sidelobes will interfere with the trapping effect produced by CABs. From Fig. 10(b), the axial confinement force exhibited by the CAB is slightly less than that of the ABB. Upon comparing Fig. 10(c) with Fig. 8(b), it is evident that the CAB possesses at least two deep potential wells, whereas the ABB exhibits only one deep potential well. Therefore, CABs are appropriate for multi-point trapping along the axis, whereas ABBs are more suited for precise single-point trapping.
Fig. 10. (a) Gradient force in the transverse scale, z = f; (b) confinement force in the axial scale, r = 0; (c) potential well corresponding to the gradient force.
As a comparison, Table 1 summarizes the trapping characteristics of autofocusing beams, which have been reported in other papers. The trapping position on the propagation axis can be adjusted by controlling the corresponding parameters of CABs [26,27]. Unlike CABs, the focal length of ABBs can be directly defined. The trapping position exhibited by ABBs can be tuned via directly adjusting the focal length of the initial field. By controlling the chirp factor and the spatial coherence length of CABs, the radiation force and potential well depth can be effectively modulated, thereby obtaining more trapping positions [25,31]. The analysis of this paper shows that ABBs excel at precise single-point trapping.
Table 1. Summary of trapping characteristics of autofocusing beams
It should be pointed out that our theoretical model is more effective for continuous monochromatic waves. In addition, the Nf utilized in this paper to characterize the diffraction ability draws on the widely used mathematical form of the Fresnel number. Nf may exceed the most primitive physical meaning of the Fresnel number.
6. Conclusion
We investigate the focusing property and radiation force of ABBs and FGBs. The electric field expressions of ABBs and FGBs are derived using the Fresnel diffraction theory, and the propagation dynamics are analyzed. In contrast to FGBs, ABBs can achieve autofocusing without an initial focusing phase. We define the focusing parameter and identify the critical Fresnel number for achieving focus. The critical Fresnel numbers for achieving focusing of the FGB and the ABB are 2 and 0.5, respectively. At the same Fresnel number, ABBs have a smaller focus shift than FGBs. In contrast to the uniform focusing of FGBs, ABBs possess the characteristic of accelerated focusing, with the beam width decreasing along the elliptic curve during the focusing process. Near the focus, the maximum intensity and power in the bucket of FGBs are higher than those of ABBs, whereas ABBs have stronger axial and transverse gradient forces than FGBs. In comparison to FGBs, ABBs create optical potential wells characterized by a stable and precise trapping region. The high peak intensity of FGBs offers advantages in laser processing and ablation, whereas the strong confinement force of ABBs provides significant benefits for optical manipulation. Furthermore, our comparative analysis between ABBs and CABs revealed that CABs are suited for multi-point trapping along the axis, whereas ABBs excel at precise single-point trapping.
Funding
Natural Science Foundation of Chongqing (Grant No. CSTB2022NSCO-MSX1027); Natural Science Foundation of Jilin Province (Grant No.20210101154JC).
Acknowledgments
We are very grateful for the aforementioned funding. We appreciate the reviewers for their valuable comments and suggestions.
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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