## Abstract

In this paper, we introduce a novel kind of abrupt autofocusing beams namely the generalized circular Airy derivative beams (CADBs) as an extension of circular Airy beam (CAB). The propagation dynamics of the CADBs is examined theoretically. Our results show that the CADBs exhibit stronger autofocusing ability than the CAB under the same condition. The physical mechanism of the abruptly autofocusing of the CADBs is interpreted by mimicking the Fresnel zone plate lens. Here, the abruptly autofocusing ability is described by a ratio *K *= *I _{fm}*/

*I*

_{0m}where

*I*and

_{fm}*I*

_{0m}correspond to the maximum intensities in the focal and the source planes, respectively. As an example, the

*K*-value of the circular Airyprime beam (CAPB, the first-order Airy derivative beam) is about 7 times of that of the CAB. In addition, the CAPB have narrower FWHM (full width at half maxima) in the focus position than the CAB, and the focal spot size of the CAPB is smaller than that of the CAB. Furthermore, we establish an optical system involving a phase-only spatial light modulator to generate the CAPB and measure its autofocusing characteristics experimentally. The measured

*K*-value is about 9.4 percentage error between theory and experiment owing to the imperfection generation of the CAPB. The proposed generalized CADBs will find applications in biomedical treatment, optical manipulation and so on.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

## 1. Introduction

Due to the lateral self-acceleration, the circular Airy beam (CAB) has unique characteristic of abruptly autofocusing without help of external optical elements [1–5]. The abruptly autofocusing denotes that its intensity increases suddenly right before the focal point while it maintains a low intensity profile until that very point. The abruptly autofocusing ability can be described by a ratio *K *= *I _{fm}*/

*I*

_{0m}where

*I*and

_{fm}*I*

_{0m}correspond to the maximum intensities in the focal and the source planes, respectively. If the first few rings of a CAB are blocked, the

*K*-value of a CAB can be increased up to a factor of 3 and is about 100 in experiment [6]. As demonstrated in experiments, the

*K*-value of a CAB can reach up to 195 [2].

*K*= 450 was experimentally measured when paraxial ring Airy beams approach the wavelength limit [7]. The abruptly autofocusing polycyclic tornado ring Airy beam even has controllable multi focus [8]. The theoretical

*K*-value of a CAB with an optical vortex of two topological charges is 96 [9]. When two opposite optical vortices are superimposed on the CAB, the

*K*-value can theoretically reaches 300 [9]. The maximum

*K*-value of annular arrayed-Airy beams carrying vortex arrays is 212 [10].

The Airy beam belongs to the typical representative of the one-dimensional caustic field, and Pearcey beams [11] are another kind of autofocusing beams that represents two-dimensional caustic field. Similar to pure Airy beams, pure Pearcey beams have infinite energy. In order to ensure finite energy, Pearcey beams are often modulated by a Gaussian factor. Auto-focusing and self-healing behavior of Pearcey beams and symmetric odd-Pearcey Gauss beams have been theoretically and experimentally demonstrated, respectively [11,12]. When a spiral phase is imposed to the circle Pearcey beams, the *K*-value can be theoretically enhanced and can easily exceed 200 [13]. Numerical simulation indicates that the *K*-value of circular chirp Pearcey Gaussian vortex beams is about 100 when the beam parameters are optimized [14]. The *K*-value of radially polarized circle Pearcey vortex beams is theoretically proved to be greater than that of the circle Pearcey vortex beams [15]. Swallowtail beams stem from high-order swallowtail catastrophe and are the representative of the multi-dimensional caustic field [16–18]. The energy of a swallowtail beam is infinite. Therefore, a swallowtail beam is usually truncated in an experiment. The abruptly autofocusing experiment manifests that the *K*-value of a truncated circular swallowtail beam is no more than 150 [19]. The abruptly autofocusing vortex beam has been interpreted from the perspective of caustics [20]. A simple but efficient method to generate abruptly autofocusing beams with arbitrary caustics has been theoretically proposed [21]. Of course, the optical researchers also designed abruptly autofocusing beams that are not based on the Airy function, the Pearcey function and the swallowtail catastrophe integral. For examples, nonparaxial vectorial abruptly autofocusing beams with structured polarization states have been designed [22], and abruptly autofocusing beams have been realized by adding a perturbation with forced symmetry into the spectral phase [23]. The abruptly autofocusing feature has broad applications in biomedical treatment [1], optical manipulation [24], nonlinear manipulation [25], free-space optical communication [26], optical trapping and guiding [27–29], light bullet [30], multi-photo polymerization [31], terahertz wave emission [32], dynamic imaging [33] and so on.

In this paper, we introduce a novel vortex-free beam family namely the generalized circular Airy derivative beams (CADBs). CADBs are circular autofocusing beams with a radial profile that is described by derivatives of the Airy function. Here we mainly concern the comparison of the autofocusing properties between the CADBs and the well-known CABs under the condition of same beam parameters. As a typical representative, the circular Airyprime beam (CAPB, the first-order CADB) demonstrates a more powerful abruptly autofocusing ability than the corresponding CAB. The *K*-value of the CAPB theoretically reaches 356.94 and achieves 323.5 in our experiment. Under the same conditions, the *K*-value of the CAB is only 52.19 in theory and 46.8 measured in our experiment.

## 2. Generalized CADBs and their propagation characteristics

The electric field of a generalized CADB in the source plane *z *= 0 is described by:

*r*is the radial coordinate, and

*φ*denotes an azimuthal angle. The

*z*-axis is the direction of beam propagation.

*r*

_{0}is the radius of the primary ring.

*w*

_{0}is a scaling factor, and

*a*is an exponential decay factor.

*Ai*

^{(}

^{n}^{)}(.) is the

*n*

^{th}-order derivative of the Airy function with respect to

*r*[34].

*n*is the order of derivative and is a non-negative integer. When

*n*= 0, the generalized CADB reduces to the familiar CAB. When

*n*= 1, the generalized CADB is referred as the CAPB, which is originated from the Airyprime function namely the first-order Airy derivative function. It must be pointed out that the previously reported Airyprime beams in Ref. [35] are pseudo Airyprime beams, whose optical field in the arbitrary transverse direction of the source plane is determined by the product of two Airyprime functions with opposite signs. The pseudo Airyprime beam is similar to but not exactly the same as the second-order elegant Hermite-Gaussian beam [35]. By performing the Airy transform of a Laguerre-Gaussian beam, a Hermite-Gaussian beam, an elegant Hermite-Gaussian beam or a Gaussian vortex beam, the output beam is a multimode beam, which includes different Airy derivative modes with different weight coefficients [36–39]. This is why we want to generate a single Airy derivative beam.

The optical field of the generalized CADB propagating in free space is found to be [40]:

*k*= 2

*π*/

*λ*with

*λ*being the optical wavelength, and

*J*

_{0}(⋅)is the zero-order Bessel function of the first kind. It is hard to find the analytical expression for the

*U*(

*r*,

*φ*,

*z*) by any means. Fortunately, the numerical simulation of Eq. (2) can be carried out by using the fast Fourier transform algorithm [41]. The light intensity of the generalized CADB propagating in free space is given by

*I*(

*r*,

*φ*,

*z*) = |

*U*(

*r*,

*φ*,

*z*)|

^{2}. Owing to the circular symmetry of the optical field, we could obtain the analytical expression for the optical field at the on-axis point, i.e.,

*r*= 0. By using the following integral formulae [34,42]:

*C*

_{n}_{, s}being given by

*n*

^{th}-order is the sum of the Airy function up to the Airy derivative function of (

*n*+1)

^{th}-order. Moreover, the weight coefficients of each component are different. When

*n*= 0, Eqs. (2) and (6) provide the optical field and the on-axis optical field of the CAB propagating in free space, respectively.

## 3. Numerical calculations and analysis

In this section, we first study the propagation dynamics of generalized CADBs through numerical examples, and then investigate the abruptly autofocusing characteristics in detail. For simplicity and without loss of generality, the wavelength of the optical field is fixed at *λ*=532 nm throughout the paper.

Figure 1 illustrates the radial intensity distribution of the CADB for *n *= 1 (referred as the CAPB) at several propagation distances during free-space propagation. The beam parameters are *r*_{0 }= 1 mm, *a *= 0.1 and *w*_{0 }= 0.1 mm in the calculation. For a comparison, the corresponding results for the CAB (*n *= 0) with the same beam parameters are shown in Fig. 2. In the source plane *z *= 0, the second ring from the beam center (*r *= 0) of the CAPB is the brightest ring whose radius and intensity are respectively 1.231 mm and 0.3094, while the first ring of the CAB is the main ring whose radius and intensity are respectively are 1.092 mm and 0.2364. The radius and intensity of the first ring of the CAPB in the source plane are 0.991 mm and 0.0675, respectively. In the source plane, the intensity of the CAPB decays more slowly alongthe radial direction than that of the CAB, resulting in more visible rings of the CAPB. Upon propagation, the adjacent rings interfere from the outermost to the inner in turn, which is caused by the lateral self-acceleration. The number of rings decreases gradually during the propagation, meanwhile the main ring shrinks gradually and finally evolves into a bright solid spot in the center, occurring the autofocusing phenomenon. When the propagation distance is at 780 mm, the maximum intensity abruptly increases to more than 110 for the CAPB and about 12 for the CAB. The full width at half maxima (FWHM) of the CAPB and the CAB at *z *= 780 mm is 0.067 mm and 0.073 mm, respectively. Therefore, the CAPB have slightly narrower FWHM than the CAB at *z *= 780 mm. The narrowing FWHM of the CAPB means that the maximum light intensity is limited to a smaller range. When the propagation distance is larger than 780 mm, the on-axis intensity of these two beams both decreases.

To clearly show the evolution of the intensity distribution of the CAPB and the CAB, we plot the two-dimensional intensity profiles of the CAPB and the CAB in different observation planes, which are shown in Figs. 3 and 4. The beam parameters used in Figs. 3 and 4 are same as those in Figs. 1 and 2. It can be seen that the central beam spot size of the CAPB is smaller than that of the CAB at *z *= 780 mm, which makes the autofocusing ability stronger. Due to a difference in the colormap limits, the differences in intensity localization at the focus of the CAPB [Fig. 3(d)] and the CAB [Fig. 4(d)] are not visually apparent. Nevertheless, from the maximum value of color bar in Fig. 3(d) and Fig. 4(d), the focusing ability of the CAPB is about 7 times of the CAB under the same condition. We want to stress here that for the CADBs with *n*≥1, they exhibit the similar abruptly autofocusing propagation features, but the focusing properties closely depend on the initial beam parameters, which we will study in detail.

The abruptly autofocusing ability and the focus position of the generalized CADBs can be judged by analyzing the characteristics of on-axis light intensity. Here, the focus position is defined as the distance from the source plane (*z *= 0) to the plane of *z* where the on-axis intensity reaches the maximum value. As the on-axis light intensity of the generalized CADBs is determined by three beam parameters *r*_{0,} *a* and *w*_{0}, the influences of these three beam parameters on the on-axis light intensity of the generalized CADBs are investigated, respectively. A parameter named the relative on-axis light intensity, defined by the ratio of the on-axis light intensity to the maximum intensity in the source plane *I*_{0m}, is used to assess the abruptly autofocusing ability. Note that the relative light intensity in the focus position reduces to the *K*-value because near/at the focus position the on-axis intensity is the intensity maxima at the beam cross-section. Figure 5 presents the relative on-axis intensities of the generalized CADBs with different *r*_{0} as a function of the axial propagation distance *z.* The other two parameters are *a *= 0.1 and *w*_{0 }= 0.1 mm in Fig. 5. With the increase of the radius of the primary ring *r*_{0}, the abruptly autofocusing ability is enhanced and the focus position is elongated. As the order of derivative *n* increases, the abruptly autofocusing ability is significantly improved. Nevertheless, the focus position is nearly independent of the order of derivative *n*. For the CAB (*n *= 0), there is an estimation equation to calculate the focus position, i.e., *z _{f }*= 4

*πw*

_{0}(

*w*

_{0}

*R*

_{0})

^{1/2}/

*λ*[2,6], where

*R*

_{0}is the radius of the first intensity maximum ring. In the case of

*r*

_{0 }= 1 mm,

*a*= 0.1 and

*w*

_{0 }= 0.1 mm, the calculated

*R*

_{0}of the CAB is about 1.092 mm, and the estimated value of the focus position

*z*is about 0.78 m. Our numerical calculation finds that the focus positions of the CADBs with

_{f}*n*= 1, 2 and 3 are about 0.778 m, 0.778 m and 0.784 m, respectively, only a very slight change. Thus, one can still use the estimation equation of the CAB to estimate the focus position of the CADBs. The

*K*-values of the CADBs with

*n*= 0, 1, 2 and 3 are 52.19, 359.04, 813.60 and 1040.60, respectively, which increase as the

*n*increases.

The effect of the exponential decay factor *a* on the relative on-axis intensity of the generalized CADBs is shown in Fig. 6 where *r*_{0 }= 1 mm and *w*_{0 }= 0.1 mm. The smaller the value of *a* is, the stronger autofocusing ability the CADBs have. The focus position is nearly unchanged at two different values of *a*. When *a *= 0.15, the *K*-values of CADBs with *n *= 0, 1, 2 and 3 are 23.33, 156.99, 438.93 and 700.12, and the corresponding focus positions are 0.782 m, 0.776 m, 0.774 m and 0.774 m. Finally, the influence of the scaling factor *w*_{0} on the relative on-axis intensity is displayed in Fig. 7 where *r*_{0 }= 1 mm and *a *= 0.1. With the decrease of the scaling factor *w*_{0}, the focus position shortens, and the abruptly autofocusing ability of the CADB with *n*≥1 is strengthened, whereas the abruptly autofocusing ability of the CAB is weakened. When *w*_{0 }= 0.08 mm, the focus positions of CADBs with *n *= 0, 1, 2 and 3 are 0.554 m, 0.552 m, 0.552 m and 0.554 m, and the corresponding *K*-values are 47.24, 386.94, 1013.65 and 1440.98.

## 4. Discussion of the physical mechanism of the abruptly autofocusing of the CADBs

In Ref. [1], the explanation for the abruptly autofocusing of the CAB is attributed to the lateral acceleration of the Airy beams themselves. Under this circumstance, large transverse velocities are attained and energy rushes in an accelerated fashion toward the focus. Li et al. found that the autofocusing ability of the CAB blocked first few rings is enhanced whereas the focus position remains unchanged [6]. In our studies, the derivative of the circular Airy function is used to replace the circular Airy function in the source’s electric field, but the abruptly autofocusing ability is still remained. This result seems to show that the autofocusing phenomenon is not unique to the CABs. Here, we present an alternative explanation for the autofocusing of the CADBs by mimicking the *Fresnel zone plate* (FZP) lens.

The FZP lens is composed of the specially designed transparent concentric rings. At the particular on-axis point, the optical path difference from the concentric rings to that point is an even or odd multiple of half wavelength. Therefore, the interference construction occurs at that point and the light is focused if the plane wave is incident. Are the intensity rings of the CAB or CADBs just located at the position of rings of the specially designed FZP lens? Let us examine the characteristics of the amplitude of the CAB and the CADBs (*n *= 1, the CAPB as an example) again, as shown in Fig. 8(a) and 8(b). The beam parameters in the calculation are *r*_{0 }= 1 mm, *a *= 0.1 and *w*_{0 }= 0.1 mm. They are all real functions and oscillate with positive and negative values along the radial directions, implying that the phase difference between two adjacent rings is *π*. The only differences between these two beams are the radius of concentric intensity rings and the amplitude decay speed along the radial direction. The amplitude decay of the CAPB is much slower than that of the CAB, and the decay speed decreases further with the increase of the derivative order.

Figure 8(c) illustrates the propagation scenario of the CAB or CAPB from the source plane to its focus position. *z _{i}* (

*i*= 1, 2, 3,…) denotes the distances from intensity rings (maximum intensity) to the focus point

*p*, given by

*z*is the focus position. In the derivation of the right side of Eq. (9), we remain the first two terms of Tylor expansions under the paraxial approximation of

_{f}*r*<<

_{i}*z*. We present in Table 1 the data for the radius of first fifteen rings

_{f}*r*, the difference between the distance from the rings to the focus position and the focus position

_{i}*z*-

_{i}*z*, the optical path difference between two adjacent rings to the focus position

_{f}*z*-

_{i}*z*

_{i}_{-1}and the deviation of optical path difference to the half wavelength, defined as

*D*= 2(Δ

*z*-

_{i}*λ*/2)/

*λ*, where Δ

*z*=

_{i }*z*-

_{i}*z*

_{i}_{-1}(

*i*= 2,3…). From Table 1, one finds that the optical path differences between two adjacent rings to the focus position are close to half wavelength of the light (

*λ*/2 = 266 nm) except for the first two rings of the CAB and the CAPB. Different from the conventional FZP lens, the phase difference between two adjacent rings of the CAB or the CAPB is

*π*. Therefore, the condition for the interference construction is that the optical path difference is an integral multiple of half wavelength, not even or odd times. Owing to that the optical path difference of first two rings deviates greatly from the half wavelength, the focusing ability is enhanced when the first two rings are blocked.

Comparing the amplitude of the CAB and the CAPB in the source plane shown in Fig. 8(a) and 8(b), the CAPB have more rings with appreciable intensity and the interference construction occurs at the focus position, resulting the stronger autofocusing ability. In addition, it could also be found from Table 1 that the interference construction at the focus position is not destroyed by making the derivative of the Airy function in the source plane (the data of other CADBs with *n *> 1 are not shown here). As a result, the focus position of the CADBs almost does not change. However, the focusing ability is enhanced because the amplitude decays slowly with the increase of derivative order *n*.

To clearly show the phase changing between two adjacent rings of the CAB or the CAPB, we plot in Fig. 8(d) the variation of the phase differences Δ*φ _{i }*= 2

*π*(

*z*-

_{i}*z*

_{i}_{-1})/

*λ*–

*π*between the

*n*ring and (

^{th}*n*-1)

*ring from the source plane to the observation point*

^{th}*p*. The term “–

*π*” in phase difference expression denotes the

*π*-phase shift between two adjacent rings. The solid line in Fig. 8(d) denotes the zero-phase difference line for comparison. One can see that except for the first three rings, the phase difference deviation from zero is relatively large, whereas the phase difference between other adjacent rings is nearly zero. From Table 1, the deviations

*D*are less than or nearly 5%. It should be emphasized here that the interference construction with different rings at the observation point

*p*still occurs because the phase differences is nearly zero except the first three rings.

## 5. Experimental results

In this section, we experimentally generate the CAPB by using a phase-only spatial light modulator (SLM) and then measure the intensity distribution and the autofocusing ability. As a comparison, the corresponding experimental results for the CAB are also presented. The schematic for the experimental setup is illustrated in Fig. 9. A linearly polarized fundamental Gaussian beam (*λ*=532 nm, beam waist size *ω*_{1 }= 0.42 mm) generated from Nd:YAG laser is expanded by a 50× beam expander (BE), reflected by a mirror (RM), and then passes through a 50: 50 intensity beam splitter (BS). The beam waist size is enlarged to 21 mm after the BE. The transmitted part arrives at the SLM which is used to modulate the amplitude and phase of the incident beam. The SLM used here is a Holoeye GAEA-VIS which has a 3840×2160 liquid crystal pixel array with each pixel size 3.74 µm×3.74 µm, and thus the effective area is 14.4 mm×8 mm. The high resolution of the SLM ensures to produce the high-quality beam with complex and fine structure.

In order to produce the CAPB or the CAB, we first perform the Fourier transform of the source’s electric field, *U _{f}*(

*ρ*,

*θ*)=

*F*[

*U*(

*r*,

*φ*, 0)] =

*A*(

*ρ*,

*θ*)exp[

*iϕ*(

*ρ*,

*θ*)], where

*F*denotes the Fourier transform.

*A*and

*ϕ*are the amplitude and phase of the Fourier transformed field. Owing to the rotational symmetry of the source’s field,

*U*is a real function in which the phase is only two binary values of 0 and

_{f}*π*, i.e., the phase difference between the positive and the negative values is

*π. U*of the CAPB and the CAB are shown in Fig. 10(a) and 10(d), respectively. Now, we assume that the transmittance of a phase-only computer-generated hologram (CGH), expressed as a function dependent on the amplitude and the phase of the encoded field, is written as

_{f}*h*(

*x*,

*y*)=exp[

*iΨ*(

*A*,

_{n}*ϕ*)], where

*Ψ*(

*A*,

_{n}*ϕ*) is the CGH phase modulation, and

*A*is the normalized amplitude of the encoded field, i.e.,

_{n}*A*=

_{n }*A*/

*A*. For brevity, the dependence of the amplitude and the phase on

_{max}*ρ*and

*θ*is omitted. To evaluate

*Ψ*(

*A*,

_{n}*ϕ*), we adopt the method used in Ref. [43]. In this case, the phase function is written as the form of

*Ψ*(

*A*,

_{n}*ϕ*)=

*f*(

*A*)sin

_{n}*ϕ*where

*f*(

*A*) is an unknown function to be determined. Hence, the CGH transmittance function can be expressed as the following form in terms of Jacobi-Anger identity [44]:

_{n}*J*is the first-kind of Bessel function of order

_{m}*m*. Suppose that the first-order series (

*m*= 1) is our desired electric field. Therefore, the encoding condition is fulfilled if the function

*f*(

*A*) is inverted from the relation

_{n}*cA*=

_{n }*J*

_{1}[

*f*(

*A*)] where

_{n}*c*is a constant. The

*f*(

*A*) can be solved numerically. Figure 10(b) and 10(e) illustrate the phase holograms

_{n}*Ψ*(

*A*,

_{n}*ϕ*) for the CAPB and the CAB by numerically solving

*f*(

*A*), respectively. However, as shown in Eq. (10), the transmittance will accompany many unwanted orders (

_{n}*m*≠1) except the desired first-order. In order to spatially separate the first-order from other orders, the phase hologram should be modified to the form as

*Ψ*(

*A*,

_{n}*ϕ*+2

*πG*2

_{x}x+*πG*) where

_{y}y*G*and

_{x}*G*are the spatial frequencies along

_{y}*x-*and

*y-*directions, respectively. This modification is to add the tilt phase exp(

*i2πmG*) on the mode order

_{x}x + i2πmG_{y}y*m*shown in Eq. (10). As a result, the modulated light containing different orders propagation from the SLM to the L

_{1}is spatially isolated. We want to emphasize here that besides the different diffraction orders produced by the hologram, The SLM itself will produce intrinsic diffraction orders owing to that there are small areas between two SLM’s adjacent pixels that can’t modulate the phase. As a consequence, the SLM acts as two-dimensional phase grating that produces diffraction orders. In order to efficiently separate the desired order from other unwanted orders, the spatial frequency

*G*and

_{x}*G*should be optimized and is chosen to be 42 mm

_{y}^{-1}and 74 mm

^{-1}, respectively, in the experiment. The modified phase holograms which are loaded on the SLM are shown in Fig. 10(c) and 10(f) for the CAPB and the CAB, respectively. Note that the color map is used to show the detail information of the modified phase hologram owing to that the spatial frequency is too high to display the detail information in gray picture. It is worth mentioning that owing to the high oscillation of the phase hologram, the sampling points in the outer part (

*r*> 3 mm) may be not enough to represent the theoretical functions, resulting in insufficient accuracy.

The modulated light reflecting from the SLM is reflected by the BS. A Fourier Lens L_{1} with the focal length *f*_{1 }= 50 cm is used to perform the Fourier transform of the modulated light in the SLM plane. In fact, the unwanted diffraction orders are separated spatially in front of L_{1}. Hence, a circular aperture (CA) is placed before the L_{1} to block other unwanted diffraction light generated by the hologram and the SLM. Both the distances from the SLM to L_{1} and from L_{1} to the Fourier plane/source plane are 50 cm. Hence, the generated CAPB/CAB source is just located at the source plane (denoted in Fig. 9). A beam profile analysis (BPA, Spiricon, SP928) is mounted on the *z*-axis translation stage to measure the intensity distribution and trace the maximum intensity at different propagation distances after the source plane.

Figures 11 and 12 show the experimental results of the two-dimensional intensity distribution of the CAPB and the CAB at several propagation distances. In the source plane, the beam profile of the CAPB consists of many thin concentric rings. From the second ring, the intensity of the rings gradually decays. The results agree well with the theoretical predictions, while the intensity on a single ring is slightly nonuniform and there are some speckle noise in the background. The beam profile of the CAB also consists of a series of thin concentric rings, but the decaying speed of the ring’s intensity is much faster than that of the CAPB. As the propagation distance increases, the number of rings gradually diminishes and a very bright spot on the beam propagation axis suddenly appears [see Figs. 11(d) and Fig. 12(d)]. The measured *K*-value at the focus position (about *z *= 0.78 m) is 323.5 for the CAPB and 46.8 for the CAB. Compare with the CAB, the CAPB indeed shows better autofocusing characteristics. Nevertheless, the *K*-value is slightly small compared to theoretical calculation which is 356.94 for the CAPB and 52.19 for the CAB. The reason may be caused by the non-uniform intensity distribution on concentric ring, which deviates the phase match condition at *z *= 0.78 m. In addition, other reasons such as inaccurate modulation of the weak intensity of the very outside rings, non-uniform intensity or phase distribution of the incident light on the SLM and the background noise also will affect the measured *K*-value.

Figure 13 represents the experimental results (circular dots) of the relative on-axis intensity against the propagation distance. For convenience of comparison, the corresponding theoretical results (solid curves) are also included. The relative on-axis intensity of the CAPB and the CAB remains around to be zero before the focus point, but it increases rapidly at a little distance before the focus point. After the focus point, moreover, the relative on-axis intensity of these two beams declines quickly. Within the allowable range of experimental error, the variation of the measured relative on-axis intensity against the propagation distance is roughly consistent with that of the theoretical relative on-axis intensity against the propagation distance.

In the above example, the focus position *z *= 0.78 m is relatively long. Some applications need a longer focus position, but some applications may also need a shorter focus position. Fortunately, the focus position can be shortened by decreasing *w*_{0}, and the corresponding abruptly autofocusing ability is strengthened, which is shown in Fig. 14. *w*_{0 }= 0.08 mm and other parameters are same as those in Figs. 11 and 12. When *w*_{0 }= 0.08 mm, the focus position is shortened from 0.78 m to 0.55 m, and the measured *K*-value is 331.8 for the CAPB and 36.14 for the CAB.

## 6. Summary

A beam family of the generalized CADBs which possesses the abruptly autofocusing ability is introduced. The effects of three beam parameters *r*_{0,} *a* and *w*_{0} on the relative on-axis intensity of the generalized CADBs are investigated, respectively. With the increase of *r*_{0}, the abruptly autofocusing ability of the generalized CADBs is enhanced and the focus position is lengthened. With the increase of *a*, the abruptly autofocusing ability of the generalized CADBs weakens, while the focus position has little response. With the decrease of *w*_{0}, the focus position shortens, and the abruptly autofocusing ability of CADB with *n*≥1 is strengthened, while the abruptly autofocusing ability of CAB is attenuated. With the increase of *n*, the abruptly autofocusing ability of the generalized CADBs is remarkably enhanced, while the focus position nearly keeps unvaried. As the generalized CADBs with *n*≥1 have the similarly abruptly autofocusing performance, the CAPB is selected as a representative of the generalized CADBs with *n*≥1 to be compared with the CAB. The abruptly autofocusing of the CAPB is theoretically and experimentally investigated. The propagation dynamic of the CAPB in free space is exhibited and compared with that of the corresponding CAB. The physical mechanism of the abruptly autofocusing of the CAPB and the CAB is interpreted by mimicking the FZP lens. The abruptly autofocusing ability of the CAPB is theoretically about 7 times of that of the CAB with the same beam parameters. Under the condition of same beam parameters, the *K*-value of the CAPB is theoretically up to 356.94 and experimentally achieves 323.5, while the *K*-value of the corresponding CAB is theoretically 52.19 and experimentally reaches 46.8. Moreover, the CAPB have narrower FWHM in the focus position than the CAB, and the focal spot size of the CAPB is smaller than that of the CAB. Owing to the imperfection generation of the CAPB, there is about 9.4 percentage error between the theoretical and the measured *K*-values. Due to the significantly enhanced abruptly autofocusing ability, the generalized CADBs could be beneficial for applications in many fields such as biomedical treatment where the effect of intended area is required while leaving any preceding tissue intact, optical manipulation where “igniting” a particular nonlinear process is needed, and microscopic imaging where the deeper penetration and the smaller focus spot are required.

## Funding

National Natural Science Foundation of China (11974313, 11874046).

## 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.

## References

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