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Abstract
The reported autofocusing ability of a ring Airyprime beam array reaches up to 8632.40, while the strongest autofocusing ability of a circular Airyprime beam (CAPB) is only 1822.49. How can the autofocusing ability of a single beam reach the autofocusing ability of a beam array? To achieve this goal, a circularly transformed Airyprime beam (CTAPB) is introduced by following two steps. First, a circular equation transformation on the two transverse coordinates in the electric field expression of a propagating Airyprime beam is performed. Then, the electric field expression of a propagating Airyprime beam is integrated over the angle. The intensity profile of a CTAPB on the initial plane changes significantly with varying the primary ring radius r0. With increasing r0, therefore, the autofocusing ability of a CTAPB undergoes a process of first increasing and then decreasing, while the focal length always increases. A CTAPB exhibits more powerful autofocusing ability than a CAPB. The maximum autofocusing ability of a CTAPB can reach up to 8634.76, which is 4.74 times that of a CAPB, while the corresponding focal length is 95.11% of a CAPB. A CTAPB on the initial plane can be approximately characterized by a ring Airyprime beam array with sufficient number of Airyprime beams. Due to the better symmetry, a CTAPB has a slightly stronger autofocusing ability than a ring Airyprime beam array and almost the same focal length as a ring Airyprime beam array. The CTAPB is also experimentally generated, and the experimental results indicate that the CTAPB has powerful autofocusing ability. As a replacement of a CAPB and a ring Airyprime beam array, this introduced CTAPB can be applied to the scenes which involve abruptly autofocusing effect.
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1. Introduction
One of the most remarkable characteristics of an Airy beam is lateral self-acceleration [1,2]. When several Airy beams form an Airy beam array, abruptly autofocusing phenomenon occurs due to mutual acceleration of evenly displaced Airy beams towards opposite directions [3–7]. Abruptly autofocusing, as the name suggests, is the automatic focusing of a laser beam without any external optical components. The characteristic of abruptly autofocusing is that a low intensity profile is maintained before and after the focal point, while the focal light intensity is suddenly increased by tens or even hundreds of times. Therefore, the researchers use the ratio of focal light intensity to initial peak light intensity to describe the autofocusing ability of autofocusing beams. Even in atmospheric turbulence [4] and self-focusing medium [5], an Airy beam array still retains abruptly autofocusing property. A circular Airy beam (CAB), which is the form of an Airy beam under the cylindrical symmetry, also exhibits abruptly autofocusing behavior [8–11]. In order to achieve strong abruptly autofocusing ability, researchers have made various improvements of CABs including modification of CABs [12–14], obstruction of CABs [15], proposal of partially coherent CABs [16,17], insertion of optical vortex into CABs [18] and introduction of chirp into CABs [19]. Due to the ability to resist atmospheric turbulence effect, abruptly autofocusing beams can be used in free space optical communication systems [20,21]. Abruptly autofocusing beams can also be applied in medical treatment [8] and other scenarios [22–33].
In order to seek further expansion of the Airy beam, we have successfully constructed an Airyprime beam model in theory and experimentally generated Airyprime beams [34,35]. Moreover, the Airyprime beam satisfies the wave equation under the paraxial condition. Airyprime beam arrays which are consisted of several Airyprime beams demonstrate superior autofocusing ability [36–38]. The reported autofocusing ability of a ring Airyprim beam array reaches up to 8632.40 [39]. However, achieving such powerful autofocusing ability requires 602 Airyprime beams in a beam array. The circular Airyprime beam (CAPB) exhibits significantly stronger autofocusing ability than a CAB [40]. As a single beam, a CAPB can achieve the strongest autofocusing ability of 1822.49 [41], however it is still very limited, and there is a significant gap between it and the maximum automatic focusing ability of an Airyprime beam array. Can the abruptly autofocusing ability of a single beam be significantly improved and achieve the autofocusing ability of a beam array? Based on the coordinate transformation and angle integration of an Airyprime beam, a novel autofocusing beam is introduced in this paper. As a single beam, the autofocusing ability of this introduced autofocusing beam can reach up to 8634.76, which is slightly stronger than that of a ring Airyprime beam array, while the focal length remains basically unchanged. Thus, it has become a reality that the autofocusing ability of a single beam can achieve that of a beam array.
2. Description of a circularly transformed Airyprime beam
A Cartesian coordinate system is established. x and y are transverse coordinates, and z is longitudinal coordinate. The plane z = 0 is defined as the initial plane. Laser beams propagate from the initial plane towards the positive direction of the z-axis. The electric field of an Airyprime beam on the plane z = 0 of this Cartesian coordinate system reads as:
A CAPB is the form of an Airyprime beam under the cylindrical symmetry. The cylindrical coordinate system is preferred for a CAPB. The initial electric field of a CAPB in the cylindrical coordinate system is described by
Based on Eq. (2), we introduce a circularly transformed Airyprime beam (CTAPB) which possesses very powerful autofocusing ability. First, we perform the following coordinate transformation:
Equation (5) can also be expressed as:
Equation (6) is a standard expression of a circle with the center coordinates (${r_0}/\sqrt 2$, ${r_0}/\sqrt 2$) and the radius r. Then, we perform the integral over the angle θ0:
Equation (7) cannot be expressed as an analytical expression. The laser beam described by Eq. (7) is defined as a CTAPB. According to Eq. (7), one can obtain the initial electric field of a CTAPB:
Because of the symmetry of the initial electric field distribution of a CTAPB, its focus must be on the axis when propagating in free space. By setting r = 0 in Eq. (7), the on-axis electric field of a propagating CTAPB has the following concise analytical form:
The light intensity of a propagating CTAPB is given by I(r, z) = |E(r, z)|2. Here, we still use Ifp/I0p to describe the autofocusing ability of a propagating CTAPB where Ifp is the foal light intensity. By drawing a curve of the on-axis light intensity |E(0, z)|2 relative to z, the point where the maximum value of the on-axis light intensity occurs is the focal point, and the maximum value of the on-axis light intensity is the focal light intensity Ifp. The plane where the focal point is located is the focal plane, and the on-axis distance of the focal point is the focal length zf.
3. Theoretical calculations and analyses
Figure 1 represent radial intensity distributions of CTAPBs with a = 0.05, w0 = 0.1 mm and different r0 on the initial plane. For ease of comparison, the initial intensity distribution of a CAPB is also demonstrated on Fig. 1(f). When r0 changes, the initial intensity distribution of a CAPB only shifts radially (it is not shown here). However, the initial intensity distribution of a CTAPB is closely related to r0. When r0 first increases, the light intensity of inner rings of a CTAPB gradually increases. When r0 = 3.20 mm, the two peak light intensities of inner and outer rings are both equal to 1. When r0 further increases from 3.20 mm, the light intensity of outer rings of a CTAPB gradually decreases. Of course, the hollow area of a CTAPB also increases radially with increasing r0.
Fig. 1. Radial intensity distributions of CTAPBs and CAPB with a = 0.05 and w0 = 0.1 mm on the initial plane: (a) A CTAPB with r0 = 1.50 mm; (b) A CTAPB with r0 = 2.50 mm; (c) A CTAPB with r0 = 3.20 mm; (d) A CTAPB with r0 = 3.31 mm; (e) A CTAPB with r0 = 4.00 mm; (f) A CAPB with r0 = 1.50 mm.
Fig. 2. Abruptly autofocusing dynamic of a CTAPB with a = 0.05, w0 = 0.1 mm and r0 = 1.50 mm upon propagation in free space. (a) z = 0.200 m; (b) z = 0.400 m; (c) z = 0.600 m; (d) z = zf = 0.802 m; (e) z = 1.000 m; (f) z = 1.166 m.
Abruptly autofocusing dynamics of a CTAPB with a = 0.05, w0 = 0.1 mm and different r0 upon propagation in free space are shown in Figs. 2 and 3. r0 = 1.50 mm in Fig. 2, and r0 = 3.31 mm in Fig. 3. λ = 532 nm is fixed in the following calculations. Upon propagation in free space, energy fluxes flow radially from the outer rings into the inner rings, which is accompanied by the gradual extinction of the outer rings. At the same time, due to lateral self-acceleration, the first inner ring that has become the dominant peak moves towards the z-axis and ultimately reaches the z-axis, resulting in an abruptly autofocusing phenomenon. After passing the focal point, the energy fluxes flow from the on-axis focal point into the outer rings, resulting in the increase of the number of outer rings, and the corresponding peak light intensity sharply decreases. When r0 is large, the energy fluxes on the outer rings take a longer time to reach the on-axis point, resulting in a longer focal length. Accordingly, z = zf = 0.802 m for r0 = 1.50 mm, while z = zf = 1.166 m for r0 = 3.31 mm.
Fig. 3. Abruptly autofocusing dynamic of a CTAPB with a = 0.05, w0 = 0.1 mm and r0 = 3.31 mm upon propagation in free space. (a) z = 0.200 m; (b) z = 0.400 m; (c) z = 0.600 m; (d) z = 0.800 m; (e) z = 1.000 m; (f) z = zf = 1.166 m.
Figure 4(a) and 4(b) present the variation of on-axis light intensities for CTAPBs and CAPBs with a = 0.05 and w0 = 0.1 mm versus z. The exponential decay factor has a significant impact on the abruptly autofocusing ability of CTAPBs. The abruptly autofocusing ability of CTAPBs decreases with increasing the exponential decay factor. Due to the expected strong autofocusin ability of CTAPBs, the exponential decay factor is chosen as 0.05. The on-axis light intensity of a CTAPB after passing through the focal point drops faster than that of a CAPB. Moreover, the on-axis light intensity of a CTAPB decreases to zero after several oscillations. Since r0 significantly affects the intensity profile of a CTAPB on the initial plane, Fig. 4(c) and 4(d) show the curves of Ifp/I0p and zf of a CTAPB versus r0. Also, the curves of Ifp/I0p and zf of a CAPB versus r0 are included in Fig. 4(c) and 4(d). The dashed lines are added for ease of explanation. With increasing r0, the automatic focusing ability of a CTAPB undergoes a process of first increasing and then decreasing, and the focal length is monotonically prolonged. If r0 = 3.31 mm is selected, the CTAPB achieves the strongest autofocusing ability of 8634.76, and the plane zf = 1.166 m is the focal plane. However, the maximum autofocusing ability of a CAPB is 1822.49, which is realized under the condition of r0 = 2.60 mm, and one has zf = 1.226 m [41]. Therefore, the maximum autofocusing ability of a CTAPB is 4.74 times that of a CAPB, while its focal length is 95.11% of a CAPB. Only when r0 ≤ 0.08 mm, a CTAPB exhibits weaker autofocusing ability than a CAPB. Moreover, a CTAPB always has a shorter focal length than a CAPB. The maximum autofocusing ability of a ring Airyprime beam array with a = 0.05 and w0 = 0.1 mm is 8632.40, which is accompanied by zf = 1.168 m [39]. Therefore, a CTAPB has a slightly stronger maximum automatic focusing ability than a ring Airyprime beam array and almost the same focal length as a ring Airyprime beam array. The powerful autofocusing ability of a CTAPB can be interpreted as follows. The integral symbol in Eq. (8) can be approximately replaced by a summation symbol. Thus, Eq. (8) can be approximately expressed as:
Fig. 4. (a) and (b) I(0, z) of CTAPBs and CAPBs with a = 0.05 and w0 = 0.1 mm as a function of z; (c) Ifp/I0p of CTAPB and CAPB as a function of r0; (d) zf of of CTAPB and CAPB as a function of r0.
The initial and the focal intensity profiles of CTAPBs with a = 0.05, w0 = 0.1 mm and different r0 are demonstrated in Fig. 5. When r0 = 2.50 mm, there is a uniform ring with the light intensity close to 1 in the intensity profile of the CTAPB on the initial plane. When r0 increases to 3.20 mm, the uniform ring with the light intensity close to 1 becomes wider. When r0 = 4.00 mm, the uniform ring with the light intensity close to 1 becomes narrower. The focal lengths of the CTAPBs with r0 = 2.50 mm, 3.20 mm and 4.00 mm are respectively 1.019 m, 1.147 m and 1.278 m, and the corresponding abruptly autofocusing abilities are 5243.89, 8601.84 and 7881.90, respectively. The focal spots of the CTAPBs with r0 = 2.50 mm, 3.20 mm and 4.00 mm all contain one bright spot and two outer rings. The initial and the focal intensity distributions of a CAPB with a = 0.05, w0 = 0.1 mm and r0 = 2.60 mm are also included in Fig. 5. The uniform ring with the light intensity close to 1 in the initial intensity profile of the CAPB is the narrowest among the first row. The comparison between focal spots in the second row of Fig. 5 denotes that only the CAPB with r0 = 2.60 mm has one outer ring. Although the autofocusing ability of the CAPB shown in Fig. 5(h) is the highest, it is much smaller than that of CTAPBs presented in Fig. 5(e)-(g).
Fig. 5. Intensity distributions of CTAPBs and CAPB with a = 0.05 and w0 = 0.1 mm on the initial (the top row) and the focal planes (the underneath row): (a) and (e) A CTAPB with r0 = 2.50 mm; (b) and (f) A CTAPB with r0 = 3.20 mm; (c) and (g) A CTAPB with r0 = 4.00 mm; (d) and (h) A CAPB with r0 = 2.60 mm.
Figure 6 demonstrates the evolutionary process of intensity profile for the CTAPB with a = 0.05, w0 = 0.1 mm and r0 = 1.50 mm during free space propagation. The theoretical peak light intensities on the observation planes z = 0.200 m, 0.400 m, 0.500 m and 0.600 m are 0.78, 1.01, 2.37 and 5.61, respectively. Before autofocusing, the peak light intensity is maintained at a low level, and even less than 1 when just leaving the initial plane. The theoretical peak light intensities on the observation planes z = 1.000 m and 1.166 m are 33.52 and 2.19, respectively. After the autofocusing phenomenon ends, the peak light intensity sharply decreases, which is accompanied by a gradual increase in the beam spot size.
Fig. 6. Intensity distributions of a CTAPB with a = 0.05, w0 = 0.1 mm and r0 = 1.50 mm on our selected planes: (a) z = 0; (b) z = 0.200 m; (c) z = 0.400 m; (d) z = 0.500 m; (e) z = 0.600 m; (f) z = zf = 0.802 m; (g) z = 1.000 m; (h) z = 1.166 m.
The evolutionary process of intensity distribution for the CTAPB with a = 0.05, w0 = 0.1 mm and r0 = 3.31 mm during free space propagation is shown in Fig. 7. The theoretical peak light intensities on the observation planes z = 0.200 m, 0.400 m, 0.500 m, 0.600 m, 0.800 m and 1.000 m are 0.97, 1.29, 2.39, 3.99, 9.38 and 24.98, respectively. The top rows of Figs. 6 and 7 denote that when two CTAPBs first leave the initial plane and begin to propagate, the peak light intensity of the CTAPB with r0 = 3.31 mm is greater than that of the CTAPB with r0 = 1.50 mm. When arriving at the observation plane z = 0.600 m, the peak light intensity of the CTAPB with r0 = 3.31 mm is smaller than that of the CTAPB with r0 = 1.50 mm. The peak light intensity is rapidly increased from 24.98 on the observation plane z = 1.000 m to 8634.76 on the focal plane z = 1.166 m, which reflects the meaning of “abruptly” very well. Figure 7 manifests that the size of the intensity profile undergoes a process of first increasing and then decreasing before abruptly autofocusing. The number of focal outer rings of the CTAPB is mainly affected by the primary ring radius r0. As the primary ring radius r0 increases, the number of focal outer rings increases from one to two, which is shown in Figs. 6(f) and 7(f).
Fig. 7. Intensity distributions of a CTAPB with a = 0.05, w0 = 0.1 mm and r0 = 3.31 mm on our selected planes: (a) z = 0; (b) z = 0.200 m; (c) z = 0.400 m; (d) z = 0.500 m; (e) z = 0.600 m; (f) z = 0.800 m; (g) z = 1.000 m; (h) z = zf = 1.166 m.
4. Experimental generation and measurement
In this section, the experimental generation of a CTAPB is realized, and the intensity profile of a CTAPB is recorded, which is used to evaluate the autofocusing ability. Figure 8 shows a schematic diagram of the experimental setup we used. The laser (Laser Quantum, model ventus) generates a seed beam namely the fundamental mode Gaussian beam. The wavelength of the seed beam is 532 nm. To make the optical path compact, a reflective mirror (RM) is used to change the propagation direction of the seed beam by 90 degrees. Secondly, we use a beam expander (BE) with a magnification of 50 times to expand the seed beam. Thirdly, a beam splitter (BS) with an intensity of 50 : 50 is employed to split the seed beam, so that half of the seed beam is incident into the phase-only spatial light modulator (SLM). The model of the SLM is Holoeye LETO-3, whose pixel size is 6.4 µm × 6.4 µm. On the basis of the method of phase plate synthesis [45], the modulation program has been upload onto the SLM. Here the SLM works at a reflective mode and modulates the seed beam into beams of various diffraction orders. The reflected beam from the SLM first propagates through the BS again and then passes through a circular aperture (CA). The CA acts as a filter, and only the first diffraction order can fortunately pass through the CA. Based on the SLM, the complex amplitude modulation technology is used to generate CTAPBs. The light intensity of the zero diffraction order is too high to be directly collected. On the contrary, the light intensity of the first diffraction is moderate and can be directly recorded. Therefore, the first diffraction order is used to generate CTAPBs. Subsequently, a Fourier transformation transforms the first diffraction order into the CTAPB by using a lens (L). Accordingly, the CTAPB represented by Eq. (8) is obtained in the real focal plane of L. By driving the beam profile analyzer (BPA) along the linear guide, the intensity profile of the CTAPB on the selected plane can be recorded. The model of the BPA is BGS-USB3-LT665, whose pixel pitch is 4.4 µm.
Figure 9 presents the phase hologram without linear phase loaded into the SLM to generate CTAPBs with a = 0.05 and w0 = 0.1 mm. r0 = 1.50 mm, 2.50 mm, 3.31 mm and 4.00 mm in Fig. 9, respectively. The relationship between the phase modulation and the gray scale in Fig. 9 is φ = 2πG/255 where G is the gray value. Figure 10 represents the initial and the focal intensity distributions of CTAPBs and CAPB that we recorded. The beam parameters are set to be a = 0.05 and w0 = 0.1 mm. Due to the high intensity of the zero diffraction order, the disturbance of zero diffraction order will result in the non-uniform distribution of experimental intensity profile of a CTAPB on and near the initial plane. The measured results of the autofocusing ability of the CTAPBs with r0 = 2.50 mm, 3.20 mm and 4.00 mm are 4883.11, 7801.79 and 7095.84, and the corresponding percentage deviations between the experimental and the theoretical autofocusing ability are 6.88%, 9.30% and 8.45%. Due to the relatively high peak light intensity, the quality of the experimental focal spot of the CTAPB is worse than that of the CAPB. The quality of the ring in the experimental focal spot of the CTAPB depends on a and r0. When a is small such as a = 0.05, due to the deceleration of decay, it leads to uneven energy distribution of the laser beam into the outer rings of the focal spot during the focusing process, resulting in additional optical distortion. An increase in r0 will lead to a deterioration in the quality of the ring, which is because that the increase of r0 means an improvement in the resolution of our experimental system.
Fig. 9. Phase hologram without linear phase of CTAPBs with a = 0.05 and w0= 0.1 mm: (a) A CTAPB with r0 = 1.50 mm; (b) A CTAPB with r0 = 2.50 mm; (c) A CTAPB with r0 = 3.31 mm; (d) A CTAPB with r0 = 4.00 mm.
Fig. 10. Recorded intensity profiles of CTAPBs and CAPB with a = 0.05 and w0 = 0.1 mm on the initial (the top row) and the focal planes (the underneath row): (a) and (e) A CTAPB with r0 = 2.50 mm; (b) and (f) A CTAPB with r0 = 3.20 mm; (c) and (g) A CTAPB with r0 = 4.00 mm; (d) and (h) A CAPB with r0 = 2.60 mm.
Experimental intensity distributions of CTAPBs with a = 0.05, w0 = 0.1 mm and different r0 on different observation planes of free space are displayed in Figs. 11 and 12. The measured autofocusing abilities of the CTAPBs with r0 = 1.50 mm and 3.31 mm are 1907.61 and 7829.86, and the corresponding percentage deviations between the experimental and the theoretical autofocusing ability are 3.51% and 9.32%. Due to the maximum size of BPA being 28 mm, the experimental measurement range of Fig. 12(d) and 12(e) is slightly smaller than the corresponding theoretical calculation range. Fortunately, only the outermost part of the light intensity distribution in Fig. 12(d) is incomplete, which does not affect the comparison between the experimental results and the theoretical calculations. During free space propagation, the influence of the zero diffraction order on the CTAPB is enhanced with the increase of the distance propagation z, leading to gradual elliptical distortion in the recorded intensity profile such as Figs. 11(g) and 12(g). Moreover, the elliptical distortion becomes more severe as the primary ring radius increases. Due to the larger primary ring radius in Fig. 12, the first diffraction order produced by the SLM is closer to the zero diffraction order and is more severely affected by the zero diffraction order, resulting in more severe elliptical distortion in Fig. 12(g). Figures 10–12 corresponds to Figs. 5–7, respectively. In a word, therefore, the experimental intensity profiles of CTAPBs coincide with the theoretical ones. Our experiments confirm that a CTAPB has a powerful autofocusing ability. Because smaller pixel sizes can provide higher spatial resolution, resulting in finer and clearer generated holograms. Therefore, choosing Holoeye GAEA-VIS in the experiment can improve the quality and the accuracy of the experiment. However, using Holoeye LETO-3 does not affect the judgment of the experiment.
Fig. 11. Recorded intensity profiles of a CTAPB with a = 0.05, w0 = 0.1 mm and r0 = 1.50 mm on our selected planes: (a) z = 0; (b) z = 0.200 m; (c) z = 0.400 m; (d) z = 0.500 m; (e) z = 0.600 m; (f) z = zf = 0.802 m; (g) z = 1.000 m; (h) z = 1.166 m.
Fig. 12. Recorded intensity profiles of a CTAPB with a = 0.05, w0 = 0.1 mm and r0 = 3.31 mm on our selected planes: (a) z = 0; (b) z = 0.200 m; (c) z = 0.400 m; (d) z = 0.500 m; (e) z = 0.600 m; (f) z = 0.800 m; (g) z = 1.000 m; (h) z = zf = 1.166 m.
5. Summary
In a conclusion, a CTAPB is introduced and obtained by two following steps. First, a circular equation transformation on the x- and y-coordinates in the electric field expression of a propagating Airyprime beam in free space is performed. Then, the integral over the angle is performed on the electric field expression of a propagating Airyprime beam after coordinate transformation. The intensity profile of a CTAPB on the initial plane changes significantly with the change of r0, which is completely different from that of a CAPB. With increasing r0, therefore, the autofocusing ability for a CTAPB undergoes a process of first increasing and then decreasing, and the focal length is monotonically extended. A CTAPB exhibits more powerful abruptly autofocusing ability than a CAPB. Under the conditions of a = 0.05 and w0 = 0.1 mm, the maximum autofocusing ability of a CTAPB can reach up to 8634.76, which is 4.74 times that of a CAPB, while the focal length is only reduced by 4.89% compared to a CAPB. A CTAPB can be approximately characterized by a ring Airyprime beam array with sufficient number of Airyprime beams. Due to the better symmetry, moreover, a CTAPB has a slightly stronger autofocusing ability than a ring Airyprime beam array and almost the same focal length as a ring Airyprime beam array. Therefore, this research makes it a reality that the autofocusing ability of a single beam can achieve or even exceed the autofocusing ability of a beam array. This CTAPB is also experimentally generated, and the experimental intensity profiles of CTAPBs are well consistent with the theoretical predictions. As a replacement of a CAPB and a ring Airyprime beam array, this introduced CTAPB can be applied to the scenes involving abruptly autofocusing effect. Because of its stronger autofocusing ability, a CTAPB has more advantages than a CAPB in resisting atmospheric turbulence. Therefore, a CTAPB can be used in free space optical communication system to further improve communication quality.
Funding
National Natural Science Foundation of China (12374281,11874046).
Disclosures
The authors declare no conflicts of interest.
Data availability
No data were generated or analyzed in the presented research.
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