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We have proposed a kind of modified circular Airy beam (MCAB) based upon a modification of the Fourier spectrum of circular Airy beams (CAB) in this paper. Unlike most abruptly autofocusing beams, the position of peak intensity of MCAB can be moved to any rings behind. Two apodization parameters are introduced to describe the propagation characteristics of MCAB. It is found that the focal position, focal trajectory and the size of focal spot do not change with the apodization parameters; but the abruptly autofocusing property will be greatly enhanced if appropriately apodization parameters are chosen. Comparing with the common CAB and the previous blocked CAB, the MCAB shows better abruptly autofocusing property. It may have more applications in various fields.
© 2015 Optical Society of America
1. Introduction
Recently, the circular Airy beam (CAB) has attracted great attentions of researchers due to its unique abruptly autofocusing property [1–6]. Without invoking any lenses or nonlinearities, the CAB can abruptly focus its energy right at the focal region while maintaining a low intensity profile before [1]. The abruptly autofocusing property makes the CAB an ideal candidate in biomedical treatment or laser ablation. Besides, the CAB can also be used in optical micromanipulation [7, 8], generating light bullet [9] and other fields [10].
The control of the abruptly autofocusing property of the CAB is of great importance in practice. Many methods have been proposed to enhance its intensity contrast or to change the focal pattern, such as adding different optical vortices [11–13], blocking front light rings [14] and so on. Meanwhile, many other abruptly autofocusing beams with special abruptly autofocusing properties have also been proposed [15–17].
The Fourier transform plays a key role in producing and modulating special laser beams. By the modification of Fourier spectrum, light beams with special property can be obtained. For example, the reduced-side lobe Airy beams [18], the curved Bessel beam [19] and other special beams, are all generated through the modulation of Fourier transform of corresponding beams.
In this paper, we propose a kind of modified circular Airy beams (MCAB) based upon the Fourier transform of the CAB. An apodization mask is added in the Fourier space to construct the theoretical model of the MCAB. The propagation characteristics of the MCAB with different apodization parameters are numerically calculated. The abruptly autofocusing property of the MCAB is theoretically analyzed and compared with the common CAB and blocked CAB (BCAB) [14]. Some useful and interesting results are found in our investigations.
2. Fourier transform of the MCAB
The electric fields of CAB at the input plane can be expressed as [1]
where Ai is the Airy function, r is the radial distance, w is the scaled parameter, a is the decay parameter, r0 is related with the initial radius of the CAB, C0 is a constant. The Fourier transform of the famous Airy beam is simply the Gaussian beam with cubic phase [20], yet the Fourier transform for circular Airy beams is not analytical. The closed form approximation of the Fourier transform of the CAB can be expressed as [3]:where k is the radial spatial frequency. Equation (2) shows the relation between the CAB and the chirped Bessel function. It also offers an effective method to produce CAB [2, 3, 9]. Thedistribution of the Fourier spectrum of CAB is shown in Fig. 1. The spectrum with low frequency mainly affects front light rings of CAB, and the spectrum with high frequency mainly affects the rings behind. As is demonstrated in [14], front rings of the CAB contribute less to the peak intensity of the focal plane, and the abruptly autofocusing property would be greatly enhanced if front rings of CAB are blocked. So it can be expected that the abruptly autofocusing property can also be enhanced by reducing the spatial spectrum with low frequency in Fourier space. Comparing with the blocking method in real space, the modification in Fourier space would produce fewer unwanted diffraction artifacts in the region of interest.
Fig. 1 Fourier transform of the modified circular Airy beam (MCAB). (a) Fourier spectrum with different β when kc = 5mm−1;(b) Fourier spectrum with different kc when β = 0.3mm. “β=0” is actually the common CAB.
We can use an apodization mask M to reduce the low-frequency component [8]:
where β controls the steepness of the apodization, and kc is the cutoff position of the spatial frequency. So the modified Fourier spectrum can be expressed as:The influences of β and kc on the Fourier spectrum of CAB are shown in Fig. 1. In our simulation, we choose r0 = 1mm, w = 10μm, a = 0.1, the wavelength λ = 1064nm, and the light power after the apodization mask is 10mW throughout this paper. It is found that the central peak of the spectrum reduces more quickly for a larger β [Fig. 1(a)], and more front peaks will reduce for a larger kc [Fig. 1(b)].
3. Propagation characteristics with different apodization parameters
The evolution of the MCAB can be obtained through the Hankel transform of UM(k):
when z = 0, Eq. (5) is the integral expression of MCAB at the initial plane. Based upon Eq. (5), we can analyze the influences of β and kc on the propagation characteristics of MCAB. Figure 2 shows the intensity patterns of MCAB with different β and kc at the initial plane when the total energy is same. Both CAB and MCAB are composed of a series of light rings. The position of each light ring remains unchanged with different β and kc, but the energies of the rings will be redistributed.Fig. 2 Intensity patterns of MCAB at the initial plane. The first row is MCAB with different β when kc = 5mm−1: (a) β = 0; (b) β = 0.3mm; (c) β = 0.6mm. The second row is MCAB with different kc when β = 0.3mm: (d) kc = 5mm−1; (e) kc = 10mm−1; (f) kc = 17mm−1.
The maximum intensity of the MCAB is smaller than that of the CAB at the initial plane, as is shown in Fig. 3. For the MCAB, a small peak will appear before the first ring because of the diffraction effect from the apodization. By increasing the value of kc, the position of the maximum intensity will be moved to the light rings behind. The second ring and the third ring become the highest ring when kc = 10mm−1, and kc = 17mm−1, respectively [Fig. 3(b)]. So theouter rings of MCAB become much more important than that of CAB.The value of β also influences the intensity distribution of the MCAB. The ratio between the highest ring and other rings can be modulated by changing the value of β.When β = 0, the MCAB becomes the common CAB; when β is too large, the diffraction effect from apodization will be very obvious.
Fig. 3 Intensity distributions of MCAB at the initial plane. (a) MCAB with different β when kc = 5mm−1; (b) MCAB with different kc when β = 0.3mm.
Based upon Eq. (5), we can numerically calculate the propagation characteristics of the MCAB. The simulation results are shown in Fig. 4. From Fig. 4, we can see that both CAB and MCAB focus at a same point. The focal position is z = 0.277m. We can define the radius of the first ring as the radius of MCAB. It is found that the radii of MCAB almost follow a same parabolic trajectory with different apodization parameters, as can be seen in Fig. 5. Although the focal position and the focal trajectory are not affected by the apodization, yet the distribution of focal intensity changes with the apodization parameters. Figure 6(a) and 6(b) show that the maximum focal intensity of MCAB is larger than that of the common CAB, and the maximum focal intensity increases with kc. The case of β is a little complicated. To obtain a large focal intensity, β should not be too small or too great [Fig. 6(a)]. The reason may be that the low-frequency component cannot be effectively reduced when β is too small, but when β is too great, the unwanted diffraction effect may be enhanced at the focal plane. In Fig. 6, we can also note that the FWHM of the central focal spot of MCAB is not changed with β or kc. So the MCAB with a larger kc and an appropriate β will autofocus more strongly.
Fig. 4 Propagation dynamics of MCAB in free space. The first row is MCAB with different β when kc = 5mm−1: (a) β = 0; (b) β = 0.3mm; (c) β = 0.6mm. The second row is MCAB with different kc when β = 0.3mm: (d) kc = 5mm−1; (e) kc = 10mm−1; (f) kc = 17mm−1.
Fig. 5 Trajectories of the radii of MCAB. (a) MCAB with different β when kc = 5mm−1; (b) MCAB with different kc when β = 0.3mm.
Fig. 6 Intensity distributions of MCAB at the focal plane. (a) MCAB with different β when kc = 5mm−1; (b) MCAB with different kc when β = 0.3mm.
We assume that I0 is the maximum intensity of the initial plane, Im is the maximum intensity of arbitrary plane along propagation. So the abruptly autofocusing property can be analyzed through the intensity contrast of Im/I0 [1]. The change of Im/I0 with z is shown in Fig. 7. From Fig. 7, we can find that both the intensity of MCAB and CAB suddenly increase at z = 0.250m, and quickly reach their maximum values at the focal point.
Fig. 7 Abruptly autofocusing property of MCAB. (a) MCAB with different β when kc = 5mm−1; (b) MCAB with different kc when β = 0.3mm.
As we have mentioned before, I0 of MCAB is smaller than that of CAB at the initial plane; but Im of MCAB is larger than that of the CAB at the focal plane. These indicate that the abruptly autofocusing property would be greatly enhanced for MCAB. Figure 7 shows that the maximum value of intensity contrast during propagation, (Im /I0)max, of MCAB is much larger than that of the CAB. For the MCAB with β = 0.3mm and kc = 17mm−1, the value of (Im /I0)max is 209, but for the CAB without apodization, this value is only 47. The value of (Im/I0)max can be controlled through β and kc. From Fig. 8(a), it is found that for the MCAB with different kc, (Im/I0)max reaches the maximum value at different β. For the beam with kc = 5mm−1, the value of β that maximizes (Im/I0)max is 0.08mm; but this value will increases to 0.14mm, when kc = 17mm−1. Figure 8(b) shows that (Im/I0)max increases with kc. So we can choose an appropriate β and a larger kc to further enhance the intensity contrast of MCAB. However, kc should not be too large. It is difficult to generate MCAB with a large kc in experiment, especially with the spatial light modulation. Practically, kc must be smaller than πL/(λf), where L is the length of the hologram on the spatial light modulation, f is the focal length of the Fourier lens.
Fig. 8 Maximum value of Im/I0 that MCAB can reach during propagation with different apodization parameters: (a) change with β when kc is different; (b) change with kc when β is different.
For the blocked circular Airy beam (BCAB), the abruptly autofocusing property is enhanced by directly blocking front light rings [14]. The autofocusing property of BCAB and MCAB are both shown in Fig. 9. For the sake of comparison, we assume that the intensity of the third ring is largest for both beams [Fig. 9(a)]. For BCAB, the first two rings are eliminated at the initial plane; for MCAB, the first two rings still exist but are reduced. They are generated from a same CAB. Figure 9(b) shows that the intensity of MCAB can be increased much more times than that of BCAB, and the abruptness of focus is more obvious for MCAB. The fact that the abruptly autofocusing property of MCAB is better than BCAB indicates a general principle: it is not necessary to completely eliminate front rings of CAB to enhance its focusing property, reducing the intensity of front rings is enough. On the other hand, in experiment, it is easier to modify the Fourier spectrum of CAB than to block front rings of CAB in real space, especially when one uses the Fourier transform method to generate abruptly autofocusing beams. So MCAB will be more useful than the common CAB and BCAB in the fields of optical micromanipulation, producing light bullet and other fields.
Fig. 9 Comparison of the MCAB with BCAB. (a) Intensity profile at the initial plane; (b) distribution of Im/I0 during propagation.CAB is the original circular Airy beams; the apodization parameters for MCAB are β = 0.3mm, kc = 17mm−1.
4. Conclusions
In summary, the propagation characteristics of MCAB is theoretically investigated in this paper. The MCAB can be generated through the modification of CAB in the Fourier space. Two apodization parameters β and kc have been introduced to describe the property of MCAB. By varying the value of the apodization parameters, the front rings of CAB can be reduced, and the position of maximum intensity can be moved to any rings behind at the initial plane. The intensity of MCAB at the focal point increases with kc. But the focal position, focal trajectory and the FWHM of the central focal spot do not change with the apodization parameters. The abruptly autofocusing property of MCAB also can be controlled through β and kc. An appropriate β and a larger kc should be chosen to further enhance the autofocusing capability of MCAB. Under the same conditions, the MCAB has better abruptly autofocusing property than the CAB and the previous BCAB, which indicates that redistributing the energies of the light rings of CAB is an effective method to enhance or modulate the abruptly autofocusing property. We believe that the investigation of MCAB is helpful for the application of abruptly autofocusing beams in various fields.
Acknowledgment
This work is supported by National Nature Science Foundation of China (NSFC) (Grant No. 11504274, and Grant No. 11474254), and Zhejiang Provincial Natural Science Foundation (Grant No. LQ16A040004).
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