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We theoretically and experimentally study the propagation characteristics of the circular Airy beam (CAB) when its first few light rings are blocked. It is shown that the focus position of the blocked CAB will remain the same, and its abruptly autofocusing property will be enhanced. Since the maximum focal intensity almost remains the same when the first ring is blocked, a better result of abruptly autofocusing property can be obtained when only the first ring is eliminated. Compared with the common CAB with a same hollow region, the intensity of the blocked CAB will focus at a different position and the intensity will be increased much larger than the common CAB.
© 2014 Optical Society of America
1. Introduction
Recently, the circular Airy beam (CAB) has attracted intense interest due to its abruptly autofocusing property [1–6]. The CAB focuses its energy right before the focal point while maintaining a low intensity profile until that very point. The abruptly autofocusing property make the CAB an ideal candidate in biomedical treatment or micro engineering with lasers [1]; the CAB can also be used to improve the conventional optical tweezers, such as to enhance the attractive force or trap particles at different positions [7, 8]. Recent investigation also demonstrates that the light bullet generated from the CAB exhibits many useful features [6].
Usually, a computer generated hologram (CGH) with amplitude and phase information loaded on the spatial light modulator (SLM) can be used to generate the CAB [3, 9]; the Fourier transformation of Bessel beams [2] or the pre-engineering method [10] also offer new approaches to generate abruptly autofocusing Airy beams. The modulation of abruptly autofocusing property is of great importance in many experiments. Imposing optical vortices has been proved to be an effective method to enhance the intensity contrast at the focus plane or to change the focal pattern [11–13]. Besides, the abruptly autofocusing property of CAB can also be controlled with its cylindrical polarization state [14].
The annular aperture is always used to increase the focal depth in the annular beam, which is very useful in optical imaging [15, 16]. But its application in abruptly autofocusing beam has never been investigated. As we have known, the CAB at the initial plane is composed of lots of light rings, and these light rings will focus at the focal plane simultaneously. So the abruptly autofocusing propagation will be changed when some of the light rings are blocked by the annular aperture. In this paper, we mainly investigate the propagation characteristics of this new kind of blocked CAB in theory and experiment. Our results show that the abruptly autofocusing property can be efficiently controlled by blocking the first few light rings of CAB.
2. Theoretical analysis
The electric field of CAB at the initial plane is expressed as [1]
where r0 is a parameter related with the radius of the CAB, x0 is a radial scale coefficient, a is a decay parameter, C is a constant related with the light power. The CAB possesses abruptly autofocusing property because of the characteristics of the Airy function. As an example the intensity distribution of a CAB at the initial plane is shown in Fig. 1. The light power is 10mW, the other parameters are r0 = 1mm, a = 0.1, x0 = 0.08mm. So the constant C is 4.45 × 103, and the power of the first three rings are 5.9mW, 1.4mW and 0.7mW, respectively. As is shown in Fig. 1, the CAB is a kind of hollow beam, and is composed of several light rings at the initial plane. Just as the main lobe of the Airy beam, the radius of the first ring of CAB also follows a parabolic trajectory before focusing [3, 17]. The focal position zf can be calculated through the expressions below,where is the wavelength, which is 1064nm in this paper; R0 is the radius of the first ring of the CAB, which is 1.07mm as we can know from Fig. 1. So this CAB will autofocus atzf = 0.275m.
Fig. 1 The intensity distribution at the initial plane. The first three rings are marked in this figure. The corresponding parameters are r0 = 1mm, a = 0.1, x0 = 0.08mm.
The annular aperture can be expressed as
when rd = 0, the annular aperture becomes a circular aperture; when ru approaches infinity, it becomes a circular plate. In this paper, rd and ru are chosen as the zero points in Fig. 1, so as to allow the whole ring to pass through the aperture. The propagation of the CAB through the annular aperture can be numerically calculated by the following Hankel transform pair [18, 19]: where k is the spatial frequency.When the first few rings of CAB are blocked, the maximum intensity at the initial plane would be decreased, which may be helpful to increase the intensity contrast of the abruptly autofocusing beam. The propagation dynamics of the common CAB and the blocked CAB are shown in Fig. 2. As is shown in Fig. 2(a), the light rings of CAB follow a parabolic trajectory before focusing. But the parabolic trajectory in Fig. 2(b) and 2(c) are a little distorted, because of the diffraction of the aperture. The comparison of Figs. 2(a) and 2(b) shows that the maximum intensity at the focus plane is a little increased when the first ring is blocked. This indicates that the first ring contributes less to the focal intensity, though it contains almost 2/3of the light power. Since the initial intensity is decreased and the focal intensity almost remains unchanged, it is expected that the abruptly autofocusing property would be greatly enhanced when the first ring of CAB is blocked. But when the first two rings are blocked, the maximum intensity at the focal plane would decrease, as is shown in Fig. 2(c). Another interesting thing we can find in Fig. 2 is that, unlike the self-healing property of the Airy beam [20], the blocked Airy rings of CAB do not appear again before the focal point.
Fig. 2 (a) Propagation dynamics of common CAB; Propagation dynamics of the blocked CAB: (b) the first ring is blocked; (c) the first two rings are blocked. The corresponding parameters are the same as Fig. 1.
Figure 3 shows the abruptly autofocusing properties of the CAB when the first few light rings are exactly blocked by different circular plates. I0 is the maximum intensity at the initial plane, Im is the maximum intensity at arbitrary planes along the propagation direction. The intensity contrast of Im/I0 can be used to describe the abruptly autofocusing property of the beam. From Fig. 3(a), we can see that the focal position will not change when several rings of the CAB are blocked. The maximum value of the Im/I0 is increased by blocking first few light rings, which means that the initial intensity can be enhanced more multiples by adding a circular plate with appropriate radius. For example, when the first ring is blocked, the maximum intensity will be increased by 126 times at the focal plane, which is much larger than that of the common CAB. For the first ring blocked CAB, the intensity begins to increase at about z = 0.245m, which is very close to that of the common CAB (z = 0.256m). When more front rings are blocked, the maximum intensity will increase much more, but the position where the light intensity begins to increase will slightly move forward. An optimum result can be obtained when only the first ring is blocked, if abruptness and intensity contrast of the autofocusing are both taken into account. For a different value of a, the abruptly autofocusing property can also be enhanced by using the blocking methods, as is shown in Fig. 3(b), where a = 0.08. A best result can also be obtained when the first ring is blocked. So the method of blocking the first ring can be considered a universal method to enhance the abruptly autofocusing property of CAB.
Fig. 3 Abruptly autofocusing property of the blocked CAB: (a) a = 0.1; (b) a = 0.08. “0” stands for the common beam; “1” stands for the first ring blocked CAB; “2” stands for the first two rings blocked CAB, so does “3”. Other parameters are the same as Fig. 1.
Although the blocking method has great influence on the abruptly autofocusing property, yet it has little influence on the transverse focal width and the axial focal depth. Figure 4 shows the distributions of transverse focal intensity and axial intensity. When a = 0.1, the transverse focal widths for the CAB and blocked CAB are all about 0.05mm, and the axial focal depths are all about 0.02m.
Fig. 4 The distributions of transverse focal intensity and axial intensity along the beam axis, when a = 0.1. Other parameters are the same as Fig. 1.
3. Experimental results
Usually, the blocked CAB can be generated with a circular plate. But in experiment it is difficult to generate the circular plate with appropriate size, so we employ the hologram method [3, 9] to generate the common CAB and the blocked CAB. According to the translation property of the Fourier Transform, we added a linear phase in the hologram to separate different diffraction orders. The experiment setup is shown in Fig. 5. The hologram is loaded on a reflection SLM (BNS, P512-1064-PCIe). The common CAB and the blocked CAB are generated at the Fourier plane of the Fourier lens. A beam profiler (Spiricon, SP620U) is used to observe the beam profile and measure the intensity distributions.
Fig. 5 Experimental setup. Laser, diode laser at 1064nm; HWP, Half wavelength plate; SLM, Spatial Light Modulator.
The intensity profiles at the initial plane are shown in Fig. 6, and the experimental results are displayed in Fig. 7, where the distributions of the intensity contrast for the common CAB, and the blocked CAB are measured in experiment. It proves that abruptly autofocusing of CAB would be enhanced when the first few rings are blocked. Although the positions wherethe light intensity begins to increase are slightly moved forward when the first few rings are blocked, yet the value of Im/I0 is greatly increased as previous theoretical simulation predicts. In practice, we can block different light rings of CAB to obtain different abruptly autofocusing beams according to actual demands. However, because of the measurement error of the beam profiler and the experiment error in the production of the blocked CAB by phase-only SLM, the peak value of blocked CAB do not agree very well with the theoretical result [Figs. 7(b) and 7(c)].
Fig. 6 Intensity profile of CAB at the initial plane in experiment:(a) common CAB; (b) first ring blocked CAB; (c) first two rings blocked CAB. The corresponding parameters are the same as Fig. 1.
Fig. 7 Intensity contrast as a function of the propagation distance measured in experiment: red solid circles is experiment values, black solid line is the theory results. (a) Common CAB; (b)first ring blocked CAB; (3) first two rings blocked CAB. The corresponding parameters are the same as Fig. 1.
The radius of the hollow region of the CAB at the initial plane, which is generally considered as the radius of the CAB, is related to the parameter r0. The abruptly autofocusing property of CAB also can be controlled by varying the radius of the CAB [1]. When the first few rings of the CAB are blocked, the radii of the beams are also changed. So it is interesting to compare the blocked CAB with the common CAB with the same radius. Figure 8(a) shows the common CAB generated in experiment, whose radius is the same as the first ring blocked CAB in Fig. 6(b). The experiment results of abruptly autofocusing property of the two beams are shown in Fig. 8(b). From this figure, we can see that the intensity of the blocked CAB increases more than the common CAB. And the blocked CAB focuses before the common CAB. These indicate that we can enhance the abruptly autofocusing property while the radius of the hollow region remains constant, accompanying with the change of the focal position.
Fig. 8 Comparison of the blocked CAB and common CAB with the same radius. (a) Intensity profile of common CAB, r0 = 1.18mm; (b) distribution of intensity contrast of the two beams. Other parameters are the same as Fig. 1.
4. Conclusion
In conclusion, we have investigated the propagation characteristics of the modulated CAB, whose front rings are blocked. It is found that the abruptly autofocusing property is enhanced when the first few rings are blocked. Because the first ring of CAB contributes little to the focal intensity, and the intensity of the initial plane will be decreased when the first ring is eliminated, the intensity contrast of blocked CAB will be obviously enhanced. A better result of abruptly autofocusing property can be obtained by only blocking the first ring of the CAB. The theoretical conclusion is verified in our experiment. Compared with the common CAB of the same hollow region, the intensity of the modulated CAB increases more at the focus plane. The blocked CAB can be applied in the fields of optical micromanipulation and biomedical treatment.
Acknowledgments
This work is supported by the National Basic Research Program of China (Grant No.2012CB921602), National Nature Science Foundation of China (Grant No. 11474254) and the program of International S&T Cooperation of China (Grant No.2010DFA04690).
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