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Abstract
We theoretically and experimentally study the propagation properties of the circular Airy beam (CAB) with a Gaussian envelope in Fourier space. The two parameters of the Gaussian function can be used to control the distribution of the spatial frequency, and thus to tune the propagation properties of this modified CAB. When the two parameters are chosen appropriately, the size of focal spot will be reduced, the maximum focal intensity and especially the abruptly autofocusing property will be greatly enhanced. Meanwhile, the focal position can remain almost the same with the common CAB. The experimental results show that the proposed beam can be generated conveniently by using the same Fourier transform method as used to generate the common CAB.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The finite-energy optical Airy beam was first experimentally observed in 2007 [1,2]. In the past 10 years, much progress has been made in Airy beam due to the rapid development of holographic methods [3–5]. Recently, some new types of Airy beam such as the fractional Airy beams [6] and the circular Airy beams (CAB) were introduced [7]. The CAB can be obtained by using of the radially symmetric Airy beam. The unique property of the CAB is that this beam can maintain quite low intensity profiles before the focal point and the maximum intensity of the beam can abruptly increase by orders of magnitude just at the focal point [7,8]. It makes the CAB an ideal candidate for biomedicine [7] and optical micromanipulation [9,10]. More recently, it has demonstrated that the CAB can also be used to laser fabrication [11] and high harmonics generation [12–14]. Hence, the topics related with the CAB received intense attention in the last few years.
The abruptly autofocusing property, as well as peak intensity, size and position of the focal spot, are of the great importance for these applications. Many efforts have been made to enhance the autofocusing properties of the CAB. Li et al found that the abruptly autofocusing property can be enhanced by blocking the first ring of the CAB [15]. Jian et al theoretically studied the propagation characteristics of the modified CAB by using a high-pass filter in Fourier space [16] It shows a better abruptly autofocusing property and a higher maximum focal intensity compared with the blocked CAB. However, the size of focal spot is not changed with different cutoff frequencies. Recently, Zhong et al indicated theoretically that the autofocusing properties of the CAB can be controlled by introducing an azimuthal modulation [17]. However, the structure of the proposed beam is much more complicated than that of the common CAB. It makes it difficult to generate the proposed beam in experiment [18].
In this paper, we propose a new approach based on the Fourier spectrum modulation to further improve the autofocusing properties of the CAB by comparing with the previous methods. The amplitude of the Fourier spectrum is modulated by a Gaussian envelope. For simplicity, this new kind of modified CAB is called GCAB in the following. Then, the propagation characteristics of the GCAB are investigated in detail. Finally, the GCAB with appropriate parameters has been generated experimentally. The experimental results agree well with the theoretical results. Therefore, it indicates that our method is an efficient way to enhance the autofocusing properties of the CAB, and is easy to achieve in experiment.
2. Theoretical analysis
The electric fields of the initial CAB in cylindrical coordinate can be expressed as [7,8]
Fig. 1. (a) Calculation results of (a) $F(k )$; (b) $Ev(k )$ (dashed line) and $|{F(k )} |$ (solid line); (c) ${F_G}(k )$ with different $\beta $ when $\gamma = 3m{m^{ - 1}}$; and (d) ${F_G}(k )$ with different $\gamma $ when $\beta = 6m{m^{ - 1}}$.
The proposed modulated spectrum in this paper can be expressed as
where C is the constant amplitude of the spectrum to keep the conservation of energy, $Ev(k )$ is the envelope function of $|{F(k )} |$ and $G(k )= \exp \left( { - \frac{{{{({k - \beta } )}^2}}}{{{\gamma^2}}}} \right)$ is the Gaussian function. In principle, the exact expression of $Ev(k )$ cannot be obtained because the analytical expression of $F(k )$ is unknown. However, in practice, the approximation of $Ev(k )$ can be obtained numerically as follow steps: Firstly, solving the differential equation $dF(k )/dk = 0$ numerically to find the positions ${k_{ei}}$ of the extreme points. Then, an array can be generated as $[{\{{{k_{ei}},|{F({{k_{ei}}} )} |} \}, i = 1,\,2,\,3 \cdots } ]$. Finally, the approximation of $Ev(k )$ can be obtained easily by using the interpolation method based on the array. Figure 1(b) shows the calculated results of $Ev(k )$. The absolute value of $F(k )$ is also presented for comparison. The influences of the two parameters $\beta $ and $\gamma $ on ${F_G}(k )$ are shown in Fig. 1(c) and 1(d), respectively. The value of $\beta $ controls the position of the center of the peak, i.e., determine the dominant frequency components. The value of $\gamma $ controls the width of the spectrum. It can be seen that the low frequency components reduce sharply due to the Gaussian envelope compared with Fig. 1(a).The propagation of the GCAB can be computed as
Fig. 2. Intensity distributions of the CAB and the GCAB at the initial plane. (a) CAB and GCAB with different $\beta $ when $\gamma = 3m{m^{ - 1}}$; (b) CAB and GCAB with different $\gamma $ when $\beta = 6m{m^{ - 1}}$.
The propagation dynamics of the CAB and the GCAB from numerical simulations are shown in Fig. 3. In order to show the waves propagation clearly, Fig. 3(a) and 3(b) were plotted in logarithmic scale. It can be seen that the intensity maxima of the CAB follow a parabolic trajectory as the wave propagates toward the focus. The focal distance of the CAB is ${f_{CAB}} = 655.8mm$. The GCAB also shows the autofocusing property. The focal distance is ${f_{GCAB}} = 653.8mm$, which is almost identical to ${f_{CAB}}$. It can be clearly seen that in Fig. 3(b) more energy of the beam converges to the focal point compared with that in Fig. 3(a).
Fig. 3. Propagation dynamics (in logarithmic scale) of the CAB and the GCAB. (a) CAB; (b) GCAB with $\gamma = 3m{m^{ - 1}}$ and $\beta = 6m{m^{ - 1}}$.
Figure 4 presents the relative focal distance difference $\eta $ as a function of $\beta $ and $\gamma $. Here, $\eta $ is defined as $\eta = \frac{{{f_{CAB}} - {f_{GCAB}}}}{{{f_{CAB}}}} \times 100\%$. It found that ${f_{GCAB}}$ will increase as $\beta $ increases. This increasing is more obvious when $\gamma $ is smaller, as shown in the upper left corner of Fig. 4. When $\gamma $ increases, the peak intensity difference between the highest ring and other rings will decrease as shown in Fig. 2(b). Hence, the influence of $\beta $ will become less important when $\gamma $ is larger. When $\beta < 7m{m^{ - 1}}$, the absolute value of the relative difference $|\eta |$ remains less than 2%. It indicates that the focal position is affected very little by the spectrum modulation when the value of $\beta $ is not too large.
Fig. 4. The relative focal distance difference between the CAB and the GCAB as a function of $\beta $ and $\gamma $.
In applications, a critical parameter is the peak intensity at the focus. Figure 5 shows the maximum focal intensity contrast between the GCAB ${({{I_{GCAB}}} )_{\max }}$ and the CAB ${({{I_{CAB}}} )_{\max }}$ as a function of $\beta $ and $\gamma $. Here, we want to mention again that the total power of each beam is set to be unit, and then the value of ${({{I_{CAB}}} )_{\max }}$ is unchanged throughout the simulations. The calculation results show that ${({{I_{GCAB}}} )_{\max }}$ can achieve 3 times of ${({{I_{CAB}}} )_{\max }}$ when $\beta = 5.2mm^{ - 1}$ and $\gamma = 2.8m{m^{ - 1}}$. It can be found that ${({{I_{GCAB}}} )_{\max }}$ is much more sensitive to $\beta $ than $\gamma $. When $\gamma $ remains unchanged, ${({{I_{GCAB}}} )_{\max }}$ will increase at first and then decrease as $\beta $ increases. A similar focal intensity change was recently observed when the parameter w of the CAB decreased [22]. It was indicated that when the focus of the nonparaxial component coincides in space with the paraxial one, it will lead to a strong enhancement of the peak intensity at the focus. Unfortunately, it can not be used to explain the autofocus behavior of the GCAB directly. Because in our simulations the propagations of the GCAB are all in the paraxial regime. However, we can still borrow this creative idea to try to understand the autofocus behavior of the GCAB.
Fig. 5. The maximum focal intensity contrast between the GCAB and the CAB as a function of $\beta $ and $\gamma $.
Here, we separated the GCAB into two components by numerically filtering the spatial spectrum of the beams. The high-frequency component corresponds to $k \ge \beta $ and the low-frequency component corresponds to $k < \beta $. In Fig. 6, simulations of three cases of the GCAB with different $\beta $ are presented. In the simulations, $\gamma $ is set to be $2.8m{m^{ - 1}}$ and remains unchanged. As we mentioned above, the value of $\gamma $ is less important for the maximum focal intensity. One can easily observe that for the case of $\beta = 5.2mm^{ - 1}$, the focus of the high-frequency component coincides in space with the low-frequency component. Under this conditions, an optimized interference can be obtained, leading to a strong enhancement of the maximum focal intensity.
Fig. 6. On-axis intensity distributions of the low-frequency, high-frequency and full- frequency components along the propagation axis with different $\beta $ when $\gamma = 2.8mm^{ - 1}$. (a) $\beta = 3.7m{m^{ - 1}}$; (b) $\beta = 5.2m{m^{ - 1}}$; (c) $\beta = 6.7m{m^{ - 1}}$.
Figures 7(a) and 7(b) show the normalized intensity distributions of the GCAB with different parameters at the focal plane. It can be seen that the spot size of the GCAB at the focal plane is smaller than that of the CAB. The FWHM value of the focal spot is affected little by $\gamma $, however, it decreases as $\beta $ increases. When $\beta = 8mm^{ - 1}$, the FWHM value is $21.5\mu m$, which is reduced by 42% compared with that of the CAB. This result is explicable, because the dominant frequency components of the GCAB will move to the higher frequency region when $\beta $ increases.
Fig. 7. Normalized intensity distributions of the GCAB with different parameters at the focal plane. (a) GCAB with different $\gamma $ when $\beta = 5.2m{m^{ - 1}}$; (b) GCAB with different $\beta $ when $\gamma = 2.8m{m^{ - 1}}$.
The abruptly autofocusing property can be described through the intensity contrast ${I_c} = {I_{\max }}(z )/{I_{\max }}(0)$, where ${I_{\max }}(z )$ is the maximum intensity of arbitrary plane along propagation and ${I_{\max }}(0)$ is the maximum intensity of the initial plane. The maximum intensity contrast ${({{I_c}} )_{\max }}$of the CAB is only 52.2 under the parameters used in this paper. Since ${I_{\max }}(0)$ of the GCAB is much smaller than that of CAB, but ${I_{\max }}(z )$ of the GCAB is larger than that of the CAB at the focal plane. It can be expected that the abruptly autofocusing property would be greatly enhanced. Figure 8 shows ${({{I_c}} )_{\max }}$of the GCAB as a function of $\beta $ and $\gamma $. It can be seen that ${({{I_c}} )_{\max }}$ of the GCAB increase dramatically as $\beta $ and $\gamma $ increase. When $\beta $ and $\gamma $ increase, more energy will transfer from inner rings to the outer rings at the initial plane. In order to keep the energy conservation, ${I_{\max }}(0)$ of GCAB will decrease sharply with the increase of $\beta $ and $\gamma $, as shown in Fig. 2. It is the main reason of such dramatic enhancement of ${({{I_c}} )_{\max }}$ shown in Fig. 8.
Fig. 8. The maximum intensity contrast ${({{I_c}} )_{\max }}$ as a function of $\beta $ and $\gamma $.
So we can choose a larger $\beta $ and a larger $\gamma $ to obtain a higher intensity contrast of the GCAB. However, the maximum focal intensity will obviously decrease when $\beta $ and $\gamma $ are too large. In addition, it also makes it difficult to generate in experiment. Thus, in practice, if abruptness and intensity of the autofocusing are both taken into account, $\beta = 5.2mm^{ - 1}$ and $\gamma = 2.8m{m^{ - 1}}$ are a good choice for this study. Under these parameters, the maximum focal intensity is 3 times that of the CAB and the maximum intensity contrast achieves 694, which is over 13 times that of the CAB.
3. Experimental results
Finally, we will show that the GCAB can be generated conveniently by using the same FT method as used to generate the common CAB. The FT method was well described in detail in [5,8]. The experiment setup is shown in Fig. 9. A He-Ne laser (632.8 nm) was used as a light source, and a phase reflection only spatial light modulator (SLM, Holoeye PLUTO-2) was employed for the phase modulation of an expanded Gaussian beam. The phase modulated reflected wavefront is then Fourier transformed by a 300mm lens. An opaque mask is used to block the undesired zero-order peak. Finally, the intensity distribution along the propagation is recorded using a CCD camera.
Based on the FT method, the maximum spatial frequency of the generated beam must be smaller than $L/({2\lambda f} )$, where L is the length of the hologram on the SLM, f is the focal length of the Fourier lens. In our experiments, $L/({2\lambda f} )\approx 23m{m^{ - 1}}$. If we set the Gaussian function $G(k )\approx 0$ when its value is less than $1/{e^4}$, the maximum spatial frequency of GCAB will be approximately equal to $\beta + 2\gamma $. Therefore, we have $\beta + 2\gamma \le L/({2\lambda f} )$.
The measured intensity distributions of CAB and GCAB with different parameters at the initial plane are shown in Fig. 10. The corresponding simulated results are also shown in Fig. 10 for comparison. It can be clearly seen that the intensity distributions of GCAB are obviously modulated by the parameters $\beta $ and $\gamma $. The experimental results agree well with the simulated results. It proves that the GCAB with different parameters can be well generated by using the FT method.
Fig. 10. Experimental (dash-dot line) and simulated (solid line) results of normalized intensity line profiles at $z = 0$ (the FT plane in Fig. 9). (a) CAB; (b) GCAB with $\beta = 4mm^{ - 1}$ and $\gamma = 3mm^{ - 1}$; (c) GCAB with $\beta = 6mm^{ - 1}$ and $\gamma = 2mm^{ - 1}$. The insert Figs are the corresponding measured 2D intensity distributions and the side lengths of insert Figs are 3mm.
The experimental and simulated results of intensity contrast $I/{I_{\max }}(0 )$ of GCAB with $\beta = 5.2mm^{ - 1}$ and $\gamma = 2.8m{m^{ - 1}}$ at different propagation planes are shown in Fig. 11, where ${I_{\max }}(0)$ is the maximum intensity of the initial plane. In Fig. 11(b), the maximum intensity contrast at $z = 450mm$ is only about 3.3, which is much less than that at the focal point as shown in Fig. 11(c). It indicates that the GCAB can maintain quite low intensity profiles before the focal point. At the focal point as shown in Fig. 11(c), the measured peak value is about 634, which is slightly smaller than that of the simulated one (about 694) due to the experimental errors. In Fig. 12, the on-axis intensity contrast distributions ${I_c}$ of generated CAB and GCAB are represent. It can be clearly seen that the intensity contrast of GCAB is greatly enhanced at the focal point as the theoretical prediction.
Fig. 11. Experimental (dash-dot line) and simulated (solid line) results of intensity contrast $I/{I_{\max }}(0 )$ of GCAB with $\beta = 5.2mm^{ - 1}$ and $\gamma = 2.8m{m^{ - 1}}$ at different propagation planes. (a) $z = 0$; (b) $z = 450mm$; (c) $z = 651mm$ (focus position). The insert Figs are the corresponding measured 2D intensity distributions. The side lengths of insert Figs in (a), (b) and (c) are 3mm, 2mm and 0.2 mm, respectively.
Fig. 12. Experimental and simulated results of on-axis intensity contrast ${I_c}$ of CAB and GCAB with $\beta = 5.2mm^{ - 1}$ and $\gamma = 2.8m{m^{ - 1}}$.
4. Conclusion
In summary, the autofocusing properties of the GCAB are theoretically and experimentally investigated in detail. The two parameters $\beta $ and $\gamma $ of the Gaussian function can be used to tune the autofocusing properties of the GCAB. The focal trajectory of the GCAB becomes flatter compared with that of the CAB. However, the focal position can remain almost the same with the CAB when $\beta $ is not too large. When $\beta$ and $\gamma $ are chosen appropriately, the focus of the high-frequency component will coincide in space with the low-frequency component, leading to a strong enhancement of the maximum focal intensity. The FWHM of the focal spot and the maximum intensity contrast will decrease and increase, respectively, as $\beta $ increases. Therefore, the proposed method can allow us to choose the appropriate parameters of the GCAB for different applications. The experimental results agree well with the theoretical results. It shows that our method is an efficient way to enhance the autofocusing properties of the CAB, and is easy to achieve in experiment.
Funding
National Natural Science Foundation of China (61975125); Natural Science Foundation of Shanghai (16ZR144600); National Key Research and Development Program of China (2017YFB0503100).
Disclosures
The authors declare no conflicts of interest.
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