Autofocusing field constructed by ring-arrayed Pearcey Gaussian chirp beams

Mingli Sun; Tong Li; Jinqi Song; Kaikai Huang; Junhui Shi; Junhui Shi; Xian Zhang; Xuanhui Lu

doi:10.1364/OE.474368

1. Introduction

For the past few years, abruptly autofocusing beams (AABs) have attracted a great of interest due to their autofocusing property [1–5]. The special autofocusing property makes AABs an idea candidate in optical trapping and biomedical treatment [6–10]. Up to now, various methods have been proposed to improve the autofocusing ability of AABs [11–13]. Pearcey beams, which possess abruptly autofocusing, form-invariance and self-healing properties during the propagation, have been theoretically and experimentally studied in recent years [14–16]. As a natural extension of Pearcey beams, the circularly symmetric Pearcey beams, i.e., circular Pearcey beams (CPBs) were proposed in 2018 [17]. Compared with other AABs (e.g. circular Airy beams), CPBs exhibit outstanding advantages of the enhanced autofocusing ability, shorter focusing distance and nonexistent oscillations after the focal plane. On the other hand, the OVs, possessing hollow-core intensity distribution and orbital angular momentum, have been extensively investigated due to their wide applications in optical tweezers, free-space information transfer, angular measurement [18–25]. AABs carrying OVs can be used to manipulate the particles whose refractive indices are lower than the surrounding medium or rotate particles as “optical spanner” [11,26]. Imposing OVs on CPBs has been theoretically studied in recent years [27,28] and Zhu [29] experimentally demonstrate vortex circular Pearcey beams recently.

In this letter, we introduce a kind of synthesized beams, named RAPGCBs, which is based on the superposition of multiple two-dimensional Pearcey beams. The analytical formula for the propagation of RAPGCBs is derived. Then the influence of synthesized number and chirp factor on the propagation characteristics are studied. The autofocusing properties of RAPGCBs carrying an on-axis OV, an off-axis OV and two OVs are investigated, respectively.

2. Theory

2.1 RAPGCBs

The electric field of RAPGCBs constructed by multiple two-dimensional Pearcey beams in the source plane can be expressed as

(1)$$U_{n 0}\left(x^{\prime}, y^{\prime}, 0\right)=\exp \left(-\frac{x^{\prime 2}+y^{\prime 2}}{w_0{^2}}\right) \times \exp \left[-i \beta\left(\frac{x^{\prime 2}+y^{\prime 2}}{p_0{^{2}}}\right)\right] \times \sum_{j=1}^n P e\left(X_j, Y_j\right)$$

where $Pe$($\cdot$) is the Pearcey integral defined by [30] $Pe(X,Y)\textrm{ = }\int_{ - \infty }^{ + \infty } {\exp [{i({{s^4} + Y{s^2} + Xs} )} ]} ds$, ${w_0}$ is the waist width of the Gaussian profile, $\beta$ is the chirp factor used to control the focal length and the autofocusing ability of the beams. Actually, the effect of the quadratic phase $\exp [{ - i\beta ({x{^{\prime2}} + y{^{\prime2}}} )/{p_0}^2} ]$ on beams is equivalent to that of a focusing lens [13]. ${p_0}$ denotes the scaling parameter that affects the amplitude distribution of input beams and $({X_j},{Y_j})$ denotes array coordinates given by

(2)$$\begin{array}{l} {X_j} = \cos \left( {2\pi \frac{{j - 1}}{n}} \right) \times \frac{{x^{\prime}}}{{{p_0}}} - \sin \left( {2\pi \frac{{j - 1}}{n}} \right) \times \frac{{y^{\prime}}}{{{p_0}}}\\ {Y_j} = \sin \left( {2\pi \frac{{j - 1}}{n}} \right) \times \frac{{x^{\prime}}}{{{p_0}}} + \cos \left( {2\pi \frac{{j - 1}}{n}} \right) \times \frac{{y^{\prime}}}{{{p_0}}}. \end{array}$$

Here, $n$ is the synthesized number of two-dimensional Pearcey beams, and $j$ = 1, 2, 3…$n$, $Pe({X_j},{Y_j})$ denotes the ${j^{th}}$ two-dimensional Pearcey beam, which can be obtained by rotating $Pe({X_1},{Y_1})$ around the center by $2\pi (j - 1)/n$ radians.

Based on Huygens-Fresnel integral,

(3)$${U_z}({x,y,z} )={-} \frac{{ik}}{{2\pi z}}\int\limits_{ - \infty }^\infty {\int\limits_{ - \infty }^\infty {{U_0}(x^{\prime},y^{\prime})\exp \{ \frac{{ik}}{{2z}}[{{(x - x^{\prime})}^2} + {{(y - y^{\prime})}^2}]\} dx^{\prime}dy^{\prime},} }$$

substituting Eq. (1) into Eq. (3), the field distribution of the RAPGCBs at z position can be obtained as

(4)$${U_{nz}}({x,y,z} )= \frac{{{G^\beta }(x,y,z)}}{{{{\left( {1 - \frac{z}{{{z_e}\chi (z )}}} \right)}^{1/4}}}} \times \sum\limits_{j = 1}^n {Pe\left( {\frac{{{X_j}^{\prime}}}{{\chi (z ){{\left( {1 - \frac{z}{{{z_e}\chi (z )}}} \right)}^{1/4}}}},\frac{{{Y_j}^{\prime} - \frac{z}{{{z_e}}}}}{{\chi (z ){{\left( {1 - \frac{z}{{{z_e}\chi (z )}}} \right)}^{1/2}}}}} \right),}$$

where

(5)$$\begin{array}{l} {G^\beta }(x,y,z) = \frac{1}{{\chi (z )}}\exp \left( { - \frac{{{x^2} + {y^2}}}{{{w_1}^2\chi (z )}}} \right),\\ \chi (z )= 1 + \frac{{iz}}{{{z_{R1}}}},\\ {z_{R1}} = \frac{{k{w_1}^2}}{2},\\ k = \frac{{2\pi }}{\lambda },\\ {z_e} = 2k{p_0}^2,\\ \frac{1}{{{w_1}^2}} = \frac{1}{{{w_0}^2}} + \frac{{i\beta }}{{{p_0}^2}},\\ {X_j}^{\prime} = \cos \left( {2\pi \frac{{j - 1}}{n}} \right) \times \frac{x}{{{p_0}}} - \sin \left( {2\pi \frac{{j - 1}}{n}} \right) \times \frac{y}{{{p_0}}},\\ {Y_j}^{\prime} = \sin \left( {2\pi \frac{{j - 1}}{n}} \right) \times \frac{x}{{{p_0}}} + \cos \left( {2\pi \frac{{j - 1}}{n}} \right) \times \frac{y}{{{p_0}}}. \end{array}$$

The propagation characteristics of RAPGCBs can be studied and analyzed by using Eq. (4). From Eq. (4), it is obviously that there are two zero points in the denominator, i.e.

(6)$$1 - \frac{z}{{{z_e}\chi (z )}}\textrm{ = }0,$$

and

(7)$$\chi (z )\textrm{ = 0}\textrm{.}$$

In fact, the distances satisfied by Eqs. (6) and (7) are the autofocusing positions of the RAPGCBs.

Figures 1(a)–1(d) display the initial intensity profiles for the beams with $n$ = 3, 4, 10 and 100, respectively, and Figs. 1(A)–1(D) are the corresponding intensity profiles in the focal plane. For $n$ = 3 and 4, the intensity profile displays triangle or quadrangle, respectively. Besides, one can find many speckles around the central bright spot due to the superposition of multiple beams. As n increases further, the speckles gradually disappear, and the intensity profile tends to be circularly symmetrical distributed, as shown in Figs. 1(c) and 1(d). Obviously, the synthesized number n affects the intensity distribution. The larger the n, the smoother the intensity distribution.

Fig. 1. Intensity profiles of RAPGCBs in the source plane and focal plane ($\beta$= 1.2): (a) (A)$n$= 3; (b) (B)$n$= 4; (c) (C)$n$= 10; (d) (D)$n$= 100. Each intensity profile is independently normalized.

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The chirp factor $\beta$ can greatly affect the autofocusing properties of RAPGCBs. Figure 2 shows the normalized intensity ${I_m}/{I_0}$ versus the propagation distance for the RAPGCBs with $n$= 100 and different values of $\beta$, where ${I_0}$ is the maximum intensity at the initial plane, and ${I_m}$ is the maximum intensity at an arbitrary propagation plane. One can see that, when $\beta$ = 1 and 2, it exhibits twice autofocusing behavior, where the first focal position is satisfied by Eq. (6) and the second focal position is satisfied by Eq. (7). However, when $\beta$ = 3 or 4, such beams only focus once obviously during the propagation. As $\beta$ increases, the focal length shortens and the two focal positions are close to each other until they overlap, which results in greatly enhanced autofocusing ability. For example, in the case of $\beta$ = 4, the value of ${I_m}/{I_0}$ increases to about 1216 in the focal plane and the focal length shortens to about 6.6 cm. The first focus is determined by both the autofocusing property of Pearcey beams and the converging effect of the chirp phase, while the second focus is mainly related with the chirp factor, i.e. the second focal length is exactly the virtual focal length of the lens phase term [13].

Fig. 2. Normalized intensity ${I_m}/{I_0}$ during the propagation of RAPGCBs with different $\beta$ ($n$= 100, $\beta$= 1, 2, 3, 4, respectively).

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2.2 RAPGCBs with single OV

The electric field of the initial RAPGCBs carrying an OV can be expressed as

(8)$$U_{n0}^v({x^{\prime},y^{\prime},0} )= {U_{n0}}({x^{\prime},y^{\prime},0} )\times \exp (im\theta ),$$

where $\theta = angle[{({x^{\prime} - {x_a}} )+ iy^{\prime}} ]$ represents the phase angle of $[{({x^{\prime} - {x_a}} )+ iy^{\prime}} ]$, m is the topological charge (TC). ${x_a}$ is the offset parameter. For the sake of simplicity, we only study the case of beams with an on-axis OV.

2.3 RAPGCBs with two OV

The initial fields of beams carrying two OVs located at $({ \pm {x_a},0} )$ can be expressed as

(9)$$U_{n0}^{vv}({x^{\prime},y^{\prime},0} )= {U_{n0}}({x^{\prime},y^{\prime},0} )\times \exp (i{m_1}{\theta _1}) \times \exp (i{m_2}{\theta _2}),$$

where ${\theta _1}$ and ${\theta _2}$ are phase angles of $[{({x^{\prime} - {x_a}} )+ iy^{\prime}} ]$ and $[{({x^{\prime} + {x_a}} )+ iy^{\prime}} ]$, respectively. ${m_1}$ and ${m_2}$ are corresponding TCs. Notice that, we set $\lambda$= 1064 nm, ${p_0}$= 0.3 mm, ${w_0}$= 3 mm throughout this paper.

3. Results and discussions

The experimental setup is shown in Fig. 3. A laser beam with $\lambda$= 1064 nm shaped by a polarization maintaining fiber (PMF) passes through a beam expander (BE). Since the spatial light modulator (SLM, BNS, 512 × 512 pixels) can only modulate a vertically polarized beam, two half-wave plate (HWP) and a polarization beam splitter (PBS) are used to generate the vertical linearly polarized beam and control the power incident on the SLM. A computer-generated hologram (CGH) encoded with the single-pixel checkerboard method [31] is loaded on the SLM. The modulated beam is reflected toward a 4$f$ imaging system made up of a couple of identical lenses with $f$= 250 mm. Then, the targeted beams will come out in the zero-order diffraction at the Fourier plane. A linear phase is introduced in the CGH to separate the modulated zero-order diffraction spot from the reflected central spot and a low-pass spatial filter that consists of an iris (I) is used to select the zero-order beam. The source plane of the target beams is just located at the output plane of the 4$f$ system. The intensity profiles at different distances are measured by a charge-coupled device (CCD, Spiricon, SP620U, pixel pitch of 4.4 µm).

Fig. 3. Schematic of the experimental setup: M, mirror; PMF, polarization maintaining fiber; BE, beam expander; HWP, half-wave plate; PBS, polarization beam splitter; SLM, spatial light modulator; L1, L2, lens; I, iris; CCD, charge-coupled device.

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Figure 4(a) exhibits the $y - z$ side-view of the RAPGCBs with $\beta$ = 1.2, $n$= 100. It shows that the arrayed beams autofocus twice at ${z_a}$= 18 cm and ${z_b}$= 22 cm, respectively. Obviously, the autofocusing ability at ${z_b}$= 22 cm is greater than that at ${z_a}$= 18 cm and the parabolic contour before focuses is not obvious because of the strong autofocusing ability. Figure 4(b) shows the normalized intensity ${I_m}/{I_0}$ of the RAPGCBs varies with the propagation distance z. As beams propagate forward, the light intensity keeps a relatively low value until the first focal plane ${z_a}$= 18 cm, where the light intensity suddenly increases with ${I_m}/{I_0}$ reaching a peak value (94). Then the peak intensity decreases slightly and reaches a much larger value (442) at the second focal plane ${z_b}$= 22 cm. After that, the light intensity rapidly decreases to a very low value without oscillation. Figures 4(c1)–4(c4) are simulation intensity distributions of RAPGCBs at several planes and the corresponding experimental results are displayed in Figs. 4(C1)–4(C4). Here, we can see that the beams show a radially symmetric solid profile at the initial plane, as displayed in Fig. 4(c1). At two focal planes, the intensity concentrates in small areas with solid spots, as shown in Figs. 4(c2) and 4(c3), and at $z$= 30cm, the intensity profile becomes larger due to the diffraction, as shown in Fig. 4(c4). One can find that the size of the second focus is smaller than that of the first one. Compared with the autofocusing circular Airy beam [1], RAPGCBs exhibit twice autofocusing feature as well as a stronger second autofocusing ability. Such beams can generate larger light intensity gradient near focal planes, which has advantages for cauterizing target tissues in laser medicine.

Fig. 4. Propagation dynamics of RAPGCBs ($\beta$= 1.2, $n$ = 100). (a) Side-view in $y - z$ plane for the propagation. (b) Normalized intensity ${I_m}/{I_0}$ during the propagation; ${z_a}$, ${z_b}$ marked with the dashed lines correspond to the locations at 18, 22 cm, respectively. (c1)–(c4) Simulated results at $z$= 0, 18, 22 and 30cm, respectively. (C1)–(C4) Experimental results at $z$= 0, 18, 22 and 30cm, respectively. Each intensity profile is independently normalized.

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Based on Eq. (8), the simulated intensity profile of beams with an on-axis OVs at $z$= 0 is displayed in Fig. 5(a1). Figure 5(a2) shows the corresponding CGH of the RAPGCBs with $m$= 1. In our experiment, an iris is located on the focal plane of the first lens L1, and the high-frequency components of the filtered target beams are inevitably cut off. By simulating our experiment generation process, when the high-frequency components are cut off, the simulated result of the initial intensity profile has a small hollow center, as shown in Fig. 5(a4). The experimental result displayed in Fig. 5(a3) agrees with the simulated result shown in Fig. 5(a4). Therefore, the low-pass filtering on the focal plane of L1 results in the hollow center of the experimental intensity profile. It should be noted that this filtering process does not affect the initial intensity distribution of RAPGCBs without OVs, which will always show solid center at the initial plane and insensitive to the loss of high-frequency components.

Fig. 5. (a1) Normalized intensity distribution of RAPGCBs with an on-axis OV at $z$= 0 ($m$= 1, $\beta$= 1.2, $n$= 100, ${x_a}$= 0). (a2) The CGH (the gray-scale level corresponds to 0-2$\pi$). (a3) Experimental result at $z$= 0; (a4) simulated result at $z$= 0 according to the experiment generation process. (b) and (c) Normalized intensity ${I_m}/{I_0}$ during the propagation for $m$= 1 and $m$= 2, ${z_1}$, ${z_2}$, ${z_3}$ correspond to the locations at 10, 22 and 30cm, respectively; (d1)–(d3) Simulated results for $m$= 1; (D1)–(D3) Experimental results for $m$= 1; (e1)–(e3) Simulated results for $m$= 2; (E1)–(E3) Experimental results for $m$= 2. The white solid curve denotes the cross line ($y$= 0). Each intensity profile is independently normalized.

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In Figs. 5(b)–5(e), the propagation characteristics of RAPGCBs with an on-axis OV is studied, in the case of $m$= 1 and $m$= 2, respectively. Compared with RAPGCBs without OV shown in Fig. 4, the autofocusing ability is decreased due to the reduced energy density by the OV in the focal plane and there is no obvious autofocusing phenomenon at $z$= 18 cm for beams with an on-axis OV, as shown in Figs. 5(b) and 5(c). Here, we can observe once autofocusing property during the propagation. The peak value of ${I_m}/{I_0}$ is 30 for $m$= 2, and it is 74 for $m$= 1. As m increases further, such as $m$= 3, 4, 5, respectively, the autofocusing ability will gradually decrease with the increase of m, which are not studied in detail here for the sake of brevity. As a result, the autofocusing ability can be enhanced with the decrease of m. This phenomenon is much different from that of circular Airy beam carrying OV [11] whose autofocusing ability is increased as m increases. Note that the autofocusing focal length remains 22 cm, which is independent with the TC. Figures 5(d1)–5(d3) and Figs. 5(e1)–5(e3) exhibit intensity distributions of RAPGCBs with $m$ = 1, 2 and corresponding profile lines at different transverse planes. The experimental results agree with the simulated results, as presented in Figs. 5(D1)–5(D3) and Figs. 5(E1)–5(E3), respectively. Obviously, the central dark area is larger for a larger m at the same plane, and it is smallest in the focal plane.

Figure 6 shows the influence of two OVs with TCs of (+ 1, + 1) or (+ 1, - 1) on autofocusing properties of the beams. Here, we set ${x_a}$ = 0.05 mm, $\beta$= 1.5 in Eq. (9). From Figs. 6(a) and 6(b), one can see that, when two positive OVs are imposed, RAPGCBs focus once, and there are two foci when two opposite OVs are imposed. In the meantime, for beams with two opposite OVs, the peak value of ${I_m}/{I_0}$ could reach approximately 680 at the second focal plane and this value is much larger than that of 117 at the first focal plane as well as that of 49 for beams carrying two positive OVs. When two positive OVs are imposed, the OVs will overlap each other and the intensity profiles are always hollow-centered during the propagation, as displayed in Figs. 6(c1)–6(c3). Figures 6(C1)–6(C3) are the corresponding experimental results. In this case, beams focus at $z$ = 18 cm. In comparison, when beams carrying two opposite OVs, the two OVs will annihilate owing to their opposite vorticity, and the intensity profiles are circular solid spots at two focal planes $z$= 15 cm and $z$ = 18 cm, as shown in Figs. 6(d2) and 6(d3). Specifically, the intensity is not evenly distributed before and after the focal planes, and the intensity of the upper parts in the planes is greater than that of the lower parts, as shown in Figs. 6(d1) and 6(d4). Our experimental results agree well with the simulation results.

Fig. 6. Propagation dynamics of RAPGCBs with two OVs ($\beta$= 1.5, $n$= 100, and ${x_a}$= 0.05mm). (a) Normalized intensity ${I_m}/{I_0}$ during the propagation for ${m_1}$= 1, ${m_2}$= 1 and ${z_1}$, ${z_2}$, ${z_3}$ marked with the dashed lines correspond to the locations at 10, 18 and 22cm. (b) Normalized intensity ${I_m}/{I_0}$ during the propagation for ${m_1}$= 1, ${m_2}$= -1 and ${z_1}$, ${z_2}$, ${z_3}$, ${z_4}$ marked with the dashed lines correspond to the locations at 10, 15, 18 and 22cm.. (c1)–(c3) Simulated results for ${m_1}$= 1, ${m_2}$= 1. (C1)–(C3) Experimental results for ${m_1}$= 1, ${m_2}$= 1. (d1)–(d4) Simulated results for ${m_1}$= 1, ${m_2}$= -1. (D1)–(D4) Experimental results for ${m_1}$= 1, ${m_2}$= -1. Each intensity profile is independently normalized.

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4. Conclusion

In conclusion, we have proposed a kind of autofocusing beams synthesized by multiple two-dimensional Pearcey beams. The general analytical formula for the propagation of RAPGCBs is presented, from which the focal positions can be obtained. For RAPGCBs, the autofocusing properties are greatly affected by the chirp factor $\beta$. As $\beta$ increases, the focal length shortens and the two focal positions are close to each other until they overlap, which results in greatly enhanced autofocusing ability. For RAPGCBs with an on-axis OV, the autofocusing ability can be increased with the decrease of m. Compared with two positive OVs, opposite OVs carried by RAPGCBs will annihilate, which results in greatly increased autofocusing ability. Our experimental results agree with the simulations. These special features can have applications in optical trapping, particle rotating and biomedical treatment.

Funding

National Key Research and Development Program of China (2017YFB0503100); National Natural Science Foundation of China (11474254, 11804298); Fundamental Research Funds for the Central Universities (2017QN81005).

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.

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Autofocusing field constructed by ring-arrayed Pearcey Gaussian chirp beams

Abstract

1. Introduction

2. Theory

2.1 RAPGCBs

2.2 RAPGCBs with single OV

2.3 RAPGCBs with two OV

3. Results and discussions

4. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (6)

Equations (9)

Optics Express