Radiation forces on a Rayleigh particle produced by partially coherent circular Airy beams

Mingli Sun; Jiahao Zhang; Nan Li; Nan Li; Kaikai Huang; Kaikai Huang; Huizhu Hu; Xian Zhang; Xuanhui Lu

doi:10.1364/OE.27.027777

1. Introduction

At present, optical trapping and micromanipulation of particles, including atoms, molecules, nanoparticles, cells and viruses [1–5], have been widely investigated since the seminal work of Ashkin [6]. In 1986, Ashkin et al. [7] demonstrated that a single-beam gradient force trap could be used for optical trapping of dielectric particles, which provided a useful method for manipulating particles. Many researches have demonstrated that various beams can be used for trapping particles, such as Laguerre Gaussian beams [8], hollow Gaussian beams [9], Bessel Gaussian beams [10], Lorentz-Gauss beams [11] and Airy beams [12]. The application of controllable light force on microparticles has played an important role in modern optics.

In the past few years, partially coherent beams have attracted a great deal of interest because of their extraordinary advantages over the completely coherent beam in many applications. For example, partially coherent beams can be used for reducing intensity fluctuations and turbulence-induced signal fading in laser communication [13,14]; increasing the conversion efficiency in the second-harmonic generation [15,16]; processing images by a scanning laser beam [17]; improving the performance of the projection systems for photolithography [18]; studying the diffraction halo function in the speckle photography [19]; solving the imaging problems [20]; investigating the imaging visibility and quality [21,22]. Since the mathematical description of genuine spatial correlation function was proposed by Gori and Santarsiero [23], more and more attention is being paid to partially coherent beams with special correlation functions and a variety of studies have been conducted both in theory and experiment, which include nonuniformly correlated beams [24–26], Bessel–Gaussian and Laguerre-Gaussian Schell-model beams [27], Hermite-Gaussian Schell-model beam [28], cosine-Gaussian Schell-model beams [29,30], partially coherent vector beam [31]. Recently, partially coherent light sources termed optical coherence grids and partially coherent circular Airy beams are also introduced theoretically [32,33]. As we know, the focusing feature and propagation characteristics of partially coherent beams can be controlled by the spatial coherent length [34,35]. Therefore, the radiation force induced by partially coherent beams can also be mainly related to their coherent lengths and various works of partially coherent beams for radiation forces on spherical particles were proposed [36–39].

Airy beam is well-known for its non-diffractive, self-healing and self-accelerating features [40–42]. Recently, N.K. Efremidis et al. [43] provided a brief overview of such beams in some theoretical and experimental developments as well as a variety of applications. In 2010, N.K. Efremidis, D.N. Christodoulides introduced the circular Airy beam which has the abruptly autofocusing property [44]. D. G. Papazoglou et al. [45] experimentally observed this kind of autofocusing beams for the first time, and P. Zhang et al. [46] demonstrated that such Airy beams could be used for trapping and guiding microparticles in experiment. However, the radiation force of PCCAB on a Rayleigh particle has not been studied before. In this paper, based on the Guassian special correlation function, we investigate the radiation force of PCCAB on a Rayleigh dielectric sphere. Besides, the influence of corresponding parameters is also analyzed as well as the stability of trapping particles. Actually, compared with the completely coherent circular Airy beam, the abruptly autofocusing property and the focal width of PCCAB can be greatly controlled by the spatial coherent length [33]. Therefore, it is an effective way to modulate the radiation force as well as the depth of potential well in optical micromanipulation.

The paper is organized as follows. We initially introduced the intensity distribution description of PCCAB based on the Huygens-Fresnel formula and plotted the propagation dynamics of PCCAB with different coherent lengths in section 2; then, based on radiation force formulas, we simulated the radiation force on a Rayleigh particle, investigating the influence of different parameters on the radiation force in section 3; the trapping stability was analyzed in section 4; finally, the summary of our results and potential applications were discussed in section 5.

2. Propagation characteristics of partially coherent circular Airy beams

The electric field of a circular Airy beam at the input plane ($z$ = 0) can be expressed as [44]:

(1)$$E(r) = C \cdot Ai(\frac{{{r_0} - r}}{w})\exp (a \cdot \frac{{{r_0} - r}}{w}),$$

where C is a constant related with the input power, $Ai$ is the Airy function, ${r_0}$ is a parameter related with the initial radius of a circular Airy beam, w is a scale factor and a represents the decay parameter which is positive so as to ensure the Airy beam is square integrable.

The cross-spectral density of the incident beam at the initial plane ($z$ = 0) takes on the form [47]:

(2)$${W_{in}}({r_1},{\theta _1},{r_2},{\theta _2}, z = 0) = E({r_1},{\theta _1}){E^\ast }({r_2},{\theta _2})\exp ( - \frac{{{r_1}^2 + {r_2}^2}}{{2{\delta _c}^2}})\exp [\frac{{{r_1}{r_2}\cos ({\theta _1} - {\theta _2})}}{{{\delta _c}^2}}],$$

where $({r_1},{\theta _1})$, $({r_2},{\theta _2})$ are two spatial points in source plane; ${\delta _c}$ is the correlation width representing the spatial coherence.

Under the paraxial approximation, according to the extended Huygens–Fresnel diffraction integral, the propagation dynamics of cross-spectral density of the PCCAB can be expressed as [48]:

(3)$$\begin{array}{l} W({\rho _1},{\varphi _1},{\rho _2},{\varphi _2}, z) = \frac{{{k^2}}}{{4{\pi ^2}{z^2}}}\int\!\!\!\int\!\!\!\int {\int_{ - \infty }^{ + \infty } {{W_{in}}} } ({r_1},{\theta _1},{r_2},{\theta _2}, z = 0) \times \exp [ - \frac{{ik}}{{2z}}({r_1}^2 + {\rho _1}^2) + \frac{{ik}}{{2z}}({r_2}^2 + {\rho _2}^2)]\\ \times \exp [\frac{{ik}}{z}{r_1}{\rho _1}\cos ({\theta _1} - {\varphi _1}) - \frac{{ik}}{z}{r_2}{\rho _2}\cos ({\theta _2} - {\varphi _2})]{r_1}{r_2}d{r_1}d{r_2}d{\theta _1}d{\theta _2}, \end{array}$$

where $({\rho _1},{\varphi _1})$, $({\rho _2},{\varphi _2})$ are two points in the output plane at a certain distance z, $k = 2\pi /\lambda$ is the wave number, $\lambda$ is the wavelength in the medium. The complicated quadruple integral of Eq. (3) can be reduced to a double integral by using formulas in [48,49]. Then, we can obtain the following propagation expression:

(4)$$\begin{array}{l} W({\rho _1},{\varphi _1},{\rho _2},{\varphi _2}, z) = \frac{{{k^2}}}{{{z^2}}}\sum\limits_{l = - \infty }^\infty {\int {\int_0^\infty {E({r_1}} } } ){E^\ast }({r_2}){J_l}(\frac{{k{r_1}{\rho _1}}}{z}){J_l}(\frac{{k{r_2}{\rho _2}}}{z}){I_l}(\frac{{{r_1}{r_2}}}{{{\delta _c}^2}}) \times \exp ( - \frac{{{r_1}^2 + {r_2}^2}}{{2{\delta _c}^2}})\\ \times \exp [ - \frac{{ik}}{{2z}}({r_1}^2 + {\rho _1}^2) + \frac{{ik}}{{2z}}({r_2}^2 + {\rho _2}^2)]{r_1}{r_2}d{r_1}d{r_2}. \end{array}$$

Substituting Eq. (1) into Eq. (4), by making ${\rho _1} = {\rho _2} = \rho$ and ${\varphi _1} = {\varphi _2} = \varphi$, the spectral intensity distribution of PCCAB $I(\rho ,\varphi , z) = W(\rho ,\varphi ,\rho ,\varphi , z)$ can be constructed at arbitrary plane. In this work, we choose ${\lambda _0}$ = 1064 $nm$, $a$ = 0.1, $w$ = 1 $\mu m$, ${r_0}$ = 10 $\mu m$. l is an integer that varies from + 10 to −10 on account of the term approaching zero with large l. The propagation medium is water with refractive index ${n_2}$ = 1.33 and the incident power is 4 W. Figure 1 shows the intensity of PCCAB in the beam center along the propagation direction with different coherent lengths ${\delta _c}$. Obviously, the abruptly autofocusing property of PCCAB is greatly affected by the spatial coherence. With the increase of the coherent length, not only the abruptly autofocusing property becomes more apparent and the intensity at the focal plane increases obviously, but also the oscillation behavior becomes obvious after the focal plane. However, the autofocusing position is hardly affected by the coherent length.

Fig. 1. The intensity propagation of PCCAB in the beam center with different coherent lengths ${\delta _c}$.

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3. Radiation forces produced by partially coherent circular Airy beams

As is well known, when the incident light is scattered by a microparticle, momentum exchange occurs between the photon and the particle, which produces light radiation force. The use of radiation forces for optical trapping has been developed and widely studied to manipulate microparticles. It is worth noting that the incident light beams are assumed to be completely coherent in most of the investigations. In practice, however, any light field is always partially coherent [47]. Moreover, an equivalence theorem was established that it is not necessary for the source to be completely spatially coherent to produce a highly directional intensity distribution and partially coherent sources could generate an optical field that has the same intensity distribution throughout the far zone [50,51]. Some authors have demonstrated that partially coherent beams could produce the same optical force as a laser beam, which can be used for optical trapping [36–39]. Due to the focusing properties are greatly affected by its spatial coherence, the depth and width of the potential well can be effectively controlled by choosing appropriate spatial coherent length. Now we study the radiation force produced by PCCAB on a Rayleigh dielectric sphere, whose radius is much smaller than the wavelength and can be treated as a point dipole that is subject to secondary radiation, scattering and Lorentz force in the field. The radiation pressure force exerted on the particle in the Rayleigh regime can be described by two types of radiation forces: the scattering and gradient forces. Both of them can be used to manipulate and trap particles. The direction of scattering force is along the propagation direction of laser beam. For a Rayleigh particle with refractive index ${n_1}$ in a surrounding medium of refractive index ${n_2}$, the scattering force can be described as [52,53]:

(5)$${\vec{F}_{sca}}(\vec{r}) = \frac{{{n_2}}}{c}{C_{pr}}I(\vec{r}){\hat{e}_z},$$

where c is the speed of light in the vacuum, ${C_{pr}}$ is the cross section for the radiation pressure of particles and ${\hat{e}_z}$ is a unit vector along with the beam propagation direction. In the Rayleigh approximation, ${C_{pr}}$ is given by [53]:

(6)$${C_{pr}} = \frac{8}{3}\pi {(kR)^4}{R^2}{(\frac{{{m^2} - 1}}{{{m^2} + 2}})^2},$$

where R is the radius of the particle, $m = \frac{{{n_1}}}{{{n_2}}}$ is the relative refractive index of the particle. For The gradient force, produced by non-uniform electromagnetic fields, can be expressed as [53]:

(7)$${\vec{F}_{grad}}(\vec{r}) = \frac{{2\pi {n_2}{R^3}}}{c}(\frac{{{m^2} - 1}}{{{m^2} + 2}})\nabla I(\vec{r}),$$

and its direction is along the gradient of light intensity. From Eqs. (5) and (7), we can see that both of the scattering force and gradient force are mainly related with the intensity distribution of the beam.

Usually, the cross-spectral density is a measure of the correlations between the fluctuations of different components at the same frequency of the light field, as expression as Eq. (2), which is given by [47]:

(8)$$W({\vec{r}_i},{\vec{r}_j},\omega ) = \langle{{E^\ast }({{\vec{r}}_i},\omega )E({{\vec{r}}_j},\omega )} \rangle ,$$

where ${\vec{r}_i},{\vec{r}_j}$ denote two points of light field and $\omega$ is the frequency. The normalized form of Eq. (8) can be expressed as [47]:

(9)$$\eta ({\vec{r}_i},{\vec{r}_j},\omega ) = \frac{{W({{\vec{r}}_i},{{\vec{r}}_j},\omega )}}{{\sqrt {S({{\vec{r}}_i},\omega )S({{\vec{r}}_j},\omega )} }},$$

where $S({\vec{r}_i},\omega )$ and $S({\vec{r}_j},\omega )$ are the general spectral density of two points in the random process. Equation (9) denotes the degree of correlation between two Fourier components at the frequency $\omega$ and follows the inequation $0 \le \eta ({\vec{r}_i},{\vec{r}_j},\omega ) \le 1$. When ${\vec{r}_i} = {\vec{r}_j} = \vec{r}$, Eq. (8) represents the average intensity at the position $\vec{r}$ and $S(\vec{r},\omega ) = W(\vec{r},\vec{r},\omega )$ in Eq. (9). The limit $\eta ({\vec{r}_i},{\vec{r}_j},\omega ) = 1$ characterizes two points which are spatially completely coherent, while $\eta ({\vec{r}_i},{\vec{r}_j},\omega ) = 0$ denotes two points are incoherent. If two points in the plane are partially coherent, the value of $\eta ({\vec{r}_i},{\vec{r}_j},\omega )$ takes on value between zero and unity. For arbitrary two different partially coherent spatial points (${\vec{r}_i} \ne {\vec{r}_j}$), the cross-spectral density is greatly influenced by the degree of coherence, which can be represented by the coherent length ${\delta _c}$ in Eq. (2). From the propagation expression of PCCAB in Eq. (4), we can clearly see that the intensity distribution of output field plane is greatly influenced by the coherent length of different points. Therefore, the scattering force and gradient force in Eqs. (5) and (7), related with the intensity distribution, are greatly influenced by the spatial coherence. Based on Eqs. (5)–(7), we can calculate the radiation force on a Rayleigh dielectric sphere produced by PCCAB. Considering the symmetry of the spherical particle, we only analyze the radius along a certain transverse direction when calculating the transverse gradient force.

Figure 2 shows the radiation forces of PCCAB with different coherent lengths. In Figs. 2(a)–2(c), we plot the distributions of the scattering force ${F_{sca}}$, the longitudinal gradient force ${F_{grad, z}}$ and the sum of them ${F_{tot, z}}$, respectively. From Figs. 2(a) and 2(b), we can see that, with the increase of the coherent length, the value of scattering force and the oscillation amplitude of longitudinal gradient force are both obviously increased on the propagation axis. For the positive ${F_{grad, z}}$, it means the direction of longitudinal gradient force is along the $+ z$ direction and it is along the $- z$ direction for the negative ${F_{grad, z}}$. In Fig. 2(b), the values of longitudinal gradient force can equal zero at several positions for different coherent lengths, which indicates that the particle could be trapped around these zero points. Considering the influence of scattering force, we plot the sum of the longitudinal scattering force and gradient force ${F_{tot, z}}$ in Fig. 2(c). And we can clearly see that there are multiple equilibrium points that could be used to trap particles. When the coherent length is too small, however, the longitudinal gradient force is difficult to overcome the forward scattering force and the total longitudinal radiation force cannot reach zero near equilibrium positions. For simplicity, we only analyze the first two equilibrium points for trapping particles at ${z_a}$ and ${z_b}$, which are represented by lines with dotted arrows in Fig. 2(c). It shows that the particle can be longitudinally trapped at ${z_a}$ and ${z_b}$ for ${\delta _c}$ = 15 $\mu m$, 20 $\mu m$, 25 $\mu m$, $\infty$. But for ${\delta _c}$ = 10 $\mu m$, the particle can be only stably trapped at ${z_a}$, since the sum of total longitudinal radiation force is greater than zero at ${z_b}$, where the equilibrium position is impossible. The PCCAB can gradually evolve into a completely coherent circular Airy beam to trap particles at more points as the coherent length increases to ${\delta _c}$ = $\infty$. Note that the coherent length less affects the possible equilibrium positions, but greatly affects the number of trapping points and the stability of trapping. Figures 2(d) and 2(e) show the transverse gradient force ${F_{grad, r}}$ at stable equilibrium points ${z_a}$ and ${z_b}$ for different coherent lengths. It’s found that the transverse gradient force is increased as the coherent length increases. Besides, the transverse gradient force is decreased and oscillates obviously as z increases from ${z_a}$ to ${z_b}$.

Fig. 2. Radiation forces produced by PCCAB with different coherent lengths exerted on a Rayleigh particle. (a) The scattering force on the z-axis; (b) the longitudinal gradient force on the z-axis; (c) the sum of the scattering force and longitudinal gradient force on the z-axis, the equilibrium points ${z_a}$ and ${z_b}$ are represented by lines with arrows in the figure; (d) the transverse gradient force at ${z_a}$ plane; (e) the transverse gradient force at ${z_b}$ plane. The parameters for calculations are ${\lambda _0} = 1064nm,\, a = 0.1, w = 1 \mu m,\, {r_0} = 10 \mu m, R = 30 nm, P = 4 W, {n_1} = 1.59$ and ${n_2}$ = 1.33.

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The radiation force produced by PCCAB can be also controlled by relevant parameters. Figures 3(a) and 3(c) show the sum of the longitudinal scattering force and gradient force by varying the scale factor w and initial radius ${r_0}$. Here, the coherent length chosen is ${\delta _c}$ = 20 $\mu m$. From Fig. 3(a), we can see that, with the increase of w, not only the trapping positions can be moved forward, but also the longitudinal radiation force for each trap is decreased. Moreover, the interval between adjacent peaks of longitudinal radiation force is broadened obviously with the increase of w. In Fig. 3(b), we plot the transverse gradient force at the first trapping points ${z_{a1}}$, ${z_{a2}}$, ${z_{a3}}$ in Fig. 3(a) for different w, respectively. It reveals that the transverse gradient force is decreased as w increases. Figures 3(c) and 3(d) show that the initial radius can also affect the trapping positions and the magnitude of radiation forces. When ${r_0}$ increases, the trapping positions will be moved forward and radiation forces are decreased.

Fig. 3. Radiation forces produced by PCCAB with different scale parameters and initial radiuses exerted on a Rayleigh particle. (a) The sum of the scattering force and longitudinal gradient force with different w on the $z$-axis; (b) the transverse gradient force at the first trapping planes ${z_{a1}}$, ${z_{a2}}$, ${z_{a3}}$ in Fig. 3(a) ; (c) the sum of the scattering force and longitudinal gradient force with different ${r_0}$ on the $z$-axis; (d) the transverse gradient force at the first trapping planes ${z_{a4}}$, ${z_{a5}}$, ${z_{a6}}$ in Fig. 3(c). Other parameters are ${\lambda _0}$ = 1064 $nm$, $a$ = 0.1, $R$ = 30 $nm$, ${\delta _c}$ = 20 $\mu m$, $P$ = 4 W, ${n_1}$ = 1.59 and ${n_2}$ = 1.33. In Figs. 3(a) and 3(b), the initial radius is ${r_0}$ = $10\, \mu m$. In Figs. 3(c) and 3(d), the scale factor is $w$ = 1 $\mu m$.

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4. Discussion on trapping stability

In order to stably trapping particles, the backward longitudinal gradient force must be greatly larger than the forward scattering force $|{F_{grad, z}}|/|{F_{sca}}|\ge 1$. From Figs. 2(a)–2(c), we can see that the magnitude of the forward scattering force is smaller than the backward longitudinal gradient force near the trapping points for different coherent lengths. Also, the buoyancy and gravity for glass microsphere here is about ${10^{ - 6}}pN$, which is smaller than the longitudinal gradient force. Moreover, the particle always suffers the Brownian motion, the potential well generated by the gradient force must be deep enough to overcome the kinetic energy of the particle. The influence of Brownian motion can be judged by the Boltzmann factor [7,53]:

(10)$${R_b} = {e^{ - {U_{\max }}/{k_B}T}} \ll 1,$$

where ${k_B}$ is the Boltzmann constant, T is the temperature of ambient medium, ${U_{\max }}$ is the maximum depth of the potential well that can be expressed as [7]:

(11)$${U_{\max }} = \frac{{2\pi {n_2}{R^3}}}{c}\left|{\frac{{{m^2} - 1}}{{{m^2} + 2}}} \right|{I_{\max }}.$$

In this work, $T$ = 300 K, and ${I_{\max }}$ denotes the maximum intensity. For the sake of simplicity, we analyze the trapping points formed by PCCAB in Fig. 2(c) at ${z_a}$ and ${z_b}$. For a glass particle, we can calculate the Boltzmann factor ${R_b}$ for different coherent lengths. When ${\delta _c}$ = 10 $\mu m$, ${R_b}$ is about $1.74 \times {10^{ - 2}}$ for trapping point ${z_a}$, judged by Eq. (10), which is close to 1 and the particle is difficult to be stably trapped. When ${\delta _c}$ = 15 $\mu m$, ${R_b}$ is about $7.02 \times {10^{ - 4}}$ for ${z_a}$ and $2.88 \times {10^{ - 2}}$ for ${z_b}$. So, the particle can be trapped at ${z_a}$ more stably. When ${\delta _c}$ = 20 $\mu m$, ${R_b}$ is about $3.02 \times {10^{ - 5}}$ and $8.36 \times {10^{ - 3}}$ for ${z_a}$ and ${z_b}$, respectively; when ${\delta _c}$ = 25 $\mu m$, ${R_b}$ is about $2.02 \times {10^{ - 6}}$ and $3.15 \times {10^{ - 3}}$ for ${z_a}$ and ${z_b}$, respectively; when ${\delta _c}$ = $\infty$, ${R_b}$ is about $1.67 \times {10^{ - 10}}$ and $7.29 \times {10^{ - 5}}$ for ${z_a}$ and ${z_b}$, respectively. Based on the analysis above, we can see that the particle can be both stably trapped at ${z_a}$ and ${z_b}$ for ${\delta _c}$ = 20 $\mu m$, 25 $\mu m$, $\infty$. And for ${\delta _c}$ = 15 $\mu m$, only point ${z_a}$ is effective for trapping the particle. Note that the trapping stability is gradually improved with the increase of the coherent length. Besides, the particle can be trapped the most stably at equilibrium points when the coherent length is ${\delta _c}$ = $\infty$, on account of the completely spatial coherence.

5. Conclusion

In summary, we have investigated the radiation force on a dielectric particle produced by PCCAB in the Rayleigh scattering regime. The results demonstrate that different coherent lengths, scale factors and initial radiuses can greatly affect the trapping properties. It shows that there are several equilibrium points for different coherent lengths, which means that the PCCAB can be used to trap and manipulate particles. It is worth mentioning that the trapping positions are less influenced by the coherent length, but the trapping stability is greatly affected by the coherent length. With the increase of the spatial coherent length, the radiation force is increased and the particle can be stably trapped at more points. Besides, the trapping positions and ranges can be controlled by different values of w and ${r_0}$. With a larger w or ${r_0}$, the trapping positions will be moved forward and radiation forces are decreased. Moreover, the trapping region is broadened with the increase of w. Our investigation of the radiation force on a Rayleigh dielectric particle produced by the PCCAB may have advantages over the completely coherent circular Airy beams in some applications. For example, the radiation force can be modulated by controlling the coherent length in optical and biomedical micromanipulation. Also, because the partially coherent beam can be used in atmospheric turbulence to reduce pointing errors and turbulence-induced signal fading, maybe there are some potential applications in optical communication combined with the particle trapping.

Funding

Ministry of Science and Technology of the People's Republic of China (2017YFB0503100); Ministry of Education of the People's Republic of China ((2016XZZX004-01, 2017QN81005, 2018FZA5002); National Natural Science Foundation of China (11474254).

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Radiation forces on a Rayleigh particle produced by partially coherent circular Airy beams

Abstract

1. Introduction

2. Propagation characteristics of partially coherent circular Airy beams

3. Radiation forces produced by partially coherent circular Airy beams

4. Discussion on trapping stability

5. Conclusion

Funding

References

Cited By

Figures (3)

Equations (11)

Optics Express