1. Introduction

Since Ashikin et al. [1,2] first demonstrated that the radiation pressure of laser light can be used to trap small particles, many novel applications have been developed in various domains of science, especially in biophysical science to trap living cells and organelles [3,4], and in atomic physics to manipulate neutral atoms [5,6]. The radiation force exerted on a particle originates from the total exchange of momentum and energy between the incident photons and particles. Conventionally, the optical traps or optical tweezers are mainly constructed by highly focused laser beam to achieve a stable single-beam axial gradient force trap. Among the single-beam gradient force traps (SGFTs), the focused Gaussian beams are used the most, which trap the high-index particles (with the refractive index higher than the ambient) in the beam while expel the low-index particles (with the refractive index lower than the ambient) out of the beam. The trapping characteristics for other SGFTs [7–15] have also been explored, such as bottle beams [7], zero-order Bessel beams [8], self-focused laser beams [9], and so on.

The concept of specular cross-spectral density function (CSD) was introduced by Gori and associates in 1988 [16]. By definition, a cross-spectral density function is to be specular if it is even with respect to $x_1$ (in one dimension), which implies that it is even also with respect to $x_2$. Some related properties of such optical fields have been studied, among which the field is spatially completely coherent between the points located symmetrically with respect to the origin of the coordinate axis. Besides, the specularity property is conserved during the field propagation. It is also proposed that specular fields can be generated starting from any partially coherent field by use of Porro-prism interferometer. In spite of the peculiar properties, little work has been done since its introduction except the study of Ponomarenko and Agrawal [17]. Not long ago, it has been experimentally demonstrated that specular and antispecular light beams [18] are generated by passing a Gaussian Schell-model (GSM) beam through a wavefront-folding interferometer (WFI).

In this paper, we consider the field generated by passing an isotropic GSM beam through a WFI. The output field of the WFI has various intensity distributions for different phase difference, including the central peak shape and doughnut shape. Then, the field is focused onto a particle by a lens and the radiation force produced by the highly focused field is investigated. It is found that the new fields can be used to trap high-index particles and the position for stable trapping can be modulated by adjusting the phase difference. Finally, we analyze the condition for stable trapping.

2. Fields through a lens

The wavefront-folding interferometer [19] illustrated in Fig. 1 is mainly used to measure the spatial coherence properties of partially coherent fields. Key elements of the interferometer are the two perpendicularly oriented right-angle prisms ${\text{PR}}_x$ and ${\text{PR}}_y$, which retroreflect the incident field in the $x$ and $y$ direction, respectively. The prisms are slightly tilted with respect to the optical axis to observe interference fringes. However, we assume the device perfectly aligned to generate a field with specular and antispecular cross-spectral density function.

 

Fig. 1 The wavefront-folding interferometer. S is the source, BS is a non-polarizing beam splitter, PR_x and PR_y are right-angle prisms.

Download Full Size | PPT Slide | PDF

Suppose that a single partially coherent beam with the scalar optical field $E_0\left(x^{\prime },y^{\prime }\right)$ is incident on the interferometer. This beam is split into two light beams by the beam splitter. Due to the folding effect of the right-angle prisms, the field $E\left(x^{\prime },y^{\prime }\right)$ at the output of the interferometer corresponds to the superposition of the two beams [18]

We consider the output field of the interferometer as a secondary source, which propagates through a lens system as shown in Fig. 2. The transfer matrix of this system is

 

Fig. 2 Schematics of the output field of a WFI focused onto a particle. $F$ is the geometrical focus.

Download Full Size | PPT Slide | PDF

Assume that the field entering the interferometer is an isotropic GSM beam [20], whose cross-spectral density function is

On substituting from Eq. (4) into Eq. (2), we can obtain the CSD of the secondary source

 

Fig. 3 Intensity distributions of the GSM, specular and anti-specular fields at the (a), (c)-(e) input plane and (b) focal plane. Figures (c)-(e) correspond to $\varphi \text{=}0$, $\varphi \text{=}\pi /2$ and $\varphi \text{=}\pi$ for different values of ${\sigma }_0$, respectively. (c)-(e) black solid curve, ${\sigma }_0/{\omega }_0\text{=}0.5$; red short dashed curve, ${\sigma }_0/{\omega }_0\text{=1}$; blue short dashed-dotted curve, ${\sigma }_0/{\omega }_0\text{=2}$.

Download Full Size | PPT Slide | PDF

By employing the generalized Huygens-Fresnel principle, the CSD of this field in any transverse plane $z>0$ is given by

3. Radiation force produced by the highly focused fields

In the limit that the radius $a$ of the particle is much smaller than the wavelength of the light, i.e., $a\le \lambda /20$, the Rayleigh approximation is applicable and the particle could be treated as a simple point dipole. Under this condition the radiation force acting on the particle can be described by two components: the scattering force and the gradient force which arise from momentum changes of the light due to the scattering by the particle and the Lorenz force acting on the particle, respectively. The scattering force is proportional to the intensity of the beam and is along the beam-propagation direction, which is given by

Substitute Eqs. (8) and (14) into Eq. (15), we can obtain the transverse component $F_{Grad,x}\left(r\right)$ and the longitudinal component $F_{Grad,z}\left(r\right)$ of the gradient force at the output plane

In Fig. 4, we demonstrate the radiation forces produced by highly focused specular ($\varphi \text{=}0$) and GSM ($\varphi \text{=}\pi /2$) beams on high-index particles ($n_p\text{=1}\text{.592}$). For the transverse gradient force $F_{Grad,x}$, the positive value means the force is along the $\text{+}x$ direction while the negative means it is along the $-x$ direction. Similarly, the positive (negative) $F_{Grad,z}$ means that the longitudinal gradient force is along the $\text{+}z$ ($-z$) direction. It can be seen from Figs. 4(a)-4(c) that there is a stable equilibrium at the focus for the high-index particles illuminated by highly focused specular or GSM beam. The radiation forces produced by the specular beam are analogous to that produced by the GSM beam due to their similar intensity profiles and evolutions. However, the transverse gradient force, the longitudinal gradient force and the scattering force produced by the specular beam are larger than that produced by the GSM beam.

 

Fig. 4 Radiation forces produced by highly focused specular ($\varphi \text{=}0$) and GSM ($\varphi \text{=}\pi /2$) beams on high-index particles ($n_p\text{=1}\text{.592}$). Black solid curve, the specular beam; red dashed curve, the GSM beam.

Download Full Size | PPT Slide | PDF

Figure 5 demonstrates the radiation forces produced by highly focused anti-specular beams ($\varphi \text{=}\pi$) on high-index particles ($n_p\text{=1}\text{.592}$) and low-index particles ($n_p\text{=1}$). For the particles with $m<1$, the longitudinal gradient force $F_{Grad,z}$ maintains zero because the doughnut shape of the anti-specular beam holds during the beam propagation. Consequently, the low-index particles accelerate toward the beam propagating direction which are drawn by the scattering force. However, for the particles with $m>1$, there are two stable equilibrium points at the positions $x=\pm 0.168\text{μm}$in the focal plane. Therefore, one can use the highly focused anti-specular beam to trap or manipulate high-index particles nearby the focus.

 

Fig. 5 Radiation forces produced by highly focused anti-specular beams ($\varphi \text{=}\pi$) on high-index particles ($m>1$) and low-index particles ($m<1$). Black solid curve for the particles with $n_p\text{=1}\text{.592}$, red dashed curve for the particles with $n_p\text{=1}$.

Download Full Size | PPT Slide | PDF

In our case, the analysis of Eq. (6) reveals that in the interval $0\le \varphi \le 2\pi /3$ the field has a central peak, but as $\varphi$ increases in the interval $2\pi /3<\varphi \le \pi$, a central dip gradually appears. Consequently, the radiation force exerted on the particles changes as we adjust the phase difference. In Fig. 6, we illustrate the radiation forces on the particles with $m>1$ for the three values of $\varphi =\text{2}\pi /3,\text{3}\pi /4,\text{5}\pi /6.$ It can be seen from Figs. 6(a) and 6(d) that for the case $\varphi =\text{2}\pi /3$, the particles with $m>1$ are stably trapped at the focus. However, as $\varphi$ increases, the trapping becomes unstable at first [see Figs. 6(b) and 6(e)] and then a new stable equilibrium is generated near the focus [see Figs. 6(c) and 6(f)]. For the case $\varphi \text{=}\text{5}\pi /6$, the particles could be trapped at the positions $x=\pm 0.15\text{μm}$in the focal plane. Thus, it can be concluded that the position for stable trapping shifts from the focus to the position nearby the focus with increasing the phase difference $\varphi$.

 

Fig. 6 Radiation forces exerted on the particles with $m>1$ for different values of $\varphi$. (a), (d), $\varphi \text{=}\text{2}\pi /3$; (b), (e), $\varphi \text{=}\text{3}\pi /4$; (c), (f), $\varphi \text{=}\text{5}\pi /6$.

Download Full Size | PPT Slide | PDF

4. Analysis of trapping stability

In Figs. 7-8, we plot the maximum transverse gradient force $F_{Grad,x}$ at two points $A,B$, the maximum longitudinal gradient force $F_{Grad,z}$ at other two points $C,D$, and the scattering force $F_{Scat}$ at points $C,D$. In order to trap particles stably, two necessary conditions must be satisfied for a single-beam trap. One necessary criterion for stable trapping is that the backward longitudinal gradient force must be greatly larger than the forward scattering force, i.e., $R={\left|F_{Grad,z}\right|}_{C,D}^{\max }/{\left|F_{Scat}\right|}_{C,D}>1$. It is obvious from Figs. 7-8 that the magnitude of the scattering force is much smaller than the axial gradient force, which is also revealed in Figs. 4-6.

 

Fig. 7 Dependence of ${\left|F_{Grad,x}\right|}_{A,B}^{\max }$ (black solid curve), ${\left|F_{Grad,z}\right|}_{C,D}^{\max }$ (red dashed curve), ${\left|F_{Scat}\right|}_{C,D}$ (blue dotted-dashed curve), and $\left|{\overrightarrow{F}}_b\right|$ (green dotted curve) on the radius $a$ for different $\varphi$. (a) $\varphi \text{=}0$; (b) $\varphi \text{=}\pi /2$; (c) $\varphi \text{=}2\pi /3$; (c) $\varphi \text{=}\pi$.

Download Full Size | PPT Slide | PDF

 

Fig. 8 Dependence of ${\left|F_{Grad,x}\right|}_{A,B}^{\max }$ (black solid curve), ${\left|F_{Grad,z}\right|}_{C,D}^{\max }$ (red dashed curve), and ${\left|F_{Scat}\right|}_{C,D}$ (green dotted-dashed curve) on ${\sigma }_0$ for different $\varphi$; the horizontal dotted lines denote $\left|{\overrightarrow{F}}_b\right|$. (a) $\varphi \text{=}0$; (b) $\varphi \text{=}\pi /2$.

Download Full Size | PPT Slide | PDF

On the other hand, the Brownian motion of the particles will strongly affect the stability of trapping when the particles are very small. The magnitude of the Brownian force is given by [21] $\left|{\overrightarrow{F}}_b\right|=\sqrt{12\pi \eta a{k}_{B}T}$, where $\eta =7.977\times {10}^{-4}\text{Pa}\cdot \text{s}$ is the viscosity for water at the room temperature $T=300K$ and $k_B$ being the Boltzmann constant. To judge whether the trapping is stable or not, we compare the radiation forces with $\left|{\overrightarrow{F}}_b\right|$. Figure 7 demonstrates the dependence of ${\left|F_{Grad,x}\right|}_{A,B}^{\max }$, ${\left|F_{Grad,z}\right|}_{C,D}^{\max }$, ${\left|F_{Scat}\right|}_{C,D}$ and $\left|{\overrightarrow{F}}_b\right|$ on the radius $a$ of the particle for different $\varphi$. It can be found that the low limit of $a$ to have a stable trapping for $\varphi \text{=0},\pi /2,2\pi /3,\pi$ is $3.8,\text{5}\text{.0}\text{,}\text{6}\text{.6,}\text{8}\text{.1}\text{nm}$, respectively. On the other hand, the up limit of $a$ is $\text{31}\text{.6}\text{nm}$ because the Rayleigh approximation is no more valid if the particle size is too large. Thus, we can conclude that the range of particle sizes to have a stable trapping can be modulated by changing the phase difference.

Figure 8 illustrates the dependence of ${\left|F_{Grad,x}\right|}_{A,B}^{\max }$, ${\left|F_{Grad,z}\right|}_{C,D}^{\max }$, and ${\left|F_{Scat}\right|}_{C,D}$ on ${\sigma }_0$ for different $\varphi$. It can be seen that the magnitude of the radiation forces, including $F_{Grad,x}$, $F_{Grad,z}$ and $F_{Scat}$, increases quickly as the correlation length ${\sigma }_0$ increases, also indicated in [22]. The cross point 1 denotes the low limit of ${\sigma }_0$ for stable trapping. It is found that the low limit of ${\sigma }_0/w_0$ to have a stable trapping for $\varphi \text{=0}$ and $\pi /2$ is 0.15 and 0.19, respectively. When ${\sigma }_0$ is smaller than the cross point 1, the beam is unable to trap the particles.

The second condition for stable trapping can be given by using the Boltzmann factor [2]:

5. Conclusions

In this paper we have considered the field generated by passing an isotropic GSM beam through a WFI. It was found that the output field of the WFI has a variety of CSD functions for different phase difference, including the specular, anti-specular and GSM forms. Then, we have investigated the radiation forces on Rayleigh dielectric particle produced by such highly focused fields. The numerical results demonstrate that the new fields can be used to trap high-index particles at the focus for the specular case and nearby the focus for the anti-specular case. It is further revealed that the position, the range of particle sizes and the low limit of correlation length for stable trapping could be modulated by adjusting the phase difference. Our results may have applications in particle trapping and manipulation.