We consider a class of fields generated by passing an isotropic Gaussian Schell-model beam through a wavefront-folding interferometer. The output field has various intensity profiles for different phase differences, including the central peak and doughnut shapes. The radiation force on a Rayleigh dielectric particle produced by the highly focused fields is investigated. 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.
© 2016 Optical Society of America
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 , zero-order Bessel beams , self-focused laser beams , and so on.
The concept of specular cross-spectral density function (CSD) was introduced by Gori and associates in 1988 . By definition, a cross-spectral density function is to be specular if it is even with respect to (in one dimension), which implies that it is even also with respect to . 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 . Not long ago, it has been experimentally demonstrated that specular and antispecular light beams  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  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 and , which retroreflect the incident field in the and 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.
Suppose that a single partially coherent beam with the scalar optical field 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 at the output of the interferometer corresponds to the superposition of the two beams Eq. (2), whatever the partially coherent field of the input is, the CSD of the field at the output is specular if and anti-specular if , where is an integer.
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
Assume that the field entering the interferometer is an isotropic GSM beam , whose cross-spectral density function isEquations (5) and (6) show that the new secondary source is no longer a GSM source, but constituted by the superposition of two sources, only one of which is the GSM source. However, in the particular case that , the new source is exactly the GSM source again, i.e., The corresponding intensity profiles are demonstrated in Fig. 3 for the three values of the parameter It is seen from Fig. 3(a) that in the specular case () the field has a sharper central peak than the GSM beam while in the antispecular case () it has a dark notch in the center. As illustrated in Figs. 3(c) and 3(e), one can find that the width of the peak or notch could be adjusted by changing the correlation width of the incident field, which differs from the GSM beam shown in Fig. 3(d).
By employing the generalized Huygens-Fresnel principle, the CSD of this field in any transverse plane is given byEq. (7) that the specularity and anti-specularity properties of the CSD are always conserved during the field propagation. The contrast of Figs. 3(a) and 3(b) indicates that the central peak distribution for the case and the dark-centered distribution for the case always hold. The beams are shape-invariant during propagation because and grow at the same rate due to the isotropy of the incident beam.
3. Radiation force produced by the highly focused fields
In the limit that the radius of the particle is much smaller than the wavelength of the light, i.e., , 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
In Fig. 4, we demonstrate the radiation forces produced by highly focused specular () and GSM () beams on high-index particles (). For the transverse gradient force , the positive value means the force is along the direction while the negative means it is along the direction. Similarly, the positive (negative) means that the longitudinal gradient force is along the () 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.
Figure 5 demonstrates the radiation forces produced by highly focused anti-specular beams () on high-index particles () and low-index particles (). For the particles with , the longitudinal gradient force 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 , there are two stable equilibrium points at the positions in the focal plane. Therefore, one can use the highly focused anti-specular beam to trap or manipulate high-index particles nearby the focus.
In our case, the analysis of Eq. (6) reveals that in the interval the field has a central peak, but as increases in the interval , 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 for the three values of It can be seen from Figs. 6(a) and 6(d) that for the case , the particles with are stably trapped at the focus. However, as 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 , the particles could be trapped at the positions 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 .
4. Analysis of trapping stability
In Figs. 7-8, we plot the maximum transverse gradient force at two points , the maximum longitudinal gradient force at other two points , and the scattering force at points . 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., . 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.
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  , where is the viscosity for water at the room temperature and being the Boltzmann constant. To judge whether the trapping is stable or not, we compare the radiation forces with . Figure 7 demonstrates the dependence of , , and on the radius of the particle for different . It can be found that the low limit of to have a stable trapping for is , respectively. On the other hand, the up limit of is 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 , , and on for different . It can be seen that the magnitude of the radiation forces, including , and , increases quickly as the correlation length increases, also indicated in . The cross point 1 denotes the low limit of for stable trapping. It is found that the low limit of to have a stable trapping for and is 0.15 and 0.19, respectively. When 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 :Equation (18) means that the potential well generated by the gradient force must be much larger than the kinetic energy of the particle. At the room temperature of for the particles with , for the specular beam, for the GSM beam. It is obvious that in our case the Brownian motion can be overcome and the particles can be stably trapped by the highly focused fields.
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.
This work was supported by the National Natural Science Foundation of China (NSFC) (11274273 and 11474253).
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