Dual optical trap created by tightly focused circularly polarized ring Airy beam

Journal of Quantitative Spectroscopy & Radiative Transfer 244 (2020) 106851

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Dual optical trap created by tightly focused circularly polarized ring Airy beam Zaili Chen, Yunfeng Jiang∗ School of Biomedical Engineering, Wenzhou Medical University, Wenzhou, China

a r t i c l e

i n f o

Article history: Received 14 October 2019 Revised 21 January 2020 Accepted 21 January 2020 Available online 23 January 2020

a b s t r a c t The dual focus property of focused circularly polarized ring Airy beam (RAB) under the action of tightly focused lens is demonstrated in this paper. The radiation forces at two foci of tightly focused RAB are calculated, the numerical results show that the particle could be longitudinally and transversely trapped at the two foci. By varying corresponding parameters, we could control the property of two traps. The trapping force increases with NA and the scaling parameter w; and an appropriate initial radius r0 is necessary for the enhancement of either trap. The two traps could move closer as w increases or r0 decreases. To realize the dual optical trap, we should choose a smaller decaying parameter a and a larger NA, or the dual optical trap would degenerate into a single optical trap. Moreover, because of the influence of the Brownian motion and the scattering force, the size of the particle should be in a special range. © 2020 Elsevier Ltd. All rights reserved.

1. Introduction The dual focus property of Janus waves was proposed in 2016 [1]. The dual focus property of focused ring Airy beams (RAB) is demonstrated in experiment and applied in strong field science for phase memory preserving [2,3]. It’s well known that RAB could abruptly autofocus at a real focus in +z direction in the free space, but it has another virtual focus in –z direction. So, under the action of a lens, two images (i.e., two distinct foci) could be formed by the real and virtual focus [1]. This is called dual focus property in this paper. The abruptly autofocusing property of RAB or other similar beams is widely investigated in these years [4–16]. But there are few studies about the dual focus property of focused RAB [17]. On the other hand, optical tweezers have become an important tool in many fields since they have been first invented in 1986 by Ashkin [18–23]. The application of spatial light modulator (SLM) in optical tweezers has further improved this technique [24–26]. By using SLM, many kinds of special laser beams could be introduced into optical tweezer system, aiming to improve its performance or realize new optical micromanipulation technique. For example, trapping or rotating particles by optical vortex [27– 29], transversely clearing particles by Airy beams [30], enhancing the trapping force by radially polarized beam [31] or abruptly autofocusing beams [32–34] and so on [35–42]. To our best knowl∗

Corresponding author. E-mail address: [email protected] (Y. Jiang).

https://doi.org/10.1016/j.jqsrt.2020.106851 0022-4073/© 2020 Elsevier Ltd. All rights reserved.

edge, there are no investigations about the application of dual focus property of focused RAB in optical tweezers. By using the dual focus property of focused RAB, we could likely simplify the setup of dual optical tweezers. Usually, the dual optical tweezer is constructed by two laser sources, or one laser source with some beam splitters [43–45]. The computer-generated hologram could be applied to create multiple optical tweezers in one plane, but not in several planes [25]. By using Bessel beam, one can trap several particles at different planes by one light beam [35]. But this trapping is only 2D, which means that the longitudinal gradient force is needed. So, the dual focus property of focused RAB might have important application in the constructing dual optical traps. Because an objective lens is always used in optical tweezers to create optical traps, we need to extend the previous dual focus property from paraxial focusing region [1] to tightly focusing region first. The previous investigation shows that different kinds of focal intensity patterns could be generated by tightly focused radially polarized RAB with optical vortex [46,47]; but the dual focus property of tightly focused RAB and its application in optical trapping needs further investigation. In this paper, we first calculate the propagation characteristics of tightly focused circularly polarized RAB, and demonstrate the dual focus property under the action of tightly focusing lens. Then, radiation forces produced by tightly focused RAB on high refractive index Rayleigh particle are calculated, and the dual optical trap at two foci are theoretically proved. The influences of corresponding parameters on the property of dual optical trap are also analyzed in detail.

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Z. Chen and Y. Jiang / Journal of Quantitative Spectroscopy & Radiative Transfer 244 (2020) 106851

kf Ez (r, φ , z ) = √ eiφ 2

 θm 0

2

sin

 θ cos θ P (θ )eikz f cos θ J1 (α )dθ ,

(3c)

where k is the wave number, α = krsin θ , zf = z − f. Substituting Eq. (2) into Eq. (3), we can get the electric fields in the output plane. The output intensity in this paper is defined as,

 2

I (r, φ , z ) = |Er |2 + Eφ  + |Ez |2 . Fig. 1. The incident RAB (a) and the tightly focusing lens system (b). The focus of the lens is f, the distance between the focus plane and output plane is zf .

2. Dual focus property of tightly focused circularly polarized RAB The left hand circularly polarized beam could be expressed as a superposition of radial and azimuthal polarized beams [48,49]:



ELHC = P (ρ )e

iϕ

eˆρ + ieˆϕ √ 2



,

(1)

where eˆρ and eˆϕ denote the radially and azimuthally direction, respectively; P(ρ ) is the pupil apodization function in the initial plane of focusing system shown in Fig. 1. We assume the incident beam is circularly polarized RAB [4] and use the formula ρ = f sin θ , so



r0 − f sin θ P (θ ) = C · Ai w





r0 − f sin θ exp a w



,

(2)

where r0 is the initial radius of RAB; a is the decaying parameter; w is the scaling parameter; f is the focal length; θ varies from 0 to θ m , where θ m is the maximum angle of NA; C is a constant related with the incident power. According to the theory of Richards and Wolf, the tightly focusing of left hand circularly polarized RAB could be expressed as [49–51],

Eφ (r, φ , z ) =

Er (r, φ , z ) =

  k f e iφ θm sin θ cos θ P (θ )eikz f cos θ [(cos θ − 1 )J2 (α ) √ 2 2 0 + (cos θ + 1 )J0 (α )]dθ , (3a)

  −ik f eiφ θm sin θ cos θ P (θ )eikz f cos θ [(− cos θ + 1 )J2 (α ) √ 2 2 0 + (cos θ + 1 )J0 (α )]dθ , (3b)

(4)

In our simulation, we choose r0 =1 mm, w = 50 μm, a = 0.08, f = 2 mm, NA=0.85, unless stated otherwise; the wavelength is 632.8 nm, the incident power is 1 W. The dual focus property of focused RAB under the tightly focusing condition is demonstrated in Fig. 2. The two foci in Fig. 2(a) are located at f1 = 1.989 mm and f2 = 2.011 mm, and the intensity distributions at the two focus planes are nearly the same Fig. 2(b). Note that positions of two foci under the tightly focusing condition is different with the paraxial results, which could be calculated by the formula in [1] (the paraxial result is that f1 = 1.982 mm, f2 = 2.018 mm). Comparing with the paraxial focusing, under the action of tightly focusing lens, f1 and f2 become closer to the focal length f. Changes of the dual focus property with different parameters are shown in Fig. 3. In this figure, we assume the focal intensity at f1 and f2 are If 1 and If 2 , respectively. And the distance between two foci is f = |f2 –f1 |. From Fig. 3(a), we can see that two foci f1 , f2 would not change with a; the two focal intensities If 1 and If 2 increase as a decreases. When a is too large, RAB tends to focus at lens focus, the intensity at z = f would exceed If 1 and If 2 when a is 0.18. With a larger NA, more Airy light rings can enter the lens, so the dual focus property would be more obvious, as we can see in Fig. 3(b). From this figure, we can also find that If 1 , If 2 both increase with NA. Fig. 3(c) shows that as r0 increases, f would decrease, and the intensity If 1 and If 2 would decrease. The influence of w is different, as shown in Fig. 3(d), as w increases, f would decrease, but the focal intensities would increase. 3. Radiation forces on Rayleigh particle at two foci of focused RAB The Rayleigh dielectric particle, whose radius ar << λ, can be simply considered as a point dipole in the light fields. The polarizability α of Rayleigh particles can be expressed as [52],

α = 4π a3r

ε p − εm , ε p + 2εm

(5)

Fig. 2. Propagation dynamics of tightly focused circularly polarized RAB (a); the intensity distributions at two focus planes. The corresponding parameters are r0 = 1 mm, w = 50 μm, a = 0.08, NA = 0.85.

Z. Chen and Y. Jiang / Journal of Quantitative Spectroscopy & Radiative Transfer 244 (2020) 106851

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Fig. 3. Changes of intensity along z axis: (a) with different a; (b) with different NA; (c) with different r0 ; (d) with different w. Other parameters are the same as Fig. 2.

where ar is the radius of the particle, ε p is the dielectric constant of the Rayleigh particle and ε m is the dielectric constant of the surrounding medium. The gradient force Fg and the scattering force Fs can be calculated by the formulae [53]:

Fg =

1 ε0 εm Re(α )∇ I, 4

(6)

Fs =

ε0 εm3 k4  2  α I, 12π

(7)



where ε 0 is the dielectric constant in vacuum. Usually, the gradient force, which is the main trapping force, directs to the focal position, and the scattering force is along the propagation direction. By substituting Eq. (4) into Eqs. (6) and (7), we can calculate radiation forces produced by focused RAB. In our numerical calculation, we choose ar = 50 nm, the surrounding medium is water (ε m = 1.33), the particle is made of glass (ε p = 1.59). From Fig. 4, we can see that the glass particle could be longitudinally and transversely trapped at z = za and z = zb , which are defined as trapping positions in this paper. These two positions are nearly the two foci of RAB in Fig. 3(a) (i.e., za ≈f1 , zb ≈f2 ). Because the scattering force is comparable with the longitudinal gradient force Fig. 4(a) and (b), though there are several equivalence positions in the longitudinal gradient force distribution Fig. 4(a), only the two foci could trap particles Fig. 4(c). We assume the maximum trapping forces along the longitudinal direction and transverse direction at za as Fzma and Frma , respectively. Similarly, the maximum trapping forces for z = zb are Fzmb and Frmb , respectively. Note that radiation force distributions are asymmetric, so the maximum trapping force should be defined as the smaller one between the positive force and the negative force in the trapping region. Fig. 4 also shows the influence of decaying parameter a on the dual optical trap. As we can see in Fig. 4(c), the position of za and zb would not change with a; and the longitudinal trapping force Fzma is larger than Fzmb when a is small (a = 0.06); but as a increases, Fzmb would exceed Fzma , for it decreases more slowly. The transverse trapping force Frma changes little with a Fig. 4(d); Frmb decreases with the increase of a, but keeps larger than Frma Fig. 4(e). So, when a is mall, the longitudinal trapping at za is stronger, but the transverse trapping of za is weaker; as a increases, the trapping force at zb becomes larger. When a is large enough,

Fig. 4. The distributions of radiation forces on glass particle with different a: (a) the longitudinal gradient force; (b) the scattering force; (c) the sum of the gradient force and the scattering force, za and zb are the two optical traps; (d) the transverse gradient force at za ; (e) the transverse gradient force at zb .

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Z. Chen and Y. Jiang / Journal of Quantitative Spectroscopy & Radiative Transfer 244 (2020) 106851

Fig. 5. Changes of trapping force with different NA: (a) the longitudinal trapping force; (b) the transverse gradient force at za ; (c) the transverse gradient force at zb .

the dual optical trap would degenerate into a single optical trap at z = zb , as we can see in Fig. 4(c) when a = 0.18. Fig. 5 shows changes of radiation forces with NA. From these figures, we can see that as NA increases, the longitudinal trapping force increases and the trapping range decreases; the transverse trapping force at zb also increases with NA. So, the optical trapping would be more stable with a high NA. Similar with the case of large a, when NA is small, there are less Airy light rings entering the objective lens, so the beam may not exhibit dual focus property. As shown in Fig. 5(a), when NA=0.75, the gradient force at za is not large enough to overcome the scattering force, so the particle could not be trapped at za . Fig. 6 shows changes of trapping forces with the initial radius r0 . From Fig. 6(a), we can find that trapping positions za and zb move closer as r0 increases; and more trapping positions would appear for small r0 . The change of trapping force with r0 is more complicated than other parameters. Generally, Fzma and Fzmb increase as r0 decreases. But for the trap zb , Fzmb when r0 =1 mm is larger than that when r0 = 0.9 mm Fig. 6(a). For the transverse trapping force at za , Frma with r0 = 1 mm is the smallest Fig. 6(b), but in the second optical trap zb , this force with r0 = 1 mm become the largest one Fig. 6(c). So, an appropriate r0 should be chosen to enhance the longitudinal and transverse trapping force in practical application. Fig. 7 shows changes of trapping forces with the scaling parameter w. As w increases, the two trapping positions would move closer, the longitudinal and transverse trapping forces at za and zb would all increase with w. 4. Stability analysis For stable trapping, the trapping force must be large enough to overcome the longitudinal scattering force, the influence of Brownian motion and the gravity of the particle [18]. As we can see in Fig. 4(c), the backward longitudinal gradient forces at za and zb are both large enough to conquer the forward scattering force. The

Fig. 6. Changes of trapping force with different r0 : (a) the longitudinal trapping force; (b) the transverse gradient force at za ; (c) the transverse gradient force at zb .

Fig. 7. Changes of trapping force with different w: (a) the longitudinal trapping force; (b) the transverse gradient force at za ; (c) the transverse gradient force at zb .

gravity Fmg in this paper is nearly 10−5 pN, which could be neglected as shown in Fig. 8(a). The influence of the Brownian motion could be described as Brownian force, which could be calculated by the formula [54],

|Fb | =



12π ηar kB T ,

(8)

Z. Chen and Y. Jiang / Journal of Quantitative Spectroscopy & Radiative Transfer 244 (2020) 106851

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Fig. 8. Longitudinal forces (a) and transverse forces (b) on particle with different size.

where η = 8.0 × 10−4 Pa s, is the viscosity of surrounding medium, ar is the radius of the particle, kB is the Boltzmann constant, T = 300 K is the temperature. The magnitude of the trapping force at za and zb , Brownian force and the gravity are compared in Fig. 8. From this figure, we can see that to realize stable dual optical trapping by tightly focused RAB, the radius of the particle should be in a special range, 10 nm < ar < 64 nm. And the longitudinal and transverse trapping forces of optical trap zb are generally larger than those of za . When ar < 10 nm, the influence of Brownian motion is rather significant; when 64 nm < ar < 75 nm, there are only one stable optical trap at zb ; when ar > 75 nm, none of the particle could be trapped because of the large scattering force. 5. Conclusions In summary, the dual focus property of circularly polarized RAB under the action of a tightly focused lens are demonstrated, and then the radiation forces produced at two foci on the high refractive index particle are calculated and analyzed detailedly in this paper. The results show that an effective dual optical trap could be formed by using focused RAB; and by varying corresponding parameters, we can control the property of the two optical traps. Comparing with the paraxial focusing, the circular polarized RAB would focus at two different positions under tightly focused conditions. As NA increases or decaying parameter a decreases, more Airy light rings could enter the lens, two focal intensities would increase and the dual focus property is more obvious. The two foci would move closer as initial radius r0 or scaling parameter w increases; the focal intensity decreases with r0 , but increases with w. The high refractive index particle could be longitudinally and transversely trapped at two focal positions, which might be useful for dual optical trapping. By varying corresponding parameters, we could control both traps. With a larger NA or a smaller a, the trapping forces at two foci would be greatly enhanced. When NA is too small or a is too large, the dual optical trap might degenerate into a single optical trap. As w or r0 increases, two trapping positions would move closer, and the trapping force increases with w. The influence of r0 on trapping forces is more complicated, an appropriate r0 should be chosen to enhance the force of either trap. The dual optical trap generated by tightly focused RAB could only trap particle whose size is in a special range. When the particle is too small, the disturbance from Brownian motion would be obvious; when the particle is too large, the scattering force would be difficult to overcome, and the first trap is more likely to be invalid. We believe that this investigation would be helpful for the application of the dual focus property of focused RAB in optical tweezers or other optical micromanipulation techniques. The re-

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