## Abstract

Morphology-dependent resonance (MDR) of the optical forces for a particle illuminated by Airy beams is investigated with respect to its internal field distribution. We find the ring structures arising from the resonance transform significantly with the parametric evolution of Airy evanescent wave, and the interference of the internal waves have a great impact on the *Q* factor and the background of the resonant peak, but it’s not proper for Airy transmitted wave. The multiple reflections of the evanescent wave between the particle and the interface are also investigated, which show significant impacts on the region where the energy concentrate in.

© 2013 OSA

## 1. Introduction

Since the first demonstration of optical trapping of a dielectric particle by Ashkin [1], optical tweezers have been widely used for manipulating micro-particles without mechanical contact [2–6]. The observed variation of the optical forces as a function of the wavelength or the size of a spherical particle shows a regular series of resonance peaks which is generally known as Morphology-dependent resonance (MDR) [7–12]. Extensive researches have been made on this phenomenon, such as MDRs in a dielectric sphere with tiny inclusions [13], in stimulated Raman scattering [14], in a bisphere system [15], and in a plane evanescent wave [16].

Recently, the Airy beam which was first experimentally realized in 2007 [17, 18], can be well applied into the conventional optical tweezers [19, 20], which can avoid divergence and diffraction within a certain propagating distance for its unique features: “non-diffracting” and transverse accelerations. A. V. Novitsky et al. analyzed the dynamics of nonparaxial Airy beams which show the role of evanescent waves [21]. And we have investigated the optical forces exerted on a Mie particle in the Airy evanescent field theoretically [22]. Here, we present a systematic study on the MDRs of the optical forces for a particle illuminated by an Airy beam passes through the interface with respect to the electric field distribution inside the particle which reflects the motions of the internal wave.

We find the ring structures arising from the resonances change significantly with different incident angle and transmitted distance for Airy evanescent wave, and the interference strength of the internal wave have a great influence on the *Q* factor and the background of the resonant peak. E.g., travelling wave patterns correspond to high *Q* peaks, while strong interference patterns for high background peaks. But the key structure of the internal field for the Airy transmitted wave won’t change. The quality factor *Q* of the resonant peaks for a damped wave is much larger than that for a propagating wave. The multiple reflections of the Airy evanescent wave between the particle and the interface are also investigated, which show significant impacts on the region where the energy concentrate in.

## 2. Theory and description

The schematic diagram of the scattering problem is shown in Fig. 1
. The origin of coordinate system coincides with the center of the spherical particle at a distance *d* from the interface. The refractive indices of the mediums below and above interface are *n*_{1} = 1.5 and *n*_{2} = 1.0, representing the glass-air interface. The refractive index of dielectric sphere is *n*_{3}. The parameters of the incident two-dimensional (2D) Airy beam are chosen as: wavelength *λ* = 750 nm, the characteristic lengths *x*_{0} = *y*_{0} = 2 μm, the aperture coefficient which determines the beam propagating distance *a*_{0} = 0.1. The Airy beam center is at (*x _{c}*,

*y*,

_{c}*z*). The input power

_{c}*P =*1 W. We take

*x*-

*z*plane as the plane of incidence and ($\widehat{x},\widehat{y},\widehat{z}$) as unit vectors.

For a perpendicular polarized 2D Airy beam in *y*-direction, the vector potential **A** can be expressed in terms of its angular spectrum representation:

*C*is the normalization factor, (

*n*) denote the dimensionless direction cosines of the wave vector

_{x}, n_{y}, n_{z}**k**along the Cartesian coordinate axes. The integration is over the domain of ${s}_{x}^{2}+{s}_{y}^{2}<1$and ${n}_{z}>0.$

_{1}The electromagnetic fields **E** and **H** can be derived through the Maxwell equations in the Gaussian system of units:

*p*polarization and

*s*polarization, respectively,

On paraxial condition, ${n}_{x}\approx \mathrm{sin}{\theta}_{1},\text{\hspace{0.17em}}{n}_{y}\approx 0,$ if ${\theta}_{1}>{\theta}_{crit},\text{\hspace{0.17em}}\left({\theta}_{crit}={\mathrm{sin}}^{-1}\left({n}_{2}/{n}_{1}\right)\approx 0.73\text{\hspace{0.17em}}\text{rad}\right),$ the parameter $\xi $ will be an imaginary number, the incident Airy beam is totally reflected on the interface and an evanescent electromagnetic wave is generated on the side of the medium *n*_{2}. In this case, the integral domain is${\left({n}_{1}/{n}_{2}\right)}^{2}\left({n}_{x}^{2}+{n}_{y}^{2}\right)>1.$

By substituting the expressions in Eq. (9)–(12) into the Arbitrary-Beam theory (ABT) [23, 24], after using a great deal of recursions and orthogonal relationships, the coefficients ${A}_{lm}$and ${B}_{lm}$ of the expansions of the Airy transmitted field can be derived as follows:

*l*th scattered electric wave and magnetic wave term by the particle respectively are found to be:

*a*,

*n*

_{3}is radius and refractive index of the particle, respectively. ${\xi}_{l}^{\left(1\right)}={\psi}_{l}-i{\chi}_{l},$${\psi}_{l}$and ${\chi}_{l}$ are the Riccati-Bessel functions.

So we derive the force expressions for the Airy transmitted wave based on ABT. It is known that the MDRs of the optical forces originate from the specific coefficients ${a}_{l}$and ${b}_{l}$of partial scattering waves in resonance, which is $\mathrm{Re}\left({a}_{l}\right)=1$and $\mathrm{Im}\left({a}_{l}\right)=0,$ or $\mathrm{Re}\left({b}_{l}\right)=1$and $\mathrm{Im}\left({b}_{l}\right)=0$ [9].

## 3. Results and discussions

The data programs were written in double-precision FORTRAN. The coefficients *A*_{lm} and *B*_{lm} appear in Eq. (13)–(14) obtained by two-dimensional integral. We computed the integral with trapezoidal method. The integral range and step size have been chosen carefully to ensure the absolute errors less than 10^{−6}. In general, the expression for the radiation force, written symbolically as ${F}_{l}={\displaystyle \sum {f}_{l}},$has to be truncated at some upper value of *l*. Our general choice was to terminate the series when $\left|{F}_{l+1}/{F}_{l}-1\right|<{10}^{-4}.$

First, we investigate the behaviors of the optical forces as a function of the particle radius in different conditions in Fig. 2
: for Airy evanescent wave, *θ*_{1} = 0.9 rad $\left({\theta}_{1}>{\theta}_{crit}\right),$ (a) *d* = *a*, *n*_{3} = 1.59, (b) *d* = *a*, *n*_{3} = 1.59 + 10^{−3}*i*, (c) *d* = 2.1*a*, *n*_{3} = 1.59; (d) for Airy transmitted wave, *θ*_{1} = 0.3 rad $\left({\theta}_{1}<{\theta}_{crit}\right),$*d* = *a*, *n*_{3} = 1.59; (e) enlarged plot of *F*_{x} of *b*_{18} in (a) with its resonant item *l* = 18. While size parameter *α* varies from 12.4 to 14.9 (1.48<*a*<1.78 μm), *x _{c} = y_{c}* = 0,

*z*= −(

_{c}*d*+

*λ*).

As we can see, all the optical force curves exhibit strong oscillations of MDR that originate from the resonance of the partial scattering waves, but the shape of them are very different. Each peak in Fig. 2 can be directly related to the first-order resonance indicated. Higher-order resonances of lower modes [9] are not observable within this range of size parameters. For Airy evanescent wave in (a), the peaks of *F*_{x} are much sharper than those for Airy transmitted waves in (d). If we define *Q* = *a*_{res}/△*a*, where *a*_{res} is the resonant radius, △*a* is the width of the radius range of which the force is half of its maximum *a*_{res.} For *b*_{19} peak in (a), *Q≈*10^{4}. This high *Q* properties of the optical forces can be utilized for accurate sorting of micro-particles, which select resonant ones, and leaving those at non-resonance untouched.

In Fig. 2(b), we added a small imaginary part in its refractive index, *n*_{3} = 1.59 + 10^{−3}*i* this time. As we can see, the absorptions generated by the complex refractive index make the peaks a considerable decline, while △*a* increased. If the imaginary part increases further, (e.g. for nickel, *n*_{3} = 1.5 + 3.1*i*), the oscillations would disappear by strong absorptions [22]. With a transmitted distance of *d* = 2.1*a* shown in Fig. 2(c), *a*_{l} peaks emerge, dominating with respect to the *b*_{l} peaks on this occasion, while the *Q* factor is much smaller, meaning a less size-selective manipulation. Figure 2(e) shows a detailed plot of *F*_{x} of *b*_{18} compared with its resonant item *l* = 18, as observed, the resonant item accounts for the majority of the peak.

The red lines show the gradient forces that act on the particle along *z* direction. As we can see in Fig. 2(a) and 2(b), the gradient force can be attractive or repulsive with the source of evanescent wave; the reason is that the optical force is a bilinear product of the incident field and the scattering coefficients *a*_{l} and *b*_{l} [9, 16]. But for transmitted wave in Fig. 2(d), both *F*_{x} and *F*_{z} are resonant scattering forces and exhibit similar shape.

In order to understand the formation of the resonant peaks, we investigate the distributions of the electric field magnitude$\left|E\right|$ of which the Airy evanescent wave interacted with the polystyrene spherical particle situating on the interface in Fig. 3
: (a) *y-z* plane (*x* = 0), (b) *x-y* plane (*z* = 0), (c) *x-z* plane (*y* = 0) are for the resonant peak *b*_{18} and (d) *x-z* plane (*y* = 0) is for non-resonant case. The unit of electric field magnitude is statvolt/cm and 1statvolt/cm = 3 × 10^{4} V/m. We can see: most energy is distributed in the incident *x*-*z* plane. With regard to *b*_{18} (*α* = 14.01, *a* = 1.6725 μm) in Fig. 3(a), almost all the incident wave is coupled into the particle which is confined near the surface by total internal reflection. The arrows denote the directions of Poynting vectors, indicating the internal wave travelling along the surface.

But for the non-resonant case shown in Fig. 3(d) (*α* = 13.56, *a* = 1.619 μm, where *F*_{z} changes its sign between *b*_{17} and *b*_{18}), a considerable part of the energy have not transmitted into the particle, and the internal field exhibit a chaotic distribution which would lead to the offset and loss of the energy, that causes the bottom of *F*_{x}. In *y*-*z* plane Fig. 3(a) and *x*-*y* plane Fig. 3(b), the internal field is mainly distributed in the upper and lower, left and right sides of the particle, respectively. Moreover, through the propagating directions of Poynting vectors, the conclusion can be drawn that the polystyrene sphere plays a focusing lens role.

As we know from Fig. 3, the occurrence of the resonant peak would accompany a ring structure of the surface wave in the internal field. So it is interesting to see how this structure varies with different incident angle and transmitted distance. Figure 4
shows the transformation of the internal field patterns of the resonant peaks for the polystyrene particle illuminated by Airy evanescent wave for *b*_{18}, 4(a)–4(c), *θ*_{1} = 0.9 rad while transmitted distance: 4(a) *d* = 1.3*a*, 4(b) *d* = 1.7*a*, 4(c) *d* = 2.1*a*; 4(d)–4(f), *d* = 2*a* with incident angle: 4(d) *θ*_{1} = 0.75 rad, 4(e) *θ*_{1} = 0.9 rad, 4(f) *θ*_{1} = 1.05 rad; 4(g) *a*_{17}, *θ*_{1} = 0.9 rad, *d* = 2.1*a*. And illuminated by Airy transmitted wave, *b*_{18}: 4(h) *θ*_{1} = 0.2 rad, *d* = *a*, 4(i) *θ*_{1} = 0.5 rad, *d* = 2*a*.

As we can see, for the damped wave, the ring structures exhibit significant changes with the variations of incident angle and transmitted distance. Specifically, in Fig. 4(a) with *d* = 1.3*a*, the internal field shows a traveling surface wave pattern. In Fig. 4(b) with *d* = 1.7*a*, the solid ring change to a lap of bigger bright spots (2*l*) [10], representing the standing wave which is caused by the interference of the counter-propagating internal waves. With the transmitted distance increase further, with *d* = 2.1*a* in Fig. 4(c), the spots become smaller, and will stabilize with further increasing of *d*. Moreover, the wave fronts can be seen clearly this time which is propagating in the positive *x*-direction, indicating a much stronger interference of the internal wave.

With the increasing of the incident angle *θ*_{1} from 0.75 to 1.05 rad, the ring structures present a different transformation process shown in Figs. 4(d)–4(f). Figure 4(g) shows the internal field distribution of *a*_{17} in Fig. 2(c). We can see, for this high background, low-*Q* peak, the spots around the surface disappear. But for illumination by the transmitted wave, as shown in Figs. 4(h) and 4(i), the internal field patterns present no change essentially, only the propagating directions vary with the incident angle. Likewise, the shape of its corresponding resonant peaks won’t change.

Next we investigate the relationships between the resonance peak and its corresponding internal field distribution. The resonant peaks of the optical forces in Figs. 5(a)
–5(d) correspond to Figs. 4(a)–4(c), and 4(g), respectively. As we can see, the resonant peak in Fig. 5(a) of which corresponding to a solid ring of travelling wave pattern has a high *Q* factor (*Q*≈1.5 × 10^{3}), and its background approaches zero. For a larger spots of standing wave pattern which originate from the weak interference of the internal wave, as shown in Fig. 5(b), the height of its corresponding peak is reduced, the width and the background increases, leading to a lower *Q* factor (*Q*≈1.37 × 10^{3}). Furthermore, with a stronger interference pattern in Fig. 5(c), its corresponding peak has a very high background which exceeding the half of its peak value, that reduce the *Q* factor greatly. In Fig. 5(d) for *a*_{17}, also exhibit a strong interference pattern, but the spots around the surface disappear, its resonant peak width is broadened further.

In a summary of the above analysis, we can see that the strong interference of the internal wave around the particle would reduce the *Q* factor of the resonant peak, and enhance the background of the optical forces.

Finally, we investigate the impacts of multiple scattering of the Airy evanescent wave between the particle and the interface on the optical forces [25] as a function of particle radius in Fig. 6
. For convenience of comparison and clarity, the particle radius is chosen to vary in the range of (0.2<*a*<0.8) μm, while *x _{c} = y_{c}* = 0,

*n*

_{3}= 1.59,

*d = a*,

*z*= −(

_{c}*d*+

*λ*),

*θ*

_{1}= 0.9 rad. The multiple scatterings considered here is up to five orders: the black lines denote the optical forces without reflection; the red, magenta, yellow, green, and blue lines represent the evanescent wave is reflected between the particle and the interface for once, twice, three, four and five times, respectively.

As we can see, the optical forces exhibit strong oscillations of MDR with the variation of particle radius, the impacts of the multiple scattering are more and more significant with the resonance enhanced, while in off-resonance regions are negligible. This is not difficult to understand: most photons of the evanescent wave are gathered in the resonance regions where the optical force is stronger, so the absorptions and dissipations from the reflections between the particle and the interface are more significant in these regions, which make the optical force reduced greatly. On the contrary, in off-resonance regions where little photons distributed in, the loss of the transmitted photons is negligible. The inset is the variation of the peaks of *F*_{z} versus the reflected times. As we can see, after experiencing five times of the reflections of the evanescent wave, the optical forces converge ultimately.

Figure 7
shows the impacts of multiple scattering on the optical forces as a function of the beam center’s displacements *x _{c}*, while

*y*

_{c}= 0,

*a*= 0.8 μm, other parameters are the same as in Fig. 6. As we can see, the optical forces exhibit strong oscillations which are corresponding to the distributions of main lobe and the sideways of the Airy evanescent field. And the effects of the multiple scattering are more significant in the regions of bigger lobes of the Airy beam. For the same reason, most photons are distributed in where big lobes located, the more serious loss of energy there lead to the results.

## 4. Conclusions

In summary, we have investigated the MDR properties of the optical forces for a Mie particle illuminated by the Airy beam transmitted through an interface. Numerical results show that the distributions of the electromagnetic field inside the particle would affect its corresponding resonant peak structure greatly. The multiple reflections of the evanescent wave between the particle and the interface are also considered. We believe that the theoretical works presented in this paper would provide better guidance on the investigations of optical micro-manipulations and near-field optics.

## Acknowledgments

We acknowledge financial supports from the Natural Science Foundation of China (grant 11074130, 61275148), Chinese National Key Basic Research Special Fund (2011CB922003), and 111 Project (B07013).

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