Electromagnetic Forces in Negatively Refracting Photonic Crystals

Оптика и лазерные технологии

А. Ang¹, S. Sukhov^1,2, A. Shalin^1,2,3

¹The International Research Centre for Nanophotonics and Metamaterials, ITMO University (St Petersburg, Russia); ²Kotel’nikov Institute of Radio Engineering and Electronics of Russian Academy of Sciences (Ulyanovsk branch) (Ulyanovsk, Russia); ³Ulyanovsk State University (Ulyanovsk, Russia)

Аннотация. Хорошо известно, что свет может создавать оптические силы, воздействующие на освещаемые им объекты. Несмотря на широкую известность этого факта, до сих пор идут споры по поводу величины и направления этих сил в материальных средах. В частности, было высказано предположение, что эти силы могут быть отрицательными (притягивающими) внутри левых метаматериалов. В данной статье мы исследуем оптические силы внутри левых фотонных кристаллов. Выполненные расчеты не обнаружили присутствия отрицательных сил.

Ключевые слова: оптические силы, оптическое манипулирование, левые среды, фотонные кристаллы.

Abstract. It is well known that light can exert optical forces on illuminated objects. In spite of this effect is well known, still there are debates about specific magnitude and direction of these forces inside material media. In particular, it was suggested that these forces become negative (attractive) inside left-handed metamaterials. Here we investigate optical forces inside left-handed photonic crystals. Performed calculations do not show the presence of negative forces.

Key words: optical forces, optical manipulation, left-handed media, photonic crystals.

Electromagnetic Forces in Negatively Refracting Photonic Crystals

Introduction

Kepler first suggested the existence of radiation pressure by noticing that the comet’s tail points away from the Sun. Later, based on the newly-developed theory of electromagnetism, Maxwell [1] showed that momentum transfer from the field to an object would result in radiation pressure, followed by experimental results from Lebedev [2] and Nichols [3]. Much later, using focused laser field, Ashkin [4] showed the possibility of trapping a small spherical particle submerged in water. Many more methods of optical manipulation were developed after initial Ashkin’s experiments and many optomechanical effects were discovered: optomechanic nonreciprocity [5], optical advection [6], transverse optical forces [7] etc. One of these new effects, negative optical forces (optical tractor beams), results in an attraction of illuminated objects by incident radiation [8].

In general, generating negative optical forces can be done using either specifically tailored laser beams or using auxiliary structures [8]. For the former method, these tailored beams are focused onto the object, but high optical intensity can be potentially damaging for the object. For the latter method, the surrounding medium changes the incident field such that movement of particles can appear opposite to the propagation of incident field. Veselago [9] proposed an example of such a medium, the left-handed metamaterial (LHM). These materials are special in a way that they have negative index of refraction so that the incident field would be negatively refracted inside (Figure 1). Another property of LHMs is that energy and phase propagate in opposite directions inside. For the case of a homogeneous LHM, this results in optical forces acting in the direction opposite to the wave propagation creating negative radiation pressure. Pendry [10] showed experimentally that left-handed materials can be created using a structure composed of an array of split-ring resonators. It is also possible to reproduce LHMs properties using a periodic array of rods known as photonic crystals (PhCs) [11]. In this paper, we investigate optical forces acting on a small particle placed inside photonic crystal. In contrast to the homogeneous LHM, photonic crystals have spaces between structure elements in which the manipulated particle can move (Figure 2).

Figure 1: Power flow (black arrows) and forces (white arrows) in free space (top half) and homogeneous LHM (bottom half) where ε=µ=-1. The color map shows the electric field; the forces are obtained using Eq. (1).

Optical forces inside photonic crystals

We consider two-dimensional PhCs composed of an array of rods in air. Small particle can be placed in between the rods to test the direction and magnitude of the optical forces (Figure 2). We assume that this probe particle has dimensions much smaller than a wavelength of incident light (dipolar approximation). In this case the calculation of optical force F is significantly simplified [12]:

               (1)

Here p is the dipole moment of the probe particle, E is the local electric field, x_i = x, y, z are the Cartesian coordinates. Thus, to determine optical force at arbitrary location, one needs to calculate corresponding electric field E. To calculate field E, we used Finite Elements Method implemented in commercial software Comsol Multiphysics 5.0.

For our numerical simulations, we used four photonic crystal structures demonstrating left-handed properties, two arranged in a square lattice [11,13], and another two arranged in a hexagonal/triangular lattice [14,15]. These PhCs are found to have counter-propagating energy and phase. They also found to negatively bend the refracted wave (Figure 1). The geometry of the problem is illustrated in Fig. 2: the semi-infinite PhC is illuminated by plane incident wave, the probe particle is placed within the air gaps of the PhC structure. Specific parameters of the PhCs can be found in the respective papers [11,13–15], however, some parameters needed to be rescaled such that the left-handed properties would be present in the frequency domain used in the simulations presented here. The details of simulations can be found in Ref. [16].

Figure 2: A two-dimensional schematic consisting of a probe particle P and the photonic crystal serving as the auxiliary structure for generating the field.

The expression (1) for the optical force includes both conservative (gradient) and the non-conservative (scattering) force [17]. The magnitude of the gradient force is proportional to the real part of the particle’s polarizability, and its direction is given by the gradient of the electric field magnitude. This force is responsible for the trapping of the probe particles; this is undesirable as we are looking for the tractor beam effect. On the other hand, the scattering force is proportional to the imaginary part of the polarizability and is responsible for the transport phenomena.

For our purposes, we want to minimize the gradient force to prevent particle trapping. As we mentioned before, minimization of gradient force requires zero real part of particle’s polarizability. To achieve this, in simulations we used silver probe particle and an incident field with frequency of 957 THz. Other materials such as boron and copper can also be used. In simulations, we used TE polarized light (electric field vector is perpendicular to the axes of rods).

Figure 3: The total force (top row) vs the power flow (bottom row) of the photonic crystals in refs. [11,13–15], respectively. The color map of the top row represents the force magnitude, and the arrow field and streamlines show the force direction. The bottom row shows the power flow direction in the arrow field and streamlines, whereas the color map shows the electric field magnitude.

The simulation results of the forces are shown at the top row of Fig. 2. While the force is not in the same direction as the power flow (Fig. 3, bottom row), it is still directed away from the interface. In other words, left-handed PhC does not show negative forces.

Conclusions

To conclude, in this paper, we have performed an investigation of the forces inside left-handed photonic crystals. These photonic crystals exhibit negative refraction and negative phase velocity. Homogeneous left-handed metamaterials with similar properties, presumably, should exert negative radiation pressure on probe particles placed inside them. We used four photonic crystal structures from literature and numerically obtained the forces acting inside. From the simulation results, we found that the radiation pressure inside the photonic crystals remains positive. Our results show that properties of left-handed photonic crystals differ from the properties of homogeneous left-handed materials.

References:

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[17]        Novotny L., Hecht B. Principles of Nano-Optics. United Kingdom: Cambridge University Press, 2006.

Сведения об авторах:

Анг Анжелин¹, студент

Сухов Сергей Владимирович^1,2, к.ф.-м.н.

Шалин Александр Сергеевич^1,2,3, д.ф.-м.н.

Организация:

¹федеральное государственное автономное образовательное учреждение высшего образования Санкт-Петербургский национальный исследовательский университет информационных технологий, механики и оптики (Университет ИТМО);

²Ульяновский филиал федерального бюджетного учреждения науки Институт радиотехники и электроники им. В.А.Котельникова РАН;

³Ульяновский государственный университет

Authors:

Ang Angeleene S., master student

Sukhov Sergey Vladimirovich, PhD

Shalin Alexander Sergeevich, Dr. Sci.

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