5.5 Theoretical Model
5.5.2 Optical Forces on Particles in the Field
A finite element method software package Comsol was used to calculate the scatte-
red electric and magnetic fields. For simplicity, a negligible power loss was assumed for the optical system after emerging from the objective lens. Additionally, it was assumed that the water–glass interface corresponded to the focal plane of the Gaus- sian beam. Figure 5.15 shows the electric field distribution incident on a sphere placed on the interface at two angles of incidence.
5.5.2.1 Maxwell Stress Tensor
To evaluate the optical force acting on particles in the optical field, the Maxwell stress tensor was calculated and integrated across the surface of the particle. In a
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Figure 5.15: [3] Electric field distribution incident on a 5 µm polystyrene sphere in water, for angles of incidence of 61o and 70o respectively. At the critical angle the large portion is coupled out of the evanescent field into a propagating beam into the fluid medium and thus exerts a large gradient force on the sphere towards the substrate. The incidence angle of 70o is interesting as the field is close to forming a whispering gallery mode and could indeed be possible by suitable choice of wavelength and sphere size, as utilised by Kuriakose et al. [78].
host material the stress-tensorT is given by [219,220]
T =D~ ×E~∗+B~ ×H~∗− 1
2
~
D·E~∗+B~ ·H~∗ (5.14)
with −→E, −→D, −→H and −→B denoting the electric field, the electric displacement, the
magnetic field and magnetic flux respectively, and∗ is the complex conjugate. Ap-
plying the constitutive relations D~ = εrε0−→E and B~ = µrµ0−→H, where εr, ε0, µr
and µ0 are the relative and vacuum permittivity and permeability respectively, the
tensor components in SI units are
Tij =εrε0EiEj∗+µrµ0HiHj∗−
1
2(εrε0EkEk∗+µrµ0HkHk∗) (5.15)
which is summed over the identical indices k. The optical average force acting on
the particle is D ~ FE= 1 2Re Z S T ·~n ds (5.16) where n is the normal vector pointing outward to the surface S of the particle.
The force was averaged over one optical cycle. Note that, due to the jump in the index of refraction, the normal fields at the surface of the particles have a discontinuity. To calculate the total optical force the external fields were used. In this case, the total force includes not only the optical force on the particle but also the optical force acting on the water interface, which is completely transmitted to the sphere due to the hydrodynamic non-slip boundary conditions at this interface. To
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validate the integrated stress tensor approach, it was compared with the integrated force density D ~ FE=−ε0 4Re Z V Ek∗Ek∇εrdv (5.17) where the integral is taken over the volumeV of the particle [219]. The permittivity εr is constant in the volume of the particle, hence the integrand is only non-zero on
the surface of the sphere and the above integral simplifies to a surface integral. Even though analytically both force calculation methods should give identical results, it was observed that this was only the case when the numerical discretisation of the problem has sufficient resolution. This important consideration allowed the validity of the numerical simulation to be verified.
5.5.2.2 Gradient and Guiding Forces
It is beneficial to consider the forces acting on a particle in an evanescent field as two components, a vertical force acting to pull (gradient force) or push (deflec- tion/guiding force) the particle to or from the substrate, and a horizontal force which guides particles along the substrate in the direction of propagation of the evanescent field. The Maxwell stress tensor can be used to calculate the forces acting on an illuminated micro-particle and predict its behaviour; indeed as a function of particle size, refractive indices, and on the effect of polarisation state of the incident field.
To test the robustness of the numerical model, the illustrative example of Almaas and Brevik [222] was chosen for comparative purposes. Their model parameters correspond to a 2µmglass sphere (n = 1.5) submerged in water (n = 1.33) above a
sapphire substrate (n = 1.75) . In this configuration, the critical angle is 51o and the
beam has a power of 150mW and a 100µmdiameter. This corresponds to a power
density of 19 M W m−2 and a force coefficient of e0E2a2 = 130f N where a is the
radius of the particle. Using these parameters in the model detailed above predicts horizontal guiding forces of Fxp= 11f N for p-polarisation and Fxs= 8.1f N for s-
polarisation, which compare respectively to 10f N and 8.1 f N as obtained in [222].
Similarly, the calculation implies particles are pulled towards the substrate with
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the data of Almaas et al. (-44 f N and -29f N) [222].
Figure 5.16: (a) Horizontal (positive range) and vertical (negative range) forces as a
function of the particle position with respect to the incoming beam, for different angles of incidence. The greyed region corresponds to the intensity profile of the beam on the interface. Note that its maximum is shifted with respect to the centre of the incident beam due to the Goos-Hänchen shift. (b) Ratio between the horizontal optical forces of the p and s polarisations as a function of the particle position for different angles of incidence.[3]
Figure 5.16 shows the main result of the simulation for the proceeding expe- rimental section, showing the horizontal and vertical optical forces acting on the particle at different positions along the middle of the TIR beam profile. The same particle size of 5 µm in diameter, in contact with the substrate and for 1 W of
optical power was used for comparison with Almaaset al. [222]. The spot size using
the TIRF is reduced to 13.4µm (see Figure 5.12) leading to a power density of 10 GW m−2 . This system produces a force coefficiente0E2a2 = 420pN which is three
thousand times larger than the Almaas et al. case. The positive part of the graph
shows the horizontal, i.e. guiding forces, while the negative part shows the vertical forces, i.e. the pull/push towards the substrate. The greyed region shows intensity of the TIR beam together with the Goos-Hänchen shift. Further, depending on the angle of incidence, position in the beam and distance from the substrate, a variable ratio between the s and ppolarisations can be observed (see Fig. 5.16b).
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beam intensity and the maximum force amplitude. This shift, similar to the Goos- Hänchen shift but more pronounced, starts before the critical angle and decreases for angles far beyond the critical angle. A further point of note is the difference in the gradient force profile between the propagation case (green curve, θ < θc) and
the evanescent cases (red and blue curves,θ > θc). Indeed, when the beam is mostly
propagating, an inversion in the vertical force occurs, which at first pulls and later pushes the particle away from the interface.
The difference in force magnitudes as a function of angle can be observed in Figure 5.17 where the vertical and horizontal forces in the centre of the incident beam as a function of the angle of incidence are plotted. As seen in Figure 5.16, close to the critical angle the propagating beams still attract a particle towards the water glass interface while guiding it very efficiently. In this case the particle acts like a small micro-lens that bends the light away from the interface (see field distribution in 5.15). This in turn, pulls the particle towards the interface, due to momentum conservation, thus enhancing the gradient force already present.
Figure 5.17: Horizontal and vertical forces as a function of the angle of incidence for a
5 µm particles and an s polarised beam.
The horizontal guiding forces were also calculated as a function of particle size in the 1 - 5µm region. Interestingly, the force was found to linearly depend on the
size of the particle, as shown in Figure 5.18. This linear relationship has its origin in the relatively large particles considered here and the small refractive index diffe-
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rence between the water, the particle and the substrate. Indeed, for high contrast small particles one would expect a complex relationship between guiding forces and particle size [222]. It is on this basis of a predicted linear relationship between the size of a particle and the guiding force that it experiences, for the sizes considered here, that the sorting experiments are conducted in Sections 5.6.2 and 5.6.3
Figure 5.18: Linear dependence of the horizontal forces for p-polarised centred beam at
x= 0 incident at θ= 61o as a function of the particle diameter.[3]