ABSTRACT
In principle, the optical forces are customarily defined by the relationship [14]
m Qn P
F c
= (7.1)
where Q is a dimensionless efficiency, nm is the index of refraction of the suspending medium, c is the speed of light and P is the incident laser power, measured at the specimen. Q represents the fraction of power utilized to exert force. For plane waves incident on a perfectly absorbing particle, Q = 1. To achieve stable trapping, the radiation pressure must create a stable, three-dimensional equilibrium. Because biological specimens are usually contained in aqueous medium, the dependence of F on nm can rarely be exploited to achieve higher trapping forces. Increasing the laser power is possible, but only over a limited range due to the possibility of optical damage. Q itself is therefore the main determinant of trapping force. It depends upon the numerical aperture (NA), laser wavelength, light polarization state, laser mode structure, relative index of refraction, and geometry of the particle. In the Rayleigh regime, trapping forces decompose naturally into two components. Since, in this limit, the electromagnetic field is uniform across the dielectric, particles can be treated as induced point dipoles. The scattering force is given by
F n c
s = (7.2)
where
8 1 3 2
m kr r
m s p
Ê ˆ-= Á ˜+Ë ¯
(7.3)
is the scattering cross section of a Rayleigh sphere with radius r. S is the time-averaged Poynting vector, n is the index of refraction of the particle, m = n/nm is the relative index, and k = 2pnm/l is the wave number of the light. Scattering force is proportional to the energy flux and points along the direction of propagation of the incident light. The gradient force is the Lorentz force acting on the dipole induced by the light field. It is given by
2grad 2F E a
= — (7.4)
where
2m m
n r m
a Ê ˆ-
= Á ˜+Ë ¯ (7.5)
is the polarizability of the particle. The gradient force is proportional and parallel to the gradient in energy density (for m > 1). Stable trapping requires that the gradient force in the zˆ-direction, against the direction of incident light, be greater than the scattering force. Increasing the NA decreases the focal spot size and increases the gradient strength [15]. To perform the proposed concept, a bright soliton pulse is introduced into the multi-stage nano ring resonators as shown in Figure 7.1, the input optical fields (Ein) of the bright and dark soliton pulses are given by an Eqs (7.6) and (7.7) as [13]
sech exp 2 D T z
E t A i t T L
w È ˘È ˘ Ê ˆ= -Á ˜Í ˙Í ˙ Ë ¯Î ˚ Î ˚
(7.6)
tanh exp 2 D T z
E t A i t T L
w È ˘È ˘ Ê ˆ= -Á ˜Í ˙Í ˙ Ë ¯Î ˚ Î ˚
(7.7)
where A and z are the optical field amplitude and propagation distance, respectively. T is a soliton pulse propagation time in a frame moving at the group velocity, T = t – b1z, where b1 and b2 are the coefficients of the linear and second order terms of Taylor expansion of the propagation constant. LD = b/T0
is the initial soliton pulse width. Where t is the soliton phase shift time, and the frequency shift of the soliton is w0. This solution describes a pulse that keeps its temporal width invariance as it propagates and is thus called a temporal soliton. When a soliton peak intensity b G/ T2 02^ h is given, then propagation time for input T0 is known. For the soliton pulse in the micro ring device, a balance should be achieved between the dispersion length (LD) and the nonlinear length (LNL= (1/GfNL), where G = n2k0, is the length scale over which dispersive or nonlinear effects makes the beam
becomes wider or narrower. For a soliton pulse, there is a balance between dispersion and nonlinear lengths, hence LD = LNL.
