2.1.3 Measurement of linear momentum

In this section, we will assume that the diameter of the particle $ d$ is larger than the wavelength $ \lambda $ of the electromagnetic field (i.e., the laser light). Thus the system falls into the ray optics regime (see Sec. 2.1.1) and the laser can be visualized as a bundle of rays that interact with the particle [86]. It allows us to calculate the Maxwell Stress Tensor by an alternative way. A priori, one might think that the results of this section are only valid when the Mie regime is valid. However, it has been check empirically that an optical tweezers instrument built under these assumptions still works properly when the approximation is not fulfilled [86,16].

As deduced from the previous section, the force exerted on an object by an electromagnetic field is equal to the net flux of linear momentum that enters the object (Eq. 2.14). The net momentum flux that crosses the surface $ A$ can be written as an integral sum of the density of flux momentum2.3 $ \vec{\phi}_\mathrm{field}$ of the electromagnetic wave, provided the elements of surface $ da$ are normal to $ \vec{\phi}_\mathrm{field}$,

$\displaystyle \vec{\Phi}_\mathrm{field}=\oint_A\vec{\phi}_\mathrm{field}~da~.$ (2.15)

We ended the previous section with the problem of measuring the total flux of momentum and now we are facing the problem of measuring the density of flux momentum. It seems we did not make much progress. However, in the ray optics regime, the laser light has a defined direction of propagation and the wave can be decomposed into rays. It allows us to write the density of flux momentum of a light wave as (see Appendix B)

$\displaystyle \vec{\phi}_\mathrm{field}=\frac{n_m}{c}\vec{S}$ (2.16)

where $ n_m$ is the index of refraction of the surrounding medium, $ c$ is the speed of light and $ \vec{S}$ is the Poynting vector. Equation 2.16 is quite relevant, since the temporal average value of the Poynting vector $ \langle\vec{S}\rangle$ is the intensity2.4 of the light $ I$ and it can be directly measured by a photodetector. Note that what we have done up to now is to write the Maxwell stress tensor for the ray optics regime (Eq. 2.16 is a restricted version of Eq. 2.8 and Eq. 2.15 is the analogous of Eq. 2.12).

Finally, combining equations 2.16, 2.15 and 2.14 we can write the total optical force exerted on an object as

$\displaystyle \vec{F}=\frac{n_m}{c}\oint_A\vec{S}~da$ (2.17)

where the integral is extended on a surface $ A$ surrounding the object. $ \vec{S}$ and the surface $ A$ must be perpendicular because Eq. 2.16 is a heuristic calculation of the Maxwell Stress Tensor that does not include the off-diagonal elements. The off-diagonal elements produce shear stresses that act parallel to the surface and they do not contribute to the total force if the surface of integration is perpendicular to the direction of the momentum flux. Finally, the Poynting vector has two contributions $ \vec{S}=\vec{S}_\mathrm{in}-\vec{S}_\mathrm{out}$ that correspond to the light entering ( $ \vec{S}_\mathrm{in}$) and the light leaving ( $ \vec{S}_\mathrm{out}$) the volume enclosed in the surface $ A$ (see Fig. 2.6b).

JM Huguet 2014-02-12