2.1.4 A force transducer

A transducer is a device that transforms a physical magnitude into another. In this section, we describe a transducer that converts the intensity of light $ \langle\vec{S}\rangle$ into a force $ \vec{F}$ according to equation 2.17. We also consider that the transducer operates under the two following experimental conditions (see Fig. 2.7a):

The incoming laser light $ \langle\vec{S}_\mathrm{in}\rangle$ is a focused spherical wave.
The outcoming light $ \langle\vec{S}_\mathrm{out}\rangle$ is spherical and emanates from the focus of the incoming light, where the trapped particle is also located.

Figure 2.7: (a) The collimated light of the laser (in red) is focused by an objective (blue lens) producing a spherical converging beam. A particle is located at the focus and the light is scattered in all directions. The outcoming light is a diverging spherical wave. (b) Spherical coordinates and element of area.

The first condition can be easily achieved by focusing a collimated laser beam with a lens. The second condition is achieved when the outcoming light is observed far away from the source. It can be shown from the GLMT (see Sec. 2.1.1) that the scattered wave has no radial components (therefore it is a spherical wave) in the so-called radiation zone, where the distance to the particle is much larger than the wavelength ( $ \vert\vec{R}\vert\gg\lambda$). Both conditions allow us to set the origin of coordinates at the focal point of the laser and write the intensity of light in terms of the spherical coordinates $ \langle\vec{S}(r,\theta,\varphi)\rangle$. Since the incoming and outcoming lights are spherical, we can define the angular intensity distribution of light2.5 (see Fig. 2.7b)

$\displaystyle \frac{I(\theta,\varphi)~\hat{r}}{r^2}=\langle\vec{S}(r,\theta,\varphi)\rangle$ (2.18)

which is independent of the radius $ r$ of observation ($ \hat{r}$ is a unit vector from the focus). Introducing Eq. 2.18 into Eq. 2.17 we can write

$\displaystyle \vec{F}=\frac{n_m}{c}\oint_A I(\theta,\varphi)\left(\hat{\imath}\...
...varphi}+\hat{\jmath}\sin{\theta}\sin{\varphi}+\hat{k}\cos{\theta}\right)d\Omega$ (2.19)

where the differential of area ( $ da=r^2d\Omega$) and the unit vector ( $ \hat{r}=\hat{\imath}\sin{\theta}\cos{\varphi}+\hat{\jmath}\sin{\theta}\sin{\varphi}+\hat{k}\cos{\theta}$) have been expressed in spherical coordinates. Again, $ I(\theta,\varphi)$ is positive for rays entering the trap and negative for rays leaving it. The measurement of $ I(\theta,\varphi)$ cannot be done directly from the emanating rays. It is necessary the project the angular distribution of intensities into a planar photodetector $ D$, which can be achieved by using a condenser lens (see Fig. 2.8a). In a perfect condenser lens of focal length $ f$, there is a univocal relation between the direction of a ray that emanates from the focus of the lens and the position of the ray at the image principal focal plane $ P$ (see Fig. 2.8a). This relationship is known as the Abbe sine condition and it is given by

$\displaystyle \rho=f n_m \sin{\theta}$ (2.20)

where $ \rho$ is the radial distance of the ray to the optical axis at the principal plane, $ n_m$ is the index of refraction of the medium, and $ \theta$ is the angle of the emanating ray. On the other hand, if all the light is collected by the condenser lens, the conservation of the energy establishes a relationship between the total power of light that emanates from the focus and the power of light collected by the lens
$\displaystyle \oint_A I(\theta,\varphi)~d\Omega$ $\displaystyle =$ $\displaystyle \int_P E(\rho,\varphi)da'$  
$\displaystyle I(\theta,\varphi)~d\Omega$ $\displaystyle =$ $\displaystyle E(\rho,\varphi) da'$ (2.21)

where $ E(\rho,\varphi)$ and $ da'=\rho~d\rho~d\varphi$ are the irradiance2.6 and the element of area at the image focal plane, respectively. In Section 2.2.1 it is shown that the focusing lens is also useful to collect the backscattered light. Therefore, the integral sum over the image principal plane $ P$ of the condenser lens is extended to the image principal plane of the focusing lens, too. By introducing Eqs. 2.21 and 2.20 into Eq. 2.19 we can write the expression for the force as:

$\displaystyle \vec{F}=\frac{1}{c}\int_P E(\rho,\varphi)\left(\hat{\imath}\frac{...
...+\hat{k}~n_m\sqrt{1-\left(\frac{\rho}{fn_m}\right)^2}\right)\rho~d\rho~d\varphi$ (2.22)

where the integration is now taken over the surface of the image principal plane ($ P$). Note that the resulting prefactor in the calculation of the $ z$ force is obtained from Eq. 2.20

$\displaystyle \cos\theta=\sqrt{1-\sin^2\theta}=\sqrt{1-\left(\frac{\rho}{f n_m}\right)^2}~.$ (2.23)

Figure 2.8: (a) Projection of the scattered light onto a photodetector. An incoming ray (in gray) is focused by the left lens and scattered by the particle. The outcoming ray is collected by the condenser lens ($ A$) and redirected to a photodetector $ D$. Actually, the condenser lens produces a change of coordinates, from spherical ( $ \theta ,\varphi $) to cylindrical ( $ \rho ,\varphi $). (b) Measurement of axial ($ z$) force. The attenuator has a transmission profile that looks like a bull's eye. An axial ray is fully transmitted to the photodetector. An off-axis ray, suffers an attenuation and its intensity is lower when it arrives at the photodetector.

Now, the forces exerted on the $ x$ and $ y$ directions (radial force) can be inferred from the measurement of a photodetector placed at the image principal plane $ P$. A Position Sensitive Detector (PSD) is a device that produces two output signals ($ D_x$,$ D_y$) proportional to the irradiance of the light ($ E(x,y)$) weighted by the relative position from the center of the detector ($ x/R_D$,$ y/R_D$) according to

$\displaystyle D_x=\Psi\int_D E(x,y)\frac{x}{R_D}~da'=\Psi\int E(\rho,\varphi)\frac{\rho \cos\varphi}{R_D}~da'$      
$\displaystyle D_y=\Psi\int_D E(x,y)\frac{y}{R_D}~da'=\Psi\int E(\rho,\varphi)\frac{\rho\sin\varphi}{R_D}~da'$     (2.24)

where $ \Psi$ is the sensitivity of the detector, $ R_D$ is the effective radius of the detector and the integrals are over the surface ($ D$) of the detector (see Appendix C). Combining Eqs. 2.24 and 2.22 we can write the force in terms of the response of the detector
$\displaystyle F_x=\frac{D_x R_D}{c\Psi f}$      
$\displaystyle F_y=\frac{D_y R_D}{c\Psi f}$     (2.25)

The force exerted on the $ z$ axis (axial force) can be measured provided that the irradiance of light is weighted by the factor of Eq. 2.23. It can be achieved by placing an attenuator in front of a photodetector with a transmission profile given by Eq. 2.23 (see Fig. 2.8b). The output signal of the photodetector is given by

$\displaystyle D_z=\Psi\int_D E(\theta,\varphi)\sqrt{1-\left(\frac{\rho}{n_m f}\right)^2}~da'$ (2.26)

and the $ z$ force can be written as

$\displaystyle F_z=\frac{D_zn_m}{c\Psi}$ (2.27)

Eqs. 2.25 allow us to directly measure the transverse force (on the $ x$-$ y$ plane) of a trapped particle, provided that all the scattered light can be collected. What is relevant from these expressions is that calibration does not depend on the refractive index of the medium ($ n_m$), the radius of the particle nor the laser power. The factors involved in Eqs. 2.25 are constant in a typical experimental setup ($ R_D$, $ f$, $ \Psi$). On the contrary, the measurement of the axial force in Eq. 2.27 depends on the refractive index of the medium and this dependency cannot be avoided. However, the error in the $ z$ force is smaller that 0.25% when the refractive index of the medium changes from $ n_m=1.334$ (pure water) to $ n_m=1.343$ (1 M NaCl buffer).

JM Huguet 2014-02-12