VI. 5 5 Experimental results
VII. 3 Modelling the angular accelerations and rotations
This was an extension to the computer modelling that was described in section VI. 4 earlier in the thesis. Using the same basis of the modelling program, it was possible to assign £fi oi orbital angular momentum to each photon and define an absorption coefficient for the sphere being trapped. In this way, the torque acting upon the sphere due to the orbital angular momentum transfer was calculated. The angular acceleration was also deduced along with the limiting or terminal angular velocity of the sphere as it rotated. The model was thus used to predict angular accelerations and limiting angular velocities for a silica sphere suspended in water, which was acted upon by optical spanners. The results were published, along with the earlier modelling results, in the Journal of Modern Optics in December 1996 [3].
VIL 3.1
Modelling the transfer of orbital angular momentum
When using a high order LG mode with i ^ 0, there must be a transfer of the angular momentum to the trapped particle due to partial absorption. If the partial absorption is small, the LG mode is transmitted through the sphere undistorted and retains its mode characteristics. The amount of orbital angular momentum transferred from the beam to the particle is simply calculated by multiplying the momentum content in the beam, i h per photon, by the number of photons that were absorbed by the particle. See Appendix III for details of the programs written for the modelling.The resultant torque on the sphere t, was calculated for different sphere
positions along the beam axis using a laser power of 1 mW. The sphere's position along the beam axis was changed so that the region of strongest absorption, and hence rotation, was found, which was expected to be at the beam focus due to the full interaction of the beam with the particle at this point. The angular acceleration A, was simply calculated from,
= — (7.1)
where I is the moment of inertia of the sphere about the beam axis.
/ = Mass - I y 2 = (7.2)
and r is the radius of the sphere and p is the sphere's density. It was more useful to investigate the terminal (or limiting) angular velocities per mW of laser power of the sphere in the optical spanners, rather than the angular accelerations it would experience. This limiting velocity coiim is governed by the viscosity 77 of the supporting medium in the sample cell [4], and was
calculated from
It is important to note here the dependence of the moment of inertia and the limiting angular velocity on the radius of the sphere. If, for example, the optical spanners were used on a particle whose radius was 1 mm instead of 1 jxm i.e. a factor of 1000 larger, the increase in laser power to achieve the same angular acceleration is phenomenal. The moment of inertia of the sphere would increase by a factor of 10^^, and to maintain the same limiting angular velocity, the laser power would have to increase from milliWatts to MegaWatts, an increase by 10^ in laser power. This demonstrates why the optical spanners would not be able to operate successfully on particles that are much larger than a few lO's of microns in size, due to large laser powers that would be required.
VII. 3.2
Modelling results
The same silica sphere of diameter 8 |Lim suspended in water, as that used in
the earlier modelling in section VI. 4 was considered in this modelling extension. The trapping wavelength was 532 nm, thus the sphere could be considered as a Mie particle. The LG mode with i = 0 (fundamental TEMo,o) possesses no orbital angular momentum and consequently cannot be used as optical spanners. An LG mode with p = 0 and £ = A was used in the modelling as this was experimentally possible to reproduce and would also provide Ah of orbital angular momentum per photon.
Two different materials for the sphere were considered. The first was simply a standard fused silica sphere which had an absorption coefficient a, of 0.2 m-i. The second sphere had a larger absorption at the wavelength being considered, a = 5700 m-i, which was typical for an absorbing Schott glass (such as BG3).
As figure Vll.b shows, the fused silica sphere (a = 0.2 m'l) can be expected to have a limiting angular velocity of only 7 x 10"^ rad s"^ mW'^, which will not be observable at all. However, for the absorbing glass with a = 5700 mr^, the lim iting angular velocity is predicted to have a value of around 0.02 rad S"^ mW'L This magnitude of rotation, which is equivalent to 3 mHz mW"i, should be observable making the optical spanners a viable tool. The modelling considered the rotation of a sphere with 1 mW of laser
power, so if a laser mode was used with higher powers the expected rotation would be increased.
As expected the region that experiences the largest angular acceleration, and hence achieves the largest angular velocity, is a region along the beam axis around the focus of the same length as the sphere's diameter. When the centre of the sphere lies outside this region, the rotational effect diminishes very quickly. a = 5700 m"^ Absorbing Glass 102 -
I
o§
a
= 0 .2 m “ J- Fused Silica 8 10 6 •2 0 2 4 -10 ■8 •6 ■4 - 10-1 -2 -3 -4 -5 -6 ^ 10 -7 -9 1/51
Î
I
Position of sphere with respect to beam focus (|im)
Figure Vll.b The predicted angular acceleration and limiting angular velocity of an 8 jim sphere trapped with a LG niode
As the torque on the sphere is proportional to the orbital angular momentum and consequently i, then the resulting angular acceleration and terminal velocity are also, to a good approximation, proportional to i.
Therefore, the higher the mode that could be used in the optical spanners, the higher the i index and consequently the larger the terminal velocity of the rotating sphere.
It is worth mentioning the absorption coefficient here. If the absorption by the sphere is too large, then the transfer of orbital angular momentum will take place but due to the lack of transmission of the rays, there will be no tweezers effect to trap the sphere. Conversely, if the absorption is very small, there will be no rotation of the sphere but the optical trap will still be improved due to the absence of the on-axis rays..
VII. 4 Experimental set-up
The details of the experimental apparatus and their arrangement do not require to be covered again as the same system was used as that described in section VI. 2 of this thesis for the operation of the improved optical tweezers. Using this set-up, an LG mode was produced and using the beam telescope, the beam size was altered so that it filled the back aperture of the microscope objective lens to give optimum axial trapping.
There was one addition to the arrangement, which was the insertion of a coloured glass filter in front of the CCD array/camera (figure VII.c). This was a piece of Schott BG38 glass which absorbed the reflected infra-red laser light, so that the image viewed on the monitor was of the trapped particle without any of the trapping laser light. This enabled the particle to be viewed without the laser light obscuring the image.
The observation of the trapping and rotation of the particle was performed by means of a monitor and video recorder as before. If the glass filter was removed, the rotation of the particle could actually be seen more easily, due to the rotation of the image of the scattered laser light.
f = 160mm Beam telescope f = 25mm cyl. x2 f = 200mm Cylindrical lens mode converter Beam steering mirror f = 80mm Incident Hermite- Gaussian mode Coloured glass filter Tube lens /= 160mm f = 100mm Sample cell array Microscope objective Backlit illumination