2.4 Investigated matter
2.4.2 Microspheres
Follow up experiments were carried out using silica9 microspheres in solu- tion from Bangs Laboratories and Duke Scientific. In this section I will briefly evaluate the key material parameters for optical binding, especially the refractive index difference (∆n) between the sphere (nsphere) and the host
medium (nhost). Accurate determination of ∆n is crucial for the subsequent
theoretical modelling of optical binding.
Refractive index measurements on silica microspheres were carried out by Bangs laboratories using index matching technique and the following three results obtained n= 1.431, 1.442 and 1.458 at λ= 588.9nm.
Duke Scientific gives the refractive index of their silica spheres to 1.40 to 1.46 atλ = 589nm(at a temperature of 23◦C).
For fused silica the refractive index dependence on λ can be approximated
9The refractive index of polystyrene isn = 1.59 at 589nm (Bangs Laboratories) and
n= 1.57 for 1064nm[11]. Polystyrene was not used, as the paraxial approximation would start to break down for high refractive index contrasts.
with the following formula (this is an approximation of the dispersion equa- tion [37], with parameters A, B from [38]):
n =A∗ " 1 + B λ 2# (2.1) n= refractive index A= 1.4485 B= 48.7436
Using equation 2.1 a refractive index of n = 1.458 at 588nm and n = 1.451 at 1070nm is obtained. The results show that there is a shift to a lower refractive index value of ∆n≈0.007.
Although the refractive index of the spheres can be more accurately calcu- lated, factors such as the storage time can also alter the refractive index of the sphere, as water is absorbed by the sphere10[39] due to its porosity and can not be fully accounted for. Hence for the simulations an estimated refractive index of n ≈ 1.41 for 1 and 3µm sphere diameter and n ≈1.42 for 5µm sphere diameter, with errors ±4% was used (n = 1.43 was used in previous work [18] for simulations with two counter-propagating free space beams).
The sphere radius rsphere has a size distribution (Bangs laboratories) such
that the standard deviation (STD) of the mean diameter for 3µm is given as< 10%. This variation will be accounted for in the next chapter to aid a comparison and error evaluation between experimental and theory.
To mimic the refractive index difference between host solution and CHO cells and to investigate the array formation for a varying refractive index of the host solution the spheres were diluted in a de-ionised (DI) water and sucrose solution [11]. The refractive index of the solution was measured with a re- calibrated Brix refractometer11.
The Brix refractometer is calibrated for the yellow sodium D-lines at 589nm12
from references [40] and [41] a conversion plot %Brix ton was obtained, see figure 2.10(red crosses). Experimentally the refractive index was measured with light at 1070nm emerging from the laser source through the Brix re- fractometer. The scale was read through an IR viewer to obtain the Brix equivalent for the used wavelength. A shift of -0.9% Brix for 1070nmfor dif- ferent sucrose solution was measured. However there is a significant error in
10The refractive index changes from 1.36 to 1.42 (at 575nm) over a period of 800h[39]. 11Brix refractometers are usually used in the Wine industry and by bee keepers to
measure the sugar content of wine or honey. It is basically an Abbe refractometer that measures the critical angle but is calibrated in %Brix rather than refractive index.
2.4. INVESTIGATED MATTER
this measurement as internal dispersion compensation optics are optimised for 589nm.
A better approximation for 1070nmcan be obtained by fitting a curve to the conversion data and linear shifting it to a given refractive index value from literature sources. The literature gives the refractive index of DI water to:
Reference n Temperature λ Pressure
IAPWS [42] 1.326 0◦ 1.013µm 1MPa
B. Richerzhagen [43] 1.325 10◦ 1.064µm n.a. B. Richerzhagen [43] 1.320 50◦ 1.064µm n.a. P. Schiebner [44] 1.328 30◦ 0.8µm n.a.
This data results in an interpolated shifted line fit in graph 2.10, where the following formula (quadratic fit) was used to calculate the refractive indexes from Brix [%] ton:
n = 6.857∗10−6∗brix2+ 1.397∗10−3∗brix+nλ(T) (2.2)
Here nλ(T) denotes the temperature (T) dependent refractive index of DI
water, that was interpolated for the graph (see figure 2.10) to about 25◦C.
Figure 2.10: Left: Graphs shows the conversion from Brix to refractive index. Crosses show the conversion data at 589nm. Solid line represents the fit through the crosses and linearly shifted to the refractive index of DI water at 1064nm(blue) and 800nm(black) at a temperature of 25◦C. Right: Here the sucrose content of two popular soft drinks is compared with A) = 11.2% Brix and B) = 10.5% Brix.
To obtain a lower refractive index than with DI water, some measurements were also carried out with D2O (heavy water) having a refractive index of
1.328 [45] for the sodium D-lines. The linear approximation (formula 2.2) from DI water gives a refractive index of 1.320 at 1070nm and similar for 800nm. The refractive index of D2O was measured to approximately -2.5%
Brix at 589nmthis converts from the formula above ton= 1.320 for 1070nm, this is in agreement with the interpolation of formula 2.2 for negative Brix values.
At this point I want to continue solely with the term ∆n = nsphere −nhost,
denoting the refractive index difference or refractive index mismatch between the sphere and the host medium. At this stage it is possible to perform mea- surements for a variable ∆n of 0.03 to 0.09 with an accuracy of±0.001. As studies with the CHO cell arrays indicated, a low refractive index dif- ference between host solution and bound matter is a key parameter in the formation of optically bound arrays. In a follow up experiment the depen- dency of ∆n to the number of spheres that can be bound before the array collapses and a chain is formed [17] was investigated. Figure 2.11 shows the maximum number of 5.17µmsized silica microspheres that form an array for different ∆n.
Figure 2.11: Array formation dependency on ∆nfor 5µmsilica spheres. Number shows maximum number of beads before the array collapses. For ∆n=0.088 the spheres have collapsed into a chain. With decreasing ∆nmore spheres can be bound before the array collapses.
Here we clearly see that the smaller the refractive index difference between matter and host solution, the more particles can form an optically bound array. Notably the refractive index difference plays an important role in the array formation.
There are further important factors we have to take care off for the error evaluations in the next chapter. For a low ∆n, the optical forces acting are weak and the array creation (after loading by the helper tweezers) was ob- served after a time scale of several seconds before the spheres reached their