Case Study CS-TC07 Rev. A

Page 2

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PROBLEMS AND GOALS

There are two main parameters in capillary-based

backscattering interferometry (BSI): the angle of the incident

light and the geometry of the capillary. Both the angle of

incident light and the geometry affect the light rays detected

by the camera and the intensity of the backscattered light.

The most common geometry for BSI is a circular hollow

glass tube. However, some experiments have carried out

BSI using semicircular microfluidic channels. While this

geometry of capillary may be easier to analyze theoretically

due to its similarities to Fabry-Perot etalons and Michelson

interferometers, the manufacturing costs can be significantly

higher and the increase in sensitivity and stability are only

minimal.1

Figure 2 shows the cross section of a basic, circular, hollow

tube geometry. Overlaid on this diagram are the different

first-order rays possible given this particular geometry of

the capillary.

Figure 2. A diagram showing the different first-order

rays that are possible in the capillary geometry of BSI.

The incident angle of a given ray ϕ

i is simply the angle to

the surface normal. In the Xu/You et al.’s methodology,

the angle of incidence is related to the viewing angle β

i

of each ray, where β

i is defined to be the angle to the

horizontal in this geometry. The example of β

6 is shown

here. The diagram is adapted from Ref. 2.2 The rays are

demonstrative and not drawn with geometric accuracy.1

Both

Figure 1 and Figure 2 show that the angle of

incidence determines what rays of light are backscattered

as well as the intensity of the captured light. In two previous

ray tracing models3,4, there is a discrepancy regarding which

rays to include. Tarigan et al.4 uses a four beam model with

rays 1, 2, 3, and 4 in

Figure 2, while Xu et al.3 and You

et al.2 use a model with all seven beams in

Figure 2. This

discrepancy of which rays are considered relevant between

the two models creates a problem of choosing the more

accurate and more sensitive model for BSI. Because rays 6

and 7 do not contribute to the signal seen at the detector,

only the omission of ray 5 will need to be justified. Models

with more than 5 rays exist, but they are confined to

commercial software.1

Critical parameters that influence the resulting fringe pattern

of the backscattering are:

r = Inner radius of the hollow glass tube capillary

R = Outer radius of the hollow glass tube capillary

n

0 = Refractive index of air/outside the capillary

n

1 = Refractive index of the capillary

n

2 = Refractive index of the solution inside the capillary

ɸ

i = Incident angle of a given ray (i) to the surface normal

β

i = Output or viewing angle of a given ray (i) to the

surface normal

A new and unified model, backed by experimental data,

needs to be created for non-commercial use that states the

effects of multiple capillary parameters. Previous models do

not account for all seven first-order rays, the polarization

state of the incident light, and the dimensions and material

of the capillary.

METHOD

Researchers from the University of Bristol, UK have

developed a new ray tracing model for analyzing fringe

patterns of a hollow glass tube capillary. Two previous

ray tracing models are combined to create one composite

model for more general use. Through this simulation model

and experiment data, researchers can better understand

and improve the sensitivity of backscattering interferometry.

The effects of the polarization of the incident light and the

characteristics of the capillary are thoroughly examined.

The experimental apparatus can be seen in

Figure 3.

A 7 mW laser passed through a filter, fiber, and a linear

polarizer to obtain a single spatial mode with a variable

linear polarization state. The light is incident onto the

circular glass capillary with outer radius R and inner radius

r with refractive index n

1. The capillary, containing a liquid of

refractive index n

2, is held on a custom aluminum stage with

constant temperature control from a Peltier thermoelectric

module controlled by Wavelength Electronics' WTC3243HB

temperature controller. The light that is backscattered

or reflected and refracted from the capillary in a direction

anti-parallel to the incident light creates an interference

pattern collected by the camera.1 The interference pattern is

recorded and can change/move laterally due to fluctuations