An Analysis of Semicircular Channel Backscattering Interferometry through Ray Tracing Simulations

Abstract

Recent backscattering interferometry studies utilise a single channel microfluidic system, typically approximately semicircular in cross-section. Here, we present a complete ray tracing model for on-chip backscattering interferometry with a semicircular cross-section, including the dependence upon polarisation and angle of incidence. The full model is validated and utilised to calculate the expected fringe patterns and sensitivities observed under both normal and oblique angles of incidence. Comparison with experimental data from approximately semicircular channels using the parameters stated shows that they cannot be explained using a semicircular geometry. The disagreement does not impact on the validity of the experimental data, but highlights that the optical mechanisms behind the various modalities of backscattering interferometry would benefit from clarification. From the analysis presented here, we conclude that for reasons of ease of analysis, data quality, and sensitivity for a given radius, capillary-based backscattering interferometry affords numerous benefits over on-chip backscattering interferometry.

Keywords: backscattering interferometry; microfluidics; ray tracing; refractive index; semicircular channel.

Figure 1

(A): A diagram showing the ray path taken when light is incident perpendicular to a chip with a semicircular channel with intersection number $i=1$. (B): The path of a ray when the incident light is oblique to the chip surface with intersection number $i=3$. Positive x is defined to be to the right.

Figure 2

A diagram showing examples of a type a, b, and c ray, highlighting the segments of the c ray that correspond to the path lengths sections $L_{0-5}$ as described in the text.

Figure 3

Graphs showing the relative amplitudes (where $I_0=1$) of type c rays for both s- and p-polarised incident light at normal incidence ($\psi =0^{\circ }$, (A)) and oblique incidence ($\psi =3^{\circ }$, (B)). The data here are taken using the standard parameters as defined in the main text. Values of $x/r$ between $\pm 1.5$ are simulated to sample the full range of values that give rise to type c rays. The dashed lines in A represent the bounds on a given intersection number, with the central section denoting $i=1$ and increasing by 1 upon crossing a line moving outwards. The sudden reduction in amplitude at the edges is due to the rays no longer entering the channel at this angle. The amplitudes of type a and b rays for p-polarised light are omitted due to their similarity with their s-polarised counterparts.

Figure 4

A graph showing the optical path length difference of a type c ray for both normal and oblique incidence ($0^{\circ }$ and $3^{\circ }$, respectively). The difference is defined to be with respect to a ray of $\psi =0$ and $x=0$ (i.e., solving Equation (18) and subtracting $2n_{2}r+2n_{1}t+n_{0}d$). Data were simulated using the parameters as set out in the main text.

Figure 5

Graphs showing the interference patterns seen on a detector using the parameters as set out in the main body of text at a distance of $1\mathrm{m}$ with the angle given from the line of $x=0$. (A) shows the interference pattern for s-polarised incident light, whereas (B) shows the pattern imaged for p-polarised light.

Figure 6

A graph showing the dechirped Fourier transform of the interference pattern seen at normal incidence. A single sharp peak in the Fourier domain is seen here for both s- (A) and p-polarised (B) incident light. A graph showing how the phase of each peak in (A,B) changes as a function of refractive index $n_2$ is shown in (C). All data were taken using the parameters set out in the main text.

Figure 7

A graph showing the dechirped Fourier transform of the interference pattern seen at an incident angle of $\psi =3{}^{\circ }$. A sharp peak in the Fourier domain as seen in Figure 6 is also seen here for both s- (A) and p-polarised (B) incident light. A graph showing how the phase of each peak in (A,B) changes as a function of refractive index $n_2$ is shown in (C). All data were taken using the other parameters set out in the main text.