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Three-dimensional ray tracing through curvilinear interfaces with application to laser Doppler anemometry in a blood analogue fluid

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Abstract

Prediction of the effects of refractive index (RI) mismatch on laser Doppler anemometer (LDA) measurements within a curvilinear cavity (an artificial ventricle) was achieved by developing a general technique for modelling the paths of the convergent beams of the LDA system using 3D vector geometry. Validated by ray tracing through CAD drawings, the predicted maximum tolerance in RI between the solid model and the working fluid was ± 0.0005, equivalent to focusing errors commensurate with the geometric and alignment uncertainties associated with the flow model and the LDA arrangement. This technique supports predictions of the effects of refraction within a complex geometry. Where the RI mismatch is unavoidable but known, it is possible not only to calculate the true position of the measuring volume (using the probe location and model geometry), but also to estimate degradation in signal quality arising from differential displacement and refraction of the laser beams.

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Notes

  1. These latter locations were (0, ± r z , z) when the probe was parallel to the x-axis, and (± r z , 0, z) when the probe was parallel to the y-axis.

  2. A capital Z represents an approximating function (in the z–r plane), as opposed to the Cartesian coordinate.

  3. Nevertheless, the model can be readily extended should more interfaces be required.

  4. The parameters pertaining to the other three beams are readily obtained by invoking symmetry about the optic axis; this gives rise to three sets of equations analogous to Eqs. 3–6, featuring the subscripts P2, P3, P4.

Abbreviations

α :

Half angle of convergence of the laser beams (in air)

B :

Basepoint vector

γ :

Scalar variable

d :

y displacement of the probe objective lens from the near wall of the VAD

d min :

The minimum separation between two rays

f :

Focal length of the LDA probe (in air)

h :

x displacement of the optic axis from the origin of the VAD coordinates

k :

z displacement of the optic axis from the origin of the VAD coordinates

μ :

Fluid viscosity

n :

Refractive index

N :

Normal vector

r :

Radial coordinate (normal to the outflow axis)

ρ :

Fluid density

R :

Equatorial radius of the VAD

s :

Displacement of a beam from the optic axis at the probe objective lens

s sub :

Displacement from the optic axis of an intersection between a laser beam and a model interface (‘sub’ = F or P)

t :

Minimum thickness of the PMMA wall

T :

Transmission vector

x :

Cartesian coordinate in the plane of the diaphragm–housing junction, parallel to the projection of the inflow axis

y :

Cartesian coordinate in the plane of the diaphragm–housing junction, normal to the projection of the inflow axis

z :

Cartesian coordinate parallel to the outflow axis

Z :

Function of r describing the profile of the axially symmetric part of the VAD cavity (subscript C indicates circular arc; subscript P indicates polynomial segment)

F:

The PMMA:fluid interface

O:

outflow tract

P:

The air:PMMA interface

S:

intersection of VAD profile functions

1:

The y-most beam in the x–y plane

2:

the y-least beam in the x–y plane

3:

the z-most beam in the y–z plane

4:

the z-least beam in the y–z plane

References

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Authors and Affiliations

  1. Graduate School of Biomedical Engineering, University of New South Wales, Sydney, NSW, 2052, Australia

    Allen H. Nugent

  2. Biofluid Mechanics Laboratory, Faculty of Engineering, University of New South Wales, Sydney, NSW, 2052, Australia

    Christopher D. Bertram

  3. Unit 6/15 Rosa Street, Oatley, NSW, 2223, Australia

    Allen H. Nugent

Authors
  1. Allen H. Nugent
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  2. Christopher D. Bertram
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Correspondence to Allen H. Nugent.

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Nugent, A.H., Bertram, C.D. Three-dimensional ray tracing through curvilinear interfaces with application to laser Doppler anemometry in a blood analogue fluid. Med Biol Eng Comput 48, 147–156 (2010). https://doi.org/10.1007/s11517-009-0511-7

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  • DOI: https://doi.org/10.1007/s11517-009-0511-7

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