# Light pipes and fiber optics

Snell's Law as given by (see Predictions of real surface properties for radiation)

$\sin \chi = \frac{{{n_1}}}{{{n_2}}}\sin \theta$

gives the angle of refraction for radiation traveling through medium 1 with refractive index n1 and encountering an interface with a second material with refractive index n2. Suppose that the radiation is traveling in glass or other transparent material and encounters a surface such as air with n2 < n1. In this case, Snell's Law predicts

$\sin \chi = (\frac{{{n_1}}}{{{n_2}}} > 1)\sin \theta \qquad \qquad(1)$

Thus, unless the angle of incidence on the interface θ < sin − 1(n2 / n1) , the angle of refraction is predicted to be χ > π/2. Practically, whenever θ > sin − 1(n2 / n1), the radiation within the higher refractive index material will undergo 100 percent reflection at the interface. This means that once radiation enters a perfectly transmitting material (a perfect dielectric) such as an optical fiber in which no radiation absorption occurs, the radiation will propagate along the fiber with the perfect wall reflectivity not allowing any losses by refraction through the fiber surface. This phenomenon is observed in a planar geometry by swimmers (and fish), who can only see through the water surface above them within a cone of angles described by

$\theta < {\sin ^{ - 1}}\left( {\frac{{{n_{air}}}}{{{n_{water}}}}} \right) = {\sin ^{ - 1}}\left( {\frac{1}{{1.33}}} \right) = {\sin ^{ - 1}}\left( {0.75} \right) = {48.6^o}$
Critical angle for view through the surface of a material with refractive index greater than surroundings.

At greater angles, the water surface appears to be a mirror (see figure). For a light-pipe or fiber optic, radiation enters the flat end of a long circular cylinder. If the refractive index ratio n2 / n1 exceeds $\sqrt 2$, all radiation entering the light-pipe end will encounter the cylindrical surface at greater than the critical angle (Qu et al., 2007b). Because of the increasing reflectivity of dielectrics with angle (see Predictions of real surface properties for radiation

${\rho _\lambda }_{,\parallel }\left( {\lambda ,\theta } \right) = {\left\{ {\frac{{{{\left( {\frac{{{n_2}}}{{{n_1}}}} \right)}^2}\cos \theta - {{\left[ {{{\left( {\frac{{{n_2}}}{{{n_1}}}} \right)}^2} - {{\sin }^2}\theta } \right]}^{1/2}}}}{{{{\left( {\frac{{{n_2}}}{{{n_1}}}} \right)}^2}\cos \theta + {{\left[ {{{\left( {\frac{{{n_2}}}{{{n_1}}}} \right)}^2} - {{\sin }^2}\theta } \right]}^{1/2}}}}} \right\}^2}$

the radiation actually entering the lightipipe end is from near-normal angles.

Qu et al. (2007a, 2007b) have analyzed the errors that arise because of various signal loss mechanisms that can occur in a light-pipe, such as the presence of very minor surface imperfections, effects of blocking and shadowing of reflected energy from the surface being measured, changing the temperature of the measured object by radiative transfer to the probe itself, and any environmental radiation that may enter the light-pipe walls and enter the internal signal path by encountering surface imperfections or scattering centers within the light-pipe.

The signal entering the detector at the end of an LPRT or optical fiber consists of emitted plus reflected energy from the object being viewed. If the environment is cold relative to the viewed object so that reflected energy can be neglected, and the detector is only sensitive in a small wavelength range around a particular wavelength λ, then the detected emission is, from eq. ${\varepsilon _\lambda }(\lambda ,T) = \frac{{{E_\lambda }\left( {\lambda ,T} \right)}}{{{E_{\lambda b}}\left( {\lambda ,T} \right)}}$ from Opaque Surface Property Definitions,

${E_\lambda }(\lambda ,{T_{act}}) = {\varepsilon _\lambda }{E_\lambda }(\lambda ,{T_{act}}) = {\varepsilon _\lambda }\frac{{2\pi {C_1}}}{{{\lambda ^5}\left[ {\exp \left( {\frac{{{C_2}}}{{\lambda {T_{act}}}}} \right) - 1} \right]}} \qquad \qquad(2)$

However, rather than the actual temperature Tact , the detector reads an apparent temperature Tapp that would appear to originate from a blackbody,

${E_\lambda }(\lambda ,{T_{app}}) = \frac{{2\pi {C_1}}}{{{\lambda ^5}\left[ {\exp \left( {\frac{{{C_2}}}{{\lambda {T_{app}}}}} \right) - 1} \right]}} \qquad \qquad(3)$

Equating the two emitted energy rates relates the actual and apparent temperatures as

${\varepsilon _\lambda }\frac{{2\pi {C_1}}}{{{\lambda ^5}\left[ {\exp \left( {\frac{{{C_2}}}{{\lambda {T_{act}}}}} \right) - 1} \right]}} = \frac{{2\pi {C_1}}}{{{\lambda ^5}\left[ {\exp \left( {\frac{{{C_2}}}{{\lambda {T_{app}}}}} \right) - 1} \right]}} \qquad \qquad(4)$

For most engineering conditions, the factor of (-1) in the denominator can be neglected relative to the exponential term, and the actual temperature in terms of the apparent absolute temperature then becomes

${T_{act}} = \frac{{{T_{app}}}}{{\left[ {1 + \frac{{\lambda {T_{app}}}}{{{C_2}}}\ln {\varepsilon _\lambda }} \right]}} \qquad \qquad(5)$

As ε λ → 1, the apparent and actual temperatures approach one another. eq. (7) can be used for many spectrally-based temperature measurements, remembering the proviso that the environmental temperature must be low in order get an accurate temperature measurement from an optical pyrometer or light-pipe radiation thermometer.

## References

Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.

Qu, Y., Puttitwong, E., Howell, J. R. and Ezekoye, O. A., 2007a, ‘‘Errors Associated with Light-pipe Radiation Thermometer Temperature Measurements,” IEEE Trans. Semiconductor Manufacturing, Vol. 20, pp. 26-38.

Qu, Y., Howell, J. R. and Ezekoye, O. A., 2007b, ‘‘Monte Carlo Modeling of a Light-pipe Radiation Thermometer,” IEEE Trans. Semiconductor Manufacturing, Vol. 20, pp. 39-50.