Question

The photograph of upconversion in Color Plate 21shows total internal reflection of the blue ray inside the cuvet. The angle of incidence of the blue ray on the wall of the cuvet is$\mathbf{~}\mathbf{55}\mathbf{°}$. We estimate that the refractive index of the organic solvent is 1.50and the refractive index of the fused-silicacuvet is 1.46. Calculate the critical angle for total internal reflection at the solvent/ silica interface and at the silica/air interface. From your calculation, which interface is responsible for total internal reflection in the photo?

Short answer

The critical angle for total internal refractive index is ${\mathrm{\theta }}_{\mathrm{critical}}=76.7°{\mathrm{\theta }}_{\mathrm{critical}}=43.6°$.

Step-by-step solution

Concept used.

The following formula may be used to calculate the critical angle for total internal reflection:$\sin \left({\mathrm{\theta }}_{\mathrm{critical}}\right)=\frac{{\mathrm{n}}_2}{{\mathrm{n}}_1}$

The critical angle.

The critical angle for the solvent/silica interaction is:

$$\begin{gathered} \sin \left({\mathrm{\theta }}_{\mathrm{critical}}\right)=\frac{1.46}{1.5}\sin \left({\mathrm{\theta }}_{\mathrm{critical}}\right) \\ =0.9733{\mathrm{\theta }}_{\mathrm{critical}} \\ =76.7° \end{gathered}$$

The critical angle for the silica/air interaction is:

$$\begin{gathered} \sin \left({\mathrm{\theta }}_{\mathrm{critical}}\right)=\frac{1}{1.45}\sin \left({\mathrm{\theta }}_{\mathrm{critical}}\right) \\ =0.6897{\mathrm{\theta }}_{\mathrm{critical}} \\ =43.6° \end{gathered}$$

The angle of incidence of the blue ray on the wall of the cuvet is$\mathbf{55}\mathbf{°}$.Which is a smaller angle than the critical angle for the solvent/silica contact and a bigger angle than the critical angle for silica/air. As a result, the photo's whole internal reflection is caused by the silica/air contact.