Pavilion - Angles Of Reflected Rays  
Click for enlarge
Figure 1

Figure 1To enlarge Figure 1 shows a flat-top gem with a typical ray reflected off both sides of the pavilion so that it is returned to the viewer. Equating angles of incidence and reflection (relative to the surface normal) gives ...



P + q1 = q2 - P .......... q1 - q2 = - 2P        (1a)
P + q3 = 180o - q2 - P .......... q3+q2 = 180o - 2P        (1b)
Adding these eliminates q2, thus .......... q1+q3 = 180o - 4P        (1c)
or .......... q1+q3 = 4(45o - P)       

Eq.(1c) shows that the angle (q1+ q3) between rays q1 and q3 is constant for each pavilion angle P; thus, as angle q1 decreases, angle q3 increases by the same amount. This means that one ray is at a maximum value when the other is at a minimum ...
q max+q min = 180o-4P .......... (1d)

  Pavilion - Limits Of Reflected Rays  

   A ray is reflected 3 only if its angle to the surface normal is more than the critical angle C; thus, for reflection to occur ...
P+qmin = C ......or...... qmin = C-P        (2a)

Substituting this into Eq.(1d) gives ...
qmax+ C-P = 180o- 4 P .....thus..... qmax = 180o-C-3P        (2b)

  Pavilion Angle Limits To Reflect All Incident Rays  

  Light incident to the table from all directions enters the stone 3 within the critical angle. For all of this light to be reflected back to the table,

qmin <= - C, qmax >= + C;

thus, per Eq.(2a,b) ...

-C >= (qmin = C-P) .....whereby..... P min = 2C       
+C <= (qmax = 180o-C-3P) .....whereby..... P max = 60o-2C/3       

For diamond, C=24.43o; this gives ...

P min = 48.86o,   P max = 43.71o

The minimum is more than the maximum, which means that there is no solution for diamond which returns all incident light to the table (a solution exists only for C <= 22.5o = synthetic rutile).

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