Calcite deviation between r-ray and o-ray

enotstrebor

One explanation of the deviation of o-ray and e-ray is done using Huygen's wave fronts which for the o-ray is a circle and for the e-ray is and ellipse. The ellipse is given by the two refraction induces of 1.658 and 1.486 (http://physics.info/refraction/)
are "a" (major axis) and "b" (minor axis) respectively.

I have read in several places that the maximum deviation of the o-ray and e-ray is about 6 degrees.

However when I do the calculation for the angular difference for the same tangent using Huygen's wave fronts (a circle and ellipse) I only get a maximum angular deviation of about 3 degrees.

Can anyone explain the difference, which obviously includes some possible error of mine?



NOTE:

As, for example, in the following equations, the square root only gives positive values the equations require the addition of "-" to put the results in the correct quadrant.

The ellipses I used had the angle from the major axis which I put along the y-axix; given the tangent angle Ang the tangent line equation is

y = Ang*x + sqrt(Ang^2*b^2+a^2)

The x coordinate of the point on the ellipse is

x = - [ sqrt( (b^4*a^2) / ( tan(Ang)^-2 a^4 + b^2*a^2 ) ) ]

where the "-" comes in because the positive slopes are to the left of zero.

One then uses the radian arcsin of "(-x/b)" to get the angle to that tangent point (TangentAngle =arcsin(-x/b)*180/\pi).

Subtracting the starting (circle tangent) Ang-TangentAngle gives you the deviation.

The maximum deviation is at ~46.568 where the asymmetry is due to the "stretching" along the "a" direction.

Any experimental measurement of the maximum deviation angle will be appreciated.