Small n Approximation for Uniaxial Retarders
(A simple way to calculate the effect of retarders on off-axis rays.)
(These equations not readily available elsewhere.)
Rotated Index Ellipsoid:
For a uniaxial retarder with the extraordinary axis along Z, the index ellipsoid is:
X 2 Y2 Z2
(eq. 1)


1.
no2 no2 ne2
To find new axes, x, y, z  , rotated  degrees about the Y axis, we use the
following transformations:
 x  cos  0  sin   X 
 y   0
1
0    Y 
  
 z   sin  0 cos    Z 
-and X   cos  0 sin    x 
Y    0
1
0    y 
  
 Z   sin  0 cos   z 
Using the second transformation above to substitute into equation 1, we get the
equation of the index ellipsoid in the new axes:
x cos   z sin 2  y 2  z cos   x sin 2  1 .
no2
no2
ne2
For a ray propagating along the (new) z -axis, the effective indices will be given
by the major and minor axes of the ellipse obtained by cutting the above ellipsoid
with the z  0 plane. Setting z  0 in the above equation and rearranging gives
the ellipse:
x2
y2

 1.

 no2
no2 ne2
 2

2
2
2
 ne cos   no sin  
Hence, the effective extraordinary index for this ray is:
ne no
n e 
(eq. 2)
n e2 cos 2   n o2 sin 2 
Expand the right hand side of equation 2 and apply the approximation,
ne  no  n 1:
1
A)
ne2 cos 2   no2 sin 2   ne2  no2   ne2  no2 cos 2
2


ne2  no2  (no  n) 2  no2
 2no2  2no n  n 2
 2no2  2no n,
B)
n  1
 2no (no  n)
 2 no ne
ne2  no2  no  n   no2
2
 2no n  n 2
C)
 2 n o  n,
n  1
Putting expression A, and approximations B & C into equation 2:
ne no
ne 
ne no  no n cos 2
ne no

1
n
cos 2
ne
no 1 

1
n
no
n
cos 2
ne
Expanding the radicals in truncated Taylor Series:


n 
n
1 
ne  no 1 
cos 2 ,
 2no  2ne

n  1
n n  no 
n 2


 no 

cos 2 
cos 2
2
2  ne 
4ne
n n

cos 2,
2
2
 no  n sin 2 
 no 
n  1
n  ne  no   n sin 2 
Hence,
(eq. 3)
Where  is the angle between the ray and optic axis of the retarder.
Retardance of Plate:
Consider a ray incident on a plane, plate retarder with angle of incidence  i . In
general, there will be two refracted rays in the plate, related by Snell’s law:
sin  i  ne sin  e  no sin  o .
Under the assumption that n  1, we can say that:
sin  e  sin(  o  )
 sin  o cos   cos  o sin 
 sin  o   cos  o ,
since   1
Substituting the above result into Snell’s law and solving for  we get:
 
n 
tan  o .
ne
(eq. 4)
Also, expanding cos  e ;
cos  e  cos o   
 cos  o cos   sin  o sin 
 cos  o   sin  o ,
  1.
Substituting  from equation 4 above:
cos  e  cos  o 
n
tan  o sin  o
ne
(eq. 5)
The retardance of a birefringent plane/parallel plate is:1
  t ne cos e  no cos o ,
where t is plate thickness. Substituting the approximation for cos  e from eq. 5:
  t ne cos  o  n tan  o sin  o  no cos  o 
 tncos  o  tan  o sin  o 

tn
.
cos  o
Hence,

1
tn
cos  o
See Principles of Optics, Born & Wolf, p.697, for example.
(eq. 6)
Summary:
Equations 3 & 6 give the retardance of an arbitrary uniaxial retarder, given that
(ne  no )  1. Summarizing the relevant relations:
n  n sin 2 
1.
2.
where:
 
tn
cos 
 is the angle between a refracted ray and the optic axis,
and,
 is the angle between the refracted ray and the normal to the plate.
In one equation:
sin 2 
(, )  0
,
0  t ne  no 
cos 