PEM200 User Manual
56
Bessel Function Ratio Methods
Most of the calibration methods described above are used to establish specific
retardation levels with precision. There is sometimes a need to extend the calibration to
other retardation settings. Using a lock-in amplifier, these extensions may be
accomplished using the ratios of Bessel functions. The method described here
assumes the use of a stable light source and optical setup. The example given uses
the half-wave condition, but the technique can be easily modified to use a Bessel
Function Zero starting point.
Consider Figure A.2 and the 2f term of equation 4. The signal voltages
f
V
2
for two
different retardation levels A
0
(1)
and A
0
(2)
are proportional to the Bessel functions of
those retardations, as shown in equation 5:
( )
( )
( )
( )
( )
( )
( )
( )
(
)
( )
( )
(
)
( )
( )
( )
( )
2
0
2
1
0
2
2
0
2
1
0
2
2
0
2
1
0
2
2
2
1
2
2
cos
2
2
cos
2
A
J
A
J
t
A
J
t
A
J
A
I
A
I
V
V
f
f
f
f
=
Ω
Ω
=
=
(5)
Assume that one of the retardation settings is the half-wave calibration setting and the
other setting is at somewhat less retardation. (Quarter-wave retardation is a good
example.) Equation 5 may be rewritten as equation 6:
( )
( ) ( )
( )
π
π
λ
λ
2
2
2
2
2
2
4
J
J
V
V
f
f
=
(6)
half-wave retardation:
π
=
0
A
quarter-wave retardation:
2
0
π
=
A
The method is simple:
1. Using an oscilloscope, determine the half-wave calibration condition.
2. Read the lock-in amplifier reading and compute the expected lock-in reading for
quarter-wave retardation.
3. Adjust the controller until this lock-in output is obtained. The modulator will then
be correctly set for quarter-wave operation.
A sample calculation is given below for quarter-wave retardation. The required Bessel
functions are:
For half-wave:
( )
485
.
0
2
=
π
J
For quarter wave:
( )
250
.
0
2
2
=
π
J
