Bessel Function Ratio Methods
PEM 100 User Manual
71
Addition of a birefringent element as shown in figure A.1, so that a significant 1f
signal at the modulator frequency is obtained, will enable determining the controller
setting for which
( )
0
0
1
=
A
J
. (This occurs for
872
.
3
0
=
A
radians.) Detection of the
null point is made with a lock-in amplifier at the modulator frequency.
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.
Summary of Contents for PEM-100
Page 1: ...PEM 100 PHOTOELASTIC MODULATOR USER MANUAL Hinds Instruments Inc P N 010 0000 021 UM Rev H ...
Page 4: ......
Page 8: ...Table Of Contents viii PEM 100 User Manual ...
Page 12: ...Table of Tables xii PEM 100 User Manual ...
Page 22: ...Unpacking the PEM 100 10 PEM 100 User Manual ...
Page 46: ...Chapter 5 PEM 100 Controller Display 34 PEM 100 User Manual ...
Page 52: ...Chapter 6 Operating the PEM 100 Controller 40 PEM 100 User Manual ...
Page 68: ...Chapter 9 Troubleshooting 56 PEM 100 User Manual ...
Page 74: ...Chapter 10 Maintenance 62 PEM 100 User Manual ...
Page 88: ...Appendix A Calibration 76 PEM 100 User Manual ...
Page 92: ...Appendix C Optical HeadSpecifications 80 PEM 100 User Manual ...
Page 100: ......