W. M. McClain, Wen-Haw Jeng, Biswajit Pati, Yaoming Shi, and Duan Tian, "Measurement of the Mueller scattering matrix by use of optical beats from a Zeeman laser," Appl. Opt. 33, 1230-1241 (1994)
A two-frequency beam from a Zeeman laser scatters elastically from an isotropic medium, such as randomly oriented viruses or other particles suspended in water. The Zeeman effect splits the laser line by 250 kHz, and beats can be seen electronically in the signal from a phototube that views the scattered light. There are independently rotatable half-wave and quarter-wave retardation plates in the incident beam and a similar pair in the observed scattered beam, plus a fixed linear polarizer directly in front of the detector. Each of the four retarders has two angular positions, providing a total of 16 possible polarization cases. For each of the 16 cases, there are three data to be collected: (1) the average total intensity of the scattered light, (2) the amplitude of the beats in the scattered light, and (3) the phase shift between the beats of the scattered light and those of a reference signal from the laser. When a singular value decomposition technique is used, these threefold redundant data are rapidly transformed into a best-fit 4 × 4 Mueller scattering matrix. We discuss several different measurement strategies and their systematic and statistical errors. We present experimental results for two kinds of particle of wavelength size: polystyrene spheres and tobacco mosaic virus. In both cases the achiral retardation element M34 of the Mueller matrix is easily measurable.
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Each retarder i has retardation δi and two rotational positions, ρi(a) and ρi(b). When the rotation angle is zero, the fast axis is vertical (i.e., perpendicular to the scattering plane); positive rotations are counterclockwise looking into the light source.
Table 2
Example Mueller Matrix on Which the Error Calculations Are Based, Showing the Perrin Symmetry
1.000
0.800
0.030
0.020
0.800
0.950
0.040
0.035
−0.030
−0.040
0.350
−0.200
0.020
0.035
0.200
0.450
Table 3
Data Expected When Measuring the Mueller Matrix of Table 2 by Using the Measurement Strategy of Table 1
See Table 1 for rotator positions a and b.
These columns are scaled with each other but are in arbitrary units because the efficiency η is unknown.
These columns are redundant with columns Vin and Vout.
Note the range of ϕ, the phase angle between incident and scattered Zeeman beats.
Table 4
Tabulated quantity (1/10)(d/dθ)log(Mij) for the Sparse Strategya
The derivative is evaluated by using the angular values of Table 1 and the Mueller matrix values of Table 2.
Angles δ are retardation angles; the two associated angles ρ are the rotation angles of the same retarder. The numbering of retarders is the same as in Table 1.
Angle ρ5 is the orientation angle of the final polarizer.
Table 5
Specification of the Diagonal Strategy, with a D+ Polarizer Preceding Retarder 1 and Following Retarder 4
Retarder
Parameter
i = 1
i = 2
i = 3
i = 4
δI
π/2
π/4
π/4
π/2
ρi(a)
π/4
π/4
π/4
π/4
ρi(b)
3π/8
3π/8
π/8
π/8
Table 6
Tabulated Quantity (1/10)(d/θ)log(Mij) for the the Diagonal Strategy
Each retarder i has retardation δi and two rotational positions, ρi(a) and ρi(b). When the rotation angle is zero, the fast axis is vertical (i.e., perpendicular to the scattering plane); positive rotations are counterclockwise looking into the light source.
Table 2
Example Mueller Matrix on Which the Error Calculations Are Based, Showing the Perrin Symmetry
1.000
0.800
0.030
0.020
0.800
0.950
0.040
0.035
−0.030
−0.040
0.350
−0.200
0.020
0.035
0.200
0.450
Table 3
Data Expected When Measuring the Mueller Matrix of Table 2 by Using the Measurement Strategy of Table 1
See Table 1 for rotator positions a and b.
These columns are scaled with each other but are in arbitrary units because the efficiency η is unknown.
These columns are redundant with columns Vin and Vout.
Note the range of ϕ, the phase angle between incident and scattered Zeeman beats.
Table 4
Tabulated quantity (1/10)(d/dθ)log(Mij) for the Sparse Strategya
The derivative is evaluated by using the angular values of Table 1 and the Mueller matrix values of Table 2.
Angles δ are retardation angles; the two associated angles ρ are the rotation angles of the same retarder. The numbering of retarders is the same as in Table 1.
Angle ρ5 is the orientation angle of the final polarizer.
Table 5
Specification of the Diagonal Strategy, with a D+ Polarizer Preceding Retarder 1 and Following Retarder 4
Retarder
Parameter
i = 1
i = 2
i = 3
i = 4
δI
π/2
π/4
π/4
π/2
ρi(a)
π/4
π/4
π/4
π/4
ρi(b)
3π/8
3π/8
π/8
π/8
Table 6
Tabulated Quantity (1/10)(d/θ)log(Mij) for the the Diagonal Strategy
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