6.1
Introduction & aims
The aim of this chapter was to see if the LCTVD could be used as part o f a full polarimeter with no moving parts. It was anticipated that polarisation o f light incident on the LCTVD could be determined simply by applying known voltage signals to the display, and measuring the intensity transmitted by an analyser. The LCTVD would induce a phase difference between the orthogonal components of the incident light, and this would enable all four Stokes vectors to be determined. The sense o f rotation of the polarisation ellipse would also be known. As mentioned in Chapter 2, polarisation can be quantified using Jones or Stokes vectors, and each method has advantages and disadvantages. Because the LCTVD had been modelled as a Jones matrix, an algorithm was first developed using Jones calculus. Then a method is described which uses Stokes vectors and a partial Mueller matrix.
This chapter will begin by analysing both methods, and presenting the results obtained using them. It will then consider the effect errors have on each method, and the sources of those errors. The advantages and disadvantages of the LCTVD polarimeter over polarimeters proposed by previous researchers will then be discussed, and the effect of polychromatic sources and oblique angles of incidence considered. The chapter will conclude with two specific applications of the device.
Chapter 6 Algorithm to determine unknown input polarisation
6.2
Jones calculus method
6.2.1 Theory
If the
R, + i M Ry \ +
LCTVD is represented by a (known) Jones matrix,
, its effect on polarised light can be written as:
Ry2 y2 R y^ + iM y, J \ + iJ 2 J -^ + iJ ^ L
+
i M N+
iP where the incident light polarisation is represented by the vector J \ + i J 23 + iJ ^
equation 6-1
, the components
of which are to be determined, and the output light is fully polarised with Jones vector components, L, M, N and P. If the analyser is fixed so that its transmission is along thex-axis.
+ M , and the y component then, from equation 6-1, the intensity measured is equal to
can be ignored. Multiplying out the left-hand side of equation 6-1, separating into real and imaginary components and squaring ‘ leads to equation 6-2. In this equation the K„ values are constants derived from the real and imaginary components of the Jones matrix for each grey level used.
+ J2 ) K.2^'^I +*>^4^)+ -^3(*^1*^3 +*^2*^4) ■^■^4(*^1*^4 ~ 2'^i) equatiou
6-2
In order to determine the 4 unknown components (7/, 7?, 7j, 7/ ), four GLs were applied (in sequence) to the LCTVD. The intensity transmitted by the analyser, JoL (where GL =255, 140, 60 and 0) was measured for each GL. This gave 4 simultaneous equations that could be solved for 7 y , 7 j and 7 / . (equation 6-3). As polarisation rather than absolute phase is
being determined, 7 ^ can be taken to be zero. 7 / therefore represents the relative phase of they
component o f the Jones vector of the input light, equation 6-3 enables the Jones vector o f the input light, and therefore the polarisation, to be calculated.
equation 6-3 ' K , ^ 2 K , K /
-I
2 255 ■+
' E ' K , K , K ,-^140
F K , K u K n J \ J ^"i"
J 2 ^ X G. / »
. _J \ J ^ — J i j y ^ H _Measuring intensity only (i.e. no phase information) lead to a sign ambiguity because Ji was determined as the square root o ïE . If J/ was negative, the phase of the x component would be different by 180°. Tliis would have the effect that the ellipticity o f the light would remain unchanged, (although its sense of rotation would be reversed), but the azimuth angle would be reflected about the co-ordinate system. It was anticipated that this ambiguity could be resolved using some other knowledge of the unknown polarisation.
6.2.2 Results
The Jones matrix algorithm worked well in theory [Blakeney et al., 1999], but was very sensitive to error. Neither the theoretical model, nor the SVD model of the LCTVD predicted the behaviour of the LCTVD sufficiently accurately for this purpose. An example of the sensitivity of the method is shown in Table 6-1 (for an input polarisation at +30° (linear)).
Error induced into modelled a e
intensity values {I2 5 5,1 n o, ho, h) (should be 30°) (should be infinity)
A ll+10% 29.606° 5839
Mixed +/-2% " 43.63° 10.24
Table 6-1 Values for or and e for input polarisation of +30° (linear) if theoretical intensities are corrupted by error.
These values were obtained using the theoretical model o f the LCTVD to calculate the intensities that would be measured for this particular input polarisation. These values were then corrupted by various errors. If the four intensities were all increased or decreased, even a comparatively large error in intensity (10%) had little effect on the polarisation which would have been calculated using these values. However, if some intensities were increased and some were decreased, even as small an error as +/2% caused an unacceptable value o f a and e to be obtained. The SVD model of the LCTVD, although it matched the experimental results more closely than the theoretical model, was not accurate enough for this method to give good results.
The poor experimental results caused by the sensitivity of the method, together with the inherent sign ambiguity, and the limitation of only completely polarised light being measured, lead to an alternative method being developed. This was based on Stokes vectors and Mueller calculus.
Chapter 6 Algorithm to determine unknown polarisation
6.3
Stokes vector method
Stokes vectors.are intensity based, as opposed to the phase based Jones vectors. They are derived from experimental measurements of intensity (s.2.3.7) and so lend themselves to situations where intensity (rather than phase) is measured.
6.3.1 Theory
The action of the LCTVD upon incident light can be written as:
m, m3
m , my mg
mg mil m,2 ^' 2
_f%l3 ^14 mi5 ^16 _ .^ 3 . _^3_
equation 6-4
Where [5'o, Sj, S2, are the Stokes components o f the input light (unknown), and [S"o, S ’l, S ’2, are the Stokes components of the output light. The matrix, tw is the Mueller matrix of
the LCTVD. Assuming the LCTVD is not a depolarising optical element, a Jones matrix can be converted to a Mueller matrix [Lu & Chipman, 1998], using the procedure laid out by Azzam [1979]. From the definitions of the Stokes vectors laid out in s.2.3.7, the intensity transmitted by a polariser with its transmission along the x axis, is:
S \ + S \ equation 6-5
If the intensities measured, using the photodiode, through the polariser (along thex axis) for each of the four LCTVD voltages are denoted by Igl, (where GL= 255, 140, 60 and 0), and the Mueller matrix components for each GL are denoted by m^Gi) then combining equation 6-4 and equation 6-5:
^ 5 ( G L ) ^ 0 ^ 6 { G L ) ^ \ ^ S { G L ) ^ 3 (G L ) ^ 0 equation 6-6
Equation 6-6 assumes that the total intensity remains the same before and after the light has passed through the LCTVD, so S ’o of the output is the same as So of the input.