From Fig. 2.13 in Chapter 2 it is evident that the linear polarisation fraction of the MSE emission is greater than, but does not necessarily dominate, the circular polarisation fraction. In some IMSE polarimeters s1 ors2 share a spatial carrier with s3 which may
potentially corrupt the linear polarisation orientation encoding, therefore the interfero- metric properties of the circular polarisation s3 spectrum are considered in this section.
Sensitivity to the circular polarisation can be achieved using a polarimeter with φ1 = 0,
as illustrated in the right of Fig. 3.11, such that the general polarimeter described by Eq. 3.9 is now sensitive to s3 at the expense of s1. In this case the detector measures
2S = Z ∞ 0 s0(ω) +s2(ω) cosφ2(ω, y) +s3(ω) sinφ2(ω, y) f(ω)dω. (3.50)
The interferogram for Rpc,M SEsinφdω is shown in the left of Fig. 3.13 and can be com-
pared to the interferogram for the linear component in Fig. 3.12 after scaling by γ/ and appropriate geometric factors. Integrating over the MSE spectrum, the signal reduces to
2S= Z ∞
0
I(ω) 1 + sin 2θσpl,M SE(ω) cosφ2(ω) +pc,M SE(ω) sinφ2(ω)
f(ω)dω = 1 +ζl sin 2θσcos(φ0+αl) + ζc ζl sin(φ0+αc) (3.51) where equivalent definitions forζcandαc, are given earlier in Eqs. 3.46 and 3.47 requiring
only a change of subscripts. However the integrand in Eq. 3.47 is now approximately purely imaginary for the circular component, because s3(ω) =pc,M SE(ω) is an odd func-
tion aboutω0. Hencethere is an approximately±90◦ phase shift between the linearαl and
circular αc terms and care must be taken to determine the sign of αc. In Fig. 2.13 where
the view has the property sign(i·B) >0 we see that s3 >0 for ω < ω0 and s3 < 0 for ω > ω0 such that the Eq. 3.47 integral gives a negative imaginary number when φ0 >0.
Therefore the phase offset for the circular component can be expressed
αc=−m±π/2 +δc (3.52)
wherem±= (−1)nsign (i.B)×φ0
,
where δc ≈ 0 and n is the number of zero crossings of ζc (not including the crossing at φ= 0). The phase of αc flips at each zero crossing of ζc as evident in Fig. 3.13. Similar
to the idealised calculation for ζl in Eq. 3.48 the value ofζc is
ζc= 2γsinψcosϕ P8 n=−8InζSnCnsin nκφ 2~ω0 P8 n=−8In (3.53)
whereCnare the circularity of the transitions given in Table. 2.11, correct to second order
inγ/. The sign of−m±in Eq. 3.52 is the same as the sign of the term inside the absolute value brackets of Eq. 3.53. For simplicity the first order γ/ circularity terms have been kept but the second order line splitting terms have been dropped in deriving Eq. 3.53.
0.0 0.5 1.0 1.5 2.0 -0.4 -0.2 0.0 0.2 0.4 3κ ϵ 2π ω0ℏϕ(rad) 1 ∫ I0 pc sin ϕ d ω Linear Circular 0.0 0.5 1.0 1.5 2.0 0.0 0.2 0.4 0.6 0.8 3κ ϵ 2π ω0ℏ ϕ(rad) Contrast ζ
Figure 3.13: (Left) Interferogram of the circular component (unscaled such that
γsinψcosϕ/= 1) for a|E|= 200MV m−1. The phase flip at the zero crossing is evident. (Right) Contrast of linear polarisation (blue) and circular polarisation (green) for the nor- malised delay. The linear contrast must be scaled by sin2ψ and the circular contrast by
γsinψcosϕ/, both must be scaled by ζS. Gridlines are included for the maximum linear
polarisation contrast and the zero crossing of the circular polarisation. Zero crossings in
ζc are evident near 0.95 and 1.6.
ζland ζcare plotted in the right of Fig. 3.13 where it is evident thatζcis significantly
less than ζl after the necessary scaling by γ/ and the sinψcosϕ geometric factors are
applied. The effective interferometric ellipticityξI for an IMSE system can now be defined
with
tan 2ξI(φ0) = ζc(φ0) ζl(φ0)
. (3.54)
The definition is comparable to Eq. 2.22 where tan 2ξ =s3/
p
s21+s22, however ξI is now
dependent on the value of φ0 unlike the polarisation orientation θσ. For the spectra in
Fig. 2.13 and at the delay that maximisesζl we have that tan 2ξI= 0.15γsincosψϕ = 0.023 or ξI = 0.67◦ when γ/= 6.6, ψ=π/2 andϕ= 0.
Now that the interferometric ellipticity has been defined Eq. 3.51 can be reduced to 2S =I0
1 +ζl sin 2θσcos(φ0(y) +αl)−m±tan 2ξIcos(φ0(y) +δc)
. (3.55) Therefore the linear polarisation and circular polarisation carriers are actually out of phase for the MSE spectrum when αl =δc (or in phase when m± =−1). This differs to a monochromatic elliptically polarised source where the carriers would be in quadrature, as seen in Eq. 3.50.
A key result is thatthe circular polarisation is ‘invisible’ to an IMSE polarimeter with
delay that satisfiesζc(φ0) = 0. For the Stark-Zeeman effect the zero crossing of the circular
polarisation contrast occurs when
φ(ω0) = 0.94×
2πω0~
3κ . (3.56)
the relative upper-state populations of the different transitions. Importantly the crossing point is close to the maximum of the linear polarisation orientation and therefore can be targeted to eliminate the sensitivity of the polarimeter to circular polarisation without significantly sacrificing the linear polarisation signal. However it should be noted that
ζc= 0 cannot be precisely satisfied across the field of view due to variations in the Doppler
shiftω0and line splitting, with an example given later in Fig. 4.16. Furthermore if some
of the half energy multiplet lies in the passband of the IMSE filter or if the filter doesn’t uniformly transmit the full energy component then δc6= 0. In this case it is necessary to
consider both the phase and magnitude of the interferometric ellipticity and not just the sign and magnitude as considered here. These complications will become clearer in the next chapter where they are applied in a realistic situation.