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Information in Polarisation Imaging

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Informational Limits in Optical Polarimetry and Vectorial Imaging

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Abstract

Information in both the natural and man-made world is frequently not spatially confined to a single point. Whilst, for example, studying the autofluorescence from a single molecule in a cell provides information with regards to that molecule, nothing is learnt about the processes and structure in the whole cell. To do so requires information to be collected from multiple locations. Such is the reason for the prevalence and success of imaging systems. In an optical context, a CCD can be used to record the intensity incident upon each pixel for instance. If located in the image plane of an optical microscope or telescope, information with regards to the object can then be extracted from the intensity readings.

Observations always involve theory.Edwin Hubble

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Notes

  1. 1.

    Paul Abbott and Peter Falloon from the Physics Department of the University of Western Australia kindly provided the Mathematica code necessary to compute Slepian’s generalised spheroidal functions, for which the author is particularly grateful.

  2. 2.

    Mathematically the same inversion procedure can be followed using integration over an infinite plane in the focal region, however here the mathematics is demonstrated using a restricted domain since such integrations are more suitable for numerical routines.

  3. 3.

    This definition of the condition number differs from that used in Chap. 5, however is more appropriate to an eigenfunction analysis.

  4. 4.

    The factor of \(i\) in Eq. (6.26) represents a global phase and can safely be ignored.

  5. 5.

    A technique, developed by the author and colleagues, capable of measuring the longitudinal orientation is presented in Chap. 8.

  6. 6.

    Whilst Eq. (6.31) strictly only compares the shape of the desired and optimised intensity profiles with no regard to the phase distribution, this is of little consequence to the results presented since only the intensity profile is deemed of any importance here.

  7. 7.

    Measurements from different positions can be stacked into a vector format and Eq. (6.35) used, however the assumption that the noise at each measurement position is independent allows simplification to Eq. (6.37), such that the dimensions of \(\mathbf{ D} \) are greatly reduced.

  8. 8.

    The Cauchy-Schwarz inequality reads

    $$\begin{aligned} \left|\int \int f({\varvec{\rho }})g({\varvec{\rho }})d {\varvec{\rho }}\right|^2 \le \int \int |f({\varvec{\rho }})|^2 d {\varvec{\rho }}\int \int |g({\varvec{\rho }})|^2 d {\varvec{\rho }}, \nonumber \end{aligned}$$

    which, with the substitutions

    $$\begin{aligned} f({\varvec{\rho }}) = \frac{1}{\sqrt{D_{\text{ im}}({\varvec{\rho }},\Omega _{\text{ ob}})}} \frac{\partial {D_{\text{ im}}({\varvec{\rho }},\Omega _{\text{ ob}})}}{\partial {w}} \,\quad \text{ and}\quad g({\varvec{\rho }}) = \sqrt{D_{\text{ im}}({\varvec{\rho }},\Omega _{\text{ ob}})},\nonumber \end{aligned}$$

    yields

    $$\begin{aligned} \left|\int \int _{\Omega _{\text{ im}}}\!\frac{\partial {D_{\text{ im}}({\varvec{\rho }},\Omega _{\text{ ob}})}}{\partial {w}}d{\varvec{\rho }}\right|^2 \le \int \int _{\Omega _{\text{ im}}} \frac{1}{D_{\text{ im}}({\varvec{\rho }},\Omega _{\text{ ob}})}\left|\frac{\partial {D_{\text{ im}}({\varvec{\rho }},\Omega _{\text{ ob}})}}{\partial {w}}\right|^2 d{\varvec{\rho }}\,\,\int \int _{\Omega _{\text{ im}}}\!D_{\text{ im}}({\varvec{\rho }},\Omega _{\text{ ob}})d{\varvec{\rho }}\nonumber \end{aligned}$$

    thus leading to Eq. (6.40).

  9. 9.

    For magnetic dipoles the far field distribution takes the form \((\mathbf{ s} _1 \times \mathbf{ p} ) \exp (ik\rho _1)/\rho _1\).

  10. 10.

    Two measurements are required such that the resulting set of linear equations are well conditioned (neglecting an ambiguity as to which quadrant the dipole lies in), when noise is absent.

  11. 11.

    Whilst the notation \(J_\gamma ^{\max }\) will be retained, it should be observed that when \(S_0\) is known a priori, the maximum Fisher information is \(2J_\gamma ^{\max }\).

  12. 12.

    The \(K\) integrals of Eqs. (6.44a–e) accommodate a defocus of the detector in the image plane. If however the dipole suffers an axial shift, \(z_{\text{ dp}}\) the exponential term \(\exp [ikz_2\cos \theta _2]\) must be modified to \(\exp [ik(z_2\cos \theta _2 - z_{\text{ dp}}\cos \theta _1)]\) (see [33]).

  13. 13.

    Malus’ law states that the intensity transmitted through a polariser with transmission axis at \(\vartheta \) when illuminated with light, linearly polarised at an angle \(\gamma \), is given by \(D\propto \cos ^2(\vartheta -\gamma )\).

  14. 14.

    The main computational restriction lies in the inversion of the associated \(14,112 \times 14,112\) FIM.

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  1. Imperial College London, London, UK

    Matthew R. Foreman

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Foreman, M.R. (2012). Information in Polarisation Imaging. In: Informational Limits in Optical Polarimetry and Vectorial Imaging. Springer Theses. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-28528-8_6

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  • DOI: https://doi.org/10.1007/978-3-642-28528-8_6

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