Modelación teórico-práctica de la propuesta

Section 1.2 gave an overview of how to describe planetary radiation, and the equa- tions and methods detailed there are sufficient for the applications considered in this work. However, a PhD thesis on polarimetry would not be complete without an exploration of the various formalisms that light and its polarisation can be described with. Equally as important is the mathematical representation of the various optical devices that are used in astronomical polarimeters, and the various components of a polarimeter are described in Section 3.1.

2.2.1 Stokes Formalism

The Stokes formalism was introduced by Stokes [1852], and it applies to intensity measurements of photon fluxes as performed by detectors in the optical regime. It can describe partial polarisation, which is useful for astronomical polarimetry because most sources have a degree of polarisation of at most a few percent. In- terference phenomena however cannot be described by the Stokes formalism. The Stokes parameters are illustrated in Figure 2.2.

- = Q

- = U

- = V

Figure 2.2: A basic diagram showing how the Stokes parameters are defined in terms of the light oscillating in various planes.

The Stokes vector is expressed as:

S= 0 B B B B @ F Q U V 1 C C C C A= 0 B B B B @ F₀0 +F₉₀0 F₀0 F₉₀0 F₄₅0 F0₄₅ F_RHC0 F_LHC0 1 C C C C A (2.1)

which spans the space of all the polarisation states of light (namely unpolarised, par- tially polarised, and fully polarised). The F component represents the incoherent sum of the signal (there are no interference e↵ects), Q and U are the di↵erences in linear polarisation states at two perpendicular planes and the V component represents the circular polarisation. F can also be expressed as F₄₅0 + F0₄₅ or

F_RHC0 +F_LHC0 . So, for example, light with a Stokes vector of S = (1,0,0,0) would be completely unpolarised, and a Stokes vector ofS= (1,1,0,0) describes light with a linear polarisation in the 0 direction. A Stokes vector of S = (1,0,0,1) would imply right-handed, circularly polarised light (clockwise as seen by the observer), and (1,0,0,-1) would imply left-handed, circularly polarised light (anti-clockwise as seen by the observer).

F =p(Q2_+U2_+V2_{) (2.2)} and for partially polarised light the flux is given by:

F p(Q2_+U2_+V2_{) (2.3)} The degree of polarisation, P, is the ratio of the flux of polarised light to the flux of the unpolarised light. This quantity can be represented in terms of the Stokes parameters by:

P =

p

Q2_+U2_+V2

F (2.4)

which in the case of only linear polarisation (V = 0), we add an L subscript to P

to indicate that only linear polarisation is present:

PL =

p

Q2_+U2

F . (2.5)

2.2.2 Mueller Calculus

The Mueller calculus employs matrix algebra to compute the signal of a beam of light propagating through one or more optical elements, usually polarisers and retarders (more details on the optics are given in Section 3.1). Conventional algebraic and trigonometric methods are cumbersome when the number of polarisers or retarders is large, fundamentally due to the complicated nature of light and its interaction with matter. The incident light is described by several parameters: amplitude, degree of polarisation, and form of polarisation (linear or circular). The Mueller calculus simplifies the calculations by condensing all of the necessary parameters describing the light into one single vector. Each optical element can also be represented by a single matrix, and the result of introducing any number of optical combinations can be found simply by multiplying the matrices of each component together. Then this matrix is multiplied by the vector specifying the incoming light, to produce the vector describing the light exiting the optical setup.

The incoming light is described by the Stokes vector (see Eqn. 2.1) and the elements present are described using the Mueller matrices, so the Stokes vector of the light

upon exiting the optical setup is given by:

Sout =M·Sin (2.6)

where M represents a Mueller matrix of an arbitrary single optical device. Sin

and Sout are the Stokes vectors describing the light upon entry to and exit from the optical device, respectively. For a system consisting of several di↵erent optical devices, the outgoing Stokes vector is given by:

Sout =Mn...·M2·M1·Sin (2.7)

where Mn...M1 are the Mueller matrices of the n elements present in the opti- cal system, with M1 representing the first element and Mn representing the last element.

The Mueller matrix is what is used to describe each optical element, e.g., a polariser, a retarder or a scatterer. These matrices are 4⇥4, containing 16 elements. Most of the elements are zero for certain ideal devices.

The Mueller matrix of an individual device indicates the composition of the device along with its azimuthal orientation. For example, the Mueller matrix of a linear polariser with a horizontal transmission axis is di↵erent from the matrix describing a similar polariser which has been turned so that its transmission axis is at 45 . Turning an optical device around such that a di↵erent face serves as the entrance face for the light may also mean a di↵erent Mueller matrix is required. Tilting a device so that light is incident at an oblique angle may also necessitate the use of a di↵erent matrix.

A Mueller matrix only describes the optical device with respect to one beam emerg- ing. For the case of a Wollaston prism, which has two emerging beams, a single Mueller matrix can only describe one of these beams. So if both beams are of inter- est, then two Mueller matrices must be used and two separate calculations must be carried out.

The standard rules of vector and matrix algebra apply for the Mueller calculus, but the following convention must also be observed: the vector which represents the incident beam must be written furthest to the right, and the matrices representing successive devices encountered must be arranged in order, so the last device to be encountered by the light would have its matrix written at the far left [Shurcli↵, 1962].

2.2.3 Jones Calculus

The Jones calculus, developed by Jones [1941], is another formalism which may be used to describe the interaction of light with an optical system. The Jones formal- ism of polarisation describes light in terms of an electric field vector, with an initial amplitude and phase. In the context of astronomical polarimetry, this formalism applies to the submillimetre/radio regime, where the detectors are antennae which directly measure the electric field vector. This treatment is derived directly from electromagnetic theory, and employs a vector to describe the incident light, with the optical device represented by a matrix, and the outgoing light ray is obtained by multiplying the vector of the incoming light with the matrix of the optical device. It is most suited for solving problems in which the phase is of importance. The Jones vector describes an incoming beam of light’s state of polarisation and ampli- tude components at a given position along the beam. In a right-handed Cartesian coordinate system, if the beam of light is travelling along thez-axis, then the Jones vector has the general form:

E= Ex(t) Ey(t) ! = Axe i(kz !t+ x) Ayei(kz !t+ y) ! (2.8)

whereExandEy are the scalar components of the instantaneous electric field vector

along the x and y axes, respectively. Ax is the maximum value of Ex, and Ay is

the maximum value of Ey. The wavenumber is represented by k (= 2⇡/ ), z is

the position along the z (propagation) axis at t = 0, ! is the angular frequency (!= 2⇡⌫) andt the time. x and y represent the initial phases of the waves.

The intensity of a beam of light in the Jones calculus is proportional to the sum of the squares of the magnitudes of the individual elements of the vector. The units of intensity or amplitude can be chosen such the constant of proportionality is unity, giving:

F =A2_x+A2_y (2.9)

Shurcli↵ [1962] gives a more detailed description of both the Jones and Mueller calculi, and provides several examples.

2.2.4 Comparison of Mueller Calculus and Jones Calculus

The Jones and Mueller calculi have much in common; each formalism describes the incoming and outgoing light with a vector, and employs matrix algebra in order to compute the final vector. Each type of calculus can be performed simply by looking up the matrices describing the various types of optical device and carrying out the calculations.

However, the Jones calculus di↵ers from the Mueller calculus in several ways:

• The Mueller calculus is based on experimental studies, whereas the Jones

formalism is derived from the classical theory of electromagnetism.

• The optical devices are described by a 2⇥2 matrix instead of a 4⇥4 in the

Jones calculus. The elements of these matrices can be complex, whereas all matrix elements are real for the Mueller calculus.

• The Jones calculus is useful for describing problems in which phase information

must be preserved, whereas the Mueller calculus pays no attention to phase.

• The Jones calculus is suitable for handling problems which involve combining

two coherent beams. The Mueller calculus is not designed to handle such scenarios.

• The Mueller calculus utilises the Stokes vector in its calculations, with the

first component of the Stokes vector being the intensity. The intensity is not directly given in the Jones calculus, the sum of the squares of each element must be computed in order to find it.