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We have seen [»] previously that a generalized electromagnetic wave, propagating along the Z axis, could be written as
which describes an elliptical path as shown in Figure 1.
I also gave in that same post some high-level description of the treatment of elliptical waves. In this post, I’m however showing a more rigorous mathematical framework to describe polarization :)
To achieve this, let’s first rewrite the previous equations as
with
where φx and φy are the absolute phase lags along the X and Y axis.
We can drop the exp(j(kz-ωt)) term as it describes the spatial and temporal variation of the wave such that the pair (A,B) represents the meaningful description of the polarization state of the light. The study of polarization is therefore the study of this pair which is precisely what R. Clark Jones did in “A New Calculus for the Treatment of Optical Systems” (Journal of the Optical Society of America, Vol. 31, No. 7 & 8). In reference to its original author, this pair of complex numbers is called a Jones vector.
In the study of polarization, it is also common to normalize the vector such that its magnitude is 1 – therefore leaving the overall magnitude of the electrical field aside which can be relegated in the expression of the electrical field. Using this notation, (1,0) represents a horizontal (X) polarization state and (0,1) represents a vertical (Y) polarization state. The term “horizontal” and “vertical” being merely naming conventions here.
As we have also seen in our [»] birefringence post, the relative phase lag between the X and Y components of the electrical field plays an important role in optics. The condition for light to be in full linear polarization state is that there is no apparent phase lag between the two components of the electrical field, whatever its frame of reference be. Any other wave will be described as in an elliptical polarization state. A convenient property of the Jones vector is that there is no apparent phase lag between the two components if the imaginary part of their product is zero. Indeed,
which is true when
This later expression can be translated in pure vectorial form using
where v is the Jones vector (A,B), v† its conjugated transpose and C the transform matrix
Indeed,
And if the Jones vector v has been transformed by a matrix M such that v’=Mv,
The unique purpose of the Jones notation, as I understand it, is to ease up computations using matrices. And, the first thing that should come to your mind in that regard is to apply coordinates changes in the form of a rotation of angle θ:
such that
and we can check that a rotation of 90° transforms a (1,0) state (horizontal) into (0,1) state (vertical).
We also have matrices for linear polarizers in either the horizontal (Mh) or vertical state (Mv):
which selects the projection along either the horizontal or vertical direction.
Indeed,
The effect can be generalized to any angle θ by transforming the previous matrices in [»] local space,
From this later equation we can deduce Malus’s law, stating that intensity of light between two crossed polarizers decreases with the square cosine of their angle. Indeed,
and the total intensity is equal to the mean squared values of the components of the electrical field
such that
Using the Jones notation also explains very odd behavior of polarizers, such as the fact that placing an extra polarizer between two crossed polarizers now let some of light pass through! Indeed, by applying a stack of polarizer rotated by an angle θ followed by a vertical polarizer on a pure horizontal polarization state, we get
and therefore
This effect is maximum at 45° where it equals 25% of the total light intensity. This effect is illustrated in Figure 2 and is a very funny trick to play at students :) The “normal” expectation being that the middle polarizer should have no effect on the transmitted intensity (but Jones calculus shows us it has!).
Another useful trick for development is that any Jones vector can be rotated by an angle ξ such that it takes the canonic form
with a and b being real numbers.
The previous equation is nothing but an ellipse of axis length a and b and axis orientation ξ. If we choose a Jones vector with a linear polarizer aligned with the local axis ξ, we get a canonical form of (a,jb) which gives respectively, after transmission through a horizontal and vertical polarizer,
The expected transmission intensities being then proportional to a² and b² respectively. The maximum achievable extinction ratio of that wave is the ratio between the two intensities transmitted at both linear polarizers
which is also the square of the ratio between the two axis of the polarization ellipse.
In conclusion, any ellipticity in a wave reduces the maximum achievable extinction ratio when passing the wave through a linear polarizer. In full linear polarization,
In full circular polarization,
Any elliptical polarization state having then a maximum achievable extinction ratio in the range ]0,∞[.
The true power of the Jones vector however comes from the fact that it is a pair of complex numbers where the phase codes for the lag of the electromagnetic wave. Following this observation, any phase lag (φx,φx) along the X and Y axis becomes
such that
But recall that the important property in polarization optics is the relative phase lag between the X and Y components of the electrical field. It is therefore more interesting to represent the effect of a phase lag η along an angle θ relative to the horizontal axis using
The birefringence effects we have seen in our [»] previous post could therefore be written in matrix form using one of the two equations here-above.
Polarization optics frequently deal with such phase lag using devices known as wave plates which are made of a birefringent material that introduces a carefully selected phase lag which is obtained by controlling the thickness of the birefringent material (note that as a consequence, waveplates are therefore designed for a specific wavelength). Since waveplates are ubiquitous in polarization optics, let’s spend some time discussing them in details.
There are three common types of waveplates:
The quarter-wave plate which introduces a phase lag of 90° (η=ϖ/2+2kϖ with k being an integer):
Quarter wave plates are used to introduce phase shifts in electromagnetic waves. They can convert linear polarization states to circular polarization states (and vice versa). They have many usages, among which is the ability to convert elliptical polarization states into (full) linear polarization states.
The math is a bit complex to solve but make uses of the expression v†(M†CM)v=0 that we developed earlier. Expressed in the canonical base of the Jones vector, the solution should be
and gives the angle of the quarter wave plate to use, relative to the canonical axis of the Jones vector.
In practice, the quarter wave plate is simply rotated until all ellipticity is removed from the electromagnetic wave. I will dedicate an entire experimental post to the usage of quarter-wave plates.
Another interesting wave plate is the half-wave plate which introduces a phase lag of 180° (η=ϖ+2kϖ with k being an integer):
which gives along the principal axis
and at 45°
The half-wave plate therefore flips the sign of the electrical field when aligned with the initial polarization (identity matrices) or perpendicular to it, but it swaps a horizontal polarization to a vertical one (and vice versa) when placed at 45° from the initial polarization state. More generally, the half-wave plate rotates the light by twice the angle it makes with the initial polarization state.
A true half-wave plate therefore does not affect the ellipticity of the electromagnetic wave and cannot be used to compensate for any phase lag as the quarter wave plate did. I’m saying here “true half-wave plate” because it is only valid if the phase shift is 180° - which happens only at the design wavelength of the wave plate. Any other wavelength will have a slight variation in the phase shift and will therefore introduce a small ellipticity in the signal.
Half-wave plates are convenient when you need to accommodate optical devices that require different polarization axis, but I don’t find them as interesting as quarter-wave plates so I won’t dig further into them. Our [»] previous post on stress-birefringence had a nice experimental illustration of chromatic half-wave plates.
In practice, there also exists a full-wave plate which introduces a phase lag of 360° (η=2kϖ with k being an integer) but it is of little interest in Jones calculus since it corresponds to the identity matrix:
Such wave plates may look irrelevant but recall that phase shifts depend on wavelength and a full wave plate only acts as a true identity matrix at the wavelength at which it has been designed but introduces small phase lag at other wavelengths with increased magnitude as we depart from the design wavelength. This effect is used in polarization in some polarization microscopes to introduce contrasts in the samples. The effect is very specific and I will not cover it in more details for now.
Jones representation is ideal for single electromagnetic waves, but it lacks a more general representation of broad monochromatic light, in particular it misses completely the case of partially polarized light that we have met in our [»] first post. [∞] Stokes representation fills this gap but at the expense of 4 components vectors. Matrix transformation of Stokes parameters is still possible, but this time via 4×4 matrices. [∞] Mueller matrices are not necessarily more complex than Jones matrices, but the fact that they are 4×4 makes computations less appealing. When dealing with single electromagnetic waves, Jones representation is therefore sufficient. Interestingly, Stokes parameters were introduced before Jones representation and Mueller then extended Jones calculus to Stokes parameters (in respectively 1852, 1941 and 1948).
The 4 components of the Stokes vector are
where we define the diagonal basis (a,b) and the circular basis (r,l) as
and
The stokes parameters S0 therefore represents the overall light intensity, S1 the tendency of light to be polarized along the horizontal or vertical axis, S2 the tendency of light to be polarized along the diagonal axes, and S3 the tendency of light to be polarized in either right-handed or left-handed circular state. It is common to normalize the Stokes vectors by the overall intensity such that polarization states of (1,0,0,0) is therefore completely unpolarized light, (1,+1,0,0) is horizontally polarized light, (1,-1,0,0) is vertically polarized light, (1,0,+1,0) and (1,0,-1,0) are diagonally polarized light and (1,0,0,+1) and (1,0,0,-1) are circularly right and left polarized light respectively.
[∞] Wikipedia also gives alternative definitions of the Stokes vector for various basis
as well as on the [∞] Poincaré sphere
where we defined
with I the total intensity, p the degree of polarization, and ψ and χ represents respectively the amount of diagonally polarized light in the linear polarization state and the amount of circularly vs. linear polarization state of the light.
The Poincaré sphere is often used to represent polarization states using normalized coordinates as it reduces the number of parameters to 3 and can therefore be represented more easily on a graph as shown in Figure 3.
We can spot a few points of interest in Figure 3. By definition, states (0,0,1) and (0,0,1) correspond to pure circular polarization states. States with no S3 component correspond to the linear polarization states which form a disk in Figure 3 – the direction of polarization depending on the amount of S1 and S2 components. Any other point on the sphere being then an elliptical polarization state. Finally, the distance to the origin represents the polarization degree of the polarization state such that the point (0,0,0) corresponds to fully depolarized light.
I will not dig further into the Poincaré sphere here.
This post should have now completely demystified polarization and how to treat polarization using mathematical representations :) There is certainly more that could be written on the topic, but I thought that this was a straightforward, coherent, way of introducing the topic.
In my next post, I will discuss experimental consequences of elliptical polarization and how to handle polarization in practice. Be sure to stay tuned for updates!
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You may also like:
[»] Polarization Part #3: Birefringence
[»] Polarization Part #2: Reflection
[»] Polarization Part #1: Scattering