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the amplitudes of the y and x components of the field. We therefore have linear polarisation as expected. Note that this implies that it is the phase shift that gives rise to the elliptical shape that E traces out.

(ii) Γ = ±π. In this case cos Γ = −1, sin Γ = 0. Thus Eq. (7.55) gives us

E_y = −E_0y E0x

E_x. (7.68)

Again, we have linear polarisation with the negative of the gradient for the Γ = 0 case.

(iii) Γ = ±π/2. Now cos Γ = 0 and sin Γ = ±1. Equation. (7.55) therefore reduces to Eq. (7.45), the equation of an ellipse in standard form.

 E_y which is the equation of a circle. Thus the polarisation is circular.

7.6 Wave plates

7.6.1 Birefringence

A retardation or wave plate is an optical element that produces some retar-dation between the orthogonal components of the wave. The physical origin of this retardation is due to the phenomenon of birefringence, in which the components of the wave see a different refractive index (and hence have a different phase velocity) depending on the orientation of the polarisation within some anisotropic material. The subject of anisotropy is dealt with in detail in Part V on Crystal Optics. Meanwhile, we shall describe a simplified picture for the sake of insight.

Just as a linear polariser has a particular transmission axis, so we can define two transmission axes perpendicular to the propagation direction for a wave plate. These are known as the fast and slow axes, corresponding to the wave speeds of the transmitted orthogonal components. Thus, the phase of the wave along the fast axis is ahead of that along the slow axis.

The speed of the waves is, of course, determined by the refractive in-dex that it sees. This is determined by a construction known as the inin-dex

ellipsoid. This concept will be fully developed in Chapter 12. For now, we shall just consider an ellipsoid defined by the equation

x² n²_o +y²

n²_o + z²

n²_e = 1. (7.71)

where n_oand n_eare called the ordinary and extraordinary refractive indices respectively. The fact that there are only two special refractive indexes reflects the fact that the crystal type Eq. (7.71) describes has only one optical axis. Hence, such a material is described as being uniaxial.

Consider that case of two orthogonally polarised plane waves, E_e and Eo propagating with the same wavevector k in this crystal, as shown in Fig. 7.7. The plane perpendicular to k intersects the index ellipsoid in an ellipse. One of the axes of this ellipse is always equal to the ordinary refrac-tive index no, the other is dependent on the extraordinary refractive index n_e and the angle of k to the extraordinary axis. Thus, one of the plane waves will travel with a wave speed v_o = c/no, the other with wave speed vo(θ) = c/n (θ). These are known as the ordinary and extraordinary waves respectively.

Figure 7.7: The index ellipsoid for a uniaxial crystal projected in 2D show-ing two orthogonally polarised plane waves, E_e and E_o propagating with the same wavevector k. The plane perpendicular to k intersects the index ellipsoid in an ellipse. One of the axes of this ellipse is always equal to the ordinary refractive index n_o, the other is dependent on the extraordinary refractive index n_eand the angle of k to the extraordinary axis.

7.6. WAVE PLATES 149 Uniaxial crystals may be further defined according to the relative sizes of ne and no. A negative uniaxial crystal has ne < no. Examples include calcite (CaCO₃) and ruby (Al₂O₃). For n_e> n_o, we have a positive uniaxial crystal, for example, quartz (SiO₂).

In a retardation plate, in a negative uniaxial crystal, the extraordinary axis is aligned with the fast axis of the plate, since c/n_e > c/n_o. For a positive uniaxial crystal, we have the opposite case and the extraordinary axis is therefore aligned with the slow axis.

The birefringence is defined as

∆n(θ) = n(θ) − n0. (7.72)

For a wave with extraordinary and ordinary components, E_eand E_o respec-tively, propagating in a direction r, we may write

E_e= E₀exp where we have used k = ω/v = nω/c. The second of these equations may be re-written

Hence, after a distance r, the ordinary wave acquires a retardation Γ (r) = ω∆n(θ)

c r. (7.76)

7.6.2 Half-wave plate

A half-wave plate introduces a retardation of Γ = ±π. Suppose, then, that the initial polarisation of a propagating EM wave is given according to Eq. (7.2) by

After transmission through a half-wave plate the new polarisation will be E1 = |E0|

 cos θ

− sin θ



. (7.78)

We can express this in terms of the action of a matrix M_π acting on E₀. Clearly, this matrix is given by

M_π =

 1 0 0 −1



. (7.79)

Thus, this has the effect of reversing the direction of y-component sin θ (where θ is the initial angle of the polarisation to the x-axis). In other words, the half-wave plate has the effect of rotating a state of linear polarisation by −2θ through the x-axis. Note that this is equivalent to a rotation of 2θ⁰ through the y-axis, where −θ⁰is the angle of the electric field vector to the y-axis. Thus, whilst the fast axis may be aligned in either the x or y directions, we may say unambiguously that

• a half-wave plate has the effect of rotating a linear state of polarisation by an angle −2θ through the fast (or slow) axis, where θ is the initial angle of the electric field vector to the fast (or slow) axis.

7.6.3 Quarter-wave plate

For a quarter-wave plate, we have Γ = ±π/2. Applying the same reasoning as before, we find that the Jones matrix for a quarter-wave plate is given by

M_±π/2=

 1 0 0 ±i



. (7.80)

Applying this to the Jones vector of Eq. (7.2) we have E₁= |E₀| Comparing this result to Eq. (7.43), we see that this yields elliptically po-larised light. In this case, the principle axes of the ellipse are aligned with the E_xand E_y axes. In the specific case where

cos θ = sin θ, (7.82)

This is the Jones vector for circularly polarised light. Specifically, as given in the last section, for Γ = π/2 we have right circular polarisation and for Γ = −π/2we have left circular polarisation. Thus

7.6. WAVE PLATES 151

• a quarter-wave plate introduces a phase shift of π/2 between the components of the optical field, producing elliptically polarised light.

In the specific case where E_x and E_y are initially equal, the quarter-wave plate produces circularly polarised light.

7.6.4 General retardation plate

A general retardation plate introduces an arbitrary phase shift Γ = ±π/2. It is straightforward to see that the Jones matrix for such a plate will be given by

M_Γ=

 1 0 0 e^iΓ



. (7.84)

7.6.5 3D glasses

Figure 7.8: 3D glasses based on circularly polarised light.

A recent use of wave-plates has been in the design of 3D glasses used in conjunction with stereoscopic filming and projection of video to give the impression of visual depth. Stereoscopic visualisation is achieved by tak-ing two images of the same scene from the position of both eyes. These images are then brought together again in such a way that the brain inter-prets them as different perspectives on the same view and producing the impression of a three dimensional scene.

One way of achieving this effect is to project the two images in oppo-sitely handed circularly polarised light. The lenses of special 3D glasses

then pass one of these projections and block the other. This is achieved by first passing the light through a quarter wave plate and then a linear polariser, the linear polarisers being rotated at 90^◦ to each other in each lens.

The two projections are right and left circularly polarised. Passing each through a quarter wave plate gives

M_π/2E₊= E₀

Note that Eq. (7.85) yields linearly polarised light aligned along the y = −x axis, whilst Eq. (7.86) gives linearly polarised light along the y = x axis.

Hence, if this is followed by a linear polariser with its transmission axis aligned along y = x, the combination will pass left circularly polarised light and block right circularly polarised light.

For the other lens, the opposite effect may be achieved either with a M_−π/2 quarter wave plate and the same orientation linear polariser or the same wave plate and an orthogonally orientated linear polariser. Note, that due to the rotational symmetry of the circularly polarised light, tilting the glasses does not effect the analysis of the light (an advantage over just using linear polarisers).

Figure 7.9: Analysis of circularly polarised light into linearly polarised light.