E00 = R−θE0. (7.23)
In this new coordinate system, the transmission axis of the polariser is aligned to the Ex0-axis. Thus, in the rotated frame, the polariser is now an linear x-polariser. To obtain the transmission through it, we therefore need to apply Px
E01= PxE00= PxR−θE0. (7.24) Finally, we need to transform back to the original coordinate system.
We do this with Rθ
E1 = RθE01 = RθPxR−θE0. (7.25) Since matrix multiplication is associative, we can encapsulate the entire process in a resultant matrix obtained by multiplying all three matrix opera-tors together
M = RθPxR−θ. (7.26)
Substituting from Eqs. (7.103) and (7.9), we have
M =
We see here an example of the way in which operations performed one after another may be encapsulated by multiplying the matrices for the indi-vidual processes together. However, it is important that this multiplication is carried out in order (later operations acting on the left), since matrix mul-tiplication is not commutative.
7.5 Elliptically polarised light
7.5.1 The retardation
The most general form of polarisation is elliptical polarisation, in which the electric field spirals around the propagation axis tracing out an ellipse. This may be understood by resolving the electric field into orthogonal compo-nents. So long as these components remain in phase, the polarisation will
Figure 7.5: Illustration of elliptically polarised light in terms of orthogonal components with a phase shift Γ between them, giving the resultant wave-form (helical line).
be linear. If, however, a phase shift is introduced on to one of the compo-nents, the polarisation will become elliptical, as in Fig. 7.5. This phase shift is known as the retardation Γ.
Writing the polarisation out in terms of Jones vectors
E =
|E0x|
eiφx 0
+ |E0y|
0 eiφy
ei(ωt−kz), (7.28) where the retardation is given by
φy− φx= Γ. (7.29)
Multiplying Eq. (7.28) through by e−iφx then gives Ee−iφx = |E0|
cos θ eiΓsin θ
ei(ωt−kz), (7.30)
where, again, θ is the angle between the electric field and the x-axis. Ab-sorbing the common phase shift into the electric field vector, the Jones vector for elliptical polarisation is
E0= |E0|
cos θ eiΓsin θ
. (7.31)
We may pause to identify some special cases of Eq. 7.31 that reduce back down to linear polarisation.
7.5. ELLIPTICALLY POLARISED LIGHT 141 (i) Γ = 0. This just yields a common phase shift on both components
and the wave remains linearly polarised. Hence
E0 = |E0|
Thus E remains linearly polarised but the direction of polarisation is rotated by an angle 2θ through the x-axis. We shall see later that this change in polarisation may be achieved by an optical element known as a half-wave plate.
7.5.2 Circular polarisation
Let us now consider the cases where Γ = ±π/2 and impose the condition θ = π/4, so that cos θ = sin θ = 2−1/2.
Asserting the propagator, the electric field is given by
E = |E0|
Considering the real part of E
Re [E] = |E0|
Thus, in space the electric field vector, E, rotates around the prop-agation axis (z-axis) in the same direction as a right-handed screw.
Hence, this is known as right circular polarisation. Note that looking along the approaching wave towards the origin, an observer would see E at a fixed point in space rotating clockwise in time.
(iv) Γ = −π/2. In this case we have
This reverses the sign of y-component from the previous case. Hence, the electric field rotates around the propagation axis in the opposite direction to right circular polarisation. Moreover, at a given point in space, E rotates anti-clockwise in time. This is therefore known as left circular polarisation.
Summarising the results for circularly polarised light,
• right circular polarised light
E+= √E0
• left circular polarised light
E−= E0
The results of the previous section held for the case where imposed the condition that amplitudes of the x and y components of the electric field were equal. If we now relax this condition, the resultant polarisation is no longer circular. Instead, for phase shifts of Γ = ±π/2, we will have
E = |E0|
cos θ
±i sin θ
eiφ. (7.43)
where φ = ωt − kz. Considering just the real part, we then have
7.5. ELLIPTICALLY POLARISED LIGHT 143
which just traces out an ellipse in either a clockwise (Γ = π/2) or anti-clockwise (Γ = −π/2) direction. Putting
E0x= |E0| cos θ,
This is an ellipse in standard form, so its principle axes are aligned to the xand y coordinate axes. Hence the maximum and minimum values of Ex are ±E0xand similarly for Ey.
7.5.4 Case: Γ is arbitrary, E0x6= E0y
In the most general case, the phase shift Γ is arbitrary and there is no restriction on the relative amplitudes of E0x and E0y. The electric field vector may now be written as
E =
Firstly, let us consider the rotation of the field vector around the propa-gation direction. From Eq. (7.47), we see that the real part of E makes an angle
θ = tan−1 E0ycos (φ + Γ) E0xcos φ
(7.48) to the x-axis. Taking the derivative with respect to time, we have
∂θ
∂t = − ωE0xE0ysin Γ
E20xcos2φ + E0y2 cos2(φ + Γ). (7.49) In any of the cases E0x = 0, E0y = 0, Γ = 0 or Γ = ±π, we have linear polarisation and, as expected, we have ∂θ/∂t = 0. In other words, the orientation of the electric field vector E stays fixed.
In other cases, the direction of rotation is given by the sign of ∂θ/∂t, i.e. anti-clockwise for ∂θ/∂t > 0 and clockwise for ∂θ/∂t < 0. Inspect-ing Eq. (7.49), we see that this is determined soley by the sign of sin Γ.
Assuming that ω, E0xand E0yare all positive, we therefore have
• 0 < Γ < π, so sin Γ > 0 and ∂θ/∂t < 0.
Rotation is thereforeclockwise.
• −π < Γ < 0, so sin Γ < 0 and ∂θ/∂t > 0.
Rotation is thereforeanti-clockwise.
Note that the direction of rotation changes as the electric field vector E passes through linear states of polarisation for Γ = 0 and Γ = π.
Figure 7.6: The polarisation ellipse for a general state of polarisation.
Let us now define
Ex = E0xcos φ (7.50)
and
Ey = E0ycos (φ + Γ) . (7.51) This gives
7.5. ELLIPTICALLY POLARISED LIGHT 145
This is the general equation of an ellipse.
Equation (7.55) is illustrated in Fig. 7.6. Note that in the Ex0-Ey0 coor-dinate system (rotated from Ex-Ey by an angle α about the z-axis), the ellipse is in standard form. The coordinate systems are therefore related by the transformation
where the matrix encapsulates a rotation by α about the origin. Expressing the products of Exand Ey in terms of the primed coordinates, we have
Ex2 = Ex02cos2α + Ey02sin2α − 2Ex0Ey0 cos α sin α, (7.57)
Ey2 = Ex02sin2α + Ey02cos2α + 2Ex0Ey0 cos α sin α, (7.58) and
ExEy = Ex02− Ey02 cos α sin α + Ex0Ey0 cos2α − sin2α . (7.59)
Since the ellipse is in standard form in the Ex0-Ey0 coordinate system, the cross-terms Ex0Ey0 must disappear. From Eq. (7.55), we therefore require
sin 2α 1 E0y2 − 1
E0x2
!
− 2 cos 2α
E0xE0ycos Γ = 0, (7.60) where we have used the identities
2 cos α sin α = sin 2α (7.61)
and
cos2α − sin2α = cos 2α. (7.62) Hence, we have
tan 2α = 2E0xE0y
E0x2 − E0y2 cos Γ. (7.63) Now, Eq. (7.63) may be re-written
tan 2α = 2 (E0y/E0x)
1 − (E0y/E0x)2 cos Γ. (7.64) Defining tan β = E0y/E0x and using the double angle identity for tan, this may be expressed as
tan 2α = 2 tan β
1 − tan2βcos Γ = tan 2β cos Γ. (7.65) Note from Eq. (7.63) that if E0x= E0y, tan 2α → ∞, so α = π/4.
7.5.5 Limiting cases
We may now apply the special conditions considered in the previous sub-sections to our general formulation for the polarisation.
(i) Γ = 0. In this case cos Γ = 1, sin Γ = 0 and Eq. (7.55) reduces to
Ey E0y − Ex
E0x
2
= 0, (7.66)
yielding
Ey = E0y
E0xEx. (7.67)
7.6. WAVE PLATES 147