Waveplates

Alexei Gilchrist

3 Polarisation

Description of waveplates in terms of Jones’ matrices.

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1 Optical Retarders

If an electromagnetic wave travels a distance \(l\) through a plate made from an isotropic medium with refractive index \(n\), it will acquire a phase delay of \(e^{i \theta}\), where \(\theta = 2 \pi n L /\lambda\). Or, slightly simpler, \(\theta = n k_0 L\) for the free-space wavevector with magnitude \(k_0=2\pi/\lambda\). This is just the usual phase delay in travelling through an isotropic medium and it would only be observable for phases acquired along different paths. This optical device is called a simple retarder.

Now imagine though that the wave sees two different refractive indices depending on polarisation. Up to a global phase we can write the acquired phase only on the vertical polarisation so that

\[\begin{equation*}a|H\rangle+b|V\rangle \rightarrow a|H\rangle+b e^{i\theta}|V\rangle\end{equation*}\]

where we have used Dirac braket notation to denote the two-element complex vector representing the polarisation, and in particular \(|H\rangle=(1\;\; 0)^T\) and \(|V\rangle=(0\;\; 1)^T\). In terms of a matrix, this transformation is represented by

\[\begin{equation*}W_{\theta} = \begin{pmatrix} 1&0\\ 0 &e^{i\theta} \end{pmatrix}\end{equation*}\]

Two particular cases are important, \(\theta = \pi\) and \(\theta = \pi/2\).

2 Half-wave plate

For \(\theta = \pi\) the phase delay is equivalent to having introduced an optical path length of half a wavelength and it forms a half-wave plate.

\[\begin{equation*}W_{\lambda/2} \equiv W_\pi = \begin{pmatrix} 1&0\\ 0 &-1 \end{pmatrix}\end{equation*}\]

We’ll use the \(W_{\lambda/2}\) notation otherwise it will become really confusing with the name “half-wave plate”.

As an optical device, the half-wave plate will covert diagonally polarised to and from anti-diagonally polarised light, and convert right-circularly polarised to and from left-circularly polarised light.

Example

\[\begin{equation*}W_{\lambda/2}|D\rangle = \begin{pmatrix} 1&0\\ 0 &-1 \end{pmatrix}\frac{1}{\sqrt{2}}\begin{pmatrix} 1\\1 \end{pmatrix} = \frac{1}{\sqrt{2}}\begin{pmatrix} 1\\-1 \end{pmatrix} = |A\rangle\end{equation*}\]

The half-wave plate will have no effect on the polarisation of horizontally or vertically polarised light, which will be obvious after a moments reflection.

3 Quarter-wave plate

For \(\theta = \pi/2\) in \(W_\theta\), the device forms a quarter-wave plate.

\[\begin{equation*}W_{\lambda/4} \equiv W_{\pi/2} = \begin{pmatrix} 1&0\\ 0 & i \end{pmatrix}\end{equation*}\]

This waveplate will now convert between diagonal and left-circularly polarised light and between anti-diagonal and right-circularly polarise light.

Example

\[\begin{equation*}W_{\lambda/4}|D\rangle = \begin{pmatrix} 1&0\\ 0 &i \end{pmatrix}\frac{1}{\sqrt{2}}\begin{pmatrix} 1\\1 \end{pmatrix} = \frac{1}{\sqrt{2}}\begin{pmatrix} 1\\i \end{pmatrix} = |L\rangle\end{equation*}\]

Again, both \(|H\rangle\) and \(|V\rangle\) are left in the same polarisation by the device.

4 Rotated Waveplates

Imagine we rotate the waveplates by and angle \(\phi\). This is equivalent to first rotating the polarisation of the light by \(-\phi\), passing through the waveplate, and the rotating it back by \(\phi\):

\[\begin{align*}W_{\lambda/2}(\phi) \equiv & R(\phi)W_{\lambda/2} R(-\phi) = \begin{pmatrix} \cos(2\phi)& \sin(2\phi)\\ \sin(2\phi) &-\cos(2\phi) \end{pmatrix}\\ W_{\lambda/4}(\phi) \equiv & R(\phi)W_{\lambda/4} R(-\phi) = \begin{pmatrix} \cos^2\phi+i\sin^2\phi & (1-i)\cos\phi\sin\phi\\ (1-i)\cos\phi\sin\phi & i\cos^2\phi+\sin^2\phi \end{pmatrix}\end{align*}\]

Now we have considerably more freedom in how we manipulate the polarisation.

Example

If we rotate a half-wave plate by \(\pi/4\) we can use it to convert between horizontally and vertically polarised light

\[\begin{equation*}W_{\lambda/2}(\pi/4) |H\rangle = \begin{pmatrix} 0&1\\ 1 &0 \end{pmatrix} \begin{pmatrix} 1\\0 \end{pmatrix} = \begin{pmatrix} 0\\1 \end{pmatrix} = |V\rangle\end{equation*}\]

In fact, it’s possible to construct any transformation of purely polarised light by a sequence:

\[\begin{equation*}W_{\lambda/4}(\phi_3) W_{\lambda/2}(\phi_2) W_{\lambda/4}(\phi_1)\end{equation*}\]