Jones Calculus - Phase Retarders

Phase Retarders

Phase retarders introduce a phase shift between the vertical and horizontal component of the field and thus change the polarization of the beam. Phase retarders are usually made out of birefringent uniaxial crystals such as calcite, MgF2 or quartz. Uniaxial crystals have one crystal axis that is different from the other two crystal axes (i.e., ni ≠ nj = nk). This unique axis is called the extraordinary axis and is also referred to as the optic axis. An optic axis can be the fast or the slow axis for the crystal depending on the crystal at hand. Light travels with a higher phase velocity through an axis that has the smallest refractive index and this axis is called the fast axis. Similarly, an axis which has the highest refractive index is called a slow axis since the phase velocity of light is the lowest along this axis. Negative uniaxial crystals (e.g., calcite CaCO3, sapphire Al2O3) have ne < no so for these crystals, the extraordinary axis (optic axis) is the fast axis whereas for positive uniaxial crystals (e.g., quartz SiO2, magnesium fluoride MgF2, rutile TiO2), ne > n o and thus the extraordinary axis (optic axis) is the slow axis.

Any phase retarder with fast axis vertical or horizontal has zero off-diagonal terms and thus can be conveniently expressed as

\begin{pmatrix} e^{i\phi_x} & 0 \\ 0 & e^{i\phi_y} \end{pmatrix}

where, and are the phases of the electric fields in and directions respectively. In the phase convention, the relative phase between the two waves when represented as suggests that a positive (i.e., > ) means that doesn't attain the same value as until a later time i.e., leads . Similarly, if i.e., >, leads . For e.g., if the fast axis of a quarter wave plate is horizontal, this suggests that the phase velocity along the horizontal direction is faster than that in the vertical direction i.e., leads . Thus, which for a quarter wave plate suggests that .

In the opposite convention, the relative phase when defined as suggests that a positive means that doesn't attain the same value as until a later time i.e., leads .

Phase retarders Corresponding Jones matrix
Quarter-wave plate with fast axis vertical

e^{i \pi/4} \begin{pmatrix} 1 & 0 \\ 0 & -i \end{pmatrix}

Quarter-wave plate with fast axis horizontal

e^{i \pi/4} \begin{pmatrix} 1 & 0 \\ 0 & i \end{pmatrix}

Half-wave plate with fast axis at angle w.r.t the horizontal axis

\begin{pmatrix} \cos2\theta & \sin2\theta \\ \sin2\theta & -\cos2\theta \end{pmatrix}
Any birefringent material (phase retarder)

\begin{pmatrix} e^{i\phi_x} \cos^2\theta+e^{i\phi_y} \sin^2\theta & (e^{i\phi_x}-e^{i\phi_y}) \cos\theta \sin\theta \\ (e^{i\phi_x}-e^{i\phi_y}) \cos\theta \sin\theta & e^{i\phi_x} \sin^2\theta+e^{i\phi_y} \cos^2\theta \end{pmatrix}

The special expressions for the phase retarders can be obtained by using the general expression for a birefringent material. In the above expression:

  • Phase retardation induced between and by a birefringent material is given by
  • is the orientation of the fast axis with respect to the x-axis.
  • is the circularity (For linear retarders, = 0 and for circular retarders, = ± /2. For elliptical retarders, it takes on values between - /2 and /2).

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