Faraday rotator

A Faraday rotator is the packaged device that puts the Faraday effect to work: a rod or plate of magneto-optic material inside a permanent magnet, with the field along the beam, cut to a length that turns the plane of linear polarization by a set angle. Almost all commercial rotators are built for 45°, because two passes then give 90°, which is what an optical isolator and a Faraday mirror need. Free-space rotators for the near infrared typically use terbium gallium garnet (TGG) in a magnet of order 1 T; fiber-optic and telecom rotators at 1310 and 1550 nm use thin films of bismuth-substituted iron garnet a fraction of a millimetre thick. The physics of the effect and the Verdet constants of common materials are covered in the Faraday effect entry; this entry covers the device.

Construction and sizing

The rotation angle is $\theta = VBL$, with $V$ the Verdet constant, $B$ the axial field and $L$ the crystal length. For TGG at 1064 nm, $V \approx 40$ rad/(T·m), so a 45° rotator ($\theta = 0.785$ rad) in a uniform 1 T field needs

$$L = \frac{0.785}{40 \times 1.0} \approx 19.6\ \text{mm}.$$

Fused silica, with $V \approx 4$ rad/(T·m) at 633 nm, would need about 20 cm in the same field, which is why rotators use terbium-containing crystals and glasses instead. The magnet is usually a ring or stacked assembly of NdFeB segments; the field inside is not uniform, so the rotation depends on the integral of $B$ along the crystal and the crystal's axial position in the housing. Many free-space rotators are tuned in manufacture, or by the user, by sliding the crystal along the magnet axis until the rotation is exactly 45° at the working wavelength.

Iron garnet films behave differently. They are ferrimagnetic, so once saturated by a modest bias magnet their rotation no longer depends on the field strength, and latching versions hold their magnetization with no external magnet at all. That makes them compact enough for the isolators built into laser diode packages, at the cost of higher absorption at shorter wavelengths.

Checking the rotation angle

The rotation is measured by placing the rotator between two polarizers, setting the first to a known axis, and rotating the second to find the transmission minimum; the angle between that minimum and the crossed position is the Faraday rotation. A power meter behind the analyzer and a fine rotation mount give accuracy of a few tenths of a degree. Turning the rotator end to end and seeing the observed sense of rotation reverse, which an optically active crystal would not do, is the quick test that the device is working as a Faraday rotator and has not been confused with an optically active crystal.

Angle errors and their consequences

In an isolator, with the polarizers fixed at 0° and 45°, a rotation error $\delta$ from 45° leaves the returning light $\delta$ away from orthogonal, so the leakage fraction is $\sin^2\delta$. Errors of 0.5°, 1° and 3° limit isolation to about 41, 35 and 26 dB. Two things commonly produce such errors.

Wavelength: in paramagnetic materials such as TGG the Verdet constant scales roughly as $1/\lambda^2$ away from absorption bands. A rotator set to 45° at 1064 nm, used at 1030 nm, rotates about $45° \times (1064/1030)^2 \approx 48.0°$, an error of 3°.

Temperature: the Verdet constant of a paramagnetic crystal falls approximately as $1/T$. A 20 K rise from 293 K lowers a 45° rotation to about 42.1° by this estimate. At high average power, absorption in the crystal also produces thermal lensing and stress birefringence that depolarizes the beam, so rotators for kilowatt-class lasers use low-absorption material, short crystals in stronger magnets, or compensation schemes in which the depolarization from two rotators partially cancels.

Applications

• Optical isolators and circulators, where the non-reciprocal 45° rotation separates forward and backward beams.
• Faraday mirrors: a 45° rotator in front of a mirror returns light rotated by 90° relative to the input at every point along the path. In a fiber interferometer this cancels the birefringence of the fiber on the return trip, so the output polarization is stable without polarization-maintaining fiber.
• Unidirectional ring lasers, where a rotator paired with a reciprocal rotator (a half-wave plate or out-of-plane beam path) gives slightly different loss for the two directions.
• Double-pass amplifiers, where the 90° round-trip rotation lets a polarizing beam splitter separate the amplified return beam from the input.

Common questions

What is the difference between a Faraday rotator and a half-wave plate?

A half-wave plate rotates linear polarization by an amount set by its orientation, and a beam retracing its path is rotated back to where it started. A Faraday rotator's sense of rotation is fixed by the magnetic field, so a return pass adds to the forward rotation. An isolator depends on this non-reciprocity.

Why 45 degrees?

A double pass through a 45° rotator gives 90°, which turns a returning beam into the orthogonal polarization, where a polarizer can block or divert it.

References: E. Hecht, Optics, 5th ed. (Pearson, 2017); B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019); M. Martinelli, Opt. Commun. 72, 341 (1989).