Optics 081 · Polarization, anisotropy, and modulation

Faraday Isolator

Why can a Faraday isolator pass light going forward but block the same light coming back? Because the sense of Faraday rotation is set by the magnetic field, not by the direction of travel. The angle grows with field strength and crystal length; set it to 45 degrees and forward light lines up with the exit polarizer. Returning light is turned another 45 degrees the same way instead of back, so it meets the entrance polarizer at 90 degrees and is blocked.

Run light both ways through a two-port isolator in 3D. A TGG rotator with signed dispersion, Jones propagation in a fixed basis and absorptive polarizers set the result; compare with a reciprocal element and freeze paired scans.

Signed Faraday rotation

\theta_F

$0.50$

Active detector / prepared input power

P_D/P_0

$50\%$

Reverse attenuation for aligned port input

D_{\leftarrow}

$0.00\pi$

Model regime $\text{valid model regime}$

Model checked against analytic results

Faraday Isolator

Signed axial field along laboratory X

B_X

-0.94\,\mathrm T

Physical rod length

L

20\,\mathrm{mm}

Monochromatic vacuum wavelength

\lambda_0

1064\,\mathrm{nm}

Right polarizer axis from Y toward Z

\beta

45^\circ

Injection: zero forward, one reverse

r

0

Medium: zero Faraday, one reciprocal reference

q

0

Assumed polarizer extinction

E_P

40\,\mathrm{dB}

Send light forward, then inject it from the other end. The fixed magnet rotates the polarization in the same direction in the lab frame both ways, so the returning light arrives at the wrong polarizer axis. Swap in a reciprocal element for comparison, flip the magnetic field and record both transmission curves.

The magnetic field is uniform and imposed, the anti-reflection faces are ideal, and the dispersion follows a room-temperature fit. Light enters each port already aligned to it, and the polarizers have an assumed finite extinction. Thermal effects and commercial performance are not modeled.

Physics tutorial

Audit the physics of Faraday Isolator

BackgroundAn ordinary optical rotator undoes its rotation when light comes back through it. A magnetized Faraday medium does not: $\theta_F=V(\lambda_0)B_XL$ . Use the same laboratory axes for both directions. At the design setting, the forward field lines up with the output polarizer, while light sent backward arrives crossed to the input polarizer.

Why it mattersWhy can a Faraday isolator pass light going forward but block the same light coming back?

Start with the essentials

Focus question: Why can a Faraday isolator pass light going forward but block the same light coming back?
One-sentence intuition: In fixed laboratory axes, the rule is $F_{\rightarrow}=F_{\leftarrow}=R(\theta_F)$ . That does not mean the rotation looks the same when you turn around and look along the other direction. A basis that follows the propagation direction also flips; flipping the basis and the material sign together would count the reversal twice.

Core mathematical model

The Verdet constant and the signed rotation

\begin{aligned}V(\lambda_0)&=\frac{4.45\times10^7}{(258.2)^2-\lambda_0^2}\,\frac{\mathrm{rad}}{\mathrm{T\,m}}\\\theta_F&=V(\lambda_0)B_XL\end{aligned}

The wavelength goes in as a number in nanometres, and the length is converted once from millimetres to metres. The published single-resonance fit for TGG is used only from 650 to 1100 nanometres at room temperature. The end faces have ideal antireflection coatings, and bulk attenuation is ignored.

A Jones-matrix chain for both directions

\begin{aligned}R(\theta)&=\begin{pmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{pmatrix}\\P(\alpha)&=R(\alpha)\begin{pmatrix}1&0\\0&\sqrt\epsilon\end{pmatrix}R(-\alpha)\\J_{\rightarrow}&=P(\beta)R(\theta_F)P(0)\\J_{\leftarrow}&=P(0)R(\theta_F)P(\beta)\end{aligned}

The components lie along fixed laboratory Y and Z, with a negative imaginary exponential time dependence. Each polarizer transmits fully along its pass axis and has an assumed ratio of blocked to passed intensity. The reciprocal comparison changes only one thing: the reverse rotation is replaced by its inverse. Moving the camera does not redefine any angle.

Power, absorption and isolation

\begin{aligned}T_{\rightarrow}&=\cos^2(\theta_F-\beta)+\epsilon\sin^2(\theta_F-\beta)\\T_{\leftarrow}&=\cos^2(\theta_F+\beta)+\epsilon\sin^2(\theta_F+\beta)\\A&=1-T\\D_{\leftarrow}&=-10\log_{10}T_{\leftarrow}\end{aligned}

These simplified formulas apply to the aligned input in each direction. They do not cover arbitrary incident polarization or the product of a round trip. At the design angle, the assumed extinction of the polarizer caps the reverse attenuation. All blocked power is absorbed, not reflected. The logarithmic quantity is the reverse attenuation relative to the reverse beam’s own incident power, not the ratio of forward to reverse power.

Common difficulties

Does any forty-five-degree rotator make an isolator?

Typical misconceptionTwo polarizers set at an angle, plus any rotator, will work as an optical diode.

Better mental modelThe reciprocal reference transmits the same forward power but undoes its rotation on the way back. Only the nonreciprocal model with a magnetic bias blocks the returning light. The blocked power is absorbed here; no drawn beam carries the missing energy away.

Run the experiment

01
Scene 1: Faraday: forward bright
Compare the forward and reverse Faraday presets. Look at the gold field direction just before the last polarizer and at the detector power.
What to observe: The beam runs along laboratory X. Field angles are measured from upward Y toward Z. The signed Verdet constant is negative in this wavelength band, so the default negative axial field gives a positive rotation. All optics and beam endpoints are in millimetres.
02
Scene 2: Faraday: reverse rejected
Choose the reciprocal preset, then flip the magnetic-field control in Faraday mode. Compare the two power curves: ordinary optical activity does not protect the input.
What to observe: Each run launches unit power aligned with the first polarizer it meets. Reverse mode is a separate injection, not the forward beam reflected back automatically. The path ends at the detector faces, and for these aligned inputs only the last polarizer absorbs power.
03
Scene 3: Reciprocal: return passes
Change the rod length and the wavelength. Tune to the design point, acquire a paired field scan, then change the live settings and check that the stored record stays the same.
What to observe: The coil is a cutaway symbol for an imposed uniform field; its shape does not predict the field. The near-infrared light and the unit field markers are drawn in false color with enlarged radii. The reciprocal comparison is an effective model, not a simulation of quartz.