Fiber Polarization Controller
Retardance per paddle from bend-induced birefringence, quarter- and half-wave loop diameters and the output polarization state of a three-paddle fiber polarization controller.
Paddle angle: the tilt of the loop plane from the x axis. The bend's fast axis lies in the loop plane. χ > 0 is right-handed (S₃ > 0), as on the site's Stokes and Mueller calculators.
Retardance per Loop
0.8822 rad
0.1404 waves; bend B = 2.309e-6
Paddle Retardances
0.281 / 0.562 / 0.281
waves for 2 / 4 / 2 loops (λ/4 = 0.25, λ/2 = 0.5)
Loop Diameter for λ/4 and λ/2
33.7 / 33.7 mm
λ/4 with 2 loops, λ/2 with 4 loops
Output Azimuth ψ
55.87°
major axis from x
Output Ellipticity χ
0.40°
right; axis ratio 0.007
Output Stokes (S₁, S₂, S₃)
-0.370, 0.929, 0.014
normalized, S₀ = 1
Model
B_bend = 0.133 (d/D)² (Ulrich et al. 1980)
Γ = 2π² · 0.133 · N d²/(λD) per paddle (Lefèvre 1980)
S_out = M(Γ₃, θ₃) M(Γ₂, θ₂) M(Γ₁, θ₁) S_in (linear retarders)
D(λ/4) = 4π · 0.133 · N d²/λ, D(λ/2) = 2π · 0.133 · N d²/λ
Each paddle is a linear retarder whose fast axis turns with the paddle. The coefficient 0.133 is for silica (photoelastic constants, Poisson ratio) and depends little on wavelength. Ignored: the twist of the fibre between paddles (a small circular retardance), fibre tension, the fibre's own birefringence, and bend loss.
Chart: S₁, S₂ and S₃ after each paddle.
Chart: output polarization ellipse (field, normalized).
Model and references
Textbook approximation. A standard approximation from the literature (for example paraxial, plane-wave or Gaussian-beam), valid within the range the page states. About the tiers
Lefèvre loop paddles: bend birefringence 0.133(d/D)², retardance 2π²·0.133·Nd²/(λD) per paddle, chained as rotated linear retarders (Mueller). Not modelled: twist between paddles, tension, bend loss.
- Lefèvre H. C. (1980). Single-mode fibre fractional wave devices and polarisation controllers. Electron. Lett. 16, 778. doi:10.1049/el:19800552
- Ulrich R., Rashleigh S. C., Eickhoff W. (1980). Bending-induced birefringence in single-mode fibers. Opt. Lett. 5, 273. B = 0.133 (r/R)². doi:10.1364/ol.5.000273
- Goldstein D. H. (2011). Polarized Light, 3rd ed. CRC Press. Mueller matrices of rotated retarders. doi:10.1201/b10436