Polarization Scrambling
Simulate polarization scrambling: how randomizing polarization state reduces residual polarization.
stage k: retarder with fast axis θk, Γk(t) = Γ0 + Γm w(fk t); a slow detector sees the time-average Stokes vector
sine: ⟨cos Γ⟩ = J₀(Γm) cos Γ0, ⟨sin Γ⟩ = J₀(Γm) sin Γ0 (J₀ = 0 at Γm = 2.4048 rad)
ramp: ⟨cos Γ⟩ = [sin(Γ0 + Γm) − sin Γ0]/Γm, ⟨sin Γ⟩ = [cos Γ0 − cos(Γ0 + Γm)]/Γm
independent stages: ⟨M3M2M1⟩ = ⟨M3⟩⟨M2⟩⟨M1⟩; locked stages: (1/T)∫₀ᵀ M3(t)M2(t)M1(t) dt over their common period T
combination n₁f₁ + n₂f₂ + n₃f₃ = 0: ⟨M3M2M1⟩ = Σj A3,jn₃A2,jn₂A1,jn₁ eij n·φ (Ak,m: Fourier coefficients of Mk in its drive phase)
With incommensurate frequencies (the defaults are 1, √2 and √5 kHz) the stage phases are independent and the drive phases φ don't matter; with locked frequencies (whole-number ratios up to 12, or a relation such as f₁ + f₂ = f₃) they do.
Stage 1
Stage 2
Residual DOP, this input (long-time average)
0.0000
0.00 % of the light stays polarized
Residual DOP, worst case over all inputs
0.0000
largest over fully polarized inputs
Residual DOP against averaging time
Residual DOP of the average over [0, T] for this input; the dashed line is the long-time value.
Long-time residual DOP against drive amplitude
The same Γm on every stage, for this input; the marker appears when all stages share one amplitude. A stage's average vanishes at the zeros of J₀.
Model and references
Exact. The equations are exact within the stated assumptions, or are a standard's tabulated values. The results are as good as the inputs. About the tiers
Ideal driven retarders; stage means in closed form (sine J₀(Γm), ramp sinc-type), multiplied for incommensurate frequencies. Locked stages (whole-number ratios up to 12) are averaged numerically over their common period, with the drive phases; three stages tied by a relation such as f₁ + f₂ = f₃ keep its Fourier terms. Relations above order 12 count as incommensurate (they matter only for large Γm or ramps). Worst case = top singular value of the 3×3 block. No drive nonlinearity or loss.
- Goldstein D. H. (2011). Polarized Light, 3rd ed., ch. Mueller Matrices for Polarizing Components, pp. 93–115. CRC Press. doi:10.1201/b10436
- Heismann F., Tokuda K. L. (1995). Polarization-independent electro-optic depolarizer. Opt. Lett. 20(9), 1008–1010. doi:10.1364/ol.20.001008
- NIST Digital Library of Mathematical Functions, §10.12, eq. 10.12.2 (Jacobi–Anger expansions). dlmf.nist.gov
- Abramowitz M., Stegun I. A. (1964). Handbook of Mathematical Functions, NBS AMS 55: Table 9.1 (values of J₀) and Table 9.5 (zeros of J₀).
- Smith O. K. (1961). Eigenvalues of a symmetric 3 × 3 matrix. Commun. ACM 4(4), 168. doi:10.1145/355578.366316