Tool · PIC characterization

Vπ Fitter

What half-wave voltage, extinction ratio and bias point does a measured modulator bias sweep imply? The fitter fits the full cosine transfer function to every point of a pasted sweep and returns Vπ, VπL, extinction ratio, bias phase and the nearest quadrature bias, with residuals; the validation table compares the fit with reading the peak-to-null distance from two points. Background: half-wave voltage, Mach-Zehnder modulator, and extinction ratio.

Measured sweep

Two columns per line: bias voltage, then transmission. Comma, tab, or space separated. Any power units work; dB is detected automatically (or force it below).

Input units

Electrode length (mm)

The example is a synthetic 1.73 V device with realistic noise, drift-free. The fit is a one-dimensional scan over Vπ with an exact linear least-squares solve at each trial (the model is linear in offset and quadrature amplitudes), then golden-section refinement. No starting guess, no divergence.

Transfer curve

Paste a sweep (or load the example) and press fit.

Residuals · data minus fit

Fitted parametersno fit yet

The transfer model is P(V) = C + B cos(πV/Vπ + φ_b). For silicon depletion devices the phase is nonlinear in voltage, so treat the result as the local Vπ over your sweep window. The ± on Vπ is one standard error from the residuals and the fit’s Jacobian: it measures the scatter that noise alone would cause, not drift during the sweep or a wrong model. Method details and pitfalls: Measuring modulator Vπ and EO bandwidth.

Checked against

These validation rows run in your browser on every load: the fitter is exercised on synthetic sweeps with known answers, including the clipped-sweep case where peak-picking fails.

Check	Expected	Computed	Tolerance

The table fits synthetic sweeps generated with Vπ = 1.73 V and a bias phase of 0.4 rad, so the right answer is known by construction, and it checks that data the fit cannot pin down is refused. A ± tolerance is the largest absolute difference that still passes, in the units of the Expected column; a percentage is relative to the expected value; “condition” means the Expected entry is itself the test, and “exact” means the outcome must match.

The model

An intensity modulator driven through its bias voltage $V$ transmits

P (V) = C + B \cos ⁣ (\frac{π V}{V_{π}} + φ_{b})

where $C$ is the mean level, $B$ the fringe amplitude and $φ_{b}$ the bias phase. At a fixed $ω = π / V_{π}$ the model is linear in $C$ and in the two quadrature amplitudes of $\cos ω V$ and $\sin ω V$ , so the tool solves those three exactly by least squares at each trial value, scans $V_{π}$ over 701 logarithmically spaced values from a tenth of the sweep span to five times it, and refines the best one by golden-section search. There is no starting guess to choose. The residuals are minimized in linear transmission, whether the data were pasted in dB or linear units. The derived quantities are

E R = 10 \log_{10} \frac{C + B}{C - B}

\frac{π V_{q}}{V_{π}} + φ_{b} = \frac{π}{2} + k π

with $V_{q}$ the quadrature bias nearest the center of the sweep, and $V_{π} L$ in V·cm when an electrode length is entered. The model assumes a linear electro-optic response, so it does not describe thermal phase shifters (phase quadratic in voltage) or carrier-depletion modulators over a range where their efficiency changes with bias; it also assumes no drift during the sweep. The fit is refused when the sweep has zero span, when the fitted fringe is smaller than three times the RMS residual, or when the fitted Vπ exceeds twice the span, since the data then cover less than half a fringe. The ± on Vπ is one standard error, $σ_{V_{π}} = V_{π} σ_{ω} / ω$ , with $σ_{ω}^{2}$ the $ω$ element of $s^{2} (J^{T} J)^{- 1}$ , where $J$ is the Jacobian of the four-parameter model at the fit and $s^{2}$ the residual sum of squares over $n - 4$ . It assumes independent residuals of equal variance, so it covers noise, not drift or a model that does not fit.

Worked example

Press load example scan: 81 points from −4 to +4 V in 0.1 V steps, synthesized from Vπ = 1.73 V, bias phase 0.4 rad and a 27 dB extinction ratio, with 0.05 dB of Gaussian noise added in dB. The fit returns Vπ = 1.7304 ± 0.0010 V (one standard error), 0.02% from the true value, a bias phase of 0.3972 rad, an extinction ratio of 27.0 dB, the nearest quadrature at 0.646 V and an RMS residual of 0.72% of the fringe. The largest and smallest samples of the same scan lie 1.80 V apart, so a peak-to-null reading would be 4.0% high. Entering a 3 mm electrode gives $V_{π} L$ = 1.7304 × 0.3 = 0.5191 ± 0.0003 V·cm. The bench procedure, including the RF methods, is in Measuring modulator Vπ and electro-optic bandwidth.

Related: the plasma dispersion calculator estimates, from the carrier density, the index change, the length for π and the loss of a silicon phase shifter.

References: A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1984), ch. 7. E. L. Wooten et al., “A review of lithium niobate modulators for fiber-optic communications systems,” IEEE Journal of Selected Topics in Quantum Electronics 6, 69–82 (2000).