Half-wave voltage (Vπ)

The half-wave voltage $V_{\pi }$ of an electro-optic modulator is the voltage that changes the optical phase in its phase shifter by $\pi$ radians, half an optical cycle. In a Mach-Zehnder modulator, where the two arms interfere at the output, a phase difference of $\pi$ turns constructive interference into destructive, so $V_{\pi }$ is the voltage that moves the output from a transmission maximum to the adjacent minimum:

$$T(V)=\frac{1}{2}\text{\hspace{0.17em}⁣}[1+\cos \text{\hspace{0.17em}⁣}(\frac{\pi V}{V_{\pi }}+{\varphi }_b)]\mathrm{.}$$

The pattern repeats every $2V_{\pi }$, and the bias phase ${\varphi }_b$ sets the operating point; intensity-modulated links usually hold the modulator at quadrature, halfway down the slope, where the response is steepest. $V_{\pi }$ is the modulator's figure of merit for drive, since the voltage swing the driver must supply, and the power it spends, scale with it.

What sets Vπ

In a phase shifter based on the Pockels effect, an applied field $E=V\mathrm{/}d$ across an electrode gap $d$ changes the index by $\mathrm{\Delta }n=\frac{1}{2}{n}^{3}rE$, and the phase accumulated over an electrode length $L$ is $\mathrm{\Delta }\varphi =(2\pi \mathrm{/}\lambda )\mathrm{\Gamma }\mathrm{\Delta }nL$, where $\mathrm{\Gamma }$ is the overlap between the applied field and the optical mode. Setting $\mathrm{\Delta }\varphi =\pi$ gives

$$V_{\pi }=\frac{\lambda d}{n^{3}r\mathrm{\Gamma }L}\mathrm{.}$$

In a push-pull Mach-Zehnder, where one drive voltage shifts the two arms in opposite directions, the phase difference doubles and $V_{\pi }$ is half this value. For x-cut thin-film lithium niobate at 1550 nm, with $n_e$ = 2.138, $r_{33}$ = 30.8 pm/V, a 5 µm gap and an overlap of 0.5, a single arm gives $V_{\pi }L$ = 5.1 V·cm and a push-pull device 2.6 V·cm, so a 15 mm push-pull modulator has a $V_{\pi }$ of about 1.7 V (electro-optic effect).

Silicon modulators change the index through the plasma-dispersion effect instead. The depletion width of a reverse-biased junction grows roughly as the square root of the voltage, so the phase is not linear in voltage and a quoted $V_{\pi }$ holds only around the bias at which it was measured.

VπL: comparing platforms

Because $V_{\pi }$ falls in inverse proportion to $L$ in a traveling-wave device, the product $V_{\pi }L$, in V·cm, characterizes the material and the waveguide cross-section rather than a particular device:

A longer electrode buys a lower $V_{\pi }$ at the cost of optical loss, which grows in proportion to $L$, and of bandwidth, since the RF loss of the electrode and any mismatch between the RF and optical velocities accumulate along it. The product of the optical loss per unit length and $V_{\pi }L$, in V·dB, is therefore quoted as a companion figure: the optical loss that comes with a phase shifter whose $V_{\pi }$ is 1 V. Hybrid designs that fill a narrow silicon slot or a plasmonic gap with an electro-optic polymer reach lower $V_{\pi }L$ still, traded against optical loss.

Drive swing and power

For on-off keying at full extinction the drive swings through a whole $V_{\pi }$. A smaller swing still modulates, with a lower extinction ratio: biased at quadrature and driven with a peak-to-peak swing of $V_{\pi }\mathrm{/}2$, a modulator reaches levels of $\frac{1}{2}[1\pm \sin (\pi \mathrm{/}4)]$ of its peak transmission, an extinction ratio of 7.7 dB and 71% of the full optical modulation amplitude (eye diagram). A random NRZ drive of peak-to-peak swing $V_{pp}$ into a traveling-wave electrode terminated in a resistance $R$ dissipates about $V_{pp}^2\mathrm{/}4R$: 45 mW for a 3 V swing into 50 Ω and 11 mW for 1.5 V. Halving $V_{\pi }$ therefore quarters the drive power at the same extinction, and a $V_{\pi }$ within the output swing of the driver electronics avoids a separate amplifier. For PAM4 the swing is usually kept to the more linear middle of the cosine, or the drive levels are predistorted, so that the four optical levels stay evenly spaced.

What changes the value

• Wavelength. $V_{\pi }$ scales with $\lambda$ at fixed $n^{3}r\mathrm{\Gamma }$, so a modulator has a higher $V_{\pi }$ at 1550 nm than at 1310 nm, by roughly the ratio of the wavelengths, 1.18, before the dispersion of the index, the coefficient and the overlap is counted.
• Frequency. At high frequency the electrode loss and the velocity mismatch reduce the phase shift per volt, so $V_{\pi }(f)=V_{\pi ,\text{DC}}\mathrm{/}\mathrm{\mid }m(f)\mathrm{\mid }$, where $m(f)$ is the normalized electro-optic response (modulation bandwidth). Datasheets distinguish the DC value, measured with a slow bias sweep, from the RF value at a stated frequency.
• Bias. The nonlinear phase of silicon modulators makes $V_{\pi }$ depend on the bias and on the size of the swing. In lithium niobate the bias point drifts slowly under a DC voltage, which moves ${\varphi }_b$ rather than $V_{\pi }$ but complicates both measurement and operation.
• Drive arrangement. Push-pull halves $V_{\pi }$ relative to driving one arm of the same length and cancels the chirp; x-cut thin-film lithium niobate gets push-pull from the electrode layout alone, with the ground electrodes on either side of the signal electrode applying opposite fields to the two arms.

Measuring it

For a Mach-Zehnder device, the bias is swept slowly over more than one full fringe and the cosine is fitted to the whole curve, which returns $V_{\pi }$, the extinction ratio and the bias phase together; reading $V_{\pi }$ as the distance between an apparent maximum and minimum uses the two flattest and least certain points. A bare phase modulator has no fringe to sweep, so it is driven with an RF tone of amplitude $V_m$ and the optical carrier watched on a spectrum analyzer: the carrier vanishes where $\pi V_m\mathrm{/}{V}_{\pi }$ = 2.405, the first zero of the Bessel function $J_0$, so $V_{\pi }=\pi V_{m,\text{null}}\mathrm{/}2.405$. The procedures, with the swept-frequency measurement of $V_{\pi }(f)$ and the usual errors, are in Measuring modulator Vπ and EO bandwidth, and the Vπ Fitter fits a pasted bias sweep.

Common questions

What is the half-wave voltage of a modulator?

The voltage that shifts the optical phase in the modulator's phase shifter by π radians. In a Mach-Zehnder modulator it is the voltage between a transmission maximum and the adjacent minimum.

How is Vπ calculated?

For a Pockels phase shifter, $V_{\pi }=\lambda d\mathrm{/}(n^{3}r\mathrm{\Gamma }L)$, with $d$ the electrode gap, $n$ the refractive index, $r$ the electro-optic coefficient, $\mathrm{\Gamma }$ the overlap of the applied field with the optical mode and $L$ the electrode length. Push-pull drive halves it.

What is VπL?

The product of the half-wave voltage and the electrode length, which stays roughly constant as the length changes and so compares platforms rather than devices. Thin-film lithium niobate and silicon depletion modulators both reach 1 to 3 V·cm.

Why is Vπ higher at high frequencies?

Electrode loss and the mismatch between the RF and optical velocities reduce the phase shift per volt as the frequency rises, so more voltage is needed for the same π shift. The RF $V_{\pi }$ equals the DC value divided by the normalized electro-optic response at that frequency.

Does push-pull drive halve Vπ?

Yes. When one drive voltage shifts the two arms of a Mach-Zehnder in opposite directions, the phase difference between them is twice that of a single driven arm, so half the voltage reaches π. It also removes the chirp from the output.

References: A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications, 6th ed. (Oxford University Press, 2007). 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 (2000). G. T. Reed, G. Mashanovich, F. Y. Gardes and D. J. Thomson, "Silicon optical modulators," Nature Photonics 4, 518 (2010). C. Wang et al., "Integrated lithium niobate electro-optic modulators operating at CMOS-compatible voltages," Nature 562, 101 (2018). L. Chrostowski and M. Hochberg, Silicon Photonics Design (Cambridge University Press, 2015).