Modulation depth

Modulation depth describes how strongly a signal is modulated: the size of the swing relative to the mean. For an optical intensity modulated as $P(t) = P_0\,[1 + m\cos\Omega t]$, the modulation depth (also called modulation index) is

$$m = \frac{P_\text{max} - P_\text{min}}{P_\text{max} + P_\text{min}},$$

which runs from 0 for unmodulated light to 1 when the minimum reaches zero. Analog links and small-signal measurements typically use $m$ of a few percent to about 0.5; a binary transmitter with an 8.2 dB extinction ratio has $m = 0.74$; light switched fully off by a chopper has $m = 1$. For phase modulation the depth is instead the peak phase excursion $\beta$ in radians.

Relation to extinction ratio

The extinction ratio $r = P_\text{max}/P_\text{min}$ and the modulation depth carry the same information:

$$r = \frac{1 + m}{1 - m}, \qquad m = \frac{r - 1}{r + 1}.$$

A depth of 0.5 gives $r = 3$, or 4.8 dB. The 8.2 dB minimum often specified for on-off keying is $r = 6.61$ and $m = 0.737$, and 20 dB is $m = 0.980$. Near full modulation $m$ changes little while $r$ changes a lot, so digital transmitters are specified by extinction ratio. The peak-to-peak swing, the optical modulation amplitude, is $P_\text{max} - P_\text{min} = 2mP_0$.

Setting the depth in common modulators

A directly modulated laser with a linear L-I curve above threshold has $m = I_p/(I_b - I_\text{th})$, for a sinusoidal current of amplitude $I_p$ around a bias $I_b$. With $I_\text{th} = 10$ mA, $I_b = 40$ mA and $I_p = 15$ mA, $m = 0.5$. Driving beyond $m = 1$ pushes the laser below threshold for part of each cycle, which clips the waveform and adds turn-on delay and distortion.

A Mach-Zehnder modulator biased at quadrature transmits $\tfrac12[1 + \sin(\pi V/V_\pi)]$, so a drive of peak amplitude $V_p$ gives

$$m = \sin\!\left(\frac{\pi V_p}{V_\pi}\right).$$

With a half-wave voltage of 4 V and $V_p = 0.4$ V, $m = 0.309$, slightly below the linear estimate $\pi V_p/V_\pi = 0.314$; the difference is the compression of the sinusoidal transfer curve that generates harmonics in analog links. An acousto-optic modulator sets the depth of the diffracted beam through the RF drive power, following the sine-squared diffraction-efficiency curve.

Measuring it

On a photodiode and oscilloscope, $P_\text{max}$ and $P_\text{min}$ are read from the waveform after the dark offset is subtracted, since the photocurrent follows the optical power with the same depth. In the frequency domain the detected signal at $\Omega$ has a current amplitude $mI_0$ about a mean photocurrent $I_0$. For $I_0 = 1$ mA and $m = 0.1$, the RF power delivered to 50 Ω is $(mI_0)^2 R/2 = 0.25$ µW, or −36.0 dBm. The laser's own noise limits the signal-to-noise ratio to $m^2/(2\,\text{RIN}\cdot B)$, as the relative intensity noise entry shows, so $m = 0.5$ with −150 dB/Hz in 1 GHz gives 51.0 dB.

Phase modulation depth

An electro-optic phase modulator imposes $\phi(t) = \beta\sin\Omega t$ with $\beta = \pi V_p/V_\pi$. The field then contains a carrier and sidebands at multiples of $\Omega$, with amplitudes given by Bessel functions:

$$e^{i\beta\sin\Omega t} = \sum_{n} J_n(\beta)\,e^{in\Omega t}.$$

The power in the carrier is $J_0^2(\beta)$ and in each first-order sideband $J_1^2(\beta)$. At $\beta = 0.3$ rad the carrier keeps 95.6% and each first sideband holds 2.2%, close to the small-signal value $(\beta/2)^2$. At $\beta = 1.08$ rad the product $J_0J_1$, which sets the error signal in Pound-Drever-Hall locking, is largest, with 53% of the power in the carrier and 22% in each first sideband; at 1.84 rad the first sidebands are largest, and at 2.405 rad the carrier vanishes. Pure phase modulation leaves the intensity unchanged until an interferometer, cavity or dispersion converts it.

Other uses of the term

In a lock-in amplifier experiment the modulation depth is the fraction of the signal that is chopped or dithered, and only that part is recovered. A saturable absorber's modulation depth is the saturable part of its loss, often around 1% for mirrors used in bulk solid-state lasers. For gratings, it is the amplitude of the periodic index change.

Pitfalls

Some texts define depth as the peak-to-peak swing over the mean, $(P_\text{max} - P_\text{min})/P_0 = 2m$, which doubles the number; quoting the formula avoids the ambiguity. A dark offset or background light on the detector lowers the apparent $m$. Optical intensity depth and RF amplitude-modulation index are different quantities when the field is modulated, because intensity is the square of the field: a field modulated with index $m_E$ gives intensity depth $2m_E/(1 + m_E^2)$.

Common questions

How do you convert modulation depth to extinction ratio?

Use $r = (1 + m)/(1 - m)$ and take $10\log_{10}r$ for decibels: $m = 0.5$ gives 4.8 dB, $m = 0.9$ gives $r = 19$, or 12.8 dB.

What is the difference between modulation depth and modulation index?

For intensity modulation they are normally the same quantity, $m$. For phase modulation, "modulation index" usually means the peak phase deviation $\beta$ in radians.

References: B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019); A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications, 6th ed. (Oxford University Press, 2007); E. D. Black, "An introduction to Pound–Drever–Hall laser frequency stabilization," Am. J. Phys. 69, 79 (2001).