Rise time (10–90%)

Rise time is the interval during which a step response climbs from 10% to 90% of the way between its initial and final levels. It is the time-domain counterpart of bandwidth: a detector, amplifier, modulator or oscilloscope with a 3 dB bandwidth $f_\text{3dB}$ has a rise time of roughly $0.35/f_\text{3dB}$, so a 20 GHz photodiode rises in about 17.5 ps and a 70 GHz sampling head in about 5 ps. Fall time is the same measurement on the falling edge, from 90% to 10%. Eye-diagram measurements often use 20% and 80% points instead, because noise and jitter blur the extremes of a transition.

Rise time and bandwidth

For a first-order low-pass response with time constant $\tau$, the step response is $1 - e^{-t/\tau}$ and the 10–90% interval is $\ln 9\cdot\tau = 2.2\,\tau$. Since $f_\text{3dB} = 1/(2\pi\tau)$,

$$t_r = \frac{\ln 9}{2\pi f_\text{3dB}} \approx \frac{0.35}{f_\text{3dB}} .$$

This is the relation derived in the RC time constant entry, where a 0.5 pF photodiode on 50 Ω has $\tau$ = 25 ps and $t_r$ = 55 ps. The factor depends on the shape of the response. A Gaussian impulse response gives 0.34; systems with a flat passband and a steep roll-off give 0.4 to 0.45, and they also overshoot. The 20–80% time is 0.63 of the 10–90% time for a single pole and 0.66 for a Gaussian response, so converting between them requires knowing the shape.

Cascaded systems

When several stages with roughly Gaussian responses are cascaded, their rise times add approximately in quadrature:

$$t_\text{sys}^2 \approx t_1^2 + t_2^2 + t_3^2 + \ldots$$

The rule is exact for Gaussian responses, whose variances add, and a working approximation for other smooth, non-overshooting responses. A photodiode with a true rise time of 20 ps, measured on a sampling head with 15 ps, shows $\sqrt{20^2 + 15^2}$ = 25 ps. The correction can be reversed, but it becomes unreliable when the instrument is about as fast as the device, since small errors in either value then dominate the difference. An instrument three times faster than the device inflates the reading by 5.4%; five times faster, by 2.0%.

Measuring lasers, modulators and detectors

A detector's rise time is measured by illuminating it with an optical step or with pulses much shorter than its response, whose recorded shape is the impulse response and whose running integral is the step response, and recording the output on a sampling oscilloscope or a fast real-time scope. Transmitter rise times are read from isolated edges in a repeating pattern. The scope's own optical or electrical rise time must then be removed by quadrature subtraction. The complementary frequency-domain measurement, with a network analyzer, gives the modulation bandwidth directly; the photodetector bandwidth procedure describes both.

Physical limits show up as distinct features of the edge. In a photodiode, the transit time of carriers across the depletion region and the RC constant set the main edge, while carriers generated outside the depletion region diffuse slowly and add a tail to the impulse response, which shows on the trailing edge of a short pulse and as a slow final approach to the settled level of a step. A directly modulated laser overshoots and rings at its relaxation-oscillation frequency after a current step; the 10% and 90% points are then taken relative to the settled levels before and after the transition.

Eye diagrams and symbol periods

In a digital link the rise time is compared with the unit interval, one symbol period. At 25 GBd the unit interval is $1/(25\ \text{GHz})$ = 40 ps (symbol rate). A system rise time of 25 ps then occupies 62.5% of each symbol, which closes the eye diagram horizontally and causes intersymbol interference: one symbol's slow edge is still settling when the next is sampled. Reference receivers limit their bandwidth to a fraction of the symbol rate, so a measured transmitter edge depends on the receiver filter as well as the transmitter.

Pitfalls

The 0.35 rule applies to bandwidth defined by the 3 dB point of the electrical power response. For photodetectors, where the electrical power varies as the square of the photocurrent, a "3 dB optical" bandwidth would correspond to the 6 dB electrical point; the conventions should be checked before converting. Ringing, reflections from unmatched cables and probe ground leads distort the edge in ways the quadrature rule does not capture. Averaging on a sampling scope smooths noise but cannot remove the broadening that timing jitter adds to an edge.

Common questions

How do you convert rise time to bandwidth?

Divide 0.35 by the rise time for a response close to single-pole or Gaussian: 17.5 ps corresponds to 20 GHz. Instruments with a sharp roll-off use 0.4 to 0.45.

How do you correct a measured rise time for the oscilloscope?

Subtract in quadrature: $t_\text{device} = \sqrt{t_\text{meas}^2 - t_\text{scope}^2}$. With 25 ps measured and a 15 ps scope, the device rise time is 20 ps.

What is the difference between rise time and pulse duration?

Rise time describes a single edge, measured between the 10% and 90% levels. Pulse duration is usually the full width at half maximum of a whole pulse. A pulse can be no shorter than about the rise time of the system that records it.

References: P. Horowitz and W. Hill, The Art of Electronics, 3rd ed. (Cambridge University Press, 2015); B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019); G. P. Agrawal, Fiber-Optic Communication Systems, 4th ed. (Wiley, 2010).