RC time constant (RC-limited bandwidth)

The RC time constant is the time a resistor–capacitor circuit needs to charge or discharge: after a step, the capacitor voltage covers 63% of the change in one $\tau = RC$ and 99% in 4.6 $\tau$. In photonics it appears wherever a capacitance must be charged through a resistance, most often the junction capacitance of a photodiode loaded by 50 Ω, or a modulator electrode driven from a 50 Ω source. Values run from picoseconds for small high-speed devices to microseconds for large detectors on high-value load resistors.

Bandwidth and rise time

An RC circuit is a first-order low-pass filter. Its response falls 3 dB below the low-frequency value at

$$f_\text{RC} = \frac{1}{2\pi R C}$$

and its 10–90% rise time after a step is $\ln 9 \cdot \tau = 2.2\,\tau$. Combining the two gives the familiar rule

$$t_r = \frac{2.2}{2\pi f_\text{RC}} \approx \frac{0.35}{f_\text{RC}} .$$

The 0.35 factor belongs to a single-pole response; instruments with a flat passband and steep roll-off have larger factors, 0.4 to 0.45.

Worked example. Model the photodiode as a current source in parallel with its capacitance $C$ = 0.5 pF, driving a 50 Ω load (the input of an oscilloscope or amplifier) with no other resistance in the loop, so $R$ = 50 Ω. Then $\tau$ = 25 ps, $f_\text{RC}$ = 6.4 GHz and $t_r$ = 55 ps. If the diode is back-terminated, with a 50 Ω resistor at the diode in parallel with the 50 Ω cable and load, the diode sees 25 Ω: the bandwidth doubles to 12.7 GHz and half of the photocurrent is lost in the termination.

Junction capacitance

For a reverse-biased diode the capacitance is mainly that of the depletion region, a parallel-plate capacitor:

$$C = \frac{\varepsilon_0 \varepsilon_r A}{W} .$$

A 30 µm diameter InGaAs PIN photodiode with a 1 µm depleted absorber and $\varepsilon_r \approx 13.9$ has $C$ = 87 fF. Into 50 Ω this gives $f_\text{RC}$ = 37 GHz, the figure used in the depletion region entry. Bond pads and the package add capacitance of the same order, so measured values are usually higher.

Combining with the transit-time limit

A photodiode is also limited by the transit time of carriers across the depletion region. A common approximation adds the two limits in quadrature:

$$\frac{1}{f_\text{3dB}^2} = \frac{1}{f_\text{RC}^2} + \frac{1}{f_\text{tr}^2} .$$

For the 30 µm diode above, the hole-limited transit bandwidth $f_\text{tr} = 0.4\,v_h/W$ is 19 GHz, and the combination is 17 GHz. Thinning the absorber shortens transit but raises $C$; the transit time entry finds the optimum width where the two limits are equal. Large-area detectors are almost always RC limited: a 1 mm diameter InGaAs diode with the same 1 µm depleted layer has 97 pF, which limits it to 33 MHz into 50 Ω.

Load resistors and transimpedance amplifiers

A larger load resistor gives more signal voltage per unit photocurrent and less thermal noise current, but the bandwidth falls in proportion. With 10 kΩ and 1 pF, $f_\text{RC}$ is 15.9 MHz and the rise time 22 ns. A transimpedance amplifier holds the diode at virtual ground so the photocurrent does not have to charge $C$ through the feedback resistor; the bandwidth is then set by the feedback resistor, the input capacitance and the amplifier's gain-bandwidth product, and can exceed the simple RC limit by a large factor.

Modulators and lasers

A lumped modulator or a directly modulated laser presents a capacitance to its 50 Ω driver, and the resulting RC limit is one term in its modulation bandwidth. Long modulators avoid it with traveling-wave electrodes, where velocity match and RF loss replace RC as the limit.

Thermal time constants

Heat flow follows the same mathematics, with thermal resistance $R_\text{th}$ in K/W in place of $R$ and heat capacity $C_\text{th}$ in J/K in place of $C$. The thermal time constant $\tau_\text{th} = R_\text{th} C_\text{th}$ sets how quickly a laser junction, a heater-driven phase shifter or a thermistor follows a change in dissipated power: an element with 1000 K/W and 1 µJ/K settles with $\tau_\text{th}$ = 1 ms.

Pitfalls

Measured rise times include the instrument: cascaded responses combine approximately in quadrature, so a 55 ps device seen through a 35 ps instrument shows about 65 ps; a scope or sampling head should therefore be several times faster than the device under test. Inductance from bond wires adds a resonance that the RC model omits, which shows as peaking in a frequency response measured with a network analyzer.

Common questions

How do I calculate the RC time constant?

Multiply the resistance in ohms by the capacitance in farads to get seconds: 50 Ω × 0.5 pF = 25 ps. Use the total resistance the capacitor sees, which for a photodiode is the parallel combination of every resistor across it.

What is the relationship between rise time and bandwidth?

For a first-order response, $t_r \approx 0.35/f_\text{3dB}$, so a 6.4 GHz RC-limited detector has a 10–90% rise time of 55 ps.

References: S. M. Sze and K. K. Ng, Physics of Semiconductor Devices, 3rd ed. (Wiley, 2007); B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019); P. Horowitz and W. Hill, The Art of Electronics, 3rd ed. (Cambridge University Press, 2015).