Integration time (exposure time)

Integration time is the interval over which a detector collects signal before the result is read out and the detector is reset. In a camera it is the exposure time of each frame, typically microseconds to seconds; in a grating spectrometer with a line sensor it is set per spectrum, often 1 ms to several seconds; in a digital multimeter or source-measure unit it is an averaging window, often set in power-line cycles (1 PLC is 20 ms at 50 Hz and 16.7 ms at 60 Hz). A pixel receiving a photon flux of 10⁴ photons per second at a quantum efficiency of 0.8 collects 80 photoelectrons in 10 ms.

Signal-to-noise ratio and integration time

The collected signal $S$ is proportional to the integration time $t$. The noise has three common parts: shot noise on the signal, $\sqrt{S}$; shot noise on the charge from dark current, $\sqrt{D}$, with $D$ also proportional to $t$; and read noise $\sigma_r$, added once per readout and independent of $t$. All in electrons,

$$\text{SNR} = \frac{S}{\sqrt{S + D + \sigma_r^2}} .$$

In the shot-noise limit the SNR grows as $\sqrt{t}$: four times the integration time doubles it. In the read-noise limit it grows in proportion to $t$. For the pixel above, with negligible dark charge and $\sigma_r$ = 3 e⁻, the noise is $\sqrt{80 + 9}$ = 9.4 e⁻ and the SNR is 8.5, close to the shot-noise value of 8.9. At 40 ms the signal is 320 e⁻ and the SNR is 17.6, a factor of 2.08. At a flux one hundred times lower, the same change takes the SNR from 0.26 to 0.92, a factor of 3.6, approaching the linear scaling of the read-noise limit.

The same accounting decides between one long exposure and several short ones. Ten frames of 1 ms summed to the same 80 e⁻ carry ten read-noise contributions, a total noise of $\sqrt{80 + 10 \times 9}$ = 13.0 e⁻ and an SNR of 6.1, against 8.5 for a single 10 ms frame. Frame summing pays off only when read noise is small against the signal per frame, which is the case for scientific CMOS sensors near 1–2 e⁻ (see CCD vs CMOS).

Noise bandwidth

Integrating a signal for a time $T$ with equal weight is a boxcar filter, whose equivalent noise bandwidth is

$$B = \frac{1}{2T} ,$$

so a 10 ms integration passes 50 Hz of white noise. This links integration time to the bandwidth-based noise formulas used for photodiodes and receivers: the shot-noise current $\sqrt{2qI B}$ evaluated with $B = 1/(2T)$ gives the same result as counting $\sqrt{S}$ electrons. A lock-in amplifier expresses the same trade through its output time constant $\tau$: a single-pole filter has a noise bandwidth of $1/(4\tau)$, equal to that of a boxcar of length $2\tau$. Halving the noise amplitude costs four times the measurement time in either form.

Saturation and dynamic range

A longer integration time raises the signal until the pixel or integrating capacitor fills. The full-well capacity sets the ceiling: a 30,000-electron well caps the single-frame shot-noise-limited SNR at $\sqrt{30{,}000}$ = 173, 44.8 dB, and the pixel in the example above would fill in 3.75 s. The ratio of full well to read noise is the sensor's dynamic range, 86 dB for a 1.5 e⁻ read noise; it is reached across a scene only when the integration time is chosen so that the brightest pixel sits just below full well. A saturated pixel reports a wrong value without warning, and in CCDs it can bloom into neighboring pixels.

Dark charge grows with integration time in the same way as signal. A dark current of 1 pA per pixel, a level reached by large InGaAs linear-array pixels at room temperature and far above that of cooled silicon sensors, is 6.24 × 10⁶ electrons per second and adds about 62,000 electrons in 10 ms, which is one reason InGaAs SWIR cameras are often cooled and run short exposures. Dark frames taken at the same temperature and integration time are subtracted to remove the mean, but the shot noise of the dark charge remains.

Spectrum analyzers and swept instruments

In swept instruments the corresponding setting is the dwell or sweep time per point. On an optical spectrum analyzer, the sensitivity setting chooses the detector bandwidth and therefore the time per point; on an electrical analyzer, a narrower resolution bandwidth or video bandwidth lowers the displayed noise floor at the cost of a slower sweep.

Common questions

Does doubling the exposure time double the signal-to-noise ratio?

Only when read noise dominates. In the shot-noise limit doubling the exposure raises the SNR by $\sqrt{2}$, about 1.41, and four times the exposure is needed to double it.

What is the difference between integration time and frame rate?

Integration time is the part of each frame period during which charge is collected. The frame period also includes readout and reset, so the integration time is at most the frame period; it can equal the frame period in sensors that integrate one frame while reading out the previous one.

Why do cameras need new calibration frames when the integration time changes?

Dark charge and some fixed-pattern offsets scale with integration time, so dark and flat-field frames taken at one setting do not correct another. Infrared cameras with nonuniformity correction tables are the most sensitive to this.

References: J. R. Janesick, Photon Transfer: DN → λ (SPIE Press, 2007); G. C. Holst and T. S. Lomheim, CMOS/CCD Sensors and Camera Systems, 2nd ed. (SPIE Press, 2011); 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).