= Photoconductor response time
<Photoconductor response time> describes how quickly the electrical signal follows a change in illumination under stated bias, temperature, background and readout conditions. A rise time, fall time and modulation bandwidth summarize different features and require definitions. The manufacturer’s PbS description notes that sensitivity, response speed and dark resistance vary with temperature, so a timing number is not independent of operating conditions.<How does a PbS photoconductive detector work? by Shimadzu Scientific Instruments>[temperature-dependence paragraph]{ref}
An original one-lifetime model takes excess carrier density $n_e$ to obey
$$
\frac{dn_e}{dt}=g(t)-\frac{n_e}{\tau}.
$$
With constant mobility and fixed detector bias, suppose the excess current is proportional to $n_e$. After a step from zero to generation $g_0$, the normalized signal is $1-e^{-t/\tau}$. The times to reach ten and ninety percent of its final value are $-\tau\ln0.9$ and $-\tau\ln0.1$, giving
$$
\boxed{t_{10\text{–}90}=\tau\ln9\approx2.1972\tau.}
$$
The time from zero to ninety percent is instead $\tau\ln10$. A quoted rise time without its thresholds is therefore incomplete.
For small sinusoidal generation changes, the normalized <frequency> response is
$$
H(f)=\frac1{1+i2\pi f\tau},\qquad
|H|=\frac1{\sqrt{1+(2\pi f\tau)^2}},\quad
\arg H=-\arctan(2\pi f\tau).
$$
The half-power <frequency> is $f_c=1/(2\pi\tau)$, so this single-pole model gives $t_{10\text{–}90}f_c=\ln9/(2\pi)\approx0.3497$. This relationship is a consequence of the stated model, not a universal identity for arbitrary detectors.
For $\tau=100\,\mathrm{\mu s}$, the rise time is about $219.7\,\mathrm{\mu s}$ and the cutoff about $1.592\,\mathrm{kHz}$. At $1\,\mathrm{kHz}$, the signal amplitude is about $0.8467$ of its low-<frequency> value and its phase lags by $32.14^\circ$. The signal can therefore have a substantial phase delay before reaching the half-power point.
A readout adds its own dynamics. If the detector response is cascaded with a separate first-order electronic response of the same time constant, the overall magnitude is $1/[1+(2\pi f\tau)^2]$. Its half-power <frequency> is $\sqrt{\sqrt2-1}/(2\pi\tau)$, about $0.6436$ times the single-pole cutoff. Assigning that combined bandwidth entirely to the material would give an incorrect carrier-lifetime estimate.
<Photoconductor rise and fall asymmetry> tests another limitation of the simple model: linear one-lifetime dynamics predict equal normalized rise and fall behavior, but trap occupancy, changing background or a nonlinear circuit can produce different transients. A long relaxation tail should be measured rather than compressed into an unsupported single time constant.
A proposed experiment switches or modulates a monitored optical source, records detector current and actual bias, and measures ten-to-ninety and ninety-to-ten crossings after settling the initial illumination state. Repeat the modulation sweep with a characterized faster readout or independently measured amplifier response. Vary temperature and optical background while holding the wavelength and spot fixed. Agreement between step and <frequency>-response fits tests the one-pole approximation; disagreement identifies missing dynamics without by itself proving which physical mechanism caused them.
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