= Dynamic range
{title2=$P_{\max}/P_{\min}$}
{wiki}
The \b[dynamic range] of a <measurement procedure> is the <ratio> between the largest and smallest positive inputs it can usefully distinguish under specified conditions. The quantity, <bandwidth>, observation interval, <measurement calibration> and acceptance criteria must accompany the number. An upper boundary might be set by <amplifier clipping>, while a lower boundary might require a specified <signal-to-noise ratio>. More displayed digits do not by themselves establish either boundary or better <measurement precision>.
Consider a <photodiode responsivity measurement> in which a <photodiode> produces <electric current> $i=R_\lambda P$ from <optical power> $P$, with <spectral responsivity> $R_\lambda$ in amperes per watt. A <transimpedance amplifier> with transimpedance $R_T$ produces an output <electric potential difference> $v=R_Ti=KP$, where $K=R_TR_\lambda$ is expressed in volts per watt. If a chosen detection criterion is $k$ times the <root mean square> output noise $\sigma_v$, and the linear output ceiling is $v_{\max}$, the <noise-limited dynamic range> is
$$
P_{\min}=\frac{k\sigma_v}{K},\qquad P_{\max}=\frac{v_{\max}}K,\qquad D=\frac{P_{\max}}{P_{\min}}=\frac{v_{\max}}{k\sigma_v}.
$$
Increasing $K$ shifts this input interval toward smaller <optical power>; it does not enlarge its <ratio> when the stipulated output noise and output ceiling remain fixed. The corresponding <input-referred noise> is $\sigma_v/K$.
An original numerical example makes the distinction concrete. Stipulate $\sigma_v=0.5\,\mathrm{mV}$ in either setting, $k=5$, $v_{\max}=1\,\mathrm V$, and no other limiting effect. A high-sensitivity setting $K_H=10^5\,\mathrm{V/W}$ covers $25\,\mathrm{nW}$ to $10\,\mathrm{\mu W}$; a lower-sensitivity setting $K_L=10^3\,\mathrm{V/W}$ covers $2.5\,\mathrm{\mu W}$ to $1\,\mathrm{mW}$. Each interval has \b[$D=400$]. With compatible <measurement calibration> and <range-overlap calibration> in their common interval, <gain ranging> covers a contiguous union from $25\,\mathrm{nW}$ to $1\,\mathrm{mW}$: \b[$D=40\,000$]. These are <optical power> ratios, so their values in <decibels> are $10\log_{10}D$, approximately $26.02\,\mathrm{dB}$ and $46.02\,\mathrm{dB}$ respectively.
\Image[/dual-range-optical-power-measurement.png]
{title=Two calibrated optical-power ranges cover a wider continuous interval}
{description=Original illustrative ranges with the same stipulated output noise and ceiling. The shaded interval is available for checking agreement between gain settings.}
The equal-noise assumption isolates the geometric effect of <gain ranging>; it is not a specification for an actual instrument. A real <transimpedance amplifier> changes several properties together: its feedback network affects <bandwidth>, noise and allowable input <electric current>. The DLPCA-200 manufacturer sheet shows a current-to-voltage converter followed by programmable amplification and gives separate tables for these dependencies.<Datasheet DLPCA-200 by FEMTO>[pp. 1–3: block diagram and gain-dependent performance tables]{ref} To test a combined <dynamic range>, vary a stable light source through the overlap, compare the two estimates of <optical power>, and allow the output to settle after switching. Agreement within the required <measurement uncertainty> is evidence that the two ranges can be joined; the mere presence of two gain switches is insufficient.
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