= Shot noise photocurrent measurement
MIT's guide uses an illuminated <vacuum photodiode>, with collecting <anode> positive relative to the emitting <cathode>. A <transimpedance amplifier> gives $V_1=R_F I$; a <multimeter> reads its mean. The fluctuation path rejects below $100\,\mathrm{Hz}$, adds roughly tenfold <voltage gain>, then nominal hundredfold preamplification and a $1$–$50\,\mathrm{kHz}$ <band-pass filter>; the readout measures <root mean square> voltage. With diode bias off, a sinusoid at the test port calibrates $g(f)$. Restore bias, retain gain settings, vary lamp brightness, and repeat both readings. The guide warns about overload; it supplies a procedure, not measured results.<Johnson Noise and Shot Noise: The Determination of the Boltzmann Constant, Absolute Zero Temperature and the Charge of the Electron by MIT Department of Physics>[pp. 3, 11–12: Fig. 1; §§V–VI, Figs. 6–7]{ref}
\Image[/shot-noise-photocurrent-apparatus-and-calibration.png]
{title=Functional photocurrent measurement: separate mean-current, fluctuation and calibration paths}
{description=Original functional diagram organizing the laboratory's measurement roles; it is not a component-level circuit reproduction. $K(f)$ includes current-to-voltage conversion and the complete fluctuation path. The dark comparison below is an additional control.}
The inference can be expressed without relying on nominal passband edges. Let $g(f)$ denote the calibrated voltage transfer after the $R_F$ conversion, so $K(f)=R_Fg(f)$ and define $\mathcal K=\int_0^\infty|K(f)|^2\,df$. Under independent arrivals and an unchanged additive background,
$$
\sigma_{v,\rm measured}^2(I)=2q\mathcal K I+\sigma_{v,\rm background}^2,\qquad
q=\frac{1}{2\mathcal K}\frac{d\sigma_{v,\rm measured}^2}{dI}.
$$
The slope of voltage <variance> against mean <electric current> can therefore estimate $q$. A free intercept accounts for a constant background; forcing the line through zero generally biases the slope. The <root mean square> readings must have their mean removed and then be squared. Subtracting two <root mean square> amplitudes does not subtract independent <variances>. Estimate the <variance> from successive records; averaging the waveforms first would suppress the fluctuation being measured.
Additional controls sharpen this interpretation. Block the <light> with detector bias, <temperature>, and gain unchanged; this preserves <dark current> and electronics while removing <photocurrent>. Repeat this dark comparison between illumination settings to check drift. If independent dark arrivals have mean magnitude $I_d$ and illuminated arrivals add $I_{\rm ph}$, their spectra add: $2q(I_d+I_{\rm ph})$. Subtracting the unchanged dark <variance> isolates $2qI_{\rm ph}\mathcal K$. This subtraction model assumes unchanged transfers and <independence>; switching off bias for gain calibration is a different control and need not preserve the dark background.
Vary measured $I_{\rm ph}$ rather than assuming lamp knob position is proportional to <photocurrent>. Check collection versus bias to identify a range where incomplete collection is negligible; a <space charge> limited tube can correlate transfers. Verify that the <transimpedance amplifier> remains linear and that its <frequency response> is unchanged over the illumination range. Changing <signal bandwidth> should change the excess <variance> through $\mathcal K$; compare inferred input spectra in several bands. Correlated fluctuations of the light source can add a contribution scaling approximately as $I_{\rm ph}^2$, while an electronics floor can hide the linear term at small <electric currents>. A fitted line over a short range alone does not establish independent transfers.
The gain calibration must include the same detector-to-readout path used for <noise>, or explicitly supply any omitted transfer. Detector <capacitance>, finite collection time, <amplifier> voltage and current <noise>, <thermal electrical noise>, pickup, clipping, and finite recording precision limit the inference. Current-dependent <electrical impedance> or heating can make a background current-dependent too. Since $\mathcal K$ contains squared gain, a small fractional error in gain produces approximately twice that fractional error in the predicted <variance> scale. Knowing a nominal gain and two cutoff settings is insufficient.
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