Photodiodes, ADCs and Measuring Light Over Three Decades

Reading to 3.0 D means measuring a signal, and then measuring one one-thousandth of it, and calling both readings good. Three decades is not a hard measurement for a silicon photodiode, which is linear over nine of them. It is a very hard measurement for everything you are going to build around one.

This page is the arithmetic that decides the instrument before any of it is bought: what the detector does, what the amplifier does to it, what the converter can and cannot resolve, and which of the four error sources — dark current, noise, quantisation and stray light — actually sets the limit. The answer is not the one most people guess, and by the end of the page you will be able to derive it yourself.

The low-voltage electronics primer is assumed here in full: Ohm’s law, the LED curve, the low-side switch, the datasheet-reading habit. None of it is repeated.

What a photodiode does, and the one thing it must not be asked to do

A photon of sufficient energy absorbed in the depletion region of a silicon junction produces an electron–hole pair, the field sweeps the two carriers apart, and the result is a current. Hamamatsu’s technical note gives the compact statement: the short-circuit current is proportional to the light level, and it stays so over a very wide range because the two terms that would spoil it — the series resistance at a few ohms and the shunt resistance at 10⁷ to 10¹¹ ohms — are negligible almost everywhere.

A photodiode can be operated three ways, and the choice is not a detail.

Open circuit. Let the diode develop a voltage across nothing. The voltage rises logarithmically with the light level, which sounds ideal for measuring density — a logarithm is exactly what a density is — and it is a trap. Hamamatsu say why in one sentence: the open-circuit voltage varies greatly with temperature, which makes it unsuitable for measurement of light level. A logarithmic sensor whose logarithm shifts with the weather is not a sensor.

Short circuit, or photovoltaic mode. Hold the diode at zero volts across its terminals and measure the current that flows. This is the linear, low-noise mode, and it is the one this instrument uses.

Reverse biased, or photoconductive mode. Apply a reverse voltage. This widens the depletion region, speeds the response and raises the upper limit of linearity, and it costs dark current and noise. It is right for a fast photometer and wrong for a densitometer, where nothing moves and the interesting signal is the smallest one.

!The trick that makes short-circuit mode practical

Holding a diode at exactly zero volts while drawing current out of it sounds contradictory. An op-amp does it. With the diode across the inverting input and a feedback resistor from output to that input, the amplifier drives its output to whatever voltage keeps the input at zero — so the diode sees no bias at all, and the whole photocurrent flows through the feedback resistor instead. Hamamatsu put it formally: the load the diode sees becomes the feedback resistance divided by the amplifier’s open-loop gain, several orders of magnitude smaller than any resistor you could fit, so the arrangement enables an ideal measurement of the short-circuit current.

That circuit is the transimpedance amplifier, and everything downstream in this instrument depends on it.

Linearity over nine orders, and where it actually ends

Hamamatsu state the range: for incident light between 10⁻¹² and 10⁻² W, a silicon photodiode’s photocurrent is linear over more than nine orders of magnitude, depending on the device and the circuit around it. The lower limit is set by the noise equivalent power — the point at which the photocurrent is buried in the detector’s own noise — and the upper limit by the load resistance, the reverse voltage and the series resistance.

Nine orders is thirty stops. Three decades of density is three orders. The detector is not remotely the limit, and this is the single most useful thing on the page: every difficulty ahead belongs to the electronics, the geometry or the light leaking in around them, and none of it belongs to the photodiode.

✕A published linearity figure that does not transfer, and what to do about it

The OPT101 datasheet quotes a nonlinearity of ±0.01 per cent of full scale — a superb figure, three orders better than anything this instrument needs. Read the test condition beside it: full-scale output = 24 V. The part will run on anything from 2.7 to 36 V, and this course runs it at 3.3 V, where the output cannot go above about 2.0 V because the datasheet also limits the output high to the supply minus 1.15 to 1.3 V.

A nonlinearity specified at one full scale is not a nonlinearity at another, and this course does not carry the figure across. Nor has it found any source that specifies photodiode-plus-amplifier linearity over the three decades a densitometer works in, at the supply voltage a Pico provides.

So the linearity of your instrument over the range that matters is unknown, and it is yours to measure. The method is the calibration experiment’s, and it is simple in principle: read a wedge whose steps are equal in density and see whether your readings step equally. Any curvature you find is the sum of your amplifier, your converter and your stray light, and separating them is what the rest of this page equips you to do.

Dark current, and the error that is bigger than it

Dark current is what the junction passes with no light on it, and it sets a floor because it is indistinguishable from signal. The datasheet numbers are worth having in front of you together, because they say something surprising.

Device and mode	Dark current	Temperature behaviour
OPT101 photodiode, 7.5 mV across it (effectively short-circuit)	2.5 pA	Doubles every 7 °C
OPT101 op-amp input bias current	165 pA	Doubles every 10 °C
BPW34, reverse-biased at 10 V	2 nA typical, 30 nA maximum	Rises steeply with temperature

Two lessons fall out. First, the price of reverse bias is written in that table: the same silicon passes nearly a thousand times more dark current at 10 V of reverse bias than at zero. Second — and this is the surprise — the OPT101’s dark current is not its problem. Through the internal 1 MΩ feedback resistor, 2.5 pA is 2.5 µV at the output. Even at 39 °C, two doublings later, it is 10 µV.

What is its problem is the output offset. The datasheet gives 5 to 10 mV, typically 7.5 mV, with a temperature coefficient of ±10 µV/°C. Follow that through the instrument you are building. Set the zero — the air reading — at 2.0 V. At 3.0 D the signal is a thousandth of that, 2.0 mV, which is smaller than the offset sitting under it. Subtract the offset once and walk away, and a 10 °C change in room temperature moves it by 100 µV: five per cent of a 2.0 mV signal, which is log₁₀(1.05) = 0.021 D of error at the top of the range and essentially none at the bottom.

That is a bigger number than the whole error budget of a good commercial instrument, and it is why the firmware on the next page takes a dark reading with every measurement rather than once at switch-on, and why the calibration experiment measures drift over half an hour rather than assuming it away. The 361T’s own specification concedes the same thing in its own units: zero stability of ±0.02 D per eight hours, and a two-minute warm-up before any figure applies.

Spectral responsivity: the detector is looking the wrong way

Silicon’s response peaks in the near infrared. OSRAM give the BPW34 a maximum sensitivity at 920 nm and a spectral range of 420 to 1120 nm at the ten per cent points. Texas Instruments give the OPT101 a responsivity of 0.45 A/W at 650 nm — a figure quoted at one wavelength because responsivity is a curve, and one point on it is not a conversion factor for anything else.

Two consequences, and the second is the nastier.

You are working on the low flank. A green LED at about 525 nm sits well down silicon’s response, where it delivers perhaps a third to a half of the current the same optical power would give at the peak. That is not fatal — it costs light, and light is cheap at the clear end — but it means the signal at 3.0 D is smaller than the naive arithmetic suggests.

Anything infrared that leaks in is amplified more than the signal. Room light, an indicator LED inside the case, a nearby remote control, the residual glow of an LED that has been switched off through a resistor rather than a hard low: any of these arrive at a wavelength where the detector is up to three times more sensitive than at your measuring wavelength, and where you cannot see them to find them. Baffling and blackening are therefore not tidiness. They are calibration.

From current to voltage, and what the amplifier costs

The transimpedance amplifier converts a photocurrent into a voltage through one resistor.

V_out = I_photo × R_f

Transimpedance

V_out is the output voltage, I_photo the photocurrent, and R_f the feedback resistor. Every design choice is in that resistor. Make it large and small currents become readable voltages; make it too large and the amplifier saturates at the clear end, the bandwidth collapses, and the resistor’s own thermal noise rises.

The OPT101 settles the question by putting the photodiode, the amplifier and a laser-trimmed 1 MΩ resistor on one die. Its headline figure, 0.45 A/W and 0.45 V/µW at 650 nm, is those two numbers multiplied. The integration buys three things worth having: no stray capacitance between diode and amplifier, a trimmed gain to ±0.5 per cent, and one part instead of five. It costs the freedom to choose R_f, a bandwidth fixed at 14 kHz, and a 2.29 × 2.29 mm active area you cannot change.

The discrete alternative — a BPW34 and an op-amp — gives you the resistor back. The BPW34’s 7.02 mm² sensitive area is a third more than the OPT101’s 5.2 mm², which matters when the aperture is small, and you can pick a feedback resistor to suit your LED and your geometry. What you take on is the layout problem the integrated part removed, and a decision to justify.

The signal chain, with the number that matters at every stage

1. 1LED on constant current — the same driver as the sensitometer; drift here is drift in every reading
2. 2Film + sampling aperture — multiplies the light by 10 to the power minus D — the only stage that carries the measurement
3. 3Stray light — ADDS light that never passed through the film; the one input that is not attenuated, and the one that caps the range
4. 4Photodiode, short-circuit mode — linear over more than nine orders; not the limit
5. 5Transimpedance amplifier — output ceiling at supply minus about 1.2 V; a 7.5 mV offset under every reading, drifting 10 µV per °C
6. 6ADC — one count is worth 10^D ÷ (N₀ ln 10) of density — a hundred times more at 3.0 D than at 1.0 D
7. 7Firmware — averages N readings for a √N gain against random noise, subtracts the dark reading, takes the logarithm

Read it as a budget rather than a circuit. Only stage 2 carries information; stages 4 to 7 can only preserve or lose it; stage 3 destroys it outright, which is why the build page spends three hours on black paint and baffles.

Quantisation: the arithmetic that decides the range

Set the instrument’s zero on air, so that the clear reading is N₀ counts. A patch of density D passes a fraction 10^−D of that light, so it reads

N = N₀ × 10^−D

Counts at density D

and inverting gives the density from the counts:

D = log₁₀(N₀ ÷ N)

Density from counts

N₀ is the count on air, N the count through the sample, and D the density. The important question is not what these say but how much density one count is worth, which is the derivative:

ΔD = 1 ÷ (N ln 10) = 10^D ÷ (N₀ × 2.3026)

Density per count

Read that second form carefully, because it is the whole design. The density resolution gets ten times worse for every unit of density. An instrument that resolves 0.0001 D on clear film resolves 0.1 D at 3.0 — a whole wedge step — with the same converter and the same everything else.

Now put numbers in it. A Raspberry Pi Pico has a 12-bit converter: 4095 counts at full scale.

Converter and range	Counts on air, N₀	Counts at 2.0 D	Counts at 3.0 D	ΔD per count at 3.0 D
Pico, 12-bit, full-scale zero	4 095	41	4.1	0.106
ADS1115, ±2.048 V, 2.0 V zero	32 000	320	32	0.0136
ADS1115, switched to ±0.256 V at the dense end	—	2 560	256	0.0017

The first row is the finding. Four counts. Not four counts of noise — four counts of everything, out of which the density is to be computed, and the difference between 2.9 and 3.1 D is less than one of them. The Pico’s converter cannot read 3.0 D from a full-scale zero, and no amount of averaging changes it, because there is nothing between count 4 and count 5 to average.

🧮It is worse than 12 bits, and the datasheet says so

The RP2040 datasheet publishes the characterisation: at 250 ksps, across fast, slow and typical silicon, the converter’s effective number of bits is 8.6 to 8.8, typically 8.7 — on a board carrying a low-noise external voltage reference, which a Pico does not have. Twelve nominal bits, fewer than nine real ones.

Turn that into counts. An ideal converter’s quantisation noise is one least-significant bit divided by the square root of twelve, so an ENOB of 8.7 in a 12-bit part means an effective step of 2^12−8.7 = 9.9 counts, and an RMS noise of about 2.8 counts. Against the 4.1 counts that 3.0 D leaves you, the noise is nearly as large as the signal.

And part of that error will not average away. The same datasheet describes the integral nonlinearity as a sawtooth rather than the expected curve, with the differential nonlinearity peaking at codes 512, 1536, 2560 and 3584 — the consequence, Raspberry Pi say, of internal capacitors of only tens of femtofarads failing to match. A sawtooth is systematic. Averaging a thousand readings taken at the same code averages the same error a thousand times.

Conclusion, and it is the reason the next page buys a separate converter: the Pico’s ADC is fine for watching an LED for drift, which is what Part XIV used it for, and it is not fit to be the reading element of a densitometer.

The ADS1115 rows fix it in two different ways, and only one of them is interesting.

Sixteen bits alone gets you to 32 counts at 3.0 D — better by eight times, and still coarse. What transforms the picture is the programmable gain: reading the dense end on the ±0.256 V range instead of ±2.048 V multiplies the counts by eight again, to 256, and drops the density resolution to 0.0017 per count. The instrument changes range in the middle of a reading session, exactly as a multimeter does.

That raises the obvious objection — does switching range not put a step in the scale? — and the datasheet answers it. Gain match between any two gain settings is 0.02 per cent typical and 0.1 per cent maximum. A tenth of a per cent of error is log₁₀(1.001) = 0.0004 D, which is forty times smaller than the repeatability a good commercial instrument claims. The step is real and it is negligible, and the way you prove that on your own unit is to read one mid-scale patch on both ranges and take the difference, which is what the calibration experiment does.

The rest of the ADS1115’s specification is quietly excellent for this job: integral nonlinearity of 1 LSB, an offset error of ±3 LSB single-ended, an offset drift of 0.005 LSB per °C, and — from the noise tables — an input-referred noise of exactly one LSB on every range at any data rate up to 128 samples per second, with the effective and noise-free resolution both reported as a full 16 bits. Slow it down and it is as good as its bits.

Noise, in the plain language the instrument needs

Three noises matter here, and only one of them can be argued with.

Johnson noise, also called thermal noise, is the random voltage generated by any resistance at any temperature above absolute zero. In this circuit it is the feedback resistor’s, and Hamamatsu’s note treats it as the noise of a resistor approximating the diode’s shunt resistance. It rises as the square root of the resistance, while the signal rises in proportion to it — so a larger feedback resistor improves the signal-to-noise ratio, up to the point where saturation or bandwidth stops you.

Shot noise is the graininess of the current itself, arising because charge comes in discrete carriers arriving at random times. It scales as the square root of the current, so it matters least where the signal is largest — and it comes from the dark current as well as the photocurrent, which is one more reason not to reverse-bias.

Quantisation noise is the converter’s, treated above.

All three are random, and random errors obey one rule that is worth more than any component choice:

σ_mean = σ_single ÷ √N

Averaging

σ_single is the spread of one reading, N the number averaged, and σ_mean the spread of their mean. Average 100 readings and the random noise falls by ten. Average 10 000 and it falls by a hundred, and you have spent a hundred times as long to gain the same factor again — which is why averaging is a cheap first factor of ten and an expensive second one.

!What averaging cannot do

It cannot remove the converter’s INL sawtooth, because that error is the same on every reading of the same code. It cannot remove the amplifier’s offset drift, because that is a slow trend rather than scatter. It cannot remove stray light, because stray light is not an error at all — it is a real signal arriving by the wrong path. And it cannot remove a lamp that is drifting, which is why the monitor photodiode exists.

Averaging fixes exactly one of the four things wrong with this instrument, and it is the one that was never going to be the limit.

Stray light: the term that caps the whole instrument

Every other error so far has been proportional, or nearly so. Stray light is not. It adds a fixed quantity of light to every reading, whatever the film is doing, because it never went through the film.

Let f be the stray light as a fraction of the clear-film signal, and zero the instrument on air with the stray light present, as you inevitably will. Then

D_measured = −log₁₀ ( (10^−D + f) ÷ (1 + f) )

What a stray-light floor does to a reading

D is the true density, f the stray-light fraction, and D_measured what the instrument reports. Let D run to infinity — put a coin over the aperture — and the reported density does not: it approaches a ceiling.

D_max ≈ −log₁₀ f

The ceiling

A stray-light floor of 0.1 per cent caps the instrument at 3.0 D. One of 1 per cent caps it at 2.0. One of 0.01 per cent caps it at 4.0. That single line does more to set the specification of a home-built densitometer than every component on the parts list, and it is why the optical head is a three-hour build with a whole stage devoted to matt black paint.

What a stray-light floor does to a scale, at three levels

• A perfect instrument
• 1 % stray light — ceiling 2.0 D
• 0.1 % stray light — ceiling 3.0 D
• 0.01 % stray light — ceiling 4.0 D

Computed from the equation above rather than measured; the shape is exact for the model, and how well the model fits your head is the first thing the calibration experiment finds out. Note the signature: a scale that is faithful for two decades and then bends over is stray light, not a bad amplifier. The curve is drawn to show the shape, not measured from a real material. Your own materials will differ, and measuring them is what the sensitometry part of the course is for.

The signature in that caption is worth memorising, because it is how you will diagnose the fault when your own instrument shows it. Stray light compresses only the top of the scale. An amplifier running out of range compresses the bottom. A wrong logarithm bends the whole thing. A drifting zero shifts it without bending it. Four faults, four different-shaped errors, and a wedge of equal steps is the tool that separates them.

For scale, a professional instrument in a metal case specifies its ambient interference as a decrease in D of less than 0.25 per cent — a figure quoted as a percentage of the reading rather than as a fraction of full scale, and one you should expect to lose badly against on your first attempt.

The other path: sensors that do the whole chain for you

Three parts put the photodiode, the amplifier, the converter and the interface on one die, and they are a legitimate route to this instrument.

Part	What it gives	What it costs
TSL2591	A stated 600 000 000 : 1 dynamic range; two channels, one full-spectrum and one infrared; gain steps of nominally 1, 24.5, 400 and 9200; integration 100 to 600 ms; 65 535 counts at 200 ms and longer; dark count typically 20 at maximum gain	Responsivity quoted only for a 4000 K white LED and an 850 nm source, so counts convert to absolute irradiance at no other wavelength; the gain steps are nominal, with the top one specified from 8500 to 9900
VEML7700	16 bits, 0 to about 140 000 lx, resolution to 0.0042 lx per step at gain ×2 and 800 ms; four gain settings including two attenuating ones; dark offset 3 steps	The headline range is a product of two settings at opposite extremes and is never available at once; the internal filtering is the maker’s, not yours
OPT3001	0.01 lux to 83 klux with automatic range selection, a 23-bit effective dynamic range, ranges matched to better than 0.2 per cent, infrared response at 850 nm of only 0.2 per cent, drift 0.01 %/°C	The photopic filter is a spectral product you did not choose and cannot remove, which is convenient for a visual density and wrong for a UV or printing one

What they simplify is real: no op-amp, no feedback resistor, no analogue layout, and in the OPT3001’s case an infrared rejection you would otherwise have to buy a filter for.

What they hide is also real, and it is the same thing in three forms. You cannot see the photocurrent. The gain and the integration time are set by writing to a register, and what comes back is a count whose relationship to the light is a chain of the manufacturer’s decisions rather than yours. When the reading misbehaves you cannot put a meter on the middle of the chain, because there is no middle. And the range switching that gives the headline dynamic range is a gain change like the ADS1115’s, but one whose matching you have no independent way to check.

✦How to choose, honestly

If your goal is a working instrument and you would rather spend the evening on the optical head than on an amplifier, buy the module. A TSL2591 behind a good aperture in a black box will out-measure a carefully built discrete chain in a leaky one, every time, because stray light beats electronics.

If your goal is to understand what a density measurement is made of, build the discrete chain, because every error in this lesson becomes a thing you can put a probe on. The course’s reference build is photodiode plus amplifier plus separate converter for that reason, and it says so rather than pretending the integrated route is inferior.

Do the maths: the table you complete

Fill this in for the parts you actually intend to buy, before you buy them. Two rows are worked; the rest is arithmetic with the equation for ΔD and nothing else. Take N₀ from the counts your zero setting gives and read ΔD at each density from 10^D ÷ (N₀ × 2.3026).

Detector and converter	N₀ on air	ΔD at 1.0	ΔD at 2.0	ΔD at 3.0	ΔD at 3.5	Verdict
Photodiode + Pico 12-bit ADC	4 095	0.0011	0.011	0.106	0.335	Unusable above about 2.0 D
Photodiode + ADS1115 at ±2.048 V, 2.0 V zero	32 000	0.00014	0.0014	0.0136	0.043	Usable to 3.0 D, coarse at the top
The same, switched to ±0.256 V above 2.0 D	—	—	—	0.0017	0.0054	Your row: is the switching worth the firmware?
TSL2591 at your chosen gain and integration
VEML7700 at your chosen gain and integration
Your own chosen combination

Then do the second calculation, which is the one people forget. Compare your ΔD at 3.0 D with your estimated stray-light fraction. If your quantisation gives 0.002 D and your black paint gives you a 0.5 per cent floor, your instrument stops at 2.3 D and the converter was wasted money. The two limits have to be brought to the same place, and the cheaper one to move is almost always the paint.

Recap

A silicon photodiode in short-circuit mode is linear over more than nine orders of magnitude, from 10⁻¹² to 10⁻² W, so the detector is never the limit in a densitometer; the open-circuit voltage is logarithmic and therefore tempting, and unusable because it moves with temperature. A transimpedance amplifier holds the diode at zero volts and converts its current through one resistor, and for the OPT101 that resistor is 1 MΩ giving 0.45 V per microwatt at 650 nm. Dark current at 2.5 pA is trivial through it; the 7.5 mV output offset and its 10 µV per °C drift are not, costing about 0.021 D at the top of the range for a 10 °C change, which is why the dark reading is retaken with every measurement. Silicon peaks at 920 nm, so a green measuring beam works on the low flank and any infrared leak is amplified more than the signal. One converter count is worth 10^D ÷ (N₀ ln 10) of density, so resolution degrades tenfold per unit of density: the Pico’s 12-bit converter leaves four counts at 3.0 D and has an effective resolution of 8.7 bits with a systematic INL sawtooth, while a 16-bit converter with programmable gain leaves 256, with a range-to-range mismatch worth only 0.0004 D. Random noise averages down as the square root of the count and systematic error does not average at all. And stray light, which adds rather than multiplies, caps the instrument at −log₁₀ f whatever else is fitted — which is why the next page is mostly about black paint.

Check your understanding

Question 1. Your zero reading on air is 30 000 counts. You read a step and get 30 counts. Your instrument is otherwise perfect. What density is that, and how much is the next count worth?

• D = 3.0, and one count is worth about 0.0145 D
• D = 3.0, and one count is worth about 0.00003 D
• D = 1000, and one count is worth 0.001 D
• D = 2.0, and one count is worth about 0.0014 D

Show the answer and why

Answer: D = 3.0, and one count is worth about 0.0145 D

The density is log₁₀(30 000 ÷ 30) = log₁₀(1000) = 3.0. The resolution is 1 ÷ (N ln 10) with N being the counts at the reading, not the counts at the zero: 1 ÷ (30 × 2.3026) = 0.0145. Option two makes the classic error of using N₀ instead of N, which gives the resolution at the clear end and flatters the instrument by a factor of a thousand. It is worth noticing what the right answer means in practice: 0.0145 D per count is comparable with a good commercial instrument's whole repeatability specification, so this is the density at which a 16-bit converter on a fixed range stops being comfortable and range switching starts to earn its keep.

Question 2. Two instruments read a wedge of equal steps. Instrument A gives equal density increments up to 2.0 D and then increments that shrink towards zero. Instrument B gives increments that are uniformly 4 per cent too small across the whole wedge. What is wrong with each?

• Both have stray light; A has more of it
• A has a stray-light floor; B has a scale error — a wrong reference or a gain error — which is a calibration problem rather than a range problem
• A has a noisy converter; B has a drifting zero
• A has a drifting zero; B has stray light

Show the answer and why

Answer: A has a stray-light floor; B has a scale error — a wrong reference or a gain error — which is a calibration problem rather than a range problem

The shapes are the diagnosis. Stray light adds a constant to every transmittance, so it does nothing at low densities where the signal is large and swamps the signal at high ones — the error grows without limit and the scale bends over towards a ceiling, which is A exactly. A uniform proportional error in every increment is a slope error: the instrument's density scale is stretched or compressed everywhere alike, which happens when the reference is wrong, when a gain is mis-set, or when the wedge's own increment is not what it is assumed to be. B is therefore fixed by calibration against a known scale, and A is fixed with black paint. Nothing about either shape suggests noise, which would appear as scatter rather than as a trend, or a drifting zero, which would shift the whole curve without changing the increments.

Question 3. You average 64 readings and the spread of your mean falls by a factor of 8, exactly as expected. You then average 4096 readings and the spread falls by only a factor of about 20 instead of 64. What is the most likely explanation?

• The converter has run out of bits
• The averaging code has a bug
• A systematic term has become dominant — the ADC's INL sawtooth, the amplifier's offset drift, or the lamp drifting over the longer measurement
• Shot noise does not obey the square-root law at high sample counts

Show the answer and why

Answer: A systematic term has become dominant — the ADC's INL sawtooth, the amplifier's offset drift, or the lamp drifting over the longer measurement

The square-root law applies only to the random part of the error. Averaging drives that part down until it reaches the level of whatever is not random, and from then on more averaging buys almost nothing — which is exactly the shape described: the law holds at 64 and has stopped holding by 4096. Three candidates account for nearly all real cases. The RP2040 datasheet's INL sawtooth is the same on every reading of the same code and cannot average away. The OPT101's offset drifts at 10 microvolts per degree C, so a long measurement is a measurement during which the offset moved. And a lamp that drifts turns your extra time into a slow trend rather than more samples of one value. The practical rule follows: find the sample count at which the law breaks and stop there, then attack the systematic term instead of the clock.

Question 4. Why does this course use the Pico's own ADC to watch the sensitometer's lamp for drift, but refuse to use it to read a densitometer?

• Because the Pico ADC is too slow for densitometry
• Because drift monitoring is a small fractional change around a large signal, where 12 nominal and 8.7 effective bits are plenty, while densitometry needs the same converter to resolve a signal a thousand times smaller
• Because the densitometer needs a differential input and the Pico has none
• Because MicroPython's read_u16 returns 16 bits and would be misleading

Show the answer and why

Answer: Because drift monitoring is a small fractional change around a large signal, where 12 nominal and 8.7 effective bits are plenty, while densitometry needs the same converter to resolve a signal a thousand times smaller

The two jobs have opposite shapes. A drift monitor sits near full scale and asks whether the reading has moved by a per cent or two — a question about the ratio of two large numbers, where an effective 8.7 bits gives roughly 400 usable levels across the range and one per cent is four of them. A densitometer must resolve the signal at 3.0 D, which is four counts from a full-scale zero and less than the converter's own 2.8-count RMS noise. Option four is worth correcting rather than dismissing: read_u16 genuinely does return a number from 0 to 65535 whatever the real resolution, and the MicroPython documentation says so, which makes it a real trap — but a trap about how the number is scaled, not the reason the converter is unsuitable.

Question 5. You measure your finished head's stray light by putting an opaque metal blank over the aperture and reading it. The reading is 0.4 per cent of your clear-film reading. What is your instrument's maximum density, and what would you have to change to gain half a decade?

• About 2.4 D; buy a converter with more bits
• About 2.4 D; reduce the stray light by about a factor of three, which is a matter of baffles, blackening and shielding rather than electronics
• About 3.0 D; the blank reading is dark current, not stray light
• It cannot be determined without knowing the converter

Show the answer and why

Answer: About 2.4 D; reduce the stray light by about a factor of three, which is a matter of baffles, blackening and shielding rather than electronics

The ceiling is minus the logarithm of the stray-light fraction: −log₁₀(0.004) = 2.40. Gaining half a decade means reaching 2.9, which needs a fraction of 10 to the power −2.9, or 0.00126 — about a third of what you have. No converter helps, because the light is genuinely arriving at the detector and is genuinely being measured; it simply did not pass through the film. Option three names the right check and draws the wrong conclusion: you must indeed separate dark current from stray light, and the way to do it is to take the blank reading twice, once with the lamp on and once with it off. The difference between those two readings is the stray light; the lamp-off reading is the dark term.

Question 6. Someone proposes using the photodiode's open-circuit voltage directly, arguing that it is logarithmic in the light level and density is a logarithm, so the conversion is free. What is the objection?

• The open-circuit voltage is logarithmic only over one decade
• It is logarithmic, but its dependence on temperature is severe enough that the scale shifts with room temperature, which is why Hamamatsu state it is unsuitable for measuring light level
• The open-circuit voltage is proportional to light, not logarithmic
• An op-amp cannot measure an open-circuit voltage

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Answer: It is logarithmic, but its dependence on temperature is severe enough that the scale shifts with room temperature, which is why Hamamatsu state it is unsuitable for measuring light level

The premise is correct and the conclusion is wrong, which is what makes the proposal attractive. The open-circuit voltage really does vary as the logarithm of the photocurrent, so a density would come out of the sensor already in the units you want. The trouble is in the same expression: the coefficient of that logarithm contains kT/q, and the saturation current in it rises exponentially with temperature, so the whole scale moves as the room warms. Hamamatsu's technical note says outright that this makes the open-circuit voltage unsuitable for the measurement of light level. Short-circuit mode gives a linear current, a stable scale and one logarithm computed in firmware where its coefficient is exact — the logarithm is the cheapest part of the whole instrument, and paying for it in analogue is a bad trade.

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What this decides for the build

The optical head that follows is built around three conclusions from this page. Stray light sets the range, so the head gets baffles, an aperture stop on the detector side, matt black everywhere and a blank test that measures the floor rather than assuming it. The detector is not the limit, so the choice between an OPT101 and a BPW34 with an op-amp is about area, layout and cost rather than about performance. And the converter is a limit that can be bought out of, which is why the electronics page fits a 16-bit part with programmable gain and uses the Pico’s own converter for nothing but the monitor channel — the job Part XIV proved it good at.