Signal Recovery with PMTs
Application Note #4
Photon Counting, Lock-In Detection, orBoxcar Averaging?

In general, the choice between boxcar averaging (gatedintegration) and lock-in detection (phase sensitive detection)is based on the time behavior of the signal. If the signal isfixed in frequency and has a 50 % duty cycle, lock-in
detection is best suited. This type of experiment commonlyuses an optical chopper to modulate the signal at some low
frequency. Signal photons occur at random times during the'open' phase of the chopper. The lock-in detects the average
difference between the signal during the 'open' phase and thebackground during the 'closed' phase.
To use a boxcar averager in the same experiment would
require the use of very long (50 % duty cycle) gates, since thephotons can arrive anywhere during the 'open' phase. Since
the gated integrator is collecting noise during this entire gate,
the signal is easily swamped by the noise. To correct for this,active baseline subtraction can be used, where an equal gatemeasures the background during the 'closed' phase of the
chopper and is subtracted from the 'open' signal. This is
identical to lock-in detection. However, lock-in amplifiers aremuch better suited to this: especially at low frequencies (longgates) and low signal intensities.
If the signal is confined to a very short amount of time, gatedintegration is usually the best choice for signal recovery. A
typical experiment might be a pulsed laser excitation where
the signal lasts for only a short time (100 ps to 1 µs), at arepetition rate of 1 Hz to 10 kHz. The duty cycle of the signal
is much less than 50 %. By using a narrow gate to detectsignal only when it is present, noise which occurs at all othertimes is rejected. If a longer gate is used, no more signal ismeasured, but the detected noise will increase. Thus, a 50 %duty cycle gate would not recover the signal well, and lock-in
detection is not suitable.
Photon counting can be used in either the lock-in or the gatedmode. Using a photon counter is usually required at very low
signal intensities, or when the use of a pulse height discriminator
to reject noise results in an increased signal-to-noise ratio (SNR).
As seen in the illustration, the crossover point between analog
detection and photon counting is never very distinct. Photon
counting works well at very low count rates because the input
discriminator virtually eliminates analog front-end noise.
Analog detection works well at large count rates since theanalog inputs do not saturate as easily as a counter. In the
middle ground, the choice should be based on SNR
considerations. At best, the achievable SNR is determined by

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Gated
Integrator
10

Lock-in
Amplifier7

Log Count Rate

Which instrument is best suited for detecting signals from a
photomultiplier tube? The answer is based on many factors
including the signal intensity, the signal's time and frequency
distribution, and the various noise sources and their timedependence and frequency distribution.

Signal Recovery Methods

Photon Counter

.01
10

-9

Log Duty Cycle

1

the statistical noise of the Poisson counting distribution. The
analog instruments degrade the SNR due to input noise.
This applications note begins by discussing photomultipliertubes and how to optimize their performance. The following
sections discuss the SNR of the various signal recovery
methods. Techniques are described which can extend theanalog instruments into the 'photon counting' regime.
However, these techniques have limits beyond which photon
counting is preferred. Experimental data illustrating theachievable signal-to-noise ratios is presented.
Using Photomultiplier Tubes
Photomultiplier tubes (PMTs) are high-gain, low-noise, lightdetectors. They can detect single photons over a spectral rangeof 180 to 900 nm. Windowless PMTs can be used from thenear UV through the X ray region, and may also be used as
particle detectors.
Photons which strike the PMT's photocathode eject anelectron by the photoelectric effect. This electron isaccelerated toward the first dynode by a potential of 100 to
400 VDC. Secondary electrons are ejected when the electron
strikes the first dynode, and these electrons are accelerated
toward the second dynode. Typically, the process continues
for 8 to 14 dynodes, each providing an electron gain of about
4 to 5, producing 106 to 107 electrons which are collected by
the anode. If these electrons arrive in a 5 ns pulse into a 50
ohm load, they will produce a 1.6 mV to 16 mV pulse.
Geometry
There are two basic geometries for photomultiplier tubes:
head-on and side-on types. The head-on type has asemitransparent photocathode and a linear array of dynodes.
The head-on types offer large photocathodes with uniform
sensitivity and lower noise. These PMTs must be operated at
a higher voltage and are usually larger and more expensive
than the side-on types. Side-on types have an opaque
photocathode and a circular cage of dynodes.

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Signal Recovery with PMTs

(Since the rise time will be limited to 1.2 ns by the 300 MHz
bandwidth of the preamplifier, do not use rise times less than
1.5 ns in this formula.)

Spectral Response
There are a variety of materials which are used as
photocathodes: the work function of the photocathode will
determine the spectral response (and will influence the darkcount rate) of the PMT. For photon counting, the figure of
merit is the "quantum efficiency" of the PMT. A 10 % quantum
efficiency indicates that 1 in 10 photons which strike thephotocathode will produce a photoelectron; the rest of theincident photons will not be detected. The quantum efficiency
is a function of wavelength, so select the PMT for the best
quantum efficiency over the wavelength region of interest.

If the PMT anode is connected via a 50 Ω cable to a muchlarger load (R >> 50 Ω), the cable termination looks like anopen circuit. All of the charge in the pulse is deposited on thecable capacitance in 5 ns. The voltage on the load will be
V=Q/C, where C=cable capacitance. This voltage will decay
exponentially with a time constant of RC, where R=loadresistance. In this case, the pulse height will be:
Amplitude (mV)= 160 × Gain (in millions)/ Cable C (pF)

Gain and Rise Time
The current gain of a PMT is a strong function of the highvoltage applied to the PMT. Very often, PMTs will be operated
well above the high voltage recommended by the
manufacturer, and thus substantially higher current gains (10×to 100× above specs). There are usually no detrimental affects
to the PMT as long as the anode current is kept well below the
rated value.

When using gated detection, it is important to select a PMTwith sufficient gain and short rise time. Large gains are
essential to both gated integrators and photon counters. For
gated integrators, the pulse rise time and width should be on
the order of the gate width (or less) so that timing information
is not lost. For photon counting, the pulse width should besmaller than the pulse-pair resolution of thecounter/discriminator to avoid saturation effects. When using
lock-in amplifiers, pulse rise time is unimportant, while highgain extends the sensitivity of the measurement.

Dark Counts
PMTs are the quietest detectors available. The primary noise
source is thermionic emission of electrons from thephotocathode, and from the first few dynodes of the electron
multiplier. PMT housings, which cool the PMT to about −20°
C, can dramatically reduce the dark counts (from a few kHz toa few Hz). The residual counts arise from radioactive decaysof materials inside the PMT, and from cosmic rays.

The criteria for a "detectable pulse" depends on the electricalnoise environment of your laboratory and the noise of yourpreamplifier. In laboratories with Q-switched lasers or pulsed
discharges, it is difficult to reduce the noise on any coaxial
cable below a few millivolts. A good, wide-bandwidthpreamplifier (such as the SR445) will have about 1.5 nV/√Hz,or about 25 µVrms over its 300 MHz bandwidth. Peak noise
will be about 2.5 times the rms noise, so it is important that the
PMT provide pulses of at least 100 µV amplitude.

PMTs which are specifically designed for photon counting
will specify their noise in terms of the rate of output pulses
whose amplitudes exceed some fraction of a pulse from asingle photon. More often, the noise is specified as an anodedark current. Assuming the primary source of dark current isthermionic emission from the photocathode, the dark countrate is given by:

Use manufacturer's specifications for the current gain and risetime to estimate the pulse amplitude from the PMT using thefollowing equation:

Dark Count (kHz)= 6 × Dark Current (nA) / Gain(millions)

Amplitude (mV) = 4 × Gain (in millions)/ Rise Time (ns)This formula assumes that the electrons will enter a 50 ohm
load in a square pulse whose duration is twice the rise time.

Photocathodephotone- HighVoltageR

8 Dynodes

Anode
Output_R

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R

R

R

R

R

R

R

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Signal Recovery with PMTs

Current for the electron multiplier is provided by the bias
network. Current drawn from the bias network will cause thedynode potentials to change, thus changing the tube gain. This
problem is of special concern in lifetime measurements. The
shape of exponential decay curves will be changed if the tube
gain varies with count rate. To be certain that this is not a
problem, repeat the measurement at half the original intensity.

PMT Base Design
PMT bases, which are designed for general purpose
applications, are not appropriate for photon counting or fast
gated integrator applications (gates <10 ns). General purpose
bases will not allow high count rates, and often cause
problems such as double counting and poor plateau
characteristics. A PMT base with the proper high voltage
taper, bypassing, snubbing and shielding is required for goodtime resolution and best photon counting performance.

The problem of gain variation with count rate is avoided if thecurrent in the bias network is about 20 times the output current
from the PMT's anode.

CAUTION! Lethal high voltages are used in PMT
applications. Use extreme caution when working with thesedevices. Only those experienced with high voltage circuits
should attempt any of these procedures. Never work alone.

Example: If a PMT is operated so that it gives 20 mV pulsesof 5 ns duration into a 50 ohm load, then the average current
at 50 MHz count rate will be 0.1 mA. If the bias resistors arechosen such that the chain current is 20 × 0.1 mA = 2 mA, thePMT's gain will remain constant vs. count rate. If this PMT isoperated at 2500 VDC, the power dissipated in this base is 5 W.

Dynode Biasing
A PMT base provides bias voltages to the PMT's photocathodeand dynodes from a single, negative, high-voltage power
supply. The simplest design consists of a resistive voltage
divider (see figure on previous page).

There are a few other methods to avoid this problem which do
not require high bias-currents. These methods depend on thefact that the majority of the output current is drawn from thelast few dynodes of the multiplier. The methods are as follows:

In this configuration, the voltage between each dynode, andthus the current gain at each dynode, is the same. Typicalcurrent gains are three to five, so there will typically be fourelectrons leaving the first dynode, with a variance of about
two electrons. This large relative variance (due to the small
number of ejected electrons) gives rise to large variations in
the pulse height of the detected signal. Since statisticalfluctuations in pulse height are caused by the low gain of thefirst few stages of the multiplier chain, increasing the gain of
these stages will reduce pulse height variations and improve
the pulse height distribution. This is important for both photon
counting and analog detection. To increase the gain of the firstfew stages, the resistor values in the bias chain are tapered up to
increase the voltage in the front end of the multiplier chain. The
resistor values are tapered slowly so that the electrostatic focusingof electrons in the multiplier chain is not adversely affected.

Photocathode(1) Replace the last few resistors in the bias chain with Zener
diodes. As long as there is some reverse current through a
Zener, the voltage across the diodes is nearly constant. This
will prevent the voltage on these stages from dropping as theoutput current is increased.
(2) Use external power supplies for the last few dynodes in themultiplier chain. This approach dissipates the least amount ofelectrical power since the majority of the output current comesfrom lower voltage power supplies. However, it is the mostdifficult to implement.
(3) If the average count rate is low, but the peak count rate ishigh, then bypass capacitors on the last few stages may be
used to prevent the dynode voltage from dropping (use 20×

8 Dynodes

Anode

100

High
Voltage

2R

1.5R

R

R

R

Vz

100

Vz

100

Vz

C/8
C/4
C/2
Note: Zener diodes may be replaced with resistors in low averagecount rate applications

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Output100

Vz

Snubber
10 Inches
RG174

C
50 Ω

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Signal Recovery with PMTs

the average output current for the chain current).
For a voltage drop of less than 1 %, the stored charge on thelast bypass capacitor should be 100× the charge output during
the peak count rate. For example, the charge output during
a 1 ms burst of a 100 MHz count rate, each with an amplitudeof 10 mV into 50 ohms and a pulse width of 5 ns, is 0.1 µC. If
the voltage on the last dynode is 200 VDC, then the bypass
capacitor for the last dynode should have a value given by:
C = 100 Q/V = 100 × 0.1 µC / 200 V = 0.05 µF
The current from higher dynodes is smaller, so the capacitors
bypassing these stages may be smaller. Only the final four orfive dynodes need to be bypassedusually with a capacitor
which has half the capacitance of the following stage. To
reduce the voltage requirement for these capacitors, they areusually connected in series. (See diagram on previous page.)
Bypassing the dynodes of a PMT may cause high frequency
ringing of the anode output signal. This can cause multiple
counts for a single photon, or poor time resolution in a gatedintegrator. The problem is significantly reduced by using
small resistors between the dynodes and the bypass capacitors,
as shown in the diagram.
Snubbing
Snubbing refers to the practice of adding a network to theanode of the PMT to improve the shape of the output pulse forphoton counting or fast gated integrator applications. This
'network' is usually a short piece of 50 ohm coax cable whichis terminated into a resistor of less than 50 ohms. Snubbing
should not be used when using a lock-in amplifier since thecurrent conversion gain of a 50 ohm resistor is very small.
There are four important reasons for using a snubber network:(1) Without some DC resistive path between the anode andground, anode dark current will charge the signal cable to afew hundred volts (last dynode potential). When the signal
cable is connected to an amplifier, the stored charge on thecable may damage the front-end of the instrument. If you
decide not to use a snubber network, install a 10 MΩ to 100 MΩ
resistor between the anode and ground to protect yourinstruments.
(2) The rise time of the output current pulse is often much
faster than the fall time. A snubber network may be used to
sharply reduce the fall time, greatly improving the pulse-pairresolution of the PMT.
(3) Ringing (with a few nanosecond period) is very common
on PMT outputs (especially if the final dynode stages are
bypassed with capacitors). A snubber network may be used tocancel these rings which can cause multiple counts from asingle photon.
(4) The snubber network will help to terminate reflections
from the input to the preamplifier.

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PMT Pulse
Without
Snubber

2 ns

PMT PulseWith
Snubber

A Slow Falltime Pulse ShapeIs Improved With A Snubber

A good starting point for a snubber network is a 10 inch pieceof RG174/U coax cable with a small 50 ohm pot connected tothe end so that the terminating impedance may be adjusted
from 0 to 50 ohms. (A 10 inch cable will have a round trip
time of about 5 nsbe sure your PMT has a rise time less than
this.) The other end of this cable is connected to the anode of
the PMT, together with the output signal cable.
Output current pulses will split: 50 % going out the signal
cable, and 50 % going into the snubber. If the snubber pot isadjusted to 50 ohms, there will be no reflection; the only effectthe snubber has is to attenuate the signal by a factor of two.
The reflection coefficient for a cable with a characteristic
impedance R0, terminated into a resistance Rt, is given by:
Reflection Coefficient = (Rt − R0) / (Rt + R0)
If the pot is adjusted to a value below 50 ohms, then some
portion of the signal will be inverted and reflected backtoward the anode. This reflected (and inverted) signal isdelayed by the round trip time in the snubber cable and sentout the signal cable. The amount of the reflection is adjusted
for the best pulse shape as shown.
The round trip time in the snubber cable may be adjusted sothat the reflected signal cancels anode signal ringing. This isdone by using a cable length with a round trip time equal tothe period of the anode ringing.

Ringing
Without
Snubber

2 ns

Snubber
Cancels
Ringing

Anode Ringing Canceled with
Snubber Cable

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Signal Recovery with PMTs

Cathode Shielding
Head-on PMTs have a semitransparent photocathode which isoperated at negative high voltage. Use care so that no objects
near ground potential contact the PMT near the photocathode.
Magnetic Shielding
Electron trajectories inside the PMT will be affected bymagnetic fields. A field strength of a few Gauss candramatically reduce the gain of a PMT. A magnetic shield
made of a high permeability material should be used to shield
the PMT.
PMT Base Summary(1) Taper voltage divider for higher gain in first stages.
(2) Bypass last few dynodes in pulsed applications.
(3) Use a snubber circuit to shape the output pulse for photon
counting or fast gated integration.
Gated Photon Counting
Gated photon counting measures the intensity of a signal bycounting the number of photons which are detected by the
PMT in a given time gate. This is sometimes referred to as a'boxcar' mode. In concept, gated photon counting is identicalto gated integration except that only PMT pulses which
exceed a certain discriminator threshold level are counted.
Due to the statistical nature of the secondary emission process,there is a distribution of signal pulse heights coming from the
PMT. There is another distribution of noise pulse heights.
Noise which results from thermionic emission from thephotocathode can not be distinguished from signal. However,noise pulses from dynode thermionic emission will have alower mean pulse-height. The PMT should be operated at
sufficient high voltage that the mean signal pulse-height iswell above the pulse height of other noise sources such aspreamp noise and EMI pickup.
There are two reasons for carefully selecting the input
discriminator level:

bandwidth. The bandwidth of a 10 pF anode with a 100 ohm
load is 300 MHz.
For convenience, 50 ohm systems are usually used. The
current pulse from the PMT travels down a 50 ohm cablewhich is terminated by the 50 ohm input impedance of apreamplifier. The attenuation of RG-58 coax cable at 300 MHz
is about 1 dB/10 foot, and does not significantly degrade
performance in this application.
To allow counting to 200 MHz, a preamplifier with abandwidth which is somewhat larger than 200 MHz is
required. The SR445 preamplifier has four 5× gain amplifiers,
each with 50 ohm input impedance and 300 MHz bandwidth.
The amplifiers may be cascaded for gains of 5, 25 or 125.
The SR400 Photon Counter can detect pulses as low as 2 mV.
To allow for some adjustment of the discriminator threshold
and to provide better noise immunity, a more practical lower
limit on pulse size is about 10 mV. The highest discriminator
level which may be set is 300 mV. The preamplifier shouldhave enough gain to amplify anode pulses to between 10 mV
and 300 mV (100 mV is a good target value).
Using the result that pulse height (in mV) is about four times
the tube gain (in millions) divided by the rise time (in ns), a
PMT with a gain of 4 million and a rise time of 2 ns will
provide 8 mV output pulses. Half of the pulse amplitude will
be lost in the anode snubber, so a gain of 25 is required toboost the output pulses to 100 mV.
Setting the Discriminator Level
There is no exact prescription for setting the discriminator
threshold; the procedure used will depend somewhat on thenature of the measurement. If dark counts are a problem thenthe discriminator level should be set higher than when drift isa concern. If the PMT is cooled (reducing thermionicemission), then a lower discriminator level is probably okay.
If the PMT has a ring on the anode signal, the discriminator
level should be set high enough so that the rings are notcounted.
The 'Correct' Way

1) To improve the signal-to-noise ratio by setting thediscriminator level above most of the noise pulses but below
most of the signal pulses.
2) To reduce drift. If the discriminator threshold is set to the
top of the signal pulse height distribution, then small changes
in the tube gain can cause a large change in the count rate.
Gain Requirement
The output of a PMT is a current pulse. This current isconverted to a voltage by a load resistor. One would like to use
a large resistor to get a large voltage pulse. However, inphoton counting it is important to maintain a high bandwidth
for the output signal. Since charge on the anode is removed by
the load resistance, smaller load resistances increase theStanford Research Systems

The tube should be operated at the maximum high voltage
recommended by the manufacturer. Use enough preamplifier
gain so that the single photon pulse-height is about 100 mV.
Provide enough light to the PMT for a count rate of a few
megahertz. Using a 300 MHz oscilloscope, adjust the snubber
termination for minimum ringing on the anode signal. Take
the pulse-height spectrum of the anode signal by scanning thediscriminator level and plotting count vs. discriminator level.
If the PMT dark count rate is a concern, you will also need totake the pulse-height spectrum of the dark count signal. It willtake much longer to take the dark count spectrum because thecount rate should be much lower. The object is to find adiscriminator level which is higher than the mean noise-pulse
height, and below the mean signal-pulse height.

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Signal Recovery with PMTs

The 'Fast and Pretty Good' Way
This technique works very well and is particularly suited for
those who do not want to make a career out of plateauing their
PMTs. The PMT should be operated at (or a bit above) therecommended maximum high voltage. Provide enoughillumination for a count rate of a few megahertz, and enoughpreamp gain to get pulse heights of about 100 mV. Using a
300 MHz oscilloscope, adjust the snubber termination
impedance for the best pulse shape. Look carefully at theanode pulse shape, and set the discriminator to a level which
is above any ringing but well below the mean pulse height. If
there is lots of EMI or amplifier noise, increase the PMT's
high voltage to increase the signal pulse height.

The SNR is just SIGNAL/(TOTAL) or:
KsT
SNR =
Define:Ns = KsT = signal photons in gate T
Nb = KbT = background photons in gate T
n = (Ks + Kb)t = total photons in time t(pulse-pair resolution)
Using these definitions, the SNR is given by:√Ns

Signal-to-NoiseSNR =The probability that n photons will be detected in a time t is
described by the Poisson distribution:

where K is the average photon rate.
The standard deviation of any measurement is √N, where N is
the number of photons detected. If many measurements ofgate width T are made, the standard deviation of the data
points will be √Ns, where Ns is the average number of photons
detected in a single measurement. The signal-to-noise ratio(SNR) is simply Ns/√Ns=√Ns. If each data point instead
consists of the sum of M measurements of gate width T, thenthe standard deviation will be √Nt, where Nt=MNs. The SNRof this measurement is √Nt=√M×√Ns and is √M better than the
single gate measurement.
Because photon counters have a maximum count rate orpulse-pair resolution limit, they can be saturated. The
probability that a photon will be counted is equal to theprobability that no photon arrived in the previous time t,where t is the pulse-pair resolution. If K is the total photon
rate, then each photon is detected with probability e−Kt. As Kt
increases, eventually the number of detected photons will
decrease as the PMT pulses are no longer distinct single
photon pulses.
Assuming that the signal photon rate is Ks and the backgroundor noise rate is Kb, then the detected signal is:
SIGNAL COUNT = KsTe−(Ks + Kb)t

[1 + Nb/Ns ]½ [1 + (Ns + Nb)n]½

The SNR is plotted below for small Nb. If Nb and n are both
small, then SNR = √Ns as discussed before. If n, the numberof photons expected to occur in one pulse-pair resolution time,is large, then SNR ≅ 1/√n and decreases with increasing count
rate.

10

Gated Photon Counting
Single GateNs
10 µs gate
1 µs gate

1
SNR

P(n,t) = (Kt)n e−Kt / n!

[(Ks + Kb)T]½ [1 + (Ks + Kb)2Tt]½

100 ns gate
10 ns gate

.1
Kb = 1000; t = 5 ns
.01
.01

.1
1
10
100
Number of Signal Photons/Gate

1000

If Nb >> Ns, then SNR ≅ Ns/√Nb. This simply means that theoutput noise is due to the statistics of the background counts
and that the signal must exceed the deviations in the
background. For example, if Nb/Ns = 10, then the SNR = 1
when the total number of signal photons counted is 10 and thetotal background count is 100. In this case, the counts frommany gates may need to be summed together to achieve this.

where T=gate width and t=pulse-pair resolution.
The total output count is:
TOTAL COUNT = (Ks + Kb)Te−(Ks + Kb)t
The deviation of the total count is:(TOTAL) = [(Ks + Kb)T]½ × [1 + (Ks + Kb)2Tt]½ × e−(Ks + Kb)t

Stanford Research Systems

Experimental data (next page) demonstrates how the SNR
grows as √N, where N is the total number of counts per data
point. An R928 photomultiplier tube with a tapered base andsnubber was used with a pulsed light source. The averagepulse height from the PMT was about 15 mV into a 50 Ω
termination. An SR400 Gated Photon Counter was used tocollect the data. The discriminator level was set to 7 mV. The
intensity of the signal was adjusted to provide an average of
one signal photon per 100 ns gate. The counts from a numberof gates, Tset, are summed together in each data point. Data isphone: (408)744-9040

Signal Recovery with PMTs

plotted for Tset = 100, 200 and 400 gates. In between each dataset, the signal was turned off and only background datacollected. For each Tset, y=average counts/data point;
dy=standard deviation; SNR=y/dy.

The total output of the gated integrator is:
OUTPUT = (1/T)(Ks + Kb)TAeR + Vn (volts)
where Kb is the background photon rate and Vn is the
input voltage noise of the gated integrator.
Pulses from a PMT have an amplitude variation due to thestatistics of the charge multiplication process. In addition, thenumber of photons detected varies as √N. Both effects aredescribed by the Polya distribution:1 ξ 2∆(K s TA = ∆(NA) = A N  (ξ − 1) 
where ξ is the dynode gain per stage.
S/N ratio for different total counts
Note that the background rate is very small and does not resultin any counts when the signal is off.
A tradeoff must be made between SNR and the length of timeeach data point takes to accumulate. The SNR for 400 gates istwice that of 100 gates, but the 400 gate points take four times
as long to acquire (SNR grows as √M where M = # of gates).
Clearly the data could be smoothed and averaged off line toimprove the SNR. However, in a scanning experiment, asignal feature may only be one or two data points. Each point
will be within one or two standard deviations of the average ofmany points, and thus, the signal may be characterized ashaving error bars of ±N.
Boxcar Averaging
Boxcar averaging, or gated integration, is an analog
measurement where the signal is averaged over a short timegate, and the results of many gates are averaged together.
Since no discriminator is used, the variation in signal pulse
height results in output deviations. Therefore, unlike thephoton counter, the output noise will be greater than the √N,due to counting statistics because of the pulse-to-pulse heightfluctuations.
The signal output of a gated integrator for a signal photon rateof Ks is:SIGNAL = (1/T) KsTAeR (volts)
where T is the gate width, A is the PMT gain, e is theelectron charge, and R is the termination resistanceof PMT.
This assumes that the PMT output pulses are shorter induration than T. (KsT) is the number of photons which arrivein the gate, and AeR/T is the voltage of a single pulse averaged
over the gate.

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As ξ becomes large, ∆(NA) approaches A√N which indicatesthat the PMT should be operated at the highest voltage (gain)
possible. The pulse-height variation results in output noise
above the √N counting noise. For example, if A=107 with
14 stages, ξ = 3.16 and ∆(NA) = 1.2 A√N.
Using the above example,
1

 1.5
2
∆(Output) = ( 2 )(K s + K b )TA 2R2e2 + ∆vn2  T
and the SNR is:

SNR =0.8 Ns
1

 Nb
2
vn2 T
1 + N + 1.5N A 2R2e2 sswhere Ns=KsT=number of signal photons which arrivein the gate, Nb=KbT=number of background photons
during the gate and Vn=input noise density (volts/√Hz).
Note that the observed input voltage noise will decrease as
1/√T as the gate width is increased.
Assume a short gate such that Nb is negligible. When thestatistical deviation in the number of photons observed (√Ns,
which yields a noise voltage equal to √Ns AeR/T) exceeds theaveraged input noise over the gate (Vn/√T), we are in the'photon counting' regime. In this case, SNR ≅ 0.8√Ns. The
0.8 factor is due to the amplitude variation of the pulses. (Ifthe tube gain is 106, the factor is 0.56) However, unlike photon
counting, there are no saturation effects at high count rates. In
addition, since there is no discriminator, smaller photon pulses
contribute to the output increasing Ns and offsetting some ofthe pulse-height variations.
When the input noise dominates, Vn/√T >> √Ns(AeR/T) and
SNR < 0.8√Ns. Note that for a constant number of photons,
making the gate longer decreases the SNR. This is because thephone: (408)744-9040

Signal Recovery with PMTs

noise voltage will decrease as 1/√T (Gaussian) while thesignal is decreasing by 1/T (linear averaging).

10Gated Integrator
Single Gate

Ns
.56 NsSNR

1.1 10 ns

.01
.01

6

100 ns
1 µs gate

A = 10Nb = 0R = 50 Ω

.1
1
10
100
Number of Signal Photons/Gate

1000

100 gate average is exponential, the number of gates required
for the signal to go from zero to full scale is 300 to 500. In thephoton counting case, the signal reaches full scale after
200 gates. Because of this, in a scanning experiment, the photon
counter could scan twice as fast and achieve twice the SNR.
Since the output noise is independent of whether there is any
signal, the boxcar SNR is dominated by the input voltage
noise. The SNR can be improved by increasing the size of the
PMT pulses. This can be done with an SR240 preamplifier
which has a lower input noise than the boxcar averager. Whenthe gate is long, simply increasing the termination resistancecan go a long way to increasing the SNR. Choose a resistancethat does not widen the PMT pulses beyond about half of thegate width. Otherwise, timing information will be lost. In thisexperiment, a 200 Ω resistance increased the pulse amplitude
by four and brought the SNR into the 'photon counting' regime
as shown below.

The SNR is plotted above for a single gate. In all cases,
averaging over M gates increases SNR by √M. This can beseen be replacing Ns with MNs and T with MT.
For large signals, it is clear that while the photon counter will
saturate, the boxcar averager will work just fine. The fact thatthe achievable SNR is less than √N is not important since theonly way to improve the SNR is to attenuate the signal to afew photons per gate and count photons for many gates. The
inconvenience of the longer measurement times usually far
offsets the small gain in SNR which would result from counting.
For small signals (from one to much fewer than one photon
per gate) and gates greater than 10 ns, photon counting is
usually better.
Experimental data acquired with an SR250 Boxcar Averager
is shown below. The same signal and PMT which was used in
the photon counting example were used here. The sensitivity
was set to 1 V / 5 mV, and the average photon pulse had an
amplitude of 15 mV. An average of 1 photon per 100 ns gateis collected and the averaging is over 100 gates.

S/N ratio for 200 Ω terminating impedance
The baseline noise is now negligible compared to the signal,and the 100-gate data now has a SNR of 25. Increasing the
averaging increases the SNR by √samples. As discussed
before, averaging over 100 samples should be compared tophoton counting for about 30 samples, since the signal rises tofull scale in fewer data points for photon counting.
Synchronous Photon Counting

S/N ratio for boxcar integration

If a signal is fixed in frequency and has a 50 % duty cycle,
then synchronous photon counting, or photon counting in a'lock-in' mode, can be used. These signals usually result fromthe use of an optical chopper or other periodic excitation. The
photon counter uses two counting channels and one PMT.
Both counting channels use the one PMT as their signal
source. The A counter counts pulses during the 'open' phase of
the chopper and thus counts signal plus background. The B
counter counts pulses only during the 'closed' cycle of the
chopper and only counts the background. The difference
between the two counts, A−B, is the signal. Accumulating dataover many cycles is required to measure the signal, since thebackground rate usually far exceeds the signal rate.

The SNR is only 5:1, while in the 100-gate photon countingcase it was 9.5:1. In fact, the boxcar is even worse. Since theStanford Research Systems

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Signal Recovery with PMTs

Assume that the total photon rate is small (K << 100 MHz)such that saturation can be ignored. Then the signal count is:
SIGNAL COUNT= KsT1
where T1=the open cycle=½ period of the chopper frequency.
The A−B count is:
A−B COUNT = (Ks + Kb)T1 − KbT2
where T2 = the closed cycle » T1.
The noise in the A−B count is:∆(A−B) = √(Ns + 2Nb)
where Ns=KsT1=the average number of signal photons
counted during T1, and Nb=KbT1=KbT2=the average numberof background photons counted during T1 or T2. Even though
KbT1=KbT2, the noise in the output due to Kb is not zero, sincein a given measurement the number of background counts
detected during T1 differs from the number detected during T2
by √2√Nb (√Nb is the uncertainty in Nb during each cycle.)
The cancellation of the background improves as the number ofbackground counts detected increases and is the dominantfactor in the SNR.

N s = average number of signal photons per cycle, the larger
the background, the longer the data acquisition will take.
Experimental data is presented in the following figures whichshow that the SNR is determined by the background rate when
Nb>Ns. The PMT and signal source from the gatedexperiments were used. The signal was gated on for 10 ms at
a repetition rate of 50 Hz (50 % duty cycle). The signal
amplitude was adjusted to provide an average of one signal
photon per cycle (50 photons/s). The background rate of darkcounts was about 100 counts/s. The SR400 Photon Counter isconfigured for counting A−B, where counter A is gated on
during the signal phase, and counter B is gated on during thebackground phase. Both counters were gated on for 9 ms.
The plots below show data accumulated for 50, 200 and
400 cycles. Note that the noise is basically independent of
whether the signal is on or off. This is because 2Nb>Ns. The
SNR improves as √M, where M=number of cycles as thedeviations in the background count become less than thesignal itself.

The SNR is:
Ns
SNR =

(Ns + 2Nb)½

If Nb << Ns, then SNR = √Ns as expected.
If Nb >> Ns , then SNR is:Ns
SNR =(2Nb)½
In this case, the noise is the uncertainty in the measurement of
the background rate. For example, if Nb/Ns=10, the SNR=1 only
after Ns=20 and Nb=200. This can be achieved by adding theresults of many cycles together or increasing the count rate.
A plot of SNR vs. total number of signal photons counted
for several values of N b/N s is shown below. For a given
10

Photon Counting in
Lock-In Mode
(M cycles)

Nb/Ns = 0Ns

Nb/Ns = 10

1
SNR

Nb/Ns = 100

.1

.01
.01

M = Ns.11
10100
Number of Signal Photons

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1000

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Signal Recovery with PMTs

The DC output of the lock-in over one cycle is:Lock-In Detection
Can a lock-in amplifier detect a single photon per reference
cycle? In the previous photon counting experiment, the signal
was about one photon per cycle at 50 cycles/second. This is a
PMT current of 50Ae where A is the PMT gain. For A=107,
the current is 80 pA (within the capability of most lock-ins).
A lock-in amplifier detects current by using a large resistor, or
current amplifier, to convert the signal current into a voltage.
When detecting single photon signals from a PMT, a wellchosen termination resistor can provide signal-to-noise ratios
governed entirely by the counting statistics of the photons.
Typically, a PMT output consists of a coaxial cable terminatedby a 50 Ω resistor. The output voltage appears across theresistor. Since the output cable is terminated in its
characteristic impedance, the output voltage pulse will be
Vo≈AeR/∆t, where ∆t is the pulse width of the PMT. For
A=107 and ∆t=5 ns, Vo≈16 mV. Now assume that one photon
is detected per cycle at exactly the same time during eachcycle. In the time domain, the signal is a periodic series ofdelta functions spaced by the reference period T, where T ismuch greater than ∆t. In the frequency domain, the signal
spectrum is a series of delta functions spaced by 1/T and
extends from DC to 1/∆t. In the case where ∆t=5 ns, thespectrum extends to 200 MHz. A lock-in amplifier which islocked to f=1/T is not suited to detecting this signal, notbecause of the amplitude, but because of the frequency spectrum.
Now suppose the output of the PMT is terminated by a highresistance (R >> 50 Ω). Because the cable is terminated in ahigh impedance, the cable can be modeled solely by its
capacitance C. The charge from the PMT pulse is deposited on
the capacitance in time t. The voltage on the cable will be
Vo=Ae/C. The charge bleeds away through R over many time
constants (τ=RC). Thus, a photon arriving at time t=0 results
in an output voltage waveform V(t)=Voe−t/τ. Note that theamplitude of the pulse does not vary with R. The large R
serves to lengthen the pulse width which changes thefrequency spectrum of the pulse. The frequency spectrum now
has components to much greater than 1/RC. If C=100 pF and
R=107 Ω, the pulse amplitude is 1.6 mV. The frequency spectrum
extends from DC to >> 1 kHz and has a larger component at
50 Hz for the lock-in to detect.
The signal output of the lock-in can be estimated byconsidering a square wave multiplier and a periodic photon
train at the reference frequency as shown below.
Reference+100
Signal
V0

T/2

T

V(t)
t0

Stanford Research Systems-1

S(t 0 ) =Vo T - t / τe dt (volts)
T ∫0Since real photons arrive at a random t0 between 0 and T/2,
the response for a random photon is:
T

S=

2 2
S(t 0 )dt 0 (volts)
T ∫0

S=

T
T
τ
4τ 2
(1− e − τ ) − 2 (1− e − 2 τ )2 (volts)
T
T

S is the response for an average of one photon arriving at a
random time during each reference cycle. If τ >> T, thenS >> 0 because the RC time constant of the PMT output
attenuates signals at the reference frequency. If τ << 0, thenS >> 0 because the signal extends to frequencies far greaterthan the reference frequency.
S maximizes for τ = T/6, at which point S=0.065Vo. The
factor 0.065 is due to the fact that the signal has manyfrequency components other than 1/T, as well as a randomly
shifting phase. Thus, the signal output of the lock-in is
SIGNAL = 0.065NsAe/C(volts)where Ns is the average number of signal photons per
cycle and C is the cable capacitance.
For T=20 ms (50 Hz), C=100 pF, R=30 MΩ, Ns=1, and
A=107, the signal will be 1 mV.
The shunt resistor method is simple and easy to implement;
however, phase information is lost. In many experiments
phase is not important. When phase measurements must be
made, a current preamplifier is used. The current
preamplifier eliminates the cable capacitance, and the
bandwidth of the amplifier is determined by the capacitanceof the current gain resistor. Since this capacitance is muchsmaller, the time constant of the amplifier output pulse ismuch shorter than the case discussed above. Assuming thatthe reference period is much longer than this time constant,
T >> τ, then the above formula applies and S≈Voτ/T and
Vo=Ae/C, where C is the capacitance of the current
conversion resistor. Since τ = RC, where R is the current
gain, then S=AeR/T which is just the average current times
the current gain resistor. The output signal will be:
SIGNAL = NsAeR/T(volts)For the conditions stated above, the signal will be 2.4 mV.
A disadvantage of this approach is that the output of thecurrent preamplifier is a pulse of much greater amplitude
and shorter duration than the simple shunt resistor. Thisrequires using higher dynamic reserve, so high-frequency
components of the pulse do not overload the amplifier.

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Signal Recovery with PMTs

The output of the lock-in when there are background photons
is:
OUTPUT=(0.065Ae/C)(Ns + N1 − N2) + Vn (volts)
where N1 is the number of background photons detectedduring the open cycle, and N2 is the number detectedduring the closed cycle. Vn is the noise voltage of thecurrent conversion resistor.

Ns

SNR =

1

2

vn2
+
+
N
N
b
2
 s
∆T(0.065Ae / C) The signal to noise ratio is:where Ns and Nb are the number of signal and backgroundphotons which occur during a lock-in output time constant∆T, and Vn=0.13√R nV/√Hz is the Johnson noise density of
the conversion resistor.
If Ns or Nb is large, the SNR is identical to the photon countingcase described in the previous section, where data is
accumulated for M cycles and ∆T >>M reference cycles. If Vn
dominates, the SNR is worse than pure counting statistics.
However, the Johnson noise of large resistors is very small
and does not limit many measurements. For example, a
30 MΩ resistor has a noise voltage of 2 µV (for ∆T=1 s),
while the signal due to 50 photons/s is 1 mV. In fact, in thisexample, as long as one background photon is detected per
second, the SNR will be dominated by counting statistics. In
all cases, the SNR increases as √∆T, where ∆T is lock-in time
constant. This is because more photons are detected, and thestatistical counting noise is reduced.
Experimental data is shown below. The signal source and
PMT were the same as in the previous photon counting
discussion. A 30 MΩ resistor was used to terminate the PMT
output. An SR530 dual phase lock-in was used to measuresignal magnitude. The resulting output was about 2 mV, which
agrees well with the calculations above for an average of one
photon per cycle. When the PMT high voltage was off, there
was no measurable output noise as expected. In all cases, the
SNR is dominated by the background count rate which
exceeded the signal rate by 2 to 3.

As seen from the data, a signal of 50 photons/s, at a referenceof 50 Hz, is easily detected with a lock-in amplifier to thesame SNR as a photon counter.
In general, large signals require the use of analog instruments
such as boxcar averagers and lock-in amplifiers. While it istrue that the theoretical achievable signal-to-noise ratio may
not be as good as the counting statistics, the practical matter isthat large signals take less time to measure to a given SNR level.
When the signal is low (less than one photon per gate or
cycle), the analog instruments, with the appropriate technique,can achieve photon counting signal-to-noise ratios. When thesignal is much lower photon counting is required.
In most experiments, the key to optimizing the measurement
will lie in factors other than signal intensity. In all cases, thePMT quantum efficiency, gain, and noise are the mostimportant factors. The initial gain from the PMT can never be
replaced as well with an amplifier. Low background or darkcount rates are essential in low-level measurements. External
noise pickup in signal cables, or unstable background countrates (such as from an unstable glow discharge), can result inlarge fluctuations in signal amplitude: far in excess of thecounting statistics. These experimental factors can be the mostimportant considerations when choosing an instrument.
For further reading:
1. Photomultiplier Handbook (pub PMT-62), RCA Corp., 1980.
2. The Art of Electronics, Horowitz and Hill, Cambridge
University Press, Cambridge, 1982.

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