Comparing detector noise specifications

Filed under: detectors — Tags: compare, Detectivity, detector, detector noise, NEP, noise, noise equivalent power, noise power, optical noise, photodetector, signal to noise ratio, SNR, Specific detectivity — Webmaster @ 1:13 am

After having explained the causes of optical noise in detectors, I’d like in this post to define the parameters that makes it possible to compare detectors noise specifications.

Signal to noise ratio

Also noted SNR or S/N. This is defined as the ratio between the signal power and the noise power. Hence:

$\frac{S}{N}=\frac{\overline{\left|i_{s} \right|^{2}}}{\overline{\left|i_{noise} \right|^{2}}}$

Understanding the meaning of this is quite straightforward: the higher this ratio, the best signal you get. At equal input signal, the detector with the highest SNR is the less noisy one. If S/N<1, you cannot see anything, if S/N>>1, the signal is easy to pick up. As such, the signal-to-noise ratio is not a usable figure of merit of a detector. It is rather a measure of how strong your signal is compared to the “sensitivity” of your detector.

However, comparing the optical power needed to get a SNR of 1 is a step in the right direction to compare detector noise. According to the optical noise models explained earlier,

$\overline{\left|i_{s} \right|^{2}}=\left(\frac{e\eta}{h\nu}\right)^{2}\overline{P_{opt}^{2}}$

$\overline{\left|i_{noise} \right|^{2}}=\left[2e\left(\overline{i_{s}}+\overline{i_{0}}\right)+\frac{4k_{B}T}{R} \right]\Delta\nu$

Obviously, the noise depends completely on the bandwidth of the detector. This is understandable: to differentiate a true experimental result from random experimental error, you need to repeat the experiment. Translated in detector terminology, to get a better signal you need to increase the integration time of the experiment (= decrease the bandwidth).

To define a good figure of merit, it needs to show the minimum detectable optical power and not to depend on the integration time. Enters the noise equivalent power.

Noise equivalent power

Also noted NEP. This is a slightly confused definition. The initial concept is to define the noise equivalent power as the optical power which will yield a signal to noise ratio of 1. This is then the limit of what can be detected. But with this definition the noise equivalent power can only be given at a specific bandwidth (Δν enters in the expression of S/N).

Since not two detectors have the same integration time, manufacturers tend to call Noise Equivalent Power the minimum detectable power per square root of bandwidth. We will note this noise equivalent power per unit of bandwidth $NEP_{\sqrt{\Delta\nu}}$ to avoid confusion. In this situation we have then:

$\left(NEP_{\sqrt{\Delta\nu}}\right)^{2}=\frac{NEP^{2}}{\Delta\nu}=\frac{\overline{P_{opt\mid S/N=1}^{2}}}{\Delta\nu}=\left(\frac{h\nu}{e\eta}\right)^{2}\left[2e\left(\overline{i_{s}}+\overline{i_{0}}\right)+\frac{4k_{B}T}{R} \right]$

this normalised $NEP_{\sqrt{\Delta\nu}}$ only depends on the detector itself (and sometime on the ambient temperature!) and is measured in $W\cdot Hz^{-1/2}$ . The smallest the NEP, the better is the detector.

Getting back to the ambient temperature issue, the fluctuations of the ambient temperature are generally too small in comparison of the absolute temperature to introduce a bias in the comparison. However, it is true that the higher the temperature, the more noisy a detector is. For that reason some high quality detector are cooled (generally thermoelectrically but sometime with cryogenic cooling).

Detectivity and Specific detectivity

The detectivity D is defined as the reciprocal of the NEP: $D=\frac{1}{NEP}$ . Since all of parameters we defined depend on the area of the detector, in some cases this introduces a bias in the detector comparison. Thus sometime is specified a specific detectivity D* (D star), defined as:

$D^{*}=\frac{\sqrt{A}}{NEP_{spectral}}$

In fairness, I have very rarely encountered people using this specific detectivity in optical detectors.

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Noise in photodetectors

Filed under: detectors — Tags: Dark current, fluctuation, Johnson, model, noise, Nyquist, photodetector, photodiode, physics, poisson distribution, shot noise, Thermal noise — Webmaster @ 12:01 am

After my recent post on power and energy meters, I’d like to speak about the different kind of noises that affect a photodetector and how to model them.

Quantum effect and noise: the shot noise.

Fundamental physics tells us the light is made of particles (photons), which are emitted by the source at random. For that reason, the amount of photons emitted by the source (sun, bulb, laser, etc.) is not constant, but exhibits detectable statistical fluctuations. And this is in a nutshell what shot noise is. Because of its nature, it does not depends on the quality of the detector and is unavoidable. However the shot noise becomes a real issue only when the optical intensity is fairly low: in this case quantum fluctuations become much more noticeable.

The random process of light emission can generally be modelled using a Poisson distribution, the properties of which are very well known. If we note p(n) the probability that n photons arrive on the detector:

$p(n)=\frac{\bar{n}^{n}exp(-\bar{n})}{n!}$

$\bar{n}=\sum_{0}^{+\infty}{np(n)}$

$\sigma_{n}^{2}=\sum_{0}^{+\infty}{(n-\bar{n})^{2}p(n)=\overline{n^{2}}-\bar{n}^{2}=\bar{n}}$

where $\sigma_{n}$ is the standard deviation. What this means is that for 100 photons arriving on the detector, the uncertainty about the number of photon is of ±10 (±10%). If the number of photon is somewhat closer to common levels, e.g. $10^{10}$ , the uncertainty becomes $\pm 10^{5}$ , which is ±0.000,01%. It then becomes obvious that the shot noise is an issue only at low light level.

Let’s find out now what would be the fluctuations of the signal current due to the shot noise. Since each photon induce a free electron with an efficiency $\eta$ , during the time $\tau$ the number of electron produced is then $\eta\bar{n}$ . Every electron contributes to the signal current $i_{s}$ for a charge $e$ , the average value of the signal current $\overline{i_{s}}$ is then:

$\overline{i_{s}}=\frac{e}{\tau}\eta\bar{n}$ .

Fluctuation in the number of photons create a fluctuation in the signal current, and those fluctuations are characterised by the standard deviation:

$\sigma_{i_{s}}^{2}=\left(\frac{e}{\tau}\right)^{2}\sigma_{\eta n}^{2}=\left(\frac{e}{\tau}\right)^{2}\eta \bar{n}$

If we note $\Delta\nu\simeq 1/2\tau$ the bandwidth of the detector, we can find the useful formula below:

$\sigma_{i_{s}}^{2}=2e\overline{i_{s}}\Delta\nu$

And this standard deviation $\sigma_{i_{s}}$ characterise the shot noise current.

Dark current

The dark current is the constant response exhibited by a detector during periods when it is not actively being exposed to light. It is sometime classified as another type of shot noise. It is also referred to as reverse bias leakage current in non optical devices and is present in all diodes. Physically, dark current is due to the random generation of electrons and holes within the depletion region of the device that are then swept by the electric field applied to the diode.

Similarly to the photon noise, it can be modelled by a Poisson distribution, with:

$\sigma_{i_{0}}^{2}=2e\overline{i_{0}}\Delta\nu$

Let’s note that $\overline{i_{0}}$ depends from many parameters, but generally speaking:

Si photodiodes : $\overline{i_{0}}$ ranges from 1 to 10 nA
Ge photodiodes: $\overline{i_{0}}$ ranges from 50 to 500 nA
InGaAs photodiodes: $\overline{i_{0}}$ ranges from 1 to 20 nA

Thermal noise

Thermal noise, also called Johnson noise or Nyquist noise is the electronic noise generated by the thermal agitation of the electrons inside an electrical conductor at equilibrium, which happens regardless of any applied voltage. It was discovered by Johnson in 1927 and explained by Nyquist.

A device (a photodiode for instance) thermal noise can be modelled as a voltage source $V_{th}(t)$ in series with an ideal resistor R. $V_{th}(t)$ has a Gaussian distribution with a mean value of zero. In this case,

$\sigma_{V_{th}(t)}^{2}=4k_{B}TR\Delta\nu$

Where $\sigma_{V_{th}(t)}$ is the standard deviation of the voltage, $k_{B}$ is the Boltzmann’s constant in joules per kelvin and T is the resistor’s absolute temperature in kelvins. It can also be modelled a current source $i_{th}$ in parallel with R:

$\sigma_{i_{th}(t)}^{2}=\frac{4k_{B}T}{R}\Delta\nu$

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NIST, Special Publications: High Accuracy Laser Power And Energy Meter Calibration Service

Buying a laser power meter: check-list

Filed under: detectors, laser beam diagnostic — Tags: beam diagnostic, check-list, detector, energy meter, how to chose, laser, photodiode, power meter, Pyroelectric, thermopile — Webmaster @ 11:21 am

Because of the wide range of power and energy meter available on the market, and even more because they tend to be not totally versatile, you need to carefully examine your needs against the capabilities of the instrument you are planning to acquire. Here is a little check list to help you decide if a laser power meter or energy meter would fit your application.

Is the meter’s calibration traceable to internationally recognized standards such as NIST?
Is your laser wavelength within the wavelength range of the power meter?
What is the power range you expect to measure (highest and lowest limit)? Does it fall within the range the power meter can measure?
What is the diameter of your beam at measurement point? Do you have any control on this (using a lens for instance)? Is the power meter aperture big enough?
What is your power density (W/cm²) and energy density (J/cm²)? Is it below the damage threshold of the power meter?
Is your laser a pulsed femtosecond? If yes you will need a flat spectral response across the laser bandwidth. This may also be the case if your laser is widely tunable and you can’t adjust the wavelength setting manually, or simply if you don’t know your wavelength.
Is your laser pulsed and do you need to measure each pulse’s energy or an average power is sufficient? If the average power is enough or if you want to measure a single pulse energy, a thermopile is better. Otherwise you will have to go for a pyroelectric sensor or a specialised photodiode
Are there a lot of vibrations in your environment? If so this would rule out a pyroelectric detector.
Most power meters are sold nowadays in a set of two separate items: a display and a sensor. Make sure you order both and that they are compatible with each other
Assess what type of display you need: do you need computer connectivity, LabView compatibility, is it to go “in the field”, do you need it wireless (yes some manufacturer do that now)…

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