# Noise-equivalent Power

Author: the photonics expert Dr. Rüdiger Paschotta

Acronym: NEP

Definition: the input power to a detector which produces the same signal output power as the internal noise of the device

Categories: light detection and characterization, fluctuations and noise

Units: W or W / Hz^{1/2}

When a photodetector does not get any input light, it nevertheless produces some noise output with a certain average power, which is proportional to the square of the r.m.s. voltage or current amplitude. The *noise-equivalent power* (NEP) of the device is the optical input power which produces an additional output power identical to that noise power for a given bandwidth (see below). If the input is interpreted as a signal, the output signal and noise powers are then identical, i.e., the signal-to-noise ratio would be 1.

The inverse of the noise-equivalent power is called the *detectivity*.

The possible signal-to-noise ratio of a measurement (for a 1-Hz bandwidth) can be estimated simply as the available input power divided by the noise-equivalent power. For that purpose, one does not need to know the detector's responsivity.

Note that the noise-equivalent power depends on the optical wavelength, since that influences the responsivity of the detector. The lowest NEP is achieved for those wavelengths where the responsivity is the highest.

## Influence of the Bandwidth

The noise power and thus also the noise-equivalent power depends on the measurement bandwidth. (For white noise, it is proportional to that bandwidth.) At a first glance, one may find it most natural to use the full detection bandwidth of the device. Then, however, the NEP would not allow a fair comparison of detectors with different bandwidth; it would be reduced if additional electronic filtering, reducing the detection bandwidth, would be applied. Therefore, it is common to assume a bandwidth of 1 Hz, which is usually far below the detection bandwidth.

Some authors specify the NEP in units of W / Hz^{1/2} rather than W (watts), as would be the usual units for a power. Effectively, they base the NEP on the square root of a power spectral density (PSD) rather than on a power. The numerical results are the same as when assuming a bandwidth of 1 Hz.

## Measurement of Noise-equivalent Power

For experimentally obtaining the noise-equivalent power, one first needs to measure the noise amplitude of the instrument output in the given noise bandwidth (e.g. 1 Hz) without any optical input. That result has to be divided by the responsivity.

For example, consider a photodetector which in the dark produces a photocurrent (dark current or current generated by its electronics) with a noise amplitude of 1 nA / Hz^{1/2}. If its responsivity is 0.5 A/W, and we consider a bandwidth of 1 Hz, the NEP is 1 nA / 0.5 A/W = 2 nW. For a larger bandwidth of 10 kHz, the noise amplitude would rise to 100 nA – not to 10,000 nA, as this scales with the square root of the bandwidth –, and the NEP would rise to 200 nW.

The square root dependence on the bandwidth may be surprising; it is related to the fact that the noise power scales linearly with the bandwidth, and is proportional to the square of the noise amplitude. In our example case, the noise power in a 1-Hz bandwidth generated by the detector current in a 50-Ω-resistor would be 50 Ω · (1 nA)^{2}, and in a 10-kHz bandwidth it would be 10,000 times larger, i.e., 50 Ω · (100 nA)^{2}.

## Optimization

Obviously, a low noise-equivalent power is desirable because that power level is about the minimum input power level which can be detected easily when averaging the signal over a time of the order of one second. Using advanced methods such as lock-in detection, one can actually detect much weaker signals, provided that these have a bandwidth far below the detection bandwidth. In effect, one restricts the detection bandwidth to a value far below 1 Hz, which also reduces the noise power with which the signal has to compete. The required averaging time is correspondingly longer.

If the responsivity of a photodetector (e.g. a photodiode) can be increased without increasing the delivered noise power, the noise-equivalent power can be reduced. However, by using an avalanche photodiode, for example, where the responsivity can be greatly enhanced due to an internal amplification mechanism, one also obtains substantially more noise, so that the noise-equivalent power may even be increased.

## More to Learn

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## Questions and Comments from Users

2023-02-10

I want to compute the NEP at a different wavelengths. Is there a way to compute the NEP at another wavelength, if the responsivity of the detector R(lambda) is known? Is it true that NEP(lambda_{1}) * R(lambda _{1}) = NEP(lambda_{2}) * R(lambda

_{2})?

The author's answer:

Yes, that's a good approach.

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2021-03-03

I would like to understand the relationship between the NEP and the signal-to-noise ratio. If the NEP is large, does the signal-to-noise ratio become small?

The author's answer:

Yes, it will. However, the signal-to-noise ratio also depends on the strength of the transmitted signal, while the NEP does not deal with that.