# Photodetector Noise

This is a continuation from the previous tutorial - **semiconductor laser characteristics**.

## Types of Photodetectors

A photodetector is a device that converts an optical signal into a signal of another form. Most photodetectors convert optical signals into electrical signals, in the form of either current or voltage, that can be further processed or stored.

All photodetectors are square-law detectors that respond to the power or intensity, rather than the field amplitude, of an optical signal.

Based on the difference in the conversion mechanisms, there are two classes of photodetectors: ** photon detectors** and

**.**

*thermal detectors*Photon detectors are ** quantum detectors** based on the

**, which converts a photon into an emitted electron or an electron-hole pair; a photon detector responds to the number of photons absorbed by the detector.**

*photoelectric effect*Thermal detectors are based on the ** photothermal effect**, which converts optical energy into heat; a thermal detector responds to the optical energy, rather than the number of photons, absorbed by the detector.

Because of difference in their fundamental mechanisms, there are a number of important differences in the general characteristics of these two classes of detectors.

The response of a photon detector is a function of optical wavelength with long-wavelength cutoff, whereas that of a thermal detector is wavelength independent.

A photon detector can be much more responsive than a thermal detector in a particular spectral region, which typically falls somewhere within the range from the near ultraviolet to the near infrared.

In comparison, a thermal detector normally covers a wide spectral range from the deep ultraviolet to the far infrared with a nearly constant response.

Photon detectors can be made extremely sensitive. Some of them have a photon-counting capability that is not possible for a thermal detector. A photon detector can be designed to have a high response speed capable of following very fast optical signals.

Most thermal detectors are relatively slow in response because the speed of a thermal detector is limited by thermalization through heat diffusion and by heat dissipation when the power of an optical signal varies.

For these reasons, photon detectors are suitable for detecting optical signals in photonic systems, whereas thermal detectors are most often used for optical power measurement or infrared imaging.

In this tutorial, only photon detectors are discussed because our major concern is with devices for photonics applications.

Photon detectors can be classified into two groups: one based on the ** external photoelectric effect** and another based on the

**.**

*internal photoelectric effect*Photodetectors based on the external photoelectric effect are ** photoemissive devices**, such as vacuum photodiodes and the photomultiplier tubes, in which photoelectrons are ejected from the surface of a photocathode.

Photodetectors based on the internal photoelectric effect are semiconductor devices, in which electron-hole pairs are generated through absorption of incident photons. A host of such devices have been developed, such as photoconductors, junction photodiodes, many photovoltaic devices, phototransistors, and charge-coupled devices.

## Photodetector Noise

Noise is one of the most fundamental phenomena in nature. It is ubiquitous. Noise in a photodetector sets the fundamental limit on the detectivity of the detector, thus determining the usefulness of a detector for particular application.

In terms of the physical nature, there are a few different types of noise for a photodetector. Two types of noise, ** quantum noise** and

**, originate from the basic physical laws of nature.**

*thermal noise*Quantum noise, described as ** shot noise** of electrons or photons in electronics and photonics, results form the statistical nature of a quantum event dictated by the uncertainty principle.

Thermal noise, known as ** Johnson noise** or

**in electronics and photonics, is the consequence of thermal fluctuations and is directly associated with thermal radiation.**

*Nyquist noise*Noise of such fundamental nature can only be minimized but can never be completely eliminated.

In terms of physical sources, the noise of a photodetector can come from the following: the detector itself, the possible amplifier used in conjunction with the detector, and the circuit used to extract the electrical signal from the detector.

Noise appears in a signal as random fluctuations about the mean value of the signal. A measured signal \(s\) has a mean value of \(\bar{s}\) defined as

\[\tag{14-1}\bar{s}=\sum_sp(s)s\]

where \(p(s)\) is the probability of the measured signal having a value \(s\) and the sum is carried out over all possible values obtained from measuring the signal.

This mean value \(\bar{s}\) is the expected value, or the ensemble average, of the variable \(s\).

The variance, or the mean square deviation, of the signal \(s\) is

\[\tag{14-2}\sigma_s^2=\overline{(s-\bar{s})^2}=\overline{s^2}-\overline{s}^2\]

The noise in a signal \(s\) can be expressed by a random variable \(s_\text{n}\) defined as

\[\tag{14-3}s_\text{n}=s-\bar{s}\]

The noise represented by the random variable \(s_\text{n}\) has a few general characteristics. As can be seen clearly from (14-3), it has a zero mean value:

\[\tag{14-4}\overline{s_\text{n}}=0\]

From (14-2) and (14-3), we find that the ** mean square value** of \(s_\text{n}\) is equal to the variance of \(s\):

\[\tag{14-5}\overline{s_\text{n}^2}=\overline{(s-\bar{s})^2}=\sigma_\text{s}^2=\overline{s^2}-\bar{s}^2\]

The mean square value of the noise in a signal is simply the mean square deviation of the signal.

Because \(\overline{s_\text{n}}=0\) but \(\overline{s_\text{n}^2}\ne0\), the average amplitude of the noise vanishes but the power of the noise does not. Therefore, the magnitude of the noise is not measured by its average value but rather by its root mean square (rms) value defined as

\[\tag{14-6}\text{rms}(s_\text{n})=\overline{s_\text{n}^2}^{1/2}\]

Noise characterized by random fluctuations is incoherent. If two or more independent noise sources, \(s_\text{n1}\), \(s_\text{n2}\), \(\ldots\), are simultaneously present in a signal \(s\), their combined effect is not found by adding their amplitudes but is obtained by adding their mean square values, or their powers:

\[\tag{14-7}\overline{s_\text{n}^2}=\overline{s_\text{n1}^2}+\overline{s_\text{n2}^2}+\ldots\]

The total noise from different independent sources then has an rms value of

\[\tag{14-8}\text{rms}(s_\text{n})=\overline{s_\text{n}^2}^{1/2}=\left(\overline{s_\text{n1}^2}+\overline{s_\text{n2}^2}+\ldots\right)^{1/2}\]

One important figure of merit for a detection system is the ** signal-to-noise ratio** (SNR or S/N). It is defined as the ratio of the power of a signal to the power of its noise or, equivalently, the ratio of the mean square of a signal to the mean square of its noise:

\[\tag{14-9}\text{SNR}=\frac{\overline{s^2}}{\overline{s_\text{n}^2}}=\frac{\overline{s^2}}{\sigma_s^2},\qquad\text{or}\qquad\text{SNR}=10\log\frac{\overline{s^2}}{\overline{s_\text{n}^2}}(\text{dB})\]

The SNR defined above is also known as the ** signal-to-noise power ratio** to be distinguished from the

**defined as**

*signal-to-noise current ratio*\[\tag{14-10}\text{SNR}_\text{current}=\frac{\bar{s}}{\overline{s_\text{n}^2}^{1/2}}=\frac{\bar{s}}{\sigma_s}\]

Without specification, however, the SNR of a detection system generally refers to the signal-to-noise power ratio defined in (14-9).

In a photodetection system, a signal can take the form of photon number or photon flux as the input optical signal. It can also take the form of photocurrent or photovoltage as the output electrical signal. Therefore, the signal \(s\) can represent photon number, photon flux, photocurrent, or photovoltage. The general characteristics discussed above for the noise \(s_\text{n}\) apply to every case.

In the following discussions of photodetector noise, we consider an input optical signal with an optical power \(P_s\). The detection system has an electrical response bandwidth of \(\Delta{f}=B\), which can effectively sample the optical signal within a rectangular time interval of

\[\tag{14-11}T=\frac{1}{2B}\]

The total number of photons received by the photodetection within this time interval is

\[\tag{14-12}\mathcal{S}=\frac{P_s}{h\nu}T=\frac{P_s}{2Bh\nu}\]

If the photodetector has a ** quantum efficiency** \(\eta_\text{e}\), the total number of charge carriers generated in the detector by the photoelectric effect upon receiving the photons within the time interval \(T\) is

\[\tag{14-13}\mathcal{N}=\eta_\text{e}\mathcal{S}=\eta_\text{e}\frac{P_s}{2Bh\nu}\]

where \(0\le\eta_\text{e}\le1\). Consequently, the photocurrent in the detector is

\[\tag{14-14}i_\text{ph}=\frac{e\mathcal{N}}{T}=2eB\mathcal{N}=\eta_\text{e}\frac{eP_s}{h\nu}\]

where \(e\) is the electronic charge. For a detector without an internal gain, the signal current is simply \(i_\text{s}=i_\text{ph}\). For a detector with an internal gain \(G\), the signal current is \(i_\text{s}=Gi_\text{ph}\).

## Shot Noise

The shot noise in a photodetector results from the quantum nature of the photons in the optical signal and that of the charge carriers generate in the detector.

Due to the quantum-mechanical probabilistic nature of photons, the photons in an optical signal are not distributed uniformly in time but arrive at the detector randomly in time. Therefore, both the power \(P_\text{s}\) of the optical signal and the number of photons \(\mathcal{S}\) received in a given time interval \(T\) fluctuate randomly around their respective average values of \(\overline{P_\text{s}}\) and \(\bar{\mathcal{S}}\).

The random fluctuations of the photons are characterized by the Poisson statistics. In any given time interval \(T\), the probability of receiving \(\mathcal{S}\) photons is given by the following Poisson probability distribution:

\[\tag{14-15}p(\mathcal{S})=\frac{\bar{\mathcal{S}}^\mathcal{S}\text{e}^{-\bar{\mathcal{S}}}}{\mathcal{S}!}\]

The mean square noise in the photon number fluctuations can then be calculated as

\[\tag{14-16}\overline{\mathcal{S}_\text{n}^2}=\sigma_\mathcal{S}^2=\sum_\mathcal{S}p(\mathcal{S})(\mathcal{S}-\bar{\mathcal{S}})^2=\bar{\mathcal{S}}\]

This photon contribution to the noise of the photodetector is independent of the physical properties of the detector because it is external to the detector. It is the ultimate lower limit of the noise in an optical detection system. It sets the fundamental limit on the detectivity of a photodetector.

The photons received by a photodetector are converted to photoelectrons or electron-hole pairs, depending on the type of the detector, through the photoelectric effect.

With a quantum efficiency \(\eta_\text{e}\), which has a value between 0 and 1, the number of photoelectrons generated is only a fraction of that of the photons received by the detector.

Because a given photon can only generate either one or no electron, but not a fraction of an electron, the photoelectric process is clearly quantum mechanical and probabilistic. The shot noise associated with this process has to be considered if the quantum efficiency is less than unity.

This effect is fully accounted for by considering the statistics of the number \(\mathcal{N}\) of charge carriers given in (14-13) that are generated with a quantum efficiency \(\eta_\text{e}\) of the photodetector.

The random fluctuations of the charge carriers generated by the photoelectric effect are also characterized by the Poisson statistics with the following probability distribution for generating a number \(\mathcal{N}\) in a time interval \(T\):

\[\tag{14-17}p(\mathcal{N})=\frac{\overline{\mathcal{N}}^\mathcal{N}\text{e}^{-\overline{\mathcal{N}}}}{\mathcal{N}!}\]

where \(\mathcal{N}=\eta_\text{e}\mathcal{S}\).

We find, through a procedure similar to that used in (14-16), that the mean square noise in the number of photo-generated carriers is

\[\tag{14-18}\overline{\mathcal{N}_\text{n}^2}=\sigma_\mathcal{N}^2=\overline{\mathcal{N}}\]

Because \(\overline{\mathcal{N}}\lt\overline{\mathcal{S}}\) if \(\eta_\text{e}\lt1\), the noise is actually reduced by an imperfect quantum efficiency.

This result seems odd. However, what really counts in a detection system is not the noise alone, but the SNR. While the noise is reduced by an imperfect quantum efficiency of \(\eta_\text{e}\lt1\), the signal is reduced even more. As a result, the SNR is lower for a detector that has a poorer quantum efficiency.

We consider there a detector without an internal gain, such that \(i_\text{s}=i_\text{ph}\). Using (14-14) and (14-18), we find the following shot current noise in the photodetector:

\[\tag{14-19}\overline{i_\text{n,sh}^2}=4e^2B^2\overline{\mathcal{N}_\text{n}^2}=4e^2B^2\overline{\mathcal{N}}=2eB\overline{i_\text{s}}\]

We then have the following mean square current fluctuations for the shot noise of a photodetector that receives an optical power \(P_\text{s}\) from an input optical signal:

\[\tag{14-20}\overline{i_\text{n,sh}^2}=2eB\overline{i_\text{s}}=2\eta_\text{e}e^2B\frac{\overline{P_\text{s}}}{h\nu}\]

From this relation and (14-5), we have

\[\tag{14-21}\overline{i_\text{s}^2}=\overline{i_\text{s}}^2+2eB\overline{i_\text{s}}\]

In practice, there are other sources that also contribute to the shot noise of a photodetector. One important source is the photons from the background radiation that impinge on the detector. The contribution of this noise source can be minimized by reducing the aperture of the detector to the minimum needed for receiving the optical signal. It cannot be completely eliminated, however, because at the very minimum there is still background thermal radiation, which can only be reduced by reducing the temperature of the environment surrounding the detector.

Another important source of shot noise is the dark current of the detector. The dark current is the current in a detector when it is not illuminated with any optical input. In a semiconductor device, it is normally caused by thermal generation of electron-hole pairs and by leakage currents due to surface defects of the device.

When these additional noise sources are considered, the total shot noise of a photodetector is given by

\[\tag{14-22}\overline{i_\text{n,sh}^2}=2eB\overline{i}=2eB(\overline{i_\text{s}}+\overline{i_\text{b}}+\overline{i_\text{d}})\]

where \(i_\text{b}\) is the photocurrent generated by background radiation and \(i_\text{d}\) is the dark current of the detector.

## Excess Shot Noise

In a photodetector, such as a photomultiplier, a photoconductor, or an avalanche photodiode, that has an internal gain, both signal and noise are amplified. For a detector that has a gain \(G\), the signal current, the background radiation current, and the dark current are all amplified by the factor \(G\):

\[\tag{14-23}i_\text{s}=Gi_\text{ph}=G\eta_\text{e}\frac{eP_\text{s}}{h\nu}\]

and

\[\tag{14-24}i_\text{b}=Gi_\text{b0},\qquad{i_\text{d}}=Gi_\text{d0}\]

where \(i_\text{b0}\) and \(i_\text{d0}\) are unamplified background and dark currents, respectively, and \(i_\text{b}\) and \(i_\text{d}\) are the amplified currents that can be directly measured externally.

The shot noise is also amplified through a process of random multiplication of the noise electrons. The statistical nature of this random multiplication process results in an ** excess noise factor**, \(F\), which is a function of the material, the structure, and the gain of a detector.

As a consequence, the mean square shot noise current for a detector with an internal gain can be expressed as

\[\tag{14-25}\overline{i_\text{n,sh}^2}=2eBG^2F(\overline{i_\text{ph}}+\overline{i_\text{b0}}+\overline{i_\text{d0}})=2eBGF(\overline{i_\text{s}}+\overline{i_\text{b}}+\overline{i_\text{d}})\]

where the excess noise factor \(F=\overline{G^2}/\overline{G}^2\) is a function of the gain.

For a detector without an internal gain, we find that \(G=1\) and \(F=1\); then, the shot noise given in (14-25) reduces to that in (14-22), as expected.

## Thermal Noise

Thermal noise results from random thermal motions of the electrons in a conductor. It is associated with the blackbody radiation of a conductor at the radio or microwave frequency range of the signal.

Because only materials that can absorb and dissipate energy can emit blackbody radiation, thermal noise is generated only by the resistive components of the detector and its circuit. Capacitive and inductive components do not generate thermal noise because they neither dissipate nor emit energy.

The energy of the thermal noise generated by a resistive element is independent of the detailed physical properties of the resistor but is dictated by the law of blackbody radiation. At a temperature \(T\), the thermal noise power in a small frequency interval of \(\text{d}f\) centered around \(f\) is

\[\tag{14-26}P_\text{n,th}(f)\text{d}f=\frac{4hf}{\text{e}^{hf/k_\text{B}T}-1}\text{d}f\]

In normal operation of most photodetectors, \(f\ll{k_\text{B}}T/h\). Thus, the frequency dependence of the thermal noise power is negligible, resulting in

\[\tag{14-27}P_\text{n,th}(f)\text{d}f\approx4k_\text{B}T\text{d}f\]

Then, the total thermal noise power for a detection system of a bandwidth \(B\) is simply

\[\tag{14-28}P_\text{n,th}=4k_\text{B}TB\]

For a resistor that has a resistance \(R\), the thermal noise can be treated as either current noise or voltage noise through the relation of \(P_\text{n,th}=\overline{i_\text{n,th}^2}R=\overline{v_\text{n,th}^2}/R\). Then, we have

\[\tag{14-29}\overline{i_\text{n,th}^2}=\frac{4k_\text{B}TB}{R}\]

and

\[\tag{14-30}\overline{v_\text{n,th}^2}=4k_\text{B}TBR\]

For an optical detection system, the resistance \(R\) is the total equivalent resistance, including the internal resistance of the detector and the load resistance from the circuit, at the output of the detector.

For a detector that has a current signal, (14-29) is used. In this case, the thermal noise is determined by the lowest shunt resistance to the detector, which is often the load resistance of the detector. The thermal noise can be reduced by increasing this resistance at the expense of reducing the response speed of the system.

For a detector that has a voltage signal, (14-30) is used. In this situation, the thermal noise is determined by the largest series resistance to the detector, which again is often the load resistance of the detector. The thermal noise can now be reduced by decreasing this resistance, but at the expense of reducing the output voltage signal.

## Signal-to-Noise Ratio

There are other noise sources, such as the \(1/f\) noise, but they are usually not important for the normal operation of photodetectors. Therefore, the total noise of a photodetector, whether it has an internal gain or not, is basically the sum of its shot noise and thermal noise:

\[\tag{14-31}\overline{i_\text{n}^2}=\overline{i_\text{n,sh}^2}+\overline{i_\text{n,th}^2}\]

A photodetector is said to function in the ** quantum regime** if \(\overline{i_\text{n,sh}^2}\gt\overline{i_\text{n,th}^2}\). A photodetector operating in the quantum regime is

**because shot noise is the primary source of noise in this regime.**

*shot-noise limited*A photodetector is in the ** thermal regime** if \(\overline{i_\text{n,th}^2}\gt\overline{i_\text{n,sh}^2}\). A photodetector operating in the thermal regime is

**because its thermal noise dominates its shot noise in this regime.**

*thermal-noise limited*For a photodetector that has no internal gain, the SNR is given by

\[\tag{14-32}\begin{align}\text{SNR}=\frac{\overline{i_\text{s}^2}}{\overline{i_\text{n}^2}}&=\frac{\overline{i_\text{ph}^2}}{2eB(\overline{i_\text{ph}}+\overline{i_\text{b}}+\overline{i_\text{d}})+4k_\text{B}TB/R}\\&=\frac{\overline{P_\text{s}^2}\mathcal{R}^2}{2eB(\overline{P_\text{s}}\mathcal{R}+\overline{i_\text{b}}+\overline{i_\text{d}})+4k_\text{B}TB/R}\end{align}\]

where \(\mathcal{R}=\eta_\text{e}e/h\nu\) is the ** responsivity** of a photodetector without an internal gain, defined in the next tutorial.

For a photodetector that has an internal gain \(G\), the SNR is

\[\tag{14-33}\begin{align}\text{SNR}=\frac{\overline{i_\text{s}^2}}{\overline{i_\text{n}^2}}&=\frac{G^2\overline{i_\text{ph}^2}}{2eBG^2F(\overline{i_\text{ph}}+\overline{i_\text{b0}}+\overline{i_\text{d0}})+4k_\text{B}TB/R}\\&=\frac{\overline{P_\text{s}^2}\mathcal{R}^2}{2eBGF(\overline{P_\text{s}}\mathcal{R}+\overline{i_\text{b}}+\overline{i_\text{d}})+4k_\text{B}TB/R}\end{align}\]

where \(\mathcal{R}=G\eta_\text{e}e/h\nu\) is the responsivity of a photodetector with an internal gain, also defined in the next tutorial.

The relations in (14-32) and (14-33) apply to photodetectors that have current signals at the output.

For a photodetector that has an output voltage signal, the SNR is defined as

\[\tag{14-34}\text{SNR}=\frac{\overline{v_\text{s}^2}}{\overline{v_\text{n}^2}}=\frac{\overline{P_\text{s}^2}\mathcal{R}^2}{\overline{v_\text{n}^2}}\]

where \(\mathcal{R}\) is the responsivity of a photodetector that has an output voltage signal defined in the following tutorial.

**Example 14-1**

A photodetector that responds to an optical signal with a photocurrent has a load resistance of \(R=50\text{ Ω}\) and a bandwidth of \(B=100\text{ MHz}\). It has a negligible background radiation current and a dark current of \(i_\text{d}=10\text{ nA}\).

(a) Find its shot noise, thermal noise, and signal-to-noise ratio when it generates a signal photocurrent of 1 μA.

(b) What are its shot noise, thermal noise, and signal-to-noise ratio when it generates a signal photocurrent of 1 mA?

**(a)**

For \(\overline{i_\text{s}}=\overline{i_\text{ph}}=1\text{ μA}\), we find that the shot noise

\[\begin{align}\overline{i_\text{n,sh}^2}&=2eB(\overline{i_\text{s}}+\overline{i_\text{d}})=2\times1.6\times10^{-19}\times100\times10^6\times(1\times10^{-6}+10\times10^{-9})\text{ A}^2\\&=3.23\times10^{-17}\text{ A}^2\end{align}\]

At \(T=300\text{ K}\), \(k_\text{B}T=25.9\text{ meV}\). The thermal noise for \(R=50\text{ Ω}\) is

\[\begin{align}\overline{i_\text{n,th}^2}&=\frac{4k_\text{B}TB}{R}=\frac{4\times25.9\times10^{-3}\times1.6\times10^{-19}\times100\times10^6}{50}\text{ A}^2\\&=3.32\times10^{-14}\text{ A}^2\end{align}\]

Thus, the total noise

\[\overline{i_\text{n}^2}=\overline{i_\text{n,sh}^2}+\overline{i_\text{n,th}^2}=3.23\times10^{-17}\text{ A}^2+3.32\times10^{-14}\text{ A}^2=3.32\times10^{-14}\text{ A}^2\]

We see that in this example the shot noise is mainly contributed by the signal photocurrent, but the shot noise is negligible compared to thermal noise.

From (14-21), we have

\[\overline{i_\text{s}^2}=\overline{i_\text{s}}^2+2eB\overline{i_\text{s}}=(1\times10^{-6})^2\text{ A}^2+3.23\times10^{-17}\text{ A}^2=1\times10^{-12}\text{ A}^2\]

We find that \(\overline{i_\text{s}^2}\approx\overline{i_\text{s}}^2\) in this example. Thus, the SNR is

\[\text{SNR}=\frac{\overline{i_\text{s}^2}}{\overline{i_\text{n}^2}}=\frac{1\times10^{-12}}{3.32\times10^{-14}}=30\]

which is 14.8 dB.

**(b)**

For \(\overline{i_\text{s}}=\overline{i_\text{ph}}=1\text{ mA}\), the shot noise

\[\begin{align}\overline{i_\text{n,sh}^2}&=2eB(\overline{i_\text{s}}+\overline{i_\text{d}})=2\times1.6\times10^{-19}\times100\times10^6\times(1\times10^{-3}+10\times10^{-9})\text{ A}^2\\&=3.2\times10^{-14}\text{ A}^2\end{align}\]

The thermal noise is the same as that found in (a): \(\overline{i_\text{n,th}^2}=3.32\times10^{-14}\text{ A}^2\). In this example, the shot noise is contributed almost entirely by the signal photocurrent and is comparable to the thermal noise. The total noise

\[\overline{i_\text{n}^2}=\overline{i_\text{n,sh}^2}+\overline{i_\text{n,th}^2}=3.2\times10^{-14}\text{ A}^2+3.32\times10^{-14}\text{ A}^2=6.52\times10^{-14}\text{ A}^2\]

We also have

\[\overline{i_\text{s}^2}=\overline{i_\text{s}}^2+2eB\overline{i_\text{s}}=(1\times10^{-3})^2\text{ A}^2+3.2\times10^{-14}\text{ A}^2=1\times10^{-6}\text{ A}^2\]

The SNR is

\[\text{SNR}=\frac{\overline{i_\text{s}^2}}{\overline{i_\text{n}^2}}=\frac{1\times10^{-6}}{6.52\times10^{-14}}=1.53\times10^7\]

which is 71.8 dB.

We see that the SNR is significantly increased by 57 dB when the signal photocurrent is increased from 1 μA to 1 mA. The reason for this significant improvement is that the detector is limited by thermal noise at low photocurrents.

The next tutorial covers the topic of **photodetector performance parameters**.