Thermal noise

Thermal noise (also called Johnson noise or Johnson-Nyquist noise) is the electrical noise produced by the random thermal motion of charge carriers (electrons) in any resistive component at finite temperature. In optical receivers, thermal noise in the transimpedance amplifier (TIA) input resistance and feedback resistor is usually the dominant noise source at moderate signal levels and high bandwidths, setting the sensitivity floor of the receiver.

Voltage and current spectral densities. A resistor $R$ at temperature $T$ produces noise with spectral densities:

$$S_V=4k_{B}TR{\text{[V}}^2\text{/Hz]}$$ $$S_I=\frac{4k_{B}T}{R}{\text{[A}}^2\text{/Hz]}$$

where $k_B=1.38\times {10}^{-23}$ J/K is the Boltzmann constant. The RMS noise voltage in bandwidth $\mathrm{\Delta }f$:

$$v_{\text{thermal}}^{\text{RMS}}=\sqrt{4k_{B}TR\mathrm{\Delta }f}\mathrm{.}$$

At $T=300$ K (room temperature), this is approximately:

$$v_{\text{thermal}}\approx 0.13\sqrt{R\mathrm{\Delta }f\mathrm{/}\text{Hz}}\text{nV},$$

for $R$ in ohms. So a 1 kΩ resistor at room temperature produces 4 nV/√Hz of voltage noise.

Typical values.

Equivalent input current noise (TIA front-end). For a transimpedance amplifier with feedback resistor $R_f$, the thermal noise of $R_f$ appears as an input-referred current noise:

$$i_{\text{noise}}^{\text{RMS}}=\sqrt{\frac{4k_{B}T}{R_f}\mathrm{\Delta }f}\mathrm{.}$$

This is the noise current that, applied at the TIA input, would produce the same output voltage as the actual $R_f$ thermal noise. For a 10 kΩ TIA at room temperature: $\sim 1.3$ pA/√Hz input-referred noise.

Shot vs thermal noise crossover. A photodetector with photocurrent $I$ has shot noise $i_{\text{shot}}^2=2qI$ and thermal noise from its load resistor or TIA $i_{\text{thermal}}^2=4k_{B}T\mathrm{/}R$. The crossover photocurrent (shot equal to thermal):

$$I_{\text{crossover}}=\frac{2k_{B}T}{qR}=\frac{52\text{ mV}}{R}\text{(at 300 K)}\mathrm{.}$$

For a 1 kΩ TIA at room temperature: $I_{\text{crossover}}=52$ μA. Below this, thermal noise dominates; above it, shot noise dominates. The Photodiode Noise Calculator finds this crossover as an optical power for a given load resistor, temperature, amplifier noise and dark current.

Noise figure. The noise figure (NF) of an amplifier or receiver characterizes how much it adds to the input shot noise:

$$\text{NF}=\frac{{\text{(input shot noise)}}^2+i_{\text{thermal}}^2+i_{\text{electronics}}^2}{(\text{input shot noise}{)}^2}\mathrm{.}$$

For a typical optical receiver: NF $\sim 3-6$ dB at moderate signal levels. The "noise-figure" specification of TIAs (typically given as input-referred noise current in pA/√Hz) is the equivalent way to express the same information.

Why thermal noise is unavoidable. Thermal noise arises from the same physics as Brownian motion: at finite temperature, every degree of freedom has $\sim k_{B}T\mathrm{/}2$ of energy. In a circuit, this energy appears as electrical fluctuations. It cannot be eliminated, only reduced: by lowering the temperature (cryogenic cooling), by narrowing the bandwidth, since the noise power grows in proportion to $\mathrm{\Delta }f$, or, for the input-referred current noise of a TIA, by raising the feedback resistance, since $4k_{B}T\mathrm{/}{R}_f$ falls as $R_f$ grows, at the price of bandwidth.

Cooled photodetectors. Some applications cool the detector to suppress thermally generated noise and dark current:

Thermal noise vs flicker (1/f) noise. Thermal noise has flat (white) spectrum: same power per Hz at all frequencies. Real devices also have 1/f flicker noise dominant at low frequencies (DC to ~kHz). For:

• Op amps: 1/f corner at 10 Hz – 1 kHz
• BJTs: 1/f corner at 1 – 10 kHz
• FETs: 1/f corner at 100 Hz – 1 MHz
• TIAs: depends on input stage

At DC measurements (chopping with a lock-in amplifier or other narrowband detection), 1/f noise is often more important than thermal noise. At high-frequency telecom rates, thermal noise dominates.

Quantum thermal noise. At very low temperatures and high frequencies ($hf\gtrsim k_{B}T$), thermal noise becomes quantum-limited:

$$S_V=4hfR\frac{1}{1-e^{-hf\mathrm{/}{k}_{B}T}}\mathrm{.}$$

In the limit $hf\gg k_{B}T$: $S_V\to 4hfR$, frequency-dependent thermal noise that does not vanish at $T=0$ (zero-point fluctuations). For optical-frequency electronics this matters; for sub-THz electronics at room temperature, $hf\ll k_{B}T$ and the classical formula applies.

TIA-input-referred noise for telecom receivers.

These TIA noise values are far above the shot-noise limit at typical telecom signal levels. The receiver is thermal-noise-limited, not shot-noise-limited.

Optical preamplification reduces effective thermal noise. An optical preamplifier (an EDFA, for example) placed directly before the detector lifts the signal above the thermal-noise floor; the receiver is then limited by signal-ASE beat noise instead, which keeps its signal-to-noise ratio within the amplifier's noise figure (3 dB at best) of the shot-noise limit. Inline EDFAs in long-haul links do a different job: they make up the fiber loss of each span, and the link is then limited by accumulated ASE, that is, by OSNR, rather than by receiver thermal noise.

References: Saleh & Teich, Fundamentals of Photonics (3rd ed., 2019), Ch. 18 (photodetection noise); Razavi, Design of Integrated Circuits for Optical Communications (2nd ed., 2012) for the IC-level treatment; Horowitz & Hill, The Art of Electronics (3rd ed., 2015), Ch. 8.