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:

$$S_V \;=\; 4 k_B T R \quad \text{[V}^2\text{/Hz]}$$ $$S_I \;=\; \frac{4 k_B T}{R} \quad \text{[A}^2\text{/Hz]}$$

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

$$v_\text{thermal}^\text{RMS} \;=\; \sqrt{4 k_B T R \, \Delta f}.$$

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

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

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

Typical values.

Resistor	Temperature	$v_\text{thermal}$ density
50 Ω	300 K	0.9 nV/√Hz
1 kΩ	300 K	4.1 nV/√Hz
10 kΩ	300 K	13 nV/√Hz
1 MΩ	300 K	129 nV/√Hz
1 kΩ	77 K (LN₂)	2.1 nV/√Hz
1 kΩ	4 K (LHe)	0.47 nV/√Hz

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:

$$i_\text{noise}^\text{RMS} \;=\; \sqrt{\frac{4 k_B T}{R_f} \, \Delta f}.$$

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 $i_\text{thermal}^2 = 4 k_B T / R$. The crossover photocurrent (shot equal to thermal):

$$I_\text{crossover} \;=\; \frac{2 k_B T}{q R} \;=\; \frac{52 \text{ mV}}{R} \quad \text{(at 300 K)}.$$

For a 1 kΩ TIA at room temperature: $I_\text{crossover} = 52$ μA. Below this, thermal noise dominates; above it, shot noise dominates.

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

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

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 $\sim k_B T / 2$ of energy. In a circuit, this energy appears as electrical fluctuations. The only way to reduce it is to lower temperature (cryogenic cooling) or to use a higher transconductance front-end (e.g., low-impedance current input).

Cooled photodetectors. Some applications use cooled detectors to reduce thermal noise:

Application	Cooling	Effect
Astronomical CCDs	$-100°$ C (Peltier or LN₂)	Sub-electron read noise
Single-photon avalanche diodes	$-50°$ C (Peltier)	Reduces dark count rate
Photoconductive detectors (LWIR)	77 K or 4 K	Operates in shot-noise-limited regime
Superconducting nanowire detectors	$<$ 2 K	Single-photon counting, sub-ps timing
Cryogenic transimpedance amplifiers	4 K	Sub-pA/√Hz input current noise

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 ($h f \gtrsim k_B T$), thermal noise becomes quantum-limited:

$$S_V \;=\; 4 h f R \, \frac{1}{1 - e^{-hf/k_BT}}.$$

In the limit $hf \gg k_BT$: $S_V \to 4 h f R$ — 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, $hf \ll k_BT$ and the classical formula applies.

TIA-input-referred noise for telecom receivers.

Receiver type	Input current noise	Equivalent NEP at 1.0 A/W responsivity
10G TIA	3 – 5 pA/√Hz	3 – 5 pW/√Hz
25G TIA	6 – 12 pA/√Hz	6 – 12 pW/√Hz
50G TIA	15 – 25 pA/√Hz	15 – 25 pW/√Hz
100G TIA	30 – 60 pA/√Hz	30 – 60 pW/√Hz

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 amplification reduces effective thermal noise. Pre-amplifying the optical signal (with an EDFA, for example) before detection elevates the signal above the thermal-noise floor; the receiver then operates closer to the shot-noise limit. This is the principal motivation for inline EDFAs in long-haul telecom: not for raw amplification, but for signal-to-thermal-noise improvement.

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.