Power spectral density (PSD)

The power spectral density (PSD) of a fluctuating quantity gives how its power, or its variance, is distributed over frequency: the amount per unit bandwidth at each frequency. Its units are those of the quantity squared per hertz, so a photocurrent noise has a PSD in A²/Hz, a voltage noise in V²/Hz, and the spectral distribution of an electrical or optical power in W/Hz or dBm/Hz (the PSD of power fluctuations, used for RIN, is in W²/Hz). The square root, in A/√Hz or V/√Hz, is the amplitude spectral density in which amplifier noise is usually specified. Typical values: the shot noise of 1 mA of photocurrent is $3.2 \times 10^{-22}$ A²/Hz (17.9 pA/√Hz); the thermal noise available from a matched source at 290 K is −174 dBm/Hz; a good single-frequency laser has a relative intensity noise of −150 to −160 dB/Hz.

Definition and one- and two-sided forms

For a stationary random signal $x(t)$, the PSD $S_x(f)$ is defined so that the variance is its integral over frequency. The Wiener–Khinchin theorem states that $S_x(f)$ is the Fourier transform of the autocorrelation function $\langle x(t)\,x(t+\tau)\rangle$. Mathematically the transform runs over positive and negative frequencies, giving the two-sided PSD. Engineering practice folds the negative frequencies onto the positive ones, giving the one-sided PSD, twice as large and defined for $f \geq 0$:

$$\sigma_x^2 = \int_0^\infty S_x^{(1)}(f)\,df .$$

The noise formulas used in photonics, $2qI$ for shot noise and $4k_BT/R$ for thermal noise, are one-sided. A factor of 2 between two sources almost always comes from this choice.

White noise examples

Shot noise has a flat (white) one-sided current PSD $S_I = 2qI$ up to frequencies set by the transit time of the carriers. Thermal noise in a resistor $R$ has $S_I = 4k_BT/R$, or as a voltage $S_V = 4k_BTR$. At 300 K a 1 kΩ resistor gives 4.07 pA/√Hz and 4.07 nV/√Hz, and a 50 Ω resistor gives 18.2 pA/√Hz, close to the shot noise of 1 mA, so a photodiode loaded directly by 50 Ω becomes shot-noise limited only above about 1 mA of photocurrent. RIN is a PSD normalized to the square of the mean power, $S_{\delta P}/\langle P\rangle^2$, in 1/Hz and quoted in dB/Hz.

Expressed as electrical power delivered by a current noise source into a 50 Ω load, $S_P = S_I R$: the 1 mA shot noise is $1.6 \times 10^{-20}$ W/Hz, or −168 dBm/Hz, 6 dB above the −174 dBm/Hz thermal reference. This is the form an electrical spectrum analyzer displays.

Integrating over bandwidth

The rms noise in a measurement is the square root of the PSD integrated over the measurement bandwidth $B$; for white noise this is $\sqrt{S\,B}$. For 1 mA of photocurrent in a 10 GHz receiver,

$$i_\text{rms} = \sqrt{2qIB} = 1.79~\mu\text{A},$$

and the shot-noise-limited electrical signal-to-noise ratio, $I^2/(2qIB)$, is 54.9 dB. For RIN of −150 dB/Hz integrated over 1 GHz, the total relative variance is $10^{-15} \times 10^{9} = 10^{-6}$ (−60 dB), an rms power fluctuation of 0.1%. Where the PSD is not flat, as with 1/f noise or the relaxation-oscillation peak of a laser's RIN, the integral is taken numerically over the measured spectrum. An instrument's noise floor is the same integral over its own noise.

Optical PSD on a spectrum analyzer

An optical spectrum analyzer reports power within its resolution bandwidth, so for broadband light such as amplified spontaneous emission the trace is a PSD in W per resolution bandwidth, usually dBm/0.1 nm. Wavelength and frequency densities are related by $|d\nu| = c\,|d\lambda|/\lambda^2$:

$$S_\nu = S_\lambda\,\frac{\lambda^2}{c} .$$

At 1550 nm, 0.1 nm corresponds to 12.48 GHz and 1 nm to 124.8 GHz. A noise level of −30 dBm/0.1 nm (1 µW per 0.1 nm, or 10 µW/nm) is therefore $8.0 \times 10^{-17}$ W/Hz, or −131.0 dBm/Hz. A narrow laser line has no meaningful PSD on this scale: it reads its full power in any resolution bandwidth wider than its linewidth.

Pitfalls

Electrical spectrum analyzers measure power in a resolution bandwidth whose equivalent noise bandwidth differs from the nominal setting, and with logarithmic averaging they under-read Gaussian noise by 2.51 dB; a noise-marker function applies both corrections. Sinusoidal tones and broadband noise scale differently with resolution bandwidth, so a spectrum containing both must be read with the bandwidth stated. RIN, dBc/Hz and dBm/Hz differ in their reference: the first two are relative to the mean power or the carrier, the last is absolute.

Common questions

What is the difference between power spectrum and power spectral density?

A power spectrum gives the power in each frequency bin of a particular analysis, so its values depend on the bin width; the PSD divides by that width, giving values that are independent of the analysis and can be integrated over any band.

How do you convert dBm/Hz to dBm?

Add $10\log_{10}B$ for white noise in bandwidth $B$: −168 dBm/Hz in 10 GHz is −68 dBm.

Why is noise quoted per root hertz?

Noise amplitudes add in quadrature, so the rms amplitude grows as $\sqrt{B}$. Quoting $\sqrt{S}$ in A/√Hz or V/√Hz lets the rms noise in any bandwidth be found by multiplying by $\sqrt{B}$.

References: B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019); L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995); P. Horowitz and W. Hill, The Art of Electronics, 3rd ed. (Cambridge University Press, 2015).