Resolution bandwidth

The resolution bandwidth (RBW) is the width of the final filter in a spectrum analyzer: the instrument tunes this filter across the span and records the power that passes at each point, so each point of the trace is the power within one RBW. In an electrical spectrum analyzer the RBW is set in hertz, from about 1 Hz to a few megahertz; in a grating optical spectrum analyzer it is set in nanometers, typically 0.02–0.1 nm for laser and WDM work and up to a few nanometers for broadband sources. The setting decides three things at once: which spectral features are separated, how much noise each point collects, and how long a sweep takes.

Optical and electrical units

An optical RBW converts to frequency as $\Delta f = c\,\Delta\lambda/\lambda^2$. At 1550 nm, 0.01 nm is 1.25 GHz, 0.02 nm is 2.50 GHz, 0.05 nm is 6.24 GHz and 0.1 nm is 12.5 GHz. Even the narrowest grating setting is therefore thousands of times wider than the linewidth of a single-frequency laser: at 0.02 nm, a 1 MHz line is displayed about 2500 times wider than it is, with the shape of the instrument's filter. Linewidths are measured instead with heterodyne methods such as delayed self-heterodyne, whose beat note is then viewed on an electrical analyzer with a kilohertz-scale RBW.

Noise scales with the RBW

A spectral line narrower than the filter passes entirely and reads the same power at any RBW. Broadband noise reads in proportion to the RBW, because the filter collects noise from its whole width. For white noise of power spectral density $S$ in dBm/Hz, the displayed level is

$$N = S + 10\log_{10}(\text{RBW}/\text{Hz})$$

For an electrical analyzer with a 15 dB noise figure, $S = -174 + 15 = -159$ dBm/Hz, giving a displayed noise floor of $-129$ dBm at 1 kHz RBW and $-139$ dBm at 100 Hz. Each tenfold reduction of the RBW lowers broadband noise by 10 dB and leaves a narrow line unchanged, so the signal-to-noise ratio of a narrow line improves by 10 dB.

The same rule applies on an OSA. OSNR is referred to 0.1 nm, so a noise level of $-40$ dBm read at 0.05 nm RBW is $-37.0$ dBm in the reference bandwidth. Dividing a noise reading by the filter's bandwidth gives a spectral density (dBm/Hz or dBm/nm), the form used for relative intensity noise: a noise reading of $-120$ dBm in 1 MHz RBW, against a 0 dBm electrical carrier, is a RIN of $-180$ dB/Hz.

The bandwidth that enters these conversions is the filter's equivalent noise bandwidth, which differs from its nominal 3 dB width. For a Gaussian-shaped filter the noise bandwidth is 1.064 times the 3 dB width, a correction of 0.27 dB; analyzers apply the appropriate factor in their noise-marker functions.

Resolving neighboring lines

Two lines of equal amplitude are separated on the trace when their spacing is at least about one RBW. Lines of unequal amplitude need more, because the weaker one must rise above the skirt of the stronger; this is set by the filter's shape factor, the ratio of its 60 dB to 3 dB widths, and on an OSA by stray light inside the monochromator. A side-mode suppression ratio measurement therefore needs an RBW well below the mode spacing. Practical cases: DWDM channels on a 50 GHz grid lie 0.40 nm apart at 1550 nm, which calls for 0.1 nm or finer when the noise between them is read, and a 1 nm RBW merges the modes of a Fabry-Perot laser spaced about 1.1 nm apart into a smooth envelope.

Sweep time

A narrow filter takes time to respond. On a swept analyzer the minimum sweep time scales as span/RBW², with a proportionality constant of order one set by the filter type, so a 1 GHz span takes of order 10 s at 10 kHz RBW and of order 1000 s at 1 kHz. Reducing the RBW tenfold to gain 10 dB of noise floor lengthens the sweep about a hundredfold. FFT-based analyzers, which compute narrow-RBW spectra from digitized time records, avoid most of this penalty over modest spans.

Pitfalls

Video bandwidth and trace averaging smooth the displayed noise but do not lower its mean level; only the RBW does that. Averaging on a logarithmic display underreads Gaussian noise by 2.51 dB relative to power averaging. Instruments that couple RBW to span automatically will change the noise level when the span is changed, so a broadband level compared between two traces must be compared at the same RBW or normalized.

Common questions

What RBW is used on an OSA?

For OSNR and WDM measurements, 0.1 nm or finer; for side-mode suppression of a DFB laser, a setting well below the mode spacing, often 0.02–0.05 nm; for broadband sources such as LEDs or ASE, a wider setting improves the noise level without losing detail.

Why does the noise floor change with the RBW?

Noise is spread over frequency, so the filter collects more of it when it is wider: the floor moves by $10\log_{10}$ of the RBW ratio, 10 dB per decade.

References: D. Derickson (ed.), Fiber Optic Test and Measurement (Prentice Hall, 1998); P. Horowitz and W. Hill, The Art of Electronics, 3rd ed. (Cambridge University Press, 2015).