Spectral efficiency

Spectral efficiency is the bit rate a channel or a fiber carries divided by the optical bandwidth it uses, in bits per second per hertz. For a WDM system the bandwidth is normally the channel spacing, so the figure includes the guard bands between channels. A 400ZR channel carrying 400 Gb/s of Ethernet in a 75 GHz slot has a spectral efficiency of 5.3 b/s/Hz, or 4.0 b/s/Hz on a 100 GHz grid; eighty 100 Gb/s channels at 50 GHz spacing reach 2 b/s/Hz. The quantity decides how much traffic one fiber band can hold: the C-band, about 4.38 THz wide, carries 58 channels of 400 Gb/s at 75 GHz, about 23 Tb/s.

Gross, net and per-polarization figures

The rate in the numerator is usually the net (payload) rate, excluding forward error correction and framing overhead. 400ZR transmits a gross line rate of 478.75 Gb/s, which over 75 GHz would be 6.38 b/s/Hz; the payload figure of 5.3 b/s/Hz is the one used to compare systems. Papers state which they mean, and a figure given without it is ambiguous by the size of the FEC overhead, roughly 15–25%.

For a single channel the upper bound set by the format is the number of bits per symbol. A signal at symbol rate $R_s$ with ideal Nyquist pulse shaping occupies a bandwidth equal to $R_s$, so its spectral efficiency is

$$\text{SE} = N_\text{pol}\,\log_2 M$$

for an $M$-point constellation on $N_\text{pol}$ polarizations. Dual-polarization 16QAM gives 8 b/s/Hz on this basis, which is the "maximum" column in tables of modulation formats. Real channels fall short of it because practical pulses have a nonzero roll-off and lasers drift: 400ZR runs at 59.84 GBd in a 75 GHz slot, leaving about 15 GHz for the spectral skirts and the frequency tolerance, and the FEC overhead takes the rest of the difference between 8 and 5.3.

The Shannon limit

For a linear channel with additive white Gaussian noise, the capacity per unit bandwidth is

$$\frac{C}{B} = \log_2(1 + \text{SNR})$$

per polarization, where SNR is the electrical signal-to-noise ratio in the signal bandwidth. At an SNR of 20 dB this is 6.66 b/s/Hz per polarization, or 13.3 b/s/Hz for two polarizations; at 10 dB it is 3.46 per polarization. Read the other way, carrying 5.33 b/s/Hz over two polarizations needs an SNR of at least 7.3 dB, and the 8 b/s/Hz of an ideal DP-16QAM channel at least 11.8 dB, before any implementation penalty.

The SNR relates to the OSNR measured in the 0.1 nm (12.5 GHz) reference bandwidth by

$$\text{OSNR} = \frac{N_\text{pol}\,R_s}{2B_\text{ref}}\,\text{SNR}$$

For a dual-polarization signal at 59.84 GBd the factor is 4.8, so OSNR in dB is SNR plus 6.8 dB. In optical fiber the SNR cannot be raised indefinitely by increasing launch power, because the Kerr nonlinearity adds distortion that grows faster than the signal; this sets a maximum practical spectral efficiency for each distance, below the linear Shannon value.

Direct detection: NRZ and PAM4

Intensity-modulated links use one real-valued dimension and one polarization. NRZ carries 1 bit per symbol and PAM4 carries 2, so a 106.25 Gb/s PAM4 lane runs at 53.125 GBd, half the symbol rate NRZ would need. In short-reach datacenter optics this halving matters mainly for the bandwidth of the lasers, modulators and electronics; the lanes sit on a sparse wavelength grid or on separate fibers, and optical spectral efficiency is seldom the constraint. In DWDM transport, where spectrum is the scarce resource, coherent formats on two polarizations are used for this reason.

The DWDM grid

The ITU DWDM grid comes in a fixed form, with 50 GHz or 100 GHz spacing, and a flexible form in which each channel occupies a slot whose width is a multiple of 12.5 GHz. The flexible grid lets the slot be matched to the signal, for example 75 GHz for a 60 GBd channel, which raises the system spectral efficiency over placing the same channel on a 100 GHz grid by a factor of 100/75. Further capacity per fiber comes from more spectrum (the C+L band) or from more spatial channels, such as parallel fibers or multi-core fiber.

Pitfalls

Spectral efficiency per channel and per fiber differ when the band is not fully used. A figure per polarization is half the dual-polarization figure. A symbol-rate bandwidth ignores pulse roll-off, so it overstates what fits in a real slot.

Common questions

What is a typical spectral efficiency for optical fiber?

Deployed DWDM systems range from about 2 b/s/Hz (100 Gb/s on 50 GHz) to 5–6 b/s/Hz for 400 Gb/s and faster coherent channels on metro distances; long-haul links use lower-order formats and lower figures.

How is spectral efficiency different from bits per symbol?

Bits per symbol counts the information in one symbol. Spectral efficiency divides the bit rate by the bandwidth occupied, so it also accounts for pulse shape, guard bands, polarization multiplexing and coding overhead.

What limits spectral efficiency in fiber?

The SNR available at the receiver, set by amplifier noise and by fiber nonlinearity at high launch power, together with the guard bands needed for filters and laser frequency tolerance.

References: C. E. Shannon, "A mathematical theory of communication," Bell System Technical Journal 27, 379–423 and 623–656 (1948); R.-J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini and B. Goebel, "Capacity limits of optical fiber networks," Journal of Lightwave Technology 28, 662–701 (2010).