Editing Noise figure (section)

==Optical noise figure==
The above describes noise in electrical systems. The optical noise figure is discussed in multiple sources.<ref name="Desurvire1994">E. Desurvire, ''Erbium doped fiber amplifiers: Principles and Applications'', Wiley, New York, 1994</ref><ref name="Haus1998">H. A. Haus, "The noise figure of optical amplifiers," in ''IEEE Photonics Technology Letters'', vol. 10, no. 11, pp. 1602-1604, Nov. 1998, doi: 10.1109/68.726763</ref><ref name="Noe2022">R. Noe, "Consistent Optical and Electrical Noise Figure," in ''Journal of Lightwave Technology'', 2022, doi: 10.1109/JLT.2022.3212936, https://ieeexplore.ieee.org/document/9915356</ref><ref name="NoeNF2023">R. Noe, "Noise Figure and Homodyne Noise Figure" Photonic Networks; 24th ITG-Symposium, Leipzig, Germany, 09-10 May 2023, pp. 85-91, https://ieeexplore.ieee.org/abstract/document/10173081, presentation https://www.vde.com/resource/blob/2264664/dc0e3c85c8e0cb386cbfa215fe499c4c/noise-figure-and-homodyne-noise-figure-data.pdf</ref><ref name="Haus2000">H. A. Haus, "Noise Figure Definition Valid From RF to Optical Frequencies", in ''IEEE Journal of Selected Topics in Quantum Electronics'', Vol. 6, NO. 2, March/April 2000, pp. 240–247</ref> Electric sources generate noise with a power spectral density, or energy per mode, equal to {{math|''kT''}}, where {{math|''k''}} is the Boltzmann constant and {{math|''T''}} is the absolute temperature. One mode has two quadratures, i.e. the amplitudes of {{math|cos}}<math>\mathrm{\omega}t</math> and {{math|sin}}<math>\mathrm{\omega}t</math> oscillations of voltages, currents or fields. However, there is also noise in optical systems. In these, the sources have no fundamental noise. Instead the energy quantization causes notable shot noise in the detector. In an optical receiver which can output one available mode or two available quadratures this corresponds to a noise power spectral density, or energy per mode, of {{math|''hf''}} where {{math|''h''}} is the Planck constant and {{math|''f''}} is the optical frequency. In an optical receiver with only one available quadrature the shot noise has a power spectral density, or energy per mode, of only {{math|''hf''/2}}. 

In the 1990s, an optical noise figure has been defined.<ref name="Desurvire1994"/> This has been called  {{math|''F''<sub>''pnf''</sub>}} for ''p''hoton ''n''umber ''f''luctuations.<ref name="Haus1998" /> The powers needed for SNR and noise factor calculation are the electrical powers caused by the current in a photodiode. SNR is the square of mean photocurrent divided by variance of photocurrent. Monochromatic or sufficiently attenuated light has a Poisson distribution of detected photons. If, during a detection interval the expectation value of detected photons is {{math|''n''}} then the variance is also {{math|''n''}} and one obtains {{math|''SNR''<sub>''pnf,in''</sub>}} = {{math|''n''<sup>2</sup>/''n''}} = {{math|''n''}}. Behind an optical amplifier with power gain {{math|''G''}} there will be a mean of {{math|''Gn''}} detectable signal photons. In the limit of large {{math|''n''}} the variance of photons is {{math|''Gn''(2''n''<sub>''sp''</sub>(''G''-1)+1)}} where {{math|''n''<sub>''sp''</sub>}} is the spontaneous emission factor. One obtains {{math|''SNR''<sub>''pnf,out''</sub>}} = {{math|''G''<sup>2</sup>''n''<sup>2</sup>/(''Gn''(2''n''<sub>''sp''</sub>(''G''-1)+1))}} = {{math|''n''/(2''n''<sub>''sp''</sub>(1-1/''G'')+1/''G'')}}. Resulting optical noise factor is {{math|''F''<sub>''pnf''</sub>}} = {{math|''SNR''<sub>''pnf,in''</sub> / ''SNR''<sub>''pnf,out''</sub>}} = {{math|2''n''<sub>''sp''</sub>(1-1/''G'')+1/''G''}}.

{{math|''F''<sub>''pnf''</sub>}} is in conceptual conflict<ref name="Noe2022" /><ref name="NoeNF2023" /> with the ''e''lectrical noise factor, which is now called {{math|''F''<sub>''e''</sub>}}:

Photocurrent {{math|''I''}} is proportional to optical power {{math|''P''}}. {{math|''P''}} is proportional to squares of a field amplitude (electric or magnetic). So, the receiver is nonlinear in amplitude. The "Power" needed for {{math|''SNR''<sub>''pnf''</sub>}} calculation is proportional to the 4th power of the signal amplitude. But for {{math|''F''<sub>''e''</sub>}} in the electrical domain the power is proportional to the square of the signal amplitude.

If {{math|''SNR''<sub>''pnf''</sub>}} is a noise factor then its definition must be independent of measurement apparatus and frequency. Consider the signal "Power" in the sense of {{math|''SNR''<sub>''pnf''</sub>}} definition. Behind an amplifier it is proportional to {{math|''G''<sup>2</sup>''n''<sup>2</sup>}}. We may replace the photodiode by a thermal power meter, and measured photocurrent {{math|''I''}} by measured temperature change <math>\mathrm{\Delta\theta}</math>. "Power", being proportional to {{math|''I''<sup>2</sup>}} or {{math|''P''<sup>2</sup>}}, is also proportional to <math>(\mathrm{\Delta\theta})</math>{{math|<sup>2</sup>}}. Thermal power meters can be built at all frequencies. Hence it is possible to lower the frequency from optical (say 200 THz) to electrical (say 200&nbsp;MHz). Still there, "Power" must be proportional to <math>(\mathrm{\Delta\theta})</math>{{math|<sup>2</sup>}} or {{math|''P''<sup>2</sup>}}. Electrical power {{math|''P''}} is proportional to the square {{math|''U''<sup>2</sup>}} of voltage {{math|''U''}}. But "Power" is proportional to {{math|''U''<sup>4</sup>}}.

These implications are in obvious conflict with ~150 years of physics. They are compelling consequence of calling {{math|''F''<sub>''pnf''</sub>}} a noise factor, or noise figure when expressed in dB.

At any given electrical frequency, noise occurs in both quadratures, i.e. in phase (I) and in quadrature (Q) with the signal. Both these quadratures are available behind the electrical amplifier. The same holds in an optical amplifier. But the direct detection photoreceiver needed for measurement of {{math|''SNR''<sub>''pnf''</sub>}} takes mainly the in-phase noise into account whereas quadrature noise can be neglected for high {{math|''n''}}. Also, the receiver outputs only one baseband signal, corresponding to quadrature. So, one quadrature or degree-of-freedom is lost.

For an optical amplifier with large {{math|''G''}} it holds {{math|''F''<sub>''pnf''</sub>}} ≥ 2 whereas for an ''e''lectrical amplifier it holds {{math|''F''<sub>''e''</sub>}} ≥ 1.

Moreover, today's long-haul optical fiber communication is dominated by coherent optical I&Q receivers but {{math|''F''<sub>''pnf''</sub>}} does not describe the SNR degradation observed in these.

Another optical noise figure {{math|''F''<sub>''ase''</sub>}} for ''a''mplified ''s''pontaneous ''e''mission has been defined.<ref name="Haus1998" /> But the noise factor {{math|''F''<sub>''ase''</sub>}} is not the SNR degradation factor in any optical receiver.

All the above conflicts are resolved by the optical in-phase and quadrature noise factor and figure {{math|''F''<sub>''o,IQ''</sub>}}.<ref name="Noe2022" /><ref name="NoeNF2023" /> It can be measured using a coherent optical I&Q receiver. In these, power of the output signal is proportional to the square of an optical field amplitude because they are linear in amplitude. They pass both quadratures. For an optical amplifier it holds {{math|''F''<sub>''o,IQ''</sub>}} = {{math|''n''<sub>''sp''</sub>(1-1/''G'')+1/''G''}} ≥ 1. Quantity {{math|''n''<sub>''sp''</sub>(1-1/''G'')}} is the input-referred number of added noise photons per mode.

{{math|''F''<sub>''o,IQ''</sub>}} and {{math|''F''<sub>''pnf''</sub>}} can easily be converted into each other. For large {{math|''G''}} it holds {{math|''F''<sub>''o,IQ''</sub>}} = {{math|''F''<sub>''pnf''</sub>/2}} or, when expressed in dB, {{math|''F''<sub>''o,IQ''</sub>}} is 3 dB less than {{math|''F''<sub>''pnf''</sub>}}. The ideal {{math|''F''<sub>''o,IQ''</sub>}} in dB equals 0 dB. This describes the known fact that the sensitivity of an ideal optical I&Q receiver is not improved by an ideal optical preamplifier.