[Noise Figure] The origin of the number 58 in the well-known 58 formula

Many engineers engaged in optical module, trunk transmission and amplifier testing are taught the 58 Formula by senior colleagues when they first start their careers to quickly calculate noise figure and OSNR. However, in practical operations, they often find that the noise figure calculated via the 58 Formula fails to match the actual measurement results from an optical spectrum analyzer (OSA). What exactly causes this discrepancy? Starting from the fundamental formulas, this paper derives the origin of the constant 58, and elaborates on the applicable scenarios of the 58 Formula as well as cases where precise algorithms must be adopted instead.

# Origin of the "58 Formula"
In optical communication systems, noise figure is a critical metric for evaluating the performance of optical amplifiers, serving as the core parameter that quantifies the extra noise generated by an optical amplifier. In accordance with the international standard "IEC 61290-3-1:2003 Optical Amplifiers – Part 3: Noise figure parameters", the noise figure formula is defined as follows:

NF(dB)=PASE(dBm)-10lgG-10lg(hv*B0)

NF(dB): Noise figure of the optical amplifier, in decibels (dB); a smaller value indicates superior noise performance of the amplifier.
PASE(dBm): Power of amplified spontaneous emission (ASE) noise from the amplifier, namely the noise floor power measured by an optical spectrum analyzer.
G: Gain in linear units, calculated as the output optical power (linear unit) divided by the input optical power (linear unit).
G(dB): Gain in logarithmic units, equal to logarithmic output optical power minus logarithmic input optical power.
h: Planck's constant.
v: Central optical frequency of the optical signal.
B0: Reference optical bandwidth adopted for noise calculation (frequency bandwidth instead of wavelength in nanometers; typically 12.5 GHz corresponding to 0.1 nm).

Further simplification of the formula yields:

NF(dB)= PASE(dBm)-G(dB)-10lg(hv*B0)

The entire final term -10lg(hv*B0) is the constant "58" commonly referred to by practitioners. This term equals 58 dBm only under specific wavelength and bandwidth conditions. When the optical signal wavelength is 1550 nm and the optical bandwidth is set to 0.1 nm, -10lg(hv*B0)=58 dBm, which gives rise to the simplified estimation formula for noise figure:

NF=58+PASE-G

Likewise, the 58 Formula for OSNR can be further deduced:

OSNR = Pout - NF - 10·log₁₀N + 58

OSNR: Single-channel optical signal-to-noise ratio (dB) after N optical amplifier spans
Pout: Single-channel fiber-input / amplifier output power (dBm)
NF: Noise figure of the amplifier (dB; average value taken for multi-stage amplifiers)
N: Number of optical amplifier spans
58: Fixed constant under standard bandwidth and wavelength conditions

# Practical Pitfall Case
The following test on the noise figure of our company's high-power polarization-maintaining SOA (JSA-BT525G35-PM) illustrates why the NF values calculated by the 58 Formula sometimes deviate from OSA measurements.

The yellow curve and cyan curve in the figure represent the input spectrum and output spectrum respectively. The parameters obtained from the figure are as follows:
WL=1551.2263 nm
Pin= -13.566 dBm
PASE= -38.656 dBm
Res= 0.069 nm (the optical bandwidth B0 in the noise figure formula)
G = 15.506 dB
NF = 5.44 dB

1. Calculation using the 58 formula:
NF=58+PASE-G=58-38.656-15.506=3.838 dB
This result deviates significantly from the value provided by the OSA, indicating obvious distortion.

This occurs because the optical bandwidth corresponding to the OSA resolution (RES) here is not 0.1 nm, so the constant term -10lg(hv·B0) is no longer equal to 58 dBm, and recalculation using the standard noise figure formula is required.
NF(dB)=PASE(dBm)-10lgG-10lg(hv·B0)
In the formula, B0 corresponds to the spectrometer resolution Res=0.069 nm, and B0=c·Res/WL², which yields a B0 value of 8.616 GHz after calculation. The frequency v=c/WL=193.548 THz, and h stands for Planck's constant. Substituting all values, the constant term -10lg(hv·B0)=59.57 dBm, which is no longer equal to 58. Further calculation of the noise figure is performed as follows:
NF(dB)=PASE(dBm)-10lgG-10lg(hv·B0)
=-38.656-15.506+59.57=5.408 dB
This value matches the output result of the spectrometer.

Core conclusion: The value 58 is not a universal constant; it is only valid for the combination of 1550 nm wavelength and 0.1 nm bandwidth. The constant must be recalculated when the wavelength or spectrometer resolution changes.

Applicability of the "58 Formula"
The above derivation shows that the value 58 corresponding to -10lg(hv·B0) is calculated under the conditions of a 1550 nm wavelength and a reference bandwidth of 12.5 GHz. In the early stage, optical amplifiers were mainly applied to trunk transmission systems operating in the C-band, where the 58 formula was applicable.
However, with the emergence of Semiconductor Optical Amplifiers (SOAs), optical amplifiers have been deployed across the 800–2000 nm range, rendering the 58 formula inapplicable.

Scenarios where the 58 formula is applicable
1. Traditional EDFA trunk DWDM systems operating near the 1550 nm C-band wavelength;
2. Rough engineering estimation of system OSNR with the industry-standard 0.1 nm reference bandwidth;
3. Linear transmission links free from optical nonlinear interference;
4. Preliminary rough performance evaluation of C-band SOAs for scheme estimation only.

Scenarios where the 58 formula cannot be directly adopted
1. Optical amplifiers operating outside the C-band: O/S/L/U band amplifiers, SOAs covering 800–1400 nm and above 1600 nm;
2. Noise figure testing via spectrometers with resolutions other than 0.1 nm (e.g., 0.069 nm, 0.05 nm as shown in the case);
3. Pure Raman distributed amplification systems, whose noise models differ fundamentally from those of EDFAs and SOAs;
4. High-speed long-haul transmission links with severe nonlinear effects requiring precise simulation;
5. Accurate laboratory component testing and factory calibration of products, where the original standard formula must be adopted.