# Decibel

Author: the photonics expert Dr. Rüdiger Paschotta

Definition: a logarithmic measure for power ratios, applied e.g. to optical powers or to noise powers

The *decibel* (dB) is often used for quantifying the gain of an amplifier or the loss of some optical element, such as an optical fiber or an optical attenuator. The number of decibels is 10 times the logarithm (to base 10) of the power amplification factor or loss factor, or alternatively 20 times the logarithm of the amplitude ratio of the electric field strengths. If there is an exponential gain coefficient <$g$> such that the power amplification factor is <$\exp(g)$>, the decibel value is <$10 \: \lg \exp(g) = 10 \: \lg(e) \cdot g ≈ 4.34 \cdot g$>. Similarly, one quantifies the insertion loss of some optical component as a decibel value: <$10 \: \lg(P_\rm{in} / P_\rm{out})$>.

Such a logarithmic quantity is useful because e.g. the decibel gain values of several amplifiers in a sequence can simply be added to obtain the total gain of the amplifier chain. Similarly, one can add up the decibel values of attenuators used in a sequence.

## Decibels in the Context of Optical Signals

The decibel is also often used in the context of transmitted signals (e.g., for optical filters) and of noise e.g. of lasers or amplifiers. In the context of optical signals, one is dealing with two different kinds of power, which should of course not be confused:

- There is the
*optical power*of a signal. In the case of direct detection with a photodetector, this is translated into an electrical photocurrent or voltage. In other cases, e.g. with modulation of the optical phase, the optical power is not directly relevant for the signal. - The
*modulation power*is proportional to the square of the signal amplitude. If the signal is an electrical voltage or current, the signal power corresponds to an electrical power delivered to a given impedance (e.g. 50 Ω). However, one may also take the signal to be an optical power, e.g. in cases with intensity (power) modulation; the signal power is then again related to the square of the signal amplitude and can thus have units of W^{2}(watts squared).

When the optical input power is doubled while the modulation remains unchanged, the signal power (modulation power) will be increased by a factor of 4. Therefore, 3 dB more optical input power leads to 6 dB more signal power.

## Frequently Used Specifications

Some frequently used related specifications are:

- dBc = dB relative to the carrier. This is used e.g. to specify the power of a sideband in a modulated signal relative to the carrier. For example, −30 dBc means that the sideband is 30 dB below the carrier, i.e., it has a 1000 times lower power.
- dBc/Hz: This is used for noise and means dBc in a 1-Hz bandwidth.
- (Of course, this does not mean that there would be twice as many dBc in a 2-Hz bandwidth, as decibels are a logarithmic measure; therefore an interpretation as “dBc per hertz” would not be appropriate!)
- Often, such specifications are calculated from measurements based on a larger bandwidth. For example, if one obtains −25 dBc in a 1-MHz bandwidth, this converts into −85 dBc in 1 Hz, i.e., −85 dBc/Hz. The 60-dB difference reflects the bandwidth reduction by a factor of 10
^{6}. - dBm = dB relative to a reference power of 1 mW. This is often used to specify absolute power levels, e.g. of the saturated output power of a fiber amplifier. For example, 23 dBm correspond to 10
^{2.3}× 1 mW = 200 mW.

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