quency component of the field at the amplifier input that is also a single frequency component of the fundamental mode of the fiber, can be written, starting from Equation 4.2 as (4.55) where indicates operators ain is the input creation operator and the field is measured, as it is generally advanta- geous in telecommunication related problems, in W-1 The frequency dependent ain contains all the amplitude fluctuations (e.g., the modulation), and the phase can be written as (4.56) We will also assume that the field bandwidth, which we will call B measured in angular frequency, is much smaller than the amplifier gain bandwidth, hypothesis matching prac- tical situations very well. Technology for Telecommunications: Optical Fibers, Amplifiers, and Passive Devices After traveling through the amplifier, following the amplifier definition and neglect- ing for the moment any noise contribution from temperature or spurious radiation, the output field frequency component can be written as a function of the amplifiers spectral gain G(w) as (4.57) where T is the delay due to the propagation through the amplifier, and with a redefinition of the time origin can be eliminated. The operator f(w) is added at the output to comply with the property that the output field has to be written as a function of the output creation operator as (4.58) and aout(a) has to commute following the rule (4.59) where + indicates the Hermitian conjugate S(@ - w') indicates the Dirac distribution The relation (4.59) requires the presence in Equation 4.57 of the term containing f(w), that physically represents the contribution of the spontaneous emission that is unavoidable at the amplifier output. Using Equations 4.57 and 4.58, the output creation operator expression can be obtained. Substituting this expression in (4.59), the following expression is obtained for the auto- commutation of the operator f(w): (4.60) In order to provide a characterization of the quantum noise, the optical power flux through the amplifier can be evaluated. Using the symm