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The propagation of light in optical fibers is governed by the equation

$i\frac{\partial A}{\partial z} = \frac{\beta_2}{2}\frac{\partial^2A}{\partial T^2} - \gamma|A|^2A$

where $A(z,T)$ represents the amplitude of the field envelope and $\beta_2$ and $\gamma$ are constants.

My textbook (Nonlinear Fiber Optics by Agarwal) says that the amplitude $A$ is independent of $T$ in case of CW radiation at the input end of the fiber ($z=0$).

Why is $A$ independent of $T$?

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3 Answers 3

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Note that $A$ is the field envelope, which means that $A$ is the average over some duration of time and interval of space. It is closely related to intensity in that respect. For CW excitation, the envelope is constant. There's no mechanism shown that would cause a time-varying envelope within the fiber. In other words the first term on the right serves as a source term, and since the source is CW, the first term on the right is zero.

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  • $\begingroup$ Just to be very clear, let me consider the image at en.wikipedia.org/wiki/Envelope_(waves)#/media/…. What you are referring to is the red and green envelopes of the sine wave right? So if the amplitude of the sinusoid is constant, then both the red and green lines become flat and hence independent of $T$? $\endgroup$
    – Paddy
    Commented Jul 24, 2020 at 17:48
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    $\begingroup$ @Paddy, in principle it just means the difference between the two lines is constant. In practice, at optical frequencies there's no way for that to happen without the two lines both being constant individually. $\endgroup$
    – The Photon
    Commented Jul 24, 2020 at 17:57
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That's the definition of "CW radiation" in this context.

CW stands for "continuous wave", but here it is being used to mean "not varying with time".

Much like in electronics the term "DC" literally means not changing direction, but has been co-opted in circuit theory to mean not changing at all.

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The $A(z,t)$ in the fibre optics equation is the $A$ in $$ E(z,t)=A(z,t) e^{i(kz-\omega t)}. $$ It does not include the $e^{-i\omega t}$ part of the CW time dependence therefore.

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