I don't get the Slowly Varying Envelope Approximation

[carllacan]

Hi.

I can't for the life of me understand the math behind the SVEA. I graphically/intuitively understand what it means that the envelope varies slowly, but I can't connect that with the mathematical expression: $$ \left \vert \frac{\partial ^2 E_0}{\partial t ^2} \right \vert << \left \vert \omega\frac{\partial E_0}{\partial t}\right \vert $$

If the field is sinusoidal then the first derivative should simply add a factor of ω , so that $$ \left \vert \frac{\partial ^2 E_0}{\partial t ^2} \right \vert = \left \vert \omega\frac{\partial E_0}{\partial t}\right \vert $$

but instead the approximation says that <<. What am I missing?

THank you for your time.

 

[blue_leaf77]

SVEA means that the envelope is almost linear everywhere with very minimal bends. That's how you come with
$$
\left \vert \frac{\partial ^2 E_0}{\partial t ^2} \right \vert << \left \vert \omega\frac{\partial E_0}{\partial t}\right \vert
$$
If the above inequality holds, you don't need to worry about the higher order derivatives, they are guaranteed to be much smaller than the first derivative.
You can imagine a function which increases monotonically and is very close to being linear for $t<0$. In $t>0$, it's also very close to linear but keeps decreasing. In these two regions, the above equality can be seen to hold with a high degree of validity. But at the turning point $t=0$, the function may bends downwards too steeply such that the second derivative cannot be neglected at this point. To avoid this, the transition must be occurring with a somehow very long duration of time such that everywhere the function's second derivative is negligible. Just imagining this picture in mind should lead you to conclude that this function must vary slowly.

If the field is sinusoidal then the first derivative should simply add a factor of ω

$E_0$ is the envelope, the sinusoidal oscillation is already excluded.

 

[carllacan]

SVEA means that the envelope is almost linear everywhere with very minimal bends. That's how you come with
$$
\left \vert \frac{\partial ^2 E_0}{\partial t ^2} \right \vert << \left \vert \omega\frac{\partial E_0}{\partial t}\right \vert
$$
If the above inequality holds, you don't need to worry about the higher order derivatives, they are guaranteed to be much smaller than the first derivative.
You can imagine a function which increases monotonically and is very close to being linear for $t<0$. In $t>0$, it's also very close to linear but keeps decreasing. In these two regions, the above equality can be seen to hold with a high degree of validity. But at the turning point $t=0$, the function may bends downwards too steeply such that the second derivative cannot be neglected at this point. To avoid this, the transition must be occurring with a somehow very long duration of time such that everywhere the function's second derivative is negligible. Just imagining this picture in mind should lead you to conclude that this function must vary slowly.

$E-0$ is the envelope, the sinusoidal oscillation is already excluded.

All clear now, thanks!