Can It Be Proven That Optics Field Amplitudes Satisfy Negative Frequencies?

[Niles]

Hi

In (quantum) optics, many authors state that the field amplitudes satisfy

<br /> E\left( { - \omega } \right) = E^* \left( \omega \right)<br />

But how is it that one can prove that this is correct? I have never seen any book do this,

 

[Nabeshin]

Isn't this trivial and not specific to quantum cases anyways? Suppose you have an oscillating E-field (of course, the same works for the B-field component),
E(\vec{r},\omega,t)=E_0(\vec{r},t)e^{-i \omega t}

Then the property you mention is trivial.

 

[Niles]

You might be right; it probably isn't related to quantum cases. Here is how I have understood it: We can generally write

<br /> E\left( {r,t} \right) = \sum\limits_{n &gt; 0} {\left( {E\left( r \right)e^{ - i\omega t} + c.c.} \right) = \sum\limits_{n,\,\,all} {E\left( \omega \right)e^{ - i\omega t} } } <br />

The last equality follows it we define

<br /> \begin{array}{l}<br /> E\left( r \right) \equiv E\left( \omega \right) \\ <br /> E\left( \omega \right)^* = E\left( { - \omega } \right) \\ <br /> \end{array}<br />

But these are just definitions. So I don't see how we can really prove that <br /> <br /> E\left( { - \omega } \right) = E^* \left( \omega \right)<br /> <br />Niles.

 

[Niles]

They say it here as well (top of page 1052): http://books.google.dk/books?id=Jj-...q=nonlinear optics negative frequency&f=false

 

[Niles]

Ok, it's pretty obvious now. Thanks.