My explanation of the effect is long a bit technical. I hope that it is intelligible.
(To keep this simple, I will ignore the phases of the waves.)
* * * Complex notation:
First, the equation for the electric field of the light is
E = A cos(kz-wt)
It's more convenient to write it as a sum of complex exponentials
E = A [exp(i(kz-wt)) + exp(i(-kz-(-w)t))] /2
E = Re( A [exp(i(kz-wt))])
By the linearity of the equations, the exp(i(kz-wt)) part and the exp(i(-kz-(-w)t)) have the same behavior, so you usually write simply
"E" = A exp(i(kz-wt))
and solve everything as if it where a complex function, but just before writing the final version or going to the laboratory you must remember that the other part was there, and that the real physical object is the real part of the function.
* * * Standard non linearity effects:
If the media is linear but not uniform, there appear other waves that travel in other direction z' (reflection/refractions). All of them have the same w.
"E_tot" = A exp(i(kz-wt)) + A' exp(i(kz'-wt))
Really all of them have two parts, one with w and the other with -w,
but usually you simply ignore that details, and only put a +cc (complex conjugate) or Re at the last minute.
If the media is no linear there can appear waves with a different frequency w'.
"E_tot" = A exp(i(kz-wt)) + A' exp(i(k'z-w't))
(There can appear more than two exponentials.)
Again they have two parts, and the real thing is the real part. It's more clear to choose w' as a positive number, because (-w') will appear in the hidden part of the equation.
E_tot = Re("E_tot")
* * * New non linearity effects in this article:
In this article they use a very sharp pulse in a very non linear material. So, from the
"E" = A exp(i(kz-wt))
part they get two new exponentials
"E_tot" = A exp(i(kz-wt)) + A' exp(i(k'z-w't)) + A'' exp(i(k_n''z-w_n''t))
where k' and w' are positive numbers as expected. But k_n'' and w_n'' are negative numbers!! They get this numbers from the same equation that has k' and w' as a solution, so all of them appear from the same mathematical term. They call this negative solution "NRR".
But it is important to remember that the original E has two exponentials, so you must repeat all the computations with the other part
"E*" = A exp(i(-kz-(-w)t))
everything is equivalent, so after some recalculations you get
"E*_tot" = A exp(i(-kz-(-w)t)) + A' exp(i(-k'z-(-w')t)) + A'' exp(i(-k_n''z-(-w_n'')t))
where every k and every w has an additional "-". They call this part "NRR* ". But now -k_n'' and -w_n'' are positive numbers. We can change the names, and call
k'' = - k_n''
w'' = - w_n''
and now k'' and w'' are positive numbers. So the first part of the solution is now
"E_tot" = A exp(i(kz-wt)) + A' exp(i(k'z-w't)) + A'' exp(i(-k''z-(-w'')t))
and the second part is
"E*_tot" = A exp(i(-kz-(-w)t)) + A' exp(i(-k'z-(-w')t)) + A'' exp(i(k''z-w''t))
And the real physical object is
E_tot = ("E_tot"+"E*_tot")/2
So we can regroup the term. We exchange the terms with k'' and w'' that have the wrong signs from on part to the other, because the sum doesn't change.
"E_totx" = A exp(i(kz-wt)) + A' exp(i(k'z-w't)) + A'' exp(i(k''z-w''t))
and the second part is
"E*_totx" = A exp(i(-kz-(-w)t)) + A' exp(i(-k'z-(-w')t)) + A'' exp(i(-k''z-(-w'')t))
and as before
E_tot = ("E_totx"+"E*_totx")/2
Now the interpretation of "E_totx" is straightforward. From the original field "E" you get three waves, with frequencies w, w' and w'', all of them positive. And in "E* _totx" is the complex conjugate part, so the final result is real.
And they can measure the three waves.
* * * Notes:
Usually, the A'' coefficient is so small that all this strange part can be ignored, but they were able to measure it in the laboratory.
One important detail is that k/w, k'/w' and k_n''/w_n''= k''/w'' are all positive, so they represent waves that travel in the same direction.
At times like this I think it's a shame that HN doesn't support TeX formatting. It's much easier to follow a mathematical argument with proper notation.
Which allow a pretty seamless interleaving of LaTeX and normal content within a page. Within a submission one need only enclose the LaTeX content in unique delimiters, like the often-used $$double-dollar-signs$$.
All this because W3C can't seem to agree on a standard way to render mathematics in the browser itself -- or more to the point, get the browser builders to agree to anything they've proposed.
My explanation of the effect is long a bit technical. I hope that it is intelligible.
(To keep this simple, I will ignore the phases of the waves.)
* * * Complex notation:
First, the equation for the electric field of the light is
It's more convenient to write it as a sum of complex exponentials By the linearity of the equations, the exp(i(kz-wt)) part and the exp(i(-kz-(-w)t)) have the same behavior, so you usually write simply and solve everything as if it where a complex function, but just before writing the final version or going to the laboratory you must remember that the other part was there, and that the real physical object is the real part of the function.* * * Standard non linearity effects:
If the media is linear but not uniform, there appear other waves that travel in other direction z' (reflection/refractions). All of them have the same w.
Really all of them have two parts, one with w and the other with -w, but usually you simply ignore that details, and only put a +cc (complex conjugate) or Re at the last minute.If the media is no linear there can appear waves with a different frequency w'.
(There can appear more than two exponentials.)Again they have two parts, and the real thing is the real part. It's more clear to choose w' as a positive number, because (-w') will appear in the hidden part of the equation.
* * * New non linearity effects in this article:In this article they use a very sharp pulse in a very non linear material. So, from the
part they get two new exponentials where k' and w' are positive numbers as expected. But k_n'' and w_n'' are negative numbers!! They get this numbers from the same equation that has k' and w' as a solution, so all of them appear from the same mathematical term. They call this negative solution "NRR".But it is important to remember that the original E has two exponentials, so you must repeat all the computations with the other part
everything is equivalent, so after some recalculations you get where every k and every w has an additional "-". They call this part "NRR* ". But now -k_n'' and -w_n'' are positive numbers. We can change the names, and call and now k'' and w'' are positive numbers. So the first part of the solution is now and the second part is And the real physical object is So we can regroup the term. We exchange the terms with k'' and w'' that have the wrong signs from on part to the other, because the sum doesn't change. and the second part is and as before Now the interpretation of "E_totx" is straightforward. From the original field "E" you get three waves, with frequencies w, w' and w'', all of them positive. And in "E* _totx" is the complex conjugate part, so the final result is real.And they can measure the three waves.
* * * Notes:
Usually, the A'' coefficient is so small that all this strange part can be ignored, but they were able to measure it in the laboratory.
One important detail is that k/w, k'/w' and k_n''/w_n''= k''/w'' are all positive, so they represent waves that travel in the same direction.