Hi Jeapil,
How can we use K^\dag instead of K^-1? I understood that we used K^\dag to construct TM in noise, but I can’t see why K^\dag operate like K^-1 and be same phase conjugation, and KK^\dag(O_foc) be called time reversal operator.
There is two questions here:
- First, why does K^\dag operates like K^-1, well it does not really works exactly as the inverse operator, but it is good enough to focus light. When you use this operator to focus light, the only thing it does, is to bring back all the components of the light in phase at this particular point. Unlike the inverse operator, it does not minimize the intensity elsewhere, which is more tricky.
You will find more information in this paper:
doi.org/10.1088/1367-2630/13/12/123021
For the discussion on why the inverse does not work in the presence of noise, we treated that in the case of imaging:
doi.org/10.1038/ncomms1078
- Secondly, why KK^\dag(O_foc)is called the time reversal operator? If you have a pulse instead of a monochromatic light, doing time reversal consist in recording the signal and send it back reversed in time. If you do the Fourier transform, you find that time reversal operation is equivalent to doing phase conjugation for each frequency. If you have only a monochromatic light, inverting time is exactly equivalent of just conjugating the phase. That is why we say that phase conjugation is the monochromatic equivalent of time reversal.
I have acquired TM and focused light through scattering medium by using E_in(input electric field) = K^\dag*E_out(pattern desired). I can focus on one point and two points arranged horizontally but can’t focus on two points arranged vertically or five points or pattern such F-shape. If you tried to focus on multiple spots vertically, can you see the similar phenomenon that failed focusing?
I am not sure of what you ask. In a random scattering media, there is no difference between vertical or horizontal, so it should be the same. But, because phase conjugation does not compensate for the losses (unlike inversion it does not amplify the weak signals), if you try to focus on two points that are not equivalent statistically, you may focus only on the one where you have more signal. For instance, if the thickness of your medium is not homogenous and you try to focus on two points where the thickness is different, you will not have the same brightens. In both points you will put back the contributions in phase, but, because the average transmission is lower where the medium is thicker, the components you put back in phase are of lower amplitude. So you have one weak spot and one bright spot. That may be what happened.
Another thing, because phase conjugation is not "perfect" like inversion, it add some noise at elsewhere than the focus point too, so if increase the number of points you want to focus on, you decrease the signal to noise ration. At some point, you will not see anything.
Best,
Sebastien