Creating Higher-order Mode Laser Beams
|HG10 mode||HG20 mode||HG11 mode|
If one uses a beam splitter to split HG beams and then overlap them together to observe the interference pattern, he would see the relative phase difference between lobes of HG modes.
I used one beam splitter infront of the laser and it splitted the beam into two branches. I used a couple of lens to amplifiy the size of one branch of beams to make it a reference beam. Then two branches need to be overlap together again. I overlaped the beam from the lower branch showed in the set up (which was not be amplified) with one lobe of reference beam. The result was showed below.
|HG10 mode||interference of HG10
||interference of HG11
||interfence of HG20
I could see phase difference between two lobes clearly in the first two pictures. One can see that how a bright horizontal fringe in one lobe conresponding to a dark one in the other one. That was caused by the phase difference between two lobes.
Light can be twisted by two cylindrical lens around its axis of propogation. If one shines a twisted light on a flat screen, he can see a light "ring". Because ligth with different phase cancel each other at the center and that makes a singularity point at the center of spot. Here I want to first introduce formulas for a Hermitian Gaussian(HG) and Laguerre-Gaussian mode. Here are equations for HG and LG which are propogate along z axis.
One can see from this equation that HG beam is living in Cartesian
coordinates. It depends on x,y and propogation direction z. At the same
time, an LG mode is depending on r (distance from the center of "ring"),
\[Phi] (angle) and propogation direction z. LG modes only excist in
From equations above,we saw that LG mode can be decompsed into HG modes. A LG(1,0) mode can be decomposed into two parts. The first one is a HG01 mode and the second is HG10 mode which is half Pi out of phase with the first HG component. This decomposition can be illuminated by graph below.
The only changes I made in this setup compared to the last one is this converter in the path of lower beam. The converter was made up by two cylindrical lens with same focal length which were seperated by double focal length diatance. Because cylindrical lens only has one direction be curved, so input beam will undergo different changes in Gouy phase (1). That can make the phase whift between two components I want. For more details about how this converter works, one can check M.Pagget's paper(2).
|Input light||Reference beam
This time, I made vortex I got from this converter and overlaped it with reference HG beam. The interference pattern was showed in the last picture. In this last picture, we can see this "fork" clearly in the center. Why there is a "fork"?
We can see in this picture, when I overlap reference beam and vortex, the vortex inferenced with two different parts of reference beam. Due to the phase difference between lobes, there will be a "phase jump" at the center which is caused by discontinuation of phase.