Tutorial Number 10
MAKING VORTEX LOOPS
In this example we are going to construct a pair of linked vortex loops. In doing so we will become acquainted with the construction of Laguerre-Gauss functions and their propagation. I have chosen to use the 512x512 screen default in the example because it is large enough to show what you need and small enough to run fast.
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In this example we are going to construct a pair of linked vortex loops. In doing so we will become acquainted with the construction of Laguerre-Gauss functions and their propagation, as well as learn some things about computation effects. If you set up the window as shown at the left you should get the results shown below. Try some variations. This beam expands with distance according to the equation (there are versions of this equation in many places this is somewhat modified from Wikipedia) and undergoes both a quadratic phase shift with distance an a mode and distance dependent "Gouy" phase shift. That the rate of change with distance depends upon the size of the original LG function is shown in the following example. |
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Set the Gauss window to look like the one at the top of this page. Make both display windows to real part and colorize (for a start). Then set the chirp window to look like the one at the left. When you close the window you will be asked to select a file which will be read by this window to fill in a different distance on each cycle. Select the text file you created above. The HOLD icon will stop execution until you press the space bar or click continue. Before you click continue you may look at the appearance of the imaginary, magnitude, and phase by choosing from the drop down menu above the main display. You should see something like the images below. |
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THE HOPF LINK The script at the left is inspired by the article by J.Romero et. al., "Entangled Optical Vortex Links" PRL 106, 100407 (2011) and previous papers by M.R.Dennis. The numbers are taken from this paper. The script forms a weighted sum of 4 LG functions all have σ=66.67 pixels z=0 these are each multiplied by an appropriate weight saved and summed. #1 l=0,n=0 weight=0.264 #2 l=0,n=1 weight=-0.628 #3 l=0,n=2 weight=0.426 #4 l=2,n=0 weight=-0.596 The weights are entered in the rescale icons beneath the Gauss icons. The chirp icon is set the same as above. As this executes you should see the two vortex pairs appear one of each pare circle the other and finally rejoin. |
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"How do I make a hologram that produces this vortex" I here you cry. The answer is in the script at the left. It does most of the work of making a field of number that can be turned into an appropriate hologram. It also checks out the result. The top four lines of the script are the same as before (you can clip and paste between scripts). The first HOLD stops to show you the result of the first four lines. Following that is a direct propagation to calculate the result of propagating 1755.4 mm. The complex conjugate produces the phase conjugate which is the hologram. Again you get to inspect the result. The script then proceeds to the familiar loop that follows the progression of the beam. The DIRECT icon is used rather than the CHIRP because the former does not calculate by convolution on a torus, thereby letting us do a large scale change to the pixel size of the light valve. Figure 5 shows the settings for the first of these DIRECT icons. The pixel size at plane 2 is selected to be the pitch of our light valve. The pixel size in P1 is made a little smaller than the example above because the resulting hologram fills the screen a little better. The distance 1755.4 mm in the calculation is selected to keep the spacial frequency at the edge within the Nyquist limit. |
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