# Laguerre-Gaussian Modes: Post #3

During my last week in the lab, I finished the fiber optic coupling part of my project. The data I collected led to more questions than answers, but that’s science. There’s a special excitement and beauty in knowing that the quest for knowledge will never truly end. This project has sparked my curiosity, and there’s so much more to learn now. Here are my activities and findings from my last days in Small Hall this July:

Early in the week, I moved the double vortex across the laser beam. I then conducted a data analysis for this experiment as well as for the single vortex case, which I had investigated the week before. I graphed the fiber power output vs. vortex position. The results matched my expectations: as the vortex moved off the beam, away from the center, the power output increased. During the experiment, the power meter fluctuated slightly. To diminish this effect, I took the ratio of the power output to input. In fact, this ratio represents the “coupling efficiency,” an indicator of how well I aligned the laser so that the maximum amount of light would make it to the end of the fiber. For a pure Gaussian beam incident on a single mode fiber, a coupling efficiency of 50% is considered reasonable.

For this experiment, the coupling efficiency was lowest when the vortex was centered on the beam. The efficiencies were 3.8% and 0.89% for the single and double vortices, respectively. When the vortex left the beam, the coupling efficiency reached 60%. This large change suggests that other modes were successfully filtered out of the fiber when the vortex was applied. The images below show the single vortex as it moved across the beam. Each image corresponds to fifteen micrometers of change in vortex position.

Moving the Single Vortex from center to 75 micrometers

Moving the Single Vortex from 90 to 165 micrometers

As an interesting side note, I learned how to distinguish between the single and double vortex. By placing a tilted lens in front of the beam, one can observe a long horizontal interference pattern. A single dark fringe in the pattern correspond to the single vortex, while two dark fringes indicate the double vortex. This technique provides a reliable method for checking which vortex is applied to the beam.

After coupling the pure vortices with the physical phase mask, I progressed to superpositions on the SLM. I concentrated on the modes l=+/-1,0, while p was fixed at 0. First, I measured the power of the light entering and exiting the fiber for the three pure modes, the pure Gaussian (|0,0>), and |+/-1,0>. Then, I superimposed |0,0> and |1,0> and measured the incoming and outgoing power for these two modes combined at different percentages. I repeated this procedure for |0,0> combined with |-1,0>.

I observed high percent errors between the measured and expected output power in both cases. For example, for |0,0> combined with |1,0> at 50%-50%, we would expect half of the power output of the pure |0,0> mode added to half of the power output of the pure |1,0> mode. I measured the output power of these pure modes to be 51.46 and 1.2 microwatts, respectively. Thus, theoretically, I should measure (0.5*51.4)+(0.5*1.2) = 26.33 microwatts. Instead, I measured 13.22 microwatts, for a 50% error. For the other percentage combinations, the error ranged from 4.37% to 64.37%. Additionally, for |0,0> combined with |-1,0>, the error ranged from 0.8%-30.53%.

In addition to the high error, we noticed some strangeness about these results. First, for |1,0>, the error was highest in general closer to 50%-50% than 90%-10%. This result alone makes sense because combinations farther from 50%-50% approach a single mode. However, the data for |1,0> and |-1,0> were quite different, when these two modes should be symmetrical. We speculated, then, that the SLM formed these mode differently. To confirm our hypothesis, we took images of these two modes and found that, indeed, the images differed quite significantly.

I consulted with Professor Novikova, and we decided to do something fun with these results. Based on the power output data, it seemed that the percentages for the superpositions, which we input into the SLM, did not match the percentages at which the light actually combined, likely due to problems with the SLM itself. So, essentially, I would act as the computer: I would send a percentage to the SLM, but then use my own judgment to compare the image of the beam to the theoretical ones and choose the best match, regardless of what I input into the SLM. Then, I would measure the power output, do some calculations, and determine how well my “guess” compared to the actual superposition the SLM generated.

For example, I thought the 50%-50% superposition of |0,0> and |1,0> more closely resembled a combination weighted as 30%-70%. I measured the output power of |0,0> and |1,0> alone as 56.5 and 2.08 microwatts, respectively. So, if the intended 50%-50% superposition did indeed turn out as 30%-70%, the expected power output should be (0.3* 56.5)+(0,7*2.08) = 18.41 microwatts. Similarly, the expected output of a 50%-50% superposition would be 29.29 microwatts. I measured 12.80 microwatts.  Thus, my guess for the superposition percentage came much closer than what I input into the SLM.

My other guesses were similarly close. Interestingly, though, farther from 50%-50%, my guesses better matched the values I input into the SLM anyway. This fact corroborates the earlier observation that the percentages farther from 50%-50% had lower percent errors. I took images of all the superpositions since this task was qualitative and rather subjective. My guesses did not yield any conclusive results other than to confirm our conviction that the SLM did not generate superpositions correctly. However, the exercise gave me an interesting insight into a task that a computer program usually performs. As Professor Novikova said, the human eye alone can be quite apt at recognizing images, calling into question our trust of machines.

After some conferring, we decided to investigate the SLM’s power calibration. We wanted to determine whether the powers of the beams entering and exiting the SLM matched. If they did not, we would know that the SLM somehow distorted the beam and required a recalibration, thus causing improper superpositions. I removed the lenses from my setup and conducted the test. However, I obtained normal results. So, we faced a bigger, unknown problem.

As a starting point, we compared our code for generating superpositions with that of our collaborators at LSU. We found no striking difference except that they added a sign change or absolute value somewhere, which caused a small change in our phase masks. In the meantime, though, we noticed that our images improved substantially without the lenses in the setup. We had added the lenses to increase the size of the beam on the SLM, but each lens contains aberrations, which compound to distort the light. So, I realigned the laser without the lenses.

I retook the coupling data twice, at an efficiency of 41% and 68%. The percent errors, however, remained high. The power output for |1,0> and |-1,0> differed drastically, but only for the more efficient coupling, which is perplexing. Additionally, the numbers for |-1,0> looked rather reasonable, while the ones for |1,0> were far off, which the highest error being 90%. The results indicate that perhaps our assumed method of calculating the expected power output using “simple math” does not accurately represent the way the modes add in space. Combined with the possibility that our SLM generates |1,0> and |-1,0> differently, this inaccuracy would contribute to the high error. Below is the data analysis for these two trials:

Data analysis for superposition of |0,0> and |+/-1,0> at a coupling efficiency of 41%

Data analysis for superposition of |0,0> and |+/-1,0> at a coupling efficiency of 68%

I tried two final tests to gather more information about the problem: First I superimposed |1,0> and |-1,0> with each other, rather than with the pure Gaussian beam. Again, the scattered results only confirmed our suspicion of the SLM’s inconsistencies. Below are the results of this trial:

Data analysis for superposition |1,0> and |-1,0> at an efficiency of 62%

Finally, we went into the Matlab code again and gave our superposition command one final tweak. We added a “multiplier” to the our equation modeling the electric field. This term can be 1, -1, i, or -i, and it describes the light’s phase. I retook the data for each multiplier. Ultimately, I got strange values again. No clear pattern emerged. We decided then that solving this problem would be a bigger project to embark upon in the future.

To conclude my project on a high note, I spent my last hour in the lab doing a fun extra experiment that involved playing with the laser modes. I added the physical phase mask I had used before to my current setup. Then, I sent superpositions to the SLM to determine the way the extra vortices on the physical phase mask would affect the combinations. I observed many cool interference patterns. Most interestingly, adding the single vortex to the mode |-1,0> caused a cancellation, which returned the light to the pure, Gaussian beam. Basically, adding two donut-shaped modes created a bright spot! I really enjoyed experimenting with random superpositions. The exercise helped me see more clearly the way the modes interact.

I have many more thoughts on this project now that it has come to a close. This post has gotten long, though, so I will save them all for my summary post, which will come soon.