This week in my optics class, I have been teaching Kramers-Kronig (KK) relations of electric susceptibility. It is fascinating to see the causality argument emerge from the relationship between the real and imaginary parts of the complex susceptibility. Whereas the time domain explanation is relatively easier to appreciate (that dissipation follows perturbation in time), for me, the frequency domain implication in KK relation is fascinating: the fact that information about the real part of the function at all frequencies can give you insight into the imaginary part at any given frequency (and vice versa) makes it such a powerful mathematical and physical tool. For example, by knowing the absorption spectrum of a medium, you can find out the refractive index of a medium at a particular frequency that is not easily accessible in experiments.
Two inferences I draw:
1) Complex analysis combined with differential calculus is one of the most beautiful and powerful mathematical tools invented, and exploring its application in experimental scenarios has made physics intriguing, useful, and profound.
2) The KK relationship shows how causality and the structure of matter are connected to each other, and by studying them, one will be able to extrapolate the idea beyond the problem at hand and apply it to a different context in physics. It just shows how ideas hop from one domain to another and how mathematics plays a critical role in intellectual arbitrage.
Real is imaginary and vice versa. Complex numbers zindabad!
Whereas Sunday was bright, sunny, and clear for outdoor activities, Monday started cloudy with a forecast of rain. I started from my living place to Kyoto University around 10 in the morning. I took the city bus, which shuttles people from the city centre to the university. Within half an hour, I was in a serene, green, and beautiful campus, typical of a Japanese university. Kyoto University has a rich blend of modern and ancient architecture, and I was not surprised to see a large maroon-coloured ark at the entrance of the university.
With Prof. Tetsuro, who hosted me at the Graduate School of Informatics at Kyoto University.
I met Tesuji Tetsuro upon arrival (our previous in-person meeting was in the 2023 Optics & Photonics Congress on optical manipulation at Yokohama). He had just arrived from his run (he is a regular marathon runner), and we had a brief chat. He had arranged an office for me to occupy for the day. We had a short discussion and thereafter went for lunch. Prof. Kazuo Aoki (Tetsuro’s erstwhile advisor at Kyoto University) accompanied us, and I was delighted to meet him. We had a delicious lunch at a small Italian restaurant.
Around 3 pm, we met at the seminar hall where I gave my talk titled Hot Brownian Dynamics Driven by Structured Light. One of the key points I emphasized in my talk was the relevance of structured light in driving Brownian dynamics of colloids. I spoke about various parts of the stochastic differential equation (see equation 1 below) that represent the dynamics of a colloidal system interacting with an external force.
A key element of my discussion was the generalized driving force on the right-hand side of the equation, where the conventional restoring force in an optical trap can be generalized to an external driving force due to structured light. This versatile force is a result of a large set of linear and angular momentum states of structured light. These states can drive soft matter, further resulting in unconventional assembly and dynamics. Furthermore, the generalized driving force can include not only the optical force but also the thermal and hydrodynamic effects initiated by optical illumination. The combination of these forces culminates in a resultant force, offering an unconventional driving mechanism to drive the structure, assembly, and dynamics of colloids and other kinds of soft matter systems, including droplets and fluids. I showed some of our experimental results related to the above-mentioned concepts with emphasis on rotational and orbital degrees of freedom. I also presented our recent results on synchronization in an optothermal trap.
campus mapnear the entrance of Kyoto University at a Japanese izakayaWith Tetsuro and some PhD students
We had a long discussion on how to measure fluid dynamic properties around such colloids, especially when there is an external perturbation force, such as a laser beam, which can itself influence the colloidal dynamics. Tetsuro also mentioned his protocols and certain simulation strategies utilized to study thermo-osmotic flows in such situations. I learned about interesting methods they have been developing to numerically simulate the interactions using differential temperatures. The strategy is interesting and deserves further attention by the community. He also showed his experimental setup and gave a tour of his laboratory facilities.
Overall, it was a long, thoughtful day with wonderful discussions on topics of common scientific interest. We ended with a delicious dinner at a Japanese izakaya, and I thank Tetsuro for his invitation and hospitality. Kyoto University has a wonderful atmosphere for research, and I hope to visit again.
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In the immediate future, designing energy-efficient computational platforms will be a necessity. Electronic transport is noisy and dissipative. Optical alternatives can be important, but challenges remain…
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