The authors derive an analytical efficiency formula for third-harmonic generation in ENZ materials using Green's tensor quantization, which reduces to classical results and matches their own ITO data only for angles above 20 degrees.
The fundamental solution of the space-time fractional diffusion equation
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abstract
We deal with the Cauchy problem for the space-time fractional diffusion-wave equation, which is obtained from the standard diffusion equation by replacing the second-order space derivative with a Riesz-Feller derivative of order alpha in (0,2] and skewness theta, and the first-order time derivative with a Caputo derivative of order beta in (0,2]. The fundamental solution is investigated with respect to its scaling and similarity properties, starting from its Fourier-Laplace representation. By using the Mellin transform, we provide a general representation of the solution in terms of Mellin-Barnes integrals in the complex plane, which allows us to extend the probability interpretation known for the standard diffusion equation to suitable ranges of the relevant parameters alpha and beta. We derive explicit formulae (convergent series and asymptotic expansions), which enable us to plot the corresponding spatial probability densities.
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physics.optics 1years
2026 1verdicts
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Quantum Theory of Third-harmonic Generation in Epsilon-Near-Zero Materials
The authors derive an analytical efficiency formula for third-harmonic generation in ENZ materials using Green's tensor quantization, which reduces to classical results and matches their own ITO data only for angles above 20 degrees.