Graduate Institute of Astrophysics, National Taiwan University Leung Center for Cosmology and Particle Astrophysics Chia-Yu Hu OSU Radio Simulation Workshop Simulation of Cerenkov signal in near-field 1

1-D Shower Model 2 Lorenz gauge Assuming longitudinal development N(z) with zero lateral width: Solving retarded time(s): t+t+ t-t- tsts

1-D Shower Model 3 Scalar Potential in Near-field: ∵ no signal before t s assuming

Single Particle 4 Infinite particle track, constant velocity (no acceleration), R = 100m Cherenkov shock

Single Particle, finite track 5  c track length = 100 m R = 100 m R = 500 m R = 2000mR = m

Single Particle, finite track 6  c +5 。 track length = 100 m R = 100 m R = 500 m R = 2000mR = m

Finite-Difference Time-Domain method We adopt the finite-difference time-domain technique, a numerical method of EM wave propagation. EM fields are calculated at discrete places on a meshed geometry. Near field pattern can be produced directly. Algorithm: 1) Initializing fields. 2) Calculate E fields on each point by the adjacent H fields. 3) Calculate H fields on each point by the adjacent E fields. 4) Go back to 2) 7 z r detector shower R  Cherenkov radiation

Maxwell equations in Cylindrical Coordinate d/dφ = 0 σ m = 0 H z = H r = E φ = 0 8

Single Particle in FDTD 9 Grid size is determined by the smallest length scale (lateral width in our case), in order to prevent from numerical dispersions.  Single point charge will lead to numerical artifacts!  Need to approximate by smooth functions (Gaussian)

2D Contour Plot 10

2D Contour Plot 11

2D Contour Plot 12

2D Contour Plot 13

2D Contour Plot 14

2D Contour Plot 15

2D Contour Plot 16

Waveform 17 track length = 4.5 m, R = 1 m

Waveform 18 track length = 4.5 m, R = 1.5 m

Waveform 19 track length = 4.5 m, R = 2 m

Waveform 20 track length = 4.5 m, R = 2.5 m

Waveform 21 track length = 4.5 m, R = 5.5 m

Towards more realistic cases LPM-elongated showers have a stochastic multi-peak structure (can be viewed as superposition of many sub-showers). Squeezing effect greatly enhances part of the shower in near-field, so the potential always shows a sharply rise and smoothly decay (with small oscillations) behavior. 22 R = 100m (near-field) R = 1000m (far-field) R = 300m E ~ eV ~ 150 m resembles the shower distribution (no squeezing) sharply rise & slowly decay five-peak structure gradually appears

Unique Signature in Near-field Due to the enhancement of squeezing effect, the waveform displays a bipolar & asymmetric feature, regardless of the difference of multi-peak structure from shower to shower. 23 R = 300mR = 200m R = 100mR = 50m

Summary We introduce an alternative calculation of coherent radio pulse based on FDTD method that is suitable in the near-field regime. It is also useful in studying the effect of space-varying index of refraction. Because of squeezing effect, the Cherenkov waveform in near-field regime presents a generic feature: bipolar and asymmetric, irrespective of the specific variations of the multi-peak structure. For ground array neutrino detectors (e.g. ARA) where the antenna station spacing is comparable to the typical length of LPM-elongated showers, the near-field effect must come into play. Detecting the transition from asymmetric bipolar to multi-peak waveform simultaneously provides confident evidence for a success detection. 24

25 Backup Slides

Graphics Processing Unit (GPU) The FDTD method is highly computational intensive, reasonable efficiency can only be obtained via parallel computing. We do it on Graphics Processing Unit (GPU), a device particularly suitable for compute-intensive, highly parallel computation. Compute Unified Device Architecture (CUDA). * Programmable framework provided by NVIDIA. * An extension of C language (easy to learn). A graphic card (NVIDIA GTX280) is capable to give a speed up 10x – 100x comparing with a single CPU. 26

Performance of GPU Calculation 27 ~10 mins~ 33 hrs

Limitations of the FDTD Method Even with the help of GPUs, the computing resources required by FDTD still challenge the state-of-the-art devices. Grid size is determined by the smallest length scale (lateral width in our case), in order to prevent from numerical dispersions. The ratio of simulation region to grid size is limited by the computer memory. (at most 8k*8k ~ 1GB) More sophisticated algorithm has to be implemented (e.g. Adaptive mesh refinement) 28

Results from FDTD: Waveform 29 More and more symmetric as the pulse propagates into the far-field regime Asymmetric bipolar waveform in near-field due to the squeezing effect The waveform in near-field generated by full 3-D simulation: bipolar & asymmetric (longitudinal contribution)

Frequency Domain (Spectrum) 30 (from top to bottom) (far-field)(near-field) Comparing spectra obtained from FDTD method (solid curves) with that from far-field formula (dashed curves). The FDTD method reproduces the far-field spectrum and provides correct solutions in near-field.

Distance Dependence 31 Cylindrical behavior (E   1/√R ) for small R Spherical behavior (E   1/R ) for large R transition at smaller R (low freq. mode diffracts more easily)

Numerical Dispersion (1) ---- (2) (2) (1) ---- (3) Numerical dispersion relation Phase velocity ---- (4)