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-

ts

1-D Shower Model

3

Scalar Potential in Near-field: ∵ no signal before ts

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 = 2000m R = 10000 m

Single Particle, finite track

6

c +5 。

track length = 100 m

R = 100 m R = 500 m

R = 2000m R = 10000 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 beproduced 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

show

er

R

Cherenkov

radiatio

n

Maxwell equations in Cylindrical Coordinate

d/dφ = 0

σm = 0

Hz = Hr = Eφ = 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 ~ 1019eV

~ 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 abipolar & asymmetricfeature, 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.

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Performance of GPU Calculation

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~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 sizeis limited by the computer memory.(at most 8k*8k ~ 1GB)

More sophisticated algorithm has to be implemented (e.g. Adaptive mesh refinement)

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Results from FDTD: Waveform

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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)

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(from top to bottom) (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

32

22

2

)(

)()(2)(

x

xxfxfxxf

x

f

)](exp[0 wtkxiff

fx

xk

fx

xikxik

2

2

2

)2

(

)2

(sin

)(

)exp(2)exp(

02

2

2

2

fc

w

x

f

---- (1)

---- (2)

(2)

(1)

0)

2(

)2

(sin

2

2

2

2

c

wx

xk

---- (3)

Numerical dispersion relationcxk

xk

V)

2(

)2

sin(

Phase velocity

---- (4)