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

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

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

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

5
Single Particle, finite track q = q c track length = 100 m
R = 100 m R = 500 m
R = 2000m R = 10000 m

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Single Particle, finite track q = q c
+5 。 track length = 100 m
R = 100 m R = 500 m
R = 2000m R = 10000 m

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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.
z detector
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) q
R r

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

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Single Particle in FDTD
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)

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2D Contour Plot

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2D Contour Plot

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2D Contour Plot

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2D Contour Plot

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2D Contour Plot

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2D Contour Plot

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2D Contour Plot

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Waveform track length = 4.5 m, R = 1 m

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Waveform track length = 4.5 m, R = 1.5 m

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Waveform track length = 4.5 m, R = 2 m

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Waveform track length = 4.5 m, R = 2.5 m

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Waveform 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).
E ~ 10 19 eV
~ 150 m
 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.
R = 100m
(near-field)
R = 300m R = 1000m
(far-field)
22 sharply rise & slowly decay five-peak structure gradually appears resembles the shower distribution
( no squeezing )

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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.
R = 50m
R = 200m R = 300m
R = 100m

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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.

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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.

Performance of GPU Calculation
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~ 33 hrs ~10 mins

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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)

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

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

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

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Numerical Dispersion f
= f
0 exp[ i ( kx
 wt )] ---- (1)
 2
 x
2 f
 2
 x
2 f
V

=

 f exp( ik
 x ) w
2 c
2
( x f
 sin(
2 k
 x
( k

2
) x
)
=
 c x
0
)

2 f (
---- (4) x )
 f (
Phase velocity x
  x )
(
 x )
2

2

(
 x ) 2 exp(
 ik
 x )
(2) f
=  sin
(
2
(
 x
2 k
 x
)
2
)
2 sin 2 (
(
 x
2 k
 x
)
2
) 2
 w
2 c
2
=
0 f
---- (2)
---- (3)
Numerical dispersion relation