CTA 報告 48: ROOT ベースの光線追跡
シミュレータの開発とその応用
奥村 曉
名古屋大学 太陽地球環境研究所
他 The CTA Consortium2012 年 9 月13 日
日本物理学会 2012 年秋季大会 @ 京都産業大学
A. Okumura et al. (2011) ICRC 2011
A. Okumura (2012) Astropart. Phys.

“Optical” Cosmic-ray Telescopes

H.E.S.S. IIMAGICSTACEE

http://pic.twitter.com/XQDY6d26

http://magic.mppmu.mpg.de

http://www.astro.ucla.edu/~stacee/

Pierre Auger Observatory

JEM EUSO

Ashra

http://www.auger.org

http://jemeuso.riken.jp

Final Optics Design by A. O.

Segmented Mirrors with Complex Configurations

H.E.S.S. I (Davies-Cotton Optical System)

STACEE (Heliostats)© H.E.S.S.

http://www.astro.ucla.edu/~stacee/

Need a large and complex optical system with segmented mirrors
Building the geometry and tracking photons in “non-sequential” mode are not easy
Existing programs’ functions are not adequate

Pixelated Focal Planes and Light Collectors

H.E.S.S. II Camera

H.E.S.S. II Winston Cones

http://www.mpi-hd.mpg.de/hfm/HESS

http://www.mpi-hd.mpg.de/hfm/HESS

The focal plane of a typical cosmic-ray telescope is pixelated with PMTs
Hexagonal Winston cones and similar ones are used to reduce dead area
Pixel information and multiple reflection on light collectors are important

Photon Timing Analysis for Large Dishes

MAGIC (Parabolic 17-m Dish)

H.E.S.S. II (Parabolic 28-m Dish)

http://magic.mppmu.mpg.de

http://www.mpi-hd.mpg.de/hfm/HESS

Signal timing is important information for gamma-ray detection
Parabola Cherenkov signal ~5 ns
Optical path diﬀerence of a 10-m Davies Cotton telescope is Δt ~ 5 ns

Connection to Air-shower and Electronics Simulations

CORSIKA MC (1 TeV Photon)

TARGET (ASIC Developed for CTA)

http://www.ast.leeds.ac.uk/~fs/photon-showers.html

Bechtol et al. (2012)

Used along with an air-shower simulator and an electronics simulator
No “Windows Only” software allowed (Linux and OS X preferred)
C++ and Python preferredExisting Ray-tracing (Particle Tracking) Software

Geant4 Optical Process (Pierre Auger)

Zemax for Optimization Study (Ashra)

Abraham et al. (2010)

by A. Okumura

Geant4 is famous and powerful, but can you use it? Very hard for C++ beginners
Zemax is a standard commercial software with a fast optimization engine, but
expensive and Windows only

Analyses with ROOT

200
180

435
30

120

2540

θ = 3.0 (deg)16-2

15-410-6

5-4-2

02

468X (arcmin)

0108642-8-6-4-202468X (arcmin)θ cross

θ max

80
60
40

Winston Cone
Quadratic Bézier

20

Cubic Bézier

00
05101520

15
10
50025
30
35
Incident Angle θ (deg)

51015

20

2530
35
Incident Angle θ (deg)(d)

R = 0.9, n = 1.550
45

θ cross

θ max

40
3530
2520
1510mm2020

ρ 1 = 20

025mm

1230

ρ 1 = 3025100mm

2R = 0.9, n = 1.5

ρ 1 = 2014352530
35
Incident Angle θ (deg)mm30-620ρ 1 = 30

4-8-81518 (c)θ = 4.0 (deg)351040mm05

θ max

ρ 1 = 20000

θ cross

45mm

5Cubic Bézier

R = 1.0, n = 1.050

ρ 1 = 3020mm

10Quadratic Bézier

20

ρ 1 = 20-640

Winston Conemm

1560-460
40

208080ρ 1 = 30

100-2Y (arcmin)

40

140

0645

160

2-88

θ = 2.0 (deg)θ maxCollection Efficiency × sinθ (%)θ = 1.0 (deg)θ cross(b)Collection Efficiency × sinθ (%)

6

R = 1.0, n = 1.0

100

Collection Efficiency (%)

Y (arcmin)

8

Collection Efficiency (%)(a)

5005101520253035
Incident Angle θ (deg)PSF of Schwarzschild-Couder TelescopeCollection Eﬃciencies of Light Collectors

A. Okumura et al. (2011) ICRC

A. Okumura (2012) arXiv:1205.3968

ROot-BAsed Simulator for ray Tracing (ROBAST)

ROOT geometry libraryfor particle tracking
and complicatedgeometry construction
Open source software
http://sourceforge.net/projects/robast/
Runs in C++/CINT/
PythonA. Okumura et al. (2011) ICRC 2011

The First Application (The Ashra Optical System)
A good example toshow how complex
geometries can bebuilt with ROOT
Lens parameters wereoptimized with Zemax
Realistic eﬀective areaof the optical system
can be estimated byconsidering shadows
Study on alignment of
non-ideal segmented
mirrors

CTA Application 1: Schwarzschild-Couder (SC) Optical System
Segmented
Secondaries
5.4 m
Focal Plane Camera
SegmentedPrimaries

The first dual-mirror
design in VHE
Proposed by CTA US
and Europe groups9.7 m

Aspherical segmented
mirrors for primaries
and secondaries
Multi-anode PMT(MAPMT) or multi-pixel
photo counter (MPPC)
array

Schwarzschild-Couder Telescope for CTA
Original design in Vassiliev et al. (2007) Astropart. Phys.

CTA Application 1: Schwarzschild-Couder Optical System
52 mm × 52 mm

SC Telescope built with ROBAST

MAPMT Array at the Focal Plane

Original design in Vassiliev et al. (2007) Astropart. Phys.

A. Okumura et al. (2011) ICRC

Easy to build aspherical segmented mirrors and a photodetector array
The ROBAST library is currently used inside the MC code developed by the CTAUS group

CTA Application 1: Schwarzschild-Couder Optical System

1520X (arcmin)

0-20-20-15-1020
15

3500
105101520
X (arcmin)

2500

102500

02000

-5

1500

-102000-15-10-50

5101520X (arcmin)

1500

-520

10

1600

105

1400

52000

15-150-20-20

1000

800
700

10600

600

-10500-15-10-50

5

10

1520
X (arcmin)-150-20-20

5101520X (arcmin)-10-50520
15500

101520X (arcmin)

1200

400

400

15-5-5300-10300

2000-10-15-10-50

5101520
X (arcmin)

5101520
X (arcmin)

0-20-20-15-100

5101520
X (arcmin)

20300

15250
1002022015200

51500-5200
180

500

5101520
X (arcmin)

100

010

θ = 4.0 °100-20-20-15-10-50

5060-5101520
X (arcmin)

20-20-20-15-10-50

5101520X (arcmin)

0-15-10161514

180
160

08
65101520
X (arcmin)-580-5-505101520
X (arcmin)-10-20-20

21.8151.61.210080-560

20-15200-20-20

101520X (arcmin)2015-15-10-50

5101520 0
X (arcmin)

05101520X (arcmin)

101055-10-100.2-15-150-20-20-15-10-50

5101520
X (arcmin)

0X (arcmin)5100

θ = 2.0 (deg)403020151010

θ = 1.2 (deg)10-50

X (arcmin)510

θ = 3.0 (deg)040045-10-101030
2502015-5-50X (arcmin)

51010-10-1010-50

X (arcmin)5100

θ = 2.2 (deg)4045502015-51030
25020
15-51030
2502015-510-50X (arcmin)510

θ = 4.0 (deg)035-10-1010-50X (arcmin)510

θ = 3.2 (deg)040-10-10
10-50

X (arcmin)

510

5020
15-540510-10-1010-50

X (arcmin)510

θ = 4.2 (deg)

MAPMT Pixel
Size (= 4‘)-20-20-15-10-50

5101520X (arcmin)015
10-501025020
15-510-10-1010-50X (arcmin)

51060

543-520

X (arcmin)

5100

Y (arcmin)

20-510-10-10-50

X (arcmin)

51000

X (arcmin)5100

θ = 1.6 (deg)45-10-10
10-50

X (arcmin)510

θ = 1.8 (deg)

540

25

020
15-51035

53530

30
25020
15-51055-10-10
10-50

X (arcmin)5100

θ = 2.6 (deg)40-10-10
10-50

X (arcmin)

5104035530
25020
15-51030250

2015-51055-10-1010-50X (arcmin)5100

θ = 3.6 (deg)350θ = 2.8 (deg)

35

5045

40-10-10
10-50

X (arcmin)5100

θ = 3.8 (deg)
35

30
30

525
20

015-5

10525
20

015
10-5-10-10
10-50

X (arcmin)510

θ = 4.4 (deg)0

4-10-10
10

3.5

5

5

3
2.5

0

2
1.5-5

10-5

0.5-10-10-50

X (arcmin)

510030

525
20

0

15-5

105

51-50

θ = 3.4 (deg)5-10-1025

35

87-55951030

5252030

05-10-1035

30250

θ = 2.4 (deg)

35

30

535

15
10555-10-1020-535

3550

θ = 1.4 (deg)

530
25040

35

55-10-10Y (arcmin)

Y (arcmin)

Y (arcmin)

Y (arcmin)-5

15

0.4-5-50

1020

0-101025

θ = 4.8 (deg)

-5-15

15

5-15
5535

50-20-20

040

020

510-5015
20
X (arcmin)

1520
X (arcmin)-10-50X (arcmin)

1200.8-1510101401-10

5540-100150

20020-54-5-5

1.4
5-15-10-10

θ = 4.6 (deg)20-10-15-150-15-540

35

50

00.6-20-20

200-10-20-20-10-10
1025-10-10250

600

θ = 1.0 (deg)

45

535

100

120

0

30

1080

20-10

15
20 0
X (arcmin)

100
0-15

10150

140

5

10300

-15

160

55

θ = 3.8 (deg)

10120

10

100

15

100

-20-20

515

50

20

0

12
5-540

0

20-5-15-20
-20

200

0-5

θ = 4.4 (deg)40-10015-520θ = 3.6 (deg)

-15

20

1080

51520X (arcmin)20

θ = 4.2 (deg)120

10-1040-15Y (arcmin)

Y (arcmin)155-1010100

6060

θ = 4.0 (deg)200-150140

120-10-15-55140

100-5-1080-10-10-15

10160-5-15-20-20-20-20

101055

θ = 3.4 (deg)105-20-200

θ = 3.2 (deg)Y (arcmin)

Y (arcmin)θ = 3.0 (deg)-50250-100

X (arcmin)-5-15300-5200

Y (arcmin)

0400

15-540θ = 2.8 (deg)15-1510800

150

Y (arcmin)-510002002050-10-1015-10-5Y (arcmin)-10-15Y (arcmin)-15-15Y (arcmin)-20-20100

20-5400

0100

0600

600

100-151520X (arcmin)

200

150-10100

5200-5200

5800

10300

0

50-20-20

350-51000

-15

020

50-10

10250
0-15

θ = 2.6 (deg)105-20-20

1400-10200-150

θ = 1.8 (deg)15θ = 2.4 (deg)

10500

0400

400
5-5-5800

θ = 2.2 (deg)Y (arcmin)

Y (arcmin)0-5

θ = 2.0 (deg)15-101800

1000

500

20-1512000Y (arcmin)-20-2051000 -10-15-20-20

20

3000
50-5205500

θ = 1.6 (deg)

3000

4000
150

θ = 1.4 (deg)Y (arcmin)

20-52025030

Y (arcmin)

10500

10030

Y (arcmin)

51000-15

θ = 0.8 (deg)540

Y (arcmin)0θ = 1.2 (deg)Y (arcmin)

Y (arcmin)θ = 1.0 (deg)-51000-510-10-10-50

X (arcmin)510

θ = 4.6 (deg)05-10-1010-50

X (arcmin)10

θ = 4.8 (deg)50-5-50

X (arcmin)

510-10-10-50

X (arcmin)Translation and rotation matrices in the ROOT geometry library are useful toevaluate the PSF worsening from mirror misalignments
Studies on tolerance analysis and mirror alignment are on going

5tion
art
sno
Demly !
nO
esoPurp
Y (arcmin)-10500-15-105040

Y (arcmin)

-15

1000

1515001500-1020

θ = 0.6 (deg)535

Y (arcmin)-20-20500

-15

2000

10Y (arcmin)

0-525030

Y (arcmin)

1520
X (arcmin)

2000

45

540

Y (arcmin)

10-5250030

5Y (arcmin)

5030003553500

Y (arcmin)

01000

2500

θ = 0.4 (deg)40

Y (arcmin)

-5500

-150

1500
-1051040

Y (arcmin)

-10

1500
-10300010

θ = 0.2 (deg)50

Y (arcmin)

-15

2000

-5

5Y (arcmin)

-20
-20

2500

03500

10Y (arcmin)

1000

-15

3000

1045

4000

Y (arcmin)

1500
-105

θ = 0.8 °4000

θ = 0.0 (deg)4500

Y (arcmin)

2000-53500

15Y (arcmin)

2000-525000101020Y (arcmin)

2500

03000

Y (arcmin)

54000

4000
3500

4500

15Y (arcmin)

3000

4000

204500
15Y (arcmin)

10450015

θ = 0.8 (deg)Y (arcmin)

53500

20Y (arcmin)θ = 0.0 °4500

Rough Mirrors + Misalignmentsθ = 0.6 (deg)Y (arcmin)1020

θ = 0.4 (deg)Y (arcmin)

20
15

Y (arcmin)

Y (arcmin)θ = 0.2 (deg)Y (arcmin)

Ideal Mirrors and Alignmentsθ = 0.0 (deg)5100

22-(13)

Hexagonal light collectors are widely used in Cherenkov telescopes to increase
the eﬀective area of the focal plane, and to reduce background stray photons

n 2 R 2 > n1 R, sinOmax,max. -A. Okumura (2012) Astropart. Phys.

(7)

Z-collector. As is well known from the theory of lenses, af number allowed by the sine inequality an ideal light

We call an optical system that has the smallest

Dead Area

system of lenses that acts as an ideal light collector
would have an f number = 0.5, a physically unrealizable
limit.4 We are therefore led to consider the possibilityof an ideal reflecting-surface collector.
The conservation of phase-space volume expressedby Eqs. (3) and (4) ensures that transmission of unwanted rays will necessarily result in reduced acceptanceLight Collector
Optical Axis ofthe Telescope

3 This result is derived by Luneburg,' also by J. Marshall, Phys.
Rev. 86, 685 (1952).
4 This follows from the connection between the f number andthe Abbe sine lawvdiscussed in many optics texts. See, for example,M. Born and E. Wolf, Principles of Optics (Pergamon Press, Ltd.,
Oxford, 1965), 3rd ed.

The traditional design is based on Winston’s idea (so-called Winston Cone)which is identical with Eq. (10) for rotationally symmetric svstems.

xldS+z Z axis

the case ofFIG. 2. Construction of an ideal light collector for0
constant index of refraction. In this example, Amas=160.

Winston (1970)

Equations (7) and (10) provide the desired restriction
on the exit aperture, imposed by the initial angulardivergence, for optical systems that are not necessarilyimageforinig.

Stray Light(> θang/2)

IDEAL LIGHT COLLECTORS

Photons Reflectedby Mirrors (< θang/2)Z2 and theer the enthe sine
he content
pace for an in three(10)2-dimensional Winston ConeLight Collectors + PMTs

PMT1Pxrays at thector. In thisace occupiedor. We have

CTA Application 2: Hexagonal Light Collector

Collection
Eﬃciency
Collection
Efficiency(%)(%)

CTA Application 2: Hexagonal Light Collector
100(a)

ρ180(b)(c)
P2(1, 1)

60
Q1L40

Ideal 2DCone
Ideal
2DWinston
WinstonCone
Axisymmetric Winston
ConeCone
Axisymmetric
Winston

2000

1015z

θmax

Q0F

Hexagonal Winston
ConeHexagonal
Winston
Cone

5B(t)

ρ2

ρ2P0(0, 0)

P1(P1r, P1z)r2025 3035 40Paraboloid & Bézier Curve (Bézier 1978)
Incident
Angleθθ(deg)(deg) A. Okumura (2012) Astropart. Phys.
Incident
Angle

Winston cone is optimized in the 2D space
Known to have good performance in the 3D space, but need better ones
Use Bézier curves instead of paraboloids

CTA Application 2: Hexagonal Light Collector
95
95

P2z

P1z

95
95
90
90

90
90

85
85

85
85

80
80

80
80

75
75

75
75

70
70

70
70

65
65

65
65

60
60

60
60

55
55

55
55

P1r

P2r

Maximize the collection eﬃciency (contour) of a cone within the cutoﬀ angleThe coordinates of two control points (P1r, P1z, P2r, P2z) are scanned in 4parameter space

CTA Application 2: Hexagonal Light Collector
Ideal Case (R = 1.0, n = 1.0)

R = 1.0, n = 1.0

θ cross

100
80
60

Cherenkov
Photons
2.4% Up

20

2530
35
Incident Angle θ (deg)2.6% Up80
60

21.9%Down40

Winston Cone
Quadratic Bézier

2000

θ max

mm

15

θ cross

ρ 1 = 20

10100mm

5

R = 0.9, n = 1.5

ρ 1 = 30

Cubic Bézier

mmmm

Quadratic Bézier

ρ 1 = 20

Winston Cone

2000Stray Photons
18.4% Down

ρ 1 = 3040

θ max

Realistic Case (R = 0.9, n = 1.5)(c)Collection Efficiency (%)

Collection Efficiency (%)(a)

Cubic Bézier
5101520253035
Incident Angle θ (deg)Achieved higher collection eﬃciency and BG reduction rate
~3% increase of Cherenkov photons, ~20% decrease of stray BG photons in
the case of cut-oﬀ angle of 30 deg

Summary

We have developed ROot-BAsed Simulator for ray Tracing(ROBAST) for optics simulations of cosmic-ray telescopes
First used in Ashra, currently used in CTA
Simulations of Schwarzschild-Couder telescope are on goingin the CTA-US team
A hexagonal light collector was successfully optimized usingROBAST
New users are very welcome!