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LISA Optics Model,Penn State,22 July,2002 GSFC JPL ESA LISA Optics in the U.S. Eugene Waluschka NASA/Goddard Space Flight Center Greenbelt, Maryland 20771
LISA Optics M odel, Penn State, 22 July, 2002
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LISA Optics in the U.S.
Eugene Waluschka
NASA/Goddard Space Flight Center Greenbelt, Maryland 20771
LISA Optics M odel, Penn State, 22 July, 2002
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Two major participants:
• JPL o Modeling wavefront quality (mid-to-high spatial
frequency) vs. telescope tilt. o Optical block bonding
• Goddard o Program office o Systems engineering
• Requirements definition • End-to-end modeling • Structural, Thermal, Optical (STOP) and self
gravity
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Laser Interferometer Space Antenna
• Counts fringes (about a million/second) • Deduce a variable strain (within the band of
interest) between freely falling proof masses • Magnitude of strain is about 10-21 • About 10 picometers • Out of 5 million kilometers.
To accomplish this, the LISA experiment has: • Three spacecraft, • Two telescopes in each spacecraft, for a total of
six identical telescopes, • Each telescope tracks a distant spacecraft and
sends and receives light (at a slightly different angle),
• Collimated (quasi) monochromatic, light centered on 1.064 microns.
• Circularly polarized beam
LISA Optics M odel, Penn State, 22 July, 2002
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Knowing the relative positions of optical elements is a good starting point.
HELIOCENTRIC COORDINATE
FRAME AND KEPLERIAN ORBITS
0k
1 ( )S t�
2 ( )S t�
3 ( )S t�
0j
0i0O
ecliptic
LISA Optics M odel, Penn State, 22 July, 2002
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A Keplerian orbit in the ecliptic is given by (from L&L Mechanics)
32
(cos )( )
( ) ( ) 1 sin ( sin )
0 0 sun
a exa
S t y a e where t eG M
ξξξ ξ ξ ξ
− = = − ⋅ = ⋅ − ⋅ ⋅
�
a is the major axis of the ellipse e is the eccentricity G is the universal constant of gravitation Msun is the mass of the sun
A complete passage round the ellipse corresponds to ξ increasing by 2π , so that when 0t = then 0ξ =and (0) ( ,0,0)S ae= − .
LISA Optics M odel, Penn State, 22 July, 2002
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Three LISA like orbits are obtained by the following rotations and time translations:
1
2
3
( ) ( ) ( )
1( ) (120 ) ( ) ( )32( ) (240 ) ( ) ( )3
y
oz y
oz y
S t R S t
S t R R S t year
S t R R S t year
β
β
β
= ⋅
= ⋅ −
= ⋅ −
� �
� �
� �
( )Z
R γ and ( )Y
R β are rotation matrices about the
heliocentric z and y axes. If 0.948oβ = then
a roughly equilateral triangle leg length and angles varying about 1%
Three Orbits
LISA Optics M odel, Penn State, 22 July, 2002
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LISA Optics M odel, Penn State, 22 July, 2002
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SPACECRAFT AND OPTICAL BENCH
1 ( )S t� � � ���
1O
Spacecraft
12O
13O
LISA Optics M odel, Penn State, 22 July, 2002
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12O
detector
laser
proof mass
telescope
towardspacecraft 2
y
z
fold
LISA Optics M odel, Penn State, 22 July, 2002
Optical Block + Telescope = Optical Assembly
LISA Optics M odel, Penn State, 22 July 2002
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Far field intensity pattern
LISA Optics M odel, Penn State, 22 July, 2002
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Far field phase variations “sensitivity analysis”
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POINT AHEAD
1S
2 12 1 12( ) ( )s sS t t S t t− ∆ − − ∆� �
2 12 1 12( ) ( )R RS t t S t t+ ∆ − + ∆� �
3 13 1 13( ) ( )s sS t t S t t− ∆ − − ∆� �
313
113
() (
)R
R
S t t S t t+ ∆ − + ∆
�
�
12 ( )tθ
23 ( )tΘ
LISA Optics M odel, Penn State, 22 July, 2002
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Computing the point ahead positions
Light transit time about 16 seconds.
2 12 1 12 12
2 12 1 12 12
( ) ( )
( ) ( )
R R R
S S S
S t t S t t c t
S t t S t t c t
+ ∆ − + ∆ = ⋅∆
− ∆ − − ∆ = ⋅∆
� �
� �
Table 1: The positions of all three when transmitting and receiving light from the other spacecraft. Receive position of spacecraft Inertial frame Send position of spacecraft
2 12 1 12( ) ( )R RS t t S t t+ ∆ − + ∆� �
1S 2 12 1 12( ) ( )S SS t t S t t− ∆ − − ∆� �
3 13 1 13( ) ( )R RS t t S t t+ ∆ − + ∆� �
1S 3 13 1 13( ) ( )S SS t t S t t− ∆ − − ∆� �
1 21 2 21( ) ( )R RS t t S t t+ ∆ − + ∆� �
2S 1 21 2 21( ) ( )S SS t t S t t− ∆ − − ∆� �
3 23 2 23( ) ( )R RS t t S t t+ ∆ − + ∆� �
2S 3 23 2 23( ) ( )S SS t t S t t− ∆ − − ∆� �
1 31 3 31( ) ( )R RS t t S t t+ ∆ − + ∆� �
3S 1 31 3 31( ) ( )S SS t t S t t− ∆ − − ∆� �
2 32 3 32( ) ( )R RS t t S t t+ ∆ − + ∆� �
3S 2 32 3 32( ) ( )S SS t t S t t− ∆ − − ∆� �
LISA Optics M odel, Penn State, 22 July, 2002
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In a local inertial frame attached to a spacecraft the motion of a distant spacecraft.
LISA Optics M odel, Penn State, 22 July, 2002
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Radial velocity of spacecraft
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Angle between two telescopes
LISA Optics M odel, Penn State, 22 July, 2002
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DISTURBANCE REDUCTION SYSTEM
18 degrees of freedom SIMULINK™ DRS
• 6 degrees of freedom for spacecraft • 6 degrees of freedom for each proof mass
LISA Optics M odel, Penn State, 22 July, 2002
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FROM LASER TO DETECTOR
Light leaving laser
[ ]( , , , )( , , , ) ( , , , ) laseri t x y z tlaser laserE x y z t A x y z t e ω φ− +=
then by tracing a sufficient number of rays, we get the outgoing wavefront at the telescope aperture.
( , , , )( , , , ) ( , , , ) outgoing
i t x y z t
outgoing outgoingE x y z t A x y z t eω φ − + =
(6)
The (far) field at the aperture of the distant spacecraft is given by
12
2( ( , , , ) )
( )min 12
( , , , )( , , , )
outgoing
R
Si x y z t
outgoingi t tRinco g far
A
E x y z t eE x y z t t A e dA
S
πφλ
ω
− +
′− +∆′ ′ ′ +∆ = ∫∫
LISA Optics M odel, Penn State, 22 July, 2002
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At the detector
[ ( ) ( , , , )]
[ ( ) ( , , , )]
( , , , )( , , , )
( , , , )
slocal local
plocal local
i t t x y z tslocal
local i t t x y z tplocal
A x y z t eE x y z t
A x y z t e
ω ϕ
ω ϕ
− ⋅ +
− ⋅ +
⋅ = ⋅
�
[ ( ) ( , , , )]
[ ( ) ( , , , )]
( , , , )( , , , )
( , , , )
sfar far
pfar far
i t t x y z tsfar
far i t t x y z tpfar
A x y z t eE x y z t
A x y z t e
ω ϕ
ω ϕ
− ⋅ +
− ⋅ +
⋅ = ⋅
�
Jones vectors to remind us of the fact that the light really is polarized.
The intensity of the light at any point (x,y,z) on the detector:
2( , , , ) ( , , , ) ( , , , ) .local farI x y z t E x y z t E x y z t scattered light+ +
� ��
LISA Optics M odel, Penn State, 22 July, 2002
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Signal extraction
Over the detector area
( ){ }2 2( ) ( ) ( ) 2 ( ) ( )cos ( ) ( ) ( )local far local far local far local farI t A t A t A t A t t t t tω ω φ φ+ + − + −�
Doppler beat note ( )local far
tω ω
−
( ) ( ) ( ) ( )local far noise signalt t t tφ φ φ φ− = +
( )noise tφ optical path noise from the sending laser to the receiving detector
( )signal tφ is the gravitational signal
LISA Optics M odel, Penn State, 22 July, 2002
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Conclusion
Goal of the optics model
guide us in the spacecraft and mission design
extend standard optical practice
“Perfect LISA” + telescope
