ray matrix approach for the real time control of …ray matrix approach for the real time control of...
TRANSCRIPT
Ray Matrix Approach for the Real TimeControl of SLR2000 Optical Elements
John J. Degnan
Sigma Space Corporation
14th International Workshop on Laser Ranging
San Fernando, Spain
7-11 June 2004
Basic 1D & 2D Ray Matrices*
d
1 d
0 1
Propagationover distance dx
Optic Axis
Thin Lensof FocalLength, f
1 0
-1/f 1
Optic Axis
Mirror Surfacewith Curvature R
1 0
-2/R 1
Optic Axis
Dielectric Interface
Optic Axis
n21 0
0 n1/n2n1
αx
I
dII
0
1D 2D
IIf
I1
0
−
IIR
I2
0
−
In
nI
2
10
0
α
x
10
01=I
y
x
y
x
α
α
*Valid only for a linear optical system. We need to perform a coordinatetransformation whenever the beam changes direction
Example of Coordinate SystemChange: Canted Mirror
mirror
Optical axis
Optical axisOff-axis
Ray
αpi+1
αpi’
S-vector into page
di
di+1
pi+1
pi'
( )
s
p
s
p
iii
ii
i
iiii
x
x
sdp
sdp
ddss
α
α
θ
1000
0100
0010
0001
ˆˆˆ
'ˆˆ'ˆ
2sin
ˆˆ'ˆˆ
111
1
11
−
−
×=
×=
×==
+++
++
Simplified SLR2000 Transceiver*
Automated Devices•Star CCD cameraperiodically updates mountmodel•3x telescope compensates forthermal drift in maintelescope focus•Beam magnifier controlslaser spot size and divergenceat exit aperture•Risley prism pair controlstransmitter point-ahead•Variable iris controls receiverfield of view (FOV)•Quadrant detector providesfine pointing corrections
*Planned modifications in red
CCDCamera
0.4 m
0.088 m5.4XBeam
Reducer
0.04 m
0.025 m
CCDSplitter
QUADMirror Transmitter
CompensatorBlock
TelescopePit Mirror
FaradayIsolator
0.080 m
0.070 m
Polarizers
Polarizer
0.085 m 0.105 m 0.350 m
0.070 m
“Zero”Wedge
Day/NightFilters
3-elementTelephoto
Lensf=0. 85 m
0.130 m
LaserTransmitter
0.213 m
Risley Prisms
Computer-controlled
Diaphragm/Spatial Filter
(Min. Range : 0.5to 6 mm diameter)
QuadrantRangingDetector
0.200 m0.051 m
3 X Telescope
f1 = -0.1 m
0.099 m
NORTH(α = 0o)
α0 = 67.4o
f2 =0.3 m
d1
d2
d3(out of page)
f1 = 0.108 m
f2 = -0.02 m
f = .025 m
d1
p1
d1
p1
d1
p1
d2
p2
Vector out of page
Vector into page
p3
s3
d2
s2'
0.250 mWorkingDistance
0.258 m
SpecialOptics2x-8xBeam
Expander
Translation Stage
Transceiver Bench Matrices
01
10 −=Λ
Λ
ΛΛ=
333.00
267.231aMTransmitter
Quadrant Detector
Star Camera0469.2
405.0559.351 Λ−
ΛΛ−=cM
General Form
ΛΛ
ΛΛ=
xx
xxx dc
baM1
Λ−Λ−
ΛΛ=
101.0392.0
57.2079.01bM
Coude Mount Matrix
C1
C2
C3Telescope
TurningMirror
FromTelescopePit Mirror
AzimuthAxis
ElevationAxis
Coude TrackingMount (not to scale)
s-Vector out of page
s -Vector into page
d4
p4
d5
p5
d6
p6
d5
p5'
d4
p4'
d4
d5
d6To Telescope
d3
d3
p3'
0.64 m
0.792 m
0.31 m
InstantaneousAzimuth
out of page
γγ
γγ
cossin
sincos
−
−−=Γ
εααγ −−= 0
Γ
ΓΓ=02
CdM
α= mount azimuth angleε = mount elevation angleα0 = azimuth angle of transceiver axis at the Coude pit mirror = 67.4o (SLR2000)dC = Coude path length = 1.742 m
Telescope Assembly Matrix
1.2819 m
1.2192 m
0.4 m
Rp = 3.048 m
Rs = - .8128 m
SecondaryMirror
Primary Mirror
TelescopeTurning Mirror
Window
ft = - 0.6 m0.6214 m 0.1905 m
From CoudeTrackingMount
SLR2000TELESCOPE
(NOT TO SCALE)
0.211 m
9.97o
s-Vector out of page
s -Vector into page
d9
p9
d8
p8
d8
p8'
Elevation Axis
d7
p7d7
p7'
d9
+ Az
+ El s9
Im
IdImM
T
TT
103 =
mT = telescope magnification = 10.16dT = 5.758 m
Total SLR2000 System Matrix
( )( )
t
xx
t
xx
xTxCxtx
xTxCxtx
xx
xxxx
m
dD
m
cC
ddddbmB
cdcdamA
DC
BAMMMM
=
=
++=
++=
−−
−=Γ
ΓΓ
ΓΓ==
γγ
γγ
sincos
cossin'
''
''123
Outgoing Rays Incoming Rays
γγ
γγ
sincos
cossin'
''
''1
−
−−=Γ
ΓΓ−
Γ−Γ=−
T
Tx
Tx
Tx
Tx
xAC
BDM
x = a Transmitter b Quadrant Detector c Star Camera
ΛΛ
ΛΛ=
xx
xxx dc
baM1
Star Calibrations
p-axis
s-axis
Optical Bench
Looking throughrear of Star Camera
p-axis
s-axis
Looking throughtelescope window
s sc
psc
αs9
Mount Elevation Axis
To Ground
αp9
CCD array
s
p
n
n
pixel
arcγγ
γεγε
ε
α
sincos
cossecsinsecsec5.0 −=
Δ
Δ
Δα = star azimuth offsetΔε = star elevation offsetnp = CCD pixel columnns = CCD pixel row
εααγ −−= 0
Quadrant Pointing Correction
p-axis
s-axis
Optical Bench
Looking throughrear of quadrant PMT
p-axis
s-axis
Looking throughtelescope window
sqd
pqd
αp9
αs9
Mount Elevation Axis
To Ground
Satellite
Q1Q2
Q3 Q4
c
c
s
p
mm
arcγγ
γεγε
ε
α
sincos
cossecsinsecsec5.10 −=
Δ
Δ
Δα = azimuth pointing correctionΔε = elevation pointing correctionpc = horizontal centroid coordinatesc = vertical centroid coordinate
εααγ −−= 0
Receiver Field of View
Da = iris diameterFOV = Full Receiver Field of View
in arcsec
FOVarc
mmDa sec
125.0=
Da
TTaTaa
TT
a
T
T
TT
T
a
a
arc
mm
rad
mxxx
rad
mx
xx
αα
α
αα
sec
125.0908.25
'908.25
'387.24'039.0
'908.250
===
Γ−=
Γ−Γ
Γ−=
rr
rr
r
r
r
r
Stepper-controlled Iris
Transmitter Point-Ahead
Optical Bench
p-axis
s-axis
Looking throughtelescope window
αs9
Mount Elevation Axis
To Ground
αp9
“Apparent”Position
(receiver axis)
Future Position(transmitter
axis)
ActualSatellitePosition
p-axis
s-axis
Wedge 1
Wedge 2
Looking through RisleyPrisms toward
telescope
ξ1ξ2
Δξε
α
γεγ
γεγτ
α
α
&
&
sincoscos
coscossin
−
−−= rT
rp
rp ms
p
αprp =Risley prism output angleprojected into p planeαsrp = Risley prism output angleprojected into s planemT = post-Risley magnification of transmitter =30.48τr = pulse roundtrip time of flight
•
•
ε
α = azimuth rate
= elevation rate
εααγ −−= 0
Computing Risley Prism Orientations
r
rT
rp
rp ms
p
τε
τα
γεγ
γεγ
ξδξδ
ξδξδα
α
&
&
sincoscos
coscossin
sinsin
coscos
2211
2211
−
−−=
+
+=
δ1 = half cone angle traced by wedge 1δ2 = half cone angle traced by wedge 2ξ1 = wedge 1 angle relative to home positionξ2 = wedge 2 angle relative to home position
( ) ( ) ( )( ) ( )( )[ ]( )ξδδδδ
ξγδγδτεξγδγδεταξ
Δ++
Δ−++Δ−+−=
cos2
cos(cossinsincoscos
2122
21
21211
rrTm &&
( ) ( )( ) ( ) ( )( )[ ]( )ξδδδδ
ξγδγδτεξγδγδεταξ
Δ++
Δ−+−Δ−+=
cos2
sinsincoscoscos.)sin(
2122
21
21211
rrTm &&
( )[ ] ( )
+−+
=−≡Δ −
21
22
21
2221
12 2
)cos(cos
δδδδτεετα
ξξξ rrTm &&
Solve above two equations for two unknown Risley orientations ξ1 and ξ2:
ξ2 = ξ1 + Δξ
Simulated LAGEOS Pass
440 460 480 500 520174.5
175
175.5Differential Wedge Angle vs Time
Δξii
deg
PCA
tii
min
440 460 480 500 520180
90
0
90
180
270
360Risley Orientations vs Time
ξ1ii
deg
ξ2ii
deg
PCA
tii
mina( ) b( )
440 460 480 500 5200.001
5 .104
0
5 .104
0.001Differential Angular Rate vs Time
Time (min)
Differential Angular Rate (deg/sec)
PCA
440 460 480 500 5200.05
0.025
0
0.025
0.05Risley Angular Rates vs Time
Time (min)
Wedge Angular Rates (deg/sec)
PCA
c( ) d( )
Azimuth- Elevation Offsets & Beam Centering
40 20 0 20 4040
20
0
20
40Transmitter Point-Ahead
Azimuth Offset (arcsec)
Elevation Offset (arcsec)
4 2 0 2 44
2
0
2
4Central Ray Trajectory on Exit Window
p-axis (increasing azimuth), cm
s-axis (increasing elevation), cm
a( ) b( )
Ray Matrices and Gaussian Beams
Paraxial ray matrix theory can be applied to gaussian beampropagation if we define the following complex parameter:
( ) ( )zj
zRzq 2)(
11
πωλ
−=
If propagation from a point z0 to z can be described by the ray matrix
DC
BAM =
then the gaussian beam properties at z are given by
( )
( )0
0
1
1
)(
1
zqBA
zqDC
zq +
+=
Controlling Beam Divergence
The ray matrix which takes the transmitter beam from theRisley Prism output to the far field is of the form
'1
0
''
'1
0
''
0lim
Γ
ΓΓ=Γ
ΓΓ=
∞→
t
tt
t
tt
r
m
m
rm
m
dm
I
rIIFF
where mt is the total transmitter magnification. From our gaussian parameter, we obtain the following for the full beam divergence
( )( )
( )( )
( )( )
2
0
02
min
2
0
02
0
112
2
+=
+==
zR
z
zR
z
zmr
r
tt λ
πωθ
λπω
ωπλω
θ
where ω(z0) and R(z0) are the beam radius and phasefront radius of curvature out of the computer-controlled telescope in the transmit path. To first order, beam divergence varies linearly withthe lens displacement from perfect focus.
Beam Divergence vs Phase FrontCurvature at Risleys
Radius of SLR2000 primary, a = 20 cmOptimum spot radius at window*,ωopt = a/1.12 = 17.9 cmPost-Risley magnification, mt = 30.48Optimum beam radius at Risley,ω(zo) = ωopt /mt = 5.9 mmMinimum Divergence,θmin = 0.388 arcsec
0 0.1 0.2 0.3 0.4 0.50
10
20
30
40
Inverse Phase Front Curvature, m^-1
Full Beam Divergence, arcsec
θ jj θmin 1π ω0
2⋅
λRinv
jj⋅
2
+⋅:=
Summary• Ray matrix approach provides us with the
mathematical tools to calculate in real time:– Scale factor and angular rotation for converting star
image offsets from center in the CCD camera toazimuth and elevation biases
– Scale factor and angular rotation for convertingquadrant centroid position to satellite pointingcorrection in az-el space
– Transmitter point ahead as a function of round triptime-of-flight and the instantaneous azimuthal andelevation angular rates
– Iris diameter (spatial filter) setting for a given receiverFOV
– Transmitter beam size divergence as a function oftransmit telescope defocus