A Tutorial on Retroreflectors and Arrays for SLR

John J. Degnan Sigma Space Corporaon ILRS Workshop, Frasca, Italy November 5, 2012

J. Degnan 2012 ILRS Workshop, Frasca, Italy 1 Corner Cube Retroreflectors

Hollow Cube Corner Solid Cube Corner

J. Degnan Cube corner retroreflectors reflect light back to the point of origin in a narrow beam. 2012 ILRS Workshop, Frasca, Italy 2 Three Types of Cube Corners Type Al Back-Coated Solid Uncoated Solid (TIR) Hollow

Frequency of Use Most Common Occasional Use Not currently used in the visible Satellite Examples Most satellites Apollo, LAGEOS, AJISAI, ETS- ADEOS RIS, REM, TES VIII Reflecvity, ρ 0.78 0.93 Can approach 1.0

Polarizaon Sensive No Yes No – metal coang Yes-dielectric coang Weight Heavy Heavy Light

Far Field Paern Wide Wide Narrow

Issues Metal coangs absorb sunlight Fewer thermal problems but Thermal heang and gradient and create thermal gradients. TIR “leaks” at incidence angles effects on joints Not as well shielded at high > 17o. Polarizaon effects altudes. reduce cross-secon by factor of 4.

J. Degnan 2012 ILRS Workshop, Frasca, Italy 3 Peak Cross-Secon of a Perfect Cube Corner

For normally incident light, a single unspoiled retroreflector (cube corner) has a peak, on-axis, opcal cross-secon defined by

2 3 4 ⎛ 4π ⎞ ⎛ 4πAcc ⎞ π ρD σ cc = ρAcc ⎜ ⎟ = ρ⎜ 2 ⎟ = 2 ⎝ Ω ⎠ ⎝ λ ⎠ 4λ where the reflecvity of the cube corner, ρ, is typically equal to 0.78 or 0.93 for aluminum-coated back faces and uncoated Total Internal Reflecon (TIR) surfaces respecvely , Acc is the collecng aperture of the corner cube, D is the cube diameter, and 4π/Ω is the on-axis reflector gain and Ω is the effecve solid angle occupied by the Far Field Diffracon Paern (FFDP) of the retroreflector. Blue = TIR; Red = Aluminum 11 1 10 × 1.5

10 1×10

9 The peak opcal cross-secon rises rapidly as the 1×10 retroreflector diameter to the fourth power. For 8 1×10 7 the popular 1.5 in (38 mm) diameter cube, the 5.8⋅10 peak cross-secon is about 5.8 x 107 m2. 7 1×10

6 1×10 Peak Cross-section,square meters Cross-section,square Peak

5 1×10 0 1 2 3 4 5 6

J. Degnan Retroreflector Diameter, inches2012 ILRS Workshop, Frasca, Italy 4 Retroreflector Far Field Diffracon Paern (FFDP)

Since the retroreflector aperture is illuminated by a uniform plane wave from the distant SLR staon, the electric field strength in the far field , E(x’,y’), is equal to the 2D Fourier Transform of the retroreflector entrance aperture as seen by the plane wave source while the Intensity distribuon is the electric field mulplied by its complex conjugate, E*(x’,y’), i.e. E(x', y') = ∫ dx∫ dy exp(ikx'x)exp(iky' y) I(x', y') = E(x', y')E *(x', y') where k = 2π/λ and λ is the opcal wavelength (532 nm). For a circular aperture we can use cylindrical coordinates

J. Degnan 2012 ILRS Workshop, Frasca, Italy 5 Retroreflector Far Field Diffracon Paern For a circular aperture, the FFDP of the reflected wave is the familiar Airy Funcon given by 2 ⎡2J1(x)⎤ σ (x) = σ cc ⎣⎢ x ⎦⎥ where J1 (x) is a Bessel funcon and the argument x is related to the off-axis angle θ by πD x = sinθ λ λ = 532 nm is the most widely used SLR laser wavelength and D is the cube aperture diameter. Normalized Cross-Section vs Angle Theta Normalized Cross-Section vs Angle Theta 1 1 1.6 3.8 3.8 7 0.9 0.1 0.8 0.7 0.01 0.6 − 3 0.5 0.5 1×10 0.4 − 4 1×10 0.3

0.2 − 5 Normalized Cross-Section Normalized Cross-Section Normalized 1×10 0.1 − 6 0 1×10 0 2 4 6 8 10 0 2 4 6 8 10

x=pi*D*sin(theta)/lambda x=pi*D*sin(theta)/lambda

The half-power and first null occur at x = 1.6 and 3.8 respecvely. For the popular 1.5 in (38 mm) diameter J. Degnan cube at 532 nm, this corresponds to θ = 7.1 and 16.9 2012 ILRS Workshop, Frasca, Italy microradians (1.5 and 3.5 arcsec) respecvely. 6 Peak Cross-Secon vs Incidence Angle (Hollow Cube vs Coated Fused Silica) At arbitrary incidence angle, the effecve area of the cube is reduced by the factor

2 −1 o η(θinc ) = (sin µ − 2 tanθref )cosθinc 13 π

where θinc is the incidence angle and θref is the 9o internal refracted angle as determined by Snell’s Law, i .e.

−1⎛ sinθinc ⎞ θref = sin ⎜ ⎟ ⎝ n ⎠ where n is the cube index of refracon. The quanty µ is given by the formula •The 50% and 0% efficiency points for fused 2 o o µ = 1− tan θref silica (n=1.455) are 13 and 45 respecvely. • The 50% and 0% efficiency points for a Thus, the peak opcal cross-secon in the hollow cube (n=1) are 9o and 31o respecvely. center of the reflected lobe falls off as •In short, hollow cubes have a narrower

2 angular response range than solid cubes. σ eff (θinc ) =η (θinc )σ cc J. Degnan 2012 ILRS Workshop, Frasca, Italy 7 Effect of Incidence Angle on the FFDP Normal Incidence Non- Normal Incidence

D Retro Face Seen by Dη(θinc) Incident Radiaon

θ = 3.8λ/πDη(θ ) Far Field θ = 3.8λ/πD inc Diffracon Paern

J. Degnan 2012 ILRS Workshop, Frasca, Italy 8 Introducon to Velocity Aberraon •If there is no relave velocity between the staon and satellite, the beam reflected by the retroreflector will fall directly back onto the staon . •A relave velocity, v, between the satellite and staon causes the reflected beam to be angularly deflected from the staon in the forward direcon of the satellite by an amount α = 2v/c. • We have seen that small diameter cubes have small cross-secons but large angle FFDPs , and therefore the signal at the staon is not significantly reduced by velocity aberraon. •Similarly, large diameter cubes with high cross- secons have small angle FFDPs, and the signal at the staon is therefore substanally reduced by velocity aberraon. •In general, the signal is reduced by half or more if

the cube diameter, Dcc, sasfies the inequality 2v x λ 1/ 2 1.6λc α = > θ1/ 2 = or D D c πD cc > 1/ 2 = cc πα J. Degnan 2012 ILRS Workshop, Frasca, Italy 9 Velocity Aberraon: General Earth Orbit J. Degnan, Contribuons of Space Geodesy to Geodynamics: Technology, Geodynamics 25, pp. 133- 162 (1993) Satellite If there is a relave velocity between the satellite and the staon, the v p coordinates of the FFDP are translated in the direcon of the velocity r qzen vector. The magnitude of the angular displacement in the FFDP is given by Station 2 2 2 α(hs ,θzen ,ω) = αmax (hs ) cos ω + Γ (hs ,θzen )sin ω where the maximum and minimum values are given by 2 2vs 2 gRE αmax (hs ) = α(hs ,0,0) = = c c R + h E s Geocenter o o o αmin (hs ) = α(hs ,70 ,90 )= αmax (hs )Γ(hs ,70 ) 2 ⎛ R sinθ ⎞ h , 1 E zen 1 Γ( s θ zen ) = − ⎜ ⎟ < ⎝ RE + hs ⎠ ∧ ∧ ∧ −1 ⎡⎛ ⎞ ⎤ Alpha: Max = Red (EL=90 deg); Min = Blue (EL=20 deg) ω = cos r x p • v 55 ⎢⎜ ⎟ ⎥ LAGEOS GNSS ⎣⎝ ⎠ ⎦ 50 v = satellite velocity at altude h s s 45 RE = Earth radius = 6378 km g = surface gravity acceleraon =9.8m/sec2 40 35 hs=satellite height above sea level 8 30 c = velocity of light = 3x10 m/sec GEO o microradiand Alpha, 25 θzen = largest satellite zenith angle for tracking = 70 r = unit vector to satellite from the geocenter 20

15 p = unit vector from staon to satellite 0.1 1 10 100 J. Degnan 2012 ILRS Workshop, Frasca, Italy 10 v= unit vector in direcon of satellite velocity Satellite Altitude, Thousands of km

Why is α Non-Zero for Geostaonary Satellites? The angular velocies about the Earth’s rotaon axis are the same for a ground staon and a Geostaonary satellite but the physical velocies are different resulng in a relave velocity between them. The common angular velocity is given by

2π 5 ω = = 7.27x10− Hz E day The physical velocity of the geostaonary satellite in the Earth-centered frame is

−5 vGEO = ωE (RE + hGEO ) = 7.27x10 (6378km + 35786km) = 3.066km/sec while the physical velocity of the ground staon due to Earth rotaon is latude dependent and given by

vstation = ωE RE cos(lat)≤ 0.464km/sec However, as shown on the previous slide, the relave velocity also depends on the observing zenith angle at the staon (assumed maximum of 70o). The resulng velocity aberraon has a very narrow range , i.e. 2v 20.29µrad ≤ α = ≤ 20.46µrad c

J. Degnan 2012 ILRS Workshop, Frasca, Italy 11 Apollo 15 Lunar Example 6 Earth-Moon Distance : REM = h+RE = 384.4 x10 m. From the previous equaons αmax =6.74 µrad or 1.40 arcsec αmin = 6.68 µrad or 1.39 arcsec at an elevaon angle of 20 degrees v = relave velocity between target and staon due to lunar orbital moon = 1km/sec However, the laer equaons ignore the small contribuon of staon moon due to Earth rotaon (~0.46 km/sec) to the relave velocity which typically reduces α to 4 or 5 µrad for LLR but is negligible for LEO to GEO satellites. If the Apollo reflector arrays are pointed at the center of the Earth, the maximum beam incidence angle on the array from any Earth staon (ignoring lunar libraon) is ⎛ R ⎞ a tan E 0.95deg θinc = ⎜ ⎟ = ⎝ REM ⎠ The unspoiled cube diameter for which the cross- secon falls to half its peak value is

D1/ 2 = 40.6mm =1.6in Typical manufacturing tolerances are 0.5 arcsec for dihedral angles and λ/10 for surface flatness.

Apollo 15 has a flat array of 300 38 mm fused quartz cubes each with an unspoiled peak cross- secon of 5.8 x 107 m2. Thus, the theorecal array cross-secon, ignoring manufacturing tolerances and local environment effects, is Σ ≅ 300(0.5)(5.8 x 107 m2)= 8.7x109 m2. According to Dave Arnold, polarizaon losses due to uncoated TIR faces reduce cross-secon by factor of 4, leaving ]Σ~2. 2x109 m2 . The tabulated ILRS value is 1.4x109 m2. J. Degnan 2012 ILRS Workshop, Frasca, Italy 12

Lunar Alternave to Apollo Array

Otsubo et al, Advances in Space Research, Vol. 45, pp. 733-740, 2010. According to the authors, simulaons indicate that a single reflector with a diameter of 150 to 250 mm has similar performance to Apollo arrays. No dihedral angle is required for small diameter reflectors (<150 mm for coated and <100 mm for uncoated and hollow reflectors) . Larger diameters required dihedral angles 0f 0.20, 0.25, and 0.35 arcsec for coated, uncoated, and hollow reflectors respecvely.

250 mm reflectors with 0.25 arcsec dihedral angles, incidence angle = 6 degrees J. Degnan 2012 ILRS Workshop, Frasca, Italy 13 GNSS and Geostaonary Satellites GNSS and Geosynchronous Satellites have features in common with LLR: 1. Their orbital altudes correspond to several Earth radii 2. They generally perform a ulitarian funcon (Earth observaon, communicaons, navigaon, etc. ) which keeps the nadir side of the satellite approximately facing the Earth CoM

3. The difference Δα = αmax - αmin is very small (see previous graph).

Their differences from LLR are : 1. The velocity aberraon α is 4 to 5 mes larger (20 to 25 µrad) 2. For a maximum zenith tracking angle of 70o, beam Incidence angles can vary from 0 to β where

⎡ RE 0 ⎤ =13.1 deg for GNSS satellites at 20,000 km β = asin⎢ sin(110 )⎥ = 8.2 deg for GEO satellites at 36,000 km ⎣RE + h ⎦

The smaller range of incidence angles implies limited pulse spreading from a flat array, especially if the array is compact in size and the retros are densely packed together to achieve the necessary cross-secon. Nevertheless, the maximum flat panel induced spreading per linear foot of array due to zenith tracking angle is sll 474 and 292 psec for GNSS and GEO satellites respecvely. This spreading can increase further if satellite atude deviaons from true nadir extend the range of incidence angles. J. Degnan 2012 ILRS Workshop, Frasca, Italy 14 Retroreflector Arrays for High Altude Satellites Ref: D.A. Arnold, “Retroreflector Studies”,

J. Degnan 2012 ILRS Workshop, Frasca, Italy 15 “Spoiled” Retroreflectors Circular Cube •“Spoiling” is used to compensate for velocity aberraon and improve the signal return from the satellite. • If we offset one or more (N = 1 to 3) of the cube angles from 90o by an amount δ, the central lobe of the FFDP splits into 2N spots. Sector • If n is the cube index of refracon, the mean angular distance of the lobe from the center of the original Airy paern increases linearly with the dihedral angle offset, δ, according to 4 γ = 6nδ 3 Hexagonal •As before, the angular size of any given lobe decreases as Cube the cube diameter gets larger. •The FFDP of each lobe is the 2D Fourier transform of an Lobe individual 60o sector. The energy distribuon is complex but has hexagonal symmetry if all δs are equal. •Furthermore, the effecve area and peak cross-secon of each lobe is reduced to

Acc 2 σ cc Aeff =η(θinc ) σ peak =η (θinc ) 2 2N (2N ) J. Degnan 2012 ILRS Workshop, Frasca, Italy 16 Clocking

Filling in circumferenal gaps between lobes can be accomplished by rotang an adjacent cube by an amount not divisible by 60o. The bigger the gap and/or the smaller the lobe diameter , the more rotaonal posions are needed. This may allow the use of larger diameter cubes which don’t overfill the annulus thereby reducing array efficiency.

0o Clocking 30o Clocking

γ = α min α min γ =

J. Degnan 2012 ILRS Workshop, Frasca, Italy 17 Signal Strength vs Zenith Angle For a constant target cross-secon, the signal strength decreases with zenith angle via two effects: Space Loss:

4 1 ⎡cos(θzen )⎤ = R4 ⎣⎢ h ⎦⎥ Atmospheric Transmission Loss:

2sec(θ zen ) T0 o o The signal strengths at θzen = 70 relave to zenith (θzen = 0 ) is therefore

4 2[sec(θ zen )−1] Standard Clear Atmosphere cos (θzen )T0 1 where, for a Standard Clear Atmosphere, the

one-way atmospheric transmission is T0 =0.8 0.1 at 532 nm. Thus, from the plot, the signal o strength at αmin (θzen = 70 ) is about two orders o 0.01 of magnitude smaller than at αmax (θzen = 0 ) so the array design should bias the return toward Relative Signal Strength Relative − 3 1×10 high zenith angles (low elevaon angles). 0 10 20 30 40 50 60 70

Zenith Angle, deg J. Degnan 2012 ILRS Workshop, Frasca, Italy 18 LEO to MEO Satellites LAGEOS Example

•Seng γ = αmin =33 µrad places the peak of the lobe a max where the energy is needed most, i.e. at low elevaon o angles. (θzen = 70 ) •The corresponding dihedral offset is given by 3α a δ = min = 7.1µrad min 4 6n •To fill the annulus quasi-uniformly in the γ = α circumferenal direcon, the transverse angular min radius of the lobe must sasfy

1.6λ παmin θ1/ 2 = > =17.3µrad πDc 6

or Dc >15.7 mm •To fill the annulus in the radial direcon, the radial angular radius of the lobe must sasfy 3.8λ θr = > αmax −αmin = 5µrad πDr

or Dr>129 mm Any reflected energy outside the annulus is wasted!

J. Degnan 2012 ILRS Workshop, Frasca, Italy 19 Retroreflector in Space

Retroreflector in Space (RIS) Effecve Diameter = 50 cm One dihedral offset One curved reflecng face Effecve Divergence 60 µrad Designed for Thermal Infrared Visible performance was poor.

J. Degnan 2012 ILRS Workshop, Frasca, Italy 20 MM ACCURACY GEODETIC SATELLITES

•Technical Challenges •The satellite must present a high enough cross-secon, consistent with its altude, to support ranging by the enre ILRS network. Velocity aberraon limits the size of the retroreflector. •To simultaneously achieve maximum range accuracy, the satellite impulse response must be made as short as possible. •Planar LLR arrays present a common face to all Earth ranging staons provided it is inially pointed approximately toward the Earth Center-of-Mass (COM). The number or density of small diameter retroreflectors (< 38 mm) can be increased without limit to increase the target cross- secon, and the near perpendicularity of the flat panel to the incoming opcal wavefronts introduces lile or no pulse spreading for maximum range accuracy. • On the other hand, geodec satellites (LAGEOS, Starlee, etc.) are in much lower orbits,have large velocity aberraons, rotate freely in space, and are spherical in shape to permit simultaneous and unbiased ranging from mulple SLR staons. •Spherical geodec satellites can be made to mimic their LLR cousins by: •Building larger supporng spheres or flat panel polyhedrons . •Improving the packing density to increase the number of retros per unit surface area and hence the effecve array cross-secon. •For spheres, limit target pulse spreading by restricng returns over a smaller range of incidence angles through the use of hollow or recessed solid retros . •For flat panel polyhedrons, ensure that returns from adjacent panels cannot be seen by the ground staon. J. Degnan 2012 ILRS Workshop, Frasca, Italy 21 MM ACCURACY GEODETIC SATELLITES J. Degnan, Contribuons of Space Geodesy to Geodynamics: Technology, Geodynamics 25, pp. 133- 162 (1993)

Basic Design Guidelines • Increase radius of sphere to beer approximate a flat surface •Recess the individual cubes in their holders to restrict the response at large incidence angles and minimize the target signature •Increase packing density of the cube corners on the satellite surface (rao of cube aperture within a given surface area). Hexagonal arrays have the greatest packing density but do not permit “clocking”. (LAGEOS has a packing density of 0.435. ) •Choose retro diameter and clocking scheme to opmally fill the annular FOV, Ω, between

αmax and αmin

J. Degnan 2012 ILRS Workshop, Frasca, Italy 22 Cross-Secon in the Large Satellite Limit (Rs>>nL) J. Degnan, Contribuons of Space Geodesy to Geodynamics: Technology, Geodynamics 25, pp. 133- 162 (1993) The opcal cross-secon in the large satellite limit is

Where ρ is the cube corner reflectance c is the speed of light Rs is the satellite radius n is the corner cube index of refracon nL is the opcal depth of the cube (face to vertex) β is the packing density (acve area/total area)

Ω Is the array FOV , ideally well matched to the annular FOV between α max and αmin

J. Degnan 2012 ILRS Workshop, Frasca, Italy 23 Impulse Response in the Large Satellite Limit (Rs>>nL) J. Degnan, Contribuons of Space Geodesy to Geodynamics: Technology, Geodynamics 25, pp. 133- 162 (1993) In the following graph, τ is a me normalized to the me it takes a light pulse to travel the diameter of the satellite, i.e. 2Rs/c. Increasing the radius of the satellite to make it appear flaer will increase the impulse response. However, narrowing the incidence angle range by using hollow cubes or recessing the solid cubes reduces the impulse response of the satellite and improve range accuracy.

Larger radius satellite with narrow incidence angle range has higher cross-section and narrower impulse response.

Larger radius satellite with same incidence angle range has much higher cross-section and broader impulse response.

J. Degnan 2012 ILRS Workshop, Frasca, Italy 24 BLITZ Retroreflector

Inner Sphere: High Index ΤΦ105 glass Outer Sphere: ΠΚ6 glass Reflecve Coang: Aluminum protected by a varnish layer Mass: 7.53 kg J. Degnan σ ~105 m2 (equivalent to a 0.3 in or 7.6 mm corner cube) 2012 ILRS Workshop, Frasca, Italy 25 Summary

mm Accuracy LEO to MEO Geodec Satellites •Large radius satellites •to beer match the incoming plane wave •Allow more reflectors in the acve area to increase cross-secon •Reduce range of accepted incidence angles to minimize satellite impulse response •Hollow cubes •Recessed hollow or solid cubes • Also incidence angles < 17 o do not leak light in solid TIR reflectors •Selecon of cube diameters and clocking to best match the “ α annulus” while favoring high zenith (low elevaon) angles is key to efficient array design

GNSS and GEO Satellites •Typically have nadir face pointed near Earth center due to other funcons (Earth observaon, communicaons, navigaos, etc.) •Flat panels OK but sll several hundred psecs of temporal spread at large zenith angles. •Range accuracy might benefit from using a small segment of a large sphere following LEO/MEO design guidelines

LLR •Small incidence angles (<1 deg ignoring lunar libraons) and velocity aberraons (<1.0 arcsec) suggest the possible use of large diameter cubes provided thermal issues on the lunar surface can be resolved. J. Degnan 2012 ILRS Workshop, Frasca, Italy 26