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) respecGvely. 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.