Laser Cross Section (LCS) (Chapter 9) EC4630 Radar and Laser Cross Section Fall 010 Prof. D. Jenn jenn@nps.navy.mil www.nps.navy.mil/jenn November 011 1
Laser Cross Section (LCS) Laser radar (LADAR), also known as light detection and ranging (LIDAR), is an active system that measures range and angle in a manner similar to microwave radar down range from time delay cross range from angle information Pros and cons: high resolution (small range and angle bins) due to narrow beams and short pulses short range because of atmospheric attenuation (limit ~ several km at surface level) Common wavelengths of operation: 10.6 mm (CO gas lasers, 10% efficiency) 1.06 mm (Neodymium YAG crystal laser, 3% efficiency) ONE-WAY ATTENUATION (db/km) WAVELENGTH FREQUENCY (GHz) November 011
GO and the Beam Expander BEAM WAIST ρ = w o GEOMETRICAL OPTICS RAYS LENS z = z o GO FOCUS Focused beam and its GO approximation LENS Beam waveguide Beam expander: (Φ is beam divergence) D in Φ in Φ out D out D D out in Φ = Φ out in November 011 3
Laser Radar System System block diagram for a coherent laser radar Receive optics and detector Half power beamwidth: 1.0λ θb D D = lens or mirror diameter November 011 4
Laser Radar Modes Possible operating modes: Point target: laser beam illuminates entire target, detector field of view (FOV) encompasses entire target (good for search and track) Extended target: partial illumination of the target, detector FOV limited to partial view of target (good for imaging) LASER/TRANSMITTER TARGET, σ WIDE FOV (DASHED) NARROW FOV (SOLID) RECEIVER/DETECTOR November 011 5
Laser Radar Equation Rθ 1 Spot area at range : Area B PG R π Received power: t t Pr = σ A r 4πR πr Area πθ Beam solid angle (sr): Ω = = B A where σ = laser cross section R 4 4π 16 Gain of the transmit antenna: Gt = = A receive optics area r = ΩA θb Include "optical efficiency" (0 Lo 1), let At = Ar, and add round trip atmospheric attenuation: OPTICS D t BEAM SPOT 8P σ tσ Lo α R Pr = Ae 3 4 r θ B π R λ R α = one way power attenuation coefficient November 011 6
Quantities and Symbols Quantity Units Description Symbol Eng. Physics Radiant flux W Rate of emission of power from a source Φ P Radiant emittance (excitance) W/m Power radiated per unit source surface area, M = dφ / ds M W Radiant intensity W/sr Radiant source power per unit solid I J (candlepower) angle I = dφ/ dω Radiant flux density W/m Poynting vector W -- Irradiance W/m Power per unit surface area E H received E = dφ / ds Radiance (brightness) W/m sr Intensity per unit area per steradian of a source L N L = I / ds = d Φ/(cos θ ds dω ) n November 011 7
Geometry for Definition of Quantities p = polarization of the receiver (measurer) q = polarization of the transmitter (source) pq, = θφ, or H,V or other polarization designation Definition of LCS: ( ) ( ) ( ) ( ) ( ) ( ) ( ) Wrp θr, φr Irp θr, φr / R Irp θr, φr σ pq θi, φi, θr, φr = lim 4πR = lim 4πR = lim 4π W θ, φ W θ, φ W θ, φ R iq i i R iq i i R iq i i Monostatic LCS ( θi = θr, φi = φr): I σ pq ( θi, φi, θr, φr ) = lim 4π W R rp iq ( θφ, ) ( θφ, ) November 011 8
LCS Comments The limiting process ( R ) is rarely satisfied at optical wavelengths. For example, using the standard far field criterion for antennas at a wavelength of 10 µm for a ½ m diameter optical system: R ff t 6 D (0.5) = = 500 km λ 10 10 Consequently the measurement of LCS cannot be decoupled from the measurement system (i.e., ladar). Therefore, measured LCS is a function of: beam profile receiver aperture and FOD detector averaging laser characteristics (temporal and spatial coherence) target surface characteristics surface roughness reflectivity (bidirectional reflectance distribution function, BRDF) LCS is still a useful quantity for characterizing a target s scattering cross section. November 011 9
Surface Reflectivity and the BRDF The reflectivity of surface materials is described by the bidirectional reflectance distribution function or BRDF. Similar to RCS, LCS decreases with the reflectivity of the surface. The BRDF of a surface is denoted by: -1 ρ ( r, θ, φ, θ, φ ) steradian pq i i r r where r is a position vector to a point on the surface (i.e., ρ is a function of position). A differential surface area ds illuminated with radiant flux density iq ( i, i ) power Wiq ( θi, φi ) cosθ i. The radiance is Lrq ( θr, φr ) = ρpqwiq ( θi, φi ) cosθi ds The differential LCS is Irp ( r, r ) Lrp ( r, r ) cos r ds d θ φ θ φ θ 4 4 4 cos cos W ( θ, φ ) W ( θ, φ ) W θ φ collects σ π π πρ θ θ ds pq pq r i iq i i iq i i November 011 10
Sample BRDFs BRDFs for white surfaces BRDFs for black surfaces From: J. C. Stover, Optical Scattering Measurement and Analysis, McGraw-Hill, 1990 November 011 11
Diffuse Scattering The Rayleigh condition is commonly used to define a rough surface at wavelength k ˆnˆn λ kˆi ˆi h 8sinψ θ ROUGH SURFACE h = average height of irregularities ψ = π / θ = grazing angle ψ h The one-way phase error due to a deviation in height h is khcos θ. As the heights of the irregularities increase, the scattering transitions from specular to diffuse. This scattering pattern has both diffuse and specular components. November 011 1
Ideal Diffuse Scattering For an ideal diffuse surface the scattering is isotropic for any angle of incidence. The scattering is constant with angle. CASE 1 INFINITE IDEAL DIFFUSE SURFACE θi θr MEASURED SIGNAL CONSTANT FOR ALL θ i For a finite sample, there may be an angle dependence due to changing projected area as illustrated in the figures. CASE FINITE IDEAL DIFFUSE SURFACE FOV < A p ˆn DETECTOR VIEW (FOV) PROJECTED AREA, A p AREA, A CASE 3 FINITE IDEAL DIFFUSE SURFACE FOV > A p VIEW (FOV) ˆn DETECTOR PROJECTED AREA, A p AREA, A November 011 13
Typical Bistatic Scatter Pattern θ s z OPPOSITION EFFECT θ = θ SPECULAR s = i i θ θi θi DIFFUSE ROUGH FLAT SURFACE Features: Uniform scattering for most angles Specular lobe may exist (given by Snell s law) Opposition effect gives a second angle of enhanced scattering in the back direction due to secondary scattering mechanisms (localized shadowing, multiple reflections, etc.) volume scattering e.g.: halo around an aircraft shadow November 011 14
Hemispherical Reflectance For an ideal diffuse surface the BRDF is a constant: ρ ( r, θ, φ, θ, φ ) ρ. Define the pq i i r r o hemispherical reflectance as the total scattered power in a hemisphere. ππ/ = ρ cosθ sinθ dθ dφ = πρ d o r r r r o 0 0 This is often a quantity that is measured for a sample. Consider a flat diffuse surface of area A. The monostatic LCS is dσ d = 4πρocos θ ds σ = 4πρ cos θ ds = 4πρ Acos θ = 4 Acos θ d o o d For a diffuse sphere of radius a (Example 9.1), the illuminated part is a hemisphere: π / o d 0 a o a d σ = 4πρ cos θ sinθ θ d 8π ρ 8π = = 3 3 A SHADOW BOUNDARY a DIFFUSE SPHERE θ ds a sinθ kˆi November 011 15
Components of LCS Empirically LCS is found to have three components: 1) specular σ s, ) diffuse σ d, and 3) projected area σ p. The total LCS is the sum: σ σs + σd + σ p Specular and diffuse components are expected from the random surface model. From Eq. (6.116): 4πA 4k δ 4πckδ c π sin θ / λ σ = e P 0 + e norm λ A 0norm 0 P0 = Specular Diffuse where δ = sum of variances of amplitude and phase errors c = correlation interval of random surface A = surface area P = P / A error free (perfectly flat surface) power scattering pattern Note that δ is a function of angle because the phase error due to surface roughness is a function of angle (see Eq. (6.99)). November 011 16
Components of LCS Measured data is found to differ from the simple specular plus diffuse behavior. This is attributed to secondary scattering effects, and the coupling between the measurement system and target. A third term, the projected area component σ, is included, where σ Acosθ. For conservation of energy, we require that the total hemispherical reflectance satisfy = s + d + p The distribution of hemispherical reflectance is often done after the fact. Recall that (from antennas) the directivity of a hemispherical cosθ power pattern is, so the projected area component is given by: dσ p = pcosθds Example: The projected area component for a sphere of radius a is a disk of radius a A = π a so the projected area LCS component is ( p ) p = pap = a p σ cosθ π cosθ p p November 011 17
Flat Plate Example (Example 9.) LCS of a L = 6 inch square plate with rms deviation of 0.001 inch at 10.6 µm. 1. Specular component: σ π (0.001)(.054) 4k δ = 4 6 10.6 10 4k δ 906 e 0 The specular component is negligible and =0.. Diffuse component: σ = 4L cos d s d θ 3. Projected area component: σ = L cosθ p Pattern is shown for p p = d = 0.5 4π A 4k e δ = ( P 0 = 1) λ norm November 011 18
Diffuse Reflections Diffuse reflected rays do not have to satisfy Snell s Law. 1. Each ray impinging on a surface at A 1 gives rise to an infinite number of diffuse rays. Some of these diffuse reflected rays hit the second surface A 3. Each one of these in turn gives rise to an infinite number of diffuse rays A 1 kˆi ˆn 1 W i ˆn I r 4. The total LCS is the total sum of all direct reflected and doubly reflected rays (and higher reflections if they exist) σ = σ1+ σ + σ3+ R 1 DIFFUSE SCATTERING POINTS A November 011 19
Corner Reflector Assume constant BRDF, ρ o. The first bounce terms: σ1= 4πρo cos θ1ds + cos θds = 4 d A1cos θ1+ Acos θ A1 A The second bounce terms 4π I σ r 1 = Wi1 I = L cosθ ds r r r A In differential form: y A 1 θ i1 θ r1 kˆi ˆx W i ŷ I r di r o i r i1 r1 r1 r1 1 i = dwi1= = R1 R1 r = ρo cosθi1cosθr1ds dw d I = ρ cosθ cosθ ds dw di L cosθ ds cosθ cosθ ds i r DIFFUSE SCATTERING POINTS R 1 θi θr A φ x November 011 0
Corner Reflector Final double bounce result: 4 d cosθ cos cos cos i θ σ = σ = θi θr r π AA 1 R1 which is easy to evaluate numerically. 1 1 1 ds1ds Example: Corner reflector with 6 inch plates (same parameters as in Example 9.) November 011 1
LCS Reduction The same techniques that are used for RCS reduction also apply to LCS reduction: 1. Shaping: a. Only effective for specular reflections. b. Diffuse scattering is only mildly dependent on angle so tilting does not reduce LCS significantly. Materials selection: a. Most effective approach in general. b. Select materials with a low BRDF (flat black finishes) 3. Active and passive cancellation a. Traditionally applied to coherent scattering mechanism b. Most LCS contributions are non-coherent November 011
Anti-reflection Coatings Thin films can be used as anti-reflection coatings (e.g., for eye glasses). The principle is based on the quarter wave transformer concept. Summing up all reflections and transmissions gives the reflectivity R and transmissivity T of the structure. They are the power reflection and transmission coefficients, respectively. ε o ε 1 ε E o θ t cos θ Γ E 1 o 1 E o τ Γ...... t ττ 1 E o ττ 1 ΓΓ 1 E o FREE SPACE MATCHING FILM TARGET BODY (LOAD MATERIAL) Reflection and transmission coefficients at the two interfaces: n Γ = τ τ n o 1 1 no + n1 n1 n n1+ n 1= 1+Γ1 = 1+Γ Γ = n = index of refraction of the layer November 011 3
General case: ( δ = kn t cosθ) Anti-reflection Coatings 1 ττ 1 Er Γ 1 +Γ Γ1Γ R i +Γ1Γ ΓΓ 1 δ Ei +Γ1Γ ΓΓ 1 Et cosδ T = = = = E 1 cos 1 cosδ For a quarter wave layer ( cosδ = 1) : ( Γ 1+Γ) ( 1+ΓΓ ) From Example 9.4 Free space to glass requires a film with n 1 = 1. Typical improvement is shown in the plot. 1 R= n = nn R= 0 1 o From D. C. Harris, Infrared Window and Dome Materials, SPIE Press November 011 4
LCS Prediction The vast majority of recent and current laser radar efforts are in the area of environmental and remote sensing, as opposed to hard target laser radar. Two older simulation packages for LCS prediction: Image resolution test panel (USAF) LCS- Laserx Image of test panel November 011 5
LSC-: Plate and Sphere Numbers are intensity levels (dots are over maximum intensity or under minimum) Notice high returns from specular points on sphere and plate November 011 96
LCS-: Missile Model Numbers are intensity levels (dots are over maximum intensity) Notice high returns from specular points and edges November 011 97