Laser Cross Section (LCS) - Naval Postgraduate School ...faculty.nps.edu/jenn/EC4630/LCSV2.pdf ·...
Transcript of Laser Cross Section (LCS) - Naval Postgraduate School ...faculty.nps.edu/jenn/EC4630/LCSV2.pdf ·...
November 2011 1
Laser Cross Section (LCS) (Chapter 9)
EC4630 Radar and Laser Cross Section
Fall 2010
Prof. D. Jenn [email protected]
www.nps.navy.mil/jenn
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Naval Postgraduate School Department of Electrical & Computer Engineering Monterey, California
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 (CO2 gas lasers, 10%
efficiency) • 1.06 mm (Neodymium YAG crystal
laser, 3% efficiency)
WAVELENGTH
FREQUENCY (GHz)
ON
E-W
AY
ATT
ENU
ATI
ON
(dB
/km
)
WAVELENGTH
FREQUENCY (GHz)
ON
E-W
AY
ATT
ENU
ATI
ON
(dB
/km
)
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GO and the Beam Expander
owρ =
oz z=
GEOMETRICAL OPTICS RAYSBEAM WAIST
GO FOCUS
LENS
LENS
inΦ outDinD
outΦinΦ outD
inDoutΦ
Focused beam and its GO approximation
Beam waveguide
Beam expander: (Φ is beam divergence)
out out
in in
DD
Φ=
Φ
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Laser Radar System
System block diagram for a coherent laser radar Receive optics and detector Half power beamwidth:
1.02
lens or mirror diameterB D
D
λθ ≈
=
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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)
TARGET, σ
RECEIVER/DETECTOR
LASER/TRANSMITTER
WIDE FOV(DASHED)
NARROW FOV (SOLID)
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Laser Radar Equation
2
2 22
2
1Spot area at range : Area Received power: 2 4 2
AreaBeam solid angle (sr): = = where laser cross section 4
Gain of the transmit ant
t tBr r
BA
PGRR P AR R
R
θπ σπ π
πθ σ
≈ =
Ω =
24 16enna: receive optics areat r
A BG Aπ
θ= = =
Ω
R
BθσBEAM SPOT
OPTICS
Dt
2 23 4 2
Include "optical efficiency"(0 1), let ,and add round tripatmospheric attenuation:
8
one way power attenuation coefficient
o t r
Rt or r
L A A
P LP A eR
ασπ λ
α
−
≤ ≤ =
=
=
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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/m2 Power radiated per unit source surface area, /M d ds= Φ
M W
Radiant intensity (candlepower)
W/sr Radiant source power per unit solid angle /I d d= Φ Ω
I J
Radiant flux density W/m2 Poynting vector W -- Irradiance W/m2 Power per unit surface area
received /E d ds′= Φ E H
Radiance (brightness)
W/m2 sr Intensity per unit area per steradian of a source
2 / /(cos )nL I ds d ds dθ= = Φ Ω
L N
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Geometry for Definition of Quantities
p = polarization of the receiver (measurer) q = polarization of the transmitter (source)
, , or H,Vor other polarizationdesignation
p q θ φ=
Definition of LCS:
( ) ( )( )
( )( )
( )( )
22 2, , / ,
, , , 4 4 4lim lim lim, , ,rp r r rp r r rp r r
pq i i r rR R Riq i i iq i i iq i i
W I R IR R
W W Wθ φ θ φ θ φ
σ θ φ θ φ π π πθ φ θ φ θ φ→∞ →∞ →∞
= = =
Monostatic LCS ( ,i r i rθ θ φ φ= = ):
( ) ( )( )
,, , , 4lim ,
rppq i i r r
R iq
IW
θ φσ θ φ θ φ π
θ φ→∞=
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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:
2 2
62 2(0.5) 500 km
10 10t
ffDRλ −= = ≈
×
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.
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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( , , , , ) steradianpq i i r rrρ θ φ θ φ′
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 iW θ φ collects power ( ), cosiq i i iW θ φ θ . The radiance is
( ) ( ), , cosrq r r pq iq i i iL W dsθ φ ρ θ φ θ= The differential LCS is
( )( )
( )( )
, , cos4 4 4 cos cos
, ,rp r r rp r r r
pq pq r iiq i i iq i i
I L dsd ds
W Wθ φ θ φ θ
σ π π πρ θ θθ φ θ φ
= = =
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Sample BRDFs
BRDFs for white surfaces
BRDFs for black surfaces
From: J. C. Stover, Optical Scattering Measurement and Analysis, McGraw-Hill, 1990
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Diffuse Scattering
The Rayleigh condition is commonly used to define a rough surface at wavelength
θh
ROUGH SURFACE
ψ
ik n
θh
ROUGH SURFACE
ψ
ikik nn
The one-way phase error due toa deviation in height is cos .As the heights of the irregularities increase, the scattering transitions from specular to diffuse. This scattering pattern has bothdif
h kh θ
fuse and specular components.
8sinaverage height of irregularities
/ 2 grazing angle
h
h
λψ
ψ π θ
≥
== − =
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Ideal Diffuse Scattering
For an ideal diffuse surface the scattering is isotropic for any angle of incidence. The scattering is constant with angle. For a finite sample, there may be an angle dependence due to changing projected area as illustrated in the figures.
CASE 1INFINITE
IDEALDIFFUSE
SURFACE
MEASURED SIGNALCONSTANT FOR
ALL θi
iθ rθ
DETECTOR
VIEW (FOV)
AREA, A
CASE 2FINITE IDEAL
DIFFUSESURFACEFOV < Ap
PROJECTEDAREA, Ap
n
DETECTOR
AREA, A
n
CASE 3FINITE IDEAL
DIFFUSESURFACEFOV > Ap
VIEW (FOV)
PROJECTEDAREA, Ap
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Typical Bistatic Scatter Pattern
z
DIFFUSE
OPPOSITIONEFFECT
ROUGH FLAT SURFACE
s iθ θ= SPECULAR s iθ θ= −
iθ iθ
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
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Hemispherical Reflectance
For an ideal diffuse surface the BRDF is a constant: ( , , , , )pq i i r r orρ θ φ θ φ ρ′ ≡
. Define the hemispherical reflectance as the total scattered power in a hemisphere.
2 / 2
0 0cos sind o r r r r od d
π πρ θ θ θ φ πρ= =∫ ∫
This is often a quantity that is measured for a sample. Consider a flat diffuse surface of area A. The monostatic LCS is
2
2 2 24 cos
4 cos 4 cos 4 cosA
d o
d o o d
d dsds A A
σ πρ θσ πρ θ πρ θ θ
== = =∫∫
For a diffuse sphere of radius a (Example 9.1), the illuminated part is a hemisphere:
/ 2
0
2
2 2 2
4 cos sin
8 83 3
d o
od
d
a a
πσ πρ θ θ θ
π ρ π
= ∫
= = ik
aθ
sina θ
DIFFUSESPHERE
SHADOW BOUNDARY
ds
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Components of LCS
Empirically LCS is found to have three components: 1) specular sσ , 2) diffuse dσ , and 3) projected area pσ . The total LCS is the sum: s d pσ σ σ σ≈ + + Specular and diffuse components are expected from the random surface model. From Eq. (6.116):
2 2 2 2 2 2
norm
2 2 2 24 sin /
02
Specular Diffuse
4 4
k cA c ke P eA
δ π θ λπ π δσλ
− − = +
norm
2
0 0
0
where sum of variances of amplitude and phase errors = correlation interval of random surface = surface area
/error free (perfectly flat surface) power scattering pattern
cA
P P AP
δ =
==
Note that 2δ is a function of angle because the phase error due to surface roughness is a function of angle (see Eq. (6.99)).
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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 pσ , is included, where cosp Aσ θ . 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 2, so the projected area component is given by:
2 cosp pd dsσ θ= Example: The projected area component for a sphere of radius a is a disk of radius a
( )2pA aπ= so the projected area LCS component is
22 cos 2 cosp p p pA aσ θ π θ= =
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Flat Plate Example
(Example 9.2) LCS of a L = 6 inch square plate with rms deviation of 0.001 inch at 10.6 µm.
1. Specular component: 2 22
42
4 kA e δπσλ
−= ( )norm0 1P =
2 26
2 2
2
4
2 (0.001)(.0254)4 410.6 10
906 0 The specular component is negligible and =0.
k
s
k
e
δ
δ
π−=
−
×
≈ → ≈
2. Diffuse component: 2 24 cosd dLσ θ=
3. Projected area component: 22 cosp pLσ θ=
Pattern is shown for
0.5p d= =
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Diffuse Reflections
Diffuse reflected rays do not have to satisfy Snell’s Law. 1. Each ray impinging on a surface at 1A gives rise to an infinite number of diffuse rays 2. Some of these diffuse reflected rays hit the second surface 2A 3. Each one of these in turn gives rise to an infinite number of diffuse rays
DIFFUSESCATTERING
POINTS
1n2n
ik
1A
2A
12R
iW
rI
4. The total LCS is the total sum of
all direct reflected and doubly reflected rays (and higher reflections if they exist)
1 2 3σ σ σ σ= + + +
November 2011 20
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Corner Reflector
Assume constant BRDF, oρ . The first bounce terms:
1 2
2 2 2 21 1 2 1 1 2 24 cos cos 4 cos coso d
A Ads ds A Aσ πρ θ θ θ θ
= + = +∫∫ ∫∫
The second bounce terms
2
212
12 2 2 2
4
cos
r
ir r r
A
IW
I L ds
πσ
θ
=
= ∫∫
In differential form:
2 2 2 2 11 1 1 1
2 1 2 212 12
2 22 1 1 2
2 2 2
cos coscos
cos coscos cos
r o i r ir r r
i i
r o i ri r
dI ds dWdI L dsdW dWR R
d I dsds
ρ θ θθ
ρ θ θθ θ
=
= = =
=×
DIFFUSESCATTERING
POINTS
x
y
1A
2A
12R
iW
x y
ik
1iθ
1rθ
2iθ 2rθ φ
rI
November 2011 21
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Corner Reflector
Final double bounce result:
1 2
2 112 21 2 2 1 22
12
4 cos coscos cosd i ri r
A Ads ds
Rθ θσ σ θ θ
π= = ∫ ∫
which is easy to evaluate numerically.
Example: Corner reflector with 6 inch plates (same parameters as in Example 9.2)
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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
2. 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
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Naval Postgraduate School Department of Electrical & Computer Engineering Monterey, California
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.
MATCHING FILM
...
...
FREE SPACE
TARGET BODY(LOAD MATERIAL)
t
t cos θ
θ
1εoε
2ε
oE1 oEΓ
1 2 oEτ τ 1 2 1 2 oEτ τ Γ Γ
21 2 oEτ Γ
MATCHING FILM
...
...
FREE SPACE
TARGET BODY(LOAD MATERIAL)
t
t cos θ
θ
1εoε
2ε
oE1 oEΓ
1 2 oEτ τ 1 2 1 2 oEτ τ Γ Γ
21 2 oEτ Γ
Reflection and transmission coefficients at the two interfaces:
11
11 2
21 2
1 12 2
11
o
o
n nn nn nn n
ττ
−Γ = −
+−
Γ = −+
= + Γ= + Γ
n = index of refraction of the layer
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Anti-reflection Coatings
General case: ( )12 coskn tδ θ=
2 22 2 2 21 2 1 2 1 2
2 2 2 2 2 21 2 1 2 1 2 1 2
2 cos1 2 cos 1 2 cos
t r
i i
E ET R
E Eτ τ δ
δ δΓ + Γ − Γ Γ
= = = =+ Γ Γ − Γ Γ + Γ Γ − Γ Γ
For a quarter wave layer ( )cos 1δ = − :
( )( )
21 2
1 22 01 21
oRR n n n
=
Γ + Γ= → =
+ Γ Γ
From Example 9.4
Free space to glass requires a film with
1 1.22n = Typical improvement is shown in the plot.
From D. C. Harris, Infrared Window and Dome Materials, SPIE Press
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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:
• LCS-2 • Laserx
Image resolution test panel (USAF)
Image of test panel
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LSC-2: 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