27
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

Laser Cross Section (LCS) - Naval Postgraduate School ...faculty.nps.edu/jenn/EC4630/LCSV2.pdf · Laser Cross Section (LCS) ... • Point target: laser beam illuminates entire target,

Embed Size (px)

Citation preview

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

November 2011 2

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

)

November 2011 3

Naval Postgraduate School Department of Electrical & Computer Engineering Monterey, California

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

November 2011 4

Naval Postgraduate School Department of Electrical & Computer Engineering Monterey, California

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 ≈

=

November 2011 5

Naval Postgraduate School Department of Electrical & Computer Engineering Monterey, California

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)

November 2011 6

Naval Postgraduate School Department of Electrical & Computer Engineering Monterey, California

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

≤ ≤ =

=

=

November 2011 7

Naval Postgraduate School Department of Electrical & Computer Engineering Monterey, California

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

November 2011 8

Naval Postgraduate School Department of Electrical & Computer Engineering Monterey, California

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→∞=

November 2011 9

Naval Postgraduate School Department of Electrical & Computer Engineering Monterey, California

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.

November 2011 10

Naval Postgraduate School Department of Electrical & Computer Engineering Monterey, California

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

= = =

November 2011 11

Naval Postgraduate School Department of Electrical & Computer Engineering Monterey, California

Sample BRDFs

BRDFs for white surfaces

BRDFs for black surfaces

From: J. C. Stover, Optical Scattering Measurement and Analysis, McGraw-Hill, 1990

November 2011 12

Naval Postgraduate School Department of Electrical & Computer Engineering Monterey, California

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

== − =

November 2011 13

Naval Postgraduate School Department of Electrical & Computer Engineering Monterey, California

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

November 2011 14

Naval Postgraduate School Department of Electrical & Computer Engineering Monterey, California

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

November 2011 15

Naval Postgraduate School Department of Electrical & Computer Engineering Monterey, California

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

sina θ

DIFFUSESPHERE

SHADOW BOUNDARY

ds

November 2011 16

Naval Postgraduate School Department of Electrical & Computer Engineering Monterey, California

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)).

November 2011 17

Naval Postgraduate School Department of Electrical & Computer Engineering Monterey, California

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= =

November 2011 18

Naval Postgraduate School Department of Electrical & Computer Engineering Monterey, California

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= =

November 2011 19

Naval Postgraduate School Department of Electrical & Computer Engineering Monterey, California

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

Naval Postgraduate School Department of Electrical & Computer Engineering Monterey, California

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

Naval Postgraduate School Department of Electrical & Computer Engineering Monterey, California

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)

November 2011 22

Naval Postgraduate School Department of Electrical & Computer Engineering Monterey, California

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

November 2011 23

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

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

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

November 2011 24

Naval Postgraduate School Department of Electrical & Computer Engineering Monterey, California

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

November 2011 25

Naval Postgraduate School Department of Electrical & Computer Engineering Monterey, California

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

November 2011 96

Naval Postgraduate School Department of Electrical & Computer Engineering Monterey, California

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

November 2011 97

Naval Postgraduate School Department of Electrical & Computer Engineering Monterey, California

LCS-2: Missile Model

• Numbers are intensity levels (dots are over maximum intensity) • Notice high returns from specular points and edges