Microwave Optics - Naval Postgraduate Schoolfaculty.nps.edu/jenn/EC4630/GOandGTD(RCS).pdf · Wavefronts are locally plane and waves are TEM 2. ... CAUSTICS B. 7. ... however they

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Microwave Optics (Chapter 5)

EC4630 Radar and Laser Cross Section

Fall 2011

Prof. D. Jenn [email protected]

www.nps.navy.mil/jenn

mailto:[email protected]

http://www.nps.navy.mil/jenn

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

Geometrical Optics (1)

Geometrical optics (GO) refers to the simple ray tracing techniques that have been used for centuries at optical frequencies. The basic postulates of GO are:

1. Wavefronts are locally plane and waves are TEM 2. The wave direction is specified by the normal to the equiphase planes (“rays”) 3. Rays travel in straight lines in a homogeneous medium 4. Polarization is constant along a ray in an isotropic medium 5. Power in a flux tube (“bundle of rays”) is conserved

Area 1 Area 2oW ds W ds⋅ = ⋅∫∫ ∫∫

6. Reflection and refraction obey

Snell’s law

The figure shows an astigmatic flux tube.

ods

dss

1R

2R

A B

CAUSTICS ods

dss

1R

2R

A B

CAUSTICS

7. The reflected field is linearly related to the incident field at the reflection point by a reflection coefficient

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We have already used GO for the simple case of a plane wave reflected from an infinite flat boundary between two dielectricsFor example, for perpendicular polarization ( E

⊥ to the plane defined by s′ and n , which is TE if ˆ ˆn z= ):

jksir esEsE −⊥⊥⊥ ′Γ= )()(

where s′ is the distance from the source to the reflection point and s the distance from the reflection point to the observation point. This has the general form of postulate 7. The curvature of the reflected wavefront determines how the power spreads as a function of distance and direction. It depends on the curvature of both the incident wavefront, iR , and reflecting surface, sR .

s′ s

Source point

Observationpoint

Reflectionpoint

n

Source point

Curvedsurface, sR

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A doubly curved surface (or wavefront) is defined by two principal radii of curvature in two orthogonal planes: ss RR 21 , for a surface or ii RR 21, for the incident wavefront. The reflected wavefront curvature ( rr RR 21 , ) can be computed by first finding the focal lengths in the principal planes. When the principal planes of the incident wavefront and surface can be aligned, then

2/1

21

2

2

12

1

22

22

12

1

22

2,1

4sinsincos

1sinsincos

11

−

+±

+= ssss

iss

i RRRRRRfθθ

θθθ

θ

where

2,1212,1

111211

fRRR iir +

+=

1 2ˆ ˆ,u u = unit vectors tangent to the surface in the two principal planes

=n surface normal at the reflection point

θ i

θ 1

θ 2

n ik

sR1 of Plane

sR2 of Plane

1u2u

θ i

θ 1

θ 2

n ik

sR1 of Plane

sR2 of Plane

1u2u

θ iθ i

θ 1θ 1

θ 2θ 2

n n ik

sR1 of Plane

sR2 of Plane

1u2u

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For an arbitrary angle of incidence and polarization, the field is decomposed into parallel and perpendicular components. The reflected field can be cast in matrix form as:

cjjks

sA

rr

rr

ii

rr ee

sRsRRR

sEsE

sEsE φ−⊥

⊥

⊥⊥⊥⊥

++

′′

ΓΓΓΓ

=

Factor Spreading ),(21

21||||||||

||

|| ))(()()(

)()(

where:

=cφ phase change when the path traverses a caustic (a point at which the cross section of the flux tube is zero)

=Γpq reflection coefficient for p polarized reflected wave, q polarized incident wave

Disadvantages of GO: 1. Does not predict the field in shadows 2. Cannot handle backscatter from flat or singly curved surfaces ( ∞=ss RR 21 or )

Parabola

Focus

Example of a caustic: the focus of a parabola. All reflected rays pass through the focus. The cross section of a tube of reflected rays is zero

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GO Examples

Example: Plane wave 1 2( )i iR R= = ∞ reflected from a plane surface 1 2( )s sR R= = ∞ . The reflected wave is plane.

1

1 2 1 21 2

1 12r ri if f R R

R R

−

= = ∞ → = = + = ∞

Example: Spherical wave 1 2( )i iR R s a′= = = reflected by a plane surface 1 2( )s sR R= = ∞ .

1

1 2 1 21 12r rf f R R aa a

− = = ∞ → = = + =

The reflected wave is spherical, which is also predicted by using image theory.

SOURCE

IMAGE (SOURCEOF SPHERICAL

WAVE)

PLANE REFLECTORa

a

SOURCE

IMAGE (SOURCEOF SPHERICAL

WAVE)

PLANE REFLECTORa

a

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RCS of a Sphere

Example: A plane wave incident on a sphere of radius a.

At the reflection point, the wave is normally incident 1cos0 =→= ii θθ and

1sinsin90 2121 ==→== θθθθ 2/1

21

2

21212,1

41111111

−

+±

+= ssssss RRRRRRf

For the sphere aRR ss == 21 and 2/21 aff == . For a plane wave ∞== ii RR 21 so that

( )2

21,21,2 1 2

/ 21 1 1 1 1 2 ( )2 2r i i

a aA sf a sR R R s

= + + = → = =

where in the denominator it was assumed that s a>> . For a PEC, 1−=Γ , and the reflected field is

2

jksi

rE a eE

s

− = −

which is, a spherical wave.

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RCS of a Sphere

From the definition of RCS, when the observer is at a distance s from the sphere 2

2 22

24

jksi

i

E a es

s aE

σ π π

−−

= =

The is the asymptotic value obtained with the Mie series.

n

nnIncident Wavefront

Reflected Wavefront

n

nnIncident Wavefront

Reflected Wavefront

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RCS of a Parabolic Reflector

A plane wave is incident on a parabolic reflector of diameter D and focal length f. The

reflector edge angle is 12 tan4efD

ψ − =

and the normal vector (for / 2eθ ψ< ) is

ˆˆ ˆcos( / 2) sin( / 2)n r ψ θ ψ= − − . For plane wave incidence 1 21 2 1 2 4

s sr r R RR R f f= = and for a

paraboloid 1 2 22

cos ( / 2)s s fR R

ψ= = . The RCS is

22 24 / 2

1 cos(2 ) efσ π θ ψθ

= < +

.

The plot is for D=5λ, f/D=0.4.

ik

ˆrk

n

2D

ψ

θ

θ ′

/2ψ

f

0 10 20 30 40 50 60 70 80 90-15

-10

-5

0

5

10

15

20

25

30

MMP OGO

, DEGREESθ

σ/λ2 ,

dB

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Geometrical Theory of Diffraction (1)

The geometrical theory of diffraction (GTD) was devised to eliminate many of the problems associated with GO. The strongest diffracted fields arise from edges, but ones of lesser strength originate from point discontinuities (tips and corners). The total field at an observation point P is decomposed into GO and diffracted components

)()()( PEPEPEGTDGOr

+= The behavior of the diffracted field is based on the following postulates of GTD:

1. Wavefronts are locally plane and waves are TEM. 2. Diffracted rays emerge radially from an edge. 3. Rays travel in straight lines in a homogeneous medium 4. Polarization is constant along a ray in an isotropic medium 5. The diffracted field strength is inversely proportional to the cross sectional area of the

flux tube 6. The diffracted field is linearly related to the incident field at the diffraction point by a

diffraction coefficient

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Define a local ray fixed coordinate system: the z axis is along the edge; the x axis lies on the face and points inward.

The internal wedge angle is π)2( n− , where n is not necessarily an integer. A knife edge is the case of 2=n .

Primed quantities are associated with the source point; unprimed quantities with the observation point. Variable unit vectors are tangent in the direction of increase (like spherical unit vectors)

Diffracted rays lie on a cone of half angle ββ ′= (the Keller cone)

x

y

z

( ′ s , ′ φ , ′ β )

(s,φ, β )

s

′ s

β

′ β

φ

′ φ

Q

(2 )n π−

β ′ˆ

φ′ φ

β

The matrix form of the diffracted field is

jks

i

i

h

s

d

d essAsEsE

DD

sEsE −

′

′ ′

′′

−

=

),(

)()(

00

)()(

φ

β

φ

β

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The diffraction coefficients, sD and hD 1, are determined by an appropriate “canonical” problem (i.e., a fundamental related problem whose solution is known).

The scattered field from an infinite wedge was solved by Sommerfeld. The infinitely long edge can represent a finite length edge if the diffraction point is not near the end.

The diffraction coefficient can be “backed out” of Sommerfeld’s exact solution because we know the GO field and the form of the GTD field from the postulates

known found

becan

Sommerfeld from

known

),()()()( jksir essAQEDPEPE

GO−′+=

The basic expressions for the diffraction coefficients of a knife edge are simple (they contain only trig functions), however they have singularities at the shadow and reflection boundaries.

More complicated expressions have been derived that are well behaved everywhere. The most common is the uniform theory of diffraction (UTD). (See Stutzman and Thiele.)

1 This notation is borrowed from optics: s = soft or parallel polarization; h = hard or perpendicular polarization. The reference plane is the plane defined by the edge and the incident ray (for incident polarization) or diffracted ray (for diffracted polarization).

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Diffraction Coefficients

These are the uniform theory of diffraction (UTD) coefficients which have be modified by the transition function F to cancel singularities that occur in the original Keller formulas at the shadow and reflection boundaries:

/ 4

,e [ ( )] [ ( )]( , , , , )

2 2 sin cos( / 2) cos( / 2)

j

s hF k a F k aD

k

π φ φφ φ β βπ β φ φ

− − +

− +

′ ′ = ±

22

| |

2

where , ( ) 2cos ( / 2), and [ ] 2 | |

sin , spherical, spherical wave incident

( )( , ) , cylindrical

1 , cylindrical and plane wavess

jx jt

xa F x j x e e dt

s ss s s

s s sA s s

s s

φ φ φ φ φ

β

ρρρ ρ

∞± ± ± −′= ± = = ∫

′ ′ ′ + ′ + ′′ = = ′+

2in , plane β

′

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GO and GTD for a Wedge

Example: scattering from a wedge for three observation points

Side view of wedge

1P

2P

3P

Source

Direct, reflected anddiffracted field present

Direct and diffracted fields are present (no point on the wedge satisfies Snell’s law)

Only diffracted field present(direct path is in the shadow)

R Q

GO and GTD rays can “mix” (reflected rays can subsequently be diffracted, etc.) to obtain: reflected – reflected (multiple reflection)

reflected – diffracted and diffracted – reflected diffracted – diffracted (multiple diffraction)

An accurate (converged) solution must include all significant contributions.

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Application of GO and GTD

Example 5.4: Scattering from a knife edge (n=2). The field components are:

cos( )

cos( )

( ) /

jki

jkr

jkd i

E eE eE E Q D e

ρ φ φ

ρ φ φ

ρ ρ

⊥

⊥

⊥

′− −

′− +

−⊥

===

where ρ is the distance from the edge to the observation point.

x

y

105φ =

REFLECTIONBOUNDARY

SHADOWBOUNDARY

255φ =

KNIFE EDGE (n = 2)

75φ′ =

50s λ′ =

E

x

y

105φ =

REFLECTIONBOUNDARY

SHADOWBOUNDARY

255φ =

KNIFE EDGE (n = 2)

75φ′ =

50s λ′ =

E

Direct, reflected Direct and Diffracted and diffracted diffracted only

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Example: Fields Scattered From a Box

f =1 GHz, −100 dBm (blue) to −35 dBm (red), 0 dBm Tx power , 1 m PEC box computed using Urbana

ANTENNA

BOX

Reflected Field Only

REFLECTED Incident + Reflected

Reflected + Diffracted

Incident + Reflected + Diffracted

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Surface Waves

GTD accounts for surface waves as combinations of rays that propagate along the surface. See example 5.5 for a strip of width w=5λ. When double diffractions are included, the traveling wave is present.

1Q 2Q

SINGLY DIFFRACTED

RAYS

1Q 2Q

DOUBLY DIFFRACTED

RAYS

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Physical Theory of Diffraction (PTD)

The PO current approximation is not accurate near edges. The ripples in the current decay with distance from the edge, as shown below. They are significant out to about 1λ from the edge. The edge (fringe) current can be found from the exact knife edge currents:

total PO edge edge total POJ J J J J J= + → = −

edgeJ

is used to obtain an edge scattered field that can be added to PO:

( ) ( ) ( )4

jkr jkgos PO edge

edge

jkE P E P e J r e dsr

ηπ

− ′ ′= − ∫∫

Adding the edge scattered fields give the same results as singly diffracted GTD rays. J

0

PHYSICAL OPTICS

POJ

DISTANCE FROM EDGE

ACTUAL (TM POLARIZATION)

J

0

PHYSICAL OPTICS

POJ

DISTANCE FROM EDGE

ACTUAL (TM POLARIZATION)

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Multiple Reflections in Layers

The figure shows a layer of material ( , )µ ε on a PEC conductor. This structure is generally referred to as a Dallenbach layer. If the material is lossless there is an infinite number of reflections that converge to give an input reflection of (see Problem 5.8):

2cos , for TM

, ,/ cos , for TE1

td d o L oin d L

tL od d o

P P Z Z ZZ ZP P

η θη θ

Γ − −Γ = Γ = = +− Γ

where , / cos , , 2 tan sin , cos .jkd t o t o oP e t P e kt Zγ θ θ θ η θ− − ∆= = = ∆ = =

A thickness of λ/4 gives cancellation of the reflections from the two interfaces (similar to the quarter wave transformer effect in transmission lines)

iθ

x

z

t

PEC

rθ

tθ ,µ ε

Reflections in layers are conveniently handled using wave matrices (see p. 354 in the book.)

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Wave Matrices For Layered Media (1)

For multilayered (stratified) media, a matrix formulation can be used to determine the net transmitted and reflected fields. The figure below shows incident and reflected waves at the boundary between two media. The positive z traveling waves are denoted c and the negative z traveling waves b. We allow for waves incident from both sides simultaneously.

Thus,

112222221111cbc

bcbτ

τ+Γ=

+Γ=

where Γ and τ are the appropriate Fresnel reflection and transmission coefficients.

11,εµ 22,εµ

22bΓ

2b

112cτ1c

11cΓ

221bτ

2c1b

0=zz

Medium 1 Medium 2

Rearranging the two equations:

212

12

12

21211 cbb

τττ Γ

+

ΓΓ−=

12

22

12

21 ττ

bcc Γ−=

which can be written in matrix form:

ΓΓ−ΓΓ−=

22

21211212

1211 11

bc

bc

τττ

This is called the wave transmission matrix. It relates the forward and backward propagating waves on the two sides of the boundary.

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As defined, c and b are the waves incident on the boundary, z = 0. At some other location, 1zz = the forward traveling wave becomes 1

1zjec β− and the backward wave

becomes 11

zjeb β . For a plane wave incident from free space onto N layers of different material ),( nn εµ and thickness ( nt ), the wave matrices can be cascaded

≡

ΓΓ=

+

+

+

+

=Φ−Φ

Φ−Φ

∏11

22211211

11

111 1

NN

NN

N

njj

n

jn

j

n bc

AAAA

bc

eeee

bc

nn

nn

τ

11,εµ 22,εµ

1Γ

NN εµ ,

z1

1

+

+

N

N

b

c

1

1

b

c

2

2

b

c

3

3

b

c

1t 2t Nt

Region 1 2 N

1τ

where, for normal incidence n n ntβΦ = is the electrical length of layer n. If the last layer extends to ∞→z then 01 =+Nb . We can arbitrarily set 0=ΦN if the transmission phase is not of interest.

The overall transmission coefficient of the layers (i.e., the transmission into layer N when

01 =+Nb ) is 1111 /1/ AccN =+ . The overall reflection coefficient is 112111 // AAcb = .

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If the incidence angle in region 1 is not normal, then nΦ must be determined by taking into account the refraction in all of the previous n −1 layers. The transmission angle for layer n becomes the incidence angle for layer n +1 and they are related by Snell’s law:

.sinsinsinsin121 121 −−====

NiNiiio θβθβθβθβ

Oblique incidence and loss can be handled by modifying the transmission and reflection formulas as listed below, where iθ is the incidence angle at the first boundary:

1−niθnn it θθ =

−1

Layer nLayer n −1

Boundaryn - 1

Boundaryn

z

( )1/ 22sinn nn o n r r itβ ε µ θΦ = −

1

1

−

−+−

=Γnn

nnn ZZ

ZZ , nn Γ+= 1τ

2sin

cosn n

n

r r in

o r i

ZZ

ε µ θ

ε θ

−= , (parallel polarization)

2

cos

sinn

n n

r in

o r r i

ZZ

µ θ

ε µ θ=

−, (perpendicular polarization)

For lossy materials: n n

nr r

oj σε εωε

→ −

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Microwave Optics - Naval Postgraduate Schoolfaculty.nps.edu/jenn/EC4630/GOandGTD(RCS).pdf · Wavefronts are locally plane and waves are TEM 2. ... CAUSTICS B. 7. ... however they