Monochromatic Radiation is Always 100% Polarized

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Ερατώ Βασιλειάδης
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1 Mnchrmatic Radiatin is lwas % Plarized Plarizatin llipse z Prpagatin Right-Hand Plarizatin v b z θ a ϕ (t) v 3 Parameters Specif llipse e.g. a, b, ϕ a, ϕ, θ v, v, θ ls, (need + r t right r left elliptical) Lec3a.3- // V

2 Let Plarizatin f Narrwband Radiatin ( t) v ˆ (t)cs v (t) and v (t) are slwl varing and randm; v, v, and δ ma be nn-zer Stkes Parameters [ ωt + φ(t) ] + v ˆ (t)cs[ ωt + φ(t) + δ(t) ] I Q U S S S [ ] v (t) + v (t) η [ ] v (t) v (t) v ( t) η v (t) cs δ(t) η [W m - ] ttal pwer -ness 45 -ness Lec3a.3- // V S3 v ( t) v (t) sin δ(t) η circularit V

3 Let % Plarized Narrwband Waves ( t) v ˆ (t)cs [ ωt + φ(t) ] + v ˆ (t)cs[ ωt + φ(t) + δ(t) ] v (t) and v (t) are slwl varing and randm; v, v, and δ ma be nn-zer δ ( t) δ and v v (t) cnstant fied ellipse, variable size (t) ls : S S + S + S 3 Therefre, an 3 Stkes parameters specif plarizatin Lec3a.3-3 // V3

4 Partiall Plarized Narrwband Radiatin Stkes Parameters I S [ v (t) + v (t) ] η [W m - ] ttal pwer Q S [ v (t) v (t) ] η -ness U S v (t) v (t) cs δ(t) η 45 -ness circularit V S 3 v (t) v (t) sin δ(t) η Nte: Fr uncrrelated waves superimpsed (+B), we have S i+b S i + S ib where i,,, 3 Fr % plarizatin, Stkes: S ; S S S 3 Therefre, fr partiall plarized wave: [S, S, S, S 3 ] [S u,,, ] + [S S u, S, S, S 3 ] where (S S u ) S + S + S 3 Lec3a.3-4 // Define percentage plarizatin S S u S % m, m V4

5 Cherenc Matri η { } { } (t) ( (t)e R (t)e R t e t e ω ω + e.g. X-plarizatin RCP (right-circular) S RC Unplarized S u V5 Lec3a.3-5 // var slwl ), t where (t) where S

6 Finding Orthgnal Plarizatin RC e.g. X-plarizatin S RCP (right-circular) Unplarized RC u S S Nte : + u RC + LC u B, and If then + B T T B r r u Therefre, we can find rthgnal plarizatin B n Lec3a.3-6 // V6

7 Plarized ntennas Define e.g. G ( θ, φ) i G ( θ, φ) i ( θ, φ) ( θ, φ ) i + { i, } {,}, { r, },{ a, b }( b a) Far Fields ; claim P rec T r t inc [m ][Wm ] [ W] fr incident plane wave Lec3a.3-7 // W

8 Plarized ntennas t ; claim P T [ W] rec r inc [m ][Wm ] fr incident plane wave S P rec [ ] fr incident unifrm plane wave n antenna Fr Ω s : P rec 4 π T r t ( θ, φ) ( θ, φ) dω Lec3a.3-8 // W

9 Measure 4 pwers; use 4 antennas W3 T Measure Plarizatin d c b a M M M M d d c b b a a a a e.g. M ( ) ˆ M Ĵ s Lec3a.3-9 //, estimate is singular? Is

10 Fr,, RC, LC POL: T Measure Plarizatin M ( ) ˆ M Ĵ s W4 Can nt distinguish vs det Fr, 45, RC, LC: det Lec3a.3- //, estimate is singular? Is k" "

11 ample f a Plarimeter Right Circular.5 MHz Dipleer 3.5 MHz.5 MHz Lcal 3 MHz Oscillatr 3 MHz [Chen, Prc. IR,, 958] Left Circular 5. MHz Lcal Oscillatr 5. MHz Lcal Oscillatr 5 MHz MHz ( ) KHz Phase Cmparatr dt ρ ρ r r ρ ρ 4 measurements 4 Stkes parameters Lec3a.3- // W5

12 ntenna Phase rrrs phase frnt Sstematic antenna phase errrs: ) pr design and fabricatin ) gravit, wind, thermal (gravit and thermal limits near arc minute) 3) feed ffset Randm antenna phase errrs: ) machine tlerances, surface rughness ) adustment errrs 3) feed ffset Lec3a.3- // X

13 amples f ntenna Phase rrrs Randm antenna phase errrs: ) matching tlerances, surface rughness ) adustment errrs 3) feed ffset 3-ft parablic reflectr antenna at NRO, Greenbank, West Virginia ) sstematic sag fi backup; ftprints n mesh ) steamrlled mesh lng waves 3) ~ new panels: θ B >. 5 arc minute Lec3a.3-3 // X

14 Tpes f ptical and radi prpagatin phase errrs Sstematic: h T(h) ρ (h) velcit f light c c < c c insphere c arth Randm phase: + amplitude? ~ RMS < λ RMS >> λ π, π, arth weak fluctuatins strng fluctuati ns vs pathlength L nλ interference and nulls Lec3a.3-4 // Thin screen (cnstant amplitude) Thick screen X3

15 ffect f Phase Variatin n Directivit aperture Fr -plarizatin: ϕ ϕ z ~ ( ϕ ϕ ) ( ), R, ( τ ) ~ ( ϕ ) D( ϕ), G ( ϕ) D ( f, θ, φ ) [ ( ) ] R π + csθ λ () τ e π ( ϕ τ) λ (,) dτ dd dτ Lec3a.3-5 // X4

16 ffect f Phase Variatin n Directivit D ( [ ( ) ] R f, θ, φ ) π + csθ λ π (+ csθ) () τ e (,) { } { (, θ, φ )} R () τ λ (,) dd D f π ( ϕ τ) λ e dτ dd ( r ) ( r τ ) ( ) ( r) r e dτ π ( ϕ τ) λ d r dτ dτ Therefre { } { ()} () R () r ( r τ R τ τ e ) Spatial statinari t : { () ( )} r r τ () () r e e { } Lec3a.3-6 // X5

17 Definitin f Characteristic Functin It is the Furier transfrm f prbabilit distributin p() (als called the mment-generating functin) [ ω ] ω p ()e d F.T.[ p ()] e Γ ( ω ;) characteristic functin f p() One use f the Furier transfrm f p() is when we seek p ( ) + n π i ( ) p ( ) p ( ) F.T. F.T.[ p ( )] n p... n i Lec3a.3-7 // X6

18 Cmputatin f { } ( ()) ω, ; ( () + ω ω ), e () τ τ Thus Γ ω { ( ) } R τ Recall : If, are GRV, then (, ω,, ) Γ ω e [ ω ω ] ω ω Here, (), () τ Therefre : { ()} ()- τ e Γ ω, ω ; (), () τ ( ) Lec3a.3-8 // X7

19 Cmputatin f ( ) (, ω,, ) Γ ω e Here, (), [ ω ω ] () τ { } R τ ω ω Since: ( ) { ()} ()- τ e Γ ω, ω ; (), () τ () φ () () () τ φ ( ) e ( ω, ω ; (), () τ ) Lec3a.3-9 // Therefre : τ Therefre e [ ] ( ) φ () ( ) () φ( ) φ ( τ) () τ φ ( τ ) φ( τ) φ() { ( )} () φ ( τ ) φ () R τ τ e φ τ R b statinari t X8

20 Cmputatin f pected Directivit π ( + cs θ) { D( f, θφ, )} e R τ e d τ dτ λ () r φ ( τ ) ( ) ( τ) φ ( τ ) φ () e σ e L π - ( ) ϕτ i φτ φ () ( ) λ da crrelatin length L f phase irregularities τ B(τ) e ( ) φ ) φ τ σ + ( e τ σ φ () σ L τ L τ τ Lec3a.3- // X9

21 e Slutin t pected Directivit φ ( τ ) φ () B(τ) σ e π (+ csθ) σ { ( )} () D f, θ, φ λ () e + B τ R () τ r da L τ + e σ e τ π ϕ τ λ dτ dτ σ { D ( f, θ, φ )} e D ( f, θ, φ ) + B( ϕ ) D ( f, θ, φ ) gain degradatin sidelbe increase B(ϕ ) λ /L ϕ Lec3a.3- // X

22 amples f Randm ntenna Surface Let b RMS surface tlerance f reflectr antenna On-ais gain f randm antenna G σ ( b π λ ) ( b 4 π λ G e G e G e ) If b b b λ λ λ 4π 6 3 G G G e new lg G (pwer shifts t sidelbes) n aperture antenna, fied illuminatin - lg G lg λ + lg 4 π e Lec3a.3- // ~minimum useful wavelength lg λ X

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