Definition of frames

Source frame

Direction of propagation

ks = {Sin[θo]Cos[ϕo],Sin[θo]Sin[ϕo],Cos[θo]};

Source basis constructed on the direction of propagation as eθ = ((∂ k)/(∂θo)) and eϕ = (1/(sin θ)) ((∂ k)/(∂ϕo)) and p=eθ

eθ = {Cos[θo]Cos[ϕo],Cos[θo]Sin[ϕo],-Sin[θo]};
eϕ = {-Sin[ϕo],Cos[ϕo],0};

D[ks, θo]
 D[ks, ϕo ]

SSB frame

Direction of propagation

k = -{Sin[θ]Cos[ϕ],Sin[θ]Sin[ϕ],Cos[θ]};

SSB basis constructed on the direction of propagation as v =-((∂ k)/(∂β)) and u = (1/(sin θ)) ((∂ k)/(∂λ))

D[k, θ]
D[k, ϕ ]

v = {-Cos[θ]Cos[ϕ],-Cos[θ]Sin[ϕ],Sin[θ]};
u = {Sin[ϕ],-Cos[ϕ],0};

Orbital angular momentum in SSB

L = {Sin[θL]Cos[ϕL],Sin[θL]Sin[ϕL],Cos[θL]};

Source frame constructed on L and k

p=Simplify[Cross[L,k]];
p//MatrixForm

Simplify[p.p]

q=Simplify[Cross[k,p]];
q//MatrixForm

Inclination

Cos inclination

FullSimplify[L.k]

Polarisation convention

polarisation : tan ψ = -((eϕ.u)/(eϕ.v))

FullSimplify[-((eϕ.u)/(eϕ.v))]

q=Cross[L,k]

FullSimplify[-((q.u)/(q.v))]

FullSimplify[Sin[θL]Sin[ϕ-ϕL](-Cos[θ] Cot[ϕ-ϕL]+Cot[θL] Csc[ϕ-ϕL] Sin[θ])]

((Cos[θL] Sin[θ]-Cos[θ] Cos[ϕ-ϕL] Sin[θL])/(Sin[θL]Sin[ϕ-ϕL]))

Eθ = Cos[ψ]U + Sin[ψ]V
Eϕ =  - Sin[ψ]U + Cos[ψ]V

Simplify[Eθ Eθ -Eϕ Eϕ]

Expand[FullSimplify[2 Eθ Eϕ ]]

2 U V (((Cos[ψ])^2)- ((Sin[ψ])^2))-((U^2) - (V^2) )2Cos[ψ] Sin[ψ]
-( (U^2)- (V^2)) Sin[2 ψ]+2 U V Cos[2ψ]

ϵp=U U - V V
ϵc = 2 U V

ep =  Cos[2 ψ] ϵp + Sin[2 ψ] ϵc
ec = - Sin[2 ψ] ϵp + Cos[2 ψ] ϵc

hij = Ap ep + Ac ec

Collect[hij,{(U^2)-(V^2),2 U V }]

ξp (Ap Cos[2 ψ]-Ac Sin[2 ψ])+ξx (Ac Cos[2 ψ]+Ap Sin[2 ψ])

2(- Ac  Cos[2 ψ]- Ap Sin[2 ψ])ξx+ (- Ap Cos[2 ψ]+Ac Sin[2 ψ])ξp

hijFinal = ξp (-Ap Cos[2 ψ]+Ac Sin[2 ψ])+2 ξx (-Ac Cos[2 ψ]-Ap Sin[2 ψ])

Collect[hijFinal/.ψ-> (π/2)-ψb//FullSimplify,{ξp,ξx}]

Polarisation alternative convention

polarisation : tan ψ = ((p.v)/(p.u))

FullSimplify[((p.v)/(p.u))]

((Cos[θL]  Sin[θ]-Cos[θ]Sin[θL] Cos[ϕ-ϕL])/(Sin[θL]Sin[ϕ-ϕL]))

P = + Cos[ψ]U + Sin[ψ]V
Q = - Sin[ψ]U + Cos[ψ]V

Simplify[P P-Q Q]

Expand[FullSimplify[2 P Q ]]

(2 (V^2)-2 (U^2)) Cos[ψ] Sin[ψ]+2 U V(((Cos[ψ])^2)- ((Sin[ψ])^2))
-( (U^2)- (V^2)) Sin[2 ψ]+2 U V Cos[2ψ]

ϵp=U U - V V
ϵc = 2 U V

ep = Cos[2 ψ] ϵp + Sin[2 ψ] ϵc
ec = -Sin[2 ψ] ϵp + Cos[2 ψ] ϵc

hij = Ap ep + Ac ec

Collect[hij,{(U^2)-(V^2),2 U V }]