Exact reflection formula

main formula - Huyghens-Fresnel equation

Eout[x,y,z]
=((I k)/(2 π))(∫_(-∞)^∞)(∫_(-∞)^∞)(Subscript[E, in])[u,v,ζ]((Exp[-I k ρ])/ρ)((z-δ)/ρ)Cos[(Subscript[θ, in])]\[DifferentialD]u\[DifferentialD]v
=((I k)/(2 π))(∫_(-∞)^∞)(∫_(-∞)^∞)(OverBar[(Subscript[E, in])])[u,v,ζ]Exp[I k δ - I k ((((u^2)+(v^2)))/(2 (Subscript[R, in])))-((((u^2)+(v^2)))/(w^2))]((Exp[-I k ρ])/ρ)((z-δ)/ρ)Cos[(Subscript[θ, in])]\[DifferentialD]u\[DifferentialD]v

(μ^2) = (((x-u))^2)+(((y-v))^2);
ρ = (Sqrt[(μ^2)+(((z-δ))^2)]);
ρ = (Sqrt[(μ^2)+(((z-δ))^2)])=z-δ + (Sqrt[(μ^2)+(((z-δ))^2)]) - (z - δ)= z - δ + ((μ^2)/((Sqrt[(μ^2)+(((z-δ))^2)]) + (z-δ)));
Cos[(Subscript[θ, in])]=(dr/dz)=(Sqrt[1-(((u^2)+(v^2))/(Rmir^2))]) for parabolic, = (Sqrt[(((Rmir^2)-2((u^2)+(v^2)))/((Rmir^2)-((u^2)+(v^2))))]) for spherical surface
Cos[(Subscript[θ, out])]=((z-δ)/ρ);

calculation

ρ-(z-δ)=((μ^2)/((Sqrt[(μ^2)+(((z-δ))^2)]) + (z-δ))); 
ρ - (z-δ) - ((μ^2)/(2 z))
δ = d + (((u^2)+(v^2))/(2 (Subscript[R, mir])));
-I k ρ + I k δ - I k ((((u^2)+(v^2)))/(2 (Subscript[R, in])))-((((u^2)+(v^2)))/((Subscript[w, in])^2))=-I k (ρ-(z-δ))-I k (z-δ) - I k n δ - I k ((((u^2)+(v^2)))/(2 (Subscript[R, in])))-((((u^2)+(v^2)))/((Subscript[w, in])^2))=-I k z +I k δ + I k δ -I k  ((μ^2)/((Sqrt[(μ^2)+(((z-δ))^2)]) + z-δ))- I k ((((u^2)+(v^2)))/(2 (Subscript[R, in])))-((((u^2)+(v^2)))/((Subscript[w, in])^2))=-I k z + I 2 k d[u,v] -((u^2)+(v^2))(I (k/2)(-(2/(Subscript[R, mir]))+(1/(Subscript[R, in])))+(1/(w^2)))-I k  ((μ^2)/((Sqrt[(μ^2)+(((z-δ))^2)]) + z-δ))=-I k z + I 2 k d[u,v] -((u^2)+(v^2))(I (k/(2 (Subscript[R, out])))+(1/(w^2)))-I k  ((μ^2)/((Sqrt[(μ^2)+(((z-δ))^2)]) + z-δ))
(1/(Subscript[R, out]))=-(2/(Subscript[R, mir]))+(1/(Subscript[R, in])) < 0 for conversing beam
d[u,v] = 0 for parabolic surface
δ(spherical)=(Subscript[R, mir]) - (Subscript[R, mir])(Sqrt[1-(((u^2)+(v^2))/((Subscript[R, mir])^2))])=(Subscript[R, mir])(1-(Sqrt[1-(((u^2)+(v^2))/((Subscript[R, mir])^2))]))=(Subscript[R, mir])((((u^2)+(v^2))/((Subscript[R, mir])^2))/(1+(Sqrt[1-(((u^2)+(v^2))/((Subscript[R, mir])^2))])))=(((u^2)+(v^2))/((Subscript[R, mir])(1+(Sqrt[1-(((u^2)+(v^2))/((Subscript[R, mir])^2))]))))
d[u,v] = (Subscript[R, mir]) - (Subscript[R, mir])(Sqrt[1-(((u^2)+(v^2))/((Subscript[R, mir])^2))])-(((u^2)+(v^2))/(2 (Subscript[R, mir]))) for spherical mirror
(Subscript[R, mir]) - (Subscript[R, mir])(Sqrt[1-(((u^2)+(v^2))/((Subscript[R, mir])^2))])-(((u^2)+(v^2))/(2 (Subscript[R, mir])))=(((u^2)+(v^2))/((Subscript[R, mir]) +(Subscript[R, mir])(Sqrt[1-(((u^2)+(v^2))/((Subscript[R, mir])^2))])))-(((u^2)+(v^2))/(2 (Subscript[R, mir])))=(((u^2)+(v^2))(2 (Subscript[R, mir])-(Subscript[R, mir])-(Subscript[R, mir])(Sqrt[1-(((u^2)+(v^2))/((Subscript[R, mir])^2))])))/(2 (Subscript[R, mir])((Subscript[R, mir]) +(Subscript[R, mir])(Sqrt[1-(((u^2)+(v^2))/((Subscript[R, mir])^2))])))=((((u^2)+(v^2))(Subscript[R, mir])(1-(Sqrt[1-(((u^2)+(v^2))/((Subscript[R, mir])^2))])))/(2 ((Subscript[R, mir])^2)(1 +(Sqrt[1-(((u^2)+(v^2))/((Subscript[R, mir])^2))]))))=((((u^2)+(v^2))(((u^2)+(v^2))/((Subscript[R, mir])^2)))/(2 (Subscript[R, mir])(((1 +(Sqrt[1-(((u^2)+(v^2))/((Subscript[R, mir])^2))])))^2)))=(((((u^2)+(v^2)))^2)/(2 ((Subscript[R, mir])^3)(((1 +(Sqrt[1-(((u^2)+(v^2))/((Subscript[R, mir])^2))])))^2)))=(1/(2 (Subscript[R, mir])))((((((u^2)+(v^2))/((Subscript[R, mir])(1 +(Sqrt[1-(((u^2)+(v^2))/((Subscript[R, mir])^2))]))))))^2)=((δ(((spherical))^2))/(2 (Subscript[R, mir])))

summary

Eout[x,y,z]=((I k)/(2 π))Exp[-I k z](∫_(-∞)^∞)(∫_(-∞)^∞)(OverBar[(Subscript[E, in])])[u,v,ζ]Exp[I 2k d - ((u^2)+(v^2))(I (k/(2 (Subscript[R, out])))+(1/(w^2)))]((Exp[-I k( ρ-(z-δ))])/ρ)((z-δ)/ρ)\[DifferentialD]u\[DifferentialD]v

one dimentional propagation to find change of variable

(∫_(-∞)^∞)Exp[-I k (((u^2)/(2R)) + ((((u-x))^2)/(2 L)))-((u^2)/(w^2))]\[DifferentialD]u=(∫_(-∞)^∞)Exp[-ϕ[u]]\[DifferentialD]u
ϕ[u] = I k (((u^2)/(2R)) + ((((u-x))^2)/(2 L)))+((u^2)/(w^2)) = ((1/(w^2))+I (k/2)((1/R)+(1/L)))(u^2)-I k (x/L)u + I ((k (x^2))/(2 L))= A (u^(2 ))-2 B u + C = A ((( u - (B/A)))^2) + C - ((B^2)/A);
A = ((1/(w^2))+I (k/2)((1/R)+(1/L))); 
(B/A)= I k (x/(2L)) / A = ((OverBar[B])/A)x;

Simplify[I k (x/(2L)) / ((1/(w^2))+I (k/2)((1/R)+(1/L)))]

(B/A)=((k R (w^2) x)/(k L (w^2)+k R (w^2)-2 I L R))=((OverBar[B])/A)x;

D = C-(B^2)/A=I ((k (x^2))/(2 L))+(((k (x/(2L))))^2)/((1/(w^2))+I (k/2)((1/R)+(1/L)))=(OverBar[D])(x^2);

Simplify[(I ((k (x^2))/(2 L))+(((k (x/(2L))))^2)/((1/(w^2))+I (k/2)((1/R)+(1/L)))) ]

D=((k (x^2) (2 R+I k (w^2)))/(2 (k L (w^2)+k R (w^2)-2 I L R)))=(OverBar[D])(x^2);

ϕ[u] = A ((( u - B/A))^2) + D;
(∫_(-∞)^∞)Exp[-ϕ[u]]\[DifferentialD]u=(∫_(-∞)^∞)Exp[- A ((( u - B/A))^2) - D]\[DifferentialD]u=Exp[-(OverBar[D]) (x^2)] (1/(Sqrt[A]))(∫_(-∞)^∞)Exp[- (α^2)]\[DifferentialD]α=Exp[-(OverBar[D]) (x^2)](Sqrt[(π/A)]);
α = (Sqrt[A])(u-B/A); u = (α/(Sqrt[A]))+(B/A)=(α/(Sqrt[A]))+((OverBar[B])/A)x;

variable change from (u,v) to (α, β)

(OverBar[(Subscript[E, in])])[u,v,ζ]=(Sqrt[(2/π)])(1/w)Exp[I ψ[u,v]];

Eout[x,y,z]
=((I k)/(2 π))(Sqrt[(2/π)])(1/w)Exp[-I k z](∫_(-∞)^∞)(∫_(-∞)^∞)Exp[I ψ + I k 2d-I k ((u^2)+(v^2)) (1/(2 (Subscript[R, ref])))-(((u^2)+(v^2))/(w^2))]((Exp[-I k (ρ - (z-δ) )])/ρ)((z-δ)/ρ)\[DifferentialD]u\[DifferentialD]v
=((I k)/(2 π))(Sqrt[(2/π)])(1/w)Exp[-I k z](∫_(-∞)^∞)(∫_(-∞)^∞)Exp[-I k ((u^2)+(v^2)) (1/(2 (Subscript[R, ref])))-Ik((μ^2)/(2 z))-(((u^2)+(v^2))/(w^2))+I k (2d+((μ^2)/(2 z)))+I ψ]((Exp[-I k (ρ - (z-δ) )])/ρ)((z-δ)/ρ)\[DifferentialD]u\[DifferentialD]v
=
((I k)/(2 π))(Sqrt[(2/π)])(1/w)Exp[-I k z]Exp[-((x^2)+(y^2))(OverBar[D])](1/A)(∫_(-∞)^∞)(∫_(-∞)^∞)Exp[I ψ+I k (2d-(ρ - (z-δ) - ((μ^2)/(2 z))))-((α^2)+(β^2))]((z-δ)/(ρ^2))\[DifferentialD]α \[DifferentialD]β

α = (Sqrt[A])(u-B/A); u = (α/(Sqrt[A]))+(B/A)=(α/(Sqrt[A]))+((OverBar[B])/A)x;

Express A, B and D by w[z] and R[z]

initial values

Rref and w are values at the starting z, and z in the rest of the formula is the difference between the initial location and the target location, i.e., z = z2-z1, Rref = R(z1), w = w(z1).

z and z0 (Rayleighy range) by w and R at z

(1/(z + I z0))=(1/R)-I (2/k)(1/(w^2)); z + I z0 = (1/((1/R)-I (2/(k (w^2)))))=(((1/R)+I (2/(k (w^2))))/((1/(R^2))+(4/((k^2)(w^4))))); 
z0= ((2/(k (w^2)))/((1/(R^2))+(4/((k^2)(w^4)))))=((2 (R^2) k (w^2))/((k^2) (w^4)+4 (R^2)));
z = ((1/R)/((1/(R^2))+(4/((k^2)(w^4)))))=((R (k^2) (w^4))/((k^2) (w^4) + 4 (R^2)));

relate (w2,R2) and (w1,R1)

(1/R2)-I (2/k)(1/(w2^2))=(1/(z1 + L +I z0)); (* L = z2-z1 *)

ComplexExpand[(1/(z + L +I z0))/.{z0->((2 (R^2) k (w^2))/((k^2) (w^4)+4 (R^2))),z ->((R (k^2) (w^4))/((k^2) (w^4) + 4 (R^2)))}]

FullSimplify[(((k^2) R (w^4))/(((k^2) (w^4)+4 (R^2)) (((((((k^2) R (w^4))/((k^2) (w^4)+4 (R^2)))+L))^2)+((4 (k^2) (R^4) (w^4))/((((k^2) (w^4)+4 (R^2)))^2)))))+(L/(((((((k^2) R (w^4))/((k^2) (w^4)+4 (R^2)))+L))^2)+((4 (k^2) (R^4) (w^4))/((((k^2) (w^4)+4 (R^2)))^2))))]

FullSimplify[((2 I k (R^2) (w^2))/(((k^2) (w^4)+4 (R^2)) (((((((k^2) R (w^4))/((k^2) (w^4)+4 (R^2)))+L))^2)+((4 (k^2) (R^4) (w^4))/((((k^2) (w^4)+4 (R^2)))^2)))))]

(1/R2)=(((k^2) (w^4) (L+R)+4 L (R^2))/((k^2) (w^4) (((L+R))^2)+4 (L^2) (R^2))); (1/(w2^2))=(( (k^2) (R^2) (w^2))/((k^2) (w^4) (((L+R))^2)+4 (L^2) (R^2)))

w2/w1 summary

((((w1/w2)))^2)=(( (k^2) (R^2) (w^4))/((k^2) (w^4) (((L+R))^2)+4 (L^2) (R^2)));
((((w2/w1)))^2)=((4 (L^2))/((k^2)(w^4)))+(((1+(L/R)))^2)=((L^2)/(k^2))((4/(w^4))+(k^2)((((1/L)+(1/R)))^2))=((L^2)/(k^2))((4/(w^4))+(k^2)((((L+R))^2)/((R^2) (L^2))));
(w2/w1)-> 1 when L-> 0

A by (w2,R2)

A = ((1/(w^2))+I (k/2)((1/R)+(1/L))); 
ComplexExpand[Abs[(1/A)]]

(1/(Sqrt[(1/4) (k^2) ((((1/L)+(1/R)))^2)+(1/(w^4))]))=2 (w1/w2) (L/k)

(1/A)=Abs[(1/A)]Exp[ -I ArcTan[ ((k (w^2))/2)((1/R)+(1/L)) ]];

dEta = ArcTan[((z+L)/z0)]-ArcTan[(z/z0)]=ArcTan[(((R (k^2) (w^4))/((k^2) (w^4) + 4 (R^2)))+L) / ((2 (R^2) k (w^2))/((k^2) (w^4)+4 (R^2)))] - ArcTan[((R (k^2) (w^4))/((k^2) (w^4) + 4 (R^2))) / ((2 (R^2) k (w^2))/((k^2) (w^4)+4 (R^2)))]

dθ = ArcTan[a+b] - ArcTan[a];

FullSimplify[TrigExpand[ Tan[ArcTan[a+b] - ArcTan[a]]]]

FullSimplify[(b/(a (a+b)+1))/.{a->(((R (k^2) (w^4))/((k^2) (w^4) + 4 (R^2)))) / (((2 (R^2) k (w^2))/((k^2) (w^4)+4 (R^2)))),b->L/ (((2 (R^2) k (w^2))/((k^2) (w^4)+4 (R^2))))}]

dEta=ArcTan[((2 L R)/(k (w^2) (L+R)))];

summary of A by (w2,R2)

A = ((1/(w^2))+I (k/2)((1/R)+(1/L))); ((k (w^2))/2)((1/R)+(1/L))=((k (w^2) (R+L))/(2 R L))
(1/A) = 2 (w1/w2)(L/k) Exp[-I ArcTan[ 1/Tan[dEta]]] =  2 (w1/w2)(L/k) Exp[-I (π/2-ArcTan[ Tan[dEta]])] =-I 2 (w1/w2)(L/k) Exp[I dEta]

D by (w2,R2)

(OverBar[D])=((k (2 R+I k (w^2)))/(2 (k L (w^2)+k R (w^2)-2 I L R)))

ComplexExpand[((k (2 R+I k (w^2)))/(2 (k L (w^2)+k R (w^2)-2 I L R)))]

FullSimplify[(((k^3) L (w^4))/(2 ((((k L (w^2)+k R (w^2)))^2)+4 (L^2) (R^2))))+(((k^3) R (w^4))/(2 ((((k L (w^2)+k R (w^2)))^2)+4 (L^2) (R^2))))+((2 k L (R^2))/((((k L (w^2)+k R (w^2)))^2)+4 (L^2) (R^2)))]

summary of D by (w2,R2)

D=((k (x^2) (2 R+I k (w^2)))/(2 (k L (w^2)+k R (w^2)-2 I L R)))=(OverBar[D])(x^2);
Re(OverBar[D]) = (((k^2) (R^2) (w^2))/((((k L (w^2)+k R (w^2)))^2)+4 (L^2) (R^2)))=(1/(w2^2))
Im(OverBar[D]) = (((k^3) (w^4) (L+R)+4 k L (R^2))/(2 (k^2) (w^4) (((L+R))^2)+8 (L^2) (R^2))) =(k/2)(1/R2)

final result

((I k)/(2 π))(Sqrt[(2/π)])(1/w)Exp[-I k z]Exp[-((x^2)+(y^2))(OverBar[D])](1/A)=((I k)/(2 π))(Sqrt[(2/π)])(1/w)Exp[-I k z]Exp[-((x^2)+(y^2))((1/(w2^2))+I (k/(2 R2)))](-I 2 (w1/w2)(L/k)Exp[I dEta])=(( L)/π)(Sqrt[(2/π)])(1/w2)Exp[-I k z]Exp[-((x^2)+(y^2))((1/(w2^2))+I (k/(2 R2)))]Exp[I dEta]

Eout using w2,R2

Eout
=((I k)/(2 π))(Sqrt[(2/π)])(1/w)Exp[-I k z]Exp[-((x^2)+(y^2))(OverBar[D])](1/A)(∫_(-∞)^∞)(∫_(-∞)^∞)Exp[I ψ + I k (2d-(ρ - (z-δ) - ((μ^2)/(2 z))))-((α^2)+(β^2))]((z-δ)/(ρ^2))\[DifferentialD]α \[DifferentialD]β
=(Sqrt[(2/π)])(1/w2)Exp[-I k z]Exp[-((x^2)+(y^2))((1/(w2^2))+I (k/(2 R2)))]Exp[I dEta] (z/π)(∫_(-∞)^∞)(∫_(-∞)^∞)Exp[I ψ +I k (2d-(ρ - (z-δ) - ((μ^2)/(2 z))))-((α^2)+(β^2))]((z-δ)/(ρ^2))\[DifferentialD]α \[DifferentialD]β
=HG00[ z,y,z] Corr;
Corr[z,y,z,win,Rmir,Rref] = (( z)/π)(∫_(-∞)^∞)(∫_(-∞)^∞)Exp[I ψ +I k (2d-(ρ - (z-δ) - ((μ^2)/(2 z))))-((α^2)+(β^2))]((z-δ)/(ρ^2))\[DifferentialD]α \[DifferentialD]β

Correction factor

Ein = (Sqrt[(2/π)])Exp[I (ψ - (((x^2)+(y^2))/(2 (Subscript[R, in])))-k z)-(((x^2)+(y^2))/((Subscript[w, in])^2))]; Eout = HG00[x,y,z] Corr;
Corr[z,y,z,win,Rmir] 
= (( z)/π)(∫_(-∞)^∞)(∫_(-∞)^∞)Exp[I k (ψ +2d-(ρ - (z-δ) - ((μ^2)/(2 z))))-((α^2)+(β^2))]((z-δ)/(ρ^2))\[DifferentialD]α \[DifferentialD]β
= (( z)/(2π))(∫_0^(2π))(∫_0^1)Exp[I k (ψ +2d-(ρ - (z-δ) - ((μ^2)/(2 z))))]((z-δ)/(ρ^2))\[DifferentialD]ϕ \[DifferentialD]ζ

calculation

ζ=Exp[-((α^2)+(β^2))];
α = (Sqrt[-Log[ζ]])Cos[ϕ]; β = (Sqrt[-Log[ζ]])Sin[ϕ];  ϕ=0~π if y=0

α = (Sqrt[A])(u-(OverBar[B])/A x); u = (α/(Sqrt[A]))+(B/A)=(α/(Sqrt[A]))+((OverBar[B])/A)x;
β = (Sqrt[A])(v-(OverBar[B])/A y); v = (β/(Sqrt[A]))+(B/A)=(β/(Sqrt[A]))+((OverBar[B])/A)y;
A = ((1/(w^2))+I (k/2)((1/R)+(1/L)))=(1/L)((L/(w^2))+I (k/2)((L/R)+1));
A -> ((I k)/(2 L))~(1/((Subscript[w, eff])^2)), (Subscript[w, eff])~(Sqrt[((2 L)/k)])
B/A=((k R (w^2) x)/(k L (w^2)+k R (w^2)-2 I L R))=((OverBar[B])/A)x;
((OverBar[B])/A)=((k R (w^2) )/(k L (w^2)+k R (w^2)-2 I L R))=(1/(1+((k (w^2)-2 I R)/(k R (w^2)))L))≈1-((k (w^2)-2 I R)/(k R (w^2)))L =1-((1/R)-2I (1/(k (w^2))))L  when L->0
(μ^2) = (((x-u))^2)+(((y-v))^2);
(μ^2) ≈((((1/R)-2I (1/(k (w^2)))))^2)(L^2)((x^2)+(y^2))+(((α^2)+(β^2))/A)=((((1/R)-2I (1/(k (w^2)))))^2)(L^2)((x^2)+(y^2))+((α^2)+(β^2))((2L)/(I k))-> ((α^2)+(β^2))((2L)/(I k))
((z(z-δ))/(ρ^2))=((z(z-δ))/((μ^2)+(((z-δ))^2)))=1+((z(z-δ)-(μ^2)-(((z-δ))^2))/((μ^2)+(((z-δ))^2)))=1+(((z^2)-δz-(μ^2)-(z^2)+2 z δ -(δ^2))/((μ^2)+(((z-δ))^2)))=1+(( δ(z -δ)-(μ^2))/((μ^2)+(((z-δ))^2)))

δ = d + (((u^2)+(v^2))/(2 (Subscript[R, mir])));
(u^2) = ((α^2)/A)+(((((OverBar[B])/A) x))^2)=(α^2) (L/k) + (x^2) when L->0, δ-> ((x^2)/(2 (Subscript[R, mir])))

(1/Rout)=-(2/(Subscript[R, mir]))+(1/(Subscript[R, in])) < 0 for conversing beam
d[u,v] = 0 for parabolic surface
d[u,v] = (Subscript[R, mir]) - (Sqrt[((Subscript[R, mir])^2)-(u^2)-(v^2)])-(((u^2)+(v^2))/(2 (Subscript[R, mir]))) for spherical mirror

ρ = (Sqrt[(μ^2)+(((z-δ))^2)])=z-δ + (Sqrt[(μ^2)+(((z-δ))^2)]) - (z - δ)= z - δ + ((μ^2)/((Sqrt[(μ^2)+(((z-δ))^2)]) + (z-δ)));
ρ-(z-δ)-((μ^2)/(2z))=((μ^2)/((Sqrt[(μ^2)+(((z-δ))^2)]) + (z-δ)))-((μ^2)/(2 z))
=(μ^2)((2z - (Sqrt[(μ^2)+(((z-δ))^2)]) -(z-δ))/(2z ((Sqrt[(μ^2)+(((z-δ))^2)]) + (z-δ))))
=(μ^2)((z+δ - (Sqrt[(μ^2)+(((z-δ))^2)]) )/(2z ((Sqrt[(μ^2)+(((z-δ))^2)]) + (z-δ))))
=(μ^2)((((((z +δ))^2)-(μ^2)-(((z-δ))^2))/(z +δ +(Sqrt[(μ^2)+(((z-δ))^2)]) ))/(2z ((Sqrt[(μ^2)+(((z-δ))^2)]) + (z-δ))))
=(μ^2)((((z +δ))^2)-(μ^2)-(((z-δ))^2))/((2z ((Sqrt[(μ^2)+(((z-δ))^2)]) + (z-δ))(z +δ +(Sqrt[(μ^2)+(((z-δ))^2)]))))
=(μ^2)(4 z δ-(μ^2))/(2z ((Sqrt[(μ^2)+(((z-δ))^2)]) + (z-δ))(z +δ +(Sqrt[(μ^2)+(((z-δ))^2)])))

Series[(μ^2)(4 z δ-(μ^2))/((2z ((Sqrt[(μ^2)+(((z-δ))^2)]) + (z-δ))(z +δ +(Sqrt[(μ^2)+(((z-δ))^2)]))))/.z->(1/ϵ),{ϵ, 0, 5}]

(( z)/π)(∫_(-∞)^∞)(∫_(-∞)^∞)Exp[-I k (1/2) δ (μ^2) (1/(z^2))-((α^2)+(β^2))]((z-δ)/(ρ^2))\[DifferentialD]α \[DifferentialD]β
((w^4)/(λ R (z^2)))=((0.01^4)/((10^(-6)) 1 (100^2)))

((0.01^4)/((10^(-6)) 1 (100^2)))

((0.001^4)/((10^(-6)) 0.023 (0.1^2)))

small z limit ???

FullSimplify[Series[(μ^2)(4 z δ-(μ^2))/((2  ((Sqrt[(μ^2)+(((z-δ))^2)]) + (z-δ))(z +δ +(Sqrt[(μ^2)+(((z-δ))^2)])))),{z,0, 2}]]

(μ^2) ≈((((1/R)-2I (1/(k (w^2)))))^2)(L^2)((x^2)+(y^2))+(((α^2)+(β^2))/A)=((((1/R)-2I (1/(k (w^2)))))^2)(L^2)((x^2)+(y^2))+((α^2)+(β^2))((2L)/(I k))-> ((α^2)+(β^2))((2L)/(I k));
(u^2) = ((α^2)/A)+(((((OverBar[B])/A) x))^2)=(α^2) (L/k) + (x^2) when L->0; δ-> ((x^2)/(2 (Subscript[R, mir])))

ρ-(z-δ)-((μ^2)/(2z))->(1/z)(-((μ^2)/2)+z ((Sqrt[(δ^2)+(μ^2)])+δ)+(z^2) (-(δ/(Sqrt[(δ^2)+(μ^2)]))-1))-> -((((α^2)+(β^2)))/(I k)) + ((x^2)/(Subscript[R, mir]))+O((L^2))=-(((α^2)+(β^2))/(I k)) + ((x^2)/(Subscript[R, mir]))

(( z)/π)(∫_(-∞)^∞)(∫_(-∞)^∞)Exp[I k (-(ρ - (z-δ) - ((μ^2)/(2 z))))-((α^2)+(β^2))]((z-δ)/(ρ^2))\[DifferentialD]α \[DifferentialD]β= (( z)/π)(∫_(-∞)^∞)(∫_(-∞)^∞)Exp[I k (-(-(((α^2)+(β^2))/(I k)) + ((x^2)/(Subscript[R, mir]))))-((α^2)+(β^2))]((z-δ)/(ρ^2))\[DifferentialD]α \[DifferentialD]β=(( z)/π)(∫_(-∞)^∞)(∫_(-∞)^∞)Exp[I k (- ((x^2)/(Subscript[R, mir])))]((z-δ)/(ρ^2))\[DifferentialD]α \[DifferentialD]β

00 mode loss : matlab result for 1934m, 5.3cm reflection z = 10cm ~ 4km

5 (10^(-5)) for spherical mirror
4 (10^(-10)) for parabloc mirror

Mode loss

Eout[x,y,z]=HG00[ z,y,z] Corr[z,y,z];
Corr[z,y,z,win,Rmir] = (z/π)(∫_(-∞)^∞)(∫_(-∞)^∞)Exp[I k (2d-(ρ - (z-δ) - ((μ^2)/(2 z))))-((α^2)+(β^2))]((z-δ)/(ρ^2))\[DifferentialD]α \[DifferentialD]β= (( z)/(2π))(∫_0^(2π))(∫_0^1)Exp[I k (2d-(ρ - (z-δ) - ((μ^2)/(2 z))))]((z-δ)/(ρ^2))\[DifferentialD]ϕ \[DifferentialD]ζ

A00 = (∫_(-∞)^∞)(∫_(-∞)^∞)Eout[x,y,z] (SuperStar[HG00])=((Abs[HG[x,y,z]])^2) Corr[x,y,z]\[DifferentialD]x \[DifferentialD]y
= (2/π)(1/((w[z])^2)) (∫_(-∞)^∞)(∫_(-∞)^∞)Exp[-2(((x^2)+(y^2))/((w[z])^2))]Corr[x,y,z]\[DifferentialD]x \[DifferentialD]y
= (2/π)(1/((w[z])^2)) (∫_(-∞)^∞)(∫_(-∞)^∞)Exp[-2(((x^2)+(y^2))/((w[z])^2))]Corr[x,y,z]\[DifferentialD]x \[DifferentialD]y
= (2/π)(1/((w[z])^2))2 π  (∫_0^∞)Exp[-2((r^2)/((w[z])^2))]Corr[r,0,z] r \[DifferentialD]r
=  (∫_0^∞)Exp[-2((r^2)/((w[z])^2))]Corr[r,0,z]  \[DifferentialD]((2(r^2))/((w[z])^2))
=  (∫_0^∞)Exp[-ξ]Corr[r,0,z]  \[DifferentialD]ξ
= (∫_0^1)Corr[r,0,z]  \[DifferentialD]ζ
ξ =2((r^2)/((w[z])^2)); r = w[z] (Sqrt[(ξ/2)]);
ζ = Exp[-ξ]; dζ=Exp[-ξ] dξ;
ξ = -Log[ζ]; r = w[z] (Sqrt[((-Log[ζ])/2)]);

Evaluation at z=0

Eout[x,y,z]=-((I k)/(2 π))(∫_(-∞)^∞)(∫_(-∞)^∞)(Sqrt[(2/π)])(1/w)Exp[I k δ - I k ((((u^2)+(v^2)))/(2 (Subscript[R, in])))-((((u^2)+(v^2)))/(w^2))]((Exp[-I k ρ])/ρ)(δ/ρ)\[DifferentialD]u\[DifferentialD]v=-((I k)/(2 π))(Sqrt[(2/π)])(1/w)(∫_(-∞)^∞)(∫_(-∞)^∞)Exp[I k (δ - ((((u^2)+(v^2)))/(2 (Subscript[R, in])))-ρ)-((((u^2)+(v^2)))/(w^2))](δ/(ρ^2))\[DifferentialD]u\[DifferentialD]v

δ = d + (((u^2)+(v^2))/(2 (Subscript[R, mir])))
ρ = (Sqrt[(μ^2)+(δ^2)])
ρ - δ = ((μ^2)/(ρ + δ))
(μ^2) = (((x-u))^2)+(((y-v))^2);

(∫_(-∞)^∞)(∫_(-∞)^∞)Exp[-I k ((((u^2)+(v^2)))/(2 (Subscript[R, in])))-((((u^2)+(v^2)))/(w^2))](δ/(ρ^2))\[DifferentialD]u\[DifferentialD]v

-I k ((((u^2)+(v^2)))/(2 (Subscript[R, in])))-((((u^2)+(v^2)))/(w^2))= -A ((u^2) + (v^2))
(A^2)  = I (k/(2 (Subscript[R, in])))+(1/(w^2)); (A^2)~ I ((2 π (10^6))/(2 0.05))+(1/(10^(-6)))=I 2 π (10^7)+(10^6)~ I (10^8);
α = A u, β = A v;
(∫_(-∞)^∞)(∫_(-∞)^∞)Exp[-I k ((((u^2)+(v^2)))/(2 (Subscript[R, in])))-((((u^2)+(v^2)))/(w^2))+I k (δ - ρ)](δ/(ρ^2))\[DifferentialD]u\[DifferentialD]v=(1/(A^2))(∫_(-∞)^∞)(∫_(-∞)^∞)Exp[-((α^2)+(β^2))+I k(δ - ρ)](δ/(ρ^2))\[DifferentialD]α\[DifferentialD]β=(1/( I (k/(2 (Subscript[R, in])))+(1/(w^2))))(∫_(-∞)^∞)(∫_(-∞)^∞)Exp[-((α^2)+(β^2))+I k(δ - ρ)](δ/(ρ^2))\[DifferentialD]α\[DifferentialD]β

5/256 (((2 π/1.064 (10^6)))^2) (0.001^8) / (0.023^6)

Transmission

main formula - Huyghens-Fresnel equation

Em[u,v]=EinT[u,v, (δ/n)]Exp[-I k n δ ]=(OverBar[E])in[u,v]Exp[-I k n δ ](Sqrt[(2/π)])(1/((Subscript[w, in])[δ/n]))Exp[-((u^2)+(v^2))(Ik (1/(2 (Subscript[R, in])[δ/n]))+(1/(((Subscript[w, in])[δ/n])^2)))]

Eout[x,y,z]
=((I k)/(2 π))(∫_(-∞)^∞)(∫_(-∞)^∞)Em[u,v,ζ]((Exp[-I k ρ])/ρ)((z-δ)/ρ)\[DifferentialD]u\[DifferentialD]v
=((I k)/(2 π))(∫_(-∞)^∞)(∫_(-∞)^∞)(OverBar[(Subscript[E, in])])[u,v,ζ]Exp[-I k n δ - I k ((((u^2)+(v^2)))/(2 (Subscript[R, in])))-((((u^2)+(v^2)))/((Subscript[w, in])^2))]((Exp[-I k ρ])/ρ)((z-δ)/ρ)\[DifferentialD]u\[DifferentialD]v
(Subscript[R, in])and (Subscript[w, in])are values on the right hand side of the lens

change of w and R by the finite lens thickness ~ d / n / R

(1/R2)=(((k^2) (w^4) (L+R)+4 L (R^2))/((k^2) (w^4) (((L+R))^2)+4 (L^2) (R^2))); (1/(w2^2))=(( (k^2) (R^2) (w^2))/((k^2) (w^4) (((L+R))^2)+4 (L^2) (R^2)))

Series[(((k^2) (w^4) (L+R)+4 L (R^2))/((k^2) (w^4) (((L+R))^2)+4 (L^2) (R^2))),{L,0,2}]

(1/(R[L]))=(1/R)+L ((4/((k^2) (w^4)))-(1/(R^2)))=(1/R)(1+(L/R)(((R^2)/((k^2) (w^4)))-1))

(((R^2)/((k^2) (w^4)))-1)/.{k-> 2 π/(10^(-6)), w->(10^(-3)), R->0.03}

Series[(( (k^2) (R^2) (w^2))/((k^2) (w^4) (((L+R))^2)+4 (L^2) (R^2))),{L,0,2}]

(1/((w[L])^2))=(1/(w^2))-((2 L)/(R (w^2)))=(1/(w^2))[1-2(L/R)]

L/R ~ d/n / R = 0.1mm / 1.45 / 3cm = (10^(-4)) / (2 (10^(-2))) = 0.5 (10^(-2))
0.1mm thick for 3cm curvature induce 0.5% change of RoC and w

sumamry of thickness effect

(1/(R[L]))=(1/R)(1+(L/R)(((R^2)/((k^2) (w^4)))-1));  (1/((w[L])^2))=(1/(w^2))[1-2(L/R)];

L/R ~ d/n / R = 0.1mm / 1.45 / 3cm = (10^(-4)) / (2 (10^(-2))) = 0.5 (10^(-2))
0.1mm thick for 3cm curvature induce 0.5% change of RoC and w

mode mixing due to change of R and w ~ d / n / R

change of RoC

((k (w^2))/4)((1/R1)-(1/R2))=((k (w^2))/4)(L/(R^2))(((R^2)/((k^2) (w^4)))-1)=L((1/(4 k (w^2)))-((k (w^2))/(4 (R^2))))

L((1/(4 k (w^2)))-((k (w^2))/(4 (R^2))))/.{k-> 2 π/(10^(-6)), w->(10^(-3)), R->0.03, L->(10^(-4))}

change of w

Series[(Sqrt[(2/(1+(w2^2)/(w^2)))(2/(1+(w^2)/(w2^2)))])-1 /.{w2->w(1+ϵ)}, {ϵ,0,2}]

00->00 when RoC and w are changed

|Amp(00->00)|=((((1+dβ))^4)/(1+((dα)^2)))
dβ ~ relative change of w ~ L/R
dα ~ relative change of RoC ~ (L/R) ((kw^2)/R)≫ (L/R)

dEta = ArcTan[(((R (k^2) (w^4))/((k^2) (w^4) + 4 (R^2)))+L) / ((2 (R^2) k (w^2))/((k^2) (w^4)+4 (R^2)))] - ArcTan[((R (k^2) (w^4))/((k^2) (w^4) + 4 (R^2))) / ((2 (R^2) k (w^2))/((k^2) (w^4)+4 (R^2)))];
Series[dEta, {L,0,2}]

dη ~ ((2 L)/(k (w^2)))

C00 = (4/((1+(w1^2)/(w^2))(1+(w^2)/(w1^2))))/(1+((((k/2)(((w^2) (w1^2))/((w^2)+(w1^2)))((1/R1)-(1/R))))^2));
Series[C00/.{w1->w, R1->R(1+ϵ)}, {ϵ, 0, 2}]
Series[C00/.{w1->w(1+ϵ), R1->R}, {ϵ, 0, 2}]

calculation

ρ-(z-δ)=((μ^2)/((Sqrt[(μ^2)+(((z-δ))^2)]) + (z-δ))); 
ρ - (z-δ) - ((μ^2)/(2 z))
δ = d + (((u^2)+(v^2))/(2 (Subscript[R, mir])));
-I k ρ - I k n δ - I k ((((u^2)+(v^2)))/(2 (Subscript[R, in])))-((((u^2)+(v^2)))/((Subscript[w, in])^2))=-I k (ρ-(z-δ))-I k (z-δ) - I k n δ - I k ((((u^2)+(v^2)))/(2 (Subscript[R, in])))-((((u^2)+(v^2)))/((Subscript[w, in])^2))=-I k z +I k δ - I k n δ -I k  ((μ^2)/((Sqrt[(μ^2)+(((z-δ))^2)]) + z-δ))- I k ((((u^2)+(v^2)))/(2 (Subscript[R, in])))-((((u^2)+(v^2)))/((Subscript[w, in])^2))=-I k z - I  k (n-1) d[u,v] -((u^2)+(v^2))(I (k/2)(((n-1)/(Subscript[R, mir]))+(1/(Subscript[R, in])))+(1/((Subscript[w, in])^2)))-I k  ((μ^2)/((Sqrt[(μ^2)+(((z-δ))^2)]) + z-δ))=-I k z - I  k (n-1) d[u,v] -((u^2)+(v^2))(I (k/(2 (Subscript[R, out])))+(1/((Subscript[w, in])^2)))-I k  ((μ^2)/((Sqrt[(μ^2)+(((z-δ))^2)]) + z-δ))
(1/(Subscript[R, out]))=((n-1)/(Subscript[R, mir]))+(1/(Subscript[R, in])) > 0 for diverging beam
d[u,v] = 0 for parabolic surface
d[u,v] = (Subscript[R, mir]) - (Subscript[R, mir])(Sqrt[1-(((u^2)+(v^2))/((Subscript[R, mir])^2))])-(((u^2)+(v^2))/(2 (Subscript[R, mir]))) for spherical mirror

summary

Eout[x,y,z]=((I k)/(2 π))Exp[-I k z](∫_(-∞)^∞)(∫_(-∞)^∞)(OverBar[(Subscript[E, in])])[u,v,ζ]Exp[- I  k (n-1) d[u,v] - ((u^2)+(v^2))(I (k/(2 (Subscript[R, out])))+(1/((Subscript[w, in])^2)))]((Exp[-I k( ρ-(z-δ))])/ρ)((z-δ)/ρ)\[DifferentialD]u\[DifferentialD]v

difference between reflection and transmission

(1/(Subscript[R, out]))=((n-1)/(Subscript[R, mir]))+(1/(Subscript[R, in])); (* (1/(Subscript[R, out]))=(n/(Subscript[R, in])) when (Subscript[R, mir])=(Subscript[R, in]) *)
(1/(R[L]))=(1/R)(1+(L/R)(((R^2)/((k^2) (w^4)))-1)); (1/((w[L])^2))=(1/(w^2))[1-2(L/R)]; L=d/n; (* thickness changes R and w *)
Exp[I 2 k d] -> Exp[-I k (n-1) d] (* d without macroscopic thickness *)

Propagation with disturbance

E[x,y,z]=E00[x,y,z]Corr[x,y,z,(Subscript[ψ, in])]=E00[x,y,z]Exp[I (Subscript[ψ, out])[x,y,z]];
Corr[x,y,z,(Subscript[ψ, in])]=(( z)/π)(∫_(-∞)^∞)(∫_(-∞)^∞)Exp[I (Subscript[ψ, in])[u,v] +I k (2d-(ρ - (z-δ) - ((μ^2)/(2 z))))-((α^2)+(β^2))]((z-δ)/(ρ^2))\[DifferentialD]α \[DifferentialD]β

Brute force

Correcgion factor

Corr[z,y,z,win,Rmir,Rref] = (( z)/π)(∫_(-∞)^∞)(∫_(-∞)^∞)Exp[I ψ +I k (2d-(ρ - (z-δ) - ((μ^2)/(2 z))))-((α^2)+(β^2))]((z-δ)/(ρ^2))\[DifferentialD]α \[DifferentialD]β

Eout[x,y,z]=(I/λ)(Sqrt[(2/π)])(1/w)Exp[-I k z](∫_(-∞)^∞)(∫_(-∞)^∞)(OverBar[(Subscript[E, in])])[u,v,ζ]Exp[I 2k d - ((u^2)+(v^2))(I (k/(2 (Subscript[R, out])))+(1/(w^2)))]((Exp[-I k( ρ-(z-δ))])/ρ)((z-δ)/ρ)\[DifferentialD]u\[DifferentialD]v

u = w ξ Cos[ϕ]; v = w ξ Sin[ϕ];
ζ = Exp[- ((u^2)+(v^2))/(w^2)] = Exp[-(ξ^2)]; dζ=-2ξ dξ Exp[-(ξ^2)];ξ = (Sqrt[-Log[ζ]]);

Exp[-(((u^2)+(v^2))/(w^2))]du dv =(w^2) Exp[-(ξ^2)]ξ dξ dϕ =(1/2)(w^2)dζ

Eout[x,y,z]=(I/λ)(Sqrt[(2/π)])(1/w)(w^2)Exp[-I k z](∫_(-∞)^∞)(∫_(-∞)^∞)(OverBar[(Subscript[E, in])])[u,v,ζ]Exp[I 2k d - ((u^2)+(v^2))(I (k/(2 (Subscript[R, out])))+(1/(w^2)))]((Exp[-I k( ρ-(z-δ))])/ρ)((z-δ)/ρ)\[DifferentialD]u\[DifferentialD]v
=((I k)/(4 π))(Sqrt[(2/π)])(Subscript[w, in]) Exp[-I k z](∫_(-π)^π)(∫_(Exp[-((a^2)/(w^2))])^1)(OverBar[(Subscript[E, in])])[u,v,ζ]Exp[I 2k d - I k (((w^2)(ξ^2))/(2 (Subscript[R, out])))]((Exp[-I k( ρ-(z-δ))])/ρ)((z-δ)/ρ)dζ dϕ

(μ^2) = (((x-u))^2)+(((y-v))^2)=(r^2)-2 r ξ Cos[ϕ]+(ξ^2);
δ = d + (((u^2)+(v^2))/(2 (Subscript[R, mir])))=d+((ξ^2)/(2 (Subscript[R, mir])));
(ρ^2) = (μ^2)+(δ^2);
ρ = (Sqrt[(μ^2)+(((z-δ))^2)])=z-δ + (Sqrt[(μ^2)+(((z-δ))^2)]) - (z - δ)= z - δ + ((μ^2)/((Sqrt[(μ^2)+(((z-δ))^2)]) + (z-δ)));
ρ-(z-δ)=((μ^2)/((Sqrt[(μ^2)+(((z-δ))^2)]) + (z-δ)));