Diffraction

Slits at [-.5, 0], [.5, 0], illuminated from [0, large negative]. In the upper half-plane, we have signal

[x, y]->

expi(w.sqrt(square(y) + square(x+.5))) / sqrt(square(y) + square(x+.5)) +
expi(w.sqrt(square(y) + square(x-.5))) / sqrt(square(y) + square(x-.5))

times some global constant about which I don't care. Now, square(x+.5) is equal to 2x+square(x-.5). So take R= x +square(x-.5) +square(y) = .25 +square(x) + square(y) to obtain the signal as

[x, R]->

expi(w.sqrt(R+x))/sqrt(R+x) + expi(w.sqrt(R-x))/sqrt(R-x)

= {sqrt(R-x).expi(w.sqrt(R+x)) +sqrt(R+x).expi(w.sqrt(R-x))} / sqrt(R.R -x.x)

Now, let U= (sqrt(R+x) +sqrt(R-x))/2 and V= ( sqrt(R+x) -sqrt(R-x) )/2 so that sqrt(R+x) =U+V and sqrt(R-x) =U-V, with 2.U.V =x, to make our signal

expi(w.U).{(U-V).expi(w.V) +(U+V).expi(-w.V)} /sqrt(R.R -x.x)

= 2.expi(w.U).{U.cos(w.V) -V.i.sin(w.V)} /sqrt(R.R -x.x)

Thus U has no impact on amplitude. The 1/(R.R -x.x) is an overall scaling with a simple global structure: for all practical purposes, we can treet it as 1/R/(x.x+y.y). Now, R, R+x and R-x are positive everywhere, so R is bigger than x. Indeed, in so far as either x or y is at all large (compared to 1), R dwarfs x. Any difference between R and x suffices to make variation in the R.sin +i.x.cos term relevant. We are thus led to examination of the lines on which w.V is constant, since crests and troughs of the signal arise when it is an even or odd multiple (respectively) of π/w.

Now, 2.V.2.U = (R+x) - (R-x) = 2.x, so V= x/U/2. At least when R dwarfs x, U is close to sqrt(R): indeed, U/sqrt(R) is 1 to within x.x/R/R/8. Thus our phase is essentially x/r/2 with r.r=x.x+y.y as usual, giving the signal as

[x,y]->

2.i.expi(w.U).sin(w.x/r/2) /r/r

Aim to get into r, (x/y) terms. R = x.x +y.y +1/4 2.U = sqrt(R+x) +sqrt(R-x) 2.V = sqrt(R+x) -sqrt(R-x) 2.U.V = x signal is 2.{U.cos(w.V) -V.i.sin(w.V)}/sqrt(R.R -x.x) r.r = x.x +y.y x = a.r, a is in (| (({-1}:≤|): ≤ :{1}) ) R = r.r +1/4 2.V = sqrt(R+a.r) -sqrt(R-a.r) R.R -x.x = square(r.r +1/4) -x.x = r.r.r.r +1/16 +r.r/2 -a.a.r.r = r.r.r.r +1/16 +(1 -2.a.a).r.r 2.V = x/U = a.r/U is roughly x/sqrt(R) or x/r = a U is roughly sqrt(R) or r. sqrt(R.R -x.x) = U.U -V.V = U.U -a.a.r.r/4/U/U = U.U.(1 -square(a.r/2/U/U)) is roughly r.r when this dwarfs 1: a.r/2/U/U is small compared to 1, so its square is tiny. primary signal is U/(U.U -V.V) = 1/U/(1 -square(a.r/2/U/U)) is roughly 1/r, with secondary V/(U.U -V.V) = a.r/U/U/U/(1 -square(a.r/2/U/U)), roughly a/2/r. The ratio secondary/primary is V/U = a.r/U/U/2, or roughly a/r/2. phase is w.V = w.x/U/2 is roughly w.a/2