According to Huygens principle, the total electric field at a point y on the screen is the superposition of wave fields from an infinite number of point sources in the aperture region. Each point s on the wave front inside the aperture ( –a/2 ≤ s ≤ a/2) is the source of a spherical wave. A distance r from the point s the electric field is due to this point sources is
dE = (Asds/r)cos(kr-ωt).
If r0 is the distance from the point s = 0 on the optical axis to a point y on the screen, then the contribution dE to the total amplitude on the screen from the point at s = 0 is
dE(y) = (Asds/r0)cos(kr0-ωt).
Here As is the amplitude per unit width and ds is the infinitely small width of a point source. For off-axis points for which s ≠ 0, the distance is longer or shorter than r0 by an amount Δ.
The contribution dE(y) to the total amplitude on the screen from an off-axis point (s ≠ 0) is
dE(y) = (Asds/(r0+Δ(s)))cos(k(r0+Δ(s))-ωt).
To find the total amplitude E(y) we have to add up the contributions from all points on the aperture. Because there are an infinite number of points, the sum becomes an integral.
E(y) = ∫-a/2+a/2(A/(r0+Δ(s)))cos(k(r0+Δ(s))-ωt)ds.
We define sinθ = Δ/s. Since r0 >> Δ, we approximate 1/(r0+Δ) with 1/r0. However we cannot drop the Δ inside the cosine function, since kΔ(s) is not necessarily much smaller than 2π.
We then have
E(y) = (As/r0)∫-a/2+a/2cos[(ksinθ)s + (kr0-ωt)]ds.
Using
,
integration yields
E(y) = (As/r0)cos(kr0-ωt)(sin(ka(sinθ)/2)/(ka(sinθ)/2) .
or. inserting k = 2π/λ,
E(y) = (As/r0)cos(kr0-ωt)(sin(πa(sinθ)/λ)/(πa(sinθ)/λ).
The function sin(x)/x = sinc(x) is called the sinc function.
The intensity is proportional to the square of the field, I(y )is proportional to E2(y). Since the square of a cosine function averages to ½, the time-averaged intensity is given by
<I(y)> = <I0>(sin2(πa(sinθ)/λ)/(πa(sinθ)/λ)2,
where <I0> is the average intensity at the center,.
The time-averaged intensity has a peak in the center with smaller fringes on the sides. For small angles we may approximate sinθ ~ θ. Then the first zeros on the sides of the central peak occur when πa(sinθ)/λ ~ πaθ/λ = π, or θ = λ/a.