2—Infinite Series 37Use this and the light amplitude is
2 A
−ksinθ
sin(
−
ka
2
sinθ
)
cos(
k(r 0 −
a
2
sinθ)−ωt
)
(2.29)
Thewaveis the cosine factor. It is a cosine of(k.distance−ωt), and the distance in question
is the distance to the center of the slit. This is then a wave that appears to be coming from the middle
of the slit, but with an amplitude that varies strongly with angle. That variation comes from the other
factors in Eq. (2.29).
It’s the variation with angle that’s important. The intensity of the wave, the power per area,
is proportional to the square of the wave’s amplitude. I’m going to ignore all the constant factors, so
there’s no need to worry about the constant of proportionality. The intensity is then (up to a factor)
I=
sin^2(
(ka/2) sinθ
)
sin^2 θ
(2.30)
For light, the wavelength is about 400 to 700 nm, and the slit may be a millimeter or a tenth of a
millimeter. The size ofka/ 2 is then about
ka/2 =πa/λ≈ 3. 0. 1 mm/ 500 nm≈ 1000
When you plot this intensity versus angle, the numerator vanishes when the argument ofsin^2 ()isnπ,
withnan integer,+,−, or 0. This says that the intensity vanishes in these directionsexceptforθ= 0.
In that case the denominator vanishes too, so you have to look closer. For the simpler case thatθ 6 = 0,
these angles are
nπ=
ka
2
sinθ≈
ka
2
θ n=± 1 ,± 2 ,...
Becausekais big, you have many values ofnbefore the approximation thatsinθ=θbecomes invalid.
You can rewrite this in terms of the wavelength becausek= 2π/λ.
nπ=
2 πa
2 λ
θ, or θ=nλ/a
What happens at zero? Use power series expansions to evaluate this indeterminate form. Thefirst term in the series expansion of the sine isθitself, so
I=
sin^2(
(ka/2) sinθ
)
sin^2 θ
−→
(
(ka/2)θ
) 2
θ^2
=
(
ka
2
) 2
(2.31)
What is the behavior of the intensitynearθ= 0? Again, use power series expansions, but keep
another term
sinθ=θ−
1
6
θ^3 +···, and (1 +x)α= 1 +αx+···
Remember,ka/ 2 is big! This means that it makes sense to keep just one term of the sine expansion
forsinθitself, but you’d better keep an extra term in the expansion of thesin^2 (ka...).
I≈
sin^2