10.3.3 Derivation of the Lλ/d Formula

In the previous section, we leant how the angle θ at which the fringes are formed is related to the wavelength of the light (and the slit separation). In practice, however, we almost never measure the angle θ for the double-slit pattern. Instead we work with the fringe separation ∆y. Between which two fringes? You might ask. Well, any two. Because the double-slit pattern (usually) has a constant fringe spacing. A little mathematics tells us why.

Let’s use θ_n to denote the angle at which the nth order bright fringe is formed. Let’s also use y_n to denote the distance between nth order and 0th order bright fringe.

Let’s start from the fact that

$d\sin {{\theta }_{n}}=n\lambda$ , $n=0,1,2...$

Rearranging the equation, we get

$\displaystyle \sin {{\theta }_{n}}=n\frac{\lambda }{d}$ , $n=0,1,2...$

For double-slits, d is of the order of 0.1 mm, and l is between 400 nm and 750 nm, making $\displaystyle \frac{\lambda }{d}$ very small. This means that the low order fringes (let’s say $n<10$ ) are all formed at very small θ.

For small values of θ, we can replace $\sin \theta$ with $\displaystyle \frac{y}{L}$ . This is because if θ is small, $\displaystyle \sin \theta \approx \tan \theta =\frac{y}{L}$ . So

$\displaystyle \frac{{{{y}_{n}}}}{L}=n\frac{\lambda }{d}$ , $n=0,1,2...$

In other words, the bright fringes are formed at

$\displaystyle {{y}_{n}}=n\frac{{L\lambda }}{d}$ , $n=0,1,2...$

The fringe separation is thus

$\displaystyle \begin{aligned}\Delta y&={{y}_{{n+1}}}-{{y}_{n}}\\&=(n+1)\frac{{L\lambda }}{d}-(n)\frac{{L\lambda }}{d}\\&=\frac{{L\lambda }}{d}\end{aligned}$

Let’s remind ourselves that the fringe separation is constant only for the fringes which are formed at small θ. However, the higher order fringes of a double-slit pattern are usually too dim to be seen anyway (due to the diffraction envelope effect, which will be discussed later). All we get to see are the lower order fringes, which are formed at small values of θ, which thus display a constant fringe separation.

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Applet

Interference of Light at a Double-Slit (Walter-Fendt)

Concept Test

2017