Eq. (24) and hyperbolas

The geometric definition

A hyperbola is the set of points whose distances to two fixed points (the foci) have a constant difference:

Eq. (24) says exactly this, with

• foci ${𝐹}_1,{𝐹}_2={𝑆}_1,{𝑆}_2$ (the two coherent sources),
• “constant” $=𝑚𝜆$ (so $2𝐴=𝑚𝜆$, one hyperbola per integer order $𝑚$).

So no calculation is needed to recognize it: each bright-fringe condition is the defining property of a hyperbola with the slits as foci. Changing $𝑚$ changes the constant, which gives a family of hyperbolas sharing the same foci.

Checking it algebraically

Following Figure 6, put the sources on the $𝑦$-axis at ${𝑆}_{1,2}=\left(0,\pm 𝑐\right)$ with $𝑐=𝑎/2$, and let $𝑑=𝑚𝜆$:

Subtracting, ${𝑠}_2^2-{𝑠}_1^2=4𝑐𝑦$. Since ${𝑠}_2-{𝑠}_1=𝑑$,

Adding and subtracting these two relations,

Square it and set equal to ${𝑥}^2+{\left(𝑦-𝑐\right)}^2$:

The $-2𝑐𝑦$ terms cancel. Collect terms:

Substituting back $𝑑=𝑚𝜆$, $𝑐=𝑎/2$: $$\frac{{𝑦}^2}{{\left(𝑚𝜆/2\right)}^2}-\frac{{𝑥}^2}{{\left(𝑎/2\right)}^2-{\left(𝑚𝜆/2\right)}^2}=1$$

This is the standard hyperbola ${𝑦}^2/{𝐴}^2-{𝑥}^2/{𝐵}^2=1$ with

Consequences

• Existence: ${𝐵}^2>0$ requires $|𝑚|𝜆<𝑎$, so only finitely many orders exist, $|𝑚|\le 𝑎/𝜆$. This is the same as $|sin𝜃|\le 1$ in Eq. (19).
• $𝑚=0$: $𝐴=0$ degenerates to the line $𝑦=0$, the perpendicular bisector of ${𝑆}_1{𝑆}_2$ (the central fringe).
• Far field: the asymptotes are $𝑦/𝐴=\pm 𝑥/𝐵$. Their angle $𝜃$ from the bisector satisfies $sin𝜃=𝐴/𝑐=𝑚𝜆/𝑎$, which is Eq. (19), $𝑎sin𝜃=𝑚𝜆$. Far from the sources the hyperbolas become straight rays.

• Rotation about the $𝑦$-axis: the setup is symmetric about the line through the sources, so the 2D curves sweep out hyperboloids of revolution. These are the nonlocalized fringe surfaces.

  • A screen $⟂$ to $𝑂𝑋$ cuts hyperbolic arcs, which look like straight fringes near the axis (${𝑦}_{𝑚}\approx 𝑚𝜆𝐿/𝑎$, Eq. 22).
  • A screen $⟂$ to $𝑂𝑌$ cuts the surfaces in circles, giving concentric ring fringes.