Why Eq. (24) Describes Hyperbolas

Young's double-slit experiment (§7.2): bright-fringe surfaces

1. The statement in the text

Eq. (24) says that the bright fringes of two coherent point sources $𝑆_{1}$ , $𝑆_{2}$ satisfy

𝑠_{2} - 𝑠_{1} = 𝑚 𝜆, 𝑚 = 0, \pm 1, \pm 2, \dots

where $𝑠_{1}$ and $𝑠_{2}$ are the distances from a point $𝑃$ to the two sources. The text then says this is “precisely the condition for a family of hyperbolic curves with parameter $𝑚$ .” The reason is just the geometric definition of a hyperbola:

A hyperbola is the set of points whose distances to two fixed points (the foci) differ by a constant.

Compare: $| 𝑠_{2} - 𝑠_{1} | = constant$ with the foci $=$ the two sources. Each integer $𝑚$ fixes a different constant $𝑚 𝜆$ , so each $𝑚$ gives a different hyperbola, all sharing the same foci $𝑆_{1}$ , $𝑆_{2}$ . (An ellipse is the same idea with the sum of distances held constant.)

2. Algebraic derivation

Put the sources on the $𝑦$ -axis (as in the text’s Fig. 6), separated by $𝑎$ :

𝑆_{1} = (0, + \frac{𝑎}{2}), 𝑆_{2} = (0, - \frac{𝑎}{2}) .

For a point $𝑃 = (𝑥, 𝑦)$ ,

𝑠_{1} = \sqrt{𝑥^{2} + {(𝑦 - \frac{𝑎}{2})}^{2}}, 𝑠_{2} = \sqrt{𝑥^{2} + {(𝑦 + \frac{𝑎}{2})}^{2}} .

Step 1 — get the sum. The difference of squares is easy:

𝑠_{2}^{2} - 𝑠_{1}^{2} = {(𝑦 + \frac{𝑎}{2})}^{2} - {(𝑦 - \frac{𝑎}{2})}^{2} = 2 𝑎 𝑦 .

Since $𝑠_{2}^{2} - 𝑠_{1}^{2} = (𝑠_{2} - 𝑠_{1}) (𝑠_{2} + 𝑠_{1})$ and $𝑠_{2} - 𝑠_{1} = 𝑚 𝜆$ ,

𝑠_{2} + 𝑠_{1} = \frac{2 𝑎 𝑦}{𝑚 𝜆} .

Step 2 — solve for each distance. Adding and subtracting,

𝑠_{1} = \frac{𝑎 𝑦}{𝑚 𝜆} - \frac{𝑚 𝜆}{2} .

Step 3 — square and compare. Squaring gives

𝑠_{1}^{2} = \frac{𝑎^{2} 𝑦^{2}}{𝑚^{2} 𝜆^{2}} - 𝑎 𝑦 + \frac{𝑚^{2} 𝜆^{2}}{4},

while from the definition $𝑠_{1}^{2} = 𝑥^{2} + 𝑦^{2} - 𝑎 𝑦 + \frac{𝑎^{2}}{4}$ . Equating (the $- 𝑎 𝑦$ terms cancel):

𝑦^{2} (\frac{𝑎^{2} - 𝑚^{2} 𝜆^{2}}{𝑚^{2} 𝜆^{2}}) - 𝑥^{2} = \frac{𝑎^{2} - 𝑚^{2} 𝜆^{2}}{4} .

Dividing by the right-hand side gives the standard hyperbola:

$\frac{𝑦^{2}}{{(𝑚 \frac{𝜆}{2})}^{2}} - \frac{𝑥^{2}}{{(\frac{\sqrt{𝑎^{2} - 𝑚^{2} 𝜆^{2}}}{2})}^{2}} = 1$

Reading it off the standard form $\frac{𝑦^{2}}{𝛼^{2}} - \frac{𝑥^{2}}{𝛽^{2}} = 1$ :

Semi-axis (vertex distance)	$𝛼 = 𝑚 \frac{𝜆}{2}$
Other semi-axis	$𝛽 = \frac{1}{2} \sqrt{𝑎^{2} - 𝑚^{2} 𝜆^{2}}$
Focal distance	$𝑐 = \sqrt{𝛼^{2} + 𝛽^{2}} = \frac{𝑎}{2}$ → the sources, as required

So the sources really are the foci, and the order $𝑚$ only controls how “open” the hyperbola is.

3. The hyperbolic-function connection

The name “hyperbolic” also shows up in the parametrization. Because ${cosh}^{2} 𝑡 - {sinh}^{2} 𝑡 = 1$ , the equation above is solved by

𝑦 = \frac{𝑚 𝜆}{2} cosh 𝑡, 𝑥 = \frac{1}{2} \sqrt{𝑎^{2} - 𝑚^{2} 𝜆^{2}} sinh 𝑡, 𝑡 \in ℝ .

(The sign of $𝑚$ picks the branch: $𝑚 > 0$ puts the curve closer to $𝑆_{1}$ , i.e. $𝑦 > 0$ ; $𝑚 < 0$ gives $𝑦 < 0$ .)

4. Consequences for the interference pattern

Finite number of fringes. We need $𝛽$ real, so $| 𝑚 | 𝜆 \leq 𝑎$ . This is just the triangle inequality $| 𝑠_{2} - 𝑠_{1} | \leq 𝑎$ , and gives $| 𝑚 | \leq \frac{𝑎}{𝜆}$ bright surfaces in total. The extreme $| 𝑚 | 𝜆 = 𝑎$ degenerates to the ray along the $𝑦$ -axis beyond each source.

The $𝑚 = 0$ surface. $𝛼 = 0$ degenerates the hyperbola into the straight line $𝑦 = 0$ (the perpendicular bisector of $𝑆_{1} 𝑆_{2}$ , the central maximum, the $𝑥$ -axis in Fig. 6).

Far-field limit recovers Eq. (19). Far from the sources the hyperbola approaches its asymptotes $𝑦 = \pm (\frac{𝛼}{𝛽}) 𝑥$ . Writing $𝜑$ for the angle of an asymptote from the $𝑦$ -axis,

tan 𝜑 = \frac{𝛽}{𝛼} = \frac{\sqrt{𝑎^{2} - 𝑚^{2} 𝜆^{2}}}{𝑚 𝜆} ⟹ cos 𝜑 = \frac{𝑚 𝜆}{𝑎} .

The text’s angle $𝜃$ is measured from the $𝑥$ -axis (the optical axis $𝑂 𝑋$ ), so $𝜃 = 90 ° - 𝜑$ and

𝑎 sin 𝜃 = 𝑚 𝜆,

which is exactly Eq. (19). So the straight-line fringes seen on a distant screen are the asymptotes of the hyperbolas; near the axis a hyperbola looks like a straight line.

Rotation about $𝑦$ . The pattern is symmetric about the line through the sources, so rotating each hyperbola about the $𝑦$ -axis gives hyperboloids of two sheets — the “bright fringe surfaces.” A screen perpendicular to $𝑂 𝑋$ cuts them in (nearly straight) hyperbolic arcs; a screen perpendicular to $𝑂 𝑌$ cuts them in concentric circles. For the latter, from the equation above,

𝑥^{2} + 𝑧^{2} = (𝑎^{2} - 𝑚^{2} 𝜆^{2}) (\frac{𝑦^{2}}{{(𝑚 𝜆)}^{2}} - \frac{1}{4}),

a circle of radius growing with the screen distance $𝑦$ .

5. Picture

Bright-fringe curves $𝑠_{2} - 𝑠_{1} = 𝑚 𝜆$ for $𝑎 = 4.5 𝜆$ (so $| 𝑚 | \leq 4$ ). Blue: $𝑚 > 0$ ; red: $𝑚 < 0$ ; black: $𝑚 = 0$ . The sources $𝑆_{1}, 𝑆_{2}$ are the common foci of every curve.

6. Summary

Eq. (24) is literally the focal definition of a hyperbola: “constant difference of distances to two fixed points.” The two sources are the foci, $𝑚 𝜆$ is the constant ( $= 2 𝛼$ , the vertex separation), and each integer $𝑚$ labels a different hyperbola. Its asymptotes reproduce the small-angle double-slit formula $𝑎 sin 𝜃 = 𝑚 𝜆$ , and rotating about the source axis gives the three-dimensional hyperboloid fringe surfaces the text describes.