Animated Solution for Physics - Wave Optics: A Young's double slit experiment uses a monochromatic source. The shape of the interference fringes formed on a screen is

Visualizing the Double Slit

When we first learn about Young's Double Slit Experiment (YDSE), we often draw it as a simple 2D diagram on a piece of paper. We draw two slits, $S_1$ and $S_2$, and a screen at a distance $D$.

However, to truly understand the shape of the interference fringes, we must step out of flatland and visualize the experiment in three dimensions. Imagine two coherent point sources emitting spherical wavefronts into the space around them. At a distance $D$, we place a flat, 2D screen to catch the light and observe the resulting interference pattern.

Path Difference

The fundamental principle of interference dictates that a bright fringe (constructive interference) occurs at any point $P$ on the screen where the light waves from $S_1$ and $S_2$ arrive exactly in phase.

Mathematically, this means the difference in the distance traveled by the two waves—known as the path difference $\Delta x$—must be an integer multiple of the wavelength $\lambda$. We can write this as:

where $n=0,1,2,\dots$ represents the order of the fringe.

The Hyperboloid

Now, let's ask a purely geometric question: What is the locus of a point $P$ in 3D space such that the difference of its distances from two fixed points ($S_1$ and $S_2$) is a constant value?

If the sum of the distances were constant, the locus would be an ellipsoid. But because the difference is constant, the locus is a hyperboloid of two sheets. For every integer value of $n$, there exists a distinct hyperboloid in space representing the region of constructive interference.

Slicing the Cone

Our observation screen is simply a flat 2D plane placed at $x=D$. To find the shape of the fringes on the screen, we just need to find the geometric intersection of the flat screen plane with the 3D hyperboloids of constructive interference.

From conic sections, we know that when a plane cuts a hyperboloid parallel to its axis of symmetry (the line joining $S_1$ and $S_2$), the resulting 2D curve is a hyperbola. Therefore, the bright and dark bands we observe on the screen are mathematically families of hyperbolas!

A Special Case

There is one beautiful exception to this rule. Consider the central maximum, where $n=0$.

For this fringe, the path difference is exactly zero, meaning $|S_{2}P-S_{1}P|=0$, or $S_{2}P=S_{1}P$. The locus of points equidistant from two fixed points in 3D space is a plane—specifically, the perpendicular bisector plane of the line segment joining $S_1$ and $S_2$.

When this bisector plane intersects our flat observation screen, the intersection of two planes is a perfectly straight line. Thus, the central fringe is a straight line, while all higher-order fringes ($n\ge 1$) bend away from it as hyperbolas.

You might occasionally encounter older textbooks or answer keys that mistakenly list the shape of the fringes as a "parabola." This is mathematically incorrect.

A parabola is defined as the locus of points equidistant from a single focus and a directrix line. A hyperbola is defined by two foci (which perfectly matches our two sources, $S_1$ and $S_2$). The physics of the path difference strictly generates a hyperbola, never a parabola.

Why do we often draw fringes as straight, parallel lines in our notebooks?

If the screen is placed very far away ($D\gg d$), and we only look at a small region near the center of the screen, the eccentricity of the hyperbolas becomes very large. In this localized region, the curves flatten out so much that they are virtually indistinguishable from straight lines. Furthermore, if the sources are long parallel slits rather than point sources, the overlapping hyperbolas from every point along the slit merge to form perfectly straight lines.

However, when asked for the exact geometric shape produced by a monochromatic point source, the rigorous and undeniable answer is a hyperbola.