Young’s double-slit experiment is one of the classic demonstrations of wave interference. When coherent light passes through two closely spaced slits, the light from the two openings spreads out and overlaps on a screen. In some places the waves arrive in phase and produce bright fringes, while in other places they arrive out of phase and produce dark fringes.
The key idea is the path difference between the two waves. If the path difference is an integer multiple of the wavelength,
\[
\delta = m\lambda,
\]
then constructive interference occurs and the fringe is bright. If the path difference is a half-integer multiple,
\[
\delta = \left(m+\frac12\right)\lambda,
\]
then destructive interference occurs and the fringe is dark.
For a screen far from the slits, the small-angle approximation is used. If the slit separation is \(d\), the screen distance is \(L\), and the screen position from the center is \(y\), then
\[
\sin\theta \approx \tan\theta \approx \frac{y}{L}.
\]
The path difference becomes
\[
\delta \approx d\sin\theta \approx d\frac{y}{L}.
\]
Setting this equal to \(m\lambda\) for bright fringes gives
\[
y_m = m\frac{\lambda L}{d}.
\]
The quantity
\[
\Delta y = \frac{\lambda L}{d}
\]
is the fringe spacing. It is the distance between adjacent bright fringes, and also between adjacent dark fringes. This formula shows the main trends clearly:
If the wavelength increases, the fringes spread farther apart. If the screen is moved farther away, the pattern also spreads out. If the slit separation increases, the fringes move closer together. These dependencies are very important in real experiments and are easy to test with a laser and a double-slit slide.
The ideal intensity pattern for equal slits is
\[
I \propto \cos^2\!\left(\frac{\delta}{2}\right).
\]
This means the intensity oscillates smoothly between maxima and minima across the screen. In the simplest model used here, the maxima are equally spaced and the central maximum occurs at
\[
y=0.
\]
Using the sample values \(\lambda=550\text{ nm}\), \(L=1\text{ m}\), and \(d=0.1\text{ mm}\), the fringe spacing is
\[
\Delta y = \frac{\lambda L}{d}
= \frac{(550\times10^{-9})(1)}{0.1\times10^{-3}}
= 5.5\times10^{-3}\text{ m}
= 5.5\text{ mm}.
\]
So the first bright fringe (\(m=1\)) is at
\[
y_1=5.5\text{ mm}.
\]
This is a convenient scale for tabletop optics experiments.
At more advanced level, finite slit width, nonuniform illumination, and diffraction envelopes are included. Those effects modify the ideal pattern but do not change the basic interference condition. This calculator uses the standard ideal two-slit model, which is the correct starting point for learning fringe spacing, fringe positions, and the dependence of the pattern on wavelength, screen distance, and slit separation.