Young’s double-slit experiment is one of the most famous demonstrations of the wave nature of light.
A coherent source illuminates two narrow slits separated by a distance \(d\). Each slit acts like a secondary source, and the two emerging waves overlap on a screen placed a distance \(L\) away.
At some points on the screen, the waves arrive in phase and reinforce each other, producing bright fringes.
At other points, they arrive out of phase and partially or completely cancel, producing dark fringes.
The key quantity is the path difference between the two rays:
\[
\Delta x = r_2-r_1.
\]
If the path difference is an integer multiple of the wavelength, the interference is constructive:
\[
\Delta x = m\lambda,
\qquad m=0,\pm1,\pm2,\dots
\]
These locations are the bright fringes. Under the standard small-angle approximation, where the screen distance \(L\) is much larger than the slit separation and the screen displacement \(y\), the bright-fringe positions are
\[
y_m = m\frac{\lambda L}{d}.
\]
Consecutive bright fringes are separated by the same distance, called the fringe spacing:
\[
\Delta y = \frac{\lambda L}{d}.
\]
This formula is especially useful because it shows immediately how the pattern changes:
a larger wavelength makes the fringes farther apart,
a larger screen distance also spreads them out,
while a larger slit separation makes them closer together.
Dark fringes occur halfway between neighboring bright fringes. Their path difference is an odd half-integer multiple of the wavelength:
\[
\Delta x = \left(m+\frac12\right)\lambda.
\]
Therefore the dark-fringe positions are
\[
y_{\text{dark}}=\left(m+\frac12\right)\frac{\lambda L}{d}.
\]
In addition to fringe position, the experiment also gives an intensity variation across the screen.
The phase difference between the two waves is
\[
\delta = \frac{2\pi\Delta x}{\lambda}.
\]
If the two slits contribute equal amplitudes, the intensity is
\[
I = I_0\cos^2\left(\frac{\delta}{2}\right),
\]
where \(I_0\) is the reference maximum intensity.
This means the pattern is not just “bright or dark”; rather, it changes smoothly from maximum to minimum as the phase difference changes.
For the sample input
\(\lambda=550\ \text{nm}\),
\(L=2\ \text{m}\),
and
\(d=0.2\ \text{mm}\),
first convert to SI units:
\[
\lambda=550\times10^{-9}\ \text{m},
\qquad
d=0.2\times10^{-3}\ \text{m}.
\]
Then the fringe spacing is
\[
\Delta y=\frac{\lambda L}{d}
= \frac{(550\times10^{-9})(2)}{0.2\times10^{-3}}
= 5.5\times10^{-3}\ \text{m}.
\]
So
\[
\Delta y = 5.5\ \text{mm}.
\]
The first bright fringe \((m=1)\) is therefore at
\[
y_1 = 1\cdot \Delta y = 5.5\ \text{mm}.
\]
The first dark fringe lies halfway between the central maximum and the first bright fringe:
\[
y_{\text{dark},1}=\frac{\Delta y}{2}=2.75\ \text{mm}.
\]
The animation in this calculator shows two rays traveling from the slits to a selected probe point on the screen.
It also displays the interference pattern on the screen and an intensity plot in normalized units.
This helps connect the geometry of path difference with the brightness variation actually seen in a laboratory laser experiment.
At more advanced university level, the ideal two-slit model is refined by including finite slit width, diffraction envelopes, partial coherence, unequal amplitudes, and phase plates.
Even so, the central lesson remains the same:
light from the two slits interferes because each path contributes a wave, and the relative phase controls what appears on the screen.
So the big formulas to remember are
\[
\Delta y=\frac{\lambda L}{d},
\qquad
y_m=m\frac{\lambda L}{d},
\qquad
I=I_0\cos^2\left(\frac{\delta}{2}\right),
\]
which together describe the spacing, location, and brightness of the double-slit interference fringes.
