KAHIBARO
Table of Contents
Seeing Light Interfere with Itself
Young's double-slit experiment is one of the most important demonstrations in wave optics. It shows that light can produce an interference pattern when it passes through two narrow slits. Instead of making only two bright patches on a screen, the light forms many alternating bright and dark bands. This behavior is a clear sign of wave superposition.
In this experiment, a light source illuminates two very narrow slits that are close together. Each slit acts like a source of waves. The waves spread out from the slits and overlap on a screen placed some distance away. At some points on the screen, the waves arrive in step and reinforce each other. At other points, they arrive out of step and weaken or cancel each other.
Path Difference and Interference
The key idea is the path difference between the two waves. If one wave travels a slightly longer distance than the other, the two waves may arrive with a phase difference.
Suppose the two slits are separated by distance $d$, the screen is at distance $L$, and a point on the screen is seen at angle $\theta$ from the centerline. The path difference between the two waves is
$$
\Delta = d \sin \theta
$$
Bright fringes appear where the path difference is an integer multiple of the wavelength $\lambda$:
$$
d \sin \theta = m\lambda
$$
where $m = 0, \pm1, \pm2, \pm3, \dots$
Dark fringes appear where the path difference is a half integer multiple of the wavelength:
$$
d \sin \theta = \left(m + \frac{1}{2}\right)\lambda
$$
with $m = 0, 1, 2, 3, \dots$
For double-slit interference,
$$
\text{bright fringes: } d\sin\theta = m\lambda
$$
$$
\text{dark fringes: } d\sin\theta = \left(m+\frac12\right)\lambda
$$
These are the central conditions for the experiment.
The Fringe Pattern on the Screen
The middle bright fringe is called the central maximum. It occurs at $\theta = 0$, where the two paths are equal. On both sides of it, bright and dark fringes appear in a regular pattern.
If the screen is far from the slits, which means $L \gg d$, then the angle is small and we may use the small angle approximation:
$$
\sin\theta \approx \tan\theta \approx \theta
$$
If $y$ is the distance from the central bright fringe to a point on the screen, then
$$
\tan\theta \approx \frac{y}{L}
$$
So the bright fringe positions are approximately
$$
y_m = \frac{m\lambda L}{d}
$$
The distance between neighboring bright fringes is called the fringe spacing:
$$
\Delta y = \frac{\lambda L}{d}
$$
This same spacing also separates neighboring dark fringes.
When the screen is far away and angles are small,
$$
y_m = \frac{m\lambda L}{d}
$$
and the fringe spacing is
$$
\Delta y = \frac{\lambda L}{d}
$$
A larger wavelength or larger screen distance gives wider fringes. A larger slit separation gives narrower fringes.
What the Experiment Shows
The experiment shows that light behaves as a wave in propagation and interference. If light were only a stream of simple particles with no wave character, the regular bright and dark interference pattern would not appear in this way.
A remarkable feature is that even when light is made extremely weak, so that photons pass one at a time, the full interference pattern still builds up gradually after many detections. This tells us that the wave nature is fundamental. In modern physics, this experiment also becomes important for understanding wave-particle duality, but that deeper interpretation belongs elsewhere.
Conditions for a Clear Pattern
To see stable interference fringes clearly, the two slits must be illuminated coherently. This means the waves from the slits must maintain a fixed phase relationship. In practice, using one source to illuminate both slits provides this coherence.
The light should also be nearly monochromatic, meaning it has a nearly single wavelength. If many wavelengths are present, each wavelength makes its own fringe pattern, and the result becomes blurred.
The slits must be narrow enough so that light spreads out after passing through them. If the slits are too wide, the overlap needed for clear interference is reduced.
Dependence on Experimental Parameters
The main quantities that control the pattern are easy to summarize.
| Quantity changed | Effect on fringe spacing |
|---|---|
| Increase $\lambda$ | Fringes spread farther apart |
| Increase $L$ | Fringes spread farther apart |
| Increase $d$ | Fringes move closer together |
This means the fringe pattern can be used to measure wavelength if $d$ and $L$ are known, or to determine slit separation if the wavelength is known.
A Simple Numerical Example
Suppose light of wavelength $\lambda = 600 \text{ nm}$ passes through slits separated by $d = 0.30 \text{ mm}$. The screen is $L = 2.0 \text{ m}$ away. The fringe spacing is
$$
\Delta y = \frac{\lambda L}{d}
$$
Convert units first:
$$
\lambda = 600 \times 10^{-9}\,\text{m}
$$
$$
d = 0.30 \times 10^{-3}\,\text{m}
$$
Then
$$
\Delta y = \frac{(600\times10^{-9})(2.0)}{0.30\times10^{-3}}
= 4.0\times10^{-3}\,\text{m}
$$
So
$$
\Delta y = 4.0 \text{ mm}
$$
The bright fringes are spaced by $4.0$ mm.
Why the Central Fringe is Bright
At the center of the screen, the distances from both slits are equal. Therefore the path difference is zero:
$$
\Delta = 0
$$
Zero is an integer multiple of the wavelength, since $0 = 0\lambda$, so the condition for constructive interference is satisfied. That is why the central fringe is bright and corresponds to $m=0$.
Relation to Real Experiments
In a real setup, the bright fringes are not all equally intense across the whole screen. Their intensity is often strongest near the center and weaker farther out. This happens because each slit also produces diffraction, and the interference pattern sits inside a broader diffraction envelope. The full treatment of that effect belongs more naturally with diffraction.
Still, the basic double-slit formulas remain the starting point for understanding the observed pattern.
Young's double-slit experiment is the classic demonstration that light produces interference.
The essential results are
$$
\Delta = d\sin\theta
$$
$$
d\sin\theta = m\lambda \quad \text{for bright fringes}
$$
$$
d\sin\theta = \left(m+\frac12\right)\lambda \quad \text{for dark fringes}
$$
and for small angles,
$$
\Delta y = \frac{\lambda L}{d}
$$
Final Picture
Young's double-slit experiment turns the abstract idea of wave interference into a visible pattern on a screen. Two narrow slits act like two coherent wave sources. Their overlap creates alternating bright and dark fringes. By measuring the fringe spacing, we can connect the geometry of the apparatus to the wavelength of light. This experiment is simple in design, but it reveals one of the deepest facts in physics, that light shows unmistakable wave behavior.