Superposition of Waves

When two waves enter the same region of space at the same time they interfere in a way that obeys the Superposition of Waves. This addition of waves creates places where the peaks line up and the resultant wave is larger, which would manifest as a bright spot for light waves. There are also regions where the peak from one wave is lined up with the trough from the other and the resultant is wave cancelation, where their amplitudes add to zero and there would be a dark spot. These constructive and destructive areas are seperated by an entire gradient of partially constructive and partially destructive. The resulting interference pattern is one of the defining features of a wave

Wave Interpretation of Light - Young's Double Slit Interference

The theory of Electromagnetic Radiation Wave Theory is one of the most tested and confirmed theories in physics. It relies heavily on the experimental fact that light has wave-like properties. These wave-like properties are displayed by the interference effects that has only been observed in wave systems. The hallmark experiment that enabled us to observe the effect was Young's Double Slit Experiment.

To observe the interference of two sources you need two, coherent, single frequency sources. Essentially you would like two identical sources. The clever way Young achieved this was by isolating a single color of light then sending that light through two small slits. Each slit acts like a new wave source due to the diffraction of light, which is the effect that light "bends" around corners and spreads out when passing through an opening. Since each slit acts like a new source, and each originated from the same source, they are two coherent, single frequency (same color) sources. The diagram below shows a snapshot of the waves as they interfere.

The modern technology of a LASER (Light Amplification through the Stimulation of Electromagnetic Radiation) has made this experiment much easier due to the very high intensity soures they provide. The experiment also enables one to measure the wavelength of visable light, something on the order of hundreds of nanometers, a size only an order of magnitude greater than the size of atoms. For nearly 100 years this was the only way to probe scales this size scale and it still remains as one of the best.

The geometry of Young's Double Slit is below.

Here the dstance between the slits is $d$ and the screen observing the interference effect is a distance $D$ away from the slits. The central maximum is located along the perpendicular bisector between the two sources where the Path Length Difference (PLD) between the two sources is zero. The condition for constructive interference is for the PLD to be an interger ($m$) multiple of the wavelength. So as you move away from the central maximum and go from constructive to destructive and back to constructive, you've increased the PLD by one whole wavelength. These bright spots are called interference fringes. Using the condition for constructive interference $PLD = m \lambda$ and the geometry that $PLD=d sin(\theta)$, you arrive at the overall condition for constructive interference.

$d sin{\theta}=m \lambda$

The distance $y$ is measured from the central maximum and can be related to the angle ($\theta$) and the distance $D$ by the equation $tan (\theta) = \frac{y}{D}$. In most cases, where $\lambda << d$ the angle is so small that $sin (\theta) \approx \theta$. Since $tan (\theta) = \frac{sin(\theta)}{\cos(\theta)}$ and $\cos(\theta) \approx 1$ for very small angles, $tan(\theta) \approx sin(\theta) \approx \theta$. This allows a more simple connection between the variables shown in the equation below, but note, this only if the angles are very small.

$y \approx \frac{m \lambda D}{d}$, if $\lambda << d$

Now, take a look at the pre-lecture reading and videos below.