A red laser pointer with a wavelength of \(635 \mathrm{nm}\) is shined on a double slit producing a diffraction pattern on a screen that is \(1.60 \mathrm{~m}\) behind the double slit. The central maximum of the diffraction pattern has a width of \(4.20 \mathrm{~cm}\) and the fourth bright spot is missing on both sides. What is the size of the individual slits, and what is the separation between them?

The phenomena of interference and diffraction are fundamental concepts in wave physics and form the basis for understanding various optical patterns created by light. Interference occurs when two or more waves overlap and combine to form a new wave pattern. Depending on the phase difference between the overlapping waves, the interference can be constructive or destructive, leading to an increase or decrease in the wave intensity respectively.

Diffraction, on the other hand, involves the bending of waves around obstacles or the spreading of waves as they pass through openings; in essence, it allows light to apparently 'turn corners'. The most intriguing effects of diffraction are observed when the size of the obstacle or opening is comparable to the wavelength of the light. In such cases, one can observe distinct patterns on a screen, characterized by a series of bright and dark fringes. An important aspect of diffraction is that it demonstrates the wave nature of light.

The double-slit experiment is a crucial demonstration of the wave-like properties of light. In this experiment, a beam of light is directed towards two very closely spaced slits. As the light passes through these slits, it diffracts and the resulting wavefronts interfere with each other.

A screen placed behind the double slits captures the resulting pattern of light, known as an interference pattern. This pattern consists of a series of light and dark fringes—an artifact of constructive and destructive interference, respectively. The central maximum is the brightest fringe located at the center of the pattern, and bright fringes (maxima) and dark fringes (minima) are symmetrically arranged about it. By carefully analyzing the spacing and location of these fringes, one can deduce information about the wavelength of light and dimensions of the slits.

The single-slit diffraction formula is key to understanding the pattern produced when light passes through a single, narrow aperture. The formula is given by:
\[ x_n = \frac{2n-1}{2} \frac{\lambda L}{a} \]
where \( x_n \) is the position of the dark fringe on a screen, \( n \) is the order of the fringe, \( \lambda \) is the wavelength of the light, \( L \) is the distance from the slit to the screen, and \( a \) is the width of the slit. This equation predicts the locations where the intensity of light is minimal or essentially zero due to destructive interference, known as dark fringes. The absence of a bright fringe in the double-slit pattern can be correlated with the position of a dark fringe in the single-slit pattern, thereby providing insights into the characteristics of the slits used in experiments.

The concept of the wavelength of light plays a central role in understanding diffraction and interference patterns. It is defined as the distance between successive crests (or troughs) of a wave and is a critical parameter in the equations that describe light's behavior in wave terms. The wavelength determines the color of light within the visible spectrum and also influences how light diffracts and interferes when it encounters obstacles or openings.

In optical experiments, the wavelength can be used to calculate the dimensions of slits or spacings in gratings that produce particular interference and diffraction patterns, as observed in the double-slit experiment. Wavelength is typically measured in nanometers (nm) or micrometers (μm), and the solution to the textbook problem uses the known wavelength of a red laser pointer to determine the slit dimensions, demonstrating the interplay between these concepts and their practical applications.