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The wavelength of light is a fundamental concept in understanding double-slit interference. It is the distance between consecutive peaks of a wave and is typically measured in nanometers (nm) for light. In this exercise, we have two wavelengths: 660 nm for red light and 470 nm for blue light. These wavelengths determine the color of the light and play a crucial role in the interference pattern observed.
Wavelengths dictate how the light waves interfere with each other when they pass through slits. Longer wavelengths, like the red light in our exercise, will spread out more and produce a broader fringe pattern. Conversely, shorter wavelengths, like blue light, are more closely spaced, leading to more narrowly spaced fringes. Thus, understanding the concept of wavelength helps explain why different colors of light form distinctive patterns when they pass through slits.

In double-slit interference, 'fringe position' refers to the specific locations on the observation screen where bright and dark bands appear. These bands, known as fringes, are the result of constructive and destructive interference of light waves. Constructive interference happens when the path difference between the two waves equals an integer multiple of the wavelength, resulting in bright fringes.
The position of these fringes is calculated using the formula: \[ y_m = \frac{m \lambda L}{d} \]where \(y_m\) is the fringe position, \(m\) is the order number of the fringe (1, 2, 3,...), \(\lambda\) is the wavelength of light, \(L\) is the distance from the slits to the screen, and \(d\) is the separation between the slits. In our problem, the first-order fringe (\(m = 1\)) positions are calculated for two different wavelengths, demonstrating how different wavelengths result in fringes appearing at different positions on the screen.

Coherent light is a key requirement for creating an interference pattern, such as in the double-slit experiment. Coherent light consists of waves that are in phase or have a constant phase difference. In simpler terms, coherent light waves travel together in a uniform manner, allowing them to consistently interfere with each other—crucial for forming clear and stable patterns of bright and dark fringes.
For the exercise, the coherent source ensures that light maintains its wave characteristics after passing through the slits. This coherence allows us to apply our formula for fringe positions confidently, as the waves maintain a predictable relationship, leading to the formation of observable interference patterns. Without coherence, the light waves would not interact in a consistent way, leading to a lack of structured patterns on the screen.