When dots are placed on a page from a laser printer, they must be close enough so that you do not see the individual dots of ink. To do this, the separation of the dots must be less than Raleigh's criterion. Take the pupil of the eye to be \(3.0 \mathrm{mm}\) and the distance from the paper to the eye of \(35 \mathrm{cm}\); find the minimum separation of two dots such that they cannot be resolved. How many dots per inch (dpi) does this correspond to?

Optical resolution is a measure of how well a particular optical system can distinguish between two points that are close together. It's an essential factor in the performance of devices like telescopes, microscopes, and printers. When we talk about resolution in the context of our eyes, we refer to the ability to perceive two distinct points rather than a single blob. Raleigh's criterion is a pivotal guideline in determining optical resolution. In essence, it defines the minimum angular separation two light sources (or dots) must have for an observer to discern them as separate. This principle is vital in many fields, including astronomy and, as our example shows, printing. For clear visualization, imagine looking at a starry sky; the stars are distant light sources. If two stars are very close together and the distance between them falls below the threshold defined by Raleigh's criterion, they would appear to us as a single star.

However, optical resolution is not just about distance; it involves the interplay of several factors, including the wavelength of light and the diameter of the optical system's aperture, which in the case of the human eye is the pupil. This interdependence is why even the sharpest eyes cannot distinguish between the fine details of a very close pair of dots on a printed page if those dots are too close together.

Calculating the minimum separation between two points, such as dots on a page, is pivotal when we aim to achieve an image or printout that appears sharp to the human eye. The concept of minimum separation is deeply tied to the optical resolution we just explored. By using Raleigh's criterion, we can formulate an equation to determine this minimum distance, effectively ensuring that the individual dots are not discernible and thus presenting a continuous image.

The calculation of the minimum separation requires us to take into account factors like the viewing distance, the wavelength of light (usually taken at peak visibility which is around 550 nm for the human eye), and the diameter of the aperture (or the pupil size in the case of human vision). By applying Raleigh's formula—factoring in the angle of separation as derived from the observer's perspective—we can compute the smallest distance at which two points are perceived as one. In practical applications, such as printing, this mathematical determination helps in creating a resolution that is within the limits of human vision, ensuring that pictures and texts are perceived as smooth and continuous.

Dots per inch (DPI) is a metric for measuring the resolution of printed images or displays. It quantifies how many dots are placed along a one-inch line and thus is a direct indicator of how detailed an image can be. The higher the DPI, the more dots can fit within the span of an inch, which generally translates to a more detailed and crisper image.

In the context of the given exercise, after finding the minimum separation that meets Raleigh's criterion, we can convert this separation distance into an easily understood DPI value by determining how many of these minimum-separated dots can be packed into an inch. This conversion is crucial because DPI is a standard measurement that describes the resolution in printing and various display systems. High DPI values are especially important in high-quality photo printing and fine art reproductions, where the slightest detail matters. For everyday applications, such as reading text on a printed page or viewing a screen, a lower DPI may suffice, as the human eye may not distinguish between ultrahigh resolutions at typical viewing distances.