The resolution of a digital camera is limited by two factors: diffraction by the lens, a limit of any optical system, and the fact that the sensor is divided into discrete pixels. Consider a typical point-and-shoot camera that has a 20-mm-focal-length lens and a sensor with 2.5μm x 2.5 μm pixels. What is the f-number of the lens for the diameter you found in part b? Your answer is a quite realistic value of the f-number at which a camera transitions from being pixel limited to being diffraction limited. For f-numbers smaller than this (larger-diameter apertures), the resolution is limited by the pixel size and does not change as you change the aperture. For f-numbers larger than this (smaller-diameter apertures), the resolution is limited by diffraction, and it gets worse as you “stop down” to smaller apertures.

Verified step by step guidance

1. Determine the f-number formula: The f-number (f/#) of a lens is defined as the ratio of the focal length (f) to the diameter of the aperture (D). The formula is: $f/\#=f/D$.
2. Identify the given values: From the problem, the focal length of the lens is $20$ mm. The diameter of the aperture (D) was found in part b, so use that value in this step.
3. Substitute the values into the formula: Replace $f$ with $20$ mm and $D$ with the diameter value from part b into the formula $f/\#=f/D$.
4. Simplify the expression: Perform the division to calculate the f-number. Ensure the units are consistent (e.g., both focal length and diameter should be in millimeters).
5. Interpret the result: The calculated f-number represents the transition point where the camera shifts from being pixel-limited to diffraction-limited. For f-numbers smaller than this, resolution is limited by pixel size, while for larger f-numbers, resolution is limited by diffraction.

Key Concepts

F-number (f-stop)

The f-number, or f-stop, is a measure of the lens aperture's size, defined as the ratio of the lens's focal length to the diameter of the entrance pupil. It indicates how much light the lens can gather; lower f-numbers correspond to larger apertures, allowing more light and shallower depth of field, while higher f-numbers indicate smaller apertures, reducing light intake and increasing depth of field. Understanding f-numbers is crucial for determining the balance between light exposure and image sharpness.

Diffraction Limit

Diffraction is a phenomenon that occurs when light waves encounter an obstacle or aperture, causing them to spread out. In optical systems, diffraction limits the resolution of images, particularly at smaller apertures where the light waves bend more significantly. The diffraction limit is a critical concept in photography, as it defines the point at which further decreasing the aperture size will not improve image sharpness, but rather degrade it due to the wave nature of light.

Pixel Limitation

Pixel limitation refers to the constraints imposed by the discrete nature of a digital camera's sensor, which is divided into individual pixels. When the resolution of an image is primarily determined by the size of these pixels, rather than the optical quality of the lens, the camera is said to be pixel limited. This concept is essential for understanding how sensor resolution interacts with lens performance, particularly in determining the effective sharpness of an image at various f-numbers.