In bright light, the human eye's pupil diameter is about 2 mm. If diffraction were the limiting factor, what's the eye's minimum angular resolution under these conditions, assuming 550 -nm light?

Understanding the diffraction limit is essential for uncovering how well an optical system can resolve detail. It's particularly important in the study of microscopes, telescopes, and even the human eye. The term 'diffraction limit' refers to the minimum distance two points can be separated by and still be distinguishable as two separate points, as opposed to one blurred point.

Diffraction is a phenomenon that occurs when a wave, such as light, encounters an obstacle or a slit that is comparable in size to its wavelength. This interaction causes the light to spread out, or 'diffract'. It imposes a fundamental limit on the resolution of the optical system, which depends on both the wavelength of light and the size of the aperture through which the light is passing.

The formula for the angular resolution in radians, considering diffraction as the limiting factor, is given by \( \theta = 1.22 \frac{\lambda}{D} \), where \( \lambda \) is the wavelength of light and \( D \) is the diameter of the aperture. The factor 1.22 is a constant that applies to circular apertures and arises from a calculation involving Bessel functions, which describe the intensity pattern of the diffracted light.

The pupil, which is the opening in the centre of the iris of the eye, plays a critical role in visual acuity. The pupil's diameter can affect the amount of light entering the eye and, importantly for our discussion, the eye's resolving power. In bright light conditions, the pupil constricts to a smaller diameter to reduce the amount of light entering the eye, while in dim light, it dilates to allow more light to enter.

The human eye normally has a pupil diameter ranging from about 2 mm in bright light to 8 mm in very dark conditions. This variation in size not only regulates light intake but also affects the image sharpness due to diffraction. A smaller pupil diameter, as encountered in bright conditions, creates a larger diffraction effect, potentially limiting the resolution of the eye.

Thus, given the diameter of the pupil, one can infer the diffraction-limited resolution of the eye. In the exercise provided, a 2 mm pupil diameter in bright light conditions is used to calculate the eye's minimum angular resolution.

The concept of wavelength is fundamental to understanding wave phenomena such as diffraction. Wavelength (represented by \( \lambda \) in equations) is the distance between successive crests (or troughs) of a wave. For light waves, which are electromagnetic waves, wavelength corresponds to their color, with longer wavelengths being red and shorter ones being blue or violet.

When discussing the diffraction limit and resolution, the wavelength of light in use can drastically impact the outcome. Shorter wavelengths can yield a higher resolution, as the diffraction pattern produced by an aperture becomes tighter. This is why blue light, with its shorter wavelength compared to red light, is used in some optical instruments for higher resolution imaging.

In the context of the human eye and the exercise provided, a standard wavelength of 550 nm (nanometers), which lies in the green portion of the visible spectrum, is used to calculate the angular resolution. This wavelength is often chosen since it's near the peak sensitivity of the human eye in daylight conditions.