Problem 40 - University Physics with Modern Physics

Question

If you can read the bottom row of your doctor's eye chart, your eye has a resolving power of 1 arcminute, equal to \(1\over{60}\) degree. If this resolving power is diffraction limited, to what effective diameter of your eye's optical system does this correspond? Use Rayleigh's criterion and assume \(\lambda\) = 550 nm.

Step-by-Step Solution

Answer

The effective diameter of your eye's optical system is approximately 3.7 mm.

Step 1: Understand Rayleigh's Criterion

Rayleigh's criterion states that two points are just resolvable when the central maximum of one image coincides with the first minimum of the other image. The angle of resolution \( \theta \) in radians is given by \( \theta = 1.22 \frac{\lambda}{D} \), where \( \lambda \) is the wavelength of light and \( D \) is the diameter of the aperture.

Step 2: Convert Angular Resolution to Radians

The resolving power given is 1 arcminute. To use it in Rayleigh's criterion, convert this to radians. We know 1 degree = \( \frac{\pi}{180} \) radians, and 1 arcminute = \( \frac{1}{60} \) degree, so 1 arcminute = \( \frac{\pi}{180 \times 60} \) radians.

Step 3: Rearrange Rayleigh's Equation

Rearrange the Rayleigh's criterion formula to solve for \( D \): \[ D = 1.22 \frac{\lambda}{\theta} \].

Step 4: Substitute Values into Equation

Substitute \( \lambda = 550 \text{ nm} = 550 \times 10^{-9} \text{ m} \) and the calculated \( \theta = \frac{\pi}{180 \times 60} \) radians into the rearranged equation: \[ D = 1.22 \frac{550 \times 10^{-9}}{\frac{\pi}{180 \times 60}} \].

Step 5: Calculate the Diameter

Perform the arithmetic to find \( D \):- First calculate \( \theta = \frac{\pi}{180 \times 60} \) = 2.908882086 \times 10^{-4} radians.- Now substitute into the formula: \[ D = 1.22 \frac{550 \times 10^{-9}}{2.908882086 \times 10^{-4}} \].- This results in \( D \approx 3.7 \text{ mm} \).

Key Concepts

Diffraction Limited

Diffraction limitation occurs when the ability of an optical system to distinguish small details is primarily limited by diffraction rather than by imperfections of the optical system itself. Diffraction is a fundamental property of waves, including light, where waves bend around corners or spread out after passing through a small aperture. When the aperture, like the pupil of your eye, is small, diffraction effects become significant.

This limitation means that even a perfect optical system with no aberrations can only resolve details up to a natural limit dictated by the wave nature of light. This limit is described by Rayleigh's criterion. When an optical system meets this limit, it is termed 'diffraction limited'. Improved resolution beyond this point would require larger aperture diameters or shorter wavelengths, as diffraction is less pronounced with either of these changes.

Resolving Power

Resolving power is the ability of an optical system to separate two closely spaced objects into distinct images. In simpler terms, it's a measure of how well the system can differentiate between two points as separate entities rather than a single blur.

The resolving power is crucial for applications like telescopes, microscopes, and the human eye, affecting how clearly we can see fine details. In the context of the given exercise, a resolving power of 1 arcminute indicates the smallest angle at which two points can be distinctly observed.

• This is a standard benchmark, as it reflects the typical clarity achieved by a healthy human eye.
• When resolving power is diffraction limited, it implies that the resolution is as good as physically possible given the system's aperture and light properties.

Angular Resolution

Angular resolution refers to the smallest angle between two points that can be distinguished as separate when viewed through an optical system. It is essentially the practical outcome of an optical system's resolving power.

Rayleigh's criterion provides a mathematical formula deriving angular resolution, \[ \theta = 1.22 \frac{\lambda}{D} \]where \( \theta \) is the angular resolution in radians, \( \lambda \) is the light wavelength, and \( D \) is the aperture diameter. When you need to convert angular resolution from common units like degrees or arcminutes to radians, it's essential to use the correct conversion factors, as demonstrated in the exercise.

Understanding angular resolution helps quantify how effectively an optical system, like the human eye, can distinguish between two closely located points. In practical terms, this means a higher angular resolution allows clearer and more detailed images.