Problem 44 - University Physics with Modern Physics

Question

A wildlife photographer uses a moderate telephoto lens of focal length 135 mm and maximum aperture \(f/\)4.00 to photograph a bear that is 11.5 m away. Assume the wavelength is 550 nm. (a) What is the width of the smallest feature on the bear that this lens can resolve if it is opened to its maximum aperture? (b) If, to gain depth of field, the photographer stops the lens down to \(f/\)22.0, what would be the width of the smallest resolvable feature on the bear?

Step-by-Step Solution

Answer

(a) 0.23 mm; (b) 1.27 mm.

Step 1: Understanding the diffraction limit

To find the smallest feature size that can be resolved, we use the Rayleigh criterion for diffraction-limited resolution, which states that the minimum resolvable feature size, \( w \), is given by \( w = \frac{1.22 \times \lambda \times D}{d} \), where \( \lambda \) is the wavelength, \( D \) is the distance between the lens and the object, and \( d \) is the diameter of the aperture.

Step 2: Calculating the aperture diameter for f/4.0

The focal length \( f = 135 \text{ mm} \), and the aperture number \( N = 4.0 \), so the diameter of the aperture \( d \) can be calculated using \( d = \frac{f}{N} = \frac{135}{4} \). Hence, \( d = 33.75 \text{ mm} \).

Step 3: Calculating the smallest resolvable feature using f/4.0

Plugging in the values, we have \( \lambda = 550 \text{ nm} = 550 \times 10^{-9} \text{ m} \), \( D = 11.5 \text{ m} \), and \( d = 33.75 \times 10^{-3} \text{ m} \). So, \( w = \frac{1.22 \times 550 \times 10^{-9} \times 11.5}{33.75 \times 10^{-3}} \approx 0.23 \text{ mm} \).

Step 4: Calculating the aperture diameter for f/22.0

For \( N = 22.0 \), the aperture diameter \( d = \frac{135}{22} \). This gives \( d = 6.136 \text{ mm} \).

Step 5: Calculating the smallest resolvable feature using f/22.0

Using the same formula as before with \( d = 6.136 \times 10^{-3} \text{ m} \), we get \( w = \frac{1.22 \times 550 \times 10^{-9} \times 11.5}{6.136 \times 10^{-3}} \approx 1.27 \text{ mm} \).

Step 6: Conclusion

The smallest feature on the bear that can be resolved when the lens is opened to its maximum aperture \( f/4.0 \) is approximately \( 0.23 \text{ mm} \), and when stopped down to \( f/22.0 \), it is approximately \( 1.27 \text{ mm} \).

Key Concepts

Diffraction-limited resolution

The ability of optical systems, like cameras and telescopes, to distinguish two closely spaced objects is influenced by the phenomenon of diffraction. This limit is known as the diffraction-limited resolution, a concept arising from the wave nature of light. When light passes through an aperture, it bends around the edges, creating an interference pattern that limits the detail that can be perceived.
Rayleigh's criterion offers a practical way to estimate this limit. According to the Rayleigh criterion, two objects are considered resolvable when the central maximum of the diffraction pattern of one image falls on the first minimum of the other. For circular apertures, this can be simplified mathematically to:

• Minimum resolvable angle: \[ \theta = 1.22 \frac{\lambda}{d} \]
• Minimum resolvable feature size: \[ w = \frac{1.22 \times \lambda \times D}{d} \]

In this context, \( \lambda \) is the wavelength of light, \( D \) is the distance to the object, and \( d \) is the aperture diameter. Understanding this concept helps photographers and scientists design systems that maximize clarity.

Aperture diameter

The aperture of a lens plays a critical role in determining the resolution of an image. It is the size of the lens opening that allows light to enter the camera or telescope. Measured in millimeters, the aperture diameter, \( d \), is vital for the Rayleigh criterion and is calculated as:
\[ d = \frac{f}{N} \]

• \( f \): focal length of the lens
• \( N \): f-number (or f-stop) that indicates the size of the aperture relative to the focal length

For example, in the exercise, at \( f/4.0 \), the aperture diameter is calculated as 33.75 mm, while at \( f/22.0 \), it shrinks to approximately 6.136 mm. A larger aperture diameter allows more light through, increasing the potential for better resolution but reducing the depth of field. Conversely, stopping down increases depth but decreases resolution.

Wavelength

Wavelength, denoted as \( \lambda \), is a fundamental characteristic of light that defines the distance between successive peaks of a wave. It is typically measured in nanometers (nm) for visible light, with 550 nm being a common reference for calculations involving typical daylight conditions.
The wavelength of light significantly impacts the diffraction-limited resolution. Longer wavelengths result in larger diffraction patterns, which can limit resolution, while shorter wavelengths enhance it. In practical terms, this is why ultraviolet or blue light, with shorter wavelengths, can achieve higher resolutions compared with red light.

• In the scenario, the wavelength is used in the formula for the minimum resolvable feature size: \[ w = \frac{1.22 \times 550 \times 10^{-9} \times 11.5}{d} \]

This formula highlights how the wavelength interacts with the aperture diameter and distance to determine resolution limits.

Focal length

Focal length, represented as \( f \), is a crucial lens specification that affects both the field of view and the size of objects captured within an image. It is the distance from the lens to the sensor (or film) where light converges to a point.

• A shorter focal length results in a wider field of view, suitable for landscape photography.
• A longer focal length, like the 135 mm used in this exercise, narrows the view but magnifies distant subjects, ideal for wildlife photography.

When considering diffraction, the focal length impacts the total light path and, consequently, the potential resolution as it interacts with the aperture setting. The combination of focal length and aperture diameter defines the f-number, influencing depth of field and clarity:
\[ N = \frac{f}{d} \]Understanding focal length aids in achieving the desired balance between subject magnification and sharpness, making it a key element in photographic and observational setups.