Question

What minimum path difference is needed to cause a phase shift by \(\pi / 4\) in light of wavelength \(700 . \mathrm{nm} \)

Short answer

Answer: The minimum path difference required is approximately 87.5 nm.

Step-by-step solution

Understand the relationship between phase angle, wavelength, and path difference

The relationship between phase shift (\(\Delta \phi\)), wavelength (\(\lambda\)), and the path difference (\(\delta\)) is given by the equation: \(\Delta \phi = \cfrac{2 \pi \delta}{\lambda}\) Here, we are given the phase shift \(\Delta \phi = \cfrac{\pi}{4}\) and wavelength, \(\lambda=700 \ \mathrm{nm}\). We will now solve for the path difference \(\delta\).

Solve for path difference

We can isolate the path difference (\(\delta\)) by rewriting the equation in Step 1: \(\delta = \cfrac{\lambda \Delta \phi}{2 \pi}\) Next, plug in the given values of phase shift and wavelength: \(\delta = \cfrac{(700 \ \mathrm{nm})\left(\cfrac{\pi}{4}\right)}{2 \pi}\)

Calculate the path difference

Now we can calculate the minimum path difference for the given situation: \(\delta = \cfrac{(700 \ \mathrm{nm})\left(\cfrac{\pi}{4}\right)}{2 \pi} = \cfrac{700}{8} \ \mathrm{nm}\) \(\delta \approx 87.5 \ \mathrm{nm}\) Therefore, the minimum path difference required to cause a phase shift of \(\pi/4\) for light with a wavelength of \(700 \ \mathrm{nm}\) is approximately \(87.5 \ \mathrm{nm}\).

Key concepts

Path Difference

In the realm of wave optics, the path difference is crucial for understanding how waves interact. Path difference refers to the difference in the distance traveled by two waves from their respective sources to a common point. This difference determines how the waves will interfere with each other - whether constructively or destructively.
When discussing light, such interactions are significant because they result in phenomena such as interference patterns. Interference can occur when two or more waves meet, and their path differences lead to phase shifts. This path difference is directly related to the phase shift between the waves. The equation for phase shift (\( \Delta \phi \)) is given by:

• \( \Delta \phi = \cfrac{2 \pi \delta}{\lambda} \)

where \( \delta \) is the path difference, and \( \lambda \) is the wavelength of the waves. In our specific example, to achieve a specific phase shift of \( \pi/4 \), we calculated the path difference to be approximately \( 87.5 \ \mathrm{nm} \).
This relationship tells us precisely how far one wave must "lead" or "lag" behind another to create a measurable phase difference.

Wavelength

Wavelength is a fundamental concept in understanding wave behavior. It is defined as the distance between two consecutive points in the same phase of a wave, such as from crest to crest or trough to trough. For light waves, wavelength determines the color of the light, and it is inversely related to frequency, meaning that as wavelength increases, the frequency decreases and vice versa.
The exercise we explored involves a wavelength of \( 700 \ \mathrm{nm} \), which falls within the visible light spectrum and corresponds to red light. The wavelength is crucial in the calculation of path difference because it directly influences the phase angle change. By knowing the wavelength, you can determine how much a wave must shift its path to achieve a desired phase shift.
For example, when calculating the path difference needed for a certain phase shift, the formula \( \delta = \cfrac{\lambda \Delta \phi}{2 \pi} \) was used. This demonstrates how wavelength (\( \lambda \)) is utilized to find the path distance that will cause a specific phase shift. Understanding the wavelength helps us predict how light will behave in various situations, such as through slit experiments or when passing through different media.

Optics

Optics is the study of light, its properties, and its interaction with matter. This field of physics broadly covers the behavior and properties of light, including its interactions with materials and the formation of images by lenses and mirrors.
The concept of path difference and phase shifts is central in optics, particularly in understanding interference and diffraction, key phenomena in wave optics. Light as a wave will bend, overlap, and interfere as it interacts with different materials. These interactions form the foundation for technologies such as lasers, optical fibers, and a wide range of optical devices.

• Interference: Occurs when two light waves converge, resulting in patterns that could be either constructive (increased intensity) or destructive (reduced intensity). The exact nature of interference depends on the phase relationship between the waves, which, as discussed, is influenced by the path difference.
• Diffraction: This refers to the bending of light waves around obstacles and openings. It is observable in situations where the size of the obstacle or opening is comparable to the wavelength of the light. Diffraction effects are critical in many optical applications.

Understanding these principles allows us to harness light for various practical applications and innovations. The equation relating path difference and wavelength to phase shifts is a key mathematical tool used in optics to solve practical problems and design sophisticated devices.