Question

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

Short answer

Answer: The minimum path length difference needed to cause a phase shift of π/4 in light with a wavelength of 700 nm is 87.5 nm.

Step-by-step solution

Write down the formula for phase shift, path length difference, and wavelength

The formula relating phase shift (\(\Delta \phi\)), path length difference (\(\Delta L\)), and wavelength (\(\lambda\)) can be written as: \(\Delta \phi = \frac{2\pi}{\lambda} \Delta L\)

Plug in the given values

Now we can plug in the given values for phase shift (\(\pi/4\)) and wavelength (\(700\, \mathrm{nm}\)): \(\frac{\pi}{4} = \frac{2\pi}{700\, \mathrm{nm}} \Delta L\)

Solve for the path length difference

Next, we need to solve for the path length difference (\(\Delta L\)): \(\Delta L = \frac{\pi}{4} \times \frac{700\, \mathrm{nm}}{2\pi}\) Notice that the \(\pi\) term cancels out: \(\Delta L = \frac{1}{4} \times \frac{700\, \mathrm{nm}}{2}\)

Calculate the value of the path length difference

Finally, we can calculate the value of the path length difference: \(\Delta L = \frac{1}{4} \times \frac{700\, \mathrm{nm}}{2} = \frac{1}{4} \times 350\, \mathrm{nm} = 87.5\, \mathrm{nm}\) So, the minimum path length difference needed to cause a phase shift of \(\pi/4\) in light of wavelength \(700\, \mathrm{nm}\) is \(87.5\, \mathrm{nm}\).