Question

A laser beam with wavelength \(633 \mathrm{nm}\) is split into two beams by a beam splitter. One beam goes to Mirror \(1,\) a distance \(L\) from the beam splitter, and returns to the beam splitter, while the other beam goes to Mirror \(2,\) a distance \(L+\Delta x\) from the beam splitter, and returns to the same beam splitter. The beams then recombine and go to a detector together. If \(L=1.00000 \mathrm{~m}\) and \(\Delta x=1.00 \mathrm{~mm},\) which best describes the kind of interference at the detector? (Hint: To doublecheck your answer, you may need to use a formula that was originally intended for combining two beams in a different geometry, but which still is applicable here.) a) purely constructive b) purely destructive c) mostly constructive d) mostly destructive e) neither constructive nor destructive

Short answer

Answer: (c) mostly constructive

Step-by-step solution

Understand the Concepts of Constructive and Destructive Interference

Constructive interference occurs when two waves add up to create a larger wave. It occurs when the difference in the path lengths is an integer number of wavelengths. Destructive interference occurs when two waves cancel each other out, resulting in a smaller wave or zero amplitude. It happens when the difference in the path lengths is an odd-integer (odd multiple) of half wavelengths.

Calculate the Path Length Difference

Calculate the path length difference ΔL to determine the difference in distance between the two beams. ΔL = 2Δx = 2 × 1.00 mm = 2.00 mm

Convert All Distances to the Same Unit and Calculate Number of Wavelengths in ΔL

Convert all distances to the same unit (meters) and calculate the number of wavelengths in the path length difference. lambda = 633 nm = 633 x 10^{-9} m ΔL = 2.00 mm = 2.00 x 10^{-3} m Number of wavelengths in ΔL = ΔL / lambda = (2.00 × 10^{-3}) / (633 × 10^{-9}) = 3162.39

Determine Interference Type

Now, we'll determine the interference type based on the number of wavelengths in the path length difference. Since 3162.39 is very close to an integer value (3162), the interference will be mostly constructive. Therefore, the correct answer is (c) mostly constructive.

Key concepts

Understanding Constructive Interference

Imagine two waves meeting and joining forces to form a larger wave. This is what constructive interference is all about. It happens when the crests (high points) and troughs (low points) of two waves align perfectly. In this way, they add up to make a wave with even bigger crests and deeper troughs.

For constructive interference to occur, the difference in the path lengths of these waves has to be a whole number of wavelengths. Think of it as waves taking an identical route and arriving together in sync.

When they overlap perfectly, their energies combine, leading to an increase in wave amplitude. This results in a brighter or more pronounced outcome, depending on the context, like in light or sound waves.

• Two waves traveling different paths.
• Arriving in phase (crests align with crests, troughs align with troughs).
• Path length difference equals integer multiples of wavelength.
• Result: Intensified wave.

Understanding Destructive Interference

Destructive interference works like a wave's worst enemy. This is because two waves meet in such a way that they cancel each other out. Imagine one wave's crest aligning with the other's trough, leading to a smaller wave or even no wave at all.

For destructive interference to work, the path length difference must be an odd-number multiple of half the wavelength. When this happens, waves meet out of sync, and their energies cancel each other.

Destructive interference is crucial in applications where noise reduction or signal cancellation is needed. It's like turning the volume down on noise by clashing it with its opposite signal.

• Two waves traveling different paths.
• Arriving out of phase (crests align with troughs).
• Path length difference equals odd multiples of half a wavelength.
• Result: Diminished or null wave.

Exploring the Path Length Difference

Path length difference refers to the additional distance one wave travels compared to another before they meet. It's the core reason behind interference patterns, as it determines how waves interact upon recombination.

In our exercise, the path length difference, (\(\Delta L\)), was calculated by considering the extra distance to Mirror 2 and back again, leading to \(\Delta L = 2\Delta x\), or 2.00 mm.

Understanding the path length difference is vital because it tells us whether the waves will reinforce each other through constructive interference or cancel out via destructive interference.

• Extra distance traveled by one wave.
• Calculates how out of sync two waves are.
• Responsible for dictating interference type.
• Core factor in determining wave interaction.

The Process of Wavelength Calculation

Calculating the wavelength is a necessary step in determining the type of interference. Wavelength is the length of one complete cycle of a wave, measured from crest to crest.

In our example, the wavelength was given as 633 nm (nanometers), which is equivalent to (\(633 \times 10^{-9}\, \text{m}\)). Converting units to meters is essential for accurate calculations.

By dividing the path length difference by the wavelength, we obtain how many wavelengths fit into that additional distance. If this number is close to an integer, constructive interference is likely. If it's nearer to a half-integer, we might expect destructive interference.

• Measure of wave cycle length.
• Essential for interference calculations.
• Converting to uniform units helps in computing paths.
• Dividing path length by wavelength guides interference prediction.