Questions: Optical Path Length and Its Role in Interference

Question 1

Two rays of light begin in phase and travel identical geometric distances of 3 cm. Ray A passes through air (n = 1.0); Ray B passes through glass (n = 1.5). What happens when they recombine?

• A. They interfere constructively because the geometric path lengths are equal
• B. They have a phase difference because Ray B accumulated more phase than Ray A
• C. Ray A accumulates more phase because air offers less resistance to the wavefront
• D. They interfere destructively because glass absorbs part of the wave amplitude

Answer: (B)

Interference depends on optical path length (OPL = n × geometric distance), not geometric distance alone. Ray B has OPL = 1.5 × 3 cm = 4.5 cm; Ray A has OPL = 1.0 × 3 cm = 3.0 cm. The optical path difference is 1.5 cm, which produces a phase difference of 2π × 1.5 cm / λ₀. Whether this causes constructive or destructive interference depends on the wavelength, but the key point is that equal geometric paths do not mean equal phase — option A is the classic misconception this topic targets.

Question 2

An anti-reflection lens coating works by causing destructive interference between light reflecting off its front and back surfaces. For this to produce perfect destructive interference (ignoring phase shifts from reflection), what must be true of the optical path difference between the two reflected rays?

• A. The OPD must equal zero, so the rays cancel by being exactly in phase
• B. The OPD must equal λ₀/2, so the rays are half a wavelength out of phase
• C. The geometric thickness of the coating must equal λ₀, the free-space wavelength
• D. The coating's refractive index must match that of the lens glass exactly

Answer: (B)

Destructive interference requires a phase difference of π (half a cycle), which corresponds to an optical path difference of λ₀/2. The coating thickness is chosen so that the ray reflecting off the back surface travels an extra OPL of λ₀/2 compared to the ray reflecting off the front surface — placing the two reflected waves half a wavelength apart and causing them to cancel. The geometric thickness needed is λ_coating/4 = λ₀/(4n), not λ₀ itself.

Question 3

Two light rays that travel the same geometric distance will generally arrive at the same phase.

• T. True
• F. False

Answer: False

Phase accumulation depends on optical path length (OPL = n × geometric path), not geometric distance alone. If the two rays pass through media with different refractive indices, they accumulate different amounts of phase even over identical geometric distances. Equal geometric paths guarantee equal phase only when both rays travel through media with the same refractive index (e.g., both in vacuum).

Question 4

Inside a medium with refractive index n = 2, light completes twice as many wave cycles per centimeter compared to traveling the same geometric distance in vacuum.

• T. True
• F. False

Answer: True

Inside a medium with n = 2, the wavelength shortens to λ_medium = λ₀/n = λ₀/2. Because there are twice as many wavelengths packed per unit length, the light completes twice as many oscillation cycles per centimeter. This is exactly what optical path length accounts for: OPL = 2 × geometric distance captures the fact that the wave accumulates phase twice as fast inside this medium.

Question 5

Why is optical path length, rather than geometric path length, the physically meaningful quantity for predicting whether two rays will interfere constructively or destructively?

Model answer: Phase accumulation — not distance traveled — determines interference. A wave's phase advances by 2π for every wavelength it travels. Inside a medium, the wavelength shortens to λ₀/n, so the wave completes more cycles per unit length. Optical path length (OPL = n × geometric distance) counts the equivalent vacuum distance that would produce the same phase advance, making it the common currency for comparing rays that have traveled through different media. Two rays interfere constructively when their OPLs differ by an integer multiple of λ₀ and destructively when they differ by a half-integer multiple — regardless of how the geometric paths compare.

The geometric path is easy to measure but physically misleading when different media are involved. OPL converts all paths to a single frame of reference — vacuum-equivalent length — which directly predicts phase. This is why OPL is the central quantity in all interference calculations: thin films, interferometers, and anti-reflection coatings are all analyzed by computing OPL for each ray path and finding their difference.