In Fig. 35-51 a, the waves along rays 1 and 2 are initially...

In Fig. $35-51 a$, the waves along rays 1 and 2 are initially in phase, with the same wavelength $\lambda$ in air. Ray 2 goes through a material with length $L$ and index of refraction $n$. The rays are then reflected by mirrors to a common point $P$ on a screen. Suppose that we can vary $L$ from 0 to $2400 \mathrm{~nm}$. Suppose 0 (nm) also that, from $L=0$ to $L_{s}=900 \mathrm{~nm}$, the intensity $I$ of the light at point $P$ Figure 35-52 Problem 87 . varies with $L$ as given in Fig. $35-52$. At what values of $L$ greater than $L_{s}$ is intensity $I$ (a) maximum and (b) zero? (c) What multiple of $\lambda$ gives the phase difference between ray 1 and ray 2 at common point $P$ when $L=1200 \mathrm{~nm}$ ?

In Fig. $35-51 a$, the waves along rays 1 and 2 are initially in phase, with the same wavelength $\lambda$ in air. Ray 2 goes through a material with length $L$ and index of refraction $n$. The rays are then reflected by mirrors to a common point $P$ on a screen. Suppose that we can vary $L$ from 0 to $2400 \mathrm{~nm}$. Suppose 0 (nm) also that, from $L=0$ to $L_{s}=900 \mathrm{~nm}$, the intensity $I$ of the light at point $P$ Figure 35-52 Problem 87 . varies with $L$ as given in Fig. $35-52$. At what values of $L$ greater than $L_{s}$ is intensity $I$ (a) maximum and
(b) zero? (c) What multiple of $\lambda$ gives the phase difference between ray 1 and ray 2 at common point $P$ when $L=1200 \mathrm{~nm}$ ?

Key Concepts

Constructive Interference

Constructive interference occurs when waves meet in phase, meaning their peaks and troughs align, resulting in an increased amplitude. This condition is achieved when the optical path difference is equal to an integer multiple of the wavelength (or equivalently, when the phase difference is a multiple of 2?). This is the principle behind the bright fringes observed in interference experiments.

Phase Difference

The phase difference refers to the relative offset between the phases of two waves at a given point. It arises due to differences in their paths (optical path difference) and is key to determining whether the interference between them will be constructive or destructive. A phase difference that is an integer multiple of 2? leads to constructive interference, while a phase difference of an odd multiple of ? causes destructive interference.

Destructive Interference

Destructive interference takes place when two waves meet out of phase, leading to the cancellation of their amplitudes. This happens when the optical path difference corresponds to an odd multiple of half wavelengths (or equivalently, when the phase difference is an odd multiple of ?). This concept explains the dark fringes or points of zero intensity in interference patterns.

Interference

Interference is the phenomenon that occurs when two or more waves overlap, resulting in a new wave pattern. The nature of the interference—whether constructive (resulting in increased amplitude) or destructive (resulting in decreased or zero amplitude)—depends on the phase relationship of the waves as they combine. This foundational concept is crucial for understanding how light behaves when it meets obstacles or passes through different media.

Optical Path Difference (OPD)

Optical path difference is the difference in the effective distance traveled by two waves, taking into account the refractive index of the medium through which the wave travels. It is determined by multiplying the physical distance by the refractive index, and it directly affects the phase difference between the waves. OPD is central to predicting the outcome of interference patterns in experiments.

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