Coherent light rays of wavelength λstrike a pair of slits separated by distance d at an angle θ1

step 1 Okay, so a path difference for any given ray is given by path difference delta, it's given by sl

Coherent light rays of wavelength λstrike a pair of slits...

Coherent light rays of wavelength $\lambda$ strike a pair of slits separated by distance $d$ at an angle $\theta_{1}$ with respect to the normal to the plane containing the slits as shown in Figure P27.14. The rays leaving the slits make an angle $\theta_{2}$ with respect to the normal, and an interference maximum is formed by those rays on a screen that is a great distance from the slits. Show that the angle $\theta_{2}$ is given by $$\theta_{2}=\sin ^{-1}\left(\sin \theta_{1}-\frac{m \lambda}{d}\right)$$ where $m$ is an integer.

Key Concepts

Double-slit Interference

This concept involves the interference pattern produced when coherent light passes through two narrow, closely spaced slits, resulting in alternating bright and dark fringes on a screen. The pattern arises because the waves originating from the two slits can interfere constructively or destructively, depending on their relative phases.

Optical Path Difference

Optical path difference refers to the difference in the distances traveled by light waves from different paths. In the context of two-slit interference, it is the varying extra distance one wave travels compared to the other, which translates into a phase difference and determines whether the interference will be constructive or destructive.

Constructive Interference Condition

For constructive interference (a bright fringe) to occur, the path difference between the two light rays must equal an integer multiple of the wavelength. This relationship is typically expressed as d(sin ?1 - sin ?2) = m?, where d is the slit separation and m is an integer. This equation forms the basis for predicting the angles at which bright interference maxima occur.

Trigonometric Relations in Interference

Trigonometric concepts are used to relate the geometrical setup of the experiment—specifically the angles of incidence and diffraction—to the interference conditions. The use of sine functions in the interference condition allows one to derive the specific emission angle (?2) for a given incident angle (?1), ultimately leading to the formula involving the inverse sine function.

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