Wave Optics

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5

Assume the single slit has a width, w

Light is diffracted as it passes through the slit and then propagates to the screen

The key to the calculation of where the fringes occur is Huygen’s principle

All points across the slit act as wave sources

These different waves interfere at the screen

For analysis, divide the slit into two parts

Slide 24

If one point in each part of the slit satisfies the conditions for destructive interference, the waves from all similar sets of points will also interfere destructively

Destructive interference will produce a dark fringe

Single-Slit Analysis: Dark Fringers

Conditions for destructive interference are

w sin θ = ±m λ (m = 1, 2, 3, … )

The negative sign will correspond to a fringe below the center of the screen

Slide 25

Single-Slit Analysis: Bright Fringes

There is no simple formula for the angles at which the bright fringes occur

The intensity on the screen can be calculated by adding up all the Huygens waves

There is a central bright fringe:

The central fringe is called the central maximum

The central fringe is about 20 times more intense than the bright fringes on either side

The width of the central bright fringe is approximately the angular separation of the first dark fringes on either sideThe full angular width of the central bright fringe = 2 λ / w

If the slit is much wider than the wavelength, the light beam essentially passes straight through the slit with almost no effect from diffraction

Slide 26

Diffraction Gratings

An arrangement of many slits is called a diffraction grating

Assumptions

The slits are narrow

Each one produces a single outgoing wave

The screen is very far away

If the slit-to-slit spacing is d, then the path length difference for the rays from two adjacent slits is

ΔL = d sinθ

If ΔL is equal to an integral number of wavelengths, constructive interference occurs

For a bright fringe, d sin θ = m λ (m = 0, ±1, ±2, …)

Slide 27

The condition for bright fringes from a diffraction grating is identical to the condition for constructive interference from a double slit

The overall intensity pattern depends on the number of slits

The larger the number of slits, the narrower the peaks

Slide 28

Example 25 .6 Diffraction of Light by a Grating

A diffraction experiment is carried out with a grating. Using light from a red laser (λ = 630 nm), the diffraction fringes are separated by h=0.15 m on a screen that is W=2.0 m from the grating. Find the spacing d between slits in the grating.