# Lecture 4: Interference and diffraction of light (I)

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Lecture 4: Interference and diffraction of light (I)
Young’s two slit experiment (Y&F 35.2) To observe interference of light one needs to set-up two independent, coherent, monochromatic (same frequency) light sources. Coherent light sources are difficult to make since light is emitted by the random agitation of atoms (we will see later on that lasers can be sources of coherent light). To achieve two coherent sources of light, we can use a common source and split it such that the light emerges from two secondary sources, so the relative phase between them is always the same. S1 S0 S2 P1X: Optics, Waves and Lasers Lectures,

P1X: Optics, Waves and Lasers Lectures, 2005-06.
Young’s two slit experiment (Cont.) Young (1800) set up an experiment to demonstrate the wave nature of light. A monochromatic beam of light is incident on a single slit S0. This acts as a source of wavefronts onto two slits S1 and S2. The coherent waves interfere with each other forming a pattern of light and dark bands on a screen some distance from the two slits. P1X: Optics, Waves and Lasers Lectures,

P1X: Optics, Waves and Lasers Lectures, 2005-06.
Geometry of Young’s two slit experiment The triangle S1S2Q means that: S1Q = d sin q If the screen is at a distance R which is much larger than the separation d of the two slits (R could be a few metres and d a few millimetres) then r1 and r2 are nearly parallel to each other and q is small: y r1 S1 S2 R q r2 Intensity d d sin q = r2- r1 Q P P1X: Optics, Waves and Lasers Lectures,

P1X: Optics, Waves and Lasers Lectures, 2005-06.
Constructive interference: bright fringes Destructive interference: dark fringes P1X: Optics, Waves and Lasers Lectures,

P1X: Optics, Waves and Lasers Lectures, 2005-06.
Example 35-1 (Y&F): In a two slit interference experiment, the slits are 0.20 mm apart, and the screen at a distance of 1.0 m. The third bright fringe (not counting the central bright fringe) is displaced by 7.5 mm. Find the wavelength of light used. Third fringe: m=3, R=1.0 m, d=0.2 mm, R>>d Another possibility: m= ym= -7.5 mm P1X: Optics, Waves and Lasers Lectures,

P1X: Optics, Waves and Lasers Lectures, 2005-06.
Example L61 (Example book): Light from a source containing 2 wavelengths of 567 nm and 486 nm illuminates a double slit arrangement with d=1.0 mm and R=1.50 m. At what distance from the central fringe on the screen will a bright fringe from one interference pattern coincide with the bright fringe from the other interference pattern. Bright fringes: P1X: Optics, Waves and Lasers Lectures,

P1X: Optics, Waves and Lasers Lectures, 2005-06.
Intensity in interference patterns (Y&F, 35.3) The waves associated with light are called electromagnetic waves. Unlike the other waves we have encountered, they don’t need a medium to propagate. They can propagate in vacuum. In the two slit experiment we have an electric field coming from S1: and another coming from S2: (frequency and amplitude are the same since the two sources are coherent). Principle of superposition: P.1463 Y&F Since: Trigonometric functions P1X: Optics, Waves and Lasers Lectures,

P1X: Optics, Waves and Lasers Lectures, 2005-06.
Since: Then: Phase difference between two waves: r2-r1 is called the path difference. New wave: P1X: Optics, Waves and Lasers Lectures,

P1X: Optics, Waves and Lasers Lectures, 2005-06.
New wave has angular frequency w and wave number k but the new amplitude is: with: EP 2p Df -2E 3p 4p p 5p 2E P1X: Optics, Waves and Lasers Lectures,

P1X: Optics, Waves and Lasers Lectures, 2005-06.
Intensity of light is the average power per unit area (energy per unit area per unit time): The intensity of light is proportional to the square of the amplitude of the electric field: I 2p Df 3p 4p p 5p I0 P1X: Optics, Waves and Lasers Lectures,

P1X: Optics, Waves and Lasers Lectures, 2005-06.
Lloyd’s mirror Lloyd’s mirror is just like Young’s two slit experiment but with only one source of light and a mirror used to create a second virtual source of light. a y r 2 S1 R I d MIRROR r1 r2 S2 The source interferes with its own image in the mirror. P1X: Optics, Waves and Lasers Lectures,

P1X: Optics, Waves and Lasers Lectures, 2005-06.
There is a phase change of p radians due to the reflection of the mirror (equivalent to a path difference of l/2). The constructive and destructive conditions are reversed with respect to Young’s experiment and there is a dark fringe next to the mirror. Constructive interference: bright fringes Destructive interference: dark fringes EP 2p Df p BEFORE MIRROR EP 2p Df p AFTER MIRROR P1X: Optics, Waves and Lasers Lectures,

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