1 P1X: Optics, Waves and Lasers Lectures, Lecture 3: Introduction to wave theory (III) o Principle of superposition: When two or more waves overlap, the resultant displacement at any point at any instant may be found by adding the instantaneous displacements that would be produced at the point by the individual waves if they were alone. Interference and coherence (Y&F 15.6, 35.1): t

2 P1X: Optics, Waves and Lasers Lectures, o Coherence: If two overlapping waves have the same frequency and have a definite constant phase between them then they are said to be coherent waves. In the previous example: o Coherent source: The source S 1 is emitting coherent waves in two dimensions outwards. All crests of the wave are said to be in phase. For example: waves on water

3 P1X: Optics, Waves and Lasers Lectures, o Interference: If two sources S 1 and S 2 emit coherent waves, the resultant pattern will be due to the interference of the overlapping waves according to the principle of superposition. The amplitude of the wave at an arbitrary point will be the sum of the two amplitudes. Point a is equidistant from S 1 and S 2 (  difference  sum of amplitudes Point b is  from S 1 and  from S 2 : waves in phase so sum of amplitudes constructive interference Point c is  from S 1 and  from S 2 : both amplitudes cancel out destructive interference

4 P1X: Optics, Waves and Lasers Lectures, o Constructive interference: Both waves are in phase so the two waves add up. CONSTRUCTIVE INTERFERENCE y T3T/4 T/2 T/4t - A A T3T/4 T/2 T/4t - A A - 2A 2A T3T/4 T/2 T/4t y T3T/4 T/2 T/4t - A A T3T/4 T/2 T/4t - A A - 2A 2A T3T/4 T/2 T/4t DESTRUCTIVE INTERFERENCE o Destructive interference: Both waves are exactly out of phase so the two waves cancel.

5 P1X: Optics, Waves and Lasers Lectures, o Constructive interference patterns: The locus of all points where one obtains constructive interference (ie. when the crests of the waves align) are called the antinodal curves. o Nodal curves are the pattern of destructive interference. antinodal curves

6 P1X: Optics, Waves and Lasers Lectures, Interference and diffraction of light i) to recognise the observed phenomena of interference and diffraction; ii) to understand Huygen's principle and its application to both geometrical and physical optics; iii) to understand interference, in particular Young's double slit experiment and Lloyd's mirror experiment, and in thin films including Newton's rings experiment; iv) to appreciate the application of the theory of interference in such areas as non-reflective and highly reflective coatings; v) to understand the limitation to resolving power of lenses due to diffraction; vi) to solve simple problems involving interference and diffraction phenomena. Objectives:

7 P1X: Optics, Waves and Lasers Lectures, Physical Optics: wave behaviour of light (Y&F ; 33.7) o Wave nature of light: In geometric optics: light considered as straight line rays Light undergoes certain phenomena that cannot be explained simply by light travelling in straight lines Light not only reflects on mirrors but also refracts in glass, water and other media Interference phenomena of light are observable every day: oil spots, soap bubbles show multi-coloured patterns Interference and diffraction of light (I) Thin film of oil illuminated by white light

8 P1X: Optics, Waves and Lasers Lectures, o Wave nature of light (cont.): Diffraction phenomena are also common for waves: an example is that sound bends around corners, due to the wave behaviour on the edges of objects. Diffraction is also visible with light on edges of sharp objects (for example, photograph of razor) o Physical optics is the study of light, taking into account its wave behaviour. Initiated by the Dutch scientist Christian Huygens in 1678 who proposed the wave theory of light, in opposition to Newton who believed light was made out of corpuscles (particles). We now know they were both right!

9 P1X: Optics, Waves and Lasers Lectures, Huygens’ Principle (Y&F 33.7) o Wave character of light: the Dutch scientist Christian Huygens believed in the wave character of light and he used this to explain reflection and refraction o Huygens’ Principle: Every point of a wavefront may be considered as the source of secondary wavelets that spread out in all directions with a speed equal to the speed of propagation of the wave. o In the figure, the new wavefront BB’ is constructed by making the surface tangent to the secondary wavelets (envelope of the wavelet), a distance r=vt from the initial wavefront AA’. o The success of Huygens principle is that it explains reflection and refraction.

10 P1X: Optics, Waves and Lasers Lectures, o Reflection by Huygens Principle:  a = incident angle  r = reflected angle Triangles OPA and OQA are equal, therefore  a =  r The reflected angle is equal to the incident angle

11 P1X: Optics, Waves and Lasers Lectures, o Refraction by Huygens Principle:  a = incident angle  b = refracted angle Snell’s law Refractive index:

12 P1X: Optics, Waves and Lasers Lectures, o Wavelength in the medium: The wavelength in the medium will depend on the refractive index. Frequency is fixed (colour of light) but speed of light in the medium changes depending on the medium. In medium a: Since: then: Therefore, at an interface between two media: Example: Light of wavelength 550 nm in air is incident onto a glass plate of refractive index n= What is the wavelength of light in the plate? n a =1 and n b =1.52: