P1X: Optics, Waves and Lasers Lectures, 2005-06. 1

Lecture 3: Introduction to wave theory Lecture 3: Introduction to wave theory (III)(III)

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):

)sin(),(

)sin(),(

2222

1111

xktAtxy

xktAtxy

)sin()sin(

),(),(),(

222111

21

xktAxktA

txytxytxy

t

P1X: Optics, Waves and Lasers Lectures, 2005-06. 2

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:

)sin()sin(),(),(),( 2121 kxtAkxtAtxytxytxy

kkk 21;21

Coherent source:• The source S1 is emitting coherent waves in two dimensions outwards.• All crests of the wave are said to be in phase.• For example: waves on water

P1X: Optics, Waves and Lasers Lectures, 2005-06. 3

Interference:• If two sources S1 and S2 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 S1 and S2 (difference sum of amplitudes• Point b is from S1 and from S2: waves in phase so sum of amplitudes constructive interference• Point c is from S1 and from S2: both amplitudes cancel out destructive interference

P1X: Optics, Waves and Lasers Lectures, 2005-06. 4

Constructive interference:Both waves are in phase so the two waves add up.

mrr 12

CONSTRUCTIVEINTERFERENCEy

T3T/4T/2T/4 t

- A

A

T3T/4T/2T/4 t

- A

A

- 2A

2A

T3T/4T/2T/4 t

y

T3T/4T/2T/4 t

- A

A

T3T/4T/2T/4 t

- A

A

- 2A

2A

T3T/4T/2T/4 t

DESTRUCTIVEINTERFERENCE

,2,1,0 m

2

112 mrr

,2,1,0 m

Destructive interference:Both waves are exactly out of phase so the two waves cancel.

P1X: Optics, Waves and Lasers Lectures, 2005-06. 5

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. Nodal curves are the pattern of destructive interference.

antinodal curves

P1X: Optics, Waves and Lasers Lectures, 2005-06. 6

Interference and diffraction of lightInterference and diffraction of light

i) to recognise the observed phenomena of interference anddiffraction;ii) to understand Huygen's principle and its application to bothgeometrical and physical optics;iii) to understand interference, in particular Young's double slitexperiment and Lloyd's mirror experiment, and in thin films includingNewton's rings experiment;iv) to appreciate the application of the theory of interference in suchareas as non-reflective and highly reflective coatings;v) to understand the limitation to resolving power of lenses due todiffraction;vi) to solve simple problems involving interference and diffractionphenomena.

Objectives:

P1X: Optics, Waves and Lasers Lectures, 2005-06. 7

Physical Optics: wave behaviour of light (Y&F 35.1-2; 33.7)

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)Interference and diffraction of light (I)

Thin film of oil illuminated by white light

P1X: Optics, Waves and Lasers Lectures, 2005-06. 8

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)

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!

P1X: Optics, Waves and Lasers Lectures, 2005-06. 9

Huygens’ Principle (Y&F 33.7)

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 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. 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’.

The success of Huygens principle is that it explains reflection and refraction.

P1X: Optics, Waves and Lasers Lectures, 2005-06. 10

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

P1X: Optics, Waves and Lasers Lectures, 2005-06. 11

Refraction by Huygens Principle:

a= incident angle b= refracted angle

OA

tv

OA

OQ aa sin

OA

tv

OA

AB bb sin

tvtv b

b

a

a sinsin

aa v

cn

bb v

cn

bbaa nn sinsin Snell’s law

Refractive index:

P1X: Optics, Waves and Lasers Lectures, 2005-06. 12

aa v

cn

aa fv

bbaa nn

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: a

a f

cn

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= 1.52. What is the wavelength of light in the plate?

na=1 and nb=1.52: nmnm

nn

b

abbba 8.361

52.1

550