Chapter 24Wave Optics

General Physics

Review – optical elementsReview – optical elements

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Optical Optical equationsequations

Sign conventionsSign conventionsp (+) real on the sidep (+) real on the side

light goes in light goes inq (+) real on the sideq (+) real on the side light comes outlight comes outf (+) for focusing opticsf (+) for focusing opticsR check sign of fR check sign of fM (+) for upright imagesM (+) for upright images

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InterferenceInterferenceSections 1 – 3Sections 1 – 3

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Wave OpticsThe wave nature of light explains

various phenomenaInterferenceDiffractionPolarization

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Principle of SuperpositionPrinciple of SuperpositionTwo waves travelling in opposite Two waves travelling in opposite

directions will pass straight through directions will pass straight through each othereach other

When the waves are on top of each When the waves are on top of each other, the two amplitudes add up to other, the two amplitudes add up to get the total amplitudeget the total amplitude

This is the cause of interference This is the cause of interference effectseffects

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Huygen’s Principle Christian Huygens (1629 – 1695) assumed that light

is a form of wave motion rather than a stream of particles

Huygens’ Principle is a geometric construction for determining the position of a new wave at some point based on the knowledge of the wave front that preceded it

All points on a given wave front are taken as point sources for the production of spherical secondary waves, called wavelets, which propagate in the forward direction with speeds characteristic of waves in that medium After some time has elapsed, the new position of the wave

front is the surface tangent to the wavelets

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Huygens’ Construction for a Plane Wave

At t = 0, the wave front is indicated by the plane AA’

The points are representative sources for the wavelets

After the wavelets have moved a distance cΔt, a new plane BB’ can be drawn tangent to the wavefronts

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Huygens’ Construction for a Spherical Wave

The inner arc represents part of the spherical wave

The points are representative points where wavelets are propagated

The new wavefront is tangent at each point to the wavelet

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Diffraction Huygens’ principle

requires that the waves spread out after they pass through narrow slits

This spreading out of light from its initial line of travel is called diffraction In general, diffraction

occurs when waves pass through small openings, around obstacles or by sharp edges

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Interference Light waves interfere

with each other much like water waves do

All interference associated with light waves arises when the electromagnetic fields that constitute the individual waves combine

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Conditions for InterferenceFor sustained interference between two

sources of light to be observed, there are two conditions which must be metThe sources must

be coherentThey must maintain

a constant phase with respect to each other

The waves must have identical wavelengths

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Producing Coherent Sources Light from a monochromatic (single

wavelength) source is allowed to pass through a narrow slit, So

The light from the single slit is allowed to fall on a screen containing two narrow slits, S1 and S2

The first slit is needed to insure the light comes from a tiny region of the source which is coherent

The waves emerging from the second screen originate from the same wave front and therefore are always in phase

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Producing Coherent SourcesCurrently, it is common to use a laser

as a coherent sourceThe laser produces an intense,

coherent, monochromatic beam over a width of several millimeters

The laser light can be used to illuminate multiple slits directly

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Young’s Double Slit Experiment

Thomas Young first demonstrated interference in light waves from two sources in 1801

The light from the two slits form a visible pattern on a screen

The pattern consists of a series of bright and dark parallel bands called fringes

Constructive interference occurs where a bright fringe appears

Destructive interference results in a dark fringe

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Resulting Interference Pattern

Active Figure: Young's Double-Slit Experiment

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Interference Patterns Constructive

interference occurs at the center point

The two waves travel the same distance Therefore, the waves

arrive in phase A bright fringe occurs

Path difference is δ = 0

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Interference Patterns, 2 The upper wave travels

one-half of a wavelength farther than the lower wave

The trough of the bottom wave overlaps the crest of the upper wave Therefore, the waves arrive

out of phase This is destructive

interference A dark fringe occurs

Path difference is δ = 1/2λ

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Interference Patterns, 3 The upper wave

travels one wavelength farther than the lower wave Therefore, the waves

arrive in phase Again, constructive

interference results A bright fringe occurs

Path difference is δ = λ

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Interference Equations

The path difference is δ = r2 – r1

If L is much greater than d (L » d) The paths are

approximately parallel

The path difference is found from the smaller triangle to be

δ = r2 – r1 = d sin θ

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Interference Equations, 2 For a bright fringe, produced by

constructive interference, the path difference must be either zero or some integral multiple of the wavelength

δ = d sin θbright = m λm = 0, ±1, ±2, … m is called the order number

When m = 0, it is the zeroth order maximumWhen m = ±1, it is called the first order maximumetc.

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Interference Equations, 3 For a dark fringe, produced by destructive

interference, the path difference must an odd half wavelength

δ = d sin θdark = (m + ½) λm = 0, ±1, ±2, …

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Interference Equations, 4 The positions of the fringes can be

measured vertically from the zeroth order maximum

y = L tan θ L sin θ for bright fringes sin θ = m λ / d for dark fringes sin θ = (m + ½) λ / d

AssumptionsL>>dd>>λ

Approximationθ is small and therefore the approximation tan θ

sin θ can be used

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Interference Equations, final

For bright fringes

For dark fringes

0, 1, 2brightLy m m

d

1 0, 1, 22darkLy m m

d

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Uses for Young’s Double Slit Experiment

Young’s Double Slit Experiment provides a method for measuring the wavelength of the light

This experiment gave the wave model of light a great deal of credibilityIt is inconceivable that particles of light

could cancel each other, as in the case of destructive interference