CHAPTER 35 : THE NATURE OF LIGHT AND THE LAWS OF GEOMETRIC OPTICS

35.1) The Nature of Light

Light – A stream of paticles that either was emitted by the object being viewed or emanated from the eyes of the viewer

Light might be some sort of wave motion

Diffraction

Young – light rays interfere with each other

The speed of light in liquids is less than its speed in air

Light was a form of high-frequency electromagnetic wave

Photoelectric effect

Quantization - photons

E = hf (35.1)h = 6.63 x 10-34 Js = Planck’s constant

Dual nature – light exhibits the characteristics of a wave in some situations and the characteristics of a particle in other situations

35.2) Measurements of the Speed of Light

• Light travels at such a high speed (c = 2.9979 x 108 m/s)

Galileo

Attempted to measure the speed of light by positioning two observers in towers separated by approximately

10 km – each observer carried a shuttered lantern – not succeed

Roemer’s MethodIn 1675 – the Danish astronomer Ole Roemer (1644 – 1710) made the first successful estimate of the speed of

light – involved astronomical observations of one of the moons of

Jupiter, Io.

Fizeau’s Method• Armand H. L. Fizeau (1819-1896)

– the successful method for measuring the speed of light by means of purely terestrial techniques – in 1849.

• Figure (35.2) – represents a simplified diagram of Fizeau’s apparatus.

• The basic procedure is to measure the total time it takes light to travel from some point to a distant mirror and back.

35.3) The Ray Approximation in Geometric Optics

Geometric Optics

Involves the study of the propagation of light, with the assumption that light travels in a fixed direction in a straight line as it passes through a uniform medium and changes its

direction when it meets the surface of a different medium or if the optical properties of the medium are nonuniform in either

space or time.

Ray approximation

The rays of a given wave are straight lines perpendicular to the wave fronts (Figure (35.3)) for a plane wave – the wave

moving through a medium travels in a straight line in the direction of its rays.

Figure (35.4b) – if the diameter of the opening is of the order of the wavelength - waves spread out from the opening in all

directions

Figure (34.54c) – if the opening is much smaller than the wavelength – the opening can be approximated as a point

source of waves

(Figure (35.4a)) – If the wave meets a barrier in which there is a circular opening whose diameter is much larger than the

wavelength– the wave emerging from the opening continues to move in a straight line (apart from some small edge effects) –

the ray approximatin is valid

Similar effects – when waves encounter an opaque object of dimension d – when << d, the object casts a sharp shadow

Ray approximation – for the study of mirrors, lenses, prisms, and associated optical instruments – such as telescopes,

cameras, and eyeglasses

35.4) Reflection

When a light ray traveling in one medium encounters a boundary with another medium – part of the incident light is

reflected

Figure (35.5)

• Figure (35.5a) – several rays of a beam of light incident on a smooth, mirror-like, reflecting surface.

• The reflected rays are parallel to each other

• The direction of a reflected ray is in the plane perpendicular to the reflecting surface that contains the incident ray.

• Reflection of light from such a smooth surface = specular reflection

• Figure (35.5b) – if the reflecting surface is rough- the surface reflects the rays not a s a parallel set but in various directions.

• Reflection from any rough surface = diffuse reflection

A surface behaves as a smooth surface as long as the surface variations are much smaller than the wavelength of the incident light

Concern only with specular reflection and use the term reflection to mean specular

reflection

Consider a light ray traveling in air and incident at an angle on a flat, smooth surface – Figure (35.6)

Incident rayNormal

Reflected ray

1'1

Figure (35.6)

• The incident and reflected rays make angles 1 and 1´

respectively, where the angles are measured from the normal to the rays.

• The normal is a line drawn perpendicular to the surface at the point where the incident ray strikes.

• The angle of reflection equals the angle of incidence :

1'1 (35.2)Law of reflection

35.5) Refraction

When a ray of light traveling through a transparent medium encounters a boundary leading into another transparent

medium – Figure (35.9)

Part of the ray is reflected and part enters the second medium

The part that enters the second medium is bent at the boundary and is said to be refracted

The incident ray, the reflected ray, and the refracted ray all lie in the same plane

The angle of refraction 2 (Figure (35.9)) – depends on the properties of the two media and on the angle of incidence

through the relationship :

constantv

v

sin

sin

1

2

1

2

(35.3)

Where v1 is the speed of light in the first medium and v2 is the speed of light in the second medium

The path of a light ray through a refracting surface is reversible – For example, the ray shown in Fig. (35.9a) travels from point a to point B. If the ray originated at B, it would travel to the left along line BA to reach

point A, and the reflected part would point downward and to the left in the glass.

Form Equation (35.3) – can infer that when light moves from a material in which its speed is high to a material in which its

speed is lower

Figure (35.10a) – The angle of refraction 2 is less than the angle of incidence 1

The beam is bent toward the normal

If the ray moves from a material in which it moves slowly to a material in whcich it moves more

rapidly – Figure (35.10b)

2 is greater than 1

The beam is bent away from the normal

Index of Refraction

The speed of light in any material is less than its speed in vacuum

Light travels at its maximum speed in vacuum

The index of refraction n of a medium = to be the ratio :

v

c

medium ain light of Speed

in vacuumlight of Speedn (35.4)

The index of refraction is a dimensionless number greater than unity because v is always less than c

n is equal to unity for vacuum

Table (35.1) The indices of refraction for various substances

As light travels from one medium to another – its frequency does not change but its wavelength does

Why ?

Consider Figure (35.13)

Wave fronts pass an observer at point A in medium 1 with a certain frequency and are incident on the

boundary between medium 1 and medium 2

The frequency with which the wave fronts pass an observer at point B in medium 2 must equal the frequency at

which they pass point A

The frequency must be a constant as a light ray passes from one medium into another.

Because the relationship v = f (Equation 16.14) must be valid in both media and because f1 = f2 = f :

11 fv 22 fv and (35.5)

Because v1 v2 – it follows that 1 2

A relationship between index of refraction and wavelength – by dividing the first Equation (35.5) by the second and then using Equation (35.4) :

2

1

2

1

2

1

2

1

n

n

n/c

n/c

v

v

(35.6)

2211 nn That gives :

If medium 1 is vacuum, or for all practical purposes air, then n1 = 1

It follows from Equation (35.6) – the index of refraction of any medium can be expressed as the ratio :

n

n

(35.7)Where = the wavelength of light in vacuum and n = the wavelength in the medium whose index of refraction is n

n > 1, n <

Replace the v2/v1 term in Eq. (35.3) with n1/n2 from Eq. (35.6) :

2211 sinnsinn (35.8)

35.6) Huygens’s Principle

Develop the laws of reflection and refraction by using a geometric method proposed by Huygens in 1678

Huygens assumed that light is some form of wave motion rather than a stream of particles

Huygens’s principle – using knowledge of an earlier wave front to determine the position of a new wave front at some instant

Huygens’s principle = all points on a given wave front are taken as point sources for the production of spherical secondary

waves, called wavelets, which propagate outward through a medium 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

Consider a plane wave moving through free space – (Figure (35.16a)

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

In Huygens’s construction – each point on this wave front is considered a point source – only three points on AA’ are shown

With these points as sources for the wavelets – we draw circles, each of radius ct (where c = speed of light in free space and t = the time of propagation from one wave front to the next

The surface drawn tangent to these wavelets is the plane BB’, which is parallel to AA’

Figure (35.16b) – shows Huygens’s construction for a spherical wave

Figure (35.16)

Huygens’s Principle Applied to Reflection and Refraction

Figure (35.18)

• Figure (35.18a) – the law of reflection.

• The line AA’ represents a wave front of the incident light.

• As ray 3 travels from A’ to C, ray 1 reflects from A and produces a spherical wavelet of radius AD.

• Radius of a Huygens wavelet is ct.

• Because the two wavelets having radii A’C and AD are in the same medium – they have the same speed c; therefore A’C = AD.

• The spherival wavelet centered at B has spread only half as far as the one centered at A because ray 2 strikes the surface later than ray 1 does.

• From huygens’s principle – the reflected wave front is CD, a line tangent to all the outgoing spherical wavelets.

• The remainder of our analysis depends on geometry – Figure (35.18b)

• The right triangles ADC and AA’C are congruent because they have the same hypotenuse AC and because AD = A’C.

• From Figure (35.18b) :

• Thus :

AC

C'Asin 1 and

AC

ADsin '

1

'11 sinsin

'11 Law of reflection

Use Huygens’s principle and Figure (35.19) – to derive Snell’s law of refraction

In the time interval t, ray 1 moves from A to B and ray 2 moves from A’ to C

The radius of the outgoing spherical wavelet centered at A is equal to v2 t

The distance A’C is equal to v1 t

Geometric considerations show that angle A’AC equals 1 and that angle ACB equals 2

From tiangles AA’C and ACB :

AC

tvsin 1

1

and

AC

tvsin 2

2

If we divide the first equation by the second :

2

1

2

1

v

v

sin

sin

But from Equation (35.4), v1 = c/n1 and v2 = c/n2 :

1

2

2

1

2

1

n

n

n/c

n/c

sin

sin

2211 sinnsinn Snell’s law of refraction

35.7) Dispersion and Prisms

• An important property of the index of refraction is that, for a given material, the index varies with the wavelength of the light passing through the material – Figure (35.20)

• This behavior is called dispersion

• Because n is a function of wavelength, Snell’s law of refraction indicates that light of different wavelengths is bent at different angles when incident on a refracting material.

• Figure (35.20) – the index of refraction generally decreases with increasing wavelength.

• Blue light bends more than red light does when passing into a refracting material.

Dispersion

• Consider what happens when light strikes a prism – Figure (35.21).

• A ray of single-wavelength light incident on the prism from the left emerges refracted from its original direction of travel by an angle = the angle of deviation.

• Suppose that a beam of white light (a combination of all visible wavelengths) is incident on a prism – Figure (35.22).

• The rays that emerge spread out in a series of colors known as the visible spectrum.

• These colors, in order of decreasing wavelength, are, red, orange, yellow, green, blue, and violet.

• The angle of deviation depends on wavelength.

• Violet light deviates the most, red the least, and the remaining colors in the visible spectrum fall between these extremes.

• Newton – each color has a particular angle of deviation and that the colors can be recombined to form the original white light.

• Formation of a rainbow.

35.8) Total Internal Reflection

• Total internal reflection – when light attempts to move from a medium having a given index of refraction to one having a lower index of refraction.

• Consider a light beam traveling in medium 1 and meeting the boundary between medium 1 and medium 2 – where n1 is greater than n2 (Figure (35.27a)).

• Various possible directions of the beam are indicated by rays 1 through 5.

• The refracted rays are bent away from the normal because n1 is greater than n2.

• At some particular angle of incidence c , called the critical angle, the refracted light ray moves parallel to the boundary so that 2 = 90o (ray 4 in Figure (35.27a))

• For angles of incidence greater than c , the beam is entirely reflected at the boundary – ray 5 in Figure (35.27a).

• This ray is reflected at the boundary as it strikes the perfectly reflecting surface.

• This ray, and all those like it, obey the law of reflection; that is , fro these rays, the angle of incidence equals the angle of reflection.

• Snell’s law of refraction – to find the critical angle.

• When 1 = c , 2 = 90o and Equation (35.8) gives :

2o

2c1 n90sinnsinn

1

2c n

nsin (for n1 > n2) (35.10)

Critical angle for total internal reflection

Only when n1 is greater than n2

Total internal reflection occurs only when light moves from a medium of a given index of refraction to a medium

of lower index of refraction

If n1 were less than n2 , Equation (35.10) would give sin c > 1 – meaningless result because the sine of an angle

can never be greater than unity

The critical angle for total internal reflection is small when n1 is considerably greater than n2

A prism and total internal reflection alter the direction of travel of a light beam

Two possibilities are illustrated in Figure (35.29a) and (35.29b)

Figure (35.29)

A common application of total internal reflection – a submarine periscope

Fiber Optics

• Application of total internal reflection is the use of glass or transparent plastic rods to “pipe” light from one place to another.

• Figure (35.31) – light is confined to traveling within a rod, even around curves, as the result of successive internal reflections.

• An optical transmission line, images can be transferred from one point to another = fiber optics.

Figure (35.31)

35.9) Fermat’s Principle

• Pierre de Fermat (1601 – 1665) developed a general principle that can be used to determine light paths.

• Fermat’s principle = when a light ray travels between any two points, its path is the one that requires the least time.

• The paths of light rays traveling in a homogeneous medium are straight lines because a straight line is the shortest distance between two points.

How Fermat’s principle can be used to derive Snell’s law of refraction.

• Suppose that a light ray is to travel from point P in medium 1 to point Q in medium 2 (Figure (35.32)), where P and Q are at perpendicular distances a and b, respectively, from the interface.

Figure (35.32)

• The speed of light is c/n1 in medium 1 and c/n2 in medium 2.

• Using the geometry of Figure (35.32) – the time it takes the ray to travel from P to Q is :

• To obtain the value of x for which t has its minimum value – take the derivative of t with respect to x and set the derivative equal to zero :

• or

2

22

1

22

2

2

1

1

n/c

)xd(b

n/c

xa

v

r

v

rt

(35.11)

0])xd(b[c

)xd(n

)xa(c

xn

])xd(b[

)1)(xd(2

2

1

c

n

)xa(

x2

2

1

c

n

)xd(bdx

d

c

nxa

dx

d

c

n

dx

dt

2/1222

2/1221

2/1222

2/1221

222221

2/1222

2/1221

])xd(b[

)xd(n

)xa(

xn

(35.12)

• From Figure (35.32) :

• Substituting into Equation (35.12) :

2/1221 )xa(

xsin

2/1222 ])xd(b[

xdsin

2211 sinnsinn Snell’s law of refraction