Chapter 35: Geometric Optics

Review Basic Geometry!

Ray Approximation

• The rays are straight lines perpendicular to the wave fronts

• With the ray approximation, we assume that a wave moving through a medium travels in a straight line in the direction of its rays

Diffraction depends on SLIT WIDTH: the smaller the width, relative to wavelength, the more bending and diffraction.

We will assume that λ<<d , where d is the diameter of the opening. This approximation is good for the study of mirrors, lenses, prisms, etc.

Light Rays: Ignore Diffraction and Interference of waves!

Reflection & Refraction

i rθ θ= 1 1 2 2sin sinn nθ θ=

Following the Reflected and Refracted Rays

•Ray is the incident ray. •Ray is the reflected ray. •Ray is refracted into the lucite. •Ray is internally reflected in the lucite. •Ray is refracted as it enters the air from the lucite. Section 35.5

Law of Reflection • The normal is a line

perpendicular to the surface – It is at the point where the

incident ray strikes the surface

• The incident ray makes an angle of θ1 with the normal

• The reflected ray makes an angle of θ1’ with the normal

Specular Reflection

• Specular reflection is reflection from a smooth surface

• The reflected rays are parallel to each other

• All reflection in this text is assumed to be specular

Diffuse Reflection

• Diffuse reflection is reflection from a rough surface

• The reflected rays travel in a variety of directions

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

Law of Reflection

i rθ θ=

Why are most materials Opaque? (Opaque – Can’t see through)

They absorb light without re-emitting it. Vibrations given by the light to their atoms and molecules are turned into random kinetic energy – they become slightly warmer.

Opacity: Mirrors Free electrons in opaque reflective surfaces

can vibrate, absorb & re-emit at any frequency.

How many times will the incident beam shown be reflected by each of the parallel

mirrors?

Refraction: Bending Light into Focus

Refraction: Bending of Light Transmitted through Materials

Light Bends because it Slows Down.

Atoms are Optical Tuning Forks Light slows down as it travels through

glass because it takes time to be absorbed and re-emitted.

Light Slows Down in Materials Light Bends Toward the Normal when going from a medium of lower refractive index to one that has a

higher refractive index and visa versa.

lower n

higher n

Index of Refraction

cnv

=

Vacuum: 1 Water: 1.33 Glass: 1.46 Diamond: 2.4

1n ≥

The Index of Refraction • Refraction: Light Bends in

Transmission • The speed of light in any

material is less than its speed in vacuum

• The index of refraction, n, of a medium can be defined as ≡ = =

speed of light in a vacuumnspeed of light in a medium n

c λv λ

in vacuumin a mediumn

λ λnλ λ

=

• For a vacuum, n = 1 – We assume n = 1 for air

also • For other media, n > 1 • n is a dimensionless number

greater than unity, not necessarily an integer

Some Indices of Refraction

Frequency Doesn’t Change!

• As light travels from one medium to another, its frequency does not change – Both the wave speed and

the wavelength do change – The wavefronts do not

pile up, nor are created or destroyed at the boundary, so ƒ must stay the same

Snell’s Law of Refraction

1 1 2 2sin sinn nθ θ=Angles are always measured from the normal.

Snell’s Law – Example

Light is refracted into a crown glass slab. n1 = 1.00 and n2 = 1.52

If θ1 = 30.0o, θ2 = ? θ2 = sin-1(n1 / n2) sin θ1 = 19.2o The ray bends toward the normal,

as expected because n2 > n1

PreLab #30

Snell’s Law of Refraction

1 1 2 2sin sinn nθ θ=In general:

1 1 2 2n nθ θ≈

12 1

2

nn

θ θ≈

2 1 1 2If , then measured from the normal!

n n θ θ> >

Study Example 35.4 Carefully! Emerging Beam is Parallel to Incident Beam but

offset distance d, called the Lateral Shift and is the subject of next week’s lab!

Fig. 35-15, p. 989

Following the Reflected and Refracted Rays

•Ray is the incident ray. •Ray is the reflected ray. •Ray is refracted into the lucite. •Ray is internally reflected in the lucite. •Ray is refracted as it enters the air from the lucite. Section 35.5

Beam & Refraction Directions

• Possible directions of the beam are indicated by rays numbered 1 through 5

• The refracted rays are bent away from the normal since n1 > n2

Total Internal Reflection

2

1

sin Cnn

θ =

2 90θ =

1 1 2 2sin sinn nθ θ=

The Critical Angle

Critical Angle • There is a particular angle

of incidence that will result in an angle of refraction of 90° – This angle of incidence is

called the critical angle, θC

21 2

1

sin (for )Cnθ n nn

= >

Fiber Optics • An application of internal reflection

• Plastic or glass rods are used to “pipe” light from one place to another

• Applications include: – medical use of fiber optic

cables for diagnosis and correction of medical problems

– Telecommunications • A flexible light pipe is called an

optical fiber • A bundle of parallel fibers

(shown) can be used to construct an optical transmission line

Critical Angle Sample Problem A ray of light, emitted by a laser located beneath the surface of an unknown liquid with air above it, undergoes total internal refection as shown. What is the index of refraction for the liquid? What is its likely identification?

Prelab Problem # 35

If you pass white light through a prism, it separates into its component colors.

R.O.Y. G. B.I.V long wavelengths short wavelengths

The index of refraction depends on WAVELENGTH.

R.O.Y. G. B.I.V long wavelengths short wavelengths

The speed and wavelength change but the FREQUENCY does NOT.

R.O.Y. G. B.I.V long wavelengths short wavelengths

Fr Frequency depends on the oscillating source!

Why does Violet Light bend more than Red Light?

Violet light slows down more because the atoms in the material are tuned to higher frequencies. As the violet light travels

through glass it takes more time to be absorbed and re-emitted.

Variation of Index of Refraction with Wavelength

• This dependence of n on λ is called dispersion

• The index of refraction for a material generally decreases with increasing wavelength

• Violet light bends more than red light when passing into a refracting material

≡ = =speed of light in a vacuumnspeed of light in a medium n

c λv λ

Refraction in a Prism

•Since all the colors have different angles of deviation, white light will spread out into a spectrum.

– Violet deviates the most.

– Red deviates the least. – The remaining colors

are in between.

Section 35.7

END

Dispersion via Diffraction

: sin , 0,1, 2,3constructive d m mθ λ= =

If you pass white light through a prism, it separates into its component colors.

R.O.Y. G. B.I.V long wavelengths short wavelengths

The index of refraction depends on WAVELENGTH.

R.O.Y. G. B.I.V long wavelengths short wavelengths

The speed and wavelength change but the FREQUENCY does NOT.

R.O.Y. G. B.I.V long wavelengths short wavelengths

Fr Frequency depends on the oscillating source!

Why does Violet Light bend more than Red Light?

Violet light slows down more because the atoms in the material are tuned to higher frequencies. As the violet light travels

through glass it takes more time to be absorbed and re-emitted.

Variation of Index of Refraction with Wavelength

• This dependence of n on λ is called dispersion

• The index of refraction for a material generally decreases with increasing wavelength

• Violet light bends more than red light when passing into a refracting material

≡ = =speed of light in a vacuumnspeed of light in a medium n

c λv λ

Angle of Deviation • Since all the colors

have different angles of deviation, white light will spread out into a spectrum – Violet deviates the most – Red deviates the least – The remaining colors are

in between

Dispersion Sample Problem The index of refraction for

violet light in silica flint glass is 1.66, and that for red light is 1.62. What is the angular dispersion of visible light passing through a prism of apex angle 60.0° if the angle of incidence is 50.0°? red (660 nm) violet (410 nm)

1 1 2 2sin sinn nθ θ=Use Snell’s Law twice and some geometry! Angles are always measured from the normal.

How are Rainbows Formed?

Dispersion: Raindrops Act like Prisms

• A ray of light strikes a drop of water in the atmosphere

• It undergoes both reflection and refraction – First refraction at the

front of the drop • Violet light will

deviate the most • Red light will deviate

the least

The Rainbow • At the back surface the light is

reflected • It is refracted again as it

returns to the front surface and moves into the air

• The rays leave the drop at various angles – The angle between the white

light and the most intense violet ray is 40°

– The angle between the white light and the most intense red ray is 42°

Observing the Rainbow

• If a raindrop high in the sky is observed, the red ray is seen • A drop lower in the sky would direct violet light to the

observer • The other colors of the spectra lie in between the red and

the violet

The droplets form a circular arc, with each droplet within the arc dispersing light and reflecting it back towards the observer with the greatest concentration of outgoing rays found at these 40-42 degree angles of deviation. Every droplet within the arc is refracting and dispersing the entire visible light spectrum (ROYGBIV).

Rainbow facts • an observer is in a position to see only a single color from any

one droplet of water. • your rainbow is slightly different from the rainbow seen by

others • your rainbow moves with you • disk within the bow is brighter because of overlapping of

multiple refractions (which don’t occur outside the disk)

A line drawn from your eye to the top of the rainbow forms a 42-degree angle with the imaginary line from the sun through your eye. (If there is a secondary rainbow, it forms an angle of 51-degrees). Because these angles determine the position of the rainbow in the sky, it will sink as the sun rises and rise as the sun sinks. At some points, the entire rainbow, not just the bottom half, will be below the horizon where you can't see it. That's why you'll never see a summer rainbow at midday.

Double Rainbow

• The secondary rainbow is fainter than the primary

• The secondary rainbow arises from light that makes two reflections from the interior surface before exiting the raindrop

• Higher-order rainbows are possible, but their intensity is low

• Halos are caused by the light of the sun or moon passing through a very thin layer of cirruform (ice-crystal) clouds in the upper atmosphere. The ice crystals refract the light of the moon, similar to the way water droplets in the lower atmosphere can refract sunlight to produce a rainbow. Just like a rainbow, strong halos can have bands of color in them, due to slightly different refractive properties of the ice crystals for different colors. Essentially, halos ARE rainbows caused by primary refraction in ice crystals.

• Some interesting facts about halos: Halos always occur exactly 22 degrees away from the sun or moon. Occasionally, intense halos can be double halos, just as intense rainbows can be doubled. Intense halos can also produce "moondogs" or "sundogs," very bright regions on the halo evenly spaced at 90 degree intervals around the halo.

Physics Fun on an Airplane Always sit on the side opposite the sun when traveling north-south!!

Thin Film Interference

If you pass white light through a prism, it separates into its component colors.

R.O.Y. G. B.I.V

spectrum

long wavelengths short wavelengths

Radiation of Visible Sunlight

Additive Primary Colors Red, Green, Blue

RGB Color Theory

Additive Complementary Colors Yellow, Cyan, Magenta

The color you have to add to get white light.

Red + Green = Yellow Blue + Green = Cyan Red + Blue = Magenta Red + Blue + Green = White

White light – yellow light = ??

White light – red light = ??

FYI: Mixing Colored Pigments Subtractive Colors

Pigments subtract colors from white light.

Yellow + Cyan = Green Cyan + Magenta = Purple Yellow + Magenta = Red Yellow + Cyan + Magenta = Black

Why are some materials colored?

Why is a Rose Red?

Colored materials absorb certain colors that resonate with their electron energy levels and

reject & reflect those that do not.

Why is the Ocean Cyan?

White light minus cyan is red. Ocean water absorbs red.

Shine cyan light on a red rose and what color do you see?

Shine cyan light on a red rose and what color do you see?

Interference in Thin Films When reflecting off a medium of greater refractive index, a

light wave undergoes a phase shift of ½ a wavelength. Wave 1 undergoes a phase shift of 180 degrees.

From Low to High, a phase change of

pi! From High to Low,

a phase change? NO!

Interference in Thin Films • The wavelength of ray 1 in the

film is λ/n • For constructive interference

2t = (m + ½) λ/n (m = 0, 1, 2 …) This takes into account both the

difference in optical path length for the two rays and the 180° phase change

• For destructive interference 2t = mλ/n (m = 0, 1, 2 …)

A thin film of gasoline floats on a puddle of water. Sunlight falls almost

perpendicularly on the film and reflects into your eyes a yellow hue. Interference

in the the thin gasoline film has eliminated blue (469nm in vacuum) from the

reflected light. The refractive indices of the blue light in gasoline and water are

1.40 and 1.33 respectively. Determine the minimum nonzero

thickness of the film. What color do you see?

Problem: Thin Films

Thin Film Interference

The light reflected from a soap bubble (n = 1.40) appears red (λ = 640 nm). What is

the minimum thickness (in nm)? a. 124 b.104 c. 114 d.134 e. 234

Galileo

In the early 17th century, many scientists believed that there was no such thing as the "speed of light"; they thought light could travel any distance in no time at all. Galileo disagreed, and he came up with an experiment to measure light's velocity: he and his assistant each took a shuttered lantern, and they stood on hilltops one mile apart. Galileo flashed his lantern, and the assistant was supposed to open the shutter to his own lantern as soon as he saw Galileo's light. Galileo would then time how long it took before he saw the light from the other hilltop. The problem was that the speed of light is simply too fast to be measured this way; light takes such a short time (about 0.000005 seconds, in fact) to travel one mile that there's no way the interval could have been measured using the tools Galileo had.

The Speed of Light? • 186,000 miles per second • 300,000 kilometers per second • 3 x 108 m/s • first successfully determined by

Danish astronomer Ole Roemer in 1675: 2.3 x 108 m/s

• First Terrestrial Measurement by Fizeau in 1849: 2.9979 x 108 m/s

• In 1926, Michelson used a rotating prism to measure the time it took light to make a round trip from Mount Wilson to Mount San Antonio in California, a distance of about 22 miles (36 km). The precise measurements yielded a speed of 186,285 miles per second (299,796 kilometres per second).

Huygens’s Principle Construction for a Plane Wave

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

• 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 passed, the new position of the wave front is the surface tangent to the wavelets

Huygens’s 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

Huygens’s Principle Prove the Laws of Reflection & Refraction

Huygens’s Principle and the Law of Reflection

• Triangle ABC is congruent to triangle ADC

• cos γ = BC / AC • cos γ’ = AD / AC • Therefore, cos γ = cos γ’

and γ = γ’ • This gives θ1 = θ1’ • This is the law of

reflection

Huygens’s Principle and the Law of Refraction

• Ray 1 strikes the surface and at a time interval Δt later, ray 2 strikes the surface

• During this time interval, the wave at A sends out a wavelet, centered at A, toward D

Huygens’s Principle and the Law of Refraction

• The wave at B sends out a wavelet, centered at B, toward C

• The two wavelets travel in different media, therefore their radii are different

• From triangles ABC and ADC, we find

1 21 2sin and sin BC v t AD v tθ θ

AC AC AC AC∆ ∆

= = = =

÷÷

= = =1 1 1 2

2 2 2 1

sin sin

v c n nv c n n =1 1 2 2 sin sin n θ n θ