Plane Refraction

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January 03 LASERS 51

Refraction at plane surfaces• Speed of light in air

– Slight digression on what is light

– Fizeau’s measurement

• Index of refraction

– speed of light in transparent materials

– dispersion

– wave picture of refraction

– ray picture, Snell’s law

• Plane refractive surfaces

– single surfaces, images – plane-parallel plate

• Critical angle

• Prisms




What is light?• Carries energy

• Exerts force on electrical charges• Wave similar to sound or water wave

– Frequency related to color (high

frequency=blue, low frequency = red)

– Wavelength also related to color (long

wavelength=red, short wavelength=blue)

– Other electromagnetic radiation behaves

similarly (radio, microwave, xrays,

gamma rays)

• Wave fronts perpendicular to rays

rays

wavefronts

Wavefront – surface of

peaks of waves

Wave

length,

λ amplitude

Wave speed, V




How fast does light travel

• Light travels in a straight line – How fast does it travel?

• Galileo’s method (1600)

– Lantern’s on neighboring hills

– One person unblocks his lantern

– When the other sees the lantern, he unblocks his

– First person sees light from the second lantern, giving

round trip time

– Doesn’t work, light too fast


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Measurement of speed of light in air • Olaf Romer (1675)

used eclipses of

Jupiter’s moons

estimatedc=2.967x1010cm/sec

• Fizeau (1849) firstearthbound

measurement

• Value in vacuum – 3x108m/sec=3.0x1010cm/sec

– 30 cm/nsec≅1 ft/nsec

Speed of light is constant

Independent of color (wavelength)

Independent of motion of source

Symbol, c (Latin, celere=speedy)



Speed of light in transparent media

• First measured byFoucault in 1850

– helped to settle

controversy aboutwave/particle nature of

light

– light travels slower indenser media

• Be careful about the

meaning of the word

“density”

mediuminspeed

in vacuumspeed refractionof index =



Dispersion

• Index of refraction (speed) changes with wavelength

• Decreases for longer wavelengths

• Small, but important effect, note y-axis doesn’t start at zero

Index of refraction of Quartz

1.454

1.456

1.458

1.460

1.462

1.464

1.466

1.468

1.470

1.472

350 400 450 500 550 600 650 700

Wavelength (nm)

R e f r a c t i v e I n d

e x

Blue Green Red



Refractive index also varies

widely between materials

Index of refraction of variousmaterials at 589 nm (d light)

Material IndexWater 1.333

Air 1.000293BK7 glass

(crown) 1.517

BaSF6 glass

(flint)1.668

FDS9 glass 1.847Quartz 1.458

Diamond 2.419



Refraction of a wavefront at plane surface

• Wavelength in higher index

material is shorter

– velocity slower but frequencyunchanged

• Wavefronts are continuous across

surface• Wave bends towards the normal

when going to a higher index

material





Graphical construction for refraction

n1

n1

n2

A

n1

n2

A

B

n2

n1

parallel

• Set a compass to radius n1 (units don’t matter!!!)

– Use this to mark off a distance n1 along input ray from point of intersection with surface

– This is point A on the diagram

• Set compass to n2

– With point at A draw arc that intersects surface normal

– Intersection point is B in diagram

• Refracted ray is parallel to line AB – Transfer a parallel line to the intersection point



Principle of reversibility• If a ray is reversed in direction:

– output angle Θ2 becomes input angle – final index, n2 becomes initial index

– angle which satisfies Snell’s law for the

final angle is then Θ1, the original inputangle

– therefore, the ray retraces its path

• The law of reflection is symmetricalwith respect to its initial and finaldirections also

Θ1 Θ2

Θ1Θ2

n2

n2

n1

)sin()sin( 1

2

12 Θ=Θ

n

n

n1 In geometrical optics, if any ray is

reversed it will retrace its original

path. This holds for all rays.)sin()sin( 21

2

1 Θ=Θ n

n



Refraction of rays from a point source

• Rays at a higher angle bend more

• From in the high index material,light appears to come from an

image point farther from the

surface than the object – if n2<n1 then the image is closer to

the surface

• The image is virtualn2

n1

R f i f f i




Refraction of rays from a point source

h

h’

R

L

R’

ΘΘ

Θ’

Θ’

Object

Image n’

n

Snell’s law:

In this approximation hn

n

h

'

' =

)sin()sin( Θ′′

=Θn

n

)cos()cos( Θ

′=Θ′

′ h

n

nh

For small angles this is

approximately

(paraxial approximation

angle in radians)

Θ′′=Θn

n

Exact result Not a perfect

image, location

depends on rayangle!



Rays out of plane give different image from in




Rays out of plane give different image from in

plane rays – astigmatism

• Observer will see both sets of rays, therefore blurryimage

• Only meridional rays appeared on previous diagram

object point

image point

for meridional rays

object point

image pointfor sagital rays

R f i h h l ll l l b




Refraction through a plane-parallel slab

• final ray parallel to the initial ray – not true if faces are not parallel

– displacement, d, proportional to

thickness, t, increases with largerindex

t

φ

φ2

φ

d

φ

φ2

β=φ1-φ2

A

BC

Dh

)cos(

)sin()sin(2

2121

φ

φ φ φ φ −=−= t hd

n2

n1

Geometry gives:

This can be expressed in different ways using trigonometry

−

−=

−=

−=

)(sin

)cos(1)sin(

)cos(

)cos(1)sin(

)cos(

)cos()sin()sin(

1

2

2

1

2

2

11

22

111

2

121

φ

φ φ

φ

φ φ

φ

φ φ φ

n

n

t n

nt t d

Last form doesn’t require calculation of φ2

I i f i h h ll l l




Imaging or focusing through a parallel plate

• Imaging – object locatedat O, appears to be

at O’

– Application, microscope

imaging through

cover glass

• Focusing – light to left

of plate is focusing towards O’

Actually focuses at O

– Application, machining laser focused through a window, for

example to prevent debris from getting on lens

• Imaging or focusing depending on direction of travel – reversibility

O’, location of focus

in absence of plate

E

n

n

t

1E

−=

t

O

n




Prisms

• Deflections at the two surfacesdon’t cancel

• Deflection angle depends on apex

angle of prism, α, not angles of base

• From geometry δ=φ1

+φ2

’-α – can be considered a function of φ1

since φ2 depends on φ1 and α is fixedfor a given prism

– depends on index of refraction andtherefore also on wavelength

α

φ1

φ1’ φ2

φ2’δ

G hi l t ti f




Graphical construction for

deviation by a prism

• Begin by finding the refraction at the first surface as before

– Before setting compass for n2, draw an arc of radius n1 centered

on A

• Through the point B draw a line perpendicular to the

second surface – the point where this line intersects the n1 circle drawn previously

is C

• The final ray exiting the prism is parallel to AC

n2

n1α

AB

C




30

35

40

45

50

55

60

65

30 40 50 60 70 80 90

Incidence angle (degrees)

D e f l e c t i o n a

n g l e

( d e g r e e

Deflection angle of an

equilateral prism with

an index of 1.517

minimum deflection angle

Deflection of a prism as a function of

incidence angle

• Minimum

deviation depends

only on index and prism apex angle




Minimum deviation formulas

• Formulae derived simply by setting φ1=φ2’ and φ1’=φ2

• Obviously, φ1’ can be found from φ1 from Snell’s law

• The details of the derivation are not too important, but

not too difficult either

Final results VALID ONLY FOR MINIMUM DEVIATION

To find minimumdeviation angle,

given index and

apex angle

To find input angle atminimum deviation angle

given index and apex

angle

= −

2sinsin 1

1

Anφ A−= 1min 2φ δ

To find index, givenminimum deviation

angle and apex angle

+

=

2sin

2sin min

A

A

n

δ




Critical angle, total internal reflection

• A ray traveling from a higher indexto a lower index bends away from the

normal

– at an angle called the critical angle, the

refracted angle is 90°, the ray travels

along the surface

– for larger incidence angles, Snell’s law

says the sine of the refracted angle is

greater than one, this is impossible

• For incident angles larger than the

critical angle, the light is completely

reflected, there is no refracted ray

n

n

critical

'

)sin( =Θ

Θn

n’

C i i t t l i t l




Common prisms using total internal

reflection

Documents

Plane Refraction