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Physics Part IA Cambridge University, 2012-2013

Notes for Waves and Quantum Waves, Part 2 Huygen's principle: Every point on a primary wavefront behaves as a source of spherical secondary wavelets, such that the primary wavefront at a later time is the envelope of these wavelets; the wavelets have the same speed and frequency as the incoming wave:

ct

ct

In the above diagrams, the blue line represents a wavefront at some time red line represents the wavefront at time

t

later, for a wave of speed

t =0 , and the

c .

This principle of wave propagation can be used to derive the laws of reflection and refraction.

vt

2vt

thr i r

-t 0 t A B vt

Physics Part IA Cambridge University, 2012-2013

thi

Reflection using Huygen's principle: Each point of contact between the wavefront and boundary produces a wavelet which emanates from that point. The wavefronts have speed wavelet produced at time

vt

radius of

at time

v , hence a

t=0 has a

t . Thus, by simple

geometry:

sin thr =

vt vt , sin thi=

AB AB

sin thr =sin thi Phase changes of reflection: When a wave travels from media of refractive index

n1 to n2 and is reflected at the

boundary, the phase changes upon reflection dependent upon the relative magnitudes of

n1 and n2 :

*

n2 >n1

*

n1 >n2 is equivalent to a 'free end': No phase change.

is equivalent to a 'fixed end': Phase change of

p .

Physics Part IA Cambridge University, 2012-2013 Refraction using Huygen's Principle: Huygen's Principle can also be used to derive the law of refraction geometrically, as follows:

v2 t r i v1 t n1 n2

th1

th2 A B

th1

th2 The first medium has refractive index

n1 and the second has

refractive index

n2

. In medium

one the wave speed is medium two it is

v2

v 1 and in

.

Thus, by simple geometry:

sin th1=

v1 t AB

Physics Part IA Cambridge University, 2012-2013

v2 t AB

sin th2=

Therefore:

sinth 1 v1 n2

= [?]

sinth 2 v2 n1 n1=c /v 1 and n2=c /v 2 . This is called Snell's

Since refractive index is defined such that Law. Total internal reflection: Rearranging Snell's Law gives:

sin th2=

n1 sin th1 n2

If a wave is travelling from a denser to a less dense medium ( increases and the

n1 >n2

), then as

th1

RHS - 1 , increasing amounts of the incident wave are reflected, until

th1

the incident wave undergoes total internal reflection at some critical value of

. At this

sin th2=1 , so for any angles larger than th1 , th2 does not exist. All energy in

value,

the wave is reflected from the boundary. Spread of white light through a triangular prism: The refractive index

n of a medium depends on the wavelength of light passing through it,

such that the refraction of white light through a triangular prism results in the splitting of this light into its component colours:

a

For an equilaterial triangle ( a=60 deg ), and assuming that the surroundings has a refractive index of

n=1 :

sinth 1 sin th4

=n ,

=n sinth 2 sin th3

And

a+

Thus, for

th1=60 deg :

th2

th3

n n=1

( p2 -th )+( p2 -th )=p [?] th =a -th

2 th1

th4

3 3

2 Physics Part IA Cambridge University, 2012-2013

*

R

(

)

-1

[?] 3 =35.8deg [?] th =60-35.8=24.2 deg [?] th =37.3 deg

th

=sin n=1.48 2 3 4 ed light ( ): 2 ( 1.48 )

*

lue light ( n=1.52 ):

B

th2=sin

Hence the angle of spread is

-1

( 2 ( [?]1.523 ) )=34.7 deg [?] th =60-34.7=25.3 deg [?] th =40.5 deg

40.5 deg-37.3 deg=3.2 deg .

3 4

Physics Part IA Cambridge University, 2012-2013

Lenses: The physics of lenses is based upon geometries of refraction of a paraxial ray which obeys the paraxial approximation for small angles,

sin th [?] th [?] tan th .

First, consider refraction at a planar surface: This is a special case of general refraction which can be described by Snell's Law:

sinth 1 n2

=

sinth 2 n1

n1 n2

th2

th1

In the small angle approximation:

y

sin th [?] th [?] tan th

O

[?] sin th1 [?]

y y ,sin th 2 [?]

d v

I v d

Therefore:

sinth 1 v v n2

= [?] =

sinth 2 d d n1

Hence an object O at distance

d

from the interface gives rise to an image

distance

v =n2 d / n1 from the interface, when observed from the other side:

*

n1 >n2

If

I at a

, as for an object in water observed from the air, the distance between the

object and the interface appears to be shortened

*

If

n2 >n1

, as for an object in air observed from the water, the distance between the

object and the interface appears to be lengthened Extend this principle to the spherical surfaces of lenses: A lens can be thought of as two successive, spherical, coaxial interfaces: interface 1 with radius of curvature a refractive index Interface 1:

u

th2

a

th1

R1 and interface 2 with radius of curvature

n2

R2 . The whole lens has

and the surrounding medium has refractive index

n1

.

Physics Part IA Cambridge University, 2012-2013

d1 S n1 n2 S' R1

b

C

th

y1 Radius of curvature

R1 :

u is the distance between the object S and the interface.

d 1 is the distance

between the image

S ' and the interface.

It can be seen geometrically that:

th=th 2+ b [?] th 2=th-b , th1=th+ a

[?]

sin th1 sin (th +a ) n 2

=

= [?] n1 (th+ a )[?] n2 (th-b ) sin th2 sin ( th-b ) n 1 tan th [?] th : a [?] y 1 /u ,

Since

[?] n1

(

b [?] y 1 /d 1 , th= y1 / R1 :

y1 y1 y y n n n -n

+

=n2 1 - 1 [?] 1 + 2 = 2 1 R u R d u d1 R1

) (

)

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