the hong kong polytechnic university optics ii----by dr.h.huang, department of applied physics1...

Optics II----by Dr.H.Huang, Department of Applied Physics 1

The Hong Kong Polytechnic University Superposition of Light Waves

Principle of Superposition:When two waves meet at a particular point in space, the resultant disturbance is simply the algebraic sum of the constituent disturbance.

Addition of Waves of the Same Frequency:

Let

We have

Resultant

interference term

Two waves in phase result in total constructive interference:

Two waves anti-phase result in total destructive interference:

111 sin tkxE 222 sin tkxE

11 kx 22 kx

tE 111 sin tE 222 sin

tkxEtE sinsin21

122122

21

2 cos2 EEEEE

2211

2211

coscos

sinsintan

EE

EE

2211

2211

coscos

sinsintan

EE

EE

122121 cos2 IIIII

2121max 2 IIIII

2121min 2 IIIII



Coherent: Initial phase difference 2-1 is constant.

Incoherent: Initial phase difference 2-1 varies randomly with time.

Phase difference for two waves at distance x1 and x2 from their sources,

in a medium:

Optical Path Difference (OPD): n(x2-x1)

Optical Thickness or Optical Path Length (OPL): nt

12121122 xxktkxtkx

12121212

22

xxnxxm



Phasor Diagram:Each wave can be represented by a vector with a magnitude equal to the amplitude of the wave. The vector forms between the positive x-axis an angle equal to the phase angle .

Suppose:

For multiple waves:

1011 sin tE 2022 sin tE

tE sin021

2202101

22021010 sinsincoscos EEEEE

202101

202101

coscos

sinsintan

EE

EE

tEtEN

iii sinsin 0

10

XYYXE tanand220

N

iiiEX

10 cos

N

iiiEY

10 sin



Example:Find the resultant of adding the sine waves:

Example:Find, using algebraic addition, the amplitude and phase resulting from the addition of the two superposed waves and , where 1=0, 2=/2, E1=8, E2=6, and x=0.

t sin201 4sin102 t 12sin103 t 32sin154 t

23.293

2cos15

12cos10

4cos100cos20

X

47.173

2sin15

12sin10

4sin100

Y

3422 YXE 30tan 1 XY 6sin34 t

111 sin tkxE 222 sin tkxE

011 kx 222 kx 10cos2 122122

21 EEEEE

87.3675.0arctancoscos

sinsinarctan

2211

2211

EE

EE

6435.0sin10 tkx



Example:Two waves and are coplanar and overlap. Calculate the resultant’s amplitude if E1=3 and E2=2.

tkxE sin11 tkxE sin22

1

1cos23223

cos222

122122

21

2

E

EEEEE

Example:Show that the optical path length, or more simply the optical path, is equivalent to the length of the path in vacuum which a beam of light of wavelength would traverse in the same time.

c

nd

nc

d

v

d

speed

distancetime



Standing Wave;Suppose two waves: and

having the same amplitude E0I=E0R and zero initial phase angles.

III tkxE sin0 RRR tkxE sin0

tkxE I cossin2 021

nodes ornodal points antinodes

Nodes at:

Antinodes at:

,....3,2,1,0,2

nnx

,....3,2,1,0,22

1

nnx



Addition of Waves of Different Frequency:

Group velocity:

dispersion relation =(k)

txkE 1111 cos txkE 2212 cos

txktxkE ggpp coscos2 121

222121

gp

kkkkv

g

gg

21

21

dk

dvg



Coherence:Frequency bandwidth:

Coherent time:

Coherent length:

21

1t

tcx

Example: (a) How many vacuum wavelengths of =500 nm will span space of 1 m in a vacuum? (b) How many wavelengths span the gap when the same gap has a 10 cm thick slab of glass (ng=1.5) inserted in it? (c) Determine the optical path difference between the two cases. (d) Verify that OPD/ is the difference between the answers to (a) and (b).

69

10210500

1hs wavelengtofnumber :)(

a

m05.110.05.190.01:)( 2211 dndnOPLb

69

101.210500

05.1hs wavelengtofnumber

OPL

m05.0105.1:)( OPDc

6659

100.2101.21010500

05.0:)(

OPDd



Example:In the figure, two waves 1 and 2 both have vacuum wavelengths of 500 nm. The waves arise from the same source and are in phase initially. Both waves travel an actual distance of 1 m but 2 passes through a glass tank with 1 cm thick walls and a 20 cm gap between the walls. The tank is filled with water (nw=1.33) and the glass has refractive index ng=1.5. Find the OPD and the phase difference when the waves have traveled the 1 m distance.

m11 ndOPL

m076.1

20.033.102.05.178.01

342512

dnddnddnOPL wga

m076.012 OPLOPLOPD

59

1052.110500

076.0hs wavelengtofnumber

OPD

radian1055.92 5

OPD


The Hong Kong Polytechnic University

Example: Show that the standing wave s(x,t) is periodic with time. That is, show that s(x,t)= s(x,t+).

Homework: 11.1; 11.3; 11.4; 11.5; 11.6

tkxEtxs cossin2,

tx

tkxE

tkxE

tkxE

tkxEtx

s

s

,

cossin2

2cossin2

cossin2

cossin2,

Superposition of Light Waves

the hong kong polytechnic university optics ii----by dr.h.huang, department of applied physics1...

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