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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
Optics II----by Dr.H.Huang, Department of Applied Physics 2
The Hong Kong Polytechnic University Superposition of Light Waves
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
Optics II----by Dr.H.Huang, Department of Applied Physics 3
The Hong Kong Polytechnic University Superposition of Light Waves
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
Optics II----by Dr.H.Huang, Department of Applied Physics 4
The Hong Kong Polytechnic University Superposition of Light Waves
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
Optics II----by Dr.H.Huang, Department of Applied Physics 5
The Hong Kong Polytechnic University Superposition of Light Waves
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
Optics II----by Dr.H.Huang, Department of Applied Physics 6
The Hong Kong Polytechnic University Superposition of Light Waves
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
Optics II----by Dr.H.Huang, Department of Applied Physics 7
The Hong Kong Polytechnic University Superposition of Light Waves
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
Optics II----by Dr.H.Huang, Department of Applied Physics 8
The Hong Kong Polytechnic University Superposition of Light Waves
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
Optics II----by Dr.H.Huang, Department of Applied Physics 9
The Hong Kong Polytechnic University Superposition of Light Waves
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
Optics II----by Dr.H.Huang, Department of Applied Physics 10
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