Download Optics I

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:
 2  E2 sin kx  t   2 
 1  E1 sin kx  t  1 
Let 1  kx   1
 2  kx   2
We have  1  E1 sin 1  t 
 2  E2 sin  2  t 
Resultant   1  2  E sin   t   E sin kx  t   
E 2  E12  E22  2E1 E2 cos 2  1 
E1 sin 1  E2 sin  2
tan  
E1 cos 1  E2 cos  2
tan  
I  I1  I 2  2 I1 I 2 cos 2  1 
E1 sin  1  E2 sin  2
E1 cos  1  E2 cos  2
interference term
Two waves in phase result in total constructive interference: I max  I1  I 2  2 I1 I 2
Two waves anti-phase result in total destructive interference: I min  I1  I 2  2 I1 I 2
Optics II----by Dr.H.Huang, Department of Applied Physics
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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,
  kx2  t   2   kx1  t   1   k x2  x1    2  1 
in a medium:  
2
m
x2  x1    2   1   2 nx2  x1    2   1 

Optical Path Difference (OPD): n(x2-x1)
Optical Thickness or Optical Path Length (OPL): nt
Optics II----by Dr.H.Huang, Department of Applied Physics
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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 .
 1  E01 sin t  1 
 2  E02 sin t   2 
Suppose:    1   2  E0 sin t   
E0 
E01 cos 1  E02 cos  2 2  E01 sin 1  E02 sin  2 2
E01 sin 1  E02 sin  2
tan  
E01 cos 1  E02 cos  2
For multiple waves:
N
   E0i sin t   i   E0 sin t   
i 1
E0 
X 2 Y2
N
X   E 0i cos  i
i 1
and
tan   Y X
N
Y   E0i sin  i
i 1
Optics II----by Dr.H.Huang, Department of Applied Physics
3
The Hong Kong Polytechnic University Superposition of Light Waves
Example:
Find the resultant of adding the sine waves:
 2  10 sin t   4
 1  20 sin t
X  20 cos 0  10 cos
Y  0  10 sin
 3  10 sin t   12  4  15sin t  2 3

2
  
 10 cos    15 cos
 29.23
4
3
 12 

2
  
 10 sin     15 sin
 17.47
4
3
 12 
E  X 2  Y 2  34
  34 sin t   6
  tan 1 Y X  30
Example:
Find, using algebraic addition, the amplitude and phase resulting from the addition of the two
superposed waves  1  E1 sin kx  t  1  and  2  E2 sin kx  t   2  , where 1=0,
2=/2, E1=8, E2=6, and x=0.
1  kx  1  0
  arctan
 2  kx   2   2
E  E12  E22  2 E1 E2 cos 2  1   10
E1 sin 1  E2 sin  2
 arctan 0.75  36.87 
E1 cos 1  E2 cos  2
  10 sin kx  t  0.6435
Optics II----by Dr.H.Huang, Department of Applied Physics
4
The Hong Kong Polytechnic University Superposition of Light Waves
Example:
Two waves  1  E1 sin kx  t  and  2  E2 sin kx  t    are coplanar and overlap.
Calculate the resultant’s amplitude if E1=3 and E2=2.
E 2  E12  E22  2 E1 E2 cos 2  1 
 32  22  2  3  2  cos   1
E 1
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.
time 
distance d
d
nd
 

speed
v cn c
Optics II----by Dr.H.Huang, Department of Applied Physics
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The Hong Kong Polytechnic University Superposition of Light Waves
Standing Wave;
Suppose two waves:  I  E0 I sin kx  t   I  and  R  E0 R sin kx  t   R 
having the same amplitude E0I=E0R and zero initial phase angles.
   1  2  2E0 I sin kx cos t

Nodes at: x  n ,
2
n  0, 1, 2, 3,....
1

x

n

Antinodes at:

 ,
2 2

nodes or
nodal points
antinodes
n  0, 1, 2, 3,....
Optics II----by Dr.H.Huang, Department of Applied Physics
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The Hong Kong Polytechnic University Superposition of Light Waves
Addition of Waves of Different Frequency:
 2  E1 cosk2 x  2t 
 1  E1 cosk1 x  1t 
   1  2  2 E1 cosk p x   p t cosk g x   g t 
p 
1  2
2
Group velocity:
g 
vg 
1  2
2
g
kg

dispersion relation =(k)
1   2
k1  k 2

vg 

k
d
dk
Optics II----by Dr.H.Huang, Department of Applied Physics
7
The Hong Kong Polytechnic University Superposition of Light Waves
Coherence:
Frequency bandwidth:    1  2
Coherent time: t 
1

Coherent length: x  ct
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).
1
6

2

10
500 10 9
(b) : OPL  n1d1  n2d2  1 0.90  1.5  0.10  1.05 m
OPL
1.05
6
number of wavelengt hs 


2
.
1

10

500 10 9
(c) : OPD  1.05  1  0.05 m
OPD
0.05
5
6
6
(d ) :


10

2
.
1

10

2
.
0

10

500 10 9
(a) : number of wavelengt hs 
Optics II----by Dr.H.Huang, Department of Applied Physics
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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.
OPL1  nd  1 m
OPL2  na d1  d 5   ng d 2  d 4   nw d 3
 1 0.78  1.5  0.02  1.33  0.20
 1.076 m
OPD  OPL2  OPL1  0.076 m
number of wavelengt hs 
  2 
OPD

OPD


0.076
5

1
.
52

10
500 10 9
 9.55 105 radian
Optics II----by Dr.H.Huang, Department of Applied Physics
9
The Hong Kong Polytechnic University Superposition of Light Waves
Example:
Show that the standing wave s(x,t) is periodic with time. That is, show that s(x,t)=
s(x,t+).
 s x, t   2E sin kx cos t
 s x, t     2 E sin kx cos  t   
 2 E sin kx cost   
 2 E sin kx cost  2 
 2 E sin kx cos t
  s  x, t 
Homework: 11.1; 11.3; 11.4; 11.5; 11.6
Optics II----by Dr.H.Huang, Department of Applied Physics
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