11

Brilliant Pattern

ProblemProblem

Suspend a water drop at the lower end Suspend a water drop at the lower end of a vertical pipe. Illuminate the drop of a vertical pipe. Illuminate the drop using a laser pointer and observe the using a laser pointer and observe the pattern created on a screen. Study and pattern created on a screen. Study and explain the structure of the patternexplain the structure of the pattern

22

Overview HypothesisHypothesis

Assumption of Experiment Experimental SetupExperimental Setup

Devices and Process InvestigationsInvestigations

Experimental Results TheoryTheory

The Physics Behind Brilliant Pattern ConclusionConclusion

33

HypothesisEffects of reflection and refraction

Diffraction

Geometrical Optics

Physical Optics

d/2

a

44

Experimental Setup

Laser Pointer

10 mw, 532 nm, Radius: 1.45 mm

1 mm

35 cm

55

Observation: 0o

66

(a) (b) (c)

(a)

(b)

(c)

Observation: 150o ~180o ; -150o~ -180o

42o

51o

77

TheoryPassing through water drop (0o)

2d

a

ir

'r

o

: angle of incidence i

: angle of refraction (air water)r

' : angle of refraction (water air)r

: angle of divergenceo

: diameter of laserd

: radius of waterdropa

88

2d

a

ir

'r

o

sin( ) 2i d a

sin(2 )cos(2 ) 1

tan(2 2r i

r ii r

f a

d f

f

99

2d

a

ir

'r

o

1 sin( ) 1.33 sin( )i r

sin( ) 2i d a

'1.33 sin( ) 1 sin( )r r

'r i

2o i r d o

Snell’s LawSnell’s Law

1010

Traveling backwards (180o)

Theory

a

i

1

2

r32d

1111

a

i

1

2

r32d

o42

o51

2

13

1 o

2 42 o

3 51

a

1

2

o3

2

4 2

180 2 6

i

r i

i r

(b) (c)

(b)(c)

1212

(a) (b) (c)

(a)

(b)

(c)

0o

42o

51o

Experimental Results

1313

Experimental Video(a)

1414

Distance between water drop and screen 3 m

Water Droplet Radius: 0.9~2 mm

Laser Pointer Radius: 1.45 mm

Wavelength: 532 nm

Experiment 2: Physical Optics

1515

2 cm

4 m

Experiment 2: Physical Optics

Diffraction

1616

Theory Huygens’s Principle

Every point on a propagating wavefront serves as the source of spherical secondary wavelets, wavelets, such that the wavefront at some later time is the envelope of these wavelets

v twavelett tv

wavelet0t

wavelet0t

v

v

v twavelet0t

wavelett t

1717

R

( , , )X Y Z

y

z

PO

dS

r

x

21

222:

2number wave:

2frequencyangular :

areaunit per source ofstength :

ZYXOPR

k

fA

Theory Huygens’s Principle

Every point on a propagating wavefront serves as the source of spherical secondary wavelets, wavelets, such that the wavefront at some later time is the envelope of these wavelets

( )

( )( )/

Aperature

i t krA

i t kRik Yy Zz RA

dE e dSr

eE e dS

R

1818

a

O R

y

z

P

q

0( )J u

1( )J u

2( )J u

u

( )2

12 ( ) ( )i t kR

A e R k a qE a J

R k a q R

2

1( ) 2 ( sin )

(0) ( sin )

I J k a

I k a

2 2

2(0)

2A a

IR

Bessel Function22 2

12

2 ( )

( )A a J k a q R

IR k a q R

1919

0( )J u

1( )J u

2( )J u

u

R

y

z

P

q

( )

( )( )/

out of aperture

i t krA

i t kRik Yy Zz RA

dE e dSr

eE e dS

R

( )

02 ( ' ) ( )i t kR

A e k a qE a a J

R R

2 2

2

02

2 ( ' )( )A a a

I J k a q RR

a

'a

2020

Theoretical Analysis

4 m

2 cm

0.9 mm

1.2 mm

1.5 mm

2.0 mm

Diameter of water drop:

2121

0.9 mm 1.2 mm 1.5 mm 2 mm

Theoretical AnalysisDiameter of water drop:

2222

0.9 mm 1.2 mm

1.5 mm 2 mm

Experimental Results

0.9 mm 1.2 mm

Experimental Results

1.5 mm 2 mm

2323

ConclusionGeometrical Optics:Geometrical Optics:

– At 0o the light passes through the water droplet and is magnified

– At 180o the light is reflected and refracted by the water droplet; thus, interference is demonstrated

– At other degrees because the light beam has attenuated its energy so patterns are unable to be observed

Wave Optics: Wave Optics: – Due to Huygens’s law, light is diffracted,

so diffraction bands are observable– As water droplets are enlarged the

distance between each band decreases

(a)

(b)

(c)

Geometrical Optics:Geometrical Optics: – At 0o the light passes through the water

droplet and is magnified– The light is reflected and refracted by

the water droplet; thus, interference is demonstrated

– At other degrees because the light beam has attenuated its energy so patterns are unable to be observed

Wave Optics: Wave Optics: – Due to Huygens’s principle, light is

diffracted, so diffraction bands are observable

– As water droplets are enlarged the distance between each band decreases

Conclusion