Team of Brazil Problem ## Title Diego de Moura, Felipe de Melo, Matheus Camacho, Thiago Bergamaschi, Thiago KalifeNakhon Ratchasima, 27 June – 4 July 2015

Team of BrazilProblem ## Title

Diego de Moura, Felipe de Melo, Matheus Camacho, Thiago Bergamaschi, Thiago Kalife Nakhon Ratchasima, 27 June – 4 July 2015Reporter: Felipe de Melo 1

BrazilProblem 14 – Circle of Light

Brazil

Problem 14 Circle of Light

Reporter: Felipe de Melo




Problem 14 Circle of Light

When a laser beam is aimed at a wire, a circle of light can be observed on a screen perpendicular to the wire. Explain this phenomenon and investigate how it depends on the relevant parameters.




• Reflection• Refraction• Snell’s Law• Interference

Theoretical Introduction

• Materials• Procedures• Data Analysis

Experiments

• Relevant parameters

Conclusion




REFLECTION

Reflection occurs on the surface of the wire

Conic formation

Incidence angleReflection angle




REFLECTION

Half of the wire surface All the circle, without

the wire shadow

Only because the wire is rounded the reflection

forms a circle




REFRACTION

Refraction occurs through the wire

Will form the front part of the circle

Higher refraction indeces will distribute the refraction along a bigger part of the circle




REFRACTIONVertical

projection plan

We will call this angle as beta

Both angles are equal




REFRACTION

Distance to wire center

Incidence angle

First angular deviation

Refraction angle

Wire radius

A laser light ray

The wire

Horizontal projection plan




REFRACTION

SNELL’S LAW

�⃗�1

�⃗�2

�⃗�

𝑛1∗𝑠𝑒𝑛 (𝜃1 )=𝑛2∗𝑠𝑒𝑛(𝜃2)

𝜃1𝜃2

Does not exist a plan that contains both at the

same time

Best way to quantify both angles is using

vectors

Incident light ray

Refracted light ray

Normal straight line in relation with the surface




𝑠2=𝑛1𝑛2

[𝑁×(−𝑁×𝑠1) ]− �̂� √1−(𝑛1𝑛2 )2

( �̂�×𝑠1 )⋅ ( �̂�×𝑠1 )

REFRACTION

VECTORIAL FORM OF SNELL’S LAW

Refracted light ray

unit vector

Wire refraction index

Air refraction index Normal unit

vector

Incident light ray unit vector




𝛼=tan−1(√1−𝜂2 (1−cos2 (𝛽 )cos2 (𝜃 ) )−𝜂 cos (𝜃 ) cos (𝛽) ) sin (𝜃 )

𝜂 sin2 (𝜃 ) cos (𝛽)+cos (𝜃)√1−𝜂2 [1− cos2 ( 𝛽) cos2(𝜃) ]

REFRACTION

Inclination in relation to horizontal axis 𝜂=

𝑛1𝑛2

lim𝜃→

𝜋2

𝛼=tan−1 √1−𝜂2

cos 𝛽𝜂The last refracted ray, with biggest angular deviation




REFRACTION

𝛼𝑚𝑎𝑥=tan−1 √1−𝜂2cos𝛽𝜂𝛼𝑚𝑎𝑥=tan

−1 √𝑛2−1cos𝛽

Assuming : wire immersed in air




0<cos 𝛽<1

REFRACTION

𝛼𝑚𝑎𝑥=tan−1 √𝑛2−1cos𝛽 0<𝛽<

𝜋2

On this interval, is a decresent

function




REFRACTION

Refraction

index

Light intensity

Inclination angle

Light intensity

Increase the area iluminated by refracted light




REFRACTION

Form primaly the front part of the circle




REFRACTION AND REFLECTION

Translucid wireAssume absorption

equals to zero

𝑇 +𝑅+𝐴=1Energy conservation

𝑇 +𝑅=1





FRESNEL’S RELATION

http://www.ece.rice.edu/~daniel/262/pdf/lecture13.pdf





From spherical geometry:

cos𝜃𝑖=cos𝜃 cos𝛽

𝛽 ≤𝜃𝑖<𝜋2

Higher values for beta will increase the reflected light intensity




CIRCLE FORMATION

𝑟 ≅htan 𝛽

Disconsidering the wire dimensions

LaserWire




CIRCLE FORMATION

Δ h=𝑑 (sin 𝛽−sin 𝛽 ′)

Wire diameter

Horizontal inclination

Horizontal inclination after

refraction

Much smaller than other experiment

dimensions

can be disconsidered




𝑤=𝑧 𝜆𝑑

INTERFERENCE

DOUBLE SLIT PATTERN OF INTERFERENCE

Wire diameterDistance

between two dark

shadows

Light wave-lengthDistance to screen




PREDICTION OF THE RELEVANT PARAMETERS

Incident angleThe measure of the radius

Light intensity distribution

Wire dimensionsThinner wires will turn easier

to see the interference

Refraction Index Light intensity distribution




EXPERIMENTS

MATERIALS

• Nylon wires (n=1.56) with different diameters (0.405mm, 0.7mm, 0.8mm)

• Copper wire of 150 • Micrometer• Laser of wave-length

532nm• Chalk powder




EXPERIMENTS

CONIC FORMATIONConic

formation

Height and radius

Light intensty

Intensity




0 5 10 15 20 250

5

10

15

20

25𝜷=45°±1°

Height [cm]

Rad

ius

[cm

]

EXPERIMENTS

RELATION BETWEEN HEIGHT AND RADIUSConic

formation

Height and radius

Light intensty

Intensity




EXPERIMENTS

LIGHT INTENSITY DISTRIBUTION

Point of maximun reflection

Point of maximun refraction +

𝛽=35 .0 °±0.5 Conic formation

Height and radius

Light intensty

Intensity




EXPERIMENTS

DIFFERENT SHAPES

It will form a circle

It will form an ellipse

It will form a parabola

It will form a hyperbole

Laser

Screen

Wire




EXPERIMENTS

DIFFERENT SHAPES

Circle

Ellipse

Parabole

Hyperbole




𝑑𝑧𝑤=𝜆

𝑑=150𝜇𝑚±5𝜇𝑚𝑧=2,79𝑚±0,05𝑚𝑤=1,00𝑐𝑚±0,05𝑐𝑚 𝜆𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙=

150∗10− 6

2,7910− 2

𝜆𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙=537𝑛𝑚±33𝑛𝑚

EXPERIMENTS

INTERFERENCE

Copper wire, only reflects the light




EXPERIMENTS

INTERFERENCE

Interference+

Reflection+

Refraction0.2 mm nylon wire




SUMMARY:

THEORETICAL INTRODUCTION




SUMMARY:

EXPERIMENTS




CONCLUSIONS

RELEVANT PARAMETERS

Increase reflectanceLaser inclination

Refraction Index

Define the shape of figureShape of the wire

Position of the surface

Visibility of interference Thinner wire diameter




BIBLIOGRAPHY

• Vector form of Snell’s Law, Available on http://www.starkeffects.com/snells-law-vector.shtml, Access on 20 April

• REFLECTIVITY; Wikipedia. Available on <http://en.wikipedia.org/wiki/Reflectivity> Access on 10 november

• FRESNEL EQUATIONS; Wikipedia. Available on <http://en.wikipedia.org/wiki/Fresnel_equations> Access on 10 november

• KONG, H. J. ; CHOI, Jin; SHIN, J. S.; YI, S. W.; JEON, B. G.; Hollow conic beam generator using a cylindrical rod and its performances.

• MV Klein & TE Furtak, Optics, 1986, John Wiley & Sons, New York; Huygens Principle

http://www.starkeffects.com/snells-law-vector.shtml



http://en.wikipedia.org/wiki/Reflectivity

http://en.wikipedia.org/wiki/Fresnel_equations

http://en.wikipedia.org/wiki/Fresnel_equations




Thank you!Thank you!




REFRACTION

𝜃=arcsin (𝑥𝑟 )𝛼=𝜃−𝜃 ′




HEIGHT VARIATION

Δ h=𝑑 (sin 𝛽−sin 𝛽 ′)

Δ h=2𝑑∗ sin 𝛽− 𝛽′

2∗ cos 𝛽+𝛽 ′

2≤1

Δ h≤2𝑑




RADIUS VARIATION (I)

Δ 𝑟=Δhtan 𝛽

Δ h=𝑑 (sin 𝛽− sin 𝛽 ′ )




RADIUS VARIATION (II)

Before refraction light velocity on this projection:

Δ 𝑟=𝑑 cos𝛽 (1− 𝜂cos𝛼 )

sin 𝛽cos𝛼

𝑛1=sin 𝛽 ′𝑛2

Δ h=𝑑 sin 𝛽 (1− 𝜂cos𝛼 )




RADIUS VARIATION (III)

Δ 𝑟=𝑑 cos𝛽 (1− 𝜂cos𝛼 )

≤1 ¿1

Δ 𝑟<𝑑

|𝛼|< 𝜋2




LIGHT INTENSITY

FRESNEL’S EQUATIONS

𝑅1 𝑠 (𝑥 )=(𝑛𝑎𝑖𝑟 cos𝜃 cos 𝛽 –𝑛𝑤𝑖𝑟𝑒√1−( 𝑛𝑎𝑖𝑟

𝑛𝑤𝑖𝑟𝑒)2

∗ (1− cos2𝜃 cos2 𝛽 )2

𝑛𝑎𝑖𝑟 cos𝜃 cos𝛽+𝑛𝑤 𝑖𝑟𝑒√1−( 𝑛𝑎𝑖𝑟


∗ (1− cos2𝜃 cos2 𝛽)2 )2

𝑅1𝑝 (𝑥)=(𝑛𝑎𝑖𝑟 √1−(( 𝑛𝑎𝑖𝑟


∗ (1−cos2𝜃 cos2 𝛽 )2)2

−𝑛𝑤𝑖𝑟𝑒cos𝜃 cos 𝛽

𝑛𝑎𝑖𝑟√1−(( 𝑛𝑎𝑖𝑟


∗ (1− cos2𝜃 cos2𝛽 )2)2

+𝑛𝑤𝑖𝑟𝑒 cos𝜃 cos𝛽 )2

Paralel polarized light

Perpendicular polarized light




LIGHT INTENSITY

FRESNEL’S EQUATIONS

Documents

Team of Brazil Problem ## Title Diego de Moura, Felipe de Melo, Matheus Camacho, Thiago Bergamaschi, Thiago KalifeNakhon Ratchasima, 27 June – 4 July 2015