Developing ideas of Developing ideas of refraction, lenses and refraction, lenses and

rainbow through the use rainbow through the use of Medieval Resources of of Medieval Resources of

Arabs and PersiansArabs and Persians

Pavlos MihasPavlos Mihas

Democritus UniversityDemocritus University

Learning to apply a law

Linear approximation

incidenceofanglerefractionofangle3

2

This approximation holds up to 40°

Quadratic approximation

angleincidencebangleincidenceanglerefraction ²0025,0

Usually students do not learn to apply general principles.In optical phenomena there is a need to learn to apply at least qualitatively Snell’s law.There are some useful approximations:

Usefulness of a law

It is good to have an idea of what are the values expected.

This helps the students to repeat a measurement.

A comparison with Historical measurements also helps to see how well they perform

So we do measurements and compare with “law – like” relations and historical data.

Historical data and methods

Ptolemy’s method: A semicircle and 3 pins

At first we put a pin B at the center and then we choose the angle of incidence where we put A. A pin C is moved along the curved side until the 3 pins overlap.

Usually it works well up to 80°.

Extensions of Ptolemy’s law

The same idea can be applied to other shapes

Which Law?PTOLEMY’S RESULTS FOR

REFRACTION FROM AIR TO WATER

Angle of Incidence

Angle of Refraction

Increase in the Angle of Refraction

Difference of Increase

0° 0° 8°00’

10° 8° 30’ 7°30’

20° 15°30’ 30’ 7°00’

30° 22°30’ 30’ 6°30’

40° 29° 30’ 6°00’

50° 35° 30’ 5°30’

60° 40°30’ 30’ 5°00’

70° 45°30; 30’ 4°30’

80° 50°

PTOLEMY’S RESULTS FOR REFRACTION FROM AIR TO

GLASS Angle of Incidence

Angle of Refraction

Increase in the Angle of Refraction

Difference of Increase

0° 0° 7°00’

10° 7° 30’ 6°30’

20° 13°30’ 30’ 6°00’

30° 19°30’ 30’ 5°30’

40° 25° 30’ 5°00’

50° 30° 30’ 4°30’

60° 34°30’ 30’ 4°00’

70° 38°30; 30’ 3°30’

80° 42°

It seems that Ptolemy gave results that have constant Differences of the Increase of the Refraction Angle

Does this hold also for his air to glass data?

Ptolemy’s results and quadratic lawy = -0,0025x2 + 0,725x

R2 = 1GLASS

y = -0,0025x2 + 0,825x

R2 = 1WATER

0

10

20

30

40

50

60

0,00 20,00 40,00 60,00 80,00

INCIDENCE ANGLE

RE

FR

AC

TIO

N A

NG

LE Angle of Refraction

WATER

Angle of RefractionGLASS

Quadratic (GLASS)

Quadratic (WATER)

s

Students measurements can be very well fit in a quadratic law

Why we have to believe Snell’s Law?

At this point we can stress the importance of theory in experiments

Scientists design their experiments according to the theory they hold.

The data they collect reflect the theory or Law they believe.

Ptolemy believed in “visual Rays” ὄψεις, while Al Haytham الهىثام believed that light comes to our eyes from the bodies.

Al Haytham’s method Al Haytham employed a

light beam. He used two diaphragms (UHF, hole on MPQN) to make a “parallel beam”

He measured the angle of deviation KCI

He did not give one general relation but some rules.

His rules are not accurate for ranges that exceed Ptolemy’s results.

Students can check Al Haytham’s rules

Checking Al Haytham’s rulesChecking Al Haytham’s rules•In rare to dense refraction, deviation angle<1/2 incidence angle

Interpretation of the LawInterpretation of the Law

Corpuscular InterpretationCorpuscular Interpretation Wave InterpretationWave Interpretation

Al Haytham’s interpretationAl Haytham’s interpretation• Al Haytham proposed a corpuscular

interpretation• He expressed the idea of vector Analysis:

There is a portion (قسط koust) of velocity in a direction parallel to the surface and a portion perpendicular to the surface .

• According to Al Haytham, light must be deviated toward the path of least resistance.

• The resistance is smallest in the direction perpendicular to the surface.

• If refraction is a weaker case of reflection, why refraction is not away from the perpendicular?

Newton’s InterpretationNewton’s Interpretation• A kind of gravitational

field exists.• This field accelerates

the particles when they enter in a “denser” medium, and decelerates them when they enter into a rarer medium.

• When the particles exit then they are decelerated.

This interpretation avoids all the difficulties that were presented by Al Haytham’s treatment. It can be used to describe the behavior of lenses and prisms.

Which model to teach?Which model to teach?• In a 8th grade Greek textbook is

presented Fermat’s principle.• In a 7th grade Albanian textbook is

presented a corpuscular theory.

Particle model for reflection in Prifti et al (2003): elastic ball on the left impinging on the floor and light on the right impinging on a mirror

Which model for refraction?Which model for refraction?• It is inconsistent to teach refraction

with a particle model

Refraction of sound in water (air in the upper part) (Prifti et al 2003). On the right is shown a wave refraction

Wave model vs. corpuscular Wave model vs. corpuscular modelmodel

• Wave model is easier to understand and gives a grasp to the interested students.

• Corpuscular model is misleading. It directs the student’s thought to classical bodies. For example the explanation of the shadow in comparison to rain or dust.

• Photon model should be taught because of quantum mechanics, but the ideas of quantum probabilities and Feynman’s path integrals are not appropriate for students of 7th of 8th grade.

Refraction and the problem Refraction and the problem of focusingof focusing

• Ibn Sahl’s law of refraction

Ibn Sahl's expression of Snell's law

Ibn Sahl’s perfect focusingIbn Sahl’s perfect focusing

Ibn Sahl’s Hyperbolic Lens

Refraction on a Sphere: Special raysRefraction on a Sphere: Special rays

• The ray that Al Haytham proposed as a limiting case was employed by Al Farisi to divide the incident rays to the exterior and interior cone.

ci

ci

ci

cinn

lim

lim(

2

1

))sin(

arcsin)sin(

arcsin

Relation of rays to the special ray of critical angle

of incidence. Rays with a larger incidence angle meet the sphere at a point nearer to the axis.

If we consider the point where the ray meets the sphere after refraction we can seeA that there is a limit for a special angle

The PARALLEL rays cut the SPECIAL ray either inside the sphere or outside.Geometrically we have

Refraction and the RainbowRefraction and the Rainbow

External and Internal cones of Al Farisi

Refraction and ObservationsRefraction and Observations• Al Haytham’s Theory about the

influence of atmospheric refraction on the observation of astronomical phenomena

• Al Haytham proposed that the atmosphere causes a refraction of the light. This causes a change in the angle where the Moon is seen.

Al Haytham’s modelAl Haytham’s model• He proposed several factors for the

change of the appearance:• Refraction by air• Possibility of existence of a layer of

vapors over the atmosphere.• Possibility of an “error of

observation”

What model for the What model for the atmosphere:atmosphere:

Atmosphere: VaporsAtmosphere: Vapors