11

Huygens’ and Fermat’s Huygens’ and Fermat’s principles (Hecht 4.4, 4.5)principles (Hecht 4.4, 4.5)

Application to reflection & refraction at an Application to reflection & refraction at an interfaceinterface

Monday Sept. 9, 2002Monday Sept. 9, 2002

22

Huygens’ wave front constructionHuygens’ wave front construction

Given wavefront at t

Allow wavelets to evolve for time Δt

r = c Δt ≈ λ

New wavefront

What about –r direction? See Bruno Rossi Optics. Reading, Mass: Addison-Wesley Publishing Company, 1957, Ch. 1,2

for mathematical explanation

Construct the wave front tangent to the wavelets

33

Plane wave propagationPlane wave propagationNew wave front is New wave front is still a plane as long still a plane as long as dimensions of as dimensions of wave front are >> wave front are >> λλIf not, edge effects If not, edge effects become importantbecome importantNote: no such thing Note: no such thing as a perfect plane as a perfect plane wave, or collimated wave, or collimated beambeam

44

Geometric OpticsGeometric OpticsAs long as apertures As long as apertures are much larger than are much larger than a wavelength of light a wavelength of light (and thus wave fronts (and thus wave fronts are much larger than are much larger than λλ) the light wave front ) the light wave front propagates without propagates without distortion (or with a distortion (or with a negligible amount)negligible amount)i.e. i.e. light travels in light travels in straight linesstraight lines

55

Physical OpticsPhysical OpticsIf, however, If, however, apertures, apertures, obstacles etc have obstacles etc have dimensions dimensions comparablecomparable to to λλ (e.g. < 10(e.g. < 1033 λλ) then ) then wave front wave front becomes distortedbecomes distorted

66

Let’s reflect for a momentLet’s reflect for a moment

77

Hero’s principleHero’s principle

Hero (150BC-250AD) asserted that Hero (150BC-250AD) asserted that the the path taken by light in going from path taken by light in going from some point A to a point B via a some point A to a point B via a reflecting surface is the shortest reflecting surface is the shortest possible onepossible one

88

Hero’s principle and reflectionHero’s principle and reflectionA B

A’

OR

O’

99

Let’s refract for a momentLet’s refract for a moment

1010

Speed of light in a mediumSpeed of light in a medium

n

cv

Light slows on entering a medium – Huygens

Also, if n → ∞ = 0

i.e. light stops in its track !!!!! See:

P. Ball, Nature, January 8, 2002

D. Philips et al. Nature 409, 490-493 (2001)

C. Liu et al. Physical Review Letters 88, 23602 (2002)

1111

Snel’s lawSnel’s law

1621 - Willebrord Snel (1591-1626) 1621 - Willebrord Snel (1591-1626) discovers the law of refractiondiscovers the law of refraction

1637 - Descartes (1596-1650) 1637 - Descartes (1596-1650) publish the, now familiar, form of the publish the, now familiar, form of the law (viewed light as pressure law (viewed light as pressure transmitted by an elastic medium)transmitted by an elastic medium)

nn11sinsin11 = n = n22sinsin22

1212

Huygens’ (1629-1695) Principle:Huygens’ (1629-1695) Principle:Reflection and Refraction of lightReflection and Refraction of light

Light slows on entering a mediumLight slows on entering a medium

Reflection and Refraction of WavesReflection and Refraction of Waves

Click on the link aboveClick on the link above

1313

Total internal reflectionTotal internal reflection

n1

n2

θC

n1 > n2

1611 – Discovered by Kepler

1414

Pierre de Fermat’s principlePierre de Fermat’s principle

1657 – Fermat (1601-1665) proposed 1657 – Fermat (1601-1665) proposed a a Principle of Least Time Principle of Least Time encompassing both reflection and encompassing both reflection and refractionrefraction

““The actual path between two points The actual path between two points taken by a beam of light is the one taken by a beam of light is the one that is traversed in the least time”that is traversed in the least time”

1515

n1

n2

Fermat’s principleFermat’s principle

n1 < n2

A

O

B

θi

θr

x

a

h

bWhat geometry gives the shortest time between the points A and B?

1616

Optical path lengthOptical path length

n1

n4

n2

n5

nm

n3

S

P

1717

Optical path lengthOptical path lengthTransit time from S to PTransit time from S to P

m

iiisnc

t1

1

m

iiisnOPL

1

P

SdssnOPL )(

P

S

dsv

cOPL

Same for all rays

1818

Fermat’s principleFermat’s principlet = OPL/ct = OPL/c

Light, in going from Light, in going from point S to P, point S to P, traverses the route traverses the route having the smallest having the smallest optical path lengthoptical path length

c

OPLt

1919

Optical effectsOptical effects

LoomingLooming

MiragesMirages

2020

Reflection by plane surfacesReflection by plane surfacesr1 = (x,y,z)

x

y

r2 = (x,-y,z)

Law of Reflection

r1 = (x,y,z) → r2 = (x,-y,z)

Reflecting through ((x,z) plane

x

y

zr2= (-x,y,z)

r3=(-x,-y,z)

r4=(-x-y,-z)

r1 = (x,y,z)

2121

n2

Refraction by plane interfaceRefraction by plane interface& Total internal reflection& Total internal reflection

n1

n1 > n2

θC

P

θ1θ1

θ1 θ1

θ2

θ2

Snell’s law n1sinθ1=n2sinθ2

2222

Examples of prisms and total Examples of prisms and total internal reflectioninternal reflection

45o

45o

45o

45o

Totally reflecting prism

Porro Prism

2323

Imaging by an optical systemImaging by an optical system

Optical

SystemO I

O and I are conjugate points – any pair of object image points - which by the principle of reversibility can be interchanged

Fermat’s principle – optical path length of every ray passing through I must be the same

2424

Cartesian SurfacesCartesian SurfacesCartesian surfaces – those surfaces Cartesian surfaces – those surfaces which form perfect images of a point which form perfect images of a point objectobjectE.g. ellipsoid and hyperboloidE.g. ellipsoid and hyperboloid