chapitre i : fermat - descartes

Ray OpticsRay OpticsErasmusErasmus MundusMundus OptSciTechOptSciTechNathalie Nathalie WestbrookWestbrook

30 30 teaching hours teaching hours ((every wednesdayevery wednesday 99--12am) 12am) includingincluding lectures, lectures, problemsproblems in class in class andand regular assignmentsregular assignments, as , as many many

labslabs as possible,as possible, tutoringtutoring((see NW’s homepage see NW’s homepage on on wwwwww..atomopticatomoptic..frfr))

Reference Reference booksbooks ((available at theavailable at the Institut d’OptiqueInstitut d’Optique librarylibrary): ): «« OpticsOptics » by E. » by E. Hecht Hecht (chap5 (chap5 Geometrical opticsGeometrical optics--paraxial theoryparaxial theory))

«« Modern Modern Optical Optical EngineeringEngineering » by W. J. Smith (» by W. J. Smith (chap chap 22--44--55--66--9)9)

MyMy lecture notes «lecture notes « Ray Ray opticsoptics » » translatedtranslated in in englishenglish, in print,, in print, also also available available on on the webpagethe webpage

Subject covered in this course:

Image formation and optical instrumentsin the paraxial approximation

Subject covered in this course:

Image formation and optical instrumentsin the paraxial approximation

Optical systemThe scene

(Light sources) Image detector

complementarycomplementary to «to « sourcessources and detectorsand detectors » in 1st» in 1st semestersemester,,and followedand followed by «by « opticaloptical designdesign » in 2nd» in 2nd semestersemester

Applications of complex optical systemsApplications of complex optical systems

Microscopy

Photography

Astronomy

Very Large Telescope with adaptive optics, Chile

SPOT satellite, earth observation

Single molecule fluorescence microscopy

Simplifications of light propagationSimplifications of light propagation

Electromagnetic waves

Ray propagation

Paraxial approximation

•Maxwell’s equations•Wavefronts•Interference•Diffraction

•Fermat’s principle•Straight trajectories (in homogenous medium)•Rays perpendicular to wavefronts•Snell’s law (loi de Descartes)•Diffraction added with stops and apertures

•Perfect imaging•Focal length, principal points•Aberrations added with 3rd order approx, wavefront deformation

λ→0

y, α→0

Ray Optics Ray Optics -- Image formationImage formation 55

Ray propagation: Fermat’s principleRay propagation: Fermat’s principle

TheThe opticaloptical pathpath lengthlength ((takingtaking intointo accountaccount thethe index of index of refractionrefraction alongalong thethe pathpath) ) isis extremum.extremum.

AA

BB ∫=B

A

ndsL

0=LδPierre de Fermat

(1601-1665)


Snell’s law (or ‘loi de Descartes’)Snell’s law (or ‘loi de Descartes’)

René Descartes (1596-1650)

( )Nininununrrr )cos()cos( 22112211 −=−

AA

BB

II

Ir

δ

Nrii11

ii22

1ur

2urnn11 nn22

1 2( )L AB n AI n IB= +

( )1 1 2 2 1 1 2 2( ) 0L AB n u I n u I n u n u Iδ δ δ δ= − = − =uruur uuruur ur uur uur

1 1 2 2n u n u aN− =ur uur uur

Fermat :Fermat :

)sin()(sin 2211 inin =in the incidence plane:

Willebrord Snell (1580-

1626)


Construction of refracted rays Construction of refracted rays

Based Based on on Huyghens’s theoryHuyghens’s theory

Index surfaces Index surfaces

Based Based on on Snell’s lawSnell’s lawVelocity Velocity surfacessurfaces

1/n1/n22 1/n1/n11 II

nn2 2 > n> n11

nn11 nn22


GlassGlass--air interface :air interface :

iicriticalcritical=42°=42°

nn1 1 > n> n22

Total internal reflectionTotal internal reflection

nn1 1 sin sin iicriticalcritical = n= n22


Snell’s law for reflectionSnell’s law for reflection

Niuurrr

)cos(221 =−

AA

II

Ir

δ

Nriiincinc

iirefref

1ur

2urnn11 nn22

in the incidence plane: incref ii −=

BB


Image formation

Quality of an optical system

Image formation

Quality of an optical system


Without adaptive optics

With adaptive optics

Stigmatism (perfect imaging) :The image of a point source is a point.

EXAMPLE: image of a star


If a system If a system is perfectly stigmatic is perfectly stigmatic for A (for A (objectobject) ) andand A’(image of A), A’(image of A), thenthen ::

AA AA ’’

nn n’n’II II ’’

II II ’’

L ( A AL ( A A ’) = Constant’) = Constant

for for any any ray ray coming fromcoming from A A passing through the optical passing through the optical system (system (Fermat’sFermat’s principleprinciple).).

Stigmatic condition in terms of optical path :


Tache d’AiryTache d’Airy

NO, because even an ideal optical system is limited by diffraction

Image of a point source : Airy function

For a point source For a point source atat infinityinfinity :: pupil entrance

'44.244.2 DfNAiry λλ ==Φ

Is perfect stigmatism really necessary?


+ there is the limitation due to the image detector :Grain size (or pixel size) of the detector

•Some optical systems do not require perfect imaging !!Lighting systems (search lights, condensers, road signs,..)

An optical system is always limited by diffraction

Why perfect stigmatism is not necessary


flat image (no flat image (no fieldfield curvaturecurvature))Constant Constant magnificationmagnification (no (no distortiondistortion))AchromatismAchromatismSufficient luminous Sufficient luminous fluxfluxUniform Uniform illuminationillumination

Other qualities of an optical system Other qualities of an optical system


No, No, unfortunatelyunfortunately !!!!!!

EvenEven a plane a plane refractiverefractive surface or a surface or a spherical mirrorspherical mirror

SameSame for simple for simple lenseslenses: : planconvex lensplanconvex lens((there is there is aabetterbetter orientation),orientation), biconvex lensbiconvex lens

Do simple systems make perfect images?

Do simple systems make perfect images?


MIROIR SPHERIQUE

C

objet � l’infini sur l’axe

DIOPTRE PLAN

n=1 n=1.5

objet

Plane Plane refractiverefractive surfacesurface Spherical mirrorSpherical mirror

Object at infinity Object at infinity on axison axis

ObjectObject

centercenter of of curvaturecurvature


YesYes!!

butbut onlyonly for a for a specific specific pairpair of of conjugate conjugate pointspoints

Are there simple optical systems that are perfectly stigmatic ?

Are there simple optical systems that are perfectly stigmatic ?


Only the Only the plane plane mirrormirror isis alwaysalways stigmaticstigmatic, , otherothermirrorsmirrors are are onlyonly stigmaticstigmatic for for specificspecific pointspoints

SphericalSpherical ((centercenter), ), parabolicparabolic ((objectobject atatinfinityinfinity), ), elliptical and hyperbolic mirrorselliptical and hyperbolic mirrors((focifoci of of thethe conicalconical formsforms))Application to Application to telescopestelescopes

Stigmatic points for mirrorsStigmatic points for mirrors


Stigmatic points for a refractivesurface

Stigmatic points for a refractivesurface

Perfect stigmatism for a refractive surface: nAI + n’IA’=K (cst)

C S1AA’n’=1

n

IA and A’ : one real and one virtual

one inside, one outside the sphere

S2S1A/S1A’= S2A/A’ S2 = n’/n

K≠0: Descartes Ovoïds

K=0: IA/IA’=cst ⇒ spherical surface


R= SC, CA =R.n’/n,CA’=R.n/n’

C SAA’n’=1

n

I

Weierstrass or aplanetic points:Weierstrass or aplanetic points:

Stigmatic points for a spherical refractive surface

Stigmatic points for a spherical refractive surface


C S

n’=1nAA’

Immersion oil : n ≈ 1,5

Problem : place the object at point A… insidethe lens !!!Problem : place the object at point A… insidethe lens !!!

Large aperture angle in the object plane, reduced after the lens

Application to microscope objectivesApplication to microscope objectives


AsphericalAspherical surfaces or surfaces or aspherical lensesaspherical lenses

Other stigmatic lenses Other stigmatic lenses


AreAre there perfect optical systemsthere perfect optical systems forforseveral several pair ofpair of conjugateconjugate points? points?

BUTBUT

We can maintain We can maintain approximate stigmatismapproximate stigmatism ::

-- eithereither in a plane orthogonal toin a plane orthogonal to thethe axis (axis (aplanetismaplanetism))

-- or or along thealong the axis (Herschel Condition)axis (Herschel Condition)

AplaneticAplanetic single surface must single surface must be be sphericalspherical

No ,No , unfortunatelyunfortunately !!!!!!


Approximate stigmatism in a plane: aplanetism

Abbe sine condition

Approximate stigmatism in a plane: aplanetism

Abbe sine condition


AA AA ’’

II’’

nn n’n’

HypothesisHypothesis : : centered optical centered optical system system perfectly perfectly stigmaticstigmatic for A for A andand AA ’’

Entrance Entrance pupilpupil

Exit Exit pupilpupilAperture Aperture

stopstop

Fermat :Fermat : I )'( ∀=cstAAL


AA

AA ’’

IIII ’’

δδAA

δδAA ’’

αααα ’’

n’n’nn Aperture Aperture stopstop

BB

BB ’’

The The system system is perfectly stigmaticis perfectly stigmatic for B for B andand B’ if :B’ if :

ThusThus ::

uuu’u’

I )'( ∀=cstBBLcstAALBBLL =−=Δ )'()'(

cstuAnuAnL =⋅+⋅−=Δ '''δδ


Abbe sine condition : a fundamental theorem for imaging optical systemsAbbe sine condition : a fundamental theorem for imaging optical systems

'sin''sin αα ynny =

AA A’A’

II II ’’δδAA

δδAA ’’

αααα ’’


yy

yy ’’

enPenP exPexP


Abbe condition for an object at infinityAbbe condition for an object at infinity

'sin'' αθ ynnh =−

AA

FF ’’II II ’’δδAA

δδAA ’’

θθ αα ’’hhyy

yy ’’PP

enPenP exPexPAperture Aperture stopstop


Approximate stigmatismalong the axis:

Herschel’s condition

Approximate stigmatismalong the axis:

Herschel’s condition


AA A’A’

IIII ’’

αααα ’’


BB B’B’

Now Now B B andand B’ are B’ are slightly displaced alongslightly displaced along thethe opticaloptical axis :axis :

ThusThus ::

δδAAδδAA ’’uu

u’u’

I )'( ∀=cstBBLcstAALBBLL =−=Δ )'()'(

0)'( ==+−=+−=⋅+⋅−=Δ

ααδδαδαδδδ

'''cos''cos'''

xnxnxnxncstuAnuAnL


Herschell conditionHerschell condition

)2/'(sin'')2/(sin 22 αδαδ xnxn =

Condition for almost perfect imaging along the optical axis:

general in satisfied both becannot conditions Herscheland Abbe '

0122

22

αα

αααα

αα

±=⇒

====⇒+

=⇒

=⇒

)'()/'cos(/)/cos(

sinsin

)/'sin(/)/sin(

cstAbbeHerschelcstα'/αAbbe

cstHerschel


Paraxial approximationParaxial approximation

Linearized form of Abbe and Herschel conditions:

' 'ny n y 'α α=2 2' ' 'n x n xδ α δ α=

Lagrange invariantLagrange invariant

Small Small objectobject AND Small aperture :AND Small aperture :

(y,(y,δδxx)) αα

Satisfied for all conjugate points!

chapitre i : fermat - descartes

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