Erasmus Mundus Mundus OptSciTech Nathalie Westbrook Ray Optics 30 teaching hours (every wednesday 9-12am) including lectures, problems in class and regular assignments,, as many labs as possible, tutoring (see NW s homepage on www.atomoptic atomoptic.fr) Reference books (available at the Institut d Optique library): «Optics» by E. Hecht (chap5 Geometrical optics-paraxial paraxial theory) «Modern Optical Engineering» by W. J. Smith (chap( 2-4-5-6-9) My lecture notes «Ray optics» translated in english,, in print, also available on the webpage
Subject covered in this course: Image formation and optical instruments in the paraxial approximation The scene (Light sources) Optical system Image detector complementary to «sources and detectors» in 1st semester, and followed by «optical design» in 2nd semester
Applications of complex optical systems Astronomy Microscopy Photography Very Large Telescope with adaptive optics, Chile Single molecule fluorescence microscopy SPOT satellite, earth observation
Simplifications of light propagation Electromagnetic waves Maxwell s equations Wavefronts Interference Diffraction λ 0 Ray propagation Fermat s principle Straight trajectories (in homogenous medium) Rays perpendicular to wavefronts Snell s law (loi de Descartes) Diffraction added with stops and apertures y, α 0 Paraxial approximation Perfect imaging Focal length, principal points Aberrations added with 3 rd order approx, wavefront deformation
Ray propagation: Fermat s principle The optical path length (taking into account the index of refraction along the path) is extremum. B B L = nds A δl = 0 A Pierre de Fermat (1601-1665) Ray Optics - Image formation 5
Snell s law (or loi de Descartes ) Fermat : LAB ( ) = nai 1 + nib 2 uruur uuruur ur uur uur δlab ( ) = nu ( ) 1 1δI nu 2 2δI= nu 1 1 nu 2 2 δi= 0 ur uur uur nu nu = an 1 1 2 2 n 1 n 2 i 1 δi r I u r 2 N r i 2 B René Descartes (1596-1650) A r nu u r 1 r n u r = ( n1 cos( i1) n2 cos( i2) )N 1 1 2 2 in the incidence plane: n 1sin( i1 ) = n2sin( i2) Willebrord Snell (1580-1626) Ray Optics - Image formation 6
Construction of refracted rays Based on Snell s law Index surfaces n 2 > n 1 Based on Huyghens s theory Velocity surfaces n 1 n 2 1/n 2 1/n 1 I Ray Optics - Image formation 7
Total internal reflection n 1 > n 2 n 1 sin i critical = n 2 Glass-air air interface : i critical =42 Ray Optics - Image formation 8
Snell s law for reflection i ref B n 1 n 2 u r 2 δi r i inc I N r A u r 1 r u r u = 1 2 in the incidence plane: r 2cos( i) N i ref = i inc Ray Optics - Image formation 9
Image formation Quality of an optical system Ray Optics - Image formation 10
Stigmatism (perfect imaging) : The image of a point source is a point. EXAMPLE: image of a star Without adaptive optics With adaptive optics Ray Optics - Image formation 11
Stigmatic condition in terms of optical path : If a system is perfectly stigmatic for A (object( object) and A (image of A), then : L ( A A ) = Constant for any ray coming from A passing through the optical system (Fermat s( principle). n n I I I I A A Ray Optics - Image formation 12
Is perfect stigmatism really necessary? NO, because even an ideal optical system is limited by diffraction Tache d Airy For a point source at infinity : Φ Airy = 2.44λ N= 2.44λ D f ' entrance pupil Image of a point source : Airy function Ray Optics - Image formation 13
Why perfect stigmatism is not necessary An optical system is always limited by diffraction + 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,..) Ray Optics - Image formation 14
Other qualities of an optical system flat image (no field curvature) Constant magnification (no distortion) Achromatism Sufficient luminous flux Uniform illumination Ray Optics - Image formation 15
Do simple systems make perfect images? No, unfortunately!!! Even a plane refractive surface or a spherical mirror Same for simple lenses: planconvex lens(there is a better orientation), biconvex lens Ray Optics - Image formation 16
Plane refractive surface Spherical mirror Object at infinity on axis center of curvature Object Ray Optics - Image formation 17
Are there simple optical systems that are perfectly stigmatic? Yes! but only for a specific pair of conjugate points Ray Optics - Image formation 18
Stigmatic points for mirrors Only the plane mirror is is always stigmatic, other mirrors are only stigmatic for specific points Spherical (center), parabolic (object at infinity), elliptical and hyperbolic mirrors (foci of the conical forms) Application to telescopes Ray Optics - Image formation 19
Stigmatic points for a refractive surface Perfect stigmatism for a refractive surface: nai + n IA =K (cst) K 0: Descartes Ovoïds K=0: IA/IA =cst spherical surface A and A : one real and one virtual one inside, one outside the sphere I S 1 A/S 1 A = S 2 A/A S 2 = n /n A S 2 A C S 1 n =1 Ray Optics - Image formation 20 n
Stigmatic points for a spherical refractive surface Weierstrass or aplanetic points: R= SC, CA =R.n /n, CA =R.n/n I A A C n Ray Optics - Image formation 21 S n =1
Application to microscope objectives Problem : place the object at point A inside the lens!!! Immersion oil : n 1,5 A A C Large aperture angle in the object plane, reduced after the lens n S n =1 Ray Optics - Image formation 22
Other stigmatic lenses Aspherical surfaces or aspherical lenses Ray Optics - Image formation 23
Are there perfect optical systems for several pair of conjugate points? No, unfortunately!!! BUT We can maintain approximate stigmatism : - either in a plane orthogonal to the axis (aplanetism( aplanetism) - or along the axis (Herschel Condition) Aplanetic single surface must be spherical Ray Optics - Image formation 24
Approximate stigmatism in a plane: aplanetism Abbe sine condition Ray Optics - Image formation 25
Hypothesis : centered optical system perfectly stigmatic for A and A Fermat : L( AA') =cst I Entrance pupil n Aperture pupil n I stop I Exit A A Ray Optics - Image formation 26
The system is perfectly stigmatic for B and B if : L( BB') =cst Thus : I Δ L = L( BB') L( AA' ) = cst B δa n α I Aperture stop I n α A A u δa u B Ray Optics - Image formation 27 Δ L = nδau + n'δa' u' = cst
Abbe sine condition : a fundamental theorem for imaging optical systems y δa n α enp I Aperture stop A A I exp α n y δa ny =n sinα ' y α' Ray Optics - Image formation 28 'sin
Abbe condition for an object at infinity y δa enp Aperture stop exp I I θ h α F A P y δa nh =n θ ' y α' Ray Optics - Image formation 29 'sin
Approximate stigmatism along the axis: Herschel s condition Ray Optics - Image formation 30
Now B and B are slightly displaced along the optical axis : L( BB') =cst Thus : I Δ L = L( BB') L( AA' ) = cst A u δa n Aperture stop I I α A α δa B B u n ΔL= nδau + n' δa' u' = cst nδxcosα + n' δx'cosα' = nδx+ n' δx' ( α= α' = 0) Ray Optics - Image formation 31
Herschell condition Condition for almost perfect imaging along the optical axis: δ 2 α n x sin ( / 2) = n' δ x' sin 2 ( α ' / 2) Herschel sin( α / 2)/sin( α'/ 2) = cst Abbe sinα'/ sinα= cst Herschel + Abbe cos( α / 2)/cos( α'/ 2) = cst = 1( α = α' = 0) α =± α ' Abbe and Herschel conditions cannot be both satisfied in general Ray Optics - Image formation 32
Paraxial approximation Small object AND Small aperture : (y,δx) α Linearized form of Abbe and Herschel conditions: nyα = n ' y ' α ' Lagrange invariant 2 2 nδ xα = n' δ x' α ' Satisfied for all conjugate points! Ray Optics - Image formation 33