Chapter 3 Geometrical Optics

Transcription

1 Chapter 3 Geometrical Optics Gabriel Popescu University of Illinois at Urbana Champaign Beckman Institute Quantitative Light Imaging Laboratory Principles of Optical Imaging Electrical and Computer Engineering, UIUC

2 Objectives Introduction to geometrical optics and Fourier optics precedes Microscopy 2

3 3. Geometrical Optics If the objects encountered by light are large compared to wavelength, the equations of propagation can be greatly simplified (λ0) i.e. the wave phenomena (scattering, interference, etc) t) are neglected In homogeneous media, light travels in straight lines rays G.O. deals with ray propagation trough optical media (eg. Imaging systems) Ob black box imaging system Im Optical Axis 3

4 3. Geometrical Optics G.O. predicts image location trough complicated systems; accuracy is fairly good Nowadays there are software programs that can run ray propagation trough arbitrary materials So, what are the laws of G.O.? 4

5 3.2 Fermat s principle a) n = constant b) n = n( r ) = function of position L B ds B A c c v vr () n nr () L Time: t nl AB v ds c dt n () s ds v c B straight line t n() s ds AB (3.) c A A 5

6 3.2 Fermat s principle Fermat s Principle is reminiscent of the following problem that you might have seen in highschool: Someone is drowning in the Ocean at point (x,y) The lifeguard at point (u,w) can travel across the beach at speed v and in the water at speed v 2. What is his best possible path? v 2 (x,y) (u,w) v 6

7 3.2 Fermat s principle Definition: S ct n() s dsoptical path length How can we predict ray bending (eg. mirage)? Fermat s Principle: Light connects any two points by a path of minimum time (the least timeprinciple) (3.2) A B B nsds ( ) 0 A (3.3) If n=constant in space, AB=line, of course 7

8 3.3 Snell s Law Consider an interface between 2 media: y B θ θ 2 X x n The rays are bent such that: n sin n X A sin 2 2 Snell s law (3.4) can be easily derived from Fermat s principle, by minimizing: S n AO n OB total path length 2 Take it as an exercise n 2 (3.4) demo available 8

9 3.3 Snell s Law n n 2 Consequences of Snell s Law: a) If n 2 >n Ѳ 2 < Ѳ (ray gets closer to normal) b)if n 2 < n quite interesting! n sin 2 sin (3.5) n 2 Ray gets away from normal n y n 2 2 n 2 < n k 2 So, if n sin 2 n 2 2 θ k θ 2 NO TRANSMISSION x k 2 k demo available 9

10 3.3 Snell s Law c The angle of incidence for which n is called critical angle sin c n 2 This is total internal reflection c) law of reflection n n Snell s law is: 2 (3.6) n r y n 2 θ t n sin n sin 2 2 (3.7) 2 (reflection law) Energy conservation: P t + P r = P i θ 2 θ i x demo available 0

11 3.4 PropagationMatrices ing.o Efficient way of propagating p grays through optical systems y Optical System O x θ θ 2 y y2 OA Optical Axis Any given ray is completely determined at a certain plane by the angle with OA, Ѳ, and height wrtoa w.r.t OA, y Let s propagate (y, Ѳ ), assume small angles Gaussian approximation

12 3.4 PropagationMatrices ing.o a) Translation 2 y y dtan 2 y θ 2 y OA d Small angles: y y d 2 0 y 2 (3.8) 2

13 3.4 PropagationMatrices ing.o a) Translation We can re write in compact form: y d y (3.9) 3

14 3.4 PropagationMatrices ing.o b)refraction spherical dielectric interface y n α Snell s law: Geometry: θ n n (3.0) y y 2 R R n n 2 n y n y R R n y R θ 2 2 α 2 C x n 2 OA 4

15 3.4 PropagationMatrices ing.o b)refraction spherical dieletric interface y y 0 n y n ( ) 2 n R n So: y2 y n n n 2 2 n R n 2 2 (3.) 5

16 3.4 PropagationMatrices ing.o b)refraction spherical dieletric interface Important: To avoid confusion between Ѳ and Ѳ angles, use sign convention. angle convention + OA Counter clock wise = positive 2. distance convention Left negative Right positive A B OA + 6

17 3.4 PropagationMatrices ing.o b)refraction spherical dieletric interface Example: R R + We found 0 n n n 2 nr n 2 2 and 0 n n n 2 nr n 2 2 Same+/ convention applies to spherical mirrors. Without sign convention, it s easy to get the wrong numbers. 7

18 3.4 PropagationMatrices ing.o c) Dielectric interface particular case of R n n 2 θ θ 2 OA 0 0 lim n n n n R 2 0 nr n n (3.2) 8

19 3.4 PropagationMatrices ing.o The nice thing is that cascading multiple optical components reduces to multiplying matrices (linear systems) Example: n n 2 n 3 n 4 A B T R T 2 R 2 T 3 R 3 yb ya T R T R T R T B A (3.3) Note the reverse order multiplication (chronological order) T 4 9

20 3.4 PropagationMatrices ing.o Note the reverse order multiplication (chronological order) T = Translation matrix = d 0 0 n n n nr n 2 2 R=refraction matrix = 2 20

21 3.5 The thick Lens t n = n 2 = B A Typical glass: n =.5 Basicopticalcomponent: component: typically 2 spherical surfaces yb ya R T R. B t A B A 0 0 t ya n n. n 0 A R nr n 2 R n M R 2 2

22 3.5 The thick Lens After some algebra: M In general t t C R n (3.4) t t ( C C CC ) C n nr 2 C n R n 2 convergence of spherical surface R > 0, R 2 < 0 C > 0 & C 2 > 0 convergent Note [C] = m = dioptries 22

23 3.5 The thick Lens Definition: t C C CC 2 2 f n (3.5) f is the focal distance of lens Eq (3.5) is the lens makers equation demo available 23

24 3.6 Cardinal points y O Image Formation Ray Tracing F F O z y O = object; O =image ; O O =conjugate points F = focal point image (image of objects from ) F = focal point object Transverse magnification: y' M y (3.6) 24

25 3.6 Cardinal points Definition: principal p planes are the conjugate planes for which M = F f H H f F H, H = principal planes f, f = focal distances! f, f measured from H 25

26 3.7 Thinlens Particular use: t 0 Transfer matrix for thin lens: t t C R n 0 lim t0 t t ( C C2) ( C C2 CC 2 ) C 2 n nr 2 Since C C ( n)( ) 2 f R R 2 (Note R > 0, R 2 < 0) 0 M = thin lens (3.7) f 26

27 3.7 Thinlens Remember other matrices: Translation: d T (3.8) 0 0 R n n n (3.9) nr n M 2 (3.20) R Refraction spherical surface: 2 Spherical mirror: (f = R/2) 27

28 3.7 Spherical Mirrors Convergent f R C f 2 Divergent 28

29 3.8 Ray Tracing thin lenses L y B A θ F f f F A y x x θ B = convergent lens; f > 0 = divergent lens; f < 0 29

30 3.8 Ray Tracing thin lenses y' y TMT x' f x ' 0 x' xy 0 0 f x' xx' x x' f f y x f f (3.2) 30

31 3.8 Ray Tracing thin lenses y' A By ' C D y' Ay B ; y can be found as: (3.22) Condition for conjugate planes: y OA y (Figure page III 3) For conjugate planes, y should be independent of angle Ѳ B = 0 i.e. stigmatism condition (points are imaged into points) We neglect geometric/chromatic aberrations 3

32 Quiz y OA y Explain how Fermat s principle works here.

33 Solution n S n S n() s ds Because all of the rays leaving a given point converge again in the image, we know from Fermat s principle that their paths must all take the same amount of time. Another way to say this is that all the paths have the same optical path length. This is because those paths that travel further in the air, have a shorter distance to travel in the more time expensive glass. If the optical path lengths were not the same, the image would not be in focus because rays from a single point would be mapped to several points.

34 So, B = Ray Tracing thin lenses xx' x x' 0 f x' x f (3.23) Eq above is the conjugate points equation (thin lens) Eq 3.22 becomes: y = ya M x' A Transverse magnification (3.24) f 34

35 3.8 Ray Tracing thin lenses Use Eq 3.23: M x' x' f x ' x x' 0 (inverted image) x M x' ' x (3.25) 2 f y'? If object and image space have different refractive indices, 3.23 has the more general form: n' n (3.26) x' x f 35

36 3.8 Ray Tracing thin lenses xx' n' f f = focal distance in air x' xnf Let s differentiate (3.26) for air, n =n=: dx' x' dx' dx' dx x 2 2 x' x 2 dx 2 M dx (3.27) Eq 3.27 says that if the object gets closer to lens, the image moves away! 36

37 3.8 Ray Tracing thin lenses y F F x x OA y y Δ OA Δ F F y demo available 37

38 3.8 Ray Tracing thin lenses What happens when x < f? 0 x' x f x' f x? F y y x F This imageisformed is by continuations of rays Sometimes called virtual images These images cannot be recorded directly (need re imaging) demo available 38

39 3.8 Ray Tracing thin lenses Other useful formulas in G.O (figure above: Δ,, Δ ) ) 2 ' f (Newton s formula) y' ' f ( lens formula ) y f (3.28) demo available 39

40 3.9 System of lenses The image through one lens becomes object for the next lens, etc x 2 L L x L x 2 L 2 y f f y F f 2 y 2 f 2 x x y = y 2 Apply lens equation repeteatly. Or, use matrices L L 2 B A A B T T 2 A, A = conjugate through L L B,B = conjugate through L 2 demo available 40

41 3.9 System of lenses Use T = T 2.T ; T matrix from 3.2 T Note: 2 ; x ' 0 f (3.29) f x f x f ' x magnification x ' M x f x x 4

42 3.9 System of lenses x f M (3.30) x ' also: M f det(t ) = T T 2. T M2 0 M 0 f 2 M 2 f M MM 2 0 M f2 fm2 MM 2 Transverse Magnification y ' f f ' y' M y ' y y' M (3.3) y' 42

43 2 lens system is equivalent to: M f f f M M M M System of lenses Microscopes achieve M=0 00 easily Can be reduced to 2 lens system Question: cascading many lenses such that M=0 6, would we be able to see atoms? Well, G.O can t answer that. So,back to wave optics 43