Chapter 3 –Geometrical Optics

Gabriel Popescu

University of Illinois at Urbana‐Champpgaign Beckman Institute

Quantitative Light Imaging Laboratory http://light.ece.uiuc.edu

Principles of Optical Imaging Electrical and Computer Engineering, UIUC ECE 460 –Optical Imaging Objectives

. Introduction to geometrical optics and Fourier optics – precedes Microscopy

Chapter 3: Imaging 2 ECE 460 –Optical Imaging 3.1 Geometrical Optics

. If the objects encountered by light are large compared to wavelength, the equations of propagation can be greatly simplified (λ0) . i.e. the wave‐phenomena (tti(scattering, itinter ference, et)tc) are neglected . In homogeneous media, light travels in straight lines  rays . G.O. deals with ray propagation trough optical media (eg. Imaging systems)

black box Ob Im Optical Axis imaging system

Chapter 3: Imaging 3 ECE 460 –Optical Imaging 3.1 Geometrical Optics

. G.O. predicts image location trough complicated syy;stems; accuracy is fairly good . Nowadays there are software programs that can run “ray propagation ” thtrough arbitrary matilterials . So, what are the laws of G.O.?

Chapter 3: Imaging 4 ECE 460 –Optical Imaging 3.2 Fermat’ s principle

a) n = constant b) n = n(r ) = function of position B ds B L

A A c c v  vr() n nr() L 1 tnL ds 1 Time: AB vc dt n() s ds vcB  straight line 1 tnsds () (3.1)  AB  c A

Chapter 3: Imaging 5 ECE 460 –Optical Imaging 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 v1 and in the water at speed v2. What is his best possible path?

(x,y)

v2

v1 (u,w)

Chapter 3: Imaging 6 ECE 460 –Optical Imaging 3.2 Fermat’ s principle

. Definition: Sctnsds () optical path length (3.2) . How can we predict ray bending (eg. mirage)? . Fermat’s Principle: . Light connects any two points by a path of minimum time (the least time principle)

B B   (3.3)   nSdS() 0  A  A

. If n=constant in space, AB=line, of course

Chapter 3: Imaging 7 ECE 460 –Optical Imaging 3.3 Snell’s Law

. Consider an interface between 2 media: y B X θ2 θ1 x

n n 1 X 2 A . The rays are “bent” such that: (3.4)

nn1122sin  sin . Snell’ s law (3.4) can be easily derived from Fermat’s principle, by minimizing:

SnAOnOB12total path‐length . Take it as an exercise

Chapter 3: Imaging demo available 8 ECE 460 –Optical Imaging 3.3 Snell’s Law nn21

. Consequences of Snell’s Law: 2

1 a) If n2 >n1  Ѳ2 < Ѳ1 (ray gets closer to normal)

b)If n2 < n1  quite interesting!

1 n1 21 sin sin  (3.5) n2  n2 < n1  Ray gets away from normal n y 1 n2

k2 θ2 θ1 x 2 k kk  1  n 1 si1in1 . So, if 122 NO TRANSMISSION n2

Chapter 3: Imaging demo available 9 ECE 460 –Optical Imaging 3.3 Snell’s Law

. The angle of incidence c for which (3.6) nn12sinc  is called critical angle . This is total internal reflection y n1 r n2 t c) law of reflection θ2 θ1 x nn21 Snell’s law is: i

 nn1122sin sin (3.7)  12  (reflection law)

. Energy conservation: Pt + Pr = Pi

Chapter 3: Imaging demo available 10 ECE 460 –Optical Imaging 3.4 Propagation Matrices in G.O

. Efficient way of pppgropagatin g rays through optical systems y

Optical System O x θ θ 1 y 2 y1 2 OA ≡ Optical Axis

. Any given ray is completely determined at a certain plane by

the angle with OA, Ѳ1 , and height wrtw.r.t OA, y1 

. Let’s propagate (y1 , Ѳ1), assume small angles Gaussian approximation

Chapter 3: Imaging 11 ECE 460 –Optical Imaging 3.4 Propagation Matrices in G.O

a) Translation y θ 2 y1 OA d

21

yyd21tan 1

Small angles:

yy21d 1 (3.8)

21101y

Chapter 3: Imaging 12 ECE 460 –Optical Imaging 3.4 Propagation Matrices in G.O

a) Translation . We can re‐write in compact form:

yy211 d     (3.9) 2101 

Chapter 3: Imaging 13 ECE 460 –Optical Imaging 3.4 Propagation Matrices in G.O

b)Refraction‐spherical dielectric interface y

α1 n  θ2 1 x n2 θ1 y α2 1  OA C R

. Snell’s law: nn11  2 2

. Geometry: 11  22  (3. 10) yy   12 R R nn1 n 12y n  y |   11 1 2 2 2 R Rn2 Chapter 3: Imaging 14 ECE 460 –Optical Imaging 3.4 Propagation Matrices in G.O

b)Refraction‐spherical dieletric interface

. So: yy210  1

nyn111 21(1) nRn22

10 yy21    nn n (3.11) 12 1  21  n22R n

Chapter 3: Imaging 15 ECE 460 –Optical Imaging 3.4 Propagation Matrices in G.O

b)Refraction‐spherical dieletric interface . Important: To avoid confusion between Ѳ and – Ѳ angles, use “sign convention”

1. angle convention + ‐

OA

. Counter clock‐wise = positive 2. distance convention Left  negative OA Right  positive A B

‐ +

Chapter 3: Imaging 16 ECE 460 –Optical Imaging 3.4 Propagation Matrices in G.O

b)Refraction‐spherical dieletric interface . Example: R R + ‐

10 10 nn n nnn We found 12 1and 21 1  nR22 n nR22 n Same +/‐ convention applies to spherical mirrors. Without sign convention, it’s easy to get the wrong numbers.

Chapter 3: Imaging 17 ECE 460 –Optical Imaging 3.4 Propagation Matrices in G.O

c) Dielectric interface – particular case of R ∞ n n 1 θ2 2 θ1 OA

1010  lim nn n n (3.12) R 12 1 1 0 nR22 n n 2

Chapter 3: Imaging 18 ECE 460 –Optical Imaging 3.4 Propagation Matrices in G.O

. The nice thing is that cascading multiple optical components reduces to multiplying matrices (linear systems) . Example:

n2 n3 n1 n4

A B

T1 R1 T2 R2 T3 R3 T4

yyB A (3.13) TRTRTRT4332211  BA  . Note the reverse order multiplication (chronological order)

Chapter 3: Imaging 19 ECE 460 –Optical Imaging 3.4 Propagation Matrices in G.O

. Note the reverse order multiplication ((gchronological order)

1 d . T = Translation matrix =  01

10 nnn . 21 1 R=refraction matrix =  nR22 n

Chapter 3: Imaging 20 ECE 460 –Optical Imaging

3.5 The thick Lens t n =1 1 n2=1 B A

R1 R2 . Typical glass: n = 1.5 n . Basic optical component: typically ‐ 2 spherical surfaces

yyBA  RTRBt A.  BA  10  10  1 t y nn111  . A n    01   A R21 nR n  M Chapter 3: Imaging 21 ECE 460 –Optical Imaging 3.5 The thick Lens

. After some algebra: tt 1 C 1 R n M  1 (3. 14) tt ()1CCCC  C 1212 2 nnR2 nn . In general C 21convergence of spherical surface R

. R1 > 0, R2 < 0  C1 > 0 & C2 > 0  convergent . Note [C] = m‐1 = dioptries

Chapter 3: Imaging 22 ECE 460 –Optical Imaging 3.5 The thick Lens

1 t . Definition: CCCC (()3.15) f 1212n

. f is the focal distance of lens

. Eq (3.15) is the “lens makers equation”

Chapter 3: Imaging demo available 23 ECE 460 –Optical Imaging

3.6 Cardinal points

Image Formation  Ray Tracing y O’ O F F’ y’

z

. 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 (3.16)

Chapter 3: Imaging 24 ECE 460 –Optical Imaging 3.6 Cardinal points

. Definition: ppprincipal planes are the conjjgugate planes for which M = 1

F’ f H, H’ = principal planes f, f’ = focal distances F f’ ! f, f’ measured from H H H’

Chapter 3: Imaging 25 ECE 460 –Optical Imaging 3.7 Thin lens

. Particular use: t  0 . Transfer matrix for thin lens: tt 1 C 1 R n 10 lim1  t0  tt()1CC12 ()1CCCC1212  C 2 nnR2 111 . Since CC12(1)( n  ) f RR12 . (Note R1 > 0, R2 < 0) 10  M = (3. 17) thin lens 1  1 f

Chapter 3: Imaging 26 ECE 460 –Optical Imaging 3.7 Thin lens

. Remember other matrices:

1 d (3.18) . Translation: T   01 10  . Refraction‐spherical surface: R  nnn (3.19) 21 1  nR22 n 10 . Spherical mirror: M  2 (3.20)  1 (f = R/2) R

Chapter 3: Imaging 27 ECE 460 –Optical Imaging 3.7 Spherical Mirrors

Convergent

R f  C f 2

Divergent

Chapter 3: Imaging 28 ECE 460 –Optical Imaging

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

Chapter 3: Imaging 29 ECE 460 –Optical Imaging 3.8 Ray Tracing – thin lenses

yy'  TMTxfx'  '  10 1'xxy  1  1  01 1  01 f xxx'' 1'xx y ff (3.21)   1 x  1 ff

Chapter 3: Imaging 30 ECE 460 –Optical Imaging 3.8 Ray Tracing – thin lenses

yABy'   ; y’ can be found as:  ' CD  yAyB'    (()3.22)

. Condition for conjugate planes: y OA y’ (Figure page III‐13) . For conjugate planes, y’ should be independent of angle Ѳ  B = 0 . iei.e. stigmatism condition (points are imaged into points) . We neglect geometric/chromatic aberrations Chapter 3: Imaging 31 Quiz

y OA y’

Explain how Fermat’s principle works here. SltiSolution

nS

n 1 Snsds  ()

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. ECE 460 –Optical Imaging 3.8 Ray Tracing – thin lenses

xx' . So, B = 0  xx '  0 f 111    (3.23) xx' f

. Eq above is the conjugate points equation (thin lens) . Eq 3.22 becomes: y’= yA x' M A 1 Transverse magnification (3.24) f

Chapter 3: Imaging 34 ECE 460 –Optical Imaging 3.8 Ray Tracing – thin lenses

. Use Eq 3.23: y'  ?

x'11 1   Mx11'   2 fxx' x' f 0 (inverted image) x x' M  (3.25) x . If object and image space have different refractive indices, 3.23 has the more general form: nn' 1  (3.26) xx' f

Chapter 3: Imaging 35 ECE 460 –Optical Imaging 3.8 Ray Tracing – thin lenses

. xxn''  f f = focal distance in air . xxn'   f . Let’s differentiate (3.26) for air, n’=n=1: dx' dx  xx'22 2 x' dx'  dx x dx'  M2 dx (3.27) . Eq 3.27 says that if the object gets closer to lens, the image moves away!

Chapter 3: Imaging 36 ECE 460 –Optical Imaging 3.8 Ray Tracing – thin lenses

y OA FF’y’

x x’

y Δ’ OA Δ F F’ y’

Chapter 3: Imaging demo available 37 ECE 460 –Optical Imaging 3.8 Ray Tracing – thin lenses

. What happens when x < f ? 111 1 11    0 ? xxf'' x fx

y y’ F x’ F

. This image is formed by continuations of rays . Sometimes called “virtual images” . These images cannot be recorded directly (need re‐imaging)

Chapter 3: Imaging demo available 38 ECE 460 –Optical Imaging 3.8 Ray Tracing – thin lenses

. Other useful formulas in G.O ((gfigure above: Δ, Δ’) . '  f 2 (Newton’s formula) (3.28) yf'' . (“lens formula”) yf

Chapter 3: Imaging demo available 39 ECE 460 –Optical Imaging 3.9 System of lenses

. The image through one lens becomes object for the next lens, x ’ etc L1 x2 L2 2

y2’ y1 f2 f2’

f1 f1’ y1’ F1

x1 x1’ y1’ = y2 . Apply lens equation repeteatly. Or, use matrices L1 L2 B A A’ B’

T1 T2 . A, A’ = conju gate throu gh L1

. B,B’ = conjugate through L2 Chapter 3: Imaging demo available 40 ECE 460 –Optical Imaging 3.9 System of lenses

. Use T = T2.T1 ; T matrix from 3.21

x1 ' 10 f1 T   ()(3.29) 1 1 x 1 ff1  . Note: x1 11 11x1    f111xx' x 1 11 1  magnification xM11'

Chapter 3: Imaging 41 ECE 460 –Optical Imaging 3.9 System of lenses x 1  11 fM 11 (3.30) x1 ' also: 1M1 y  f1  ' y' f f '

det(T1) = 1 y'  TTT21. Transverse Magnification M  MM2100 y  11 11  '1y    f22MfM 11  yM'  MM12 0  M 11 1 (3.31)  ffMMM21212 Chapter 3: Imaging 42 ECE 460 –Optical Imaging 3.9 System of lenses . 2‐lens system is equivalent to: 11M 1 ff212 fM

MMM12

. Microscopes achieve M=10‐100 easily . Can be reduced to 2‐lens system . Question: cascading many lenses such that M=106, would we be able to see at?toms? . Well, G.O can’t answer that. . So, back to wave optics

Chapter 3: Imaging 43