Chapter 1. Ray Optics

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1 Chapter 1. Ray Optics

2 Postulates of Ray Optics n c v A ds B

3 Reflection and Refraction

4 Fermat s Principle: Law of Reflection Fermat s principle: Light rays will travel from point A to point B in a medium along a path that minimizes the time of propagation. AB ( ) ( ) ( ) ( ) OPL = n x + y y + n x + y y B (x 3, y 3 ) Fix x, y, x, y θ r θ i (0, y 2 ) dopl 1 1 n 2 y y n 2 y y 1 ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) AB = 0 = dy x1 + y2 y1 x3 + y3 y2 0 ( ) ( ) ( ) ( ) ( ) n y2 y1 n y3 y2 = x + y y x + y y y 0 = nsinθ nsinθ i r A (x 1, y 1 ) x sinθ = sinθ i i r r θ = θ : Law of reflection

5 Fermat s Principle: Law of Refraction Law of refraction: ( ) ( ) ( ) ( ) OPL = n x x + y + n x x + y AB i t Fix x, y, x, y (x 1, y 1 ) y A x θ i (x 2, 0) ( ) i ( ) d OPL 1 1 n 2 x x n 2 x x ( ) ( ) i( ) ( ) ( ) ( )( ) 2 1 t 3 2 AB = 0 = + dy x2 x1 + y1 x3 x2 + y3 0 n i ( ) ( ) t ( ) ( ) ( ) n x2 x1 n x3 x2 = x x + y x x + y n t 0 = n sinθ n sinθ i i t t θ t n sinθ = n sinθ i i t t (x 3, y 3 ) nθ = nθ : Law of refraction i i t t in paraxial approx.

6 Refraction Snell s Law : n i sin θ = i n t sinθ t ni nt < 0????

7 Negative index of refraction : n < 0 RHM N > 1 LHM N = -1

8 Principle of reversibility

9 Reflection in plane mirrors

10 Plane surface Image formation

11 Total internal Reflection (TIR)

12 Imaging by an Optical System

13 Cartesian Surfaces A Cartesian surface those which form perfect images of a point object E.g. ellipsoid and hyperboloid O I

14 Imaging by Cartesian reflecting surfaces

15 Imaging by Cartesian refracting Surfaces

16 Approximation by Spherical Surfaces

17 Reflection at a Spherical Surface

18 Reflection at Spherical Surfaces I Reflection from a spherical convex surface gives rise to a virtual image. Rays appear to emanate from point I behind the spherical reflector. Use paraxial or small-angle approximation for analysis of optical systems: 3 5 ϕ ϕ sinϕ = ϕ + L ϕ 3! 5! 2 4 ϕ ϕ cosϕ = 1 + L 1 2! 4!

19 Reflection at Spherical Surfaces II II Considering Triangle OPC and then Triangle OPI we obtain: θ = α + ϕ 2θ = α + α Combining these relations we obtain: α α = 2ϕ Again using the small angle approximation: h h h α tanα α tanα ϕ tanϕ s s R

20 Reflection at Spherical Surfaces III Image distance s' in terms of the object distance s and mirror radius R: h h h = 2 = s s R s s R At this point the sign convention in the book is changed! = s s R The following sign convention must be followed in using this equation: 1. Assume that light propagates from left to right. Object distance s is positive when point O is to the left of point V. 2. Image distance s' is positive when I is to the left of V (real image) and negative when to the right of V (virtual image). 3. Mirror radius of curvature R is positive for C to the right of V (convex), negative for C to left of V (concave).

21 Reflection at Spherical Surfaces IV s = R f = 2 R < 0 f > 0 R > 0 f < 0 The focal length f of the spherical mirror surface is defined as R/2, where R is the radius of curvature of the mirror. In accordance with the sign convention of the previous page, f > 0 for a concave mirror and f < 0 for a convex mirror. The imaging equation for the spherical mirror can be rewritten as = s s f

22 Reflection at Spherical Surfaces VII Real, Inverted Image Virtual Image, Not Inverted s m > f = > s f s s = < 0 s 0 s m < f = < s f s s = > 0 s 0

23 Refraction

24 Prisms

25 Beamsplitters

26 Spherical boundaries and lenses At point P we apply the law of refraction to obtain n sinθ = n sinθ Using the small angle approximation we obtain nθ = n θ Substituting for the angles θ 1 and θ 2 we obtain n ( α ϕ) = n ( α ϕ) 1 2 n 2 > n 1 Neglecting the distance QV and writing tangents for the angles gives n h h n h h = s R s R 1 2

27 Refraction by Spherical Surfaces Rearranging the equation we obtain n n n n = s s R Using the same sign convention as for mirrors we obtain n1 n2 n2 n1 + = = s s R P P : power of the refracting surface n 2 > n 1

28 Example : Concept of imaging by a lens

29 Thin (refractive) lenses

30 The Thin Lens Equation I n 1 n 1 n 2 O' C 1 O C 2 V 1 V 2 For surface 1: n n n n + = s s R s 1 t s' 1

31 The Thin Lens Equation II II For surface 1: n n n n + = s s R For surface 2: n n n n + = s s R Object for surface 2 is virtual, with s 2 given by: s2 = t s 1 For a thin lens: t s, s s = s Substituting this expression we obtain: n1 n2 n2 n1 n1 n1 n2 n1 n1 n2 + + = + = + = P + P s s s s s s R R

32 The Thin Lens Equation III Simplifying this expression we obtain: ( n n ) = s1 s 2 n1 R1 R2 For the thin lens: s = s s = s 1 2 ( n n ) s = s n1 R1 R2 The focal length for the thin lens is found by setting s = : s ( n n ) = = = s f n1 R1 R2

33 The Thin Lens Equation IV In terms of the focal length f the thin lens equation becomes: = s s f The focal length of a thin lens is positive for a convex lens, negative for a concave lens.

34 Image Formation by Thin Lenses Convex Lens m s = s Concave Lens

35 Image Formation by Convex Lens Convex Lens, focal length = 5 cm: h o F RI F h i = s f s f =+ 5 cm s =+ 9 cm s = m = s s =

36 Image Formation by Concave Lens Concave Lens, focal length = -5 cm: h o h i F VI F = s f s f = 5 cm s =+ 9 cm s = m = s s =

37 Image Formation: Two-Lens System I 60 cm s f s f s s f 1 1 = = f1 = + 15 cm s1 = + 25 cm s1 = = f2 = 15 cm s2 = s 2 = s f s m = m m = 1 2

38 Image Formation: Two-Lens System II II 7 cm = f1 =+ 3.5 cm s1 =+ 5.2 cm s 1 = s f s = f2 = cm s2 = s 2 = s f s m = m m = 1 2

39 Image Formation Summary Table

40 Image Formation Summary Figure

$Vergence and refractive power : Diopter 1 1 1 + = s s f D > 0$ $wavefront at the lens Refracting power (P) Diopter (D) : unit of$

41 Vergence and refractive power : Diopter = s s f D > 0 reciprocals D < 0 V + V ' = P 1m 0.5m 2 diopter 1 diopter 1 m -1 diopter Vergence (V) : curvature of wavefront at the lens Refracting power (P) Diopter (D) : unit of vergence (reciprocal length in meter)

42 Two more useful equations P= P1+ P2 + P3+L

43 2-12. Cylindrical lenses

44 Cylindrical lenses Top view Side view

45 D. Light guides

47 1-3. Graded-index (GRIN) optics

Rays in heterogeneous media The optical path length between two points x 1 and x 2 through which a ray passes is Written in terms of parameter s, Because the optical

48 Rays in heterogeneous media The optical path length between two points x 1 and x 2 through which a ray passes is Written in terms of parameter s, Because the optical path length integral is an extremum (Fermat principle), the integrand L satisfies the Euler equations. For an arbitrary coordinate system, with coordinates q1, q2, q3,

49 GRIN In Cartesian coordinates so the x equation is Similar equations hold for y and z. In Cartesian Coordinates with Parameter σ = s. Ray equation Paraxial Ray Equation ds ~ dz

50 GRIN slab : n = n(y) Derivation of the Paraxial Ray Equation in a Graded-Index Slab Using Snell s Law The two angles are related by Snell s law,

51 Ex GRIN slab with Assuming an initial position y(0) = y o, dy/dz = θ o at z = 0,

52 GRIN fibers

53 1.4 Matrix optics : Ray transfer matrix In the par-axial approximation,

54 What is the ray-transfer matrix

55 How to use the ray-transfer matrices

56 How to use the ray-transfer matrices

57 ( y o, α o ) Translation Matrix ( y 1, α 1 ) L α = α y = y + L tanα y + Lα () 1 ( ) ( 0) y ( 1) y = y + L α = + α α y1 1 L y0 1 x1 x0 y0 α = 0 1 = α 0 1 α 1 0 0

58 α = θ φ = θ α = θ φ = θ θ = α + y R y R y R Refraction Matrix y=y Paraxial Snell ' s Law : nθ = n θ y n y n y y 1 n n α = θ = θ = α + = 1 y + α R n R n R R R n n ( 1) y ( 0) y = + α 1 0 y y Concave surface : R < 0 1 n n α = 1 α Convex surface : R > 0 R n n

59 Reflection Matrix y y y α = θ φ = θ α = θ + φ = θ + θ = α + R R R Law of Reflection : θ = θ y y 2 α = θ + = θ + = α + R R R () 1 y ( 0) y = + 2 α = y + R () 1 α α y y=y 1 0 y y 2 α = 1 α R

60 Thick Lens Matrix I Refraction at first surface 1 0 y y0 y0 n n n = M1 α = 0 α 0 n L 1 : L α1 nr L 1 y2 1 t y1 y1 Translation from 1st surface to 2 nd surface : M 2 α = = α 1 α 1 Refraction at second surface y 1 0 y M y = = : α nl n nl nr 2 n α α

61 Thick Lens Matrix II II Thick lens matrix : M = M M M t M = nl n n L n nl n 0 1 nr n nr n 2 L 1 L Assuming n = n : ( ) ( ) t n nl tn nr n M = n n n L 1 L L L n nl n nr2 n nr L 1 nl t n nl tn 1 + nr L 1 nl = nl n t( n nl ) n nl nl n 1+ + t + 1 nr2 nlr1 nr1 nl R2

63 Thin lens matrix : 1 0 M = nl n n R2 R 1 Thin Lens Matrix The thin lens matrix is found by setting t = 0: but 1 nl n 1 1 = f n R R 1 2 n L M 1 0 = 1 1 f

64 Summary of Matrix Methods

65 Summary of Matrix Methods

66 System Ray-Transfer Matrix y1 α 1 y α 2n+ 2 2n+ 2 Introduction to Matrix Methods in Optics, A. Gerrard and J. M. Burch

67 System Ray-Transfer Matrix Any paraxial optical system, no matter how complicated, can be represented by a 2x2 optical matrix. This matrix M is usually denoted M A B = C D : system matrix A useful property of this matrix is that Det M = AD BC = n n 0 f where n 0 and n f are the refractive indices of the initial and final media of the optical system. Usually, the medium will be air on both sides of the optical system and n M AD BC n 0 Det = = = 1 f

68 Significance of system matrix elements The matrix elements of the system matrix can be analyzed to determine the cardinal points and planes of an optical system. y f A B y α f 0 = C D α 0 y = Ay + Bα α f f = Cy Dα 0 0 Let s examine the implications when any of the four elements of the system matrix is equal to zero. D=0 : input plane = first focal plane A=0 : output plane = second focal plane B=0 : input and output planes correspond to conjugate planes C=0 : telescopic system

69 D=0 A=0 B=0 C=0

70 System Matrix with D=0 Let s see what happens when D = 0. y f A B y α f 0 = C 0 α 0 y = Ay + Bα α f f = Cy When D = 0, the input plane for the optical system is the input focal plane.

71 Ex) Two-Lens System Input Plane f 1 = +50 mm f 2 = +30 mm Output Plane F 1 F 1 F 2 F 2 r q = 100 mm s T 1 R 1 T 2 R 2 T y f y0 1 s 1 q 1 r M M T3R2T2R1T α = f α = = f 2 f 1 q qr r r+ q 1 s 1 q 1 s f1 f M 1 1 r 1 1 = = f 2 f1 f 1 r 1 f2 + 1 f1 f 1

72 M 1 s q qr 1 r+ q f f 1 1 = T3R2T2 R1T 1 = = q 1 1 qr r 1 r + q + 1 f2 f1 f1 f2 f1 f1 q+ s s q qr r+ q qr r 1 1 r+ q s 1+ f1 f2 f1 f1 f2 f2 f1 f1 = 1 q 1 1 qr r 1 r+ q + 1 f2 f1 f1 f2 f1 f1 < check! > 1 qr r D = r+ q + 1 = 0 f2 f1 f1 f2 f1+ q f1 r = q f f 1 2 H H ƒ 1 ƒ 2 F F r ( 30)( 50) ( 100)( 50) + = = mm r d h ƒ ƒ s d ff 1 2 = + f = f f f f f f + f d s P P 2 h = d = d f f 2 f2 d f1f2 f1d r = f h= f = f2 f1+ f2 d

73 System Matrix with A=0, C=0 When A = 0, the output plane for the optical system is the output focal plane. y f 0 B y0 α = f C D α 0 y = Bα α f f 0 = Cy + Dα 0 0 When C = 0, collimated light at the input plane is collimated light at the exit plane but the angle with the optical axis is different. This is a telescopic arrangement, with a magnification of D = α f /α 0. y f A B y0 α = f 0 D α 0 y = Ay + Bα α f f = Dα 0 0 0

74 System Matrix with B=0 When B = 0, the input and output planes are object and image planes, respectively, and the transverse magnification of the system m = A. y f A 0 y0 α = f C D α 0 y α f f = Ay 0 = Cy + Dα 0 0 y m= A= y f 0

76 Ex) Two-Lens System with B=0 Object Plane f 1 = +50 mm f 2 = +30 mm Image Plane F 1 F 1 F 2 F 2 r q = 100 mm s ( ) ( ) T 1 R 1 T 2 R 2 T 3 qr r+ q qr r+ q qr r f1 B = r+ q s 1+ = 0 s = f r q qr r 1 f2 f2 f1 f f f f f f1 f2 r+ q f2qr r f1 f2 f2q + f1 f2q = = f r+ q qr+ f f f r r f q+ f + f q f f ( ) ( ) m q+ s s q = A = 1 1 f1 f2 f1

ROINN NA FISICE Department of Physics

ROINN NA FISICE Department of 1.1 Astrophysics Telescopes Profs Gabuzda & Callanan 1.2 Astrophysics Faraday Rotation Prof. Gabuzda 1.3 Laser Spectroscopy Cavity Enhanced Absorption Spectroscopy Prof. Ruth