Chapter 1. Ray Optics

Chapter 1. Ray Optics

2

cn

v

Postulates of Ray Optics

A Bds

3

Reflection and Refraction

7

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.

2 2 2 2

1 2 1 3 3 2

1 1 3 3

2 1 3 2

2 2 2 22

1 2 1 3 3 2

2 1 3 2

2 2 2 2

1 2 1 3 3 2

, , ,

1 12 2 1

2 20

0

0 sin sin

sin sin

AB

AB

i r

i r

OPL n x y y n x y y

Fix x y x y

n y y n y ydOPL

dy x y y x y y

n y y n y y

x y y x y y

n n

x

y

(x1, y1)

(0, y2)

(x3, y3)

r

i

A

B

i r : Law of reflection8

Fermat’s Principle: Law of Refraction

2 2 2 2

2 1 1 3 2 3

1 1 3 3

2 1 3 2

2 2 2 22

2 1 1 3 2 3

2 1 3 2

2 2 2 2

2 1 1 3 2 3

, , ,

1 12 2 1

2 20

0

0 sin sin

sin sin

AB i t

i tAB

i t

i i t t

i i t t

OPL n x x y n x x y

Fix x y x y

n x x n x xd OPL

dy x x y x x y

n x x n x x

x x y x x y

n n

n n

Law of refraction:

x

y

(x1, y1)

(x2, 0)

(x3, y3)

t

i

A

ni

nt

i i t tn n : Law of refractionin paraxial approx.9

Refraction –Snell’s Law :

???? nn ti 0

ttii nn sinsin

10

Reflection in plane mirrors

11

Plane surface – Image formation

12

Total internal Reflection (TIR)

13

Imaging by an Optical System

14

A Cartesian surface – those which form perfect

images of a point object

E.g. ellipsoid and hyperboloid

Cartesian Surfaces

O I

15

Imaging by Cartesian reflecting surfaces

16

Imaging by Cartesian refracting Surfaces

17

Approximation by Spherical Surfaces

18

Reflection at a Spherical Surface

19

Reflection at Spherical Surfaces I

Use paraxial or small-angle approximationfor analysis of optical systems:

3 5

2 4

sin3! 5!

cos 1 12! 4!

L

L

Reflection from a spherical convex surface gives rise to a virtual image. Rays appear to emanate from point I behind the spherical reflector.

20

Reflection at Spherical Surfaces II

Considering Triangle OPC and then Triangle OPI we obtain:

2

Combining these relations we obtain:

2

Again using the small angle approximation:

tan tan tanh h h

s s R

21

Reflection at Spherical Surfaces III Image distance s' in terms of the object distance s and mirror radius R:

1 1 22

h h h

s s R s s R

At this point the sign convention in the book is changed !

1 1 2

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).

22

Reflection at Spherical Surfaces IV

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

1 1 1

s s f

2

s

Rf

R < 0f > 0

R > 0f < 0

23

Reflection at Spherical Surfaces VII

1 1 10

0

s fs f s

sm

s

1 1 10

0

s fs f s

sm

s

Real, Inverted Image Virtual Image, Not Inverted

24

Refraction

25

Prisms

27

Beamsplitters

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Spherical boundaries and lenses

n2 > n1

At point P we apply the law of refraction to obtain

1 1 2 2sin sinn n

Using the small angle approximation we obtain

1 1 2 2n n

Substituting for the angles 1 and 2 we obtain

1 2n n

Neglecting the distance QV and writing tangents for the angles gives

1 2

h h h hn n

s R s R

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Refraction by Spherical Surfaces

n2 > n1

Rearranging the equation we obtain

Using the same sign convention as for mirrors we obtain

1 2 1 2n n n n

s s R

1 2 2 1n n n nP

s s R

P : power of the refracting surface

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Example : Concept of imaging by a lens

31

Thin (refractive) lenses

32

The Thin Lens Equation I

O

O'

t

C2

C1

n1n1

n2

s1

s'1

1 2 2 1

1 1 1

n n n n

s s R

V1 V2

For surface 1:

33

The Thin Lens Equation II

1 2 2 1

1 1 1

n n n n

s s R

For surface 1:

2 1 1 2

2 2 2

n n n n

s s R

For surface 2:

2 1s t s

Object for surface 2 is virtual, with s2 given by:

2 10t s s

For a thin lens:

1 2 2 1 1 1 2 1 1 21 2

1 1 1 2 1 2 1 2

n n n n n n n n n nP P

s s s s s s R R

Substituting this expression we obtain:

34

The Thin Lens Equation III

2 1

1 2 1 1 2

1 1 1 1n n

s s n R R

Simplifying this expression we obtain:

2

2

2

1

1

1 1

1 1 1 1s

n n

ss

ss

ns

R R

For the thin lens:

2 1

1 1 2

1 11 1n n

f n R Rs

s

The focal length for the thin lens is found by setting s = ∞:

35

The Thin Lens Equation IV

In terms of the focal length f the thin lens equation becomes:

1 1 1

s s f

The focal length of a thin lens is

positive for a convex lens,

negative for a concave lens.

36

Image Formation by Thin Lenses

Convex Lens

Concave Lens

sm

s

37

Image Formation by Convex Lens

1 1 15 9f cm s cm s

s f s

m s s

Convex Lens, focal length = 5 cm:

F

F

ho

hi

RI

38

Image Formation by Concave Lens

Concave Lens, focal length = -5 cm:

1 1 15 9f cm s cm s

s f s

m s s

FF

ho

hi

VI

39

Image Formation: Two-Lens System I

1 11 1 1

1 1 1 1 1

2 2 2

2 2 2

1 2

1 1 115 25

1 1 115

s ff cm s cm s

s f s s f

f cm s ss f s

m m m

60 cm

40

Image Formation: Two-Lens System II

1 1 1

1 1 1

2 2 2

2 2 2

1 2

1 1 13.5 5.2

1 1 11.8

f cm s cm ss f s

f cm s ss f s

m m m

7 cm

41

Image Formation Summary Table

42

Image Formation Summary Figure

43

Vergence and refractive power : Diopter

1 1 1

s s f

'V V P

reciprocals

Vergence (V) : curvature of wavefront at the lens

Refracting power (P)

Diopter (D) : unit of vergence (reciprocal length in meter)

D > 0

D < 0

1m

0.5m2 diopter

1 diopter

1 m

-1 diopter

44

1 2 3P P P P L

Two more useful equations

45

2-12. Cylindrical lenses

46

Cylindrical lenses

Top view

Side view

47

D. Light guides

48

1-3. Graded-index (GRIN) optics

51

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,

2

1

),,(

t

t

dttqqLAction

0),,(

2

1

t

t

dttqqLAction

0

ii q

L

q

L

dt

d

Lagrange’s equations

52

GRIN

In Cartesian Coordinates with Parameter s = s .

In Cartesian coordinates so the x equation is

Similar equations hold for y and z .

: Ray equation

Paraxial Ray Equation ds ~ dz

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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,

n=n(y): paraxial ray equation

54

Ex. 1.3-1 GRIN slab with

Assuming an initial position y(0) = yo, dy/dz = o at z = 0,

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GRIN fibers

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1.4 Matrix optics : Ray transfer matrix

In the par-axial approximation,

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What is the ray-transfer matrix

58

How to use the ray-transfer matrices

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How to use the ray-transfer matrices

60

Translation Matrix

1 0 1 0 0 0 0

1 0 0

1 0 0

0 01 1 0

0 01

tan

1

0 1

1 1

0 1 0 1

y y L y L

y y L

y

y yy L x x

( yo, o )

( y1, 1 )

L

Refraction Matrix

' :

11

y

R

y

R

y

R

Paraxial Snell s Law n n

y n y n y y n ny

R n R n R R R n n

1 0

1 0: 0

11 : 0

y y

y y Concave surface Rn n

Convex surface RR n n

y=y’

62

Reflection Matrix

:

2

1 0

21

1 0

21

y y y

R R R

Law of Reflection

y yy

R R R

y y

yR

y y

R

y=y’

63

Thick Lens Matrix I

0 01

1

0 01

1

1 0

:L

L L

y yyRefraction at first surface Mn n n

n R n

2 1 1

2

2 1 1

11 2 :

0 1

y y ytTranslation from st surface to nd surface M

3 2 2

3

3 2 2

2

1 0

:L L

y y yRefraction at second surface Mn n n

n R n

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Thick Lens Matrix II

1

2

1

1

2 1 1 2

:

11 0

1

1 1

L

L L

L L

L

L L

L

L L

LL L L

L L

Assuming n n

t n n t n

n R nM n n n

n n nn R n

n R n

t n n t n

n R n

t n nn n n n n nt

n R n R n R n R

2 1

1 0 1 01

0 1L L L

L L

tM n n n n n n

n R n n R n

3 2 1:Thick lens matrix M M M M

65

Thin Lens Matrix

2 1

1 2

:

1 0

1 11

1 1 1

1 0

11

L

L

Thin lens matrix

M n n

n R R

n nbut

f n R R

M

f

The thin lens matrix is found by setting t = 0:

nL

67

Summary of Matrix Methods

68

Summary of Matrix Methods

69

System Ray-Transfer Matrix

Introduction to Matrix Methods in Optics, A. Gerrard and J. M. Burch

1

1

y

2 2

2 2

n

n

y

73

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

: system matrixA B

MC D

A useful property of this matrix is that

0Detf

nM AD BC

n

where n0 and nf 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

0Det 1f

nM AD BC

n

74

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.

0

0

f

f

y yA B

C D

Let’s examine the implications when any of the four elements of the system matrix is equal to zero.

0 0

0 0

f

f

y Ay B

Cy D

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 system75

D=0 A=0

B=0 C=0

76

System Matrix with D=0

Let’s see what happens when D = 0.

0

0

0 0

0

0

f

f

f

f

y yA B

C

y Ay B

Cy

When D = 0, the input plane for the optical system is the input focal plane.

77

Ex) Two-Lens System

f1 = +50 mm f2 = +30 mm

q = 100 mmr s

InputPlane

OutputPlane

F1 F2F1 F2

T1 R1 R2T3T2

0

3 2 2 1 1

0

2 1

1

2 1 1 2

1 0 1 01 1 1

1 11 10 1 0 1 0 1

11 0 1 1 01 1 1

1 1 11 1 10 1 0 1 0 1

f

f

y y s q rM M T R T R T

f f

q q rr qr

f fs q sM r

f f f f

1

1 1

11

r

f f

78

1 1

3 2 2 1 1

2 1 1 2 1 1

1 2 1 1 2 2 1 1

2 1 1 2 1 1

11

0 1 1 1 11 1

1 1 1

1 1 11 1

q q rr q

f fsM T R T R T

q q r rr q

f f f f f f

q s s q q r r q q r rr q s

f f f f f f f f

q q r rr q

f f f f f f

2 1 1

2 1 1

1 2

11 0

30 50 100 50175

100 50 30

q r rD r q

f f f

f f q fr

q f f

r mm

ƒ1 ƒ2

d

H H’

F F’

ƒ ƒ’

s’s

h

r

1 2

1 2 1 2 1 2

1 1 1

f fdf

f f f f f f f d

2

2

f

fd

P

Pdh

2 1 2 1

2 1 2

f d f f f dr f h f

f f f d

< check! >

79

System Matrix with A=0, C=0

0

0

0

0 0

0f

f

f

f

y yB

C D

y B

Cy D

When A = 0, the output plane for the optical system is the output focal plane.

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 = f/0.

0

0

0 0

0

0

f

f

f

f

y yA B

D

y Ay B

D

80

0

0

0

0 0

0

0f

f

f

f

f

y yA

C D

y Ay

Cy D

ym A

y

When B = 0, the input and output planes are object and image planes, respectively, and the transverse magnification of the system m = A.

System Matrix with B=0

81

Ex) Two-Lens System with B=0

f1 = +50 mm f2 = +30 mm

q = 100 mmr s

ObjectPlane

ImagePlane

F1 F2F1 F2

T1 R1 R2T3T2

1

1 2 2 1 1

2 2 1 1

1 2 2 1 2 2 1 2

1 1 2 2 1 2 1 1 2

1 2 1

1 0

1

1 1

q rr q

q r r q q r r fB r q s s

r q q r rf f f f f

f f f f

f f r q f qr r f f f q f f q

f r q q r f f f r r f q f f q f f

q s s qm A

f f f

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Chapter 1. Ray Optics - webpage.khu.ac.ir

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