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(1)

Chapter 2.

Geometrical Optics Chapter 2.

Geometrical Optics

Light Ray : the path along which light energy is transmitted from one point to another in an optical system.

Speed of Light : Speed of light (in vacuum): a fundamental (or a “defined”) constant of nature given by c = 299,792,458 meters / second = 186,300 miles / second.

Index of Refraction

(2)

Optical path length Optical path length

The optical path length in a medium is the integral of the refractive index and a differential geometric length:

b

OPL = ∫

a

n ds a ds b

n1

n4 n2

n5 nm n3

=

=

m

i

i i

s c n

t

1

1

=

=

m

i

i i

s n OPL

1

=

P

S

n s ds

OPL ( ) =

P

S

v ds OPL c

S

P

(3)

Reflection and Refraction

Reflection and Refraction

(4)

Plane of incidence

Plane of incidence

(5)

Fermat’s Principle: Law of Reflection 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 2

2 1 2 1 3 3 2

2 1 3 2

2 2 2 2

1 2 1 3 3 2

, , ,

1 1

2 2 1

2 2

0

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 y

dOPL

dy x y y x y y

n y y n y y

x y y x y y

n θ n θ

θ θ

= + + +

= = +

+ +

=

+ +

=

=

x y

(x1, y1)

(0, y2) (x3, y3)

θ

r

θ

i

A B

i r

θ θ =

: Law of reflection

(6)

Fermat’s Principle: Law of Refraction 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 2

2 2 1 1 3 2 3

2 1 3 2

2 2 2 2

2 1 1 3 2 3

, , ,

1 1

2 2 1

2 2

0

0

0 sin sin

sin sin

AB i t

i t

AB

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 x

d OPL

dy x x y x x y

n x x n x x

x x y x x y

n n

n n

θ θ

θ θ

= + + + −

= = +

+ +

=

+ +

=

=

Law of refraction:

x y

(x1, y1)

(x2, 0)

(x3, y3) θt

θi A

n

i

n

t

i i t t

n θ = n θ

: Law of refraction in paraxial approx.

(7)

Refraction –Snell’s Law :

????

n

n

i

×

t

< 0

t t

i

i

n

n sin θ = sin θ

(8)

Negative index of refraction : n < 0

RHM

N > 1

LHM

N = -1

(9)

Principle of reversibility

Principle of reversibility

(10)

2-4. Reflection in plane mirrors

2-4. Reflection in plane mirrors

(11)

Plane surface – Image formation

Plane surface – Image formation

(12)

Total internal Reflection (TIR)

Total internal Reflection (TIR)

(13)

2-6. Imaging by an Optical System

2-6. Imaging by an Optical System

(14)

Cartesian Surfaces Cartesian Surfaces

• A Cartesian surface – those which form perfect images of a point object

• E.g. ellipsoid and hyperboloid

O I

(15)

Imaging by Cartesian reflecting surfaces

Imaging by Cartesian reflecting surfaces

(16)

Imaging by Cartesian refracting Surfaces

Imaging by Cartesian refracting Surfaces

(17)

Imaging by Cartesian refracting Surfaces Imaging by Cartesian refracting Surfaces

The optical path length for any path from Point O to the image Point I must be the same by Fermat’s principle.

( )

2

2 2 2 2 2

constant

constant

o o i i o o i i

o i o i

o o i i

n d n d n s n s

n x y z n s s x y z

n s n s

+ = + =

+ + + + − + +

= + =

The Cartesian or perfect imaging surface is a

paraboloid in three dimensions.

Usually, though, lenses have spherical surfaces

because they are much easier to manufacture.

ÆSpherical approximation or, paraxial approx.

(18)

Approximation by Spherical Surfaces

Approximation by Spherical Surfaces

(19)

2-7. Reflection at a Spherical Surface

2-7. Reflection at a Spherical Surface

(20)

Reflection at Spherical Surfaces I Reflection at Spherical Surfaces I

Use paraxial or small-angle approximation for analysis of optical systems:

3 5

2 4

sin 3! 5!

cos 1 1

2! 4!

ϕ ϕ

ϕ ϕ ϕ

ϕ ϕ ϕ

= − + − ≅

= − + − ≅

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.

(21)

Reflection at Spherical Surfaces II Reflection at Spherical Surfaces II

Considering Triangle OPC and then Triangle OPI we obtain:

θ α ϕ = + 2 θ α α = +

Combining these relations we obtain:

α α − = − 2 ϕ

Again using the small angle approximation:

tan h tan h tan h

s s R

α ≅ α ≅ α ≅ α ≅ ϕ ≅ ϕ ≅

(22)

Reflection at Spherical Surfaces III Reflection at Spherical Surfaces III

Image distance s' in terms of the object distance s and mirror radius R:

1 1 2

h h 2 h

ss = − Rss = − 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).

(23)

Reflection at Spherical Surfaces IV 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

f R

= ∞

= −

R < 0 f > 0

R > 0 f < 0

(24)

Reflection at Spherical Surfaces V Reflection at Spherical Surfaces V

C F

O I

1

2

3

I' O'

Ray 1: Enters from O' through C, leaves along same path Ray 2: Enters from O' through F, leaves parallel to optical axis Ray 3: Enters through O' parallel to optical axis, leaves along

line through F and intersection of ray with mirror surface

1 1 1

7 8 / 2

? ?

s cm R cm f R

s f s

s m s

s

= = + = − = −

′ = = − ′ =

V

s s’

(25)

Reflection at Spherical Surfaces VI Reflection at Spherical Surfaces VI

C F

O 1

2 3

I

I' O'

1 1 1

17 8 / 2

s cm R cm f R

s f s

s m s

s

= + = − = − = = − =

′ = = − ′ =

V

(26)

Reflection at Spherical Surfaces VII Reflection at Spherical Surfaces VII

1 1 1

0

0

s f

s f s

m s

s

> ⇒ = − >

= − ′ <

1 1 1

0 0

s f

s f s

m s

s

< ⇒ = − <

= − ′ >

Real, Inverted Image Virtual Image, Not Inverted

(27)

2-8. Refraction at a Spherical Surface 2-8. Refraction at a Spherical Surface

n2 > n1

At point P we apply the law of refraction to obtain

1

sin

1 2

sin

2

n θ = n θ

Using the small angle approximation we obtain

1 1 2 2

n θ = n θ

Substituting for the angles θ1 and θ2 we obtain

( ) ( )

1 2

n α ϕ − = n α ϕ

Neglecting the distance QV and writing tangents for the angles gives

1 2

h h h h

n n

s R s R

⎛ ⎞

⎛ − ⎞ = ⎜ − ⎟

⎜ ⎟ ′

⎝ ⎠ ⎝ ⎠

(28)

Refraction by Spherical Surfaces II Refraction by Spherical Surfaces II

n2 > n1

Rearranging the equation we obtain

Using the same sign convention as for mirrors we obtain

1 2 1 2

n n n n

s s R

− = −

1 2 2 1

n n n n

s s R P

+ = − =

P : power of the refracting surface

(29)

Refraction at Spherical Surfaces III Refraction at Spherical Surfaces III

C O

I

I' O'

1 2

1 2 2 1

7 8 1.0 4.23

' ?

s cm R cm n n

n n n n

s s R s

= = + = =

+ = − ⇒ =

V θ1

θ2

(30)

Example 2-2 : Concept of imaging by a lens

Example 2-2 : Concept of imaging by a lens

(31)

2-9. Thin (refractive) lenses

2-9. Thin (refractive) lenses

(32)

The Thin Lens Equation I The Thin Lens Equation I

O O'

t C2

C1

n1 n1

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 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 1

s = − t s′

Object for surface 2 is virtual, with s

2

given by:

2

,

1 2 1

t s s ′ ⇒ s = − s

For a thin lens:

1 2 2 1 1 1 2 1 1 2

1 2

1 1 1 2 1 2 1 2

n n n n n n n n n n

P P

s s s s s s R R

− −

+ − + = + = + = +

′ ′ ′ ′

Substituting this expression we obtain:

(34)

The Thin Lens Equation III The Thin Lens Equation III

(

2 1

)

1 2 1 1 2

1 1 n n 1 1

s s n R R

− ⎛ ⎞

+ ′ = ⎜ ⎝ − ⎟ ⎠

Simplifying this expression we obtain:

(

2

)

2

2 1

1

1 1

1 1 1 1

s n n

s s

s s

s n

R R

′ ′

= = − ⎛ ⎞

+ ′ = ⎜ ⎝ − ⎟ ⎠

For the thin lens:

(

2 1

)

1 1 2

1 1

1 n n 1

f n R R

s = ∞ ⇒ s =

− ⎛ ⎞

= ⎜ − ⎟

⎝ ⎠

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

(35)

The Thin Lens Equation IV 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 Image Formation by Thin Lenses

Convex Lens

Concave Lens

m s

s

= − ′

(37)

Image Formation by Convex Lens Image Formation by Convex Lens

1 1 1

5 9

f 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 Image Formation by Concave Lens

Concave Lens, focal length = -5 cm:

1 1 1

5 9

f cm s cm s

s f s

m s s

= − = − = + ⇒ ′ =

= − ′ =

F F ho

hi VI

(39)

Image Formation: Two-Lens System I Image Formation: Two-Lens System I

1 1

1 1 1

1 1 1 1 1

2 2 2

2 2 2

1 2

1 1 1

15 25

1 1 1

15 s f

f cm s cm s

s f s s f

f cm s s

s f s

m m m

= = = + = + =

= = − = =

= =

60 cm

(40)

Image Formation: Two-Lens System II Image Formation: Two-Lens System II

1 1 1

1 1 1

2 2 2

2 2 2

1 2

1 1 1

3.5 5.2

1 1 1

1.8

f cm s cm s

s f s

f cm s s

s f s

m m m

= = + = + =

= = + = =

= =

7 cm

(41)

Image Formation Summary Table

Image Formation Summary Table

(42)

Image Formation Summary Figure

Image Formation Summary Figure

(43)

Vergence and refractive power : Diopter 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

0.5m 2 diopter

1 diopter

1 m -1 diopter

(44)

1 2 3

P = + P P + +L P

Two more useful equations

Two more useful equations

(45)

2-12. Cylindrical lenses

2-12. Cylindrical lenses

(46)

Cylindrical lenses Cylindrical lenses

Top view

Side view

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