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 byc = 299,792,458 meters / second = 186,300 miles / second.

Index of Refraction

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

aOPL n ds= ∫ a b

ds

n1

n4

n2

n5

nm

n3

∑=

=m

iiisnc

t1

1

∑=

=m

iiisnOPL

1

∫=P

SdssnOPL )( ∫=

P

S

dsvcOPL

S

P

Reflection and RefractionReflection and Refraction

Plane of incidencePlane of incidence

Fermat’s Principle: Law of ReflectionFermat’s Principle: Law of ReflectionFermat’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 21 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 21 2 1 3 3 2

, , ,

1 12 2 12 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 ydOPLdy x y y x y y

n y y n y y

x y y x y y

n nθ θ

θ θ

= − + − + − + −

− − −= = +

+ − + −

− −= −

+ − + −

= −

⇒ =x

y

(x1, y1)

(0, y2)

(x3, y3)

θr

θi

A

B

i rθ θ= : Law of reflection

Fermat’s Principle: Law of RefractionFermat’s Principle: Law of Refraction( ) ( ) ( ) ( )

( ) ( )

( ) ( )

( )( )

( ) ( )( )

( ) ( )( )

( ) ( )

2 2 2 22 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 22 1 1 3 2 3

, , ,

1 12 2 12 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 OPLdy 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

ni

nt

i i t tn nθ θ= : Law of refractionin paraxial approx.

Refraction –Snell’s Law :

???? nn ti 0

Negative index of refraction : n < 0

RHM N > 1

LHMN = -1

Principle of reversibility Principle of reversibility

2-4. Reflection in plane mirrors 2-4. Reflection in plane mirrors

Plane surface – Image formation Plane surface – Image formation

Total internal Reflection (TIR) Total internal Reflection (TIR)

2-6. Imaging by an Optical System 2-6. Imaging by an Optical System

Cartesian SurfacesCartesian Surfaces

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

• E.g. ellipsoid and hyperboloid

O I

Imaging by Cartesian reflecting surfaces Imaging by Cartesian reflecting surfaces

Imaging by Cartesian refracting Surfaces Imaging by Cartesian refracting Surfaces

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.

( )22 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 surfacesbecause they are much easier to manufacture.

Spherical approximationor, paraxial approx.

Approximation by Spherical SurfacesApproximation by Spherical Surfaces

2-7. Reflection at a Spherical Surface2-7. Reflection at a Spherical Surface

Reflection at Spherical Surfaces I 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!

ϕ ϕϕ ϕ ϕ

ϕ ϕϕ

= − + − ≅

= − + − ≅

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.

Reflection at Spherical Surfaces II Reflection at Spherical Surfaces II

Considering Triangle OPC and then Triangle OPI we obtain:

2θ α ϕ θ α α′= + = +

Combining these relations we obtain:

2α α ϕ′− = −Again using the small angle approximation:

tan tan tanh h hs s R

α α α α ϕ ϕ′ ′≅ ≅ ≅ ≅ ≅ ≅′

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 22h h hs s R s s R− = − ⇒ − = −

′ ′

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

1 1 2s 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).

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 1s s f+ =

′

2

sRf

= ∞

= −

R < 0f > 0

R > 0f < 0

Reflection at Spherical Surfaces V Reflection at Spherical Surfaces V

CFO I

12

3

I'

O'

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

line through F and intersection of ray with mirror surface

1 1 17 8 / 2

? ?

s cm R cm f Rs f s

ss ms

= = + = − = −′

′′ = = − =

Vs s’

Reflection at Spherical Surfaces VI Reflection at Spherical Surfaces VI

CFO

1

23

I

I'

O'

1 1 117 8 / 2s cm R cm f Rs f s

ss ms

= + = − = − = = − =′

′′ = = − =

V

Reflection at Spherical Surfaces VII Reflection at Spherical Surfaces VII

1 1 1 0

0

s fs f s

sms

> ⇒ = − >′

′= − <

1 1 1 0

0

s fs f s

sms

< ⇒ = − <′

′= − >

Real, Inverted Image Virtual Image, Not Inverted

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 1 2 2sin sinn nθ θ=Using the small angle approximation we obtain

1 1 2 2n nθ θ=

Substituting for the angles θ1 and θ2 we obtain

( ) ( )1 2n nα ϕ α ϕ′− = −Neglecting the distance QV and writing tangents for the angles gives

1 2h h h hn ns R s R

⎛ ⎞⎛ ⎞− = −⎜ ⎟⎜ ⎟ ′⎝ ⎠ ⎝ ⎠

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 2n n n ns s R

−− =

′

1 2 2 1n n n n Ps s R

−+ = =

′

P : power of the refracting surface

Refraction at Spherical Surfaces III Refraction at Spherical Surfaces III

CO

I

I'

O'

1 2

1 2 2 1

7 8 1.0 4.23

' ?

s cm R cm n nn n n n ss s R

= = + = =−

+ = ⇒ =′

V

θ1

θ2

Example 2-2 : Concept of imaging by a lensExample 2-2 : Concept of imaging by a lens

2-9. Thin (refractive) lenses2-9. Thin (refractive) lenses

The Thin Lens Equation I The Thin Lens Equation I

O

O'

t

C2

C1

n1 n1n2

s1s'1

1 2 2 1

1 1 1

n n n ns s R

−+ =

′

V1 V2

For surface 1:

The Thin Lens Equation II The Thin Lens Equation II

1 2 2 1

1 1 1

n n n ns s R

−+ =

′

For surface 1:

2 1 1 2

2 2 2

n n n ns s R

−+ =

′

For surface 2:

2 1s t s′= −

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

2 1 2 1,t s s 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 n P Ps s s s s s R R

− −+ − + = + = + = +

′ ′ ′ ′

Substituting this expression we obtain:

The Thin Lens Equation III The Thin Lens Equation III

( )2 11 2 1 1 2

1 1 1 1n ns s n R R

− ⎛ ⎞+ = −⎜ ⎟′ ⎝ ⎠

Simplifying this expression we obtain:

( )22

21

1

1 1

1 1 1 1sn n

ss

ss

ns

R R′ ′= =

− ⎛ ⎞+ = −⎜ ⎟′ ⎝ ⎠

⇒

For the thin lens:

( )2 11 1 2

1 11 1n nf n R R

ss

= ∞ ⇒ =′

− ⎛ ⎞= −⎜ ⎟

⎝ ⎠

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

The Thin Lens Equation IV The Thin Lens Equation IV In terms of the focal length f the thin lens equation becomes:

1 1 1s s f+ =

′

The focal length of a thin lens is

positive for a convex lens,

negative for a concave lens.

Image Formation by Thin LensesImage Formation by Thin Lenses

Convex Lens

Concave Lens

sms′

= −

Image Formation by Convex LensImage Formation by Convex Lens

1 1 1 5 9f cm s cm ss f s

m s s

′= − = + = + ⇒ =′

′= − =

Convex Lens, focal length = 5 cm:

F

F

ho

hi

RI

Image Formation by Concave LensImage Formation by Concave LensConcave Lens, focal length = -5 cm:

1 1 1 5 9f cm s cm ss f s

m s s

′= − = − = + ⇒ =′

′= − =

FF

hohi

VI

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

1 11 1 1

1 1 1 1 1

2 2 22 2 2

1 2

1 1 1 15 25

1 1 1 15

s f f cm s cm ss f s s f

f cm s ss f sm m m

− ′= − = = + = + ⇒ =′

′= − = − = ⇒ =′

= =

60 cm

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

1 1 11 1 1

2 2 22 2 2

1 2

1 1 1 3.5 5.2

1 1 1 1.8

f cm s cm ss f s

f cm s ss f s

m m m

′= − = + = + ⇒ =′

′= − = + = ⇒ =′

= =

7 cm

Image Formation Summary TableImage Formation Summary Table

Image Formation Summary FigureImage Formation Summary Figure

Vergence and refractive power : DiopterVergence and refractive power : Diopter

1 1 1s 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

1 2 3P P P P= + + +L

Two more useful equationsTwo more useful equations

2-12. Cylindrical lenses2-12. Cylindrical lenses

Cylindrical lensesCylindrical lenses

Top view

Side view