Geometrical optics: rays and images

A ray is an imaginary line along the direction of propagation of the light wave: this line is perpendicular to the wave front

If description in terms of rays is enough,it is geometrical optics.

Reflection and refraction; index of refraction; Snell’s law.Image of an object in an optically more dense medium

sin sin

incident water refracted air

water air

water water air air

n n n n

n nn nθ θ

= =

>=

Image by a plane mirror

incident reflectedθ θ=

Sign rules for objects and images

0; ' 0 's s s s> < = −

plane mirror

'ymy

=

Lateral magnification

Caution: lateral magnification can be less than 1, (m<1), and even negative!

m is negative for inverted images

Image by a spherical mirror

2

tan tan tan'

'

h h hs s R

h h hs s R

φ α θ β φ θα β φ

α β φδ δ δ

α β φ

= + = ++ =

= = =− − −

= = =

1 1 2+ =s s' R

Image by a spherical mirror;paraxial approximation, focal point and focal length

paraxial rays; paraxial approximation

1 1 2 ' / 2'

s Rs R

+ = =∞

Rf =2

focal point and focal length

Image by a spherical mirror; spherical aberration; focal point and focal length

paraxial approximation

1 1 2 / 2'

f Rs s R+ = =

Concave mirror: focal point

1 1 1's

+ =∞ f

Rf =2

Concave mirror: image of an extended object

' 'y smy s

= = −

Attention: always find two most convenient rays to find an image of a point

Concave mirror: image of an extended objectExample 34.1

' 'y smy s

= = −

Concave mirror when s<f; plane mirror as a limiting case of a concave mirror (f→∞)

' ' 1y smy s

= = − =1 1 2 1 1 0 | ' | | |

' 's s f

s s R s s+ = ⇒ + = = <

Convex mirror: image of an extended object

1 1 2 0, ' 0' | |

s ss s R+ = ⇒ > <

−

' 'y smy s

= = −

Convex mirror: virtual focal point

1 1 2 0, ' 0' | |

s ss s R+ = ⇒ > <

−

' 'y smy s

= = −

Convex mirror: virtual focal point; f<0

1 1 2 0, ' 0' | |

s ss s R+ = ⇒ > <

−

' 'y smy s

= = −

Convex mirror: virtual focal point; f<0example 34.3

1 1 1' | |s s f

+ =−

' ' 1.76 0.023475

' 0.0234 160 3.8

y smy s

y my cm cm

−= = − = − =

= = × =

1.81 1 1 1 1' | | 1.8 75' 1.76

f cm

s f ss

= −

= − = − −−

= −

Principle rays for concave and convex mirrors

Principle rays for concave and convex mirrors

Spherical refracting image

1 1 2' | | '

a b b an n n ns s R s s R

−+ = − ⇒ + =

' ' '//

a b

b a

y n s s nmy n s s n

= = − = −

Caution: spherical mirror and spherical refractions act very differently

Plane refractive image as a limit of spherical refracting image

1 1 2' | | '

1 1: 0 0' '

a b b a

a b

n n n ns s R s s R

n nRs s s s

−+ = − ⇒ + =

→∞ + = + =

' ' 1a

b

y n smy n s

= = − =

Caution: don’t confuse lateral magnification, m=1, with other directions.

Plane refractive image as a limit of spherical refracting image

Plane refractive image as a limit of spherical refracting imageExample 34.7

1 1.330 0' ' 2.00

' 1.5

a bn ns s s

s m

+ = + =

= −

Note that although image is inside water, one has to use n of the air: nb =n(air)=1 !

Spherical refracting image;Example 34.5

'1 1.52 1.52 1 ' 11.3

8 ' 2' 1.00 11.3 0.93

1.52 8.0

a b b a

a

b

n n n ns s R

s cmcm s cm

n s cmmn s cm

−+ =

−+ = =

+×

= − = − = −×

Spherical refracting image;Example 34.6

'1.33 1.52 1.52 1.33 ' 21.38 ' 2

' 1.33 ( 21.3) 2.331.52 8.0

a b b a

a

b

n n n ns s R

s cmcm s cm

n s cmmn s cm

−+ =

−+ = = −

+× −

= − = − = +×

Lenses; converging lenses

1 1 1' | |s s f

+ =

f1 =f2

Thicker in the middle

Lenses; converging lenses

Caution: left hand remains left

Lenses; diverging lenses

Thinner in the middle

1 1 1' | |s s f

+ =−

Lens maker’s equation for thin lens

1 2 1 2

1 1 1 1 1 1( 1) ( 1)| | | |

n nf R R f R R

⎛ ⎞ ⎛ ⎞= − − ⇒ = − +⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

Caution: sign of a radius, R1 or R2, can be negative!

R2 <0R1 >0

Double convex

1 1 1' | |s s f

+ =

Lens maker’s equation for thin lens

1 2 1 2

1 1 1 1 1 1( 1) ( 1)| | | |

n nf R R f R R

⎛ ⎞ ⎛ ⎞= − − ⇒ = − − +⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

Caution: sign of a radius, R1 or R2, can be negative!

1 1 1' | |s s f

+ =−

R1 <0R2 >0

Double concave

Lenses: principal rays

Lenses: principal rays

Two lenses: image of an image

Eye as an optical system

Eye as an optical system: farsighted/nearsighted eyes

Eye as an optical system;correction of nearsighted eye

Eye as an optical system;correction of farsighted eye

Eye as an optical system;correction of farsighted eye

Example 34.13 the near point is 100cm (i.e., much further than 25cm). What contact lenses are required? Caution: contact lenses, rather than glasses.Image is on incoming side of the source, i.e., negative!

1 1 1' | |

25 ' 10033 1/ 0.33 3.0

s s fs cm s cmf cm diopters

+ =

= = −= = +

Example 34.14 the far point is 50cm (i.e., very short). What glasses are required? Caution: glasses, rather than contact lenses.Image is on incoming side of the source, i.e., negative!

1 1 1' | |

' (50 2) 4848 1/ 0.48 2.1

s s fs s cm cmf cm diopters

+ =−

= ∞ = − − = −= − − = −

Eye as an optical system;correction of nearsighted eye