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### Geometrical Optics

Refraction, reflection at a spherical/planar interface

Hecht, Chapter 5

Wednesday Sept. 11, 2002

Independence of path

For any rays traveling from point S to another point P in an optical system the optical path lengths are identical!!

y

x

y

x

Reflection by plane surfacesr1 = (x,y,z)

r2= (-x,y,z)

r1 = (x,y,z)

r3=(-x,-y,z)

r4=(-x-y,-z)

r2 = (x,-y,z)

Law of Reflection

r1 = (x,y,z) → r2 = (x,-y,z)

Reflecting through (x,z) plane

θ2

θ2

n1

θ1

θ1

θ1

θC

θ1

Refraction by plane interface& Total internal reflectionn1 > n2

P

Snell’s law n1sinθ1=n2sinθ2

Fermat’s principle

- Light, in going from point S to P, traverses the route having the smallest optical path length
- More generally, there may be many paths with the same minimum transit time, e.g. locus of a cartesian surface

System

Imaging by an optical systemO and I are conjugate points – any pair of object-image points - which by the principle of reversibility can be interchanged

I

O

Fermat’s principle – optical path length of every ray passing through I must be the same

Cartesian Surfaces

- A Cartesian surface – those which form perfect images of a point object
- E.g. ellipsoid and hyperboloid

O

I

Cartesian refracting surface

- Surface ƒ(x,y) will be cartesian for points points O and I if…

___________________________________

- The equation defines an ovoid of revolution for a given s, s’
- Equality means all paths are equal (i.e. for all x,y)
- We then have perfect imaging by Fermat’s principle
- But we can see that the surface will be cartesian for one set of s, s’ (no too useful)

Paraxial ray approximation

- We would like a single surface to provide imaging for all s, s’.
- This will be true if we place certain restrictions on the bundle of rays collected by the optical system
- Make the PARAXIAL RAY APPROXIMATION
- Assume y << s,s’ (i.e. all angles are small)
- x << s, s’ (of course)

Paraxial ray approximation

- All distances measured from V (i.e. assume x=0)
- All angles are small

sinα ≈ tan α ≈ α ; cos α = 1

- Snell’s law

nθ = n’θ’

Refraction at spherical interfaces

- Light travels left to right
- V = origin – measure all distances from here
- R = positive to the right of V, negative to the left
- S = positive for real objects (i.e. one to the left of V), negative for virtual
- S’ = positive for real image (to right of V), negative for virtual images
- Heights – y,y’ – positive up, negative down

θ2

α

Ф

s’

s

Refraction at a spherical interface: Paraxial ray approximationy

C

Note: small angles means that s + x ≈ s

α + Ф = θ1

Refraction at a spherical interface: Paraxial ray approximation

- Snell’s law

____________________________

- Leads to…

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