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Geometrical Optics. Refraction, reflection at a spherical/planar interface Hecht, Chapter 5 Wednesday Sept. 11, 2002. Same for all rays. Independence of path. For any rays traveling from point S to another point P in an optical system the optical path lengths are identical!! . z. y. x. y.

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Geometrical Optics

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Geometrical optics l.jpg

Geometrical Optics

Refraction, reflection at a spherical/planar interface

Hecht, Chapter 5

Wednesday Sept. 11, 2002


Independence of path l.jpg

Same for all rays

Independence of path

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


Reflection by plane surfaces l.jpg

z

y

x

y

x

Reflection by plane surfaces

r1 = (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


Refraction by plane interface total internal reflection l.jpg

n2

θ2

θ2

n1

θ1

θ1

θ1

θC

θ1

Refraction by plane interface& Total internal reflection

n1 > n2

P

Snell’s law n1sinθ1=n2sinθ2


Examples of prisms and total internal reflection l.jpg

Examples of prisms and total internal reflection

45o

45o

45o

Totally reflecting prism

45o

Porro Prism


Fermat s principle l.jpg

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


Imaging by an optical system l.jpg

Optical

System

Imaging by an optical system

O 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 l.jpg

Cartesian Surfaces

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

  • E.g. ellipsoid and hyperboloid

O

I


Cartesian refracting surface l.jpg

Surface ƒ(x,y)

Cartesian refracting surface

n’>n

n

P(x,y)

n’

y

x

I

O

s

s’


Cartesian refracting surface10 l.jpg

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 l.jpg

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 approximation12 l.jpg

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’


Slide13 l.jpg

θ1

θ2

α

Ф

α’

R

s

s’

n

n’

V

O

C

I


Refraction at spherical interfaces l.jpg

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


Refraction at a spherical interface paraxial ray approximation l.jpg

θ1

θ2

α

Ф

s’

s

Refraction at a spherical interface: Paraxial ray approximation

y

C

Note: small angles means that s + x ≈ s

α + Ф = θ1


Refraction at a spherical interface paraxial ray approximation16 l.jpg

θ2

Ф

α’

Refraction at a spherical interface: Paraxial ray approximation

I

C

α’ + θ2 = Ф


Refraction at a spherical interface paraxial ray approximation17 l.jpg

Refraction at a spherical interface: Paraxial ray approximation

  • Snell’s law

    ____________________________

  • Leads to…


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