Chapter 26 Geometrical Optics Snell’s Law Thin Lens Equation

1) Index of Refraction, n Speed of light is reduced in a medium Air Water4/3 Glass1.5 Diamond2.4

2) Snell’s Law a) Reflection and Transmission Transmitted ray light splits at an interface

Transmitted ray

(b) Refraction: Transmitted ray is bent at interface toward normal if n increases

away from normal if n decreases

toward normal if n increases c) Derivation of Snell’s Law

Example: Rear-view mirror

Example: Apparent Depth

 x d For small angles, d’ 

3) Total internal reflection a) The concept For small values of  1, light splits at an interface

For larger values of  1,  2 > 90º and refraction is not possible Then all light is reflected internally Note: this is only possible if n 1 > n 2 

b) Critical incident angle Snell’s law:

Some critical angles Water-air: 49º Glass - air: 42º Diamond - water: 33º Diamond - air: 24º Why diamonds sparkle

c) Prisms (glass-air critical angle = 45º)

Prisms in binoculars –Longer light path –Image erect

d) Fibre optics Low loss transmission of light, encoded signals.

Fibre optic bundles, coherent bundles Imaging applications: endoscopy

4) Dispersion Index of refraction depends on wavelength

Rainbow

Sun Dogs (parhelia)

5) Image Formation a) Seeing an object Diffuse reflection

b) Image formation with a pinhole Diffuse reflection Diffuse reflection screen

Characteristics of pinhole imaging –Infinite depth of field (everything in focus) –Arbitrary magnification –Low light (increasing size produces blurring) Diffuse reflection screen Diffuse reflection

c) Ideal lens

Characteristics of the ideal lens –All rays leaving a point on object meet at one point on image –Only one perfect object distance for selected image distance (limited depth of field -- better for smaller lens)

6) Thin lenses a) Converging - thicker in the middle

(i) Parallel coaxial rays converge at focus Reversible

(ii) Symmetric - rays leaving focal point emerge parallel (f’ = f)

(iii) Ray through centre undeviated

Summary of ray tracing rules for converging lens

b) Diverging - thinner in the middle

(i) parallel, coaxial rays diverge as if from focus Reversible

(ii) symmetric - rays converging toward focus emerge parallel

(iii) ray through centre undeviated

Summary of ray-tracing rules for diverging lens

c) Real lenses: - usually spherical surfaces - approximate ideal lens for small angles (paraxial approximation)

7) Image Formation with thin lenses (ray tracing) (a)Converging lens - real image Use 2 of 3 rays:

camera /CCD sensor

(b) Converging lens - virtual image

(c) Diverging lens - virtual image

8) Thin Lens Equation a) The equation

b) Sign Convention (left to right) (i)Focal Length: f > 0 converging f < 0 diverging (ii) Object distance d o > 0 left of lens (real; same side as incident light) d o < 0 right of lens (virtual; opposite incident light) (iii) Image distance d i > 0 right of lens (real; opposite incident light) d i < 0 left of lens (virtual; same side as incident light) (iv) Image size h i > 0 erect h i < 0 inverted

c) Lateral magnification Definition: From geometry (and sign convention):

9) Compound Lenses Image of first lens is object for the second lens. Apply thin lens equation in sequentially.

Overall magnification is the product:

Example: Problem Find final image and magnification.

10) Vision and corrective lenses a) Anatomy of the eye

120 x 10 6 rods - detect intensity: slow, mono, sensitive 6 x 10 6 cones - detect frequency: R nm, G nm, B nm

b) Optics - Accomodation: focal length changes with object distance - near point: nearest point that can be accomodated - normally < 25 cm - far point: furthest point that can be accomodated - normally ∞

c) Myopia - far point < ∞ - near-sighted (far-blind) - correction: object at ∞ --> image at far point

Correction: object at ∞ --> image at far point (ignoring the eye-lens distance)

d) Refractive Power For a far point of 50 cm, f = -50 cm, Lens prescription: -2

e) hyperopia (hypermetropia) - near point > 25 cm - far-sighted (near-blind) - correction: object at 25 cm --> image at near point

Correction: object at 25 cm --> image at near point (ignoring the eye-lens distance) For near point of 40 cm, f = 66 cm Power = (reading glasses)

Examples: Problem Age 40: f = 65.0 cm --> NP’ = 25.0cm Age 45: NP’ --> 29.0 cm (a) How much has NP (without glasses) changed? (b) What new f is needed? Problem FP = 6.0 m corrected by contact lenses. (Find f) An object (h = 2.0 m) is d = 18.0 m away. (a)Find image distance with lenses. (b)Find image height with lenses.

11) Angular Magnification a) Angular size

b) Angular magnification

12) Magnifier With magnifier: (Magnifier allows object to be close to the eye)

Without magnifier: Highest magnification (d i = -N): Lowest magnification (d i = -∞): (tense eye) (relaxed eye) (Magnification quoted with N = 25 cm, for relaxed eye)

Example: Problem Farsighted person has corrective lenses with f = 45.4 cm. Maximum magnification of a magnifier is 7.50 (normal vision). What is the maximum magnification of the magnifier for the farsighted person without lenses?

13) Compound Microscope Simple magnifier: M = N/f –to increase M, decrease f –practical limits to decreasing f (and therefore size): small lens difficult to manufacture and use increases aberrations Microscope introduces an additional lens to form a larger intermediate image, which can be viewed with a magnifier

L Magnification: For image at ∞, d i2 = f e For max M, d o1  f o For d i2 = ∞, d i1 + f e = L

Example: Problem Microscope with f o = 3.50 cm, f e = 6.50 cm, and L = 26.0 cm. (a) Find M for N = 35.0 cm. (b) Find d o1 (if first image at F e ) (c) Find lateral magnification of the objective.

14) The Astronomical Telescope Magnifier requires d o ∞ for stars Introduce objective to form nearby image, then use magnifier on the image

Magnification: Long telescope, small eyepiece

Example: Problem Yerkes Observatory: f o = 19.4 m, f e = 10.0 cm. (a) Find angular magnification. (b) If h o = 1500 m (crater), find h i, given d o = 3.77 x 10 8 m (c) How close does the crater appear to be.

Galilean Telescope (Opera glasses)

Reflecting Telescope