Chapter 26

Geometrical OpticsSnell’s Law

Thin Lens Equation

1) Index of Refraction, n

Speed of light is reduced in a medium

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n =Speed of light in a vacuum

Speed of light in medium=

c

v

Air 1.000293

Water 4/3

Glass 1.5

Diamond 2.4

2) Snell’s Law

a) Reflection and Transmission

Transmittedray

light splits at an interface

Transmittedray

(b) Refraction: Transmitted ray is bent at interface

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θ1 ≠ θ2

toward normal if n increases

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θ1 ≠ θ2

away from normalif n decreases

toward normalif n increases

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θ1 ≠ θ2

c) Derivation of Snell’s Law

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sinθ2 =λ 2

h

€

sinθ1

v1

=sinθ2

v2

€

n1 sinθ1 = n2 sinθ2

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sinθ1 =λ1

h

€

=v1 f

h

€

=v1

hf

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=v2 f

h

€

=v2

hf

€

but v = c /n

Example: Rear-view mirror

Example: Apparent Depth

θ

x

d

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tanθ1 =x

d

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d tanθ1 = ′ d tanθ2

For small angles,

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sinθ ≅ tanθ

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→ ′ d = dsinθ1

sinθ2

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′ d = dn1

n2

d’

θ

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tanθ2 =x

′ d

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so d sinθ1 = ′ d sinθ2

3) Total internal reflection

a) The conceptFor small values of θ1, light splits at an interface

For larger values of θ1, θ2 > 90º and refraction is not possible

Then all light is reflected internally

Note: this is only possible if n1 > n2

θ

b) Critical incident angle

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If θ1 = θc, then θ2 = 90º

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n1 sinθc = n2 sinπ

2Snell’s law:

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sinθc =n2

n1

(n1 > n2)

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)

Diffusereflectionscreen

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 imageUse 2 of 3 rays:

camera

/CCD sensor

(b) Converging lens - virtual image

(c) Diverging lens - virtual image

8) Thin Lens Equation

a) The equation

€

1

f=

1

di

+1

do

b) Sign Convention (left to right)(i) Focal Length:

f > 0 convergingf < 0 diverging

(ii) Object distancedo > 0 left of lens (real; same side as incident light)do < 0 right of lens (virtual; opposite incident light)

(iii) Image distancedi > 0 right of lens (real; opposite incident light)di < 0 left of lens (virtual; same side as incident light)

(iv) Image sizehi > 0 erecthi < 0 inverted

c) Lateral magnification

Definition:

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m =hi

ho

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From geometry (and sign convention):

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m =−di

do

9) Compound Lenses

Image of first lens is object for the second lens.

Apply thin lens equation in sequentially.

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m =hi2

ho1

Overall magnification is the product:

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=hi2

ho2

ho2

ho1

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=hi2

ho2

hi1

ho1

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=m1m2

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m = m1m2

Example: Problem 26.66

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f1 = 9.0 cm

f2 = 6.0 cm

d = 18.0 cm

Find final image and magnification.

10) Vision and corrective lenses

a) Anatomy of the eye

120 x 106 rods - detect intensity: slow, mono, sensitive

6 x 106 cones - detect frequency: R - 610 nm, G - 560 nm, B - 430 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

€

1

f=

1

do

+1

di

€

1

f=

1

∞+

1

−FP

€

f = −FP

(ignoring the eye-lens distance)

d) Refractive Power

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Refractive power in diopters =1

f (in meters)

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

€

1

f=

1

do

+1

di

€

1

f=

1

25 cm+

1

−NP

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f =(25cm)NP

NP − 25cm> 0

(ignoring the eye-lens distance)

For near point of 40 cm, f = 66 cmPower = + 1.5 (reading glasses)

Examples:

Problem 26.73Age 40: f = 65.0 cm --> NP’ = 25.0cmAge 45: NP’ --> 29.0 cm(a) How much has NP (without glasses) changed?(b) What new f is needed?

Problem 26.75FP = 6.0 m corrected by contact lenses. (Find f)An object (h = 2.0 m) is d = 18.0 m away. • Find image distance with lenses.• Find image height with lenses.

11) Angular Magnification

a) Angular size

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θ =h

d

b) Angular magnification

€

M =′ θ

θ=

Angular size with optical device

Angular size without optical device

12) Magnifier

€

′ θ =ho

do

With magnifier:

€

where 1

f=

1

do

+1

di

€

so ′ θ = ho

1

f−

1

di

⎛

⎝ ⎜ ⎞

⎠ ⎟

(Magnifier allows object to be close to the eye)

Without magnifier:

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θ =ho

N

€

so M =′ θ

θ= N

1

f−

1

di

⎛

⎝ ⎜ ⎞

⎠ ⎟

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We had ′ θ = ho

1

f−

1

di

⎛

⎝ ⎜ ⎞

⎠ ⎟

Highest magnification (di = -N):

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M =N

f+1

Lowest magnification (di = -∞):

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M =N

f

(tense eye)

(relaxed eye)

(Magnification quoted with N = 25 cm, for relaxed eye)

Example:

Problem 26.82Farsighted 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

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L

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M =hi1 fe

ho N

Magnification:

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M =′ θ

θ

€

=moMe

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M =−di1

do1

⎛

⎝ ⎜ ⎞

⎠ ⎟N

fe

⎛

⎝ ⎜ ⎞

⎠ ⎟

For image at ∞, di2 = fe

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=hi1 di2

ho N

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=hi1

ho

⎛

⎝ ⎜ ⎞

⎠ ⎟N

fe

⎛

⎝ ⎜ ⎞

⎠ ⎟

For max M, do1 fo

For di2 = ∞, di1 + fe = L

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M =−N

fe

⎛

⎝ ⎜ ⎞

⎠ ⎟L − fe

fo

⎛

⎝ ⎜ ⎞

⎠ ⎟

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do1 ≅ fo

Example:Problem 26.88Microscope with fo = 3.50 cm, fe = 6.50 cm, and L = 26.0 cm.

(a) Find M for N = 35.0 cm.

(b) Find do1 (if first image at Fe)

(c) Find lateral magnification of the objective.

14) The Astronomical Telescope

• Magnifier requires do < f, but do -> ∞ for stars

• Introduce objective to form nearby image, then use magnifier on the image

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For do ≈ ∞, di ≈ fo

Magnification:

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M =′ θ

θ

€

=hi fe

ho do

€

=hi fe

−hi fo

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M =− fo

fe

Long telescope, small eyepiece

Example:Problem 26.94Yerkes Observatory: fo = 19.4 m, fe = 10.0 cm.

(a) Find angular magnification.

(b) If ho = 1500 m (crater), find hi, given do = 3.77 x 108 m

(c) How close does the crater appear to be.

Galilean Telescope (Opera glasses)

Reflecting Telescope