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Lecture 36. Spherical and thin lenses. Spherical lens sculpture. Thin lenses. Refraction in a spherical surface. n 1. n 2. h. s > 0. s ’ > 0. R > 0. Paraxial approximation. Magnification (spherical refracting surface). n 1. n 2. y. y ’. s > 0. s ’ > 0. s = 14 cm R = –14 cm

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Lecture 36

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Lecture 36

Spherical and thin lenses

Spherical lens sculpture

Thin lenses


Refraction in a spherical surface

n1

n2

h

s > 0

s’> 0

R > 0


Paraxial approximation


Magnification (spherical refracting surface)

n1

n2

y

y’

s > 0

s’> 0


s = 14 cm

R = –14 cm

s’=–14 cm

Or:

Fish appears in the center but 33% larger!

In-class example: Fish bowl

A spherical fish bowl has a 28.0 cm diameter and a fish at its center. What is the apparent position and magnification of the fish to an observer outside of the bowl?

A. s’ = –7 cm, m = 2.0

B. s’ = +7 cm, m = –2.0

C. s’ = –14 cm, m = 1.3

D. s’ = +14 cm, m = –1.3

E. s’ = –14 cm, m = 1.0


Lenses: through two spherical surfaces

nout

nout

Do the calculation twice, once for each surface.

nin

R1

R2

Overall effect

(combination of nin, nout, R1and R2)


Thin lens model

  • In this model

    • thickness of material <<distances to objects, images

    • angles tend to be small

    • consider doubly effective refraction at center of lens

Everything can then be described in terms of two focal points:

f: focal distance

|f |

|f |

F2

F1


Lensmaker’s equation

If we analyze a thin lens in terms of the two spherical surfaces it is made of (in the paraxial approximation), we obtain:

Proof: see book

Remember:

R > 0 if center of curvature is on the same side of surface as outgoing rays


R1

R2

diverging

Example: Diverging lens

A double-concave lens is made of glass with n = 1.5 and radii 20 cm and 25 cm. Find the focal length.

R1 = –20 cm

R2 = +25 cm

Note: If we reverse the lens (R1 = –25 cm and R2 = +20 cm), the result is the same.


Principal rays

Of the many possible rays you could draw, 3 are very useful

  • parallel to axis refracts through focus

2) through center (no net refraction due to symmetry)

3) through focus refracts parallel to axis


DEMO: Converging lens

Typical shapes:

Converging lens

Object at infinity forms a real image at F2

(observer sees rays appearing to originate from point F2)

Focal length f > 0

Small |f| more converging


DEMO: Diverging lens

Typical shapes:

Diverging lens

Object at infinity forms a virtual image at F2

(observer sees rays as if coming from point F2)

Focal length f < 0

Small |f| more diverging


DEMO: Smiley on bulb.

Example: Converging lens

Where does the image of the arrow form?

F

F


Eye intercepts reflected rays that come from a point of the image on screen

If we place a screen

at image location

Diffuse reflection

from screen

F

F


Converging or diverging “power” (in diopters = m-1 ):

The formulae…

y

F

F

y’

f

f

s

s’

Valid for both convergent and diverging thin lenses (and mirrors) in the paraxial limit


Example: Diverging lens

Where is the image if this is a lens of -10 diopters?

30 cm

Virtual, smaller, upright image


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