Physics 1700

Geometric Optics

(rough draft)

W. Pezzaglia

Updated: 2012Aug21

Geometric Optics

A. Focal Length

B. Lens Law, object & image

C. Optical Instruments

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A. Focal Length

1. Converging Lens (or Mirror)

2. Diverging Lens (or Mirror)

3. Lens Maker Equation

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1. Converging Systems

• Converging:o Concave mirror o Convex lens

• Light parallel to principle axis converges at the “focal point”.

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2. Diverging Systems

• Diverging:o Convex mirror o Concave lens

• Focal length is “negative”

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3. Lens Maker Equation

• Focal length “f” of a mirror with radius of curvature “R”:

• Focal length “f” of a lens of index “n” with radius of curvature “R” (in air)

• For Lens “f” of index n2 submerged in medium of index n1:

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1

2

nn

Rf

n

Rf

Rf

3. Lens Maker Equation

• The more curved a lens or mirror, the shorter the focal length.

• A bigger index of refraction makes a shorter focal length lens.

• A lens underwater will have a longer focal length than above water. Hence you become “farsighted” underwater.

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B. Lens Law

1. Lens Equation

2. Virtual images

3. Lens combination

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1. Lens Equation (Gauss)

• Distance to object “p”

• Distance to image “q”

• If p>f then image will be “real” and inverted.

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qpf

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2. Virtual Image

• For converging lens if p<f then image will be “virtual” and erect. [Magnifying glass]

• [Diverging systems only can make virtual images.]

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3. Lens Combination

• Two lenses (close together) act as one lens with focal length given by:

• Example: eyeglasses. Farsighted eye’s lens focal length is too long. Adding an extra lens shortens it.

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C. Optical Instruments

1. The Eye

2. Telescopes

3. Microscopes

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1a Structure of eye

• Fovea of eye has best resolution (this is what you are using to read), which only has cones (color vision)

• Optic nerve creates a blind spot

• Lens is flexible

• Peripheral part of retina is mostly “rods” which are black & white but can see much fainter light (more than 10x)

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1b Color Perception

• Eye has 3 types of color receptors (blue green red).

• Hence using just these three colors can “trick” eye into seeing nearly all colors (can’t do violet)

• Your computer screen has only these 3 colors!

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1c Eye: Farsighted

• Hyperopia, also known as farsightedness

• Cannot see clearly closer than “near point”

• Correct with a converging lens

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1c Eye: Nearsighted

• Myopia or nearsighted

• Cannot see clearly past “far point”

• Correct with diverging lens (focal length equal to negative of far point)

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1c Astigmatism

• Focal length horizontal is not same as vertical

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1c Astigmatism18

Generically, means you can focus on one axis, but the other is out of focus.

1c Astigmatism19

1d Binocular Vision20

•Two eyes see slightly different images (due to parallax).•Brain puts it together to “see” 3D.

•Can “trick” brain into seeing 3D by giving each eye a different picture (stereographs). Modern 3D movies do it using polarization glasses.

Columbian Exposition 1893, stereograph, viewer

1d. 3D Vision 21

1d. 3D Vision 22

1e. Magnitudes and Brightness 43

1.Magnitude Scale:Hipparchus of Rhodes (160-127 B.C) assigns “magnitudes” to stars to represent brightness. The eye can see down to 6th magnitude

1e Herschel Extends the Table 44

William Herschel (1738-1822) extended the scale in both directions

1e Herschel-Pogson Relation 45

Herschel’s measurements suggested a 1st magnitude star is 100x more luminous that a 6th magnitude one. Norman Pogson (1854) showed that this is because the eye’s response to light is logarithmic rather than linear.

22.5

r-2.5Logm

C.3b Gemini 26Mag2

Mag3

Mag4

Mag6

• Our new telescope!• Meade 14" LX200-ACF (f/10)

Advanced Coma-Free w/UHTC #1410-60-03

• USD$ 6,999.00• Schmidt-Cassegrain Focus• Aperture 14 inch (356 mm)• Focal Length 3556 mm (f/10)• Resolution 0.321”

2. Telescopes 27

2a Light Gathering Power

• Light Gathering Power (aka Light Amplification) is proportional to area of mirror

• More light, can see fainter (moredistant) objects.

• Aperture=diameter of objective mirror (for our telescope its 14 inches=356 mm)

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eye than more35206

356

eye Aperture

objective ApertureLGP

2

2

mm

mm

2b Limiting Magnitude

• Consulting table, a LGP of 3520 translates to more than 8.5 magnitudes. Exact calculation:

• Limiting magnitude of naked eye is +6, can see over 8000 stars.

• hence looking through scope we can see up to +14.8 or over 100,000,000 stars in our galaxy, and over 1000 galaxies!

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87.8)3520(5.2 Logm

2c. Basic Telescope Design

Refractor Telescope (objective is a lens)

Focal Length of Objective is big

Focal Length of eyepiece is small

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2d Chromatic Aberration 4

Different colors are bent (refracted) at slightly different angles, with red light focusing the furthest away. Difficult to get a sharp color picture. Problem with refracting telescopes

2e Newtonian Reflecting Telescope

• Objective is a mirror, no chromatic problems!

• Focal length is approximately the length of tube

• Light is directed out the side for the eyepiece

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2f Schmidt Cassegrain Reflecting Telescope

• Cassegrain focus uses “folded optics” so the focal length is more than twice the length of tube

• Light path goes through hole in primary mirror

• Schmidt design has corrector plate in front

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2. Magnification Power

The overall magnification of the angular size is given by the ratio of the focal lengths

For our scope (F=3556 mm), with a 26 mm eyepiece, the power would be:

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eyepiece

objective

F

FPower

13726

3556

mm

mmPower

2g. Too Much Magnification?• Extended objects (planets, nebulae, galaxies) when

magnified more will appear fainter

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Low Power High Power

2h. Too Much Magnification?• Too much magnification and you won’t see it at all!

• Its about surface brightness! Your eye can’t see below a certain amount.

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Low Power High Power

• Creates a BIG virtual image 25 cm away from the eye.

3. Microscopes (briefly) 37

D. References

• Lick Observatory Video: http://www.youtube.com/watch?v=g63lk_0kngk

• Video: Mechanical Universe #40, Optics • Stereographs: http://www.library.hbs.edu/hc/pc/stereograph.html• Stereograph: http://www.sun.travel.pl/stereoscope&page=6

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