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Refractive Optics Chapter 26

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Page 1: Refractive Optics Chapter 26. Refractive Optics  Refraction  Refractive Image Formation  Optical Aberrations  The Human Eye  Optical Instruments

Refractive Optics

Chapter 26

Page 2: Refractive Optics Chapter 26. Refractive Optics  Refraction  Refractive Image Formation  Optical Aberrations  The Human Eye  Optical Instruments

Refractive Optics

RefractionRefractive Image FormationOptical AberrationsThe Human EyeOptical Instruments

Page 3: Refractive Optics Chapter 26. Refractive Optics  Refraction  Refractive Image Formation  Optical Aberrations  The Human Eye  Optical Instruments

Refraction

Refractive IndexSnell’s Law

Total Internal Reflection Polarization Longitudinal Focus Shift Dispersion

Page 4: Refractive Optics Chapter 26. Refractive Optics  Refraction  Refractive Image Formation  Optical Aberrations  The Human Eye  Optical Instruments

Refraction: Refractive Index

Speed of light in vacuum: c = 3.00×108 m/s

Speed of light in anything but vacuum: < c

Index of refraction:

n is a dimensionless ratio ≥ 1

v

cn

Page 5: Refractive Optics Chapter 26. Refractive Optics  Refraction  Refractive Image Formation  Optical Aberrations  The Human Eye  Optical Instruments

Refraction: Refractive Index

Index of refraction:

n depends on: material wavelength of light

v

cn

Page 6: Refractive Optics Chapter 26. Refractive Optics  Refraction  Refractive Image Formation  Optical Aberrations  The Human Eye  Optical Instruments

Refraction: Snell’s Law

When light passes from one material into another:

2211 sinsin nn

2 (angle of refraction)

1

(angle of incidence)

n = n1

n = n2

Page 7: Refractive Optics Chapter 26. Refractive Optics  Refraction  Refractive Image Formation  Optical Aberrations  The Human Eye  Optical Instruments

Refraction: Snell’s Law

When light passes from a less-dense (lower index) medium into a more-dense (higher index) medium, the light bends closer to the surface normal.

2211 sinsin nn

Page 8: Refractive Optics Chapter 26. Refractive Optics  Refraction  Refractive Image Formation  Optical Aberrations  The Human Eye  Optical Instruments

Refraction: Snell’s Law

1

2

2

1

cn1

T

Tn2

c

x

2211

2211

2211

22

11

sinsin

sin

1

sin

1

sinsin

sin sin

nn

nn

n

cT

n

cTx

x

Tnc

x

Tnc

Page 9: Refractive Optics Chapter 26. Refractive Optics  Refraction  Refractive Image Formation  Optical Aberrations  The Human Eye  Optical Instruments

Snell’s Law: Total Internal Reflection

Consider light passing from a more-dense medium into a less-dense one (example: from water into air).

The angle of refraction is larger than the angle of incidence.

Page 10: Refractive Optics Chapter 26. Refractive Optics  Refraction  Refractive Image Formation  Optical Aberrations  The Human Eye  Optical Instruments

Snell’s Law: Total Internal Reflection

If the angle of incidence is large enough, the angle of refraction increases to 90°

1

2 = 90°

n = n2

n = n1

Page 11: Refractive Optics Chapter 26. Refractive Optics  Refraction  Refractive Image Formation  Optical Aberrations  The Human Eye  Optical Instruments

Snell’s Law: Total Internal Reflection

At that point, none of the light is transmitted through the surface. All of the light is reflected (total internal reflection). The angle of incidence for which this happens is called the critical angle.

C C

n = n2

n = n1

Page 12: Refractive Optics Chapter 26. Refractive Optics  Refraction  Refractive Image Formation  Optical Aberrations  The Human Eye  Optical Instruments

Snell’s Law: Total Internal Reflection

We can easily calculate the critical angle by

imposing the additional condition

on Snell’s Law:

902

)(for arcsin

90sinsin

211

2

221

nnn

n

nnn

C

C

Page 13: Refractive Optics Chapter 26. Refractive Optics  Refraction  Refractive Image Formation  Optical Aberrations  The Human Eye  Optical Instruments

Snell’s Law: Polarization

We can calculate the angle of incidence for light entering a more-dense medium from a less-dense medium so that the reflected and refracted rays are perpendicular:

B B

2

n = n2

n = n1

Page 14: Refractive Optics Chapter 26. Refractive Optics  Refraction  Refractive Image Formation  Optical Aberrations  The Human Eye  Optical Instruments

Snell’s Law: Polarization

By inspection of our drawing, we see that the perpendicularity of the reflected and transmitted rays requires that:

B B

2

B

B

90

90

2

2

n = n1

n = n2

Page 15: Refractive Optics Chapter 26. Refractive Optics  Refraction  Refractive Image Formation  Optical Aberrations  The Human Eye  Optical Instruments

Snell’s Law: Polarization

Snell’s Law:

Substitute for 2:

Bnn sinsin 122

BB nn sin90sin 12

Page 16: Refractive Optics Chapter 26. Refractive Optics  Refraction  Refractive Image Formation  Optical Aberrations  The Human Eye  Optical Instruments

Snell’s Law: Polarization

Snell’s Law:

Substitute for 2:

angle-difference identity:

Bnn sinsin 122

BB nn sin90sin 12

sincoscossin)sin(

1

2

2

1

2

1

arctan

1tan sincos90sin

n

n

n

n

n

n

B

BBBB

Page 17: Refractive Optics Chapter 26. Refractive Optics  Refraction  Refractive Image Formation  Optical Aberrations  The Human Eye  Optical Instruments

Snell’s Law: Polarization

B is called Brewster’s angle.

1

2arctann

nB

Page 18: Refractive Optics Chapter 26. Refractive Optics  Refraction  Refractive Image Formation  Optical Aberrations  The Human Eye  Optical Instruments

Snell’s Law: Polarization

When light is incident on a dielectric at Brewster’s angle: the reflected light is linearly polarized, perpendicular to the

plane of incidence the transmitted light is partially polarized, parallel to the

plane of incidence

B B

2

1

2arctann

nB n = n1

n = n2

Page 19: Refractive Optics Chapter 26. Refractive Optics  Refraction  Refractive Image Formation  Optical Aberrations  The Human Eye  Optical Instruments

Snell’s Law: Longitudinal Focus Shift

Rays are converging to form an image:

Page 20: Refractive Optics Chapter 26. Refractive Optics  Refraction  Refractive Image Formation  Optical Aberrations  The Human Eye  Optical Instruments

Snell’s Law: Longitudinal Focus Shift

Insert a window: the focus is shifted rightward (delayed)

Page 21: Refractive Optics Chapter 26. Refractive Optics  Refraction  Refractive Image Formation  Optical Aberrations  The Human Eye  Optical Instruments

Snell’s Law: Longitudinal Focus Shift

The amount of the longitudinal focus shift:

d

t

n = n2 n = n1

tn

nn

2

12d

Page 22: Refractive Optics Chapter 26. Refractive Optics  Refraction  Refractive Image Formation  Optical Aberrations  The Human Eye  Optical Instruments

Snell’s Law: Longitudinal Focus Shift

If an object is immersed in one material and viewed from another: “apparent depth”

n = n1

n = n2

d

d’d

n

nd

1

2'

Page 23: Refractive Optics Chapter 26. Refractive Optics  Refraction  Refractive Image Formation  Optical Aberrations  The Human Eye  Optical Instruments

Snell’s Law: Longitudinal Focus Shift

The longitudinal focus shift and apparent depth relationships presented:

are paraxial approximations. Even flat surfaces exhibit spherical aberration in converging or diverging beams of light.

dn

nd

1

2'tn

nn

2

12d

Page 24: Refractive Optics Chapter 26. Refractive Optics  Refraction  Refractive Image Formation  Optical Aberrations  The Human Eye  Optical Instruments

Snell’s Law: Dispersion

As we noted earlier, the index of refraction depends on: the material the wavelength of the light

The dependence of refractive index on wavelength is called refractive dispersion.

Page 25: Refractive Optics Chapter 26. Refractive Optics  Refraction  Refractive Image Formation  Optical Aberrations  The Human Eye  Optical Instruments

Snell’s Law: Dispersion

If each wavelength (color) has a different value of n, applying Snell’s law will give different angles of refraction for a common angle of incidence.

Page 26: Refractive Optics Chapter 26. Refractive Optics  Refraction  Refractive Image Formation  Optical Aberrations  The Human Eye  Optical Instruments

Refractive Image Formation: Lenses

Just as we used curved (spherical) mirrors to form images, we can also use windows with curved (spherical) surfaces to form images.

Such windows are called lenses.

A lens is a piece of a transmissive material having one or both faces curved for image-producing purposes. (A lens can also be a collection of such pieces.)

Page 27: Refractive Optics Chapter 26. Refractive Optics  Refraction  Refractive Image Formation  Optical Aberrations  The Human Eye  Optical Instruments

Refractive Image Formation: Lenses

Lens forms (edge views)

Positive: center thicker than edge

Negative: edge thicker than center

biconvex plano-convex positive meniscus

negative meniscusplano-concavebiconcave

NEGATIVE FORMS

POSITIVE FORMS

Page 28: Refractive Optics Chapter 26. Refractive Optics  Refraction  Refractive Image Formation  Optical Aberrations  The Human Eye  Optical Instruments

Refractive Image Formation: Lenses

Positive: also called “converging”

Negative: also called “diverging”

biconvex plano-convex positive meniscus

negative meniscusplano-concavebiconcave

NEGATIVE FORMS

POSITIVE FORMS

Page 29: Refractive Optics Chapter 26. Refractive Optics  Refraction  Refractive Image Formation  Optical Aberrations  The Human Eye  Optical Instruments

Refractive Image Formation: Lenses

Real image formation by a positive lens:

f(focal length)

focal point

optical axis

Page 30: Refractive Optics Chapter 26. Refractive Optics  Refraction  Refractive Image Formation  Optical Aberrations  The Human Eye  Optical Instruments

Refractive Image Formation: Lenses

Positive lens, do > 2f:

Page 31: Refractive Optics Chapter 26. Refractive Optics  Refraction  Refractive Image Formation  Optical Aberrations  The Human Eye  Optical Instruments

Refractive Image Formation: Lenses

Positive lens, do = 2f:

Page 32: Refractive Optics Chapter 26. Refractive Optics  Refraction  Refractive Image Formation  Optical Aberrations  The Human Eye  Optical Instruments

Refractive Image Formation: Lenses

Positive lens, f < do < 2f:

Page 33: Refractive Optics Chapter 26. Refractive Optics  Refraction  Refractive Image Formation  Optical Aberrations  The Human Eye  Optical Instruments

Refractive Image Formation: Lenses

Positive lens, do = f:

Page 34: Refractive Optics Chapter 26. Refractive Optics  Refraction  Refractive Image Formation  Optical Aberrations  The Human Eye  Optical Instruments

Refractive Image Formation: Lenses

Positive lens, do < f:

Page 35: Refractive Optics Chapter 26. Refractive Optics  Refraction  Refractive Image Formation  Optical Aberrations  The Human Eye  Optical Instruments

Refractive Image Formation: Lenses

Negative lens, do >> f:

Page 36: Refractive Optics Chapter 26. Refractive Optics  Refraction  Refractive Image Formation  Optical Aberrations  The Human Eye  Optical Instruments

Refractive Image Formation: Lenses

Negative lens, do > f:

Page 37: Refractive Optics Chapter 26. Refractive Optics  Refraction  Refractive Image Formation  Optical Aberrations  The Human Eye  Optical Instruments

Refractive Image Formation: Lenses

Negative lens, do < f:

Page 38: Refractive Optics Chapter 26. Refractive Optics  Refraction  Refractive Image Formation  Optical Aberrations  The Human Eye  Optical Instruments

Refractive Image Formation: Lenses

How are the conjugate distances measured?

“Thin lens:” a simplifying assumption that all the refraction takes place at a plane in the center of the lens.

dodi

Page 39: Refractive Optics Chapter 26. Refractive Optics  Refraction  Refractive Image Formation  Optical Aberrations  The Human Eye  Optical Instruments

Refractive Image Formation: Lenses

A better picture: “thick lens:”

The conjugate distances are measured from the principal points.

principal planes

principal points

do di

Page 40: Refractive Optics Chapter 26. Refractive Optics  Refraction  Refractive Image Formation  Optical Aberrations  The Human Eye  Optical Instruments

Refractive Image Formation: Lenses

A catalog example:

Image from catalog of Melles Griot Corporation

Page 41: Refractive Optics Chapter 26. Refractive Optics  Refraction  Refractive Image Formation  Optical Aberrations  The Human Eye  Optical Instruments

Refractive Image Formation: Lenses

The lens equation:

Magnification:

Combinations: one lens’s image is the next lens’s object.

fdd io

111

o

i

d

dm

Page 42: Refractive Optics Chapter 26. Refractive Optics  Refraction  Refractive Image Formation  Optical Aberrations  The Human Eye  Optical Instruments

Refractive Image Formation: Lenses

Sign conventions Light travels from left to right Focal length: positive for a converging lens; negative for

diverging Object distance: positive for object to left of lens

(“upstream”); negative for (virtual) object to right of lens Image distance: positive for real image formed to right of lens

from real object; negative for virtual image formed to left of lens from real object

Magnification: positive for image upright relative to object; negative for image inverted relative to object

Page 43: Refractive Optics Chapter 26. Refractive Optics  Refraction  Refractive Image Formation  Optical Aberrations  The Human Eye  Optical Instruments

Aberrations

Image imperfections due to surface shapes and material properties.

Not (necessarily) caused by manufacturing defects.

A perfectly-made lens will still exhibit aberrations, depending on its shape, material, and how it is used.

Page 44: Refractive Optics Chapter 26. Refractive Optics  Refraction  Refractive Image Formation  Optical Aberrations  The Human Eye  Optical Instruments

Aberrations

The basic optical aberrations Spherical aberration: the variation of focal length with ray

height Coma: the variation of magnification with ray height Astigmatism: the variation of focal length with meridian Distortion: the variation of magnification with field angle

Chromatic: the variation of focal length and/or magnification with wavelength (color)

Page 45: Refractive Optics Chapter 26. Refractive Optics  Refraction  Refractive Image Formation  Optical Aberrations  The Human Eye  Optical Instruments

Lens Power

The reciprocal of the focal length of a lens is called its power.

This isn’t power in the work-and-energy sense. It really means the efficacy of the lens in converging rays to focus at an image. It can be positive or negative. If thin lenses are in contact, their powers may be added.

Unit: if the focal length is expressed in meters, the power is in diopters (m-1).

fP

1

Page 46: Refractive Optics Chapter 26. Refractive Optics  Refraction  Refractive Image Formation  Optical Aberrations  The Human Eye  Optical Instruments

The Human Eye

Horizontal section of right eyeball (as seen from above).

Illustration taken from Warren J. Smith, Modern Optical Engineering, McGraw-Hill, 1966)

Page 47: Refractive Optics Chapter 26. Refractive Optics  Refraction  Refractive Image Formation  Optical Aberrations  The Human Eye  Optical Instruments

The Human Eye

Characteristics Field of view (single eye): 130° high by 200° wide Field of view both eyes simultaneously: 130° diameter Visual acuity (resolution): 1 arc minute Vernier acuity: 10 arc seconds accuracy; 5 arc seconds

repeatability Spectral response: peaks at about = 0.55 m (yellow-

green). Response curve closely matches solar spectrum. Pupil diameter: ranges from about 2 mm (very bright

conditions) to about 8 mm (darkness).

Page 48: Refractive Optics Chapter 26. Refractive Optics  Refraction  Refractive Image Formation  Optical Aberrations  The Human Eye  Optical Instruments

The Human Eye

Function Image distance is nearly fixed (determined by eyeball

shape and dimensions Viewing objects significantly closer than infinity:

accommodation Far point: the farthest-away location at which the relaxed

eye produces a focused image (normally infinity) Near point: the closest location at which the eye’s ability to

accommodate can produce a focused image (“normal” near point is 25 cm for young adults)

Page 49: Refractive Optics Chapter 26. Refractive Optics  Refraction  Refractive Image Formation  Optical Aberrations  The Human Eye  Optical Instruments

The Human Eye

Defects and Problems Myopia (nearsightedness)

Too much power in cornea and lens (or eyeball too long)

Far point is significantly closer than infinity

Corrected with diverging lens (negative power)

Page 50: Refractive Optics Chapter 26. Refractive Optics  Refraction  Refractive Image Formation  Optical Aberrations  The Human Eye  Optical Instruments

The Human Eye

Defects and Problems Hyperopia (farsightedness)

Too little power in cornea and lens (or eyeball too short)

Near point is significantly farther away than 25 cm

Corrected with converging lens (positive power)

Page 51: Refractive Optics Chapter 26. Refractive Optics  Refraction  Refractive Image Formation  Optical Aberrations  The Human Eye  Optical Instruments

The Human Eye

Defects and Problems Astigmatism

Different radii of curvature in horizontal and vertical meridians of the cornea

More power in one meridian than the other

Corrected with oppositely-astigmatic lens (toroidal surface)

Page 52: Refractive Optics Chapter 26. Refractive Optics  Refraction  Refractive Image Formation  Optical Aberrations  The Human Eye  Optical Instruments

The Human Eye

Defects and Problems Presbyopia (“elderly vision”)

Significantly decreased accommodation

Normal effect of aging (lens hardens, becomes difficult to squeeze)

Requires positive-power correction for near vision

Page 53: Refractive Optics Chapter 26. Refractive Optics  Refraction  Refractive Image Formation  Optical Aberrations  The Human Eye  Optical Instruments

Optical Instruments

Angular Size of Objects and Images

Angular size is the angle between chief rays from opposite sides or ends of the object.

Angular sizes of object and image are equal.

ho

do

Page 54: Refractive Optics Chapter 26. Refractive Optics  Refraction  Refractive Image Formation  Optical Aberrations  The Human Eye  Optical Instruments

Optical Instruments

Angular Size of Objects and Images

small-angle approximation:

ho

do

(radians) o

o

d

h

Page 55: Refractive Optics Chapter 26. Refractive Optics  Refraction  Refractive Image Formation  Optical Aberrations  The Human Eye  Optical Instruments

Optical Instruments

Angular Size of Objects and Images

The larger is, the more retinal pixels (rod and cone cells) are covered by the image. (Better, more detailed picture.)

ho

do

Page 56: Refractive Optics Chapter 26. Refractive Optics  Refraction  Refractive Image Formation  Optical Aberrations  The Human Eye  Optical Instruments

Optical Instruments

Angular Size of Objects and Images

Optical instruments present an image to the eye that has a larger angular size than it would without the instrument.

ho

do

Page 57: Refractive Optics Chapter 26. Refractive Optics  Refraction  Refractive Image Formation  Optical Aberrations  The Human Eye  Optical Instruments

Optical Instruments: Simple Magnifier(“Magnifying Glass”)

Enlarged virtual image of object has larger angular size

oo

o

o

d

N

Nh

dh

M

M

Page 58: Refractive Optics Chapter 26. Refractive Optics  Refraction  Refractive Image Formation  Optical Aberrations  The Human Eye  Optical Instruments

Optical Instruments: Simple Magnifier(“Magnifying Glass”)

The value of do depends on di (how the person uses the magnifier).

Image at infinity:

Image at near point:

f

NM

1f

NM

Page 59: Refractive Optics Chapter 26. Refractive Optics  Refraction  Refractive Image Formation  Optical Aberrations  The Human Eye  Optical Instruments

Optical Instruments: Compound MicroscopeCompound microscope:

consists of an objective lens and an eyepiece.

Illustration from the online catalog of Melles Griot Corporation.

Page 60: Refractive Optics Chapter 26. Refractive Optics  Refraction  Refractive Image Formation  Optical Aberrations  The Human Eye  Optical Instruments

Optical Instruments: Compound MicroscopeMagnification (“official” Cutnell & Johnson version):

where: fo is the objective focal length

fe is the eyepiece focal length

L is the distance between objective and eyepiece

N is the near point distance

eo

e

ff

NfLM

Page 61: Refractive Optics Chapter 26. Refractive Optics  Refraction  Refractive Image Formation  Optical Aberrations  The Human Eye  Optical Instruments

Optical Instruments: Compound MicroscopeMagnification (useful):

where: Mo is the objective magnification

Me is the eyepiece magnification, and the eyepiece and objective are separated by the mechanical tube

length for which they were designed (if not, the image quality will be poor anyway). 160 mm is standard in the U.S.

eoMMM

Page 62: Refractive Optics Chapter 26. Refractive Optics  Refraction  Refractive Image Formation  Optical Aberrations  The Human Eye  Optical Instruments

Optical Instruments: Telescope

Objective lens forms real image of distant object (at infinity)

Eyepiece acts as simple magnifier: presents enlarged virtual image of real image, located at infinity.

Page 63: Refractive Optics Chapter 26. Refractive Optics  Refraction  Refractive Image Formation  Optical Aberrations  The Human Eye  Optical Instruments

Optical Instruments: Telescope

Telescopes are afocal: both object and image are located at infinity.

entrancepupil

exitpupil

fo

fe

focalplane

Page 64: Refractive Optics Chapter 26. Refractive Optics  Refraction  Refractive Image Formation  Optical Aberrations  The Human Eye  Optical Instruments

Optical Instruments: Telescope

The magnification is the ratio of the objective to eyepiece focal lengths.

entrancepupil

exitpupil

fo

fe

focalplane

e

o

f

fM

Page 65: Refractive Optics Chapter 26. Refractive Optics  Refraction  Refractive Image Formation  Optical Aberrations  The Human Eye  Optical Instruments

Optical Instruments: Telescope

Here is a common reflecting form: Newtonian

Page 66: Refractive Optics Chapter 26. Refractive Optics  Refraction  Refractive Image Formation  Optical Aberrations  The Human Eye  Optical Instruments

Optical Instruments: Telescope

Another widely-used reflecting form: Cassegrain

Page 67: Refractive Optics Chapter 26. Refractive Optics  Refraction  Refractive Image Formation  Optical Aberrations  The Human Eye  Optical Instruments

Optical Instruments: Telescope

Astronomical refracting telescope: has inverted image

entrancepupil

exitpupil

fo

fe

focalplane

Page 68: Refractive Optics Chapter 26. Refractive Optics  Refraction  Refractive Image Formation  Optical Aberrations  The Human Eye  Optical Instruments

Optical Instruments: Telescope

Galilean refracting telescope: has upright image

Page 69: Refractive Optics Chapter 26. Refractive Optics  Refraction  Refractive Image Formation  Optical Aberrations  The Human Eye  Optical Instruments

Optical Instruments: Telescope

Erecting relay lens configuration. Has upright image and a place to put a reticle. Rifle scopes, spotting scopes, etc.