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Refractive Optics
Chapter 26
Refractive Optics
RefractionRefractive Image FormationOptical AberrationsThe Human EyeOptical Instruments
Refraction
Refractive IndexSnell’s Law
Total Internal Reflection Polarization Longitudinal Focus Shift Dispersion
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
Refraction: Refractive Index
Index of refraction:
n depends on: material wavelength of light
v
cn
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
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
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
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.
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
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
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
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
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
Snell’s Law: Polarization
Snell’s Law:
Substitute for 2:
Bnn sinsin 122
BB nn sin90sin 12
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
Snell’s Law: Polarization
B is called Brewster’s angle.
1
2arctann
nB
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
Snell’s Law: Longitudinal Focus Shift
Rays are converging to form an image:
Snell’s Law: Longitudinal Focus Shift
Insert a window: the focus is shifted rightward (delayed)
Snell’s Law: Longitudinal Focus Shift
The amount of the longitudinal focus shift:
d
t
n = n2 n = n1
tn
nn
2
12d
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'
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
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.
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.
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.)
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
Refractive Image Formation: Lenses
Positive: also called “converging”
Negative: also called “diverging”
biconvex plano-convex positive meniscus
negative meniscusplano-concavebiconcave
NEGATIVE FORMS
POSITIVE FORMS
Refractive Image Formation: Lenses
Real image formation by a positive lens:
f(focal length)
focal point
optical axis
Refractive Image Formation: Lenses
Positive lens, do > 2f:
Refractive Image Formation: Lenses
Positive lens, do = 2f:
Refractive Image Formation: Lenses
Positive lens, f < do < 2f:
Refractive Image Formation: Lenses
Positive lens, do = f:
Refractive Image Formation: Lenses
Positive lens, do < f:
Refractive Image Formation: Lenses
Negative lens, do >> f:
Refractive Image Formation: Lenses
Negative lens, do > f:
Refractive Image Formation: Lenses
Negative lens, do < f:
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
Refractive Image Formation: Lenses
A better picture: “thick lens:”
The conjugate distances are measured from the principal points.
principal planes
principal points
do di
Refractive Image Formation: Lenses
A catalog example:
Image from catalog of Melles Griot Corporation
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
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
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.
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)
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
The Human Eye
Horizontal section of right eyeball (as seen from above).
Illustration taken from Warren J. Smith, Modern Optical Engineering, McGraw-Hill, 1966)
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).
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)
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)
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)
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)
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
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
Optical Instruments
Angular Size of Objects and Images
small-angle approximation:
ho
do
(radians) o
o
d
h
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
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
Optical Instruments: Simple Magnifier(“Magnifying Glass”)
Enlarged virtual image of object has larger angular size
oo
o
o
d
N
Nh
dh
M
M
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
Optical Instruments: Compound MicroscopeCompound microscope:
consists of an objective lens and an eyepiece.
Illustration from the online catalog of Melles Griot Corporation.
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
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
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.
Optical Instruments: Telescope
Telescopes are afocal: both object and image are located at infinity.
entrancepupil
exitpupil
’
fo
fe
focalplane
Optical Instruments: Telescope
The magnification is the ratio of the objective to eyepiece focal lengths.
entrancepupil
exitpupil
’
fo
fe
focalplane
e
o
f
fM
Optical Instruments: Telescope
Here is a common reflecting form: Newtonian
Optical Instruments: Telescope
Another widely-used reflecting form: Cassegrain
Optical Instruments: Telescope
Astronomical refracting telescope: has inverted image
entrancepupil
exitpupil
’
fo
fe
focalplane
Optical Instruments: Telescope
Galilean refracting telescope: has upright image
Optical Instruments: Telescope
Erecting relay lens configuration. Has upright image and a place to put a reticle. Rifle scopes, spotting scopes, etc.
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