Transcript Refractive Optics

Refractive Optics
Chapter 26
Refractive Optics
Refraction
Refractive Image Formation
Optical Aberrations
The Human Eye
Optical Instruments
Refraction
Refractive Index
Snell’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:
c
n
v
n is a dimensionless ratio ≥ 1
Refraction: Refractive Index
Index of refraction:
n depends on:
material
wavelength of light
c
n
v
Refraction: Snell’s Law
When light passes from one material into
another:
n = n1
n = n2
2 (angle of refraction)
1
(angle of incidence)
n1 sin 1  n2 sin 2
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.
n1 sin 1  n2 sin 2
Refraction: Snell’s Law
c
T
n1
sin 1 
x
c
T
n2
sin  2 
x
cT
cT
x

n1 sin 1 n2 sin  2
1
c
n1 T
c
n2 T
1
2
x
2
1
1

n1 sin 1 n2 sin  2
n1 sin 1  n2 sin  2
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°
2 = 90°
n = n2
n = n1
1
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.
n = n2
n = n1
C
C
Snell’s Law: Total Internal Reflection
We can easily calculate the critical angle by
imposing the additional condition 2  90
on Snell’s Law:
n1 sin C  n2 sin 90  n2
 n2 
C  arcsin   (for n1  n2 )
 n1 
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
n = n1
n = n2
2
Snell’s Law: Polarization
By inspection of our drawing, we see that the perpendicularity of
the reflected and transmitted rays requires that:
 B   2  90
 2  90   B
B
B
n = n1
n = n2
2
Snell’s Law: Polarization
Snell’s Law:
n2 sin  2  n1 sin  B
Substitute for 2:
n2 sin 90  B   n1 sin  B
Snell’s Law: Polarization
Snell’s Law:
n2 sin  2  n1 sin  B
Substitute for 2:
n2 sin 90  B   n1 sin  B
angle-difference identity: sin(    )  sin  cos   cos  sin 
n1
sin 90   B   cos B  sin  B
n2
 n2 
 B  arctan  
 n1 

n1
tan  B  1
n2
Snell’s Law: Polarization
 n2 
 B  arctan  
 n1 
B is called Brewster’s angle.
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
 n2 
 B  arctan  
 n1 
B
B
n = n1
n = n2
2
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:
t
n = n2
 n2  n1 
t
d  
 n2 
n = n1
d
Snell’s Law: Longitudinal Focus Shift
If an object is immersed in one material and viewed
from another: “apparent depth”
n = n2
n2
d ' d
n1
n = n1
d’
d
Snell’s Law: Longitudinal Focus Shift
The longitudinal focus shift and apparent depth
relationships presented:
 n2  n1 
t
d  
 n2 
n2
d ' d
n1
are paraxial approximations. Even flat surfaces exhibit
spherical aberration in converging or diverging beams
of light.
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
POSITIVE FORMS
Lens forms (edge
views)
Positive: center
thicker than edge
biconvex
plano-convex
biconcave
plano-concave
positive meniscus
Negative: edge thicker
than center
NEGATIVE FORMS
negative meniscus
Refractive Image Formation: Lenses
POSITIVE FORMS
Positive: also called
“converging”
biconvex
plano-convex
biconcave
plano-concave
positive meniscus
Negative: also called
“diverging”
NEGATIVE FORMS
negative meniscus
Refractive Image Formation: Lenses
Real image formation by a positive lens:
optical axis
f
(focal length)
focal point
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?
do
di
“Thin lens:” a simplifying assumption that all the
refraction takes place at a plane in the center of the
lens.
Refractive Image Formation: Lenses
A better picture: “thick lens:”
principal planes
do
di
principal points
The conjugate distances are measured from the
principal points.
Refractive Image Formation: Lenses
A catalog example:
Image from catalog of Melles
Griot Corporation
Refractive Image Formation: Lenses
The lens equation:
1
1 1
 
do di
f
Magnification:
di
m
do
Combinations: one lens’s image is the next lens’s
object.
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.
1
P
f
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).
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 l = 0.55 mm (yellowgreen). 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.

ho

do
Angular sizes of object and image are equal.
Optical Instruments
Angular Size of Objects and Images

ho

do
small-angle approximation:  
ho
(radians)
do
Optical Instruments
Angular Size of Objects and Images

ho

do
The larger  is, the more retinal pixels (rod and cone cells) are
covered by the image. (Better, more detailed picture.)
Optical Instruments
Angular Size of Objects and Images

ho

do
Optical instruments present an image to the eye that has a
larger angular size than it would without the instrument.
Optical Instruments: Simple Magnifier
(“Magnifying Glass”)
Enlarged virtual image
of object has larger
angular size

M

ho
M
ho
do
N
N

do
Optical Instruments: Simple Magnifier
(“Magnifying Glass”)
The value of do depends on
di (how the person uses
the magnifier).
Image at infinity:
N
M
f
Image at near point:
N
M  1
f
Optical Instruments: Compound
Microscope
Compound microscope:
consists of an
objective lens and an
eyepiece.
Illustration from the online catalog of
Melles Griot Corporation.
Optical Instruments: Compound
Microscope
Magnification (“official” Cutnell & Johnson version):
M

L  f e N

fo fe
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
Optical Instruments: Compound
Microscope
Magnification (useful):
M  MoMe
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.
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.
f
o
fe
’

focal
plane
exit
pupil
entrance
pupil
Optical Instruments: Telescope
The magnification is the ratio of the objective to eyepiece focal
lengths.
f
o
fe
’

focal
plane
exit
pupil
entrance
pupil
fo
M 
fe
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
fo
fe
’

focal
plane
exit
pupil
entrance
pupil
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.