Transcript Imaging condition

Lenses and imaging







Huygens principle and why we need imaging
instruments
A simple imaging instrument: the pinhole camera
Principle of image formation using lenses
Quantifying lenses: paraxial approximation & matrix
approach
“Focusing” a lens: Imaging condition
Magnification
Analyzing more complicated (multi-element) optical
systems:
– Principal points/surfaces
– Generalized imaging conditions from matrix formulae
MIT 2.71/2.710
09/10/01 wk2-a-1
The minimum path principle
∫Γ n(x, y, z) dl
Γ is chosen to minimize this
“path” integral, compared to
alternative paths
(aka Fermat’s principle)
Consequences: law of reflection, law of refraction
MIT 2.71/2.710
09/10/01 wk2-a-2
The law of refraction
nsinθ = n’sinθ’
MIT 2.71/2.710
09/10/01 wk2-a-3
Snell’s Law of Refraction
Ray bundles
point
source
spherical
wave
(diverging)
point
source
very-very
far away
plane
wave
MIT 2.71/2.710
09/10/01 wk2-a-4
Huygens principle
Each point on the wavefront
acts as a secondary light
source emitting a spherical
wave
The wavefront after a short
propagation distance is the
result of superimposing all
these spherical wavelets
optical
wavefronts
MIT 2.71/2.710
09/10/01 wk2-a-5
Why imaging systems are needed




Each point in an object scatters the incident illumination into a
spherical wave, according to the Huygens principle.
A few microns away from the object surface, the rays emanating
from all object points become entangled, delocalizing object details.
To relocalize object details, a method must be found to reassign
(“focus”) all the rays that emanated from a single point object into
another point in space (the “image.”)
The latter function is the topic of the discipline of Optical Imaging.
MIT 2.71/2.710
09/10/01 wk2-a-6
The pinhole camera
The pinhole camera blocks all but one ray per object point from reaching
the image space ⇒ an image is formed (i.e., each point in image space
corresponds to a single point from the object space).
 Unfortunately, most of the light is wasted in this instrument.
 Besides, light diffracts if it has to go through small pinholes as we will see
later; diffraction introduces artifacts that we do not yet have the tools to quantify.

MIT 2.71/2.710
09/10/01 wk2-a-7
Lens: main instrument for image
formation
Point source
(object)
Point image
The curved surface makes the rays bend proportionally to their distance
from the “optical axis”, according to Snell’s law. Therefore, the divergent
wavefront becomes convergent at the right-hand (output) side.
MIT 2.71/2.710
09/10/01 wk2-a-8
Analyzing lenses: paraxial ray-tracing
Free-space
propagation
Refraction at
air-glass
interface
MIT 2.71/2.710
09/10/01 wk2-a-9
Free-space
propagation
Free-space
propagation
Refraction at
glass-air
interface
Paraxial approximation /1

In paraxial optics, we make heavy use of the following approximate
(1st order Taylor) expressions:
sinε ≒ ε≒ tanε
cosε ≒ 1
where ε is the angle between a ray and the optical axis, and is a small
number (ε<<1 rad). The range of validity of this approximation
typically extends up to ~10-30 degrees, depending on the desired
degree of accuracy. This regime is also known as “Gaussian optics.”
Note the assumption of existence of an optical axis (i.e., perfect
alignment!)
MIT 2.71/2.710
09/10/01 wk2-a-10
Paraxial approximation /2
Apply Snell’s law as if
ray bending occurred at
the intersection of the
axial ray with the lens
Ignore the distance
between the location
of the axial ray
intersection and the
actuall off-axis ray
intersection
axial ray
MIT 2.71/2.710
09/10/01 wk2-a-11
Valid for small curvatures
& thin optical elements
Example: one spherical surface,
translation+refraction+translation
MIT 2.71/2.710
09/10/01 wk2-a-12
Translation+refraction+translation /1
Starting ray: location x0 direction α0
Translation through distance D01 (+ direction):
-------------------------------------------------------Refraction at positive spherical surface:
MIT 2.71/2.710
09/10/01 wk2-a-13
Translation+refraction+translation /2
Translation through distance D12 (+ direction):
--------------------------------------------------------Put together:
MIT 2.71/2.710
09/10/01 wk2-a-14
Translation+refraction+translation /3
MIT 2.71/2.710
09/10/01 wk2-a-15
Sign conventions for refraction





Light travels from left to right
A radius of curvature is positive if the surface is convex towards the
left
Longitudinal distances are positive if pointing to the right
Lateral distances are positive if pointing up
Ray angles are positive if the ray direction is obtained by rotating the
+z axis counterclockwise through an acute angle
MIT 2.71/2.710
09/10/01 wk2-a-16
On-axis image formation
All rays emanating at x0 arrive at x2
irrespective of departure angle α0
∂x2 / ∂α0
=0
“Power” of the spherical
surface [units: diopters, 1D=1m-1]
MIT 2.71/2.710
09/10/01 wk2-a-17
Magnification: lateral (off-axis), angle
Lateral
Angle
MIT 2.71/2.710
09/10/01 wk2-a-18
Object-image transformation
Ray-tracing transformation
(paraxial) between
object and image points
MIT 2.71/2.710
09/10/01 wk2-a-19
Image of point object at infinity
MIT 2.71/2.710
09/10/01 wk2-a-20
Point object imaged at infinity
MIT 2.71/2.710
09/10/01 wk2-a-21
Matrix formulation /1
translation by
distance D10
form common to all
MIT 2.71/2.710
09/10/01 wk2-a-22
refraction by
surface with radius
of curvature R
ray-tracing
object-image
transformation
Matrix formulation /2
Refraction by spherical surface
Translation through uniform medium
MIT 2.71/2.710
09/10/01 wk2-a-23
Power
Translation+refraction+translation
MIT 2.71/2.710
09/10/01 wk2-a-24
Thin lens
MIT 2.71/2.710
09/10/01 wk2-a-25
The power of surfaces

Positive power bends rays “inwards”
Simple spherical Plano-convex
refractor (positive)
lens

Bi-convex
lens
Negative power bends rays “outwards”
Simple spherical
Plano-concave
refractor (negative)
lens
MIT 2.71/2.710
09/10/01 wk2-a-26
Bi-concave
lens
The power in matrix formulation
(Ray bending)= (Power)×(Lateral coordinate)
⇒ (Power) = −M12
MIT 2.71/2.710
09/10/01 wk2-a-27
Power and focal length
MIT 2.71/2.710
09/10/01 wk2-a-28
Thick/compound elements:
focal & principal points (surfaces)
Note: in the paraxial approximation, the focal & principal surfaces are flat (i.e.,
planar). In reality, they are curved (but not spherical!!).The exact calculation is
very complicated.
MIT 2.71/2.710
09/10/01 wk2-a-29
Focal Lengths for thick/compound
elements
generalized
optical
system
EFL: Effective Focal Length (or simply “focal length”)
FFL: Front Focal Length
BFL: Back Focal Length
MIT 2.71/2.710
09/10/01 wk2-a-30
PSs and FLs for thin lenses
1/(EFL) ≡ P = P1 + P2
The principal planes coincide with the (collocated) glass surfaces
The rays bend precisely at the thin lens plane (=collocated glass surfaces &
PP)


(BFL) = (EFL) = (FFL)
MIT 2.71/2.710
09/10/01 wk2-a-31
The significance of principal planes /1
MIT 2.71/2.710
09/10/01 wk2-a-32
The significance of principal planes /2
MIT 2.71/2.710
09/10/01 wk2-a-33
Imaging condition: ray-tracing


Image point is located at the common intersection of all rays which
emanate from the corresponding object point
The two rays passing through the two focal points and the chief ray
can be ray-traced directly
MIT 2.71/2.710
09/10/01 wk2-a-34
Imaging condition: matrix form /1
MIT 2.71/2.710
09/10/01 wk2-a-35
Imaging condition: matrix form /2
Imaging condition: 1
Output coordinate x´ must not
depend on entrance angle γ
MIT 2.71/2.710
09/10/01 wk2-a-36
Imaging condition: matrix form /3
Imaging condition:
MIT 2.71/2.710
09/10/01 wk2-a-37
S’/n’ + S/n – PSS’/nn’ = 0
system immersed in air,
n=n’=1;
power P=1/f
1/S + 1/S’ = 1/f
n/S + n’/S’ = P
Lateral magnification
MIT 2.71/2.710
09/10/01 wk2-a-38
Angular magnification
MIT 2.71/2.710
09/10/01 wk2-a-39
Generalized imaging conditions
image
Power:
Imaging condition:
Lateral magnification:
Angular magnification:
MIT 2.71/2.710
09/10/01 wk2-a-40
system
matrix
object
P = -M12 ≠ 0
M21 = 0
mX = M22
Ma = n/n’M11