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Lenses and Imaging (Part I)
• Why is imaging necessary: Huygen’s principle
– Spherical & parallel ray bundles, points at infinity
• Refraction at spherical surfaces (paraxial approximation)
• Optical power and imaging condition
• Matrix formulation of geometrical optics
• The thin lens
• Surfaces of positive/negative power
• Real and virtual images
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Parabloid mirror: perfect focusing
(e.g. satellite dish)
f: focal length
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Lens: main instrument for image
formation
air
glass
air
optical
axis
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.
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Why are focusing instruments
necessary?
•
•
•
•
Ray bundles: spherical waves and plane waves
Point sources and point images
Huygens principle and why we can see around us
The role of classical imaging systems
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Ray bundles
point
source
point
source
very-very
far away
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spherical
wave
(diverging)
wave-front
⊥ rays
wave-front
⊥ rays
plane
wave
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
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Why are focusing instruments
necessary?
incident light
...
is scattered by
the object
object
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Why are focusing instruments
necessary?
incident light
...
is scattered by
the object
object
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⇒need optics to “undo”
the effects of scattering,
i.e. focus the light
image
Ideal lens
air
glass
air
optical
axis
Point source
(object)
Point image
Each point source from the object plane focuses onto a point
image at the image plane; NOTE the image inversion
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Summary:
Why are imaging systems 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.
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The ideal optical imaging system
optical
elements
object
image
Ideal imaging system:
each point in the object is mapped
onto a single point in the image
Real imaging systems introduce blur ...
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Focus, defocus and blur
Perfect focus
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Defocus
Focus, defocus and blur
Perfect focus
Imperfect focus
“spherical
aberration”
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Why optical systems do not focus
perfectly
• Diffraction
• Aberrations
• However, in the paraxial approximation to Geometrical
Optics that we are about to embark upon, optical systems
do focus perfectly
• To deal with aberrations, we need non-paraxial
Geometrical Optics (higher order approximations)
• To deal with diffraction, we need Wave Optics
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Ideal lens
air
glass
air
optical
axis
Point source
(object)
Point image
Each point source from the object plane focuses onto a point
image at the image plane
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Refraction at single spherical surface
for each ray, must calculate
• point of intersection with sphere
• angle between ray and normal to
surface
• apply Snell’s law to find direction of
propagation of refracted ray
R: radius of
spherical
surface
center of
spherical
surface
medium 1
index n,
e.g. air n=1
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Paraxial approximation /1
• In paraxial optics, we make heavy use of the following approximate
(1st order Taylor) expressions:
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” or
“paraxial optics.”
Note the assumption of existence of an
optical axis (i.e., perfect
alignment!)
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Paraxial approximation /2
Ignore the distance
between the location
of the axial ray
intersection and the
actual off-axis ray
intersection
Apply Snell’s law as if
ray bending occurred at
the intersection of the
axial ray with the lens
Valid for small curvatures
& thin optical elements
axial ray
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Refraction at spherical surface
Refraction at positive spherical surface:
Power
x: ray height
α: ray direction
R: radius
of curvature
optical axis
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Propagation in uniform space
Propagation through distance D:
x: ray height
α: ray direction
optical axis
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Paraxial ray-tracing
air
glass
Free-space
propagation
Free-space
propagation
Refraction at
air-glass
interface
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air
Free-space
propagation
Refraction at
glass-air
interface
Example: one spherical surface,
translation + refraction + translation
R: radius of
spherical
surface
Paraxial rays
(approximation
valid)
center of
spherical
surface
Non-paraxial ray
(approximation
gives large error)
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medium 1
index n,
e.g. air n=1
Translation + refraction + translation /1
Starting ray: location x0 directionα0
Translation through distance D01 (+ direction):
Refraction at positive spherical surface:
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Translation + refraction + translation /2
Translation through distance D12 (+ direction):
Put together:
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Translation + refraction + translation /3
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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
optical axis +z
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On-axis image formation
Point
object
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Point
image
On-axis image formation
All rays emanating at arrive at x2
irrespective of departure angleα0
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On-axis image formation
All rays emanating at arrive at x2
irrespective of departure angleα0
“Power” of the spherical
surface [units: diopters, 1D=1 m - 1 ]
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Off-axis image formation
Point
object
(off-axis)
optical
axis
Point
image
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Magnification: lateral (off-axis), angle
Lateral
Angle
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Object-image transformation
Ray-tracing transformation
(paraxial) between
object and image points
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Image of point object at infinity
Point
image
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Image of point object at infinity
object
at ∞
image
: image focal length
Note:
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ambient refractive index
at space of point image
1/Power
Point object imaged at infinity
Point
object
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Point object imaged at infinity
Image
at ∞
object
: object focal length
Note:
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ambient refractive index
at space of point image
1/Power
Image / object focal lengths
Point
image
object
at ∞
Image
at ∞
Power
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Point
object
Power
Matrix formulation /1
translation by
Distance D 01
form common to all
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refraction by
surface with radius
of curvature R
ray-tracing
object-image
transformation
Matrix formulation /2
Refraction by spherical surface
Power
Translation through uniform medium
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Translation + refraction + translation
translation
by D12
result…
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Refraction translation
by r.curv. R by D01
Thin lens in air
Objective: specify input-output relationship
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Thin lens in air
Model: refraction from first (positive) surface
+ refraction from second (negative) surface
Ignore space in-between (thin lens approx.)
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Thin lens in air
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Thin lens in air
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Thin lens in air
Power of the
first surface
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Power of the
second surface
Thin lens in air
thin lens
thin lens
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Lens-maker’s
formula
Thin lens in air
thin lens
thin lens
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Ray bending is proportional
to the distance from the axis
The power of surfaces
• Positive power bends rays “inwards”
Simple spherical
refractor (positive)
Plano-convex
lens
Bi-convex
lens
• Negative power bends rays “outwards”
Simple spherical
refractor (negative)
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Plano-convex
lens
Bi-convex
lens