lenses and imaging (part i) why is imaging necessary: huygen’s principle – spherical &...
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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 imageformation
air
opticalaxis
airglass
Point source(object)
Point image
The curved surface makes the rays bend proportionally to their distancefrom 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 instrumentsnecessary?
• 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
pointsource
sphericalwave
(diverging)wave-front
⊥ rays
wave-front ⊥ rays
pointsource
very-veryfar away plane
wave
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Huygens principle
Each point on the wavefrontacts as a secondary light sourceemitting a spherical wave
The wavefront after a shortpropagation distance is theresult of superimposing allthese spherical wavelets
opticalwavefronts
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Why are focusing instrumentsnecessary?
incident light
...is scattered by
the object
object
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Why are focusing instrumentsnecessary?
incident light
...is scattered by
the object
objectimage
⇒need optics to “undo”the effects of scattering,i.e. focus the light
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Ideal lens
air
opticalaxis
airglass
Point source(object)
Point image
Each point source from the object plane focuses onto a pointimage 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.
The ideal optical imaging system
opticalelements
object image
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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 Defocus
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Focus, defocus and blur
Perfect focus Imperfect focus
“sphericalaberration”
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Why optical systems do not focusperfectly
• 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
opticalaxis
airglass
Point source(object)
Point image
Each point source from the object plane focuses onto a pointimage 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 tosurface• apply Snell’s law to find direction ofpropagation of refracted ray
R: radius ofsphericalsurface
center ofsphericalsurface
medium 1index 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 smallnumber (ε «1 rad). The range of validity of this approximationtypically extends up to ~10-30 degrees, depending on the desireddegree of accuracy. This regime is also known as “Gaussian optics” or“paraxial optics.”
Note the assumption of existence of an optical axis (i.e., perfectalignment!)
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Paraxial approximation /2
Ignore the distancebetween the location
of the axial rayintersection and theactual off-axis ray
intersection
Apply Snell’s law as ifray bending occurred at
the intersection of theaxial ray with the lens Valid for small curvatures
& thin optical elements
axial ray
off-axis r
ay
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Refraction at spherical surface
Refraction at positive spherical surface:
Power x: ray heightα: ray direction
R: radiusof curvature
optical axis
off-axis r
ay
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Propagation in uniform space
Propagation through distance D:
x: ray heightα: ray direction
optical axis
off-axis ray
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Paraxial ray-tracing
air airglass
Free-spacepropagation
Refraction atair-glassinterface
Free-spacepropagation
Free-spacepropagation
Refraction atglass-airinterface
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Example: one spherical surface,translation + refraction + translation
Non-paraxial ray(approximation
gives large error) medium 1index n,
e.g. air n=1
center ofsphericalsurface
Paraxial rays(approximation
valid)
R: radius ofsphericalsurface
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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
Pointobject
Pointimage
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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 sphericalsurface [units: diopters, 1D=1 m - 1 ]
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Off-axis image formation
Pointobject
(off-axis)
Pointimage
opticalaxis
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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
Pointimage
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Image of point object at infinity
objectat ∞
image
: image focal length
Note:
ambient refractive indexat space of point image
1/Power
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Point object imaged at infinity
Pointobject
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Point object imaged at infinity
object
Imageat ∞
: object focal length
Note:
ambient refractive indexat space of point image
1/Power
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Image / object focal lengths
Pointobject
Imageat ∞
objectat ∞
Pointimage
Power Power
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Matrix formulation /1
translation byDistance D 01
refraction bysurface with radius
of curvature R
ray-tracingobject-image
transformationform common to all
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Matrix formulation /2
Refraction by spherical surface Power
Translation through uniform medium
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Translation + refraction + translation
translation by D12
Refraction by r.curv. R
translation by D01
result…
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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
Thin lens in air
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Thin lens in air
Power of thefirst surface
Power of thesecond surface
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Thin lens in air
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Lens-maker’sformula
thin lens
thin lens
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Thin lens in air
thin lens
thin lens Ray bending is proportionalto the distance from the axis
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The power of surfaces
• Positive power bends rays “inwards”
Simple sphericalrefractor (positive)
Plano-convexlens
Bi-convexlens
• Negative power bends rays “outwards”
Simple sphericalrefractor (negative)
Plano-convexlens
Bi-convexlens