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Lenses and imaging
MIT 2.71/2.71009/10/01 wk2-a-1
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
The minimum path principle
MIT 2.71/2.71009/10/01 wk2-a-2
(aka Fermat’s principle) Consequences: law of reflection, law of refraction
∫Γ n(x, y, z) dl
Γ is chosen to minimize this“path” integral, compared to
alternative paths
The law of refraction
MIT 2.71/2.71009/10/01 wk2-a-3
nsinθ = n’sinθ’ Snell’s Law of Refraction
Ray bundles
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pointsource
planewave
pointsource
very-veryfar away
sphericalwave
(diverging)
Huygens principle
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Each point on the wavefrontacts as a secondary light source emitting a spherical wave
The wavefront after a shortpropagation distance is theresult of superimposing allthese spherical wavelets
opticalwavefronts
Why imaging systems are needed
MIT 2.71/2.71009/10/01 wk2-a-6
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 pinhole camera
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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.
Lens: main instrument for imageformation
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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.
Point source(object)
Point image
Analyzing lenses: paraxial ray-tracing
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Free-spacepropagation
Refraction atair-glassinterface
Free-spacepropagation
Free-spacepropagation
Refraction atglass-airinterface
Paraxial approximation /1
MIT 2.71/2.71009/10/01 wk2-a-10
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 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.”
Note the assumption of existence of an optical axis (i.e., perfectalignment!)
Paraxial approximation /2
MIT 2.71/2.71009/10/01 wk2-a-11
Valid for small curvatures& thin optical elements
Ignore the distancebetween the location
of the axial rayintersection and theactuall off-axis ray
intersection
off-axis ra
y
axial ray
Apply Snell’s law as ifray bending occurred atthe intersection of theaxial ray with the lens
Example: one spherical surface,translation+refraction+translation
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Translation+refraction+translation /1
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Starting ray: location x0 direction α0
Translation through distance D01 (+ direction):
--------------------------------------------------------
Refraction at positive spherical surface:
Translation+refraction+translation /2
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Translation through distance D12 (+ direction): ---------------------------------------------------------
Put together:
Translation+refraction+translation /3
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Sign conventions for refraction
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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
On-axis image formation
MIT 2.71/2.71009/10/01 wk2-a-17
All rays emanating at x0 arrive at x2
irrespective of departure angle α0
∂x2 / ∂α0
= 0
“Power” of the sphericalsurface [units: diopters, 1D=1m-1]
Magnification: lateral (off-axis), angle
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Lateral
Angle
Object-image transformation
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Ray-tracing transformation(paraxial) between
object and image points
Image of point object at infinity
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Point object imaged at infinity
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Matrix formulation /1
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translation bydistance D10
refraction bysurface with radius
of curvature R
form common to allray-tracing
object-imagetransformation
Matrix formulation /2
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PowerRefraction by spherical surface
Translation through uniform medium
Translation+refraction+translation
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Thin lens
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The power of surfaces
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Positive power bends rays “inwards”
Negative power bends rays “outwards”
Simple sphericalrefractor (positive)
Plano-convexlens
Bi-convexlens
Simple sphericalrefractor (negative)
Plano-concavelens
Bi-concavelens
The power in matrix formulation
MIT 2.71/2.71009/10/01 wk2-a-27
(Ray bending)= (Power)×(Lateral coordinate)
⇒ (Power) = −M12
Power and focal length
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Thick/compound elements:focal & principal points (surfaces)
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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 isvery complicated.
Focal Lengths for thick/compoundelements
MIT 2.71/2.71009/10/01 wk2-a-30
generalizedopticalsystem
EFL: Effective Focal Length (or simply “focal length”)FFL: Front Focal LengthBFL: Back Focal Length
PSs and FLs for thin lenses
MIT 2.71/2.71009/10/01 wk2-a-31
The principal planes coincide with the (collocated) glass surfaces The rays bend precisely at the thin lens plane (=collocated glass surfaces &
PP)
1/(EFL) ≡ P = P1 + P2 (BFL) = (EFL) = (FFL)
The significance of principal planes /1
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The significance of principal planes /2
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Imaging condition: ray-tracing
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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
Imaging condition: matrix form /1
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Imaging condition: matrix form /2
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Imaging condition: 1Output coordinate x´ must notdepend on entrance angle γ
Imaging condition: matrix form /3
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Imaging condition: S’/n’ + S/n – PSS’/nn’ = 0
system immersed in air,n=n’=1;
power P=1/f
n/S + n’/S’ = P
1/S + 1/S’ = 1/f
Lateral magnification
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Angular magnification
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Generalized imaging conditions
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Power:
Imaging condition:
Lateral magnification:
Angular magnification:
image systemmatrix
object
P = -M12 ≠ 0
M21 = 0
mX = M22
Ma = n/n’M11
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