Lenses and Imaging (Part I)web.mit.edu/2.710/Fall06/2.710-wk2-b-ho.pdf · 09/14/05 wk2-b-13 Focus,...
Transcript of Lenses and Imaging (Part I)web.mit.edu/2.710/Fall06/2.710-wk2-b-ho.pdf · 09/14/05 wk2-b-13 Focus,...
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Lenses and Imaging (Part I)• Review: paraboloid reflector, focusing• Why is imaging necessary: Huygens’ principle
– Spherical & parallel ray bundles, points at infinity• Refraction at spherical surfaces (paraxial approximation)
– Optical power of surfaces: positive, negative surfaces• Paraxial ray-tracing
– Matrix formulation of paraxial geometrical optics• Imaging condition• Object/image at infinity
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Parabloid mirror: perfect focusing
s(x) x
z
(e.g. satellite dish)
( )f
xxs4
2
=f
f: focal length
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Lens: main instrument of 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.
air glass air
opticalaxis
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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
pointsource
sphericalwave
(diverging)
… …∞
pointsource
very-veryfar away
planewave
wave-front⊥ rays
wave-front⊥ rays
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Huygens principle
opticalwavefronts
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
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Why are focusing instruments necessary?
objectobject
...is scattered by
the object
incident light
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Why are focusing instruments necessary?
objectobject
...is scattered by
the object
incident light
imageimage
⇒need optics to “undo”the effects of scattering,
i.e. focus the light
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Ideal lens
Point sources(object)
Point images
Each point source from the object planeobject plane focuses onto a point image at the image planeimage plane; NOTENOTE the image inversionthe image inversion
air glass air
opticalaxis
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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
opticalelements
Ideal imaging system:each point in the objectobject is mapped
onto a single point in the imageimage
objectobject imageimage
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 do notnot focus focus perfectlyperfectly
•• DiffractionDiffraction•• AberrationsAberrations
• However, in the paraxial approximation to Geometrical Optics that we are about to embark upon, optical systems do focus perfectlydo 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
Point sources(object)
Point images
Each point source from the object planeobject plane focuses onto a point image at the image planeimage plane
air glass air
opticalaxis
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Refraction at single spherical surface
medium 1index n,
e.g. air n=1
01D 12D
medium 2index n´,
e.g. glass n´=1.5
R
center ofsphericalsurface
R: radius ofsphericalsurface
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
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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 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!)
εεε tansin ≈≈
εε2111 +≈+
1cos ≈ε
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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 actual off-axis ray
intersection
axial ray
off-axis ra
yValid for small curvatures
& thin optical elements
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Refraction at spherical surface
optical axis
off-axis ra
y
Refraction at positive spherical surface: ( )111
11
xRn
nnnn
xx
⎥⎦⎤
⎢⎣⎡
′−′
−′
=′
=′
αα
1α
1α′
1x 1x′
x: ray heightα: ray direction
n n’
R: radiusof curvature
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Refraction at spherical surface
optical axis
off-axis ra
y
Refraction at positive spherical surface: ( )111
11
xRn
nnnn
xx
⎥⎦⎤
⎢⎣⎡
′−′
−′
=′
=′
αα
1α
1α′
1x 1x′
x: ray heightα: ray direction
n n’
R: radiusof curvature
PowerPower
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Propagation in uniform space
off-axis ray
Propagation through distance D:
1α1α′
1x
1x′n
optical axis
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111
ααα
=′+=′ Dxx
D
x: ray heightα: ray direction
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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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The power of surfaces
• Positive power bends rays “inwards” (converging bundle)
• Negative power bends rays “outwards” (diverging bundle)
Simple sphericalrefractor (positive)
1 n
R>0
R<0
Simple sphericalrefractor (negative)
1 n
–
( ) ( )( ) 0
1leftright >++−
=−
≡R
nR
nnP
( ) ( )( ) 0
1leftright <−+−
=−
≡R
nR
nnP
+
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The power of surfaces
• Positive power bends rays “inwards” (converging bundle)
• Negative power bends rays “outwards” (diverging bundle)
+
n 1
R<0
Simple sphericalrefractor (negative)
–
( ) ( )( ) 0
1leftright >−−−
=−
≡R
nR
nnP
( ) ( )( ) 0
1leftright <+−−
=−
≡R
nR
nnP
R>0
n 1
Simple sphericalrefractor (positive)
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Paraxial ray-tracing
air glass air
Free-spacepropagation
Free-spacepropagation
Refraction atair-glassinterface
Refraction atglass-airinterface
Free-spacepropagation
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Example: one spherical surface,translation+refraction+translation
medium 1index n,
e.g. air n=1
Paraxial rays(approximation
valid)
Non-paraxial ray(approximation
gives large error)
01D 12D
medium 2index n´,
e.g. glass n´=1.5
R
center ofsphericalsurface
R: radius ofsphericalsurface
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Translation+refraction+translation /1
01D 12D
0x 1x′ 2x10 αα =
21 αα =′
Starting ray: location direction0x 0α
Translation through distance (+ direction):01D01
00101
ααα
=+= Dxx
Refraction at positive spherical surface: ( )111
11
xRn
nnnn
xx
⎥⎦⎤
⎢⎣⎡
′−′
−′
=′
=′
αα
1x
n n´
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Translation+refraction+translation /2
01D 12D
0x 1x′ 2x10 αα =
Translation through distance (+ direction):12D12
11212
ααα
′=
′+= Dxx
Put together:
1x
21 αα =′n n´
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Translation+refraction+translation /3
( ) ( )0
120
120112012 1 x
RnDnn
RnDDnn
nnDDx ⎥⎦
⎤⎢⎣⎡ +
′′−
+⎥⎦⎤
⎢⎣⎡
′′−
+′
+= α
( )00
012 x
Rnnn
RnDnn
nn
⎥⎦⎤
⎢⎣⎡
′′−
+⎥⎦⎤
⎢⎣⎡
′′−
+′
= αα
01D 12D
0x 1x′ 2x10 αα =
1x
21 αα =′n n´
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(Paraxial) Ray-tracing in general
inxoutx
inαoutαnin nout
OPTICALSYSTEM
For an arbitrary ray entering with• lateral departure displacement xin wrt the optical axis• angle of departure αin wrt the optical axis• in a departure space of refractive index nin
determine the ray’s• lateral arrival displacement xout wrt the optical axis• angle of arrival αout wrt the optical axis• in an arrival space of refractive index nout
upon exiting the optical system
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Matrix formulation /1
01
00101
ααα
=+= Dxx
( )111
11
xRnnn
nn
xx
⎥⎦⎤
⎢⎣⎡
′′−
+′
=′
=′
αα
02 xmx x=
0021 αα αmxf
+′
−=
refraction bysurface with radius
of curvature R
ray-tracingobject-image
transformation
translation bydistance 01D
form common to all
in22in21out
in12in11out
xMMxxMM
+=+=
ααα
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Matrix formulation /2
in22in21out
in12in11out
xMMxxMM
+=+=
ααα
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛
in
inin
2221
1211
out
outout
xn
MMMM
xn αα
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟
⎠
⎞
⎜⎜
⎝
⎛ −′−=⎟⎟
⎠
⎞⎜⎜⎝
⎛′′′
1
1
1
1
10
1x
nR
nn
xn αα( )
111
11
xRnnn
nn
xx
⎥⎦⎤
⎢⎣⎡
′′−
+′
=′
=′
αα
01
00101
ααα
=+= Dxx
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟
⎠
⎞
⎜⎜
⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛
0
001
1
1
101
xn
nD
xn αα
Refraction by spherical surface
Translation through uniform medium
Power
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Translation+refraction+translation
( ) ( )0
120
120112012 1 x
RnDnn
RnDDnn
nnDDx ⎥⎦
⎤⎢⎣⎡ +
′′−
+⎥⎦⎤
⎢⎣⎡
′′−
+′
+= α
( )00
012 x
Rnn
RDnnnn ⎥⎦
⎤⎢⎣⎡ ′−
+⎥⎦⎤
⎢⎣⎡ ′−
+=′ αα
01D 12D
0x 1x′ 2x10 αα =
1x
21 αα =′n n´
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛ ′
0
0
01122
2
by ntranslatio
r.curv.by refraction
by ntranslatio
xn
DRDxn αα
result…
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Matrix ray-tracing in general
inxoutx
inαoutαnin nout
OPTICALSYSTEM
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛
in
inin
2221
1211
out
outout
xn
MMMM
xn αα
⎟⎟⎠
⎞⎜⎜⎝
⎛=
2221
1211
MMMM
M is the “system matrix,” found as the product, in reverse order, of the matricesof the elements (free space, spherical surfaces)comprising the system
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On-axis image formation
Pointobjectobject
Pointimageimage
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On-axis image formation
All rays emanating at arrive at irrespective of departure angle
0x 2x
0α00 21
0
2 =⇔=∂∂ Mxα
01D 12D
00 =x 2x0α
2αn n´
( ) [ ] 00120112
012 xRn
DDnnn
nDDx L+⎥⎦⎤
⎢⎣⎡
′−′
−′
+= α
Imaging conditionImaging condition
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On-axis image formation
Rnn
Dn
Dn −′
=+′
0112“Power” of the spherical
surface [units: dioptersdiopters, 1D=1 ]-1m
01D 12D
00 =x 2x0α
2αn n´
All rays emanating at arrive at irrespective of departure angle
0x 2x
0α00 21
0
2 =⇔=∂∂ Mxα
Imaging conditionImaging condition
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Off-axis image formation
Pointobjectobject
(off-axis)
Pointimageimage
opticalaxis
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Magnification: lateral (off-axis), angle
01D 12D
00 =x
2x
0α2α
n n´
01
1212
0
2 ..... 1' D
Dnn
nD
Rnn
xxmx ′
−==+′−
==
0α∆
2α∆
12
01
0
2
DDm −=
∆∆
=αα
α
LateralLateral
AngleAngle
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Object-image transformation
01D 12D
00 =x
2x
0α2α
n n´0α∆
2α∆
02 xmx x=
0021 xf
m′
−= αα α Paraxial ray-tracingtransformation betweenobject and image points
condition) (imaging 0 subject to 21 =M
21
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Matrix ray-tracing: summary
inxoutx
inαoutαnin nout
OPTICALSYSTEM
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛
in
inin
2221
1211
out
outout
xn
MMMM
xn αα
• Power• Imaging condition• Lateral magnification• Angular magnification
12M021 =M
01121=M
M021
21=MM
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Image of point object at infinity
Pointimageimage
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Image of point object at infinity
∞=01D 12D
0x02 =x
00 =α
0x
2αn n´
length focal image : 1212
fnn
RnDR
nnDn ′≡
−′′
=⇒−′
=′
objectat ∞
image
Note: nn
Rnnn
Rnf−′
×′≡−′
×′=′
ambient refractive indexat space of point image
1/Power
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Point object imaged at infinity
Pointobjectobject
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Point object imaged at infinity
∞=12D01D
00 =x2x
02 =α0αn n´
length focalobject : 01
01
fnn
RnDR
nnDn
≡−′
=⇒−′
=
object
imageat ∞
Note: nn
Rnnn
Rnf−′
×≡−′
×=
ambient refractive indexat space of point object
1/Power
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Image / object focal lengths
Pointobjectobject
Pointimageimage
Objectat ∞
Imageat ∞
Power1
×=
−′×=
n
nnRnf
Power1 ×′=
−′×′=′
n
nnRnf