16 Lens
Transcript of 16 Lens
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Lenses
Optics, Eugene Hecht, Chpt. 5
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Lenses for imaging
Object produces many spherical waves
scattering centers
Want to project to different location
Object is collection
of scattering centers
Lens designed to project
and reproduce scattering centers
Diverging spherical waves
Converging spherical waves
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Plane wave approximation
Distant object
Radius of curvature large
Approximate by plane wave
Image approximately at focal plane
Distant object gives plane waves
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Lenses for collimation Convert diverging spherical wave to plane wave
Plane wave like spherical wave with infiniteradius of curvature
First step toward imaging
plane wave like intermediate
To flatten wavefront
distance from S to D must be constant
independent of A
Use Snells law and geometry
Result is equation of hyperbola
ni li + nt lt = const
ntni
lilt
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Spherical lenses
Object distance
Image distanceVertex
Optic axis
Collimation
Focussing
Hyperbolic and elliptical lenses hard to make
Spherical lenses easy to make
Good enough approximation in many cases Example: condition for imaging
path lengths from object to image are equal
n1l0 + n2li = const
From geometry:
Paraxial approximation:
o
o
i
i
io
snsn
R
nn
1221 1
R
nn
s
n
s
n
io
1221
First focal length
= object focal length
R
nn
nfo
22
1
Second focal length
= image focal length
Rnn
n
fi22
2
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Real lenses High index material finite
Two radii of curvature Lensmakers formula
Focal length
Thin lens equation
21
2
111
11
RRn
ss io
21
2
111
1
RRn
f
fss io
111
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Variable focal length
Positive and negative lens combos
Effective focal length (L1 first)
Long focal-length lenses Curvature of incoming light becomes important
Result: Lens does not behave as expected
Solution: Variable focal length
Achromats
Different wavelength dispersions Dispersion ratio = 1/ (focal length ratio)
All colors focus at same point
)( 21
21
ffd
fdf
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Types of lenses
Focal length general case
Special case -- double convex
21
2
111
1
RRn
f
21
2
111
1
RRn
f
Sign conventions for radii
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Lens aberrations
Focusing or collimating
hyperbolic lens shape is ideal
Spherical lens shape
gives insufficient refraction near edges
use plano-convex
Face flat toward spherical wavefront extra refraction
spherical wave on flat interface
Why not double convex ?
Computer solution
plano convex better
only for collimation/focusing
4fimaging
double convex better
symmetry argument
Additional refraction
when spherical wave
encounters planar boundary
Refraction angle
too shallow
Hyperbolic lens best
Aberration reduction
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Non-axial focusing Extended object
Light enters lens from several angles Focus to points on sphere
Approximate by plane
Focal plane
Parallel ray focus to points on sphere
Focal plane
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Basic lens ray tracing tricks
1. Rays through lens center undeflected
2. Rays parallel to optic axis
go through focal point
3. Parallel rays go to point on focal plane
f f
1
2
3
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Lens alignment Position important
Angle less important
slightly changes focal length in one dimension
aberration
Use translation mount instead of tilt plate
ff f f
Lens translation Lens tilt
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Lenses for imaging Single lens -- image
Two lenses -- depends on seperation Interesting case -- telescope
equal focal lengths
4fimaging
unequal focal lengths
magnification =f2/f1 transverse = longitudinal
fss io
111
f
so si
o
iT
s
sM 2
TL MM
ff f f
f1f1 f2 f2
4 fimaging Imaging telescope
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Imaging: transparent vs. scattering objects Scattering object acts as array of sources
image is replica -- one or two lenses
4fconfiguration puts image at a distance w/o magnification -- relay lenses
Transmission object -- curvature important
4fconfiguration better
Scattering
Transmission
f ff f 2f 2f
ff f f
illumination4 fimaging 2 fimaging
2f2f
illumination
illum.
illum.
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Beam expanders Analogous to 4fimaging
wavefront curvature preserved magnification is focal length ratio
independent of lens spacing
Two types
Galilaen and spatial-filter arrangements
Galilaen easier to to set and maintain alignment
Spatial-filter arrangement
Galilaen
- f1
f2
d
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Alignment of telescope Need both tilt and translation (2 lenses)
first tilt to correct far field spot position second translate to center spot in output lens
interate
focus to adjust collimation
Tilt to correct far-field alignment
Far-field
alignment
Translate to center spot in output lens
center spot
Focus to
set collimation
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Spatial filters Laser beam intensity noise
can view as interference of intersecting beamlets Example: beamsplitter
front surface 4% reflection
4% intensity = 20% field
reflected field modulated between 0.8 and 1.2
intensity modulation between 0.64 and 1.4
large effect
Lens converts angle to position
use pinhole to filter out one position
Result is spatial filter
beamsplitter
destructive
f f
Pinhole
apertureAberrated
laser beam
Cleaned
laser beam
Sources of laser aberrations
Spatial filter for laser beam cleanup
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Spatial filter alignment Standard alignment procedure
Translate pinhole aperture until light comes through
Difficult procedure
usually no light until position almost perfect
random walk in 2D not efficient
Solution:
Defocus input lens
larger spot at aperture easy to align
Refocus input lens
spot at aperture shrinks
fine tune alignment
Iterate
f f
Pinhole
apertureAberrated
laser beam
Cleaned
laser beam
Spatial filter alignment:
Translate pinhole
until light comes through
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Problem with spatial filter design Pinhole and output lens define alignment for rest of system
Translating pinhole destroys alignment
Better option:
Translate input lens
Leave output fixed -- alignment reference for rest of system
independent of changes in laser input
f f
PinholeapertureAberrated
laser beam
Cleaned
laser beam
Better spatial filter alignment technique:
Translate lens instead of pinhole
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Resolution of lenses First find angular resolution of aperture
Like multiple interference
Diffraction angles: d sin q = n l Diffraction halfwidth (resolution of grating): N d sin q1/2= l
Take limit as d --> 0, but N d = a (constant) Diffraction angle: sin q = n l / d
only works for n = 0, q = 0 -- (forward direction)
Angular resolution: sin q1/2= l/ N d = l/ D Lens converts angle resolution to position resolution
x1/2 =fl / D(n = 1)
circular lens: x1/2 = 1.22 fl / D
d
qPath
difference
d sin q = n l
Path difference
N d sin q1/2= n l
N d = D
D
f
2 x1/2
Lens resolutionLike array
of sources
limit of zero
separation
Grating resolution
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More on lens/aperture resolution
Lens exchanges angle for position
Fourier transform
Lens is rectangular aperture
F.T. of rectangle is sinc(x) = sin(x)/x
D
f
2 x1/2 =2.44fl / DLens resolution
Like array
of sources
limit of zero
separation Sinc function
Airy disk =
2-D Sinc function
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Lens formulae
F-number: F/# = (M+1)f/ D, (M is magnification)
Numerical aperture: NA = n sin f , (n is refractive index)
for small angles NA = D/2f= 1/(2 F#)
Focal spot size x1/2 = 1.22fl / D = 1.22l F# = 1.22 l 2/NA
Depth of focus z = 1.22 x 4l (f/D)2
cos f small angles z = 1.22 l /NA2
z
x1/2D
f
f
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Lens example
Microscope objectives
Spot size = 1.22 l / (2 NA)
NA = n D / 2f= n sin f
Example:
NA = 1.3, spot size: x1/2 = l / 2
Microscope objectives
z
x1/2D
f
f
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Review Gaussian beams
Zero order mode is Gaussian Intensity profile:
beam waist: w0
confocal parameter:z
far from waist
divergence angle
22 /2
0
wreII
2
2
0
0 1
w
zww
l
l
2
0wzR
0w
zw
l
00
637.02
ww
l
l
Gaussian propagation
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Lens resolution with laser light
(Gaussian beams) Laser beam diameter is effective lens diameter: D = 2w
Fourier transform of Gaussian is Gaussian
Standard lens Gaussian
Aperture size D 2w
Focal spot size 1.22fl / D w0 = (4/)fl / 2w = 1.27 fl / 2w
Depth of focus 1.22l (2f/ D)2 z= 1.27l (2f/2w)2
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Fresnel lenses
Start with conventional lens
Constrain optical thickness to be modulo l
Advantage -- thinner and lighter
Fresnel vs conventional lens
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Other fresnel lenses
Spherical waves intersect plane Phase depends on distance from
optic axis
Block out negative phase regions
Fresnel lens construction
Block out
one phase
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Graded index (GRIN) lens Glass rod with radial index gradient
Quadratic gradient -- high index in center like lens
optical path length varies quadratically from center
Periodic focusing laser spot size varies sinusoidally with distance
index
Radialposition
GRIN rod lens GRIN fiber coupler
epoxy
GRIN periodic focusing
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Lenses as Fourier transformers
Angle at front focal plane --> position at back focal plane Position at front focal plane --> angle at back focal plane
Angle maps to position Position maps to angle
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Fourier transform example 4fconfiguration -- transform plane in center
Fourier transform of letter E
Fourier transform of mesh
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Lenses as retro-reflectors
Angle of input
defines position in focal plane
Mirror in focal plane
converts position back to angle at output
Output angle = input angle
translations still possible
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Other retro-reflectors
Right angle reflectors, 90
reflection angles complementary, add 90
Net result is 180 reflection
translation can still occur -- off axis
Corner cube