Lenses
Optics, Eugene Hecht, Chpt. 5
Lenses for imaging
• Object produces many spherical waves
– scattering centers
• Want to project to different location
Lens designed to project
and reproduce scattering centers
Object is collection
of scattering centers
Diverging spherical waves
Converging spherical waves
Plane wave approximation
•
•
•
•
Distant object
Radius of curvature large
Approximate by plane wave
Image approximately at focal plane
Distant object gives plane waves
Lenses for collimation
• Convert diverging spherical wave to plane wave
– Plane wave like spherical wave with infinite
radius of curvature
• First step toward imaging
– plane wave like intermediate
• To flatten wavefront
– distance from S to D must be constant
– independent of path A
• Use Snell’s law and geometry
– Result is equation of hyperbola
– ni li + nt lt = const
lt
li
ni
nt
Spherical lenses
• Hyperbolic and elliptical lenses hard to make
• Spherical lenses easy to make
– Good enough approximation in many cases
Collimation
First focal length
= object focal length
fo 
n1
R
n2  n1
• Example: condition for imaging
– When path lengths from object to image are equal
– n1 l0 + n2 li = const
• From geometry: n1
o

n2 1  n2 si n1so 

 

 i R   i
 o 
• Paraxial approximation (s ~ l):
n1 n2 n2  n1
 
so si
R
Focussing
Second focal length
= image distance
fi 
Optic axis
Vertex
Object distance
Image distance
n2
R
n2  n1
Real lenses
• High index material finite
• Two radii of curvature
• Lensmakers formula
1
1 1
1 

  n2  1  
so si
 R1 R2 
• Focal length
1
1
1 
 n2  1  
f
 R1 R2 
• Thin lens equation
1 1 1
 
so si f
Variable focal length
•
Positive and negative lens combos
– Effective focal length (L1 first)
f 2 d  f1 
d  ( f1  f 2 )
•
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
d
Types of lenses
• Focal length
– general case
1
1
1 

 n2  1  
f
 R1 R2 
• Special case -- double convex
1
1
1 
 n2  1  
f
 R1 R2 
Sign conventions for radii
Lens aberrations
Aberration reduction
• Focusing or collimating
– hyperbolic lens shape is ideal
• Spherical lens shape
– gives insufficient refraction near edges
– use plano-convex
Hyperbolic lens best
• Face flat toward spherical wavefront
– extra refraction
– spherical wave on flat interface
• Why not double convex ?
• Computer solution
Refraction angle
too shallow
– plano convex better
– only for collimation/focusing
• 4 f imaging
– double convex better
– symmetry argument
Additional refraction
when spherical wave
encounters planar boundary
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
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
3
1
2
f
f
Lens alignment
• Position important
• Angle less important
– slightly changes focal length in one dimension
– aberration
• Use translation mount instead of tilt plate
Lens translation
f
f
Lens tilt
f
f ’
Lenses for imaging
• Single lens -- image
• Two lenses -- depends on seperation
• Interesting case -- telescope
so
si
MT  
– equal focal lengths
• 4 f imaging
– unequal focal lengths
f
f
f
f
f1
4 f imaging
M L  M T2
1 1 1
 
so si f
• magnification = f2/f1
• transverse = longitudinal
f
si
so
f1
f2
Imaging telescope
f2
Imaging: transparent vs. scattering objects
• Scattering object acts as array of sources
– image is replica -- one or two lenses
– 4 f configuration puts image at a distance w/o magnification -- “relay” lenses
• Transmission object -- curvature important
– 4 f configuration better
illumination
4 f imaging
2 f imaging
illumination
Scattering
f
f
f
f
2f
2f
Transmission
illum.
illum.
f
f
f
f
2f
2f
Beam expanders
• Analogous to 4 f imaging
– 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
f2
Galilaen
- f1
d
Spatial-filter arrangement
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
Focus to
set collimation
Far-field
alignment
Tilt to correct far-field alignment
center spot
Translate to center spot in output lens
Spatial filters
• Laser beam intensity noise
– can view as interference of intersecting beamlets
• Example: beamsplitter
Sources of laser aberrations
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
destructive
– use pinhole to filter out one position
• Result is spatial filter
Spatial filter for laser beam cleanup
Pinhole
aperture
Cleaned
laser beam
Aberrated
laser beam
f
f
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
Spatial filter alignment:
Translate pinhole
until light comes through
Pinhole
aperture
Cleaned
laser beam
Aberrated
laser beam
f
f
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
Better spatial filter alignment technique:
Translate lens instead of pinhole
Pinhole
aperture
Cleaned
laser beam
Aberrated
laser beam
f
f
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 = f l / D (n = 1)
– circular lens: x1/2 = 1.22 f l / D
Lens resolution
Like array
of sources
limit of zero
separation
Grating resolution
Path difference
N d sin q1/2 = n l
q
2 x1/2
Path
difference
d sin q = n l
d
D
f
Nd=D
More on lens/aperture resolution
• Lens exchanges angle for position
– Fourier transform
• Lens is rectangular aperture
Airy disk =
2-D Sinc function
– F.T. of rectangle is sinc(x) = sin(x)/x
Lens resolution
Like array
of sources
limit of zero
separation
2 x1/2 =2.44 f l / D
D
f
Sinc function
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 = 1.22 f l / D = 1.22 l 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
f
D
f
x1/2
Lens example
• Microscope objectives
– Spot size = 1.22 l / (2 NA) = 0.61 l /NA
– NA = n D / 2 f = n sin f
• Example:
– NA = 1.3, spot size: x1/2 = l / 2
z
D
f
f
x1/2
Microscope objectives
Review Gaussian beams
• Zero order mode is Gaussian
• Intensity profile: I  I e 2 r 2 / w2
0
• beam waist: w0
 lz 
w  w0 1   2 
 w0 
2
• confocal parameter: z
w02
zR 
l
• far from waist
w
lz
w0
• divergence angle
2l
l

 0.637
w0
w0
Gaussian propagation
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 1.22 f l / D
w0 = (4/) f l / 2w = 1.27 f l / 2w
Depth of focus 1.22 l (2f / D)2
z = 1.27 l (2f /2w)2
Divergence
1.22 l / D
q  1.27 l / 2w
Fresnel lenses
• Start with conventional lens
• Constrain optical thickness to be modulo l
• Advantage -- thinner and lighter
Fresnel vs conventional lens
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
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
Radial position
GRIN rod lens
GRIN fiber coupler
epoxy
index
GRIN periodic focusing
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
Fourier transform example
• 4 f configuration -- transform plane in center
Fourier transform of mesh
Fourier transform of letter “E”
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
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