1. Optics, Eugene Hecht, Chpt. 5
Lenses
Optics, Eugene Hecht, Chpt. 5

2. Lens designed to project and reproduce scattering centers
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

3. Plane wave approximation
Distant object
Radius of curvature large
Approximate by plane wave
Image approximately at focal plane
Distant object gives plane waves

4. 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 A
Use Snell’s law and geometry
Result is equation of hyperbola
ni li + nt lt = const lt li nt ni

5. Spherical lenses Hyperbolic and elliptical lenses hard to make
First focal length
= object focal length
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
n1 l0 + n2 li = const
From geometry:
Paraxial approximation:
Collimation
Object distance
Image distance
Vertex
Optic axis
Second focal length
= image focal length
Focussing

6. Real lenses High index material finite Two radii of curvature
Lensmakers formula
Focal length
Thin lens equation

7. 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

8. Types of lenses Focal length Special case -- double convex
general case
Special case -- double convex
Sign conventions for radii

9. Lens aberrations Focusing or collimating Spherical lens shape
Additional refraction
when spherical wave
encounters planar boundary
Refraction angle
too shallow
Hyperbolic lens best
Aberration reduction
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
4 f imaging
double convex better
symmetry argument

10. Non-axial focusing Extended object Focus to points on sphere
Light enters lens from several angles
Focus to points on sphere
Approximate by plane
Focal plane
Focal plane
Parallel ray focus to points on sphere

11. 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

12. Lens alignment Position important Angle less important
slightly changes focal length in one dimension
aberration
Use translation mount instead of tilt plate
Lens translation
Lens tilt f f f
f ’

13. Lenses for imaging Single lens -- image
Two lenses -- depends on seperation
Interesting case -- telescope
equal focal lengths
4 f imaging
unequal focal lengths
magnification = f2/f1
transverse = longitudinal f so si f f f f f1 f1 f2 f2
4 f imaging
Imaging telescope

14. 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
4 f imaging
2 f imaging
illumination
illumination f
2 f
Scattering
Transmission f
2 f
illum.
illum.

15. Beam expanders Analogous to 4 f imaging Two types
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
Galilaen
- f1 f2 d
Spatial-filter arrangement

16. 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
Tilt to correct far-field alignment
Far-field
alignment
Translate to center spot in output lens
center spot

17. Spatial filters Laser beam intensity noise Example: beamsplitter
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
Sources of laser aberrations
beamsplitter
destructive
Spatial filter for laser beam cleanup
Pinhole
aperture
Cleaned
laser beam
Aberrated
laser beam f f

18. 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

19. 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

20. 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
Grating resolution d q
Path
difference
d sin q = n l
Path difference
N d sin q1/2 = n l
N d = D D f
2 x1/2
Lens resolution
Like array
of sources
limit of zero
separation

21. 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
Airy disk =
2-D Sinc function
Lens resolution
Like array
of sources
limit of zero
separation
2 x1/2 =2.44 f l / D
Sinc function D f

22. 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
x1/2 D f

23. Lens example Microscope objectives Example:
Spot size = 1.22 l / (2 NA)
NA = n D / 2 f = n sin f
Example:
NA = 1.3, spot size: x1/2 = l / 2 z
x1/2 D f

24. Review Gaussian beams Zero order mode is Gaussian Intensity profile:
beam waist: w0
confocal parameter: z
far from waist
divergence angle
Gaussian propagation

25. 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 w
Focal spot size f l / D w0 = (4/p) f l / 2w = 1.27 f l / 2w
Depth of focus l (2f / D)2 z = 1.27 l (2f /2w)2

26. Fresnel lenses Start with conventional lens
Constrain optical thickness to be modulo l
Advantage -- thinner and lighter
Fresnel vs conventional lens

27. 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

28. 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
GRIN fiber coupler
index
Radial position
GRIN rod lens
epoxy
GRIN periodic focusing

29. 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

30. Fourier transform example
4 f configuration -- transform plane in center
Fourier transform of mesh
Fourier transform of letter “E”

31. 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

32. 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