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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