finite optics

8/8/2019 Finite Optics

1/22

103

ECE 5616 OE System Design

Robert McLeod

Optics of finite sizeIntroduction

Design of ideal imaging systems with geometrical optics Finite optics: stops, pupils, and windows

Up to now, all optics have been infinite intransverse extent. Now well change that.

Types of apertures: edges of lenses,intermediate apertures (stops).

Two primary questions to answer: What is the angular extent (NA) of the lightthat can get through the system. Aperture =largest possible angle for object of zeroheight. Depends on where the object islocated.

What is the largest object that can get throughthe system = Field. At the edge of the field,the angular transmittance is one half of the on-axis value.

Will find each of these with two particularrays, one for each of above.

Will find two specific stops, one of whichlimits aperture and one which limits field.

The conjugates to these stops in objectand image space are important and gettheir own names.

Finally, this will allow us to understandthe total power efficiency of the system.


2/22

104


Robert McLeod

The aperture stopand the paraxial marginal ray.


Aperture stop

Paraxial marginal ray(PMR)

D

A diaphragm that

is not the aperturestop

Launch an axial ray, the paraxial marginal ray , from the object. Increase the ray angle until it just hits some aperture. This aperture is the aperture stop. The sin of is the numerical aperture . The aperture stop determines the system resolution, lighttransmission efficiency and the depth of field/focus.


3/22

105


Robert McLeod

PupilsThe images of the aperture stop

The entrance (exit) pupil is the image of the aperture stop in object(image) space.

OShea 2.4, M&M chapter 5


Axial rays at the object (image) appear to enter (exit) the system entrance (exit) pupil. When you look into a camera lens, it is the pupil you see (the image of the stop). Both pupils and the aperture stop are conjugates.

Aperture stop and entrance pupil

Paraxial marginal ray(PMR)

Exit pupil

D

A diaphragm that

is not the aperturestop


4/22

106


Robert McLeod

WindowsImages of the field stop


Aperture stopField stop

C h i e f r a y

Launch an axial ray, the chief ray , from the aperture stop. Increase the ray angle until it just hits some aperture. This aperture is the field stop. The angle of the chief ray in object space is the angular field of view. The height of the chief ray at the object is the field height. The chief ray determines the spatial extent of the object whichcan be viewed. When that object is very far away, it isconvenient to use an angular field of view. When the field stop is not conjugate to the object, vignettingoccurs, cutting off ~half the light at the edge of the field.


5/22


6/22

108


Robert McLeod

Numerical apertureThe measure of angular bandwidth

sinNA n Definition of numerical aperture.Note inclusion of n in this expression.

NA is the conserved quantity in Snells Law because it representsthe transverse periodicity of the wave:

NAn f k x x00

2sin2 2 ===

F

D 2= Paraxial approximation

D

F NA

F

=

2

1# Definition of F-number

Paraxial approximation

Therefore NA/ 0 equals the largest spatial frequency that can betransmitted by the system.Note that NA is a property of cones of light, not lenses.


#6.00 F NAr

= Radius of Airy disk


7/22

109


Robert McLeod

Effective F#Lens speed depends on its use


D

D

Infinite conjugate condition

D f NA

F

=

=2

1#

( )( ) #1

1#

F M D

M f Dt F i

=

==

Finite conjugate condition

t

or( )( ) M o

i

NA NA

M NA NA11

1

=

=

( )

( ) #1

1#

1

1

F

D

f

Dt

F

M

M o

=

== t


8/22

110


Robert McLeod

Depth of focusDependence on F#

D

z

( )( )2*

2*

**

2.1#4

#2

NAF

NAF z

==

==

First eq. is detector limited, second is diffraction limitedThus as we increase the power of the system (NA increases)the depth of field decreases linearly for a fixed resolvable spotsize or quadratically for the diffraction limit r 0.


M&M 5.1.3

From geometry.F*# = F o# or F i# depending

on which space were in

If = 2r 0= Airy disk diameter

Exit pupil

Your detection system has a finite resolution of interest (e.g. adigital pixel size) =

Limited object DOF

Source: Wikipedia


9/22


10/22

112


Robert McLeod

VignettingLosing light apertures or stops


Take a cross-section through theoptical system at the plane of anyaperture. Launch a bundle of raysoff-axis and see how they get

through this aperture:

Aperture radius = a

Ray bundle of radius y

Unvignetted Central ray height, y

y ya +

Aperture radius = a

Ray bundle of radius y

Fully vignettedCentral ray height, y

y ya

Example of vignetting


11/22

113


Robert McLeod

Pupil matchingOff-axis imaging & vignetting

f 1f 2

A.S.

Exit pupil of telescope (image of A.S. in image space) matchesentrance pupil of following instrument (eye).


Good place for a field stop if the object is at infinity.

Note that the entrance pupil in the system above will appear to beellipsoidal for off-axis points. Thus even in this well-designed,aberration-free case, off-axis points will not be identical on axis.

Vignetting: Further, we might find extreme rays at off-axis pointsare terminated. This loses light (bad) but also loses rays at extremeangles, which might limit aberrations (good).

f 1

f 2A.S.


12/22

114


Robert McLeod

Telecentricity

f 1 f 2

Aperture stop


In a telecentric system either the EP or the XP is located at infinity.The system shown above is doubly telecentric since both the EP andthe XP are at infinity. All doubly telecentric system are afocal.

When the stop is at the front focal plane (just lens f2 above) the XPis at infinity and slight motions of the image plane will not changethe image height.

When the stop is at the back focal plane (just lens f1 above) the EPis at infinity and small changes in the distance to the object will notchange the height at the image plane.

Telecentricity is used in many metrology systems.


13/22

115


Robert McLeod

Detailed exampleHolographic data storage

ob ec t

x y

reference

S LM

Holographic

M ateria l

Reference

Telescope

B.S .

Galvo

Mirror

Correlation

Im a g in g L e n s C C D

SLMImaging

Lens

Linear CCD

Object beam shutter

Were going to design the imaging path

Design of ideal imaging systems with geometrical optics Example


14/22

116


Robert McLeod

1 to 1 imagingAperture in Fourier plane


DSLM =10 mmp = 10 m

SLMLens

Aperture CCD

DLens =10 mmf = 20 mm

DAp= 2 mm D CCD =10 mm

40 mm 20 mm 20 mm

Material

Fourier diffraction calculation for a random pixel pattern.

S L M

C C D

L e n s


15/22

117


Robert McLeod

Finding the aperture stopTrace the paraxial marginal ray (PMR)

DSLM =10 mmp = 10 m

SLM Lens Aperture CCD

DLens = 10 mmf = 20 mm


=

+

+

k

k k

k

k

u yt

u y

101

1

1

=

k k

k k

k

u

y

u

y

1

01

01220PMRy

.2.08.08y/r

-.05-.05-.05.05.05PMRu

-.01-.01-.01.01.01u

0.2.4.40y

32110Surf

Aperture stop


0 1 2 3


16/22

118


Robert McLeod

What does the aperture stop do?

Limits the NA of the system

81

)2(205

)1(2 / * === M f

D NA

What is the lens effective NA?

However we have stopped the system down to NA = 2/40=1/20

* NA

NA


Pixel size = 10 m201

105.

=== p

NA pix

The aperture stop thus:1. Determines the system field of view2. Controls the radiometric efficiency3. Limits the depth of focus4. Impacts the aberrations5. Sets the diffraction-limited resolution

rAS= 1 mmy=2 mm

40 mm


17/22

119


Robert McLeod

Find the entrance pupil(image of aperture stop in object space)


Aperture stop andexit pupil

f t t

111

10

=+20

11

20

1

1

=+t

=1

t

Entrance pupil

Whats that mean? If the entrance pupil is at infinity, then everypoint on the object radiates into the same cone:


18/22

120


Robert McLeod

Find the field stopTrace the chief ray (PPR)

DSLM =10 mmp = 10 mm

SLM Lens Aperture CCD

DLens = 10 mmf = 20 mm


=+

+

k

k k

k

k

u

yt

u

y

10

1

1

1

=

k k

k k

k

u

y

u

y

1

01

0.04.04y/r

-50555PPRy

-.01-.01-.0100u

-.20.2.2.2y32110Surf

Field stop


Entrance and exit windows at field stop (since it is lens).


19/22


20/22

122


Robert McLeod

Thicken the lensesGaussian design

DSLM =10 mmp = 10 m

SLM

Lens

Aperture CCD

DLens =12 mmF = 20 mm


40 mm 20 mm 20 mm5

mm

P P

Design of ideal imaging systems with geometrical optics Summary


21/22

123


Robert McLeod

Optical invariantaka Lagrange or Helmholtz invariant


yunun

ynuun

=

=

Write the paraxial refraction equations for the marginal ray (PMR)and chief or pupil ray (PPR):

( ) ( ) yu yun yu yun =With a bit of algebra:

( ) yu yun H =

So

is conserved.

At the object (or image) of limited field diameter L:

angleraymaximumfield,of edge,0 === u y y

spotsobj N L

r

L NA ynu H

226.

2 0

===

Thus we have found the information capacity of the opticalsystem, aka the space-bandwidth product:

objspots H N 2

=


22/22


Optical invariantfor the holographic storage example

pix

SLM

SLM

spots

N

p D

p D

NA y

H N

=

=

=

=

0

0

0

00

0

2

2

Design of ideal imaging systems with geometrical optics Summary

Definition of optical invariant

Stopped-down NAStopped-down NA

So we have correctly designed this system to transmit theproper number of resolvable spots.

finite optics

Documents