•Design of ideal imaging systems with geometrical optics
–Finite optics: stops, pupils, and windows
ECE 5616 OE System Design
Optics of finite size
Introduction
• Up to now, all optics have been infinite in
transverse extent. Now we’ll change that.
• Types of apertures: edges of lenses,
intermediate apertures (“stops”).
• Two primary questions to answer:
– What is the angular extent (NA) of the light
that can get through the system. “Aperture” =
largest possible angle for object of zero
height. Depends on where the object is
located.
– What is the largest object that can get through
the system = “Field”. At the edge of the field,
the angular transmittance is one half of the onaxis value.
• Will find each of these with two particular
rays, one for each of above.
• Will find two specific stops, one of which
limits aperture and one which limits field.
• The conjugates to these stops in object
and image space are important and get
their own names.
• Finally, this will allow us to understand
the total power efficiency of the system.
Robert McLeod
103
•Design of ideal imaging systems with geometrical optics
–Finite optics: stops, pupils, and windows
ECE 5616 OE System Design
The aperture stop
and the paraxial marginal ray.
Aperture stop
Paraxial marginal ray
(PMR)
α
D
A diaphragm that
is not the aperture
stop
• 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 α is the numerical aperture.
• The aperture stop determines the system resolution, light
transmission efficiency and the depth of field/focus.
Robert McLeod
104
•Design of ideal imaging systems with geometrical optics
–Finite optics: stops, pupils, and windows
ECE 5616 OE System Design
Pupils
The images of the aperture stop
Exit pupil
Aperture stop and entrance pupil
Paraxial marginal ray
(PMR)
α
D
A diaphragm that
is not the aperture
stop
The entrance (exit) pupil is the image of the aperture stop in object
(image) space.
• 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.
Robert McLeod
O’Shea 2.4, M&M chapter 5
105
•Design of ideal imaging systems with geometrical optics
–Finite optics: stops, pupils, and windows
ECE 5616 OE System Design
Windows
Images of the field stop
Aperture stop
Field stop
f ray
Chie
• 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 which
can be viewed. When that object is very far away, it is
convenient to use an angular field of view.
• When the field stop is not conjugate to the object, vignetting
occurs, cutting off ~half the light at the edge of the field.
Robert McLeod
106
•Design of ideal imaging systems with geometrical optics
–Finite optics: stops, pupils, and windows
ECE 5616 OE System Design
Field stops and windows
Definitions
Entrance Window
Aperture stop
Field stop
&
Exit Window
Chief ray
M&M refer to the chief ray as the paraxial pupil ray (PPR)
The entrance (exit) window is the image of the field stop in object
(image) space.
Robert McLeod
107
•Design of ideal imaging systems with geometrical optics
–Finite optics: stops, pupils, and windows
ECE 5616 OE System Design
Numerical aperture
The measure of angular bandwidth
NA ≡ n sin α
D2
=
F
1
F#≡
2 NA
F
=
D
r0 = 0.6
λ
NA
Definition of numerical aperture.
Note inclusion of n in this expression.
Paraxial approximation
Definition of F-number
Paraxial approximation
≈ λ F#
Radius of Airy disk
NA is the conserved quantity in Snell’s Law because it represents
the transverse periodicity of the wave:
k x = 2π f x =
2π
λ0
n sin α =
2π
λ0
NA
Therefore NA/λ0 equals the largest spatial frequency that can be
transmitted by the system.
Note that NA is a property of cones of light, not lenses.
Robert McLeod
108
•Design of ideal imaging systems with geometrical optics
–Finite optics: stops, pupils, and windows
ECE 5616 OE System Design
Effective F#
Lens speed depends on its use
Infinite conjugate condition
1
F#=
2 NA
f
=
D
−α
D
Finite conjugate condition
t ′ f (1 − M )
=
D
D
= (1 − M )F #
Fi#=
f (1 − M1 )
t
F #= =
D
D
= (1 − M1 )F #
o
or
Robert McLeod
−α
t
D
t’
NAi = NA (1 − M )
NAo = NA (1 − M1 )
109
•Design of ideal imaging systems with geometrical optics
–Finite optics: stops, pupils, and windows
ECE 5616 OE System Design
Depth of focus
Dependence on F#
Your detection system has a finite resolution of interest (e.g. a
digital pixel size) = ρ
Exit pupil
Limited object DOF
ρ
D
δz
*
δz = 2 ρ F # =
(
*
)
2
ρ
From geometry.
F*# = Fo# or Fi# depending
on which space we’re in
NA*
= 4λ F # = 1.2
Source: Wikipedia
λ
(NA )
* 2
If ρ = 2r0
= Airy disk diameter
First eq. is “detector limited”, second is “diffraction limited”
Thus as we increase the power of the system (NA increases)
the depth of field decreases linearly for a fixed resolvable spot
size ρ or quadratically for the diffraction limit r0.
Robert McLeod
M&M 5.1.3
110
•Design of ideal imaging systems with geometrical optics
–Finite optics: stops, pupils, and windows
ECE 5616 OE System Design
Hyperfocal distance
Nominal focus
Near point
Far point
Important for fixed-focus systems
Entrance pupil
ρ
D
δz
Depth of Field
Note how each focus falls within
the acceptable blur ρ
The object distance which is perfectly in focus on the detector
is defines the nominal system focus plane.
When this plane is positioned such that the far point goes to
infinity, then the system is in the “hyperfocal” condition and
the nominal focus plane is called the “hyperfocal distance”.
This is very handy for fixed focus cameras – you can take a
portrait shot (if the person is beyond the near point) out to a
landscape and not notice the defocus.
Robert McLeod
111
•Design of ideal imaging systems with geometrical optics
–Finite optics: stops, pupils, and windows
ECE 5616 OE System Design
Vignetting
Losing light apertures or stops
Example of vignetting
Take a cross-section through the
optical system at the plane of any
aperture. Launch a bundle of rays
off-axis and see how they get
through this aperture:
Ray bundle of radius y
Unvignetted
a≥ y + y
Central ray height, y
Aperture radius = a
Ray bundle of radius y
Central ray height, y
Fully vignetted
a≤ y − y
Robert McLeod
Aperture radius = a
112
•Design of ideal imaging systems with geometrical optics
–Finite optics: stops, pupils, and windows
ECE 5616 OE System Design
Pupil matching
Off-axis imaging & vignetting
Exit pupil of telescope (image of A.S. in image space) matches
entrance pupil of following instrument (eye).
f1
f2
A.S.
Good place for a field stop if the object is at infinity.
Note that the entrance pupil in the system above will appear to be
ellipsoidal 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 points
are terminated. This loses light (bad) but also loses rays at extreme
angles, which might limit aberrations (good).
f1
f2
A.S.
Robert McLeod
113
•Design of ideal imaging systems with geometrical optics
–Finite optics: stops, pupils, and windows
ECE 5616 OE System Design
Telecentricity
Aperture stop
f1
f2
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 and
the XP are at infinity. All doubly telecentric system are afocal.
When the stop is at the front focal plane (just lens f2 above) the XP
is at infinity and slight motions of the image plane will not change
the image height.
When the stop is at the back focal plane (just lens f1 above) the EP
is at infinity and small changes in the distance to the object will not
change the height at the image plane.
Telecentricity is used in many metrology systems.
Robert McLeod
114
•Design of ideal imaging systems with geometrical optics
–Example
ECE 5616 OE System Design
Detailed example
Holographic data storage
B .S .
Object beam shutter
S LM
o b jec t
referen c e
SLM
Imaging
Lens
x
y
R efe ren ce
Tel es co pe
ϕ
Ho l o g rap h ic
Ga lvo
M at eria l
M i rro r
Linear CCD
C o rrel ati o n
Im a g in g Le n s
CCD
We’re going to design the imaging path
Robert McLeod
115
•Design of ideal imaging systems with geometrical optics
–Example
ECE 5616 OE System Design
1 to 1 imaging
Aperture in Fourier plane
Lens
SLM
Aperture
CCD
Material
40 mm
DSLM=10 mm
p = 10 µm
20 mm
DLens=10 mm
f = 20 mm
20 mm
DAp= 2 mm
DCCD=10 mm
Fourier diffraction calculation for a random pixel pattern.
CCD
Lens
SLM
Robert McLeod
116
•Design of ideal imaging systems with geometrical optics
–Example
ECE 5616 OE System Design
Finding the aperture stop
Trace the paraxial marginal ray (PMR)
0
1
2
3
SLM
Lens
Aperture
CCD
 yk   1
u ′  = − φ
 k  k
0  y k 
1 u k 
 y k +1  1 t k′   y k 
 u  = 0 1   u ′ 
 k 
 k +1  
DSLM=10 mm
p = 10 µm
DLens= 10 mm
f = 20 mm
DAp= 2 mm
DCCD=10 mm
Aperture stop
Surf
0
1
1’
2
3
y
0
.4
.4
.2
0
u’
.01
.01
-.01
-.01
-.01
.08
.08
.2
y/r
PMR
y
0
2
2
1
0
PMR
u’
.05
.05
-.05
-.05
-.05
Robert McLeod
117
•Design of ideal imaging systems with geometrical optics
–Example
ECE 5616 OE System Design
What does the aperture stop do?
Limits the NA of the system
NA*
y=2 mm
rAS= 1 mm
NA
40 mm
What is the lens effective NA?
NA* =
D/2
5
1
=
=
f (1 − M ) 20(−2) 8
However we have stopped the system down to NA = 2/40=1/20
Pixel size = 10 µm
NApix =
λ
p
=
.5
1
=
10 20
The aperture stop thus:
1. Determines the system field of view
2. Controls the radiometric efficiency
3. Limits the depth of focus
4. Impacts the aberrations
5. Sets the diffraction-limited resolution
Robert McLeod
118
•Design of ideal imaging systems with geometrical optics
–Example
ECE 5616 OE System Design
Find the entrance pupil
(image of aperture stop in object space)
Aperture stop and
exit pupil
Entrance pupil
1 1 1
− + =
t0 t1 f
1 1 1
+ =
20 t1 20
t1 = ∞
What’s that mean? If the entrance pupil is at infinity, then every
point on the object radiates into the same cone:
Robert McLeod
119
•Design of ideal imaging systems with geometrical optics
–Example
ECE 5616 OE System Design
Find the field stop
Trace the chief ray (PPR)
SLM
Lens
Aperture
CCD
 yk   1
u ′  = − φ
 k  k
0  y k 
1 u k 
 y k +1  1 t k′   y k 
 u  = 0 1   u ′ 
 k 
 k +1  
DSLM=10 mm
p = 10 mm
DLens= 10 mm
f = 20 mm
DAp= 2 mm
DCCD=10 mm
Field stop
Surf
y
0
.2
1
.2
1’
.2
2
0
3
-.2
u’
0
0
-.01
-.01
-.01
.04
.04
0
5
5
0
y/r
PPR
y
5
Robert McLeod Entrance
-5
and exit windows at field stop (since it is lens).
120
•Design of ideal imaging systems with geometrical optics
–Example
ECE 5616 OE System Design
Required lens diameter
And why windows should be at images
Object vignetted at edges
inal ra
Extreme marg
y
NApix
Note that the extreme marginal ray makes the same angle with the
chief ray (PPR) as the paraxial marginal ray (PMR) makes with the
axis. Thus to fit the extreme marginal ray through an aperture, we
require:
D ≥ 2( y + y ) = 2(2 + 5) = 12
• Now the camera is the field stop and the entrance window is the
SLM.
• The optical system now captures the same light from each SLM
pixel and it can be adjusted for all pixels via the aperture stop.
• Independently, we can adjust the field size.
Robert McLeod
M&M 5.1.8
121
•Design of ideal imaging systems with geometrical optics
–Summary
ECE 5616 OE System Design
Thicken the lenses
Gaussian design
Lens
P
P’
SLM
Aperture
40 mm
DSLM=10 mm
p = 10 µm
Robert McLeod
5
mm
20 mm
DLens=12 mm
F = 20 mm
CCD
20 mm
DAp= 2 mm DCCD=10 mm
122
•Design of ideal imaging systems with geometrical optics
–Finite optics: stops, pupils, and windows
ECE 5616 OE System Design
Optical invariant
aka Lagrange or Helmholtz invariant
Write the paraxial refraction equations for the marginal ray (PMR)
and chief or pupil ray (PPR):
n′u ′ − nu = −φ y
n′u ′ − nu = −φ y
With a bit of algebra:
n′(u ′ y − u ′ y ) = n(u y − u y )
So
H = n(u y − u y )
is conserved.
At the object (or image) of limited field diameter L:
y = 0, y = edge of field, u = maximum ray angle
H obj
L
λ L λ
= nu y = NA = .6
≈ N spots
2
r0 2 2
Thus we have found the information capacity of the optical
system, aka the space-bandwidth product:
N spots =
Robert McLeod
2
λ
H obj
123
•Design of ideal imaging systems with geometrical optics
–Summary
ECE 5616 OE System Design
Optical invariant
for the holographic storage example
N spots ≈
=
=
2H
λ0
2 y0 NA0
Definition of optical invariant
λ0
DSLM
λ0
λ0
p
Stopped-down NA
DSLM
=
p
= N pix
So we have correctly designed this system to transmit the
proper number of resolvable spots.
Robert McLeod
124