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Lenses and Imaging (Part II)
• Reminders from Part I• Surfaces of positive/negative power• Real and virtual images• Imaging condition• Thick lenses• Principal planes
MIT 2.71/2.71009/20/04 wk3-a-1
The power of surfaces
•• Positive Positive power : exiting rays convergeconverge
•• NegativeNegative power : exiting rays diverge
+Simple spherical
refractor (positive)
1 n
R>0 + +R>0
R<0
n11
n
R>0
1 1
+ NPlano-convex
lensBi-convex
lens
diverge
MIT 2.71/2.71009/20/04 wk3-a-2
R<0
Simple sphericalrefractor (negative)
1 n
–Plano-concave
lens
n1
1
R>0
R<0
n
R<0
1 1
– – –NBi-concave
lens
Thin lens in air
P
P′
n=1n=1
⎟⎟⎠
⎞⎜⎜⎝
⎛
in
in
xα
⎟⎟⎠
⎞⎜⎜⎝
⎛
out
out
xα
n
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛ −=⎟⎟
⎠
⎞⎜⎜⎝
⎛
in
inlensthin
out
out
10 1
xP
xαα
MIT 2.71/2.71009/20/04 wk3-a-3
( ) ⎟⎠⎞
⎜⎝⎛
′−−=
′−
+−
=′+=RR
nR
nR
nPPP 11111lensthin
Lens-maker’sformula
Thin lens in air
P
P′
n=1n=1
⎟⎟⎠
⎞⎜⎜⎝
⎛
in
in
xα
⎟⎟⎠
⎞⎜⎜⎝
⎛
out
out
xα
n
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛ −=⎟⎟
⎠
⎞⎜⎜⎝
⎛
in
inlensthin
out
out
10 1
xP
xαα
inout
inlensthin inout xx
xP=
−=αα Ray bending is proportionalproportionalto the distanceto the distance from the axis
MIT 2.71/2.71009/20/04 wk3-a-4
Positive thin lens in air
object at ∞
Ray bending is proportionalproportionalto the distanceto the distance from the axis
MIT 2.71/2.71009/20/04 wk3-a-5
Positive thin lens in air
0 , inin =αxinlensthin out
inout ,xP
xx−=
=α
0lensthin >P
fthin lens as a “black box” RealReal image
lensthin out
in 1 P
fxf =⇒−=α
Focal pointFocal point = image of an object at ∞Focal lengthFocal length = distance between lens & focal pointMIT 2.71/2.710
09/20/04 wk3-a-6
Negative thin lens in air
object at ∞
Ray bending is proportionalproportionalto the distanceto the distance from the axis
MIT 2.71/2.71009/20/04 wk3-a-7
Negative thin lens in air
VirtualVirtual image
lensthin
1P
f =
f
0 0lensthin
<⇒⇒<
fP
(to the “left”“left”)
object at ∞
still applies, now with
MIT 2.71/2.71009/20/04 wk3-a-8
Imaging condition: ray-tracing
MIT 2.71/2.71009/20/04 wk3-a-9
2nd FP
1st FP
n=1n=1
object
image
chief ray
• Image point is located at the common intersection of all rays whichemanate from the corresponding object point
• The two rays passing through the two focal points and the chief raycan be ray-traced directly
Imaging condition: ray-tracing
MIT 2.71/2.71009/20/04 wk3-a-10
F’A F
C
D
L
N
MB
x ff
x’
• (ABF)~(FLN) and (F’CD)~(MLF’) are pairs of similar triangles
(CD)(LN) (ML)(AB) C)F(
(CD))F(L
(LM) (FL)(LN)
(AF)(AB)
==′
=′
=
2fxx =′)F(L (FL) C)F( (AF) ′=′
Imaging condition: matrix method
MIT 2.71/2.71009/20/04 wk3-a-11
2nd FP
object
image
chief ray1st FP
• Location of image point must be independent of ray departure angle at the object
n=1 n=1
Imaging condition: matrix method
ss’
lensobject
image
⎟⎟⎠
⎞⎜⎜⎝
⎛
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
′−
′−′+
−−=⎟⎟
⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛ −⎟⎟⎠
⎞⎜⎜⎝
⎛′
=⎟⎟⎠
⎞⎜⎜⎝
⎛
in
in
in
in
out
out
1
11
101
10
11101
xfs
fssss
ffs
xsfsxααα
=0MIT 2.71/2.71009/20/04 wk3-a-12
Imaging condition: matrix method
ss’
lensobject
image
fssfssss 111 0 =
′+⇔=
′−′+ Imaging condition
(aka Lens Law)
MIT 2.71/2.71009/20/04 wk3-a-13
Imaging condition: matrix method
ss’
lensobject
image
ss
fsM
xxM
′−=
′−=
=
1
:ionmagnificat Lateral
T
in
outT
inoutin
in
out
out 1 10
11x
fsx
xfsff
s
x ⎟⎟⎠
⎞⎜⎜⎝
⎛ ′−=⇒⎟⎟
⎠
⎞⎜⎜⎝
⎛
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
′−
−−==⎟⎟
⎠
⎞⎜⎜⎝
⎛ αα
MIT 2.71/2.71009/20/04 wk3-a-14
Real & virtual images
2nd FP
objectimage
1st FP
MIT 2.71/2.71009/20/04 wk3-a-15
2nd FP
objectimage
1st FP
+ +
image: real & inverted; MT<0 image: virtual & erect; MT>1
1st FP2nd FP
objectimage
1st FP2nd FP
image ––object
image: virtual & erect; 0<MT<1 image: virtual & erect; 0<MT<1
The thick lensair air
glass
Rays bend in “two steps”
MIT 2.71/2.71009/20/04 wk3-a-16
The thick lensair air
glass
Equivalent to a thin lens placed “somewhere” within the thick element.
The location of this “equivalent thin lens” isthe Principal PlanePrincipal Plane of the thick element
MIT 2.71/2.71009/20/04 wk3-a-17
The thick lens
MIT 2.71/2.71009/20/04 wk3-a-18
airglass
R
R′−)(
air n=1n=1⎟⎟⎠
⎞⎜⎜⎝
⎛
in
in
xα
⎟⎟⎠
⎞⎜⎜⎝
⎛
out
out
xα
n
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟
⎠
⎞
⎜⎜
⎝
⎛ −−
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛⎟⎟
⎠
⎞
⎜⎜
⎝
⎛′
−−=⎟⎟
⎠
⎞⎜⎜⎝
⎛
in
in
out
out
10
11101
10
11xR
n
ndR
n
xαα
The thick lens
MIT 2.71/2.71009/20/04 wk3-a-19
airglass
R
R′−)(
air n=1n=1⎟⎟⎠
⎞⎜⎜⎝
⎛
in
in
xα
⎟⎟⎠
⎞⎜⎜⎝
⎛
out
out
xα
n
( ) ( )
⎟⎟⎠
⎞⎜⎜⎝
⎛
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
−−
⎥⎦
⎤⎢⎣
⎡′
−+⎟⎠⎞
⎜⎝⎛
′−−−
′−
+=⎟⎟
⎠
⎞⎜⎜⎝
⎛
in
in
2
out
out
11
111111
xRd
nn
nd
RnRdn
RRn
Rd
nn
xαα
The thick lens: powerair air
glass
⎟⎟⎠
⎞⎜⎜⎝
⎛
in
in
xα
⎟⎟⎠
⎞⎜⎜⎝
⎛
out
out
xα
( ) ( )
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
⎟⎠⎞
⎜⎝⎛ −−
⎥⎦
⎤⎢⎣
⎡′
−+⎟⎠⎞
⎜⎝⎛
′−−−
=⎟⎟⎠
⎞⎜⎜⎝
⎛⇒⎟⎟
⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛
xRd
nn
xRnR
dnRR
n
xxx 11
111 10
2
out
out
in
in αα inout Px−=α
Object at infinityObject at infinity
MIT 2.71/2.71009/20/04 wk3-a-20
The thick lens: powerair air
glass
⎟⎟⎠
⎞⎜⎜⎝
⎛
in
in
xα
⎟⎟⎠
⎞⎜⎜⎝
⎛
out
out
xα
( ) ( )⎥⎦
⎤⎢⎣
⎡′
−+⎟⎠⎞
⎜⎝⎛
′−−=
RnRdn
RRnP
2111 1 ( ) ( )⎥⎦
⎤⎢⎣
⎡′
−+⎟⎠⎞
⎜⎝⎛
′−−=
RnRdn
RRn
f
2111 11
f: EffectiveEffective Focal LengthPowerMIT 2.71/2.71009/20/04 wk3-a-21
The very thick lensair air
glass
Funny things happening:rays divergediverge upon exiting from the element, i.e.
too much positive power leading to a negative element!MIT 2.71/2.71009/20/04 wk3-a-22
The thick lens: back focal length
MIT 2.71/2.71009/20/04 wk3-a-23
air airglass
⎟⎟⎠
⎞⎜⎜⎝
⎛
in
in
xα
⎟⎟⎠
⎞⎜⎜⎝
⎛
out
out
xα
z
⎟⎟⎠
⎞⎜⎜⎝
⎛
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
−−
−′
−+
⎟⎟⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛
in
in
out
out
11
111
101
xRd
nn
nd
fRd
nn
zxαα
The thick lens: back focal length
MIT 2.71/2.71009/20/04 wk3-a-24
air airglass
⎟⎟⎠
⎞⎜⎜⎝
⎛
in
in
xα
⎟⎟⎠
⎞⎜⎜⎝
⎛
out
out
xα
z
⎟⎠⎞
⎜⎝⎛ −−=⇒=
Rd
nnfzx 110out
z: BackBack Focal Length
Focal Lengths & Principal Planes
generalized optical system(e.g. thick lens,
multi-element system)
2nd PS
2nd FP
1st FP
1st PS
EFL
BFL
EFL
FFL
EFL: Effective Focal Length (or simply “focal length”)FFL: Front Focal LengthBFL: Back Focal LengthFP: Focal Point/Plane PS: Principal Surface/Plane
MIT 2.71/2.71009/20/04 wk3-a-25
PSs and FLs for thin lenses
1P 2P
0=lD
glass, index n
n n´
21(EFL)1 PPP +=≡ (FFL) (EFL)(BFL) ==
• The principal planes coincide with the (collocated) glass surfaces• The rays bend precisely at the thin lens plane (=collocated glass surfaces & PP)
MIT 2.71/2.71009/20/04 wk3-a-26
The significance of principal planes /1
MIT 2.71/2.71009/20/04 wk3-a-27
generalizedoptical system
thin lensof the same power
located at the 2nd PSfor rays passing through 2nd FP
1st PS
2nd PS
2nd FP
1st FP
The significance of principal planes /2
MIT 2.71/2.71009/20/04 wk3-a-28
generalizedoptical system
thin lensof the same power
located at the 1st PPfor rays passing through 1st FP
1st PS
2nd PS
2nd FP
1st FP
