optical design : back to basics
TRANSCRIPT
Optical design : back to basics
Geometrical optics
• Fermat’s principle : – Principle of least time : the path taken by light
travelling from A to B through an optical system will be such that the time of travel is a minimum
– Exemple : the straight line for a homogeneous medium
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Geometrical optics
• Optical Path Length :
[ ] ( ) ×==B
A
dszyxnABL ,,
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Geometrical optics
• Wavefront : – All the points that have equal optical path length
from the source
– Exemple : point source in a homogeneous medium : all the wavefront are spherical, centered on the source
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Geometrical optics
• Malus theorem : the rays are normal to wavefronts
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Geometrical optics• Stigmatism :
– Rigorous : the image of a point (A) is a point (A’). So all the rays are converging through the same point, so the wavefront is spherical !
Fermat’s principle
– approximated : paraxial or gaussian optics• All the angles are small. The Snell-Descartes
relationship is linear (1st order or geometrical optics) :
2211 αα ×=× nn
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[ ] cteAA =′
Geometrical optics
• Principal planes
• Focal lengths :
– Objet :
– Image :
• Power :
HFf =
FHf ′′=′
f
n
f
nP −=
′′
=
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A
B
F
H H’ F’ A’
B’
+
Geometrical optics
• Conjugate equations :
– Newton’s equation :
– Descartes’ equation :
2f
n
nffAFFA ′×
′−=′×=′′×
f
n
x
n
x
n
′′
+=′′
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AHxHAx ′′=′= ,
Geometrical optics
• Magnifications :
– Transverse :
– Angular :
– Axial :
x
x
n
n
y
yg y
′×
′=
′=
ygn
n
x
xg
1
′=
′=
′=
αα
α
2
yx gn
n
dx
xdg ×
′=
′=
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BAyABy ′′=′= ,
System of two separated components
• Object focal length:
• Gullstrand’s equation :
2,1 ygff ×=
2121 f
n
f
n
n
e
f
n
f
n
f
n
′′′
×′′
×′
−′′′
+′′
=′′′
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Formula of the thin lens
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R2R1
( ) ( )�����
0 0
21
2
21
111 1
1
≈≈
−+
−−=
′esi
RnR
en
RRn
fe
Pupils
star (at infinity)
lens
eyepiece
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Pupils
star (at infinity)
F
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Pupils
star (at infinity)
F
intermediate pupil ou
stop
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Pupils
star (at infinity)
F
stop
field ray
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Pupils
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Vignetting
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Geometrical optics and optical design, P. Mouroulis and J. Macdonald
Aperture and field
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marginal ray
pupil (aperture stop)
sensor
chief ray
α’θ
α’ = aperture (limited by the pupil)θ = field (limited by the sensor)
Interests of the pupil concept
• Link with the definition of field rays
• Link with the notion of vignetting(radiometry of the system)
• Link with the wavefront aberrations (W) which are defined in a pupil plane, by convention
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Other definitions
• Numerical aperture :
• Aperture number (F or f/# or N) :
α ′′= sinnNA
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PupilEntranceD
fN
at Object Aplaneticsin n 2
1 ′=
′′=
∞α
Stigmatism for a small volume, aplanetism
• Hypothesis : A et A’ are conjugated.
• What about B et B’, near A et A’ ?
• What about C et C’, near A et A’ ?
A
B
A’
B’
C
C’
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Stigmatism for a small volume, aplanetism
• Necessary and suffisant condition for the planes containing AB = y et A’B’ = y’ to be conjugated = optical sine theorem = Abbe’s equation :
• The system is aplanetic
A
B
A’
B’
αα sinsin ××=′×′×′ ynyn
α α’
P
P’
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Stigmatism for a small volume, aplanetism
• Necessary condition for C et C’ being conjugated = Herschel’s equation :
A A’C C’
2sin
2sin 22 αα ××=
′×′×′ dxnxdn
α α’
P
P’
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dx dx’
Stigmatism for a small volume, aplanetism
• The 2 relationships from Abbe and Herschel are not compatible (except for ) : one have to choose !
A
B
A’
B’
C C’
α α’
P
P’
αα ′=
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-40
-35
-30
-25
-20
-15
-10
-5
0
• The one for which the image of a point is a point !
• In real life, due to diffraction and optical aberrations, the image of a point is in the best case an Airy pattern
What is a perfect optical system ? ( ) ( )
( ) ( )rr
OrI systeminvariant -spacelinear : HP
ion.magnificat e transvers image, rI objet, rO
y
′∗
′=′
=′=
PSFg
g y
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•
...)distorsion curvature, field m,astigmatis coma, ab, (spherical
saberration geometricstigmatism no
Non-stigmatism : the culprits
• Angles of incidence :
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2211 sinsin inin =
stigmatismorder)(1 optics lgeometrica 2211 = stinin
( )λn
lateral) and nal(longitudi saberration chromatic
If very small :
Real life : not so small :
The spherical mirror
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( )
−=′′i
RAF
cos
11
2F’
A’C
i
et faibles
1
sin sin
sin sin
sin
The last relationship is true if and only if the
image principal "plane" is a sphere of radius f’
Hence : sin
n n
Entrance
ny n y
nh n y n f
h
f
Dh
f
α θ
α αθ α θ α
α
α
′= =
′ ′ ′=′ ′ ′ ′ ′ ′⇔ − = =
′⇔ − =′
′ = − = −′
1
2 2
Pupil
f N= −
′
Aplanetism
A’=F’
B’
α’H H’
h
f’
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EP
y θ α
Y’
θθ
object at ∝
h