Optics 214
LaGrange Invariant
3/13/2023
2023 Optics 214 - John P. Bowen
Definition of Lagrange Invariant:
Any surface
Lagrange invariant
is constant through
the system. It can
be determined by
tracing both an aray and a b-ray:
ub
yb
ya
ua
H = n ( ua yb − ub ya )
index = n
3/13/2023
Optics 214 - John P. Bowen
Lagrange Invariant:
Pupil or stop
The Lagrange
invariant can be
easily calculated at
special planes in
the system, where
either ya or yb is
zero:
H stop = n ( −ub ya )
Object or image plane
index = n
ya
ub
yb
ua
H pupil = n ( −ub ya )
H object = n ( ua yb )
index = n
H image = n ( ua yb )
3/13/2023
Optics 214 - John P. Bowen
index = n
Lagrange Invariant:
It is a key property of a lossless
optical system that you can’t
change the optical invariant.
This means if you know H at one
plane, you know H at every other
plane in the system.
H is a measure of how much
information capacity a lens has, and
how hard it is to make – higher H
means more information (spots) for
a given lens.
3/13/2023
Optics 214 - John P. Bowen
Lagrange Invariant:
An example:
EPD
Someone asks you to design a
lens with:
q
EPD = 20 mm,
h=image height=10 mm,
NA=0.25 (ua’)
FOV = field angle=q = 10
degrees.
3/13/2023
h
u a
f
Optics 214 - John P. Bowen
Lagrange Invariant:
EPD
h
q
There is a problem:
u a
H at pupil = (20 mm/2)*(tan(10
deg) = 1.76 mm
H at image = 10 mm * 0.25 = 2.5
mm
These are incompatible – Either
the field needs to made smaller,
or the field angle needs to be
made larger.
3/13/2023
Optics 214 - John P. Bowen
Lagrange Invariant:
We have not discussed the
Lagrange invariant in three
dimensions. This is called
etendue.
In particular, the onedimensional formula shown here
could not handle non-rotationally
symmetric optics, like cylinder
lenses. In that case, you have to
keep track of the invariant in
both the x and y directions.
3/13/2023
Optics 214 - John P. Bowen