Proteep C.V. Mallik
OPTI 689 Summary
12/8/04
1
OPTI 689: Final Report
Introduction:
I am writing a summary of what I have learnt in the Hamiltonian Optics course. I took
this course because I felt that it would give me a new way to look at optics. It surely did
that. I learnt a new approach to deriving several of the most important concepts in
geometrical optics. The design examples helped me understand how we made certain
conclusions. I find the mathematical construct of Hamiltonian optics to be a very
powerful tool, even though it is difficult to use for multiple-surface systems.
1. Hamilton’s Principle:
Modified from Fermat’s Principle, it states that a ray between 2 points has a stationary
path compared to any other ray between the same 2 points. Also, the path will vary if the
end points vary by a small amount.
2. Fermat’s Principle:
States that light always travels along the shortest possible path. Or rather, the optical
path length is the smallest. This is incomplete, and must be used along with Hamilton’s
principle for completeness.
3. Snell’s Law:
For a ray A incident on a refracting surface with surface normal n̂ and emerging ray B
Snell’s Law says that:
ni A  nr B  nˆ  0 ; ni and nr are incident and emergent refractive indices respectively.


This law can be derived from Fermat’s Principle using a diffractive optical element
(DOE) as a phase object.
The above 3 laws are consistent only when they are modified to Hamilton’s Principle of
stationary optical path.
4. Euler’s Equations:
Parametrizing any curve between 2 points A and B one can define a stationary integral.
This is governed by Fermat’s Principle. For any general integral to be stationary one can
define Euler’s Equations. Applying these equations to Fermat’s Principle we get the
Eikonal equation, which is also the ‘characteristic’.
5. Ray Vector:
This is a vector indicating the propagation direction with magnitude equal to the local
index of refraction.
Fig 1: (From class notes).
Proteep C.V. Mallik
OPTI 689 Summary
12/8/04
2
6. Hamiltonian Point Characteristic:
For a point P in object space the surface with constant optical path length from it is
defined by the point characteristic. From Fermat’s Principle it can also be shown that
rays are always perpendicular to the wavefront. By varying the point characteristic one
can derive Hamilton’s Principle of Varying Action.
7. Hamiltonian Mixed Characteristic:
From the point characteristic for 2 fixed planes one can derive the mixed characteristic of
the 1st type and the 2nd type. The 2 types arise from whether one defines the optical path
length in object space or image space respectively. Also, when Z = Z0 the 1st type mixed
characteristic fails to determine the ray vector in object space. Similarly, when Z = Z1 the
2nd type mixed characteristic fails to determine the ray vector in image space.
8. Hamiltonian Angle Characteristic:
The angle characteristic is the optical path length from Q0 to Q1. It isn’t suitable for use
in afocal systems. In fact:
Point characteristic cannot be used when second plane is image plane
Mixed characteristic cannot be used when one plane is coincident with focal plane
Angle characteristic cannot be used when dealing with afocal systems
9. Free Space Parameters:
The above 3 characteristics can be found for propagation of a ray in free space.
10. Stratified Media:
For propagation through a stratified medium one can derive the expression of the mixed
characteristic. Making some simplifications one can find the transverse ray aberration,
and this turns out to be an expression for spherical aberration.
11. Spherical Mirror:
The angle characteristic can be derived for an incident ray and the reflected ray from a
spherical mirror. The ray coordinates at 2 planes (object and mirror) can be calculated.
The angle characteristic may be expanded in a power series to finally obtain the wellknown Gaussian equations in geometrical optics.
12. Refracting Spherical Surface:
Similarly one can find the angle characteristic for a spherical refracting surface. It is very
similar to the angle characteristic for the mirror, except that the refractive index needs to
be taken into account in the present case.
13. Perfect Conjugate Points and Cartesian Ovals:
One can geometrically derive Snell’s law. This is a sufficient condition for perfect
imaging. 2 examples can be investigated here, where refractive index outside Cartesian
oval is greater than inside it (hyperbola) and vice versa (ellipse).
14. Aplanatic Points of a Sphere:
The aplanatic points of a sphere are easily found using the geometry used in the above
case.
15. Abbe Sine Condition:
Cartesian ovals can image only 1 point perfectly. But to image a small field of view
perfectly, the Abbe Sine Condition needs to be satisfied. The Abbe Sine Condition
corrects for all orders of coma in a system.
Proteep C.V. Mallik
OPTI 689 Summary
12/8/04
3
Fig 2: (From class notes).
Using principle planes or surfaces the Sine Condition can be derived for both finite and
infinite conjugate cases.
16. Taylor Expansion of Mixed Characteristic:
The mixed characteristic depends on only 3 quantities: h2 ,  2 and h . . Taylor
expanding we can derive the Sine Condition. The magnification factor is a constant for
the condition to hold, ie, all linear transverse field dependent aberrations are corrected.
Lateral aberrations may still exist.
17. Use Sine Condition to Derive 2 Aspheric-element Aplanatic System:
A system corrected both for coma and spherical aberration is called aplanatic. The Sine
Condition can be used to make such a system. Off-axis aberrations can be controlled by
on-axis properties of the system.
18. Telescopes:
Telescopes are aplanatic systems. The Sine Condition is satisfied for the RitcheyChretien design, which ensures that the zonal focal length is the same for each pupil
zone. The zonal magnification is also satisfied. Thus both Snell’s law and the Sine
Condition are satisfied.
19. Coddington Equations:
To derive these one needs to understand and define principle and parabasal rays. To use
an axi-symmetric system one can use the approximation that the surfaces are torroidal.
By finding the general expression for the OPD between the principle and any other day,
one can Taylor expand it and derive the Coddington equations. This gives us expressions
for the sagittal and tangential radii of curvature.
The Ritchey-Common test is an application of these equations, and one can use it to find
the astigmatism of a flat surface.
20. Pupil Astigmatism Condition:
For axi-symmetric systems the mixed characteristic is used for finite conjugate systems
and the angle characteristic is used for infinite conjugate systems. Quadratic field
dependent aberrations can be corrected using this condition.
M() = constant, if Sine Condition is satisfied.
a). s = t/Cos2 = constant implies that quadratic field dependent aberrations are corrected.
Proteep C.V. Mallik
OPTI 689 Summary
12/8/04
4
b). t/Cos2 = constant implies that quadratic field dependent aberrations are corrected in
the tangential plane.
c). s = constant implies that all aberrations of the form W2n0 are corrected.
d). s = t/Cos2 implies that all aberrations of the form W2n2 are corrected.
21. Difference Between Coddington Equations and Pupil Astigmatism
Condition:
Coddington equations can be used to find where the tangential and sagittal rays intersect
in the image plane. These give information about aberrations for any field point, but only
for those with quadratic field dependence. The pupil astigmatism condition looks at
bundles of rays that all go through the object point. Each point in pupil has its own ray
bundle. This is used to determine the quadratic field dependent aberrations.
22. Concentric and Retroreflective Systems:
Design examples we went through in class really brought out the significance of the pupil
astigmatism conditions. How one can control astigmatism by appropriately placing the
stop was also significant.
23. 3 and 4 Surface Systems:
We also studied 3 and 4 surface systems with design examples.
24. Plane Symmetric Systems:
We also studied plane symmetric systems and found Hamilton’s characteristic and the
pupil astigmatism conditions for both.
Conclusions:
A lot of material was covered in this course. I would have preferred a few more hw
assignments, especially design problems so that I could understand the concepts better
and put them to practical use.