PHYS 420 Project II :

Off-axis Analysis for Simple Optical Systems

Official Report

(Jeremy) Yu Gong

4 October 2014

The purpose of study and analyzing an off axis optical system under simple settings is to help

having a more generalized ideas of the structure of image and object systems, defining certain

transformations from object space and image spaces, and sometimes creating open windows for

possible shortcuts in practical calculations. The major principle of approaching an off axis

analysis for simple optical system is that, for any off axis optical systems, they could be

transformed back into an in-line optical system; and to understand the structure of any off axis

optical system, is of having general ideas of their corresponding in-line systems in the first place.

We thus began our approach by study a two dimensional in-line structure from point object to

point image, then generalize this pair into the functions for their paths; more importantly, we

may lately be capable of solving for any paths in any off axis systems by taking advantages from

the previous works.

1. Point Analysis in a Simple In-line Structure

As for any complicated optical systems, one shall approach it by reducing it into the

simple system as it can be; relevantly, our analysis will start by setting up the simplest

optical system in two-dimensions, and to firstly analyze its point to point relations. As

the following figure indicates, a simple optical system is formed by a lens-object

combination, an image is thus generated correspondingly.

Base on the conjecture equation, we shall easily have:

Where x and x' represent the in-line position on the optical axis, respectively.

Meanwhile, we could rearrange the above equation, and define certain middle terms

that:

Now to derive out the expression of their vertical position requires to consider the

point magnifications, for instance, due to the positions on the optical axis for the object

and image pair, we could obtain the following relations:

A coordinates transformation is thus formed as:

Where, its anti transformation is defined to be:

The expression for the Anti-middle term is found to be:

By taking their differential forms, we shall have that:

Therefore, we shall have that:

Meanwhile, their anti-transformation is that:

( For complete proof and analysis, see: "Reference and Analysis", Page 1-3 )

2. Path Equation

Consider the linear object structure in the following figure:

One may easily notice that, a linear object does not always have a linear image while it

has certain angular difference than 90 degrees to the optical axis, to explain it, however,

we may refer to the vertical relation from our previous analysis, for instance:

By taking its differential form, we shall have that:

Now consider that, the objective vertical position and the magnification are both

functions of in-line positions on the optical axis, while the in-line position is a (anti-

transforming) function of the in-line position on the optical axis for the image;

mathematically, we shall denote that:

Consequently, our previous differential form should have become as follows:

Meantime, consider an explicit form of the objective path function, so that we can

(always) denote it in the following manner:

It is quite an ambiguous move since we are not yet sure of having a differential

equation with respect of relations from an implicit path function; however, an explicit

from may have us certain advantage of generalizing all, for instance, if by considering

it, the following relation always holds:

The above relation is always true with, or without an implicit from of the objective

path, while substituting the y for:

The eventually from of the differential equation all be yielded as:

We name it the "Path Equation" in image space, while, the anti-transformation for any

point relation still holds, which allows us of rearranging the above equation in object

space, therefore, we shall have:

( For complete proof and analysis, see: "Reference and Analysis", Page 3-7 )

3. Off-axis Path Equation and Applications

Since we have discovered the general construction for the path functions in-between

object and image spaces, we are now, able to verify the nature of the image path, by

applying certain path functions in object spaces.

One typical instance, as we have mentioned in the previous section, is a linear path

with a non-90-degree angle to the in-line axis:

Rearranging its form and substituting in the path equation, we shall have a simple

solution, that:

Which yields a simple curve listed in the following figure:

The Above picture assumes to have a focal length of 6.4 units, while the slop is started at

8.0 units away from the lens center, having a slop of -6.4; and its image, on the other

hand has a curved path.

One thing deserves to be mentioned before we go through further applications for the

path equation, that by taking a close look at the homogenous solution for the path

equation, one may easily obtain a linear function that:

Meanwhile, the condition for the homogenous solution is that we have a free path for the

position of x, which physically means that the objective path function is purely

depending on the value of y, indeed, summarizing the above all, it simply indicates that,

all parallel rays must go through the focal point.

Slightly more complicated path function could be applied to the object space while

remains yielding analytical solutions, for instance, assumingly we have a circle centered

at certain point several unit distance away from the optical center, where:

We shall then solve for its path function in the image space, that:

In practice, if by letting the center fixed at 8.0 units away from the optical center, with

its radius of 1.0 unit, focal length of 6.0 units, we will have a thin mushroom-lip path as

its image, shows as follows:

A more complicated case would be applied as an elliptical path in object space, assume

that, we have a generalized path function:

Where its close form of differential part is:

By substituting it into the path equation, we shall have its solution as:

The image has quite a strange look while we applied a, and b are 3.0 and 6.0 units,

respectively; focal length of 6.0 units on the optical axis. The following figure shows a

"fat" mushroom-hat looking of the image:

( For complete proof and analysis, see: "Reference and Analysis", Page 8, Page 9-12 )

It is easier for us to derive the whole expression for an off-axis system for now:

considering an off-axis system with a certain reference angle respect to the in-line

optical axis, as the following picture shows:

Thus a coordinate-transformation is to be formed as:

While the anti-transformation could presented by the invert matrix of the above one,

where:

More importantly, the point transformation still holds in in-line systems, thus:

Now we substitute the expression of formal in-line image space with its off-axis image

space, where the linear algebra should have brought us to:

Now we apply the invert matrix for the x and y coordinates, we shall therefore have:

It is noticeable that, the above calculation is merely implying the transformation matrix

to the original function space, which, if we note any of the above two matrix as A,

there is:

It is also obvious that, any off-axis is indeed, an in-line optical system, which we are

always able to transform from one space to another. The following figure is for such

instance of a relatively perpendicular linear object in off-axis, and a linear non-

perpendicular object on the in-line optical axis:

( For complete proof and analysis, see: "Reference and Analysis", Page 9, 12-14 )

4. Conclusion

As a conclusion, we have firstly defined the point object-image transformation, so that,

for any point (x,y) on the object space, there is its image (x', y'), where they must

following the restrain that:

And:

Where:

We have also defined the path equation, such that, for any path functions: P(x, y)=0 in

the object space, there are path functions for their images, where:

In image-image space:

in Object-image space:

Nonetheless, we have defined the transformation for any off-axis optical system, which

allows any off axis system of transforming into the in-line optical system, and the

transformation shows as follows:

Where:

And:

PHYS 420 Project III :

Off axis Analysis and the Focal Aberration

Official Report

(Jeremy) Yu Gong

25 October 2014

The purpose of designing and the construction of varies of optical system to verify the

existence and behavior of an close-focal, off axis object to image pair, is to further realize the

applications for the path equations in conditions of different objective forms, meanwhile to

physically capture images that are generated from certain objects, from varies angles of visions.

The above experimental analysis would provide us a general idea of how an off axis optical

system works, and potential serves as proofs that the existence of certain aberrations from the

any off axis structures.

1. Introduction

The off axis structure usually cause a shifting of focal point along the vertical

plane respect to the optical axis, which in turn, a resultant comatic aberration will occur

on the image plane. When the reference center of an object is relatively close to the focal

point, the appearance of the image, however, would be much different than a comatic

aberration would have been able to describe. In general, the image of a close-to-focal, of

axis object would be sharply curved, and yields completely a different looking path

function than simple have a resultant comatic image on its reference plane; for instance, a

linear object is set straightly upward with respect to its off axis, while sitting it close to

the focal point, its image would have an abrupt curvature at its image center; more

importantly, one may not able to visualize its image on any image planes that are set off

axis to the optical systems, instead, the resultant image may have its occupation along the

optical axis on the image side to the convex lens.

Further more, the close-to-focal, off-axis system has certain comparisons to the

spherical aberration, as one may visualize the images share somehow similarities with

one to the other, for instance, they have the same curvatures (shape expansion) among the

center region of the object; however, one should have easily noticed, the level of the

curvatures and the curves of the image while viewing from side-ways would have them

differed. Another fact is that, the spherical aberration could be reduced or even avoided

by reducing the thickness of the lens or by the replacement of a Cartesian oval surface,

respectively. The close-to-focal, off-axis optical system, or rather to denote it as the

"focal aberration", on the other hand, could not be avoided by any means in optical

system that have focal points.

One may refer to the following figures as a general condition of how those two

aberrations might differ.

The above figure shows a straight vertical object and its image for a thick lens

system, while the curvature caused by the spherical aberration almost in opposite

direction with respect to the curvature of the image caused by the focal aberration, shown

in the following figure:

As the figure above indicates, the focal aberration happens only the object (linear)

is in an off axis position, thus to break the symmetric of with respect to the on-axis of the

optical system. More importantly, even all rays as in the above figure, have passed

through exactly the same focal, the curvature happens still. Overall, the spherical

aberration is caused by differentiating the focal lengths that are relative to varies of

vertical positions of the object; the focal aberration is caused by the off axis arrangements

in any optical systems.

Nonetheless, one shall realize that the focal aberration is unavoidable for any

optical systems with focal lengths.

( Additional information, see, "Reference and Analysis", Page 8)

2. Theory, Approximations, and Applications

As we aim to solve for the path function of the image that is generated by the off

axis object, and are able to approach it based on the results we have found in our previous

project.

Recall the path equation for any object in an off axis optical system, with its path

function of P(x, y)=0; we therefore, assume the vertical and horizontal position on the

image space as y' and x, thus, there always is:

Where:

As the object reference is quite close to the focal point, consequently its image

reference must be much further away from the focal point on the other side, therefore,

certain approximation may take in places that:

Assuming:

The first approximation we may apply is by considering:

We shall thus obtain our first level approximation that:

Where the formal terms in path equations have been reduced as:

One shall now consider a further approximation that:

The higher order terms from the previous level of approximation must be reduced

into:

As we now have the above two levels of approximation for the path equations, we

are then able to obtain certain applications from our previous approximations. before the

analysis starts, we need to as well re-denote the expressions for the magnification, in both

of levels of approximations:

Under condition of the first level approximation:

Under condition of the second level approximation:

Now we are able to directly dive into the homogenous solutions of the above two

path equation approximations, while the third term vanishes. One must notice that, the

physical application for the vanishing of the third term happens when either parallel rays

go through the principle plane, or the source object is fixed at infinity. In the meantime,

as we have assumed that the focal aberration happens at a close-to-focal reference

condition, the only cause of the homogenous solution, therefore, is the parallel rays that

go through the principle plane.

By applying the vanishing of the third term of both of the levels of

approximations, we shall solve for their solution sets, so that:

for the first level approximation:

for the second level approximation:

One must notice that the homogenous solution for the first level approximation

might look as if a straight line through the center, while having a wide range of

applicability; however, as for the solution itself, it is not too well consistent with the

physical nature of optics. Moreover, it would provide a tedious calculation for further

solving of the path functions for the image plane, due to the none-exact forms of the

differential equation.

We therefore, consider a more practical choice of the second level approximation,

that for a linear function in object space, we may obtain its image path function, as we

assume that, for the object path function:

Resultantly, its image function has a simple form that:

Now for any enclosed path function that has a symmetric nature, we assume for a

general form of its path function that:

By applying the magnification approximation for the partial differential form of P

with respect to x, and y, we shall have the following exact forms for y', and x:

Solving for the above equations, we shall have that:

( For complete proof, see, "Reference and Analysis", Page 1-5)

3. Structure of Optical System and Measurements

To construct an optical system for verifying the focal aberration is extremely

simple, as one requires only a convex lens with a short focal length and objects in certain

shape (path function).

For instance, we have tested and captured pictures of images for a linear fiber, a

circle made by blue thin wire, and a ellipse made by the same thin blue wire. In the

meantime, as we have obtained the path function for the image space in the previous part,

it is quite easy for comparing the shapes from a physical image and the image generated

by path function simulations:

The following figures show images of a linear object that has an angular

difference to the optical axis, and simulations based on the solutions for its path function

in image space:

Linear Objects:

linear object and image in its front site

Part of the image from simulations and physical image from side view

Circular Objects:

Circular object and image in a slight side-view

Part of the image from simulations and physical image from side view

Elliptic Objects:

Elliptic object

Part of the image from simulations and physical image from side view

Much easier recording method could be made for the images by a hologram, with

proper settings and adjustments, a simple holographic structure could be arranged as

following figure shows:

The above figure shows for a simple set up on the surface of a sand box, while

one may have to add the obstacle between the laser source and the lens surface, since we

would to reduce as much possible reflection or illumination lights from any surface other

than the object itself. The holographic plate could be placed as close as the reference

beam may reach, since holograms record only the interference patterns of assigned object,

relevantly leaving a clear focused image unnecessary.

The exposure time for a regular PG-01 holographic plate/film is roughly about 22

seconds, under a 5mW reference laser; during the experiment, we have set the distance

from the object to the convex lens no further than 10 cm, where its focal length was less

than 7 cm.

(Additional holographic structure and designs, see "Reference", Page 6-8)

4. Conclusions and Future Topic

Focal aberration happens whenever an optical system has any focal lengths, as we

have introduced in previous sections, focal aberration could not be avoided by choice of

the focal lenses, however is capable of being reduced when object distance is relatively

far from its focal point.

There are certain similarities between spherical aberrations and focal aberrations,

meanwhile, the spherical aberrations could be easily reduced, and could be avoided by

choosing of a Cartesian oval surface. The focal aberration, on the other hand, is caused by

off axis structures of the object, which could not be avoided, and is quite noticeable when

the object is relatively close to the focal point.

The path equation for any focal aberrations could be further deduced into a proper

approximation form, such that:

While its magnification is reduced into:

Applications for certain regular forms of objects could be solved from the path

equation approximation, which are listed below:

Parallel lights:

Linear objects that has a path function of:

has its path function for the image:

Any enclosed path function with symmetric forms of:

has their path function for the image:

The above resultant solutions for path functions in image space are

experimentally verified as suited for physical images under proper conditions.

Nonetheless, a hologram could be taken as an advanced method for recording the

physical images from the experiment.