Important Points from Course

ECEN 4616/5616 4/19/2013

Important Points from Course:

Overview:

The basic methods of optical design:
Historical Research – What has been done before that you can use? While it is necessary to understand the past, it is wasteful to re-invent it.
First Order Layout – Use your knowledge of previous designs and the behavior of optical components (lenses, prisms, etc) to create a rough layout of idealized components that will approximately do the job.
Gaussian Optics – Use the theorems and methods developed in first order optics to refine the layout and determine the basic parameters (focal lengths, aperture sizes, stops).
Degrees of Freedom – When transitioning to finite optical elements, put in enough elements so that it will be possible to refine the system adequately. For example:
Field curvature cannot be corrected without a mix of positive and negative lenses or mirrors – this is from first order optics (Petzval Theorem).
Color aberrations cannot be corrected (except for an all-mirror system) without using glasses with different V-numbers.
Heuristic Refinement – The system is evaluated by simulating the propagation of light (by ray tracing or beam propagation algorithms). Successive changes in the system are made and evaluated to search for a better system. While the tedious part of this job is now done by computer, a good design does not often come automatically. The designer still must:
Create a merit function defining the analysis algorithms and results that constitute a “good” design. This often changes throughout the process.
Decide what parts of the system can be varied in the heuristic search and how those variables should be constrained. This is also subject to change during the design process.
Shepard the process through sticking points to reach an acceptable solution. This often requires changes in the merit function and sometimes changes in the basic system layout.
The Tools of Optical Design:
General knowledge of Optical Components and their approximations:
The thin lens approximation, imaging equation and lensmaker’s formula (including mirrors).
A general knowledge of various optical methodologies, such as holography, coded-aperture imaging, computed tomography, interferometry, etc. When you need one of these, you can then research it in more detail – but if you never heard of it before, you will not know that you need to look into it.

(It is worth ordering a catalog from a major company such as Melles Griot – not only is it a valuable reference on the variety of optical components and systems available, but they also include enough tutorial text to constitute a good, condensed textbook.)

Knowledge of Light Propagation Approximations:
The ray approximation to light propagation (Geometrical Optics); Postulates, theorems and limitations.
The methods of graphical ray-tracing which allow you to quickly sketch out optical systems without doing calculations.
The postulates, basic equations and theorems of Paraxial Optics (Gaussian Optics). In general, optical systems must be designed to work in the paraxial approximation before they can be made to work with real components.
The basics of Fourier optics and beam propagation – where they can be used with geometrical optics (PSF, MTF calculations for example), and when they need to replace ray optics (zone plates and holography, for example).
Familiarity with computer optical design programs:

In today’s world, no one will ask you to design an optical system without access to such a program. You need to be aware of the methods of use, the limitations, and pitfalls of such programs. In particular, you need to be aware that these programs are not “automatic”, but need to be carefully guided, debugged, and prodded to reach a satisfactory solution. A general knowledge of how they work, and a willingness to read the manual is generally more important than skill with a particular program.

Paraxial Optics Review

Coordinates:

Coordinates for ray tracing are local Cartesian coordinates, centered on the intersection of the surface with the z-axis.

All distances are directed (signed), and measured from the coordinate origin. (i.e., points above and to the right of the origin are positive distances, points to the left and below are negative distances.)

Rays are always directed lines. Ray angles are either measured w.r.t. the z-axis, or the surface normal (as appropriate).

All angles are acute. If the rotation direction from the reference (z-axis or surface normal) to the ray is CCW, the angle is positive. If CW, the ray is negative.

All angles are equivalent to slopes (tangents) for purposes of ray propagation, and sines, for purposes of refraction.

The distance from surface n to surface n+1 is a property of surface n and is a positive number designated .

Example:

Notation Rules:

Unprimed rays, angles and distances are before the ray is refracted by the surface, primed values are after.

Paraxial Assumptions:

Surface sag is ignored as negligible.
angles sines tangents

Positive Values:

Angles:
Lengths:

Negative Values:

Angles:
Lengths:

Geometric relationships (taking the sign rules into account):

Snell’s Law (Paraxial)

Relates the quantities to the left of the surface with those to the right:

Ray Trace Equations:

If we substitute for the incident and refracted angles in Snell’s Law their expressions in terms of lengths, but leave the ray angles, we arrive at:

Where is the surface power

and is the surface curvature

If we trace through sequential surfaces numbered we can derive the ray transfer equation from the ray’s direction and height and the distance, di, between surface i and surface i+1:

Thin Lens in Air: If we trace a ray through two surfaces, assuming the distance between them is negligible (d=0), we get the thin lens equation:

Where is the power of the thin lens

Imaging Equations:

If we again start with the Paraxial Snell’s Law expression, but substitute for all the angles in terms of the relevant distances, we can derive a similar set of equations which trace the position of images through a surface, lens, or series of lenses:

Imaging through a surface:

Imaging through a thin lens:

and image transfer between lenses:

Analysis of the image equations leads to the concept of focal length (=1/K).

Multiple Lenses, Thick Lenses, etc:

Iteratively applying the above equations to multiple surfaces or lenses allows the calculation of the thick lens equation, the lens combination equation, and the derivation of the concept and usage of Principle Planes.

See notes from 1/18/2013 and 1/23/2013 for details.

Summary of Paraxial Optics:

Tracing Equations:

Refract at a surface

Refract through a thin lens in air

Transfer
Imaging Equations:

Image location from a surface

Image location from a thin lens in air

Image transfer

Definitions:

Power of a surface

Power of a thin lens in air

Focal length

curvature is inverse radius of surface

Tracing Graphical Rays:

Ray parallel to axis:

, since

Hence the ray passes through the focal point.

Ray passing through focal point:

, since

Hence the ray leaves the lens parallel to the axis.

Ray passing through lens vertex:

, since

Hence the ray is undeviated.

Fourier Optics, The Plane Wave Spectrum and Beam Propagation:

The (scalar) plane wave solution to Maxwell’s equations has useful properties for describing the propagation of EM fields:

The plane wave solution has the same amplitude, frequency, and propagation direction (in free space) everywhere.
“Propagation” of a plane wave involves simply changing it’s phase by inserting a different position vector, into the expression , where .

Thus, an EM field can be simply calculated at any point in space, provided it can be described as a collection of plane waves – the “Plane Wave Spectrum” (PWS).

A connection between plane waves and the Fourier Transform can be made by looking at the intersection of a plane wave with a plane surface. For simplicity, we will set t = 0 and look at the intersection at the x,y,z = 0 plane. The scalar expression for a plane wave at this plane and time is:

When (the real part of) this function is plotted, it is seen to be a 2-D spatial frequency:

This can be made explicit by an appropriate variable change, u kx/2, v ky/2, resulting in the expression:

whereu, v are spatial frequencies.

The Fourier Transform decomposes functions on a plane into a set of spatial frequencies:

Hence, the Fourier Transform can decompose a EM field (at t = 0) on an x-y plane into a set of spatial frequencies, each of which can be associated with a specific plane wave (given the wavelength, ) with amplitude equal to the spatial frequency’s amplitude and wave vector , where:

The spatial frequency that represents this plane wave’s intersection with any other x-y plane down the z-axis is simply:

Hence, any monochromatic E-M field defined on a x-y plane (such as a pupil function or an illuminated aperture) can be propagated to another parallel x-y plane at distance z from the first plane, by:

Decomposing the function into spatial frequencies by the Fourier Transform.
Finding the k vector of the plane wave associated with each spatial frequency (the plane wave whose intersection with the plane is the given spatial frequency).
Changing the phase of each spatial frequency by the factor .
Inverse Fourier Transforming the re-phased spatial frequencies to find the new E-M field distribution.

(Of, course, taking due account of the organization and characteristics of the FFT, if done discretely.)

This technique can be used for many optical design and evaluation tasks:

Evaluation and design of zone plates.
Decoding optical holograms.
Evaluation and design of very small lenses and optical elements – too small for the ray approximations of geometrical optics to apply.
Evaluation of diffraction effects of apertures in optical systems.

Digital Filtering:

Most imaging today eventually produces pixilated images that can be manipulated by a computer. The field of digital image enhancement is extremely broad, but some basic principles hold throughout:

An image filter is easiest to visualize in the frequency (Fourier) domain: Any digital image can be decomposed by the DFT into a finite set of spatial frequencies. Each of these frequencies can be multiplied by a scalar gain factor and, potentially, a phase factor to adjust their phase. A filter in the frequency domain, therefore, is simply a matrix of (possibly complex) gain factors – one for each spatial frequency in the image. (Note that the spatial frequencies of an image have no relationship to the Plane Wave Spectrum – they are simply intensity distributions that an intensity image can be decomposed into. The PWS is based upon amplitude and phase distributions of EM fields in space.)
A space-domain (convolution) filter can be derived from the frequency filter by inverse Fourier Transforming the frequency-domain filter array. If the space filter consists of a few finite values surrounded by near-zero values, there is no penalty (and a speed advantage) to truncating it to the non-zero area.
Random noise in the image will result in spurious data in the spatial frequency domain. Any filter that amplifies spatial frequencies will also amplify noise. Linear filters will not increase the SNR of an image.
Spatial frequencies missing from the image cannot be recovered by linear filtering. This kind of “image restoration” is done by using a heuristic search to find an object “consistent with” the partial information. There is no guarantee that a “restored image” is correct.

We covered a specific example of a linear filter designed to boost the MTF of an optical system with a cubic wavefront distortion designed to produce an extended depth of field:

To invert the imaging process, , using the filter operation , we calculate the filter by: , where W is a band limiting matrix (maximum limits on each spatial frequency boost, determined by the physical band limit of the optics) and  is a small number designed to prevent division by zero.

Aberrations in Optical Systems

Key Points:

Aberrations are deviations from the ideal optical system.
Aberrations can be described by ray trace errors – Transverse Ray Aberration is the distance that a ray strikes the image plane from the ideal image point.
Aberrations can be described by wavefront errors at the exit pupil – The ZemaxWavefront map is the deviation of the wavefront at the exit pupil (in waves) from a perfect sphere centered on the ideal image point.
Since rays travel perpendicular to wavefronts, the Transverse Ray Aberration is directly proportional to the difference in slope between the exit pupil wavefront and the ideal sphere.

NOTE: Wavefront and ray errors are in some ways equivalent and Zemax’s Default Merit Function gives you the choice to minimize either ray errors or wavefront errors. Highly aberrated systems, however, will not optimize nearly as well using wavefront errors as when using ray errors.

Aberrations in First-Order Optics:

First order (or Paraxial) Optics describes optical systems in which the sine function can be well approximated by the angle itself. In this approximation, optical systems are linear systems and hence many useful theorems and formulas can be derived.

First order optics can be extended to finite ray angles by a suitable re-definition of the angle variable of a ray with respect to the z-axis as the tangent of the angle w.r.t. the z-axis. This allows first order optics to describe ideal (or perfect) optical systems and is extremely useful for deciding on the general layout of an optical system.

Certain aberrations arise in the first order approximation, and must be corrected in the first order system before there is any hope of producing a corrected real system. These are:

Chromatic aberrations. Since the index of refraction changes the same in first order optics as in finite optics, the same changes of focal length and magnification with wavelength occur. These aberrations must be corrected at the first order (paraxial) level before there is any hope of correcting them in a finite ray-angle system. Except for mirrors (which have no chromatic aberrations), this correction requires use of at least two glasses with different V-numbers (rates of index change with wavelength).
Image plane curvature (Petzval curvature). A quasi-paraxial derivation shows that an optical system can only image a plane onto a plane if it contains equal powers of negative and positive elements – otherwise, a plane will be imaged onto a curved surface. It is still possible for a system to have net positive or negative power, even if there is exactly the same total negative and positive power in the system – this is because of the theorem that the power a surface contributes to the whole system is proportional to its power times the height of the marginal ray at that surface. (The marginal ray is the ray from a given object point that just touches the edge of the stop.) Hence, if the marginal ray has less height at the negative surfaces than the positive ones, the system can have net positive power.

Third-Order Optics:

“Third Order Optics” describes optical systems where the ray angles of incidence don’t exceed values for which the third-order Taylor expansion is a good approximation to the sin function: . In the third-order approximation, families of aberrations can be described that are independent of each other, and the total aberration of the optical system can be described as the sum of the aberrations of each element or surface.

Even in systems where the third-order approximation isn’t valid, the third-order aberrations must be corrected for there to be any hope of designing a good system. In other words, “higher-order” aberrations generally cannot be used to compensate for uncorrected third-order ones.

The third-order aberrations are generally called Seidel aberrations after Ludwig von Seidel, who first described them in 1857. Historically, the Seidel aberrations were very important because they could be derived from Paraxial ray traces. With today’s computers, this is much less important, and few will ever perform these calculations manually.

The Seidel aberrations are:

Spherical Aberration: The change in power of a surface with ray height
Coma: The power asymmetry in an off-axis pencil or rays – the rays at large angles to the z-axissee a different magnification than the rays more parallel to the z-axis. This typically results in elongated and blured PSFs with.
Astigmatism: A fan of off-axis rays perpendicular to the z-axis (tangential) has a different focal length then a fan pointing at the z-axis (sagittal). This is a common aberration of the eye and is corrected by including a cylindrical lens power in eyeglasses.
Field Curvature: While this aberration is derived strictly from first-order ray tracing, like the other Seidel aberrations it can be described as a sum of the aberration from each individual surface.
Distortion: The change in magnification with object distance from the z-axis. This results in images having a “pin-cushion” or “barrel” shape.

Zemax will produce a very useful diagram of the surface-by-surface and total sum of these aberrations, using the “Analysis – Aberration Coefficients – Seidel Diagram” choice of analysis window. This diagram also includes “lateral color” and “axial color” aberration, which can also be separated into individual surface contributions.