Aberrations and their Correction Aberrations in an optical system refers to the system’s deviation from ideal Gaussian (linear) behavior. We will only deal with small errors in an optical system that is already producing a reasonable image. This means that wavefronts will deviate from the ideal (usually spherical, for an imaging system) by only a few wavelengths. It was demonstrated about 150 years ago that wavefront errors in an optical system greater than λ/4 result in a noticeable degradation in performance. (This is one of two optical results known as the “Rayleigh Criteria”.) Hence, ground glass, structural glass bricks and shower doors won’t be considered suitable for analysis by aberration theory.

Zemax Wavefront Analysis Window (Analysis-Wavefront-Wavefront Map) The wavefront error map is given at the exit pupil, and referenced to the image surface. How does a ray trace program derive this information?

Wavefront Errors and Ray Tracing Stop Entrance Pupil Exit Pupil Ideal Spherical Wavefront Actual, Abberrated Wavefront Ray trace programs will report “Wavefront Errors” at the Exit Pupil of a system. The errors are the difference between the actual wavefront and the ideal spherical wavefront converging on the image point.

Relation between Ray Error and Wavefront Error TRA In a system with aberrations, rays from different positions in the exit pupil may miss the ideal image position by various amounts when they reach the image plane. This is known as “Transverse Ray Aberration”, or ‘TRA’. Zemax will plot a “ray fan” (Analysis – Fans – Ray Aberration): This is, for each field point, a plot of the TRA for rays that pass through different positions in the Exit Pupil. This plot is a slice through the pupil in the x-direction plotted against TRA in the x-direction, and a similar plot in the y-direction.

Zemax does not normally propagate waves through an optical system, but uses a geometric relationship between Ray errors and wavefront errors to calculate the wavefront error map: Since rays are always perpendicular to wavefronts, a wavefront tilt error (at some location in the pupil) corresponds directly to a Transverse Ray error (TRA) at the image plane. The wavefront aberration, W, is the sag difference between the actual wavefront and the ideal spherical wavefront centered on the paraxial image point. The angle, a, between the ideal ray and the real ray is just slope of W,, and the Transverse Ray Aberration is: R Note that, while this method accounts for wave aberrations at the exit pupil (and can, therefore, be used to derive some image aberrations) it takes no account of wave diffraction effects throughout the rest of the optical system, as a true wave propagation method would.

By tracing a number of rays through the pupil, and measuring the TRA of each, we can get the slope of W; then by numerical integration, we can find the wavefront aberrations, W. Somewhat remarkably, therefore, we can find W to a small fraction of a wavelength without having to track the path length of the rays we use to find it. (This is the way a Shack- Hartmann wavefront sensor works.) A wavefront sensor can be used to measure the wavefront distortions introduced by an optical system. If you start with a diffraction limited point source (a light source focused through a small pinhole, say), the wave front sensor will measure the distortions of the collimated wave. Since the optics are reversable, this also gives the distortions present when light is focused by the system. Wavefront Sensor Dif.Lim Point Source

Measurements of wavefront distortion can be done by interferometry. This requires that the light used be coherent (e.g., from a laser). A Shack- Hartmann wavefront sensor, however, can work in incoherent and even broad-band light: A Shack-Hartmann wavefront sensor consists of an array of micro-lenses focusing portions of an incoming wavefront onto position-sensitive detectors. The local tilt in the wavefront causes the focused spots to shift accordingly. Once an array of wavefront tilts has been calculated, numerical integration gives the wavefront shape.

Modeling Secret Lens Systems A problem that sometimes arises is that a component of a larger system, which you are trying to model, is proprietary. For example, high end microscope objectives can easily cost $10,000 or more and getting the prescription for these objectives is essentially impossible. For example, here at CU we are interested in extending the DOF of microscopes by introducing particular wavefront distortions which allow the in-focus image to be recovered by computer processing over a wide range of defocus. Camera Computer Distorted Waveplate (Exaggerated) Microscope objective Wavefront distorted to increase DOF (+ unknown dist from M.O.) While we can assume that the microscope objective’s distortions are minimal in-focus, its distortions from an out-of-focus point are unknown and we therefore can’t model their interaction with the waveplate.

Modeling Secret Lens Systems The problem can be solved by using a wavefront sensor to actually measure the out-of-focus distortions over the DOF range of interest. The microscope objective is then modeled in Zemax as a combination of an ideal (Paraxial) lens and a “Grid Phase Surface” (described in Ch. 13 of the Manual: “Surface Types”) which simply adds a numerically specified phase to the output of the Paraxial lens: The phase data is entered into the “Extra Data Editor” via the tools-import diaglog.:

‘Low Order’ Aberrations Any classification of aberrations must be somewhat arbitrary, as there are obviously an infinite number of ways that optical systems can fail to be ideal. There is both historical and practical interest in generating a classification of “Low Order” aberrations, where “Low Order” is defined by several different methods. Historically, these are called “Third-Order Aberrations”, although the reason is somewhat obscure. (There are no “Second or First Order Aberrations”, for example.) The historical significance is that, in the third order approximation, aberrations cold be calculated using only data from a Gaussian ray trace through the ideal model of a system – a huge savings in effort before computers were available. Additionally, the total 3 rd order aberration of a system is simply the linear sum of the contributions from each surface, and they can be corrected by combining surfaces with opposite signs of aberrations. Their significance today is that, unless a system is corrected to 3 rd order, it is unlikely to be possible to correct ‘higher-order’ aberrations, so it is necessary to start by insuring that the low order aberrations are taken care of.

The Monochromatic Aberrations of a Circularly Symmetric System The Wave Aberration Polynomial Approach: An aberrated system has an aberrated wavefront. If we describe this wavefront with a (2D) polynomial expansion, we can select the lower order terms as our primary aberrations. Limiting the analysis to circularly symmetric systems with smooth surfaces, and naming the polynomial as ‘W’, the analysis goes like this: 1.W describes the wavefront in the pupil, so must be a function of the pupil coordinates, call them x,y ⇒ W = W(x,y), or W(r,Ф) in polar coordinates. 2.Since the aberrations change with field point, W must also be a function of the object point, u,v, or τ,θ ⇒ W = W(r, Ф, τ,θ ). Since the system is circularly symmetric, we can always pick the coordinates such that one of these angles = 0, say Ф. 3.The requirement for smoothness (no cusps, etc) limits the possible terms to r 2,τ 2, and rτcos(θ). Hence It’s common to write this as: Where m,l,n now can’t be arbitrary integers.

The two forms for W: Imply that l = 2i+k, m=2j+k, n=k. Using i,j,k = [1 2 3] gives only 5 terms that describe aberrations which cannot be ‘cured’ by re-defining the coordinates or re-focusing the image plane: 0 W 40 r4r4 Spherical aberration 1 W 31  r 3 cos  Coma 2 W 22  2 r 2 cos 2  Astigmatism 2 W 20 2r22r2 Field Curvature 3 W 11  3 rcos  Distortion Symbol PolynomialName

The Monochromatic Aberrations of a Circularly Symmetric System The Ray Approach: We make the following assumptions: 2) The sag of a sphere is: 3) The Aberrations are small enough that a real ray trace (via the approximations above) is “substantially the same” as a Gaussian ray trace – the ray paths don’t differ enough to change the system layout. With these assumptions, Philipp Ludwig von Seidel (in 1857) showed that: 1.The aberrations of an optical system are the sum of the aberrations due to each surface, independently. 2.The surface aberrations could be calculated using the data from a Gaussian (e.g., ideal system) ray trace. (This may be where the term “3 rd order aberrations” came from)

What the Low Order Aberrations look like Spherical Aberration: Different heights on a spherical lens have different focal lengths. Polynomial Description: This is the only monochromatic aberration that exists on axis. Coma: For off-axis objects, the rays through different heights on a spherical lens will have different magnifications (and hence, arrive at different points on the image plane). Polynomial Description:

Astigmatism: Rays in the plane of the Z-axis (tangential plane) have a different focal length than rays in the perpendicular plane (sagittal plane, which intersects the axis at the entrance pupil). Only off-axis astigmatism is present in a lens with spherical surfaces, but on-axis astigmatism can be caused by cylindrical lenses. Polynomial Description: Field Curvature: A flat object plane is imaged to a curved image plane. Polynomial Description: (The text, p.210, has this wrong – probably a typo.)

Distortion: The magnification changes with increasing field angles. Object Images Barrel Distortion (+) Pincushion Distortion (-) Polynomial Description:

Zemax Analysis of Third Order Aberrations Zemax will report (list or graphically) the 3 rd order aberrations of a system surface-by-surface and the sum. This allows one to see which surfaces are problematical, and how they may be corrected. Following is a layout of the “Tessar” lens in the Zemax samples directory:

This is the graphical surface-by-surface plot of the 5 Seidel third-order aberrations of the lens, plus two chromatic aberrations. Analysis window accessed by choosing “Analysis – Aberration Coefficients – Seidel Diagram” Note that the sum of all aberrations is quite small (right-hand column), even though the aberrations of some surfaces are large.

Total (3 rd order) aberrations are the sum of the aberrations of each element This is one of the most important results of 3 rd order aberration theory. While total aberrations are a complicated (and nonlinear) function of all the elements of a system, the low order aberrations can be shown to be a simple linear sum of the low order aberrations of each surface or lens. This allows us to correct for, say positive spherical aberration, by introducing an element with the appropriate amount of negative spherical aberration somewhere else in the system. We can control the power added (or subtracted) from the system by placing the corrector element in an appropriate place: L2L2 L1L1 L2L2 L1L1 L 2 has large effect on system power. L 2 has small effect on system power. But, both have similar effects on 3 rd order aberrations.

Sum of 3 rd Order Ray Aberrations The following figure shows a system with 3 components (surfaces, thin lenses, etc.) with the Chief ray (through the center of the stop) and a ray passing through some point of the stop off center where we want to know the aberration. Each ray is traced from the object point, O, to the corresponding image point, I. Since the wave aberration at each pupil height is the difference between the ray path length traced at that height, and the chief ray (through the center of the pupil), the total wave aberration, W, at the height of the ordinary ray is: W = [OABCDEI] – [OABCDEI] = {[OAB] – [OAB]} + {[BCD] – [BCD]} + {[DEI] – [DEI]} = W(L1) + W(L2) + W(L3) i.e., the sum of the ray aberrations from each lens individually.

Sum of 3 rd Order Wave Aberrations In the paraxial approximation, we characterized the effect of a lens as just putting a variable delay on an incoming wave front, which is a function of the optical thickness of the glass at each point. This also works as a 3 rd order approximation, because we are considering the change of direction of a ray to have a significant effect only on the wavefront error, and an insignificant effect on its future path. For the wave 3 rd order approximation, this means that we can consider deviations from the perfect spherical wave to propagate essentially the same from surface to surface (except for the increase or decrease in curvature with distance): Hence, in the 3 rd order approximation, the wave aberrations of each element sum to give the wave aberrations of the total system.