By finding how the chromatic aberration changes with the change in R2, we can get our desired chromatic aberration. We will also use the lens bending technique to get the desired spherical aberration.
Harting Zeitschrift fiir Instrumentenkunde, vol. Even as recent as , achromatic doublet glass choice has been studied. First, the calculations from the results we got from step 2: Here are the iterations I did for the shape, as I changed R2 and adjusted R4 to get the desired focal length and colour correction. In this case, it is optimizing the focal length difference between the d line and the other lines, g , C , and F.
And the results for the colour correction are as follows:. First, setting up the 1st order system and normalization: Next, the YNU raytrace: Finally, the evaluation of the 3rd order aberrations:.
These will come up later. As a prerequisite, lens bending does not change the focal length of the chromatic aberration. Lens bending is changing the curvature of the surfaces of the lens while not changing the power and therefore focal length of the lens. By lens bending the negative lens alone, we can overcorrect the spherical aberration in the negative lens surfaces 3 and 4 to balance the entire system while not impacting the other aberrations.
Look, the spherical aberration SA is better already. By lens bending the entire system, the coma can be controlled, and by increasing the front curvature the coma is under corrected. This obviously changes the spherical aberration, but the spherical aberration can be corrected by lens bending the negative lens to cancel it out. If we do both bending iteratively, the coma and spherical aberration will converge to the minimum. I set up a simple system in the software and ran an optimization.
I plugged in the numbers for the radius of curvature that Zemax calculated, and looked at the spreadsheet. This leads me to think that when evaluating for 3rd order aberrations, the spreadsheet method and the Zemax optimization method are identical. Separated telescope objective, designed by the spreadsheet method, evaluated in Zemax Click to enlarge. Separated telescope objective, designed in Zemax, evaluated in Zemax Click to enlarge.
What do you think? I think that the spreadsheet method with 3rd order aberration correction does remarkably well. But I think we learned a lot more by doing the calculations by hand. We got deep into the process of lens design with the telescope objective, while looking at the key aberrations. This already gives us a leg up on what to look for in the lens system overall, and the effects of each surface going forward.
For a camera lens, the primary difference with the telescope objective is the field of view. The main effects of the higher field of view is the astigmatism and the field curvature. With the aplanatic lens in the previous chapter, the uses are for telescope objectives or a collimator, and has a maximum field of about one degree or so.
With a half angle larger than 10 degrees, the astigmatism and field curvature correction becomes important. A lens that has both of these aberrations corrected is called an Anastigmat lens. As you can see in the diagram, the point of focus is different in the two planes, the tangential and the sagittal. The front defocus and the back defocus differs so much that the out of focus or bokeh becomes very unpleasant to the eye, especially the back focus.
For cinematic lenses, the astigmatism is almost always corrected more so than typical camera lenses. The causes of astigmatism is the change in refractive power between the two planes, since a pencil of rays at an angle pass through a different portion of the lens in the meridional and the sagittal planes.
On the optical axis, the lens is rotationally symmetric, so the pencil of rays are all in the meridional plane, and astigmatism does not occur. A simple but effective schematic and concise description on astigmatism can be found in the Astrophysics section in Hyper Physics website for Georgia State University.
I like the word Anastigmat. The image plane is not on a singular plane, and the image plane is curved, hence the name. That means that with a lens that has field curvature, if the lens is focused at the center of the image, the corners of the image are out of focus, and vice versa.
Conversely to astigmatism, the front and back out of focus are the same, as they are circular. Although the spherical aberration is also a circular defocus, it does not change from the center of the picture to the outer corner edges.
As far as the cause of field curvature, for a positive lens, the focus point off axis is closer to the lens than the focus point of the optical axis. This is expressed mathematically with the Petzval sum.
A Viennese mathematician Joseph Petzval discovered this phenomenon in For a thin single lens the equation is simplified to. What this means, conceptually, is that the Petzval sum is decided by the power of the lenses and their index of refraction. Historically, the Petzval sum is given as a number when the focal length of the lens is normalized to 1. The Petzval sum is said to represent the curvature of the field in the paraxial domain. The revolutionary lens that Petzval designed, which is immortalized as the Petzval lens , ironically does not satisfy the Petzval condition.
In fact, it is a very good example of a lens that does not satisfy the Petzval condition. But imperfect optics is not always a bad thing, and this lens was sought out for portraiture. It has recently been revived in the form of the Lomography New Petzval 85 Art Lens , which I think is very interesting as a lens designer. This is possible for an Anastigmat, but it is the direct opposite of the colour correction and spherical aberration correction that we apply by using the high index, high dispersion lens for the negative flint lens for an Aplanat.
Which is to say that a doublet lens has more field curvature than a singlet lens. This can be done with one positive lens and one negative lens. To correct astigmatism, the lens and stop must be farther from each other. To summarize, to have a well corrected lens that has a field of view, we now know that there needs to be some thickness along the optical axis, compared to the entire length of the optical system.
It is best explained with a lattice like structure that looks like a barrel or looks like a pincushion, hence their names. There are some distinct differences between the properties of distortion compared to other aberrations. A bit of history about distortion, in old cinematography, the best cine lenses still had some measure of distortion, especially for wide angle lenses. A lot of scenes where the main protagonist walks away down a straight path had to be shot parallel to the field.
Also, some scenes had benches, for example, that were bent the opposite way from the distortion so that the image would be straight. Distortion is when the ideal height of the image is located in a different place on the image plane. If the scale tips to the left, the distortion is positive and therefore pincushion shaped, and the opposite is the barrel distortion.
Higher orders of aberration affect this qualitative thought process, but the idea holds true for most cases. Therefore it is easy to imagine that symmetric systems like the Double Gauss lens has small distortion, a telephoto Iens has pincushion distortion, and wide angle retrofocus lenses have barrel distortion.
Finding a suitable value for distortion is tricky because depending on the optical system, there are different requirements. Landscape photographs or distance metering devices require high distortion correction, if objects such as railways need to be imaged in a straight line. Also for photographs, even if the absolute value of distortion is small, a non-smooth transition of the distortion can look very unpleasant to the eye.
In this case, not only is the absolute value of the distortion cared for, but the change in distortion also needs to be taken into account. Of course, a lot of digital cameras have software distortion manipulation, so all of this may be moot. If the optical system has a positive lens after the stop, after the marginal rays pass the lens, the shorter the wavelength is, the more it is refracted towards the image plane. The index of refraction for shorter wavelengths is larger in the material.
This effect is the transverse chromatic aberration. If we are photographing on a plane, we would see the transverse chromatic aberration most clearly at the edges. Not to be confused, the lens may have longitudinal chromatic aberration as well, at a point off-axis. Similarly to distortion, the correction of transverse chromatic aberration can be done by making the optical system symmetric with the stop in the center.
The amount of refraction in the rays depending on the wavelength are balanced. In summary, the longitudinal chromatic aberration is corrected with a positive lens and a negative lens, while the transverse chromatic aberration can be corrected with positive lenses only if they are placed symmetrically from the stop.
Optical lens design is founded on ray tracing. We can discuss OTF Optical Transfer Function and wavefront aberrations, but very often raytracing is at the core of the design. We humans are inherently good at recognizing patterns, much more than computers. Therefore, an expert lens designer using a relatively primitive computer can usually provide a better lens design than a novice optical lens designer using a supercomputer.
Pattern recognition for lens design includes the cross-sectional diagram of the lens, the ray path diagrams, and the aberration diagrams. Certainly computer programs do it for us now, and understanding the process can help our lens design process many fold.
For a triplet, three wavelengths are sufficient. For telephoto lenses and large focal length range zoom lenses however, four wavelengths d , g , F , C are needed. Some cases use five wavelengths, including IR. Although we will not go deep into this, for modern faster F-number lenses like F1. In our current example, we have an angle of As we will see later, this corresponds to the diagonal position of I think the triplet is the perfect lens to explore lens design.
Although it is not the first photographic lens designed, almost all modern lenses can be traced back to the triplet. Therefore, studying the triplet carefully can provide the basis for most modern photographic lens design. Just like that, we can now qualitatively and conceptually dissect the lens and figure out the expected performance without opening any software.
This prevents blindly optimizing a non-winning design concept, by tackling lens design without knowledge and just the software. Now, onto our triplet. How are we going to rectify the problems above and make a compete lens that corrects all of the seven deadly aberrations? Take the separated telescope objective and split the positive lens, put it at the end, and make it a triplet.
So, the monochromatic third order aberrations and the two chromatic aberrations can all be corrected, and we finally have an example that has a reasonable field of view and relatively fast F-number as well. The triplet later can evolve to an Ernostar type, which leads to a Sonnar type. The triplet can also evolve into the Tessar type, and most notably the Double Gauss type. The triplet at its simplest form does not take thick lenses into account, and can be designed with thin lenses.
This allows for lens bending that is sufficiently correct, and the lack of cemented surfaces make the triplet a very good tool for learning lens design. Max Berek, A. Conrady, B. Johnson, Robert E. Hopkins, Rudolf Kingslake, Fumio Kondo, all giants in lens design, have written about their methods for improving the triplet design. Even though classified as a triplet, in their examples there are some highly asymmetrical types, large aperture types, with all kinds of variations. The spherical aberration is corrected as a polynomial function, and the coma can be corrected as a linear equation, via lens bending.
This can only be solved iteratively, and this can be applied to more compact lenses and zoom lenses albeit partially, and is very useful.
Also, we can divide the properties of the lens into the lens bending portion of the element, and the power portion of the element. Lens bending does not affect the power of the lens, so they can be evaluated and corrected separately. Lens bending corrects one set of aberrations, and the power of the lens corrects a different set of aberrations, so we can see the properties of the lens clearly.
One more thing, optimization using a computer is almost a brute force approach, while thinking about the 3rd order aberrations and the thin lens equations is a more artisanal approach, looking through the system properties and figuring out the uniqueness of our design. Looking at the cross sectional diagram and the ray diagram, using our pattern recognition and intuition as human beings can be a very powerful lens design process.
Optimization with a computer is also powerful, but in a different way. The biggest choice in lens design. Even for a relatively simple lens like the telescope objective, we needed to choose the glass carefully. For a triplet, with just one more lens, it becomes critical. The danger in using the computer optimization for glass selection has to do mostly with the price if the lens, where it will choose the lens based on it giving the best answer as the merit function decreases to a minimum.
Then you would have gotten a material that has an index of refraction of say, 2. However, we can face some problems in choosing glass automatically. The aforementioned cost is one. Density, therefore the weight of the lens, making it heavier AND more expensive is another. Some lenses have absorption in the blue spectrum which is manageable if the lens is relatively thin, but makes the image yellow if it is too thick.
Other lenses are soft and scratch easily, therefore are not suitable for the front most lens, like in the Leitz Summar lens.
Some lenses have a lower melting point Tg and hard coating is difficult. Glass that cracks when polishing. Glass that expands more with temperature, compared to other glasses.
There are many lenses to choose from for the crown lens, but the choice of the lenses determines the Petzval sum , so making a mistake here will cost us dearly as we go forward. Usually, if the positive lens has a higher index of refraction, the negative lens will also have a higher index of refraction.
This allows the spherical aberration, coma, astigmatism, to be corrected as well. Once the positive lens is chosen, we choose the negative lens, but the Petzval sum is calculated much later in the process. That means we have to use our experience or experience from others to choose wisely at this stage. If we look closely at the glass we would see amazing performance for the glass Taylor could use.
This is a triplet with an F-number of 1. A creative choice of glass is chosen, and significant optical poser is needed for the two positive lenses. Interesting that the index of refraction is almost the same for all three lenses. This is a triplet that is color corrected extremely well US Looking at different glass would be a lot of fun. At this point, choosing the glass looks like a random process.
The Petzval sum: If we can make the Petzval sum small we can design a lens that has a larger field of view, but since each lens power increases, it is difficult to design a large aperture lens. The amount of aperture we can afford depends on the index of refraction of the glass. A high index glass allows for larger aperture, because the spherical aberrations are smaller due to the larger radius of curvature with respect to the lens power.
On the other hand, a large aperture lens has shallower depth of field, so the field curvature and hence the Petzval sum must be small. If the field of view is small, the Petzval sum can be large, and larger apertures are possible. In any case, the Schwarzschild solution is a method derived by Karl Schwarzschild , a German physicist and astronomer.
At the time, he felt Germans were a step behind the English as far as optics went. We set a fixed focal length, and correct the longitudinal chromatic aberration, and construct three equations for the Petzval sum, and solve for the three values of power. The three equations are total power, Petzval sum, and longitudinal chromatic aberration. This sort of solution is quite simple for the modern computer. This is a useful method for choosing the position and the power of the relay lens for a zoom lens as well.
The only issue here is how to choose y2 and y3, namely the height of the marginal ray at the second and third lenses. Usually, a normal field of view, a slightly longer e1 compared to e2, or a slightly shorter e1 compared to e2 is preferred. If e2 is larger, the stop can be placed in between the negative lens and the last lens, and the back focal length can be longer, giving more room for the lens barrel.
We use these to set the focal length, the longitudinal chromatic aberration, the Petzval sum, the total thickness, and the ratio of the lens separation.
The three lens bending does not change the focal length, the longitudinal chromatic aberration, the Petzval sum, the total thickness, and the lens separation ratio. From the symmetry of the system, the lateral chromatic aberration, and the distortion we will assume is good enough for now due to the symmetry , and is corrected enough at this point to be able to go forward with the lens design. The symmetry of the system allows for correction of the off-axis aberrations.
The first lens is larger than the third lens and can correct the spherical aberration the most effectively. As a side note, this is true for most lens types except for retrofocus lenses. In the simulation, you can view the behavior and path of light rays after the reflection from the spherical mirror.
At any time, you can change the distance and size of the object along with the focal point of rays to create different scenarios. As soon as you change any parameter an immediate change in the behavior of light can be observed.
This software also provides options to simulate the focusing of light in ideal and actual scenarios to analyze the difference between both situations. Plus, some additional options including Show Image to view the point where image will be formed , Cental Ray to highlight the central ray , Parallel Ray to highlight the parallel ray with the ground , and Multiple Rays to view the path and behaviour of multiple rays at a time are also available.
Geometric Optics is a free web-based optics simulation software for Windows. It is also a Java based software that requires both Java and Adobe Flash player to work.
This simulator is used to create various scenarios in order to analyze and understand the behavior of light , refraction , and optics. In order to simulate the optics and light behavior, it provides an object and a convex lens in the simulation.
The rays of light that go from object to convex lens create an upside-down image of the object on the other side of the lens. You can change the position of the object to see the changed output image in the real time. There are more handy parameters that you can vary to make changes in the simulation like curvature radius , diameter , refractive index of the lens , marginal rays , principal rays , etc.
According to changes in parameters, you can observe the corresponding change in the output simulation. OpticalRayTracer is yet another free Java based optics simulation software for Windows. Using this simulator, you can easily simulate the optics and effects of optics of different lenses like convex , concave , flat , curve , etc.
In it, you can create combinations of various lenses and mirrors to create custom optics simulation. Plus, properties of the lenses and mirrors like radius , edge thickness , center thickness , position , curvature class , hyperbolic factor , etc.
Opto-Mechanical Systems Design. Optical System Design with the WinLens suite. Back to main navigation menu Back to start of content. Clip 1. Clip 2. Clip 3. Clip 4. We discuss the various tools that provide a paraxial and seidel based analysis of a lens system.
Clip 5. Clip 6. Clip 7. Clip 8. Clip 9. Clip All these quantities are easily analyzed and tied back to your key system performance metrics.
OSLO enables development of specialized design and analysis methods, without sacrificing any of the typical features for optimization, analysis and tolerancing that designers require to solve problems.
OSLO is the leader in flexibility for optical design programs. Text output functions have fully addressable data output cells that make extracting data trivial for analysis routines in the included scp macro language, ccl programming languages, dll implementation, or external interfacing through dynamic data exchange.
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It also restricts you to working with systems that have up to 10 surfaces. OSLO EDU gives you the basic ability to layout, edit, optimize, analyze, tolerance, and save a wide range of optical systems.
Feature Highlights. Compare Editions. See which edition meets your design needs. Current Releases. Go to the current release for downloading. Optical System Setup.
Set up an optical system for optimization or analysis. Surface Types. View the many types of optical surface supported by OSLO. User Interface. Discover the versatile user interface. OSLO offers many optimization methods to solve your design problem. Source and Illumination Analysis. Analyze image quality with point sources, simulate real sources, or calculate vignetting. Import files from other optical design software and export CAD files. Advanced Features. Advanced features are available to design unusual optical systems.
OSLO has powerful features for designing zoom or other multi-configuration systems. Complete and thorough tolerance analysis ensures your design can be built cost-effectively.
Standard Analysis. Standard analysis that every optical designer needs is available here. Many examples for getting started in designing with OSLO and writing macro programs are provided. Detailed tutorials are available to get you up to speed.
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