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ECEN 4616/5616 Optoelectronic Design Class website with past lectures, various files, and assignments: http://ecee.colorado.edu/ecen4616/Spring2014/ To view video recordings of past lectures, go to: http://cuengineeringonline.colorado.edu and select “course login” from the upper right corner of the page. Lecture #33: 4/7/14
Complex Refractive Index It is customary to describe the absorbance of a material through the imaginary part of a complex index of refraction, If we use the definition of the wave vector as: Then the equation of a plane wave traveling down the Z-axis will be: So, the wave amplitude decays with the decay constant: And the (intensity) absorption coefficient is:
Zoom Lenses A “Zoom” lens is a lens whose focal length is variable. There are several sub-categories: 1.‘Varifocal’ lenses are those that have to be re-focused after changing the focal length. 2.‘Parfocal’ lenses maintain the image in focus as the focal length is changes. Parfocal lenses are harder to design, due to the extra constraints, and therefore were developed later. Varifocal lens designs (for telescopes) have been described since 1834. The first patented parfocal design was in 1902 (US Pat. #638,788). The zoom technique is similar to ones still used today. With the proliferation of auto-focus cameras, varifocal lenses are coming back in style. The relaxation of the need to maintain focus allows larger zoom ratios, and the camera’s autofocus makes the lens-camera system behave as if the lens were parfocal.
Zoom Lenses Another distinction between zoom lenses is between “Optical Compensation” and “Mechanical Compensation”. This is a somewhat arbitrary distinction: If the lens is zoomed by a single motion of a lens or group of lenses, it is called Optically Compensated. If multiple groups move in multiple directions or at multiple speeds, the lens is called Mechanically Compensated. Optically Compensated Zoom Lens: Two lens groups move as a single unit. Mechanically Compensated Zoom Lens: Lenses move in different directions and at different speeds.
Zoom Lenses Optical compensation requires only one linear movement be used to change the len’s focal length. This constraint makes the lens harder to design than if all lenses are allowed to move arbitrarily. Why bother designing an optically compensated system? One reason that is valid would be simplicity of the zoom mechanism, but the original reason was that it was thought that the pin and slot cam system used for mechanically compensated designs would eventually wear and the movements would become inexact. A high-ratio zoom lens with three moving groups: The motion of the three groups: The pin, slot, and cam system that accomplishes the motions: In fact, cam systems are perfectly reliable and are universally used today.
Zoom Lenses A closer look at the cam system: The three lens groups (center row) slide within a sleeve (bottom). The lens mounts have cam levers that protrude through slots in the sleeve that allow them to slide back and forth over the required range, while maintaining axial alignment. The lens+sleeve assembly fits into a housing (top) with curved slots that engage the cam levers. When the sleeve is rotated w.r.t. the housing, the lens groups are driven in a pattern determined by the shape of the curved slots. The assembled lens looks like this:
Lesson for Engineers (from the Optical/Mechanical Zoom issue) Why did optical designers decide that cam systems would be unreliable? This was outside their area of expertise, and in fact they were mistaken as virtually every zoom lens today uses cams to drive diverse motions of the elements. The time and effort wasted spent trying to avoid cam systems in zoom lenses was completely unnecessary, had the optical designers simply consulted with the appropriate mechanical engineers. This is a problem that occurs all the time, at a unknown but high cost to companies and society. A good example is the development of the world’s most accurate gravitometer – an instrument which promises to give us insight (and perhaps images) into the Earth’s interior. The instrument in question was developed in Boulder at the Joint Institute for Laboratory Astrophysics (JILA): The instrument measures the acceleration of gravity to an accuracy of 2 µGal → 20 µm/s 2, and a repeatability of 1 µGal. This is ~ 10 -12 of the Earth’s gravity field.
The Drop Gravitometer This is one of the most sensitive physical measurements ever made. The instrument can detect the change if: 1.You raise or lower it by 1 cm. 2.There are two people in the upstairs office instead of one. 3.There is snow on the roof or not. In fact, measurements of gravity at a fixed location at this level of precision show nearly continuous changes. The largest are caused by tidal forces – primarily the Moon and Sun positions. Many of the other changes can be correlated with the weather (high pressure means more mass of air overhead to counteract the Earth’s gravity). There are other, slow changes, however, for which no explanation is known for sure. One possibility is that they measure slow convection currents in the Earth’s mantle moving different densities of molten rock around. One common use of these instruments is, of course, the location of underground reservoirs of oil and gas. With enough long term measurements from enough places around the Earth, we may eventually be able to map and ‘image’ the convection currents in the interior of the Earth.
The Drop Gravitometer So, how does this amazing instrument work? Naturally, it’s an electro-optical device:
The Drop Gravitometer (What was the Problem?) A falling mirror (actually a corner-cube retro-reflector) is measured via an interferometer as it falls in an evacuated chamber. The measurement is so potentially sensitive that even the residual air friction from a normal hard vacuum was enough to corrupt the data. When the vacuum was reduced to the level that potentially the measurement wouldn’t be affected, the moving parts began to vacuum weld themselves together: Vacuum (or cold) welding is the inverse process of fracturing; Just as a single piece of material (metal, say) can fracture into two separate pieces, two pieces of compatible materials can bond into a single piece – if they are brought together without other materials intervening. This is what can happen in an ultra-high vacuum environment. In air, two pieces of metal will have oils or other materials (even air molecules) adhering to the surfaces. These will eventually sublimate away in a high vacuum, leaving the metallic surface bare. When two such metal surfaces come together, they can bond into a single piece.
The Drop Gravitometer (What was the Solution?) The scientists building the prototype device (none of whom were mechanical engineers) considered putting the reflector cube in a glass cage which would be moved by motors to track the fall, thus isolating the cube from the “wind” of moving in the vacuum chamber. They decided that this would be prohibitively difficult to do. Instead, they spent a year trying to solve the cold welding problem, but failing. Thus, their problem was solved in a few days for a few hundred dollars (plus the engineer’s fee). Finally, as a last resort, they brought in a consulting mechanical engineer who told them that the linear motors and controllers used on 10s of thousands of automated assembly lines world-wide were easily capable of performing the required motions (plus extra features, like sharp takeoffs and soft landings). Nobody is expert in everything – recognize your limitations and seek help when appropriate.
Zoom Lens Systems The simplest (but still useful) zoom system is suggested by the combination of powers formula for lenses: By changing the distance between two lenses, we can change the combination power from the sum of the two to zero. Here is a pair of 40 mm fl paraxial lenses in three configurations: fl: 33mm fl: 27mm fl: 21mm Lens 1 Motion Lens 2 Motion In order to make this zoom system telecentric in image space, the stop also moves. With two positive lenses, the longer focal length is limited by the back lens reaching the focal plane.
Zoom Lens Systems It can be hard to come up with good starting systems for zoom lenses. One solution is to look through the patent literature, or other databases of lens designs:
Zoom Lens Systems For paraxial zoom models, it is possible to derive a set of algebraic equations (from the lens combination formula) to allow selection of zoom lens starting points. For example, for the two lens zoom, say we want the focal length of the combination to vary from 10mm to 30mm as the separation between lenses is changed from 5mm to 25mm. The we solve the set of equations: i.e., one lens has a focal length of 20 mm and the other has a focal length of 15 mm, in either order. The journal literature contains numerous examples of this technique carried out to systems of 3 and 4 lenses, with a number of constraints such as constant distance between object and image, and corrected Seidel aberrations: 1)Method of zoom lens designMethod of zoom lens design 2)Paraxial analysis of four-component zoom lens with fixed distance between focal pointsParaxial analysis of four-component zoom lens with fixed distance between focal points 3)Mechanically compensated zoom lenses with a single moving elementMechanically compensated zoom lenses with a single moving element
Zoom Lens Systems A good way to design a zoom system is by combining an “Afocal Zoom” (essentially, a variable power telescope) with a fixed lens. One of the simplest afocal zoom systems is a triplet, where the total power is zero: Afocal Zoom: L2 changes the telescopic power, L1 moves slightly to maintain the afocal condition. An afocal zoom added to any fixed lens produces a zoom lens with a fixed BFL. The actual focal lengths of the zoom are the power change of the afocal system times the focal length of the fixed lens.
Modifying a Sample Zoom System There is a zoom lens system in Zemax’s “Samples/Sequential/Zoom Systems” directory: This system has 6 lenses, two of which are doublets, but is the functional equivalent of the two-lens zoom system first shown, since two groups of 3 lenses move with respect to each other, just as our two paraxial lenses did.
Example Zoom You would think that 2 singlets and 1 doublet should be able to make a good replacement for each paraxial lens, but looking at the LDE, we see that three of the surfaces in the system are also aspheric. The Even Asphere surface is defined as the Sag equation for the conic sections plus a polynomial in even powers of the distance from the Z-axis, ‘r’.
Example Zoom We expected this lens to work well, but inspection of the “Configuration Matrix” spot diagram shows something appears to be wrong with Configuration 3: The ‘Through-Focus” spot diagram for configuration #3 indeed shows that the system is out of focus by ½ mm in configuration #3.
Example Zoom The Multi-Configuration Editor (MCE) controls the spacing between the two lens groups and the distance to the image plane for the 3 zoom positions: We will use the “Quick Focus” tool, once while in every configuration: All three thicknesses change.
Example Zoom The Through-Focus spot diagram now looks much better. The MTFs for Configurations 2 and 3 look OK, but something is going wrong with configuration #1:
Example Zoom The spot diagram for configuration #1, field 3 (the off axis field) shows a significant amount of Lateral Color aberration (different magnification for different wavelengths). This is confirmed by the “Lateral Color” analysis window under “Analysis- Miscellaneous-Lateral Color”:
Example Zoom Given that this lens is simply an elaboration of the two-element zoom; has 6 elements including 2 doublets; and has 3 aspheric surfaces; the performance is not impressive, even after we re-focused it. We will try to optimize this lens (and remove the aspheric surfaces while doing it): The first thing we do is to tidy up the focal lengths. We set EFFL commands in the MFE to get the configurations to 30, 50, & 70 mm focal lengths: Make the BFL and the spacing between lens groups variable:
Example Zoom After a short optimization, the Configurations now represent 30, 50, & 70 mm fl: While not absolutely necessary, this was an easy first step.
Example Zoom Simply thowing away the aspheric coefficients (by converting the surfaces to “Standard” type), results in a serious loss of performance: Trying to recover by optimizing the curvatures of the three converted surfaces gets some performance back, and results in a strange system where the entire zoom range is achieved with a 2 mm shift between lens groups. This is interesting, at the least, so let’s find out what we can get by optimizing the rest of the curvatures. (This kind of drastic action will often result in errors in the Merit Function (TIR, missed surfaces, etc.) A slower, but safer method is to gradually reduce the values of the aspheric components while re-optimizing the suface curvatures.)
Example Zoom After a rather lengthy optimization (there are now 20 variables), we get back to a system at least as good as the original, but without the aspheric surfaces: We still have a Lateral Color problem in Configuration #1 (30mm fl), however, so let’s adjust the merit function and try to fix it:
Example Zoom We add “LACL” (Lateral Color) operands to the merit function: As expected, Lateral Color is worst for Configuration #1: Why do we use wavelengths 2 & 3 in the LACL parameters? Because the red and blue wavelengths have the greatest spread in the analysis window: And those are # 2 & 3 in the wavelength data window.
Example Zoom Another lengthy optimization solves the Lateral Color Problem: But creates other problems – overlapping lens edges and a negative focal distance in Configuration #1:
Example Zoom The Default Merit Function is modified to include glass and air thickness constraints, as well as increasing it’s weight to 1000 (to encourage moving to a realistic system) (1) All the thicknesses are made variable as well. We now have 32 variables. (1) The specific values used in the Default Merit Function are the result of several iterations of trial and error – these parameters got to the best system.
Example Zoom After optimizing, all three zoom configurations are diffraction limited: We have made the beginning system much better, and dumped the aspheric surfaces as well.
Example Zoom A note about long optimizations with many variables When optimizing in a high-dimensional space, the optimizer often ‘plateaus’ – that is, stops improving for a while. This causes the “Automatic” optimization choice to stop. If you pick “Inf Cycles”, however, the optimizer will continue and often reach the edge of the plateau and then continue improving at a good rate. Selecting the “Auto Update” box will slow the optimizer down by a large amount, as it has to update every analysis window you have open every cycle. A better strategy is to leave the box unchecked, so the optimizer goes at maximum speed and check the box periodically (while the optimizer is running) to see how things are progressing.
Aspheric Lenses Why you probably shouldn’t use aspheres: The mathematical formula used for aspheric surfaces has a polynomial in r 2 added to the conic surface formula. As we have seen, almost any combination of curvature, c, and conic constant, k, will result in a “lens-like” surface. However, only an extremely small volume of the α n (polynomial) parameter space can be described as “lens-like”. (Most polynomials will “wiggle” – or diverge -- in a most un-lens-like manner.) This gives the optimizer a nearly impossible task – find a tiny spot in the parameter space representing the solution surrounded by a vast multi-dimensional volume of extremely bad configurations. The successful use of aspheres requires a great deal of innovative “Merit Function Engineering”. It is usually much easier to use spline curves, either by using Zemax’s spline surfaces, or by controlling Zemax with Matlab and fitting the test spline surface to an aspheric polynomial before entering it into Zemax.