Mathcad Ray Tracing Programming
Blue- ray Disc Clip Information file. Color binary screen image (1st Reader)Color definitions (Photostyler). C class definition file. Ray Tracing C. Question about Ray Tracing and Matrices. (self.GraphicsProgramming) submitted 12 months ago by CokeCoding. Hello everyone, hope this is the right place to post this question. I'm building my first and very simple ray tracing program and I'm facing problems in comprehending transformations (matrices multiplication).
. What it is: A sophisticated, cross-platform virtual optical bench.
Written in: Java. Works with: Windows, Linux, Macintosh, etc.OpticalRayTracer is a free cross-platform application that analyzes systems of lenses and mirrors.
It uses optical principles and a virtual optical bench to predict the behavior of many kinds of ordinary and exotic lens types as well as flat and curved mirrors. OpticalRayTracer includes an advanced, easy-to-use interface that allows the user to rearrange the optical configuration by dragging objects around using the mouse.OpticalRayTracer fully analyzes lens optical properties, incuding refraction and dispersion. The dispersion display uses color-coded light beams to simplify interpretation of the results.Recent OpticalRayTracer versions allow the creation of mirrors, flat and curved. In modern optical designs, mirrors often produce better results than lenses, for example in astronomical instruments. Such instruments can be roughed out in OpticalRayTracer's virtual workbench.Educators take note: OpticalRayTracer has significant educational potential in the teaching of basic optical principles, and has some entertaining and game-like behaviors to hold the student's attention.OpticalRayTracer includes a detailed tutorial/help file to assist the user in getting started in this interesting activity, and is also available. OpticalRayTracer is © Copyright 2017,. All rights reserved.OpticalRayTracer is released under the.OpticalRayTracer is also Careware ,unless this kind of idea makes you crazy, in which case OpticalRayTracer is free (e.g.
OpticalRayTracer is a Java application, so it requires a Java runtime engine in order to run. To pick up your free Java runtime. Version 9.6. Updated the bundled help document to emphasize the presence of the program's right-click context menu that includes many important control functions. Version 9.5. Fixed some cross-platform configuration file read/write issues (configurations can now be seamlessly transferred between platforms). Version 9.4.
Changed Windows launch procedure after discovering that Microsoft has abandoned support for Java. Version 9.3. Fixed a serious bug that would freeze the application if the same number were entered for both 'X Source Plane' and 'X Beam Rotation Plane'. Now, instead of freezing, the application rings a bell and waits for the human to sort it out.
Version 9.2. Fixed a subtle bug that prevented timely optical component updates after certain program parameters were changed, added a reset button that resets only program values, not optical components. Version 9.1. By user request, added a simple command-line interface to support test automation methods. Version 9.0.
Trapped an on-axis overflow error condition in the planar curvature class. Version 8.9. Changed the control layout, revised the help file for greater clarity, added a graphic image that shows how a lens is defined by overlapping spheres. Version 8.8. Recoded the management of the planar curvature class, added element center thickness to the configuration class (meaning it's now included in clipboard data and the initialization file), edited the help file to reflect recent changes and to simplify the dispersion experiment. Version 8.7. Added a display of element-center thickness, corrected an inconsistency in how element edge thickness was specified.
Version 8.6. Cleaned up on-axis solution code, solely to avoid ugly code, no change in behavior.
Version 8.5. Made further changes to address more on-axis edge cases. Version 8.4. Coded a more efficient on-axis overflow method, tuned for each element class. Added 'Common Problems' section to help file. Version 8.3.
Fixed a bug in the parabolic curvature class that created a non-physical on-axis deflection caused by floating-point overflow. Version 8.2.
Added a planar curvature class to avoid problems associated with using large spherical radii to imitate planar surfaces. Version 8.1. Fixed a bug in the spherical lens model that failed to detect a small subset of ray-surface intersections, changed how minimum lens thickness is computed to prevent an optical problem, updated the help file content. Version 8.0. Recoded much of the basic ray tracing engine to use vectors instead of scalars, with a big improvement in stability and accuracy. A number of difficult edge cases are solved by this change.
Version 7.9. Finally sorted out the Snell's Law lens refraction computation issues. OpticalRaytracer's refraction results are now as precise as entered lens data allow.
Version 7.8. Revised the Snell's Law code block to make displayed results more closely approximate exact Snell's Law computations. The Snell's Law computations are still approximate, but the outcomes are now closer to reality. Version 7.7. Fixed a subtle error in how a certain kind of multiple lens refraction is tallied — how multiple transitions into and out of a single medium's acute surface are computed.
Version 7.6. Fixed a number of uncovered and reported bugs, made a few minor improvements, changed the lens surface detection method. Version 7.5. Fixed a number of reported bugs having to do with edge cases in the behavior of lenses, refactored the control manager class for better comprehensibility. Version 7.4.
Added parabolic curvature to the supported lens/mirror curvature types, changed the user interface, cleaned up and refactored much of the code structure to address the much larger code base recently created. Version 7.3.
Recoded the spherical and hyperbolic generators for more shared structure and code, improved the Python/SymPy function generators to make future changes easier. Version 7.2. Replaced the spherical model algorithm with one that's more efficient and that shares some mathematical and technical aspects in common with the hyperbolic model.
Resolved a reported locale issue. Version 7.1. Improved the algorithms responsible for managing surfaces to avoid some annoying and unphysical edge-case behaviors. Version 7.0. Fixed a bug related to parabolic curves and mirrors, fixed another that prevented correct operation at user-entered angles that were multiples of 90 degrees, changed to a more robust hyperbolic generation scheme.
Version 6.9. Solved some technical issues with rotating reflecting surfaces, explored the approximate parabolic curve option accessible by very large number entries to the hyperbolic curvature control. Version 6.8. Created a scheme to prevent the sides of lenses from crossing over, which can be very confusing as well as unphysical. Fixed a serious bug in how reflections were calculated. Version 6.7.
Fixed a bug that prevented creation of concave hyperbolic lenses. Version 6.6. Refactored the hyperbolic model for greater accuracy and efficiency, added fields to data table, added the ability to copy individual table lines to system clipboard, cleaned up small bugs in other areas. Version 6.5.
Recoded the spectral dispersion algorithm to agree more closely with the underlying physics. Version 6.4. Improved table formatting for both the internal and HTML exported tables, changed the table data format and layout.
Version 6.3. Replaced the HTML table display with one based on a Swing table component for much better update speed. Version 6.2. Replaced intersection dots with arrows, which tell the user the direction the beam is heading.
Reworked the dispersion computation to better reflect the underlying physics. Fixed a number of bugs that came up in testing. Version 6.1. Fixed a regression/bug, introduced in recent major code changes, that prevented creation of concave lenses. Version 6.0. Optimized a critical part of the hyperbolic model code, fixed a bug that affected treatment of a lens that's half spherical and half hyperbolic. Version 5.9.
Replaced the hyperbolic lens model with a much better one, easier to explain technically, and the hyperbolics can now be translated and rotated into any position. Refactored much of the code base for greater clarity and efficiency. Version 5.8. Added a few more controls, arranged for a child help window to be cloneable/detachable from the main app window for more convenient interactive browsing.
Version 5.7. Implemented keyboard controls so users without pointing devices can use the program. Using a mouse or other pointing device is much easier, but using only a keyboard is now possible.
Version 5.6. Fixed two issues, one having to do with the location of lenses and mirrors created from the context menu, the other having to do with the behavior of the hyperbolic lens profile algorithm.
Version 5.5. Fixed a number of small bugs, changed how optical objects are named to assure that new and copied objects have an automatically assigned unique name. Version 5.4. Changed the virtual space configuration to eliminate a barrier that no longer served a purpose, changed configuration controls, cleaned up some minor issues. Version 5.3. Fixed some Windows-related and locale-related issues, added a tab that displays a table of ray trace data, solved a keyboard function problem, made a number of other improvements.
Version 5.2. Fixed an issue with the new proximity detector, added an input for surface epsilon to allow users to fine-tune the ray tracer's response for special applications, added surface normal angle to the ray data table and dialog. Version 5.1. Replaced the lens proximity detector algorithm, responsible for deciding whether a light beam has intersected with a lens. The new method is much improved.
Version 5.0. Added absorbers (optical components that stop light rays) and a text naming feature for each optical component. Version 4.9. Changed the dispersion beam alpha value to reflect that of the default tracing beam alpha, edited the help file, made a number of other small changes. Version 4.8.
Solved a cross-platform plain-text format compatibility issue that came up when trying to share lens descriptions between Linux and Windows. Version 4.7.
Restored a documentation example that shows how to correct chromatic aberration, and changed how the image-to-clipboard function works — the new version preserves the aspect ratio of the original display while keeping a uniform width. Version 4.6.
Enabled alpha (transparency) setting for system color selections, fixed an inconsistency in the undo/redo system. Version 4.5. Replaced my original circle-line intersection algorithm with a faster, more efficient one. Version 4.4. Fixed a bug in the ray generating algorithm that prevented generating just one ray and that sometimes failed to generate the specified number of rays. Version 4.3. Added line information features.
Users can export a CSV-formatted table of information about all generated lines to the system clipboard, and can get details about a particular line by double-clicking the display near the line of interest. Version 4.2. Recoded the ray tracing logic to make lenses symmetrical, so that light can pass through in both directions. Version 4.1. Fixed a bug that prevented correct mathematical handling of off-axis lenses in some circumstances. Version 4.0. The version number jump results from a complete rewrite and change of development environment (i.e.
The new version has many new features and abilities — mirrors as well as lenses, off-axis light paths, improved import and export options, better user interface and controls, and many smaller improvements. Version 3.3. Fixed a bug that would freeze the appplication if the source Y start and end values were equal. Version 3.2. At the suggestion of a user, added an option to rotate light beams with respect to the X origin rather than the X target plane, increased the beam rotation rate of change when adjusted using the mouse wheel, fixed additional small bugs. Version 3.1. Fixed a library issue caused by combining the application/applet versions of this program into one JAR file.
Version 3.0. Rewrote this program so it can be run as a Java applet as well as an application — there is now a 'live' running copy of OpticalRayTracer at the top of this page. Version 2.9.
Fixed some small annoying bugs, updated help file. Version 2.8. Fixed a bug that made lenses twice as thick as the user specified. Version 2.7. Replaced string-based keyboard identifiers with keycodes for the benefit of non-English users. Version 2.6. Fixed a clipboard copy/paste bug that affected only Windows users.
Version 2.5. Fixed a number of annoying small bugs that cropped up in connection with recent changes. Version 2.4. Fixed a subtle bug that prevented correct tracing through very large lenses having slight curvature. For this particular activity users may want to delete the user configuration file located at (user directory)/.OpticalRayTracer/OpticalRayTracer.ini to permit lenses with thinner edges and/or centers. Version 2.3.
Fixed a bug brought on by the new locale-processing code. Version 2.2. Recoded this application to make it locale-immune. Version 2.1. Added some scaling options to the mouse/keyboard gesture recognizer.
Version 2.0. After years of dealing with essentially insoluble C installation issues, gave up and completely rewrote OpticalRayTracer in Java.
The new version is better than the old in every way, and it is platform-portable. Version 1.2. Updated source to allow compilation on 64-bit systems and with newer compilers. Version 1.1. Updated source to conform to gcc 4.1 requirements.
Version 1.0. Initial public release.
To understand particle accelerators and their components, we need to study and solve the equations of motion of charged particles in external electromagnetic fields. In their most general form, these equations contain terms involving all orders in the coordinates and velocities of the particles ,. However, if the particles are confined to small distances from a reference orbit and with small angles relative to the same orbit, a linear equation of motion resembling a simple pendulum (or mass-on-a-spring) equation results. This is the basis for the paraxial approximation presented in section. The 'spring constant' in the equation of motion can be generalized to include a piecewise focusing function whose form depends on the type of external fields present. In addition to the paraxial approximation, the focusing elements of particle accelerators can be treated as thin or thick lenses, depending on whether the focal length is large or comparable to the physical axial extent of the external fields.
We discuss thin and thick lenses in sections. For both thin and thick lenses, a matrix description of particle motion is both convenient and powerful. 1.1. Paraxial approximation. Where primes indicate derivatives with respect to the axial distance ' s' measured along the reference trajectory.
For now, we consider just one of the transverse coordinates for particle motion, say the horizontal component (or radial in a local curvilinear coordinate system).Note that equation , known as Hill's equation, is a homogeneous, second-order ODE with a coefficient, the focusing function, which is not constant in general. The solution will depend on the initial conditions, and on the form of, but not on any higher orders of either. This latter condition is the essence of the paraxial approximation; it is the equivalent of using θ instead of sin θ in Snell's law of standard optics.The general solution of equation is studied in great detail in many textbooks (e.g.
The solution can be written as a superposition of cosine-like C( s) and sine-like S( s) functions. The space spanned by the coordinates for an ensemble of particles is called trace space or, more commonly, phase space. Strictly speaking, this phase space would be the projection on the plane of the full phase space. In the simple situations whereby no radiation, acceleration, or particle losses occur, the area in phase space is conserved under transformations represented by M. The latter statement is known as Liouville's Theorem, which we will revisit in chapter.
As a consequence of this theorem, the determinant of M is equal to 1, i.e.The simplest case of a focusing function would be the 'point' lens, a mathematical construct whereby. Δ represents the Dirac delta function and f is a constant. By integration of equation with this focusing function, it is easy to see that f represents the focal length of a zero-length lens located at s = 0.
Figure illustrates the zero-length or point lens. If the extent of a real lens, as quantified by the effective length l eff of the profile of the focusing function, is much smaller than the focal length, then the lens is considered to be thin. (The point lens introduced above is an extreme example of a thin lens because it remains so for any f 0).
A useful construction that applies especially to lenses whose profiles are not dominated by fringe fields, i.e. Extended 'tails', is the hard-edge model. Figure shows an example of an actual axial profile (of a magnetic field) related to and the corresponding hard-edge model. Therefore, the condition for thin lens is.
By this criterion, a very strong lens can have a focal length comparable to its effective length, in which case the lens is not thin; otherwise, the lens can be weak and have a very long focal length, satisfying the condition of a thin lens.Ray tracing for a thin lens proceeds just as for the point lens (figure ), so, for example, the exit x-coordinate of a given ray is unchanged after traversing the lens. Thus, the basic transfer matrix that connects ray coordinates on both sides of the lens is the same as in equation. 1.3. Thick lens. Thick lenses in charged-particle optics can be treated just as thick lenses in standard optics. Concepts such as principal planes, nodal points, etc apply to magnetostatic or electrostatic lenses as well.
The principal planes of a thick lens or system of lenses are two hypothetical planes that connect locations of unit magnification, i.e. An object located at one plane is imaged at the second plane and with the same size.
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Figure illustrates the primary and secondary principal planes, H and, for a system of two focusing thin lenses separated by a distance d and having the same focal lengths. Ray tracing is done by using the method of parallel rays.
This quantity is positive, indicating a convergent equivalent thick lens for the two-lens system if f 1 d. Thus, we arrive at the important result that the combination of two lenses, one convergent (positive) and the other divergent (negative), and having the same strengths leads to net focusing if their separation is smaller than the positive focal length f 1. This has an important implication for strong focusing in particle accelerators, as discussed in chapter.Figure shows the action of the focusing–defocusing lens combination for two values of their separation d. The geometrical construction for finding the principal planes uses the method of parallel rays, as discussed in the standard optics book by Jenkins and White , but examination of the figures may reveal the basic ideas. Note that the principal planes lie outside the space between the lenses in figure (a), unlike figure for a pair of converging lenses with the same separation as in figure (a). Figure 1.4. Combination of convergent (lens 1) and divergent (lens 2) lenses with the same strengths.
In (a) the spacing d is smaller than the individual focal lengths ( f 1 = −f 2 0); in (b) the spacing d is larger than the individual focal lengths. Note how the principal planes H and H' are found from the parallel ray technique.Download figure:Additional relationships between the elements of the matrix in equation and important optical properties of a general lens system are discussed in the article by Halbach and in appendix D of. Computer resources.
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Matrix multiplication, both symbolically and numerically, can be readily implemented in either Mathcad or Matlab or their freeware counterparts Smath Studio and Octave (see appendix). Many standalone optics design programs are also available, but most are either too expensive or for professional use. Some software of this type is available as packages for codes like Mathematica (e.g. Optics Lab).We found the free program Optgeo (see appendix) to be instructive and relatively easy to use; in particular, the ray tracing in figures and can be reproduced without too much effort. We include two Optgeo files in the book's website.
Alternatively, for Mathcad users we recommend a very well designed program for 2D optics called 2D Optical Ray Tracer, which is written by Valter Kiisk of the University of Tartu (Estonia) and available online for free. We include in our website a version with input that reproduces closely the results shown in figure (a). It is important to keep the lens thicknesses small relative to their focal lengths for comparison with figure, but we can also explore easily thick-lens combinations.
Figure illustrates typical output. Figure 1.5. Graphics from 2D optics Mathcad program 2D Optical Ray Tracer by Valter Kiisk. Distances are in mm.Download figure:Another program that we found useful to illustrate the two-thin lens combination is Ray Optics for Android devices (there is a 'Pro' version that adds details of the calculations).
Figure shows a typical screen from Ray Optics. Note that the object is on the right and that ray tracing goes from right to left, unlike the rest of the figures in the chapter (however, object and image can be exchanged). Figure 1.6. Screen from the free Ray Optics app for the Android OS. Magnification is close to unity in the example shown.Download figure:RayLab, a free app for Apple's IOS devices, looks very professional and rather challenging as an educational tool for geometrical optics.Thin magnetostatic lenses can be implemented in several of the most popular codes for particle accelerator design. (In fact, in some cases thin lenses are required for some operations such as tracking in the code MAD).
We include in our website three example files for the program Winagile (see appendix). The first file illustrates the implementation of one thin quadrupole lens; the other two files are examples of two thin-quadrupole combinations. The files illustrate the concepts of this chapter and also serve as a preamble to quadrupole focusing (chapter ) and to alternating gradient, or strong focusing (chapter ). In the appendix we indicate how to download, install and run Winagile. In figure we show a screen from Winagile for trajectory tracking with a two thin-quadrupole combination.
Figure 1.7. Trajectory tracking in Winagile for a combination of two thin quadrupoles of opposite strengths. The quadrupoles are located at s = 20 and 25 cm, while their individual focal lengths are f 1 = −f 2 = 10 cm. See also chapters and for additional important considerations.Download figure:Finally, 'Thin Lens Simulation with Python' (Frant'z blog, April 2011), a Python program for ray tracing through a system of four lenses, can be used to learn both optics and Python (see appendix).