June 14, 2015
June 14, 2015
June 17, 2015
Computers in Education
26.1683.1 - 26.1683.11
Using Student Knowledge of Linear Systems Theory to Facilitate the Learning of Optical EngineeringExtended AbstractThis paper will present a highly effective method of teaching a subject (optical engineering) whichis new to students from various engineering disciplines. In particular, this method leverages exist-ing student knowledge of linear systems theory to facilitate their learning of this new subject morequickly and intuitively, and makes extensive use of MATLAB plots and simulations as a primarytool. The speciﬁc challenge was the need for students from various engineering disciplines to de-velop, in only a single course, a practical working knowledge of optical engineering to supporttheir research efforts. That is, many research projects were relying on digital cameras and otherimaging systems to obtain critical data, yet the students had no background in optical engineering.Therefore, the ability to design an appropriate imaging setup, or to know what limitations shouldbe taken into account when interpreting image data from existing setups, was completely lacking.Without any background in optical engineering, common errors or misconceptions could result inthe students’ research data being tainted or even useless.In order to optimize the learning environment for engineering students, the various theories ofhow best to teach adults (andragogy) can be studied and taken into account.1 One apropos an-dragological learning theory is that of constructionism, which builds on Piaget’s highly regardedepistemological theory of constructivism.2–4 The most salient aspect of constructionism that ap-plies to the subject of this paper was well described by Ausubel in reference 5: “If I had to reduce all educational psychology to just one principle, I would say this: The most important single factor inﬂuencing learning is what the learner already knows. Ascertain this and teach him/her accordingly” (p. iv).That is, how easily our students can learn a new topic depends to a large degree on their priorknowledge, or what might be called their existing cognitive frameworks. This comes as no surpriseto most experienced professors, but it can be reassuring to know that the method has a strongfoundation in educational psychology.The students who needed this introduction to optical engineering included those from electrical,computer, mechanical, civil, and chemical engineering. Some had been brieﬂy exposed to tradi-tional ﬁrst-order optics in a sophomore physics class, but most had not. However, all the studentshad a background in linear systems theory that included convolution, impulse response, transferfunction, Fourier transforms, and so forth, whether applied to electrical circuits, mechanical sys-tems, or chemical processes. They also had experience with MATLAB to varying degrees. Thatwas an existing cognitive framework that could be leveraged for more effective learning.The authors have many years of experience taking advantage of existing cognitive frameworks, ac-tive learning methods, and experiential exercises using both MATLAB and C to teach digital signalprocessing topics to students whose background was often just a Signals and Systems course.6–12Signals and Systems is the electrical and computer engineering focused version of linear systemstheory. Leveraging more general linear systems theory to teach a much broader audience of stu-dents the topic of optical engineering was a new challenge.The key, as described above, is to make the best use of what the learner already knows. Ratherthan taking the more traditional (and lengthy) route of ﬁrst teaching theoretical optics (ﬁrst-orderray tracing, interference, Rayleigh-Sommerfeld diffraction, Fresnel diffraction, Fraunhofer diffrac-tion, third-order aberrations, higher-order aberrations, etc.) then following that with application-oriented optical engineering, we chose to go directly to practical optical engineering by way oflinear systems theory and what is often called Fourier optics.13 That is, we quickly made the linkof what they already knew as an impulse response to an optical point spread function (PSF). TheFourier transform of the impulse response is the transfer function, and the Fourier transform of thePSF is likewise the optical transfer function (OTF), the magnitude of which is the very importantmodulation transfer function (MTF). The issue of aberrations can be presented as simple phasedeviations using MATLAB scripts that apply Zernike polynomials to the OTF phase term. Add inan algebraic treatment of concepts such as aperture, sensor, and pixel size, depth of ﬁeld, ﬁeld ofview, reﬂection, refraction, etc., and the students quickly have a very practical working knowledgeof optical engineering (via a single course) that is sufﬁcient to support their research efforts.The full paper will provide more complete illustrations of how these topics are taught, with ex-ample ﬁgures and MATLAB techniques, along with student feedback. As an example of howMATLAB helps facilitate the course, the Fourier transform pair relationship between the PSF andMTF is shown in Fig. 1. A comparison of the PSF and MTF before and after aberrations as simu-lated with MATLAB is shown in Fig. 2.The course that we created, using these methods, has been taught twice so far to students from elec-trical, computer, mechanical, civil, and chemical engineering. A textbook was written expressly tosupport this course, and will be published soon.References  H. Gardner, The Development and Education of the Mind: The Selected Works of Howard Gardner. World Library of Educationalists, New York: Routledge, 2006.  J. Piaget, The Psychology of Intelligence. New York: Routledge, 1950.  I. Harel and S. Papert, Constructionism. New York: Ablex Publishing, 1991. 1 0.9 1 0.8 0.8 0.7 0.6 0.6 PSF PSF 0.4 0.5 0.2 0.4 0.3 0 0.2 2 0 2 0.1 0 −2 −2 0 0 1 2 3 λ /D 1 0.9 1 0.8 0.8 0.7 mag OTF 0.6 0.6 mag OTF 0.4 0.5 0.2 0.4 0.3 0 1 0.2 1 0 0.1 0 0 −1 −1 0 0.5 1 D/λFigure 1: The PSF (top) and MTF (bottom) from an optical system with a circular aperture. no aberration with aberration 340 340 330 330 320 320 310 310 320 340 320 340 PSF PSF 600 600 400 400 200 200 200 400 600 200 400 600 MTF MTFFigure 2: Comparison contour plots of the PSFs and MTFs without and with a combination ofcoma in x and astigmatism in y. Note that to show sufﬁcient detail, the PSF plots (top row) are“zoomed in” compared to the scale shown for the MTF plots (bottom row).  M. Cakir, “Constructivist approaches to learning in science and their implications for science peda- gogy: A literature review,” International Journal of Environmental & Science Education, vol. 3, no. 4, pp. 193–206, 2008.  D. P. Ausubel, J. D. Novak, and H. Hanesian, Educational Psychology: A Cognitive View. Geneva, IL (USA): Holt McDougal, 2nd ed., 1978.  C. H. G. Wright and T. B. Welch, “Teaching real-world DSP using M ATLAB,” ASEE Comput. Educ. J., pp. 1–5, January–March 1999.  T. B. Welch, M. G. Morrow, and C. H. G. Wright, “Teaching practical hands-on DSP with MATLAB and the C31 DSK,” in Proceedings of the 2000 ASEE Annual Conference, June 2000. Paper 1320-03.  T. B. Welch, C. H. G. Wright, and M. G. Morrow, “Poles and zeroes and M ATLAB, oh my!,” ASEE Comput. Educ. J., pp. 70–72, April–June 2000.  C. H. G. Wright, T. B. Welch, D. M. Etter, and M. G. Morrow, “Teaching DSP: Bridging the gap from theory to real-time hardware,” ASEE Comput. Educ. J., pp. 14–26, July–September 2003. C. H. G. Wright, M. G. Morrow, M. C. Allie, and T. B. Welch, “Using real-time DSP to enhance student retention and engineering outreach efforts,” ASEE Comput. Educ. J., pp. 64–73, October–December 2008. T. B. Welch, C. H. G. Wright, and M. G. Morrow, “The DSP of money,” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing, pp. 2309–2312, Apr. 2009. T. B. Welch, C. H. G. Wright, and M. G. Morrow, Real-Time Digital Signal Processing: From MAT- LAB to C with C6x DSPs. Boca Raton, FL (USA): CRC Press, 2nd ed., 2012. J. W. Goodman, Introduction to Fourier Optics. New York: McGraw-Hill, 2nd ed., 1996.
Wright, C. H. G., & Welch, T. B., & Morrow, M. G. (2015, June), Using Student Knowledge of Linear Systems Theory to Facilitate the Learning of Optical Engineering Paper presented at 2015 ASEE Annual Conference & Exposition, Seattle, Washington. 10.18260/p.25019
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