Student Preferences of Ray Tracing Methods for Understanding Geometrical Optics

Transcription

1 Student Preferences of Ray Tracing Methods for Understanding Geometrical Optics Samuel Steven, Sarah Walters, and Anthony Yee, University of Rochester Introduction Optical engineering students at the University of Rochester are taught ray tracing, calculating the path of light rays through an optical system, in the undergraduate Geometrical Optics course (OPT 241). The method traditionally used to teach ray tracing employs what is known as a Brick Diagram 1. This visual tool is a table with a column representing each surface in the optical system. One can put in known parameters for each surface, calculate initial quantities in the ray tracing equations, and quickly calculate the path of light through the optical system by simple mathematical operations between quantities in the table (see appendix). Alternatively, the student can choose to recursively apply the ray trace equations to each surface by hand without using the visual aid. The ray tracing equations can also be represented in matrix form. The matrix method is more practical in today s world as computer based ray-trace programs, including optical design software, are programmed to trace rays using matrices 2,3. This method is the preferred teaching tool in the University s graduate geometrical optics course. Although linear algebra is not a prerequisite for the undergraduate Geometrical Optics course, only basic matrix operations are needed to calculate the path of light rays through the system. However, matrix math can be daunting for students that are unfamiliar with it 4. In addition, students varied learning styles must be taken into account. The Brick Diagram is conducive to visual and sequential learners, whereas matrices are more useful for global and intuitive learners 5,6.

2 The aim of this research is to determine whether matrices should be taught as the primary method for tracing rays. In the Fall 2012 course, the matrix method was incorporated into the curriculum, but the Brick Diagram was still taught as the primary method for ray tracing. This research focuses specifically on determining the students comfort level with basic matrix math, their understanding of using matrices for ray tracing problems, and whether the students feel they could achieve the same level of understanding Geometrical Optics if they had learned how to ray trace using only the matrix method. Ultimately, we suggest based on our findings whether the matrix method of ray tracing should supplant the Brick Diagram as the primary way of teaching ray tracing in the undergraduate Geometrical Optics course. Methodology The subjects used for our study were 40 students in the Fall 2012 Geometrical Optics course. The majority of students were sophomores, although a few freshman were taking the course a year early, and a small number of upperclassmen were taking the course as an engineering elective. We gathered data by giving two sets of surveys and a ray tracing problem which required the students to use only the matrix method. Figure 1. Outline of information gathered through two surveys and a ray tracing problem.

3 The first survey, which collected information about students background and comfort level with matrix math, was given at the beginning of the semester when the students had just been introduced to the matrix method. This allowed us to understand their relative outlook toward matrices in general (see Figure 1). The second survey was given at the end of the course after the students had used the brick diagrams extensively but also had some practice with the matrix method, primarily in workshops. This survey focused on obtaining the students preferences for a ray tracing method (i.e. when presented with a ray trace problem, which method did they feel most comfortable using) as well as determining which method most benefited their understanding of geometrical optics principles. This survey also contained a ray tracing problem which had to be solved using the matrix method, which revealed students comfort level as well as their ability to use the matrix method. Through analysis of the students responses, we were able to connect the two surveys together to evaluate their full understanding of matrices. Results An important investigation for the analysis of our question was the amount of experience students had with matrix math and how comfortable they felt with matrix math. After looking at the results of the students comfort level along with their experience in matrix math we could see that there was a correlation between the students comfort level and their previous math experience (see Figure 2). The graph on the left shows the different math experience for the 29 students who felt comfortable using matrices and the graph on the right shows the varying matrix experience for the 11 students who didn t feel comfortable using matrices.

4 Students' Matrix Math Exposure for high comfort level Students' Matrix Math Exposure for low comfort level At Least College High School No Experience Limited College and/or HS Experience No Experience (a) (b) Figure 2. Students Matrix Math Exposure for students who reported (a) a high comfort level of matrix math and (b) students who reported a low comfort level of matrix math. Limited college and/or high school experience indicates that the student was introduced to the basics of matrix math whereas an exposure in at least college specifies that the student has taken a linear algebra course. Those with limited college and/or high school experience were only introduced to the basics of matrices and thus it makes sense that they felt less comfortable using the matrix method. Most preferred method Least preferred method 5% 27% 39% Ray Tracing Brick Diagram 73% 56% Matrix Method Design Software Figure 3. Students preferred method and least preferred method to understanding a ray tracing problem. The students reported using the brick diagram and ray tracing by hand as their preferred methods and the matrix method and the optical design software their least preferred methods with regard to helping them understand geometrical ray tracing (see Figure 3). This is not

5 surprising since the brick diagram is the emphasized method in the undergraduate course. What is unanticipated is that the matrix method is the least preferred method over the design software. This can be correlated to the fact that the design software had several workshops using the design software and was a method the class had recently learned. After noting the students lack of preference for the matrix method, the results of questioning them about the potential of the matrix method in helping them understand geometrical optics was surprising (see Figure 4). 52% of students conveyed that given more instruction and practice in the matrix math they would be able to achieve a similar understanding of geometrical optics without learning the brick diagram, 35% thought they maybe could, and 13% thought they wouldn t be able to Potential of the Matrix Method for Understanding Geometrical Optics 52% 35% % Yes Maybe No Figure 4. Students opinion on whether or not they would be able to understand geometrical optics only learning the matrix method. This quite clearly shows that students believe that they could learn geometrical optics in only learning the matrix method for ray tracing, which shows that perhaps not only the students comfort level, but also the amount the material that is focused on in class has an effect on the students preference of ray tracing method. Combining all of these results together, we see that

6 students in the class have experience with matrices, and whether or not they feel comfortable with matrix math, many students believe they could learn geometrical optics using matrices. Conclusions The majority of students felt that they would be able to sufficiently understand geometrical optics if taught only the matrix method. Nonetheless, a non-negligible amount of students were not confident that they would reach the same level of understanding geometrical optics. Students preferred ray trace method did not tend to correlate with previous education in matrix math, so requiring a linear algebra course as a prerequisite would likely not increase students preference for the matrix method. Therefore, we conclude that the use of Brick Diagrams to teach ray tracing should be continued; however, the matrix method should be taught alongside this method and given more emphasis in order to expand students understanding of geometrical optics by using both methods. Further research could be carried out to test our hypothesis that the matrix method should be given more emphasis in the undergraduate Geometrical Optics course. In the future, the course could be taught with a greater emphasis on the matrix method and we could survey the class similarly, noting if the students find the matrix method useful and voluntarily use it to solve ray tracing problems. The results of this analysis would provide further insight into the matrix method as a useful tool for teaching undergraduate level ray tracing.

7 References 1. Bentley, J. (2010, February 4). The Brick Diagram [PDF document]. 2. Almeida, J. (2008, February 2). Programming matrix optics into Mathematica. At URL: 3. Ibrahim, D. (2011). Engineering simulation with MATLAB: improving teaching and learning effectiveness. Procedia Computer Science, 3, Firouzian, S., Ismail, Z., Rahman, R. A., & Yusof, Y. M. (2012). Mathematical Learning of Engineering Undergraduates. Procedia-Social and Behavioral Sciences, 56, Felder, R. M., & Soloman, B. A. (2000). Learning styles and strategies. At URL: 6. Heyen, G., & Kalitventzeff, B. (1999). Spreadsheet based teaching aids in chemical engineering education. Computers & Chemical Engineering, 23, S629-S632.

8 Appendix Example Brick Diagram

9 OPT 241 Survey Name: Class Year: Intended Major(s): Question 1 What is your primary reason for taking OPT 241? Question 2 a) Required for major b) Required for minor c) Required for cluster d) Other (please explain): List any other optics classes that you have taken so far (e.g. OPT 101): Question 3 Which of the following best describes your use of matrices in high school classes? Question 4 a) I did not learn matrix math in high school b) I learned basic matrix operations in high school classes c) I learned basic matrix math for a standardized test, but not formally in high school classes d) I learned more advanced matrix math in high school (i.e. beyond basic operations such as multiplication and addition) What college classes, if any, have you taken/are taking that incorporate the use of matrices? Examples include MTH 165 and CSC 160 (or another Matlab course) Question 5 On a scale of 1 to 5, rate your comfort level using matrices: Not at all comfortable Somewhat comfortable Very comfortable

10 Question 1 OPT 241 Survey II After learning several methods of solving geometrical optics problems using ray tracing, which method do you feel most comfortable using? e) Ray trace equations by hand f) Matrices g) Brick Diagram Why is this your preferred method (i.e. easiest to understand, fastest, etc.)? Question 2 Rank the following methods in the order in which they helped you understand geometrical ray tracing (1 the most helpful, 4 the least): Question 3 Ray trace equations by hand Matrices Brick Diagram Code V What best describes the role of using Code V as a tool for aiding your understanding of geometrical optics? e) It was a useful tool that benefited my understanding of ray tracing principles f) It was fun/cool to use, but didn t really benefit my understanding of ray tracing principles g) It had a negative effect Comment: Question 4 Consider the following lens system in air: Surface # Radius (mm) Thickness (mm) Index of refraction For a 20 mm diameter object located 50 mm to the left of the 1st surface, use the Matrix method to find the paraxial image location, the paraxial image height and the magnification of the system.

11 n n y u u y S S 1 -S k 0 S k+1 Given a ray height (y) and index/angle product (nu) at some arbitrary initial surface (S 0 ) the ray's height (y') and index/angle product (n'u') at some other arbitrary surface (S k+1 ) can be calculated as follows (Note: S 0 and S k+1 are not necessarily the object and image surfaces!): y' y Tk RkTk 1Rk 1 R2T1 R1T 0 n' u ' nu y ' A B y n' u ' C D nu R j 1 0 j 1 T j t j ' 1 n j ' 0 1 (hint: E = , F = , G = , H = ) Question 5 Would you feel confident solving the above problem with a Brick Diagram? Given more instruction and practice in the matrix method, do you feel that you could achieve the same understanding of geometrical optics without learning the Brick Diagram method? Explain.