1

INTRODUCTION

In general, teaching has not changed over the centuries. Figure 1 shows a scene from a classroom from about 1350 CE; the image is a reproduction of a parchment painting rendered by the artist Laurentius de Voltolina. It depicts a university lecture during the fourteenth century at the University of Bologna in Italy.[1] It shows students talking to each other or falling asleep while the teacher lectures. This scene depicted will be very familiar to those of us who teach/lecture. John Amos Comenius (1592-1670) who was a Czech teacher, writes in the frontispiece to the Didactica Magna (1628) “Let the long and the short of our didactic be to investigate and discover the means for teachers to teach less, and learners to learn more.”[2] According to Albert Camus, “Some people talk in their sleep, lecturers talk while other people sleep.”[3] So, the teacher acts as a presenter of facts and the students are passive absorbers of this knowledge.

Figure 1:

Henry of Germany lecturing at the University of Bologna. Painting by Laurentius de Voltolina (1350).

There is considerable evidence that lectures are ineffective in teaching STEM (Science, Technology, Engineering, Mathematics) subjects. Numerous studies have examined the effectiveness of active learning.[5] In a recent meta- analysis of undergraduate STEM education, Freeman et al. [4] found that the failure rate was 55% higher in traditional lecturing method than in the active learning modes. One of the methods to incorporate active learning in the classical lecture format is the interactive lecture demonstration.[6] The other ways of active learning include visual learning, problem solving or problem based learning, cooperative learning, role playing, peer education, games, computer based education, and simulations.[7] A more recent development is the flipped classroom paradigm [8] wherein students read or study about an assigned topic by themselves and classroom contacts are utilized to solve problems or to do what would be part of traditional homeworks. In this paper we deal with another important component of active learning, namely, computer simulations.

2

COMPUTER SIMULATIONS

Over the past couple of decades, there has been a great increase in the use of technology in STEM education. More specifically information technology has enabled great advances in teaching.[9] The mass availability of computers, laptops, and mobile devices such as iPads, tablets and the like has led to the implementation of computer simulations in a variety of science subjects. Simulations have been around ever since the advent of computers, and computer simulations have become available for a wide range of science subjects.[10]. According to de Jong and van Joolingen [10], a computer simulation is “a program that contains a model of a system (natural or artificial; e.g., equipment) or a process.” Computer simulations offer many advantages including the fact they are often less expensive to create and construct than their real life counterparts. They can remove danger or uncertainty from the situation and explore hypothetical scenarios and hypotheses. They can be paused so that students can assess the results. In recent times, experimental situations have become complex; in addition, in traditional situations only few students can be trained at a time. In these situations, computer simulations offer a great alternative.

What are some of the criteria for a good computer simulation for classroom use? A number of guidelines for “best practice” can be found in the literature.[11] Some of the criteria that are relevant for learning via simulation include:

In the use of computer simulations, there are two important questions, namely: (1) How can they enhance traditional education, and (2) how can they be optimally used to improve learning outcomes. Rutten et al. [13] have analyzed results from education research over a period of a decade (2001-2010) to answer these questions. They also find that computer simulations can enhance traditional instruction, especially with laboratory activities. However, they note that the use of computer simulations has been approached without consideration of the possible impact of teacher support, the lesson scenario, and the computer simulation’s place within the curriculum. Well-designed computer simulations provide a model of those elements most relevant to the problem at hand and the learning objective [14]. In addition, “they inform the instructor and the learner of aspects of the real-life system or process that have been simplified” or eliminated.[15]

How can simulations be applied to optics education? The earliest published literature on the application of computer simulation to optics was published by Eylon et al. In 1996.[16] These authors used a simulation called RAY and studied the potential of simulations to improve optics learning in a high school classroom. RAY was a learning environment that included a flexible ray tracing simulation, graphic tools, and task authoring facilities. They showed the importance of three factors to students' understanding of concepts and their ability to use the ray model: the computerized environment (versus written instruction of similar kind); a task design that addresses directly conceptual difficulties in ray tracing; and the explicit reformulation of ideas.

Carnicer et al. [17] at the University of Barcelona developed an educational optical software package for undergraduate students. It consisted of a web based textbook with several applets for illustrating the theory to simplify the teaching tasks in the classroom. The applets were written in the Java language using the Java Network Launching Protocol. These programs could also used as a method for self-learning in an on-line environment.[17] These applets can be downloaded from: http://www.ub.edu/javaoptics/index-en.html. Foley et al. at Mississippi State University created WebTOP, a three-dimensional, web-based, interactive computer graphics system that covered waves and optics. The subject areas include waves, geometrical optics, reflection and refraction, polarization, interference, diffraction, lasers, and scattering. WebTOP was developed with VRML, JAVA, JavaScript and VRML’s JavaEAL.[18]. The website for this is: http://webtop.msstate.edu/. There are other optics simulation programs available on the web. Notable amongst them is PHET (http://phet.colorado.edu). Founded in 2002 by Nobel Laureate Carl Wieman, the PhET Interactive Simulations project at the University of Colorado, Boulder makes available for free interactive math and science simulations. The Light and Radiation page (http://phet.colorado.edu/en/simulations/category/physics/light-and-radiation) has over 20 different simulations covering a wide variety of optical phenomena. The programs are well documented and the source code is available. The Wolfram demonstration project, (http://demonstrations.wolfram.com/) boasts of having over 10,000 interactive demonstrations covering a whole gamut of areas in engineering, math, physics, business and social systems, etc. There are over 200 optics demonstrations available as of May 2016. This is a crowd sourced project – any user of Mathematica software can write programs and submit for publication on the website. These demonstrations run from simple to very complex.

More recently, our group, presented a paper on simulations of diffraction using the Python languag e.[19] The precursor to this was the development of “Python Optics” [20]. Python has numerous advantages, especially for a novice programmer (vide infra Section 3).

3

OPTICS SIMULATION WITH PYTHON

3.4

Examples

In this article, we illustrate the great advantages offered by Python and all the new tools accompanying it, using examples from traditional topics in geometric and physical optics. We start with a simple demonstration of determining the positions of the cardinal planes for a lens system with many refracting surfaces. Then, a simple demonstration of how we can simulate light diffraction produced by slits and circular diaphragms in the approximation of the far field – the so-called Fraunhofer diffraction is provided. For diffraction by rectangular aperture, we choose the simplest case of the diffraction by a single slit because it is a well known problem and our goal here is not to describe the complexity of the physical phenomena. In the case of diffraction by circular apertures, we consider the Rayleigh criteria because it concerns a wide range of applications. Simultaneously, we demonstrate how this numerical environment (Jupyter/IPython) provide teachers with a set of rich tools that help to generate documents for education and outreach programs.

3.4.1

Cardinal planes

We begin with a demonstration of determining the positions of the cardinal planes for an optical system with an arbitrary number of refracting surfaces. In general, one first needs to determine the front vertex power (FVP), back vertex power (BVP) and the nominal equivalent power (NEP) of the system. This could be done by defining functions for do this as shown in Figure 2.

Figure 2:

Sample code demonstrating functions definitions in Python. Note the usage of Jupyter. The various boxes with names line 'In [1]:' denote the various 'cells' in the Jupyter Python Notebook. A cell is nothing but a block of code that is executed at one go.

We could use the IPython widgets to get input from the users as shown in Figure 3. With these input values, we can then use the functions as defined in Figure 2 to calculate the various quantities that are of interest to us. This is demonstrated in Figure 4. With the calculated values, one can then create the canvas as demonstrated in Figure 5 that results in a display as shown in Fig 6. As a demonstration we have used the Gullstrand Schematic Eye Model that models the optical portion of the human eye as a system of four refracting surfaces. In this particular demonstration, the Principal Planes are denoted as P1 and P2, the Focal Planes as F1 and F2 and the nodal planes as N1 and N2. Users can change the input values in Figure 3 and rerun the code to see how the positions of the cardinal planes change.

Figure 3:

It is easy to get user interaction using the ipywidgets modude. Note that we begin with importing the required modules (ipywidgets and Ipython.display in this case). Note that creating the widgets, arranging them and displaying are all done using simple one line commands.

Figure 4:

Demonstration of simple commands and function calls in Python in the Jupyter notebook.

Figure 5:

Demonstration of code to draw the various surfaces and the cardinal planes.

Figure 6:

The output of the codes using user input shown in Figures 2-5.

3.4.2

Diffraction by a single slit

The choice of the diffraction by a single slit is very useful because it allows us to simply show how easy it becomes to interactively simulate a well-known function – in this case the sinc function involving a few physical parameters. That is to say that we can reproduce observable phenomena by varying dimensions or distances. Figure 7 below shows a schematic of diffraction of light by single slit indicating the main parameters that influence the intensity distribution observed on a screen.

Figure 7:

Diffraction obtained by a single slit with a width (d) observed on screen at a distance (D) from the slit.

The expression for the intensity of light observed on the screen (E) along the X direction for a monochromatic light with wavelength (λ) is given by:

To simulate equation (1), we write a function “Diff_1S(lamda=600, b=0.1*1.E-3, D = 1)”. The default parameters of this function are: λ= 600 nm, b = 10^-03 m and D = 1 m. This function generates a plot of the intensity of the diffracted monochromatic light beam as function of ‘X’ coordinate of the screen (Figure 9). The plot changes if we change the values of the defaults parameters. We can generate a Jupyter/IPython notebook and load this program and run it to generate the plot. We can also use sliders to vary the three parameters (λ, b and D) in order to simulate various configurations we want to test. The program and the output are shown in Figures 8 and 9, respectively. Sliders are imported from the ipywidgets package. IPython widgets are interactive HTML widgets for Jupyter notebooks and the IPython kernel.

Figure 8:

Jupyter/IPython notebook cells showing codes that we need to write and execute in order to have figure 9 as output.

Figure 9:

Diffraction pattern we obtained with the sliders associated with the three parameters (λ, b, and D).

3.4.3

Rayleigh criterion

For the second case, we consider the diffraction by a circular diaphragm; we focus on the resolution issue. As for the previous case all the work can be done inside Jupyter/IPython notebook. The three figures 10-12 demonstrate this.

Figure 10.

Diffraction by circular diaphragm of radius (r) for two sources separated by a distance d.

Figure 11:

Jupyter/IPython notebook cells showing codes that we need to write and execute in order to have Figure 12 as output.

Figure 12.

Intensity distribution obtained with the parameter’s values indicated on the sliders.

The Python program is as as shown in Figure 11 and the the output in Figure 12.

We can check easily the Rayleigh criterion by dragging the slider associated with (d) or (λ) and noting the values corresponding to the situation where the minimum of the red curve is aligned with the maximum of the blue curve.

4

DISCUSSION

Simulations have a major impact on learning and understanding science concepts. We have described in this paper some uses of the Python programming language to teach optics via simulations. As noted previously, Python and associated programs are open source and free, thus making them very cost effective for use in developing nations or in situations where cost of software is a consideration. In addition, Python is an object-oriented programming language. Known for its easily readable syntax, uncluttered visual layout and extensibility to other programming languages, Python can be used to develop for a variety of uses including the Web, GUI and other applications that need a programmable interface. Python can be introduced in workshops to both teachers and students to learn physics at high school and lower grades. [21] We are currently working on a suite of Python programs to teach introductory optics and these will be published elsewhere.[22]