2.1

Mathematical analogy

The first activity of Concept Connectivity begins with the search and recognition of mathematical analogies. To elaborate, there often exists “mathematical similarities” between distinct physical concepts. These mathematical similarities are concretized in the form of analogies, which are conceptual tools that are commonly used in STEM education to help students learn unfamiliar knowledge, concepts, or phenomena by relating them to existing ones that the individuals have had prior experience with.⁹ Generally speaking, analogies are well-known and effective tools for teaching science-related topics as they improve a student’s comprehension and retention of abstract concepts. For this reason, analogies—specifically mathematical analogies—have played an important role in our previous works on relativistic analogues.^4-8

As an example, we have previously reported the similarity between the equations of (i) the direction angle of Relativistic Aberration of Light (RAL) from the STR and (ii) the phase response of a microring resonator (MRR) implemented with an All-Pass Filter (APF) configuration as shown in Fig. 2(c).⁵ Note that we refer to the latter as the RAL-APF for simplicity. Summarily speaking, we note that both equations shown in Fig. 2 use the expression 2 · tan⁻¹(X), and it is a simple matter of equating both expressions such that that θ = τ₁.

Figure 2.

(a) Example of Concept Connectivity where the equations of RAL of STR and MRR-based APF are mathematically the same, (b) the physical representation of RAL, (c) the physical implementation of MRR-based APF circuit (c), and (d) the full circuit of RAL in STR is implemented with Mach-Zehnder Interferometer (MZI)-assisted RAL-APF.

2.2

Physical or Engineering Adjustment

In our previous example concerning the RAL-APF, we note that since the relationship between the quantities 𝚽 and 𝚿_RAL-_APF are exact, they can be simulated immediately. Although the direction angle 𝚿_RAL-_APF of the RAL can be measured physically, it must be noted that the output phase response 𝚽 of the RAL-APF cannot be measured directly. Thus, there is a need for the second procedure, which we call the “physical or engineering adjustment”. This challenges researcher/students to use their engineering knowledge and skills to think of an “engineering scheme” that must be implemented to obtain the measured quantity indirectly. It must be noted that all our relativistic analogues are mostly based on the same key technology– the all-pass filter (APF) element which can be implemented in many different platforms. In bulk optical platforms, this can be generated by a Thin Film-based Gires-Tournois Interferometer (GTI). In a photonic platform, this can be implemented with microring resonator (MRR)-based photonic integrated circuits (PIC). In an electronic platform, this can be obtained using an operational amplifier (OP-Amp) based electronic integrated circuits (EIC). Despite their exact mathematical similarities, the physical or operational actions are not exact since the output phase response of the MRR and GTI cannot be measured directly. Hence, we need to perform the “Physical and engineering adjustment scheme”.

2.3

Platform Selection and Technological Implementation Platforms

The next step is the selection of an appropriate implementation platform and its corresponding physical or engineering adjustment scheme to physically represent the said analogy. We refer to this implementation platform as an analogue, which refers to any technological device or system that can mimic the behaviors and properties of a concept or phenomenon. As mentioned previously, we have developed different relativistic analogues to study the following relativistic effects in our earlier works: the (i) RAL, (ii) TRA (or Thomas Angle), and (iii) EVA.^4-8

The challenge to researchers/students is to use their engineering knowledge and skills to conceive of an engineering scheme for “Phase-to-Intensity conversion for the optics and photonics platform cases” that must be implemented to obtain the measured quantity indirectly. In the case of the photonic platform, the “engineering adjustment” can be accomplished by incorporating the RAL-APF circuit into one of the arms of a Mach-Zehnder Interferometer (MZI) as shown Fig. 2(d) and Fig. 3(a). We refer to this implementation as the MZI-assisted RAL-APF or MZI-RAL-APF. Thus, it is possible to experimentally verify the behavior of the RAL through the implementation of MZI-RAL-APF. In the case of bulk optics, this can be accomplished with the use of a Michelson Interferometer (MI) as shown in Fig. 3(a). The diagrams of the resultant relativistic analogues are shown in Fig. 3(a-c) for different platforms.

Figure 3.

Relationship between the different relativistic analogues and their respective technology platforms.

2.4

Growing list of relativistic analogues

One small caveat in the previous activity on developing an analogy between two distinct concepts is that students or researchers must initially be familiar with both concepts, as the effectiveness of the resulting analogy is dependent on the depth of their knowledge of both concepts and their underlying principles. It is also worth noting that an analogy between two distinct concepts cannot always be readily translated into a corresponding analogue, especially if both concepts are theoretical in nature. Moreover, the more complex the analogy, then the greater the specifications required for the analogue or implementation platform will be. Thus, the goal of the researcher is to find an existing analogue (e.g., device, system, etc.) that can serve as the physical manifestation of an established analogy. In our previous works, we have reported various relativistic analogues for a particular relativistic effect, which are all summarized in Table 1.

Table 1.

Summary of relativistic analogues using Photonics, Optics and Electronics platforms.

3.1

Concept Connectivity as a teaching tool for multi-disciplinary topics

As can be seen in Table 1, these analogues are implemented with any one of the three technological platforms – namely: (i) photonic integrated circuits (PICs) using microring resonator (MRR) waveguide as the primary element, (ii) free-space optic using the thin-film (TF)-based mirror as the key element, and (iii) electronics using the operational amplifier (OP-AMP)-based phase shifter as the core element. In principle, these three key elements can mimic the function of a mathematical entity called an all-pass filter (APF) as mentioned earlier. Again, it brings together the different fields of Optics, Photonics, and Electronics. More importantly, we can use their interrelationship to link these field to one another.

In STEM education, this exercise is especially significant on three fronts. First, students can understand and establish the unlikely “concept connection” between two distinct phenomena, namely the (i) RAL—a relativistic phenomenon that is highly abstract in nature, and (ii) the RAL-APF circuit—a physical object. Second, although students mentally know that there is a link, with this proposed framework, they are also exposed to a direct example of emerging engineering fields like PICs. Clearly, students can easily recognize the clear link between the (i) science of RAL in the STR with (ii) the engineering field of PICs. Third, students can be challenged to apply their engineering skills to resolve how to demonstrate this concept (for example: how to come up with or design a phase-to-intensity conversion scheme with the use of MZI-assisted RAL-APF). In short, we formed a new bridge between the fields of photonics in engineering and the STR in physics, which opens the door to many new and exciting applications.

3.2

Pedagogical benefits of Concept Connectivity in optics and photonics education

Concept Connectivity has direct benefits for optics and photonics education. In particular, since most of our relativistic analogues are based on optical and photonic technologies, both physics and engineering students are exposed to devices such as microring resonators, all-pass filter (APF) circuits, and photonic integrated circuits. To understand their connection to their intended relativistic effect, students are tasked to first familiarize themselves with the very basics of optics and photonics (e.g., the nature and properties of light). Once this is accomplished, they are required to undergo practical training in the operation of the aforementioned devices. Thus, through Concept Connectivity, students not only learn about the STR in physics and its relativistic effects, but they also acquire knowledge and experience in optics and photonics as well, which broadens their skills and critical thinking.

3.3

Electronic Platform as Low-Cost Issue implementation platform and its impact on pedagogical teaching

In the previous sections, it must be noted that most analogies we established are implemented through optical, and photonic technologies. However, the most cost-effective platform is based on electronic circuits. In our previous work,⁸ we reported an electronic analogue for the RAL (see Electronic Analogue I in Table 1) that we refer to as the RAL-EC, whose schematic design can easily be replicated using simple components as shown in Fig. 4a.

Figure 4.

Schematic diagram of the RAL-EC (Left) and its experimental set-up equivalent (Right).

We emphasized in our previous work that the RAL-EC could be used as a teaching tool through which students can learn about the RAL. Specifically, the RAL-EC can be used to ground the abstract and theoretical nature of the RAL into more tangible and digestible knowledge that students can easily assimilate. This could be accomplished by first showing how the RAL-EC operates through a laboratory experiment. Once the students are familiar with this device, the instructor can then establish the mathematical analogy between it and the RAL for the students to see how they are connected. In this manner, the student’s learning experience is reinforced as they are engaged through a more practical and “hands-on” approach, as opposed to the typical teaching approach of directly explaining what the RAL is.