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Relativistic Aberration of Light Mimicked by Microring Resonator Optik - International Journal for Light and Electron Optics 183 (2019) 82–91 Contents lists available at ScienceDirect Optik journal homepage: www.elsevier.com/locate/ijleo Original research article Relativistic aberration of light mimicked by microring resonator- based optical All-Pass Filter (APF) T ⁎ Benjamin Dingela,b, , Aria Buenaventurac, Annelle Chuac, Nathaniel J.C. Libatiqueb,d, Koji Murakawae a Nasfine Photonics Inc., Painted Post, NY 14870, USA b Ateneo Innovation Center, Ateneo De Manila University, Quezon City, Philippines c Physics Department, School of Science and Engineering, Ateneo De Manila University, Quezon City, Philippines d Computer Eng.’s Department, School of Science and Eng.’s., Ateneo De Manila University, Quezon City, Philippines e College of General Education, Osaka Sangyo University, 3-1-1 Nakagaito, Daito, Osaka 574-8530, Japan ARTICLE INFO ABSTRACT Keywords: We report a photonic circuit analogue of the relativistic aberration of light (AL) phenomenon in Photonic Integrated Circuits Special Relativity (SR) using an All-Pass Filter (APF). The APF is one of the key building blocks in Special Relativity (SR) Photonic Integrated Circuits (PICs). We investigate its phase characteristics for AL in detail. We Optical Analogue also compare the similarities and differences between the current work and our previously re- Einstein Velocity Addition (EVA) ported Thomas-Effect-inspired analogue since both analogues used the APF as their photonic Aberration of light circuit representations. This analogue strengthens the novel linkage between SR and PICs, and supports the concept of the Special-Relativity-on-a-Chip. This highlights the potential of the APF as a flexible, building block for a multipurpose circuit that combines two or more SR analogues leading toward a programmable SR circuit. Lastly, this opens up a new application space for PICs beside the traditional optical telecommunications and sensing applications. 1. Introduction Optical analogues are powerful tools to realize both experimentally and analytically key phenomena in Quantum Mechanics (QM) [1,2], General Relativity (GR) [3–5], and Special Relativity (SR) [6,7] that are oftentimes difficult to accomplish in their respective original settings. Recently, we reported an optical analogue between the Thomas rotation angle effect (or Thomas Effect) found in SR, and the phase response of an optical All-Pass Filter (APF) in Photonic Integrated Circuits (PICs or photonic chips) [6]. The Thomas Effect is one of the counter-intuitive phenomena in SR, whereas the APF is one of the key building blocks in PICs. PICs are the optical counterpart of the electronic circuits. Furthermore, we also introduced an optical analogue for Einstein Velocity Addition (EVA) [7] which utilized an optical add-drop-filter (OADF), another key building block in PICs. This particular analogue is significant because EVA is one of the key concepts in SR that underlies many relativistic effects. These two reported analogues lead to a bigger concept of what we called Special-Relativity-on-a-Chip. It is a new platform to explore both experimentally and analytically the analogous behaviors of SR in a single chip. It can validate known SR phenomena and probe still unexplored aspects of SR. It is based on the close mathematical similarity between (i) the equation of the EVA in the complex plane formulation proposed by Vigoureux [8–11] for SR, and (ii) the equation of the complex electric field output signal ⁎ Corresponding author. E-mail address: [email protected] (B. Dingel). https://doi.org/10.1016/j.ijleo.2018.12.149 Received 12 December 2018; Accepted 27 December 2018 0030-4026/ © 2019 Elsevier GmbH. All rights reserved. B. Dingel, et al. Optik - International Journal for Light and Electron Optics 183 (2019) 82–91 Table 1 Partial list of SR phenomena and their photonic circuit analogues. Special Relativity (SR) phenomena/central concepts Equivalent Photonic Integrated Circuits (PICs) analogues Remark References 1 Thomas Rotation Angle Effect All-Pass Filter (APF) with τ = V1 V2 SR Phenomenon [6] 2 Einstein Velocity Addition Add-Drop Filter (ADF) with τ1 = V1 and τ2 = V2 SR Central Concept [7] 3 Aberration of Light All-Pass Filter (APF) with τ = V1 SR Phenomenon This paper 4 Doppler Shift No circuit analogues reported SR Phenomena 5 Length Contraction 6 Time Dilation 7 Others from a coupled microring resonator (MRR)-based PICs [12–14]. Special-Relativity-on-a-Chip has two challenges. The first challenge is to develop “building blocks” for different SR phenomena that can be interconnected and/or combined to model a more advanced SR system. It is well-known that richer and bizarre SR behaviors are produced when different relativistic effects are combined. For example, the Penrose–Terrell effect [15–18] and the relativistic beaming effect [19,20] are the results of the combination of at least two or more SR phenomena. Table 1 summarizes some of these SR phenomena and their photonic circuits analogues that include, but are not limited to (i) Thomas Effect, (ii) EVA, (iii) aberration of light (AL), (iv) time dilation, (v) Lorentz (length) contraction, and (vi) Doppler shift to name a few. The second challenge is to develop “building blocks” that are multi-purpose and can be applied to multiple SR phenomena without too much complicated re- engineering efforts. Such building blocks are optimal and cost-effective. In this paper, our goal is two-fold. First, we report a new optical analogue to increase the collection of known photonic circuit analogues of SR into three. In particular, we present the analogue for the relativistic aberration of light (AL) using an all-pass-filter (APF) circuit based on microring resonator (MRR) [12–14,21,22]. We refer to this particular APF as the AL-Inspired APF or (ALI-APF for short). This will not only add to previously reported SR analogues [6,7] but also demonstrate that the APF is a multi-purpose building block since it is used previously in the reported Thomas Effect analogue [6]. Consequently, our second goal is to compare the similarities and differences between these two analogue circuits. Here, the analogy between the relativistic aberration of light (AL) and all-pass-filter (APF) circuit is established by equating two pairs of parameters namely; (1) the relativistic normalized velocity V1 in SR with the transmission coupling coefficient τ1 of the directional coupler in the APF (or simply V1 = τ1), and (2) the “generating angle” θ in the SR's Einstein velocity addition with the single-pass round-trip phase shift ϕ of the APF (or simply θ = ϕ). Note that the information of the aberration of light is encoded in the phase component of the optical output signal so it is not directly measurable! Therefore, we also introduce a phase-to-intensity converter circuit to quantify and measure the AL. The pro- posed photonic circuit is a balanced Mach–Zehnder interferometer (MZI) in which the ALI-APF is positioned in one of the arms of the MZI. We refer to this MZI as AL-Inspired MZI or (ALI-MZI). As mentioned earlier, this paper also compares the similarities and differences between (i) the present ALI-MZI circuit, and the Thomas-Inspired-MZI (TI-MZI) circuit reported in ref [6] since they used the same building block. Then, we briefly discuss the potential for a hybrid circuit that gives flexibility and programmability for the Special Relativity-on-a-Chip. The outline of this paper is as follows. In Section 2, we review the necessary background information related to EVA for two non- parallel velocities, and discuss the AL phenomenon. In Section 3, we discuss the optical APF, derive its phase response, and establish the analogy with the AL. In Section 4, we discuss the characteristics of the AL. In Section 5, we compare (i) the ALI-APF with the TI- APF analogues, and (ii) ALI-MZI with the TI-MZI circuits. Finally, the main conclusions are given in Section 6. 2. Aberration of light and Einstein Velocity Addition (EVA) The aberration of light (AL) can be derived from different methods such as: (i) wavefront approach using Huygen principle [23], (ii) Lorentz transformation [24], Einstein velocity addition (EVA) using complex plane formulation [8–11] when one of the two velocities is assumed to be traveling with the speed of light, and (iii) other approaches. Here, we use the EVA method [8–11] to derive the AL. First, we give a brief overview of EVA, and its key equations before discussing its application to the AL. A detailed discussion of EVA can be found in refs [6–11]. 2.1. EVA: three-coordinate frame (Σ, Σ′, and Σ″) EVA is a fundamental concept in SR that is used as a general framework to explain many relativistic phenomena, including the AL. Fig. 1a shows the typical illustration of the three-coordinate frame (Σ, Σ′, and Σ″) used to discuss the Einstein velocity addition of two non-collinear relativistic velocities, V¯1 (black arrow) and V¯2 (blue arrow). The angle θ2 is formed between velocities V¯1 and V¯2 which is referred to as the “generating angle” in SR literature. For simplicity, we take the direction of velocity V¯1 to be oriented with respect to frame Σ at some arbitrary positive coordinate-axis so that θ1 =0. Detailed definitions of velocities V¯1 and V¯2 are given in the next section. As shown in Fig. 1b, the addition of two relativistic velocities V¯1 and V¯2 leads to a resultant third velocity, V¯¯1 ⊕ V2 (red arrow) where the symbol ⊕ is the Einstein addition operation. Geometrically speaking, the three velocities V¯1, V¯2, and V¯¯1 ⊕ V2 form a 83 B. Dingel, et al. Optik - International Journal for Light and Electron Optics 183 (2019) 82–91 Fig.
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