Optik - International Journal for Light and Electron Optics 183 (2019) 82–91

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Optik

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Original research article Relativistic 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 (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 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) [24], Einstein velocity addition (EVA) using complex plane formulation [8–11] when one of the two velocities is assumed to be traveling with the , 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

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Fig. 1. The three-coordinate frame (Σ, Σ′, and Σ″) (a) used to discuss Einstein velocity addition (EVA) for two velocities, V¯1 and V¯2 to get the resultant velocity, V1 ⊕ V2 (b).

(hyperbolic) triangle [25,26] with vertices “A”, “B”, and “C”. The two important results of EVA are contained in (i) the magnitude

|VV12⊕ |, and (ii) the directional angle, Ψ12⊕ . The magnitude of the resultant velocity, |VV12⊕ | is denoted by the length of the red arrow line in Fig. 1b. One of the internal angles of the triangle is the directional angle, Ψ12⊕ of V¯¯1 ⊕ V2. It is the angle at which the observer A (at rest frame Σ) perceives “C” at frame Σ″. In addition, EVA also leads to a reference frame rotation known as the Thomas effect that will be discussed in Section 5.

2.2. EVA: Complex Plane Formulation

In typical SR literature, the resultant velocity V¯¯1 ⊕ V2 can be obtained only after a lengthy derivation from the Lorentz trans- formation [27,28]. To simplify this derivation, Vigoureux [8–11] proposed a complex plane formulation of EVA with the following definition of the resultant velocity W12⊕ =⊕VV¯¯ 1 2as,

VV¯¯+ VVe+ +−iθ()21 θ iΨ12⊕ 12 12 +iθ1 W12⊕⊕==We 12 = e +−iθ()21 θ 1 + VV¯1* ¯2 1 + VV12 e (1) where the symbol * stands for the conjugate operation, and the complex velocities V¯12and V¯ are defined as

β β ++iθ111 iθ ++iθ 22 iθ2 V¯11==Ve tanh eVVe, ¯22 ==tanh e 2 2 (2) where the magnitudes of their velocities are given by the terms V1, and V2, while their directions are indicated by the parameters θ1, and θ2 respectively. The terms β1 and β2 in Eq. (2) are the rapidity values associated with the speeds v1 and v2 and are defined as

−−1 v1 1 v2 β1 = tanh⎛ ⎞ ,β2 = tanh⎛ ⎞ . ⎝ c ⎠ ⎝ c ⎠ (3)

Note that the magnitudes of the complex velocities V1 and V2 are limited to the closed interval [−1,1] since the speed of light c is the normalizing factor of the velocities. Letting θ = θ2 − θ1 and noting that we set θ1 = 0, the magnitude W1⊕2 of W1⊕2 in Eq. (1) can be obtained as

VV2 ++2 2cos[] VVθ ¯¯* 1 2 12 W12⊕⊕==WW 12 12⊕ 2 2 . 12cos[]++VV1 2 VV12 θ (4)

While the directional angle, Ψ12⊕ of W1⊕2 is derived from tan[ΨWW12⊕⊕⊕ ]= Im [ ¯12]/Re [ ¯ 12] such that it can be expressed as

2 (1− VV1 )2 sin[ θ ] tan[Ψ12⊕ ] = 2 2 . VV1 (1+++2 ) (1 VVθ1 )2 cos[ ] (5)

Eqs. (4) and (5) are the same expressions obtained in typical SR literature using Lorentz transformation after a lengthy derivation. It is worth noting the ease in deriving Eqs. (4) and (5) using the complex plane formulation. For the last three decades, it has been studied by numerous authors. It has found many applications related to multilayer thin films [29–32], multiple quantum wells [33], polarization studies [34,35], and group theory [10,36] to name a few.

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Fig. 2. The three-coordinate frame (Σ, Σ′, and Σ″) used to discuss the aberration of light from EVA where one of the velocities, V¯2 is equal to 1, meaning speed of light c.

2.3. Aberration of light (AL)

The aberration of light (AL) is a SR phenomenon about the direction of light propagation as seen from two different reference frames. It can be considered as the correction to be applied between two observers in relative motion [37,38]. As mentioned earlier, we will derive AL from EVA using the complex plane formulation of Vigoureux [8–11] when one of the two velocities, V¯2 is assumed to be travelling with the speed of light c.

Consider the scenario given in Fig. 2. An observer A located at initial rest frame Σ detects a ray of light at an angle θ3 coming from a distant light source C located at reference frame Σ″. Another observer B (moving at velocity V¯1 with respect to frame Σ at angle θ1 = 0) observes the same distant ray of light at direction angle θ2 in its own referential frame Σ′. Note that the AL is about the measured angles of the light ray observed from two observers at two different reference frames (Σ and Σ′) coming from the same light source at frame Σ″. This scenario resembles the EVA illustration in Fig. 1b except that (i) the directions of the velocities V¯2 and V¯1 ⊕ V¯2 are reversed, and (ii) the velocity V2 is equal to 1 (i.e. the light source is transmitting a signal travelling at the speed of light c so that β2 = ∞ in Eq. (3), making V2 = 1 in Eq. (2)). This allows us to apply all previous findings or equations from EVA to AL, as reported and discussed by Vigoureux [8].

From Eq. (4), we observe that when the normalized velocity, V2 is equal to 1, the magnitude of the resultant velocity W1⊕2, (as noticed by observer A) becomes unity regardless of the value of V1. This is a confirmation of one of the SR tenets that the speed of light is the same for any (as perceived by both observer A at frame Σ and observer B at frame Σ′). The AL as perceived by observer A is quantitatively described by the directional angle, Ψ12⊕ in Eq. (5) when V2 =1.Todifferentiate it from the A θ θ − θ θ normal directional angle, Ψ12⊕ , we denote the AL as Ψ1⊕2. Noting that the angle = 2 1 with 1 = 0, then from Eq. (5), the AL is simply expressed as

A −1 ⎡ sin [θ ] ⎤ Ψ12⊕ = tan ⎣⎢ ab11(cos[])+ θ ⎦⎥ (6) where

2 (1+ V1 ) 2V1 a1 = 2 ,andb1 = 2 (1− V1 ) (1+ V1 ) (7) In other SR literatures, AL is also expressed in different forms as [37–39]

A sin[θ ] sin[Ψ12⊕ ] = abθ11(1+ cos[ ]) (8)

A bθ1 + cos[ ] cos[Ψ12⊕ ] = 1cos[]+ bθ1 (9) These expressions can also be obtained directly from Eq. (6) with some simple algebraic manipulations. In this paper, we derive an alternative expression for the AL that is instrumental in suggesting the existence of an optical analogue in PICs. By noting that V2 = 1, the expression W¯1+2 in Eq. (1) can be rearranged to appear as

−iθ A 1 + Ve1 W ==WeiΨ12⊕ e+iθ 12⊕⊕ 12 +iθ 1 + Ve1 (10) θ θ − θ θ A with = 2 1 and 1 = 0. Obviously, its magnitude, Ψ12⊕ is equal to unity while its phase component (or the AL term), Ψ1⊕2 from Eq. (10) can be expressed as

A −1 ⎡ Vθ1 sin[ ] ⎤ Ψθ12⊕ =−2tan ⎣⎢ 1cos[]+ Vθ1 ⎦⎥ (11) Eq. (11) is instrumental in gaining physical insights that lead to the development of the optical analogue for the AL in PICs.

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Fig. 3. Schematics of a typical MRR-based APF (a), and ALI-APF (b).

3. Optical circuit analogue of the aberration of light

In section 3.1, we first review the technical details and operations of the optical APF in PICs, and introduce its basic mathematical transfer function expression. Next, in Section 3.2, we present the proposed APF inspired by the AL. Then in Section 3.3, we establish the formal analogy between the aberration of light (AL) and the optical APF.

3.1. Optical All-Pass Filter (APF)

The optical APF is an essential building block in PICs. Functionally speaking, it passes all signal frequencies but modifies their phase responses. The behavior of the phase-vs-frequency response exhibits a linear to strongly nonlinear profile depending on the APF's parameters. Fig. 3a shows the typical lossless APF implemented with an optical microring resonator (MRR) having a circumference length L. The MRR is attached to a straight waveguide bus by a directional coupler (DC) [12–14,21,22]. This structure is readily fabricated in PICs and used in many applications in optical telecommunications. Fig. 3b shows our proposed APF configuration to mimic the response of the AL. It has an extra π-shift plate inside the ring resonator compared with the typical APF. This π-phase shifter can easily be replaced by an appropriate MRR with an additional length ΔL, corresponding to a π-shift measured at some center wavelength λ0 and valid around an interval [λa, λb]. We will refer to this particular MRR-based APF as the aberration of light (AL)-Inspired APF (or ALI-APF). The π-phase shifter shown in Fig. 3b is introduced into the MRR in order to establish simple and direct mathematical comparison A fi between (i) the AL, Ψ1⊕2 as de ned by Eq. (11), and (ii) the phase response of the ALI-APF that will be derived in Section 3.2 as Eq. (13).

3.2. AL-Inspired All-Pass Filter (ALI-APF)

As shown in Fig. 3b, an incident beam Ein is injected into port 1 of the ALI-APF, and the output signal ET1 exits at port 2 after circulating many times around the MRR. The time spent circulating around the MRR depends on the DC parameter τ1. The output complex electric field ET1 of the ALI-APF is given as [12–14,21,22]

+iϕ′ +iϕ ET11iΦAL τe− τe1 + EeT1 ===+iϕ′ +iϕ Ein 1 − τe1 1 + τe1 −iϕ 1 + τe1 +iϕ = +iϕ e 1 + τe1 (12) where the phase term ϕ′ =(ϕ + π) is the overall single-pass round-trip phase shift of the MRR with ϕ = kL, k is the propagation constant given as k =2πneff/λ, λ is the wavelength, and neff is the effective index of the waveguide. The π term in ϕ′ (ϕ + π) is due to τ ffi 2 1/2 the phase shifter, and the parameter 1 is the transmission coupling coe cient of the DC with κ1 =−(1τ1 ) as its cross-coupling ffi 2 2 coe cient. The DC is assumed lossless so that τκ1 +=1 1. Note that |ET1|/|Ein| is equal to unity, thus making the term ET1 in Eq. (12) simply equal to its phase response ΦAL. It follows immediately from Eq. (12) that the phase response ΦAL can be expressed simply as

τϕsin[ ] Φϕτ(, )=− ϕ 2tan−1 ⎡ 1 ⎤ AL 1 ⎢ ⎥ ⎣ 1cos[]+ τϕ1 ⎦ (13) which has the same form as in Eq. (11) which describes the AL.

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Fig. 4. shows the aberration of light (AL) as a function of the generating angle θ under various parameter conditions of V1 (a), and as a function of velocity V1 under different conditions of the generating angle θ.

3.3. Establishing the formal analogy

The perfect similarity between Eqs. (11) and (13) is the obvious reason for the analogue between the AL and the phase response of APF. To formalize the analogy, we make these two parameter associations namely;

(i) ϕ θ. The single-pass round-trip phase shift ϕ of the APF is associated with the “generating angle θ” in Einstein velocity addition, and

(ii) τ1 = V1. The transmission coupling coefficient τ1 of the DC is equated with the relativistic velocity V1.

4. Properties of the aberration of light (AL)

4.1. Aberration of light from the ALI-APF

A ϕ Fig. 4a shows the behavior of the aberration of light (AL), Ψ1⊕2 as a function of the phase shift of ALI-APF (or equivalently the generating angle θ in EVA for SR) under various parameter conditions of τ1 = V1 (0.1, 0.3, 0.5, 0.7, 0.9, 0.99) based on Eq. (13). It gives three noticeable observations. First, the profile is linear (or straight line) when the parameter value of V1 (or τ1) is very fi A small (for example V1 < 0.1). This implies that the measured aberration is very small. Second, the pro le of the AL, Ψ1⊕2 starts to change from straight line to curve line when V1 increases toward unity. This change starts to be visible when V1 = τ1 = 0.3. Third, the fi A fl ϕ “ ” θ τ pro le of the Ψ1⊕2 attens to near-zero across a wide range of the phase shift (or generating angle ) when V1 = 1 is near unity. This is very clear especially for the cases of V1 = τ1 = 0.9 and V1 = τ1 = 0.99. This near zero-saturation implies that the resultant velocity has entered the range when the relativistic effect from velocity V1 is profound. From a physics perspective, these three observations indicate that we can make an important connection about the “physical mechanism” of this behavior. Simply put, the increase in the relativistic velocity of V1 is functionally equivalent to the increase in the phase dispersion in ALI-APF circuit. The strength of the phase dispersion is dictated by the parameter τ1 in PICs-based APF. From a device operational perspective, tuning the phase shift ϕ of the ALI-APF from −π to π is equivalent to scanning the “generating angle” θ from −π to π in SR. In PICs, this tuning operation is accomplished in many different ways. Two straightforward methods are (i) to use a tunable laser as light source, or (ii) to thermally tune the MRR in the ALI-APF. A τ ff fi θ Fig. 4b shows the behavior of the AL, Ψ1⊕2 as a function of velocity V1 (or 1) for di erent xed values of the generating angle (or ϕ = 0, 0.125π, 0.25π, 0.5π, 0.625π, 0.75π, 0.875π, 0.937π). One noticeable observation from Fig. 4b is that when θ = 0.625π the resultant profile shows a nearly straight line. This is important because it captures a one-to-one correspondence relationship between A fi Ψ1⊕2 and V1 within a speci c range. Operationally speaking, varying the value of the velocity V1 from 0 to 1 is equivalent to tuning the value of the parameter τ1 in ALI-APF. In PICs, there are several MRR-based circuit designs with tunable τ1 capability that have

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Fig. 5. Schematic of ALI-MZI using balanced MZI with ALI-APF in one arm of MZI, (a). The intensity response of ALI-MZI as functions of the generating angle θ, (b) and velocity V1 (c) under different fixed conditions. been reported [40,41]. These designs can be employed for this purpose.

4.2. AL-Inspired Mach-Zehnder Interferometer (ALI-MZI)

Since the information related to AL is contained in the phase component of the output complex electric field, ET1 in the ALI-APF, it is not directly measurable! Therefore, we need to convert this phase-encoded-AL into intensity by using an additional photonic circuit. Fig. 5a shows such a photonic circuit. It consists of a balanced MZI with a 50:50 power splitter (PS) and a 50:50 power combiner (PC). The ALI-APF element is positioned in one of the arms of the MZI. We will refer to this configuration as the AL-Inspired MZI (or ALI-MZI).

An input signal, Ein is injected into the ALI-MZI and divided by a 1 × 2 PS (or a 2 × 2 coupler) into two beams of equal intensities. Both signals travel along their respective arm path-lengths L1 and L2. The path-length L1 is attached to the input port of the ALI-APF while its output port is connected to one of the input ports of the PC (or coupler). The path-length difference of the two arms of the

MZI is set to zero (ΔL12 = L1 − L2 = 0). Then, both beams are coupled at the PC to give the output electric field Eout as

Eϕout () 1 =++(exp[iβL21 ] exp[ i ( βL ΦAL )]) Ein 2 (14)

The electric field output of MZI is detected by a photodetector that converts it into intensity. Assuming Ein = 1 and since L1 = L2, the corresponding intensity Iout (=Eout Eout*) is then given as

1 2 ΦAL Iϕout ()=+{1 Cos[ΦAL ] } = Cos ⎡ ⎤. 2 ⎣ 2 ⎦ (15)

Clearly, this photonic circuit converts the phase-encoded AL, ΦAL(ϕ θ, τ1 V1) into intensity that can be measured directly. Fig. 5b shows the intensity response of the ALI-MZI as a function of the single-pass phase shift for different values of τ1 (or V1 = 0.1, 0.3, 0.5, 0.7, 0.9, 0.99). On the other hand, Fig. 5c shows the intensity response of ALI-MZI as a function of the coupling coefficient, τ1 (or velocity V1) for different values of the phase shift ϕ (or the generating angle θ = 0.5π, 0.625π, 0.75π, 0.825π, 0.875π). It shows a near linear intensity profile when ϕ is within the range of [0.625π, 0.75π]. To obtain the actual AL information, it is clear that we only need to perform the inverse operation of Eq. (15) from the measured intensity.

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5. Comparison between ALI-APF and TI-APF

As mentioned earlier, the second objective of this paper is to compare the ALI-APF analogue with another photonic circuit analogue that mimics a SR phenomenon called the Thomas Effect reported in our previous paper [6]. Both analogues use the APF as their photonic circuit representations. It demonstrates the multi-purpose building block of APF for Special-Relativity-on-a-Chip. Here, we will briefly review the Thomas-Inspired-APF (TI-APF) analogue before comparing it with ALI-APF. Detailed information about the TI-APF is found in Ref. [6].

5.1. Thomas rotation angle, ε

Briefly speaking, the Thomas Rotation Angle Effect (or Thomas Effect for short) is one of the counter-intuitive phenomena in SR. It is the spatial (or coordinate axis) rotation of the observed reference frame due to the addition of two successive velocities travelling in non-collinear directions. It is defined as the net frame rotation after its trajectory returns back to its initial frame [6].

Mathematically speaking, it is expressed as the ratio between W¯2⊕1 and W¯1⊕2. Since |W¯1⊕2| is equal to |W¯2⊕1|, this ratio is simply a phase-only function given as

+iθ W21⊕ 1 + VVe eiε(, θV12 V )==12 −iθ W12⊕ 1 + VVe12 (16) which leads to the Thomas rotation angle ε as

−1 ⎡ VV12sin [ θ ] ⎤ εθVV(,12 )= 2tan ⎣⎢ 1cos[]+ VV12 θ ⎦⎥ (17) The photonic circuit analogue of Thomas Effect is depicted in Fig. 6a. It is an APF configured as TI-APF.

5.2. Comparison between TI-APF and ALI-APF

The similarities and differences between the TI-APF and the ALI-APF analogues occur in two fronts as depicted in Fig. 6. In terms of the similarity, Fig. 6a shows that the TI-APF has the same structure as the ALI-APF which is depicted in Fig. 6c. It indicates that the APF circuit is a very rich platform and able to accommodate at least two different SR phenomena. On the other hand, there are two major differences between the TI-APF and the ALI-APF. First, the parameter associated with the transmission coupling coefficient, τ is not the same! In the TI-APF case, the parameter τ is associated with the product of the two magnitudes of the relativistic velocities V¯1 and V¯2 (or τ = V1 V2) whereas for the ALI-APF, the parameter τ is associated with only the relativistic velocity V¯1 (or τ1 = V1). Second, the output phase response of the ALI-APF as given by Eq. (13) has the same exact form A compared with the directional angle Ψ1⊕2 of EVA in SR as given by Eq. (11). It is a perfect and complete one-on-one analogue relationship. On the other hand, this observation does not hold true for the Thomas rotation angle, ε(θ, V1V2) as given in Eq. (17).

Fig. 6. The structural similarities and differences between the photonic circuit analogues of Thomas Effect and Aberration of light.

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Clearly, there is only a partial analogue connection between Eqs. (17) and (13)! Fig. 6b and d shows the configurations of the TI-MZI and the ALI-MZI, respectively. Both configurations are based on the same MZI structure whose one arm contained either the TI-APF or the ALI-APF element. The role of the MZI for both cases is to convert the phase information carried by the light signal into intensity for direct measurement. The main difference between these two configurations is clearly the addition of the extra path length difference, ΔL found in the case of the TI-MZI structure. The role of this term ΔL is very important since it accounts for the linear term ϕ (=kΔL) that need to be removed from the TI-APF response to mimic the Thomas Effect. For detailed discussion, please refer to Ref. [6].

5.3. Multipurpose TI-MZI/ALI-MZI

The close similarity between the TI-MZI and the ALI-MZI configurations suggests the potential for a single circuit with a re- configurable property. Here, the MZI can “select” to function either as the TI-MZI or the ALI-MZI. Note that the TI-MZI configuration can be considered as a more general case compared with the ALI-MZI since the TI-MZI reduces to the ALI-MZI configuration (i) when velocity V2 is equal to 1, and (ii) parameter ΔL is set to zero. With tools to fabricate large-scale waveguide chips becoming more available, and the rapid development in programmable PICs the photonic chip is an ideal platform to generate different circuit analogues to support Special Relativity-on-a-Chip.

6. Conclusion

We presented an optical analogue to relativistic aberration of light (AL) using a photonic-chip based All-Pass Filter (APF). We investigated its phase characteristics in detail. This analogue increases the number of reported photonics circuit analogues for SR into three phenomena. This strengthens the novel linkage between SR and PICs, and enforces the idea of a Special Relativity-on-a-Chip. We also compare the differences and similarities between the (i) new AL optical analogue and (ii) the previously reported analogue dealing with Thomas Effect in SR since both analogues use the APF as their photonic circuit representations. This indicates the potential for a multipurpose and flexible photonic circuit that combines different SR analogues toward a programmable Special Relativity-on-a-Chip. It also opens up a new application space for PICs besides the traditional telecommunications and sensing ap- plications.

Acknowledgements

BD acknowledges Nicholas Madamopoulos for his helpful discussion. One of the authors (NL) is grateful for support from the ADMU Graciano Uy Liongsin Professorial Chair endowment.

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