Poynting Vector of Electromagnetic Waves in Media of Inhomogeneous Refractive Index

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1 Poynting Vector of lectromagnetic Waves in Media of Inhomogeneous Refractive Index Jinsik MOK 1, Hyoung-In L, 1. SUNMOON University, Department of Industrial and Management ngineering; 70 Sunmoon-ro 1, Tangjeong-myeon, Asan, Choongnam, Republic of Korea;. SOUL NATIONAL University, Research Institute of Mathematics; 599 Gwanak-ro, Gwanak-gu, Seoul, 0886 Republic of Korea ( HILSAM@NAVR.COM) Abstract Poynting vector is a key property of electromagnetic waves as an energy-flow vector. Poynting vector in spatially uniform dielectric media is well known in an analytic form. In general, Poynting vector can be decomposed into al and spin parts. One way of controlling the direction of Poynting vector is to impose external potential onto the media under consideration. We are here to derive analytic expression of Poynting vector in the presence of spatially inhomogeneous refractive index, which serves as such a potential. For realizing proper profiles of refractive index, thermo-optic effect is proposed and the resulting alteration in the expression for Poynting vector is discussed. Keywords: lectromagnetic Waves; Poynting Vector; Spatial Inhomogeneity; Refractive Index; Angular Momentum; Thermo-Acoustic ffect 1. Introduction Rotational electromagnetic waves are of increasing interests to optics community concerning wireless communications, holography, optical manipulations among other application areas [1]. Poynting vector as the energy-flow vector is indicative where electromagnetic energy is directed [,3]. The trajectory traced out by Poyning vector is helical or conical for instance in the azimuthally rotational waves. Since Poynting vector is decomposable into al and spin parts in general, such a decomposition provides us with a far-reaching advantage in analysing and utilizing azimuthally rotational waves in the presence of axial propagations. For instance, in terms of optical manipulation, the al part is responsible for the rotations of small particles or molecules around a certain spatially fixed cylindrical axis, whereas the spin part is responsible for the interaction between electromagnetic waves and a small particle so that the latter undergoes spinning around its own bodily axis of rotation [4]. Therefore, controlling the direction of Poynting vector is increasingly important in nanodevices where electromagnetic energy is desired to be transported only in a certain direction, for instance, in solar cells. One way of controlling Poynting vector is to introduce spatial inhomogeneity into the refractive indices of media [5-7]. When electromagnetic waves undergo both azimuthal rotations and axial propagations along the cylindrical axis such as in optical fibers [7], the radial component of Poynting vector can be added by inhomogeneous refractive indices. Here we are to analytically show how the al and spin parts are affected by such spatial inhomogeneity. To do so, we need only elementary vector calculus for deriving key analytic formulas. Instead, we will focus more on the physical implications of the derived relations and how to implement such inhomogeneity in refractive indices [6]. AdvNanonergy: 017: 1(1):34-38 ISSN: X 34

2 . Fundamental Conservation Law Consider the electric field vector and magnetic field vector H of electromagnetic waves. When suitably scaled, Maxwell s equations can be written as follows [3]. n H i, ih n0. (1) n 0, H 0 n0 Here, both field vectors are assumed to be time-periodic with the temporal phase factor expi where is a dimensionless time and i 1. Notice that the temporal frequency has been included in the definition of. Furthermore, n x, y, z is a generic spatially inhomogeneous refractive index, which is assume positive in our study, namely, n x, y, z 1. Besides, n0 1 is a prescribed reference refractive index. As usual, and are spatial divergence and curl operators, In q. (1), Ampère s and Faraday s laws are displayed on the first row, whereas Gauss s laws for electric and magnetic fields are displayed on the second row. Notice in the case of homogeneous refractive index with n x, y, z n0 that Ampère s law and Gauss s law for electric field are simplified to H i and 0, In fact, the purpose of this study is to examine the effects of non-unity factor n x, y, z n0 1 for spatially inhomogeneous media. For positive refractive index, the well-known energy conservation for Maxwell s equations presented in q. (1) reads as follows [3]. w Poyn P 0. () Here, the energy density w and Poynting vector (a.k.a. energy-flow vector) respectively as follows. Poyn P are defined 1 n n0 w H n0 n. (3) Poyn P Re H Here, the superscript indicates a complex conjugate, while magnitude squared. Besides, refers to the Re is the realpart operator. For later use, we list that the imaginary-part operator. 3. Separation of Poynting Vector Im is Firstly we establish the following relation. Poyn 1 P Im. (4) n0 1 Im H H n To this goal, consider the following two expressions for Poynting vector Poyn Re, which are obtained by P H processing Ampère s and Faraday s laws, Poyn n0 P Im H H n. (5) Poyn P Im Therefore, q. (4) is obtained by averaging the two in q. (5) based on the spirit of electricmagnetic democracy or duality []. Next we decompose the two vectors in q. (5) further into terms involving curl operators and the remaining ones. Im Im ln n 1 Im Im. (6) Im H H 1 Im H H Im H H Here in q. (6), we stress that the right-hand sides of Im and Im H H incorporate the respective Gauss s laws n n 0 0 and H 0 shown in q. (1). We remark that the derivation in q. (6) of the first equation for Im requires a much more care than that of the second part for Im H H. A key step is to recognize from n n 0 0 that n n 0. AdvNanonergy: 017: 1(1):34-38 ISSN: X 35

3 Let us explain in more detail differential operators on the right-hand sides of q. (6) in terms of summation symbols in Cartesian coordinates. 3 j j eˆ i i, j1 xi 3 ln n lnn i i1 xi. (7) Here, e ˆi is the unit vector in the i -th direction of Cartesian coordinates. Now, Im is the famous operator of al angular momentum (OAM). The magnetic part H H is defined in a similar way. The most important addition in q. (6) is the convective-derivative term Im ln n which stems from the inhomogeneous refractive index. The convection operator is familiar from fluid dynamics or plasma physics [3]. 4. Formulas for Poynting Vector Combing qs. (4) and (6), let us summarize various formulas discussed so far for Poynting vector. 1 n0 S Im H H n Poyn 1 Im n0 P H H. (8) n 1 1 Im S ln n Therefore, the light spin S defined here makes a solenoidal contribution 1 S to Poynting vector Because n Poyn P through the curl operator. ln 0 for a uniform refractive index n x, y, z n0, we can recover from q. (8) the following well-known formula []. 1 S Im H H. (9) Poyn 1 1 P Im H H S, Resultantly, the most conspicuous alteration due to inhomogeneous refractive index is the 1 additional term Im ln n for Poynting vector. Last but not the least, we find the following equalities. 1 Im Im ln n Im Im 1 H H Im H H. (10) Here, the convective derivative is defined in accordance to q. (7). The magnetic part H H is defined in a similar way. Let us now define the al part the spin part addition to the convection part P and spin P of angular momentum in conv P as follows. 1 n0 P Im H H n spin 1 P S. (11) conv 1 1 P Im Im H H Therefore, Poynting vector presented in q. (8) can be concisely decomposed in the following two different ways. conv spin 1 P P Im ln n Poyn spin 1 Im P P P ln n. (1) P P conv Of course, for homogeneous media with n x, y, z n0, q. (1) is simplified as follows. Poyn spin conv P P P P P. (13) 5. Thermo-Optic ffects From the perspective of materials processing, a question arises as to how to implement inhomogeneous refractive index n x, y, z for the benefit of optical applications. To this goal, let us consider a simple thermo-optic effect. For the sake of simplicity, let us focus on cylindrical wave AdvNanonergy: 017: 1(1):34-38 ISSN: X 36

4 configurations with the radial coordinate r x y and the azimuthal angle tan. 1 yx Consider an azimuthally periodic profile of refractive index depending on r,. In the absence of the dependence on the axial coordinate z, we may set, F rcos. (14) n r Suppose that the refractive index is proportional to temperature around a certain reference temperature [7]. A steady-state temperature distribution is governed by the following Laplace equation. 1 T 1 T r 0. (15) r r r r By assuming azimuthal periodicity with the phase factor expi with being integer, we obtain the temperature profile as a linear combination of r cos and r cos. In consideration of q. (14), we can therefore suppose that the radial function Fr should be a linear combination of r and r. Fig. 1. An example of refractive index profile on the transverse plane Figure 1 displays an example of refractive index profile on the transverse plane. n r, r 1cos4 n 0 r for r for r 1. (16) 1 Here, the reddish and bluish colours correspond to positive and negative values of nn 0 1. The darker the colours, the larger the values of nn 0 1 become. Notice that nn 0 1 in the far field as r. With this refractive index, we can evaluate the following convection derivative. 1 ln n ln ln r n n. (17) r r Here, 1 r for r 1 ln n cos 4 3 r n r for r 1. (18) 4 r for r 1 ln n sin 4 n r for r 1 7. Discussion Inhomogeneous media could give rise to an 1 additional non-trivial term Im ln n to Poynting vector of electromagnetic waves. This term contains the spatial gradient ln n of the logarithm of refractive index of media under study. The logarithmic function ln n is AdvNanonergy: 017: 1(1):34-38 ISSN: X 37

5 characteristic of two-dimensional potential, since nr, depends only on the planar coordinates. This inhomogeneous refractive index introduces an imbalance to the conventional breakdown of Poynting vector into its al and spin parts as Poyn al spin 1 P P P Im ln n. 1 The additional term Im ln n renders unclear the measurements of Poynting vector based on the knowledge on its two parts. From an opposite perspective, this additional term could give a clue to better measurements of the two parts. Likewise, this additional term can help to identify the spin- interactions between the two parts of electromagnetic waves. In this respect, the refractive index nr,, being the transverse refractive index profile (TRIP), may be perturbed only a little from the reference value n 0. This perturbation in the refractive index is called often the shallow refractive index modulation (SRIM) [6], thereby being more amenable to practical fabrication. Oue way of achieving inhomogeneous refractive index is the static method, where refractive index is modulated once and for all during the fabrication stage of optical structures. On the other hand, dynamical modulation via high-rate optical pulses can in principle vary the refractive index dynamically [6]. 8. Conclusions We have shown analytically how the conventional formula for Poynting vector is altered in the presence of inhomogeneous profiles of refractive index. Afterwards, we have shown how the refractive index can be varied by thermos-optic effects, albeit in a rough way. The incorporation of inhomogeneous refractive indices in dielectric materials can offer potentials of better optical manipulation and more subtile characterization of light spins and angular momentum. Acknowledgements This study has been supported by the National Research Foundation (NRF) of Republic of Korea (Grant Number: NRF-015R1D1A1A ). References [1] Jones PH, Maragò OM, Volpe G. Optical Tweezers: Principles and Applications, Cambridge University Press, 015. [] Bliokh KY, Nori F. Transverse and longitudinal angular momenta of light, Physics Reports 015; 59:1-38 [3] Landau LD, Pitaevskii LP, Lifshitz M. lectrodynamics of Continuous Media, Second dition, Butterworth-Heinemann, U.S.A., [4] O'Neil AT, MacVicar I, Allen L, and Padgett MJ, Intrinsic and xtrinsic Nature of the Orbital Angular Momentum of a Light Beam, Phys. Rev. Lett. 00, 88: [5] Wesfreid J, Burgos A, Mancini H, Quel. Calculation of the focal length of a thermal lens inside a laser cavity, Optics Comm. 1977; 1 (3): [6] Kartashov YV, Vysloukh VA, Torner L. Rotating surface solitons, Opt. Lett. 007; 3: [7] Dong L. Thermal lensing in optical fibers, Opt. xpress 016; 4: AdvNanonergy: 017: 1(1):34-38 ISSN: X 38