Methodological derivation of the eikonal equation

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1. Introduction One of the foundations of the simulation program we employ for modeling optical phenomena is the eikonal model [1, 2]. While this model is wellknown, the derivation process is somewhat intricate [3, 4]. In the renowned monograph by Born and Wolf [5], the derivation appears almost like a form of physical magic (here are Maxwell’s equations, a bit of magic, and voila, we have the eikonal equation). We decided to delve deeper into the derivation of the eikonal equation. We employed analytical methods to derive the eikonal equation from Maxwell’s equations in a medium without currents and charges. The process involves analyzing differential equations and applying methods of mathematical analysis. A brief outline of our study is presented in the scheme shown in the figure 1. Maxwell's equations Maxwell's equations without currents and charges Derivation of the wave equation Helmholtz equation for a monochromatic harmonic wave Monochromatic harmonic field Field Oscillation Assumption Solution of the Helmholtz equation at plane wave Finding a solution in the form of a plane wave High Frequency Wave / shortwave radiation Eikonal equation Figure 1. Paper structure 1. Article structure In section 1.2, we present basic notations and conventions used in the article. In section 2, fundamental relationships for Maxwell’s equations are introduced. In section 3, the wave equation is derived from Maxwell’s equations. Next, in section 4, the eikonal equation is obtained from the wave equation. The transformations are performed using vector formalism. In section 5, the same is done based on covariant tensor formalism. 2. Notations and conventions 2. The primary mathematical framework used in the article is the vector analysis (a brief overview is given in Appendix) and tensor analysis. A. V. Fedorov et al., Methodological derivation of the eikonal equation 401 3. We will adhere to the following conventions. Greek indices (α, β) will refer to the four-dimensional space, with the component values as follows: α = 0, 3. Latin indices from the middle of the alphabet (i, j, k) will refer to the three-dimensional space, with the component values as follows: i = 1, 3. 4. The CGS symmetrical system [6] is used for notating the equations of electrodynamics. 1. Introduction Consider Maxwell’s equations in vector-differential form: 1 ∂D 4π ∇ × H - c = ∂t 1 ∂B j, (1) c ∇ × E + c = 0, (2) ∂t where ∇ · D = 4πρ, (3) ∇ · B = 0, (4) § E(r, t) = E(x, y, z, t) is electric field strength vector; § H(r, t) = H(x, y, z, t) is magnetic field strength vector; § D(r, t) = D(x, y, z, t) is electric field induction vector; § B(r, t) = B(x, y, z, t) is magnetic field induction vector; § j(r) = j(x, y, z) is external electric current density (current strength per unit area); § ρ(r) = ρ(x, y, z) is electric charge density; § c is vacuum speed of light; §r = (x, y, z)T is radius vector of a point, written in Cartesian coordinates. Let us briefly describe the physical meaning of each of Maxwell’s equations: § equation (1) means that electric current and a change in electric induction generate a solenoidal magnetic field, that is a field whose field lines twist into a vortex along the vector indicating the direction of the current; § equation (2) means that a change in time of a magnetic field generates an electric field; § equation (3) means that the electric charge is the source of electrical induction; § equation (4) means that there are no free magnetic poles (only magnetic dipoles have been experimentally discovered, magnetic monopoles are not known to science). The following relations, called material equations, are also valid: j = σE, D = εE, B = µH, where σ(r) is conductivity, ε(r) is permittivity, and µ(r) is permeability. In an isotropic medium, ε and µ are scalar quantities, but in the general case they are tensor quantities. 402 DCM&ACS. 2023, 31 (4) 399-418 A medium is called isotropic if its physical properties do not depend on direction. The term comes from the Greek words «izos» (ισος): equal, identical, similar and “tropos” (τροπος): direction, character. In electrodynamics, the isotropy of a medium is associated with the same values of ε(r) and µ(r) in all directions. Magnetic permeability characterizes the magnetic properties of a medium (substance). If µ /= 1, then the substance is called magnetic, if µ > 1 - paramagnetic, if µ < 1 - diamagnetic. Next, we will consider a medium that does not conduct electricity, that is, σ = 0, and also free from currents, that is, j = 0 and ρ = 0, then Maxwell’s equations simplified somewhat: 1 ∂D ∇ × H - c = 0, (5) ∂t 1 ∂B ∇ × E + c = 0, (6) ∂t ∇ · D = 0, (7) ∇ · B = 0. (8) 2. Wave equation 1. Derivation of the wave equation from Maxwell’s equations We will assume that ρ = 0 and j = 0 and consider the equations (5) and (6): 1 ∂D ∇ × H - c = 0, ∂t 1 ∂B ∇ × E + c = 0. ∂t We use the material equations D = εE and B = µH and take into account the dependence of the permittivity and permeability on coordinates: ε = ε(x, y, z) and µ = µ(x, y, z). 1 ∂ µ ∂H 1 1 ∂H ⇒ ∇ × ∇ × E + c ∂t (µH) = 0 ⇒ ∇ × E + c = 0 E + ∂t µ c = 0. ∂t Apply the curl operator ∇× to the resulting equation: ( 1 \ ∇ × µ ∇ × E 1 ∂H c + ∇ × ∂t = 0. (I ) ( I I) Let us first consider term (II) of this equation. The time derivative can be taken out from under the sign of the rotor operator: ∂H ∂ ∇ × ∇ × = ( H). ∂t ∂t A. V. Fedorov et al., Methodological derivation of the eikonal equation 403 Due to (5) we get: ∂ (∇ × H) = ∂ 1 ∂D = 1 ∂2D . ∂t ∂t c ∂t c ∂t2 We use the material equation D = εE and write as follows: ∂2D ∂2 (ε(r)E) ∂2E 1 ∂H ε ∂2E ∂t2 = ∂t2 = ε ∂t2 ⇒ c ∇ × = ∂t c2 ∂t2 . To simplify term (I), we use the relation ∇ × f v = f ∇ × v + ∇f × v, where f (x, y, z) is a scalar function, and v(x, y, z) is a vector field. Using this relation, term (I) is expanded as follows: ( 1 \ ∇ × µ ∇ × E 1 µ = ∇ × (∇ × E) + ( 1 \ ∇ µ, ∇ × E . ( I. a) ( I. b) 2 In turn, to simplify term (I.a) we use the identity ∇ × ∇ × v = ∇(∇ · v) - ∇ v, where ∇2 is the Laplace operator. 1 1 1 2 µ ∇ × (∇ × E) = µ ∇(∇ · E) - µ ∇ E. To simplify the expression ∇(∇ · E) we apply the identity ∇ · (f v) = f ∇ · v + (∇f, v) to Maxwell’s equation (7), replacing induction D with tension using the material equation D = εE: ∇ · D = ∇ · (εE) = ε∇ · E + (E, ∇ε) = 0 ⇒ 1 1 1 ( 1 \ ∇ ⇒ ∇ · E = - ε (∇ε, E) ⇒ µ ∇(∇ · E) = - µ ∇ ( ε, E) . ε As a result, the term (I.a) was transformed to the following form: 1 1 1 ( 1 \ 2 ∇ µ ∇ × (∇ × E) = - µ ∇ By combining (I) and (II) we write: ( ε, E) ε - µ ∇ E. 1 2 1 ( 1 \ ( 1 \ ε ∂2E - µ ∇ E - µ ∇ ∇ ( ε, E) ε + ∇ µ, ∇ × E + c2 ∂t2 = 0, ( I. a) ( I. b) ( I I) 2 ( 1 \ ( 1 \ µε ∂2E -∇ E - ∇ ∇ ( ε, E) ε + µ ∇ µ, ∇ × E + c2 ∂t2 = 0, 2 µε ∂2E ( 1 \ ( 1 \l ∇ E - c2 ∂t2 + ∇ ∇ ( ε, E) ε - µ ∇ µ, ∇ × E = 0. 404 DCM&ACS. 2023, 31 (4) 399-418 A completely similar equation can be obtained for the magnetic field strength vector H. In the case of an isotropic medium, that is, ε = µ = const the additional term taken in square brackets vanishes and we obtain the wave equation: 2 εµ ∂2E ∇ E - c2 ∂t2 = 0, 2 εµ ∂2H ∇ H - c2 ∂t2 = 0. We can introduce the quantity v = c/√εµ - the speed of the electromagnetic wave in the medium. 2. The case of a plane wave Consider the wave equation: 2 1 ∂2U ∇ U - v2 ∂t2 = 0. Let’s consider an electromagnetic wave that propagates in the direction s, where s = (sx, sy , sz ) - some unit vector ( s = 1) fixed direction. Any solution of this equation, having the form U = U((r, s), t) is a plane wave, since at every moment of time the vector U is constant in the plane (r, s) = -d, where |d| is the distance from the plane to the origin. The expression (r, s) = -d is actually a normal plane equation, where the vector s acts as the unit normal vector. Let’s write it in Cartesian coordinates: sxx + sy y + sz z + d = 0. The wave equation can be simplified by introducing a new coordinate system. Since the intensity vector of a plane wave entirely depends only on the distance d, we can choose a new coordinate system with axes Oξ, Oη, Oζ so that the Oζ axis is directed along the vector s, and the origin coincides with the previous Cartesian system Oxyz. Then, the coordinate along the Oζ axis will depend on the previous coordinates according to the formula ζ(x, y, z) = (r, s) = sxx + sy y + sz z, while ξ and η do not depend on the previous coordinates and can be chosen arbitrarily, for example, so that the coordinate system Oξηζ is right-handed (see the figure 2). The replacement of differential operators is carried out using the Jacobian matrix as follows: ∂ ∂ξ ∂η ∂ζ ∂ ∂ξ ∂η ∂ζ ∂x ∂x ∂x ∂x ∂ξ ∂x ∂x ∂x ∂ ( ∂(ξ, η, ζ) \T ∂ξ ∂η ∂ζ ∂ ∂ξ ∂η ∂ζ ∂y = ∂y ∂y ∂y ∂η ; ∂(x, y, z) = ∂y ∂y ∂y . ∂ ∂ξ ∂η ∂ζ ∂ ∂ξ ∂η ∂ζ ∂z ∂z ∂z ∂z ∂ζ ∂z ∂z ∂z Since ξ = const and η = const, and ζ = (r, s), then: A. V. Fedorov et al., Methodological derivation of the eikonal equation 405 ∂ ∂ s ∂ x ∂ ∂ = sx , ∂x 0 0 s ∂ξ ∂ζ ∂x ∂ζ x ∂ ∂ ∂ ∂ ∂ ∂y = 0 0 sy = sy =⇒ = sy , ∂η ∂ζ ∂y ∂ζ ∂ 0 0 sz ∂ s ∂ ∂ ∂ ∂z ∂ζ z ∂ζ ∂z = sz ∂ζ . Figure 2. Plane (r, s) = const, where s is a unit vector indicating the direction of propagation of the electromagnetic wave. New coordinate axes are chosen so that the vector s is the unit vector of the Oζ axis. The other two axes Oξ and Oη are chosen arbitrarily and form a right-handed coordinate system Oξηζ The Laplace operator after replacing coordinates is transformed to the following form: 2 2 ∂2U 2 ∂2U 2 ∂2U 2 2 2 ∂2U ∂2U ∇ U = sx ∂ζ2 + sy ∂ζ2 + sz ∂ζ2 = (sx + sy + sz ) The wave equation simplifies: ∂ζ2 = ∂ζ2 . ∂2U 1 ∂2U ∂ζ2 - v2 ∂t2 = 0. We perform another substitution p = ζ - vt and q = ζ + vt, which leads to the following transformation of the differential operators: 406 DCM&ACS. 2023, 31 (4) 399-418 ∂ ∂ζ ( ∂(ζ, t) \T ∂ ∂p ∂p ∂ζ ∂q ∂ ∂ζ ∂p ∂ = ∂(p, q) ∂ = ∂p ∂q ∂ = ∂t ∂q ∂t ∂t ∂q ∂ ∂ ∂ ∂ ∂ ∂ I 1 1l ∂p + ∂p ∂q ∂ζ = ∂p + ∂q , = ∂ = ⇒ -v v ∂ ∂ + v -v ∂q ∂p ∂q ∂ ∂t ∂ = -v∂p ∂ + v . ∂q The second derivatives are expressed through the new variables as follows: ∂2 ∂2 ∂ ∂ ∂2 ∂ζ2 = ∂p2 + 2 ∂p ∂q + ∂q2 , ∂ ∂ 2 ( 2 2 ∂ ∂ ∂2 \ ∂t2 = v ∂p2 - 2 ∂p ∂q + ∂q2 . When the operators are substituted into the wave equation, it is simplified as follows: ∂2U 1 ∂2U ∂ζ2 - v2 ∂t2 = ∂2U = ∂p2 ∂2U + 2 ∂p∂q ∂2U + ∂q2 - 1 ( ∂2U - v2 2 v2 ∂p2 ∂2U ∂p∂q ∂2U \ + ∂q2 = ∂2U = 4 ∂p∂q = 0 ⇒ ∂2U ∂p∂q = 0 . The general solution of the transformed wave equation is the function U = U1(p) + U2(q) = U1 ((r, s) - vt) + U2 ((r, s) + vt) . Another approach to the solution uses separation of variables. We will look for the solution in complex form U(r, t) = U0(r)e-iωt. When substituting into the wave equation, we obtain: ∂2U 2 ∂t2 = -ω e-iωt U0(r), ∇2U = e- iωt 2 ∇ U0(r), 2 1 ∂2U 2 ω2 ∇ U - v2 ∂t2 = 0 =⇒ ∇ U0 + v2 U0 = 0. Let’s introduce some scalar quantities: wave number k = ω/v, k0 = ω/c, wave vector k = ks. Let us recall that c - the speed of light in a vacuum, v - the speed of an electromagnetic wave in a medium, n = √εµ - the refractive index of the medium, s - direction of wave propagation. The velocities v A. V. Fedorov et al., Methodological derivation of the eikonal equation 407 and c are related by the relations v = c/√εµ = c/n, so the wave number can also be written as k = ω/v = ω√εµ/c = k0n. Now the equation for U0 can be rewritten as: (∇2 + k2) U0 = 0. This equation is called Helmholtz equation (homogeneous Helmholtz equation). In the general case, its solution can be expressed in special functions, but in the case of a plane wave, the general solution can be written in the following form: U0(r) = u0(r)eik(s,r) = u0(r)eik0n(s,r). 3. Derivation of the eikonal equation We will also consider a strictly monochromatic harmonic wave, the intensity vectors of which can be written in the following form: E(r, t) = E0(r)e-iωt, H(r, t) = H0(r)e-iωt, where r = (x, y, z)T - radius vector of a point in space in a Cartesian coordinate system, ω - cyclic frequency. We also introduce the quantity k0 = ω/c = 2π/λ0, where λ0 is the wavelength in vacuum. Let’s substitute expressions for a monochromatic wave into Maxwell’s equations. We sequentially calculate all differential operators: ∇ × H = ∇ × (H0e-iωt) = e-iωt∇ × H0, ∇ × E = ∇ × (E0e-iωt) = e-iωt∇ × E0. Using the material equations D = εE and B = µH we replace D and B everywhere through E and H, taking into account that ε(r) = ε(x, y, z) and µ(r) = µ(x, y, z): ∇ · D = ∇ · (ε(x, y, z)E) = e-iωt∇ · (εE0), ∇ · B = ∇ · (µ(x, y, z)H) = e-iωt∇ · (µH0) . Let us replace D and B also in the expressions for derivatives, taking into account that ε and µ do not depend on time, as well as E0 with H0 from the formulas E(x, y, z, t) = E0(x, y, z)e-iωt , H(x, y, z, t) = H0(x, y, z)e-iωt: ∂D ∂ = ∂t ∂t (εE0e-iωt) = ε(x, y, z)E0(x, y, z) ∂e-iωt ∂t = -iεωE0e-iωt, ∂B ∂ = ∂t ∂t (µH0e-iωt) = µ(x, y, z)H0(x, y, z) ∂e-iωt ∂t = -iµωH0e-iωt. Let’s substitute the resulting expressions into the equation (5): 1 ∂D ✟iω✟t ω ✟iω✟t ∇×H- c ⇒ = 0 ✟e- ∂t ∇×H0+iε c E0✟e- = 0 ⇒ ∇ × H0 + iεk0E0 = 0 , 408 DCM&ACS. 2023, 31 (4) 399-418 then into the equation (6): 1 ∂B ✟iω✟t ω ✟iω✟t ∇×E+ c ⇒ = 0 ✟e- ∂t ∇×E0-iε c H0✟e- = 0 ⇒ ∇ × E0 - iεk0H0 = 0 , into the equation (7): ∇ · D = 0 ⇒ e-iωt∇ · (εE0) = 0 ⇒ ∇ · (εE0) = 0 , and finally into the equation (8): ∇ · B = 0 ⇒ e-iωt∇ · (µH0) = 0 ⇒ ∇ · (µH0) = 0 . As a result, the system of equations (5)-(8) takes the following simplified form: ∇ × H0 + iεk0E0 = 0, ∇ × E0 - iµk0H0 = 0, ∇ · (εE0) = 0, ∇ · (µH0) = 0. Let us make another simplification by assuming that E0(x, y, z) = e(x, y, z) exp (ik0u(x, y, z)) = e(r) exp (ik0u(r)) , H0(x, y, z) = h(x, y, z) exp (ik0u(x, y, z)) = h(r) exp (ik0u(r)) , (9) where u(x, y, z) = u(r) is a scalar real function called optical path, and e and h - vector position functions. Let’s calculate the differential operators again, this time from E0 and H0, using the formulas (18): ∇ × H0 = ∇ × (eik0u(r)h(r)) = eik0u(r)∇ × h + ∇(eik0u(r)) × h. The gradient of the function eik0u(r) is calculated as follows: ∇(eik0u(r)) = ( ∂eik0u(r) ∂x ∂eik0u(r) , ∂y ∂eik0u(r) \ , = ∂z ( ∂u(x, y, z) = ik0eik0u(r) , ∂x ∂u(x, y, z) , ∂y ∂u(x, y, z)\ = ∂z = ik0eik0u(r)∇u(x, y, z). As a result, the term ∇ × H0 of the first equation of the system (9) takes the form: ∇ × H0 = (∇ × h + ik0∇u × h)eik0u(r). (10) In a completely similar way, we obtain the expression for ∇ × E0 in the second equation of the system (9): ∇ × E0 = (∇ × e + ik0∇u × e)eik0u(r). (11) A. V. Fedorov et al., Methodological derivation of the eikonal equation 409 The computation of divergence is somewhat more complicated because the formula (18) will have to be applied twice. The first time we use it to write down the expression ∇ · εE0: ∇ · εE0 = ε(r)∇ · E0 + (∇ε, E0) . Next, we use it to calculate ∇ · E0, where instead of E0 we substitute the expression E0 = e(r) exp (ik0u(r)): ∇ · E0 = ∇ · 1e(r)eik0u(r)l = eik0u(r)∇ · e + (∇ (eik0u(r)) , e) = = eik0u(r)∇ · e + ik0eik0u(r) (∇u, e) = (∇ · e + ik0(∇u, e))eik0u(r), (∇ε, E0) = (∇ε, e)eik0u(r). As a result, the third equation of the system (9) takes the form: ∇ · (ε(r)E0(r)) = [ε(r)∇ · e(r) + ik0ε(r)(∇u(r), e(r)) + (∇ε(r), e(r))]eik0u(r). In a completely similar way, we obtain an expression for the magnetic field strength, that is, the fourth equation of the system (9): ∇ · (µ(r)H0(r)) = [µ(r)∇ · h + (∇µ(r), h) + ik0µ(r)(∇u(r), h)]eik0u(r). After substitution into Maxwell’s equations, we obtain: ∇ × H0 + iεk0E0 = 0 ⇒ ∇ × h + ik0∇u × h +iεk0e = 0 ⇒ ( 1 0) 1 0 ⇒ ∇u × h + εe = - ik ∇ × h, ∇ × E0 - iµk0H0 = 0 ⇒ ∇ × e + ik0∇u × e -iµk0h = 0 ⇒ ( 1 1) 1 0 ⇒ ∇u × e - µh = - ik ∇ · (εE0) = 0 ⇒ ε∇ · e + ik0ε(∇u, e) + (∇ε, e) = 0, ∇ × e, 1 0 ik0ε(∇u, e) = -(∇ε, e) - ε∇ · e = 0 ⇒ (∇u, e) = - ik ( 1 \ ε ∇ε, e l + ∇ · e . Since ( ∂ ln ε ∂ ln ε ∂ ln ε \ 1 ( ∂ε ∂ε ∂ε \ 1 ∇(ln ε) = , , ∂x ∂y ∂z 1 ∇ = , , o ∂x ∂y ∂z = ε, ε 0 (∇u, e) = - ik ((∇(ln ε), e) + ∇ · e). 410 DCM&ACS. 2023, 31 (4) 399-418 Calculations for the magnetic field are carried out in a completely similar way, resulting in the fourth equation: 1 0 (∇u, h) = - ik ((∇(ln µ), h) + ∇ · h), 1 ∇u × h + εe = - ik0 1 ∇ × h, ∇u × e - µh = - 1 ik0 ∇ × e, (12) (∇u, e) = - ik0 1 ((∇ (ln ε) , e) + ∇ · e) , 0 (∇u, h) = - ik ((∇ (ln µ) , h) + ∇ · h) . The third and fourth equations from this system follow from the first two. This can be proven by scalarly multiplying the first two equations by ∇u and using the fact that the result of a vector product is orthogonal to both of its factors: (∇u, ∇u × h) +ε(∇u, e) = 0 ⇒ (∇u, e) = 0. = 0 We consider only the first two equations. Let’s express h from the second equation through u and e and substitute it into the first: 1 h = µ ∇u × e ⇒ ∇u × ( 1 \ µ ∇u × e + εe = 0 ⇒ ∇u × ∇u × e + εµe = 0. For the vector product the following identity holds: a × b × c = b(a, c) - c(a, b) from which it follows 2 ∇u × ∇u × e = ∇u(∇u, e) - e(∇u, ∇u) = ∇u(∇u, e) - e ∇u , 2 ∇u(∇u, e) - e ∇u + εµe = 0. From the third equation of the system (12) it follows that (∇u, e), therefore 2 -e ∇u 2 + εµe = 0 ⇒ e ∇u = εµe. Equating the coefficients in front of the vector e and taking into account that n(r) = jε(r)µ(r) we write the equation: 2 ∇u = n2(r), (13) which is the eikonal equation. The function u(r) = u(x, y, z) is also called eikonal, and the surfaces u(x, y, z) = const - geometric wave fronts. A. V. Fedorov et al., Methodological derivation of the eikonal equation 411 In component form in Cartesian coordinates equation (13) becomes: ( ∂u \2 ∂x ( ∂u \2 + ∂y ( ∂u \2 + ∂z = ε(x, y, z)µ(x, y, z) = n2(x, y, z), 2 ∇u = (∇u, ∇u) = ( ∂u \2 ∂x ( ∂u \2 + ∂y ( ∂u \2 + . ∂z 4. Derivation of eikonal in covariant form Let us demonstrate the derivation of the eikonal equation using the tensor formalism. 0. Vector operators in covariant form Vector operators in covariant form: - ∇V-is covariant derivative with respect to the vector field v; ∂ § ei = ∂xi is coordinate basis, ∇ ei = ∇j ; § εijk = εijk is Levi-Civita symbol; 1 | § eijk = j g|εijk , eijk = j εijk are alternating tensors (Levi-Civita |g| tensors); § ∇f = ∇if = ∂if , f is scalar field; 1 √ g - ∇ · f = ∇iV i = √ ∂i( gV i); § x = (x1, x2, x3)T is contravariant vector; 1 - ∇ × V g = eijk ∇j Vk = eijk ∂j Vk = √ εijk ∂j Vk . 1. Maxwell’s equations without currents and charges The strength of the electric and magnetic fields in the form of a covector (denoted by “∼” above the letter, the designations can be changed), and D and B are vectors: E˜ = (E1, E2, E3), H˜ = (H1, H2, H3), D = (D1, D2, D3)T , B = (B1, B2, B3)T . Material equations: Bi = µij Hj , Di = εij Ej . Vector, covector fields: E˜( x, t), H˜ ( x, t), D (x˜, t), B (x˜, t). Tensor fields: µij ( x), εij ( x). 412 DCM&ACS. 2023, 31 (4) 399-418 2. Vector-differential form of writing Maxwell’s equations ∇ × H 1 ∂D - = 0, c ∂t ∇ × E + 1 ∂B = 0, c ∂t ∇ · D = 0, ∇ · B 1 = 0; 1 dBi ijk √gε ∂j Ek + c dt = 0, 1 ijk 1 dDi √gε dj Hk - c dt = 0, 1 √ i √g∂i( gD ) = 0, 1 √ √g ∂i( gBi) = 0. 3. Monochromatic harmonic wave Assumption No. 1: Monochromatic harmonic wave: Ek = E0k e-iωt, Hk = H0k e-iωt, Dk = εklEl = εklE0le-iωt, k kl B = µ Hl = µkl H0k e -iωt, dDi d ij iωt) ij dt = dt (ε d i E0j e- = -iωε E0j , 0j dB = (µij H e-iωt) = -iωµij H0j , dt dt ∂i(√gD ) = ∂i(√gε E0j e ) = e ∂i(√gε E0j ), i ij -iωt -iωt ij ∂i(√gBi) = ∂i(√gµij H0j e-iωt) = e-iωt∂i(√gµij H0j ). Formulas: 1 √gε ijk ∂j E0k - ik0µij H0j = 0, (14) 1 √gε ijk ∂j H0k + ik0εij E0j = 0, (15) ∂i(√gεij E0j ) = 0, (16) ∂i(√gµij H0j ) = 0. (17) A. V. Fedorov et al., Methodological derivation of the eikonal equation 413 Assumption No. 2: E0k = ek eik0u(--x), H0k = hk eik0u(--x), where u( x) is an eikonal: ∂j E0k = (∂j ek )eik0u + ek eik0uik0∂j u = (∂j ek + ik0ek ∂j u)eik0u, ∂j H0k = (∂j hk )eik0u + hk eik0uik0∂j u = (∂j hk + ik0hk ∂j u)eik0u. From the equation (16): ∂i(√gεij ej eik0u) = ∂√g = ∂xi εij ej eik0u + √ ∂εij g ∂xi ej eik0u + √ gεij ∂ej ∂xi eik0u + √ gεij ej ik0eik0u ∂u ∂xi = = (∂i√gεij ej + √g∂iεij ej + √gεij ∂iej + ik0√gεij ej ∂iu)eik0u = 0, ∂i√gεij ej + √g∂iεij ej + √gεij ∂iej + ik0√gεij ej ∂iu = 0, √gεij e ∂ u = -1 0 j i ik (∂i √ gεij ej + √ g∂iεij ej + √ gεij ∂iej ). Similarly from the equation (17): √gµij hj ∂iu = - 1 ik0 (∂i √ gµij hj + √ g∂iµij + √ gµij ∂ihj ) . 1 k Provided that λ is small, ω is large, ⇒ k0 is large, and 0 obtain: is small ⇒ we to transform: √gεij ej ∂iu = 0, √gµij hj ∂iu = 0; εij ej ∂iu = 0, µij hj ∂iu = 0 1 √gε ijk (∂j ek +ik0ek ∂j u)-ik0µij 1 hj = 0 ⇒ -√gε ijk ek ∂j u+µij 1 0 h0 = ik 1 √gε ijk ∂j ek , 1 √gε ijk 1 ∂j hk + √gε ijk ik0hk ∂j u + ik0εij ej = 0, 1 √gε ijk hk ∂j u + εij 1 0 ej = - ik 1 √gε ijk ∂j hk . Maxwell’s equations are reduced to the following form: εijk ek ∂j u - √gµij hj = 0, √ εijk hk ∂j u + gεij ej = 0, ij ε ej ∂iu = 0, µij hj ∂iu = 0, 414 DCM&ACS. 2023, 31 (4) 399-418 where εijk is Levi-Civita symbol, εij is permittivity, subject to k0 → ∞. From the first equation we express hj and substitute it into the second: -1 µli ε ijk -1 ij ek ∂j u - √gµli µ hj = 0. Let’s make the replacement: µ-1µij = gj : µ-1 ijk li l j -1 ijk 1 -1 ijk li ε ek ∂j u - √ggl hj = 0 ⇒ √ghl = µli ε ek ∂j u ⇒ hl = √g µli ε ek ∂j u. We transform the indices to substitute into the second equation: 1 1 lmn kl hk = √gµ- ε en∂mu, εijk 1 -1 lmn √ ij √gµkl ε en∂mu∂j u + gε ej = 0, kl ε εijk µ-1 lmn en∂mu∂j u + gεij ej = 0, kl ε εijk µ-1 lmn ∂mu∂j uen + gεin en = 0. The eikonal equation (13) takes the form: gij ∂iu∂j u = εij µij . 5. Conclusion We hope that our work clarifies the process of derivation of the eikonal equation. And allows us to better understand the hierarchy of models in electrodynamics in general, and in optics in particular. The questions of solving the eikonal equation [7, 8] we left outside the boundaries of our consideration. Appendix. Vector analysis If at each point P of a certain spatial region of the Euclidean space Rn some scalar or vector quantity is associated, then they say that a field (scalar or vector ). · Examples of vector fields include the velocity field v(x, y, z), the force field F(x, y, z), the electrical intensity field E(x, y, z). · Examples of scalar fields: temperature field T (x, y, z), electric potential field ϕ(x, y, z). Everywhere below we consider a three-dimensional point Euclidean space on which a Cartesian coordinate system is introduced. We denote the vectors (basis vectors) of this coordinate system as (ex, ey , ez ). The coordinates of a point are specified by the radius vector r = (x, y, z)T , which is plotted from the origin O. Along with notation of coordinates x, y, z, it is sometimes convenient to use indices: x1, x2, x3, and also write the radius vector in the form x = (x1, x2, x3)T . Index notation makes it possible to briefly write A. V. Fedorov et al., Methodological derivation of the eikonal equation 415 formulas using the summation sign Σ, which is especially convenient if a nonunit metric is used. A scalar field in some region of space R3 is a real-valued function f : f : R3 → R, f (x, y, z) = f (r) ∈ R. In turn, a vector field in a region of space R3 is a vector-valued function V: V : R3 → R3, V(x, y, z) = V(r) = Vx(r)ex + Vy (r)ey + Vz (r)ez ∈ R3. The Gradient of the scalar field f (r) is a vector calculated in Cartesian coordinates as follows: ( ∂f ∂f ∂f \ ( ∂ ∂ ∂ \ ∇f (x, y, z) = gradf (x, y, z) = , , ∂x ∂y ∂z , ∇ = , , . ∂x ∂y ∂z The sign nabla ∇ denotes the Hamiltonian vector differential operator. In order to emphasize its “vectority”, the symbol ∇ is written in bold. To simplify the presentation, we made some inaccuracies in the presentation, which should be mentioned separately. · Strictly speaking, the gradient is a covector. Our definition reflects this by writing the vector components in a row rather than a column. · The definition of the gradient is based on the Cartesian coordinate system. A more general definition should be given in a componentless form. The scalar field f (r) generates a vector field ∇f , which characterizes the direction of the greatest change in the scalar field f (r). Divergence of the vector field V = (Vx, Vy , Vz )T is a scalar, calculated in Cartesian coordinates as follows: ∇ · V = divV = ∂Vx + ∂x ∂Vy + ∂y ∂Vz = ∂z 3 '\" ∂V i ∂xi . i=1 Here “·” denotes the scalar multiplication operation ∇ · V = (∇, V). The Rotor of a vector field V is a vector calculated in Cartesian coordinates as follows: ex ey ez ∇ × V = ∂/∂x ∂/∂y ∂/∂z = Vx Vy Vz ( ∂Vz = ∂y ∂Vy \ - ∂z ex + ( ∂Vx ∂z ∂Vz \ - ∂x ey + ( ∂Vy ∂x ∂Vx \ - ∂y ez . Highlight also that the rotor is not a vector in the strict sense. In classical vector analysis it is called a pseudovector, but a deeper geometric meaning is revealed only when tensor algebra is involved, where the rotor can be represented either as a 2-form or as a bivector. 416 DCM&ACS. 2023, 31 (4) 399-418 Also, when writing the wave equation, the Laplace operator will be used, which is written in the following form: 2 ∂2 ∂2 ∂2 ∇ = (∇, ∇) = ∂x2 + ∂y2 + ∂z2 . We will also need the following two relations [5]: ∇ × f V = f ∇ × V + ∇f × V, ∇ · f V = f ∇ · V + (∇f, V). (18) A vector field is called potential if there exists a scalar field f (x, y, z) such that ( ∂f ∂f ∂f \ V = ∇f = , , , ∂x ∂y ∂z df = Vx dx + Vy dy + Vz dz . In turn, a vector field is called solenoidal (tubular) if there exists a vector field U such that V = ∇ × U.

About the authors

Arseny V. Fedorov

RUDN University

Author for correspondence.
Email: 1042210107@rudn.ru
ORCID iD: 0000-0002-3036-0117

PhD student of Probability Theory and Cyber Security

6 Miklukho-Maklaya St., Moscow, 117198, Russian Federation

Christina A. Stepa

RUDN University

Email: 1042210111@pfur.ru
ORCID iD: 0000-0002-4092-4326

PhD student of Probability Theory and Cyber Security

6 Miklukho-Maklaya St., Moscow, 117198, Russian Federation

Anna V. Korolkova

RUDN University

Email: korolkova_av@rudn.ru
ORCID iD: 0000-0001-7141-7610

Candidate of Sciences in Physics and Mathematics, Associate Professor of Department of Probability Theory and Cyber Security

6 Miklukho-Maklaya St., Moscow, 117198, Russian Federation

Migran N. Gevorkyan

RUDN University

Email: gevorkyan_mn@rudn.ru
ORCID iD: 0000-0002-4834-4895

Candidate of Sciences in Physics and Mathematics, Associate Professor of Department of Probability Theory and Cyber Security

6 Miklukho-Maklaya St., Moscow, 117198, Russian Federation

Dmitry S. Kulyabov

RUDN University; Joint Institute for Nuclear Research

Email: kulyabov_ds@rudn.ru
ORCID iD: 0000-0002-0877-7063
Scopus Author ID: 35194130800
ResearcherId: I-3183-2013

Doctor of Sciences in Physics and Mathematics, Professor of Department of Probability Theory and Cyber Security of Peoples’ Friendship University of Russia named after Patrice Lumumba (RUDN University); Senior Researcher of Laboratory of Information Technologies, Joint Institute for Nuclear Research

6 Miklukho-Maklaya St., Moscow, 117198, Russian Federation; 6 Joliot-Curie St., Dubna, 141980, Russian Federation

References