Review of Electromagnetic Fields

ORMAT 1. Review of electromagnetic fields

1.1 Maxwell’s Equations

· Review the features of Maxwell’s Equations in differential forms.

· Discuss these equations in different representations: space-time (r, t), space-frequency (r, ω), wavevector-frequency (k, ω).

1.1.1 Maxwell’s Eqs. in the space-time representation

· The first equation, Faraday’s induction law, establishes that a varying magnetic field produces a rotating electric field,

· E(r, t) is the electric field (V/m) and B(r,t) the magnetic induction or magnetic flux density (Wb/m2).

· Ampere’s circuital law states a magnetic field can be generated by existing currents and by varying electric fields,

· H(r,t) is the magnetic field intensity (A/m), D(r,t) the electric induction or flux density (C/m2), Js(r,t) the source current density (A/m2), and Jc(r,t) the conduction current density (A/m2).

· Gauss’s law for the electric and magnetic fields:

· ρ(r, t) is the volume charge density (C/m3), and

, 4)

· which establishes the non-existence of magnetic charge.

· In addition, the following constitutive relations apply:

· ε the dielectric permittivity, μ the magnetic permeability, and s the conductivity

· Generally, ε, μ, and σ, are inhomogeneous (r-dependent) and anisotropic (direction-dependent, tensors). They can also be field-dependent for nonlinear media.

· For linear, homogeneous, and isotropic media, ε, μ, and σ are simple scalars. The relative permittivity er and permeability μr are defined with respect to the vacuum values, ε0=10-9/36π F/m and N/A2.

1.1.2 Boundary conditions

· To solve Maxwell’s equations for fields propagating between two media, boundary conditions for both the electric and magnetic fields are necessary.

Figure 1. Fields at the boundary between two media

· 4 equations describe the relationship between the tangent components of E and H and the normal components of D and B in the two media (Fig. 1), as follows:

· n is the unit vector normal to the interface, Js is the surface current, and ρs is the surface charge density. In the absence of free charge and currents, the boundary conditions simply imply that the tangential fields (E, H) and normal inductions (D, B) are conserved across an interface.

1.1.3 Maxwell’s equations in the space-frequency representation (r, w)

· For broad-band fields, the spatial behavior of each temporal frequency is of interest. To obtain Maxwell’s equation in the space-frequency representation, we use the differentiation property of Fourier transforms,

· F is the Fourier transform operator (in time), F(t) is an arbitrary time-dependent vector, and F(ω) its Fourier transform.

· We will denote the Fourier transform of a function by the same symbol, with the understanding that F(t) and F(ω) are two distinct functions.

· Taking the Fourier transform of Eqs. 1-4 and using the differentiation property in Eq. 7, we can rewrite the Maxwell’s equations in the (r, ω) representation,

· The constitutive relations take the form

· Even for homogeneous media ε is a function of frequency, which establishes the dispersion relation associated with the medium. The boundary conditions apply for each individual frequency,

10)

1.1.4 The Helmholtz equation

· Eliminating H, B, and D from Eqs. 1-5, we obtain an equation in E(r, t), the wave equation. The Helmholtz equation can be obtained in the (r, ω) representation, as follows.

· First let us consider linear, isotropic, and charge-free media (r=0). Eqs. 8a-d simplify to

11)

· Applying the curl operator to Eq. 11a, we obtain

12)

· Using the identity and the fact that for charge-free media, we obtain the Helmholtz equation:

13)

· In Eq. 13a, the Laplace operator applied to vector E is defined as .

· Equation 13b establishes the frequency dependence of the wavenumber b(ω), the dispersion relation of the medium. Typical biological structures are characterized by constant μ, the permittivity ε and conductivity σ can have strong dependence on frequency.

· In Cartesian coordinates, the vector Eq. 13a becomes three scalar equations.

14)

· All the equations are of the same form, referred to as the scalar wave equation,

· Assuming solutions of the form , we obtain three 1D equations, which eventually lead to the plane wave solution of the form

15)

· Decomposing the wavevector into its real and imaginary parts,

, 16)

· We can rewrite Eq. 15a as

17)

· In Eq. 17 the real and imaginary parts of k capture the refraction (phase term) and absorption/gain (amplitude term) of the medium.

· In lossless media, . In dielectric media, at optical frequencies and equals the refractive index, thus

18)

· is the wavenumber in vacuum.

1.1.5 Maxwell’s equations in the (k, ω) representation

· Often we deal with optical fields with broad angular spectra. These can be decomposed in wavevectors k of different directions. The modulus of each k-vector is defined by the dispersion relation, depending on the material properties (ε, μ, σ) and optical frequency ω.

· The normal representation of the fields is in the k-ω space. A differentiation property similar to Eq. 7 holds for the Ñ operator,

19)

· The (k, ω) representation of Maxwell’s Equations is obtained by the spatial Fourier transformation of Eqs. 11a-d.

· For media of no free charge (ρ=0) or currents (J=0), these equations are

20)

· Equations 20a-d describe the propagation of frequency component ω and plane wave of wavevector k. Eqs. 20c-d establish that and . Generally μ is a scalar, i.e. B||H, but ε is a tensor, i.e. D is not necessarily parallel to E.

· Thus we see that, such that

21)

· Equations 20a-c show that H, D, and k are mutually orthogonal vectors.

Figure 1-2. Mutually orthogonal set of vectors

· For isotropic media, D||E, such that H, E, and k are also mutually orthogonal.

· The characteristic impedance of the medium is defined as

22)

· Using Eq. 15b to express k for media with non-zero conductivity, we obtain

23)

· For non-conducting media (electric insulators),. For isotropic media, connects the moduli of E and H directly (from Eq. 19b),

24)

1.1.6 Phase, group, and energy velocity

Figure 3. Phase and group velocity.

· Consider the electric field associated with a light beam propagating along the +z direction with an average wavevector <k>.

· As shown in Fig. 3, the temporal signal has a slow modulation (envelope) due to the superposition of different frequencies, and a fast sinusoidal modulation (carrier), at the average frequency <ω>,

25)

· Thus the phase delay of the field is

26)

· The phase velocity is associated with the advancement of wave fronts, for which j=constant. Differentiating Eq. 26, we obtain

27)

· Thus, the phase velocity is

28)

· Equation 28 represents the zeroth order approximation in the expansion:

29)

· Equation 29 defines the group velocity, vg, at which the envelope of the light signal propagates. The group and phase velocities are equal in non-dispersive media ().

· We prove that the group velocity defines the energy velocity of the field. The energy velocity is the ratio between the Poynting vector S and electromagnetic volume density U,

30)

· We now differentiate Eqs. 20a-b, as follows

31)

· The meaning of δk and δω is that of a spread in k-vectors (directions) and optical frequency. Dot-multiply Eq. 31a by H and Eq. 31b by E,

32)

· Adding the two Eqs. 32a-b, we obtain

33)

· using the fact that the products and vanish.

· Finally, using the definition in Eq. 30a, we find

34)

· Equation 34 establishes the result that the electromagnetic energy flows at group velocity; it is apparent from Eq. 29 that vg can exceed the speed of light in vacuum c, in special circumstances, where (anomalous dispersion).

· This does not pose a conflict with the postulate of the relativity theory, which states that the signal velocity, at which information can be transmitted via electromagnetic fields, is bounded by c.

1.1.7 The Fresnel Equations

· Consider light propagation at the interface between two media (Fig. 4). We will derive the expressions for the field reflection and transmission coefficients.

Figure 4. Two media of refractive index n1 and n2 separated by the x-y plane. The subscripts i, t, r refer to incident, transmitted, and reflected.

· In Figure 4, the x-z plane is the plane of incidence (plane of the paper), i.e. the plane defined by the incident wavevector and normal () at the interface.

· The tangential components of the fields and normal components of inductions are conserved in the absence of surface currents and charges (Eqs. 6a-d),

35)

· For ρ=0 and J=0, Maxwell’s Eqs. in the k-ω representation yield (Eqs. 19)

36)

· Expanding the cross products in Eqs. 36a-b, the problem breaks into two independent cases: a) transverse electric (TE) mode, when E is perpendicular to the plane of incidence (E||y) and b) transverse magnetic (TM) mode, when H||y.

i) TE mode (E||y)

· If E||y, the boundary conditions for the tangent E-field and normal B-field

37)

· Using Eq. 36b to express Eq. 37b in terms of Ey components, we can rewrite the system of equations as

38)

· Eqs. 38a-b must hold for any incident field Eyi, we obtain the following result

39)

· Eq. 39 establishes Snell’s law,

40)

· using that the wavevector k in a medium of refractive index n relates to the wavevector in vacuum as k=nk0.

· Eq. 40 implies kzi=-kzn.

· To obtain the field reflection coefficient, we use the continuity of tangent H components,

41)

· which can be expressed in terms of E components (via Eq. 36b),

42)

· Finally, combining Eqs. 42 and 37a to solve for the E field transmission and reflection coefficients, we obtain

43)

ii) TM mode (H||y)

· Using the analog equations to the TE mode (Eqs. 37a-b), the conservation of Hy components and normal D components, we find that kx=const. for TM as well.

· To obtain the field reflection and transmission coefficients, we use the conservation of both the tangents fields components, Ex and Hy,

44)

· The factor occurs due to the 1/ε factor in Eq. 36b (ε=n2). Thus the H field reflection and transmission coefficients for the TM mode are

45)

· We expressed rTM and tTM in terms of H fields to emphasize the symmetry with respect to the TE case. Of course, the quantities can be further expressed in terms of E fields via . Conservation of energy is satisfied in both cases,

46)

· Together, Eqs. 43a-b and 45a-b, the Fresnel equations, provide the reflected and transmitted fields for an arbitrary incident field. Because of the polarization dependence of the reflection and refraction coefficient, polarization properties of light can be modified via reflection and refraction. In the following we discuss two particular cases that follow from the Fresnel equations, where the transmission or reflection coefficients vanish.

1.1.8 Total internal reflection

· Setting tTE=tTM=0 yields the same condition for “no transmission” in both TE and TM modes, transmission coefficient vanishes, kzt=0. Thus,

47)

· using kx=constant (Eq. 38), kxt=kxi. The transmission vanishes for

48)

· θc is the critical angle at which total internal reflection takes place. Total internal reflection can occur for both TE and TM polarizations, the only restriction being that n2<n1.

· For angles of incidence that are larger than the critical angle, θi>θc, the field reflection coefficient becomes

49)

· .

· For θi>θc, the reflection coefficient is purely imaginary, the power is 100% reflected, but the reflected field is shifted in phase by 2fTE.

· Similarly, for the TM mode we obtain

50)

· Since fTM and fTE have different values, total internal reflection can be used to change the polarization state of optical fields.

· The transmitted plane wave has the form

51)

· Equation 51 indicates that the field in medium 2 is decaying exponentially.

Figure 1-5. Evanescent field decaying exponentially with depth z.

· Thus the field is significantly attenuated over a distance on the order of 1/kzt, i.e. the field does not propagate, or is evanescent.

1.1.9 Transmission at Brewster angle

· Another case of Fresnel’s equations is when the reflection coefficient vanishes.

· For the TE mode, we have

52)

· For TE polarization, the only way to obtain maximum transmission through an interface is when there is no refractive index contrast between the two media, the trivial solution.

· For the TM mode the situation is very different:

53)

· The condition is satisfied simultaneously with Snell’s law, such that we have

54)

· Multiplying Eqs. 54a and 54b side by side, we obtain

55)

· The angle of incidence at which rTM=0, referred to as the Brewster angle, is defined by combining Eq. 55 and Eq. 54a

56)

· Unlike with total internal reflection, where the transmission can vanish for both polarizations, the reflection can only vanish in the TM mode.

· The absence of reflection at the Brewster angle for TM polarization can be understood by the absence of radiation by an (induced) dipole along its axis (Fig. 6). The concept of induced dipoles followed by re-radiation is essential for the Lorentz model of light-matter interaction, as detailed in the next section.