Energy flux of a plane wave near a planar interface

When a plane wave strikes a planar interface between two media with refractive indices n₁ and n₂, some energy gets reflected back into the first medium, while some gets transmitted to the second medium. In this section, we want to derive the energy balance for a number of different materials and angles of incidence. We will start in the upper halfspace by considering the initial field E₁⁺ and the reflected field E₁⁻:

E₁⁺ = ( E_peˆ⁺_p + E_seˆ_s) exp(iq_xx+iq_yy+iw₁z),

E₁⁻ = (r_pE_peˆ⁻_p +r_sE_seˆ_s) exp(iq_xx+iq_yy−iw₁z). (2.142) Since the energy flux is related to the square of the amplitude of the electric field, we cannot simply calculate S for the incident and reflected field separately and then add them up. Instead, we have to determine the total electric field E1 =E₁⁺+E₁⁻ and the either real (propagating PW) or purely imaginary (evanescent wave). By inserting the definitions

one can easily show that:

ˆ

and we already know that ˆes= ˆes. We are interested in the fraction of energy that is reflected or transmitted at the interface, i.e. in the flux perpendicular to the surface.

Therefore, we now proceed to determine the z-component of the Poynting vector. As we have just seen, the z-components of all mixed p-waves are zero, while e.g.

ˆ

e_z·( ˆe⁺_p ×eˆ_s) = ˆe_z·( ˆe⁺_p ×eˆ_s) = eˆ_z·eˆ⁺_k =w/k, ˆ

e_z·( ˆe_s×eˆ⁻_p) = ˆe_z·( ˆe_s×eˆ⁻_p) = −ˆe_z·eˆ⁻_k =w/k etc. (2.146) Inserting into equation (2.143), this leads to the much simpler formula:

S1,z = c

8πk_ve^−2Im(w1)z Re

Epk1(Ep+rpEp)w1

k₁ +Esk1(Es−rsEs)w1

k₁

−r_pE_pk₁(E_p+r_pE_p)w1

k₁ +r_sE_sk₁(E_s−r_sE_s)w1

k₁

= c

8πk_ve^−2Im(w1)z Re

w1|Ep|² 1− |rp|²−2iIm[rp]

+w₁|E_s|² 1− |r_s|²+ 2iIm[r_s] , (2.147) where we took into account that k₁ ∈R. Let us now interpret this result, first for pure propagating p- or s-waves with w₁ ∈ R, E_s = 0 or E_p = 0. We start by comparing with the flux of the same wave in medium 1 without the presence of the interface, S_z,p/s^FS = (cw₁)/(8πk_v)|E_p/s|². Then,

S_1,z,p/s^prop = (1− |r_p/s|²)S_z,p/s^FS . (2.148) This is an expression for the net flux inz-direction, but we can interpret it as the sum of the fluxes of two waves: The incident wave with flux S_z,p/s^FS in the positive z-direction and the reflected wave with flux |r_p/s|²S_z,p/s^FS in the negative z-direction. Thus, the reflectance R – the ratio of the reflected to the initial intensity – of a propagating p- or s-wave is given by

R_p/s=|r_p,s|². (2.149)

On the other hand, if we assume that medium 2 is a lossless material, too, and that n2 > n1 (i.e. w2 ∈R), the flux in medium 2 is given by:

S₂^prop = cn₂

8π |t_pE_p|²+|t_sE_s|² ˆ

e_k2, (2.150)

which we found by settingE0 = tpEpeˆp2+tsEseˆs2 in equation (2.130). When comparing the total amplitudes of the Poynting vectors of the incident and the transmitted waves, i.e. the total intensities of the plane waves, we find the transmittance T:

Tp/s =|tp,s,|². (2.151)

Since there are no energy sources or sinks directly at the interface, the flux through the surface – i.e. the z-component of the fluxes – has to be identical in both media. In the second medium, this flux is given for pure p- and s-waves by

S_2,z,p/s^prop = c

8πk_vw₂|t_p,s|² |E_p,s|² = w2

w₁|t_p,s|² S_z,p/s^FS . (2.152) Thus, the condition S_1,z,p/s^prop =^! S_2,z,p/s^prop is identical to the relation

1−R_p/s =^! w₂

w₁T_p/s. (2.153)

This formula can be found in many standard textbooks. We can confirm that it is true by inserting the definitions of r_s/p and t_s/p from equation (2.103).

If the incident wave in medium 1 is evanescent, i.e. w₁ = iw⁰⁰,0 < w⁰⁰ ∈ R, equation (2.147) gives a different result:

S_1,z^evan = cw⁰⁰

4πk_ve^−2w00z |Ep|²Im[rp] +|Es|²Im[rs]

(2.154) We can directly draw some interesting conclusions. Firstly, if no interface was present, both r_p and r_s would be zero, which would lead to S_1,z^evan = 0 – exactly the result we found for evanescent waves in section 2.4.2. Secondly, a closer examination of the expressions for the two reflection coefficients

r_p = (n₂/n₁)−(w₂/w₁)

(n₂/n₁)²+ (w₂/w₁) and r_s = 1−(w₂/w₁)

1 + (w₂/w₁) (see 2.103)

reveals that for realn1, n2 and imaginaryw1, the imaginary part ofrp, rsis only nonzero for real w₂. Physically, this means that the combination of an incident and a reflected evanescent wave can only transport energy along z if the wave becomes a propagating wave in the second medium. This can also be seen by expressing the conservation of energy at the interface (z = 0) as:

2w⁰⁰Im(r_p/s)=^! Re(w₂)|t_p,s|². (2.155) This formula can usually not be found in textbooks, since they seldomly treat evanescent waves except for total internal reflection. In this context, it does not make sense to talk about reflectivity, since the incident evanescent wave would not transport energy in free space, anyway. We illustrate the results found so far in figure 2.15, which shows the fluxes S1,z and S2,z at an air/water interface for a wide range of q-values. Up to the black vertical line at q= k₁, the waves are propagating in both air and water. Directly at this line, w₁ = 0 and the wave is parallel to the interface. In this case, no energy is transported perpendicular to the interface. Between the black and orange vertical lines, the waves are evanescent in air but propagating in water, leading to a large energy flux from one medium to the other. At even larger q, the waves are evanescent in both media and no energy is transported along z anymore.

0 0.2 0.4 0.6 0.8 1 1.2 1.4

0 1 2 3 4 5

fluxSalongz[c/8π]

q/k₁[−]

S_1,z, p-wave S_2,z, p-wave S_1,z, s-wave S_2,z, s-wave

(a) Interface between air (n₁ = 1) and water (n₂ = 1.33).

0 0.05 0.1 0.15 0.2

0 1 2 3 4 5

fluxSalongz[c/8π]

q/k₁[−]

S_1,z, p-wave S_2,z, p-wave S_1,z, s-wave S_2,z, s-wave 0

10 20 30

0 1 2

(b) Interface between air (n1 = 1) and silver (n2 = 0.06 + 4.11iatλ= 612 nm).

Figure 2.15: Energy fluxSz parallel to thez-axis at both sides of an interface for p-and s-waves, plotted for various values of the component q of the wavevector parallel to the interface. The perfect match of the corresponding pairs of fluxes in medium 1 and 2 proves the conservation of energy at the interface. The inset in (b) only shows S_2,z because a dashed line would have been badly visible, S_1,z =S_2,z in this case, too.

The black vertical line atq =k₁ denotes the transition from propagating to evanescent waves in air. The orange vertical line in (a) denotes the transition from propagating to evanescent waves in water.

Finally, we want to investigate what happens if a plane wave reaches the interface between a lossless dielectric (n₁ ∈ R) and a metal (n₂ = n⁰₂ + in⁰⁰₂). In this case, w₂ = w₂⁰ +iw₂⁰⁰ is complex, w⁰₂, w₂⁰⁰ 6= 0. For propagating waves in the first medium, equations (2.148) and (2.149) which describe the reflectance are still valid. For evanescent waves in the same medium, i.e. w₁ = iw₁⁰⁰, we find that Im(r_p/s) is always nonzero – all evanescent waves can couple to the metal. This is in contrast to a dielectric halfspace where energy can only be transported if a propagating wave exists. Here, energy can be dissipated as heat instead, as we described at the end of the previous section. An example can be found in figure 2.15, which shows the fluxesS_1,z andS_2,z at an air/silver interface. Again, propagating waves in medium 1 are those up to the black vertical line at q = k₁. Directly at w₁ = 0, all energy is transported parallel to the interface and no energy enters the metal, the fluxes are zero. All evanesent waves with q > k₁ can couple to the metal, unlike at the air/glass interface. One prominant feature appears aroundq ≈1.07k₁, where the flux increases dramatically (see inset of figure 2.15). This is due to the excitation of surface plasmon polaritons (SPPs). An introduction to this phenomenon can be found for example in [55, 56], we will restrict ourselves to a short description here:

At the interface between a metal and a dielectric, the “sea” of conduction electrons – which is sometimes interpreted as a plasma – can be forced to oscillate collectively,

leading to an oscillation of the polarization. This resulted in the name surface plasmon polaritons. Mathematically, SPPs are described by electromagnetic waves which decay exponentially in both the positive and the negative z-direction. By considering the boundary conditions derived at the beginning of section 2.3, one finds that SPPs cannot exist for s-waves when both media are non-magnetic [56]. Therefore, no strong feature appears in the yellow/blue line representing the energy flux of s-waves in figure 2.15.

For p-waves, a dispersion relation can be derived [56]:

q_SPP² =k_v² n²₁·n²₂

n²₁+n²₂. (2.156)

At the air/silver interface atλ = 612 nm, this corresponds toq_SPP ≈(1.61 + 0.004i)k_v ≈ (1.07 + 0.002i)k₁. Thus, evanescent waves with q close to this value can couple to SPPs, leading to an enhanced energy flux. The fact that q_SPP has a (small) imaginary component suggests that SPPs have a finite propagation length along the interface – again, the kinetic energy of the electrons ultimately gets converted to heat.