Fresnel equations

8.5.1 s and p-type polarisation

Before proceeding, we briefly introduce a common scheme for denoting the polarisation of waves propagating across a dielectric boundary. The notation is only valid for linearly polarised light and is given in reference to the plane of incidence defined by the incident wave-vector and the normal to the boundary.

8.5. FRESNEL EQUATIONS 167

Figure 8.4: Sketch showing the orthogonal components of the polarisation parallel (p) and perpendicular (s) to the plane of incidence (the plane con-taining the incident and reflected ray of light).

s-polarised

This refers to light polarised perpendicular to the plane of incidence.

The ‘s’ stands for the German word senkrecht.

This is also often referred to as transverse electric (TE). Note that for linearly polarised light, the electric field will be perpendicular, or transverse, to the plane of incidence.

p-polarised

This refers to light polarised parallel to the plane of incidence.

This is also often referred to as transverse magnetic (TM). Note that for linearly polarised light, the magnetic field will be perpendicular, or transverse, to the plane of incidence.

8.5.2 Derivation of the Fresnel equations s-polarised light

Figure 8.5 illustrates s-type polarisation (with the electric field E normal to the page) showing the magnetic field B, with the incident light taken to be from the medium with refractive index n₁. From the continuity of the H_k at the boundary, we have

Hsicos θi− H_srcos θr= Hstcos θt. (8.34) Similarly, the continuity of B_⊥gives

Figure 8.5: s-type polarisation (with the electric field E normal to the page) showing the magnetic field B.

B_sisin θ_i+ B_srsin θ_r= B_stsin θ_t. (8.35) Using θ_i = θ_r = θ₁ and θ_t= θ₂, we can rewrite these expressions, putting Hin terms of B in Eq. (8.34) via the relative permeability. This gives

1

µ₁(B_si− B_sr) cos θ₁= 1

µ₂B_stcos θ₂ (8.36) and

(Bsi+ Bsr) sin θ1 = Bstsin θ2. (8.37) Now, we may use Snell’s Law to substitute for sin θ₂ so that Eq. (8.37) becomes

(Bsi+ Bsr) = Bst

n1

n2

. (8.38)

Eliminating B_stfrom Eqs. (8.36) and (8.38) gives µ2

The ratio B_sr/B_si is therefore Bsr

B_si = n⁰₁cos θ1− n⁰₂cos θ2

n⁰₁cos θ₁+ n⁰₂cos θ₂. (8.41)

8.5. FRESNEL EQUATIONS 169 where n⁰_i= ni/µi. Now, from Eqs. (6.26) and (6.51) of Chapter 6, we have

|B| = nk0

ω |E| . (8.42)

Since ω is the same in either medium we have B_sr

Bsi

= E_sr Esi

(8.43) and thus the reflection coefficient for s-polarised light is

r_s= Esr In this case, we have

Est

Figure 8.6: p-type polarisation, in which E is parallel to the plane of inci-dence.

p-polarised light

In this case, we consider p-type polarisation in terms of the electric field.

Figure 8.6 illustrates the situation. Now, the continuity of E_kat the boundary gives

E_picos θ_i− E_prcos θ_r = E_ptcos θ_t (8.52) and the continuity of D⊥gives

Dpisin θi+ Dprsin θr= Dptsin θt. (8.53) Using θ_i = θ_r = θ₁ and θ_t = θ₂, we can rewrite these expressions, with Eq. (8.53) in terms of E, as

(E_pi− E_pr) cos θ₁ = E_ptcos θ₂ (8.54) and

₁(E_pi+ E_pr) sin θ₁= ₂E_ptsin θ₂. (8.55) Using Snell’s Law in conjunction with

n₁ n2

= n₂₁µ₁ n12µ2

, (8.56)

Eq. (8.55) becomes

₁(E_pi+ E_pr) = ₂E_ptn₁ n2

= E_ptn₂₁µ₁ n1µ2

. (8.57)

8.5. FRESNEL EQUATIONS 171 Hence

(E_pi+ E_pr) = E_ptn₂µ₁

n₁µ₂ = E_ptn⁰₂

n⁰₁, (8.58)

where, again, n⁰_i = n_i/µ_i.

Eliminating Eptfrom Eqs. (8.54) and (8.58), we have (Epi− E_pr)cos θ1

cos θ2

= (Epi+ Epr)n⁰₁

n⁰₂, (8.59)

which may be rearranged to give

rp= E_pr Epi

= n⁰₂cos θ1− n⁰₁cos θ2

n⁰₂cos θ1+ n⁰₁cos θ2

. (8.60)

Eliminating E_pr from Eqs. (8.54) and (8.58), we have 2Epi= Ept

 n⁰₂

n⁰₁ +cos θ2

cos θ₁



(8.61) or

t_p= Ept

E_pi = 2n⁰₁cos θ1

n⁰₂cos θ₁+ n⁰₁cos θ₂. (8.62)

Figure 8.7: Graph of the s and p-polarised reflection and transmission co-efficients for light incident from the lower n1 medium. Note that rp goes through zero at the Brewster angle θ_B.

8.5.3 Alternative forms

Eliminating the refractive indices using Snell’s Law the Fresnel equations for the reflection and transmission coefficients for s and p-polarised light may be given in the following alternative forms.

s-polarised

These results are plotted in Fig.8.7 for light incident from the n1 (lower re-fractive index) medium. Note that r_s is negative over every angle of inci-dence and r_p becomes negative at oblique angles. This corresponds to a phase change of π as the light passes from the lower refractive medium to the higher. In general, it is possible for the coefficiencts to exhibit other phase changes and must therefore be considered to be complex quantities.

8.5. FRESNEL EQUATIONS 173 8.5.4 Brewster angle

Equation (8.65) for the reflection coefficient rp may be written as

rp = tan (θi− θ_t)

tan (θi+ θt) = sin (θi− θ_t) cos (θi+ θt)

sin (θi+ θt) cos (θi− θ_t). (8.67) Now when θ_i + θ_t = π/2, cos (θ_i+ θ_t) = 0, so r_p = 0. In other words, no p-polarised light is reflected. The angle of incidence θ_Bat which this occurs is called the Brewster angle. This is a case of polarisation on reflection, since the reflected light will all be s-polarised.

We may derive an expression for the Brewster angle by noting that, using Snell’s Law, θ_B+ θt= π/2implies

Taking the sin of this, we have

n₁ Rearranging and taking the arctangent of both sides then gives us our re-quired result

The phenomenon of polarisation by reflection at the Brewster angle gives us a strategy for reducing glare. For example, Fig. 8.8 illustrates the case on a sunny day when light reflected from the road or bonnet of a car can have an adverse effect on visibility for the motorist. Most of the reflected light will be s-polarised (aligned parallel to the ground). Polarising sunglasses are usually designed with the transmission axis in the vertical direction. Hence, such glasses will cut out much of the glare.

Occasionally, car wind screens are also treated with a polarising film.

So long as the transmission axes of the screen and glasses are aligned, there will be little problem. However, if the driver were to rotate his or her lenses by 90^◦then the polarisers would become crossed and almost all the light would be blocked!

Figure 8.8: On a sunny day, there may be significant glare from the road or the bonnet of a car. Since this is reflected light, it will tend to be s-polarised.

This glare can then be greatly reduced by wearing polarising sunglasses to block the s-polarised component.

(a) (b)

Figure 8.9: An example of the low reflection coefficient near the Brewster angle for p-polarised light. In (a), the window is practically opaque due to the bright reflection from the Sun. In (b), however, a polarising filter is used on the camera to block s-polarised light. Since the amplitude of p-polarised light is already low, most of the light seen from the window is transmitted through it from the room inside.

Polarizing filter on camera

A similar application is shown in Fig. 8.9. Here, a polarising filter attached to a camera blocks most of the reflected light from a window. This has the result that most of the light from the window is now transmitted from inside the room. In other words, the interior of the room is now made visible.