Irradiance

In Chapter 6, we found that the time averaged Poynting vector in an isotropic medium is given by

hSi = ¹₂kEˆ ₀² ₀ µµ0

1/2

. (8.87)

This may be written in terms of the speed of light c = (ε₀µ₀)^−1/2 and the refractive index n = (εµ)^1/2to obtain

hSi = nE₀² 2µµ₀c

k.ˆ (8.88)

Note that this gives the intensity of the light or, equivalently, the irradiance.

We can therefore write Eq. (8.88) as

hSi = I ˆk. (8.89)

8.8.1 Reflectance and transmittance

Figure 8.12 shows a beam of light incident on the boundary between media of refractive indices n₁ and n₂. Also shown is the reflected beam with θ_r = θ_i as required by the Law of Reflection. The intensity over the surface at A with normal vector n is given by

I_A= −I_ikˆ_i· n = I_icos θ_i. (8.90) This also equals

8.8. IRRADIANCE 179

Figure 8.12: A beam of light incident on the boundary between media of refractive indices n₁ and n₂ together with the reflected beam. Note that θ_r = θ_i as required by the Law of Reflection. The area over which the incident intensity is spread is denoted by A and has normal vector n.

I_A= I_rkˆ_r· n = I_rcos θ_r. (8.91) We may then define the reflectance R as the ratio of the reflected and incident intensities.

R = Ircos θr

I_icos θ_i = Ir

I_i = E0r

E_0i

2

. (8.92)

Recalling the results of the previous section, we may express this in terms of the reflection coefficient r

R = r². (8.93)

An example of intensity reflectance is shown in Fig. 8.13. Note that the reflectance is low for small angles, only climbing significantly beyond the Brewster angle. This has a familiar consequence when viewing a reflec-tive surface from a low angle and the surface mirrors almost perfectly the surrounding environment (see Fig. 8.14).

The transmittance T may be defined as the ratio of the transmitted in-tensity to the incident inin-tensity. By a similar argument as before, we then have

T = I_tcos θ_t

I_icos θ_i = n₂cos θ_t n₁cos θ_i

 E_0t E_0i

2

. (8.94)

Figure 8.13: Chart of the intensity reflectance for s and p-polarised light.

Here, the refractive index of the incident medium n₁ = 1and the transmitted medium has n2 = 1.33(note that this is the refractive index of water).

Again, we may express this in terms of the transmission coefficient t

T = n₂cos θ_t

n₁cos θ_it². (8.95)

The conservation of energy

Imposing the principle of the conservation of energy (energy in = energy out), we have

Iicos θi = Ircos θr+ Itcos θt (8.96) or, dividing by I_icos θi

1 = Ircos θr

I_icos θ_i + Itcos θt

I_icos θ_i. (8.97)

Thus,

R + T = 1. (8.98)

8.9. TOTAL INTERNAL REFLECTION 181

Figure 8.14: View of a water surface from a low angle. Most of the incident light at such an angle is reflected.

8.9 Total internal reflection

In Chapter 4 we encountered the phenomenon of total internal reflection in which there is no transmission from a higher refractive index medium to a lower for incident angles less than the critical angle. Here, we shall modify this conclusion slightly to show that there is, in fact, a decaying wave transmitted into the lower refractive index medium.

Let us reconsider Fig. 8.3. We shall consider the case where n₁ > n2, so the total internal reflection will take place on the n₁side of the boundary.

Now the y component of k_tis given by

k_ty = k_tcos θ_t= k_t 1 − sin²θ_t
1/2

. (8.99)

Using Snell’s Law, this becomes

kty = ktcos θt= kt 1 − n1

n₂

2

sin²θt

!1/2

. (8.100)

This expression equals zero (implying zero transmission) when sin θ_i= n₂

n1

= sin θ_c, (8.101)

where θ_cis the critical angle. So, for θ_i> θc, we have

kty = ikt

 n1

n₂

2

sin²θt− 1

!1/2

≡ iγ_ty. (8.102) In general, we can write the transmitted wave as

E_t= E_t0e^{i(ωt−ktxx−ktyy)}. (8.103) Substituting iγ_tyfor k_ty, we have

E_t= E_t0e^{i(ωt−ktxx−iktyy)}, (8.104) which becomes

E_t= E_t0e^{i(ωt−ktxx)+γtyy}. (8.105) Since we have set the problem up so that the transmitted wave travels in the −ve y-direction, this represents an exponentially decaying wave. This is known as the evanescent wave.

8.9.1 Optical coupling in waveguides

(a) (b)

Figure 8.15: Optical coupling in two adjacent waveguides (n₂ > n₁). If the waveguides are close together, the evanescent wave may overlap the adjacent guide, leading to a coupling of power from one guide to the other.

The existence of the evanescent wave has a real-world application in optical coupling. Figure 8.15 shows an example for two adjacent slab waveguides.

As is suggested in the diagram, the electric field is not entirely contained within a given waveguide, with the evanescent wave spreading out from it.

If the waveguides are close enough, there may be significant overlap of the evanescent wave in the adjoining guide. It can be shown that this leads to a coupling of power between the guides.

A common example of an optical coupler arises in fibre optics. In this case, the central cores of two optical fibres are brought close together lead-ing to power transfer between the fibres (see Fig 8.16).

8.10. SUMMARY 183

Figure 8.16: Fibre optic coupler. Note that (as is common) the second input port has been removed.

Another use of couplers is in interferometry, when optical couplers are used as the basis of the fibre optic Mach-Zehnder interferometer . In this case, the power from an input fibre is divided equally between to output fibres. A phase shift is then applied to one arm of the Mach-Zehnder in-terferometer before the power is recombined (with interference effects) at a second coupler.

8.10 Summary

• Boundary conditions

The boundary conditions of the electromagnetic fields at interfaces between media of different refractive indices.

E_k is continuous across a boundary

H_k is continuous across a boundary.

When j_f 6= 0, we have the modified result that

H_2k− H_1k= j_f, (8.106)

D_⊥is continuous across a boundary.

If the free charge density is not zero and we have

D1⊥− D_2⊥= σf, (8.107)

where σ_f is the surface charge density .

B⊥is continuous across a boundary.

This result is always true.

• Reflection and refraction

The Laws of Reflection and Refraction may be derived by considering wavevector.

• Polarisation of incident wave – s-polarised

This refers to light polarised perpendicular to the plane of inci-dence. The ‘s’ stands for the German word senkrecht.

– p-polarised

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

• Fresnel equations

The Fresnel equations for the reflection and transmission coefficients rand t.

– s-polarised

∗ Reflection

r_s= n₁cos θ_i− n₂cos θ_t

n₁cos θ_i+ n₂cos θ_t = −sin (θ_i− θ_t)

sin (θ_i+ θ_t). (8.108)

∗ Transmission

ts= 2n1cos θi

n₁cos θ_i+ n₂cos θ_t = 2 cos θisin θt

sin (θ_i+ θ_t). (8.109) – p-polarised

∗ Reflection

r_p = n₂cos θ_i− n₁cos θ_t

n₂cos θ_i+ n₁cos θ_t = tan (θ_i− θ_t)

tan (θ_i+ θ_t). (8.110)

8.10. SUMMARY 185

∗ Transmission

tp= 2n1cos θi

n₂cos θ_i+ n₁cos θ_t = 2 cos θisin θt

sin (θ_i+ θ_t) cos (θ_i− θ_t). (8.111) – Brewster angle

When the angle of incidence equals the Brewster angle, the re-flected light is entirely s-polarised.

θ_B = tan⁻¹ n₂ n1



. (8.112)

– Phase change on reflection s-polarised light incurs a phase change of π on reflection from a lower refractive index medium for every angle of incidence. rp-polarised light also has a phase change of π when the angle of incidence is greater than the Brewster angle.

• Time reversibility

Use of the principle of time reversibility for analysis of reflection and transmission coefficients.

r⁰ = −r. (8.113)

t⁰t = 1 − r². (8.114)

• Stokes treatment

Alternative derivation of the results of time reversibility via the Stoke’s treatment.

• Irradiance

Analysis of power reflection and transmission between different me-dia.

– Reflectance

R = r². (8.115)

– Transmittance

T = n2cos θt

n1cos θi

t². (8.116)

– Conservation of energy

R + T = 1. (8.117)

• Total internal reflection

Wavevector analysis of total internal reflection to obtain the evanes-cent wave.

The evanescent wave decays as e^−γy, where γ is the imaginary wavevec-tor.

– Optical couplers

The evanescent wave may be exploited to obtain power trans-mission between adjacent waveguides.

Part IV

Geometrical Optics

187

9. Fermat’s Principle

9.1 General remarks

Although the subject of geometrical optics could be approached from wave optics, it is more typically presented in terms of ray tracing. For this, the appropriate theoretical tool is Fermat’s Principle.

This is a variational principle similar to Hero of Alexandria’s assertion that light travels by the shortest geometrical path. In fact, in an isotropic and homogeneous media, Fermat’s Principle reduces to Hero’s. More gen-erally, however, Fermat’s Principle states that light travels the path of short-est time. This is rendered in terms of the optical path length, which, al-though having dimensions of space, is actually proportional to the propa-gation time.

Using Fermat’s Principle, we begin in this chapter by deriving

• The Law of Rectilinear Propagation

• The Law of Reflection

• The Law of Refraction (Snell’s Law)

9.2 Learning objectives

The aims of this section are to gain understanding of

• Optical path length

• Fermat’s Principle

• Application of this principles to find:

– The Law of Rectilinear Propagation – The Law of Reflection

– The Law of Refraction (Snell’s Law)

• Perfect imaging: imaging all rays perfectly from a point or plane onto another point or plane

• Application of Fermat’s Principle to analyse 189

– Perfect mirrors – Perfect lenses

• The concept of curvature

• Finding the curvature of a surface

In document Optics (Page 178-190)