Teóricos y prácticos

7.7 Analysis of polarised light

7.7.1 Linear analyser

In addition to its role preparing linear states of polarisation, a linear polariser may also be used to analyse light in a general state of polarisation into its linear components. In this role, it is often referred to as an analyser.

Consider light in a general state of polarisation E = |E₀|

 cos θ₁cos ωt sin θ₁cos (ωt + Γ)



. (7.87)

This is to be passed through an analyser with its transmission axis at an angle θ₂ to the x-axis. The unit vector in this direction is then

ˆ p =

 cos θ₂ sin θ₂



. (7.88)

The analyser will only pass the component of light polarised in this direction.

Thus, the amplitude of the transmitted light E_p is found by taking the dot product

Ep= E · ˆp = |E0| [cos ωt cos θ₁cos θ2+ cos (ωt + Γ) sin θ1sin θ2] . (7.89) Putting

E_0x = |E₀| cos θ₁, E_0y = |E₀| sin θ₁,

(7.90) this may be rewritten

E_p = E_0xcos ωt cos θ₂+ E_0ycos (ωt + Γ) sin θ₂. (7.91) The intensity of the transmitted light will be proportional to the squared modulus of this, that is

|E_p|² = E²_0xcos²ωt cos²θ2+ E_0y² cos²(ωt + Γ) sin²θ2+ + 2E0xE0ycos ωt cos (ωt + Γ) cos θ2sin θ2] .

This expression is still time dependent. Averaging over a period of the optical oscillation T = 2π/ω, we have

D The time-averaged intensity may then be given as

I (θ) = I0 cos²θ + r²sin²θ + 2r cos θ sin θ cos Γ
, (7.92) where r = E_0y/E0x.

Consider the case where the incident light is linearly x-polarised, that is E_0y= 0. In this case, Eq. (7.92) reduces to

I (θ) = I0cos²θ. (7.93)

This is known as Malus’ Law for the transmission of linearly polarised light.

For circularly polarised light, r = 1 and Γ = ±π/2. In this case, we have I (θ) = I0 cos²θ + sin²θ
= I₀. (7.94) In other words, the time-averaged intensity is constant.

7.8 Summary

• Linear polarisation

Light polarised in a fixed direction all along the propagation direction is said to be linearly polarised. For light travelling in the z-direction, the general expression for linearly polarised light is

E₀= E₀

7.8. SUMMARY 155

• Retardation

In anisotropic media, orthogonal components of the light may see different refractive indices. This introduces a phase shift between the components known as the retardation Γ.

• Circular polarisation

If the y component of a linearly polarised plane wave is multiplied by a factor e^i±π/2, then it will acquire a phase shift of Γ = ±π/2. When E_0x= E_0y, this leads to circularly polarised light. There are two cases to consider:

– Γ = π/2 right circularly polarised

In this case, the electric field vector at a particular point along the z-axis rotates in a clockwise direction in the E_x-E_yplane. This is known as right circularly polarised light.

E₊ = E₀

– Γ = −π/2 left circularly polarised

In this case, the electric field vector at a particular point along the z-axis rotates in an anti-clockwise direction in the E_x-E_y plane.

This is known as left circularly polarised light.

E−= E₀

In the general case, light is elliptically polarised, with the components of the electric field E_xand E_y satisfying

 Ey

The rotation of the electric field vector about the propagation direction depends on the retardation Γ.

– Case: 0 < Γ < π. Rotation is clockwise.

– Case: −π < Γ < 0. Rotation is anti-clockwise.

The Jones vector for the general case is

A Jones matrix represents the operation of an optical element on a state of polarisation (represented by a Jones vector). Examples cov-ered are

– Linear polariser (with the transmission axis at an angle θ to the E_xaxis)

– Rotation of a state of polarisation by an angle θ

Rθ=

– General retardation plate - phase shift = Γ

MΓ=

 1 0 0 e^iΓ



. (7.106)

• Analysis of polarised light

A linear polariser is an optical element that passes only linearly po-larised light.

Such an optical element may also be used to analyse light into its lin-early polarised components. It may then be refered to as an analyser.

7.8. SUMMARY 157 For the general case of arbitrarily polarised light passing through an analyser with its transmission at an angle θ to the Ex axis, the time-averaged intensity of the transmitted light is given by

I (θ) = I0 cos²θ + r²sin²θ + 2r cos θ sin θ cos Γ
, (7.107) where r = E_0y/E0x.

• Malus’ Law

For linearly x-polarised light, the transmitted intensity through the analyser is given by

I (θ) = I₀cos²θ. (7.108) This is known asMalus’ Law for the transmission of linearly polarised light.

8. The Fresnel Equations

8.1 General remarks

A crucial application of optics involves the reflection and transmission of light at the boundary between media of different refractive indices. Al-though, strictly, this violates our notion of homogeneity, we may still think of the media involved being locally homogeneous.

The quantitative model of reflection and transmission arises out of Maxwell’s equations and are generally referred to as the Fresnel equations. The start-ing point for this analysis is to understand the boundary conditions of the different electromagnetic fields E, H, D and B, which we revise in the first section of this Chapter. Thereafter, however, we shy away from a full derivation of the Fresnel equations, referring the reader to standard texts on electromagnetism, and simply quote the results.

In the later sections, we extend our treatment to consider irradiance, i.e. the incident, reflected and transmitted intensities. Along the way, we shall also consider reflection and transmission from a wavevector picture, obtaining alternative derivations of the Laws of Reflection and Refraction, as well as predicting the existence of an evanescent wave in the case of total internal reflection.

The results obtained here contain plenty of interesting physics. Some of the possible applications are mentioned briefly along the way. Other applications that we shall cover later include

• Thin-film interference

• Anti-reflection coatings

• Resonant cavities

8.2 Learning objectives

• Boundary conditions

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

159

• Reflection and refraction

Reflection and refraction via wavevector.

• Fresnel equations

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

– Brewster angle

• Time reversibility

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

• 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.

• Total internal reflection

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

– Optical couplers