Finite Beams

Abstract

Chapters 1–18 have dealt with the reflection of monochromatic plane waves from planar interfaces. The previous chapter discussed the reflection of electromagnetic pulses and of quantum particle wavepackets. Here we shall consider transversely finite beams. The simplest beams to consider are those bounded in space but still monochromatic. These can be viewed as a superposition of plane waves of the same frequency but differing propagation directions. We shall find, accordingly, that the reflection of beams depends on the angular dependence of the reflection amplitude. Particularly important is the case of total reflection, where all the frequency and angle dependence is contained in the phase of the reflection amplitude, since its modulus is then unity. The variation of the s and p phases with angle of incidence is discussed in Appendix 1, and applied to calculation of the lateral beam shift in Sect. 20.2. Section 20.3 gives analytic results for the reflection of Gaussian beams. Appendix 2 summarizes the polarization properties of finite beams. We start by reviewing the properties of finite beams.

Appendices

Appendix 1: Total Internal Reflection : The r _s , r _p Phases and Their Difference

In Sect. 20.2 we considered the reflection of bounded beams, with emphasis on the problem of beam shift. The latter depends on the variation of the phase of the reflection amplitude with the angle of incidence, and is greatest near the critical angle where the derivative becomes infinite. We shall give examples of the angular dependence of the phases of $r_{s}$ and $r_{p}$, and then show that a square root singularity at the critical angle is universal for non-absorbing profiles.

Reflection at a sharp boundary. The s and p reflection amplitudes for a step profile located at $z = 0$ are given by (1.13) and (1.31):

$$ r_{s} = \frac{{q_{1} - q_{2} }}{{q_{1} + q_{2} }},\quad r_{p} = - \frac{{Q_{1} - Q_{2} }}{{Q_{1} + Q_{2} }} . $$

(20.61)

When medium $1$ is optically denser $\left( {\varepsilon_{1} > \varepsilon_{2} } \right), q_{1} > q_{2}$ and $r_{s}$ has zero phase (all phases are modulo $2\pi$) up to the critical angle $\theta_{c}$, where $q^{2} = \left( {\omega /c} \right)^{2} \left( {\varepsilon_{2} - \varepsilon_{1} \sin^{2} \theta_{1} } \right)$ passes through zero and $q_{2}$ changes from real to imaginary:

$$ q_{2} = i\left| {q_{2} } \right| = \frac{\omega }{c}\left( {\varepsilon_{1} \sin^{2} \theta_{1} - \varepsilon_{2} } \right)^{{\frac{1}{2}}} ,\quad [\theta_{1} \ge \arcsin \left( {\varepsilon_{2} /\varepsilon_{1} } \right)^{{\frac{1}{2}}} \,=\, \theta_{c} ] $$

(20.62)

Beyond this angle of incidence $\left| {r_{s} } \right| = 1$ and

$$ \delta_{s} = - 2{ \arctan }\left( {\left| {q_{2} } \right|/q_{1} } \right) , $$

(20.63)

where

$$ \frac{{\left| {q_{2} } \right|}}{{q_{1} }} = \left\{ {\cos^{2} \theta_{c} \tan^{2} \theta_{1} - \sin^{2} \theta_{c} } \right\}^{{\frac{1}{2}}} = \left\{ {\left( {1 - \frac{{\varepsilon_{2} }}{{\varepsilon_{1} }}} \right)\tan^{2} \theta_{1} - \frac{{\varepsilon_{2} }}{{\varepsilon_{1} }}} \right\}^{{\frac{1}{2}}} . $$

(20.64)

We note the square root singularity at $\theta_{c}$, which leads to an infinite value of $\text{d}\delta_{s} /\text{d}\theta_{1}$ at $\theta_{c}^{ + }$: in terms of $\Delta \theta = \theta_{1} - \theta_{c}$ this is

$$ \delta_{s} = - 2\left( {\frac{{4\varepsilon_{2} }}{{\varepsilon_{1} - \varepsilon_{2} }}} \right)^{{\frac{1}{4}}} (\Delta \theta )^{{\frac{1}{2}}} + O(\Delta \theta ). $$

(20.65)

The s wave phase decreases monotonically from $0$ at $\theta_{c}$ to ${-}\pi$ at grazing incidence, approaching ${-}\pi$ linearly in the glancing angle $\gamma = \frac{\pi }{2} - \theta_{1}$:

$$ \delta_{s} = - \pi + 2\left( {\frac{{\varepsilon_{1} }}{{\varepsilon_{1} - \varepsilon_{2} }}} \right)^{{\frac{1}{2}}} \gamma + O\left( {\gamma^{2} } \right)\,. $$

(20.66)

The p wave phase is zero from normal incidence to the Brewster angle $\theta_{B} = { \arctan }\left( {\varepsilon_{2} /\varepsilon_{1} } \right)^{{\frac{1}{2}}}$ where $Q_{1} = Q_{2}$ and $r_{p}$ changes sign. In the interval $\theta_{B} < \theta_{1} < \theta_{c}$ we can set $\delta_{p}$ equal to $+ \pi$ or $- \pi$. We take $\delta_{p} = - \pi$, this choice being dictated by continuity of the phase as a function of interfacial thickness, as the next example will make clear. Beyond $\theta_{c}$ the p wave phase is (from (20.31) with $Q_{2} = i\left| {Q_{2} } \right|$)

$$ \delta_{p} = - \pi - 2{ \arctan }\frac{{\left| {Q_{2} } \right|}}{{Q_{1} }} = - \pi - 2{ \arctan }\frac{{\varepsilon_{1} \left| {q_{2} } \right|}}{{\varepsilon_{2} q_{1} }}. $$

(20.67)

The strength of the square root singularity is thus larger for the p phase shift by the factor $\varepsilon_{1} /\varepsilon_{2}$:

$$ \delta_{p} = - \pi - 2\frac{{\varepsilon_{1} }}{{\varepsilon_{2} }}\left( {\frac{{4\varepsilon_{2} }}{{\varepsilon_{1} - \varepsilon_{2} }}} \right)^{{\frac{1}{4}}} (\Delta \theta )^{{\frac{1}{2}}} + O(\Delta \theta ). $$

(20.68)

The inverse factor applies as $\delta_{p}$ tends to $- 2\pi$ at grazing incidence:

$$ \delta_{p} = - 2\pi + 2\frac{{\varepsilon_{2} }}{{\varepsilon_{1} }}\left( {\frac{{\varepsilon_{1} }}{{\varepsilon_{1} - \varepsilon_{2} }}} \right)^{{\frac{1}{2}}} \gamma + O(\gamma^{2} ). $$

(20.69)

Figure 20.4 shows $\delta_{s}$ and $\delta_{p}$ for the sharp boundary between two media, and also for a homogeneous layer between the same two media (the latter to be discussed shortly).

The ellipsometric ratio $r_{p} /r_{s}$ is equal to $\exp i(\delta_{p} - \delta_{s} )$ for $\theta_{1} > \theta_{c}$. The phase difference $\Delta = \delta_{p} - \delta_{s}$ is given by

$$ \Delta = - \pi + 2\left( {{ \arctan }\frac{{\left| {q_{2} } \right|}}{{q_{1} }} - { \arctan }\frac{{\varepsilon_{1} }}{{\varepsilon_{2} }}\frac{{\left| {q_{2} } \right|}}{{q_{1} }}} \right). $$

(20.70)

The phase difference has an extremum at the angle of incidence

$$ \theta_{m} = { \arctan }\left( {\frac{{2\varepsilon_{2} }}{{\varepsilon_{1} - \varepsilon_{2} }}} \right)^{{\frac{1}{2}}} . $$

(20.71)

For comparison we list the tangents and sines of $\theta_{B} , \theta_{c}$ and $\theta_{m}$:

$$ \tan^{2} \theta_{B} = \frac{{\varepsilon_{2} }}{{\varepsilon_{1} }} , \;\;\;\tan^{2} \theta_{c} = \frac{{\varepsilon_{2} }}{{\varepsilon_{1} - \varepsilon_{2} }} ,\;\;\; \tan^{2} \theta_{m} = \frac{{2\varepsilon_{2} }}{{\varepsilon_{1} - \varepsilon_{2} }}, $$

(20.72)

$$ \sin^{2} \theta_{B} = \frac{{\varepsilon_{2} }}{{\varepsilon_{1} + \varepsilon_{2} }} , \;\;\;\sin^{2} \theta_{c} = \frac{{\varepsilon_{2} }}{{\varepsilon_{1} }} ,\;\;\; \sin^{2} \theta_{m} = \frac{{2\varepsilon_{2} }}{{\varepsilon_{1} + \varepsilon_{2} }} . $$

(20.73)

At the extremum the phase difference $\Delta = \delta_{p} - \delta_{s}$ is given by

$$ \Delta _{m} = 4\theta_{B} - 2\pi , $$

(20.74)

and the ratio of the reflection amplitudes takes the value

$$ \frac{{r_{p} }}{{r_{s} }} = \frac{{\varepsilon_{1}^{2} + \varepsilon_{2}^{2} - 6\varepsilon_{1} \varepsilon_{2} + i4\left( {\varepsilon_{1} \varepsilon_{2} } \right)^{{\frac{1}{2}}} (\varepsilon_{1} - \varepsilon_{2} )}}{{\left( {\varepsilon_{1} + \varepsilon_{2} } \right)^{2} }}\quad (\theta_{1} = \theta_{m} ). $$

(20.75)

At $\theta_{m}$ the trajectory of $r_{p} /r_{s}$ in the complex plane is farthest to the right on the unit circle. The phase difference $\Delta = \delta_{p} - \delta_{s}$ is shown in Fig. 20.5, together with that for a homogeneous layer.

Fig. 20.5

Dependence of $\Delta = \delta_{p} - \delta_{s}$ on the angle of incidence, the parameters being as in Fig. 20.4. Normal incidence is at left, glancing incidence at right. Dashed curve glass|air; solid curve glass|water|air. The homogeneous film $r_{p} /r_{s}$ ratio was shown in Fig. 2.8

Reflection phases for a homogeneous layer . For a layer of dielectric constant $\varepsilon$ and of thickness $\Delta z$, the s and p reflection amplitudes are given by (2.52) and (2.68):

$$ r_{s} = \frac{{q\left( {q_{1} - q_{2} } \right)c + i\left( {q^{2} - q_{1} q_{2} } \right)s}}{{q\left( {q_{1} + q_{2} } \right)c - i\left( {q^{2} + q_{1} q_{2} } \right)s}} , $$

(20.76)

$$ - r_{p} = \frac{{Q\left( {Q_{1} - Q_{2} } \right)c + i\left( {Q^{2} - Q_{1} Q_{2} } \right)s}}{{Q\left( {Q_{1} + Q_{2} } \right)c - i\left( {Q^{2} + Q_{1} Q_{2} } \right)s}} . $$

(20.77)

Here $q^{2} = \left( {\omega /c} \right)^{2} \left( {\varepsilon - \varepsilon_{1} \sin^{2} \theta_{1} } \right)$ and $c = \cos q\Delta z, s = \sin q\Delta z, Q = q/\varepsilon$; the film extends from $z = 0$ to $\Delta z$. When $\varepsilon_{1} > \varepsilon > \varepsilon_{2}$ we have to consider three ranges of $\theta_{1} : \theta_{1} \le \theta_{c} = \arcsin \left( {\varepsilon_{2} /\varepsilon_{1} } \right)^{{\frac{1}{2}}} , \theta_{c} \le \theta_{1} \le \theta_{c}^{\prime} = { \arcsin }\left( {\varepsilon /\varepsilon_{1} } \right)^{{\frac{1}{2}}}$, and $\theta_{1} \ge \theta_{c}^{\prime}$. In the first range $q_{1} ,q_{2} , q$ are all real, in the second $q_{2} = i\left| {q_{2} } \right|$ and $q_{1} , q$ are real, and in the third $q_{2} = i\left| {q_{2} } \right|$ and $q = i\left| q \right|$. Of particular interest to the beam shift to be discussed in the next section is the behaviour of the phases for $\theta_{1}$ slightly above $\theta_{c}$ We again find a square root singularity, with

$$ \delta_{s} \left( {\theta_{1} } \right) = \delta_{s} \left( {\theta_{c} } \right) - \frac{{2\left| {q_{2} } \right|/q_{1c} }}{{1 - \frac{{\varepsilon_{1} - \varepsilon }}{{\varepsilon_{1} - \varepsilon_{2} }}\sin^{2} q_{c}\Delta z}} + O(\left| {q_{2} } \right|^{2} ), $$

(20.78)

$$ \delta_{p} \left( {\theta_{1} } \right) = \delta_{p} \left( {\theta_{c} } \right) - \frac{{2\left| {Q_{2} } \right|/Q_{1c} }}{{1 - \left( {1 - \frac{{\varepsilon_{1}^{2} (\varepsilon_{1} - \varepsilon )}}{{\varepsilon^{2} (\varepsilon_{1} - \varepsilon_{2} )}}} \right)\sin^{2} q_{c}\Delta z}} + O(\left| {q_{2} } \right|^{2} ), $$

(20.79)

where

$$ q_{1c}^{2} = \frac{{\omega^{2} }}{{c^{2} }}\left( {\varepsilon_{1} - \varepsilon_{2} } \right),\quad q_{c}^{2} = \frac{{\omega^{2} }}{{c^{2} }}\left( {\varepsilon - \varepsilon_{2} } \right), $$

(20.80)

The phases at the critical angle are given by

$$ \delta_{s} \left( {\theta_{c} } \right) = 2{ \arctan }\left\{ {\left( {\frac{{\varepsilon - \varepsilon_{2} }}{{\varepsilon_{1} - \varepsilon_{2} }}} \right)^{{\frac{1}{2}}} \tan q_{c}\Delta z} \right\}, $$

(20.81)

$$ \delta_{p} \left( {\theta_{c} } \right) = - \pi + 2{ \arctan }\left\{ {\frac{{\varepsilon_{1} }}{\varepsilon }\left( {\frac{{\varepsilon - \varepsilon_{2} }}{{\varepsilon_{1} - \varepsilon_{2} }}} \right)^{{\frac{1}{2}}} \tan q_{c}\Delta z} \right\}. $$

(20.82)

The numerators in (20.78) and (20.79) are the sharp boundary values, and have been expressed in terms of $\Delta \theta = \theta_{1} - \theta_{c}$ in (20.65) and (20.68). For the s wave the coefficient of $(\Delta \theta )^{{\frac{1}{2}}}$ is larger for the homogenous layer than for the Fresnel case; for the p wave it can be larger or smaller, depending on the dielectric constants.

Figure 20.4 shows the s and p phase shifts and Fig. 20.5 their difference for $\left( {{\omega \mathord{\left/ {\vphantom {\omega c}} \right. \kern-0pt} c}} \right){\Delta z}\,\text{ = }\,\frac{1}{2}$. Note that there is no square root singularity in $\delta_{s}$ or $\delta_{p}$ at $\theta^{\prime}_{c}$ where q passes through zero, the s and p phases having the variation $\delta \, = \,\delta \left( {\theta^{\prime}_{c} } \right) + O\left( {q^{2} } \right)$ with

$$ \delta_{s} \left( {\theta_{c}^{\prime} } \right) = - 2{ \arctan }\left\{ {\frac{{\left[ {\left( {\varepsilon - \varepsilon_{2} } \right)/\left( {\varepsilon_{1} - \varepsilon } \right)} \right]^{{\frac{1}{2}}} }}{{1 + \left( {\varepsilon - \varepsilon_{2} } \right)^{{\frac{1}{2}}} \left( {\omega /c} \right)\Delta z }}} \right\}, $$

(20.83)

$$ \delta_{p} \left( {\theta_{c}^{\prime} } \right) = - \pi - 2{ \arctan }\left\{ {\frac{{\frac{{\varepsilon_{1} }}{{\varepsilon_{2} }}\left[ {\left( {\varepsilon - \varepsilon_{2} } \right)/\left( {\varepsilon_{1} - \varepsilon } \right)} \right]^{{\frac{1}{2}}} }}{{1 + \frac{\varepsilon }{{\varepsilon_{2} }}\left( {\varepsilon - \varepsilon_{2} } \right)^{{\frac{1}{2}}} \left( {\omega /c} \right)\Delta z }}} \right\}. $$

(20.84)

Thus $\theta_{c}^{\prime}$ is not a true critical angle, even though $q$ makes a sharp right-angle turn in the complex plane at $\theta_{c}^{\prime}$, just as $q_{2}$ does at $\theta_{c}$.

Total Reflection by the Hyperbolic Tangent Profile

For $\theta_{1} < \theta_{c}$ the phase for the s polarization is given by (2.89); as $\theta_{c}$ is approached from below the phase tends without singularity to

$$ \delta_{s} \left( {\theta_{c} } \right) = 2\mathop \sum \limits_{n = 1}^{\infty } { \arctan }\left\{ {\frac{{2y_{1c}^{3} }}{{n(n^{2} + 3y_{1c}^{2} )}}} \right\}, $$

(20.85)

where $y_{1c} = q_{1c} a, a$ being the length characterizing the thickness of the profile, and $q_{1c} = \left( {\omega /c} \right)\left( {\varepsilon_{1} - \varepsilon_{2} } \right)^{{\frac{1}{2}}}$. For $\theta_{1} > \theta_{c}$ an analysis based on (2.84) and using the infinite product representation of the gamma function (2.85) gives

$$ \delta_{s} = 2\mathop \sum \nolimits_{1}^{\infty } { \arctan }\left\{ {\frac{{2y_{1} (y_{1}^{2} + \left| {y_{2} } \right|^{2} )}}{{n(n^{2} + 3y_{2}^{2} - \left| {y_{2} } \right|^{2} )}}} \right\} - 2\arctan \left( {\frac{{\tan \pi \left| {y_{2} } \right|}}{{\tanh \pi y_{1} }}} \right), $$

(20.86)

where $y_{1} = q_{1} a, \left| {y_{2} } \right| = \left| {q_{2} } \right|a$. Thus there is again a $\left| {q_{2} } \right|$ term in the phase just above the critical angle:

$$ \delta_{s} = \delta_{s} \left( {\theta_{c} } \right) - \frac{{2\pi a\left| {q_{2} } \right|}}{{\tanh \pi aq_{1c} }} + O(q_{2}^{2} ). $$

(20.87)

The expression (20.86) tends to the sharp profile result (20.63) as $a \to 0$. For large interfacial thickness the coefficient of $\left| {q_{2} } \right|$ in (20.86) tends to $- 2\pi a$, and the strength of the square root singularity is then proportional to the thickness. At grazing incidence $\delta_{s} \to - \pi$ as before.

The above examples are sufficient to make it plausible that the $\left( {\theta_{1} - \theta_{c} } \right)^{{\frac{1}{2}}}$ singularity in the phase shift is a universal property. We shall give a proof for the restricted class of finite-ranged profiles, for which the s wave reflection amplitude is given by (2.25), which we write in the form

$$ r_{s} = \frac{{q_{1} q_{2} A + iq_{1} B + iq_{2} C - D}}{{q_{1} q_{2} A + iq_{1} B - iq_{2} C + D}}\,. $$

(20.88)

(We again set $z_{1} = 0$: the inhomogeneity extends from $z = 0$ to $\Delta z$; the substrate has dielectric function $\varepsilon_{2}$.) When $\theta_{1} > \theta_{c}$ we have $q_{2} = i\left| {q_{2} } \right|$ and

$$ r_{s} = \frac{ - \alpha + i\beta }{\alpha + i\beta },\;\;\; \alpha = \left| {q_{2} } \right|C + D,\;\;\; \beta = q_{1} (\left| {q_{2} } \right|A + B). $$

(20.89)

Thus $\delta_{s} = 2{ \arctan }\left( {\alpha /\beta } \right)$. The leading terms in $\alpha /\beta$ near $\theta_{c}$ are

$$ \frac{\alpha }{\beta } = \frac{D}{{q_{1} B}} - \frac{{\left| {q_{2} } \right|}}{{q_{1} }}\left( {\frac{W}{B}} \right)^{2} + O\left( {\left| {q_{2} } \right|^{2} } \right), $$

(20.90)

where $W$ is the Wronskian of the solutions of the wave equation; we have used the identity $AD - BC = W^{2}$ equation (2.31). This shows that all such profiles have a term linear in $\left| {q_{2} } \right|$, with negative coefficient, leading to a square root singularity :

$$ \delta_{s} = 2{ \arctan }\left( {\frac{D}{{q_{1} B}}} \right)_{c} - \frac{{2\left| {q_{2} } \right|}}{{q_{1c} }}\frac{{\left( {W/B} \right)_{c} }}{{1 + \left( {D/q_{1} B} \right)_{c}^{2} }} + O\left( {\left| {q_{2} } \right|^{2} } \right). $$

(20.91)

(A similar result may be written down for $\delta_{p}$, using (2.40), (2.48) and (2.49).) For the homogeneous layer, with solutions $\sin qz$ and $\cos qz$ in $0 \le z \le\Delta z$, we have $W = q, B = q\cos q\Delta z, D = q^{2} \sin q\Delta z$, and (20.91) gives the results contained in (20.78) and (20.81).

Appendix 2: Polarization of Electromagnetic Beams

In most of the book we have considered two linearly polarized waves, the $s$ and $p$ polarizations of Sects. 1.1 and 1.2. By these designations we mean that the electric vector $\mathbf{E}$ is respectively perpendicular and parallel to the plane of incidence. For plane waves, the corresponding magnetic vector $\mathbf{B}$ is respectively parallel and perpendicular to the plane of incidence; $\mathbf{E}$ and $\mathbf{B}$ are perpendicular to the wavevector, as well as to each other. Plane waves with $\mathbf{E}$ and $\mathbf{B}$ both circularly polarized were the eigenstates propagating within chiral media, Chap. 18. Here we shall discuss the most general polarization of a coherent monochromatic electromagnetic wave, and gives examples of the polarization properties of electromagnetic beams.

A coherent monochromatic light beam is specified by electric and magnetic vectors varying in space and harmonically in time. In general the polarization properties of the electric and magnetic vectors differ from each other, in contrast to the plane wave idealization. Most polarizers act on the electric field, and most detectors sense the electric field, so it is conventional to refer to the polarization of the electric field as the polarization. For monochromatic waves of angular frequency $\omega$ we can write

$$ \mathbf{E}\left( {\mathbf{r},t} \right) = {\text{Re}}\,\left\{ {\mathbf{E}(\mathbf{r}){\text{e}}^{ - i\omega t} } \right\} = \mathbf{E}_{r} \left( \mathbf{r} \right)\cos \omega t + \mathbf{E}_{i} (\mathbf{r})\sin \omega t, $$

(20.92)

where $\mathbf{E}_{r} \left( \mathbf{r} \right)$ and $\mathbf{E}_{i} (\mathbf{r})$ are the real and imaginary parts of the complex electric field vector $\mathbf{E}(\mathbf{r})$. The magnetic field is expressed in terms of the real and imaginary parts of the complex vector $\mathbf{B}(\mathbf{r})$ in the same way. For a plane wave in vacuum we have $\mathbf{E}\left( \mathbf{r} \right) = \mathbf{E}_{0} {\text{e}}^{{i\mathbf{k} \cdot \mathbf{r}}} ,\mathbf{ B}\left(\mathbf{r} \right) = k^{ - 1} \mathbf{k} \times \mathbf{E}(\mathbf{r})$ where $k = \omega /c$ and the wavevector $\mathbf{k}$ defines the direction of propagation. If the constant vector $\mathbf{E}_{0}$ is real (or more generally, if its real and imaginary parts are collinear), it defines the direction of linear polarization. If the complex vector $\mathbf{E}_{0}$ has equal and perpendicular real and imaginary parts, as in the plane wave $\mathbf{E}\left( \mathbf{r} \right) = E_{0} {\text{e}}^{ikz} (\hat{\mathbf{x}} \pm i\hat{\mathbf{y}})$, the physical electric vector $\mathbf{E}\left( {\mathbf{r},t} \right)$ rotates at any point in space with angular frequency $\omega$, and the wave is circularly polarized. The most general case is that of elliptic polarization, in which the endpoint of the vector $\mathbf{E}\left( {\mathbf{r},t} \right)$ describes an ellipse in time $2\pi /\omega$, as we shall now show.

For any $\mathbf{E}\left(\mathbf{r} \right) = \mathbf{E}_{r} \left(\mathbf{r} \right) + i\mathbf{E}_{i} (\mathbf{r})$ one can write

$$ \mathbf{E}_{r} + i\mathbf{E}_{i} = (\mathbf{E}_{1} + i\mathbf{E}_{2} ){\text{e}}^{i\gamma } , $$

(20.93)

and $\gamma$ can be chosen so that the real vectors $\mathbf{E}_{1}$ and $\mathbf{E}_{2}$ are perpendicular. This value of $\gamma$ and the components $\mathbf{E}_{1}$ and $\mathbf{E}_{2}$ are given by

$$ \tan 2\gamma = \frac{{2\mathbf{E}_{r} \cdot \mathbf{E}_{\mathbf{i}} }}{{E_{r}^{2} - E_{i}^{2} }} , $$

(20.94)

$$ \mathbf{E}_{1} = \mathbf{E}_{r} \cos \gamma + \mathbf{E}_{i} \sin \gamma ,\;\;\; \mathbf{E}_{2} = \mathbf{E}_{i} \cos \gamma - \mathbf{E}_{r} \sin \gamma . $$

(20.95)

Thus the physical electric field can be written as

$$ \mathbf{E}\left( {\mathbf{r},t} \right) = {\text{R}}{\text{e}}\left\{ {(\mathbf{E}_{1} + i\mathbf{E}_{2} ){\text{e}}^{i\gamma - i\omega t} } \right\} = \mathbf{E}_{1} \left(\mathbf{r} \right)\cos (\omega t - \gamma ) + \mathbf{E}_{2} (\mathbf{r})\sin (\omega t - \gamma ). $$

(20.96)

When $\gamma$ is given by (20.94), the components $\mathbf{E}_{1}$ and $\mathbf{E}_{2}$ are orthogonal, and have magnitudes given by

$$ \left( {\begin{array}{*{20}c} {E_{1}^{2} } \\ {E_{2}^{2} } \\ \end{array} } \right) = \frac{1}{2}\left\{ {E_{r}^{2} + E_{i}^{2} \pm \left[ {\left( {E_{r}^{2} - E_{i}^{2} } \right)^{2} + 4\left( {\mathbf{E}_{r} \cdot \mathbf{E}_{i} } \right)^{2} } \right]^{{\frac{1}{2}}} } \right\} . $$

(20.97)

From (20.96), $E_{1}$ and $E_{2}$ give the lengths of the semiaxes of the polarization ellipse.

For linear polarization $E_{2} = 0$; the condition for linear polarization is therefore that $\mathbf{E}_{r} , \mathbf{E}_{i}$ be collinear,

$$ E_{r}^{2} E_{i}^{2} = \left( {\mathbf{E}_{r} \cdot \mathbf{E}_{i} } \right)^{2} .\quad ({\text{linear polarization}}) $$

(20.98)

For circular polarization $E_{1}^{2} = E_{2}^{2}$, for which we need $\mathbf{E}_{r}$ and $\mathbf{E}_{i}$ to be perpendicular and equal in magnitude:

$$ \mathbf{E}_{r} \cdot \mathbf{E}_{i} = 0\ {\text {and }}\ E_{r}^{2} = E_{i}^{2} .\quad ({\text{circular polarization}}) $$

(20.99)

One spatial function can define the local degree of linear polarization (Lekner 2003), namely

$$ \Lambda \left(\mathbf{r} \right) = \frac{{E_{1}^{2} - E_{2}^{2} }}{{E_{1}^{2} + E_{2}^{2} }} = \frac{{\left[ {\left( {E_{r}^{2} - E_{i}^{2} } \right)^{2} + 4\left( {\mathbf{E}_{r} \cdot \mathbf{E}_{i} } \right)^{2} } \right]^{{\frac{1}{2}}} }}{{E_{r}^{2} + E_{i}^{2} }} = \frac{{\left| {\mathbf{E}^{2} \left(\mathbf{r} \right)} \right|}}{{\left| {\mathbf{E}\left(\mathbf{r} \right)} \right|^{2} }} . $$

(20.100)

$\Lambda \left(\mathbf{r} \right)$ is unity when the real and imaginary parts of $\mathbf{E}\left(\mathbf{r} \right) = \mathbf{E}_{r} \left(\mathbf{r} \right) + i\mathbf{E}_{i} (\mathbf{r})$ are collinear (the linear polarization condition), and zero when the circular polarization conditions are met. Equivalently, the eccentricity $e$ of the polarization ellipse provides the same information:

$$ {\text{e}}^{2} = 1 - \frac{{E_{2}^{2} }}{{E_{1}^{2} }} = \frac{{2\Lambda }}{{1 +\Lambda }} . $$

(20.101)

This has the same values as $\Lambda$ of unity and zero for the limiting cases of linear and circular polarizations. Yet more polarization measures exist, namely the Hurwitz (1945) ratio $2E_{1} E_{2} /(E_{1}^{2} + E_{2}^{2} )$ and the Stokes parameters (Born and Wolf 1999, Sects. 1.4.2, 10.8.3)

$$ S_{0} = E_{r}^{2} + E_{i}^{2} ,\;\;\; S_{1} = E_{r}^{2} - E_{i}^{2} ,\;\;\; S_{3} = 2\mathbf{E}_{r} \cdot \mathbf{E}_{i} ,\;\;\; S_{4} = 2\left[ {E_{r}^{2} E_{i}^{2} - \left( {\mathbf{E}_{r} \cdot \mathbf{E}_{i} } \right)^{2} } \right]^{{\frac{1}{2}}} . $$

(20.102)

The relation between the degree of linear polarization $\Lambda$ and the Stokes parameters is $\Lambda ^{2} = 1 - S_{3}^{2} /S_{0}^{2}$.

The remainder of this Appendix gives specific examples of polarization properties of finite monochromatic electromagnetic beams. A broader range of topics may be found in Swindell (1975), a collection of reprints of fundamental papers on polarized light with commentary, and in monographs by Collett (1992), Huard (1997) and Brosseau (1998).

Examples of Exactly and Approximately Linearly Polarized Beams

The simplest example is the TM (transverse magnetic) beam, for which the vector potential is given by the first entry in (20.26), $\mathbf{A} = A_{0} (0,0,\psi )$. For this beam $\mathbf{B}$ is transverse to the propagation direction (here along the $z$ axis):

$$ \mathbf{B} = \nabla \times \mathbf{A} = A_{0} \left( {\partial_{x} \psi , - \partial_{y} \psi ,0} \right). $$

(20.103)

When $\psi$ is independent of the azimuthal angle $\phi$, the complex fields are

$$\begin{aligned} \mathbf{B}(\mathbf{r}) &= A_{0} \left( {\sin \phi \partial_{\rho } \psi , - \cos \phi \partial_{\rho } \psi ,0} \right),\\ \mathbf{E}\left(\mathbf{r} \right) &= \frac{{iA_{0} }}{k}(\cos \phi \partial_{\rho } \partial_{z} \psi ,\sin \phi \partial_{\rho } \partial_{z} \psi , \partial_{z}^{2} \psi + k^{2} \psi ).\end{aligned} $$

(20.104)

If we take $A_{0}$ real, and write the complex wavefunction $\psi (\rho ,z)$ as $\psi_{r} + i\psi_{i}$, the real and imaginary parts of $\mathbf{B}(\mathbf{r})$ are both proportional to ($\sin \phi , - \cos \phi ,0)$, and are thus collinear. The magnetic field is therefore everywhere linearly polarized. The electric field is elliptically polarized, in general.

The dual of the TM beam under the transformation $\mathbf{E} \to \mathbf{B}, \mathbf{B} \to - \mathbf{E}$ (one of a set of duality transformations that leave the free space Maxwell equations unchanged) is the TE beam, transverse and linearly polarized in its electric field. The electric field lines are circles concentric with the beam axis (see Fig. 20.1 of Lekner 2003, for example). However, both the TM and the TE beams disappear in the plane-wave limit: as $\psi \to \exp ikz$ the electric and magnetic fields in both the TM and the TE beams tend to zero.

A beam which does have a plane-wave limit is the ‘LP’ beam, with vector potential $\mathbf{A} = A_{0} (\psi ,0,0)$. The magnetic and electric fields are

$$ \mathbf{B} = \nabla \times \mathbf{A} = A_{0} \left( {0,\partial_{z} \psi , - \partial_{y} \psi } \right), $$

(20.105)

$$ \mathbf{E} = \frac{i}{k}\nabla \left( {\nabla \cdot \mathbf{A}} \right) + ik\mathbf{A} = \frac{{iA_{0} }}{k}(\partial_{x}^{2} \psi + k^{2} \psi ,\partial_{x} \partial_{y} \psi ,\partial_{x} \partial_{z} \psi ) $$

(20.106)

In the plane wave limit $\psi \to \exp ikz,\mathbf{B} \to ikA_{0} \left( {0,1,0} \right), \mathbf{E} \to ikA_{0} \left( {1,0,0} \right)$, which is the textbook linearly polarized plane wave with $\mathbf{E}$ and $\mathbf{B}$ transverse and mutually perpendicular. But note the quotes around ‘LP’: this beam is linearly polarized only in the plane-wave limit, as we shall now see. We again consider beams with $\psi$ is independent of the azimuthal angle $\phi$, for simplicity. Then

$$ \mathbf{B} = ik\left( {0,\partial_{z} \psi , - \sin \phi \partial_{\rho } \psi } \right), $$

(20.107)

$$ \mathbf{E} = \frac{{iA_{0} }}{k}(\cos^{2} \phi \partial_{\rho }^{2} \psi + \sin^{2} \phi \rho^{ - 1} \partial_{\rho } \psi + k^{2} \psi ,\\ \sin \phi \cos \phi [\partial_{\rho }^{2} \psi - \rho^{ - 1} \partial_{\rho } \psi ], \cos \phi \partial_{\rho } \partial_{z} \psi ). $$

(20.108)

Neither $\mathbf{E}$ nor $\mathbf{B}$ have real and imaginary parts collinear in general. The electric field in the $x = 0$ plane ($\cos \phi = 0)$ is linearly polarized along the $x$ direction, as can be seen from (20.108). The polarization measure $\Lambda$ is given in  (20.27) and is plotted for the $\psi_{00} = j_{0} (kR)$ beam in Fig. 20.3 of Lekner (2003) for $kb = 2$, and in Fig. 20.6 for $kb = 6$. The polarization is linear at the beam centre, and $\Lambda \approx 1$ in the central region $\rho \ll b$, but, remarkably, there are areas of approximately circular polarization in the outer part of the beam.

Fig. 20.6

Degree of linear polarization $\Lambda$ in the focal plane $z = 0$ of an ‘LP’ beam with $\psi = \psi_{00} , kb = 6$. The light shading corresponds to linear polarization, dark to circular polarization ($\Lambda \to 1$ and $\Lambda \to 0$, respectively). The lateral extent is $\left| {kx} \right| \le 9, \left| {ky} \right| \le 9$

Approximately Circularly Polarized Beams

We wish to construct beams which in the plane wave limit have the circularly polarized electric field

$$ \mathbf{E}\left(\mathbf{r} \right) = E_{0} {\text{e}}^{ikz} (1,i,0), \,\\ \mathbf{E}\left( {\mathbf{r},t} \right) = {\text{Re}}\left( {\mathbf{E}\left( \mathbf{r} \right){\text{e}}^{ - i\omega t} } \right) = E_{0} (\cos \left( {kz - \omega t} \right), - \sin \left( {kz - \omega t} \right),0). $$

(20.109)

The vector potential $\mathbf{A} = k^{ - 1} E_{0} (i\psi , - \psi ,0)$ gives the complex fields

$$ \mathbf{B} = k^{ - 1} E_{0} \left[ {\partial_{z} ,i\partial_{z} , - (\partial_{x} + i\partial_{y} )} \right]\psi , $$

$$ \mathbf{E} = E_{0} \left[ {1 + k^{ - 2} \partial_{x} \left( {\partial_{x} + i\partial_{y} } \right), i + k^{ - 2} \partial_{y} \left( {\partial_{x} + i\partial_{y} } \right), k^{ - 2} \partial_{z} \left( {\partial_{x} + i\partial_{y} } \right)} \right]\psi . $$

(20.110)

(We have used the fact that $\psi$ satisfies the Helmholtz equation (20.1).) Both the magnetic and the electric fields are therefore circularly polarized, with positive helicity, in the plane wave limit $\psi \to \exp ikz$. Figure 20.7 shows the polarization measure $\Lambda$ for the electric field of (20.110), with $\psi = \sin kR/kR, kb = 6$, which has the focal plane zeros shown in Fig. 20.1. We note that the dark central part is circularly polarized, but the outer region of the beam (where the intensity is very low) there are circles of linear polarization. More analytic detail may be found in Section 4 and Appendix B of Lekner (2003).

Fig. 20.7

Degree of linear polarization $\Lambda$ in the focal plane of a ‘CP’ beam with $\psi = \psi_{00} , kb = 6$. The light shading corresponds to linear polarization, dark to circular polarization ($\Lambda \to 1$ and $\Lambda \to 0$, respectively). The lateral extent is $\left| {kx} \right| \le 9, \left| {ky} \right| \le 9$. The beam is completely circularly polarized on the axis, and approximately so in the central dark region. However, there are circles of exactly linear polarization in the outer part

These examples illustrate the theorems (ii) and (iii) of Sect. 20.1, and show that finite beams are quite different from the textbook plane waves, not just in having longitudinal components, but also in their polarization properties.