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Sign Conventions in Wave Optics

Definition: conventions concerning the signs of phase terms in wave optics

German: Vorzeichenkonventionen in der Wellenoptik

Category: general opticsgeneral optics

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Cite the article using its DOI: https://doi.org/10.61835/k9a

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In wave optics, different authors may use different sign conventions concerning the evolution of the optical phase of light waves. These not only directly affect equations describing things like plane waves, but also equations concerning quantities like the complex refractive index. Errors and substantial confusion can result from combining equations based on different sign conventions. This article clarifies that issue.

Sinusoidally oscillating quantities such as an electric field strength <$E(t)$> are often described with a phasor, i.e., with a complex amplitude from which we can get the time-dependent quantity by multiplication with an oscillating factor and taking the real part:

$$E(t) = {\rm Re}(E \: e^{\pm i \omega t})$$

where <$E$> on the right side is the phasor and one of the two signs may be used:

  • In electrical engineering, it is common to use the plus sign, and also to use the symbol <$j$> for the imaginary unit:
$$E(t) = {\rm Re}(E \: e^{+ j \omega t})$$
  • Here, in the complex plane, the phasor rotates in counter-clockwise direction. Any increase of the complex phase of a phasor means a phase advance, i.e., that field maxima are reached at earlier times.
  • In physics, however, it is more common to use the opposite sign and the imaginary unit <$i$>:
$$E(t) = {\rm Re}(E \: e^{-i \omega t})$$
  • Here, in the complex plane the phasor rotates in clockwise direction, and an increase in the complex phase of a phasor means a phase delay.

Here, the physical quantity of interest is always a real part of a complex quantity, i.e., only the projection of the complex value to one axis.

Although it would be hard to research the exact history behind the origins and adoption of such conventions, the reason might be explained as follows:

  • Engineers often deal with a time dependence only, and then prefer the positive sign.
  • Physicists often deal with waves, where there is also a spatial term <$k \: z$>. For a wave propagating in positive <$z$> direction, the sign of the spatial term must be opposite to that of the temporal term – for example, <$(k \: z - \omega \: t)$> – because only then we have wavefronts with increasing position coordinate <$z$> for increasing time <$t$>. If the spatial aspect is of main interest and the time <$t$> does no more occur in equations because one works with phasors (e.g., in the context of monochromatic Gaussian beams), it can be preferable to have the positive sign for the spatial term. For example, a plane wave propagating in <$z$> direction then has a spatially dependent phasor
$$E(z) = E \: e^{i k z} $$
  • where engineers have <$-j \; k \; z$> instead of <$+i \; k \; z$>.

Actually, engineers may also deal with waves, e.g. with microwaves. However, the spatial aspects is often of lower importance.

This encyclopedia is consistently using the physicists' sign convention because the scientific literature seems to prefer that, e.g. in the important context of Gaussian beams. Note, however, that by far not all physicists use physicists' convention, particularly in the context of optics, although it would be more consistent e.g. with common equations in quantum mechanics. For example, some early texts on Gaussian beams and laser resonators [1, 2] already used the engineering convention, which was then adopted by many others.

Fortunately, texts using the engineering convention also very often use the imaginary unit <$j$>, and then one can often just replace <$j$> with <$-i$> to transform the equations into those for the physicists' convention, or vice versa. Mathematically, however, <$i$> and <$j$> are both the imaginary unit, i.e., they are equal. Also note that for many equations it is not obvious how to convert them. The following sections treat some important cases.

Gaussian Beams

With physicist's convention, the complex electric field amplitude (phasor) can be written as

$$E(r,z) = {E_0}\frac{w_0}{w(z)}\;\exp \left( { - \frac{r^2}{w{(z)^2}}} \right)\;\exp \left(i\left[k \: z - \arctan \frac{z}{z_{\rm R}} + \frac{k \: r^2}{2 R(z)} \right] \right)$$

which would be

$$E(r,z) = {E_0}\frac{w_0}{w(z)}\;\exp \left( { - \frac{r^2}{w{(z)^2}}} \right)\;\exp \left(-j\left[k \: z - \arctan \frac{z}{z_{\rm R}} + \frac{k \: r^2}{2 R(z)} \right] \right)$$

for engineers.

For such equations with phasors, the conversion between the conventions is easy, since the phasor according to one convention is always the complex conjugate of that for the other convention.

However, some other equations are less obvious to treat:

The Gaussian <$q$> Parameter

With the engineering convention, the complex <$q$> parameter is often used to write the electric field amplitude as

$$E(r,z) \propto \exp \left[ {-j k r^2 / 2 q(z)} \right],$$

where <$\frac{1}{q} = \frac{1}{R} - j \frac{\lambda}{\pi w^2}$>. With the physicists' convention, we write

$$E(r,z) \propto \exp \left[{+i k r^2 / 2 q(z)} \right]$$

and then must change the sign of the imaginary part of <$1 / q$>:

$$\frac{1}{q} = \frac{1}{R} + i \frac{\lambda}{\pi w^2}$$

because only this leads to the Gaussian function for <$E$>. (The real part of <$1 / q$> does not change its sign, as the phasor must be turned into its complex conjugate.) So effectively, the <$q$> parameter is turned into its complex conjugate (just as the phasors), and we can again apply the rule that <$j$> needs to be replaced with <$-i$>.

Within paraxial optics, the effect of an optical element on a beam can be described with an ABCD matrix. With the engineering convention, the <$q$> parameter after an optical element with such a matrix can be calculated:

$$q' = \frac{{Aq + B}}{{Cq + D}}$$

With physicist's sign convention, we have <$q$> turned into its complex conjugate. Therefore, we can use a similar rule for applying an ABCD matrix, just taking the complex conjugate of that matrix. In numerous instances, the matrix is real and therefore does not change. However, there are cases such as a Gaussian aperture, for example, where at least one matrix component can be complex or even completely imaginary, and those components need to be conjugated to get correct results.

The Complex Refractive Index

Light propagation through an absorbing optical medium can be described with a complex refractive index. That refractive index also turns into its complex conjugate when using the other sign convention:

  • With physicists' convention, the electric field amplitude evolves in proportion to <$\exp(i \; n \; k \; z)$>, where <$k$> is the wavenumber in vacuum. A decaying field amplitude (in <$z$> direction) then requires a positive imaginary component of the refractive index. For a medium with gain, the imaginary component would turn negative.
  • With the engineering convention, the field amplitude evolves in proportion to <$\exp(-j \; n \; k \; z)$>, and absorption requires a negative imaginary part of <$n$>.

More to Learn

Encyclopedia articles:

Bibliography

[1]H. Kogelnik and T. Li, “Laser beams and resonators”, Appl. Opt. 5 (10), 1550 (1966); https://doi.org/10.1364/AO.5.001550
[2]A. E. Siegman, Lasers, University Science Books, Mill Valley, CA (1986)

(Suggest additional literature!)

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