2.2: Spatial modes

Total internal reflection is not the only condition for guiding, however. Light is a wave that has physical extent in all dimensions. Thus, as demonstrated in Figure \(\PageIndex{1}\), light that has reflected from the top interface interferes with light that has reflected from the bottom interface. If the interference is not constructive, then the amplitude of the light wave will decrease with distance traveled, and eventually disappears entirely. More specifically, after two reflections, the phase change of the wave along the axis of the waveguide should be an integer multiple of \(\pi\). When this condition is satisfied, the light occupies a spatial mode of the waveguide.

Figure \(\PageIndex{1}\): The brown wavefronts are for a ray traveling from top to bottom, and the pink wavefronts are for a ray traveling from bottom to top. Note how there are three lines down the waveguide where the pink and brown wavefronts cross. Also note how the wavefront at a reflection point overlaps perfectly with the wavefront from the parallel ray before the reflection. These are all conditions to be a spatial mode of the waveguide.

The picture in Figure \(\PageIndex{1}\) is complicated by the fact that total internal reflection introduces an angle-dependent phase shift to the light wave on reflection. However, in what follows, we first apply a constant phase shift of \(\pi\) (thus treating the reflection as mirror-like), and then afterwards account for an angular dependent phase shift.

Note

If you are not familiar with the wavefront picture above, it can be a bit confusing. The arrowed line represents the direction of travel for light (a light ray). The lines at right angle represent the peaks in amplitude, which are separated by one wavelength. This picture, then represents a frozen moment in time and allows us to compare the phase relationship between multiple waves, or, in this case, the wave with itself.

Mirror-like reflections

A geometric solution to the phase consistency problem is presented in Figure \(\PageIndex{2}\). The line \(\overline{AB}\) represents the path the ray would have taken if there was no reflection, while path \(\overline{AC}\) represents the path the light has actually taken. For self consistency the two paths can only differ by integer multiples of \(2\pi\). The accumulated phase over \(\overline{AB}\) is \(\phi_{AB} = \frac{2\pi\overline{AB}}{\lambda}\), where \(\lambda\) is the wavelength of the light. Likewise, the accumulated phase over \(\overline{AC}\) is \(\phi_{AC} = \frac{2\pi\overline{AC}}{\lambda}\). The difference between the two is \(\Delta\phi = \frac{2\pi\bar{AC}}{\lambda} - \frac{2\pi\overline{AB}}{\lambda} - 2\pi\), where the \(2\pi\) accounts for the additional phase shift due to two reflections. The difference must be \(\Delta\phi = 2m\pi\) where \(m\) is an integer and it doesn't really matter where we start \(m\), so we can subtract the \(2\pi\) and reset \(m\), leaving \(m = \frac{1}{\lambda}\left(\overline{AC}- \overline{AB}\right)\), where \(m = 1, 2, ...\).

Figure \(\PageIndex{2}\): A geometric view of the self consistency problem. The solid green lines represent the path of the ray in the waveguide. The dashed green line represents the path that the ray would have taken if it had not been reflected. The two paths \(\overline{AB}\) and \(\overline{AC}\) should be an integer number of wavelengths.

\[\overline{AC} = \frac{d}{\sin\gamma} \label{eq:AC}\]

where \(\gamma\) is the complementary angle to \(\theta\). The triangle \(\overline{ABC}\) allows us to solve for the angles that meet the self-consistency condition.

\[\begin{align} \cos 2\gamma &= \frac{\overline{AB}}{\overline{AC}}\\ \cos 2\gamma &= 1-2\sin^2\gamma\\ 1-2\sin^2\gamma &= \frac{\overline{AB}}{\overline{AC}}\\ \overline{AC} - 2\overline{AC}\sin^2\gamma &= \overline{AB}\\ \overline{AC}-\overline{AB} &= 2\overline{AC}\sin^2\gamma\\ \overline{AC}-\overline{AB} &= \frac{2d\sin^2\gamma}{\sin\gamma}\\ m\lambda &= 2d\sin\gamma\\ \sin\gamma_m &= m\frac{\lambda}{2d} \label{eq:mirrormodes} \end{align}\]

Equation \ref{eq:mirrormodes} represents the set of angles that satisfy the self consistency relationship for any given wavelength. By setting \(\sin\gamma_m\) = 1 and solving for \(m\), we find that the maximum number of modes supported by the waveguide is

\[M \underset{\cdot}{=} \frac{2d}{\lambda}\label{eq:modeNumber}\]

where \(M\) is rounded down to the nearest integer. Note that this also means that, for this type of waveguide, there is a minimum dimension to enable a single mode to exist. If \(M=0\) that indicates that, for that wavelength, no propagating modes exist and no light will be transported by the waveguide. In this case, an attempt to couple light into the waveguide would result in a very strong scattering from the front face.

Later on, we will find that the number of modes is a severe limiting factor in the performance of an optical communications system. Indeed, for long distance communications, \(M=1\), otherwise known as single mode operation, is an important criteria.

Angle-dependent phase shift

Snell's law does not tell the full story for refraction and reflection. When light is reflected from an interface, it also undergoes a phase shift, as illustrated in Figure \(\PageIndex{3}\).

Figure \(\PageIndex{3}\): Reflection coefficient and phase change for the case of the electric field parallel (brown) to the interface (TE mode) and perpendicular (purple) to the interface (TM mode).

Note that the phase shift also depends on the orientation of the electric field with respect to the interface. In what follows, we only discuss the case for the electric field parallel to the interface, called transverse electric (TE) modes (when the magnetic field is parallel to the interface, this is called the transverse magnetic (TM) mode). The phase shift due to reflection is given by

\[\tan\frac{\phi_r}{2} = \sqrt{\frac{\sin^2\gamma_c}{\sin^2\gamma}-1} \label{eq:phaseShift}\]

where \(\gamma_c\) is the complementary angle to the critical angle, \(\theta_c\), and \(\phi_r\) is the phase shift due to reflection. Note that equation \ref{eq:phaseShift} is not valid for \(\gamma>\gamma_c\), where, instead, one must use the Fresnel amplitude reflection coefficients, that were used to in Figure \(\PageIndex{3}\).

The self-consistency expression given in equation \ref{eq:mirrormodes} becomes

\[\tan\left(\frac{\pi d}{\lambda}\sin\gamma - \frac{m\pi}{2}\right) = \sqrt{\frac{\sin^2\gamma_c}{\sin^2\gamma}-1} \label{eq:dielectricModes}\]

where \(m = 0, 1, 2, ...\). Equation \ref{eq:dielectricModes} is best solved graphically by looking for intercepts between the left hand side and right hand side, as shown in Figure \(\PageIndex{4}\). The total number of TE modes supported by a waveguide is given by

\[M \dot{=} \frac{2d\sin\gamma_c}{\lambda}\label{eq:dielectricModeNumber}\]

which is related to the NA by \(M \dot{=} \frac{2d}{\lambda_0}\textrm{NA}\), where \(M\) is now always rounded up, and \(\lambda_0\) denotes the wavelength in vacuum. In contrast to the mirror-like case, there is no cutoff for any wavelength. However, to achieve single mode operation, the dimensions have to be quite close to the operational wavelength.

In order to know precisely which mode is being discussed, a labeling scheme is used that tells us the polarization of the mode and its spatial character. For a slab waveguide, as discussed here, the modes are denoted as TE\(_0\), TE\(_1\), TE\(_2\), ... and TM\(_0\), TM\(_1\), TM\(_2\), ... for the transverse electric and transverse magnetic modes.

Figure \(\PageIndex{4}\): Graphical solution to identifying spatial modes in a dielectric waveguide. The crossing points between the purple (right hand side of equation \ref{eq:dielectricModes}) curve and the other curves (left hand side of equation \ref{eq:dielectricModes}).

Going beyond slab waveguides

So far, we have looked at a waveguide that is an infinite wide slab so that we only have to deal with a single dimension. For square waveguides, the mathematics remains the same, but now there are modes associated with both spatial dimensions, thus, if the axis of propagation of a waveguide is along the \(z\) axis, then the two dimensions \(d_x\), and \(d_y\) each support \(M_x\) and \(M_y\) modes (calculated from equation \ref{eq:dielectricModeNumber}). The modes along each axis are independent of each other, so the total number of modes, \(M\) is all possible combinations \(M = M_xM_y\).

Optical fibers are cylinders, meaning that the geometric approach used above cannot be used. The electric and magnetic fields are never quite parallel to the interface of an optical fiber, resulting in so-called linearly polarized (LP) modes that are denoted by two integers \(l\) and \(m\). As with the dielectric slab waveguide, it is simplest to find the modes by graphically finding the intercepts between two equations (both Bessel's functions) that represent the electric field in the core and the cladding of the fiber.

Even the total number of modes supported by an optical fiber can be difficult to calculate. However, we are mostly interested in whether the fiber is single mode, multi-mode, and occasionally few-mode. To estimate this, the \(V\) parameter, which (sort of) represents the mismatch between light propagation speed in the core and cladding materials.

\[V=\frac{2\pi r_{co}}{\lambda_0}\textrm{NA} \label{eq:vParam}\]

where \(r_{co}\) is the radius of the core. Single mode operation is guaranteed for \(V<\)2.405. For multimode operation, the total number of modes can be estimated from \(M\approx\frac{4V^2}{\pi^2}\).