**Table of Contents:**

**PROPAGATION PARAMETERS**

In the third section of the “Classical Optics” portion of this website, we derived the propagation parameters for a fiber optic waveguide from Maxwell’s equation, so I would recommend looking there first if you haven’t already to find out more about how some of the results included below were derived!

A very simple explanation of a fiber optic waveguide is that it consists of a central core of higher refractive index than two surrounding cladding regions. Fibers of this type are called “step-index”, as opposed to graded index, or GRIN fibers. GRIN characteristics are covered in Section 4 of “Classical Optics”, so check that out also if you are curious! The guiding properties of step index fibers i.e. how well the central mode is confined depends on the parameters of the fiber itself. Some of the main parameters are the core and cladding index differences, typically put into a parameter as:

\begin{equation} \delta = \frac{n_1 – n_2}{n_1} \end{equation}

and the normalized frequency V, which we derived in the “Classical Optics” portion both using boundary conditions and the transfer matrix approach. The normalized frequency, or V-number, is given by:

\begin{equation} V = k_0a(n_1^2-n_2^2)^{1/2} \end{equation}

where $k_0 = 2\pi/\lambda$ and is the vacuum k-vector, $a$ is the core radius, and $\lambda$ is the wavelength of light. The V-number determines the number of modes that propagate in a fiber, and it is derived from exact solutions to the eigenvalue equation for fibers (discussed more in the next section) that V < 2.405 is the cutoff for single mode fibers.

**MATERIAL AND FABRICATION**

Most fibers in use today use silica, or $SiO_2$ molecules, as their cladding region materials and the core region “host” material. The reason that the core silica region is differentiated also has to do with how this region has a higher refractive index: the core is doped with free ions that are then commonly used for laser and amplifier operation. The specific dopants are chosen for the application.

**OPTICAL LOSSES **

An important equation to know, especially for fiber optic telecom systems but also for high power applications, is the rate at which light is absorbed in a fiber. The total power absorption is given as:

\begin{equation} P_T = P_0e^{(-\alpha{L})} \end{equation}

where $\alpha$ is the fiber loss. If you are to look at the fiber specifications of any fiber you may work with or order for your lab, the units of absorption are normally given in dB/m. Thus, the absorption in dBs is given as:

\begin{equation} \alpha_{dB} = -\frac{10}{L}log[\frac{P_T}{P_0}] = 4.343\alpha \end{equation}

It is important to know that the fiber loss coefficient $\alpha$ depends on the the wavelength of light propagating through. The figure below shows this explicitly.

A couple important things of note here. First, the minimum of absorption is at ~1.55 $\mu{m}$ with $\alpha_{dB} \approx$ 0.2 dB/dm. This is the main reason why long distance communications systems use 1.55 micron. As noted in the above figure, much of the absorption is attributable to OH ion impurities in the fiber, as OH absorbs light very well at these wavelengths.

Another important thing to note, though, is the high absorption around 1.38 $\mu{m}$. This is the result of Rayleigh scattering, which arises from random density fluctuations in the fiber that of course then changes the refractive index such that a lot of light scattered. The Rayleigh scattering loss is intrinsically estimated to be:

\begin{equation} \alpha_R = C/\lambda^4 \end{equation}

which means that the loss goes up quartically with smaller wavelengths. The reason this is significant is that 1.33 $\mu{m}$ corresponds with the zero dispersion length, which is also important for optical communications because dispersion can distort a signal. As this is a region of high loss, though, these benefits must be weighed for different applications.

**CHROMATIC DISPERSION**

Another aspect of propagation within a fiber that is highly dependent on the wavelength of the light is the amount of dispersion that the pulses of light see. For this spectral dependence, we call this *chromatic dispersion*. On a quantum mechanical level, the reason for time dependence of the frequency in dispersion has to do with the resonance frequencies of electrons in a material: different frequencies will strike a different amount of resonance. The amount of chromatic dispersion is typically first explored by writing out the frequency dependence of the refractive index, given by the Sellmeier equation:

\begin{equation} n^2(\omega) = 1 + \sum^m{j=1} \frac{B_j\omega^2_j}{\omega_j^2-\omega^2} \end{equation}

where $\omega_j$ is the frequency of the $j^{th}$ material resonance and $B_j$ is the strength of the electronic response of the material at that resonance frequency. $B_j$ parameters can only be determined analytical through quantum mechanical theory, so typically in practice people just use the tabulated values of $B_j$ for the material of interest.

The amount of dispersion seen by an optical pulse is a critical component particularly for ultrashort pulse applications, where a time delay of any spectral components can lead to catastrophic pulse degradation. Additionally, for communications systems, the amount of dispersion in a pulse can lead to detrimental effects on the fidelity of the communication line. Since the propagation of modes in space is given by $\beta$, the k-vector that is typically arbitrarily chosen to align with the z-axis, we can find the effects of the phase delay and thus pulse broadening by Taylor expanding the propagation vector about the central frequency $\omega_0$:

\begin{equation}\beta(\omega) = n(\omega)\frac{\omega}{c} = \beta_0 + \beta_1(\omega – \omega_0) + \frac{1}{2}\beta_2(\omega-\omega_0)^2 + … \end{equation}

where

\begin{equation} \beta_m = \begin{pmatrix} \frac{d^m\beta}{d\omega^m}_{\omega = \omega_0} \end{pmatrix}~~(m=0,1,2, …) \end{equation}

The concept of dispersion in general will be covered in much more detail in the next two sections (Sections 2 and 3), but for now we will just state that the pulse envelope, which is what modulates the carrier wave and gives it its pulse shape, moves at the group velocity $v_g = \beta_1^{-1}$, while the other higher order $\beta$ terms are responsible for broadening the envelope. The group velocity and $\beta_2$ can be found as:

\begin{equation} \beta_1 = \frac{1}{c}[n+\omega\frac{dn}{d\omega}] = \frac{n_g}{c} = \frac{1}{v_g} \end{equation}

\begin{equation} \beta_2 = \frac{1}{c}[2\frac{dn}{d\omega}+\omega\frac{d^2n}{d\omega^2}] = \frac{\omega}{c}\frac{d^n}{d\omega^2} = \frac{\lambda^3}{2\pi{c^2}}\frac{d^2n}{d\lambda^2} \end{equation}

where $n_g$ is the group index. The dispersion vs wavelength is plotted below.

There are several important things to note in this plot shown here. First, the zero dispersion wavelength, i.e. where $\beta_2 = 0$, occurs at $\approx~1.33~\mu{m}$. It is necessary to keep in mind though that while $\beta_2$ may be zero here, the higher order dispersion terms i.e. $\beta_3$ and higher would now need to be considered as they would have stronger effects on the pulse duration. Second, there is also a contribution to the total amount of dispersion by the waveguide itself, which can be seen in the above image. This arises from a effective modal index that is slightly lower than the material index $n(\omega)$ that is $\omega$-dependent. This is discussed in some of the references (1-2) below, but for our purposes here we will just state that the net effect, as can be seen above, is to shift the zero dispersion wavelength to slightly higher wavelengths.

Commonly in optics literature, a dispersion parameter “D” is used instead of the $\beta_2$; it is related to $\beta_2$ as:

\begin{equation} D = \frac{d\beta_1}{d\lambda} = -\frac{2\pi{c}}{\lambda^2}\beta_2 = -\frac{\lambda}{c}\frac{d^2n}{d\lambda^2} \end{equation}

A cool thing about the waveguide dispersion (and a testament to how smart certain humans who came before us were) is that it depends almost exclusively on fiber design parameters, and so it can be used to shift the zero-dispersion wavelength to longer wavelengths in the vicinity of the minimum absorption value 1.55 $\mu{m}$. These are called dispersion-shifted fibers and have potential applications in optical communications systems. This is most basically achieved through the use of multiple cladding layers, the exact necessary composition of which can be found by starting from electromagnetic theory. Dispersion shifted fibers are discussed in further detail in reference (3) below.

Since we will be talking in later sections a lot about the balance of dispersion and nonlinearity, it is important to know the two different regimes that dispersion can take. First, let’s look at the dispersion parameter some more:

\begin{equation} \beta_2 = \frac{d\beta_1}{d\omega} = \frac{d}{d\omega}[\frac{1}{v_g}] = -\frac{1}{v_g^2}\frac{dv_g}{d\omega} \end{equation}

As we can see from this equation, the effect of dispersion on a pulse is to make its instantaneous frequency (the rate of change of the group velocity) dependent on the wavelength of light. If the instantaneous frequency was zero, meaning that the group velocity was constant along the pulse due to equal material resonances for different wavelengths, then the total amount of dispersion would also be zero. A nonzero instantaneous frequency change of the group velocity means that the material resonances where the light is propagating are not equal. This in turn makes the index of refraction dependent on the wavelengths present in the pulse and thus different wavelength components in the pulse are delayed in time by different amounts than other wavelength components. If you didn’t get lost in all of those weeds, it should now be obvious that the apt name for the $\beta_2$ parameter is “group velocity dispersion”, or GVD. The notation in the field is that for $\lambda < \lambda_D$ such that $\beta_2 > 0$, the dispersion regime is known as “normal dispersion”. Normal dispersion means that the higher frequency (blue-shifted), lower wavelength components travel slower than the lower frequency (red-shifted), higher wavelength components. The “anomalous dispersion” regime occurs when $\lambda > \lambda_D$ such that $\beta_2 < 0$ and would thus have the opposite effect: lower frequency, higher wavelength components travel slower than the higher frequency, shorter wavelength components. This regime will become important when discussing solitons.

Final note about chromatic dispersion for now, but an important concept for when we discuss cross-phase modulation is the concept of pulse “walk-off” due to pulses travelling at different speeds in a fiber. Basically, if you have two or more overlapped pulses with different central wavelengths interacting nonlinearly, the effect of dispersion could be to shift the amount of overlap between the pulses such that you have “walk-off”. As should be expected by this point, there is of course a parameter for that creatively named the “walk-off parameter” $d_{12}$, and it is given as:

\begin{equation} d_{12} = \beta_1(\lambda_1)-\beta_1(\lambda_2) = v_g^{-1}(\lambda_2) \end{equation}

where $\lambda_1$ and $\lambda_2$ denote the central wavelengths of the respective pulses. For a pulse of width $T_0$, the walk-off length is defined as:

\begin{equation} L_W = T_0/|d_{12}| \end{equation}

**MODAL BIREFRINGENCE**

The secret about single mode fibers is that they are not truly single mode, as the fiber can support two degenerate, orthogonally polarized modes. This happens due to random changes in the cylindrical shape of the fiber or in small amounts of change in the birefringence. The net effect is then to “mix” these two modes such that a pulse originally launched in with a defined polarization soon then reaches an arbitrary state of polarization a short length through the fiber. In applications that require the polarization to remain unchanged within the fiber, there are “polarization-preserving” fibers that work by intentionally adding in much birefringence such that these random fluctuations do not affect propagation through the fiber.

**REFERENCES:**

$^1$ Marcuse, Dietrich. *Light Transmission Optics. –*. Van Nostrand Reinhold, 1972.

$^2$ Snyder, Allan W., and Love, John D. *Optical Waveguide Theory*. Springer US, 1984.

$^3$ B. Ainslie and C. Day, “A review of single-mode fibers with modified dispersion characteristics,” in *Journal of Lightwave Technology*, vol. 4, no. 8, pp. 967-979, August 1986, doi: 10.1109/JLT.1986.1074843.