# Linear Lightwave Propagation in an Optical Fiber

This is a continuation from the previous tutorial - ** Fourier transform for periodic signals**.

## 1. Physical Structure of a Telecommunication Optical Fiber

Optical fibers are fabricated by first depositing high-purity silica soot, doped with germania (GeO_{2}) to raise the index of refraction or fluorine (F) to lower the index of refraction, to form a core rod of 1 cm or more in diameter and 1 m or more in length.

Fabrication methods include processes known in the industry as ‘‘modified chemical vapor deposition’’ (MCVD), outside vapor deposition (OVD), vapor axial deposition (VAD), and plasma chemical vapor deposition (PCVD).

The MCVD, OVD, and VAD methods involve two steps of deposition and subsequent sintering of oxide soot formed by flame hydrolysis, while the PCVD method produces oxide layers directly in one step. The core rod comprises both the raised index light-guiding core and the portion of the cladding where significant optical power propagates, representing on the order of 10% of the total cross-sectional area of glass.

The core rod plus overcladding glass forms a preform. The overclad typically comprises silica of lower purity that may be derived from deposition of flame hydrolysis soot in the form of OVD, plasma deposition, or sol-gel casting.

This material may be either deposited directly onto the core or else formed separately as a tube that is subsequently collapsed onto the core rod. In either case, the preform is drawn down at approximately 2200°C to a 125-μm diameter optical fiber at speeds greater than 10 m/sec and coated with both a primary and a secondary acrylate ultraviolet (UV)-cured polymer before take-up on a bobbin.

The coating serves to preserve strength by protecting the glass surface from particles, to provide some limited protection from environmental moisture, and to provide mechanical protection from stresses that cause microbending losses. The light-guiding core itself comprises the inner 8- to 10-micron diameter of the 125-μm OD glass fiber.

## 2. Linear Lightwave Propagation in an Optical Fiber

### 2.1 Electromagnetic Preliminaries

Any treatment of light guiding in a fiber must begin with the Maxwell equations and describe their solution to some degree of mathematical detail.

Many excellent treatments of dielectric waveguides exist, and the reader would benefit by consulting one or more of these. We draw heavily on Buck’s treatment.

The Maxwell equations in MKS units can be written as

\[\tag{2.1}\nabla\times\pmb{E}=-\frac{\partial\pmb{B}}{\partial{t}}\qquad\nabla\times\pmb{H}=\pmb{J}+\frac{\partial\pmb{D}}{\partial{t}}\qquad\nabla\cdot\pmb{D}=\rho_\text{free}\qquad\nabla\cdot\pmb{B}=0\]

where \(\pmb{D}=\epsilon\pmb{E}\) and \(\pmb{B}=\mu\pmb{H}\), where \(\epsilon\) and \(\mu\) are the permittivity and permeability, respectfully, of the medium.

In a source-less medium, \(\pmb{J}=0\) and \(\rho_\text{free}=0\). Using standard manipulations, the wave equations for propagating \(\pmb{E}\) and \(\pmb{H}\) fields can be derived from the Maxwell equations as

\[\tag{2.2}\nabla^2\pmb{E}-\mu\epsilon\frac{\partial^2\pmb{E}}{\partial{t^2}}=0\qquad\text{and}\qquad\nabla^2\pmb{H}-\mu\epsilon\frac{\partial^2\pmb{H}}{\partial{t^2}}=0\]

The formulas in Eq. (2.2) are each three-wave equations, one for each vector component of \(\pmb{E}\) and \(\pmb{H}\). Assuming time harmonic fields, we may generally write (for a wave propagating in the \(z\) direction)

\[\tag{2.3}E=E_0\exp[j(\omega{t}\pm\beta{z}+\phi)]\]

where \(\beta=\omega/v=\omega{n}/c\) is the propagation constant, or the phase shift per length, of a sinusoidal wave measured along the \(z\) axis, and \(v=1/\sqrt{\mu\epsilon}\) is the wave velocity.

The explicit form of the time dependence can be used to simplify the form of the Maxwell equations to

\[\tag{2.4}\nabla\times\pmb{E}=-j\omega\mu\pmb{H}\qquad\nabla\times\pmb{H}=j\omega\epsilon\pmb{E}\qquad\nabla\cdot\epsilon\pmb{E}=0\qquad\nabla\cdot\mu\pmb{H}=0\]

Defining \(k=\omega\sqrt{\mu\epsilon}\), we can derive the wave equation in phasor form as the vector Helmholtz equations:

\[\tag{2.5}\nabla^2\pmb{E}+k^2\pmb{E}=0\qquad\text{and}\qquad\nabla^2\pmb{H}+k^2\pmb{H}=0\]

The wave number \(k\) has units of \(m^{-1}\) and is a property of the material layer. The wave vector \(\pmb{K}\) points in the direction of energy flow and has magnitude \(|\pmb{K}|=k\).

The propagation constant \(\beta\) will be used to refer to the rate of accumulation of phase with distance for the electromagnetic wave we seek to calculate. In the case of a plane electromagnetic wave in a uniform, linear, and isotropic medium of dielectric constant, the propagation constant is simply \(\beta=k=\omega{n}/c=nk_0\) where \(k_0=2\pi/\lambda\) and \(\lambda\) is the wavelength of light in vacuum.

In a waveguide, however, each region \(i\) will be characterized by index \(n_i\), the magnitude of the wave vector in each region \(i\) will be \(|\pmb{K}_i|=n_ik_0\), and the Helmholtz equations require solution in each region with matching of boundary conditions at the interfaces. We will always choose the direction of propagation in a guide to be along the \(z\) direction.

### 2.2 Intuition from the Slab Waveguide

Some intuition can be gained by considering the ray optics picture of a simple slab waveguide. An electromagnetic wave in a guide will spread over at least two regions (core and cladding). The given optical properties of the waveguide structure are characterized by \(k_i\), while the characteristics of the propagating wave for which we seek a solution are described by the propagation constant \(\tilde{\beta}\).

Figure 2.1 shows a step-index waveguide supporting a guided mode, with index \(n_1\gt{n_2}\). The index level of the doped glass is frequently characterized by \(\Delta=\frac{n_1^2-n_2^2}{2n_1^2}\approx\frac{n_1-n_2}{n_1}\) given in percent. The use of \(\Delta\) references doped structures in the waveguide core to the cladding index \(n_2\), without regard to the actual value of \(n_2\). In more complex designs, the value of \(\Delta\), for additional doped layers adjacent to the central core, are defined analogously, and \(\Delta\) can be positive or negative.

In Fig. 2.1, propagation occurs along the \(z\) axis (into the page), and the guiding structure confines light in the \(x\) direction. The forward propagation constant \(\beta\) for a guided mode along the \(z\) axis is constrained by the relation \(n_2k_0\lt\beta\lt{n_1}k_0\), because the mode spreads over both the core and the cladding region.

Figure 2.2 illustrates the geometries of the wave vectors in the two regions of a slab waveguide in the ray optics picture, where rays reflect and transmit according to the Fresnel equations at the interfaces.

For waveguides oriented along the \(z\) axis, we will write the wave vector in each region as \(\pmb{K}_i=\kappa_i\pmb{e}_{\pmb{x}}+0\pmb{e}_{\pmb{y}}+\beta\pmb{e}_{\pmb{z}}\) so that in each region, \(\kappa_i^2+\beta^2=n_i^2k_0^2\), \(\kappa_i=n_ik_0\cos\theta_i\), and \(\beta=n_ik_0\sin\theta_i\).

Although the guided mode propagates only in the \(z\) direction, it is common and convenient to refer to \(\kappa\) and \(\beta\) as the transverse and forward propagation constants, respectively. Intuitively, the propagation constant \(\beta\) must be the same across all regions of a waveguide, while the transverse propagation constants \(\kappa_i\) will differ and may be imaginary.

Mathematically, \(\beta\) must be the same across all regions because of the requirement that the tangential components of the fields must be continuous across the interfaces between the regions.

The electric field \(E_1\) in the guiding region 1 assumes the form

\[\tag{2.6}E_1\sim\exp(\mp{j}\kappa_1x)\exp(-j\beta{z})\]

where "\(-\)" corresponds to upward propagation and "\(+\)" corresponds to downward propagation along the \(x\) direction in Fig. 2.2.

According to Snell’s law, \(n_1\sin\theta_1=n_2\sin\theta_2\), total internal reflection will occur when \(\theta_1\gt\theta_\text{c}=\sin^{-1}(n_2/n_1)\). For \(\theta_1=\theta_\text{c}\), \(\theta_2\rightarrow90^\circ\), \(\kappa_2=0\), and \(\pmb{K}_2\) tilts over to lie along the \(z\) axis with \(\beta=n_2k_0\).

A guided mode will propagate under the condition of total internal reflection when \(\theta_1\gt\theta_\text{c}\) so that \(\beta\gt{n_2}k_0\). In this case, \(\kappa_2\) becomes imaginary, and we can write \(\kappa_2\rightarrow-j\gamma_2\), where the decay constant \(\gamma_2\) is a real number so that \(\gamma_2=j\kappa_2=(\beta^2-n_2^2k_0^2)^{1/2}\).

Then the electric field \(E_2\) in the cladding region 2 assumes the form

\[\tag{2.7}E_2\sim\exp(\mp\gamma_2x)\exp(-j\beta{z})\]

To form a propagating mode, the upward traveling waves represented by \(\exp(\mp{j}\kappa_1x)\) in Eq. (2.6) must be in phase after traversing the waveguide in region 1, including phase shifts caused by reflection from the interfaces, and the total phase shift from (1) to (2) in Fig. 2.2 must be an integral multiple of \(2\pi\).

The phase shifts are determined from the Fresnel equations for the transverse electric (TE) and transverse magnetic (TM) cases, where the electric (magnetic) field oscillates entirely in the transverse (\(xy\)) plane for TE (TM).

This transverse resonance condition fixes the values of \(\beta\) and \(\gamma\), leading to one or more guided modes that propagate along the \(z\) axis, form a standing wave along the \(x\) axis in region 1, and decay exponentially in region 2.

The same result can be obtained by solving the wave equation over the regions and matching boundary conditions at the interfaces. The guided mode is said to be cutoff when \(\theta_1\le\theta_\text{c}\), where \(\kappa_2\) is real and \(\beta\le{n_2}k_0\). An unguided wave below cutoff that nevertheless meets the transverse resonance conditions is sometimes known as a ‘‘leaky wave.’’

### 2.3 Optical Fiber: A Cylindrical Waveguide

The ray optics analysis for a cylindrical waveguide such as an optical fiber is complicated by the existence of skew rays in three dimensions, which propagate helically and do not cross the fiber axis.

Here, we outline the solution of the field equations to derive the form of the fiber modes in cylindrical coordinates and illustrate their properties. For a step-index optical fiber of the basic form illustrated in Fig. 2.1 with core index \(n_1\) and radius \(a\), we can assume field solutions of the form:

\[\tag{2.8}\pmb{E}=\pmb{E}_0(r,\phi)\exp(-j\beta{z})\qquad\pmb{H}=\pmb{H}_0(r,\phi)\exp(-j\beta{z})\]

Again, each vector Helmholtz equation in Eq. (2.5) represents three scalar equations, for a total of six. We can solve for one field component, such as \(E_z\), however, and then use the Maxwell equations to derive the others. Substituting these, the Helmholtz equation for the electric field becomes

\[\tag{2.9a}\nabla^2_tE_{z1}+(n_1^2k_0^2-\beta^2)E_{z1}=0\qquad\text{ for }r\le{a}\]

and

\[\tag{2.9b}\nabla^2_tE_{z2}+(n_2^2k_0^2-\beta^2)E_{z2}=0\qquad\text{ for }r\ge{a}\]

where the transverse Laplacian \(\nabla^2_t\) includes only the radial and angular derivatives. For notational convenience, we define the transverse propagation constants

\[\tag{2.9c}\beta_{t1}^2=(n_1^2k_0^2-\beta^2)\qquad\text{ and }\qquad\beta_{t2}^2=(n_2^2k_0^2-\beta^2)\]

which play the same role as \(\kappa_i\) in the slab waveguide discussion. As in the case of the slab waveguide \(\beta_{t2}\) is imaginary for a guided mode since \(\beta\gt{n_2}k_0\).

Writing the solution in the form \(E_z=R(r)\Phi(\phi)\exp(-j\beta{z})\), and performing the standard separation of variables, one finds that \(\Phi(\phi)=\sin(q\phi)\), where \(q\) takes integer values and is identified as the azimuthal or angular mode number.

For real \(\beta_{t1}\) (in the core region \(r\le{a}\)) the radial solutions are \(J_q(\beta_tr)\), the ordinary Bessel functions of the first kind. For imaginary \(\beta_{t2}\) (in the cladding region \(r\ge{a}\)), the radial solutions are \(K_q(|\beta_t|r)\), the modified Bessel functions that monotonically approach zero for large value of the argument.

The normalized transverse propagation and decay constants are defined, respectively, as

\[\tag{2.10a}u=\beta_{t1}a=a(n_1^2k_0^2-\beta^2)^{1/2}\qquad\text{and}\qquad{w}=|\beta_{t2}|a=a(\beta^2-n_2^2k_0^2)^{1/2}\]

Then the complete solution can be written as

\[\tag{2.10b}E_z=AJ_q(ur/a)\sin(q\phi)\exp(-j\beta{z})\qquad\text{for }r\le{a}\]

\[\tag{2.10c}E_z=CK_q(wr/a)\sin(q\phi)\exp(-j\beta{z})\qquad\text{for }r\ge{a}\]

Note that the \(\phi\) and \(z\) dependences are identical in the core and cladding, as expected by intuition, with the solutions differing only in the radial dependence.

The form of the solution for \(E_r\), \(E_\phi\), \(H_r\), \(H_z\), and \(H_\phi\) can be derived from the solution for \(E_z\) using the Maxwell equations.

The solutions are characterized by \(q\), the mode order, or azimuthal mode number, which takes on integer values \(q=0,1,2,\) and so on. The eigenvalues are the unique sets of \(u\), \(w\), and \(\beta\), which match boundary conditions requiring continuity of the tangential field components at \(r=a\), thus determining the modes supported by the waveguide.

The resulting eigenvalues are numbered by the mode rank, \(m\), or radial mode number, which takes integer values \(m=1,2,3,\) and so on.

The transverse electric or \(\text{TE}_{0m}\) set of modes has components \(E_z=0\) (by definition) and \(E_\phi\), \(H_z\), and \(H_r\ne0\). The transverse magnetic or \(\text{TM}_{0m}\) set of modes has \(H_z=0\), and \(E_z\), \(E_r\), and \(H_\phi\ne0\).

Modes with \(q\ne0\) are labeled \(\text{EH}_{qm}\) or \(\text{HE}_{qm}\), which physically correspond to skew rays in the ray optics picture.

### 2.4. The Linearly Polarized Mode Set \(\text{LP}_{lm}\)

The problem of solving for the eigenvalues is greatly simplified by the weak guidance approximation \(n_1\approx{n_2}\). Cabled telecommunications fibers invariably have values of \(\Delta\lt1\%\), although dispersion compensating fibers may have values of \(\Delta\) as high as \(2\%\).

This approximation is excellent for fibers with \(\Delta\lt1\%\), but it may be used with reasonable results for many fiber designs with \(\Delta\sim2\%\). The weak guidance approximation also aids in grouping degenerate modes, which have the same value of \(\beta\) but slightly different field configurations, to form a set of modes referenced by the notation \(\text{LP}_{lm}\) that are linearly polarized in the transverse plane.

These modes are natural modes for describing the fiber, given that communications lasers typically emit linearly polarized light that maintains its polarization in a fiber in the absence of perturbations.

The new mode number \(l\) is introduced as follows:

\[\begin{align}&1\qquad&&\text{for }\text{TE}_{0m}\text{ or }\text{TM}_{0m}\\l=&q+1\quad&&\text{for }\text{EH}_{qm}\\&q-1\quad&&\text{for }\text{HE}_{qm}\\\end{align}\]

Using the full set of vector field expression for \(E\) and \(H\) to match boundary condition at \(r=a\), and using the weak guidance approximation, the eigenvalue equation to be solved is

\[\tag{2.11}u\frac{J_{l-1}(u)}{J_l(u)}=-w\frac{K_{l-1}(w)}{K_l(w)}\]

It is also helpful to introduce the normalized spatial frequency \(V\), or \(V\) number:

\[\tag{2.12}V=(u^2+w^2)^{1/2}=ak_0(n_1^2-n_2^2)^{1/2}=n_1ak_0\sqrt{2\Delta}\]

The definition of \(V\) shows that possible values of \(u\) and \(w\) lie on a circle of radius \(V\), which can be related through Eq. (2.10a) to the fundamental parameters of the waveguide.

Larger values of \(V\), due to greater index contrast, shorter wavelength, or larger core, lead to the possibility of more guided modes in the waveguide. A mode is said to be cutoff when it ceases to be confined to the waveguide, that is, when the field in the cladding region 2 ceases to be an evanescent wave.

Cutoff, thus, occurs as \(\beta\rightarrow{n_2}k_0\). Near cutoff, \(w\rightarrow0\) and the evanescent wave extends farther into the cladding. Beyond cutoff, the propagation constant \(\beta_{t2}\) becomes real, the transverse field in the cladding begins to propagate, and the solution becomes a leaky wave rather than a guided mode.

The cutoff condition for modes are, thus, found by setting \(w=0\), in which case Eq. (2.11) reduces to \(V\frac{J_{l-1}(V)}{J_l(V)}=0\), showing that the zeros of the Bessel functions give the conditions for mode cutoff.

The \(\text{LP}_{01}\) mode is simply the \(\text{HE}_{11}\) mode and is not cutoff at any wavelength. The \(\text{LP}_{11}\) mode is composed of the \(\text{TE}_{01}\), \(\text{TM}_{01}\), and \(\text{HE}_{21}\) modes, and cuts off at a \(V\) number of 2.405, the first zero of \(J_0\).

The \(\text{LP}_{21}\) and \(\text{LP}_{02}\) modes both cut off at \(V=3.832\), the zero of \(J_{\pm1}\).

The intensity patterns of the \(\text{LP}_{lm}\) modes are given by \(I_{lm}=E_{lm}E_{lm}^*\) and can be expressed as

\[\tag{2.13a}I_{lm}=I_0J_l^2\left(\frac{ur}{a}\right)\cos^2(l\phi)\qquad\text{ for }r\le{a}\]

\[\tag{2.13b}I_{lm}=I_0\left(\frac{J_l(u)}{K_l(w)}\right)^2K_l^2\left(\frac{wr}{a}\right)\cos^2(l\phi)\qquad\text{ for }r\ge{a}\]

The physical interpretation of \(l\) and \(m\) can now be readily understood. The integer value of \(m\ge1\) gives the number of intensity maxima that occur along a radius. A higher value of \(m\) means a higher value of \(u\) for a given \(V\), resulting in more radial oscillations in the intensity pattern.

The value of \(l\) is one-half the number of azimuthal maxima in the intensity pattern. Thus, \(\text{LP}_{01}\), the fundamental mode, has no azimuthal variation, is maximum on the fiber axis at \(r=0\), and decays monotonically as \(r\rightarrow\infty\).

### 2.5. Finite Element Analysis for Waveguide Calculations

Index profiles for real fibers may be much more complex than the idealized step-index waveguide of Fig. 2.1. Even matched clad fibers that are step index in principle will depart from an idealized step in various ways, depending on the details of the particular manufacturing process.

More complex fiber designs may comprise grading of the refractive index, as well as multiple index layers, including those with index below that of the (nominally) pure silica cladding. The field solutions and propagation constants for LP modes of complex fiber designs are usually calculated numerically using finite element methods. All subsequent results presented are calculated numerically using FEM methods.

The important optical properties of a fiber that define its utility in a particular application include attenuation, mode field diameter, effective area, cutoff, dispersion, and bending losses.

The radial \(\text{LP}_{01}\) mode for the step-index fiber is near-Gaussian, approaching cutoff. The fiber mode field diameter (MFD) and effective area \(A_\text{eff}\) are defined by

\[\tag{2.14}MFD^2=2\frac{\displaystyle\int_0^\infty|E(r)|^2r\text{d}r}{\displaystyle\int_0^\infty\left|\frac{\text{d}E}{\text{d}r}(r)\right|^2r\text{d}r}\]

and

\[\tag{2.15}A_\text{eff}=2\pi\frac{\left[\displaystyle\int_0^\infty|E(r)|^2r\text{d}r\right]^2}{\displaystyle\int_0^\infty|E(r)|^4r\text{d}r}\]

respectively.

The MFD and \(A_\text{eff}\) are inherently wavelength dependent and increase toward longer wavelengths. It is intuitive that longer wavelengths of light will be less confined by the waveguide than shorter wavelengths.

From the point of view of physical optics, this is intimately related to the fact that an aperture (the waveguide) of diameter \(d=2a\) diffracts light more strongly as \(\lambda\rightarrow2a\).

The wavelength dependence can also be understood from considering the analogy between confinement of light in a region of elevated refractive index and the trapping of a particle in a potential well in mechanics.

In particular, the time-independent Schrödinger equation of quantum mechanics is also a scalar Helmholtz equation of the form of one of the field components of Eq. (2.9a). The term \(n_1^2k_0^2=(2\pi)^2n_1^2/\lambda^2\) corresponds to the potential well depth in the Schrödinger equation.

As wavelength \(\lambda\) becomes longer, \(n_1^2k_0^2\) decreases (analogous to a more shallow potential well depth), leading to weaker confinement of light and larger \(A_\text{eff}\) (analogous to a smaller binding energy and spreading of the wave function outside the well). The wave function, the square of which describes the spatial probability distribution for the quantum particle, is analogous to electric field, and the binding energy in a potential well corresponds to values of \(\beta^2\) for a guided mode.

Figure 2.3 shows the calculated wavelength dependence of MFD and \(n_\text{eff}\) for a step-index fiber of \(\Delta=0.366\%\) and core radius \(a=4.8\) microns, representative of a typical matched clad fiber design with properties compliant to ITU G.652.

The theoretical cutoff values for this fiber design are 1350 nm for the LP_{11} mode and 840 nm for the LP_{02}. The practical cutoff for the LP_{11} mode will be at a wavelength shorter than 1350 nm.

The MFDs at 1310 and 1550 nm for a commercial matched clad fiber are typically specified as \(9.2\pm0.4\) μm and \(10.4\pm0.4\) μm, respectively. It can be seen in Fig. 2.3 that as the MFD increases, then \(n_\text{eff}\) (shown here as the difference in \(n_\text{eff}\) and the cladding index, \(n_\text{eff}-n_2\)) decreases as the mode spreads outside the germanium-doped core of diameter 9.6 microns.

The effective index of the fundamental mode is ultimately an average of the waveguide refractive indices weighted by the distribution of optical power. The variation in MFD or \(A_\text{eff}\) with wavelength is important in understanding nonlinear effects that impact system performance.

The decrease of the effective index with wavelength is closely correlated with tendency for increased macrobending and microbending loss.

The next tutorial introduces ** what is a laser**.