Theory

Fiber optics rely on the principles of both ray and wave optics.  The first concept that is necessary to understand is the principle of total internal reflection as it relates to the numerical aperture, NA, of the fiber.  The numerical aperture, the ratio W/L as shown below, describes the maximum angle of incident light that will be totally internally reflected.

Incident light that converges with a NA less than .16 will be internally reflected as shown. 

Incident light that converges with a NA greater than .16 will not be internally reflected as shown. 

The numerical aperture is dependent on the differences in the indices of refraction between the core and the cladding.

The mathematics underlying the propagation of light in a fiber can be simplified by considering light traveling through a dielectric slab, since Cartesian coordinates are much easier to work with than cylindrical.  The light must follow Maxwell's equations of the form:

curl E = - i*w*m*H

curl H = i*w*m*E

and can be described by Helmholtz's equations:

ѲE+k²E=0

ѲH+k²H=0

which indicate that E and H can be represented by sine and cosine functions.

The mathematics become slightly more complicated with a fiber optic cable, but the solutions in cylindrical coordinates maintain this relationship.

When solved, solutions in cylindrical coordinates are separable, differential equations of the form:

E_z(r,j) = R(r)*F(f)

Setting both sides equal to a constant:

r²*R''/R + r*R'/R + k_c²*r² = v²

- F''/F = v²

Since they are separable, these equations can be solved independently, yielding:

F(f)= A*sin (v*f) + B*cos(v*f)

R(r) = C*J*(k*r) + D*Y*(k*r)   for k² > 0, where k² = w²*m*e

R(r) = C'*I*(|k|*r)+D'*K*(|k|*r) for k² < 0

The solutions to R(r) are known as Bessel functions.  Due to boundary conditions, only the J and K terms will be kept.  

Click on Image to view Bessel functions.

http://www.seas.ucla.edu/ch109/

Note that the Y functions tend towards infinity at x = 0 (we are not coupling an infinite amount of light into the fiber), and the I functions tend toward infinity as x tends towards infinity (we do not observe light intensity increasing as it propagates through the fiber).

Therefore, the solutions are:

u(r) µ J_l(k_t*r) for r < a and

u(r) µ K_l(g*r) for r > a

where a is the radius of the core, and l is the order of the Bessel function.  The terms k and g determine the rate of change of u(r) in the core and the cladding, respectively. (Saleh and Teich, 1991)

These terms can be normalized as follows:

X = k_t*a,     Y = g*a

where X²+Y² = V²

Back to fiber optics...

At the interface of the core and the cladding, J(r) must equal K(r), which mandates several boundary conditions ensuring that the E and H fields match up at the interface for each of the coordinates (z,q,f).

The resulting system of equations can only be solved graphically.  The graphical solutions represent the mode cutoffs for the different modes that can propagate in the fiber for any given V, where V is a convenient parameter determined by the properties of the fiber and wavelength of incident light. 

V = 2*p/l*a*NA

The graph below is an example of a set of solutions for one of many applicable functions. 

(Yariv, 1985, p. 63)

The intersections represent the V numbers at which these two modes turn on in the fiber.

The following table presents the first ten cutoff frequencies in a step-index fiber, as well as their fundamental modes.

(Buck, 1989)

The following chart diagrams the electric and magnetic fields for eight fundamental modes.  Click on image to view expanded form. 

(Buck, 1989, pp. 31-32)

The graphical solutions represent the V numbers at which certain modes turn on.  The plot below displays the regimes in which the first for LP modes are sustainable as a function of V.  The Y axis plots b, which is related to the properties of the fiber and to the free wavenumber, k_f, of the incoming light.

(Newport, 1999, p.13)

Each sustainable mode represents a unique E field distribution in the fiber. 

Setup and Procedure

Many thanks to the following for lending their expertise to this page:

J.A. Buck, (1989)

Davis, (1996)

Saleh & Teich, (1991)

Yariv, (1985)

Newport (1999)