### Mode theory for Cylindrical Waveguide

. To analyze the optical fiber propagation mechanism within a fiber, Maxwell equations are to solve subject to the cylindrical boundary conditions at core-cladding interface. The core-cladding boundary conditions lead to coupling of electric and magnetic field components resulting in hybrid modes. Hence the analysis of optical waveguide is more complex than metallic hollow waveguide analysis.
. Depending on the larger E-field or H-field, the hybrid modes are HE or EH modes. The two lowest order modes are HE and TE.
1. Overview of Modes
. The order states the number of field zeros across the guide. The electric fields are not completely confined within the core i.e. they do not go to zero at core-cladding interface and extends into the cladding. The low order mode confines the electric field near the axis of the fiber core and there is led penetration into the cladding. While the high order mode distribute the field towards the edge of the core fiber and penetrations into the cladding. Therefore cladding modes also appear resulting in power loss.
. In leaky modes the fields are confined partially in the fiber core attenuated as they propagate along the fiber length due to radiation and tunnel effect.
. Therefore in order to mode remain guided, the propagation factor must satisfy the condition.

Where,
n1 = Refractive index fiber core
n2 = Refractive index of cladding
K=Propagation constant = 2π/λ
. The cladding is used to prevent scattering loss that results from core material discontinuities. Cladding also improves the mechanical strength of fiber core and reduces surface contamination. Plastic cladding is commonly used. Materials used for fabrication of optical fibers are silicon dioxide (SiO2), boric oxide-silica.
2. Summary of Key Modal Concepts
. Normalized frequency variable, V is defined as

λ= Free space wavelength.

The total number of modes in a multimode fiber is given by

3. Wave Propagation
3.1Maxwell's Equations
Maxwell's equation for non-conducting medium :

Where,
E and H are electric and magnetic field vectors.
D and B are corresponding flux densities.
. The relation between flux densities and field vectors :

Where,
ε0    is vacuum permittivity
µ0   is vacuum permeability.
P     is induced electric polarization.
M    is induced magnetic polarization (M = 0, for non-magnetic silica glass).
P and E are related by:

Where,
x is linear susceptibility.
Wave equation :

Fourier transformer of E (r, t)

Where,

n is refractive index.
α is absorption coefficient.

. Both n and α are frequency dependent. The frequency dependent of n is called as chromatic dispersion or material dispersion..For step index fiber,
. For step index fiber,

4. Fiber Modes
Optical mode : an optical mode is a specific solution of the wave equation that satisfy boundary conditions. There are three types of fiber modes.
a) Guided modes
b) Leaky modes
. For fiber optic communication system guided mode is used for signal transmission. Considering a step index fiber with core radius 'a'.
The cylindrical co-ordinates ρ, Φ and can be used to represent boundary conditions.

. The refractive index 'n' has values

. The general solution for boundary condition of optical field under guided mode is infinite at ρ = 0 and decay to zero at ρ = ∞. using Maxwell's equation in the core region.

The cut-off condition is defined as –

It is also called as normalized frequency.
. The refractive index of graded index fiber decreases continuously towards its radius from the fiber axis and that for cladding is constant.
.The refractive index variation in the core is usually designed by using power law relationship.

Where,
r     = Radial distance from fiber axis
n1   = Refractive index core
n2   = Refractive index of cladding and
α     = The shape of the index profile
. For graded index fiber, the index difference Δ is given by,

. In graded index fiber the incident light will propagate when local numerical aperture at distance r from axis, NA(r) is axial numerical aperture NA(0).The local numerical aperture is given as,

. The axial numerical aperture NA(0) is given as,

Hence NA for graded index decreases to zero as it moves from fiber axis to core-cladding boundary.
The variation of NA for different values of α is shown in Fig. 1.

. The number of modes for graded index fiber is given as,