Beam Propagation Method - Principles

Principles

BPM is generally formulated as a solution to Helmholtz equation in a time-harmonic case,

(\nabla^2 + k_0^2n^2)\psi = 0

with the field written as,

.

Now the spatial dependence of this field is written according to any one TE or TM polarizations

\psi(x,y) = A(x,y)exp(+jk_o\nu y) ,

with the envelope

A(x,y) following a slowly varying approximation,
\frac{\partial^2( A(x,y) )}{\partial y^2} = 0

Now the solution when replaced into the Helmholtz equation follows,

\leftA(x,y) = \pm 2 jk_0 \nu \frac{\partial A_k(x,z)}{\partial z}

With the aim to calculate the field at all points of space for all times, we only need to compute the function for all space, and then we are able to reconstruct . Since the solution is for the time-harmonic Helmholtz equation, we only need to calculate it over one time period. We can visualize the fields along the propagation direction, or the cross section waveguide modes.

Read more about this topic:  Beam Propagation Method

Famous quotes containing the word principles:

    “The mode of founding a college is, commonly, to get up a subscription of dollars and cents, and then, following blindly the principles of a division of labor to its extreme,—a principle which should never be followed but with circumspection,—to call in a contractor who makes this a subject of speculation,... and for these oversights successive generations have to pay.”
    —Henry David Thoreau (1817–1862)

    “... the history of the race, from infancy through its stages of barbarism, heathenism, civilization, and Christianity, is a process of suffering, as the lower principles of humanity are gradually subjected to the higher.”
    —Catherine E. Beecher (1800–1878)

    “Struggle is the father of all things.... It is not by the principles of humanity that man lives or is able to preserve himself above the animal world, but solely by means of the most brutal struggle.”
    —Adolf Hitler (1889–1945)