The one dimensional array (or chain) of the waveguides in the linear approximation represents the system for discrete diffraction realization. In this paper we will consider the binary linear waveguide array. The unit cell contains two kind of the waveguide, i.e., A-type and B-type. In slowly varying envelopes of the electric field of the continue wave radiation in waveguide is governed by the system of equations for coupled waves [1,2]. If we would like to consider only discrete diffraction [2,3] then the nonlinear properties of the waveguides may be neglected. It cab be considered as the first approximation.
The system of the equations describing the electromagnetic wave propagation in the coupled wave approximation takes the following form
(1) where and are normalized amplitudes of the electric field in waveguide from -th unit cell, is the normalized coordinate . We assume that the phase mismatch is zero.
2. Analytical solution of the base equations
To obtain the solution of the system of equations (1) we can use the generation function method. Let us introduce the following functions
Using the equations (1) we can find the equations for the generation functions and :
From (1) it follows equation , where . The solution of this equation takes the form
The expression for the generation function follows from the first equation of (2). If the initial conditions for the amplitudes and are known, we can define the initial conditions for the generation function
The initial conditions for and allows us to determine integration constants and . Thus the generation function can be written as
(4) where . Using the orthogonality condition
the solution of the initial system of equations (1) can be written as
3. Particular examples of the field distribution over waveguides
Let us consider the following initial condition that corresponds to strong focusing radiation at : and . In this case and . By the use the expressions (3), (4) and (5) we can write
To compute the integrals in these expressions we may use the formula by Anger, which in this case is looking like
Substitution of the equation (6) into (5) results in the following expression (here we will use the term )
Hence, at n=0 we have . At it follows that
The second equation of (1) may be used to obtain the amplitudes . By using the expression (8) for , one can write
the equation for can be rewritten as
It follows that
Thus the distribution of the field amplitudes over waveguides in array under considered initial conditions is presented by the expressions (8) and (9). These equations describe the discrete diffraction in binary waveguide array under the condition of the strong focusing at .
Let us consider the case where the both waveguides in unit cell are illuminated. Initial conditions are following and . In this case we have and . The generation function can be written as
Using the (5) we can write
Two incoming here the integrals have been found previously
The third integral can be defined by the similar way. It results in following expression
So can immediately write down expressions for the distributions of the electric field amplitudes over waveguides for the selected initial conditions
where . The third terms in these expressions represent the interference phenomena in waveguide array.
This distribution of field strengths describes discrete diffraction in a binary array of waveguides. For more complex cases of the initial conditions the expressions for the electric fields in waveguides contain terms that account for interference fields in the neighboring waveguides. It should be noted that if the initial conditions are selected as and , then the diffraction is absent. The equations (1) show that under these conditions the fields in waveguides are invariants. However, the flat band [5,6,7] in the spectrum of the linear waves is absent, as the number of nodes in the unit cell is less than three. In the spectrum there are only two branches that meet dispersive waves along the chain.