#### 1. Introduction

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

#### 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

(2)

From (1) it follows equation

The expression for the generation function

The initial conditions for

(3)

(4)
where

the solution of the initial system of equations (1) can be written as

(5)

#### 3. Particular examples of the field distribution over waveguides

Let us consider the following initial condition that corresponds to strong focusing radiation at

To compute the integrals in these expressions we may use the formula by Anger, which in this case is looking like

(6)

(7)

Substitution of the equation (6) into (5) results in the following expression (here we will use the term

Hence, at *n*=0 we have

(8)

The second equation of (1) may be used to obtain the amplitudes

As

the equation for

It follows that

(9)

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

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

#### 4. Conclusion

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