and

#### 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

iζAn=Bn+Bn1,iζBn=An+An+1,

(1) where An and Bn are normalized amplitudes of the electric field in waveguide from n -th unit cell, ζ is the normalized coordinate [1]. 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

PA(ζ,y)=nAn(ζ)eiyn,PB(ζ,y)=nBn(ζ)eiyn

Using the equations (1) we can find the equations for the generation functions PA and PB :

iζPA=(1+eiy)PB,iζPB=(1+eiy)PA

(2)

From (1) it follows equation ζ2PA+Ω2PA=0 , where Ω2=4cos2(y/2) . The solution of this equation takes the form

PA(ζ,y)=C1eiΩ(y)ζ+C2eiΩ(y)ζ

The expression for the generation function PB follows from the first equation of (2). If the initial conditions for the amplitudes An and Bn are known, we can define the initial conditions for the generation function

PA(0,y)=PA0=nAn(0)eiyn,PB(0,y)=PB0=nBn(0)eiyn.

The initial conditions for PA and PB allows us to determine integration constants C1 and C2 . Thus the generation function can be written as

PA(ζ,y)=PA0(y)cosΩ(y)ζieiyPB0(y)sinΩ(y)ζ,

(3)

PB(ζ,y)=PB0(y)cosΩ(y)ζieiyPA0(y)sinΩ(y)ζ,

(4) where Ω=2cos(y/2) . Using the orthogonality condition

ππeikydy=2πδ(k),

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

2πAn(ζ)=ππeinyPA(ζ,y)dy,2πBn(ζ)=ππeinyPB(ζ,y)dy.

(5)

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

Let us consider the following initial condition that corresponds to strong focusing radiation at ζ=0 : An(0)=A0δn0 and Bn(0)=0 . In this case PA0=A0 and PB0=0 . By the use the expressions (3), (4) and (5) we can write

2πAn(ζ)=A0ππeinycosΩ(y)ζdy,2πBn(ζ)=iA0ππeinyiy/2sinΩ(y)ζdy.

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

cos(ηcosy/2)=J0(η)+2k=1(1)kJ2k(η)cosky,

(6)

sin(ηcosy/2)=2k=1(1)kJ2k1(η)cos[(2k1)y/2].

(7)

Substitution of the equation (6) into (5) results in the following expression (here we will use the term η=2ζ )

2πAn(ζ)=A0J0(η)ππeinydy+A0k=1(1)kJ2k(η)ππeinyeiky+eikydy.

Hence, at n=0 we have A0(ζ)=A0J0(2ζ) . At n1 it follows that

An(ζ)=(1)nA0J2n(2ζ).

(8)

The second equation of (1) may be used to obtain the amplitudes Bn(ζ) . By using the expression (8) for An(ζ) , one can write

An+1+An=A0(1)nJ2n(η)J2n+2(η).

As

dJn(z)dz=Jn1(z)Jn+1(z),

the equation for Bn(ζ) can be rewritten as

iζBn=2iddηBn=A0(1)nddηJ2n+1.

It follows that

Bn(ζ)=iA0(1)nJ2n+1(2ζ).

(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 ζ=0 .

Let us consider the case where the both waveguides in unit cell n=0 are illuminated. Initial conditions are following An(0)=A0δn0 and Bn(0)=B0δn0 . In this case we have PA0=A0 and PB0=B0 . The generation function can be written as

PA(ζ,y)=A0cosΩ(y)ζieiyB0sinΩ(y)ζ,
PB(ζ,y)=B0cosΩ(y)ζieiyA0sinΩ(y)ζ.

Using the (5) we can write

2πAn(ζ)=A0ππeinycosΩ(y)ζdyiB0ππeiny+iy/2sinΩ(y)ζdy,
2πBn(ζ)=B0ππeinycosΩ(y)ζdyiA0ππeinyiy/2sinΩ(y)ζdy

Two incoming here the integrals have been found previously

ππeinycosΩ(y)ζdy=2πJ0(η)δn0+(1)nJ2n(η),
ππeinyiy/2sinΩ(y)ζdy=2π(1)nJ2n+1(η).

The third integral can be defined by the similar way. It results in following expression

ππeiny+iy/2sinΩ(y)ζdy=2π(1)n+1J2n1(η),n1.

So can immediately write down expressions for the distributions of the electric field amplitudes over waveguides for the selected initial conditions

An(ζ)=A0J0(2ζ)δn0+(1)nA0J2n(2ζ)+i(1)nB0J2n1(2ζ),
Bn(ζ)=B0J0(2ζ)δn0+(1)nB0J2n(2ζ)+i(1)nA0J2n+1(2ζ),

where n=±1,±2,... . The third terms in these expressions represent the interference phenomena in waveguide array.

#### 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 An(0)=(-1)nA0 and Bn(0)=(-1)nB0 , 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.

#### Acknowledgement

We are grateful to Prof. I.R. Gabitov, Dr. E.V. Kazantseva, Dr. V. A. Patrikeev, Dr. N.V. Bikov for enlightening discussions. This research was supported by the Russian Basic Research Foundation (Grant No 15-02-02764).

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