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One of the methods for calculating electromagnetic wave dispersion in multi-layer structures is the transfer matrix method. In this paper, we use the transfer matrix method for second harmonic generation in a nonlinear multilayer structure. The nonlinear photonic crystals investigated in this paper are as one-dimensional multi-layered structures including ferroelectric materials such as LiTaO
_{3}. Our goal is to investigate the effect of the disorder on the transmission spectrum of electromagnetic waves. Our results showed that positional disorder has different effects on the transmitting band and the gap band. The disorder in the transmitting band reduces the transmission coefficient of the waves and increases the transmission coefficient of the waves in the gap band. Such work has not yet been done on nonlinear photonic crystals producing the second harmonic.

In recent decades, the study of the transmission of waves in random systems has been highly studied because it plays an important role in understanding the optical, mechanical, magnetic, and electrical properties of materials. The most important theory in this case is Anderson’s theory. This theory is about electrons crossing in random systems [

A weak disorder in 3D devices does not turn all of the extended modes into localized modes. That is; there is a series of transmission modes in the system. For strong disorder, due to the incoherent interference of waves, transmitted modes are exponentially localized. For one-dimensional systems, Anderson’s theory shows that all the passing modes are exponentially localized for any degree of disorder [

There are several models for one dimensional multi-layer system in which exponential localization is violated. For example a one-dimensional system with the periodic period consisting of two parts is known as a random-dimer model [

A periodic structure with a transmission spectrum that does not have any propagating modes in certain frequency regions is called the photonic crystal [

In a binary multi-layer structure made up of dielectric slabs, in which the optical lengths of the layers are equal, putting the disorder in the order of the layers leads to the appearance of necklace modes in the photonic band gap [

Photonic crystals contain ferroelectric materials, due to the unique properties that compensate for the phase mismatch in the second harmonic generation, which have been much considered. Second harmonic generation is investigated in multilayer nonlinear structures including ferroelectrics such as LiNbO_{3}, KTiOPO_{4} and strontium barium niobate [

But until now, the effect of the disorder on the second harmonic generation (SHG) in the photonic crystals including ferroelectric materials has not been studied. We investigate the effects of positional disorder on the transmission spectrum of one-dimensional structures composed of lithium tantalite (LiTaO_{3}). We study a binary disordered system composed of N periods. Using the transfer matrix method, we draw the transmission spectrum of the electromagnetic wave as a function of the incoming wave frequency. We study the effect of the positional disorder on the transmission of an electromagnetic wave that collides vertically on the surface of the structure, so that during the initial wave propagation in the nonlinear structure, a second harmonic wave is also produced.

This paper is organized as follows in Section 2 we focus on the details of the model and we derive the related transfer matrix. We also explain how we calculate the Transmission coefficient. In Section 3 we present and discuss our numerical results. Finally, Section 4 ends the paper with a summary of our results.

In this section, we study a multi-layered nonlinear structure, which changes the nonlinear parameter sign periodically in layers. In this structure, the scales of linear and nonlinear parameters are the same. The nonlinear structure is divided into N periods of thickness d, and each period is divided into two homogeneous sections,

It is assumed that an electromagnetic wave with frequency ω incident from the left-hand side of the system and propagates in the direction of the z axis, so that the polarization of the electric field is in the x-axis direction. Our numerical calculations are based on the nonlinear transfer matrix method (TMM) [

In the first stage of the second harmonic generation, the fundamental and second harmonic fields follow the following equations:

∇ × ( ∇ × E 2 ω ) − ε 2 ω ( 2 ω ) 2 c 2 E 2 ω = μ 0 ( 2 ω ) 2 P N L 2 ω ∇ × ( ∇ × E ω ) − ε ω ω 2 c 2 E ω = μ 0 ω 2 P N L ω (1)

In which, ε ω , ε 2 ω , P N L ω , and P N L 2 ω are permittivity and nonlinear polarization at FF and SH frequencies. Because the nonlinear process is weak. As a result, the nonlinear process has no significant effect on the intensity of the fundamental pump field (Given that the material we have used in this paper is ferroelectric (LiTaO_{3}), whose second-order non-linear coefficient χ ( 2 ) is 13.8 pm/V. Therefore, in our calculations, we used the source approximation without decreasing (strong source approximation) to solve non-linear equations and

obtaining the transmission coefficient. That is, the second-order nonlinear coefficient is small, so the nonlinear process does not have a significant effect on the initial wave intensity). So here we are using the non-depleted pump wave approximation, E 2 ω ≪ E ω . Therefore, the above equations for the m-th layer are as follows

[ d 2 d z 2 + ( k m ( f ) ) 2 ] E m f ( z ) = 0 [ d 2 d z 2 + ( k m ( s ) ) 2 ] E m s ( z ) = − μ 0 ( 2 ω ) 2 P N L 2 ω (2)

where k m ( f ) = n m ( f ) k 0 ( f ) , k m ( s ) = n m ( s ) k 0 ( s ) , k 0 ( f ) = ω / c , and k 0 ( s ) = 2 ω / c . and also, n m ( f ) and n m ( s ) show the refractive index of the pump waves and the second harmonic, respectively in the mth slab; c, the speed of light is in vacuum

The first equation of coupling Equations (2) gives the fundamental electric field in the mth layer.

E m ( f ) = E m f + e i ( k m ( f ) ( z − z m − 1 ) − ω t ) + E m f − e − i ( k m ( f ) ( z − z m − 1 ) − ω t ) (3)

where z 0 is zero, z m = z m − 1 + d m and d m is the thickness of the mth layer. E m ( f ) + is the magnitude of forward plane wave and E m ( f ) − is the magnitude of backward plane wave at the left interface of the mth layer.

According to the continuity conditions on the boundary of the layers, we have the following matrix relation between the electric and magnetic fields passing through odd layer to even layer:

( E 2 m − 1 f + E 2 m − 1 f − ) = 1 2 n I ( 1 ) ( n I ( 1 ) + n I I ( 1 ) n I ( 1 ) − n I I ( 1 ) n I ( 1 ) − n I I ( 1 ) n I ( 1 ) + n I I ( 1 ) ) ( E 2 m f + E 2 m f − ) = T 12 ( E 2 i f + E 2 i f − ) (4)

Using the matrix of dynamics and the matrix of propagation that are as follows.

D m = ( 1 1 n m ( f ) − n m (f) )

P m = ( exp ( i k m ( f ) d m ) 0 0 exp ( − i k m ( f ) d m ) )

The general transfer matrix for the multi-layer structure is as follows

T = D 0 − 1 ( D I I P I I D I I − 1 D I P I D I − 1 ) N D 0 (5)

This transfer matrix connects the fields at the beginning and the end of the structure, as follows

( E t f + E t f − ) = T ( E 0 f + E 0 f − ) (6)

For harmonic fields that is produced in the structure, we use Equation (2) with P N L 2 ω ( z , t ) = ε 0 χ m ( 2 ) [ E m ( f ) ( z ) ] 2 exp ( − i 2 ω t ) that yields to:

( E m ( s ) + E m ( s ) − ) = G 0 − 1 S G 0 ( E m − 1 ( s ) + E m − 1 ( s ) − ) + G 0 − 1 ( N I I B I F I − S B I ) A I ( ( E 2 m − 1 f + ) 2 ( E 2 m − 1 f − ) 2 ) + G 0 − 1 ( N I I − S ) ( C I E 2 m − 1 f + E 2 m − 1 f − 0 ) + G 0 − 1 ( B I I F I I − N I I B I I ) A I I ( ( Ω 2 m + ) 2 ( Ω 2 m − ) 2 ) + G 0 − 1 ( I − N I I ) ( C I I E 2 m f + E 2 m f − 0 ) (7)

In which:

A l = − 4 μ ε 0 χ l ( 2 ) ω 2 k l ( s ) 2 − 4 k l ( f ) 2 , C l = − 4 μ ε 0 χ l ( 2 ) ω 2 k l ( s ) 2

G l = ( 1 1 n l ( s ) − n l ( s ) ) , B l = ( 1 1 2 k 0 ( f ) n l ( f ) k 0 ( s ) − 2 k 0 ( f ) n l ( f ) k 0 (s) )

Q i = ( exp ( i k l ( s ) d i ) 0 0 exp ( − i k l ( s ) d l ) )

F l = ( exp ( i 2 k l ( f ) d l ) 0 0 exp ( − i 2 k l ( f ) d l ) ) l = I , I I

S = G I I Q I I G I I − 1 G I Q I G I − 1 , N I I = G I I Q I I G I I − 1

In this section, we determined the method of obtaining the amplitude of the second harmonic field in the nonlinear multi-layer structure. We can not actually measure this quantity, but the intensity of the second harmonic is measurable (In this paper, which is a theoretical work, the intensity of the waves is obtained using the electromagnetic magnitude (intensity is proportional to the square of the field magnitude). The purpose of writing the phrase “measurable intensity of electromagnetic waves” is that in the laboratory, researchers measure the intensity of electromagnetic waves. In fact, this sentence is presented for comparison with laboratory work). Using the intensity of waves, we have the following definitions for the nonlinear reflectance and nonlinear transmittance:

R N L = I h a r m r ( I p u m p ) 2 T N L = I h a r m t ( I p u m p ) 2 (8)

where I h a r m t , r and I p u m p are the intensities of the reflected or transmitted harmonic and pump fields, respectively.

In our structure, it is assumed that the nonlinear photonic crystal are composed of periodically poled lithium tantalite (LiTaO_{3}) crystal, whose nonlinear susceptibility is χ ( 2 ) = 13.8 pm / V and The refractive index for lithium tantalite (LiTaO_{3}) is given by the following dispersion formula [

n 2 ( λ , T ) = A + B + b ( T ) λ 2 − [ C + c ( T ) ] 2 + E λ 2 − F 2 + D λ 2 (9)

where A = 4.5284 , B = 7.2449 × 10 − 3 , C = 0.2453 , D = − 2.3670 × 10 − 2 , E = 7.7690 × 10 − 2 , F = 0.1838 , b ( T ) = 2.6794 × 10 − 8 ( T + 273.15 ) 2 , c ( T ) = 1.6234 × 10 − 8 ( T + 273.15 ) 2 .

Periodic polar ferroelectric (LiTaO_{3}) crystals are described by modulations of the nonlinear susceptibilities. An optoelectronic effect is used to produce the photonic band gap structure. This effect is the origin of the modulation of the refractive index of the layers in the photonic crystal. If the electric field E is applied in the direction of the optical axis of the material, correction of refractive index and new refractive index (in odd layers) are as follows:

n 1 = n + Δ n 1 = n − 1 2 n 3 r 33 E (10)

where r 33 is the optoelectronic coefficient of the material and E is the electric field amplitude. Similar to the nonlinear optical coefficient, the optoelectronic coefficient of the material in different layers has different signs. So in reverse (even) layers the optoelectronic coefficient changes sign and the refractive index in these layers is written as follows:

n 2 = n + Δ n 2 = n + 1 2 n 3 r 33 E (11)

We investigate a random binary structure constructed of N lithium tantalite (LiTaO_{3}) layers. The positional disorder in the structure means the probability that the ith layer is Layer I or Layer II is equal. In order to have an overall picture of the role played by disorder, and to illustrate the effect of disorder in the nonlinear multi-layered structure, we compare the transmission spectrum of a periodic structure with alternating layers I and II with a disordered structure, as shown in

The transmission spectrum for periodic structures is a sequence of transmitting and stop bands. In the transmitting band, there is a frequency in which the optical length of each layer is an integer multiple of half the wavelength in the vacuum. In other words, the phase change in each layer is equal to mπ, that m is the member of the integers. The nonlinear multi-layer structure for this incoming wave frequency is completely transparent. In the center of the stop bands, there are frequencies in which the optical length of each layer is odd multiples of one quarter wavelength. As a result, the phase change in these layers is equal to ( m + 1 / 2 ) π . For the random system, to calculate the Transmission coefficient quantity, averaging operations are performed on 3000 systems with different ordering (

In order to see the different effects of positional disorder in the transmitting band and the forbidden band, we plot the average transmission coefficient around the half- and quarter-wavelength modes as a function of the disorder intensity. In

coefficient in random structures, the averaging is performed on 3000 structures in different order.

For the average transmission coefficient around the half-wavelength resonance, when the disorder increases, the transmission coefficient decreases, so Anderson’s localization is established (

While

We studied the transmittance spectra in a nonlinear structure composed of tantalum lithium (LiTaO_{3}). This system is a binary structure with N layer. The effect of disorder on the transmitting and forbidden bands is quite distinct. For the transmission band, the disorder in the frequency at which the coefficient of transmission is equal to 1 does not affect. At this frequency, the transfer matrix of each layer is equal to the unit matrix (I). The lack of sensitivity of this fully resonance mode is similar to the violation of Anderson’s theory in the random-dimer model. However, the width of the transmittance band in a disordered system is narrower than the width of the transmittance band of the periodic system. For the gap band, disorder lead to the appearance of several modes in the center of the band. These modes are called necklace modes, which are based on the hybridization of degenerate modes localized. The transmittance in the stop (gap) band increases with the disorder intensity. The transmittance in the transmitting band decreases with the disorder intensity.

The authors declare no conflicts of interest regarding the publication of this paper.

Omid, B., Abdolrahim, B. and Ali, B. (2019) Disorder Effect on the Transmission of Second Harmonic Waves in One-Dimensional Periodically Poled LiTaO_{3}. Journal of Modern Physics, 10, 432-442. https://doi.org/10.4236/jmp.2019.104028