Wave electric fields in an open waveguide with a plate consisting of two dielectric layers with a corrugated interface

G- Jitter Effects on Chaotic

2. Wave electric fields in an open waveguide with a plate consisting of two dielectric layers with a corrugated interface

The generator circuit considered here is an open waveguide, which includes a thin dielectric plate consisting of two dielectric layers with a dielectric interface, periodically corrugated in the transverse direction of the waveguide. An electron conducting channel is included in one of the dielectric layers of the plate close to the corrugated dielectric interface (Figure 2), and the electrons drift in the conducting channel in the transverse direction of the waveguide (in the direction perpendicular to the direction of wave propagation).

We assume that the conducting channel is sufficiently thin (its thickness is much smaller than the corrugation amplitude). In this case, the problem of calcu- lating of electric fields in the waveguide, and the problem of calculating of interac- tions of electrons with electromagnetic waves can be solved separately. In this section, we solve the problem of calculating of electric fields without electrons. In the next section, we introduce electrons, which interact with wave electric fields, into the picture.

Figure 2.

Plate profile of an open waveguide. The plate consists of two dielectric layers with permittivitiesε₁(light gray) andε₂(dark gray) having a periodically corrugated dielectric interface. Electrons drift with a mean velocityv0

in a semiconductor layer 1. E0is an electric field of a principal wave of the waveguide (TE-wave).

Advances in Condensed-Matter and Materials Physics - Rudimentary Research to Topical …

Electromagnetic waves, including waves in the terahertz frequency range, can propagate in the proposed open waveguide. The waveguide contains a dielectric plate of thickness 2a consisting of two dielectric layers with different permittivities

ε₁andε₂(ε₁< ε₂). For sufficiently thin plate, a< <c=ω, and for not too large values

ofε₁andε₂( ε_i<10) the electromagnetic wave energy is mostly concentrated in the region outside the plate. In the waveguide, an electromagnetic wave propagates with the velocity close to the speed of light.

Let us consider the case of a TE wave, in which there is an antinode of the electric field in the center of the cross section of the plate (where the dielectric plate is) [7] (Figure 1).

For a thin plate (a< <2πc=ω) with a flat dielectric interface, the electric field of the TE wave is directed along the X-axis, and is given by

E_TEð Þ ¼r, t e_xE_xð Þ �y exp ikzð �iωtÞ, (1) The field Excan be represented as:

Ex≈Eincosð Þ,χy  if j jy ≤a, and

Ex≈Eoutexpð�s yj jÞ, if j jy ≥a (2) Here, exis a unit vector along the x-axis, Eout≈Ein¼E0,χand s are transverse components of the wave vector in a dielectric and vacuum, respectively. The values χand s are related by saffiffiffiffiffiffiffiffiffiffiffiffiε₁�ε₂ ¼¹ε�χa�tg ð Þχa andð Þχa ²þð Þsa ²¼^ω2_c2^a2ðε�1Þ, whereε¼ p is the average value of permittivity of the dielectric plate.

However, the electric field Exof the TE-wave is uniform for a homogeneous dielectric plate, as well as multi-layer plates with a flat dielectric interface. In this case, the effective interaction between electron and the wave should be absent.

We propose a scheme of a waveguide with a thin multilayered dielectric plate, and the interface between two dielectric layers is a corrugated surface (Figure 1).

The waveguide with this plate can also serve as a waveguide for the TE wave, and the wave propagation speed along the Z-axis is still being close to the speed of light.

But in this case, in the vicinity of corrugated dielectric interface, an additional surface wave is induced in a field of the TE wave. The induced inhomogeneous electric field of the surface wave is comparable with the electric field of the volume electromagnetic wave only in a very narrow region near the dielectric interface.

And the inhomogeneous electric field decreases exponentially (in Y-direction) with distance from the interface. The induced electric field is inhomogeneous in the Z- direction of the waveguide, and electrons also drift in X-direction in quasi-two- dimensional conductance channel, located in the zone of the corrugation

(Figure 2). Such electrons can interact with an inhomogeneous field of the wave.

Let the coordinates of the dielectric interface in the waveguide cross section vary according to a periodic law, for example, to the law y xð Þ ¼R cos kð cxÞ. Here kc¼2π=L, and the corrugation amplitude R is significantly smaller than the plate thickness, R< <2a, and the corrugation period L is much less than the plate width.

We accept the condition

k_c> >k, (3)

and, as will be clear from the following, it is worth considering the case kcR�1, in which the interaction of an electromagnetic wave with electrons (and the ampli- fication of this wave) will be most effective. We assume that the conducting A Compact Source of Terahertz Radiation Based on an Open Corrugated Waveguide DOI: http://dx.doi.org/10.5772/intechopen.89692

A brief description of the terahertz generator, based on the interaction of elec- trons in a quantum well with an electromagnetic wave of a corrugated waveguide, was previously presented in our works [7, 8].

In the second section of this chapter, we present calculations of the induced inhomogeneous electric field of the wave near the corrugated dielectric interface. In the third section, we use a hydrodynamic approach to describe a system of charged carriers, which drift in a quantum well and interact with the inhomogeneous elec- tric field of the wave in the zone of the corrugation. We define the parameters of the system under which the amplification of the electromagnetic field is the most effective. In the fourth section, we give a brief conclusion.

2. Wave electric fields in an open waveguide with a plate consisting of two dielectric layers with a corrugated interface

The generator circuit considered here is an open waveguide, which includes a thin dielectric plate consisting of two dielectric layers with a dielectric interface, periodically corrugated in the transverse direction of the waveguide. An electron conducting channel is included in one of the dielectric layers of the plate close to the corrugated dielectric interface (Figure 2), and the electrons drift in the conducting channel in the transverse direction of the waveguide (in the direction perpendicular to the direction of wave propagation).

We assume that the conducting channel is sufficiently thin (its thickness is much smaller than the corrugation amplitude). In this case, the problem of calcu- lating of electric fields in the waveguide, and the problem of calculating of interac- tions of electrons with electromagnetic waves can be solved separately. In this section, we solve the problem of calculating of electric fields without electrons. In the next section, we introduce electrons, which interact with wave electric fields, into the picture.

Figure 2.

Plate profile of an open waveguide. The plate consists of two dielectric layers with permittivitiesε₁(light gray) andε₂(dark gray) having a periodically corrugated dielectric interface. Electrons drift with a mean velocityv0

in a semiconductor layer 1. E0is an electric field of a principal wave of the waveguide (TE-wave).

Advances in Condensed-Matter and Materials Physics - Rudimentary Research to Topical …

Electromagnetic waves, including waves in the terahertz frequency range, can propagate in the proposed open waveguide. The waveguide contains a dielectric plate of thickness 2a consisting of two dielectric layers with different permittivities

ε₁andε₂(ε₁< ε₂). For sufficiently thin plate, a< <c=ω, and for not too large values

ofε₁andε₂( ε_i<10) the electromagnetic wave energy is mostly concentrated in the region outside the plate. In the waveguide, an electromagnetic wave propagates with the velocity close to the speed of light.

Let us consider the case of a TE wave, in which there is an antinode of the electric field in the center of the cross section of the plate (where the dielectric plate is) [7] (Figure 1).

For a thin plate (a< <2πc=ω) with a flat dielectric interface, the electric field of the TE wave is directed along the X-axis, and is given by

E_TEð Þ ¼r, t e_xE_xð Þ �y exp ikzð �iωtÞ, (1) The field Excan be represented as:

Ex≈Eincosð Þ,χy  if j jy ≤a, and

Ex≈Eoutexpð�s yj jÞ, if j jy ≥a (2) Here, exis a unit vector along the x-axis, Eout≈Ein¼E0,χand s are transverse components of the wave vector in a dielectric and vacuum, respectively. The values χand s are related by saffiffiffiffiffiffiffiffiffiffiffiffiε₁�ε₂ ¼¹ε�χa�tg ð Þχa andð Þχa²þð Þsa ²¼^ω_c²2^a2ðε�1Þ, whereε¼ p is the average value of permittivity of the dielectric plate.

However, the electric field Exof the TE-wave is uniform for a homogeneous dielectric plate, as well as multi-layer plates with a flat dielectric interface. In this case, the effective interaction between electron and the wave should be absent.

We propose a scheme of a waveguide with a thin multilayered dielectric plate, and the interface between two dielectric layers is a corrugated surface (Figure 1).

The waveguide with this plate can also serve as a waveguide for the TE wave, and the wave propagation speed along the Z-axis is still being close to the speed of light.

But in this case, in the vicinity of corrugated dielectric interface, an additional surface wave is induced in a field of the TE wave. The induced inhomogeneous electric field of the surface wave is comparable with the electric field of the volume electromagnetic wave only in a very narrow region near the dielectric interface.

And the inhomogeneous electric field decreases exponentially (in Y-direction) with distance from the interface. The induced electric field is inhomogeneous in the Z- direction of the waveguide, and electrons also drift in X-direction in quasi-two- dimensional conductance channel, located in the zone of the corrugation

(Figure 2). Such electrons can interact with an inhomogeneous field of the wave.

Let the coordinates of the dielectric interface in the waveguide cross section vary according to a periodic law, for example, to the law y xð Þ ¼R cos kð cxÞ. Here kc¼2π=L, and the corrugation amplitude R is significantly smaller than the plate thickness, R< <2a, and the corrugation period L is much less than the plate width.

We accept the condition

k_c> >k, (3)

and, as will be clear from the following, it is worth considering the case kcR�1, in which the interaction of an electromagnetic wave with electrons (and the ampli- fication of this wave) will be most effective. We assume that the conducting A Compact Source of Terahertz Radiation Based on an Open Corrugated Waveguide DOI: http://dx.doi.org/10.5772/intechopen.89692

channel is thin enough (h< <R, where h is the channel width), so that the electrons in the channel do not deform the electromagnetic fields in the dielectric.

In such waveguide, the electric field of the wave can be represented as

E r, tð Þ ¼E_TE ðr, tÞ þF r, tð Þ, where E_TE ðr, tÞis given by (1) and (2) [7]. The electric field F r, tð Þof the wave, which is induced near the corrugation, is comparable with the TE wave field only in a very narrow region (with a width of about R) near the dielectric interface. And F r, tð Þis significantly less than ETE ðr, tÞoutside this region.

Let us proceed to calculation of the electric field F r, tð Þ.The field F r, tð Þis caused by influence of the electric field ETE ðr, tÞof the TE-wave on the dielectric interface. However, due to the smallness of the corrugation size is compared to the electromagnetic wavelengthλ¼2π=χ(λ > >R, λ > >L), the TE wave’s electric field can be considered as a potential at distances typical for corrugation. Let us denote this field by E0, where E0cosð Þχy ≈E₀. Since the corrugated dielectric interface is homogeneous along the Z-axis, the problem of calculating F r, tð Þis effectively two-dimensional one [7].

The problem can be described by the scalar potentialϕ^{ð Þi}, which satisfies to the Laplace equation. We denote the potential asϕ^{ð Þ1}ðx, yÞinside the dielectric layer with the permittivityε₁, and asϕ^{ð Þ2}ðx, yÞinside the layer with the permittivityε₂ [7]. The functionsϕ^{ð Þ1}ðx, yÞandϕ^{ð Þ2}ðx, yÞsatisfy to the Laplace equation

Δϕ^{ð Þ1}ðx, yÞ ¼0 andΔϕ^{ð Þ2}ðx, yÞ ¼0 within its domain. Then the boundary condi- tions take the form

ϕ^{ð Þ1}ðx, yÞ��c¼ϕ^{ð Þ2}ðx, yÞ��_c (4) ε₁∂ϕ^{ð Þ1}ðx, yÞ

∂n ^c¼ε₂∂ϕ^{ð Þ2}ðx, yÞ

∂n

��

��

�

��

��

�_c: (5)

The conditions (4)–(5) represent the continuity of the tangential electric field and normal component of an electric induction vector in a point with coordinates ðx, yÞon the corrugation dielectric interface [7].

Since the corrugated dielectric interface is described by a periodic function of x, the electrostatic potential functionsϕ^{ð Þi}ðx, yÞcan be written as the sum of spatial harmonics:

ϕ^{ð Þi}ðx, yÞ ¼X^∞

m¼0

B^{ð Þ}_mⁱ cos mkð _cxÞexpð�mk_cj jyÞ

þX^∞

m¼0

A^{ð Þ}_mⁱ sin mkð cxÞexpð�mkcj jyÞ þE0x

(6)

In our problem, y xð Þ ¼R cos kð cxÞis a continuously differentiable function. In this case, expressions (6) represent a complete set of basis functions [9] for poten- tialsϕ^{ð Þi} and for electric fields E^{ð Þi} ¼ �∇ϕ^{ð Þi}. The expansion coefficients A^{ð Þ}_mⁱ and B^{ð Þ}_mⁱ of the sum (6) are found from the boundary conditions (4)–(5) at the point x, yð Þ ¼

x, cos kcx

ð Þ[7]. For the boundary described by y xð Þ ¼R cos kð cxÞ, this system is consistent, if the coefficients B^{ð Þ}_mⁱ ¼0, and A^{ð Þ}_mⁱ 6¼0. Then for the calculation of coefficients A^{ð Þ}_mⁱ we have limited the harmonic expansion to the seventh term.

According to our calculations, whenε₂=ε₁≥3, taking into account the seven expan- sion terms is sufficient for calculating the coefficients, and the error will not exceed a few percent.

Advances in Condensed-Matter and Materials Physics - Rudimentary Research to Topical …

The resulting expression for the x-component E^{ð Þ}_x¹ of the electric field E^{ð Þ1} ¼

�∇ϕ^{ð Þ1} induced in the upper layer near the corrugation (at y>R cos kð _cxÞ) can be represented as

E^{ð Þ}_x¹ðx, yÞ ¼E0

X^∞

m¼1

C^{ð Þ}_m¹e^imkcxe^{�mkcj jy} (7)

The coefficients C^{ð Þ}_m¹, which characterize the contribution of mth harmonic to the electric field induced above the corrugation, depend on the shape and size of corrugation and, moreover, on the parametersε₁andε₂. We carried out numerical analysis on an example of a structure with a plate of quartz (ε1= 4.5) and sapphire (ε2= 9.3). In the case, when the dielectric boundary is described by

y xð Þ ¼R cos kð cxÞ, each of the coefficients C^{ð Þ}_m¹ can be represented as a function of kcR. Figure 3 shows the calculated dependences C^{ð Þ}₁¹ðkcRÞ, C^{ð Þ}₂¹ðkcRÞand C^{ð Þ}₃¹ðkcRÞ.

For kcR≥1, the value of C^{ð Þ}₁¹ is much greater than C^{ð Þ}₂¹ and C^{ð Þ}₃¹. The coefficients C^{ð Þ}_m¹ for m>3 are much less than C^{ð Þ}₁¹, C^{ð Þ}₂¹ and C^{ð Þ}₃¹.

Figure 4 shows the first harmonic of electric field E^{ð Þ}_x¹=E0at y¼R as a function of k_cx for different values of the corrugation amplitude (k_cR¼0:3, 1:2, 2). At

y¼R, the amplitude of E^{ð Þ}_x¹=E0takes the maximum value for kcR≈ 1:2. At the maximum we have E^{ð Þ}_x¹=E0=0.15 and C^{ð Þ}₁¹ ≈0:5. At the corrugation height R≈1:2�

kc

ð Þ^�1the amplitude of the first harmonic E^{ð Þ}_x¹ of the electric field reaches its maximum value. The first harmonic electric field of the corrugation is smaller for both the higher and lower corrugation amplitude. Indeed, for lower corrugation heights (kcR< < 1), the induced electric field is small because of weak polarization of charges on the dielectric interface with weak corrugation. In the case of higher corrugation heights (k_cR> > 1), the charge-induced field largely decays at y≈R.

Thus, we have the expression for the x-component of the first harmonic electric field of a wave at y¼R (in the region where the conducting channel is)

F^{ð Þ}_x¹ ¼E0C^{ð Þ}₁¹ expð�kcRÞexp ikð cxþikz�iωtÞ: (8)

Figure 3.

Coefficients, characterizing a contribution of the first C^{ð Þ}₁¹ (black line), second C^{ð Þ}₂¹ (blue line) and third C^{ð Þ}₃¹ (green line) harmonics to the induced electric near the corrugation, as a function k_cR; R is a corrugation amplitude, k_c¼2π=L, L is a corrugation period.

A Compact Source of Terahertz Radiation Based on an Open Corrugated Waveguide DOI: http://dx.doi.org/10.5772/intechopen.89692

channel is thin enough (h< <R, where h is the channel width), so that the electrons in the channel do not deform the electromagnetic fields in the dielectric.

In such waveguide, the electric field of the wave can be represented as

E r, tð Þ ¼E_TE ðr, tÞ þF r, tð Þ, where E_TE ðr, tÞis given by (1) and (2) [7]. The electric field F r, tð Þof the wave, which is induced near the corrugation, is comparable with the TE wave field only in a very narrow region (with a width of about R) near the dielectric interface. And F r, tð Þis significantly less than ETE ðr, tÞoutside this region.

Let us proceed to calculation of the electric field F r, tð Þ.The field F r, tð Þis caused by influence of the electric field ETE ðr, tÞof the TE-wave on the dielectric interface. However, due to the smallness of the corrugation size is compared to the electromagnetic wavelengthλ¼2π=χ(λ > >R, λ > >L), the TE wave’s electric field can be considered as a potential at distances typical for corrugation. Let us denote this field by E0, where E0cosð Þχy ≈E₀. Since the corrugated dielectric interface is homogeneous along the Z-axis, the problem of calculating F r, tð Þis effectively two-dimensional one [7].

The problem can be described by the scalar potentialϕ^{ð Þi}, which satisfies to the Laplace equation. We denote the potential asϕ^{ð Þ1}ðx, yÞinside the dielectric layer with the permittivityε₁, and asϕ^{ð Þ2}ðx, yÞinside the layer with the permittivityε₂ [7]. The functionsϕ^{ð Þ1}ðx, yÞandϕ^{ð Þ2}ðx, yÞsatisfy to the Laplace equation

Δϕ^{ð Þ1}ðx, yÞ ¼0 andΔϕ^{ð Þ2}ðx, yÞ ¼0 within its domain. Then the boundary condi- tions take the form

ϕ^{ð Þ1}ðx, yÞ��c¼ϕ^{ð Þ2}ðx, yÞ��_c (4) ε₁∂ϕ^{ð Þ1}ðx, yÞ

∂n ^c¼ε₂∂ϕ^{ð Þ2}ðx, yÞ

∂n

��

��

�

��

��

�_c: (5)

The conditions (4)–(5) represent the continuity of the tangential electric field and normal component of an electric induction vector in a point with coordinates ðx, yÞon the corrugation dielectric interface [7].

Since the corrugated dielectric interface is described by a periodic function of x, the electrostatic potential functionsϕ^{ð Þi}ðx, yÞcan be written as the sum of spatial harmonics:

ϕ^{ð Þi}ðx, yÞ ¼X^∞

m¼0

B^{ð Þ}_mⁱ cos mkð _cxÞexpð�mk_cj jyÞ

þX^∞

m¼0

A^{ð Þ}_mⁱ sin mkð cxÞexpð�mkcj jyÞ þE0x

(6)

In our problem, y xð Þ ¼R cos kð cxÞis a continuously differentiable function. In this case, expressions (6) represent a complete set of basis functions [9] for poten- tialsϕ^{ð Þi} and for electric fields E^{ð Þi} ¼ �∇ϕ^{ð Þi}. The expansion coefficients A^{ð Þ}_mⁱ and B^{ð Þ}_mⁱ of the sum (6) are found from the boundary conditions (4)–(5) at the point x, yð Þ ¼

x, cos kcx

ð Þ[7]. For the boundary described by y xð Þ ¼R cos kð cxÞ, this system is consistent, if the coefficients B^{ð Þ}_mⁱ ¼0, and A^{ð Þ}_mⁱ 6¼0. Then for the calculation of coefficients A^{ð Þ}_mⁱ we have limited the harmonic expansion to the seventh term.

According to our calculations, whenε₂=ε₁≥3, taking into account the seven expan- sion terms is sufficient for calculating the coefficients, and the error will not exceed a few percent.

Advances in Condensed-Matter and Materials Physics - Rudimentary Research to Topical …

The resulting expression for the x-component E^{ð Þ}_x¹ of the electric field E^{ð Þ1} ¼

�∇ϕ^{ð Þ1} induced in the upper layer near the corrugation (at y>R cos kð _cxÞ) can be represented as

E^{ð Þ}_x¹ðx, yÞ ¼E0

X^∞

m¼1

C^{ð Þ}_m¹e^imkcxe^{�mkcj jy} (7)

The coefficients C^{ð Þ}_m¹, which characterize the contribution of mth harmonic to the electric field induced above the corrugation, depend on the shape and size of corrugation and, moreover, on the parametersε₁andε₂. We carried out numerical analysis on an example of a structure with a plate of quartz (ε1= 4.5) and sapphire (ε2= 9.3). In the case, when the dielectric boundary is described by

y xð Þ ¼R cos kð cxÞ, each of the coefficients C^{ð Þ}_m¹ can be represented as a function of kcR. Figure 3 shows the calculated dependences C^{ð Þ}₁¹ðkcRÞ, C^{ð Þ}₂¹ðkcRÞand C^{ð Þ}₃¹ðkcRÞ.

For kcR≥1, the value of C^{ð Þ}₁¹ is much greater than C^{ð Þ}₂¹ and C^{ð Þ}₃¹. The coefficients C^{ð Þ}_m¹ for m>3 are much less than C^{ð Þ}₁¹, C^{ð Þ}₂¹ and C^{ð Þ}₃¹.

Figure 4 shows the first harmonic of electric field E^{ð Þ}_x¹=E0at y¼R as a function of k_cx for different values of the corrugation amplitude (k_cR¼0:3, 1:2, 2). At

y¼R, the amplitude of E^{ð Þ}_x¹=E0takes the maximum value for kcR≈ 1:2. At the maximum we have E^{ð Þ}_x¹=E0=0.15 and C^{ð Þ}₁¹ ≈0:5. At the corrugation height R≈1:2�

kc

ð Þ^�1the amplitude of the first harmonic E^{ð Þ}_x¹ of the electric field reaches its maximum value. The first harmonic electric field of the corrugation is smaller for both the higher and lower corrugation amplitude. Indeed, for lower corrugation heights (kcR< < 1), the induced electric field is small because of weak polarization of charges on the dielectric interface with weak corrugation. In the case of higher corrugation heights (k_cR> > 1), the charge-induced field largely decays at y≈R.

Thus, we have the expression for the x-component of the first harmonic electric field of a wave at y¼R (in the region where the conducting channel is)

F^{ð Þ}_x¹ ¼E0C^{ð Þ}₁¹ expð�kcRÞexp ikð cxþikz�iωtÞ: (8)

Figure 3.

Coefficients, characterizing a contribution of the first C^{ð Þ}₁¹ (black line), second C^{ð Þ}₂¹ (blue line) and third C^{ð Þ}₃¹ (green line) harmonics to the induced electric near the corrugation, as a function k_cR; R is a corrugation amplitude, k_c¼2π=L, L is a corrugation period.

A Compact Source of Terahertz Radiation Based on an Open Corrugated Waveguide DOI: http://dx.doi.org/10.5772/intechopen.89692

The optimal corrugation amplitude R and period L, in which the first harmonic amplitude of the non-uniform electric field takes its maximum, are determined by kcR≈ 1:2 (where kc¼2π=L).