Abstract

We study the diffraction produced by a PT -symmetric volume Bragg grating that combines modulation of refractive index and gain/loss of the same periodicity with a quarter-period shift between them. Such a complex grating has a directional coupling between the different diffraction orders, which allows us to find an analytic solution for the first three orders of the full Maxwell equations without resorting to the paraxial approximation. This is important, because only with the full equations can the boundary conditions, allowing for reflections, be properly implemented. Using our solution we analyze the properties of such a grating in a wide variety of configurations.

© 2015 Optical Society of America

1. Introduction

Relatively recently it has been discovered that light propagation in an artificial meta-material can be strongly modified, to the extent that this material can become one-way invisible by controlling the Parity-Time (PT)-symmetry. Such unidirectional invisibility has been predicted [1] for diffraction on a complex refractive index perturbation profile: Δñ = Δn0exp(2πjz/Λ), which can be realized in practice as the combination of an index grating (real grating) and a balanced gain/loss grating (imaginary grating) using the Euler relation exp(2πjz/Λ) = cos(2πz/Λ) + j sin(2πz/Λ). It has been shown in the case of a one-dimensional PT symmetric grating that when a beam of light is incident on one side of such a meta-material it is transmitted without any reflection, absorption or phase modulation, which amounts to unidirectional invisibility of the medium [1, 2].

PT -symmetric gratings have been extensively studied in one-dimensional structures like waveguides [15], whereas only a few papers [69] have addressed diffraction on PT -symmetric gratings in free-space configuration or two-dimensional geometries, as in the case of computer-generated holograms. In these publications the diffractive properties were analyzed on the basis of coupled wave differential equations in which second-order derivatives were neglected. Such an approach is justified for one-dimensional gratings in optical waveguides where the gratings represent weak modulation of the refractive index (its real and/or imaginary part) without any significant changes in its average value in the grating portion of the waveguide. In the case of slab gratings, illustrated in Fig. 1, neglecting the second derivatives of the field amplitudes is equivalent to neglecting the boundary effects, i.e. the bulk diffracted orders are retained while the waves produced at the boundaries are eliminated. Such an approximation could lead to significant errors. In the case of PT -symmetric gratings, where the diffraction modes have a very unusual interaction mechanism, it is very important to study how the slab boundaries affect the diffraction and how they affect invisibility in the two-dimensional PT -symmetric volume grating.

 

Fig. 1 (a) Planar slanted grating of the index (black color fringes) and gain/loss (red color fringes) modulation and (b) non-slanted grating

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We have therefore analyzed diffraction from such a slab by using the full, second-order Maxwell equations. In Sec. 3 we study a two-mode solution valid for angles near Bragg incidence. This applies for an arbitrary ratio between the index and gain/loss modulations, allowing us to track properties from standard index grating to a PT symmetric grating at the symmetry-breaking point. Then in Sec. 4 we specialize to this latter grating. Due to the particular directed structure of the coupled equations we are able to derive analytic expressions for the first three diffractive orders, S0, S1 and S2. In the following sections 5–7 we use these expressions to analyze the properties of the PT -grating in a variety of different configurations characterized by the values of the background diffractive index within and on either side of the slab, including a possible reflective layer at the back of the slab. A discussion of the general properties of this type of grating along with our conclusions is given in Sec. 8.

2. Second-order coupled-mode equations

In this paper we study the diffraction characteristics of active holographic gratings as a gain/loss modulation in combination with traditional index gratings. The slanted grating is assumed to be composed of modulation of the relative dielectric permittivity

ε(x,z)=ε2+Δεcos(K(xsinφ+zcosφ))
and modulation of gain and loss
σ(x,z)=Δσsin(K(xsinφ+zcosφ))
in the region from z = 0 to z = d with the same spatial frequency shifted by a quarter of period Λ/4 (K = 2π/Λ) with respect to one another, where ε2 is the average relative permittivity in the grating area, Δε is the amplitude of the sinusoidal relative permittivity, Δσ is the amplitude of the gain/loss periodic distribution, and φ is the grating slant angle. Unlike traditional modulation of the refractive index, Eq. (2) describes modulation of its imaginary part, so we will call the grating of Eq. (1) the real grating, and the grating described by Eq. (2) the imaginary one. The fact that ε is symmetric while σ is antisymmetric ensures that ñ is PT -symmetric, satisfying ñ(−x, −z) = ñ(x, z)*. Figure 1 shows the generalized model of the hologram grating used in our study. It covers the case of free-space to free-space diffraction as well as planar slab holograms. The propagation constant k(x, z) inside the grating slab is spatially modulated and related to the relative permittivity ε(x, z) and the gain/loss distribution σ(x, z) by the well-known formula
k2(x,z)=k02ε(x,z)jωμσ(x,z),
where μ is the permeability of the medium, ω is the angular frequency of the wave and k0 = ω/c is the wave-vector in free space, related to the free-space wavelength λ0 by k0 = 2π/λ0.

Equations (1)(3) can be combined in the following form:

k2(x,z)=k22+2k2(κexp(jK.r)+κ+exp(jK.r)),
where k2=k0(ε2)12 is the average propagation constant and r⃗ is the coordinate vector. The coupling constants κ+ and κ are
κ±=14(ε2)12(k0Δε±cμΔσ)
They can take quite different values, unlike the situation with only real or imaginary gratings, where the coupling constants are always equal, at least in magnitude.

In the two unmodulated regions, z < 0 and z > d, where we assume uniform permittivity ε1 and ε3, respectively, the assumed solutions of the wave equation for the normalized electric fields are, for z < 0 (incident and reflected waves):

E1(x,z)=exp[jk1(xsinθ+zcosθ)]++m=Rmexp[j{(k2sinθmKsinφ)x(k12(k2sinθmKsinφ)2)12z}]
and for z > d (transmitted waves)
E3(x,z)=m=Tmexp[j{(k2sinθmKsinφ)x+(k32(k2sinθmKsinφ)2)12(zd)}]
The total electric field in the hologram region 0 < z < d is the superposition of multiple waves:
E2(x,z)=m=Sm(z)exp[j(k2sinθmKsinφ)x],
where k1=k0(ε1)12, k3=k0(ε3)12, θ′ is the angle of incidence in Region 1, and θ is the angle of refraction in Region 2, related to each other by k1 sin θ′ = k2 sin θ. In these equations Rm, and Tm are the amplitudes of the m-th reflected and transmitted waves and are to be determined. Sm(z) is the amplitude of the m-th wave in the modulated region and is to be determined by solving the wave equation for an incident plane wave with TE polarization (i.e. electric field perpendicular to the plane of incidence)
2E2(x,z)+k02ε(x,z)E2(x,z)=0
To find Sm(z), Eqs. (1) and (8) are substituted into Eq. (9), resulting in the system of coupled-wave equations [10, 11]:
d2Sm(z)dz2+[k22(k2sinθmKsinφ)2]Sm(z)++2k2[κejKzcosφSm+1(z)+κ+ejKzcosφSm1(z)]=0
This set of coupled-wave equations contains no first-derivative terms. In addition, Eqs. (10) are nonconstant-coefficient differential equations due to the presence of z in the coefficients of the Sm−1 and Sm+1 terms.

From now on we will restrict ourselves to the case of an unslanted grating, taking φ = π/2. In this case the fringes are perpendicular to the slab boundaries z = 0 and z = d, cf. Fig. 1(b), and the equations become constant-coefficient differential equations.

For θ near the (first) Bragg angle θB, given by K = 2k2 sin θB, only the zeroth-order and the first-order diffraction modes are coupled strongly to each other. Retaining only these two modes, Eqs. (10) become:

d2S0(z)dz2+k22cos2θS0(z)+2k2κS1(z)=0d2S1(z)dz2+[k22(k2sinθK)2]S1(z)+2k2κ+S0(z)=0
At the exact Bragg condition, when θ = θB, the coupled equations reduce to the following form in terms of the dimensionless coordinate u = k2z:
d2S0(u)du2+cos2θBS0(u)+ξ1S1(u)=0d2S1(u)du2+cos2θBS1(u)+ξ2S0(u)=0
where
ξ1=2κ/k2andξ2=2κ+/k2
The coupled Eqs. (12) can be decoupled by switching to
V0=S0+ξ1/ξ2S1andV1=S0ξ1/ξ2S1,
when the equations become
d2V0(u)du2+ρ12V0(u)=0d2V1(u)du2+ρ22V1(u)=0
where
ρ1=(cos2θB+ξ1ξ1)12andρ2=(cos2θBξ1ξ1)12.
Then S0 and S1 are given by S0(u)=12(V0(u)+V1(u)) and S1(u)=12ξ2/ξ1(V0(u)V1(u)), where V0(u) and V1(u) have the solutions
V0(u)=Aejρ1u+Bejρ1uV1(u)=Cejρ2u+Dejρ2u
in which the constants A, B, C and D are to be found from the boundary conditions.

These require that the tangential electric and tangential magnetic fields be continuous across the two boundaries (z = 0 and z = d). For the H-mode polarization discussed in this paper, the electric field only has a component in the y-direction and so it is the tangential electric field directly. The magnetic field intensity, however, must be obtained through the Maxwell equation. The tangential component of H is in the x-direction and is thus given by Hx = (−j/(ωμ0))∂Ey/∂z.

In the approximation of keeping only the two modes S0(u) and S1(u) the four quantities to be matched and the resulting boundary conditions are

  • a) tangential E at z = 0:
    1+R0=S0(0),R1=S1(0)
  • b) tangential H at z = 0 :
    k2S0(0)=j(k12k22sin2θ)12(R01),k2S1(0)=j[k12(k2sinθK)2]12R1
  • c) tangential E at z = d:
    T0=S0(d),T1=S1(d)
  • c) tangential H at z = d:
    k2S0(d)=j(k32k22sin2θ)12T0,k2S1(d)=j[k12(k2sinθK)2]12T1

3. Two-mode solution for θ = θB

Taking S0(u) and S1(u) as given in Eqs. (17), the boundary conditions (18)(21) lead to the following eight equations for the eight unknown constants: A, B, C, D, R0, R1, T0 and T1.

R0+ξR1+1=A+B
αB(R0+ξR11)=ρ1(AB)
R0ξR1+1=C+D
αB(R0ξR11)=ρ2(C+D)
T0+ξT1=Aejρ1ud+Bejρ1ud
T0ξT1=Cejρ2ud+Dejρ2ud
βB(T0+ξT1)=ρ1(Aejρ1udBejρ1ud)
βB(T0ξT1)=ρ2(Cejρ2udDejρ2ud)
where ξ = √(ξ1/ξ2), αB = √(ε1/ε2 − sin2 θB), βB = √(ε3/ε2 − sin2θB) and ud = k2d.

Solving these equations, we find the following expressions for the zeroth- and first-order reflection coefficients:

R0=12(F(ρ1)+F(ρ2)),R1=12ξ(F(ρ1)F(ρ2))
where, for m = 1, 2,
F(ρm)=(ρmβB)(αB+ρm)ejρmud+(ρm+βB)(αBρm)ejρmud(ρmβB)(αBρm)ejρmud+(ρm+βB)(αB+ρm)ejρmud
The transmission coefficients are expressed in terms of G(ρ1) and G(ρ2) as
T0=12(G(ρ1)+G(ρ2)),T1=12ξ(G(ρ1)G(ρ2)),
where
G(ρm)=4ρmαB(ρmβB)(αBρm)ejρmud+(ρm+βB)(αB+ρm)ejρmud
It should be clear that variation of the asymmetry coefficient ξ from 1 to 0 describes the transition from a traditional index grating, analyzed by Kong [10], to a PT -symmetric one which reaches its balanced form at ξ = 0, as shown in Figs. 2(a)–2(d) for ξ = 1 (magenta, dot-dashed curves), ξ = 0.5 (green, dashed curves), ξ = 0.25 (blue, dotted curves) and finally the PT -symmetric case (red, solid) for the slab with ε2 = 2.4 in air, ε1 = ε3 = 1.

 

Fig. 2 Two-mode solution for incidence at the first Bragg angle θ′B: transmission and reflection coefficients as functions of the grating strength for different values of ξ=ξ1/ξ2, where ξ = 1 (magenta, dot-dashed) corresponds to a traditional index grating and ξ = 0 (red, solid) describes an ideal balanced PT -symmetric grating. The other values shown are ξ = 0.5 (green, dashed) and ξ = 0.25 (blue, dotted).The remaining parameters are ε1 = 1, ε2 = 2.4, ε3 = 1, d = 8 μm, Λ= 0.5 μm, λ0=0.6328 μm.

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Unlike the solution for the PT -symmetric grating obtained through the first-order coupled wave equations [8], which provides only the transmission coefficients, with |T0| = 1 and T1ξ2ud when θ = θB, our solution shows significant intensities in the zeroth and first reflective orders. In fact, power redistribution results in a normalized power reduction in |T0|2 from 1 to 0.83, with |R0|2 = 0.17. As expected, the mode-coupling nature in PT -symmetric gratings does not provide any amplification for the zeroth orders either in transmission or reflection. However, the first diffraction orders exhibit linear growth in amplitude, quadratic in power, before they reach gain saturation.

The assumptions of neglecting the second derivatives of field amplitudes and neglecting boundary effects transform the problem into a filled-space problem [11], like a grating filling all space with imaginary boundaries at z = 0 and z = d. In our second-order derivative solution we can approach such a regime by putting ε1 = ε2 = ε3. Indeed, as we can see in Fig. 3, the zeroth-order transmission amplitude returns to unity (red solid line in Fig. 3(a)) with practically no reflection (Fig. 3(c)). Reflected light in the first order is also practically negligible (Fig. 3(d)). Note that the power supplied to T1 comes from the active grating, not at the expense of the zeroth-order diffraction, which still satisfies |R0|2 + |T0|2 = 1.

 

Fig. 3 Two-mode solution for incidence at the first Bragg angle θ′B: transmission and reflection coefficients as functions of the grating strength for the filled-space configuration ε1 = ε2 = ε3 = 2.4. The remaining quantities are the same as in Fig. 2.

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4. Analytic solution for balanced PT -symmetric grating for arbitrary angle of incidence

The expressions (23)(26) for diffraction in transmission and reflection obtained in the previous section are valid only for θ = θB but for arbitrary ξ1, ξ2, thus including the perfectly balanced PT -symmetric grating as well as the unbalanced one. In fact, these expressions even cover the case of a purely imaginary grating of gain/loss modulation with no index grating in the slab (ξ1 = −ξ2).

In this section we extend our analysis of the balanced PT -symmetric grating (ξ1 =0), but with arbitrary angle of incidence. In that case the coupled wave Eqs. (11) are

d2S0(u)du2+cos2θS0(u)=0
d2S1(u)du2+[1(2sinθBsinθ)2]S1(u)+ξ2S0(u)=0
The first Eq. (27a) for the zeroth-order amplitude S0(u) (non-diffracted light) is decoupled from the second equation for the first-order amplitude S1(u). We therefore have a solution for S0(u) of the form
S0(u)=A0ejucosθ+B0ejucosθ
Applying the boundary conditions (18)(21) we can find T0 and R0 and the constants A0 and B0:
T0=4α0cosθ(α0+cosθ)(β0+cosθ)ejudcosθ(α0cosθ)(β0cosθ)ejudcosθ
R0=(α0cosθ)(β0+cosθ)ejudcosθ+(α0+cosθ)(β0cosθ)ejudcosθ(α0+cosθ)(β0+cosθ)ejudcosθ(α0cosθ)(β0cosθ)ejudcosθ
A0=T02(cosθβ0cosθ)ejudcosθB0=T02(cosθ+β0cosθ)ejudcosθ
where α0 = √(ε1/ε2 − sin2 θ) and β0 = √(ε3/ε2 − sin2 θ).

Equation (27b) is an inhomogeneous second-order differential equation for S1(u), whose solutions can be found as a sum of the general solution of the homogenous equation, (S1)H and a particular solution (S1)I of the inhomogeneous equation. The solution of the homogeneous equation is

(S1(u))H=C1ejη1u+D1ejη1u
where η1 = √[1 − (2 sin θB − sin θ)2] and C1 and D1 are constants to be determined. The particular solution can be found using the method of undetermined coefficients. We write
(S1(u))I=A1ejucosθ+B1ejucosθ
and find that A1 = x1A0 and B1 = x1B0, where
x1=ξ24sinθB(sinθBsinθ)
Applying the boundary conditions (19)(22) we can find T1, R1, C1 and D1. The rather lengthy expressions thus obtained for T1 and R1 can be expressed in a condensed form using the functions
f(a,b,c):=(a+b)(ac)(1ej(ab)ud)(ab)(a+c)(1ej(a+b)ud)
g(a,b,c):=(a+b)(ac)(ejbudejaud)(ab)(a+c)(ejbudejaud)
h(a,b,c):=(a+b)(a+c)ejaud(ab)(ac)ejaud
In this notation, and with the definitions η0 = cos θ, αm = √[ε1/ε2 − (2msin θB − sin θ)2] and βn = √[ε3/ε2 − (2msin θB − sin θ)2],
T1=1h(η1,α1,β1)[f(η1,η0,α1)A1+f(η1,η0,α1)B1]
R1=1h(η1,α1,β1)[g(η1,η0,β1)A1+g(η1,η0,β1)B1]
The coefficients C1 and D1 are given by
C1=1h(η1,α1,β1){A1[(α1η0)(β1η1)ejη1ud(α1+η1)(β1+η0)ejη0ud]+B1[(α1+η0)(β1η1)ejη1ud(α1+η1)(β1η0)ejη0ud]}
and
D1=1h(η1,α1,β1){A1[(α1η0)(β1+η1)ejη1ud(α1η1)(β1+η0)ejη0ud]+B1[(α1+η0)(β1+η1)ejη1ud(α1η1)(β1η0)ejη0ud]}

It is important to emphasize that the mode coupling in a PT -symmetric grating has a unidirectional nature, with energy flowing from lower order to higher order modes: from zeroth order to first order, from first order to second order and so on. With such a type of coupling it is relatively easy to find practically any higher diffraction order analytically. Here we exploit this feature to derive explicit expressions for the second-order reflection and transmission coefficients.

The equation for the second-order mode has the following form:

d2S2(u)du2+[1(4sinθBsinθ)2]S2(u)+ξ2S1(u)=0
Using the same approach as to Eq. (27b), the solution is again sought as a sum of the general solution of the homogeneous equation and a particular solution of the inhomogeneous equation. The solution of the homogeneous equation is
(S2(u))H=E2ejuη2+F2ejuη2
where η2 = √[1 − (4sin θB − sin θ)2] and E2 and F2 are constants to be determined. The particular solution of the differential equation can again be found using the method of undetermined coefficients. Writing
(S2(u))I=C2ejuη1+D2ejuη1+A2ejucosθ+B2ejucosθ,
we find C2 = x3C1, D3 = x2D1, A2 = x2A1 and B2 = x2B1, where
x2=ξ28sinθB(2sinθBsinθ),x3=ξ24sinθB(3sinθBsinθ)
Applying the boundary conditions at u = 0 and u = ud we can find T2, R2, E2 and F2. With the help of the functions defined in Section 4, we can express T2 and R2 in a rather compact form:
T2=1h(η2,α2,β2)[f(η2,η0,α2)A2+f(η2,η0,α2)B2+f(η2,η1,α2)C2+f(η2,η1,α2)D2]
R2=1h(η2,α2,β2)[g(η2,η0,α2)A2+g(η2,η0,α2)B2+g(η2,η1,α2)C2+g(η2,η1,α2)D2]
Similar expressions can be obtained for E2 and F2, but are not needed for our present purposes. They would be needed for the calculation of the third-order reflection and transmission coefficients.

In subsequent figures we will display the diffraction efficiencies rather than the squared moduli of the diffraction coefficients. The diffraction efficiency for the ith order is defined as the diffracted intensity of this order divided by the input intensity. We normalized the amplitude of the incident plane wave to one. The diffraction intensities in Regions 1 and 3 are therefore

DERm=Re(αmα0)|Rm|2DETm=Re(βmα0)|Tm|2

5. Filled-space PT -symmetric grating

As a first check of our solution we will consider the particular case of the so-called filled-space grating, when the dielectric permittivity to the left and right of the slab is equal to the average dielectric permittivity of the slab: ε1 = ε2 = ε3. This configuration should provide a solution that is very close to that of the first-order coupled wave equations.

Indeed, when ε1 = ε2 = ε3, then α0 = β0 = cos θ, so that A0 = 0 and B0 = 1, R0 = 0 and T0 = ejud cosθ. With no reflections from the slab boundaries the non-diffracted wave passes through the slab without any attenuation/amplification and without any phase modulation, in accordance with the invisibility property.

On the other hand, the first-order diffraction occurs with strong amplification, as is seen from Fig. 4(a). For ε1 = ε2 = ε3 the expressions for T1 and R1 in Eqs. (36) and (37) simplify to

T1=x1(η1+η0)2η1(ejη1udejη0ud)=ξ2(η1+cosθ)(ejη1udejη0ud)8η1sinθB(sinθBsinθ)
R1=x1η1η02η1(1ej(η0+η1)ud)=ξ2(η1cosθ)(1ej(η0+η1)ud)8η1sinθB(sinθBsinθ)
These peak at the Bragg angle, where their values are
T1=jξ2ud2cosθBejudcosθBR1=jξ2sin(udcosθB)2cos2θBejudcosθB
The first-order diffraction amplitude T1 grows linearly with the grating strength ξ2ud, with amplification close to 800% for the parameters chosen in Fig. 4. This linear growth in amplitude is a characteristic of PT -symmetric structures at their breaking point. We should remember that the PT -symmetric grating is an active structure: even though the average gain/loss is zero, external energy must be supplied to provide its functionality. R1 is not zero, but is small even at the resonance, with a diffraction efficiency of less than 0.1%. %.

 

Fig. 4 Filled-space configuration (ε1 = ε2 = ε3 = 2.4) : diffraction efficiency in (a) first and (b) second orders in transmission as a function of the internal angle of incidence for Λ= 0.5 μm (red, solid), Λ= 0.75 μm (blue, dashed), Λ= 1.0 μm (magenta, dot-dashed). The other parameters are d = 8 μm, λ0=0.633 μm.

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6. Symmetric slab configuration

In this section we compare the transmission and reflection characteristics of the filled-space PT -symmetric grating without reflections from the slab boundaries (ε1 = ε2 = ε3 = 2.4) and the real configuration of the slab in air (ε1 = ε3 = 1, ε2 = 2.4). The results are presented in Fig. 5.

 

Fig. 5 Symmetric vs. filled-space configuration: diffraction efficiency in transmission ((a), (c), (e)) and reflection ((b), (d), (f)) as a function of the internal angle of incidence θ for ε1 = ε3 = 1, ε2 = 2.4 (blue, dashed), ε1 = ε2 = ε3 = 2.4 (red, solid). The other parameters are d = 8μm, Λ= 0.75 μm and λ0=0.633 μm.

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The reflections from the front and back surfaces of the slab significantly change the spectral characteristics of the zeroth-order transmission (Fig. 5(a)) and reflection (Fig. 5(b)). These two plots cover the range −41° < θ < 41° for the internal incident angle ( arcsin(ε1/ε2)=40.2°), which corresponds to the range −90° < θ′ < 90° for the external incident angle. The effect of invisibility of the PT -symmetric grating for zeroth-order transmission (red solid horizontal line in Fig. 5(a)) is strongly distorted by interference of the light reflected from the slab surfaces, as shown by the blue dashed curve. The effect is stronger for larger incident angles. Similarly the zeroth-order reflected light emerges with increasing intensity for larger incident angles (Fig. 5(b)).

The angular spectra for the first-order diffracted light are presented in Fig. 5(c) in transmission and Fig. 5(d) in reflection. As can be seen, the reflection from the slab boundaries leads to a significant increase in the reflected first diffraction order (Fig. 5(d)) along with a rather small decrease of the transmitted light in that order (Fig. 5(c)).

7. Asymmetric slab configurations

In many practical applications the slab supporting the PT -symmetric grating might be very thin and fragile and need to be attached to a substrate. Such a situation leads to different dielectric permittivity from the left and right sides of the slab, ε1ε3. Such a practical requirement might result in the input light incident from the substrate side or from the air side, as shown in Fig. 6(a) and Fig. 6(b) respectively.

 

Fig. 6 Asymmetric configurations when the input light comes from (a) the substrate side and (b) the air side.

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7.1. Light incident from the substrate side: ε3 = 1

The geometry presented in Fig. 6(a) has been analyzed, with the results shown in Fig. 7.

 

Fig. 7 Light incident from substrate: transmission (red, solid) and reflection (blue, dashed) angular spectra for zeroth order (a) and (b), first-order (c) and (d) and second-order light (e) and (f) as functions of the internal angle of incidence. In the left-hand panels ε1 = ε2 = 2.4, while in the right-hand panels ε1 = 2.0, ε2 = 2.4. The remaining parameters are ε3 = 1, d = 8 μm; Λ = 0.75 μm; λ =0.633 μm and ξ = 0.04.

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We consider the cases when the permittivity of the substrate and the average permittivity of the slab are the same, ε1 = ε2, (Figs. 7(a), (c) and (e)), and when they are different (Figs. 7(b), 7(d) and 7(f)). Comparing Figs. 7(a) and 7(b) with Figs. 5(a) and 5(c) one can see a significant difference in the angular spectral behavior in zeroth order. Equations (29) and (30) simplify significantly for ε1 = ε2, when α0 = cos θ, so that

T0=(2cosθcosθ+β0)ejudcosθ
R0=(cosθβ0cosθ+β0)e2judcosθ,
which are basically the Fresnel coefficients. The transmission rapidly decreases to zero for |θ| > θTIR ≡ arcsin(√(ε3/ε2)), the angle at which total internal reflection occurs at the second surface. Reflection in zeroth order is close to zero (4.6%), ( R0=(ε2ε3)/(ε2+ε3)), for normal incidence and then rapidly increases to 100% for |θ| > θTIR. Introduction of the second reflective interface with ε1ε2, (Fig. 7(b)), produces a weak rippling effect on the transmission and reflection spectra.

There is no significant difference in transmission and reflection of the first and second diffraction orders between the configuration of Figs. 7(a), (c) and (e) and that of Figs. 7(b), 7(d) and 7(f). It seems clear that it is reflection from the interface between the slab and Region 3 that produces the major contribution to the reflective diffraction. If that is the case, then intuitively the reflective diffraction orders can be significantly reduced by illuminating from the air, as shown in Fig. 6(b).

7.2. Light incident from the air: ε1 = 1

Indeed, in this set-up the reflection is practically invisible (blue dashed curves) in Fig. 8 for the first and second diffractive orders. Even the reflection from the slab-substrate interface where ε2ε3 does not contribute in any significant way to the reflective diffraction orders, Fig. 8(b) and Fig. 8(d). The second-order diffraction is also negligible compared to the first order, so that practically all the light diffracted by the PT -symmetric volume grating goes into the first transmissive diffraction order.

 

Fig. 8 Light incident from the air: transmission (red, solid) and reflection (blue, dashed) angular spectra for first-order (a) and (b) and second-order diffracted light (c) and (d) as functions of the internal angle of incidence. In the left-hand panels ε2 = ε3 = 2.4, while in the right-hand panels ε2 = 2.4, ε3 = 2.0. The remaining parameters are ε1 = 1, d = 8 μm; Λ = 0.75 μm; λ =0.633 μm and ξ = 0.04.

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7.3. Reflective set-up

To conclude this analysis of the PT -symmetric transmission grating we propose a method to reverse its first transmission order into reflection. This can be done by placing an aluminum layer between the slab and the substrate. If this aluminum layer is of the order of one micron in thickness then any influence of the substrate will be shielded. Such a structure can be accurately simulated by assigning the dielectric permittivity of Region 3 the aluminum permittivity at λ0=0.633 μm, namely ε3 = −54.705 + 21.829 j. The results are depicted in Fig. 9. As can be seen, the reflection now becomes dominant in zeroth order, as well as for first-order diffraction. The peak in the reflection spectrum (dashed blue curve) is now at least an order of magnitude stronger than that in the transmission spectrum (solid red curve).

 

Fig. 9 Reflective set-up (ε1 = 1.0, ε2 = 2.4, ε3 = −54.7+21.83 j): transmission (red, solid) and reflection (blue, dashed) angular spectra for zeroth-order (a) and first-order diffracted light (c) as functions of the internal angle of incidence for d = 8 μm, Λ = 0.75 μm, λ0=0.633 μm, ξ = 0.004.

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8. Discussion

In a normal (index) grating, the situation is symmetric between θ and −θ. Thus, near θ = θB, the first two modes excited are S0 and S1, while near θ = −θB, the first two modes excited are S0 and S−1. More generally θ ↔ −θ corresponds to m ↔ −m, where m labels the diffraction order.

However, in a balanced PT grating, this symmetry is lost because the index modulation has an inbuilt direction (it is symmetric under PT, but not under P itself). So in the situation we have been describing in the bulk of the paper, illustrated in Fig. 10(a), light incident near the first Bragg angle produces strong signals in first-order diffraction, particularly in transmission. In contrast, for incidence at θ near −θB, as illustrated in Fig. 10(b), there is essentially no diffraction.

 

Fig. 10 Prominent modes of the PT -symmetric grating for incidence at different angles and from different sides: (a) from the left near the first Bragg angle θB; (b) from the left near −θB; (c) from the right near θB; (d) from the right near −θB.

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The PT grating is also left-right asymmetric, even when ε1 = ε3. In Fig. 10(c), light incident in the reverse direction of the transmitted beam in Fig. 10(a) does not produce the mirror-image of Fig. 10(a), but rather that of Fig. 10(b). Likewise for Fig. 10(d), which is the mirror-image of Fig. 10(a) rather than Fig. 10(b). There is yet another type of asymmetry of the PT grating. We have called the grating “balanced” when the perturbation of the refractive index is Δñ = Δn0e2πjz, resulting in ξ1 = 0 and the consequences explored in the paper. However, if the phase of the gain/loss modulation relative to the index modulation is reversed, Δñ instead becomes Δñ = Δn0e−2πjz and the roles of ξ1 and ξ2 are interchanged, so that now ξ2 = 0. In that case the first mode to be excited is m = −1, and the coupled equations for S0 and S−1 become

d2S0(u)du2+cos2θS0(u)=0
d2S1(u)du2+[1(2sinθB+sinθ)2]S1(u)+ξ1S0(u)=0,
giving a strong excitation of S−1 near θ = −θB. Thus the symmetry θ → −θ of a normal index grating is regained provided that the phase of the gain/loss modulation is reversed at the same time.

Another prominent characteristic of a balanced PT grating in the paraxial approximation is invisibility [6, 8], that is to say that the transmission coefficient T0 of the undiffracted wave is unity. However, this approximation cannot account for reflections at the boundaries. The main part of our paper has been to find analytic solutions of the full second-derivative equations for the first three diffractive orders. With the help of these solutions we have analyzed diffraction from the slab in a variety of different configurations. The invisibility property has been shown to hold only in the filled-space situation, when the background refractive indices are the same, but when this is not the case the reflections produced by the second-order equations result in a significant reduction of |T0|2. The linear rise with grating strength of the first-order transmission amplitude, already seen in paraxial approximation, persists when the full second-order equations are used, making T1 by far the strongest signal for the range of parameters we considered.

In Sections 5, 6 and 7 we considered a variety of configurations of the slab in terms of the different background relative permittivities ε1, ε2, ε3, showing in detail how the transmitted and reflected light was affected by these different parameters. In the last subsection of Sec. 7 we showed how a reflective layer at the back of the slab could turn it into a reflective grating, with a strong reflection coefficient R1.

A PT -symmetric volume grating is a structure with many interesting and unusual properties, which can only be fully analyzed using the second-order Maxwell equations that we have treated here.

References and links

1. L. Poladian, “Resonance mode expansions and exact solutions for nonuniform gratings,” Phys. Rev. E 54, 2963–2975 (1996). [CrossRef]  

2. M. Kulishov, J. M. Laniel, N. Bélanger, J. Azaña, and D. V. Plant, “Nonreciprocal waveguide Bragg gratings,” Opt. Express 13, 3068–3078 (2005). [CrossRef]   [PubMed]  

3. M. Greenberg and M. Orenstein, “Irreversible coupling by use of dissipative optics,” Opt. Lett. 29, 451–453 (2004). [CrossRef]   [PubMed]  

4. S. Longhi, “Invisibility in PT -symmetric complex crystals,” J. Phys. A 44, 485302 (2011). [CrossRef]  

5. H. F. Jones, “Analytic results for a PT -symmetric optical structure,” J. Phys. A 45, 135306 (2012). [CrossRef]  

6. M. V. Berry, “Lop-sided diffraction by absorbing crystals,” J. Phys. Math. Gen. 31, 3493–3502 (1998). [CrossRef]  

7. Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. Christodoulides, “Unidirectional invisibility induced by PT -symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011). [CrossRef]  

8. M. Kulishov and B. Kress, “Free space diffraction on active gratings with balanced phase and gain/loss modulations,” Opt. Express 20, 29319–29328 (2012). [CrossRef]  

9. L. Feng, X. Zhu, S. Yang, H. Zhu, P. Zhang, X. Yin, Y. Wang, and X. Zhang, “Demonstration of a large-scale optical exceptional point structure,” Opt. Express 22, 1760–1767 (2014). [CrossRef]   [PubMed]  

10. J. A. Kong, “Second-order coupled-mode equations for spatially periodic media,” J. Opt. Soc. Am. 67, 825–829 (1976). [CrossRef]  

11. T. K. Gaylord and M. G. Moharam, “Planar dielectric grating diffraction theories,” Appl. Phys. B 28, 1–14 (1982). [CrossRef]  

References

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  1. L. Poladian, “Resonance mode expansions and exact solutions for nonuniform gratings,” Phys. Rev. E 54, 2963–2975 (1996).
    [Crossref]
  2. M. Kulishov, J. M. Laniel, N. Bélanger, J. Azaña, and D. V. Plant, “Nonreciprocal waveguide Bragg gratings,” Opt. Express 13, 3068–3078 (2005).
    [Crossref] [PubMed]
  3. M. Greenberg and M. Orenstein, “Irreversible coupling by use of dissipative optics,” Opt. Lett. 29, 451–453 (2004).
    [Crossref] [PubMed]
  4. S. Longhi, “Invisibility in PT -symmetric complex crystals,” J. Phys. A 44, 485302 (2011).
    [Crossref]
  5. H. F. Jones, “Analytic results for a PT -symmetric optical structure,” J. Phys. A 45, 135306 (2012).
    [Crossref]
  6. M. V. Berry, “Lop-sided diffraction by absorbing crystals,” J. Phys. Math. Gen. 31, 3493–3502 (1998).
    [Crossref]
  7. Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. Christodoulides, “Unidirectional invisibility induced by PT -symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
    [Crossref]
  8. M. Kulishov and B. Kress, “Free space diffraction on active gratings with balanced phase and gain/loss modulations,” Opt. Express 20, 29319–29328 (2012).
    [Crossref]
  9. L. Feng, X. Zhu, S. Yang, H. Zhu, P. Zhang, X. Yin, Y. Wang, and X. Zhang, “Demonstration of a large-scale optical exceptional point structure,” Opt. Express 22, 1760–1767 (2014).
    [Crossref] [PubMed]
  10. J. A. Kong, “Second-order coupled-mode equations for spatially periodic media,” J. Opt. Soc. Am. 67, 825–829 (1976).
    [Crossref]
  11. T. K. Gaylord and M. G. Moharam, “Planar dielectric grating diffraction theories,” Appl. Phys. B 28, 1–14 (1982).
    [Crossref]

2014 (1)

2012 (2)

2011 (2)

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. Christodoulides, “Unidirectional invisibility induced by PT -symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref]

S. Longhi, “Invisibility in PT -symmetric complex crystals,” J. Phys. A 44, 485302 (2011).
[Crossref]

2005 (1)

2004 (1)

1998 (1)

M. V. Berry, “Lop-sided diffraction by absorbing crystals,” J. Phys. Math. Gen. 31, 3493–3502 (1998).
[Crossref]

1996 (1)

L. Poladian, “Resonance mode expansions and exact solutions for nonuniform gratings,” Phys. Rev. E 54, 2963–2975 (1996).
[Crossref]

1982 (1)

T. K. Gaylord and M. G. Moharam, “Planar dielectric grating diffraction theories,” Appl. Phys. B 28, 1–14 (1982).
[Crossref]

1976 (1)

Azaña, J.

Bélanger, N.

Berry, M. V.

M. V. Berry, “Lop-sided diffraction by absorbing crystals,” J. Phys. Math. Gen. 31, 3493–3502 (1998).
[Crossref]

Cao, H.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. Christodoulides, “Unidirectional invisibility induced by PT -symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref]

Christodoulides, D.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. Christodoulides, “Unidirectional invisibility induced by PT -symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref]

Eichelkraut, T.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. Christodoulides, “Unidirectional invisibility induced by PT -symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref]

Feng, L.

Gaylord, T. K.

T. K. Gaylord and M. G. Moharam, “Planar dielectric grating diffraction theories,” Appl. Phys. B 28, 1–14 (1982).
[Crossref]

Greenberg, M.

Jones, H. F.

H. F. Jones, “Analytic results for a PT -symmetric optical structure,” J. Phys. A 45, 135306 (2012).
[Crossref]

Kong, J. A.

Kottos, T.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. Christodoulides, “Unidirectional invisibility induced by PT -symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref]

Kress, B.

Kulishov, M.

Laniel, J. M.

Lin, Z.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. Christodoulides, “Unidirectional invisibility induced by PT -symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref]

Longhi, S.

S. Longhi, “Invisibility in PT -symmetric complex crystals,” J. Phys. A 44, 485302 (2011).
[Crossref]

Moharam, M. G.

T. K. Gaylord and M. G. Moharam, “Planar dielectric grating diffraction theories,” Appl. Phys. B 28, 1–14 (1982).
[Crossref]

Orenstein, M.

Plant, D. V.

Poladian, L.

L. Poladian, “Resonance mode expansions and exact solutions for nonuniform gratings,” Phys. Rev. E 54, 2963–2975 (1996).
[Crossref]

Ramezani, H.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. Christodoulides, “Unidirectional invisibility induced by PT -symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref]

Wang, Y.

Yang, S.

Yin, X.

Zhang, P.

Zhang, X.

Zhu, H.

Zhu, X.

Appl. Phys. B (1)

T. K. Gaylord and M. G. Moharam, “Planar dielectric grating diffraction theories,” Appl. Phys. B 28, 1–14 (1982).
[Crossref]

J. Opt. Soc. Am. (1)

J. Phys. A (2)

S. Longhi, “Invisibility in PT -symmetric complex crystals,” J. Phys. A 44, 485302 (2011).
[Crossref]

H. F. Jones, “Analytic results for a PT -symmetric optical structure,” J. Phys. A 45, 135306 (2012).
[Crossref]

J. Phys. Math. Gen. (1)

M. V. Berry, “Lop-sided diffraction by absorbing crystals,” J. Phys. Math. Gen. 31, 3493–3502 (1998).
[Crossref]

Opt. Express (3)

Opt. Lett. (1)

Phys. Rev. E (1)

L. Poladian, “Resonance mode expansions and exact solutions for nonuniform gratings,” Phys. Rev. E 54, 2963–2975 (1996).
[Crossref]

Phys. Rev. Lett. (1)

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. Christodoulides, “Unidirectional invisibility induced by PT -symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref]

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Figures (10)

Fig. 1
Fig. 1 (a) Planar slanted grating of the index (black color fringes) and gain/loss (red color fringes) modulation and (b) non-slanted grating
Fig. 2
Fig. 2 Two-mode solution for incidence at the first Bragg angle θ′B: transmission and reflection coefficients as functions of the grating strength for different values of ξ = ξ 1 / ξ 2, where ξ = 1 (magenta, dot-dashed) corresponds to a traditional index grating and ξ = 0 (red, solid) describes an ideal balanced PT -symmetric grating. The other values shown are ξ = 0.5 (green, dashed) and ξ = 0.25 (blue, dotted).The remaining parameters are ε1 = 1, ε2 = 2.4, ε3 = 1, d = 8 μm, Λ= 0.5 μm, λ0=0.6328 μm.
Fig. 3
Fig. 3 Two-mode solution for incidence at the first Bragg angle θ′B: transmission and reflection coefficients as functions of the grating strength for the filled-space configuration ε1 = ε2 = ε3 = 2.4. The remaining quantities are the same as in Fig. 2.
Fig. 4
Fig. 4 Filled-space configuration (ε1 = ε2 = ε3 = 2.4) : diffraction efficiency in (a) first and (b) second orders in transmission as a function of the internal angle of incidence for Λ= 0.5 μm (red, solid), Λ= 0.75 μm (blue, dashed), Λ= 1.0 μm (magenta, dot-dashed). The other parameters are d = 8 μm, λ0=0.633 μm.
Fig. 5
Fig. 5 Symmetric vs. filled-space configuration: diffraction efficiency in transmission ((a), (c), (e)) and reflection ((b), (d), (f)) as a function of the internal angle of incidence θ for ε1 = ε3 = 1, ε2 = 2.4 (blue, dashed), ε1 = ε2 = ε3 = 2.4 (red, solid). The other parameters are d = 8μm, Λ= 0.75 μm and λ0=0.633 μm.
Fig. 6
Fig. 6 Asymmetric configurations when the input light comes from (a) the substrate side and (b) the air side.
Fig. 7
Fig. 7 Light incident from substrate: transmission (red, solid) and reflection (blue, dashed) angular spectra for zeroth order (a) and (b), first-order (c) and (d) and second-order light (e) and (f) as functions of the internal angle of incidence. In the left-hand panels ε1 = ε2 = 2.4, while in the right-hand panels ε1 = 2.0, ε2 = 2.4. The remaining parameters are ε3 = 1, d = 8 μm; Λ = 0.75 μm; λ =0.633 μm and ξ = 0.04.
Fig. 8
Fig. 8 Light incident from the air: transmission (red, solid) and reflection (blue, dashed) angular spectra for first-order (a) and (b) and second-order diffracted light (c) and (d) as functions of the internal angle of incidence. In the left-hand panels ε2 = ε3 = 2.4, while in the right-hand panels ε2 = 2.4, ε3 = 2.0. The remaining parameters are ε1 = 1, d = 8 μm; Λ = 0.75 μm; λ =0.633 μm and ξ = 0.04.
Fig. 9
Fig. 9 Reflective set-up (ε1 = 1.0, ε2 = 2.4, ε3 = −54.7+21.83 j): transmission (red, solid) and reflection (blue, dashed) angular spectra for zeroth-order (a) and first-order diffracted light (c) as functions of the internal angle of incidence for d = 8 μm, Λ = 0.75 μm, λ0=0.633 μm, ξ = 0.004.
Fig. 10
Fig. 10 Prominent modes of the PT -symmetric grating for incidence at different angles and from different sides: (a) from the left near the first Bragg angle θB; (b) from the left near −θB; (c) from the right near θB; (d) from the right near −θB.

Equations (63)

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ε ( x , z ) = ε 2 + Δ ε cos ( K ( x sin φ + z cos φ ) )
σ ( x , z ) = Δ σ sin ( K ( x sin φ + z cos φ ) )
k 2 ( x , z ) = k 0 2 ε ( x , z ) j ω μ σ ( x , z ) ,
k 2 ( x , z ) = k 2 2 + 2 k 2 ( κ exp ( j K . r ) + κ + exp ( j K . r ) ) ,
κ ± = 1 4 ( ε 2 ) 1 2 ( k 0 Δ ε ± c μ Δ σ )
E 1 ( x , z ) = exp [ j k 1 ( x sin θ + z cos θ ) ] + + m = R m exp [ j { ( k 2 sin θ m K sin φ ) x ( k 1 2 ( k 2 sin θ m K sin φ ) 2 ) 1 2 z } ]
E 3 ( x , z ) = m = T m exp [ j { ( k 2 sin θ m K sin φ ) x + ( k 3 2 ( k 2 sin θ m K sin φ ) 2 ) 1 2 ( z d ) } ]
E 2 ( x , z ) = m = S m ( z ) exp [ j ( k 2 sin θ m K sin φ ) x ] ,
2 E 2 ( x , z ) + k 0 2 ε ( x , z ) E 2 ( x , z ) = 0
d 2 S m ( z ) d z 2 + [ k 2 2 ( k 2 sin θ m K sin φ ) 2 ] S m ( z ) + + 2 k 2 [ κ e j K z cos φ S m + 1 ( z ) + κ + e j K z cos φ S m 1 ( z ) ] = 0
d 2 S 0 ( z ) d z 2 + k 2 2 cos 2 θ S 0 ( z ) + 2 k 2 κ S 1 ( z ) = 0 d 2 S 1 ( z ) d z 2 + [ k 2 2 ( k 2 sin θ K ) 2 ] S 1 ( z ) + 2 k 2 κ + S 0 ( z ) = 0
d 2 S 0 ( u ) d u 2 + cos 2 θ B S 0 ( u ) + ξ 1 S 1 ( u ) = 0 d 2 S 1 ( u ) d u 2 + cos 2 θ B S 1 ( u ) + ξ 2 S 0 ( u ) = 0
ξ 1 = 2 κ / k 2 and ξ 2 = 2 κ + / k 2
V 0 = S 0 + ξ 1 / ξ 2 S 1 and V 1 = S 0 ξ 1 / ξ 2 S 1 ,
d 2 V 0 ( u ) d u 2 + ρ 1 2 V 0 ( u ) = 0 d 2 V 1 ( u ) d u 2 + ρ 2 2 V 1 ( u ) = 0
ρ 1 = ( cos 2 θ B + ξ 1 ξ 1 ) 1 2 and ρ 2 = ( cos 2 θ B ξ 1 ξ 1 ) 1 2 .
V 0 ( u ) = A e j ρ 1 u + B e j ρ 1 u V 1 ( u ) = C e j ρ 2 u + D e j ρ 2 u
1 + R 0 = S 0 ( 0 ) , R 1 = S 1 ( 0 )
k 2 S 0 ( 0 ) = j ( k 1 2 k 2 2 sin 2 θ ) 1 2 ( R 0 1 ) , k 2 S 1 ( 0 ) = j [ k 1 2 ( k 2 sin θ K ) 2 ] 1 2 R 1
T 0 = S 0 ( d ) , T 1 = S 1 ( d )
k 2 S 0 ( d ) = j ( k 3 2 k 2 2 sin 2 θ ) 1 2 T 0 , k 2 S 1 ( d ) = j [ k 1 2 ( k 2 sin θ K ) 2 ] 1 2 T 1
R 0 + ξ R 1 + 1 = A + B
α B ( R 0 + ξ R 1 1 ) = ρ 1 ( A B )
R 0 ξ R 1 + 1 = C + D
α B ( R 0 ξ R 1 1 ) = ρ 2 ( C + D )
T 0 + ξ T 1 = A e j ρ 1 u d + B e j ρ 1 u d
T 0 ξ T 1 = C e j ρ 2 u d + D e j ρ 2 u d
β B ( T 0 + ξ T 1 ) = ρ 1 ( A e j ρ 1 u d B e j ρ 1 u d )
β B ( T 0 ξ T 1 ) = ρ 2 ( C e j ρ 2 u d D e j ρ 2 u d )
R 0 = 1 2 ( F ( ρ 1 ) + F ( ρ 2 ) ) , R 1 = 1 2 ξ ( F ( ρ 1 ) F ( ρ 2 ) )
F ( ρ m ) = ( ρ m β B ) ( α B + ρ m ) e j ρ m u d + ( ρ m + β B ) ( α B ρ m ) e j ρ m u d ( ρ m β B ) ( α B ρ m ) e j ρ m u d + ( ρ m + β B ) ( α B + ρ m ) e j ρ m u d
T 0 = 1 2 ( G ( ρ 1 ) + G ( ρ 2 ) ) , T 1 = 1 2 ξ ( G ( ρ 1 ) G ( ρ 2 ) ) ,
G ( ρ m ) = 4 ρ m α B ( ρ m β B ) ( α B ρ m ) e j ρ m u d + ( ρ m + β B ) ( α B + ρ m ) e j ρ m u d
d 2 S 0 ( u ) d u 2 + cos 2 θ S 0 ( u ) = 0
d 2 S 1 ( u ) d u 2 + [ 1 ( 2 sin θ B sin θ ) 2 ] S 1 ( u ) + ξ 2 S 0 ( u ) = 0
S 0 ( u ) = A 0 e j u cos θ + B 0 e j u cos θ
T 0 = 4 α 0 cos θ ( α 0 + cos θ ) ( β 0 + cos θ ) e j u d cos θ ( α 0 cos θ ) ( β 0 cos θ ) e j u d cos θ
R 0 = ( α 0 cos θ ) ( β 0 + cos θ ) e j u d cos θ + ( α 0 + cos θ ) ( β 0 cos θ ) e j u d cos θ ( α 0 + cos θ ) ( β 0 + cos θ ) e j u d cos θ ( α 0 cos θ ) ( β 0 cos θ ) e j u d cos θ
A 0 = T 0 2 ( cos θ β 0 cos θ ) e j u d cos θ B 0 = T 0 2 ( cos θ + β 0 cos θ ) e j u d cos θ
( S 1 ( u ) ) H = C 1 e j η 1 u + D 1 e j η 1 u
( S 1 ( u ) ) I = A 1 e j u cos θ + B 1 e j u cos θ
x 1 = ξ 2 4 sin θ B ( sin θ B sin θ )
f ( a , b , c ) : = ( a + b ) ( a c ) ( 1 e j ( a b ) u d ) ( a b ) ( a + c ) ( 1 e j ( a + b ) u d )
g ( a , b , c ) : = ( a + b ) ( a c ) ( e j b u d e j a u d ) ( a b ) ( a + c ) ( e j b u d e j a u d )
h ( a , b , c ) : = ( a + b ) ( a + c ) e j a u d ( a b ) ( a c ) e j a u d
T 1 = 1 h ( η 1 , α 1 , β 1 ) [ f ( η 1 , η 0 , α 1 ) A 1 + f ( η 1 , η 0 , α 1 ) B 1 ]
R 1 = 1 h ( η 1 , α 1 , β 1 ) [ g ( η 1 , η 0 , β 1 ) A 1 + g ( η 1 , η 0 , β 1 ) B 1 ]
C 1 = 1 h ( η 1 , α 1 , β 1 ) { A 1 [ ( α 1 η 0 ) ( β 1 η 1 ) e j η 1 u d ( α 1 + η 1 ) ( β 1 + η 0 ) e j η 0 u d ] + B 1 [ ( α 1 + η 0 ) ( β 1 η 1 ) e j η 1 u d ( α 1 + η 1 ) ( β 1 η 0 ) e j η 0 u d ] }
D 1 = 1 h ( η 1 , α 1 , β 1 ) { A 1 [ ( α 1 η 0 ) ( β 1 + η 1 ) e j η 1 u d ( α 1 η 1 ) ( β 1 + η 0 ) e j η 0 u d ] + B 1 [ ( α 1 + η 0 ) ( β 1 + η 1 ) e j η 1 u d ( α 1 η 1 ) ( β 1 η 0 ) e j η 0 u d ] }
d 2 S 2 ( u ) d u 2 + [ 1 ( 4 sin θ B sin θ ) 2 ] S 2 ( u ) + ξ 2 S 1 ( u ) = 0
( S 2 ( u ) ) H = E 2 e j u η 2 + F 2 e j u η 2
( S 2 ( u ) ) I = C 2 e j u η 1 + D 2 e j u η 1 + A 2 e j u cos θ + B 2 e j u cos θ ,
x 2 = ξ 2 8 sin θ B ( 2 sin θ B sin θ ) , x 3 = ξ 2 4 sin θ B ( 3 sin θ B sin θ )
T 2 = 1 h ( η 2 , α 2 , β 2 ) [ f ( η 2 , η 0 , α 2 ) A 2 + f ( η 2 , η 0 , α 2 ) B 2 + f ( η 2 , η 1 , α 2 ) C 2 + f ( η 2 , η 1 , α 2 ) D 2 ]
R 2 = 1 h ( η 2 , α 2 , β 2 ) [ g ( η 2 , η 0 , α 2 ) A 2 + g ( η 2 , η 0 , α 2 ) B 2 + g ( η 2 , η 1 , α 2 ) C 2 + g ( η 2 , η 1 , α 2 ) D 2 ]
DER m = Re ( α m α 0 ) | R m | 2 DET m = Re ( β m α 0 ) | T m | 2
T 1 = x 1 ( η 1 + η 0 ) 2 η 1 ( e j η 1 u d e j η 0 u d ) = ξ 2 ( η 1 + cos θ ) ( e j η 1 u d e j η 0 u d ) 8 η 1 sin θ B ( sin θ B sin θ )
R 1 = x 1 η 1 η 0 2 η 1 ( 1 e j ( η 0 + η 1 ) u d ) = ξ 2 ( η 1 cos θ ) ( 1 e j ( η 0 + η 1 ) u d ) 8 η 1 sin θ B ( sin θ B sin θ )
T 1 = j ξ 2 u d 2 cos θ B e j u d cos θ B R 1 = j ξ 2 sin ( u d cos θ B ) 2 cos 2 θ B e j u d cos θ B
T 0 = ( 2 cos θ cos θ + β 0 ) e j u d cos θ
R 0 = ( cos θ β 0 cos θ + β 0 ) e 2 j u d cos θ ,
d 2 S 0 ( u ) d u 2 + cos 2 θ S 0 ( u ) = 0
d 2 S 1 ( u ) d u 2 + [ 1 ( 2 sin θ B + sin θ ) 2 ] S 1 ( u ) + ξ 1 S 0 ( u ) = 0 ,

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