Analysis of PT -symmetric volume gratings beyond the paraxial approximation

Mykola Kulishov; H. F. Jones; Bernard Kress

doi:10.1364/OE.23.009347

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 [1–5], whereas only a few papers [6–9] 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) = ε_{2} + Δ ε cos (K (x sin φ + z cos φ))

and modulation of gain and loss

σ (x, z) = Δ σ sin (K (x sin φ + z cos φ))

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

k^{2} (x, z) = k_{0}^{2} ε (x, z) - j ω μ σ (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:

k^{2} (x, z) = k_{2}^{2} + 2 k_{2} (κ^{-} exp (j \vec{K} . \vec{r}) + κ^{+} exp (- j \vec{K} . \vec{r})),

where

k_{2} = k_{0} {(ε_{2})}^{\frac{1}{2}}

is the average propagation constant and r⃗ is the coordinate vector. The coupling constants κ⁺ and κ⁻ are

κ^{\pm} = \frac{1}{4 {(ε_{2})}^{\frac{1}{2}}} (k_{0} Δ ε \pm c μ Δ σ)

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):

\begin{array}{l} E_{1} (x, z) & = & exp [- j k_{1} (x sin θ^{'} + z cos θ^{'})] + \\ + & \sum_{m = - \infty}^{\infty} 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})}^{\frac{1}{2}} z}] \end{array}

and for z > d (transmitted waves)

E_{3} (x, z) = \sum_{m = - \infty}^{\infty} 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})}^{\frac{1}{2}} (z - d)}]

The total electric field in the hologram region 0 < z < d is the superposition of multiple waves:

E_{2} (x, z) = \sum_{m = - \infty}^{\infty} S_{m} (z) exp [- j (k_{2} sin θ - m K sin φ) x],

where

k_{1} = k_{0} {(ε_{1})}^{\frac{1}{2}}

,

k_{3} = k_{0} {(ε_{3})}^{\frac{1}{2}}

, 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 R_m, and T_m are the amplitudes of the m-th reflected and transmitted waves and are to be determined. S_m(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)

\nabla^{2} E_{2} (x, z) + k_{0}^{2} ε (x, z) E_{2} (x, z) = 0

To find S_m(z), Eqs. (1) and (8) are substituted into Eq. (9), resulting in the system of coupled-wave equations [10, 11]:

\begin{array}{l} \frac{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 \end{array}

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 S_m−1 and S_m+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 θ_B, given by K = 2k₂ sin θ_B, only the zeroth-order and the first-order diffraction modes are coupled strongly to each other. Retaining only these two modes, Eqs. (10) become:

\begin{array}{r} \frac{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 \\ \frac{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 \end{array}

At the exact Bragg condition, when θ = θ_B, the coupled equations reduce to the following form in terms of the dimensionless coordinate u = k₂z:

\begin{array}{l} \frac{d^{2} S_{0} (u)}{d u^{2}} + {cos}^{2} θ_{B} S_{0} (u) + ξ_{1} S_{1} (u) & = & 0 \\ \frac{d^{2} S_{1} (u)}{d u^{2}} + {cos}^{2} θ_{B} S_{1} (u) + ξ_{2} S_{0} (u) & = & 0 \end{array}

where

ξ_{1} = 2 κ^{-} / k_{2} and ξ_{2} = 2 κ^{+} / k_{2}

The coupled Eqs. (12) can be decoupled by switching to

V_{0} = S_{0} + \sqrt{ξ_{1} / ξ_{2}} S_{1} and V_{1} = S_{0} - \sqrt{ξ_{1} / ξ_{2}} S_{1},

when the equations become

\begin{array}{l} \frac{d^{2} V_{0} (u)}{d u^{2}} + ρ_{1}^{2} V_{0} (u) & = & 0 \\ \frac{d^{2} V_{1} (u)}{d u^{2}} + ρ_{2}^{2} V_{1} (u) & = & 0 \end{array}

where

ρ_{1} = {({cos}^{2} θ_{B} + \sqrt{ξ_{1} ξ_{1}})}^{\frac{1}{2}} and ρ_{2} = {({cos}^{2} θ_{B} - \sqrt{ξ_{1} ξ_{1}})}^{\frac{1}{2}} .

Then S₀ and S₁ are given by

S_{0} (u) = \frac{1}{2} (V_{0} (u) + V_{1} (u))

and

S_{1} (u) = \frac{1}{2} \sqrt{ξ_{2} / ξ_{1}} (V_{0} (u) - V_{1} (u))

, where V₀(u) and V₁(u) have the solutions

\begin{array}{l} 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} \end{array}

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 H_x = (−j/(ωμ₀))∂E_y/∂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: $1 + R_{0} = S_{0} (0), R_{1} = S_{1} (0)$
b) tangential H at z = 0 : $k_{2} S_{0}^{'} (0) = j {(k_{1}^{2} - k_{2}^{2} {sin}^{2} θ)}^{\frac{1}{2}} (R_{0} - 1), k_{2} S_{1}^{'} (0) = j {[k_{1}^{2} - {(k_{2} sin θ - K)}^{2}]}^{\frac{1}{2}} R_{1}$
c) tangential E at z = d: $T_{0} = S_{0} (d), T_{1} = S_{1} (d)$
c) tangential H at z = d: $k_{2} S_{0}^{'} (d) = - j {(k_{3}^{2} - k_{2}^{2} {sin}^{2} θ)}^{\frac{1}{2}} T_{0}, k_{2} S_{1}^{'} (d) = - j {[k_{1}^{2} - {(k_{2} sin θ - K)}^{2}]}^{\frac{1}{2}} T_{1}$

3. Two-mode solution for θ = θ_B

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.

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}})

where ξ = √(ξ₁/ξ₂), α_B = √(ε₁/ε₂ − sin² θ_B), β_B = √(ε₃/ε₂ − sin²θ_B) and u_d = k₂d.

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

R_{0} = \frac{1}{2} (F (ρ_{1}) + F (ρ_{2})), R_{1} = \frac{1}{2 ξ} (F (ρ_{1}) - F (ρ_{2}))

where, for m = 1, 2,

F (ρ_{m}) = \frac{(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}}}

The transmission coefficients are expressed in terms of G(ρ₁) and G(ρ₂) as

T_{0} = \frac{1}{2} (G (ρ_{1}) + G (ρ_{2})), T_{1} = \frac{1}{2 ξ} (G (ρ_{1}) - G (ρ_{2})),

where

G (ρ_{m}) = \frac{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}}}

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 $ξ = \sqrt{ξ_{1} / ξ_{2}}$ , 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.

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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 T₁ ∝ ξ₂u_d when θ = θ_B, 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 = θ_B 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

\frac{d^{2} S_{0} (u)}{d u^{2}} + {cos}^{2} θ S_{0} (u) = 0

\frac{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

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

S_{0} (u) = A_{0} e^{j u cos θ} + B_{0} e^{- j u cos θ}

Applying the boundary conditions (18)–(21) we can find T₀ and R₀ and the constants A₀ and B₀:

T_{0} = \frac{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} = \frac{(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} = \frac{T_{0}}{2} (\frac{cos θ - β_{0}}{cos θ}) e^{- j u_{d} cos θ} B_{0} = \frac{T_{0}}{2} (\frac{cos θ + β_{0}}{cos θ}) e^{j u_{d} cos θ}

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

{(S_{1} (u))}_{H} = C_{1} e^{j η_{1} u} + D_{1} e^{- j η_{1} u}

where η₁ = √[1 − (2 sin θ_B − 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

{(S_{1} (u))}_{I} = A_{1} e^{j u cos θ} + B_{1} e^{- j u cos θ}

and find that A₁ = x₁A₀ and B₁ = x₁B₀, where

x_{1} = \frac{ξ_{2}}{4 sin θ_{B} (sin θ_{B} - sin θ)}

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) (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}}

In this notation, and with the definitions η₀ = cos θ, α_m = √[ε₁/ε₂ − (2msin θ_B − sin θ)²] and β_n = √[ε₃/ε₂ − (2msin θ_B − sin θ)²],

T_{1} = \frac{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} = \frac{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}]

The coefficients C₁ and D₁ are given by

\begin{array}{l} C_{1} = \frac{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}}]} \end{array}

and

\begin{array}{l} D_{1} = \frac{- 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}}]} \end{array}

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:

\frac{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

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

{(S_{2} (u))}_{H} = E_{2} e^{j u η_{2}} + F_{2} e^{- j u η_{2}}

where η₂ = √[1 − (4sin θ_B − 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

{(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},

we find C₂ = x₃C₁, D₃ = x₂D₁, A₂ = x₂A₁ and B₂ = x₂B₁, where

x_{2} = \frac{ξ_{2}}{8 sin θ_{B} (2 sin θ_{B} - sin θ)}, x_{3} = \frac{ξ_{2}}{4 sin θ_{B} (3 sin θ_{B} - sin θ)}

Applying the boundary conditions at u = 0 and u = u_d 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:

T_{2} = \frac{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} = \frac{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}]

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

{DER}_{m} = Re (\frac{α_{m}}{α_{0}}) {| R_{m} |}^{2} {DET}_{m} = Re (\frac{β_{m}}{α_{0}}) {| T_{m} |}^{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 = e^{−ju_d cos_θ}. 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

T_{1} = x_{1} \frac{(η_{1} + η_{0})}{2 η_{1}} (e^{- j η_{1} u_{d}} - e^{- j η_{0} u_{d}}) = ξ_{2} \frac{(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} \frac{η_{1} - η_{0}}{2 η_{1}} (1 - e^{- j (η_{0} + η_{1}) u_{d}}) = ξ_{2} \frac{(η_{1} - cos θ) (1 - e^{- j (η_{0} + η_{1}) u_{d}})}{8 η_{1} sin θ_{B} (sin θ_{B} - sin θ)}

These peak at the Bragg angle, where their values are

\begin{array}{l} T_{1} & = & - j \frac{ξ_{2} u_{d}}{2 cos θ_{B}} e^{- j u_{d} cos θ_{B}} \\ R_{1} & = & - j ξ_{2} \frac{sin (u_{d} cos θ_{B})}{2 {cos}^{2} θ_{B}} e^{- j u_{d} cos θ_{B}} \end{array}

The first-order diffraction amplitude T₁ grows linearly with the grating strength ξ₂u_d, 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 ( $arcsin (\sqrt{ε_{1} / ε_{2}}) = 40.2 °$ ), 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, ε₁ ≠ ε₃. 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

T_{0} = (\frac{2 cos θ}{cos θ + β_{0}}) e^{- j u_{d} cos θ}

R_{0} = (\frac{cos θ - β_{0}}{cos θ + β_{0}}) e^{- 2 j u_{d} cos θ},

which are basically the Fresnel coefficients. The transmission rapidly decreases to zero for |θ| > θ_TIR ≡ 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%), (

R_{0} = (\sqrt{ε_{2}} - \sqrt{ε_{3}}) / (\sqrt{ε_{2}} + \sqrt{ε_{3}})

), for normal incidence and then rapidly increases to 100% for |θ| > θ_TIR. Introduction of the second reflective interface with ε₁ ≠ ε₂, (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 ε₂ ≠ ε₃ 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 = θ_B, the first two modes excited are S₀ and S₁, while near θ = −θ_B, 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 −θ_B, 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 θ_B; (b) from the left near −θ_B; (c) from the right near θ_B; (d) from the right near −θ_B.

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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 = Δn₀e^2πjz/Λ, 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/Λ 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

\frac{d^{2} S_{0} (u)}{d u^{2}} + {cos}^{2} θ S_{0} (u) = 0

\frac{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,

giving a strong excitation of S₋₁ near θ = −θ_B. 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]

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Analysis of PT -symmetric volume gratings beyond the paraxial approximation

Abstract

1. Introduction

2. Second-order coupled-mode equations

3. Two-mode solution for θ = θ_B

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

5. Filled-space PT -symmetric grating

6. Symmetric slab configuration

7. Asymmetric slab configurations

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

7.2. Light incident from the air: ε₁ = 1

7.3. Reflective set-up

8. Discussion

References and links

Cited By

Figures (10)

Equations (63)

Optics Express

Abstract

1. Introduction

2. Second-order coupled-mode equations

3. Two-mode solution for θ = θB

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

5. Filled-space PT -symmetric grating

6. Symmetric slab configuration

7. Asymmetric slab configurations

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

7.2. Light incident from the air: ε1 = 1

7.3. Reflective set-up

8. Discussion

References and links

Cited By

Figures (10)

Equations (63)

Optics Express

3. Two-mode solution for θ = θ_B

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

7.2. Light incident from the air: ε₁ = 1