## 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: Δ*ñ* = Δ*n*_{0}exp(2*πjz*/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(2*πjz*/Λ) = cos(2*πz*/Λ) + *j* sin(2*πz*/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.

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, *S*_{0}, *S*_{1} and *S*_{2}. 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

*z*= 0 to

*z*=

*d*with the same spatial frequency shifted by a quarter of period Λ/4 (

*K*= 2

*π*/Λ) with respect to one another, where

*ε*

_{2}is the average relative permittivity in the grating area, Δ

*ε*is the amplitude of the sinusoidal relative permittivity, Δ

*σ*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 where

*μ*is the permeability of the medium,

*ω*is the angular frequency of the wave and

*k*

_{0}=

*ω/c*is the wave-vector in free space, related to the free-space wavelength

*λ*

_{0}by

*k*

_{0}= 2

*π/λ*

_{0}.

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

*r⃗*is the coordinate vector. The coupling constants

*κ*

^{+}and

*κ*

^{−}are

In the two unmodulated regions, *z* < 0 and *z* > *d*, where we assume uniform permittivity *ε*_{1} and *ε*_{3}, respectively, the assumed solutions of the wave equation for the normalized electric fields are, for *z* < 0 (incident and reflected waves):

*z*>

*d*(transmitted waves)

*z*<

*d*is the superposition of multiple waves:

*θ′*is the angle of incidence in Region 1, and

*θ*is the angle of refraction in Region 2, related to each other by

*k*

_{1}sin

*θ′*=

*k*

_{2}sin

*θ*. In these equations

*R*, and

_{m}*T*are the amplitudes of the

_{m}*m*-th reflected and transmitted waves and are to be determined.

*S*(z) is the amplitude of the

_{m}*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) To find

*S*(

_{m}*z*), Eqs. (1) and (8) are substituted into Eq. (9), resulting in the system of coupled-wave equations [10, 11]:

*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*= 2

*k*

_{2}sin

*θ*, only the zeroth-order and the first-order diffraction modes are coupled strongly to each other. Retaining only these two modes, Eqs. (10) become:

_{B}*θ*=

*θ*, the coupled equations reduce to the following form in terms of the dimensionless coordinate

_{B}*u*=

*k*

_{2}

*z*:

*S*

_{0}and

*S*

_{1}are given by ${S}_{0}(u)=\frac{1}{2}\left({V}_{0}(u)+{V}_{1}(u)\right)$ and ${S}_{1}(u)=\frac{1}{2}\sqrt{{\xi}_{2}/{\xi}_{1}}\left({V}_{0}(u)-{V}_{1}(u)\right)$, where

*V*

_{0}(

*u*) and

*V*

_{1}(

*u*) have the solutions

*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*/(

*ωμ*

_{0}))

*∂E*.

_{y}/∂zIn the approximation of keeping only the two modes *S*_{0}(*u*) and *S*_{1}(*u*) the four quantities to be matched and the resulting boundary conditions are

- b) tangential
*H*at*z*= 0 :$${k}_{2}{S}_{0}^{\prime}(0)=j{\left({k}_{1}^{2}-{k}_{2}^{2}{\text{sin}}^{2}\theta \right)}^{\frac{1}{2}}({R}_{0}-1),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{k}_{2}{S}_{1}^{\prime}(0)=j{\left[{k}_{1}^{2}-{\left({k}_{2}\text{sin}\theta -K\right)}^{2}\right]}^{\frac{1}{2}}{R}_{1}$$ - c) tangential
*H*at*z*=*d*:$${k}_{2}{S}_{0}^{\prime}(d)=-j{\left({k}_{3}^{2}-{k}_{2}^{2}{\text{sin}}^{2}\theta \right)}^{\frac{1}{2}}{T}_{0},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{k}_{2}{S}_{1}^{\prime}(d)=-j{\left[{k}_{1}^{2}-{\left({k}_{2}\text{sin}\theta -K\right)}^{2}\right]}^{\frac{1}{2}}{T}_{1}$$

## 3. Two-mode solution for *θ* = *θ*_{B}

_{B}

Taking *S*_{0}(*u*) and *S*_{1}(*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*, *R*_{0}, *R*_{1}, *T*_{0} and *T*_{1}.

*ξ*= √(

*ξ*

_{1}/

*ξ*

_{2}),

*α*= √(

_{B}*ε*

_{1}/

*ε*

_{2}− sin

^{2}

*θ*),

_{B}*β*= √(

_{B}*ε*

_{3}/

*ε*

_{2}− sin

^{2}

*θ*) and

_{B}*u*=

_{d}*k*

_{2}

*d*.

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

*m*= 1, 2,

*G*(

*ρ*

_{1}) and

*G*(

*ρ*

_{2}) as

*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

*ε*

_{2}= 2.4 in air,

*ε*

_{1}=

*ε*

_{3}= 1.

Unlike the solution for the *PT* -symmetric grating obtained through the first-order coupled wave equations [8], which provides only the transmission coefficients, with |*T*_{0}| = 1 and *T*_{1} ∝ *ξ*_{2}*u _{d}* when

*θ*=

*θ*, our solution shows significant intensities in the zeroth and first reflective orders. In fact, power redistribution results in a normalized power reduction in |

_{B}*T*

_{0}|

^{2}from 1 to 0.83, with |

*R*

_{0}|

^{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 *ε*_{1} = *ε*_{2} = *ε*_{3}. 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 *T*_{1} comes from the active grating, not at the expense of the zeroth-order diffraction, which still satisfies |*R*_{0}|^{2} + |*T*_{0}|^{2} = 1.

## 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

*ξ*

_{1},

*ξ*

_{2}, 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 (

*ξ*

_{1}= −

*ξ*

_{2}).

In this section we extend our analysis of the balanced *PT* -symmetric grating (*ξ*_{1} =0), but with arbitrary angle of incidence. In that case the coupled wave Eqs. (11) are

*S*

_{0}(

*u*) (non-diffracted light) is decoupled from the second equation for the first-order amplitude

*S*

_{1}(

*u*). We therefore have a solution for

*S*

_{0}(

*u*) of the form Applying the boundary conditions (18)–(21) we can find

*T*

_{0}and

*R*

_{0}and the constants

*A*

_{0}and

*B*

_{0}:

*α*

_{0}= √(

*ε*

_{1}/

*ε*

_{2}− sin

^{2}

*θ*) and

*β*

_{0}= √(

*ε*

_{3}/

*ε*

_{2}− sin

^{2}

*θ*).

Equation (27b) is an inhomogeneous second-order differential equation for *S*_{1}(*u*), whose solutions can be found as a sum of the general solution of the homogenous equation, (*S*_{1})* _{H}* and a particular solution (

*S*

_{1})

*of the inhomogeneous equation. The solution of the homogeneous equation is*

_{I}*η*

_{1}= √[1 − (2 sin

*θ*− sin

_{B}*θ*)

^{2}] and

*C*

_{1}and

*D*

_{1}are constants to be determined. The particular solution can be found using the method of undetermined coefficients. We write and find that

*A*

_{1}=

*x*

_{1}

*A*

_{0}and

*B*

_{1}=

*x*

_{1}

*B*

_{0}, where

*T*

_{1},

*R*

_{1},

*C*

_{1}and

*D*

_{1}. The rather lengthy expressions thus obtained for

*T*

_{1}and

*R*

_{1}can be expressed in a condensed form using the functions

*η*

_{0}= cos

*θ*,

*α*= √[

_{m}*ε*

_{1}/

*ε*

_{2}− (2

*m*sin

*θ*− sin

_{B}*θ*)

^{2}] and

*β*= √[

_{n}*ε*

_{3}/

*ε*

_{2}− (2

*m*sin

*θ*− sin

_{B}*θ*)

^{2}],

*C*

_{1}and

*D*

_{1}are given by

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:

*η*

_{2}= √[1 − (4sin

*θ*− sin

_{B}*θ*)

^{2}] and

*E*

_{2}and

*F*

_{2}are constants to be determined. The particular solution of the differential equation can again be found using the method of undetermined coefficients. Writing

*C*

_{2}=

*x*

_{3}

*C*

_{1},

*D*

_{3}=

*x*

_{2}

*D*

_{1},

*A*

_{2}=

*x*

_{2}

*A*

_{1}and

*B*

_{2}=

*x*

_{2}

*B*

_{1}, where

*u*= 0 and

*u*=

*u*we can find

_{d}*T*

_{2},

*R*

_{2},

*E*

_{2}and

*F*

_{2}. With the help of the functions defined in Section 4, we can express

*T*

_{2}and

*R*

_{2}in a rather compact form:

*E*

_{2}and

*F*

_{2}, 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 *i*th 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

## 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: *ε*_{1} = *ε*_{2} = *ε*_{3}. This configuration should provide a solution that is very close to that of the first-order coupled wave equations.

Indeed, when *ε*_{1} = *ε*_{2} = *ε*_{3}, then *α*_{0} = *β*_{0} = cos *θ*, so that *A*_{0} = 0 and *B*_{0} = 1, *R*_{0} = 0 and *T*_{0} = *e*^{−jud 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 *ε*_{1} = *ε*_{2} = *ε*_{3} the expressions for *T*_{1} and *R*_{1} in Eqs. (36) and (37) simplify to

*T*

_{1}grows linearly with the grating strength

*ξ*

_{2}

*u*, with amplification close to 800% for the parameters chosen in Fig. 4. This linear growth in amplitude is a characteristic of

_{d}*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.

*R*

_{1}is not zero, but is small even at the resonance, with a diffraction efficiency of less than 0.1%. %.

## 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 (*ε*_{1} = *ε*_{2} = *ε*_{3} = 2.4) and the real configuration of the slab in air (*ε*_{1} = *ε*_{3} = 1, *ε*_{2} = 2.4). The results are presented in Fig. 5.

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 (
$\text{arcsin}\left(\sqrt{{\epsilon}_{1}/{\epsilon}_{2}}\right)=40.2\xb0$), 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, *ε*_{1} ≠ *ε*_{3}. 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.

#### 7.1. Light incident from the substrate side: ε_{3} = 1

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

We consider the cases when the permittivity of the substrate and the average permittivity of the slab are the same, *ε*_{1} = *ε*_{2}, (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 *ε*_{1} = *ε*_{2}, when *α*_{0} = cos *θ*, so that

*θ*| >

*θ*≡ arcsin(√(

_{TIR}*ε*

_{3}/

*ε*

_{2})), the angle at which total internal reflection occurs at the second surface. Reflection in zeroth order is close to zero (4.6%), ( ${R}_{0}=\left(\sqrt{{\epsilon}_{2}}-\sqrt{{\epsilon}_{3}}\right)/\left(\sqrt{{\epsilon}_{2}}+\sqrt{{\epsilon}_{3}}\right)$), for normal incidence and then rapidly increases to 100% for |

*θ*| >

*θ*. Introduction of the second reflective interface with

_{TIR}*ε*

_{1}≠

*ε*

_{2}, (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: ε_{1} = 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 *ε*_{2} ≠ *ε*_{3} 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.

#### 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 *λ*_{0}=0.633 *μ*m, namely *ε*_{3} = −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).

## 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

*S*

_{0}and

*S*

_{1}, while near

*θ*= −

*θ*, the first two modes excited are

_{B}*S*

_{0}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.

The *PT* grating is also left-right asymmetric, even when *ε*_{1} = *ε*_{3}. 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 Δ*ñ* = Δ*n*_{0}*e*^{2πjz/Λ}, resulting in *ξ*_{1} = 0 and the consequences explored in the paper. However, if the phase of the gain/loss modulation relative to the index modulation is reversed, Δ*ñ* instead becomes Δ*ñ* = Δ*n*_{0}*e*^{−2πjz/Λ} and the roles of *ξ*_{1} and *ξ*_{2} are interchanged, so that now *ξ*_{2} = 0. In that case the first mode to be excited is *m* = −1, and the coupled equations for *S*_{0} and *S*_{−1} become

*S*

_{−1}near

*θ*= −

*θ*. Thus the symmetry

_{B}*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 *T*_{0} 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 |*T*_{0}|^{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 *T*_{1} 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 *ε*_{1}, *ε*_{2}, *ε*_{3}, 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 *R*_{1}.

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.

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