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
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  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.
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 permittivityEq. (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
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):Eqs. (1) and (8) are substituted into Eq. (9), resulting in the system of coupled-wave equations [10, 11]: 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:Eqs. (12) can be decoupled by switching to
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
3. Two-mode solution for θ = θB
Solving these equations, we find the following expressions for the zeroth- and first-order reflection coefficients: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.
Unlike the solution for the PT -symmetric grating obtained through the first-order coupled wave equations , which provides only the transmission coefficients, with |T0| = 1 and T1 ∝ ξ2ud 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 , 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.
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) areEq. (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 (18)–(21) we can find T0 and R0 and the constants A0 and B0:
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(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
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: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
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
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−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.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%. %.
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.
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 ( ), 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.
7.1. Light incident from the substrate side: ε3 = 1
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 thatFig. 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.
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).
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.
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/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
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]
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]