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Abstract
The boundary problem of dynamical Bragg diffraction of a chirped optical pulse in a dispersive quasi-PT-symmetric photonic crystal (PhC) in the Laue geometry (“on transmission”) is solved by the analytical spectral method. It is shown that, in a quasi-PT-symmetric medium, in which an inhomogeneous spectral line width is much larger than the spectrum of investigated field, the exceptional-point (EP) condition is realized in a wide continuous frequency range, i.e., so-called broadband exceptional-point (BEP) condition takes place. If the Bragg condition is satisfied in a much narrower spectral range than the pulse spectrum, it leads to dramatic changes in the propagation dynamics and parameters of broadband chirped pulses in a quasi-PT-symmetric PhC. Indeed, for a positive Bragg angle of incidence in the case of diffraction in the Laue geometry, the entire spectrum of a broadband chirped pulse fulfills the BEP condition. The diffractionally reflected wave is absent in the BEP regardless of whether the Bragg condition is satisfied, and the pulse propagates as in a homogeneous conservative medium, i.e., without diffraction, gain and loss – unidirectional invisibility. When the sign of the angle of incidence changes, a unidirectional enhancement of the chirped diffracted pulse is observed in that part of it whose frequency simultaneously satisfies both the BEP condition and the Bragg condition. The rest part of the pulse, for which the Bragg condition is not satisfied, propagates as in the case of a positive angle of incidence - there is no diffracted wave. With a smooth change in the angle of incidence of the chirped pulse, a change in frequency that satisfies the Bragg condition occurs and, as a consequence, a smooth change appears in the frequency of the amplified output pulse, as well as in its duration and transverse size. It is also shown that the dispersion of the group velocity of the pulse is suppressed in the frequency range of the BEP condition. Therefore, all its frequency components propagate at a speed close to the speed of light in a conservative homogeneous medium.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
In the last decade, the concept of parity-time (PT) symmetry, introduced for the first time in quantum mechanics for non-Hermitian systems [1–3], is actively used in optics of non-conservative media, in which the optical properties are characterized by PT symmetry [4–7]. In such media, the spatially distributed complex dielectric permittivity is a PT-symmetric function of the coordinate, $\varepsilon (x) = {\varepsilon ^ \ast }( - x)$, where the real part $\textrm{Re} \varepsilon (x)$ is an even function, and the imaginary part ${\mathop{\textrm{Im}}\nolimits} \varepsilon (x)$ describing the gain and loss of the medium is an odd function. It has been shown theoretically and experimentally that in PT-symmetric media with gain and loss, stationary waves with real wave vectors and constant amplitudes - PT-symmetric modes [8–12] - can propagate, and there is also an exceptional point (EP) of the value of the gain-loss parameter, at which the PT-symmetric mode is broken and a transition to the PT-asymmetric solution occurs [13–14]. Under the condition of spectral singularity [15–16], a laser effect and coherent perfect absorption [17–20] were observed. Of particular interest are periodic media, or PT-symmetric PhCs, in which new PT-symmetric effects appear, associated with the translational symmetry of the structure. First of all, these are Bloch oscillations [21–22] and the effect of unidirectional Bragg reflection, or unidirectional invisibility, at the EP of spontaneous breaking of the PT-symmetric mode [23–28].
Recently it was shown [29–31] that PT-symmetric optical effects can be observed not only for monochromatic radiation or in the approximation of a dispersionless medium, but also in the case of propagation of short pulses in dispersive structure, if we use the so-called quasi-PT symmetry condition. In a quasi-PT-symmetric medium, where the width of the inhomogeneously broadened spectral line of resonant medium is much greater than the width of the incident pulse spectrum, it is possible to restore the PT-symmetric properties of a dispersive medium for broadband radiation. The term “quasi-PT symmetry” means that light-matter interaction in such medium is not exactly PT-symmetric for each spectral field component, but the average dispersion of the complex dielectric permittivity is significantly suppressed. In quasi-PT-symmetric PhCs, a number of optical effects have been described for short laser pulses spatially localized in the medium at a certain time: unidirectional reflection of short pulses in the Bragg scheme [29,30], as well as unidirectional Bragg-induced pulse splitting in Laue geometry [31], however, the propagation of a chirped pulse in the Laue diffraction scheme has not yet been studied.
In this work, we consider the frequency dependences of the dynamics and parameters of chirped pulses in a dispersive quasi-PT-symmetric PhC under dynamical diffraction in the Laue geometry (“on transmission”). It is shown that if in a quasi-PT-symmetric medium the EP condition is satisfied for the central frequency of the pulse, then it is also valid with high accuracy and in a wide frequency range of the chirped pulse spectrum - BEP condition. At the same time, the condition of selective Bragg reflection in a PhC is usually satisfied for a narrow frequency range. Therefore, the dynamics of different parts of a chirped pulse propagating in a quasi-PT-symmetric PhC will differ significantly not only depending on the sign of the angle of incidence of radiation on the structure, but also on the frequency of a particular spatial region of the pulse. At a positive angle of incidence, there exist an effect of unidirectional invisibility, when a broadband pulse spatially localized in a dispersive quasi-PT-symmetric PhC will propagate under the BEP condition as in a conservative homogeneous medium without diffraction scattering, gain, and loss. When the sign of the angle of incidence changes, the emission of an amplified diffracted pulse is observed in that part of the spectrum that satisfies the Bragg condition - a unidirectional enhancement for the chirped pulse. The frequency and transverse size of this enhanced pulse change smoothly with a change in the angle of incidence of radiation on the structure. Under the BEP condition, there is a weak dispersion of the group velocity; therefore, all spatial regions of the chirped pulse propagate at close velocities, and the pulse shape does not undergo significant changes.
The paper is organized as follows. In Sec. 2, we estimate the frequency band within which BEP condition is satisfied in a quasi-PT-symmetric medium. The chirped pulse dynamical Bragg diffraction in the vicinity of a BEP condition is described shortly in Sec. 3. In Sec. 4, we show unidirectional invisibility and enhanced reflection of a short chirped pulse. The results are summarized in the Conclusion.
2. Broadband exceptional-point condition under quasi-PT-symmetry
In the vicinity of EP, which is determined by the ratio of the real and imaginary parts of the dielectric permittivity [see Eqs. (1), (2) and (8)], the PT-symmetric mode is broken and one of the most interesting optical effect for PT-symmetric periodic structure - unidirectional enhanced Bragg reflection - is observed [23–31]. We will show in this Section that the frequency band within which the BEP condition is satisfied in a quasi-PT-symmetric medium can significantly exceed the spectrum width of a short chirped pulse, making the structure suitable for exploring chirped pulse dynamics.
Let a short chirped Gaussian optical pulse with a central frequency ${\omega _0}$ and with a dimensionless spectral width $\delta \omega = (2/\tau )\sqrt {1 + {\xi ^2}} /{\omega _0}$, where $\tau$ is the pulse duration and $\xi$ is the chirp parameter, be incident onto a one-dimensional dispersive quasi-PT-symmetric PhC under Laue scheme of Bragg diffraction [Fig. 1(a)]. In this scheme, “on transmission”, the reciprocal lattice vector $\textbf{h}$ is oriented along the PhC surface and the forward and diffracted waves propagate in the PhC without diffraction Bragg reflection on the surface. Dielectric permittivity of the structure is
Fig. 1. (a) Schematic illustration of the Laue geometry of Bragg diffraction. $\theta > 0$ and $\theta < 0$ are two cases of pulse incidence onto the PhC, ${E_0}$ and ${E_h}$ are the fields of forward and diffracted waves. (b) Frequency dependence of $|{\hat{\varepsilon }_I}(\Omega )|$, where $\Omega = \omega - {\omega _0}$, at $\gamma _2^\ast{=} 0.005$ (line 1) and $\gamma _2^\ast{=} 0.15$ (line 2). Gaussian pulse spectrum under $\tau = 1\;\textrm{ps}$, central wavelength ${\lambda _0} = 0.8\;\mathrm{\mu}\textrm{m}$ and $\xi = 20$ (line 3, top axis). Frequency dependence of ${\hat{\varepsilon }_R}(\Omega )$ at $\gamma _2^\ast{=} 0.005$ (line 4) and $\gamma _2^\ast{=} 0.15$ (line 5). The function of the spectral line $g({\omega _0} - {\omega ^{\prime}_0})$ (line 6). Graphs 3 and 6 are normalized to the maximum. (c) Function of deviation from the exceptional point $[\varepsilon ^{\prime} - {\hat{\varepsilon }_R}(\Omega )]/\varepsilon ^{\prime}$ vs the wave frequency and $\gamma _2^\ast $. Parameters: $\varepsilon ^{\prime} = 0.005$, $\hat{\varepsilon }({\omega _0}) = 0.004999$, center of inhomogeneous broadening ${\lambda _{0g}} = 0.8\;\mathrm{\mu}\textrm{m}$, ${\gamma _2} = 2/T_2^{}{\omega _0} = 0.005$, $\gamma _2^\ast{=} 0.15$.
From expressions (1) and (2), it can be seen that due to dispersion, an odd component, $- {\hat{\varepsilon }_I}(\omega )\sin (hx)$, appears in the real part of the permittivity function. This leads to violation of the PT-symmetry condition, which requires the function $\textrm{Re} \varepsilon (x,\omega )$ to be even with respect to the spatial variable. The use of the condition of quasi-PT symmetry, when the inhomogeneous width of the spectral line $\gamma _2^ \ast $ significantly exceeds the spectral width of the pulse, $\gamma _2^\ast > > \delta \omega$, makes it possible to reduce the PT-asymmetric part of the dielectric constant, $|{\hat{\varepsilon }_I}(\omega )|\sim \delta \omega /\gamma _2^\ast < < 1$, and thereby restore the PT symmetric properties of the medium in a wide continuous spectral interval [29–31].
In Fig. 1(b), it is seen that with an increase in the value $\gamma _2^\ast $ from 0.005 to 0.15, the PT-asymmetric part of the permittivity, ${\hat{\varepsilon }_I}(\omega )$, decreases by almost 30 times (lines 1 and 2) in the region of the spectral width of the chirped pulse (line 3). In addition, this figure clearly shows how, upon passing to the quasi-PT-symmetric case with an increasing of $\gamma _2^\ast $, the PT-symmetric component of the permittivity ${\hat{\varepsilon }_R}(\omega )$, which sharply changed with the frequency at $\gamma _2^\ast{=} 0.005$ (line 4), becomes a constant value at $\gamma _2^\ast{=} 0.15$ (line 5) in a wide frequency region of the chirped pulse (line 3). This means that if the EP condition is satisfied in a quasi-PT-symmetric medium, for example, for the central frequency ${\omega _0}$ of the pulse, $\varepsilon ^{\prime} - {\hat{\varepsilon }_R}({\omega _0}) = 0$ [see Eqs. (8) and (11) below], then it is also fulfilled with high accuracy and for the entire spectrum of the pulse - BEP condition. From a comparison of the plots of the inhomogeneously broadened spectral line of the medium $g({\omega _0} - {\omega ^{\prime}_0})$ (line 6) and the spectrum of the pulse (line 3), it can be seen that the quasi-PT symmetry condition $\delta \omega /\gamma _2^\ast < < 1$ is well satisfied for the selected parameters. As shown in Fig. 1(c), the spectral region, where the BEP condition exists, grows is linearly with increasing inhomogeneous width $\gamma _2^\ast $. Hence it follows that for a broadband chirped pulse the BEP condition can be satisfied for the entire spectrum of the pulse, while the Bragg condition is satisfied only for a certain narrow frequency range (see Fig. 2 below). Therefore, it can be expected that enhanced unidirectional Bragg reflection will be observed only for that part of the chirped pulse that simultaneously satisfies these two conditions. For the other part of the pulse, for which only the BEP condition is fulfilled and the Bragg condition is not satisfied, unidirectional Bragg reflection will not be observed when the sign of the angle of incidence changes.
3. Chirped pulse dynamical Bragg diffraction in the vicinity of a broadband exceptional-point condition
Here we describe shortly the solution of dynamical diffraction problem for a chirped pulse in a quasi-PT-symmetric PhC in the Laue geometry [Fig. 1(a)]. Let an s-polarized chirped pulse
To solve the boundary problem of dynamical Bragg diffraction of a pulse in a linear dispersive medium, we will use the analytical spectral method [30,33,34]. In this method, the field of the incident pulse Eqs. (3) and (4) will be represented as a two-dimensional Fourier integral (in space and in time). Next, we solve the Bragg diffraction problem for a single spectral component, i.e. a plane monochromatic wave, and carry out Fourier synthesis to find the pulse complex field $E(x,z,t)$ at each point of the medium at any time instant.
The spectral component of the field in a dispersive medium satisfies the Helmholtz equation
where $E(x,z,\omega ) = (1/2\pi )\int_{ - \infty }^\infty {E(x,z,t)\exp (i\omega t)dt}$, $k = \omega /c$, $\varepsilon (x,\omega ) = 1 + 4\pi \int_0^\infty {\chi (x,\tau ^{\prime})\exp (i\omega \tau ^{\prime})d\tau ^{\prime}}$, $\chi (x,\tau ^{\prime})$ is the real function of dielectric susceptibility, $\tau ^{\prime}$ is the delay time.In a periodic medium near the Bragg condition $2{k_0}\sin {\theta _B} = sh$, where ${\theta _B}$ is the Bragg angle, the two-wave approximation of Bragg diffraction theory is fulfilled [34], then the field in the medium can be represented in the form of two strongly coupled waves - transmitted, ${E_0}(x,z,t)$, and diffractionally reflected, ${E_h}(x,z,t)$, waves:
whereHere $g = 0,\;h$; $s = 1$ if $\theta > 0$ and $s ={-} 1$ if $\theta < 0$ [Fig. 1(a)], $K = {k_x} - {k_{0x}}$, $\Omega = \omega - {\omega _0}$, ${A_0}(K,\Omega )$ and ${A_h}(K,\Omega )$ are the amplitudes of the spectral components of transmitted and diffracted waves, respectively. Due to the conservation of the tangential components of the wave vectors on the boundary $z = 0$, the x-projection of the wave vectors of the transmitted waves within the medium is ${q_{0x}}(K) = {k_x} = {k_{0x}} + K$.
From the requirement for the existence of nontrivial solutions for the field Eq. (6), after its substitution in Eq. (5), we obtain the following dispersion relations for z-projections of the wave vectors of the transmitted and diffracted waves of two eigenmodes, called the Borrmann, $q_{0z}^{(1)}$, and anti-Borrmann, $q_{0z}^{(2)}$, modes:
From the conditions of continuity of the x-projections of the vectors of the electric and magnetic fields at the boundary $z = 0$, it is easy to obtain the amplitudes of the fields of direct and diffracted waves of the Borrmann $(j = 1)$ and anti- Borrmann $(j = 2)$ modes:
Here
From expressions (10), a simple estimate of the field amplitudes ${A_{gj}}$ depending on the sign of the angle of incidence $\theta$ at weak Fresnel reflection, ${r_F} < < 1$, at the boundary $z = 0$ can be obtained. The amplitudes of the diffracted waves are written in the form
Since at $\theta > 0$ the value $s = 1$, then in EP, for example, at the central frequency $\omega = {\omega _0}$, the magnitude ${\varepsilon _s} = {\varepsilon _1} \equiv [\varepsilon ^{\prime} - {\hat{\varepsilon }_R}({\omega _0})]/2 = 0$ and from Eq. (11) it can be seen that there are no diffracted waves, $A_{h2}^ +{=} - A_{h1}^ +{=} 0$. The index “$+$” corresponds to the case $\theta > 0$. In the case of a negative angle of incidence $\theta < 0$ (index “$-$”), $s ={-} 1$, and in EP the value ${\varepsilon _s} = {\varepsilon _{ - 1}} \equiv [\varepsilon ^{\prime} + {\hat{\varepsilon }_R}({\omega _0})]/2 \ne 0$, ${\varepsilon _{ - s}} = {\varepsilon _1} = 0$, therefore, the amplitudes $A_{h2}^ -{=} - A_{h1}^ -{=} ({\varepsilon _{ - 1}}/2|{{\alpha_s}} |){A_{in}}$ (11) increase near the Bragg condition ${\alpha _s} \to 0$. The physical reason of this asymmetry of the optical response is explained by asymmetry of the alternation of combinations of maxima and minima of the real and imaginary parts of the permittivity function when the sign of the angle of incidence changes.
Amplitudes of transmitted waves ${A_{0j}}$ Eq. (10) in EP do not depend on the magnitude of the deviation from the Bragg condition $|{{\alpha_s}} |$:
In Section 2, it was shown that in the quasi-PT-symmetric case, the following condition is satisfied for the real part of the permittivity in a wide frequency region of the chirped pulse spectrum: ${\hat{\varepsilon }_R}(\omega ) \approx {\hat{\varepsilon }_R}({\omega _0})$ [Fig. 1(b)]. Therefore, the above results for the amplitudes of transmitted and diffracted waves remain valid and for other frequencies. In other words, the EP condition can be satisfied for the entire spectrum of the pulse, which allows us to speak about the phenomenon of a BEP condition. However, for the simultaneous fulfillment of the Bragg condition ${\alpha _s} = 0$ at some given frequency ${\omega ^{\prime}_B} \ne {\omega _B}$, the angle of incidence should be changed by the value $\Delta \theta$ determined by the relation $2({\omega ^{\prime}_B}/c)\sin ({\theta _B} + \Delta \theta ) = sh$.
The total field of the pulse at each point of the PhC at any time is given by the following expression
From expressions (13) and (14), taking into account Eq. (10), it is easy to find the transmission and reflection spectra at the PhC output boundary $z = L$ for incident plane monochromatic waves, i.e. modules of transmission coefficient
Here the superscripts “$\pm$” correspond to the positive and negative signs of the angle of incidence $\theta$, respectively; $|T(\omega )|\equiv |{T^ + }(\omega )|= |{T^ - }(\omega )|$.
4. Unidirectional invisibility and enhanced reflection of a short chirped pulse under the Laue scheme of diffraction
In this Section, we consider the transmission, reflection and propagation of a short chirped pulse in quasi-PT-symmetric PhC.
Figures 2(a)–2(c) represent the transmission $|T(\omega )|$ and reflection $|{R^ \pm }(\omega )|$ spectra, Eqs. (15) and (16), in a wide spectral range, which exceeds the spectral domain of the incident chirped pulse with duration $\tau = 1\;\textrm{ps}$ and the chirp parameter $\xi = 20$ (dashed line 1). Dashed blue lines 2 describe the case when the wave frequency $\omega = {\omega _0}$ $(\Omega = 0)$ corresponds to the Bragg condition - the angles of incidence $\theta = {\theta _B} ={\pm} {30^0}$, ${\lambda _0} = d = 0.8\;\mathrm{\mu}\textrm{m}$. Solid red lines 3 correspond to the case of deviation of the angles $\theta ={\pm} {29.9^0}$ from the exact Bragg values. Line 4 in Fig. 2(c) shows the fulfillment of the BEP condition, i.e. small deviation of the value $\varepsilon ^{\prime} - {\hat{\varepsilon }_R}(\omega ) < < \varepsilon ^{\prime}$ from the exact exceptional point, $\varepsilon ^{\prime} - {\hat{\varepsilon }_R}(\omega ) = 0$, within the spectrum of the chirped pulse. This line 4 is a cross-section of the graph in Fig. 1(c) by the plane $\gamma _2^\ast{=} 0.15$.
With a negative Bragg angle of incidence ${\theta _B} ={-} {30^0}$, Fig. 2(a), the PT-symmetric effect of enhanced reflection is observed, $|{{R^ + }} |\approx 14$ (line 2), in a narrow frequency range near the Bragg condition $\Omega = 0$. When the angle of incidence deviates by $\Delta \theta = \theta - {\theta _B} = {0.1^0}$, the reflection spectrum shifts (line 3) in such a way that the maximum reflection falls on another frequency $\omega = {\omega ^{\prime}_B}$, where ${\omega ^{\prime}_B} = ch/2|{\sin \theta } |$, for which this angle, $\theta ={-} {29.9^0}$, is the Bragg one. The corresponding point ${\Omega ^{\prime}_B}/{\omega _0} = ({\omega ^{\prime}_B} - {\omega _0})/{\omega _0} = 0.3\%$ is marked on the axes in Figs. 2(a)–2(f) with red circles. We emphasize that due to BEP condition [line 4 in Fig. 2(c)], the shift of the reflection curve in frequency relative to the value $\Omega = 0$ does not lead to a significant change in the amplification of the diffracted wave [lines 2 and 3 in Fig. 2(a)]. No reflection is observed far from the Bragg frequency, ${R^ - } \approx 0$. When the sign of the angle of incidence changes, $\theta > 0$ Fig. 2(b), unidirectional invisibility is observed when the enhanced diffracted wave is not excited, ${R^ + } < < 1$, even at frequencies near the Bragg condition.
Fig. 2. (a), (b) Spectra of reflection $|{R^ \mp }(\omega )|$ Eq. (16) and (c) of transmission $|T(\omega )|$ Eq. (15) at different signs of the angle of incidence $\theta < 0$ and $\theta > 0$ for the Bragg angles $\theta = {\theta _B} ={\mp} {30^0}$ (lines 2) and at $\theta ={\mp} {29.9^0} \ne {\theta _B}$ (lines 3). Lines 1 is the spectrum of the chirped wave packet normalized to the maximum. Line 4 in (c) shows the fulfillment of the BEP condition. (d) Frequency dependence of the real, $\textrm{Re} q_{0z}^{(1,2)}$ (lines 1 and 2), and imaginary parts, ${\mathop{\textrm{Im}}\nolimits} q_{0z}^{(1,2)}$ (lines 3 and 4), of z-projections of the wave vectors of the Borrmann (red lines 1 and 3) and anti-Borrmann (blue lines 2 and 4) modes. The quantities $q_{0z}^{(1,2)}$ are normalized to ${k_0}$. (e), (f) Modules of the amplitudes $|{A_{01,2}}(\omega )|$ of the transmitted Borrmann (lines 1) and anti-Borrmann (lines 2) modes, as well as the modules of the corresponding fields, $|{E_{01,2}}|= |{A_{01,2}}(\omega )\exp (iq_{0z}^{(1,2)}L)|$, and their sum (curve 3). PhC parameters: $L = 0.75\;\textrm{mm}$, ${\varepsilon _0} = 1.3$. Other parameters are as in Fig. 1(c).
The transmission coefficient $T(\Omega )$ Eq. (15), Fig. 2(c), does not change when the sign of the angle of incidence is changed, ${T^ + }(\Omega ) = {T^ - }(\Omega ) \equiv T(\Omega )$, and is equal to unity under exact Bragg condition if the Fresnel reflection at the boundary is not taken into account. The curve $T(\Omega )$ is symmetric if the Bragg condition is satisfied at the frequency corresponding to the exact EP, ${\omega _B} = {\omega _0}$, (line 2). A significant decrease in the quantity $T(\Omega )$ when deviating from EP, $|\Omega |/{\omega _0} > 0.5\%$, and the asymmetry of the spectrum $T(\Omega )$ (line 3) is due to the gain and loss of the Borrmann and anti-Borrmann modes, which arise as a result of the violation of the exact condition of PT symmetry in the dispersive medium. Indeed, as it follows from dispersion relations Eq. (8), the small imaginary part of the quantity ${\varepsilon _s}(\omega ){\varepsilon _{ - s}}(\omega )\sim i{\hat{\varepsilon }_R}(\omega ){\hat{\varepsilon }_I}(\omega )$ in a quasi-PT-symmetric medium leads to the complexity of the propagation constants $q_{0z}^{(1,2)}$ of the Borrmann and anti-Borrmann modes.
In Figs. 2(d)–2(f), it is shown the frequency dependences of the propagation constants $q_{0z}^{(1,2)}$ calculated by formulas (8) and transmitted field modules in the case of the Bragg condition shifted relative to zero: ${\Omega ^{\prime}_B} = {\omega ^{\prime}_B} - {\omega _0} \ne 0$, $\theta ={-} {29.9^0}$. The amplitudes of the transmitted Borrmann ${A_{01}}(\omega )$ and anti-Borrmann ${A_{02}}(\omega )$ modes, as well as the corresponding fields, ${E_{01,2}} = {A_{01,2}}(\omega )\exp (iq_{0z}^{(1,2)}L)$, were calculated by Eqs. (10) and (8). As seen from the Fig. 2(d), the real parts of the propagation constants, $\textrm{Re} q_{0z}^{(1,2)}$, change smoothly and insignificantly (lines 1 and 2), while the imaginary parts, ${\mathop{\textrm{Im}}\nolimits} q_{0z}^{(1,2)}$, change sharply, more than twice, and with a change in signs (lines 3 and 4). It follows from formulas (12) that in the BEP condition region at $\Omega < {\Omega ^{\prime}_B}$, there is only a passing anti-Borrmann mode with an amplitude ${A_{02}} = {A_{in}}$ (${A_{01}} = 0$), line 2 in Fig. 2(e), which is absorbed since ${\mathop{\textrm{Im}}\nolimits} q_{0z}^{(2)} > 0$ [line 4 in Fig. 2(d)]. With an increase in the frequency, $\Omega > {\Omega ^{\prime}_B}$, the already nonzero Borrmann mode will be absorbed, ${A_{01}} = {A_{in}}$ (${A_{02}} = 0$), line 1 in Fig. 2(e), since ${\mathop{\textrm{Im}}\nolimits} q_{0z}^{(1)} > 0$ [line 3 in Fig. 2(d)]. Near the Bragg condition, $\Omega = {\Omega ^{\prime}_B}$, the amplitudes of both waves are not equal to zero, ${A_{01}} = {A_{02}} = {A_{in}}/2$, and the gain and loss of the corresponding modes increase strongly [lines 1 and 2 in Fig. 2(f)] due to the increase in the modules of the quantities $|{\mathop{\textrm{Im}}\nolimits} q_{0z}^{(1,2)}|$ [Fig. 2(d), lines 3 and 4]. Therefore, as a result, the Borrmann mode is strongly absorbed [Fig. 2(f), line 1], and the anti-Borrmann mode is strongly enhanced [Fig. 2(f), line 2]. Thus, the sum of the modules of the fields [Fig. 2(f), line 3] and, accordingly, the transmission spectrum $T(\Omega )$ [Fig. 2(c), line 3] will be asymmetric functions with local maximum and minimum. Small-scale oscillations in the plots in Figs. 2(a)–2(c) arise as a result of the interference of the Borrmann and anti-Borrmann modes at $\textrm{Re} [q_{0z}^{(1)} - q_{0z}^{(2)}] \ne 0$. Note that the quantity ${\Omega ^{\prime}_B} = {\omega ^{\prime}_B} - {\omega _0}$ is a parameter of the asymmetry of the transmission spectrum $T(\Omega )$, when ${\Omega ^{\prime}_B} = 0$ the spectrum is symmetric [line 2 in Fig. 2(c)]. If dispersion is neglected, the transmission coefficient is $T(\Omega ) = 1$ at any frequency.
Thus, the spectra of the reflection coefficients ${R^ \pm }(\omega )$ of the PhC in the BEP change insignificantly both in shape and magnitude when the value of the angle of incidence of the wave on the PhC changes [lines 2 and 3 in Figs. 2(a) and 2(b)]. The transmission spectrum $T(\Omega )$ changes particularly in shape, becomes asymmetric, but the value of the transmission coefficient does not change so significantly. Thus it follows that in a quasi-PT-symmetric medium, it is possible to observe the PT-symmetric effects of unidirectional enhanced Bragg reflection and unidirectional invisibility for a broadband chirped spatially localized pulse with changing in the angle of incidence on the structure.
In Fig. 3, we show the dynamics of a chirped spatially localized Gaussian pulse, which intensity $|E(x,z,t){|^2}$ is computed using Eqs. (13), (14), and (10), for different signs of the angle of incidence, as well as the spectra of direct and diffractionally reflected pulses at the exit of the PhC, $z = L$, with changing in the detuning $\Delta \theta $ from the angle of incidence $\theta ={\pm} {30^0}$, $|T(\Delta \theta ,\Omega )|$ Eq. (15) and $|{R^ \pm }(\Delta \theta ,\Omega )|$ Eq. (16), respectively.
Fig. 3. (а) Intensity $I(x,z,t) = |E(x,z,t){|^2}$ Eq. (13) of a chirped spatially localized Gaussian pulse at different points in time t (indicated in the figures) at $\theta ={+} {30^0} > 0$, ${\lambda _0} = {\lambda _B} = 0.8\;\mathrm{\mu}\textrm{m}$. (b) Spectrum, $|T(\Delta \theta ,\Omega )|$ Eq. (15), of transmitted pulse at the PhC exit $z = L$ vs detuning $\Delta \theta $. (c), (d) Intensity $I(x,z,t)$ Eq. (13) of the chirped pulse at $\theta ={-} {30^0} < 0$. (c) ${\lambda _0} = 0.8\;\mathrm{\mu}\textrm{m}$; (d) ${\lambda _0} = 0.806\;\mathrm{\mu}\textrm{m}$. (f) Spectrum of an enhanced diffractionally reflected chirped pulse (wave packet) $|{R^ - }(\Delta \theta ,\Omega )|$ Eq. (16) at the exit from the PhC $z = L$ at the angle of incidence $\theta ={-} {30^0}$. Parameters of the incident pulse (4): $A = 1$,$\tau = 1\;\textrm{ps}$, ${r_0} = 60\;\mathrm{\mu}\textrm{m}$, $\xi = 20$. The PhC parameters are as in Fig. 2.; arrows indicate the directions of propagation of the incident and output pulses.
At $\theta > 0$, the intensity of the pulse field changes insignificantly compared to the incident pulse [Fig. 3(a)]. The diffraction reflected field is small, which is consistent with the plots of the transmission and reflection spectra in Figs. 2(b) and 2(c). The observed weak decrease in the field is due to the Fresnel reflection at the PhC boundary $z = 0$. The maximum of the spectrum $|T(\Delta \theta ,\Omega )|$ of the transmitted pulse (spatially unlimited in transverse direction wave packet), Fig. 3(b), shifts as the angle of incidence $\Delta \theta = \theta - {\theta _B}$ changes, since the frequency corresponding to the Bragg condition shifts. A small enhancement of the field is due to a small violation of the PT symmetry condition when the frequency $\omega $ of a part of the chirped pulse deviates from the central frequency ${\omega _0}$ ($\Omega \ne 0$) of the pulse, which agrees with the transmission spectrum, line 3 in Fig. 2(c). With a significant deviation from the Bragg condition, $|\Delta \theta |> {0.5^0}$, the spectrum coincides with the spectrum of the incident pulse.
The dynamics of the chirped pulse changes dramatically when the sign of the angle of incidence changes, $\theta < 0$. Figures 3(c) and 3(d) show snapshots of a chirped spatially localized pulse $|E(x,z,t){|^2}$ Eq. (13) for two different values of the central wavelengths of the pulse: ${\lambda _0} = 0.8\;\mathrm{\mu}\textrm{m }$ is equal to the Bragg wavelength [Fig. 3(c)] and ${\lambda _0} = 0.806\;\mathrm{\mu}\textrm{m}$ is deviated from the Bragg wavelength [Fig. 3(d)]. For the chosen angle $\theta ={-} {30^0}$ and period of the structure $d = 0.8\;\mathrm{\mu}\textrm{m }$, the Bragg condition is realized for the wavelength ${\lambda _B} = d = 0.8\;\mathrm{\mu}\textrm{m }$, therefore, in Fig. 3(c) an amplified diffracted pulse is emitted from the central region of the pulse, where ${\lambda _0} = {\lambda _B}$, which agrees with the reflection spectrum in Fig. 2(a) (curve 2). Due to the BEP condition, the radiation of those pulse regions whose frequencies do not satisfy the Bragg condition, i.e. $\lambda \ne {\lambda _B}$, propagates as PT-symmetric modes (without gain and loss), but also without diffraction reflection [the frequency range $|\Omega /{\omega _0}|> 0.5\%$ in Fig. 2(a)]. For a pulse with a shifted central wavelength ${\lambda _0} = 0.806\;\mathrm{\mu}\textrm{m}$, Fig. 3(d), the Bragg wavelength $\lambda = {\lambda _B} = 0.8\;\mathrm{\mu}\textrm{m }$ is shifted relative to the center of the pulse. Therefore, the enhanced diffracted pulse is emitted from another region of the chirped pulse, closer to its leading edge, but the central wavelength of the radiation of the output enhanced pulse remains unchanged.
As follows from the graphs of the spectra of enhanced monochromatic waves in Fig. 2(a), a change in the angle of incidence of the monochromatic wave on the structure leads to a shift in the frequency of the amplified radiation. A similar frequency shift is observed for the chirped pulse as a whole. Figure 3(e) represents the dynamics of a spatially localized pulse in the medium, for which the angle of incidence is equal $\theta ={-} {29.8^0}$, the central wavelength ${\lambda _0} = 0.8\;\mathrm{\mu}\textrm{m}$ equal to the Bragg wavelength and the other parameters coincide with the pulse parameters in Fig. 3(c). Since a decrease in the angle of incidence leads to an increase in the Bragg frequency, the frequency of the amplified pulse also increases. Comparing Fig. 3(e) and Fig. 3(c), it can be seen that in Fig. 3(e), a shift of the region of generation of amplified diffractionally reflected radiation downward along the chirped pulse is observed.
Figure 3(f) shows the spectra of the enhanced diffracted pulse $|{R^ - }(\Delta \theta ,\Omega )|$ at the PhC output, $z = L$, depending on the magnitude of the angle of incidence shift $\Delta \theta = \theta - {\theta _B}$. Here, an input pulse is chirped Gaussian wave packet ${A_{in}}(x,0,t) = A\exp [ - {(c{t} - x\sin \theta /c{\tau})^2}(1 - i\xi )]$, i.e. spatially unlimited in transverse direction pulse with the spectrum ${A_{in}}(\Omega ) = {\tilde{A}_0}\exp [ - {(\Omega /\Delta \Omega )^2}(1 + i\xi )]$, where ${\tilde{A}_0} = \tau A/[2\sqrt {\pi (1 - i\xi )} ]$, and $\Delta \Omega = (2/\tau )\sqrt {1 + {\xi ^2}}$. As can be seen from the figure, the frequency of the amplified pulse changes smoothly with a change in the angle of incidence $\Delta \theta $ of the chirped pulse.
The transverse size (in the direction of the x axis) of the outgoing enhanced diffracted pulse in vacuum depends linearly on the PhC thickness and increases with increasing Bragg angle [Figs. 3(c) and 3(d)]. The duration of this pulse is determined by the size of the selective Bragg reflection region of the chirped pulse, in which the amplified pulse is generated, and as well as by the z-projection of the group velocity:
Fig. 4. (a) Dependences of the group velocities $V_z^{(1,2)}(\Delta \theta ,\Omega )$ of the Borrmann ($V_z^{(1)}$, surface 1) and anti-Borrmann ($V_z^{(2)}$, surface 2) modes on the deviation of the frequency and angle of incidence from the Bragg values. (b) Sections of surfaces $V_z^{(1,2)}(\Delta \theta ,\Omega )$ by plane $\Delta \theta = 0$ ($\theta ={-} {30^0}$). The PhC parameters are as in Fig. 2.
It is clearly seen from Fig. 4, that in the spectral region $|\Omega /{\omega _0}|< 2\%$, where the BEP condition is well satisfied for the chirped pulse [see Fig. 2(c), lines 1 and 4], the dispersion of the group velocities is small. Moreover, the group velocity changes weakly both with a change in frequency and with a change in the angle of incidence of the wave. Therefore, the group velocities of different parts of the chirped pulse and of different components of the spatial pulse spectrum are close to Eq. (18), and the profiles of the transmitted and enhanced diffracted pulses are very slightly distorted along the z-axis during propagation in the medium [see Fig. 3(c)–3(e), at t = 0.6 ps and 2.4 ps]. The pulses do not spread out, but remain spatially localized at any time in a structure with a thickness of about ten pulse lengths.
5. Conclusion
To conclude, the linear problem of propagation of a chirped spatially localized optical pulse in a dispersive quasi-PT-symmetric PhC under dynamic Bragg diffraction in the Laue geometry was solved by the analytical spectral method. The existence of a broadband exceptional-point condition in the spectral domain of a chirped pulse in a quasi-PT-symmetric medium is shown. Due to the BEP condition, the chirped pulse, firstly, can be spatially localized at a certain point in time in a quasi-PT-symmetric medium. Secondly, during its propagation, the effect of unidirectional Bragg reflection is observed only for that part of the pulse whose frequency corresponds to the Bragg condition. Another part of the pulse propagates as a PT-symmetric mode under BEP condition without diffraction reflection, gain, and loss. This makes it possible to smoothly change the frequency, duration, and transverse size of the output enhanced pulse by changing the angle of incidence of radiation on the structure. The grating-induced group velocity dispersion in the BEP condition region is small and does not distort the shape of the transmitted and amplified pulses in a PhC with a thickness of about ten pulse lengths. The asymmetry of the transmission spectrum of a dispersive quasi-PT-symmetric PhC is predicted. This asymmetry arises due to the asymmetry of gain and loss of the field when the Bragg frequency is shifted relative to the frequency corresponding to the exact EP. The predicted effects are preserved if to use a structure with piecewise-homogeneous multilayers.
Funding
Russian Foundation for Basic Research (18-02-00556-a).
Acknowledgments
The authors are grateful to V. V. Konotop for useful discussions.
Disclosures
The authors declare no conflicts of interest.
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