Abstract

Diffraction of a three-dimensional (3D) spatiotemporal optical pulse by a phase-shifted Bragg grating (PSBG) is considered. The pulse diffraction is described in terms of signal transmission through a linear system with a transfer function determined by the reflection or transmission coefficient of the PSBG. Resonant approximations of the reflection and transmission coefficients of the PSBG as functions of the angular frequency and the in-plane component of the wave vector are obtained. Using these approximations, a hyperbolic partial differential equation (Klein–Gordon equation) describing a general class of transformations of the incident 3D pulse envelope is derived. A solution to this equation is found in the form of a convolution integral. The presented rigorous simulation results fully confirm the proposed theoretical description. The obtained results may find application in the design of new devices for spatiotemporal pulse shaping and for optical information processing and analog optical computing.

© 2016 Optical Society of America

1. Introduction

Optical devices implementing required temporal and spatial transformations of optical signals are of great interest for a wide range of practical applications including all-optical information processing and analog optical computing. Among the most important operations of analog processing of optical signals are the operations of temporal and spatial differentiation and integration [1]. For the implementation of these operations, various resonant structures including Bragg gratings [2–9], diffraction gratings [10], SPP reflectors [11], microring resonators [12–14] and nanoresonators [15–17] were proposed. The utilization of resonant structures for spatial and temporal differentiation (integration) is possible due to the fact that the Fano profile describing the reflection (transmission) coefficient of the structure in the vicinity of the resonance can approximate the transfer function of a differentiating or integrating filter [10].

For spectral filtering and temporal transformations of optical pulses, phase-shifted Bragg gratings (PSBG) are widely used. PSBG consist of two symmetrical Bragg gratings separated by a phase-shift (or defect) layer and provide zero reflectance (and unity transmittance) at a certain frequency or incidence angle [5, 7]. This effect is caused by the excitation of an eigenmode of the structure localized at the defect layer. The frequency corresponding to the zero reflection (and to the excitation of the eigenmode) lies in the photonic bandgap of the Bragg grating. Such a spectrum enables the use of PSBG for temporal differentiation of the pulse envelope in reflection, and for temporal integration of the pulse envelope in transmission [3–6]. The PSBG application for spatial integration and differentiation of the profile of a monochromatic optical beam was for the first time considered in the recent works by some of the present authors [7, 8]. The mentioned spatial operations can be implemented at oblique incidence (the angle of incidence of the beam has to coincide with the zero-reflectance angle). At normal incidence, PSBG enables the optical computation of the spatial Laplacian of the incident beam profile [9].

In the mentioned works, temporal and spatial transformations of the incident beam were considered separately. Namely, the spatial operations in [7–9] were studied in the case of a monochromatic incident beam, while the temporal transformations in [2–6] were investigated without taking into account the transformations of the spatial structure of the incident optical pulse. In this regard, the description of spatiotemporal transformations of optical pulses implemented by PSBG in the general 3D case is of great interest.

In the present work, a theoretical description of the diffraction of a 3D optical pulse by a PSBG is presented for the first time. Resonant approximations of the reflection and transmission coefficients of the PSBG as functions of the angular frequency and the in-plane wave vector component are obtained. It is shown that at normal incidence the transformation of the pulse envelope can be described by a hyperbolic partial differential equation, which can be reduced to the Klein–Gordon equation, and a solution to this equation is found. The presented rigorous simulation results fully confirm the proposed theoretical description.

2. Pulse envelope transformation upon diffraction by a multilayer structure

Consider normal incidence of a 3D optical pulse on a multilayer structure consisting of homo-geneous layers (Fig. 1). Neglecting the dispersion of the superstrate material, we can write the incident pulse field as a plane wave expansion:

E(x,y,z,t)=P(x,y,z,t)exp{iω0cnsupziω0t}=V(kx,ky,ω)exp{ikxx+ikyyikzzi(ω+ω0)t}dkxdkydω,
where P(x,y,z,t) is the pulse envelope, ω0 is the central frequency of the pulse, nsup is the refractive index of the superstrate and substrate, k=(kx,ky,kz) is the wave vector, where kz=((ω+ω0)/c)2nsup2k2, k=(kx,ky) is the in-plane wave vector, and V(kx,ky,ω) is the spatiotemporal spectrum of the pulse envelope at z=0. We assume that V(kx,ky,ω) is nonzero in a bounded domain: |k|kmax, |ω|Ω.

 

Fig. 1 Incident pulse Pinc(x,y,z,t) and reflected pulse Pref(x,y,z,t) upon diffraction by a multilayer structure.

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The field of the incident pulse at the upper interface of the structure (at z=0) has the form

E(x,y,0,t)=v(x,y,t)exp{iω0t},
where
v(x,y,t)=P(x,y,0,t)=V(kx,ky,ω)exp{ikxx+ikyyiωt}dkxdkydω
is the incident pulse envelope at z=0. We assume that the incident pulse is linearly polarized along the x axis. In this case, the scalar function E(x,y,0,t) in Eq. (2) defines the x-component of the electric field Ex, while the y-compenent at z=0 equals zero (Ey0). It is possible to form such wave packet as a superposition of TE- and TM-polarized plane waves [9, 18, 19]. The function V(kx,ky,ω) in Eq. (3) represents the envelope spectrum of the Ex component. The spectra of the remaining electromagnetic field components can be expressed through the function V(kx,ky,ω) [9, 18, 19].

Envelopes of the Ex component of the reflected and transmitted pulses at the upper and lower interfaces of the multilayer structure can be written as

uR(x,y,t)=V(kx,ky,ω)R˜(kx,ky,ω+ω0)exp{ikxx+ikyyiωt}dkxdkydω,
uT(x,y,t)=V(kx,ky,ω)T˜(kx,ky,ω+ω0)exp{ikxx+ikyyiωt}dkxdkydω,
where R(kx,ky,ω+ω0) and T(kx,ky,ω+ω0) are the reflection and transmission coefficients which are expressed through the reflection and transmission coefficients of TE- and TM-polarized plane waves incident on the structure [9, 18, 19]:

R˜(kx,ky,ω+ω0)=RТE(kx,ky,ω+ω0)+kx2kx2+ky2[RТM(kx,ky,ω+ω0)RТE(kx,ky,ω+ω0)],T˜(kx,ky,ω+ω0)=TТE(kx,ky,ω+ω0)+kx2kx2+ky2[TТM(kx,ky,ω+ω0)TТE(kx,ky,ω+ω0)].

Equations (3)–(5) imply that the reflected and transmitted pulse envelopes correspond to the result of the transformation of the incident pulse envelope by linear systems with the following transfer functions (TF):

HR(kx,ky,ω)=R˜(kx,ky,ω+ω0),HT(kx,ky,ω)=T˜(kx,ky,ω+ω0).

The obtained TF of Eqs. (6), (7) are the 3D generalization of the TF describing temporal transformations of an optical pulse [2, 4, 5, 10] and spatial transformations of an optical beam [7–9].

3. Resonant representation of the PSBG spectra

From this point on we will study the reflection and transmission coefficients at fixed polarization (ТЕ or ТМ) unless stated otherwise. Since the following inference is not dependent on polarization type, the subscripts TE and TM will be omitted. For the considered structure (Fig. 1), the reflection and transmission coefficients at fixed polarization depend only on two variables: the incident light frequency ω, and the angle between the normal (z axis) and the wave vector. Thus, these coefficients can be considered as functions of the in-plane wave vector length |k|=kx2+ky2:

R(kx,ky,ω)=R(kx2+ky2,0,ω),T(kx,ky,ω)=T(kx2+ky2,0,ω).

One can expect significant change in the shape of the envelopes of the reflected and transmitted pulses in the case when the coefficients R(kx2+ky2,ω),T(kx2+ky2,ω) are rapidly varying in the vicinity of the point (0,0,ω0). Rapid change in the transmission and reflection coefficients is typical for PSBG [5, 7, 9], which are widely used as narrow-band spectral filters operating in transmission. PSBG provides zero reflection at some predetermined angle of incidence θ0 (at kx2+ky2=k0nsupsinθ0, where k0=ω/c is the wavenumber) simultaneously for TE- and TM-polarizations of the incident wave. PSBG consists of two symmetrical Bragg gratings (BGs) separated by a phase-shift (defect) layer (Fig. 1). In the simplest case, the BG layers have identical optical thickness:

n˜1h1=n˜2h2=λB/4,
where n˜i=ni2nsup2sin2(θ0), i=1,2, ni, hi are the refractive indices and thicknesses of the BG layers, respectively, and λB=2πc/ωB is the Bragg wavelength. If the optical thickness of the defect layer n˜hdef is equal to λB/2, where n˜=ndef2nsup2sin2θ0, ndef being the refractive index of the defect layer, then the PSBG reflection coefficient vanishes at the wavelength λB and angle of incidence θ0 [5, 7, 9]. Let us note that this reflection zero is located at the center of the first photonic bandgap of the Bragg grating. The emergence of a reflection zero at the center of the bandgap has resonant nature and is associated with the excitation of a mode localized at the defect layer.

To describe the diffraction of an optical pulse by a PSBG, let us obtain approximate representations of the structure's reflection and transmission coefficients in the vicinity of normal incidence (kx=ky=0) and the resonance frequency corresponding to the mode of the defect layer. According to Eq. (8), it is sufficient to consider the two-dimensional case corresponding to the condition ky0 at fixed polarization (ТЕ or ТМ). For this case, resonant representations of the reflection and transmission coefficients as functions of the angular frequency ω and the wave vector component kx=k0nsupsin(θ) were obtained in [20]:

R(kx,ω)=γRvg2kx2(ωωzR)(ωωz2R)vg2kx2(ωωp)(ωωp2),T(kx,ω)=γTvg2kx2(ωωzT)(ωωz2T)vg2kx2(ωωp)(ωωp2),
where vg corresponds to the group velocity of an eigenmode of the structure; γR and γT are the non-resonant (far-from-resonance) reflection and transmission coefficients; ωp,ωp2 are the complex angular frequencies of the eigenmodes of the structure (poles of the reflection and transmission coefficients at kx=0), ωzR,ωz2R and ωzT,ωz2T are the zeros of the reflection and transmission coefficients, respectively. Despite the fact that the resonant approximations (10) were written for a subwavelength diffraction grating in [20], they were obtained from rather general considerations and also apply to multilayer structures including PSBG.

Representations (10) assume the existence of two poles ωp,ωp2. In the case of a diffraction grating, these poles correspond to even and odd modes [20]. In the case of PSBG, a single mode localized at the defect layer is considered. In this case, assuming the complex frequency of this mode is ωp, the quantity ωp* should be used as the second pole’s frequency in Eq. (10). Indeed, in the complex ω-plane each pole ωp has a corresponding pole ωp2=ωp* with negative real part [21]. This is a consequence of the following general relations for the reflection and transmission coefficients:

R(ω*)=R(ω)*,T(ω*)=T(ω)*.

Expressions (11) reflect the fact that if some field distribution satisfies Maxwell’s equations at a frequency ω, then its complex conjugate satisfies Maxwell’s equations at the frequency ω*. This holds true for the case of lossless materials. It follows from Eqs. (10) and (11) that ωz2T=(ωzT)*, ωz2R=(ωzR)*. Since the PSBG has a real-valued zero in the reflection spectrum (ωzR), the last expression can be rewritten as ωz2R=ωzR. Taking into account the above considerations, Eqs. (8), (10) lead to the following resonant representations of the PSBG spectra:

R(kx,ky,ω)=R(kx2+ky2,ω)=γRvg2(kx2+ky2)(ωωzR)(ω+ωzR)vg2(kx2+ky2)(ωωp)(ω+ωp*),T(kx,ky,ω)=T(kx2+ky2,ω)=γTvg2(kx2+ky2)(ωωzT)(ω+ωzT*)vg2(kx2+ky2)(ωωp)(ω+ωp*).
Representations (12) can be considered as a generalization of the Fano profile. Indeed, at kx=ky=0 Eqs. (12) are reduced to the conventional Fano profile [20, 21]. At a fixed frequency ω, Eqs. (12) become the known approximations of the reflection and transmission coefficients as functions of kx (or of kx, ky) in the vicinity of normal incidence [7, 9]. Representations (12) also generalize the resonant approximations obtained in [20, 22, 23] to the general 3D case when the reflection and transmission coefficients are considered as functions of the angular frequency ω and the wave vector components kx,ky.

Equations (12) were obtained at a fixed polarization (ТМ or ТЕ). Note that the reflection and transmission coefficients for both polarizations are identical at normal incidence (kx=ky=0):

RTM(0,ω)=RTE(0,ω),TTM(0,ω)=TTE(0,ω).
According to Eq. (13), the approximations (12), which are valid in the vicinity of normal incidence, also coincide for TE- and TM-polarizations. This means that the reflection and transmission coefficients in (6) within the framework of this approximation can be rewritten as functions of two variables: R˜(kx,ky,ω)=R(kx2+ky2,ω), T˜(kx,ky,ω)=T(kx2+ky2,ω).

4. Transformation of the pulse envelope upon diffraction by a PSBG

On the basis of the arguments presented in the end of the previous section, we further assume that the reflection and transmission coefficients are identical for both polarizations in some neighborhood of the normal incidence. In this case, substituting Eq. (12) into Eq. (7), we obtain explicit expressions for the PSBG transfer functions:

HR,T(kx,ky,ω)=HR,T1(kx,ky,ω)HR,T2(kx,ky,ω),
where
HR,T1(kx,ky,ω)=γR,T[vg2(kx2+ky2)(ω+ω0ωzR,T)(ω+ω0+ωzR,T*)],
HR,T2(kx,ky,ω)=[vg2(kx2+ky2)(ω+ω0ωp)(ω+ω0+ωp*)]1,
and the indices R, T in (14)–(16) correspond to reflection and transmission, respectively. The TF (14) describes a general class of transformations of the envelope of an incident 3D pulse, which can be implemented by a PSBG. The connection between the Fourier spectra of the incident pulse (V(kx,ky,ω)) and of the reflected (transmitted) pulse (UR,T(kx,ky,ω)) has the form

UR,T(kx,ky,ω)=HR,T(kx,ky,ω)V(kx,ky,ω).

Let us recall that the TF (14) describes the transformation of the envelope of the Ex component of the incident pulse. If the reflection (transmission) coefficients for both polarizations are identical, no polarization conversion occurs, i.e. the reflected and transmitted pulses remain x-polarized. In this case, all the other electric and magnetic field components of the reflected (transmitted) pulse can be expressed through the functions UR,T(kx,ky,ω) [9, 18, 19].

For the description of the transformations corresponding to the TF of Eq. (14), let us write a differential equation corresponding to this TF. To do this, let us rewrite Eq. (17) in the following form:

[HR,T2(kx,ky,ω)]1UR,T(kx,ky,ω)=HR,T1(kx,ky,ω)V(kx,ky,ω).
Applying the inverse Fourier transform to the left and right parts of Eq. (18), we obtain the following differential equation for the envelope of the reflected (transmitted) pulse uR,T(x,y,t):
2uR,Tt2vg2ΔuR,T2i(ω0iImωp)uR,Tt(ω0ωp)(ω0+ωp*)uR,T=fR,T,
where fR,T(x,y,t) is the inverse Fourier transform of the right part of Eq. (18):
fR,T(x,y,t)=γR,T(2vt2vg2Δv2i(ω0iImωzR,T)vt(ω0ωzR,T)(ω0+ωzR,T*)v),
and Δ=2/x2+2/y2 is the Laplace operator with respect to the spatial coordinates. Let us note that the function fR,T(x,y,t) in Eqs. (19), (20) corresponds to the result of the transformation of the incident pulse envelope v(x,y,t) by a linear system with the TF HR,T1(kx,ky,ω) (15). For low-quality-factor resonances, the function HR,T2(kx,ky,ω) in Eq. (16) is slowly varying, and the main contribution to the output signal is provided by the system with the TF HR,T1(kx,ky,ω) [22, 23]. In this case, the transformation of the envelope v(x,y,t) is defined solely by Eq. (20). Equation (20) demonstrates that the PSBG enables the optical implementation of several important differential operators including the computation of the Laplacian with respect to spatial coordinates (the second term in Eq. (20)) and the computation of the first temporal derivative (the third term in Eq. (20)). Let us note that the computation of the second-order temporal derivative (the first term in Eq. (20)) cannot be implemented since the magnitude of the first term is small as compared to the term corresponding to the first derivative. Thus, Eq. (20) generalizes spatial and temporal transformations of optical pulses and beams discussed in [4, 5, 9]. In the mentioned works, the operations of computing the spatial Laplacian and the first temporal derivative were considered separately for monochromatic optical beams and optical pulses, respectively.

It is easy to see that the obtained Eq. (19) is a hyperbolic partial differential equation. The solution to this equation is derived in Appendix A in the following form:

uR,T(x,y,t)=+++fR,T(ξ,η,τ)h(xξ,yη,tτ)dξdηdτ,
where the function h(x,y,t) is nonzero at x2+y2vgt and is given by
h(x,y,t)=exp{iω0t+Imωpt}cos(t2(x2+y2)/vg2Reωp)2πvg2t2(x2+y2)/vg2,x2+y2vgt.
Let us note that h(x,y,t) is the Green’s function of Eq. (19) and corresponds to the impulse response function of a linear system with the transfer function HR,T2(kx,ky,ω) (Eq. (16)). The obtained solution has a clear physical interpretation. The integration in Eq. (21) is performed over the area located inside the cone x2+y2=vgt. This area can be considered as an analogue of the light cone: it includes the points, the perturbation from which can reach the considered point in a time t. The quantity vg has the meaning of the group velocity of the eigenmode of the structure. Assuming that this velocity does not exceed the speed of light, the impulse response function h(x,y,t) satisfies the relativistic causality condition. In that context, it is interesting to mention that Eq. (19) in the canonical form corresponds to the Klein–Gordon equation (Eq. (24) in Appendix A), which is the relativistic generalization of the Schrödinger equation [24]. Note that the impulse response of the linear system given in [23] and described by the Schrödinger equation does not satisfy the relativistic causality condition.

5. Analysis of the resonant approximations accuracy

Expressions (19)–(22) were obtained using resonant approximations (12). In this regard, let us examine the accuracy of these approximations in the case of PSBG. We denote by Np the number of periods of BG constituting the PSBG. The total number of PSBG layers amounts to Nl=4Np+1. The number of layers Nl determines the quality factor of the resonance. Using the rigorous coupled-wave analysis (RCWA) technique for multilayer systems [25], let us calculate the PSBG spectra at Nl=9 (Np=2) and Nl=29 (Np=7) corresponding to low quality factor and high quality factor resonances, respectively. The following values of the refractive indices of the BG layers, defect layers, superstrate and substrate were used in the simulations: n1=2.4547 (TiO2), n2=1.4446 (SiO2), ndef=n1, nsup=nsub=1. The thicknesses of the layers were calculated from Eq. (9) for the Bragg wavelength λB=1500nm and the incidence angle θ0=0.

PSBGs are widely used as spectral filters operating in transmission. The increase in the number of layers leads to the increase in the resonance quality factor, and, consequently, to the decrease in the spectral width of the transmission band. In this regard, it is worth studying the accuracy of the resonant approximation (12) of the transmission coefficient in the case of a large number of layers (at a large quality factor). The left half of Fig. 2(a) (kx<0) shows the PSBG transmission spectrum at Nl=29 (Np=7) calculated using RCWA in the case of TM-polarization. A sharp transmission maximum caused by the excitation of an eigenmode in the defect layer is evident in Fig. 2(a). The right half of Fig. 2(а) (kx>0) shows the PSBG transmission spectrum calculated using the resonant approximation (12) with the following parameters: ωp=1255.770.08ips1, vg=159μm/ps, ωzT=966.84+801.35ips1, γT=0.99104. The complex frequency of the eigenmode ωp was found as a pole of the scattering matrix of the structure at normal incidence (kx=ky=0) [26]. The considered case corresponds to a high-quality-factor resonance with the quality factor Q=(Reωp)/(2Imωp)=7848.6. The approximation parameters γT, ωzT and vg were obtained using an optimization procedure aimed at minimizing the difference between the transmission coefficient calculated using RCWA and the approximate expression (12) at several characteristic points of the spectrum. The normalized root-mean-square deviation (NRMSD) of the spectra in Fig. 2(a) amounts to 2.1%. In the case of TE-polarization the NRMSD is 2.0% for the considered example. Resonant representation (12) provides similar accuracy also when used for the approximation of the reflection spectra.

 

Fig. 2 (a) Transmission spectra |T(kx,ω)| of the PSBG at Nl=29 calculated using RCWA (left half, kx<0) and using the resonant approximation (12) (right half, kx>0) in the case of TM-polarization. (b) Reflection spectra |R(kx,ω)| of the PSBG at Nl=9 calculated using RCWA (left half, kx<0) and using the resonant approximation (12) (right half, kx>0) in the case of TM-polarization.

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The most interesting operations (optical computation of the first-order temporal derivative and spatial Laplacian) are performed by the PSBG in reflection at a low number of layers (at a low quality factor of the resonance). Therefore, it is worth studying the accuracy of the resonant approximation of the reflection coefficient (12) at a low number of layers. The left half of Fig. 2(b) (kx<0) shows the PSBG reflection spectrum at Nl=9 (Np=2) calculated using RCWA in the case of TM-polarization, while the right half (kx>0) shows the reflection spectrum calculated using the resonant approximation (12). In this case, the approximation parameters take the following values: ωzR=2πc/λB=1255.77ps1, ωp=1255.7717.57ips1 (the resonance quality factor amounts to Q=35.7), γR=0.9953, vg=158μm/ps. The NRMSD value of the spectra in Fig. 2(b) amounts to 1.7%. Similarly to the previous case, the NRMSD value in the case of TE-polarization is slightly less and amounts to 1.5%. Similar accuracy is obtained for the representation of the transmission spectra.

6. Examples of the pulse envelope transformation

The results of the previous section demonstrate that the resonant approximations (12) represent the PSBG reflection and transmission spectra with high accuracy. Let us now study the accuracy of the obtained model (21), (22) for the calculation of the envelope of the reflected (transmitted) pulse in the case of Gaussian incident pulse v(x,y,t)=exp{(x2+y2)/σ2t2/σt2}. In particular, we compare the results of Eqs. (21), (22) with the envelopes of the reflected (transmitted) pulse calculated using RCWA [25] and Eqs. (4)–(6). The integrals in Eq. (4)–(5) were calculated numerically using fast Fourier transform on a discrete mesh with steps ht and hx=hy.

In subsection 6.1 we consider the general spatiotemporal transformation implemented by a PSBG. The particular cases of this general transformation are the operations of temporal differentiation and of the calculation of the Laplace operator, which are considered in subsections 6.2 and 6.3. In subsection 6.4 we consider a notable transformation of a gaussian pulse to an annular-shaped pulse.

6.1 High-Q resonance: spatiotemporal transformation in transmission

In this subsection we consider a general spatiotemporal transformation implemented by PSBG with high-Q resonance. Figure 3(b) demonstrates the envelope of the transmitted pulse formed upon diffraction of a Gaussian pulse with the parameters σt=1ps, σ=2μm, central frequency ω0=ωzR=1255.77ps1 (λ0=1.5μm) (Fig. 3(a)) by a PSBG with the number of layers Nl=29. The left half of Fig. 3(b) (x<0) shows the absolute value of the envelope calculated using RCWA and Eqs. (4), (6). The right half (x>0) shows the absolute value of the envelope calculated using the proposed analytical model (12), (21), (22). The discretization steps are ht=0.02psand hx=0.06μm. The NRMSD of the distributions in Fig. 3(b) is equal to 2.2%, which confirms high accuracy of the proposed model for PSBG consisting of a large number of layers (in the case of high quality factor resonance). The insets in Figs. 3(a) and 3(b) show the isosurfaces of the incident and transmitted pulses, respectively.

 

Fig. 3 (a) Envelope of the pulse incident on a PSBG with Nl=29 layers. (b) Absolute value of the envelope of the transmitted pulse calculated using RCWA and Eqs. (5), (6) (left half, x<0) and using the proposed analytical model (12), (21), (22) (right half, x>0). The insets show the corresponding isosurfaces of the incident and transmitted pulses in (x,y,t) space.

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Let us note that the computational cost of the PSBG spectra estimation and the solution of the 3D pulse diffraction problem using Eqs. (12), (21) is much smaller comparing to the direct solution of Maxwell’s equations using the RCWA technique. Indeed, for the calculation of the parameters of the resonant approximations (12) or the convolution kernel (22), it is sufficient to rigorously solve the diffraction problem only for several points in the (kx,ky,ω) space. Moreover, calculating the parameters of the convolution kernel (22) once allows one to simulate the diffraction of an arbitrary pulse.

6.2 Low-Q resonance: temporal differentiation in reflection

As mentioned above, the operations of the optical temporal differentiation and the computation of the spatial Laplace operator are carried out by PSBG in reflection at a small number of periods (at a low quality factor of the resonance). Let us investigate the possibility of performing these operations. Note that the temporal differentiation of optical pulses was previously considered without taking into account the spatial structure of the pulse [3–5]. Figure 4(b) shows the envelope of the reflected pulse formed upon diffraction of a Gaussian pulse with the parameters σt=2ps, σ=20μm, ω0=ωzR=1255.77ps1 (λ0=1.5μm) (Fig. 4(а)) by a PSBG with the number of layers Nl=9. In the left half of Fig. 4(b) (x<0), the absolute value of the envelope calculated using the RCWA technique and Eqs. (4), (6) is shown. The right half (x>0) shows the absolute value of the envelope calculated using the proposed analytical model (12), (21), (22).

 

Fig. 4 (a) Envelope of the pulse incident on a PSBG with Nl=9 layers. (b) Absolute value of the envelope of the reflected pulse calculated using RCWA and Eqs. (4), (6) (left half, x<0) and using the proposed analytical model (12), (21), (22) (right half, x>0). (c) Absolute value of the temporal derivative of the incident pulse. The insets show the corresponding isosurfaces of the incident and reflected pulses in (x,y,t) space.

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The discretization steps are ht=0.02psand hx=0.15μm. The NRMSD value of the distributions in Fig. 4 amounts to 2.4%, which confirms high accuracy of the proposed model also for PSBG consisting of a small number of layers (the case of a low-quality-factor resonance).

In the considered example, the pulse full width at the level of e1 equals to 2σ=40μm at t=0. At such a width, the envelope of the reflected pulse (Fig. 4(b)) is already close to the analytically computed derivative v(x,y,t)/t of the envelope of the incident Gaussian pulse (Fig. 4(c)).

6.3 Low-Q resonance: spatial Laplacian in reflection

Figure 5(b) shows the envelope of the reflected pulse formed upon diffraction of a Gaussian pulse with the parameters σt=80ps, σ=4μm, central frequency ω0=ωzR=1255.77ps1 (λ0=1.5μm) (Fig. 5(а)) by a PSBG with the number of layers Nl=9. As before, the halves of Fig. 5(b) represent the envelopes calculated using RCWA (left half, x<0) and the proposed analytical model (12), (21), (22) (right half, x>0). The discretization steps are ht=0.8psand hx=0.02μm. The NRMSD value of the distributions in Fig. 5(b) amounts to 2.0%, which, again, confirms good accuracy of the presented model. In the considered example, the duration of the incident pulse at the level of e1 equals to 2σt=160ps at x=y=0 (Fig. 5(а)). At such a duration, the envelope of the reflected pulse (Fig. 5(b)) is close to the Laplacian of the incident pulse envelope Δv(x,y,t) (Fig. 5(c)). Let us note, that optical computation of the spatial Laplace operator was previously considered for monochromatic beams [9].

 

Fig. 5 (a) Envelope of the pulse incident of a PSBG with Nl=9 layers. (b) Absolute value of the reflected pulse envelope calculated using RCWA and Eqs. (4), (6) (left half, x<0) and using the proposed analytical model (12), (21), (22) (right half, x>0). (c) Laplacian of the envelope of the incident pulse. The insets show the corresponding isosurfaces of the incident and reflected pulses in (x,y,t) space.

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6.4 Low-Q resonance: spatiotemporal transformation in reflection

We now consider the diffraction of a Gaussian beam with the parameters σt=8ps, σ=5μm (Fig. 6(а)). At these parameters, one should expect both temporal and spatial transformation of the envelope of the reflected pulse. Let us choose the central frequency of the pulse ω0 so that the pulse envelope is substantially changed upon reflection. As noted above in Section 4, in the case of a low quality factor resonance, the envelope transformation can be approximated using Eq. (20), and uR(x,y,t)fR(x,y,t). Substituting the envelope of the incident Gaussian pulse v(x,y,t)=exp{(x2+y2)/σ2t2/σt2} into Eq. (20), we obtain at x=y=t=0:

fR(0,0,0)=4vg2σ2+|ωz|22σt2ω02.
According to Eq. (23), at the central frequency of the pulse ω0=4vg2/σ22/σt2+|ωz|2=1257.36ps1 (λ0=1.4981μm) one can expect a significant change in the envelope shape. Indeed, the point x=y=t=0 is the location of the maximum of the incident pulse envelope v(x,y,t)(v(0,0,0)=1), while the magnitude of the reflected pulse envelope vanishes at this point (if the approximation (20) holds).

 

Fig. 6 (a) Envelope of the pulse incident on a PSBG with Nl=9 layers. (b) Absolute value of the envelope of the reflected pulse calculated using RCWA and Eqs. (4), (6) (left half, x<0) and using the proposed analytical model (12), (21), (22) (right half, x>0). (c) Estimate of the envelope of the reflected pulse obtained using Eq. (20). The insets show the corresponding isosurfaces of the incident and reflected pulses in (x,y,t) space.

Download Full Size | PPT Slide | PDF

Figure 6(b) shows the absolute values of the envelopes calculated using RCWA and Eqs. (4), (6) (left half, x<0) and using the proposed analytical model (12), (21), (22) (right half, x>0). The NRMSD value of the distributions in Fig. 6(b) amounts to 1.6%. Figure 6 demonstrates a significant change in the envelope shape v(x,y,t): the envelope of the reflected pulse takes approximately annular shape. This transformation is also approximately described by the function fR(x,y,t) in Eq. (20) (Fig. 6(c)). The discretization steps are ht=0.08psand hx=0.02μm. The NRMSD value of the distributions in Figs. 6(b) and 6(c) does not exceed 10%, which demonstrates the possibility of using the simple expression (20) for the estimation of the envelope of the reflected (transmitted) pulse envelope at a small number of layers.

7. Conclusion

In the present work, we have theoretically described the diffraction of a 3D spatiotemporal optical pulse by a phase-shifted Bragg grating. The transformation of the incident pulse envelope was described in terms of linear system theory. It was shown that the transfer function of the linear system is determined by the reflection (transmission) coefficient of the PSBG. We have obtained resonant approximations of the reflection and transmission coefficients of the PSBG as functions of the angular frequency and the in-plane component of the wave vector. On the basis of the proposed approximations, we obtained a partial differential equation of hyperbolic type describing the transformation of the envelope of the incident optical pulse upon diffraction by a PSBG. This differential equation can be reduced to the Klein–Gordon equation. An analytical solution to this equation in the form of a convolution integral was found. The presented simulation results obtained using the rigorous coupled-wave analysis technique fully confirm the theoretical description.

The presented theoretical description determines the class of the spatiotemporal transformations of 3D optical pulses that can be implemented by a PSBG. The results of the present work may find application in the design of novel planar devices for all-optical information processing and analog optical computing.

Appendix A Solution of the differential equation

In the canonical form, Eq. (19) can be rewritten as follows:

2g(x,y,t)t2vg2Δg(x,y,t)+(Reωp)2g(x,y,t)=f(x,y,t),
where
f(x,y,t)=γR,T(2pt2vg2Δp+(ReωzR,T)2p)exp{ImωzR,TtImωpt},
uR,T(x,y,t)=g(x,y,t)exp{iω0t+Imωpt},v(x,y,t)=p(x,y,t)exp{iω0t+ImωzR,Tt}.
Hyperbolic partial differential Eq. (24) coincides with the Klein–Gordon equation [24]. In order to solve it, let us first set zero initial conditions g|t=0=g/t|t=0=0. Let us note that for an incident pulse with a limited duration one can always satisfy the zero initial conditions [22]. Now the problem (24) can be easily solved with the Fourier transform method [27]. The Fourier spectra of the functions g(x,y,t) and f(x,y,t) with respect to x and y have the form

G(kx,ky,t)=12πg(ξ,η,t)exp{ikxξ+ikyη}dξdη,F(kx,ky,t)=12πf(ξ,η,t)exp{ikxξ+ikyη}dξdη.

Applying Fourier transform (27) to both sides of Eq. (24), we reduce the initial problem to an ordinary second-order differential equation

2G(kx,ky,t)t2+(vg2(kx2+ky2)+(Reωp)2)G(kx,ky,t)=F(kx,ky,t)
with the following initial contions: G(kx,ky,t)|t=0=G(kx,ky,t)/t|t=0=0. The solution to Eq. (28) can easily be obtained in the following form:
G(kx,ky,t)=0tw(kx,ky,tτ)F(kx,ky,τ)dτ,
where w(kx,ky,τ)=sin(τvg2(kx2+ky2)+(Reωp)2)/vg2(kx2+ky2)+(Reωp)2.

The solution to the initial problem (24) is found by applying inverse Fourier transform to Eq. (29):

g(x,y,t)=12πG(kx,ky,t)exp{ikxxikyy}dkxdky=1(2π)20t[f(ξ,η,τ)[w(kx,ky,tτ)exp{ikx(ξx)+iky(ηy)}dkxdky]dξdη]dτ.

Using the identity [28]

12π+sin(yk2+q2)k2+q2exp(ikx)dk={12J0(qy2q2),|x|<y,0,|x|>y,
we can integrate Eq. (30) over kx:

g(x,y,t)=14πvg0tdτ×f(ξ,η,τ)[J0(vg2(tτ)2(ξx)2(Reωp)vg22+ky2)exp{iky(ηy)}dky]dξdη.

Finally, using the identity [28]

12π+J0(dk2+q2)exp(iky)dk={1πcos(qd2y2)d2y2,|y|<d,0,|y|>d,
we can integrate Eq. (32) over ky and obtain the following solution:
g(x,y,t)=12πvg20tdτcos(Re[ωp](tτ)2((ξx)2+(ηy)2)/vg2)(tτ)2((ξx)2+(ηy)2)/vg2f(ξ,η,τ)dξdη.
Substituting Eq. (25) into Eq. (34), we obtain the solution to the original Eq. (19) in the form of Eqs. (21), (22).

Funding

Russian Science Foundation Grant (14-19-00796).

References and Links

1. A. Silva, F. Monticone, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Performing mathematical operations with metamaterials,” Science 343(6167), 160–163 (2014). [CrossRef]   [PubMed]  

2. R. Slavík, Y. Park, M. Kulishov, and J. Azaña, “Terahertz-bandwidth high-order temporal differentiators based on phase-shifted long-period fiber gratings,” Opt. Lett. 34(20), 3116–3118 (2009). [CrossRef]   [PubMed]  

3. L. M. Rivas, S. Boudreau, Y. Park, R. Slavík, S. Larochelle, A. Carballar, and J. Azaña, “Experimental demonstration of ultrafast all-fiber high-order photonic temporal differentiators,” Opt. Lett. 34(12), 1792–1794 (2009). [CrossRef]   [PubMed]  

4. N. K. Berger, B. Levit, B. Fischer, M. Kulishov, D. V. Plant, and J. Azaña, “Temporal differentiation of optical signals using a phase-shifted fiber Bragg grating,” Opt. Express 15(2), 371–381 (2007). [CrossRef]   [PubMed]  

5. M. Kulishov and J. Azaña, “Design of high-order all-optical temporal differentiators based on multiple-phase-shifted fiber Bragg gratings,” Opt. Express 15(10), 6152–6166 (2007). [CrossRef]   [PubMed]  

6. N. Quoc Ngo, “Design of an optical temporal integrator based on a phase-shifted fiber Bragg grating in transmission,” Opt. Lett. 32(20), 3020–3022 (2007). [CrossRef]   [PubMed]  

7. L. L. Doskolovich, D. A. Bykov, E. A. Bezus, and V. A. Soifer, “Spatial differentiation of optical beams using phase-shifted Bragg grating,” Opt. Lett. 39(5), 1278–1281 (2014). [CrossRef]   [PubMed]  

8. N. V. Golovastikov, D. A. Bykov, L. L. Doskolovich, and E. A. Bezus, “Spatial optical integrator based on phase-shifted Bragg gratings,” Opt. Commun. 338, 457–460 (2015). [CrossRef]  

9. D. A. Bykov, L. L. Doskolovich, E. A. Bezus, and V. A. Soifer, “Optical computation of the Laplace operator using phase-shifted Bragg grating,” Opt. Express 22(21), 25084–25092 (2014). [CrossRef]   [PubMed]  

10. D. A. Bykov, L. L. Doskolovich, and V. A. Soifer, “Single-resonance diffraction gratings for time-domain pulse transformations: integration of optical signals,” J. Opt. Soc. Am. A 29(8), 1734–1740 (2012). [CrossRef]   [PubMed]  

11. Z. Ruan, “Spatial mode control of surface plasmon polariton excitation with gain medium: from spatial differentiator to integrator,” Opt. Lett. 40(4), 601–604 (2015). [CrossRef]   [PubMed]  

12. W. Liu, M. Li, R. S. Guzzon, E. J. Norberg, J. S. Parker, M. Lu, L. A. Coldren, and J. Yao, “A fully reconfigurable photonic integrated signal processor,” Nat. Photonics 10(3), 190–195 (2016). [CrossRef]  

13. T. Yang, J. Dong, L. Lu, L. Zhou, A. Zheng, X. Zhang, and J. Chen, “All-optical differential equation solver with constant-coefficient tunable based on a single microring resonator,” Sci. Rep. 4, 5581 (2014). [PubMed]  

14. J. Wu, P. Cao, X. Hu, X. Jiang, T. Pan, Y. Yang, C. Qiu, C. Tremblay, and Y. Su, “Compact tunable silicon photonic differential-equation solver for general linear time-invariant systems,” Opt. Express 22(21), 26254–26264 (2014). [CrossRef]   [PubMed]  

15. M. Ferrera, Y. Park, L. Razzari, B. E. Little, S. T. Chu, R. Morandotti, D. J. Moss, and J. Azaña, “All-optical 1st and 2nd order integration on a chip,” Opt. Express 19(23), 23153–23161 (2011). [CrossRef]   [PubMed]  

16. N. L. Kazanskiy, P. G. Serafimovich, and S. N. Khonina, “Use of photonic crystal cavities for temporal differentiation of optical signals,” Opt. Lett. 38(7), 1149–1151 (2013). [CrossRef]   [PubMed]  

17. N. L. Kazanskiy and P. G. Serafimovich, “Coupled-resonator optical waveguides for temporal integration of optical signals,” Opt. Express 22(11), 14004–14013 (2014). [CrossRef]   [PubMed]  

18. S. M. Sepke and D. P. Umstadter, “Exact analytical solution for the vector electromagnetic field of Gaussian, flattened Gaussian, and annular Gaussian laser modes,” Opt. Lett. 31(10), 1447–1449 (2006). [CrossRef]   [PubMed]  

19. G. Zhou, “The analytical vectorial structure of a nonparaxial Gaussian beam close to the source,” Opt. Express 16(6), 3504–3514 (2008). [CrossRef]   [PubMed]  

20. D. A. Bykov and L. L. Doskolovich, “ω–kx Fano line shape in photonic crystal slabs,” Phys. Rev. A 92(1), 013845 (2015). [CrossRef]  

21. V. Lomakin and E. Michielssen, “Transmission of transient plane waves through perfect electrically conducting plates perforated by periodic arrays of subwavelength holes,” IEEE Trans. Antenn. Propag. 54(3), 970–984 (2006). [CrossRef]  

22. N. V. Golovastikov, D. A. Bykov, L. L. Doskolovich, and V. A. Soifer, “Spatiotemporal Optical Pulse Transformation by a Resonant Diffraction Grating,” J. Exp. Theor. Phys. 121(5), 785–792 (2015). [CrossRef]  

23. N. V. Golovastikov, D. A. Bykov, and L. L. Doskolovich, “Spatiotemporal pulse shaping using resonant diffraction gratings,” Opt. Lett. 40(15), 3492–3495 (2015). [CrossRef]   [PubMed]  

24. G. Baym, Lectures on Quantum Mechanics (Benjamin, 1969).

25. M. G. Moharam, D. A. Pommet, E. B. Grann, and T. K. Gaylord, “Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach,” J. Opt. Soc. Am. A 12(5), 1077–1086 (1995). [CrossRef]  

26. S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66(4), 045102 (2002). [CrossRef]  

27. T. Myint-U and L. Debnath, Linear Partial Differential Equations for Scientists and Engineers (Birkhäuser, 2007).

28. A. Erdélyi, Tables of Integral Transforms (McGraw-Hill, 1954), Vol. I.

References

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  • |

  1. A. Silva, F. Monticone, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Performing mathematical operations with metamaterials,” Science 343(6167), 160–163 (2014).
    [Crossref] [PubMed]
  2. R. Slavík, Y. Park, M. Kulishov, and J. Azaña, “Terahertz-bandwidth high-order temporal differentiators based on phase-shifted long-period fiber gratings,” Opt. Lett. 34(20), 3116–3118 (2009).
    [Crossref] [PubMed]
  3. L. M. Rivas, S. Boudreau, Y. Park, R. Slavík, S. Larochelle, A. Carballar, and J. Azaña, “Experimental demonstration of ultrafast all-fiber high-order photonic temporal differentiators,” Opt. Lett. 34(12), 1792–1794 (2009).
    [Crossref] [PubMed]
  4. N. K. Berger, B. Levit, B. Fischer, M. Kulishov, D. V. Plant, and J. Azaña, “Temporal differentiation of optical signals using a phase-shifted fiber Bragg grating,” Opt. Express 15(2), 371–381 (2007).
    [Crossref] [PubMed]
  5. M. Kulishov and J. Azaña, “Design of high-order all-optical temporal differentiators based on multiple-phase-shifted fiber Bragg gratings,” Opt. Express 15(10), 6152–6166 (2007).
    [Crossref] [PubMed]
  6. N. Quoc Ngo, “Design of an optical temporal integrator based on a phase-shifted fiber Bragg grating in transmission,” Opt. Lett. 32(20), 3020–3022 (2007).
    [Crossref] [PubMed]
  7. L. L. Doskolovich, D. A. Bykov, E. A. Bezus, and V. A. Soifer, “Spatial differentiation of optical beams using phase-shifted Bragg grating,” Opt. Lett. 39(5), 1278–1281 (2014).
    [Crossref] [PubMed]
  8. N. V. Golovastikov, D. A. Bykov, L. L. Doskolovich, and E. A. Bezus, “Spatial optical integrator based on phase-shifted Bragg gratings,” Opt. Commun. 338, 457–460 (2015).
    [Crossref]
  9. D. A. Bykov, L. L. Doskolovich, E. A. Bezus, and V. A. Soifer, “Optical computation of the Laplace operator using phase-shifted Bragg grating,” Opt. Express 22(21), 25084–25092 (2014).
    [Crossref] [PubMed]
  10. D. A. Bykov, L. L. Doskolovich, and V. A. Soifer, “Single-resonance diffraction gratings for time-domain pulse transformations: integration of optical signals,” J. Opt. Soc. Am. A 29(8), 1734–1740 (2012).
    [Crossref] [PubMed]
  11. Z. Ruan, “Spatial mode control of surface plasmon polariton excitation with gain medium: from spatial differentiator to integrator,” Opt. Lett. 40(4), 601–604 (2015).
    [Crossref] [PubMed]
  12. W. Liu, M. Li, R. S. Guzzon, E. J. Norberg, J. S. Parker, M. Lu, L. A. Coldren, and J. Yao, “A fully reconfigurable photonic integrated signal processor,” Nat. Photonics 10(3), 190–195 (2016).
    [Crossref]
  13. T. Yang, J. Dong, L. Lu, L. Zhou, A. Zheng, X. Zhang, and J. Chen, “All-optical differential equation solver with constant-coefficient tunable based on a single microring resonator,” Sci. Rep. 4, 5581 (2014).
    [PubMed]
  14. J. Wu, P. Cao, X. Hu, X. Jiang, T. Pan, Y. Yang, C. Qiu, C. Tremblay, and Y. Su, “Compact tunable silicon photonic differential-equation solver for general linear time-invariant systems,” Opt. Express 22(21), 26254–26264 (2014).
    [Crossref] [PubMed]
  15. M. Ferrera, Y. Park, L. Razzari, B. E. Little, S. T. Chu, R. Morandotti, D. J. Moss, and J. Azaña, “All-optical 1st and 2nd order integration on a chip,” Opt. Express 19(23), 23153–23161 (2011).
    [Crossref] [PubMed]
  16. N. L. Kazanskiy, P. G. Serafimovich, and S. N. Khonina, “Use of photonic crystal cavities for temporal differentiation of optical signals,” Opt. Lett. 38(7), 1149–1151 (2013).
    [Crossref] [PubMed]
  17. N. L. Kazanskiy and P. G. Serafimovich, “Coupled-resonator optical waveguides for temporal integration of optical signals,” Opt. Express 22(11), 14004–14013 (2014).
    [Crossref] [PubMed]
  18. S. M. Sepke and D. P. Umstadter, “Exact analytical solution for the vector electromagnetic field of Gaussian, flattened Gaussian, and annular Gaussian laser modes,” Opt. Lett. 31(10), 1447–1449 (2006).
    [Crossref] [PubMed]
  19. G. Zhou, “The analytical vectorial structure of a nonparaxial Gaussian beam close to the source,” Opt. Express 16(6), 3504–3514 (2008).
    [Crossref] [PubMed]
  20. D. A. Bykov and L. L. Doskolovich, “ω–kx Fano line shape in photonic crystal slabs,” Phys. Rev. A 92(1), 013845 (2015).
    [Crossref]
  21. V. Lomakin and E. Michielssen, “Transmission of transient plane waves through perfect electrically conducting plates perforated by periodic arrays of subwavelength holes,” IEEE Trans. Antenn. Propag. 54(3), 970–984 (2006).
    [Crossref]
  22. N. V. Golovastikov, D. A. Bykov, L. L. Doskolovich, and V. A. Soifer, “Spatiotemporal Optical Pulse Transformation by a Resonant Diffraction Grating,” J. Exp. Theor. Phys. 121(5), 785–792 (2015).
    [Crossref]
  23. N. V. Golovastikov, D. A. Bykov, and L. L. Doskolovich, “Spatiotemporal pulse shaping using resonant diffraction gratings,” Opt. Lett. 40(15), 3492–3495 (2015).
    [Crossref] [PubMed]
  24. G. Baym, Lectures on Quantum Mechanics (Benjamin, 1969).
  25. M. G. Moharam, D. A. Pommet, E. B. Grann, and T. K. Gaylord, “Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach,” J. Opt. Soc. Am. A 12(5), 1077–1086 (1995).
    [Crossref]
  26. S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66(4), 045102 (2002).
    [Crossref]
  27. T. Myint-U and L. Debnath, Linear Partial Differential Equations for Scientists and Engineers (Birkhäuser, 2007).
  28. A. Erdélyi, Tables of Integral Transforms (McGraw-Hill, 1954), Vol. I.

2016 (1)

W. Liu, M. Li, R. S. Guzzon, E. J. Norberg, J. S. Parker, M. Lu, L. A. Coldren, and J. Yao, “A fully reconfigurable photonic integrated signal processor,” Nat. Photonics 10(3), 190–195 (2016).
[Crossref]

2015 (5)

N. V. Golovastikov, D. A. Bykov, L. L. Doskolovich, and E. A. Bezus, “Spatial optical integrator based on phase-shifted Bragg gratings,” Opt. Commun. 338, 457–460 (2015).
[Crossref]

Z. Ruan, “Spatial mode control of surface plasmon polariton excitation with gain medium: from spatial differentiator to integrator,” Opt. Lett. 40(4), 601–604 (2015).
[Crossref] [PubMed]

D. A. Bykov and L. L. Doskolovich, “ω–kx Fano line shape in photonic crystal slabs,” Phys. Rev. A 92(1), 013845 (2015).
[Crossref]

N. V. Golovastikov, D. A. Bykov, L. L. Doskolovich, and V. A. Soifer, “Spatiotemporal Optical Pulse Transformation by a Resonant Diffraction Grating,” J. Exp. Theor. Phys. 121(5), 785–792 (2015).
[Crossref]

N. V. Golovastikov, D. A. Bykov, and L. L. Doskolovich, “Spatiotemporal pulse shaping using resonant diffraction gratings,” Opt. Lett. 40(15), 3492–3495 (2015).
[Crossref] [PubMed]

2014 (6)

2013 (1)

2012 (1)

2011 (1)

2009 (2)

2008 (1)

2007 (3)

2006 (2)

S. M. Sepke and D. P. Umstadter, “Exact analytical solution for the vector electromagnetic field of Gaussian, flattened Gaussian, and annular Gaussian laser modes,” Opt. Lett. 31(10), 1447–1449 (2006).
[Crossref] [PubMed]

V. Lomakin and E. Michielssen, “Transmission of transient plane waves through perfect electrically conducting plates perforated by periodic arrays of subwavelength holes,” IEEE Trans. Antenn. Propag. 54(3), 970–984 (2006).
[Crossref]

2002 (1)

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66(4), 045102 (2002).
[Crossref]

1995 (1)

Alù, A.

A. Silva, F. Monticone, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Performing mathematical operations with metamaterials,” Science 343(6167), 160–163 (2014).
[Crossref] [PubMed]

Azaña, J.

Berger, N. K.

Bezus, E. A.

Boudreau, S.

Bykov, D. A.

Cao, P.

Carballar, A.

Castaldi, G.

A. Silva, F. Monticone, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Performing mathematical operations with metamaterials,” Science 343(6167), 160–163 (2014).
[Crossref] [PubMed]

Chen, J.

T. Yang, J. Dong, L. Lu, L. Zhou, A. Zheng, X. Zhang, and J. Chen, “All-optical differential equation solver with constant-coefficient tunable based on a single microring resonator,” Sci. Rep. 4, 5581 (2014).
[PubMed]

Chu, S. T.

Coldren, L. A.

W. Liu, M. Li, R. S. Guzzon, E. J. Norberg, J. S. Parker, M. Lu, L. A. Coldren, and J. Yao, “A fully reconfigurable photonic integrated signal processor,” Nat. Photonics 10(3), 190–195 (2016).
[Crossref]

Dong, J.

T. Yang, J. Dong, L. Lu, L. Zhou, A. Zheng, X. Zhang, and J. Chen, “All-optical differential equation solver with constant-coefficient tunable based on a single microring resonator,” Sci. Rep. 4, 5581 (2014).
[PubMed]

Doskolovich, L. L.

Engheta, N.

A. Silva, F. Monticone, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Performing mathematical operations with metamaterials,” Science 343(6167), 160–163 (2014).
[Crossref] [PubMed]

Ferrera, M.

Fischer, B.

Galdi, V.

A. Silva, F. Monticone, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Performing mathematical operations with metamaterials,” Science 343(6167), 160–163 (2014).
[Crossref] [PubMed]

Gaylord, T. K.

Gippius, N. A.

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66(4), 045102 (2002).
[Crossref]

Golovastikov, N. V.

N. V. Golovastikov, D. A. Bykov, and L. L. Doskolovich, “Spatiotemporal pulse shaping using resonant diffraction gratings,” Opt. Lett. 40(15), 3492–3495 (2015).
[Crossref] [PubMed]

N. V. Golovastikov, D. A. Bykov, L. L. Doskolovich, and V. A. Soifer, “Spatiotemporal Optical Pulse Transformation by a Resonant Diffraction Grating,” J. Exp. Theor. Phys. 121(5), 785–792 (2015).
[Crossref]

N. V. Golovastikov, D. A. Bykov, L. L. Doskolovich, and E. A. Bezus, “Spatial optical integrator based on phase-shifted Bragg gratings,” Opt. Commun. 338, 457–460 (2015).
[Crossref]

Grann, E. B.

Guzzon, R. S.

W. Liu, M. Li, R. S. Guzzon, E. J. Norberg, J. S. Parker, M. Lu, L. A. Coldren, and J. Yao, “A fully reconfigurable photonic integrated signal processor,” Nat. Photonics 10(3), 190–195 (2016).
[Crossref]

Hu, X.

Ishihara, T.

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66(4), 045102 (2002).
[Crossref]

Jiang, X.

Kazanskiy, N. L.

Khonina, S. N.

Kulishov, M.

Larochelle, S.

Levit, B.

Li, M.

W. Liu, M. Li, R. S. Guzzon, E. J. Norberg, J. S. Parker, M. Lu, L. A. Coldren, and J. Yao, “A fully reconfigurable photonic integrated signal processor,” Nat. Photonics 10(3), 190–195 (2016).
[Crossref]

Little, B. E.

Liu, W.

W. Liu, M. Li, R. S. Guzzon, E. J. Norberg, J. S. Parker, M. Lu, L. A. Coldren, and J. Yao, “A fully reconfigurable photonic integrated signal processor,” Nat. Photonics 10(3), 190–195 (2016).
[Crossref]

Lomakin, V.

V. Lomakin and E. Michielssen, “Transmission of transient plane waves through perfect electrically conducting plates perforated by periodic arrays of subwavelength holes,” IEEE Trans. Antenn. Propag. 54(3), 970–984 (2006).
[Crossref]

Lu, L.

T. Yang, J. Dong, L. Lu, L. Zhou, A. Zheng, X. Zhang, and J. Chen, “All-optical differential equation solver with constant-coefficient tunable based on a single microring resonator,” Sci. Rep. 4, 5581 (2014).
[PubMed]

Lu, M.

W. Liu, M. Li, R. S. Guzzon, E. J. Norberg, J. S. Parker, M. Lu, L. A. Coldren, and J. Yao, “A fully reconfigurable photonic integrated signal processor,” Nat. Photonics 10(3), 190–195 (2016).
[Crossref]

Michielssen, E.

V. Lomakin and E. Michielssen, “Transmission of transient plane waves through perfect electrically conducting plates perforated by periodic arrays of subwavelength holes,” IEEE Trans. Antenn. Propag. 54(3), 970–984 (2006).
[Crossref]

Moharam, M. G.

Monticone, F.

A. Silva, F. Monticone, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Performing mathematical operations with metamaterials,” Science 343(6167), 160–163 (2014).
[Crossref] [PubMed]

Morandotti, R.

Moss, D. J.

Muljarov, E. A.

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66(4), 045102 (2002).
[Crossref]

Norberg, E. J.

W. Liu, M. Li, R. S. Guzzon, E. J. Norberg, J. S. Parker, M. Lu, L. A. Coldren, and J. Yao, “A fully reconfigurable photonic integrated signal processor,” Nat. Photonics 10(3), 190–195 (2016).
[Crossref]

Pan, T.

Park, Y.

Parker, J. S.

W. Liu, M. Li, R. S. Guzzon, E. J. Norberg, J. S. Parker, M. Lu, L. A. Coldren, and J. Yao, “A fully reconfigurable photonic integrated signal processor,” Nat. Photonics 10(3), 190–195 (2016).
[Crossref]

Plant, D. V.

Pommet, D. A.

Qiu, C.

Quoc Ngo, N.

Razzari, L.

Rivas, L. M.

Ruan, Z.

Sepke, S. M.

Serafimovich, P. G.

Silva, A.

A. Silva, F. Monticone, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Performing mathematical operations with metamaterials,” Science 343(6167), 160–163 (2014).
[Crossref] [PubMed]

Slavík, R.

Soifer, V. A.

Su, Y.

Tikhodeev, S. G.

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66(4), 045102 (2002).
[Crossref]

Tremblay, C.

Umstadter, D. P.

Wu, J.

Yablonskii, A. L.

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66(4), 045102 (2002).
[Crossref]

Yang, T.

T. Yang, J. Dong, L. Lu, L. Zhou, A. Zheng, X. Zhang, and J. Chen, “All-optical differential equation solver with constant-coefficient tunable based on a single microring resonator,” Sci. Rep. 4, 5581 (2014).
[PubMed]

Yang, Y.

Yao, J.

W. Liu, M. Li, R. S. Guzzon, E. J. Norberg, J. S. Parker, M. Lu, L. A. Coldren, and J. Yao, “A fully reconfigurable photonic integrated signal processor,” Nat. Photonics 10(3), 190–195 (2016).
[Crossref]

Zhang, X.

T. Yang, J. Dong, L. Lu, L. Zhou, A. Zheng, X. Zhang, and J. Chen, “All-optical differential equation solver with constant-coefficient tunable based on a single microring resonator,” Sci. Rep. 4, 5581 (2014).
[PubMed]

Zheng, A.

T. Yang, J. Dong, L. Lu, L. Zhou, A. Zheng, X. Zhang, and J. Chen, “All-optical differential equation solver with constant-coefficient tunable based on a single microring resonator,” Sci. Rep. 4, 5581 (2014).
[PubMed]

Zhou, G.

Zhou, L.

T. Yang, J. Dong, L. Lu, L. Zhou, A. Zheng, X. Zhang, and J. Chen, “All-optical differential equation solver with constant-coefficient tunable based on a single microring resonator,” Sci. Rep. 4, 5581 (2014).
[PubMed]

IEEE Trans. Antenn. Propag. (1)

V. Lomakin and E. Michielssen, “Transmission of transient plane waves through perfect electrically conducting plates perforated by periodic arrays of subwavelength holes,” IEEE Trans. Antenn. Propag. 54(3), 970–984 (2006).
[Crossref]

J. Exp. Theor. Phys. (1)

N. V. Golovastikov, D. A. Bykov, L. L. Doskolovich, and V. A. Soifer, “Spatiotemporal Optical Pulse Transformation by a Resonant Diffraction Grating,” J. Exp. Theor. Phys. 121(5), 785–792 (2015).
[Crossref]

J. Opt. Soc. Am. A (2)

Nat. Photonics (1)

W. Liu, M. Li, R. S. Guzzon, E. J. Norberg, J. S. Parker, M. Lu, L. A. Coldren, and J. Yao, “A fully reconfigurable photonic integrated signal processor,” Nat. Photonics 10(3), 190–195 (2016).
[Crossref]

Opt. Commun. (1)

N. V. Golovastikov, D. A. Bykov, L. L. Doskolovich, and E. A. Bezus, “Spatial optical integrator based on phase-shifted Bragg gratings,” Opt. Commun. 338, 457–460 (2015).
[Crossref]

Opt. Express (7)

Opt. Lett. (8)

N. V. Golovastikov, D. A. Bykov, and L. L. Doskolovich, “Spatiotemporal pulse shaping using resonant diffraction gratings,” Opt. Lett. 40(15), 3492–3495 (2015).
[Crossref] [PubMed]

S. M. Sepke and D. P. Umstadter, “Exact analytical solution for the vector electromagnetic field of Gaussian, flattened Gaussian, and annular Gaussian laser modes,” Opt. Lett. 31(10), 1447–1449 (2006).
[Crossref] [PubMed]

N. L. Kazanskiy, P. G. Serafimovich, and S. N. Khonina, “Use of photonic crystal cavities for temporal differentiation of optical signals,” Opt. Lett. 38(7), 1149–1151 (2013).
[Crossref] [PubMed]

Z. Ruan, “Spatial mode control of surface plasmon polariton excitation with gain medium: from spatial differentiator to integrator,” Opt. Lett. 40(4), 601–604 (2015).
[Crossref] [PubMed]

N. Quoc Ngo, “Design of an optical temporal integrator based on a phase-shifted fiber Bragg grating in transmission,” Opt. Lett. 32(20), 3020–3022 (2007).
[Crossref] [PubMed]

L. L. Doskolovich, D. A. Bykov, E. A. Bezus, and V. A. Soifer, “Spatial differentiation of optical beams using phase-shifted Bragg grating,” Opt. Lett. 39(5), 1278–1281 (2014).
[Crossref] [PubMed]

R. Slavík, Y. Park, M. Kulishov, and J. Azaña, “Terahertz-bandwidth high-order temporal differentiators based on phase-shifted long-period fiber gratings,” Opt. Lett. 34(20), 3116–3118 (2009).
[Crossref] [PubMed]

L. M. Rivas, S. Boudreau, Y. Park, R. Slavík, S. Larochelle, A. Carballar, and J. Azaña, “Experimental demonstration of ultrafast all-fiber high-order photonic temporal differentiators,” Opt. Lett. 34(12), 1792–1794 (2009).
[Crossref] [PubMed]

Phys. Rev. A (1)

D. A. Bykov and L. L. Doskolovich, “ω–kx Fano line shape in photonic crystal slabs,” Phys. Rev. A 92(1), 013845 (2015).
[Crossref]

Phys. Rev. B (1)

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66(4), 045102 (2002).
[Crossref]

Sci. Rep. (1)

T. Yang, J. Dong, L. Lu, L. Zhou, A. Zheng, X. Zhang, and J. Chen, “All-optical differential equation solver with constant-coefficient tunable based on a single microring resonator,” Sci. Rep. 4, 5581 (2014).
[PubMed]

Science (1)

A. Silva, F. Monticone, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Performing mathematical operations with metamaterials,” Science 343(6167), 160–163 (2014).
[Crossref] [PubMed]

Other (3)

T. Myint-U and L. Debnath, Linear Partial Differential Equations for Scientists and Engineers (Birkhäuser, 2007).

A. Erdélyi, Tables of Integral Transforms (McGraw-Hill, 1954), Vol. I.

G. Baym, Lectures on Quantum Mechanics (Benjamin, 1969).

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Figures (6)

Fig. 1
Fig. 1 Incident pulse P inc ( x,y,z,t ) and reflected pulse P ref ( x,y,z,t ) upon diffraction by a multilayer structure.
Fig. 2
Fig. 2 (a) Transmission spectra | T( k x ,ω ) | of the PSBG at N l =29 calculated using RCWA (left half, k x <0 ) and using the resonant approximation (12) (right half, k x >0 ) in the case of TM-polarization. (b) Reflection spectra | R( k x ,ω ) | of the PSBG at N l =9 calculated using RCWA (left half, k x <0 ) and using the resonant approximation (12) (right half, k x >0 ) in the case of TM-polarization.
Fig. 3
Fig. 3 (a) Envelope of the pulse incident on a PSBG with N l =29 layers. (b) Absolute value of the envelope of the transmitted pulse calculated using RCWA and Eqs. (5), (6) (left half, x<0 ) and using the proposed analytical model (12), (21), (22) (right half, x>0 ). The insets show the corresponding isosurfaces of the incident and transmitted pulses in ( x,y,t ) space.
Fig. 4
Fig. 4 (a) Envelope of the pulse incident on a PSBG with N l =9 layers. (b) Absolute value of the envelope of the reflected pulse calculated using RCWA and Eqs. (4), (6) (left half, x<0 ) and using the proposed analytical model (12), (21), (22) (right half, x>0 ). (c) Absolute value of the temporal derivative of the incident pulse. The insets show the corresponding isosurfaces of the incident and reflected pulses in ( x,y,t ) space.
Fig. 5
Fig. 5 (a) Envelope of the pulse incident of a PSBG with N l =9 layers. (b) Absolute value of the reflected pulse envelope calculated using RCWA and Eqs. (4), (6) (left half, x<0 ) and using the proposed analytical model (12), (21), (22) (right half, x>0 ). (c) Laplacian of the envelope of the incident pulse. The insets show the corresponding isosurfaces of the incident and reflected pulses in ( x,y,t ) space.
Fig. 6
Fig. 6 (a) Envelope of the pulse incident on a PSBG with N l =9 layers. (b) Absolute value of the envelope of the reflected pulse calculated using RCWA and Eqs. (4), (6) (left half, x<0 ) and using the proposed analytical model (12), (21), (22) (right half, x>0 ). (c) Estimate of the envelope of the reflected pulse obtained using Eq. (20). The insets show the corresponding isosurfaces of the incident and reflected pulses in ( x,y,t ) space.

Equations (34)

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E( x,y,z,t )=P( x,y,z,t )exp{ i ω 0 c n sup zi ω 0 t } = V( k x , k y ,ω )exp{ i k x x+i k y yi k z zi( ω+ ω 0 )t }d k x d k y dω ,
E( x,y,0,t )=v( x,y,t )exp{ i ω 0 t },
v( x,y,t )=P( x,y,0,t )= V( k x , k y ,ω )exp{ i k x x+i k y yiωt }d k x d k y dω
u R ( x,y,t )= V( k x , k y ,ω ) R ˜ ( k x , k y ,ω+ ω 0 )exp{ i k x x+i k y yiωt }d k x d k y dω ,
u T ( x,y,t )= V( k x , k y ,ω ) T ˜ ( k x , k y ,ω+ ω 0 )exp{ i k x x+i k y yiωt }d k x d k y dω ,
R ˜ ( k x , k y ,ω+ ω 0 )= R ТE ( k x , k y ,ω+ ω 0 )+ k x 2 k x 2 + k y 2 [ R ТM ( k x , k y ,ω+ ω 0 ) R ТE ( k x , k y ,ω+ ω 0 ) ], T ˜ ( k x , k y ,ω+ ω 0 )= T ТE ( k x , k y ,ω+ ω 0 )+ k x 2 k x 2 + k y 2 [ T ТM ( k x , k y ,ω+ ω 0 ) T ТE ( k x , k y ,ω+ ω 0 ) ].
H R ( k x , k y ,ω )= R ˜ ( k x , k y ,ω+ ω 0 ), H T ( k x , k y ,ω )= T ˜ ( k x , k y ,ω+ ω 0 ).
R( k x , k y ,ω )=R( k x 2 + k y 2 ,0,ω ),T( k x , k y ,ω )=T( k x 2 + k y 2 ,0,ω ).
n ˜ 1 h 1 = n ˜ 2 h 2 = λ B /4,
R( k x ,ω )= γ R v g 2 k x 2 ( ω ω z R )( ω ω z2 R ) v g 2 k x 2 ( ω ω p )( ω ω p2 ) , T( k x ,ω )= γ T v g 2 k x 2 ( ω ω z T )( ω ω z2 T ) v g 2 k x 2 ( ω ω p )( ω ω p2 ) ,
R( ω * )=R ( ω ) * ,T( ω * )=T ( ω ) * .
R( k x , k y ,ω )=R( k x 2 + k y 2 ,ω )= γ R v g 2 ( k x 2 + k y 2 )( ω ω z R )( ω+ ω z R ) v g 2 ( k x 2 + k y 2 )( ω ω p )( ω+ ω p * ) , T( k x , k y ,ω )=T( k x 2 + k y 2 ,ω )= γ T v g 2 ( k x 2 + k y 2 )( ω ω z T )( ω+ ω z T* ) v g 2 ( k x 2 + k y 2 )( ω ω p )( ω+ ω p * ) .
R TM ( 0,ω )= R TE ( 0,ω ), T TM ( 0,ω )= T TE ( 0,ω ).
H R,T ( k x , k y ,ω )= H R,T 1 ( k x , k y ,ω ) H R,T 2 ( k x , k y ,ω ),
H R,T 1 ( k x , k y ,ω )= γ R,T [ v g 2 ( k x 2 + k y 2 )( ω+ ω 0 ω z R,T )( ω+ ω 0 + ω z R,T* ) ],
H R,T 2 ( k x , k y ,ω )= [ v g 2 ( k x 2 + k y 2 )( ω+ ω 0 ω p )( ω+ ω 0 + ω p * ) ] 1 ,
U R,T ( k x , k y ,ω )= H R,T ( k x , k y ,ω )V( k x , k y ,ω ).
[ H R,T 2 ( k x , k y ,ω ) ] 1 U R,T ( k x , k y ,ω )= H R,T 1 ( k x , k y ,ω )V( k x , k y ,ω ).
2 u R,T t 2 v g 2 Δ u R,T 2i( ω 0 iIm ω p ) u R,T t ( ω 0 ω p )( ω 0 + ω p * ) u R,T = f R,T ,
f R,T ( x,y,t )= γ R,T ( 2 v t 2 v g 2 Δv2i( ω 0 iIm ω z R,T ) v t ( ω 0 ω z R,T )( ω 0 + ω z R,T* )v ),
u R,T ( x,y,t )= + + + f R,T ( ξ,η,τ )h( xξ,yη,tτ )dξdη dτ ,
h( x,y,t )= exp{ i ω 0 t+Im ω p t }cos( t 2 ( x 2 + y 2 ) / v g 2 Re ω p ) 2π v g 2 t 2 ( x 2 + y 2 ) / v g 2 , x 2 + y 2 v g t.
f R ( 0,0,0 )= 4 v g 2 σ 2 + | ω z | 2 2 σ t 2 ω 0 2 .
2 g( x,y,t ) t 2 v g 2 Δg( x,y,t )+ ( Re ω p ) 2 g( x,y,t )=f( x,y,t ),
f( x,y,t )= γ R,T ( 2 p t 2 v g 2 Δp+ ( Re ω z R,T ) 2 p )exp{ Im ω z R,T tIm ω p t },
u R,T ( x,y,t )=g( x,y,t )exp{ i ω 0 t+Im ω p t }, v( x,y,t )=p( x,y,t )exp{ i ω 0 t+Im ω z R,T t }.
G( k x , k y ,t )= 1 2π g( ξ,η,t )exp{ i k x ξ+i k y η }dξdη , F( k x , k y ,t )= 1 2π f( ξ,η,t )exp{ i k x ξ+i k y η }dξdη .
2 G( k x , k y ,t ) t 2 +( v g 2 ( k x 2 + k y 2 )+ ( Re ω p ) 2 )G( k x , k y ,t )=F( k x , k y ,t )
G( k x , k y ,t )= 0 t w( k x , k y ,tτ )F( k x , k y ,τ )dτ ,
g( x,y,t )= 1 2π G( k x , k y ,t )exp{ i k x xi k y y }d k x d k y = 1 ( 2π ) 2 0 t [ f( ξ,η,τ ) [ w( k x , k y ,tτ )exp{ i k x ( ξx )+i k y ( ηy ) }d k x d k y ]dξdη ]dτ .
1 2π + sin( y k 2 + q 2 ) k 2 + q 2 exp( ikx )dk ={ 1 2 J 0 ( q y 2 q 2 ), | x |<y, 0, | x |>y,
g( x,y,t )= 1 4π v g 0 t dτ × f( ξ,η,τ )[ J 0 ( v g 2 ( tτ ) 2 ( ξx ) 2 ( Re ω p ) v g 2 2 + k y 2 )exp{ i k y ( ηy ) }d k y ]dξdη.
1 2π + J 0 ( d k 2 + q 2 )exp( iky )dk ={ 1 π cos( q d 2 y 2 ) d 2 y 2 , | y |<d, 0, | y |>d,
g( x,y,t )= 1 2π v g 2 0 t dτ cos( Re[ ω p ] ( tτ ) 2 ( ( ξx ) 2 + ( ηy ) 2 ) / v g 2 ) ( tτ ) 2 ( ( ξx ) 2 + ( ηy ) 2 ) / v g 2 f( ξ,η,τ )dξdη .

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