Group delay of electromagnetic pulses through multilayer dielectric mirrors beyond special relativity

In this Letter, we investigate the group delay of optical pulses through multilayer dielectric mirrors (MDM) combined with gravitational wave (GW). We find that the delay increases linearly with MDM length for the transmitted wave packet, whereas the Hartman effect disappears. Our study provides insight into the nature of both quantum tunnelling and GW.


GW.
The length of time during which quantum particles tunnel through a barrier has attracted considerable attention for both fundamental and technological reasons since the 1930s 1-6 . Hartman calculated the tunneling of a wavepacket through a rectangular potential barrier 3 and found that group delay becomes constant as barrier length increases. This phenomenon, known as the Hartman effect, implies that for sufficiently large barriers, the effective group velocity of a particle may be superluminal. Although a number of experiments have reported observations of electromagnetic waves propagating with "superluminal tunneling velocities," the definition of tunneling time and its exact physical meaning based on experimental results remain under heated debate [4][5][6] . A large number of tunneling time definitions have been proposed, including group delay or phase time 3 , dwell time 6 , Larmor times 11,12 , and Büttiker-Landauer time 13 . Winful recently proposed that the group delay in tunneling represents a lifetime of stored energy escaping through both sides of the barrier and does not represent a transit time 6,14 . Thus, the issues of superluminality, causality, or the speed of information transfer do not even arise.
Beyond Schrödinger's nonrelativistic quantum mechanics, the group delay for Dirac particles traveling through a potential well was also studied by using Dirac's fully relativistic quantum theory 15 . The behavior of Dirac particles is found to be the same as that in nonrelativistic quantum mechanics. Liu et al. recently studied the effect of the electromagnetic fields of the group delay of electrons and found that the group delay of the transmitted wave packet increases linearly with barrier length for the transmitted wave packet 16 . This peculiar tunneling effect is attributed to current leakage in a time-dependent barrier generated via the electromagnetic fields 13,16 . If the quantum fluctuation or the zero-point field is considered, all potential barriers are combined with electromagnetic fields. Thus, in the framework of quantum field theory, the Hartman effect of electron tunneling may disappear because of the existence of electromagnetic vacuum fields.
However, in photon tunnelling, a number of theories based on the special relativis-tic covariant Maxwell equations have proven that the group delay of photons becomes constant as the length of optical structures increases [4][5][6][17][18][19][20] . However, whether the Hartman effect still exists when a more accurate theory (e.g. general relativity theory) remains unclear. One of the unique predictions in general relativity theory is the existence of gravitational wave (GW) [7][8][9][10] . If optical structures such as a multilayer dielectric mirror (MDM) (i.e. a one-dimensional photonic crystal) is irradiated by GW, the center frequency and the width of the photonic bandgap of the MDM will vary with the GW.
Similar to the electron tunnelling in a time-dependent barrier, variations of the photonic bandgap will result in an additional leak photon current. Such current may propagate at the speed of light. Thus, the Hartman effect may also be absent in photon tunnelling within the framework of general relativity theory.
In this paper, we investigate the effect of GW on the group delay of photon passing through MDM. Our simulation shows that with a thick MDM, group delay increases linearly with increasing barrier width. The group velocity is approximately 2.95 × 10 8 m/s, slightly less than the speed of light in in vacuum. Superluminality or causality no longer occurs. We also find that the group delay of tunneling photons is sensitive to GW.
Our study may facilitate further understanding of both quantum tunnelling and GW as well as provide a different method for the detection of GW. In particularly, the MDM comprises alternating dielectric layers and vacuum layers [see Fig. 1(a)]. All layers are nonmagnetic (µ = 1), and the thicknesses of the dielectric layers (vacuum layers) satisfy where ε 1 is the permittivity of dielectric layers, λ 0 is the center frequency of the input electromagnetic pulse, and ζ 2 is a positive integer.
The group delay of tunneling photons is generally more sensitive to GW at large ζ 2 .
We set ζ 2 = 5 in this paper, unless otherwise specified. The electromagnetic pulse is incident along the normal of the surface of MDM, and the propagation direction of the plane-polarized GW is parallel to the surface of MDM.
When the polarization GW occurs, the layer spacing of dielectric layers will vary with the GW. Similar to the case of two masses separated by a distance D along the Z direction that are coupled by a lossy spring, the equation for the differential motion of the masses in the GW becomes 7, 21, 22 where z R is the proper relative displacement of the two masses, h 22 is the the perturbation matrix (tensor) element resulting from the GW; and ω m0 and Q are the natural frequency and the q-factor of the lossy spring, respectively. If the dielectric layers in MDM are free and under zero-velocity initial conditions, the equation of differential motion can be described by 7, 21, 22 where A GW is the GW amplitude. Thus, in the propagation of electromagnetic waves, the permittivity distribution will also change with time. To study such a time-dependent photon scattering process, we employ the finite-difference time-domain (FDTD) method to solve the time-dependent Maxwell equations numerically 23,24 . FDTD is a timedomain technique that can provide animated displays of electromagnetic field movement through various models, e.g., the photonic crystals, radar, etc. In the FDTD method, the one-dimensional Maxwell equations are replaced by a finite set of finite differential where E x (H y ) is the electric field (magnetic field) of the electromagnetic wave, (i, k) = (i△x, k△t) denote a grid point of the space and time; and for any function of space and time F (i△x, k△t) = F k (i), Υ GW depends on the GW wave-induced displacement.
According to Eq. 2, in the vacuum layers Υ GW = 1 + A GW cos(ω GW t)/2. The thickness changes of the dielectric layers are minimal because the natural frequency of dielectric layer is nonresonant with the frequency of GW, and the thickness the dielectric layers is considerably less than that of the vacuum layers with large L 2 . Thus, we set Υ GW = 1 in dielectric layers. At the input boundary, a Gaussian electromagnetic wave packet is injected. The wave function at the input boundary is set as However, the influence of the GW on the group delay of a transmitted wave packet differs from the given case. As shown in Fig. 1(c), for the GW amplitude A GW = 1 × On the other hand, for a non-strictly periodic GW emitted by various sources(e.g., the chaos compact binary system 26 ), the time-dependent variations attributed to the GW will also result in an additional leak photon current, thus modifying the group delay.
From Fig. 1(c), we can also find that the group velocity of the additional leakage current Finally, we discuss the experimental realization of our theoretical predication. Although a strong GW is used in the numerical calculation, our results show that the group delay of the tunneling photons is sensitive to GW and that sensitivity can be increased for a relatively low-frequency GW or for thick vacuum layers. Owing to ultrafast laser technology, an extremely high time resolution can be achieved. Thus, the detection of the effect of the GW on group delay may be feasible. Notably, a pulsar emits a beam of electromagnetic radiation with ultra-high accuracy and stability. Thus, when the pulsar electromagnetic radiation tunnels through a strong GW radiation source such as the black hole binary, the variation of the group delay should be detectable through astronomical observation.
In conclusion, we have calculated the group delay of optical pulses through MDM combined with GW. We found that the group delay increases linearly with MDM length for the transmitted wave packet. The Hartman effect disappears. This peculiar tunneling effect is attributed to the additional current leakage attributed to the GW-induced variations of the photonic bandgap. We also show that the group delay of the tunneling photons is sensitive to GW. For a relatively low-frequency GW or thick vacuum layers, the sensitivity can be enhanced remarkably. Our study provides insight into the nature both of the quantum tunnelling and GW as well as a novel process by which to detect the GW.