A novel approach to the application of an adaptive pre-shaping algorithm for ultrashort pulse distortion compensation during the propagation in AZO/ZnO multilayered metamaterials (thickness 300-700 nm) at the epsilon-near-zero spectral point is investigated. We show that using the Broyden-Fletcher-Goldfarb-Shanno algorithm to minimize the residual between frequency-resolved optical gating traces of the distorted output pulse and the zero phase pulse of 100 fs duration can yield increased output pulse field strength and a central frequency shift towards the epsilon-near-zero spectral point, which can be of future use for applications in ultrafast communication, signal processing, and super resolution imaging.
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The recent development of plasmonic materials for optical-based devices has yielded novel concepts such as negative refraction , hyperlenses , and epsilon-near-zero (ENZ) metamaterials . ENZ metamaterials, materials with permittivity |ɛ| << 1, resulted in applications such as all-optical switching , radiation phase tailoring , and deformation-free propagation . Metamaterials have also shown potential for light control on length scales smaller than the wavelength . In the context of imaging applications, metamaterials with negative refractive index were proposed for super-resolution imaging . However, any appreciable optical losses or deviations in the material’s permittivity from ɛ=-1 could suppress the resolution. Recently, it has been suggested that shaping the light pulse envelope allows for the dramatic improvement of the performance of the metamaterial-based super-resolution imaging systems in the presence of material loss .
Aluminum-doped zinc oxide (AZO) as an ENZ metamaterial is a promising alternative to noble metals due to its lower optical losses . The multilayered AZO/ZnO metamaterial has demonstrated a strong potential for precise engineering of the optical properties at the ENZ spectral point [11–14]. In particular, varying fabrication parameters of multilayered AZO/ZnO structures (deposition temperature, AZO period, total sample thickness) allow for tuning the plasma frequency, ωp and the damping frequency, γ. . Recent studies also show that AZO exhibits a strong increase in the Kerr nonlinear refractive index at the ENZ spectral point . In general, ENZ materials support various other enhancement mechanisms [16–20]. Our numerical study of ultrashort pulse propagation in multilayered AZO/ZnO showed a “soliton-like” propagation of a 100 fs Gaussian pulse at the ENZ spectral point . However, an ultrashort pulse propagating through distances larger than ∼100 nm experiences a temporal pulse distortion which could potentially be a bottleneck for applications in super resolution imaging.
It has been shown that ENZ regime supports the enhancement of the longitudinal component of the electric field for a TM-polarized wave. [22,23]. It resulted in the availability of the high local fields and allowed for the enhancement of the weak nonlinear-optical effects. As a result, ENZ material boosts the efficiency of the second and third harmonic generation [24–26].
The research of the ultra-short pulse shaping in the presence of nonlinearity and higher-order chromatic dispersion lead to a variety of novel techniques and applications for practical laser devices [27–29]. The development of compact ultrashort pulse sources requires an extensive knowledge of fundamental nonlinear optical phenomena which occur when laser dimensions are reduced to the size comparable to wavelengths. E.g., excessive self-phase-modulation leads to pulse’s phase distortions and, eventually, the pulse may lose coherence and break up. Finding ways to manage nonlinearity, which generally limits the energy of ultrashort pulses, and control the shape of the phase in time domain, allowed the dramatic development of the compact ultrafast fiber laser sources at near infrared .
Pulse pre-shaping is a highly efficient approach to the pulse’s phase control which has been used in a number of applications in the past few decades. For example, spectrally pre-shaped pulses were used in order to prevent gain shaping and nonlinear distortions in fiber amplifiers . Pulse pre-shaping has also been applied for distortion-less transmission of ultrashort pulses through optical fibers . In addition, spectral pre-shaping can be used for the ultrashort-pulse compression [32,33], to control the multiphoton molecular excitation , and maximizing output pulse energy . Many of these studies experimentally adjusted multi-element versions of liquid crystal modulators to generate pre-shaped pulses in real time.
Here, we present the results of a finite-difference-time-domain numerical study for an initial 100 fs Gaussian pulse with a central frequency beyond epsilon-near-zero point during propagation in AZO/ZnO metamaterial with various optical losses. In this work, we propose to use an adaptive algorithm for the distortion compensation of the ultrashort pulse propagating through the AZO/ZnO multilayered structures at the ENZ spectral point.
2. Adaptive pre-shaping approach
A schematic of an adaptive pre-shaping approach is shown in Fig. 1. The finite-difference time-domain (FDTD) numerical method was used to study ultrashort pulse propagation at the ENZ spectral region. A home-build interface was written in Python to access the MIT (Massachusetts Institute of Technology) electromagnetic equation propagation (MEEP) software . The FDTD simulations used an initial Gaussian pulse of 100 fs full width at half maximum (FWHM) duration with zero phase. The frequency-resolved optical gating (FROG) trace of the output pulse is calculated after each iteration and compared to the FROG trace of an initial zero-phase pulse. The adaptive algorithm is then used to predict a 5th-degree polynomial phase for the incoming pulse at the next iteration to minimize pulse distortions. The goal of this study is to develop an adaptive pre-shaping approach for a fast and efficient initial pulse phase optimization for the future experimental control of ultrashort laser pulse at the ENZ spectral point (e.g., using liquid crystal modulators).
3. Ultrashort pulse propagation in the presence of optical loss
The possibility of using plasmonic materials in practical photonic devices comes with significantly increased propagation losses, which can be detrimental for many applications. Therefore, before using adaptive pre-shaping approach, we investigated ultrashort pulse propagation in the presence of various optical losses. The effective medium approximation [37,38] was used to model in-plane and out-of-plane optical permittivity of the AZO/ZnO multilayered structure. The model parameters (AZO/ZnO period thickness: 10 nm, AZO fill factor: 0.4) were chosen to match the experimental data from our previous study . To estimate the optical permittivity, ɛ, for AZO, we used the Drude model:
It is important to note that in the near infrared spectral region the epsilon-near-zero dispersion is observed only in the case of TM-polarized (p-polarized) incident light because only in this polarization is the electric field component of the incident light able to probe both in-plane and out-of-plane components of the dielectric function. In the simulations, the angle of incidence was assumed to be 00 for simplicity (normal incidence). However, in the experiment, the angle of incidence could be varied to enhance the ENZ permittivity effect due to out-of-plane permittivity.
Our previous study  shows that AZO’s damping frequency γ has the highest impact on the optical losses. The estimated transmittance through the AZO/ZnO metamaterial with various γ is presented in Fig. 2(a). Increased propagation losses make it challenging to use even relatively thin AZO/ZnO samples (400-700 nm) for many applications. For example, increase Drude damping frequency from γ=1011 Hz to γ=5 × 1012 Hz leads to the significant decrease in transmittance for AZO/ZnO film (Fig. 2(a)). On the other hand, FDTD numerical simulations show that the lower propagation loss comes at the expense of increased ultrashort pulse distortion (Fig. 2(b)) as compared to ultrashort pulse propagation in the AZO/ZnO with higher Drude damping frequency (Fig. 2(c,d)). Therefore, reducing the propagation loss is a “double-edge sword” which require techniques for pulse distortion compensations at the ENZ spectral point.
4. Spectral tuning at the epsilon-near-zero spectral point
The atomic layer deposition (ALD) and pulsed laser deposition (PLD) fabrication methods offer also an opportunity to tune spectral position of the ENZ point for out-of-plane permittivity by adjusting the doping concentration and/or thickness of the AZO/ZnO metamaterial during fabrication [11–14], which could potentially be useful for finding an optimal initial central frequency. Our FDTD numerical calculations indicate that spectral tuning of the initial Gaussian pulse with FWHM duration of 100 fs propagating through AZO/ZnO sample with the lowest losses (γ=1011 Hz) will significantly affect output pulse temporal profile (Fig. 3). The initial pulse is tuned in such a way that the central frequency is shifted from below ENZ (positive optical permittivity) to beyond ENZ (negative optical permittivity) spectral point (Fig. 3(a-c)). The results (Fig. 3(g-i)) show that the output pulse experiences a spectral shift toward higher frequencies (lower wavelengths) as the pulse propagates through the material. Calculated from the FROG trace (Fig. 3(g)) spectral shift for the ultrashort pulse propagated through 500 nm of the Al:ZnO/ZnO sample is Δv∼0.015 × 1014 Hz. Additional numerical simulations show that increasing optical loss (Drude damping frequency changes from γ=1011 Hz to γ=5 × 1012 Hz) leads to an even larger spectral shift (Δv∼0.020 × 1014 Hz.) for an initial ultrashort pulse with a central frequency beyond ENZ.
5. Results and discussion
We used FDTD simulations to investigate the propagation of a 100 fs Gaussian pulse with zero initial phase and central frequency at ENZ frequency (1.686 × 1014 Hz (1778 nm)) through various distances of the AZO/ZnO metamaterial with lowest optical losses (γ=1011 Hz) (Fig. 4). The resulting output electric fields have a strong side-band which translates to a stretched FROG trace in the negative time-delay space. As the pulse travels between 300 and 700 nm, the distortions become more significant (Fig. 4(a-c)). For the pulse propagated through 700 nm, the resulting FROG trace, Fig. 4(f), has been stretched significantly in the negative time-delay space. As expected from results above (Fig. 3), the central frequency of the FROG traces for these distances is higher than the initial central frequency, which occurred at the ENZ, 1.686 × 1014 Hz (1778 nm).
For practical applications, we must have AZO/ZnO metamaterial with the lowest propagation losses and operate at the frequencies close to ENZ. In our study, we aim to find the optimal temporal phase that would allow to pre-shape input pulse in order to reduce strong distortions (Fig. 4) of the output pulse introduced during propagation in AZO/ZnO metamaterial with the lowest losses (Fig. 4). First, we started by inverting the output pulse’s temporal phase using a Fourier transform and then pre-shaping incoming pulse using that phase. However, this simplified approach did not result in a zero-phase output pulse. Therefore the adaptive pre-shaping approach (Fig. 1(b)) was introduced next. As an initial guess for the pre-shaped pulse, we used a linear temporal phase with a negative slope. This initial phase was approximated using a 5th-degree polynomial (Fig. 1(a)). A wrapper for the temporal phase was written in Python. Then the FDTD numerical simulations were used to obtained output pulse temporal shape and FROG trace was calculated after propagation in AZO/ZnO metamaterial (Fig. 1(b)).
The Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm  was used to minimize the difference between FROG traces for the distorted output pulse and zero phase pulse. To constraint the phase shape, the coefficients of the 5th-degree polynomial were optimized while the exponential terms were kept constant. Since propagation at the ENZ frequency can blue shift the output pulse’s central frequency (Fig. 4), it is not possible to deﬁne the residual for optimization as a simple subtraction of the output pulse’s FROG from a FROG of a 100 fs zero-phase pulse (represented by equidistant concentric circles) centered at the ENZ frequency. In order to find optimal temporal phase for next iteration, we used the distances between the equidistant concentric circles (distances between points: 7 and 8, 8 and 9, 9 and 10, 10 and 7) and absolute position in time/frequency of points 1-6 in the FROG trace of the zero-phase pulse as a reference (Fig. 1(c)). These values are then subtracted from the same time delay and frequency differences in the output pulse’s FROG to define the residual at any central frequency. In total, the residual was made up of 10 points of reference and there were 6 unknowns in the 5th-degree polynomial.
The results of FDTD simulations using adaptive pre-shaping approach are shown in Fig. 5. First, the BFGS algorithm, a well-established method for parameter estimation, was used to investigate the phase space for ultrashort pulses propagating through 300 nm. As the residual decreased, constraints on the coefficients of the 5th-degree polynomial were introduced. These coefficients were allowed to vary within two orders of magnitude of their initial value to allow for a sufficient parameter space. Then the L-BFGS-B algorithm  was used to find final temporal phase for a pre-shaped pulse. The L-BFGS-B algorithm allowed us to impose parameter limitations on the coefficients of the 5th-degree polynomial for a final phase optimization. The tolerance level was set to 10−3 to limit the time spent in each run. Once the final phase shape for the 300 nm propagation distance was found, we used its 5th-degree polynomial as the starting point for the next propagation distance, 500 nm. That process was repeated again for 700 nm. This optimization approach is similar to similarity-based learning approaching .
Figure 6 shows the final temporal phase used as the initial temporal phase for the pre-shaped pulse for each propagation distance. While it is not possible to predict how exactly the parameters of the polynomial would affect pulse propagation, using the L-BFGS-B algorithm resulted in an over an order of magnitude smaller residual compared to the approach of using an initial phase (Fig. 1(a)) only. The obtained 5th-degree polynomial allowed for the compensation of the temporal stretching of the pulse. More complex materials could potentially require a use of higher than five number of terms which is possible with the L-BFGS-B algorithm.
The results presented in Fig. 5 show that the resulting pulse maintains a 100 fs FWHM pulse duration and has an increased field strength compared to Fig. 4. It took about 30-60 iterations to find optimal phase value. Table 1 shows the final terms for the 5th-degree polynomial used as the initial temporal phase for the pre-shaped pulse for each propagation distance. The resulting FROG traces for each distance (Fig. 5(d-f)) are relatively equidistant concentric circles, which indicates a zero-phase pulse. It is important to note that nonlinearity of the AZO/ZnO metamaterial at ENZ spectral point could potentially affect optimization results, which will be a subject of our future research.
In general, the presence of high optical loss, mainly due to absorption, makes it more challenging to compensate for pulse distortions. As it was outlined above, AZO’s damping frequency γ has the highest impact on the optical losses. For the current study, metamaterials with high and low transmittance (low and high absorption loss) were chosen (Fig. 2(a)). The results in Fig. 2 show that the higher transmittance (∼0.95) comes at the expense of increased ultrashort pulse distortion. On the other hand, metamaterials with much lower transmittance (∼0.4) allow for ultrashort pulse propagation without sideband accumulation but with lower output intensity (Fig. 2(d)). Our proposed approach to the application of an adaptive pre-shaping algorithm for ultrashort pulse distortion compensation during the propagation in AZO/ZnO multilayered metamaterial at the epsilon-near-zero spectral point allows for a distortion-free ultrashort pulse propagation with high output intensity.
It is important to note that ENZ material with a very low intrinsic loss has a very low group velocity of electromagnetic wave propagation . This effect of low group velocity bears some analogy to the manipulation of the speed of light using various optical systems [44,45]. The low group velocity leads to a slow light propagation effect which could enhance the frequency conversion efficiency in the ENZ material. As a result, the adaptive pre-shaping approach investigated in this work could be used for the enhancement of the weak nonlinear optical phenomena at the ENZ spectral point. The advantages of the ENZ materials, such as the availability of the high local fields and the fact that frequency mixing could be done without phase-matching, allow the increasing of the efficiency of nonlinear optical processes.
In summary, the results of the FDTD numerical study show that initial pulse with central frequency beyond ENZ spectral point experiences spectral self-tuning during propagation in AZO/ZnO. The resulting spectral shift and accumulated pulse distortions strongly depend on optical loss. This study demonstrated that this novel approach to the adaptive pre-shaping using BFGS algorithm can be used for significant reduction of the pulse distortions and increased output pulse field strength for a 100 fs Gaussian pulse propagated through AZO/ZnO metamaterial (thickness 300-700 nm). The adaptive pre-shaping approach, using an initial phase that crosses zero in the first temporal quarter of the pulse as initial guess for BFGS algorithm, provides a computationally effective method of determining optimal temporal phase shape before future experimental implementation. The results of this study have potential applications in super-resolution imaging, ultrafast optical communications, and signal processing.
National Science Foundation (GRFP 1321850); San Diego State University (San Diego State University UGP Grant (242518)).
Priscilla Kelly gratefully acknowledges the financial support from National Science Foundation (NSF) (Graduate Research Fellowship Program 1321850). The authors acknowledge S. G. Johnson who made MEEP freely available to the community.
The authors declare that there are no conflicts of interest related to this article.
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