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Abstract
Recently, metasurfaces capable of manipulating the amplitude and the phase of an incident wave in a broad frequency band have been employed for femtosecond optical pulse shaping purposes. In this study, we introduce a phase-only pulse shaper based on an all-dielectric CMOS-compatible polarization-insensitive metasurface, composed of Si nano cylinders sitting on a fused silica substrate. The required phase profile of the metasurface for desired waveforms are calculated using an iterative Fourier transform algorithm, and the performance of the pulse shaper metasurface in implementing the phase masks was assessed using full-wave simulations. Such approach for realizing a polarization-insensitive metasurface-based phase-only pulse shaper has never been investigated to the best of our knowledge. It is demonstrated that the simulated results of the proposed metasurface-based pulse shaper is in great agreement with the results of the algorithm, while exhibiting a very high transmission efficiency. This work indicates yet another exciting but not fully examined application of meta-structures that is the optical pulse shaping.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Manipulating the temporal profile of optical pulses has been always of interest for researchers in different areas such as communication systems [1–3], quantum-state systems [4,5], coherent control of molecular and chemical systems [6–9], laser wakefield accelerators [10], and etc. To this end, various strategies have been adopted so far. Fiber-Bragg gratings [11–13], nonlinear interactions [14–16], and spatial filtering [17,18] are among this list. Spatial filtering, is the most common approach in the free space laser pulse shaping systems. The implementation setup for spatial filtering employs a 4f-line and a spatial mask. The 4f-line consists of two sets of diffraction grating-lens or diffraction grating-mirror that angularly disperse and collect the frequency components of the input and the output pulses respectively, which enables the mask to modulate the optical properties of the pulse at each frequency. Liquid crystal spatial light modulators (LC-SLMs) are a typical choice for spatial masks as they are programmable, rendering the possibility to modify both the amplitude and the phase of the electromagnetic wave, and are commercially available [18]. A growing appeal for a new generation of spatial masks capable to withstand high peak powers while offering a high spectral resolution, and operating in a large bandwidth has led to the introduction of the metasurface-based spatial masks for pulse shaping purposes [19–21].
Metasurfaces, composed of engineered subwavelength resonators, have emerged as a promising candidate to reinvent different optical and quasi-optical components possessing immense features such as structural flatness and overcoming the diffraction limit. A variety of applications for metasurfaces have been investigated so far. Invisibility cloaks [22], Analogue computing operators [23,24], broadband meta-lenses [25,26], perfect absorbers [27], filters [28], broadband meta-holograms [29], flattop beam-shapers [30], sensors [31], polarization convertors [32], and optical pulse shapers [19–21] are some examples. Plasmonic nano antennas exhibiting an inherent phase induction at resonance, with the help of accumulated geometrical phase, establish a large group of metasurfaces and have been studied extensively [33–35]. The main drawbacks of such structures are the high optical loss, low scattering amplitude and concomitant low efficiency. On the other hand, all-dielectric metasurfaces not only are capable of providing efficient phase and amplitude modulation at a specific frequency, they can also be designed for broadband operation [21,25,26,29]. Thus, all-dielectric metasurfaces composed of low-loss high refractive index materials were brought to the spotlight.
The principal method for designing a pulse shaper spatial mask, under the assumption of a linear system is to calculate the transfer function of the system, and then implement it. Many desired waveforms, require simultaneous modulation of the amplitude and the phase of the transfer function, which can be delivered by a properly designed metasurface [19–21]. In [19] a plasmonic metasurface was used for expanding and compressing ultrafast optical pulses. The adopted approach by Rahimi and Şendur, is to engineer the propagation characteristics of the metasurface composed of identical resonant nano antennas so that they can implement the desired transfer function. Since all the unit cells of the structure are the same, there is no need for decomposing the frequency components of the optical wave. However, this approach concentrates on designing the whole transfer function at once and offers a very low versatility for obtaining different transfer functions. In [20] a design procedure for femtosecond CDMA based on dielectric metasurfaces have been proposed. However, this work merely focused on implementing a 4f-line setup for CDMA systems based on metasurfaces and was not fully developed. Divitt et al. utilized a 4f-line setup composed of a pair of grating-mirror and a metasurface-based spatial mask to perform ultrafast pulse shaping [21]. The mask comprises a silicon-on-insulator (SOI) metasurface integrated with an aluminum wire-grid polarizer. The wire-grid polarizer in the proposed structure is used to solve the challenging issue of simultaneously but independently controlling the amplitude and the phase of the propagated wave through the metasurface. The indispensable presence of the polarizer not only diminishes the efficiency, it also complicates fabrication process along with rendering the structure, polarization sensitive. In addition to the polarizer, a half-wave plate condition needs to be satisfied in order to decouple the control of the amplitude and the phase modulation. This, inevitably requires a lot of optimizations on the geometrical features of the structure at each frequency.
To overcome the above-mentioned challenges in regards to employing an amplitude-modulating metasurface for pulse shaping, we address an alternative approach. Gerchberg-Saxton (GS) iterative algorithm and its modified versions in spatial or temporal domains [36–40] offer an energy-efficient and easy-to-implement solution for designing phase-only elements. By repeatedly using the Fourier and inverse Fourier transforms to convert the signals to input/output domains and satisfying the existing constraints at each domain, the algorithm finally converges to a phase profile that can generate the best possible output. Although trading the quality of the output with the benefits of using a phase-only modulator is unavoidable, these algorithms remain very popular in various applications such as optical tweezer traps [41], diffractive spectral filters [42], imaging applications [43], synthetizing multi-pulse temporal waveforms [44], and etc.
In this paper, we report on designing and implementing an efficient, polarization-insensitive, CMOS-compatible, phase-only-modulating optical pulse shaper spatial mask. The proposed spatial mask is implemented with a metasurface consisting of Si nano cylinders sitting on a fused silica substrate. In section 2, we discuss the temporal Gerchberg-Saxton (TGS) iterative algorithm used to calculate the phase profile of the metasurface mask (meta-mask). In section 3, the proposed metasurface is introduced and studied. The meta-mask’s capability for realizing the calculated phase profile and the desired outputs is then investigated in section 4. A simulation approach for assessment of the whole pulse shaping system is then presented in section 5. Additionally, a brief discussion on a typical fabrication process for the proposed structure is presented in section 6. Finally, some concluded remarks are given in section 7.
2. Calculating the phase profile of a phase-only pulse shaper
It has been shown that using a phase-only modulator instead of incorporating both phase and amplitude modulation not only facilitates the design and implementation challenges, it also improves the power efficiency [36–40]. The design procedure for a phase profile that can generate an output of an arbitrary transfer function as accurate as possible is discussed in this section. Gerchberg and Saxton proposed an algorithm in [36] that benefits an iterative Fourier transform method to determine the converting spatial phase profile of an imaging setup in which the intensity profiles at image and diffraction planes are known. Temporal versions of the GS algorithm (TGS) also have been introduced and exploited extensively [39,40]. In TGS, instead of dealing with features of an image at different spatial coordinates at image and diffraction planes, one works with temporal and spectral components of optical pulses. The goal is to calculate a spectral phase profile (${\phi _{TGS}}$) that can map an input optical pulse into a desired temporal shape. Under assumption of a linear system, and the electric fields as input and output, we have:
The inputs of the algorithm are the absolute values of the spectral amplitude of the input optical field ($|{E_{in}}(\omega )|$), the temporal amplitude of the desired output optical field ($|{{\cal E}_{desire}}(t )|$), and an initial random guess of the phase under study (${\phi _0}(\omega )$). First, we combine the spectral amplitude of the input field and the initial random phase to get complex spectral function of $|{E_{in}}(\omega )|\textrm{exp}({j{\phi_0}(\omega )} )$. By taking an inverse Fourier transform (IFT) of this function, we end up in temporal domain where we will replace the amplitude of the resultant signal with that of the desired optical field. Then a Fourier transform (FT) of the shaped signal would take us back to the spectral domain. Now we swap the amplitude of the obtained function with input optical field’s spectral amplitude.
Now one can iterate the steps by transforming the modified spectral function into temporal form. By repeating these steps, the algorithm finally converges to a spectral phase solution. The algorithm flowchart in Fig. 1 illustrates the TGS procedure. It is worth mentioning that different random initial guesses could lead to different solutions possibly at different convergence rates [39]. The mathematical presentation for the $n$th iteration can be expressed as:
in which $FT$ and $F{T^{ - 1}}$ denote Fourier and inverse Fourier transforms respectively and $\phi ({n,\omega } )$ is described as:
Fig. 1. Flowchart diagram of temporal Gerchberg-saxton algorithm for pulse shaping. An initial spectral random phase, temporal amplitude of the desired waveform, and spectral amplitude of the input optical pulse are inputs to the algorithm and a final spectral phase achieved by sufficient rounds of the iterative procedure, is the output of the algorithm for finding a phase-only solution of arbitrary pulse shaping problems.
After enough iterations ($n \ge N$) we will have $\phi ({n,\omega } )= \phi ({n + 1,\omega } )= {\phi _{TGS}}(\omega )$, and N is a large enough number that the convergence of the algorithm is certain. In our study we assumed $N = 200$ so that for any desired output waveform and initial guess, the convergence is assured. To assess the accuracy of the TGS solutions we define an error criterion at each iteration as [40]:
Fig. 2. (a) Normalized spectral magnitudes of electric fields of the input Gaussian pulse of $25\; fs$ FWHM, the output waveform containing two split Gaussian pulses of $50\; fs$ FWHM and peak separation of $150\; fs$, and the required transfer function to perform this task. (b) The obtained spectral phase profile by the TGS algorithm for a phase-only solution of the problem. (c) Normalized temporal intensity profiles of the input, desired, and the algorithm result. (d) Error criterion of the TGS algorithm versus the iteration number.
The envelopes of the input and desired optical intensities together with the solution provided by TGS algorithm are presented in Fig. 2(c). As expected, the solution provided by the algorithm follows the desired response immensely. The calculated error criterion at each iteration depicted in Fig. 2(d) indicates the convergent behavior of the algorithm. In this example, after about 60 iterations, the algorithm stagnates and the error criterion remains at 0.001.
3. Proposed metasurface for implementing the phase-only modulator
An all-dielectric metasurface suitable for implementing the obtained phase profile from the TGS algorithm is introduced here. The metasurface is composed of Si nano cylinders of the height H and the diameter $D$, arranged periodically in a square lattice of the periodicity $p$. A fused silica substrate of the thickness t supports the nano cylinders at the bottom. Schematics of a unit cell of the metasurface is shown in Fig. 3(a). Circular cross section of the nano cylinders, renders their operation polarization-insensitive, and a more precise fabrication accessible compared to unit cells with sharp edges. As we know by proper design and changing the dimensions of the unit cells, one can receive full $2\pi $ phase shifts at different frequencies [26,29,30]. However, what designing an efficient phase-only optical pulse shaper capable of generating arbitrary waveforms takes, is the possibility of receiving full $2\pi $ phase shifts at each wavelength in a large bandwidth, while keeping the transmission amplitude always constant and as high as possible.
Fig. 3. (a) A unit cell of the proposed metasurface composed of a Si nano cylinders of the height H and diameter $D$, placed on top of a fused silica substrate of the thickness t arranged in a square lattice of the periodicity $p$. (b) The optimized lattice periodicity profile at different wavelengths that enables one to avoid weak resonances of the structure and receive high transmission amplitudes as high as possible. The approximated expressions for the lattice periodicity are indicated in this figure at three regions.
Although all-dielectric metasurfaces can provide phase shifts even when the structure is off-resonance, weak resonances can happen when the phase shift is near $2\pi $. This leads to unwanted dips in the transmission amplitude spectrum. Plus, material losses and their dispersive behavior also hinders one to attain the mentioned conditions. To address these challenges, we varied the lattice periodicity ($p$) in order to optimize the structure response in terms of transmission amplitude modulation in the wavelength range of $700\; nm$ to $900\; nm$ with the center wavelength of $800\; nm$. The chosen bandwidth of operation is a sub-band of near infrared (NIR) regime, which has numerous applications such as near infrared spectroscopy [45], nano processing [46], biomedical applications [47,48], mass spectroscopy [49], and etc. The most widely used source for generating near infrared femtosecond pulses is Ti-Sapphire laser. The reason for choosing $800\; nm$ as the central wavelength of our design, is that the Ti-sapphire laser operates most efficiently at wavelengths near 800 nm [50]. By sets of simulations on the unit cell, the best possible choices for p at each wavelength has been calculated and represented in Fig. 3(b).
To facilitate our design, we approximated the resultant values for p with three relations at three wavelength regions; $p = 283\; nm$ for $700nm \le \lambda \le 800\; nm$, $p = ({283 + 0.34 \times ({\lambda - 800} )} )\; nm$ for $800nm < \lambda \le 880\; nm$, and $p = ({310.2 + 0.46 \times ({\lambda - 880} )} )nm$ for $880nm < \lambda \le 900\; nm$. After determining the proper values of $p$, we performed a full unit cell analysis on the structure to examine the changes of the complex transmission coefficient versus different cylinder diameters at different wavelengths.
We assumed the height of the nano cylinders to be $H = 680\; nm$, and the substrate thickness to be $t = 3\; \mu m$. It is worth mentioning that such a thin substrate cannot provide mechanical stability for a fabricated prototype, and is prone to Fabry-Perot effects which slightly affect the transmission amplitude. Therefore, a substrate thickness in the order of a few hundred microns is preferred for fabrication purposes. However, our studies shows that assuming a thin layer of an almost dispersion-less glass such as fused silica as the substrate, not only generates analogous results compared to a thicker substrate, it also reduces the computational costs noticeably. Optical properties of the Si and fused silica were taken from [21] and all of the results were obtained by full-wave simulations using COMSOL Multiphysics. Figure 4 shows the phase and the amplitude of the transmission coefficient for the unit cells of the structure at different wavelengths and different values of the cylinder diameter. Since the required range of cylinder diameters to achieve full $2\pi $ phase shifts before $\lambda = 800\; nm$ and after that differ noticeably, the results are presented in two wavelength intervals. Figures 4(a) and 4(c) show the phase of the transmission coefficients and Figs. 4(b) and 4(d) show the transmission amplitude in these two intervals. Clearly, full $2\pi $ phase shifts can be realized at any wavelength by varying the cylinder diameter. Moreover, very high transmission amplitudes for almost any desirable phase shift at any wavelength is achievable. Choosing the cylinder diameters in such a way that they can provide the required phase shift while having the highest transmission amplitude as possible, facilitates satisfying the aforementioned conditions.
Fig. 4. Unit cell analysis of the metasurface for $H = 680\; nm$ and $t = 3\; \mu m$ and for different cylinder diameters at different wavelengths. (a) Phase shift and, (b) Transmission amplitude of the structure at the first wavelength interval $700nm \le \lambda \le 800\; nm$. (c) Phase shift and, (d) Transmission amplitude of the structure at the second wavelength interval $800nm \le \lambda \le 900\; nm$. Since the required range of cylinder diameter to obtain full $2\pi $ phase-shifts differ in these to intervals, two separate sets of results are presented.
4. Realizing the phase-only pulse shaper using the proposed metasurface
Validity of the proposed pulse shaping approach that is to implement a phase-only modulating mask using metasurfaces is investigated in this section. As mentioned before, the behavior of metasurfaces against the amplitude of incident waves, leads to unwanted amplitude modulation. Even so, by carefully selecting the metasurface parameters to get a desired phase shift at each wavelength while having the highest possible transmission amplitude brings us closer to having an ideal phase-only modulating system.
Three arbitrary examples are presented in this section to evaluate the performance of the proposed pulse shaping method. For the first example, we consider triangular waveform as the desired output which is of interest in plasma wake-field applications [10]. The goal is to transform a Gaussian input pulse of FWHM $25\; fs$, same as the example in section 2, into a triangular waveform of the base width of $275\; fs$. Running the TGS algorithm gives the required phase profile for performing this task. The proposed metasurface in the previous section can be used to realize this profile. By determining a proper cylinder diameter at each wavelength using unit cell analysis presented in Figs. 4(a) and 4(c), the desired phase response can be formed with high precision. One must keep in mind to always select values of cylinder diameters so that the transmission amplitudes are as high as possible. Figures 4(b) and 4(d) help the designer to pick the best values if several choices for cylinder values become available. Figure 5(a) depicts the generated phase profile by TGS algorithm and the phase profile provided by the metasurface. As expected, the metasurface is able to, excellently, produce the required phase shifts given by the algorithm. Selected values of cylinder diameters and respective transmission amplitudes are presented in Figs. 5(b) and 5(c) respectively.
Fig. 5. (a) The phase profile calculated by the TGS algorithm and the realized phase shift profile by the metasurface for converting a Gaussian optical pulse of $25\; fs$ FWHM into a triangular pulse of the base width of $275\; fs$. (b) The obtained cylinder diameter profile for the Gaussian-to-Triangular pulse shaper at different wavelengths. (c) The observed transmission amplitude of the pulse shaper for the designed phase and diameter profiles. (d) Normalized temporal intensity profiles of the desired pulse, the resultant output pulse generated by the TGS algorithm, and that of the realized pulse shaper by the metasurface.
Although due to non-ideal behavior of metasurfaces, variations in the transmission amplitude spectrum are inevitable; the transmission amplitude is very high and hardly drops below 0.75 in this example. Combining the spectral phase and the spectral amplitude profiles given in Figs. 5(a) and 5(c), build the metasurface transfer function obtained by unit cell analysis. Using Eq. (1), one can calculate the waveform generated by the metasurface. Figure 5(d) shows the output temporal intensity profiles of the metasurface design based on unit cell analysis, TGS algorithm and the desired pulse shape. As we can observe, the metasurface output remarkably follows the algorithm’s response, and both are in great agreement with the desired output in terms of the base width and the arms’ slopes. Since some unwanted ripples in the transmission amplitude spectrum of the metasurface are present, a bit inconsistency in its results is predictable.
The power efficiency of the metasurface output can be easily calculated by integrating the temporal intensity profile as $76.6\%$. It is worth mentioning that if one runs the algorithm with different initial guesses, different phase profiles, different diameter profiles, hence slightly different output waveforms are achievable. In fact, this is yet another advantage of the proposed design approach; that is the solution is not always unique. This is in particular useful in implementing the pulse shaper profile using a metasurface. One can run the algorithm several times and receive different acceptable solutions with different properties such as power efficiency or implementation complexity.
In digital communication systems, having rectangular pulses has numerous applications. Hence, as the second example we investigate converting a Gaussian pulse same as what we had before, into a rectangular pulse of the width $170\; fs$. The phase profile provided from TGS algorithm along with the solution of the metasurface are shown in Fig. 6(a). Similar to previous example, cylinder diameter profile and the calculated transmission amplitude profile can be determined which are not shown here. The desired output intensity profile, the output of the TGS algorithm and that of the metasurface are presented in Fig. 6(b). Evidentially, metasurface and TGS algorithm results are in excellent agreement except for a small mismatch in plateau levels stemmed from unwanted amplitude modulation of the metasurface. Nevertheless, the power efficiency for this example is $84.9\%.$
Fig. 6. (a) The obtained spectral phase solution by the TGS algorithm and realized by the metasurface for transforming a Gaussian pulse of $25\; fs$ FWHM into a rectangular pulse of the width $170\; fs$. (b) Normalized intensity profiles of the desired rectangular pulse, the given output pulse of a phase-only modulation by TGS algorithm and the obtained output result of the designed metasurface. (c) Calculated modulating phase profile for converting a Gaussian pulse of $25\; fs$ FWHM into three Gaussian pulses of $60\; fs$ FWHM with descending peak values and centers at −210, −90, and 90 femtoseconds. (d) Normalized intensity profiles of the desired, TGS algorithm, and the metasurface output pulses.
In the final example we consider transforming the same input Gaussian pulse to a set of three Gaussian pulses of FWHM $60\; fs$ with descending peak values and centers at −210, −90, and 90 femtoseconds. This example demonstrates the flexibility of the proposed pulse shaping approach in expanding, splitting and controlling the peak values of Gaussian pulses much better. The calculated phase profile of TGS algorithm and the implemented phase shifts by the metasurface are plotted in Fig. 6(c). Figure 6(d) yields the desired output intensity profile along with the calculated results.
Considering the complexity of the target waveform, the resultant output of the metasurface remarkably follows the desired form and yields $83\%$ power efficiency.
5. Simulation of the actual pulse shaping setup
An actual pulse shaping setup used in real world applications comprises a spatial mask that performs the required modulation frequency by frequency, two diffraction gratings in charge of angularly dispersing/collecting frequency components of the optical pulses, and two mirrors/lenses that spatially direct each frequency component to a specific area. This setup is often referred to as 4f-line system in the pulse shaping literature. The mask design was described and studied in the previous sections. To examine the performance of the whole pulse shaping setup, one must fabricate the meta-mask, arrange the entire setup precisely, and do the measurement. This approach is time-consuming, not cost-efficient, and basically prone to design and measurement errors. Instead, a new approach for assessing the performance of the pulse shaping system is introduced in this section that relieves one from all the unnecessary challenges before experimentally validating their design.
In this approach the two sets of grating-mirrors or grating-lenses are modeled with proper excitation and detection ports in a simulation environment. At each wavelength, the exciting field emitting from the excitation port is assumed to be a Gaussian beam with its respective spot size. In this study, we assumed that ports are modeling the incoming wave bouncing off a parabolic mirror with a numerical aperture $0.03$ ($NA = 0.03$). The spot size of the Gaussian beam then can be calculated using $1.22\lambda /NA$ in which $\lambda $ is the operation wavelength. To more accurately model what happens in reality, the two ports and the mask must reside in the Rayleigh range of the Gaussian beam. At each wavelength, a supercell of identical unit cells of the metasurface whose diameters are determined from the designed profile, is placed at the beam waist of the incident wave. The number of unit cells placed in front of the port is different for each wavelength and is proportional to the beam spot size. A 3D schematic of the proposed setup operating at an arbitrary wavelength ${\lambda _a}$ and a side-view of the setup are presented in Figs. 7(a) and 7(b) respectively. In this case, ${N_a}$ cylinders of the diameter ${D_a}$ are placed in front of the excitation port. For the sake of simplicity, we assumed the ports and the structure to be periodic in lateral direction, as if we are to perform pulse shaping in a semi-infinite space. By calculating the S-parameters of the structure at each frequency and combining the results, one can form a more realistic transfer function for the structure and hence the output waveforms.
Fig. 7. (a) 3D schematics of the proposed setup for simulating the pulse shaping system operating at an arbitrary wavelength ${\lambda _a}$. In this setup, the excitation port radiates a Gaussian beam of the wavelength ${\lambda _a}$, onto the structure composed of a linear array of nano cylinders of proper diameter ${D_a}$ placed on the substrate. The upper and lower sides of the setup are considered to be perfectly matched layers in order to avoid any unwanted reflections into the simulation environment and properly model the existing open boundaries at the two ends of the pulse shaper. The two side walls of the simulation box parallel to yz plane are considered as periodic planes. (b) A side view of the proposed setup. The Gaussian spatial profile of the incident beam and its Rayleigh range are schematically indicated. As mentioned in the text, the metasurface containing ${N_a}$ cylinders would be placed at the beam waist of the Gaussian beam.
To this end, we used COMSOL Multiphysics to validate the results of an arbitrary example given in section 2, which is splitting a Gaussian input pulse of FWHM $25\; fs$ into two $150\; fs$ separated Gaussian pulses of $50\; fs$ FWHM. The required phase profile for this case was shown in Fig. 2(b). Similar to the design procedure we proposed in the previous section, an approximation of this phase profile, proper cylinder diameter profile and the subsequent transmission amplitudes of the metasurface are obtained based on the unit cell analysis (UCA), and are plotted in Figs. 8(a-c).
Fig. 8. (a) The phase profile realized by unit cell analysis (UCA) and the modeled pulse shaping system (MPSS) based on the proposed approach for converting an $25\;fs$ FWHM Gaussian input pulse into two split Gaussian pulses of $50\;fs$ FWHM and peak separation of $150\;fs$. (b) The obtained cylinder diameter profile for performing the desired conversion. (c) The subsequent transmission amplitude of the pulse shaper for the designed phase and diameter profiles obtained from UCA and MPSS. (d) Normalized temporal intensity profiles for the desired optical pulse, the resultant response of the UCA and that of MPSS.
To simulate this pulse shaping system based on the proposed method, we handpicked ${N_f} = 400$ uniformly spaced frequency points in the spectrum to be studied. For all of these points, a full setup containing input/output ports of the proper features and a supercell of the metasurface with predetermined cylinder diameters acting as the spatial mask are considered. By analyzing the S-parameters of the system at these points, the phase and the transmission amplitude of the modeled pulse shaping system (MPSS) by the proposed approach are formed and plotted in Figs. 8(a) and 8(c) respectively. The agreement between the UCA results and that of the MPSS is predictable, and some existing discrepancies stem from simulation errors and the fact that a unit cell simulation differs from the whole-system-simulation in small details such as boundary conditions and mutual coupling of the adjacent elements. Forming the MPSS transfer function and using Eq. (1), gives the output response of the system to the input optical pulse.
The desired waveform intensity profile, the output intensity calculated from the transfer function realized by UCA, and the output intensity of MPSS are shown all together in Fig. 8(d). Results of the unit cell analysis design and simulation of the entire system agree greatly with each other and clearly with the desired output. The conversion efficiencies obtained from UCA and MPSS results, are $79.4\%$ and $75.8\%$ respectively. Some negligible differences in these waveforms can be ascribed to limited number of frequency points, a small mismatch between the realized transfer function by UCA and by MPSS, and at last the accuracy of the designed phase by the TGS algorithm.
6. Typical fabrication process of the proposed metasurface
After assessing the performance of the proposed pulse shaping metasurface, a brief discussion on a typical fabrication process of such structures is necessary. The proposed metasurface is basically composed of an array of Si nano wires sitting on a fused silica substrate. Hence, the fabrication includes two main steps; first, a Si wafer is deposited onto the substrate utilizing low pressure chemical vapor deposition (LPCVD) technique, one of the most versatile methods in the electronic industry which was used in a similar work [21]. Then, the Si wafer needs to be patterned in order to form the nano wires. Since a Si wafer is already present, a top-down method capable of etching nano wires from a bulk Si layer in a subtractive manner is preferred. Also, top-down methods offer promising solutions in fabrication of complex nanostructures rather than bottom-up growth methods [51].
Before etching the Si into nano wires preferably with room temperature technique known as pseudo-Bosch process [51,52], a patterning mask needs to be developed. The mask would protect desired regions on the Si against being etched. We suggest aluminum for the mask material since it has a small milling rate, and the aspect ratio of the designed nano pillars is high [21,51].
To craft the mask, first, a layer of an electron beam resist such as poly-methyl methacrylate (PMMA) should be spin-coated on the Si layer and be baked at an appropriate condition. The final thickness of the resist is around 200 nm [52,53]. Then, the metasurface pattern is written on the resist using electron beam lithography (EBL) which provides very high resolution and is commonly used [21,51]. After developing the resist by a standard solvent such as methyl isobutyl ketone (MIBK), the Al mask which is usually referred to as hard-mask, is fashioned by electron-beam-evaporation followed by lift-off technique. Typical thickness of the Al hard-mask is in the order of few tens of nanometers.
Now the structure is ready for the final steps that include etching the Si in order to form the nano wires. Inductively-coupled-plasma reactive ion etching (ICP-RIE) which is a room temperature RIE process and is suitable for non-isotropic etching, with a mixture of SF6 gas for etching and C4F8 gas for passivation, is a preferred choice [21,52,53]. Most RIE processes use a passivation gas to protect the side walls of the nano wires from excessive etching process [51]. Finally, the etched structure should be washed from the post-etch residue and the Al hard-mask. Schematics of the steps of the whole fabrication process for the proposed metasurface as discussed above, is depicted in Fig. 9.
Fig. 9. Schematics of a typical fabrication process for the proposed metasurface. (a) A silicon layer is deposited onto the fused silica substrate by LP-CVD method. (b) A PMMA electron beam resist is spin-coated onto the structure and baked. (c) The desired pattern is written on the PMMA layer by EBL. (d) The PMMA layer is developed by a suitable solvent such as MIBK. (e) Al hard-mask is applied by electron-beam-evaporation. (f) Lift-off is achieved and the mask development is finalized. (g) ICP-RIE process etches the exposed Si regions. (h) The etch residue and the mask are washed off in order to finalize the fabrication process of the proposed structure.
Obviously, there is more to fabrication considerations and challenges such as balancing the chemical and the physical etching that happens in ICP-RIE [51], charge dissipation and accurate beam dosage at EBL step [51,54], avoiding the bunching phenomena for adjacent nano wires [53], and etc. However, these topics wouldn’t reside in the scope of this paper, and we refer the readers to other studies such as [21,51–54] for further details.
7. Conclusion
A new approach for designing metasurface-based spatial masks for optical pulse shaping purposes have been introduced in this paper. In this approach, we rely on an iterative Fourier transform method known as Gerchberg-Saxton in order to calculate the required phase profile of a phase-only modulating mask for translating an input optical pulse into a desired form. Then we showed that by carefully designing and selecting the geometrical aspects of an all-dielectric polarization-insensitive metasurface composed of Si nano cylinders sitting on a fused silica, it is possible to achieve full $2\pi $ phase control at each frequency in a large frequency bandwidth while keeping the transmission coefficient near unity and yet almost modulation free. Next, several examples such as shaping a Gaussian pulse of FWHM $25\; fs$ into a triangular pulse, a rectangular pulse, three separate Gaussian pulses with descending peak values, and two split Gaussian pulses have been investigated based on the responses provided by unit cell analysis of the designed metasurface which resulted in excellent outputs and provided efficiencies of $76.6\%$ $84.9\%$, $83\%$ and $79.4\%$ respectively. For the last case, we verified the performance of the pulse shaper by a new approach that is modeling the whole 4f-line setup with input/output ports and a meta-mask at sampled frequencies. It has been seen that combining the data from studies at all of the frequency samples, would create a highly satisfactory output waveform of $75.8\%$ efficiency. At the end, a brief description in regard to a typical fabrication process of the metasurface is presented. The proposed method, enables the opportunity of designing broadband, polarization-insensitive, high-efficiency and easy-to-implement phase-only spatial meta-masks for the interesting task that is optical pulse shaping and can be used as an excellent candidate in free space communications and computing systems.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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