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Designing an Integrated All-Optical Analog to Digital Converter

We present the procedure for designing a high speed and low power alloptical analog to digital converter (AO-ADC), by integrating InGaAsP semiconductor optical amplifier (SOA) with InP based photonic crystal (PhC) drop filters. The self-phase modulation in the SOA can shift the frequency of the Gaussian input pulse. The two output PhC based drop filters are designed to appropriately code the frequency-shifted analog signals by the SOA, converting them to four desired digital output levels. Our numerical results show that in an appropriately designed AO-ADC, the center wavelength (1572 nm) of an amplitude modulated Gaussian pulse of 1.8 ps width and 1.56 pJ energy can be shifted by 6.7 nm, by the SOA, and then be quantized and coded to four digital levels (00, 01, 10, and 11). The two point-defect PhC drop filters, compensating the effect of the frequency shift by SOA, minimize the AO-ADC integral and differential nonlinearity errors.


I. INTRODUCTION
Continuous nature of the real world signals makes an analog to digital converter (ADC) an inseparable part of any digital process. An electronic ADC faces with a few challenging issues, such as the jitter of the sampling window and the ambiguity of the comparator. To overcome these issues, researchers have proposed semi-optical ADC [1,2]. Moreover, various all-optical ADCs (AO-ADCs) have also been proposed in this regard [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. An AO-ADC consists of a mode-locked laser, generating high speed pulses with ultra-low jitters, as the sampling part, an optical quantizer, and an optical coder. The optical ADC designers have substituted the latter two optical parts for the electronic parts in a semioptical ADC to enhance its conversion speed [3].
The Kerr effect has been rigorously used in the design of the AO-ADCs. In the early '90s, a 3bit AO-ADC was developed experimentally, using the cross-phase modulation of a 200-m single mode fiber loop employed as a nonlinear interferometer [4]. A decade later, a 4-bit ADC based on hard limiters was presented [5]. The hard limiters consist of alternating layers of materials having different Kerr nonlinearities. Other techniques such as self-frequency shifting [6,7], slicing supercontinuum spectrum [8], and self-phase modulation (SPM) [9] in a long linear or nonlinear fiber were also used to design AO-ADCs. Later, employing the self-frequency shift and SPM induced spectral compression in a highly nonlinear fiber, a 4-bit ADC operating at 100 Gb/s were designed [10]. They employed the self-frequency shifting to quantize the analog signal and used the spectral compression to improve the resolution. Nevertheless, all these fiber-based ADCs, reported so far, have exploited long fibers that require high input power that makes it unsuitable for integration. Following the same trend, by using the multistage approach for SPM-based spectral compression in fiber and improving the spectral compression ratio more than twice, a 5-bit AO-ADC was achieved, a few years later [11]. Then, replacing the high-nonlinear fiber with a shorter photonic crystal fiber an AO-ADC with the 6-bit resolution was achieved [12]. Recently, a group has employed intensity-towavelength conversion by soliton selffrequency shift and experimentally demonstrated a 40-Gs/s AO-ADC. [13].
Meanwhile, the integrable optical ADC based on a semiconductor optical amplifier (SOA) was first reported in [14]. In this design, the cross-gain modulation of SOA was used to quantize a four-level signal at 20 Gs/s and code it into 2 bits. However, as the number of the quantization levels (N) is increased for high-resolution applications, this design faces two problematic issues: (i) the complexity of the ADC structure, requiring 2N+1 SOAs; (ii) the quantization levels becomes hardly distinct. In an alternative design, another research group has exploited polarization switches based on the nonlinear polarization rotation in SOA to achieve a 3-bit ADC operating at hundreds of Gs/s [15]. Later on, using multiple rectangular bandpass filters to digitize the chirped probe signal passing through a quantum dot (QD)-SOA, another research group has revealed a two-bit SOAbased AO-ADC, operating at 10 Gs/s with an 8-level optical quantization [16].
Furthermore, an alternative method to achieve an integrable all-optical ADC is to exploit the Kerr nonlinearities in a photonic crystal (PhC). In this regard, exploiting two-channel drop filters coupled with a cavity-based wavelength selective reflector, a PhC-based 2 bit ADC with no coder has been proposed [17]. Another group has reported the design of a two-bit PhC-based ADC with a coder, employing a nonlinear demultiplexer to quantize the input analog signal and a coder to convert the quantized levels into a 2-bit binary code [18,19]. With the aid of an appropriate reflector in conjunction with a hexagonal PhC waveguide, a 2-bit AO-ADC has been designed without the need for any coder [20]. The authors have shown their design could improve operation speed, without discussing the decrease in its nonlinearities.
In this paper, we are introducing an integrated two-bit optical ADC, consisting of an SOA as a nonlinear optical quantizer integrated with two-channel drop filters devised on a single PhC platform to act as an optical coder. Unlike the fiber-based ADCs, the new design is integrable to InP-based photonic chips and can operate with much less optical input power. Moreover, it is possible to increase its resolution by altering the configuration of the optical coder without a need for an extra SOA, consuming less cross-sectional area and requiring less complicated electronic circuitry. This integrated ADC, unlike the PhC-based ADCs, can operate with no need for exploiting a nonlinear PhC. Besides, the integration of SOA and PhC helps to decrease the ADCs integral nonlinearity (INL) and differential nonlinearity (DNL) errors.
The rest of this paper is organized as follows. Section 2 is devoted to a description of the proposed integrated ADC structure and the basic design principles. In Section 3, we present the simulation results and the related discussions. Finally, we conclude the paper with conclusive remarks in Section 4.

II. INTEGRATED OPTICAL ADC
A. Proposed Structure Figure 1 illustrates a 3D schematic representation of the proposed integrated Downloaded from ijop.ir at 10:17 +0430 on Tuesday July 27th 2021 optical ADC, composed of an InP-based traveling wave (TW) SOA, whose active region is made of In 1−x Ga x As y P 1−y , integrated with two drop filters devised on an InP pillar PhC platform. Details of the SOA and PhC physical and geometrical parameters are given in Section 3.

B. Operating principle
The operating principles of the proposed integrated optical ADC are based on the SPM nonlinear effect in the SOA. Consider a given modulated Gaussian pulse chain as an input optical signal, as schematically illustrated in Fig. 2. The signal at the SOA output, due to the SOA long gain recovery time, exhibits a frequency chirp that depends on the input pulse energy. After passing through the PhCbased drop filters, the chirped signal is quantized into a four-level coded signal. The PhC-based filters are appropriately designed to produce the Gray-code (i.e., 00, 01, 11, and 10). For this purpose, the filter bandwidth determines the pulsewidth of the input pulse. Moreover, to have four distinct output levels, the design is aimed to obtain the maximum possible shift for any given pulsewidth.

C. SPM-Induced frequency chirp
The SPM nonlinearity of an SOA, due to its long gain recovery time, is sensitive to the input pulse amplitude. Pulse propagation models in SOA has already been developed by others [20][21][22]. Employing the model developed by [20], one can describe the evolution of an optical pulse in an SOA by the following three coupled equations: where z and t are the space and time coordinates, g, and P and φ represent the gain, and the pulse amplitude and phase at position z at any instant , c , V G , α int , and α are the spontaneous carrier lifetime, signal group velocity in the active medium, internal loss, and the linewidth enhancement factor. Moreover, is the small-signal gain with Γ, a N , N 0 , and I, representing the confinement factor, gain coefficient, transparency carrier density, and the injection current. Moreover, for an active layer of length l, width w, and thickness d the transparency current is I 0 =qVN 0 /τ c with q and V=wdl being the electron charge and active layer volume. Furthermore, is the saturation energy, where ħ, ω 0 , and σ=wd/Γ are the reduced Planck's constant, pulse radian frequency, and the mode crosssection.
Integrating (1a)-(1c) over the SOA length and ignoring the internal loss for the case in which α int ≪g, we get: represents the integral gain at any instant , on the given temporal Gaussian input pulse of energy E in and time constant 0 , After solving (3a) numerically for G(τ), one can obtain the output pulse spectrum, by (3b) and its Fourier transform.
Any change in the phase with time corresponds to a chirp in the output signal frequency, Using the relation between the wavelength (λ) and radian frequency (5) It has already been shown that for an input temporal pulse of width ∆ 1/2 =2(2ln2) 1/2 0 , the larger the unsaturated single-pass gain of the amplifying cavity (i.e., G 0 = exp (g 0 l)( the larger the shift in the wavelength of the chirped signal (∆λ out ). Moreover, for an SOA under a certain biasing condition that leads to the maximum G 0 , one may increase the shift in the chirped signal wavelength, using a narrower temporal pulse [20]. However, for the latter case, an increase in ∆λ out can be achieved at the expense of an increase in the signal frequency or wavelength bandwidth (∆ω 1/2 or ∆λ 1/2 ). Recalling that for a Gaussian temporal pulse of given ∆ 1/2 , the wavelength bandwidth is given by ∆λ 1/2 =−(λ 2 /c) ∆ω 1/2 = −0.44(λ 2 /c∆ 1/2 ), according to the timebandwidth product limitation for an ultra-short Gaussian pulse. Hence, for an appropriately designed PhC drop filter to be able to code the chirped signals correctly, one needs to choose a Gaussian pulse of suitable width.

D. PhC Filter
To provide a coded digital output, the SOA output should be integrated into two appropriately designed PhC based bandpass filters, with relatively low-quality factors and wide passbands, overlapping over the desired frequency range. Each designed filter is composed of a point defect cavity coupled to a line defect waveguide, similar to that of [17], devised within a pillar PhC platform on the same InP substrate used for SOA.

III. RESULTS AND DISCUSSION
The 2D finite difference time domain (FDTD) numerical method is employed to simulate the dynamic behavior of the integrated optical ADC of Fig. 1.
First, using (3)-(6) we calculate the amount of the redshift (∆λ out ) versus the normalized pulse energy (E in /E sat ) for different temporal pulses of widths 1 ps ≤∆ 1/2 ≤ 10 ps, as illustrated in Fig 3(a). As can be observed from this figure, the redshift, for any given pulsewidth, is a sublinear function of the input pulse energy. Furthermore, for given input energy, the shorter the temporal input pulse the larger the redshift experienced by the output signal. Moreover, careful inspection reveals that at any given energy, the SOA SPM incurs a redshift on the output pulse signal that increases linearly with an increase in the input pulse bandwidth (∆λ 1/2 ). Fig. 3(b) illustrates such linear dependence at four given input energies (E in /E sat =0.005, 0.01, 0.015, and 0.02).  Fig. 3(a), dividing the input energy range 0 ≤ E in /E sat ≤0.02 into for equal steps (∆E in /E sat =0.005), the corresponding redshifts (Δ 1,2,3,4 ) can represent an example of four quantization levels of a 4-level quantization scheme shown in Table 2. From the linear relation between ∆λ out and ∆λ 1/2 (Fig 3(b)), one can conclude that the ratios Δ i / Δ j (i,j=1, 2, 3, 4) should be almost the same for any given input pulsewidth. On the other hand, an ideal ADC produces the digital outputs for equal ranges of the input pulse energy. To have a linear ADC, the bandpass filters should be designed in such a way that they compensate for the nonlinear relation between ∆λ out and E in (Fig. 3(a)). Figure 4 illustrates how this compensation can be applied schematically. The solid curves in the lower parts of the figure represent the output chirped-pulse shifts as the inputs to the bandpass filter and the dashed and the doted-dashed curve represent the bandpass filter output transmission spectra, such that the given ratios between Δ 2,3,4 and Δ 1 sustain at 50% transmission, set as the digital bit change (0→1) recognition level.

B. PhC filter design
Since the SOA is based on an InP substrate, we consider a cubic PhC of lattice constant a made by an array of InP pillars of radii r=0.17a. Knowing the relative permittivity of InP at 1550 nm;i.e., ε r~1 0 [23]-we have calculated the PhC dispersion curves for a transverse magnetic (TM) polarized incident optical beam, employing the plane wave expansion (PWE) method. The results are shown in Fig. 5. As can be observed from this figure, there is a photonic bandgap (PBG) in the normalized frequency range of 0.35≤c/a≤0.47. As an example for a=630 nm, the PBG falls within the wavelength range of 1.3 µm to 1.8 µm. Since the drop bandpass filters required for dropping fast and chirped pulses need not be highly selective wavelength filters with wide bandwidths, the point defect PhC drop filters are good enough. The schematic of a twochannel drop filter with a W1-bus waveguide is shown in Fig. 6.
Considering the radii of the point defects in the outputs 1 and 2 to be r 1 =0.1r and r 2 =0.112r, we have calculated the transmission, using a 1.8 ps TM polarized Gaussian input pulse.   This figure also shows that the designed filter provides the desired ∆ 2 /∆ 3 ratio. As seen in this figure, ∆ 2 represents the wavelength range over which output 1 is ON(1) and output 2 is OFF(0) whereas ∆ 3 represents the range over which both outputs are ON. Knowing the values ∆ 2 and ∆ 3 and expecting the ratios of ∆ i /∆ j to be as close as possible to those of determined. Notice, the transmission spectra are depicted only over a wavelength range that equals the sum of the four ∆ i values. According to the value of ΣΔ i~6 .7 nm, this design is capable of properly coding pulses of 2-nm bandwidths (Fig 3.b).

C. ADC output
Considering 1.8 ps Gaussian pulses of 1572 nm center wavelengths with different energies launched into the designed ADC input, we have calculated the wavelength shifts experienced by the ADC output, while neglecting the coupling loss between SOA and PhC filter. The wavelength shift versus the normalized energy is illustrated in Fig. 8. This figure shows that redshifts obtained for the pulses whose normalized input energies are E in /E sat =0.005, 0.01, 0.015, and 0.02 are the desired shifts that we have aimed for.
Launching four chains of 1.8-ps Gaussian pulses with the same center wavelength ( =1572 nm) but different energies (E in /E sat =0.00125, 0.00625, 0.0125, and 0.02) four digital state of the ADC output is examined. The simulation results show that the wavelength shifts incurred on the input pulses of given energies, by the SOA, coincide those of Fig. 8, resulting in four different pulses with center wavelengths of =1573.95, 1576.83, 1578.08, and 1578.7 nm. Then, the PhC filter outputs code these four pulses digitally as desired. The time responses of the PhC filter outputs are illustrated in Fig.9, showing the four digital states 00, 01, 11, and 10, considering the 37.5% transmission level (horizontal dots) as the threshold. As can be seen from this figure, the filter response time is shorter than the repetition rate; but the gain recovery time of the SOA is a limiting parameter. Using the data given in Table 1, the transparency current is I 0 =47 mA. The corresponding gain recovery time is found to be ~100 ps (see Fig. 10). In other words, the proposed AO-ADC sampling rate is ~10Gs/s. The threshold we defined for the filters, the differences between the ratios of Δ i /Δ j in the PhC filters and the SOA wavelength shift are the sources of errors in the proposed ADC.
These errors make the transfer function of the proposed ADC mismatch with the nominal one. The numerically calculated ADC transfer function is depicted in Fig. 11 and compared with the ideal one.  Finally, we have compared the parameters obtained for the proposed ADC with other optical ADC structures reported in the past decade, as tabulated in Table 3.