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Wavelength-selective directional couplers with dissimilar waveguides are designed for ultrafast optical differentiation within the femtosecond regime (corresponding to processing bandwidths > 10 THz). The theoretically proposed coupler-based differentiators can be produced by wavelength matching of the propagation constants of two different waveguides in the coupler at the center wavelength. A single directional coupler can be designed to achieve either a 2nd-order differentiator or a 1st-order differentiator by properly fixing the product of coupling coefficient and coupling length of the coupler. We evaluated the differentiation errors (~2%) and energetic efficiency (~11% for 1st order differentiation) of the designed optical differentiators through numerical simulations. The proposed design has a strong potential to provide a feasible solution as an integrated differentiation unit device for ultrafast optical signal processing circuits.
©2011 Optical Society of America
1. Introduction
Recently, high-speed optical signal processing, including optical differentiation and integration, has attracted an increasing interest for pulse shaping or sensing applications that use optical signals [1–5]. All-optical temporal differentiators, which provide the derivative of the time-domain complex envelope of an arbitrary input optical signal, have been designed by use of an integrated-optic transversal filter [1], fiber gratings (long-period fiber gratings or fiber Bragg gratings) [2–4], and two-arm interferometers [6]. These devices may find important applications as basic building blocks in ultrahigh-speed all-optical analog/digital signal processing circuits [1]. A practically useful optical differentiation scheme needs to provide a set of performance specifications in terms of simplicity, stability, compactness, and large bandwidth. Most of the components previously proposed to build up high-speed optical differentiators in ultrafast optical signal processing are based on micro-optic structures with wavelength-selective components like interference filters or gratings. Despite the fact that the previously proposed schemes offer a different range of advantages for optical differentiation, these devices are generally difficult to produce (e.g. customized integrated-optic transversal filters or fiber gratings) and/or align (e.g. bulk-optics interferometers). In addition, previous designs can hardly offer processing bandwidths above a few THz. Therefore, alternative devices like optical waveguide couplers are especially attractive because of their wider availability and easy integration: waveguide couplers can be produced in a variety of technologies (e.g. in fiber or integrated waveguide platforms) and they offer inherently low transmission losses. A wavelength-selective directional coupler [7–10] is an important passive optical device for coupling light from one or several waveguide inputs to one or several waveguide outputs (achieved in integrated optics or fiber optics).
In this paper, wavelength-selective directional couplers made of non-identical waveguides in an integrated optic structure, providing a large bandwidth (> 10 THz) and compactness, are proposed and numerically demonstrated as ultrafast optical differentiators. The design specifications of this novel optical differentiation scheme are derived here for the first time to our knowledge.
2. Operating principle
First, we recall the general operation principle of first- or second-order optical differentiators [1–4]. The temporal operation of a first-order optical differentiator can be mathematically described as v(t) ∝ ∂u(t)/∂t, where u(t) and v(t) are the temporal optical waveforms (complex envelopes) at the input and at the output of the system, respectively, and t is the time variable. In the frequency domain, V(ω)∝-jωU(ω), where U(ω) and V(ω) are the complex spectra of u(t) and v(t), respectively (ω is the base-band frequency, i.e., ω = ωopt-ω0, where ωopt is the optical frequency and ω0 is the central optical frequency of the signals). Thus first-order optical differentiation can be implemented by use of a (wavelength-selective) optical filter with a transfer function H1 ∝ -jω, which depends linearly on frequency (along the whole bandwidth of the signal to be processed). A key feature of a first-order optical differentiator is that it must introduce a π phase shift exactly at the signal’s central frequency, ω0. Similarly, the transfer function of a second-order differentiator is a parabolic function of frequency, H2 = [H1]2 ∝ -ω2.
In what follows, we show that a directional coupler composed by two dissimilar waveguides located close to each other can be designed to provide the above defined spectral transfer functions of first-order and second-order optical differentiators. This is somehow expected if one considers that the coupled-mode equations for directional waveguide couplers are very similar to those of uniform long-period fiber gratings (LPFG), which have been previously designed for first and second-order temporal differentiation [2]. However, in general, the bandwidth of the direction couplers can be significantly larger than that of the fiber gratings, so we anticipate that one can obtain highly ultrafast optical differentiators using coupler devices, potentially enabling unprecedented operation bandwidths (processing speeds). Considering a 2 × 2 directional waveguide coupler made of two dissimilar waveguides, the theoretical approach for the analysis of a directional coupler for optical differentiation is given as follows, compared to a conventional LPFG-based differentiator [2]. The spectral transmission responses of waveguide 1 and 2 in a directional coupler (assuming that the input pulse is launched into the waveguide 1) are [11]
andwhere κ and L are the coupling coefficient (coupling strength per unit length) and the coupling length of the coupler, respectively. Detuning factor (Δβ) denotes Δβ = β1−β2 where β1 and β2 are the propagation constants for the waveguide 1 and 2 in the coupler, respectively; and j is the imaginary unit. We remind the reader that the propagation constants of the waveguides depend on the optical frequency (so-called dispersion characteristics of the waveguides) in a different fashion. Note also that the detuning factor of the coupler does not contain the grating period factor, in contrast to the case of an LPFG-based device. Transmission functions in Eqs. (1) and (2) can be approximated in the vicinity of the resonance frequency ω0 (where β1≈β2) by the two first non-zero terms of the respective Taylor expansions:andThe detuning factor in the close vicinity of ω0 can also be expanded in a Taylor series and approximated by Δβ(ωopt) ≈σ1(ωopt - ω), where σ1 = (1/2)[∂β1(ω0)/∂ω- ∂β2(ω0)/∂ω]. The maximum coupled power can be obtained at the resonance frequency ω = ω0, where β1(ω0) = β2(ω0), for κL = m(π/2), with m = 1, 3, 5,… In this case, it follows from Eq. (3) that the transfer function H1 ≈-jω, corresponding to a 1st-order optical differentiator. Similarly, if the coupler parameters are fixed to achieve full energy re-coupling from the waveguide 2 into the waveguide 1, i.e., κL = mπ, with m = 1, 2, 3, …, then it follows from expression (4) that H2 ≈-ω2, corresponding to the spectral response of a 2nd-order optical differentiator. Thus, the wavelength-selective directional coupler can be designed to operate as a 1st-order or as a 2nd-order optical differentiator. We reiterate that as a main advantage, the spectral bandwidth of directional couplers is generally larger than that of the fiber gratings. Moreover, the LPFG provides the 2nd order optical differentiator by cladding-mode full coupling with κL = mπ [2], but this approach is not practically feasible because it would be extremely challenging to extract the information from the light propagating through the cladding mode. In contrast, our proposed coupler-based differentiator can provide the second-order differentiation in the fundamental mode of an optical waveguide.3. Numerical analysis
The detuning factor in the close vicinity of λ0 (corresponding to ω0) can also be approximated by Δβ ≈η(λ−λ0), where the coupling slope is defined as a function of the wavelength derivatives of the modal propagation constants of each waveguide in the coupler at the center wavelength λ0, η = ∂β1(λ0)/∂λ-∂β2(λ0)/∂λ [9]. Our theoretical predictions were confirmed by means of numerical simulations. The input signal was a transform-limited super-Gaussian optical pulse centered at the coupler’s resonance wavelength of 1550 nm and with unity peak intensity, that is, u(t) = exp[-(ln(2)/2)(2t/τ0)Q], of order Q = 3 and a full-width at half maximum (FWHM) time duration τ0 = 3 ps.
Figure 1(a) depicts the schematic concept diagram of 1st- and 2nd-order optical differentiatiors with κL = π/2 and κL = π, respectively (an input pulse with a transform-limited Gaussian shape is assumed in the illustration). Figure 1(b) shows the amplitudes of the spectral transmission functions corresponding to (i) the core mode of the bypass waveguide 1 of a directional coupler with a coupling length of 8 mm and κL = π/2, i.e. designed to operate as a first-order optical differentiator (dotted curve) and (ii) the core mode of the crossover waveguide 2 of a directional coupler with the same coupling length and κL = π, i.e. designed to operate as a second-order optical differentiator (dashed curve). The coupling slope (η) was 0.02μm−2 in the simulation. The field spectrum of the optical differentiation result was obtained by multiplying the complex spectral transfer function of the optical differentiator under evaluation by the spectrum of the input super-Gaussian input pulse with a center wavelength of 1.55 μm, also shown in Fig. 1(b) (solid curve). Note that the center wavelengths of the input pulse spectrum and devices’ transfer functions (devices resonance wavelengths) should be matched. The temporal differentiated pulse can be retrieved by inverse Fourier transformation of the transmission spectral response of the differentiator.
Fig. 1 (a) Schematic concept diagrams of 1st- and 2nd- order optical differentiators with κL = π/2 and κL = π, respectively (an input pulse with a transform-limited Gaussian shape is assumed in the illustration). (b) Amplitude of the spectral transmission function in the bypass waveguide 1 of a 8 mm directional coupler with the coupling slope of 0.02 μm−2 and κL = π/2 (dot curve) and in the crossover waveguide 2 of a 8-mm directional coupler with the coupling slope of 0.02 μm−2 and κL = π (dashed curve). The solid curve represents the energy spectrum of a 3-ps super-Gaussian input pulse with a center wavelength of 1.55 μm (corresponding to 193.55 THz), which was obtained by Fourier transformation of the considered input pulse.
Figure 2(a) and 2(b) presents the results that correspond to the coupler with κL = π/2 for 1st-order temporal differentiator and with κL = π for 2nd-order temporal differentiator, respectively. As predicted, the optical differentiations by the coupler agree very well with the exact numerical differentiations of the input temporal waveform. The maximum deviation (error) of the 1st-order temporal differentiated waveform by the coupler in the simulation from the exact numerical differentiation of the input pulse is estimated to ~1.98% (Fig. 2(a)) and the error for second-order temporal differentiator is about 2.67% (Fig. 2(b)). It is worth noting that the non-derivative output pulses from the other waveguide of the differentiators are equal to the input pulse shape, which means that these outputs could be potentially re-used in the optical signal processing circuit.
Fig. 2 Temporal waveform (dashed curves) corresponding to the 3-ps super-Gaussian optical pulse launched at the input of the simulated couplers. Output waveform (a) in the waveguide 1 of the coupler with κL = π/2 [transfer function (dotted curve) in Fig. 1] and (b) in the waveguide 2 of the coupler with κL = π [transfer function (dashed curve) in Fig. 1]. Ideal first (left) and second (right) time derivatives of the input pulse are represented with open circles.
The coupling coefficient κ was assumed to be independent of the longitudinal spatial coordinate. Its wavelength dependence was neglected relative to the stronger wavelength dependence of the propagation constants. Notice that optical differentiation is achieved over a limited bandwidth near the coupler resonance frequency ω0 (input optical pulse signal is assumed to be centered at ω0). To be more concrete, the operation bandwidth of the designed optical differentiators is approximately given by the 3dB (i.e. FWHM) spectral bandwidth of the corresponding coupler transmission response. This represents a very reasonable approximation considering that for the 1st-order differentiator, the deviation of the device’s transfer function from a linear fitting curve along this defined bandwidth region is estimated to be less than 1%; similarly, for the 2nd-order differentiator, the relative error of the device’s transfer function, in comparison with the ideal parabolic fitting curve, over the entire defined bandwidth region keeps less than 2.7%.
Figure 3 shows the 3dB-bandwidth of the transfer functions of 1st-(open circle) and 2nd-(open square) order coupler-based differentiators with respect to the coupling length (η = 0.02 μm−2) (a) and coupling slope (L = 8 mm) (b). Practically feasible values for these two parameters have been considered in these evaluations (assuming implementation in a fiber-type waveguide) [7]. The 3dB-bandwidth (in THz units) of a wavelength-selective directional coupler’s spectral response can be approximated by the following expression, extracted from a nonlinear curve fitting of the data in Fig. 3 [9]:
where we recall that L is the coupling length and c is the light speed in free space. λ0 is the center wavelength of the coupler’s transfer function. The coupling slope η represents the slope difference (in unit of μm−2) between the propagation constants of the two waveguides of the coupler. p is a parameter resulting from the curve fitting of the numerically estimated 3dB bandwidth with respect to the coupling length, L and slope η (data shown in Fig. 3). Referring to the report [8], the differentiation bandwidth of the proposed coupler devices is inversely proportional to both the coupling length and the propagation constants’ slope difference (η). The estimated bandwidths from Eq. (5) with p = 3.5 for 1st-order differentiation and p = 6.5 for 2nd-order differentiation, depending on the coupling length and coupling slope, are also presented in Fig. 3 (solid curves). The bandwidth of the 2nd-order differentiator is ~1.86 times larger than that of the 1st-order differentiator, assuming that the couplers have the same parameters in terms of L, η, and center wavelength, λ0. As evidenced by these results, the coupler-based differentiator can easily provide operation bandwidths in the tens of THz range (corresponding to the femtosecond regime), compared to the few THz operation bandwidths (sub-picosecond regime) that can be offered by LPFG-based differentiators in the best case.Fig. 3 (a) 3dB-bandwidth of the transfer functions of 1st-(open circle) and 2nd-(open square) order coupler-based differentiators with respect to coupling length (for η = 0.02 μm−2), together with the estimated bandwidth from Eq. (5) with p = 3.5 for 1st-order differentiation and p = 6.5 for 2nd-order differentiation. (b) 3dB-bandwidth of the transfer functions of 1st-(open circle) and 2nd-(open square) order coupler-based differentiators with respect to the coupling slope (for L = 8 mm), together with the estimated bandwidth from Eq. (5) with p = 3.5 for 1st-order differentiation and p = 6.5 for 2nd-order differentiation.
4. Discussions on practical design issues
In the coupler, the coupling coefficient (κ) mainly depends on the spacing between the cores of the two dissimilar waveguides in the coupler. To satisfy the condition κL = π/2 or π, as required for 1st- and 2nd- order differentiation, respectively, the coupling coefficient can be easily increased by closing the cores up and then the coupling length can be shortened – a higher coupling coefficient translates into a shorter coupling length, which in turns results in a broader operation bandwidth, see Eq. (5). In addition, the coupling slope (η) is also easily controllable in optical fiber or planar waveguide platforms, enabling a further increase of the processing bandwidth, as suggested by Eq. (5). In contrast, in the case of an LPFG-based differentiator, the index modulation, which is proportionally related to the coupling coefficient of the LPFG, cannot be easily increased as it gets quickly saturated depending on the photo-sensitivity of the used material. This represents one of the main limitations on the processing bandwidth offered by LPFG-based differentiators. Concerning the coupler devices proposed here, it should be emphasized that wide-band optical couplers with a 3-dB bandwidth of about 100 nm (corresponding to ~12.5 THz in the frequency domain at a central wavelength @1.55µm) and beyond have already been experimentally constructed [7].
As a critical factor for practical applications of the proposed optical differentiation devices, one should consider the energetic efficiency of the devices, which is the ratio between the temporal differentiated pulse power and the input pulse power. This ratio generally depends on the ratio between the bandwidth of the optical differentiator and the input pulse bandwidth, as expected for any optical notch filter. The energetic efficiencies with respect to the optical differentiator bandwidth for 1st-order and 2nd-order differentiation (devices designed above), considering a super-Gaussian input pulse with a temporal width of 1 ps, were estimated as shown in Fig. 4(a) and 4(b), respectively. The efficiency of the 1st-order differentiation is higher than that of the 2nd-order differentiation; both efficiency curves exponentially decrease with the device operation bandwidth. Nonetheless, the bandwidth of the differentiator should not be made narrower than that of the input pulse to be processed as in this case, a larger error would be induced in the differentiation process.
Fig. 4 Energetic efficiency with respect to the differentiation bandwidth for (a) 1st-order and (b) 2nd-order differentiator in case of the super-Gaussian input pulse with the temporal width of 1 ps (corresponding to 1 THz).
From Fig. 4 the efficiencies of a device having a bandwidth of 1 THz are 11% and 3.7% for 1st-order and 2nd-order differentiation, respectively. When two 1st-order optical differentiators are connected in series for 2nd-order differentiation, the efficiency is estimated to 1.21%, being smaller than the efficiency of 3.7% for a single 2nd-order differentiator. Thus, not only our proposed design would allow one to construct a 2nd-order differentiator using a single coupler device (resulting in a simpler, more compact and less costly solution) but this solution also offers a significant advantage in terms of energetic efficiency. To give another example, in the case of three 1st-order differentiators connected in series for 3rd-order differentiation, it is also estimated that the energetic efficiency for the considered case would be ~0.133%, much smaller than the efficiency of 0.41% resulting from a concatenation in series of the here-designed 1st-order and 2nd-order differentiators.
5. Conclusion
In summary, we have proposed and numerically demonstrated the use of wavelength-selective directional couplers with dissimilar (fiber or slab-) waveguides as ultrafast optical differentiators. A wavelength-selective coupler can be designed to construct either a 1st-order differentiator or a 2nd-order differentiator by properly fixing the coupling strength – length product in the coupler. The design rules and performance specifications of the proposed devices have been discussed and analyzed. A coupler-based differentiator has the potential to provide an unparalleled performance in terms of operation bandwidth (>10 THz), which can be easily controlled by the coupling length of the coupler and the (coupling) slope of the propagation constant difference between the two dissimilar waveguides. This simple and practical solution is feasible for implementation in a fiber type or planar waveguide device.
Acknowledgments
This research was supported by a grant from the Korea Research Foundation funded from the Korean Government (MOEHRD, Basic Research Promotion Fund) (KRF-2009-0064373).
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