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The femtosecond pulse-shaping capabilities of wavelength-selective directional couplers are investigated. Numerical results show that, depending on the coupling length and coupling coefficient, one can achieve very different temporal shapes at the output of the directional couplers. For instance, temporal re-shaping of Gaussian-like pulses into Hermite-Gaussian pulses, parabolic pulses, square temporal waveforms and sequences of equalized multiple pulses with time widths down to the femtosecond range can be achieved using readily feasible fiber/waveguide designs. The detrimental influence of the second-order variation of the detuning factor in these pulse shapers is also numerically investigated.
© 2016 Optical Society of America
1. Introduction
Femtosecond pulse shaping has attracted an increased interest for its important applications in the area of ultrahigh speed communications, optical signal processing and computing, analysis of ultrafast nonlinear processes, and biomedical imaging, among many others. In the context of optical communications and signal processing, pulse shaping down to the femtosecond regime is needed as the bit rates are pushed into the Tb/s range. Femtosecond optical pulses with customized temporal shapes can be generated based on linear spectral shaping using a spatial light modulator (SLM) [1–3 ]. The major advantage of an SLM-based pulse shaping system is that the system can be programmed to generated different shapes with a large flexibility. The major limitations of an SLM-based pulse shaping system are associated with its implementation involving free-space optics, i.e. large size and relatively poor stability. Alternatively, picosecond or sub-picosecond pulse shaping can be implemented using linear optical filtering with a fiber-optic or integrated-waveguide device. In particular, grating-assisted mode coupling devices have proved particularly successful for this purpose [4–15 ].
Ultrafast pulse shaping based on grating-assisted mode coupling offers many practical advantages, e.g., smaller size, low cost, improved stability and higher potential for integration, being fully compatible with fiber-optics devices and systems. Specifically, a fiber Bragg grating (FBG) is an example of contra-directional couplers which has been widely used to implement pulse shaping [4–10 ]. However, due to the limited bandwidth of practical FBGs, these devices are better suited for synthesizing temporal features above a few tens of picoseconds.
On the other hand, grating-assisted co-directional couplers, such a long period fiber/waveguide grating, can generally offer spectral filtering bandwidths up to a few THz. Generation of interesting optical pulse shapes with time features down to the sub-picosecond range has been theoretically predicted [11] and experimentally demonstrated [12] using this approach. Still, the need for a properly customized grating variation along the waveguide may limit the accessibility to this solution.
A wavelength-selective directional coupler [16–20 ] is a well-known passive optical device for coupling light from one or several waveguide inputs to one or several waveguide outputs (achievable in integrated optics or fiber optics). In these devices, the waveguide coupling is not assisted by a grating-like perturbation, which translates into simpler structures from the design and fabrication viewpoints. The bandwidth of the notch response (e.g.V-shape) of a wavelength-selective directional coupler can be extremely broad, easily up to a few tens THz [16–20 ]. Based on this feature, wavelength-selective directional couplers made of non-identical integrated/fiber waveguide have been proposed to realized optical time-domain differentiators with processing bandwidths larger than 10 THz [21]. Moreover, this proposed approach has been experimentally proven, demonstrating an all-fiber optical differentiator with processing bandwidth of ~25 THz (~200nm) [22]. In this paper, wavelength-selective directional couplers are numerically investigated to evaluate their potential for femtosecond optical pulse shaping, for the first time to our knowledge. Our results demonstrate the capabilities of this simple and practical fiber/integrated solution to implement a set of relevant pulse shaping operations (e.g., generation of Hermite-Gaussian, parabolic, flat-top pulses, and multiple pulse waveforms from Gaussian-like inputs) with unprecedented time resolutions, down to the range of a few tens femtoseconds, using readily feasible devices.
2. Theory
Figure 1 shows the proposed femtosecond pulse shaping device using a wavelength-selective directional coupler operating in the linear regime. We assume that the input pulse is launched into waveguide (I) and that there is no input signal directed into waveguide (II). In the coupling region, the gap between the two waveguides is small so that the evanescent field generated by one waveguide could excite an optical field in the other waveguide. Assuming a monochromatic input wave of radial frequency ω [analytic representation , where t is the time variable], the output signals from waveguide (I) [i.e. ] and from waveguide (II) [i.e. ] at the device output can be obtained from the following equations [23]
andIn the above equations, , where denotes coupling coefficient (coupling strength per unit length) and denotes the difference of the propagation constants between the two waveguides. and represent the propagation constants of waveguides (I) and (II), respectively. Equation (1) and (2) actually provide the spectral transfer functions for the two outputs of the directional coupler when operated in the linear regime. These functions depend on the frequency variable ω through the detuning parameter δ, i.e. frequency dependence on the propagation constants and . For a broadband input signal (e.g. optical pulse), the output complex spectrum at each port [A(z,ω) and B(z,ω)] can be directly calculated by Eq. (1) and (2) with A(0,ω) being the complex spectrum of the input signal. The corresponding time-domain variations can be obtained by Fourier transforming the complex spectrum. The propagation constant in each waveguide can be expanded in a Taylor series around the central frequency of the input pulses to be reshaped, which is made to coincide with the coupler resonance frequency (). In particular, , where is the group velocity, and is the first order group velocity dispersion (GVD). In addition, from the expanded function, we also have . The coupling slope is determined by the frequency derivations of the propagation constants of each waveguide in the coupler at the center frenquency [23]; this parameter is inversely proportional to the operation bandwidth of the directional coupler [21]. From Eq. (1) and (2) , it can be anticipated that the variations in the device’s coupling coefficient and coupling length will impact its spectral transfer functions. Thus, as shown in what follows, these parameters can be properly designed to achieve different pulse-shaping operations.
Fig. 1 Schematic showing the femtosecond pulse shaper based on a wavelength-selective directional coupler.
3. Optical pulse shaping capabilities of wavelength-selective directional couplers
In the analysis reported in what follows, we assume the coupling slope of all the tested directional couplers to be = 25.5 ps/m and the device coupling coefficient to be = 1.26 × 103/m, which are practically feasible values, e.g. in fiber-coupler implementations [21,22 ]. In the following simulations, different pulse shapes are obtained by changing the device coupling length. Equivalent pulse shaping operations could be obtained by varying the coupling coefficient instead. In all the reported simulations, the input optical pulse is assumed to be Gaussian with FWHM time-width of 10 fs, centered at a frequency of 193.54 THz (i.e. 1550nm).
3. A. Pulse shaping in waveguide (I)
The pulse shaping capability of a directional coupler in the waveguide (I) is first investigated. As shown in Fig. 2(a) , when the coupling strength κz = π/2, the spectral response corresponding to a first-order differentiator with an operation bandwidth about 37.5 THz is achieved. The spectrum of the transform-limited Gaussian optical pulse with a time-width of 10 fs is then shaped by the directional coupler, as shown in Fig. 2(b). When the carrier wavelength matches the notch wavelength of the directional coupler, the optical pulse is temporally differentiated [22]. An odd-symmetry Hermite-Gaussian pulse, consisting of two individual Gaussian-like pulses with a discrete π phase shift between them, is then generated at the output of waveguide (I).
Fig. 2 Hermite-Gaussian pulse generation at the output of waveguide (I) when κz = π/2 (i.e., first-order differentiator): (a) spectral transfer function of the directional coupler, (b) spectrum of the output pulse and (c) output pulse temporal waveform.
In addition, similar to the technique reported in Ref [12], when the carrier wavelength of the input optical pulse is shifted from the notch wavelength of the differentiator device by ~62.5nm, a slightly-chirped flat-top optical pulse with a time-width of 14.5 fs (FWHM) is generated at the output of waveguide (I), as shown in Fig. 3(c) . This approach enables the synthesis of flat-top intensity waveforms of different durations by use of input Gaussian-like pulses with different time-widths [12].
Fig. 3 Flat-top pulse generation based on a wavelength-shifted first-order differentiator at the output of waveguide (I): (a) spectral transfer function of the directional coupler, (b) spectrum of the output pulse and (c) output pulse temporal waveform.
3. B. Pulse shaping in waveguide (II)
The pulse shaping capability of the directional coupler in the waveguide (II) is also investigated as follows. When the coupling strength κz = 2.23, an optical pulse with a shape close to an ideal parabolic waveform is generated. As can be seen in Fig. 4(c) , the time-width of the generated parabolic pulse is 17 fs. When κz is increased to 2.37, a transform-limited flat-top pulse with a time-width of 10 fs is generated, as shown in Fig. 5 . From Fig. 5(b) we can see that the spectrum of the ultrashort pulse after propagation through the directional coupler shows the expected sinc-like shape.
Fig. 4 Parabolic pulse generation for κz = 2.23 based on a directional coupler in the waveguide (II), the green-dash line is an ideal parabolic pulse: (a) spectral transfer function of the directional coupler, (b) spectrum of the output pulse and (c) output pulse temporal waveform.
Fig. 5 Flat-top pulse generation for κz = 2.37 based on a directional coupler in the waveguide (II): (a) spectral transfer function of the directional coupler, (b) spectrum of the output pulse and (c) output pulse temporal waveform.
A second-order differentiator with an operation bandwidth of ~200 nm (i.e. 25 THz) is also realized when κz = π. In this case, the resonance spectrum depends on the square of the frequency deviation around the notch frequency , as necessary for second order differentiation [12,21 ]. A second order differentiated optical pulse (i.e. high order Hermite-Gaussian pulse waveform) is generated at the output of the waveguide (II), as shown in Fig. 6(c) . When the coupling strength κz is further increased (κz = 6.12), the input Gaussian optical pulse is divided into three consecutive pulses equalized in amplitude. The inter-pulse time spacing is about 10 fs, as shown in Fig. 7(c) .
Fig. 6 Second-order differentiator for z = π based on a directional coupler in the waveguide (II): (a) spectral transfer function of the directional coupler, (b) spectrum of the output pulse and (c) output pulse temporal waveform.
Fig. 7 Input single pulse is divided into three pulses based on a directional coupler in the waveguide (II) when κz = 6.12, the input optical pulse is divided into three consecutive pulses: (a) spectral transfer function of the directional coupler, (b) spectrum of the output pulse and (c) output pulse temporal waveform.
In the above numerical simulations, the detuning factor is assumed to depend linearly on the detuning frequency (ps2/m), see equations above. This approximation has been proven valid over very broad bandwidths (hundreds of nanometers) in practical fibers and integrated waveguide configurations [17–22 ]. Still, to get a deeper insight into the validity of this approximation, we have also investigated the influence of high order variations of the detuning factor in the coupler-based pulse-shaping performance. As a relevant example, Fig. 8 shows the simulation results of a flat-top pulse shaper (operation in waveguide (II)-Fig. 5) based on a directional coupler with a second order variation detuning factor . It can be seen from Fig. 8 that the output flat-top pulse is nearly undistorted when the second order coefficient is increased from 0 ps2/m to 6.35 × 1017 ps2/m. When the coefficient is further increased to 6.35 × 1019 ps2/m, the sinc-like spectrum of the output pulse becomes increasingly asymmetrical, which leads to the observed degradations in the generated flat-top temporal waveform. In addition, the optical dispersion for propagation in the device is another factor which may distort the output pulse. However, the influence of this dispersion could be simply compensated using a suitable length of dispersion compensation fiber (DCF) after the device.
Fig. 8 Simulated results of a flat-top pulse shaper based on a directional coupler with a second-order variation detuning factor. (a) spectrum of the output pulse and (b) output pulse temporal waveform.
4. Conclusion
In this paper, the femtosecond pulse shaping capabilities of wavelength-selective directional couplers have been investigated. We have demonstrated that depending on the coupling length and coupling coefficient, one can achieve very different temporal shapes at the output of a simple directional coupler. For instance, temporal re-shaping of Gaussian-like pulses into Hermite-Gaussian pulses, parabolic pulses, square temporal waveforms and sequences of equalized multiple pulses with time widths down to the femtosecond range can be achieved using readily feasible fiber/waveguide designs. Our results are interesting from both physical and applied perspectives, showing the strong potential of directional couplers for optical pulse manipulation applications into the femtosecond regime. The results presented in this paper should stimulate future research in this direction.
Acknowledgments
This work was supported by the National Natural Science Foundation of China under 61377002, 61321063, 61522509, 61535012 and 61090391. This work was also partly supported by Beijing Natural Science Foundation 4152052, in part by the National High-Tech Research and Development Program of China under 2015AA017102 and the funding from Natural Sciences and Engineering Research Council of Canada (NSERC). Ming Li was supported in part by the Thousand Young Talent program.
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