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A superluminal space-to-time mapping process is reported and numerically validated in grating-assisted (GA) co-directional couplers, e.g. fiber/waveguide long-period gratings (LPGs). We demonstrate that under weak-coupling conditions, the amplitude and phase of the grating complex apodization profile of a GA co-directional coupling device can be directly mapped into the device’s temporal impulse response. In contrast to GA counter-directional couplers, this mapping occurs with a space-to-time scaling factor that is much higher than the propagation speed of light in vacuum. This phenomenon opens up a promising new avenue to overcome the fundamental time-resolution limitations of present in-fiber and on-chip optical waveform generation (shaping) and processing devices, which are intrinsically limited by the achievable spatial resolution of fabrication technologies. We numerically demonstrate the straightforward application of the phenomenon for synthesizing customized femtosecond-regime complex optical waveforms using readily feasible fiber LPG designs, e.g. with sub-centimeter resolutions.
©2013 Optical Society of America
1. Introduction
Grating-assisted (GA) coupling devices, e.g. fiber/waveguide short-period (Bragg) gratings or long-period gratings, have been extensively used and studied for a very wide range of applications based on linear optical filtering [1]. These devices have proved especially useful for processing and synthesis of ultrafast time-domain optical waveforms [2]. In contrast to conventional schemes based on the use of spatial diffraction gratings combined with spatial light modulation [3], a GA coupling device can be easily implemented in fiber-optics or on chip integrated-waveguide technologies. A variety of well-established methods [4–10] are now available to design the grating perturbation, particularly the period and coupling-strength variations, namely the grating complex apodization profile, along the light propagation direction, so that the device can provide the desired target linear spectral or temporal response. Arbitrary spectral/temporal responses can be achieved in GA counter-directional couplers, e.g. fiber/waveguide Bragg gratings (BGs) operating in reflection, or in GA co-directional couplers, e.g. fiber/waveguide long-period gratings (LPGs), working in the cross-coupling operation mode.
In all GA coupling–based devices, the processing or synthesis of faster temporal features necessarily requires the use of smaller spatial resolutions in the grating complex apodization profile. This space-time relationship is more evident in GA counter-directional couplers working in reflection under weak-coupling conditions, so-called first-order Born approximation [4–7]. In this case, the output time-domain filter response (complex envelope) is directly proportional to the complex grating apodization profile with a space-to-time scaling factor directly determined by the classical light propagation laws through the medium.
This phenomenon provides a very straightforward mechanism to design the GA counter-directional couplers, e.g. BG structures, for customized synthesis or processing of optical waveforms [4–7]. However, as expected for any light propagation – based process, in this scheme, the ratio (v) between the space (Δz) and time (Δt) variables is necessarily lower than the propagation speed of light in vacuum (c) [4–7], i.e. v = Δz/Δt<c, see the case of BG in Fig. 1 . As a result, the achievable temporal resolution must be necessarily larger than the light-wave propagation time through the minimum spatial feature of the grating apodization profile, as determined by the spatial resolution of the used technology. For instance, fiber BG devices for optical signal processing/synthesis are limited to temporal resolutions of at least several picoseconds, considering a typically feasible sub-millimetre resolution for the fiber grating apodization profiles [4–7].
Fig. 1 Illustration of the space-to-time mapping phenomena in GA counter- and co-directional couplers (fiber BG and LPG cases, respectively).
In this work, we show that a similar space-to-time mapping phenomenon can be observed under weak-coupling conditions (Born approximation) in the case of co-directional GA coupling filters, e.g. long period gratings (LPGs), see illustrations in Fig. 1. In contrast to counter-directional GA coupling devices, we demonstrate that the space-to-time mapping speed (v = Δz/Δt) in a co-directional GA coupling device can be significantly higher than the speed of light in vacuum. We refer to this new finding as superluminal space-to-time mapping. An obvious consequence of this phenomenon is that co-directional GA coupling devices can be designed for processing/synthesis of optical waveforms with temporal features orders of magnitude faster (shorter) than those achievable using counter-directional coupling devices assuming the same practical spatial resolution limitations during grating fabrication, see Fig. 1. We numerically demonstrate the capability of the superluminal space-to-time mapping phenomenon in LPGs for synthesizing user-defined ultrafast optical waveforms with time features <100 femtoseconds using readily feasible and simple fiber LPG designs, with sub-centimetre spatial resolutions.
2. Theoretical derivation
Let us assume R(z) and S(z) being the complex spectral amplitudes of the two co-directional coupled modes in a GA co-directional coupler. They can be linked by the standard coupled-mode equations [11]:
k(z) is the complex coupling coefficient defined as k(z) = |k(z)|exp[jφ(z)], where |k(z)| accounts for variations in the coupling strength along the waveguide axial (light-propagation) direction (z) and φ(z) accounts for local grating-period variations along z. Notice that in our notation, 0≤z≤∆z, where ∆z is the total length of the GA co-directional coupler device. δ(ω) is the frequency deviation from the phase matching condition defined as:where βR(ω) and βS(ω) are the propagation constants of the two interacting modes, i.e. R-mode and S-mode respectively, along the propagation direction z, and Λ is the reference grating period, i.e. typically the grating period at the central point of the grating. ω is the angular baseband frequency variable defined by ω = ωopt-ω0, where ωopt is the optical angular frequency variable. The used variable Frequency or f hereafter is the baseband frequency defined as f = ω/2π. ω0 is the phase-matched angular frequency at the central grating period and it can be obtained from δ(0) = 0 [12]:In Eq. (3), ∆N = neff1-neff2, where neff1 and neff2 are the average effective refractive indices of the two interacting modes, i.e. R-mode and S-mode respectively, around ω0. Without loss of generality, in our derivation we assume that the S-mode is faster than the R-mode, i.e. neff1>neff2. By using Eq. (3), Eq. (2) can be re-written asNotice that Eq. (4) assumes a linear dependence of the waveguide propagation constants with frequency, i.e. no modal dispersion. This assumption allows one to simplify the derivation of the first-order Born approximation, as shown below. Our numerical simulations with realistic GA co-directional couplers (LPG), considering the actual dispersion curves of the interacting waveguide modes, prove a broad validity of the first-order Born approximation derived under the assumption of non-dispersive modes. The local spectral transfer function Hlocal(ω,z) of the GA co-directional coupler along z in the cross-coupling operation mode is defined asIn this notation, R0(ω) is the input signal to be processed and it is launched into the R-mode at z = 0, whereas S(ω,z) is the output signal received at the S-mode along z. The first-order Born approximation of the coupled-mode Eq. (1) is obtained under the weak-coupling condition defined as |Hlocal(ω,z)|2<<1. In this case, the signal in the R-mode along z, i.e. R(ω,z), can be approximated byThe linear spectral phase factor accounts for the non-dispersive propagation delay of the R-mode along the device. By replacing R0(ω) from Eq. (6) into Eq. (5), Hlocal(ω,z) can be re-written asThe couple-mode equation system in Eq. (1) can be reduced to a single differential equation involving the defined local spectral transfer function. These derivations can be greatly simplified by use of a reference time frame moving at the speed of light through the S-mode. This normalization is easily implemented by multiplying Hlocal(ω,z) in Eq. (7) by the linear spectral phase factor exp(jωzneff2/c). In this caseUsing the results in Eqs. (4) and (8), the coupled-mode equations in Eq. (1) can be reduced to the following differential equation:Under the weak-coupling condition, |Hlocal(ω,z)|2<<1, Eq. (9) can be approximated as equation:The local spectral transfer function Hlocal(ω,z) at any arbitrary location z = z0 (0≤z0≤∆z) can be obtained by integrating the above Eq. (10) from z = 0 to z = z0, asHence, the cross-coupling spectral transfer function H(ω) at the output end of the GA co-directional coupler device is Hlocal(ω,z0) evaluated at z0 = ∆z:On the other hand, based on the Fourier transform theory, H(ω) can be also expressed as below:where h(t) is the complex envelope of the temporal impulse response of the coupling device in the cross-coupling operation mode and Δt is the full time-width of this impulse response. By comparing Eqs. (12) and (13) it can be inferred that:with ∆t = ∆z·∆N/c. Therefore under weak-coupling conditions, the cross-coupling temporal impulse response of the GA co-directional coupler, e.g. core-to-cladding transfer function in a fiber LPG, is approximately proportional to the variation of the complex coupling coefficient k(z) as a function of the grating propagation distance z after a suitable space-to-time scaling. In particular, the space-to-time mapping speed (v), is obtained as v = c/∆N. Clearly, ΔN can be made much smaller than 1 and consequently, the resulting speed can be made significantly higher than the speed of light in vacuum.Figure 2 illustrates the principle of the obtained space-to-time mapping law in GA co-directional couplers as compared with the case of GA counter-directional coupling devices. The described phenomenon enables a direct mapping of the grating apodization profile along the device’s temporal impulse response, see illustration in Fig. 1. This leads to straightforward, simple grating designs providing a desired linear temporal response. This design strategy should be particularly useful for devices aimed at re-shaping an ultra-short optical pulse (temporal impulse launched at the device input). Most importantly, on the basis of the superluminal space-to-time scaling law, ultrashort temporal features can be achieved using greatly relaxed spatial resolutions, i.e. time features much faster than those intrinsically imposed by the fundamental light propagation laws through the medium, by properly designing the effective refractive index difference between the two interacting modes.
Fig. 2 Illustration of the temporal resolutions (dt) that are associated to a prescribed spatial resolution (dz) in the grating apodization profile for the GA co-directional and counter-directional cases. In each case, the temporal resolution is defined by the difference between the arrival times of the impulses coupled at the input and output ends of the corresponding spatial-resolution element.
An important practical consideration concerns the fact that the input and output signals must be carried by two different waveguide modes. In an integrated-waveguide approach [13], the device could be practically implemented by using two physically separated single-mode optical waveguides. All-fiber LPGs are typically based on coupling between the core mode (easily extracted from the fiber) and a cladding mode. The signal in the cladding mode can be efficiently extracted using several different techniques, e.g. a core-mode blocker combined with a short, strong uniform LPG inducing undistorted cladding-to-core coupling over the entire spectral bandwidth of interest [14], or by splicing a suitably misaligned fiber.
3. Numerical comparison between co-directional and counter-directional cases
Figure 3 presents results on a numerical comparison between the space-to-time mapping phenomena in GA co-directional and counter-directional couplers, using realistic fiber BG and LPG specifications. We assume a fiber BG working in reflection and a fiber LPG working in the cross-coupling operation mode, both made in standard single-mode fiber (Corning SMF28). The (uniform) grating period for the LPG is assumed to be Λ = 430μm, which corresponds to coupling of the fundamental core mode into the LP06 cladding mode at a central wavelength of 1550nm [15]. The BG has a period of 528nm, also corresponding to a Bragg reflection wavelength of 1550nm [12]. The average effective refractive index of the propagating mode in the BG is neff = 1.4684 and for the LPG: neff1 = 1.4684 and neff2 = 1.4648 [15]. Table 1 shows the estimated space-to-time mapping speeds for these two examples.
Fig. 3 An illustration of the speed difference of the two pulse shaping approaches based on space-to-time mapping in realistic fiber BGs and LPGs for a target optical OOK bit stream pattern generation.
Table 1. The estimated space-to-time mapping speeds for the considered BG and LPG made in SMF28 fiber.
We also assume that the two considered BG and LPG devices have the same length of 10cm and they are both identically spatially-apodized for generation of a target optical binary-intensity (on-off keying, OOK) bit stream pattern, as shown in Fig. 3 (spatial bit period ~1.67cm). In both cases the amount of peak coupling coefficient is assumed to be low enough to satisfy the weak-coupling condition. Based on the space-to-time mapping theory, by launching an ultra-short (i.e. <200fs) optical pulse spectrally centered at 1550nm into the considered optical filters, the target bit stream pattern will be generated at the output port of both filters as shown by the h(t) traces in Fig. 3. As it can be seen in Fig. 3, according to the space-to-time mapping rates calculated in Table 1, the speed of the generated bit stream pattern by the LPG device is nearly 1,000 faster than that generated by the BG filter.
4. Numerical simulations
To validate the anticipated space-to-time mapping process in GA co-directional couplers, we have numerically simulated two different fiber LPG designs, as shown in Fig. 4(a) and Fig. 5(a) , working in the cross-coupling operation mode. The spectral and temporal responses of the fiber LPGs have been simulated using coupled-mode theory combined with a transfer-matrix method [12]. The LPG design parameters are those defined above. In the numerical simulations, the actual wavelength dependence has been considered for the effective refractive indices of the two coupled modes [15]: neff1(λ) = 1.4884-0.031547λ + 0.012023λ2 for the core mode and neff2(λ) = 1.4806-0.025396λ + 0.009802λ2 for the LP06 cladding-mode, where 1.2<λ<1.7 is the wavelength variable in μm.
Fig. 4 Proof-of-concept numerical simulation for (a) three LPGs with the same length of 5cm and different amounts of kmax. The corresponding spectral power responses (b) and temporal impulse response amplitudes (c) of the LPGs.
Fig. 5 Simulation results of a designed fiber LPG (a) for generation of a 4-symbol optical 8-QAM (b) data stream pattern, i.e. 4137, with a speed of 4Tsymbol/s from an input ultra-short optical Gaussian pulse with a (full width at 10% of the peak amplitude) duration of 100fs (@1550nm). (c) The spectral power response of the designed LPG. (d) The output temporal amplitude and phase response to the mentioned ultra-short optical input pulse in the linear regime.
The two reported designs are as follows.
- (ii) A 8.3cm LPG, with an apodization profile including two different amplitude levels and three different discrete phase-shift levels along the grating length, as shown in Fig. 5(a). The LPG is specifically designed for generation of a 4-symbol data stream pattern, under a conventional optical 8-quadrature-amplitude-modulation (8-QAM) coding format (see Fig. 5(b)), with a target speed of 4Tsymbol/s, from an input ultra-short optical Gaussian pulse with a (full width at 10% of the peak amplitude) duration of 100fs.
Figure 4 and Fig. 5 validate the space-to-time mapping process defined by Eq. (14): under weak coupling strength condition, the complex (amplitude and phase) profile of the grating apodization profile is mapped into the LPG's temporal impulse response with the superluminal space-to-time scaling factor predicted in Table 1 (LPG). Both figures clearly prove that femtosecond time features can be achieved using readily feasible (sub-)centimeter spatial resolutions along the grating apodization profiles. Figure 4 shows that deviations with respect to a precise space-to-time mapping process are more significant as the coupling strength is increased. A fairly high accuracy is still achieved even when the cross-coupling transmission power peak is as high as ~40%.
5. Conclusion
We have demonstrated that a space-to-time mapping process occurs in GA co-directional couplers working in the cross-coupling operation mode under weak coupling conditions. The most striking feature of this process is that this mapping is not limited by classical light propagation laws, as it can take place at an equivalent superluminal speed. This mechanism opens up a promising new avenue to overcome the fundamental time-resolution limitations of present in-fiber and on-chip optical waveform linear generation (shaping) and processing devices, which are intrinsically limited by the achievable spatial resolution of available fabrication technologies. Our simulations of realistic fiber LPG devices have confirmed that customized complex optical temporal waveforms, with femtosecond range resolutions, could be easily generated and/or processed using fiber grating profiles compatible with typical resolutions, tens to hundreds of microns, of fiber grating writing techniques.
Acknowledgments
This research was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC), le Fonds Québécois de la Recherche sur la Nature et les Technologies (FQRNT), and Institut National de la Recherche Scientifique – Energie, Matériaux et Télécommunications (INRS-EMT).
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