Open Access
Add to BibSonomyAbstract
Deeply saturated fiber optic parametric amplifiers can have very high performance. While it’s a common practice to model the fiber as a longitudinally homogenous entity, we show that the inhomogeneous nature of the fiber leads to a greater performance level which is neither accessible nor accountable using the homogenous model. This indicates that some experimental results cannot be predicted using the homogenous fiber model, even in principle. Consequently, future studies on the performance limit of the system will have to include an inhomogeneous fiber model.
© 2014 Optical Society of America
1. Introduction
Optical parametric amplifiers (OPAs) have been studied since the sixties, initially in bulk crystals [1] and later in waveguides [2]. One of the OPA numerous advantages [3,4] over conventional optical amplifiers [5–7] is its ultrafast response time [3], which originates from the non-resonant optical nonlinearity [8,9]. The parametric effect can be described in terms of frequency conversion of pump photons to signal/idler photons [10], where OPAs with higher efficiency will deliver more pump power to the signal and idler. In fact, a high efficiency (HE) OPA can be saturated using a very weak signal and cause the pump to deplete [11]. The ability of ultrafast, direct manipulation of light using a rapidly varying, weak control beam is of great importance in both science and engineering.
A common way to estimate the efficiency of a parametric amplifier is according to its nonlinear Figure of Merit (FoM), given by the product of the pump power, the amplifier’s length L, and its nonlinear coefficient γ [12]. Although high FoM can be achieved using a pulsed pump, it will be beneficial from a standpoint of both energy and practicality to limit the pump to moderate power levels consequently supporting CW pump operation. In fact, a highly efficient OPA with an increased value of γL product will reduce the power requirement on the pump and will enable new functionalities.
The fabrication of materials and structures which are characterized by high γL product is technologically challenging. The advancement of material science and processing capabilities lead the creation of controlled and high quality structures with improved characteristics. Consequently, optical waveguide technology demonstrated lower propagation loss [13], higher nonlinearities [14], and improved dispersion engineering [15]. A variety of platforms have been studied to date; among the dominant ones are Photonic Crystal Fiber (PCF), chalcogenide, silicon, and silica [16]. PCF platform, although being successful in the visible spectrum, has yet to demonstrate a good fiber that can operate in the telecom band because the design requirements impose a complex fabrication process [17]. When limiting the pump to moderate power levels or alternatively searching for a high γL material, at least currently, the leading platform to explore HE OPA is a high confinement silica fiber, commonly known as Highly Non Linear Fiber (HNLF) [18].
While FoM dictate the gain of an OPA, its operation band is set by dispersion. The dispersion of conventional HNLFs has been studied extensively [19–21]. HNLF local dispersion has a strong impact on the OPA behavior. Indeed, due to its relative high confinement, nanometer–scale variation in the fiber transverse geometry drastically affects its local zero–dispersion wavelength (ZDW) [18]. The perturbations of the HNLF core size along the fiber are caused by finite fabrication tolerances, and translate to ZDW fluctuations along the fiber, effectively making it inhomogeneous [22]. Various studies have shown that ZDW fluctuations reduce the performance of unsaturated (low FoM) OPA by degradation of its bandwidth (BW), gain, and gain-equalization [19,23].
Given this concern, more recently, the affect of fiber inhomogeneity on a high FoM OPA was investigated [24]. Its two main findings were: 1) Uniform, high-FoM OPAs can have very high performance; unfortunately, however, when considering physical (non-uniform) fibers, the quality of currently available fibers is not sufficient and significantly reduces the likelihood to observe it experimentally. 2) Remarkably, high FoM OPAs exhibit non-reciprocity. Both findings require attention to the fiber inhomogeneity, which is normally treated as uniform.
At the same time, in spite of the fiber inhomogeneity, when inspecting the distribution of a large ensemble of seemingly identical inhomogeneous fibers, we are able to identify a few members which do provide high performance [24]. This special group contains a variety of members possessing different magnitudes of average core radius variations, which implies the existence of ZDW profiles that are able to positively influence the mixing process. This contradicts the common notion that a uniform fiber, i.e. with a constant ZDW, is the most desirable profile. Although it can be argued that this previously held rule is true for the case of a low FoM OPA, it is not clear whether a certain ZDW profile, i.e. not uniform, would improve the response of a deeply saturated OPA, in particular the effect of signal–induced pump depletion [25].
The report presents an investigation in which an attempt was made to find a unique dispersion profile. This was done by optimizing the fiber profile showing that non-uniform fibers are able to outperform a uniform fiber in the case of a high FoM FOPA. We initially provide details on the problem (Section 2) and set a benchmark (Section 3). Then, various profile optimization methods and their respective results are presented (Section 4), and these are followed by conclusions (Section 5).
2. High efficiency OPA
Following the introduction we describe a highly efficient (FoM 7) OPA comprised of a weak CW signal coupled to a HNLF pumped by a CW beam. The fiber based OPA (FOPA) under investigation is operated in deep saturation. Operating the FOPA in deep saturation requires positioning the wavelength of the pump in resonance with respect to the fiber dispersion properties. Then, launching a weak signal into the FOPA may induce pump depletion, as schematically illustrated in Fig. 1.
Fig. 1 An illustration of signal-induced pump depletion in a fiber optics parametric amplifier with a high FoM, operated in deep saturation. When the weak signal enters to the saturated amplifier, the amplification of the signal is accompanied by generation of multiple idlers and noise, all of which draw power from the pump. At the output of the amplifier, the band pass filter blocks the amplified signal and the other byproducts, and shows the depleted pump.
A practical example can be a FOPA that is pumped by 1 W beam, and have HNLF with a nonlinear coefficient of 20 W−1km−1, and a length of 500 m which results in a FoM of 10.
In fact, even in the absence of a signal, a FOPA with a high FoM operated in deep saturation depletes the pump as a result of vacuum noise amplification, i.e. efficient generation of parametric fluorescence [26]. In a case the FoM is low, the FOPA will not be saturated and the output spectrum will show modulation instability (MI) side bands around the pump [27]. As the FoM increases, the MI bands draw more power from the pump, and generate higher orders of MI bands, hence broadening the total BW and increasing the power occupied by noise. After the FoM becomes sufficiently high, the noise will contain a significant portion of the spectral power and the pump will be depleted. A simulation illustrating the process is shown in Fig. 2(a) and its corresponding trace (Signal OFF) at Fig. 2(b).
Fig. 2 Power analysis of a typical pump depletion effect in a highly saturated FOPA: (a) The optical spectrum (50 GHz RBW) at the output of the HNLF in the absence of a signal. Each trace corresponds to the matching color point in (b) (signal OFF), describing the FOPA output spectrum at FoM of 6.8 (blue), 7.7 (green), 8.5 (red), and 9.3 (turquoise). (b) The dependence of the pump power at the output of the amplifier as a function of the FoM in the presence and absence of the input signal using a 50 GHz band pass filter. The first and second strong contrasts are located at FoM of 8.6 and 11.3, respectively. (c) The distribution of the optical power at the output of the HNLF as a function of the FoM in the presence of the signal. The figure shows the power distribution between the pump power, integrated noise power, and the sum of signal and idlers power. The CW components were measured within a 50 GHz BW. The FoM was expressed in terms of the effective length of the fiber, Leff = [1-exp(-α L)]/α, where α is the propagation loss. The FoM was modified by extending the length of a uniform fiber, and was simulated using the following parameters: γ = 17.66 W−1km−1, α = 0.69 dB/km, ZDW at 1550 nm with a dispersion slope of 44.89·10−3 ps/nm2-km, pump wavelength at 1567.06 nm, signal wavelength at 1573.17 nm, and pump and signal input power levels of 30 and −35 dBm.
The pump of a FOPA of sufficiently high FoM can be further depleted by introducing a weak signal, as shown in Fig. 2(b). However, as the FoM increases, so does the total noise power, as shown by Fig. 2(c), and therefore the reduction in the pronunciation of the signal–induced pump depletion effect, as shown in Fig. 2(b). Indeed, each FOPA, depending on its fiber’s parameters and input condition, has an optimal FoM which will generate a maximal increase in the pump depletion when a signal is introduced, i.e. maximal signal–induced pump depletion.
Interestingly, Figs. 2(b) and 2(c) show the pump’s nearly periodic dependence on the FoM at its saturated regime. This behavior is known as the Fermi, Pasta, and Ulam recurrence phenomenon [28], which describes the periodical evolution of a dispersive, nonlinear wave system. Specifically, a MI system comprising of discrete spectral modes driven by a pump wave (fundamental mode) transfers the pump’s power to higher order modes, but will return, after sufficiently long evolution, to its initial state. Nevertheless, this work differs in three fundamental aspects: We specifically operate in the limit of weak input signal, where noise cannot be neglected [24], use an inhomogeneous FOPA model, and include noise, in a spectrally uninhibited system, which lead to spectrally-continuous broadband noise generation.
For clarity we shall make use of the following notations: P denotes the pump power; PON and POFF denote the pump output power in the presence and absence of the signal, respectively; The pump depletion represents the ratio between the pump input and output power, P(0)/P(L), where L denotes the length of the amplifier; the signal–induced pump depletion is represented in terms of extinction–ratio (ER), and is calculated according to the ratio of POFF/PON.
The amount of pump ER depends on the quality of the parametric interaction. As pump and signal are launched into the FOPA, the system evolves and power is exchanged between the pump, signal, newly generated idler(s), and noise. The quality of this interaction will determine amount of pump ER. The strength of the interaction between various participants also depends on the phase matching of the particular parametric process. For example, the phase matching of the process describing the photon exchange between the pump, signal and 1st order idler depends on their linear (dispersion-induced) wavenumber mismatch given by the following expression [29],
which can be expressed using the HNLF’s dispersion, including its ZDW,Subscripts P, S, I, and 0 denote the pump, signal, idler, and zero–dispersion, respectively. S denotes the dispersion slope at the ZDW, and λ and k denote the wavelength and wavenumber respectively. Along the fiber, changes in its transverse geometry generate local ZDW. Therefore, according to (2), the wavenumber mismatch acquires spatial dependence which will consequently affect the parametric process [30]. In practice, for a system which is not spectrally inhibited, a FOPA with good ER will tend to generate many idlers and have a wide (noise) BW. This will result in many simultaneous processes, including mixing between different spectral components of the noise, and will not be practical to trace each individually.
So far, it has been a common practice to analyze FOPAs as linear, three-wave systems, in which the rate of power transfer between pump and signal is controlled using the wavenumber mismatch, and is maximized when linear and nonlinear contributions to the phase mismatch compensate each other. In the linear regime, initially (when signal power can be approximated as constant) the idler growth is maximized when Δβ = 0; later, when signal and idler powers are comparable, the rate of power transfer is maximized when Δβ = −2γP [29]. After sufficient amplification of signal and idler powers, when the pump can no longer be approximated as constant but still assuming a three-wave system, the nonlinear contribution to the phase mismatch will constantly vary as it will also depend on the signal and idler powers. At this nonlinear regime, the wavenumber mismatch will need to be modified in order to maintain the power transfer, which will eventually reach Δβ ≈-γP/2 [29,31]. By optimizing local dispersion of a non-uniform fiber, the local, linear wavenumber can be adjusted to optimally compensate the nonlinear phase mismatch, thus maintaining the pump depletion which otherwise–and specifically in the case of a uniform fiber–will be phase-matched limited. In practice, however, the system is complex, involving many simultaneous processes, and the focus is on pump’s ER rather than signal’s gain.
Consequently this complicates the search for a unique dispersion profile as one cannot simply rely on enhancing a specific process. By using the standard Non Linear Schrödinger Equation (NLSE), it is possible to account all aforementioned parametric interactions simultaneously [32]. The equation can be expressed in the following scalar compact form,
where the dispersion has a spatial dependence to account for the fiber inhomogeneity. The fiber characteristics were described using the following parameters: ZDW profile, λ0(z), dispersion slope S, propagation loss α, nonlinear coefficient γ, and length L. Indeed, it is possible to consider higher dispersion–orders terms; however, they were neglected for simplicity.During the search for a unique ZDW profile various profiles were simulated. Each profile under test was divided and simulated as a collection of many short uniform segments. For each segment, the dispersion parameters of the NLS equation in its explicit form (second- and third- order-dispersion) were evaluated using S and local λ0. The length of each segment was set to correspond a 0.005 nm average shift in the ZDW with respect to its adjacent segments but was limited to no shorter than 1% of LNL where LNL = (γP(0))−1. As implied by [33], in a FOPA which is driven using CW fields, the ZDW variations are effectively averaged on length scales which are much shorter than LNL, which can be treated as a uniform segment. Once the optimizer was done maximizing the ER, the solution, i.e. the optimized ZDW profile, was validated. In the validation process, the optimized profile was simulated multiple times using different segment sizes and ensured the ER converges to the declared value as the segment size becomes shorter.
Another type of uncertainty concerning the validity of the solution originates from vacuum noise. The simulated ER is influenced by the noise field realization at the input. Namely, various noise realizations will generate different values of ER. The uncertainty of the ER is inversely related to the length of the simulation time–window. As a result of averaging, a longer time–window will reduce any uncertainty associated with the noise realization. As part of the validation process, the performance level was guaranteed by performing multiple simulations using random noise seeds to quantify the level of uncertainty in the calculated ER due to noise.
It’s important to bear in mind that the reported performance of a FOPA with good ER, is limited by the BW of the optical filter at the receiver; then, introducing the signal will tend to bring the pump closer to the (amplified) noise floor. Consequently the simulated value of PON will be limited by the optical BW. For example, narrowing the BW will reduce the amount of integrated noise power reaching the detector and vice versa, therefore affecting both the ER and the noise performance. In this report the investigation focused on the static (CW) response of the amplifier and the pump detection was simulated using a 50 GHz optical filter followed by a power meter.
3. The optimization criterion
The question of whether a high FoM inhomogeneous FOPA can outperform a homogenous one requires establishing a benchmark. This was done by optimizing FOPA based on uniform fiber model which resulted in an ultimate performance limit and will be used in the next section, both for comparison and as a starting point to find an optimal fiber profile.
Initially the parameters of a high FoM homogenous FOPA were optimized using realistic HNLF values [18] to provide the highest possible ER that can be reached using this type of fiber model. Following the optimization, if the signal wavelength will be detuned from its optimal value the ER will decrease, thus enabling an estimate of the BW of the device. In a similar manner, one can have such a discussion about the stability of the optimal solution to other parameters. The goal then is to find an upper bound to the ER, a so–called performance budget from which any parameter deviation causes a reduction.
When declaring a benchmark it is important to state the value of certain parameters to create a common ground for future comparisons. In the framework of an ideal three wave model [29], the problem of depleting the pump is reduced to three parameters: normalized phase matching, FoM, and the ratio between the pump power and the signal power, from which the ER can be calculated [24,34]. Therefore, when defining the benchmark level, in addition to declaring the signal input power and the maximized ER, it is necessary to declare the pump input power. As indicated by the three wave model, for a given FoM and signal input power, it would be easier to saturate (deplete) a weaker rather than a stronger pump. For this report the pump and signal powers were set to 30 dBm and −35 dBm, respectively, and will remain as such for the entire report.
After setting the system power levels, the remaining FOPA parameters were optimized. The optimization was made within the normal working range of the HNLF parameters: L ~500 m, 10 < γ < 30 W−1km−1, 0.5 < α < 1.0 dB/km, and 0.03 < S < 0.05 ps/(nm2 km). λ0 was set to 1550 nm, where λS and λP were treated as free parameters. The FOPA modeling was done using NLSE [32], injected vacuum noise was included [35], and the model was solved using the split-step algorithm [32]. Matlab optimization toolbox was used to perform the optimization using the conventional simplex algorithm [36]. An initial guess was provided to the optimizer: The fiber parameters were set to the middle values of the normal working range, where the initial signal and pump wavelengths were set to λP = λ0 + 10 nm and λS = λP + 10 nm.
The optimization yielded an intermediate solution of 500 m long fiber which in principle corresponded to the location of the ER’s “first peak” (FP), as illustrated at Fig. 2(b). The length of the fiber was then prolonged (swept) to find the corresponding position of the second peak which provided a higher ER. The parameters of the system were then loaded as an initial guess for a second round of optimization, to ultimately yield a maximized ER of 12.88 dB for the following set of values: γ = 17.66 W−1km−1, L = 672.3 m, α = 0.69 dB/km, S = 44.89·10−3 ps/nm2-km, λP - λ0 = 17.06 nm, λS - λP = 6.11 nm. This level of ER and its corresponding parameters of system serve as the benchmark and will be compared against that of an inhomogeneous FOPA.
4. Unique zero dispersion wavelength longitudinal profile
This section describes a search for a unique dispersion profile; during which the fiber’s parameters: α, S, and γ remained set to the corresponding values of the benchmark. Thus ensuring the fiber remains of the same type while the ZDW (global and local) is being modified. Regarding the length of the fiber, the FoM of the benchmark is relatively high (11.26) which makes it computationally expensive to work with. Therefore, the length of the fiber was shortened (swept) in order to locate the ER’s FP. It was found at fiber length of 507.5 m, provides an ER of 11.64 dB and corresponds to a FoM of 8.62. In general, the higher the FoM, the longer the time it takes to simulate the system. Due to limited computational power, one profile optimization was made using a fiber length corresponding to the benchmark (672.3 m) while all others were made based on the length corresponding to the FP (507.5 m).
During the search for an optimal fiber profile, it may be necessary to reposition the pump and signal wavelengths. By fixing the pump wavelength to have it serves as a reference for the signal wavelength and the ZDW profile, insignificant wavelengths drift were prevented during the optimization process. The other parameters of the system, excluding L, were set according to the benchmark and the optimization algorithm remained the same as described in Sec. 3. The following subsections will each describes the ZDW profile representation, the initial guess, and the associated optimization result.
4.1 Equally spaced static grid
In the following description, the ZDW profile of the fiber was represented by six equally spaced nodes, as illustrated by Fig. 3. The ZDW profile was defined by interpolation of the nodes using piecewise–cubic method. This interpolation method kept the profile smooth and monotonous between each of the neighboring nodes, with no over–shoot.
Fig. 3 Illustration of ZDW profile representation of a fiber using “static grid”. The profile is represented by six equally spaced nodes where the first and last nodes are positioned at the entrance and exit of the fiber, respectively. The full profile description is given by interpolating the ZDW between the nodes. Each of the double headed arrows represents a degree of freedom. The signal wavelength (λS) remains a free parameter, and a reference is formed by keeping the pump wavelength (λP) fixed.
In the first profile optimization of the static grid, the benchmark settings were used as the initial guess that was provided to the optimizer. Namely, the signal wavelength was set such that λS - λP = 6.11 nm, L = 672.3 m, and each of the six nodes was positioned at wavelength 1550 nm. The optimization yielded a maximized ER of 13.31 dB (6-Benchmark) with its associated profile shown in Fig. 4. Afterwards, a similar optimization was made where the FP settings were used as the initial guess (namely, L was set to 507.5 m). This optimization resulted in ER of 17.75 dB (6-First peak) with its associated profile presented in Fig. 4.
Fig. 4 A plot of the optimized ZDW profiles. The dashed line shows the profile of the benchmark. The blue trace (round markers) and green trace (square markers) describe solutions which were made using the six nodes static grid profile representation in the case that the initial guess of the optimizer was set as the benchmark and first peak settings, respectively. The red trace (triangular markers) describes a solution which was made using the ten nodes static grid profile representation in the case that the initial guess of the optimizer was set as the green trace. The markers represent the nodes while traces show the interpolated profile. The pump (dash-dot) and the signal (solid) positions are described by the two uppermost horizontal lines. The optimizations resulted in a practically identical signal wavelength.
The finite number of nodes previously described was chosen somewhat arbitrarily; additionally, the grid is equally spaced and fixed. It is then plausible that the optimal solution attained under those constraints is in fact an approximation of a more successful solution. Therefore the number of nodes (degrees of freedom) was increased to ten (eleven) which was then used to optimize the profile. The grid was made in a similar manner to that presented in Fig. 3, using spacing of L/9. Also, in order to gain access to new possible solutions, the number of nodes was chosen such that the grid will be displaced relative to the six nodes grid. The initial guess of the optimizer was set to be an interpolation of the latter solution. The ten nodes optimization resulted in an ER of 19.18 dB (10-First peak) with its associated profile also presented in Fig. 4.
Better solution may exist: The usage of a fixed grid representation confines the solution of the optimizer to a continuous profile and of a specific predetermined length. It is possible that an optimal solution may include discontinuities in the ZDW profile with a corresponding optimal length, as it is likely to depend on the shape of the profile. Such a solution will be effectively described by concatenation of two or more fibers.
4.2 Dynamic grid with variable fiber’s length
In this subsection, the profile optimization was done on a dynamic grid. This profile was described using six nodes as before, only in this section their spatial location was considered a variable. Only the first node was spatially fixed since it indicates the beginning of the fiber. This type of representation has a total of 12 parameters to optimize. In addition to being able to simulate a bigger variety of curves in comparison to the previous representation, the length of the fiber was also a variable. Figure 5 illustrates the dynamic grid structure.
Fig. 5 Illustration of ZDW profile representation of a fiber using “dynamic grid”. The profile is represented by six nodes where the first and last nodes are positioned at the entrance and exit of the fiber, respectively. Each of the nodes, except the first one which is positioned at the entrance of the fiber, is free to be shifted both laterally and vertically. The full profile description is given by interpolating the ZDW nodes. The position of the signal (λS) remains a free parameter; however, the wavelength of the pump (λP) is fixed and acts as a reference.
Two optimization attempts were made using this profile representation. In the first case, the initial guess of the optimizer was set to the fixed grid six node solution (6-First peak); whereas in the second case the initial guess was set as FP. This resulted in an ER of 19.65 dB (6 nodes – FP) and 18.29 dB (FP), respectively. Their associated profiles are shown in Fig. 6.
Fig. 6 A plot of the optimized ZDW profiles. The dashed line shows the profile of the benchmark. The upper horizontal line (dash-dot) represents the wavelength of the pump. The green trace (triangular markers) describes a solution made by a dynamic grid representation in the case that the initial guess of the optimizer was set as FP. The blue trace (round markers) describes a solution made by optimizing a dynamic grid in the case that the initial guess of the optimizer was set to the solution achieved by the six node static grid (6-First peak). Each of the markers represents a node.
The latter result effectively reduced the solution from six to four nodes, thus simplifying and smoothing the solution while keeping a relatively high level of performance. The former result, in comparison to the ten node solution, was able to achieve a similar ER level while having a relatively different profile.
The optimized profiles have demonstrated a rapid change in the ZDW at the last 100 m of the fiber, a distance which corresponds to approximately twice LNL (LNL = 57 m). Such fast changes tend to average; therefore it is plausible a simpler solution may be found in the form of discontinuity. Additionally, if one would attempt to generate such a profile by applying a varying tension across the fiber [19], a slowly varying profile would be considered more desirable.
4.3 Bandwidth analysis
Deeply saturated FOPA exhibit a tradeoff between the pump’s ER and the BW of the effect [29]. This created an interest to know whether the enhancement of the ER shown by the optimized profiles came at the expense of the BW of the effect. This was investigated by simulating the dependence of the ER on the signal wavelength. As the signal detune from the optimal position the pump’s ER will decrease; thus providing a measure to the BW of the signal induced pump depletion effect. Figure 7 shows the results of such analysis.
Fig. 7 A simulation of the pump’s ER dependence on the signal wavelength for different optimized ZDW profiles; the signal position is shown with respect to the pump wavelength.
It is clearly shown that the BW of the effect was not compromised. In fact for a given ER level of 10 dB the BW broadened from 0.65 nm (Benchmark) to 1.44 nm (10-FP). Even though the accuracy of this estimation is coarse, it indicates the optimized solution can perform well above unphysical uniform fiber model and that opportunities lay in the inhomogeneous fiber model.
4.4 Noise properties
The optimizer interacts with the simulator by running the simulation repeatedly using a different settings each time to eventually maximize the ER. The same noise realization was used during the simulations in order to assist the optimizer to converge to a solution. For each of the reported solutions a certain level of ER was stated; in practice the pump output power has a certain mean and variance levels. For each of the solutions – i.e. optimized profiles – a histogram was generated using random noise realizations. Each of the histograms consisted of 300 samples. The simulations were performed using a time–window of 29/(150 GHz), and a sampling–rate of 28 × (150 GHz). Table 1 shows a summary of the noise performance of the main solutions.
Table 1. Summary of the optimized FOPA performance.
The results remain sufficiently stable to distinguish between the qualities of different solutions. In comparison to the benchmark, the optimized profiles keep the pump less depleted in the absence of a signal and more depleted in the presence of it, thus achieving better ER. Additional improvement in the ER can be made in exchange to reducing the BW of the filter which will lower PON.
5. Conclusions
Deeply saturated high FoM FOPA exhibit a highly sensitive signal–induced pump depletion effect. This report explored the use of local dispersion to enhance the effect and challenged the conception that a longitudinally uniform fiber is the most desirable structure. In fact, optical fibers are commonly treated in the literature as a longitudinally homogenous medium. Previous studies showed that such approximation is not valid for HNLF based FOPA, and even more so for those operating in deep saturation.
The investigation initiated by posing a question of whether a high FoM FOPA which is based on inhomogeneous fiber can have a higher ER in comparison to homogenous one, assuming both fibers are of the same type. A benchmark based on the homogenous FOPA model was established by optimizing the parameters of the system and resulted in an ER of 12.88 dB, effectively posing an upper limit to the achievable ER using this model. The ER levels which were achieved afterward, using the inhomogeneous fiber model, resulted in significant improvement of several decibels, demonstrating ER levels which can only be accounted by an inhomogeneous FOPA model. This is attributed to the fiber’s ability to better maintain a constructive interaction (toward high ER) by having the local, linear phase mismatch vary in accordance with the nonlinear phase mismatch.
As this is the first study on the enhancement of signal–induced pump depletion of high FoM FOPA using local dispersion engineering, it is emphasized on maximizing the pump ER. Future reports may include a broader search with constraints on the profile simplicity, the BW of the effect, performance stability to profile perturbations, and noise performance, to name a few. New profile solutions may be found through the development of the optimizer, e.g. the type of optimization algorithm, fiber representation (degrees of freedom), initial guess (basin of attraction), and the effect of noise which is averaged to a degree within the limits of the available computational power. Modification of these factors could lead to additional solutions.
This work has showed, with regards to deeply saturated FOPAs, that opportunities are available if one is able to abandon the uniform fiber model. Practical FOPA designs may use heterogeneous fibers and include higher–order dispersion, all together provide access to a greater inventory of fibers and will lead to new capabilities.
Acknowledgment
This research is based on work supported by the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA).
References and links
1. P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7(4), 118–119 (1961). [CrossRef]
2. R. H. Stolen and A. Ashkin, “Optical Kerr effect in glass waveguides,” Appl. Phys. Lett. 22(6), 294–296 (1973). [CrossRef]
3. J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8(3), 506–520 (2002). [CrossRef]
4. S. Radic, “Parametric signal processing,” IEEE J. Sel. Top. Quantum Electron. 18(2), 670–680 (2012). [CrossRef]
5. A. K. Srivastava, Y. Sun, J. L. Zyskind, and J. W. Sulhoff, “EDFA transient response to channel loss in WDM transmission system,” IEEE Photon. Technol. Lett. 9(3), 386–388 (1997). [CrossRef]
6. C. J. Chen and W. S. Wong, “Transient effects in saturated Raman amplifiers,” Electron. Lett. 37(6), 371–373 (2001). [CrossRef]
7. G. Eisenstein, R. S. Tucker, J. M. Wiesenfeld, P. B. Hansen, G. Raybon, B. C. Johnson, T. J. Bridges, F. G. Storz, and C. A. Burrus, “Gain recovery time of traveling-wave semiconductor optical amplifiers,” Appl. Phys. Lett. 54(5), 454–456 (1989). [CrossRef]
8. R. H. Stolen, J. P. Gordon, W. J. Tomlinson, and H. A. Haus, “Raman response function of silica-core fibers,” J. Opt. Soc. Am. B 6(6), 1159–1166 (1989). [CrossRef]
9. D. W. Hall, M. A. Newhouse, N. F. Borrelli, W. H. Dumbaugh, and D. L. Weidman, “Nonlinear optical susceptibilities of high-index glasses,” Appl. Phys. Lett. 54(14), 1293 (1989). [CrossRef]
10. R. H. Stolen and J. E. Bjorkholm, “Parametric amplification and frequency conversion in optical fibers,” IEEE J. Quantum Electron. 18(7), 1062–1072 (1982). [CrossRef]
11. P. A. Andrekson, H. Sunnerud, S. Oda, T. Nishitani, and J. Yang, “Ultrafast, atto-Joule switch using fiber-optic parametric amplifier operated in saturation,” Opt. Express 16(15), 10956–10961 (2008). [CrossRef] [PubMed]
12. J. H. Lee, T. Tanemura, K. Kikuchi, T. Nagashima, T. Hasegawa, S. Ohara, and N. Sugimoto, “Experimental comparison of a Kerr nonlinearity figure of merit including the stimulated Brillouin scattering threshold for state-of-the-art nonlinear optical fibers,” Opt. Lett. 30(13), 1698–1700 (2005). [CrossRef] [PubMed]
13. G. Li, J. Yao, H. Thacker, A. Mekis, X. Zheng, I. Shubin, Y. Luo, J.-H. Lee, K. Raj, J. E. Cunningham, and A. V. Krishnamoorthy, “Ultralow-loss, high-density SOI optical waveguide routing for macrochip interconnects,” Opt. Express 20(11), 12035–12039 (2012). [CrossRef] [PubMed]
14. S. D. Le, M. Gay, L. Bramerie, M. Costa e Silva, K. Lenglé, T. Chartier, M. Thual, J. C. Simon, L. Brilland, D. Méchin, P. Toupin, and J. Troles, “Wavelength conversion in a highly nonlinear chalcogenide microstructured fiber,” Opt. Lett. 37(22), 4576–4578 (2012). [CrossRef] [PubMed]
15. M. Pu, H. Hu, C. Peucheret, H. Ji, M. Galili, L. K. Oxenløwe, P. Jeppesen, J. M. Hvam, and K. Yvind, “Polarization insensitive wavelength conversion in a dispersion-engineered silicon waveguide,” Opt. Express 20(15), 16374–16380 (2012). [CrossRef]
16. B. J. Eggleton, B. Luther-Davies, and K. Richardson, “Chalcogenide photonics,” Nat. Photonics 5(3), 141–148 (2011).
17. Y. Wang, X. Zhang, X. Ren, L. Zheng, X. Liu, and Y. Huang, “Design and analysis of a dispersion flattened and highly nonlinear photonic crystal fiber with ultralow confinement loss,” Appl. Opt. 49(3), 292–297 (2010). [CrossRef] [PubMed]
18. M. Hirano, T. Nakanishi, T. Okuno, and M. Onishi, “Silica-based highly nonlinear fiber and their application,” IEEE J. Sel. Top. Quantum Electron. 15(1), 103–113 (2009). [CrossRef]
19. E. Myslivets, C. Lundström, J. M. Aparicio, S. Moro, A. O. J. Wiberg, C.-S. Bres, N. Alic, P. A. Andrekson, and S. Radic, “Spatial equalization of zero dispersion wavelength profiles in nonlinear fibers,” IEEE Photon. Technol. Lett. 21(24), 1807–1809 (2009). [CrossRef]
20. E. Myslivets, N. Alic, J. R. Windmiller, and S. Radic, “A new class of high-resolution measurements of arbitrary-dispersion fibers: localization of four-photon mixing process,” J. Lightwave Technol. 27(3), 364–375 (2009). [CrossRef]
21. S. Moro, E. Myslivets, J. R. Windmiller, N. Alic, J. M. Chavez Boggio, and S. Radic, “Synthesis of equalized broadband parametric gain by localized dispersion mapping,” IEEE Photon. Technol. Lett. 20(23), 1971–1973 (2008). [CrossRef]
22. M. Karlsson, “Four-wave mixing in fibers with randomly varying zero-dispersion wavelength,” J. Opt. Soc. Am. B 15(8), 2269–2275 (1998). [CrossRef]
23. F. Yaman, Q. Lin, S. Radic, and G. P. Agrawal, “Impact of dispersion fluctuation on dual-pump fiber-optic parametric amplifiers,” IEEE Photon. Technol. Lett. 16(5), 1292–1294 (2004). [CrossRef]
24. R. R. Nissim, E. Myslivets, and S. Radic, “Performance limits of inhomogeneous fiber optic parametric amplifiers operated in saturated regime,” J. Lightwave Technol. 32(21), 3552–3559 (2014). [CrossRef]
25. C. J. McKinstrie, G. G. Luther, and S. H. Batha, “Signal enhancement in collinear four-wave mixing,” J. Opt. Soc. Am. B 7(3), 340–344 (1990). [CrossRef]
26. N. Butcher and D. Cotter, The Elements of Nonlinear Optics (Cambridge University, 1991), pp. 227.
27. K. Tai, A. Hasegawa, and A. Tomita, “Observation of modulational instability in optical fibers,” Phys. Rev. Lett. 56(2), 135–138 (1986). [CrossRef] [PubMed]
28. G. Van Simaeys, Ph. Emplit, and M. Haelterman, “Experimental demonstration of the Fermi-Pasta-Ulam recurrence in a modulationally unstable optical wave,” Phys. Rev. Lett. 87(3), 033902 (2001). [CrossRef] [PubMed]
29. G. Cappellini and S. Trillo, “Third-order three-wave mixing in single-mode fibers: exact solutions and spatial instability effects,” J. Opt. Soc. Am. B 8(4), 824–838 (1991). [CrossRef]
30. C. J. McKinstrie and G. G. Luther, “Solitary-wave solutions of the generalised three-wave and four-wave equations,” Phys. Lett. A 127(1), 14–18 (1988). [CrossRef]
31. M. E. Marhic, K. K. Y. Wong, M. C. Ho, and L. G. Kazovsky, “92% pump depletion in a continuous-wave one-pump fiber optical parametric amplifier,” Opt. Lett. 26(9), 620–622 (2001). [CrossRef] [PubMed]
32. G. P. Agrawal, Nonlinear Fiber Optics (Academic, 1995).
33. M. Farahmand and M. de Sterke, “Parametric amplification in presence of dispersion fluctuations,” Opt. Express 12(1), 136–142 (2004). [CrossRef] [PubMed]
34. P. Kylemark, H. Sunnerud, M. Karlsson, and P. A. Andrekson, “Semi-analytic saturation theory of fiber optical parametric amplifiers,” J. Lightwave Technol. 24(9), 3471–3479 (2006). [CrossRef]
35. E. Brainis, D. Amans, and S. Massar, “Scalar and vector modulation instabilities induced by vacuum fluctuations in fibers: Numerical study,” Phys. Rev. A 71(2), 023808 (2005). [CrossRef]
36. J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence behavior of the Nelder-Mead simplex algorithm in low dimensions,” SIAM J. Optim. 9(1), 112–147 (1999). [CrossRef]
