# Patent

## Title
Methods and systems for nonlinear optical wave-mixing

## Abstract

A system for conversion or amplification using quasi-phase matched nonlinear optical wave-mixing comprises a first radiation source for providing a pump radiation beam, a second radiation source for providing a signal radiation beam, and a bent structure for receiving the pump radiation beam and the signal radiation beam. The radiation propagation portion of the bent structure is made of a uniform nonlinear optical material and the radiation propagation portion comprises a dimension taking into account the spatial variation of the nonlinear optical susceptibility along the radiation propagation portion as experienced by radiation travelling along the bent structure for obtaining quasi-phase matched nonlinear optical wave-mixing in the radiation propagation portion. The dimension thereby is substantially inverse proportional with the linear phase mismatch for the nonlinear optical process. The system also comprises an outcoupling radiation propagation portion for coupling out an idler radiation beam generated in the bent structure.

## Background
FIELD OF THE INVENTION
The present invention relates to the field of optics. More particularly, the present invention relates to methods and systems for Raman converters, parametric converters and parametric amplifiers with cavity enhancement and with quasi-phase-matching.
BACKGROUND OF THE INVENTION
Over the past several decades there has been growing interest in the development of devices based on second-order nonlinear effects such as sum-frequency generation (SFG) and difference frequency generation (DFG) and devices based on third-order nonlinear effects such as Raman-resonant four-wave-mixing (FWM) and Kerr-induced four-wave-mixing (FWM). SFG, DFG and Kerr-induced FWM are parametric light-matter interactions that are not resonant with a material level and that are used in parametric converters and parametric amplifiers. Raman-resonant FWM is a light-matter interaction that is perfectly resonant or almost perfectly resonant with a characteristic energy level of the material such as a vibrational energy level and that is used in Raman converters. SFG, DFG and Kerr-induced FWM involve a pump radiation beam at frequency ωp, a signal radiation beam at frequency ωs, and an idler radiation beam at frequency ωi. Raman-resonant FWM involves a pump radiation beam at frequency ωp, a Stokes radiation beam at frequency ωstokes that is lower than the pump frequency, and an anti-Stokes radiation beam at frequency ωanti-stokes that is higher than the pump frequency. One also uses the terms signal and idler for the Stokes and anti-Stokes radiation beams, respectively, or vice versa, and uses ωs and ωi to denote their frequencies. Due to the wavelength versatility offered by SFG, DFG, Raman-resonant FWM and Kerr-induced FWM, these processes feature a multitude of application possibilities in different domains. One important example thereof are Raman converters, parametric converters and parametric amplifiers based on silicon which have attracted much attention because of their potential for application in optical communication systems.
Basically, Raman-resonant FWM and Kerr-induced FWM are interactions between two pump photons, one signal photon and one idler photon, and the frequencies of these photons ωp, ωs and ωi satisfy the relation 2ωp−ωs−ωi=0. For Raman-resonant FWM in silicon one has in addition that |ωp−ωs|=2π×15.6 THz, which is the Raman shift of silicon. In the case of SFG and DFG there is an interaction between 1 pump photon, one signal photon and one idler photon, and the frequencies of these photons ωp, ωs and ωo satisfy the relation ωp+ωs=ωi for SFG and ωp−ωs=ωi for DFG. The efficiency of all processes depends on the pump intensity and on the processes' phase mismatch. The linear part Δklinear of the phase mismatch for Raman-resonant FWM and Kerr-induced FWM is given by

Δklinear=2kp−ks−ki 

where k{p,s,a}=ω{p,s,a}×n{p,s,a}/c are wave numbers with n{p,s,a} representing the effective indices of the pump, signal and idler waves, respectively. One can also write Δklinear as Δklinear=−β2(Δω)2−1/12β4(Δω)4 where β2=d2k/dω2 is the Group velocity dispersion (GVD) at the pump wavelength, β4=d4k/dω4 is the fourth-order dispersion at the pump wavelength, and Δω is the frequency difference between the pump and signal waves. For SFG the linear part Δklinear of the phase mismatch is given by

Δklinear=kp+ks−kt 

For DFG the linear part Δklinear of the phase mismatch is given by

Δklinear=kp−ks−ki 

The total phase mismatch for these processes also contains a nonlinear part that is function of the pump intensity, but since linear phase mismatches are considered here that are mostly much larger than the nonlinear part of the phase mismatch, the latter can be neglected in the remaining part of this text.
Due to their nonlinear nature, all above mentioned processes perform best at high optical intensities. These can be obtained by tightly confining the light for example in a nanowire waveguide and also by employing ring structures, whispering-gallery-mode disk resonators, or any other resonator structure in which the incoming light waves are resonantly enhanced. Regarding the requirement of having a small effective phase mismatch for the wavelength conversion processes, much progress has been made over the past several years. Regarding silicon-based converters, by engineering the dispersion of a silicon nanowire waveguide one can obtain phase-matched Kerr-induced FWM in the near-infrared for pump-signal frequency shifts with an upper limit of 52 THz (i.e., pump-signal wavelength differences up to 418 nm in the near-infrared region).
Notwithstanding the broad applicability of this phase-matched conversion technique, there are circumstances, applications, and materials where an alternative approach can be useful. First of all, not all materials used for SFG, DFG, Raman-resonant or Kerr-induced FWM are as easily workable as silicon to fabricate waveguide structures, which implies that not all materials can benefit from the waveguide-based phase-matching technique outlined above. Furthermore, even if one considers only a material such as silicon for which the waveguide-based phase-matching technique described above is well developed, it is important to know that, although the phase-matching bandwidth of the silicon nanowire referred to above is more than wide enough to enable phase-matched Raman-resonant FWM in the near-infrared at a pump-signal frequency shift of 15.6 THz, the dispersion-engineered geometry of the waveguide is such that one crucial advantage of using nanowires cannot be fully exploited. The particular advantage that cannot be fully exploited in that case is the possibility of having a fast recombination of two-photon-absorption-created free carriers at the waveguide boundaries. Indeed, the nanowire referred to above exhibits a relatively large free carrier lifetime (τeff≈3 ns), yielding substantial free carrier absorption (FCA) losses in the waveguide, and this decreases the wavelength conversion efficiency. Although these losses could be reduced by implementing around the nanowire carrier-extracting p-i-n diodes connected to a power supply, the advantage of using the low-cost intrinsic silicon-on-insulator platform would then be lost. Thus, in case one aims at keeping the fabrication cost and packaging cost as low as possible by opting for the intrinsic silicon-on-insulator platform, the challenge will be to enable efficient Raman-resonant FWM in the near-infrared wavelength domain using a nanowire that is not dispersion-engineered in a way that leads to substantial FCA losses.
Even if the increase in fabrication complexity and in cost when implementing carrier-extracting p-i-n diodes around the silicon nanowire as outlined above would not be considered as an issue, another challenge regarding silicon-based converters is in need of a solution. On one hand, the realization of phase-matched Kerr-induced FWM in silicon for pump-signal frequency shifts up to 52 THz in the near-infrared wavelength region fulfills the requirements of most wavelength conversion applications. On the other hand, specific applications in domains such as spectroscopy, sensing, industrial process control, environmental monitoring, biomedicine, and also telecommunications could benefit from wavelength conversion possibilities in both the near-infrared range and the mid-infrared region beyond 2 μm. These wavelength conversion possibilities should all ideally be available in one device, and preferably based on Kerr-induced FWM, which offers more wavelength flexibility than Raman-resonant FWM. It is extremely challenging, however, to engineer the dispersion of a silicon waveguide such that phase-matched Kerr-induced FWM is obtained for large pump-signal frequency shifts both in the near- and mid-infrared spectral regions. One approach to circumvent this problem of dispersion engineering would be to use the Kerr-induced FWM scheme based on two different pump frequencies, but the requirement of having a second pump frequency close to the desired idler frequency might be difficult to meet in the mid-infrared region, as widely tunable and practical mid-infrared pump sources are not so common yet. As such, for these specific applications the challenge will be to enable efficient, single-pump Kerr-induced FWM for a large pump-signal frequency shift in a spectral domain where the dispersion characteristics of the silicon nanowire are not optimally engineered for phase-matched Kerr-induced FWM. Also, if one aims at realizing a silicon-based source based on Kerr-induced FWM that, by changing just one parameter, can generate radiation at different wavelengths spread in the near- and mid-infrared spectral region, then one should use a technique different from phase matching. The development of such discretely-tunable sources would represent an important step in the search for low-cost, compact, room-temperature light sources tunable in the near- and mid-infrared. Such devices are still scarce nowadays but highly desirable for many applications, ranging from telecommunications and industrial process control, to environmental monitoring and biomedical analysis.
Also for SFG and DFG in silicon nanowires, there are dispersion engineering issues. To establish SFG and DFG in silicon, one usually applies strain on a silicon nanowire to induce the second-order nonlinearity that is needed for these processes and hence make the nanowire a quadratically nonlinear optical medium. Because of the very large pump-signal frequency shifts typically used in SFG and DFG, it is practically impossible to engineer the dispersion of the strained silicon nanowires in such a way that phase-matched SFG or DFG is obtained. Hence, it is challenging to achieve efficient SFG or DFG using only dispersion engineering.
One suggestion has been to establish quasi-phase-matching for SFG, DFG, Raman-resonant FWM or Kerr-induced FWM using heterogeneous materials. This traditional quasi-phase-matching technique for these nonlinear processes can be understood as follows: In case nothing is done about the phase mismatch, the idler intensity for radiation would continuously oscillate along the propagation path between a maximal value and zero, as the phase-mismatch-induced dephasing of the fields—this dephasing evolves periodically with the propagation distance—causes the nonlinear optical processes to either increase or decrease the idler intensity along the propagation path. When using traditional quasi-phase-matching for these processes, one adjusts the propagation regions behind the positions of maximal idler intensity, so that one does not have a total drop down of the idler intensity in these regions but at the same time the fields' dephasing, accumulated up to the positions of maximal idler intensity, can evolve back to zero in these adjusted regions. Hence, after traversing these adjusted areas the idler intensity can start growing again towards a maximum. The type of “adjustment” that needs to be applied to these propagation regions is that the susceptibility should be made zero there for the Raman-resonant or Kerr-induced FWM processes or reversed in sign for the SFG and DFG processes, so that these nonlinear processes cannot establish a decrease of the idler intensity in these areas whereas the fields' dephasing can still evolve back to zero. This is done using a heterogeneous conversion medium of which the characteristics are periodically manipulated. This is a complex approach and disadvantageous from a practical point of view.
SUMMARY OF THE INVENTION
It is an object of embodiments of the present invention to provide efficient methods and systems for nonlinear optical wave mixing, such as for example SFG, DFG and FWM like for example Raman-resonant FWM and/or Kerr-induced FWM. It is an advantage of at least some embodiments of the present invention that efficient nonlinear optical wave mixing, such as for example SFG, DFG and FWM, e.g. Raman-resonant FWM and/or efficient Kerr-induced FWM, can be obtained at wavelengths suitable for use in telecommunication.
Nonlinear optical wave mixing may in one example be a third order process, such as four wave mixing, or in another example be a second order process, such as three wave mixing.
It is an advantage of at least some embodiments of the present invention that efficient closed structure silicon Raman converters such as e.g. ring or disc silicon Raman converters, efficient closed structure silicon parametric converters such as e.g. ring or disc silicon parametric converters, and efficient closed structure silicon parametric amplifiers, e.g. ring or disc silicon parametric amplifiers are provided as well as corresponding methods.
It is an advantage of at least some embodiments according to the present invention that methods and systems are provided that allow obtaining relatively high efficiencies using cavity-enhanced quasi-phase matched wave mixing such as cavity-enhanced quasi-phase-matched DFG, cavity-enhanced quasi-phase-matched SFG, cavity-enhanced quasi-phase-matched Raman-resonant FWM and/or using cavity-enhanced quasi-phase-matched Kerr-induced FWM in a silicon resonator. Such cavity-enhanced quasi-phase matched wave mixing may in one example be a third order process, such as four wave mixing, or in another example be a second order process, such as three wave mixing.
It is an advantage of at least some embodiments according to the present invention that cavity-enhanced quasi-phase-matched wave mixing, such as cavity-enhanced quasi-phase-matched SFG, cavity-enhanced quasi-phase-matched DFG, cavity-enhanced quasi-phase-matched Raman-resonant FWM, cavity-enhanced quasi-phase-matched Kerr-induced FWM, in a resonator structure, e.g. silicon ring or disc resonator, can provide both a small effective phase mismatch and a high pump intensity, resulting in a boost of the nonlinear optical wave mixing efficiency, e.g. SFG efficiency, DFG efficiency, Raman-resonant FWM efficiency and or of the Kerr-induced FWM efficiency. This is especially advantageous for those cases where phase-matched wave mixing, e.g. phase-matched SFG, DFG, Raman-resonant FWM or Kerr-induced FWM performs badly, i.e. in the cases where the group velocity dispersion at the pump wavelength is large and/or the frequency difference between the pump and the signal is large.
It is an advantage of embodiments according to the present invention that efficient nonlinear optical wave matching, e.g. efficient SFG, efficient DFG, efficient Raman-resonant FWM or efficient Kerr-induced FWM can be obtained for any value of the linear phase mismatch Δklinear.
It is an advantage of embodiments according to the present invention that no dispersion engineering of the medium where the light propagates is required to obtain efficient SFG, efficient DFG, efficient Raman-resonant FWM or efficient Kerr-induced FWM.
It is an advantage of embodiments according to the present invention that it is not necessary to use only materials out of which waveguides can be fabricated and that other structures such as whispering-gallery-mode discs and discs with a rim polished according to a polygone shape can also be used.
It is an advantage of at least some embodiments according to the present invention that when working with a waveguide its geometry can be chosen freely, that the waveguide can be a nanowire, and that the nanowire dimensions can be chosen such that the free-carrier lifetime in the nanowire is short so that the free carrier absorption losses can be kept low.
It is an advantage of at least some embodiments of the present invention that one can use the low-cost intrinsic silicon-on-insulator platform without carrier-extracting p-i-n diodes connected to power supplies to obtain efficient nonlinear optical wave-mixing, such as efficient optical four wave mixing or efficient optical three wave mixing like efficient SFG, efficient DFG, efficient Raman-resonant FWM or efficient Kerr-induced FWM.
It is an advantage of at least some embodiments according to the present invention that suitable conditions for quasi-phase matched wave mixing, such as for quasi-phase-matched SFG, for quasi-phase-matched DFG, for quasi-phase-matched Raman-resonant FWM and for quasi-phase-matched Kerr-induced FWM are obtained in a uniform medium. More particularly it is an advantage that the obtained system and method is relatively simple and does e.g. not require active periodical adaptation of the nonlinear optical properties of a medium.
It is an advantage of at least some embodiments of the present invention that efficient, single-pump Kerr-induced FWM can be obtained for a large pump-signal frequency shift in a spectral domain where the dispersion characteristics of the medium where the light propagates are not optimally engineered for phase-matched Kerr-induced FWM.
It is an advantage of at least some embodiments of the present invention that discretely-tunable silicon-based sources could be provided, resulting in a low-cost, compact, room-temperature light sources tunable in the near- and mid-infrared. Such devices are still scarce nowadays but highly desirable for many applications, ranging from telecommunications and industrial process control, to environmental monitoring and biomedical analysis.
It is an advantage of at least some embodiments according to the present invention that a high pump intensity in the Raman converter, in the parametric converter, and in the parametric amplifier does not need to be provided using a high-power pump, but that the pump is resonantly enhanced in the medium for obtaining a sufficiently high pump power. Alternatively or in addition thereto, the signal input power, and the idler power also can be resonantly enhanced regardless the value of the linear phase mismatch Δklinear, resulting in high intensities being achieved. In other words also the signal input power initially inputted does not need to be a high signal input power pump.
The above objective is accomplished by a method and device according to the present invention.
The present invention relates to a system for conversion or amplification using quasi-phase matched nonlinear optical wave mixing, the system comprising a first radiation source for providing a pump radiation beam, a second radiation source for providing a signal radiation beam, and a bent structure for receiving the pump radiation beam and the signal radiation beam, wherein a radiation propagation portion, e.g. waveguiding portion, of the bent structure is made of a uniform material, e.g. a uniform quadratically nonlinear optical material, a uniform Raman-active or uniform Kerr-nonlinear material, and wherein the bent structure comprises a dimension taking into account the spatial variation of the nonlinear susceptibility along the radiation propagation portion as experienced by radiation travelling along the bent structure for obtaining quasi-phase-matched nonlinear optical wave mixing in the radiation propagation portion, the dimension being substantially inverse proportional with the linear phase mismatch for nonlinear optical wave mixing and an outcoupling radiation propagation portion, e.g. waveguiding portion, for coupling out an idler radiation beam generated in the bent structure. It is an advantage of embodiments according to the present invention that quasi-phase matching conditions can be achieved.
The nonlinear optical wave mixing may be a third order process, such as four wave mixing, or a second order process such as three wave mixing. Three wave mixing may comprise sum-frequency generation or difference frequency generation.
The system for conversion or amplification may be a system for Raman conversion, parametric conversion or parametric amplification. It is an advantage of at least some embodiments according to the present invention that the structure may be closed, so that the structure is a resonator allowing to establish cavity enhancement. In at least some embodiments according to the present invention, the closed structure may be a ring structure or a disc structure, such as for example a circular ring, an elliptical ring, a rectangular ring, an octagonal ring, a circular disc, an elliptical disc, a rectangular disc, an octagonal disc, etc. The structure also may be an open structure such as a snake-like structure, a sickle structure, or a spiral structure.
It is an advantage of at least some embodiments according to the present invention that a relatively simple system can be obtained allowing quasi-phase-matched nonlinear optical wave mixing, e.g. quasi-phase-matched sum-frequency generation (SFG), quasi-phase-matched difference frequency generation (DFG), quasi-phase-matched Raman-resonant FWM or quasi-phase-matched Kerr-induced FWM. It is an advantage that no active alteration of the nonlinear optical properties of the structure are required for accurate operation.
The bent structure may be a ring structure or disc structure.
The ring structure or disc structure may be circular, and the radius R of the ring structure or disc structure may be determined substantially inverse proportional with the linear phase mismatch for nonlinear optical wave mixing. The radius R thereby may be defined as the distance from the center of the circle to the central longitudinal axis in the circular radiation propagation area. The linear phase mismatch thereby may be defined by the following equations:

Δklinear=kp+ks−ki for SFG, 

Δklinear=kp−ks−ki for DFG, and Δklinear=2kp−ks−ki for Raman-resonant FWM and Kerr FWM. 

The radius R of the circular ring structure may be determined by the radius R being substantially equal to a factor s, equal to +1 or −1, times four divided by the linear phase mismatch for Raman-resonant FWM or divided by the linear phase mismatch for Kerr-induced FWM, i.e. it substantially fulfills relation




  
    R
    =
    
      s
      ⁢
      
        4
        
          Δ
          ⁢
          
            
          
          ⁢
          
            k
            linear
          
        
      
    
  
  ,




with s being a factor equal to +1 or −1 so that R has a positive value, and Δklinear being linear phase mismatch for Raman-resonant FWM or being the linear phase mismatch for Kerr-induced FWM. The radius R of the circular ring structure may be determined by the radius R being substantially equal to a factor s, equal to +1 or −1, times one divided by the linear phase mismatch for SFG or divided by the linear phase mismatch for DFG, i.e. it substantially fulfills relation





  R
  =
  
    s
    ⁢
    
      1
      
        Δ
        ⁢
        
          
        
        ⁢
        
          k
          linear
        
      
    
  




with s being a factor equal to +1 or −1 so that R has a positive value, and Δklinear being linear phase mismatch for SFG or being the linear phase mismatch for DFG. With substantially being equal to or substantially fulfilling the relation there is meant that advantageously the radius is equal or the relation is fulfilled, but that a deviation on the design rule is allowed wherein the quasi-phase-matched SFG efficiency, quasi-phase-matched DFG efficiency or quasi-phase-matched FWM efficiency is still high due to the explored effects. E.g. for a deviation of 5% on the design rule—this is a value which certainly lies within the fabrication tolerances that can be achieved nowadays—a quasi-phase-matched nonlinear optical wave mixing efficiency of 0.8 times the maximal efficiency at zero deviation may still be guaranteed. For a deviation of 10% on the design rule, a quasi-phase-matched nonlinear optical wave mixing efficiency of 0.5 times the maximal efficiency at zero deviation may still be guaranteed. For a deviation of 20% on the design rule, a quasi-phase-matched nonlinear optical wave mixing efficiency of 0.3 times the maximal efficiency at zero deviation may still be guaranteed. For deviations larger than 25% on the design rule, the quasi-phase-matched efficiency might become smaller than 0.2 times the maximal efficiency at zero deviation, and the quasi-phase-matching approach might not be interesting any longer.

The bent structure may have an inscribed circle and/or circumscribed circle having a radius inversely proportional to the linear phase mismatch for nonlinear optical wave mixing, e.g. SFG, DFG, FWM.
The bent structure may have an average radius inversely proportional to the linear phase mismatch for nonlinear optical wave mixing, e.g. SFG, DFG, FWM.
The system furthermore may be adapted to provide a pump radiation beam with wavenumber kp and a signal radiation beam with wavenumber ks, and result in an idler radiation beam with wavenumber ki, so that at least one of these beams is at resonator resonance, e.g. ring or disc resonance. In ring resonators for example at least one of the beams' wavenumbers may yield, when multiplying with R, an integer number.
The system may comprise a heating and/or cooling means and a temperature controller for controlling the temperature so that at least one of the pump radiation, the signal radiation and the idler radiation is at resonator resonance.
The uniform medium may be a Raman-active medium, and the process may be a quasi-phase-matched Raman-resonant FWM process.
The uniform medium may be a Kerr-nonlinear material and the process may be a quasi-phase-matched Kerr-nonlinear FWM-process.
The uniform medium may be a quadratically nonlinear optical material and the process may be a quasi-phase-matched SFG or DFG process.
It is an advantage of at least some embodiments according to the present invention that quasi-phase-matched nonlinear optical wave mixing, e.g. nonlinear optical four wave mixing or nonlinear optical three wave mixing such as quasi-phase-matched SFG and/or quasi-phase-matched DFG and/or quasi-phase-matched Raman-resonant FWM and/or quasi-phase-matched Kerr-induced FWM in a uniform medium such as a silicon ring can be obtained since it does not require special techniques to periodically adapt the nonlinear optical characteristics of the medium.
The uniform medium may be a crystalline material. The uniform medium may be (100) grown silicon, germanium, GaAs, InGaAs, diamond, and other semiconductor materials. The uniform medium may be SiN, Ba(NO3)2, CaCO3, NaNO3, tungstate crystals, BaF2, potassium titanyl phosphate (KTP), potassium dihydrogen phosphate (KDP), LiNbO3, deuterated potassium dihydrogen phosphate (DKDP), lithium triborate (LBO), barium borate (BBO), bismuth triborate (BIBO), LiIO3, BaTiO3, yttrium iron garnet (YIG), AlGaAs, CdTe, AgGaS2, KTiOAsO4 (KTA), ZnGeP2 (ZGP), RBTiOAsO4 (RTA), and other crystals. The uniform medium may be silicon covered by a thin layer of another uniform material such as a uniform graphene layer or a uniform graphene layer through which an electric current is flowing.
It is an advantage of embodiments according to the present invention that nonlinear optical wave mixing processes, such as SFG and/or DFG and/or Raman-resonant FWM and/or Kerr-induced FWM, can be established for wavelengths suitable for e.g. telecommunication.
The (100) grown silicon may be a silicon on insulator waveguide.
It is an advantage of at least some embodiments according to the present invention that an easily manufacturable system can be used for obtaining Raman converters, parametric converters or parametric amplifiers.
A controller may be provided for tuning the system with respect to an output wavelength, an output power or an obtained bandwidth.
The system may be adapted for selecting a TE or TM output by selecting respectively a TE or TM input. It is an advantage of embodiments according to the present invention that the polarization of the output is the same as the polarization of the input of the Raman converter, of the parametric converter, and of the parametric amplifier, and thus that no additional polarization filter is required for obtaining a particular polarized output.
The present invention also relates to a method for obtaining conversion or amplification, using quasi-phase-matched nonlinear optical wave mixing, the method comprising receiving a pump radiation beam and a signal radiation beam in a bent structure, a radiation propagation portion, e.g. waveguiding portion, of the bent structure being made of a uniform nonlinear optical material and having a dimension taking into account the spatial variation of the nonlinear optical susceptibility along the radiation propagation portion as experienced by radiation travelling along the bent structure for obtaining quasi-phase-matched nonlinear optical wave mixing in the radiation propagation portion, the dimension being substantially inverse proportional with a linear phase mismatch for nonlinear optical wave mixing, obtaining an idler radiation beam by interaction of the pump radiation beam and the signal radiation beam and coupling out an idler radiation beam from the bent structure. Conversion or amplification may be any of Raman conversion, parametric conversion or parametric amplification. The nonlinear optical wave mixing may be a SFG process, a DFG progress, a Raman-resonant FWM process or a Kerr-induced FWM process.
The bent structure may for example be a ring or disc structure, where the pump radiation beam and the signal radiation beam propagate in the ring or disc structure, whereby the ring or disc structure is circular and has a radius R determined substantially inverse proportional with a linear phase mismatch for quasi-phase-matched nonlinear optical wave mixing, e.g. quasi-phase-matched SFG, quasi-phase-matched DFG, quasi-phase-matched Raman-resonant FWM or quasi-phase-matched Kerr-induced FWM.
For three wave mixing processes such as the SFG and DFG processes, the pump radiation beam and the signal radiation beam may be guided in a circular ring structure having a radius substantially fulfilling the relation




  R
  =
  
    s
    ⁢
    
      1
      
        Δ
        ⁢
        
          
        
        ⁢
        
          k
          linear
        
      
    
  




with s being a factor equal to +1 or −1 so that R has a positive value, and Δklinear being the linear phase mismatch for SFG or being the linear phase mismatch for DFG. With substantially fulfilling the relation there is meant that advantageously the relation is fulfilled, but that a deviation on the design rule is allowed wherein the quasi-phase-matched SFG efficiency or quasi-phase-matched DFG efficiency is still high due to the explored effects.

For the FWM processes, the pump radiation beam and the signal radiation beam may be guided in a circular ring structure having a radius substantially fulfilling the relation




  
    R
    =
    
      s
      ⁢
      
        4
        
          Δ
          ⁢
          
            
          
          ⁢
          
            k
            linear
          
        
      
    
  
  ,




with s being a factor equal to +1 or −1 so that R has a positive value, and Δklinear being the linear phase mismatch for Raman-resonant FWM or being the linear phase mismatch for Kerr-induced FWM. With substantially fulfilling the relation there is meant that advantageously the relation is fulfilled, but that a deviation on the design rule is allowed wherein the quasi-phase-matched FWM efficiency is still high due to the explored effects.

The method may comprise letting the radiation beams propagate in the ring or disc structure and obtaining ring or disc resonance for at least one of the different radiation beams.
The method may comprise adjusting the in- and/or outcoupling efficiency for adjusting the cavity-enhancement of the radiation beams inside the ring or disc structure.
The method may comprise tuning the system with respect to an output wavelength, an output power or an obtained bandwidth.
The present invention also relates to a method for designing a converter or amplifier using quasi-phase-matched nonlinear optical wave mixing, the converter or amplifier using a pump radiation beam and a signal radiation beam, the method comprising selecting a bent structure suitable for quasi-phase-matched nonlinear optical wave mixing comprising selecting a uniform material for a radiation propagation portion of the bent structure and selecting at least one dimension of the radiation propagation portion taking into account the spatial variation of the nonlinear optical susceptibility along the radiation propagation portion as experienced by radiation travelling along the bent structure. The dimension thereby is substantially inverse proportional with the linear phase mismatch for nonlinear optical wave mixing.
The present invention also relates to a computer program product for, when executed on a computer, performing a method and/or controlling a system as described above. The present invention also relates to a data carrier carrying such a computer program product or to the transmission of such a computer program product over a wide or local area network.
Particular and preferred aspects of the invention are set out in the accompanying independent and dependent claims. Features from the dependent claims may be combined with features of the independent claims and with features of other dependent claims as appropriate and not merely as explicitly set out in the claims. These and other aspects of the invention will be apparent from and elucidated with reference to the embodiment(s) described hereinafter.



BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1A illustrates a schematic top-view representation of a Raman converter, a parametric converter or a parametric amplifier based on a (100) grown silicon ring, according to an embodiment of the present invention.
FIG. 1B illustrates a schematic representation of a Raman converter, a parametric converter or a parametric amplifier based on a whispering-gallery-mode disc where the light is coupled in the disc and out of the disc via a buried waveguide and where the light travels around in the disk close to its rim, and wherein quasi-phase matching according to an embodiment of the present invention can be obtained.
FIG. 1C illustrates a schematic top-view representation of a Raman converter, a parametric converter or a parametric amplifier based on an octagonally polished disc where the light is coupled in the disk and out of the disk via free space and where the light travels around in the disc close to its rim through reflection on each of the eight facets of the disk, and wherein quasi-phase-matching according to an embodiment of the present invention can be obtained.
FIG. 1D illustrates a schematic top-view representation of a Raman converter, a parametric converter or a parametric amplifier based on an open, sickle-shaped structure, the contours of which are along a circular ring, and wherein quasi-phase-matching according to an embodiment of the present invention can be obtained.
FIG. 1E illustrates a schematic top-view representation of a Raman converter, a parametric converter or a parametric amplifier based on an open, snake-shaped structure, the contours of which are along a circular ring and wherein quasi-phase-matching according to an embodiment of the present invention can be obtained.
FIG. 1F illustrates a schematic top-view representation of a Raman converter, a parametric converter or a parametric amplifier based on an open, sickle-shaped structure, the contours of which are along an octagon and wherein quasi-phase-matching according to an embodiment of the present invention can be obtained.
FIG. 1G illustrates a schematic top-view representation of a Raman converter, a parametric converter or a parametric amplifier based on an open, snake-shaped structure, the contours of which are along an octagon and wherein quasi-phase-matching according to an embodiment of the present invention can be obtained.
FIG. 1H illustrates a schematic top-view representation of a Raman converter, a parametric converter or a parametric amplifier based on an open, double-spiral-shaped structure (not drawn to scale), the contours of which are along a circular ring and wherein quasi-phase-matching according to an embodiment of the present invention can be obtained.
FIG. 1I illustrates a schematic top-view representation of a Raman converter, a parametric converter or a parametric amplifier based on an open, double-spiral-shaped structure, the contours of which are along a rectangle and wherein quasi-phase-matching according to an embodiment of the present invention can be obtained.
FIG. 1J illustrates a schematic top-view representation of a parametric converter or a parametric amplifier based on an open, double-spiral-shaped structure (not drawn to scale), the contours of which are along a circular ring and the top surface of which is covered by a uniform sheet of material through which a current is flowing, and wherein quasi-phase-matching according to an embodiment of the present invention can be obtained.
FIG. 2 illustrates (a) pump, (b) signal, (c) idler intensities in a ring Raman converter with the intensity values at a distance of 0 mm (2.1 mm) corresponding to |A3|2(|A4|2) in FIG. 1A, as can be obtained in an embodiment according to the present invention. The solid and dashed lines correspond to the quasi-phase-matched Raman converter pumped with 20 mW and to the perfectly phase-matched Raman converter pumped with 5 mW, respectively.
FIG. 3 illustrates the signal-to-idler conversion efficiency of the quasi-phase-matched ring Raman converter and of the perfectly phase-matched ring Raman converter as a function of pump input power.
FIG. 4 illustrates the signal-to-idler conversion efficiency of the quasi-phase-matched ring Raman converter and of the perfectly phase-matched ring Raman converter with different ring circumferences as a function of κp2=κs2=κi2.
FIG. 5 illustrates (a) pump, (b) signal, (c) idler intensities in a parametric ring converter with λp=1.6 μm with the intensity values at a distance of 0 μm (157 μm) corresponding to |A3|2(|A4|2) in FIG. 1A, as can be obtained in an embodiment according to the present invention.
FIG. 6 illustrates (a) pump, (b) signal, (c) idler intensities in a parametric ring converter with λp=1.8 μm with the intensity values at a distance of 0 μm (157 μm) corresponding to |A3|2(|A4|2) in FIG. 1A, as can be obtained in an embodiment according to the present invention.
FIG. 7 illustrates the signal-to-idler conversion efficiency of the quasi-phase-matched ring parametric converter and of the coherence-length-dependent ring parametric converter with different ring circumferences as a function of κp2=κs2=κi2.
FIG. 8 illustrates (a) pump, (b) signal, (c) idler intensities in the ring of the quasi-phase-matched ring parametric converter with κj2=0.02, and with the intensity values at a distance of 0 μm (267 μm) corresponding to |A3|2(|A4|2) in FIG. 1A.
FIG. 9 illustrates (a) pump, (b) signal, (c) idler intensities in the ring of the coherence-length-dependent ring parametric converter with L=π/|4Δk|=8.4 μm, with κj2=0.01, and with the intensity values at a distance of 0 μm (8.4 μm) corresponding to |A3|2(|A4|2) in FIG. 1A.
FIG. 10 illustrates the idler intensity in the spiral of the quasi-phase-matched SFG converter, as can be obtained in an embodiment according to the present invention.
FIG. 11 illustrates the idler intensity in the spiral of the quasi-phase-matched DFG converter, as can be obtained in an embodiment according to the present invention.
FIG. 12 illustrates a computing system as can be used in embodiments of the present invention for performing a method of resonating, converting or amplifying.



The drawings are only schematic and are non-limiting. In the drawings, the size of some of the elements may be exaggerated and not drawn on scale for illustrative purposes.
Any reference signs in the claims shall not be construed as limiting the scope.
DETAILED DESCRIPTION OF ILLUSTRATIVE EMBODIMENTS OF THE INVENTION
The present invention will be described with respect to particular embodiments and with reference to certain drawings but the invention is not limited thereto but only by the claims. Furthermore, the terms first, second, third and the like in the description and in the claims, are used for distinguishing between similar elements and not necessarily for describing a sequence, either temporally, spatially, in ranking or in any other manner. It is to be understood that the terms so used are interchangeable under appropriate circumstances and that the embodiments of the invention described herein are capable of operation in other sequences than described or illustrated herein.
It is to be noticed that the term “comprising”, used in the claims, should not be interpreted as being restricted to the means listed thereafter; it does not exclude other elements or steps. It is thus to be interpreted as specifying the presence of the stated features, integers, steps or components as referred to, but does not preclude the presence or addition of one or more other features, integers, steps or components, or groups thereof. Thus, the scope of the expression “a device comprising means A and B” should not be limited to devices consisting only of components A and B. It means that with respect to the present invention, the only relevant components of the device are A and B.
Reference throughout this specification to “one embodiment” or “an embodiment” means that a particular feature, structure or characteristic described in connection with the embodiment is included in at least one embodiment of the present invention. Thus, appearances of the phrases “in one embodiment” or “in an embodiment” in various places throughout this specification are not necessarily all referring to the same embodiment, but may. Furthermore, the particular features, structures or characteristics may be combined in any suitable manner, as would be apparent to one of ordinary skill in the art from this disclosure, in one or more embodiments.
Furthermore, while some embodiments described herein include some but not other features included in other embodiments, combinations of features of different embodiments are meant to be within the scope of the invention, and form different embodiments, as would be understood by those in the art. For example, in the following claims, any of the claimed embodiments can be used in any combination.
In the description provided herein, numerous specific details are set forth. However, it is understood that embodiments of the invention may be practiced without these specific details. In other instances, well-known methods, structures and techniques have not been shown in detail in order not to obscure an understanding of this description.
Where in embodiments of the present invention reference is made to a uniform Raman-active material, reference is made to a material or medium wherein the Raman susceptibility has a non-zero value.
Where in embodiments of the present invention reference is made to a uniform Kerr-nonlinear material or medium, reference is made to a material or medium wherein the Kerr susceptibility has a non-zero value.
Where in embodiments of the present invention reference is made to a uniform quadratically nonlinear material or medium, reference is made to a material or medium wherein the second-order susceptibility has a non-zero value.
Where in embodiments according to the present invention reference is made to a quasi-phase-matched (QPM) nonlinear optical wave mixing process, such as for example QPM SFG, QPM DFG, QPM Raman-resonant FWM or QPM Kerr-induced FWM, reference is made to a nonlinear optical wave mixing process where quasi-phase-matching in embodiments of the present invention is obtained in a non-traditional way, namely using a uniform medium. Nonlinear optical wave mixing may encompass for example four wave mixing or three wave mixing. For example, QPM SFG, QPM DFG, QPM Raman-resonant FWM or QPM Kerr-induced FWM can be obtained for any value of the linear phase mismatch Δklinear. When the value of the linear phase mismatch Δklinear for these nonlinear optical processes is negligibly small (Δklinear≈0) and the processes take place in a device that is not designed for QPM operation, the process is called to be perfectly phase-matched (PPM). When the product of the linear phase mismatch and the propagation distance d has an absolute value smaller than pi (|Δklinear|·d<π) and the nonlinear optical process takes place in a device that is not designed for QPM operation, the process is said to feature coherence-length-dependent operation. A process is not referred to as quasi-phase matched (QPM), as perfectly phase-matched (PPM) or as coherence-length dependent (CLD) in case none of the above approaches apply. To understand how PPM operation can be obtained for example for FWM, one needs to take into account that for FWM Δklinear can be written as Δklinear=−β2(Δω)2−1/12β4(Δω)4. This relation shows that one can establish PPM operation at large |Δω|-values in e.g. a silicon waveguide by engineering the β2 and β4-factors, i.e. the dispersion of the silicon waveguide. As the term function of β4 in the formula above generally is less important than the term function of β2, this dispersion engineering implies that one should establish β2=0 at the preferred pump wavelength, which then corresponds to the so-called zero-dispersion wavelength (ZDW).
Where in embodiments of the present application reference is made to a bent structure, reference is made to a non-straight structure. The latter also may be expressed as a structure wherein the propagation direction of propagating radiation is altered. The latter may for example be a curved structure, such as for example a circular, elliptical, or spiral structure, or a broken structure, such as for example an octagonal shaped structure or a rectangular shaped structure.
Where in embodiments of the present invention reference is made to a radiation propagation portion, reference may be made to a medium that allows propagation of radiation, and that for example can be a waveguide or a medium that allows free-space radiation propagation.
In a first aspect, the present invention relates to methods and systems for performing conversion or amplification using QPM nonlinear optical processes, more particularly nonlinear optical wave mixing processes. Such nonlinear optical processes encompass e.g. four wave mixing processes as well as three wave mixing processes such as SFG, DFG, Raman-resonant FWM and Kerr-induced FWM. The methods and systems for performing conversion or amplification may be methods and systems for performing Raman conversion, for performing parametric conversion or for performing parametric amplification. The system according to embodiments of the present invention comprises a first radiation source for providing a pump radiation beam and a second radiation source for providing a signal radiation beam. The system furthermore comprises a bent structure for receiving the pump radiation beam and the signal radiation beam, wherein a radiation propagation portion, e.g. a waveguide portion of the bent structure is made of uniform material. With uniform material there is meant that the material is a uniform nonlinear optical material. Nevertheless, due to the bending or curvature of the radiation propagation structure, radiation travelling through the bent structure will not see a uniform nonlinear optical susceptibility, but will see a variation therein. More particularly, whereas the material is uniform in a laboratory reference system fixed to the system, a variation in the nonlinear susceptibility is present felt by the radiation travelling through the bent structure, depending on the polarization of the radiation and the orientation of the principle crystal axes of the material used, e.g. the silicon material or silicon covered by graphene. According to embodiments of the present invention, the dimensions of the bent structure are selected taking into account the spatial variation of the susceptibility along the bent structure as experienced by the radiation travelling along the bent structure so that non-traditional QPM SFG, DFG, or FWM is obtained in the bent structure made of a uniform material. The bent structure thus may be any structure allowing to change or alter, e.g. curve, the propagation direction of the radiation, such that a variation in susceptibility is felt by the radiation. In advantageous embodiments, the bent structure may be a closed structure, such as for example a ring structure or disc structure. Such ring or disc structure may for example be a circular ring, an elliptical ring, an octagonal ring, a rectangular ring, a circular disc, an elliptical disc, an octagonal disc or a rectangular disc and the properties of the closed structure may be selected such that at least one of the radiation beams is enhanced. Alternatively, the structure may be an open structure wherein a change is induced in the propagation direction of the radiation such that a variation in susceptibility is felt by the radiation. An example thereof could be a sickle-shaped structure, a snake-shaped structure, or a spiral-shaped structure, the contours of which are along a circular ring, an octagon, or another type of polygon. A number of particular examples is shown in FIG. 1B to FIG. 1J.
As indicated, a dimension of the bent structure is selected so that QPM FWM is obtained in the bent structure made of a uniform material. The typical dimension of a structure may be an average length of a radiation propagation portion, e.g. waveguide portion, of the bent structure, but also may be for example a radius of the bent structure, an average radius of the bent structure, a radius of an inscribed circle or incircle of the structure, a radius of a circumscribed circle or circumcircle, etc. In some embodiments, a dimension also may be an average radius of curvature. If for example the average length is used, the average length of the radiation propagation part of the bent structure may be in a range between 1 μm and 10 cm.
According to embodiments of the present invention, a dimension of the bent structure or more particularly the radiation propagation portion thereof is such that it is substantially inverse proportional with the linear phase mismatch for SFG, DFG or FWM. The linear phase mismatch for SFG equals the pump wavenumber plus the signal wavenumber minus the idler wavenumber, the linear phase mismatch for DFG equals the pump wavenumber minus the signal wavenumber minus the idler wavenumber, and the linear phase mismatch for FWM equals two times the pump wavenumber minus the signal wavenumber minus the idler wavenumber. These linear phase mismatches indicate how fast the dephasing of the different fields first grows and then returns to zero again in a periodic way along the propagation path. The typical dimension of the bent structure may be inversely proportional to the linear phase mismatch for SFG, DFG or FWM. In other words




  
    Typical
    ⁢
    
      
    
    ⁢
    dimension
    ⁢
    
      
    
    ⁢
    of
    ⁢
    
      
    
    ⁢
    the
    ⁢
    
      
    
    ⁢
    bent
    ⁢
    
      
    
    ⁢
    structure
  
  =
  
    f
    ⁡
    
      (
      
        1
        
          Δ
          ⁢
          
            
          
          ⁢
          
            k
            linear
          
        
      
      )
    
  




In some embodiments according to the present invention, a closed loop structure is used and the structure is adapted for enhancing at least one and advantageously a plurality or more advantageously all of the radiation beams in the closed loop structure. Nevertheless, also open structures or open loop structures are envisaged. The system furthermore comprises an outcoupling radiation propagation portion, e.g. a waveguide, for coupling out an idler radiation beam generated in the bent structure.
The material used may be any type of material providing a uniform material, i.e. a uniform quadratically nonlinear material, a uniform Raman-active and/or uniform Kerr-nonlinear material. One example of a material that could be used is silicon, but there exist many other materials that could also be employed. Other materials having the same crystal structure as silicon typically also can be used. Further examples of materials that can be used are silicon nitride (SiN) and crystalline materials belonging to the m3m point-symmetry group or a similar symmetry group, for example other semiconductors such as germanium, GaAs, InGaAs, diamond, and other crystals such as Ba(NO3)2, CaCO3, NaNO3, tungstate crystals, BaF2, potassium titanyl phosphate (KTP), potassium dihydrogen phosphate (KDP), LiNbO3, deuterated potassium dihydrogen phosphate (DKDP), lithium triborate (LBO), barium borate (BBO), bismuth triborate (BIBO), LiIO3, BaTiO3, yttrium iron garnet (YIG) crystals, AlGaAs, CdTe, AgGaS2, KTiOAsO4 (KTA), ZnGeP2 (ZGP), RBTiOAsO4 (RTA) Also silicon covered by a thin layer of another uniform material such as uniform graphene with or without an electric current flowing through the graphene sheet could be employed.
The structure may be made in a plurality of ways. It may be processed on a substrate, it may be fabricated using different techniques such as CMOS technology, electron beam lithography, photolithography, chemical vapour deposition (CVD), low-pressure chemical vapour deposition (LPCVD), pulsed laser deposition (PLD), plasma enhanced chemical vapour deposition (PECVD), thermal oxidation, reactive-ion etching, focused ion beam, crystal growth, epitaxial growth, sputtering, flux pulling method from a stoichiometric melt, and polishing.
As indicated above, the system comprises a first and second radiation sources for generating a pump radiation beam and a signal radiation beam. Such radiation sources typically may be lasers, although embodiments of the present invention are not limited thereto. The type of lasers selected may depend on the application. Some examples of lasers that could be used are semiconductor lasers, solid-state lasers, fiber lasers, gas lasers, . . . . The required output power and wavelength of e.g. the pump laser depends on the output that one wants to obtain, e.g. of the output power one expect from the converter or amplifier.
In some embodiments, the system also may comprise a controller for controlling the system, e.g. the first radiation source and the second radiation source, and environmental conditions of the system, so as to be able to slightly tune the system. In one embodiment, a heating and/or cooling means, e.g. heater and/or cooler, may be present for controlling the temperature of the system and in this way also properties of the system. In an advantageous embodiment, the controller may be adapted so that defined conditions for obtaining cavity-enhanced quasi-phase-matched SFG, cavity-enhanced quasi-phase-matched DFG, or cavity-enhanced quasi-phase-matched FWM, such as a well-controlled temperature, are maintained in the system. Such a controller may operate in an automated and/or automatic way. The controller may be implementing predetermined rules or a predetermined algorithm for controlling the system, or it may be adapted for using a neural network for controlling the system. The controller may comprise a memory for storing data and a processor for performing the steps as required for controlling. The controller may be computer implemented. Whereas in the present aspect, the controller is described as a component of the system, in one aspect, the present invention also relates to a controller as such for performing a method of controlling a system for operating in quasi-phase-matched SFG conditions, quasi-phase-matched DFG conditions, or quasi-phase-matched FWM conditions.
In some embodiments, the system also may comprise a feedback system, providing parameters for checking whether the appropriate conditions are fulfilled and for reporting corresponding information. Such information may for example be transferred to the controller and used by the controller for adjusting or correcting the conditions.
In some embodiments, the resonator, converter or amplifier is adapted for providing a given polarization mode. It thereby is an advantage that no filter means is required for obtaining the polarization mode, as the polarization mode is not altered by the structure.
By way of illustration and for the ease of explanation, embodiments of the present invention not being limited thereto, some features and aspects will now further be described with reference to QPM Raman-resonant FWM and to QPM Kerr-induced FWM in a circular ring structure, and with reference to QPM SFG and QPM DFG in a spiral structure. These provide, without embodiments of the present invention being bound by theory, a possible explanation of the features of the obtained structures. QPM Raman-resonant FWM and QPM Kerr-induced FWM is discussed in a (100) grown ring-shaped silicon-on-insulator (SOI) waveguide. A system according to such embodiments is illustrated by way of example in FIG. 1A. The material used has a uniform Raman-active medium for the Raman-resonant FWM process and a uniform Kerr-nonlinear medium for Kerr-induced FWM process, with respect to a laboratory reference system coupled to the system. However, as TE-polarized pump, signal, and idler waves propagate along the ring, with their polarization always perpendicular to their local direction of propagation, the fourth rank Raman tensor and the fourth rank Kerr tensor, that are uniform in the laboratory frame, are position dependent in a reference frame defined by the direction of propagation and the polarization. This leads to a spatial periodic variation of the Raman susceptibility and of the Kerr susceptibility around the ring, and this variation can be used to design a ring with QPM Raman-resonant FWM or a ring with QPM Kerr-induced FWM. Taking into account that the variation of the Raman susceptibility and of the Kerr susceptibility as experienced by the TE-polarized fields in the (100) grown silicon ring is proportional to cos2 (2θ) with θ defined as in FIG. 1a, the condition for QPM Raman-resonant FWM in the ring or the condition for QPM Kerr-induced FWM in the ring is given by




  
    
      
        
          
            Δ
            ⁢
            
              
            
            ⁢
            
              k
              linear
            
          
          =
          
            s
            ⁢
            
              4
              R
            
          
        
        ,
      
    
    
      
        (
        
          1
          ⁢
          a
        
        )
      
    
  




where s=±1 so that R has a positive value, and R is the ring radius in case of a circular ring. Important to know is that even if this quasi-phase-matching condition is not exactly fulfilled, for example due to small deviations of R, the quasi-phase-matching efficiency will still be high.

It is to be noticed that this approach can also be used for any other Raman-active medium with the same crystal symmetry as silicon, for any other Kerr-nonlinear medium with the same crystal symmetry as silicon, and for some Raman-active media and/or Kerr-nonlinear media with a crystal symmetry similar to that of silicon. Hence, many crystals can be used, some examples of which are SiN, germanium, GaAs, InGaAs, diamond, Ba(NO3)2, CaCO3, NaNO3, tungstate crystals, BaF2, potassium titanyl phosphate (KTP), potassium dihydrogen phosphate (KDP), LiNbO3, deuterated potassium dihydrogen phosphate (DKDP), lithium triborate (LBO), barium borate (BBO), bismuth triborate (BIBO), LiIO3, BaTiO3, yttrium iron garnet (YIG) crystals. In addition to achieving QPM FWM, one wants to design the ring so that all waves involved in the Raman-resonant FWM process and in the Kerr-induced FWM process are resonantly enhanced in the ring; this will lead to high intensities in the ring even for low intensity input waves. Complete resonant enhancement occurs when the values of k{p,s,i}R correspond to integer numbers. It is remarked that if kpR and ksR have integer values and if in addition the quasi-phase-matching condition expressed above is fulfilled, then kiR will also correspond to an integer number. As such, whereas for ring converters based on the principle of “coherence-length-dependent (CLD) operation” it is not possible to have the pump, signal and idler waves all at ring resonances in the presence of a non-zero Δklinear, such a triply-resonant condition at a non-zero Δklinear does become possible when using QPM operation. It is also noted that in most cases the free spectral range of the ring will be quite small, so that a small temperature tuning will suffice to guarantee that the pump and signal waves, and automatically also the idler wave, are at ring resonances. Using temperature tuning, one can also compensate for phase-shifting phenomena that might occur in the silicon medium, such as self- and cross-phase modulation.
QPM SFG and DFG are discussed in a spiral-shaped waveguide. A system according to such embodiments is illustrated by way of example in FIG. 1H to 1J. In these drawings, different spiral shapes are shown, the spiral shaped structure in one example being more circular shaped and in another example being more rectangular shaped. In FIG. 1J the spiral-shaped waveguide is a spiral-shaped silicon-on-insulator (SOI) waveguide covered by a uniform sheet of graphene through which an electric current is flowing. Because of the electric current, the graphene top sheet acquires a strong second-order nonlinearity, and the waves propagating in the spiral-shaped waveguide feel the presence of this second-order nonlinearity due to the interaction of their evanescent tails with the graphene top layer. This second-order nonlinearity is uniform with respect to a laboratory reference system coupled to the system. However, as TE-polarized pump, signal, and idler waves propagate along the spiral-shaped SOI waveguide, with their polarization always perpendicular to their local direction of propagation, the second-order nonlinearity tensor, that is uniform in the laboratory frame, is position dependent in a reference frame defined by the direction of propagation and the polarization. This leads to a spatial periodic variation of the second-order susceptibility around the spiral, and this variation can be used to design a spiral with QPM SFG or a spiral with QPM DFG. Taking into account that the variation of the second-order susceptibility as experienced by the TE-polarized fields is proportional to cos(θ) with θ defined as the angle between the local field polarization and the direction of the current flow, the condition for QPM SFG in the ring or the condition for QPM DFG in the ring is given by




  
    
      
        R
        =
        
          s
          ⁢
          
            1
            
              Δ
              ⁢
              
                
              
              ⁢
              
                k
                linear
              
            
          
        
      
    
    
      
        (
        
          1
          ⁢
          b
        
        )
      
    
  




where s=±1 so that R has a positive value, and R is the average radius of the spiral. Important to know is that even if this quasi-phase-matching condition is not exactly fulfilled, for example due to small deviations of R, the quasi-phase-matching efficiency will still be high.

It is to be noticed that this approach can also be used for any other quadratically nonlinear medium with the same crystal symmetry as graphene through which an electric current is flowing, and for some quadratically nonlinear media with a crystal symmetry similar to that of graphene through which an electric current is flowing. Hence, many crystals can be used, some examples of which are SiN, GaAs, InGaAs, Ba(NO3)2, CaCO3, NaNO3, tungstate crystals, BaF2, potassium titanyl phosphate (KTP), potassium dihydrogen phosphate (KDP), LiNbO3, deuterated potassium dihydrogen phosphate (DKDP), lithium triborate (LBO), barium borate (BBO), bismuth triborate (BIBO), LiIO3, BaTiO3, yttrium iron garnet (YIG) crystals, AlGaAs, CdTe, AgGaS2, KTiOAsO4 (KTA), ZnGeP2 (ZGP), RBTiOAsO4 (RTA).
The amplifier or converter according to embodiments of the present invention may also provide the functionality of a resonator, embodiments not being limited thereto. By way of illustration, embodiments of the present invention not being limited thereto, the present invention now will be further illustrated with reference to particular embodiments, illustrating some features and advantages of embodiments according to the present invention. Without wishing to be bound by theory, a mathematical suggestion of how the principles of embodiments of the present invention could be explained also is provided.
In a first particular embodiment, reference is made to a QPM Raman-resonant FWM system based on a silicon ring resonator. The system of the example shown thereby is not only adapted for QPM Raman-resonant FWM, but also illustrates that advantageously use can be made of cavity enhancement effects and of the free choice of the waveguide geometry when using quasi-phase matching.
In this embodiment two comparisons will be made between QPM silicon Raman ring converters and PPM Raman converters. To do this, first a modeling formalism for Raman converters is introduced.
Without restricting the general validity of the results, focus is made on quasi-continuous-wave operation and on operation at exact Raman resonance. Assuming that nk/nl≈1 (for k,l=p,s,i) and that Kerr-induced FWM in silicon is negligible at the considered working point of exact Raman resonance, the equations expressing the steady-state spatial variation of the slowly-varying pump, signal and idler field amplitudes Ap(ζ), As(ζ), Ai(ζ) in the SOI ring Raman converter are given by




  
    
      
        
          
            
              ∂
              
                A
                p
              
            
            
              ∂
              ζ
            
          
          =
          
            
              
                
                  ω
                  p
                
                
                  ω
                  s
                
              
              ⁢
              
                
                  g
                  R
                
                2
              
              ⁢
              
                
                  ρ
                  ⁡
                  
                    (
                    θ
                    )
                  
                
                ⁡
                
                  [
                  
                    
                      
                        
                          
                          
                            A
                            i
                          
                          
                        
                        2
                      
                      ⁢
                      
                        A
                        p
                      
                    
                    -
                    
                      
                        
                          
                          
                            A
                            s
                          
                          
                        
                        2
                      
                      ⁢
                      
                        A
                        p
                      
                    
                  
                  ]
                
              
            
            -
            
              
                γ
                p
              
              ⁢
              
                A
                p
              
            
          
        
        ,
      
    
    
      
        (
        2
        )
      
    
  
  
    
      
        
          
            
              ∂
              
                A
                s
              
            
            
              ∂
              ζ
            
          
          =
          
            
              
                
                  g
                  R
                
                2
              
              ⁢
              
                
                  ρ
                  ⁡
                  
                    (
                    θ
                    )
                  
                
                ⁡
                
                  [
                  
                    
                      
                        
                          
                          
                            A
                            p
                          
                          
                        
                        2
                      
                      ⁢
                      
                        A
                        s
                      
                    
                    +
                    
                      
                        A
                        p
                        2
                      
                      ⁢
                      
                        A
                        i
                        *
                      
                      ⁢
                      
                        e
                        
                          iΔ
                          ⁢
                          
                            
                          
                          ⁢
                          
                            k
                            linear
                          
                          ⁢
                          ζ
                        
                      
                    
                  
                  ]
                
              
            
            -
            
              
                γ
                s
              
              ⁢
              
                A
                s
              
            
          
        
        ,
      
    
    
      
        (
        3
        )
      
    
  
  
    
      
        
          
            
              ∂
              
                A
                i
              
            
            
              ∂
              ζ
            
          
          =
          
            
              
                -
                
                  
                    ω
                    i
                  
                  
                    ω
                    s
                  
                
              
              ⁢
              
                
                  g
                  R
                
                2
              
              ⁢
              
                
                  ρ
                  ⁡
                  
                    (
                    θ
                    )
                  
                
                ⁡
                
                  [
                  
                    
                      
                        
                          
                          
                            A
                            p
                          
                          
                        
                        2
                      
                      ⁢
                      
                        A
                        i
                      
                    
                    +
                    
                      
                        A
                        p
                        2
                      
                      ⁢
                      
                        A
                        s
                        *
                      
                      ⁢
                      
                        e
                        
                          iΔ
                          ⁢
                          
                            
                          
                          ⁢
                          
                            k
                            linear
                          
                          ⁢
                          ζ
                        
                      
                    
                  
                  ]
                
              
            
            -
            
              
                γ
                i
              
              ⁢
              
                A
                i
              
            
          
        
        ,
      
    
    
      
        (
        4
        )
      
    
  




where ζ=Rθ and A{p,s,i} is normalized such that |A{p,s,i}|2 corresponds to intensity. The function ρ(θ) will be specified further on. The terms containing ejΔklinearζ express the Raman-resonant FWM interaction, and the terms proportional to |A{p,s}|2 A{s,p} and |A{p,i}|2 A{i,p} describe two accompanying Raman processes. The coefficient gR is the Raman gain coefficient of silicon and γ{p,s,i} describe the optical losses in the SOI waveguide. At near-infrared operation wavelengths, which are considered in this embodiment for the Raman converters, the latter receive contributions from linear propagation losses, two-photon absorption (TPA) and TPA-induced free carrier absorption (FCA). At the entry point of light into the ring from the channel one has θ=0 (see FIG. 1). Coupling from the channel to the ring is described in the usual way,





  
    
      
        
          
            (
            
              
                
                  
                    A
                    
                      j
                      ⁢
                      
                        
                      
                      ⁢
                      2
                    
                  
                
              
              
                
                  
                    A
                    
                      j
                      ⁢
                      
                        
                      
                      ⁢
                      3
                    
                  
                
              
            
            )
          
          =
          
            
              (
              
                
                  
                    
                      σ
                      j
                    
                  
                  
                    
                      i
                      ⁢
                      
                        
                      
                      ⁢
                      
                        κ
                        j
                      
                    
                  
                
                
                  
                    
                      i
                      ⁢
                      
                        
                      
                      ⁢
                      
                        κ
                        j
                      
                    
                  
                  
                    
                      σ
                      j
                    
                  
                
              
              )
            
            ⁢
            
              (
              
                
                  
                    
                      A
                      
                        j
                        ⁢
                        
                          
                        
                        ⁢
                        1
                      
                    
                  
                
                
                  
                    
                      
                        A
                        
                          j
                          ⁢
                          
                            
                          
                          ⁢
                          4
                        
                      
                      ⁢
                      
                        exp
                        ⁡
                        
                          (
                          
                            i
                            ⁢
                            
                              
                            
                            ⁢
                            
                              k
                              j
                            
                            ⁢
                            L
                          
                          )
                        
                      
                    
                  
                
              
              )
            
          
        
        ,
      
    
    
      
        (
        5
        )
      
    
  




with j=p,s,i, with the positions of the fields (1)-(4) indicated in FIG. 1, and with L=2πR. One can consider real-valued coupling constants σj, κj that satisfy the relation σj2+κj2=1

One now can solve numerically equations (2) to (5) to make two comparisons: On one hand, to illustrate the effect of cavity enhancement in the QPM silicon Raman ring converters, a concrete QPM Raman ring converter configuration and a theoretical one-dimensional PPM Raman converter without losses will be compared. On the other hand, to illustrate the effect of having a free choice for the waveguide geometry in QPM silicon Raman ring converters, a concrete QPM Raman ring converter configuration and a concrete PPM Raman ring converter, both with losses also will be compared.
For the first exemplary comparison in the first embodiment, one considers for the QPM Raman ring converter configuration a TE-polarized pump input and a TE-polarized Stokes-shifted signal input with a frequency difference corresponding to the exact Raman resonance: ωp=1.22×1015 rad/s (λp=1.55 μm), ωs=1.12×1015 rad/s (λs=1.686 μm). This leads to a generated idler wave with angular frequency ωi=1.32×1015 rad/s (λi=1.434 μm). The system may have a structure as illustrated in FIG. 1. At these near-infrared operation wavelengths, the Raman gain coefficient gR of silicon equals 20×10−9 cm/W. As there are no dispersion engineering constraints for the QPM converter, one is free to choose the waveguide geometry for both the ring and the channel. When taking a nanowire of 300-nm height and 500-nm width, the free carrier lifetime τeff will be as short as 500 ps. Because of the oblong core of the nanowire, TM fields generated through spontaneous Raman scattering in the ring are for the large part coupled out after each roundtrip, and cannot build up in the ring. In case an oxide cladding is used for the nanowire, the dispersion D=−2πcβ2/λ2 in the nanowire for the TE-polarized pump field at λp=1.55 μm equals 1000 ps/(nm*km), yielding Δk=2kp−ks−ka=122 cm−1. For s=+1, it is found that the condition (1) is met for R=328 μm, corresponding to a ring circumference L of 2.1 mm. For the remaining device parameters the following values were taken: κp2=κs2=κi2=0.05 (in line with reported values), waveguide modal area Aeff=0.20 μm2, linear loss α=1 dB/cm, two-photon absorption coefficient β=0.7×10−11 m/W, free carrier absorption efficiency φ=6×10−10, Ip,in=1×1011 W/m2, Is,in=1×108 W/m2, and ρR(θ)=cos2(2θ) along the ring as specified earlier on. One then can numerically solve equations (2) to (5) for the QPM ring Raman converter. The solid lines in FIG. 2 parts (a)-(c) show the steady-state distributions along the ring of the pump, signal and idler intensities, respectively. Using equation (5), one finds from FIG. 2(c) that Ii,outring=2.15×108 W/m2. The conversion efficiency thus is larger than unity, i.e. larger than 0 dB. For comparison, a one-dimensional PPM Raman converter with equal length would yield Ii,out1D=(ωi/ωs)2(gR2/2)2Ip,in2Is,inL1=6.13×104 W/m2. The enhancement factor Ii,outring/Ii,out1D for the QPM ring converter with losses included compared to the one-dimensional PPM converter without losses thus equals 3.5×103, which is very large. In conclusion, the idler output intensity of a QPM silicon ring Raman converter can easily become 3×103 times larger than that of a one-dimensional PPM Raman converter of equal length. Taking into account the quadratic dependence of the latter's output on the pump input, this also implies that the QPM ring Raman converter needs a 50 times smaller pump input intensity than the one-dimensional PPM Raman converter to produce the same idler output. Furthermore, signal-to-idler conversion efficiencies larger than unity can be obtained using relatively low pump input intensities. These improvements in conversion performance substantially expand the practical applicability of Raman converters in different application domains.
For the second comparison in the first embodiment, the same QPM Raman ring converter configuration is considered as described above. Instead of comparing it to a theoretical one-dimensional PPM Raman converter without losses as was done above, it is compared to a concrete PPM ring Raman converter with losses included and where Δklinear=0 is obtained by dispersion engineering the nanowire. For the latter device, the same parameter values are adopted as for the QPM converter described above, including L=2.1 mm and κp2=κs2=κi2=0.05, except for the following: Δklinear=0 cm−1, ρR(θ)=1, and τeff=3 ns. Again one assumes all three waves to be at ring resonances. Now one can numerically solve equations (2)-(5) for both Raman converter configurations, while assuming a signal input power of 20×10−6 W, corresponding to Is,in=1×108 W/m2, and while varying the pump input power between 2 mW and 20 mW in steps of 1 mW—this corresponds to Ip,in ranging from 1×1010 W/m2 to 1×1011 W/m2 in steps of 0.5×1010 W/m2. The reason for choosing a variable pump input is that for the PPM converter with large τeff the nonlinear losses will become significant already at low pump powers, whereas for the QPM converter with small τeff the onset of nonlinear losses will occur at higher pump powers. For the pump power levels of 20 mW (5 mW), the steady-state distributions along the ring of the pump, signal and idler intensities in the QPM (PPM) converter are represented by the solid (dashed) lines in FIG. 2(a)-(c). FIG. 3 shows the steady-state conversion efficiencies Ii,out/Is,in at the different pump levels for the QPM and PPM devices. FIG. 3 shows that at pump input powers up to 7 mW the PPM CARS converter has higher conversion efficiencies than the QPM CARS device, whereas for higher pump powers the QPM converter outperforms the PPM converter. This can be explained as follows: At very low pump powers the nonlinear losses in both converter types are low, and so the operation point is situated quite close to the lossless, small-signal regime, where the PPM converter performs much better than its QPM counterpart. Starting from pump powers of a few mW, however, the PPM converter, which exhibits a relatively large free carrier lifetime, is subjected to substantial pump-power-dependent nonlinear losses, which is not the case for the QPM converter. As a result, the conversion efficiency of the PPM device saturates at a value of −3 dB for a pump power of 5 mW, whereas that of the QPM converter continues to grow for increasing pump power, exceeding a value of +3 dB at a pump power level of 20 mW. Hence, in case no carrier-extracting p-i-n diodes are used, the QPM converter can outperform the PPM converter by as much as 6 dB. Also, FIG. 3 shows that starting from pump powers as low as 11 mW the QPM device can establish conversion efficiencies larger than 0 dB. Taking into account that the best-performing silicon Raman converter demonstrated thus far is a channel waveguide converter that, when excited with extremely high-energy pump pulses with peak intensities of 2×1013 W/m2, produces a signal-to-idler conversion efficiency of 58% or −2.4 dB, it is found that the QPM ring converter presented here could considerably outperform this record demonstration both in terms of conversion efficiency and in terms of minimizing the required pump input intensity. This is partially due to the fact that the QPM ring device can benefit from cavity enhancement in the ring which the channel waveguide converter cannot, and partially because of the non-traditional quasi-phase-matching mechanism itself, which appears in the ring made of uniform silicon provided that the ring circumference is properly chosen.
It is pointed out that for the QPM device the TPA losses will also undergo a periodic variation proportional to (0.88+0.12 cos2(2θ)) in the ring, but as the varying part of the TPA losses is small compared to the constant part, this variation only has a small influence, as simulations that are not presented here in detail confirm. In the second embodiment it will be shown, however, that an equally small variation of the effective Kerr nonlinearity γ does suffice to effectively establish quasi-phase-matching in a parametric converter, since this variation establishes a phase effect rather than an intensity loss effect.
One might question whether the coupling coefficients κp2=κs2=κi2=0.05 assumed here yield the highest conversion efficiencies for the QPM and PPM Raman devices, and whether the ring circumference L=2.1 mm calculated for the QPM converter is the most optimal ring circumference for the PPM converter as well. When varying the coupling coefficients using intermediate steps of 0.005 for the QPM converter with L=2.1 mm and for the PPM converter with different ring circumferences (see FIG. 4), it is indeed observed that for both converters the coupling coefficients have a significant influence on the attainable conversion efficiencies and that for the PPM converter the value of the ring circumference is important as well. FIG. 4 shows that the QPM converter performs best for the coupling coefficients κp2=κs2=κi2=0.045−a value close to the value of 0.05 which was already chosen for the illustration and that the PPM converter features the highest performance for κp2=κs2=κi2=0.075 and L=1.3 mm. It is pointed out that the maximal conversion efficiency of the PPM converter is only 1 dB higher than the efficiency obtained earlier on for κp2=κs2=κis=0.05 and L=2.1 mm, so the general performance tendencies of the PPM converter as outlined above remain valid.
In conclusion for the present examples, since for a QPM Raman ring converter the nanowire geometry can be chosen such that the FCA losses are minimal, the device should, when considering actual converter operation with losses included, substantially outperform a PPM Raman ring converter based on a dispersion-engineered nanowire of the type presented earlier in the literature. It is remarked that the latter comparison holds provided that both devices are fabricated using the low-cost intrinsic silicon-on-insulator platform without carrier-extracting p-i-n diodes. Furthermore, the QPM Raman ring converter should significantly outperform the best-performing silicon Raman converter demonstrated thus far, as it is able to establish signal-to-idler conversion efficiencies larger than 0 dB at modest pump powers. Such high performance, combined with the fact that no dispersion engineering is required and that the device can be realized in the low-cost intrinsic silicon-on-insulator platform, show the potentialities of QPM Raman wavelength conversion in silicon rings.
In a second particular embodiment, reference is made to a QPM Kerr-induced FWM system based on a silicon ring resonator. The system of the example shown thereby is not only adapted for QPM Kerr-induced FWM, but also illustrates that advantageously use can be made of cavity enhancement effects and that efficient conversion can be established for a large pump-signal frequency shift in a spectral domain where the dispersion characteristics of the silicon waveguide are not optimally engineered for PPM Kerr-induced FWM. As mentioned above, the condition for QPM Kerr-induced FWM in the ring is given by 35 EMBED Equation. DSMT4




  
    
      Δ
      ⁢
      
        
      
      ⁢
      
        k
        linear
      
    
    =
    
      s
      ⁢
      
        4
        R
      
    
  
  ,




where s=±1 so that R has a positive value, and R is the ring radius in case of a circular ring. Taking into account that Δklinear≈−β2(Δω)2, one finds that this quasi-phase-matching condition can be fulfilled even if the pump-signal frequency shift is large and if one works in a spectral domain where the dispersion characteristics of the silicon waveguide are not optimally engineered for PPM Kerr-induced FWM. Furthermore, the relation Δklinear≈−β2(Δω)2 also indicates that, for a given value of R, the quasi-phase-matching condition (1) can be fulfilled for different combinations of β2 and Δω. Thus, for a ring resonator with a ring radius R and with a properly designed, non-constant dispersion profile, one can convert via QPM Kerr-induced FWM a fixed signal frequency ωs to various idler frequencies ωi spread over the near- and mid-infrared range, by changing only the pump frequency ωp. Finally, if R is chosen to be small to keep the device compact, one finds that Δω can be large also if β2 is large.

As also mentioned above, the quasi-phase-matching condition expressed above complies with the condition for having the pump field, the signal field and the idler field at ring resonances. The fact that efficient non-traditional quasi-phase-matching can be combined with cavity enhancement for all three fields in the ring resonator is an important advantage, since for Kerr-induced FWM with “phase-matched operation” one can obtain cavity enhancement for all three fields only if the pump wavelength is close to the ZDW, i.e. only if one has PPM operation. Otherwise one has CLD operation in a doubly-resonant condition rather than in a triply-resonant condition. It also can be remarked that, for QPM Kerr-induced FWM, the varying Kerr susceptibility in the ring does not reach zero as minimal value, which is not ideal. However, since it can be combined with cavity enhancement for all three fields also if the GVD at the pump wavelength has a large absolute value and/or the frequency difference between the pump and signal is large, QPM Kerr-induced FWM can in those circumstances establish efficiencies that are relatively high compared to the efficiencies achieved with CLD Kerr-induced FWM.
In this embodiment two comparisons will be made between QPM ring-based parametric converters and CLD ring-based parametric converters in a spectral domain where the dispersion characteristics of the silicon waveguide are not optimally engineered for PPM Kerr-induced FWM. A one-dimensional CLD parametric converter is not explicitly considered in this comparison to demonstrate the effect of the cavity enhancement of the QPM ring-based parametric converter, as this would yield results along the same lines as those obtained for the first comparison in the previous embodiment on Raman converters. To compare QPM ring-based parametric converters and CLD ring-based parametric converters, a modeling formalism for parametric converters is first introduced. Without restricting the general validity of the results, focus is made on (quasi-)continuous-wave operation. Assuming that nk/nl≈1 (for k,l=p,s,i), the equations expressing the steady-state spatial variation of the slowly-varying pump, signal and idler field amplitudes Ap(ζ), As(ζ), Ai(ζ) in the parametric converter are given by




  
    
      
        
          
            
              ∂
              
                A
                p
              
            
            
              ∂
              ζ
            
          
          =
          
            
              
                
                  iγ
                  ⁡
                  
                    (
                    θ
                    )
                  
                
                ⁡
                
                  [
                  
                    
                      
                        
                        
                          A
                          p
                        
                        
                      
                      2
                    
                    +
                    
                      2
                      ⁢
                      
                        
                          
                          
                            A
                            s
                          
                          
                        
                        2
                      
                    
                    +
                    
                      2
                      ⁢
                      
                        
                          
                          
                            A
                            i
                          
                          
                        
                        2
                      
                    
                  
                  ]
                
              
              ⁢
              
                A
                p
              
            
            +
            
              2
              ⁢
              
                
              
              ⁢
              
                iγ
                ⁡
                
                  (
                  θ
                  )
                
              
              ⁢
              
                A
                p
                *
              
              ⁢
              
                A
                s
              
              ⁢
              
                A
                i
              
              ⁢
              
                e
                
                  
                    -
                    iΔ
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    k
                    linear
                  
                  ⁢
                  ζ
                
              
            
            -
            
              
                Γ
                p
              
              ⁢
              
                A
                p
              
            
          
        
        ,
      
    
    
      
        (
        6
        )
      
    
  
  
    
      
        
          
            
              ∂
              
                A
                s
              
            
            
              ∂
              ζ
            
          
          =
          
            
              
                
                  iγ
                  ⁡
                  
                    (
                    θ
                    )
                  
                
                ⁡
                
                  [
                  
                    
                      
                        
                        
                          A
                          s
                        
                        
                      
                      2
                    
                    +
                    
                      2
                      ⁢
                      
                        
                          
                          
                            A
                            p
                          
                          
                        
                        2
                      
                    
                    +
                    
                      2
                      ⁢
                      
                        
                          
                          
                            A
                            i
                          
                          
                        
                        2
                      
                    
                  
                  ]
                
              
              ⁢
              
                A
                s
              
            
            +
            
              
                iγ
                ⁡
                
                  (
                  θ
                  )
                
              
              ⁢
              
                A
                p
                2
              
              ⁢
              
                A
                i
                *
              
              ⁢
              
                e
                
                  iΔ
                  ⁢
                  
                    
                  
                  ⁢
                  
                    k
                    linear
                  
                  ⁢
                  ζ
                
              
            
            -
            
              
                Γ
                s
              
              ⁢
              
                A
                s
              
            
          
        
        ,
      
    
    
      
        (
        7
        )
      
    
  
  
    
      
        
          
            ∂
            
              A
              i
            
          
          
            ∂
            ζ
          
        
        =
        
          
            
              
                iγ
                ⁡
                
                  (
                  θ
                  )
                
              
              ⁡
              
                [
                
                  
                    
                      
                      
                        A
                        i
                      
                      
                    
                    2
                  
                  +
                  
                    2
                    ⁢
                    
                      
                        
                        
                          A
                          p
                        
                        
                      
                      2
                    
                  
                  +
                  
                    2
                    ⁢
                    
                      
                        
                        
                          A
                          s
                        
                        
                      
                      2
                    
                  
                
                ]
              
            
            ⁢
            
              A
              i
            
          
          +
          
            
          
          ⁢
          
            
              iγ
              ⁡
              
                (
                θ
                )
              
            
            ⁢
            
              A
              p
              2
            
            ⁢
            
              A
              s
              *
            
            ⁢
            
              e
              
                iΔ
                ⁢
                
                  
                
                ⁢
                
                  k
                  linear
                
                ⁢
                ζ
              
            
          
          -
          
            
              Γ
              i
            
            ⁢
            
              
                A
                i
              
              .
            
          
        
      
    
    
      
        (
        8
        )
      
    
  




where ζ=Rθ, γ(θ)=n20ξKρ(θ)(ωp/c) is the effective nonlinearity, n20 is the Kerr-nonlinear refractive index along the [011] direction, ξK=5/4, and A{p,s,i} is normalized such that |A{p,s,i}|2 corresponds to intensity. The function ρ(θ) will be specified further on. The first terms containing the square brackets at the right hand side of Eqs. (6)-(8) correspond to Kerr-induced self- and cross-phase modulation, and the terms containing ejΔklinearζ express the actual Kerr-induced FWM interaction. The coefficients Γ{p,s,i} represent the optical losses in the SOI waveguide. In the near-infrared spectral domain, the latter receive contributions from linear propagation losses, two-photon absorption (TPA) and TPA-induced free carrier absorption, but in the mid-infrared spectral domain, τ{p,s,i} only receives contributions from linear propagation losses. At the entry point of light into the ring from the channel one has θ=0 (see FIG. 1). Coupling from the channel to the ring is described in the usual way,





  
    
      
        
          
            (
            
              
                
                  
                    A
                    
                      j
                      ⁢
                      
                        
                      
                      ⁢
                      2
                    
                  
                
              
              
                
                  
                    A
                    
                      j
                      ⁢
                      
                        
                      
                      ⁢
                      3
                    
                  
                
              
            
            )
          
          =
          
            
              (
              
                
                  
                    
                      σ
                      j
                    
                  
                  
                    
                      i
                      ⁢
                      
                        
                      
                      ⁢
                      
                        κ
                        j
                      
                    
                  
                
                
                  
                    
                      i
                      ⁢
                      
                        
                      
                      ⁢
                      
                        κ
                        j
                      
                    
                  
                  
                    
                      σ
                      j
                    
                  
                
              
              )
            
            ⁢
            
              (
              
                
                  
                    
                      A
                      
                        j
                        ⁢
                        
                          
                        
                        ⁢
                        1
                      
                    
                  
                
                
                  
                    
                      
                        A
                        
                          j
                          ⁢
                          
                            
                          
                          ⁢
                          4
                        
                      
                      ⁢
                      
                        exp
                        ⁡
                        
                          (
                          
                            i
                            ⁢
                            
                              
                            
                            ⁢
                            
                              k
                              j
                            
                            ⁢
                            L
                          
                          )
                        
                      
                    
                  
                
              
              )
            
          
        
        ,
      
    
    
      
        (
        9
        )
      
    
  




with j=p,s,i, with the positions of the fields (1)-(4) indicated in FIG. 1, and with L=2πR. One can consider real-valued coupling constants σj,κj is that satisfy the relation σj2+κj2=1.

One now can numerically solve Eqs. (6)-(9) to make two comparisons: a comparison is made between a concrete near-infrared-pumped QPM ring-based parametric converter and a concrete near-infrared-pumped CLD ring-based parametric converter, both for the case that the dispersion characteristics of the silicon waveguide in the near-infrared domain are not optimally engineered for PPM Kerr-induced FWM. On the other hand, a comparison is made between a concrete mid-infrared-pumped QPM ring-based parametric converter and a concrete mid-infrared-pumped CLD ring-based parametric converter, both for the case that the dispersion characteristics of the silicon waveguide in the mid-infrared domain are not optimally engineered for PPM Kerr-induced FWM.
For the first comparison in the second embodiment, a near-infrared-pumped QPM ring-based parametric converter is initially considered with the following parameter values: ωp=1.18×1015 rad/s (λp=1.6 μm), ωs=1.45×1015 rad/s (λs=1.3 μm), ωi=9.06×1014 rad/s (λi=2.08 μm), Δklinear=1606 cm−1 (corresponding to a dispersion parameter of 1600 ps/(nm*km) at λp), n20=6.5×10−18 m2/W, Ip,in=6×1010 W/m2, Is,in=1×108 W/m2, Ii,in=0 W/m2, waveguide modal area A=0.09 μm2, linear loss α=0.9 dB/cm, two-photon absorption coefficient β=0.7×10−11 m/W, free carrier absorption efficiency φ=6×10−10, effective free carrier lifetime τeff=0.1 ns, κp=0.14, κs=0.10, κi=0.17, and ρ(θ)=(0.88+0.12 cos2(2θ)) along the ring. When implementing the value for Δklinear in the quasi-phase-matching condition with s=+1, one obtains that quasi-phase-matching is obtained for a ring radius R=25 μm, which corresponds to a ring circumference of 157 μm. FIG. 5 parts (a)-(c) show the steady-state distributions along the ring of the pump, signal and idler intensities, respectively, as obtained by numerically solving equations (6) to (9) for this converter. Using Eq. (9) one can derive from FIG. 5 part (c) that Ii,out=5×104 W/m2. This corresponds to a signal-to-idler conversion efficiency of −33 dB for this QPM parametric converter with λp=1.6 μm, λs=1.3 μm, λi=2.08 μm. Taking into account that this conversion efficiency is of the same order of magnitude as the conversion efficiencies for CLD Kerr-induced FWM in a silicon ring with the same dispersion parameter but with much smaller pump-signal frequency differences as can be found in literature, one finds that this QPM cavity-enhanced converter has a relatively good performance.
To demonstrate that also efficiencies higher than −33 dB could be reached while still pumping in the near-infrared region, one now considers a QPM parametric converter that is pumped at another near-infrared pump wavelength where the nonlinear refractive index is larger than in the previous case. More specifically, a QPM parametric converter is considered with the following parameter values: ωp=1.05×1015 rad/s (λp=1.8 μm), ωs=1.32×1015 (λs=1.43 μm), ωi=7.76×1014 (λi=2.43 μm), Δklinear=1606 cm−1 (corresponding to a dispersion parameter of 1600 ps/(nm*km) at λp as in the previous case), n20=12×10−18 m2/W, two-photon absorption coefficient β=0.5×1011 m/W, and free carrier absorption efficiency φ=(1.8/1.55)2×6×10−10. For all other parameters, the same values are taken as in the previous case. FIG. 6 parts (a)-(c) show the steady-state distributions along the ring of the pump, signal and idler intensities, respectively, as obtained by numerically solving Eqs. (6)-(9) for this converter. Using Eq. (9) one can derive from FIG. 6 part (c) that Ii,out=1.2×105 W/m2. This corresponds to a signal-to-idler conversion efficiency of −29 dB.
For the second comparison in the second embodiment, a mid-infrared-pumped QPM parametric ring converter with a TE-polarized pump input at ωp=8.20×1014 rad/s (λp=2.3 μm) and a TE-polarized signal input at ωs=9.87×1014 rad/s (λs=1.91 μm) is considered, which feature a large pump-signal angular frequency difference of 2π×26.6 THz. This leads to a generated idler wave at ωi=6.52×1014 rad/s (λi=2.89 μm). It is remarked that one of the most interesting application domains for wavelength conversion towards mid-infrared idler wavelengths around 3 μm is spectroscopy, as many substances are highly absorbing in that spectral range. At a pump wavelength of 2.3 μm, the Kerr-nonlinear refractive index n20 of silicon along the [011] direction equals approximately 8×10−14 cm2/W. A nanowire is assumed here which is dispersion-engineered such that its ZDW is situated in the near-infrared, more specifically at 1.5 μm, and which features a sufficiently large cross-section to have most of the mode energy at λp=2.3 μm confined in the nanowire core. The nanowire under consideration has a height of 516 nm and a width of 775 nm and has an oxide cladding. For such a waveguide geometry the ZDW is indeed situated at 1.5 μm, and that the dispersion D at λp=2.3 μm equals approximately 1200 ps/(nm*km). The latter value yields Δk=2kp−ks−ka=940 cm−1. For s=+1, it is found that the condition (5) is met for a ring circumference L of 267 μm. The remaining device parameters are: Aeff=0.4 μm2, α=3 dB/cm, and ρ(θ)=(0.88+0.12 cos2(2θ)) along the ring. In these simulations coupling coefficients are considered ranging from κp2=κs2=κi2=0.06 to a low value of κp2=κs2=κi2=0.01, with intermediate steps of 0.005. Since at mid-infrared operation wavelengths the multi-photon absorption and the associated free carrier absorption are negligible in silicon, the free carrier lifetime is not of importance here and the only losses that need to be taken into account in Γ{p,s,i} are the linear losses.
Since the same type of nanowire are considered for the mid-infrared-pumped CLD parametric ring converter configuration, for this converter the parameter values are adopted as described above, except that in this case ρ(θ)=1 along the ring. Furthermore, different values are considered for the ring circumferences. It is also pointed out that for the CLD converter it is considered that the pump and signal waves to be at ring resonances and the idler wave to be detuned from ring resonance with the detuning given by kiL=ΔkL+2πs.
Equations (6) to (9) are numerically solved for the two mid-infrared-pumped parametric converter setups using a signal input power of 1 mW (Is,in=2.5×109 W/m2) and a fixed pump input power of 40 mW (Ip,in=1×1011 W/m2). The reason for taking a fixed pump input power is that in this comparison the pump-power-dependent nonlinear losses are negligible for both of the converters. The signal-to-idler conversion efficiencies of the QPM converter and of the CLD converter with different ring circumferences are shown in FIG. 7 as a function of the coupling coefficients κj2 (j=p,s,i). For the working point of maximal conversion efficiency, the steady-state distributions along the ring of the pump, signal and idler intensities in the QPM (CLD) converter are shown in FIG. 8 (FIG. 9).
When comparing the graphs of FIG. 7, it can be seen that the maximally attainable conversion efficiency of the QPM converter, which equals −26.7 dB, is almost 10 dB, i.e. one order of magnitude, larger than the corresponding value of the CLD converter, equal to −36.1 dB which is obtained for an extremely small ring circumference L=π/|4Δklinear|=8.4 μm. In other words, the QPM parametric converter is able to outperform the CLD parametric converter by as much as 10 dB.
In conclusion, the QPM parametric conversion method offers a feasible and competitive solution when efficient conversion needs to be achieved in the presence of a large-valued Δklinear, i.e. in the presence of a large-valued GVD at the pump wavelength and/or a large frequency difference between pump and signal. The predicted QPM parametric conversion efficiencies of the order of −33 dB, −29 dB, and −26.7 dB in the near- and mid-infrared spectral domains are high enough to generate microwatts of idler output power, which is a sufficiently high power level for the considered application domains such as spectroscopy. Finally, one has to keep in mind that this QPM parametric conversion method only offers efficient conversion for one specific set of pump, signal, and idler wavelengths, as the ring circumference has to be chosen in function of the phase mismatch between these wavelengths. So, the use of the QPM parametric conversion method presented here should be considered in the following context: in case one works with relatively small wavelength spacings yielding moderate |Δklinear| values, one can rely on CLD parametric conversion, but at the specific set of (widely spaced) wavelengths for which the ring circumference allows quasi-phase-matching, one gets due to QPM parametric conversion a much larger conversion efficiency “for free.” Therefore, if this specific set of wavelengths is often used in the application under consideration, the QPM parametric conversion method presented here can be of great value. In a third particular embodiment, reference is made to a QPM SFG system based on a spiral-shaped silicon waveguide covered by a graphene sheet through which an electrical current is flowing. In this embodiment the performance is calculated for a QPM SFG system based on a spiral-shaped silicon waveguide covered by a graphene sheet. To do this, first a modeling formalism for SFG converters is introduced. Without restricting the general validity of the results, focus is made on quasi-continuous-wave operation. The equations expressing the steady-state spatial variation of the slowly-varying pump, signal and idler field amplitudes Ap(ζ), As(ζ) Ai(ζ) in the parametric SFG-based converter are given by:




  
    
      
        
          
            d
            
              A
              p
            
          
          
            d
            ζ
          
        
        =
        
          
            i
            ⁢
            
              
                2
                ⁢
                
                  ω
                  p
                
              
              
                
                  n
                  p
                
                ⁢
                c
              
            
            ⁢
            
              d
              eff
            
            ⁢
            
              ρ
              ⁡
              
                (
                θ
                )
              
            
            ⁢
            
              A
              i
            
            ⁢
            
              A
              s
              *
            
            ⁢
            
              e
              
                
                  -
                  iΔ
                
                ⁢
                
                  
                
                ⁢
                
                  k
                  linear
                
                ⁢
                ζ
              
            
          
          -
          
            
              Γ
              p
            
            ⁢
            
              A
              p
            
          
        
      
    
    
      
        (
        10
        )
      
    
  
  
    
      
        
          
            d
            
              A
              s
            
          
          
            d
            ζ
          
        
        =
        
          
            i
            ⁢
            
              
                2
                ⁢
                
                  ω
                  s
                
              
              
                
                  n
                  s
                
                ⁢
                c
              
            
            ⁢
            
              d
              eff
            
            ⁢
            
              ρ
              ⁡
              
                (
                θ
                )
              
            
            ⁢
            
              A
              i
            
            ⁢
            
              A
              p
              *
            
            ⁢
            
              e
              
                
                  -
                  iΔ
                
                ⁢
                
                  
                
                ⁢
                
                  k
                  linear
                
                ⁢
                ζ
              
            
          
          -
          
            
              Γ
              s
            
            ⁢
            
              A
              s
            
          
        
      
    
    
      
        (
        11
        )
      
    
  
  
    
      
        
          
            d
            
              A
              i
            
          
          
            d
            ζ
          
        
        =
        
          
            i
            ⁢
            
              
                2
                ⁢
                
                  ω
                  i
                
              
              
                
                  n
                  i
                
                ⁢
                c
              
            
            ⁢
            
              d
              eff
            
            ⁢
            
              ρ
              ⁡
              
                (
                θ
                )
              
            
            ⁢
            
              A
              p
            
            ⁢
            
              A
              s
            
            ⁢
            
              e
              
                iΔ
                ⁢
                
                  
                
                ⁢
                
                  k
                  linear
                
                ⁢
                ζ
              
            
          
          -
          
            
              Γ
              i
            
            ⁢
            
              A
              i
            
          
        
      
    
    
      
        (
        12
        )
      
    
  




where ζ represents the propagation distance along the spiral, deff is the effective second-order nonlinearity, and A{p,s,i} is normalized such that 2ε0n{p,s,i}c|A{p,s,i}|2 corresponds to intensity. The function ρ(θ) defines the variation of the second-order susceptibility along the graphene-covered silicon spiral as experienced by the TE-polarized fields, and is, as specified earlier on, given by ρ(θ)=cos θ with θ defined as the angle between the local field polarization and the direction of the current flow. The terms in Eqs. (10)-(12) containing ejΔklinearζ express the actual SFG interaction. The coefficients Γ{p,s,i} represent the optical losses in the graphene-covered silicon waveguide. These receive contributions from linear propagation losses, two-photon absorption (TPA) and TPA-induced free carrier absorption. In this embodiment one considers for the QPM SFG converter configuration a TE-polarized pump input and a TE-polarized signal input which both have the same wavelength, namely λp=λs=2.34 μm, and which both can be provided by the same laser providing an input intensity Ip,in. This leads to a generated idler wave with wavelength λi=1.17 μm. The system may have a structure as illustrated in FIG. 1J. When taking an SOI ridge waveguide with a central section of 220 nm height and 1170 nm width, and two side sections of 150 nm height and 180 nm width, and when tuning the chemical potential of the graphene sheet on top to 0.3 eV, the linear phase mismatch between the fundamental pump/signal TE-mode and the 7th-order idler TE-mode is given by Δklinear=kp+ks−ki=−2×104 m−1 so that QPM can be obtained if the spiral has an average radius equal to





  R
  =
  
    
      -
      
        1
        
          Δ
          ⁢
          
            
          
          ⁢
          
            k
            linear
          
        
      
    
    =
    
      50
      ⁢
      
        
      
      ⁢
      
        μm
        .
      
    
  




When sending a current density of 103 A/m through the graphene sheet, deff ̃100×10−12 m/V. For the remaining device parameters the following values were taken: waveguide modal area Aeff=0.5 μm2, linear loss α=50 dB/cm, effective two-photon absorption coefficient β=25×10−11 m/W, effective free carrier absorption efficiency φ=6×10−12, effective free carrier lifetime τeff=0.5 ns, Ip,in=2×1011 W/m2. It is pointed out that the relatively small effective free carrier absorption efficiency is due to the fact that at the considered pump wavelength only the graphene sheet contributes to free carrier generation, and only a small fraction of these free carriers effectively diffuse to the silicon waveguide. One then can numerically solve equations (10) to (12) for the QPM SFG converter. FIG. 10 shows the distribution along the spiral for the idler intensity. The output idler intensity emerging from the spiral Ii,out thus equals 2.2×108 W/m2 corresponding to a SFG efficiency (expressed in terms of power) given by Pi,out/Pp,in=1.1×10−3 W W−1 over a propagation distance of only 1.3 mm. The intensity distribution for the idler in FIG. 10 shows an oscillating build up typical for QPM and similarly to the QPM Raman converter and QPM Kerr converter presented in, respectively, embodiments 1 and 2. As is the case for the QPM Raman converter and QPM Kerr converter in, respectively, embodiments 1 and 2, the QPM concept yields also for the SFG converter considered here performance advantages compared to other phase-matching techniques. It should be noted that whereas the spiral has been designed such that it has an average radius R=50 μm, the actual bend radius along the spiral-shaped waveguide is not constant and varies over a range of values around 50 μm. This means that the QPM scheme used here can also work effectively for values of Δklinear not exactly equal to but around





  
    
      -
      
        1
        R
      
    
    =
    
      
        -
        2
      
      ×
      
        10
        4
      
      ⁢
      
        m
        
          -
          1
        
      
    
  
  ,




i.e. it can also work effectively for pump and idler wavelengths not exactly equal to but around 2.34 μm and 1.17 μm, respectively. In a fourth particular embodiment, reference is made to a QPM DFG system based on a spiral-shaped silicon waveguide covered by a graphene sheet through which an electrical current is flowing. In this embodiment the performance is calculated for a QPM DFG system based on a spiral-shaped silicon waveguide covered by a graphene sheet. To do this, first a modeling formalism for DFG converters is introduced. Without restricting the general validity of the results, focus is made on quasi-continuous-wave operation. The equations expressing the steady-state spatial variation of the slowly-varying pump, signal and idler field amplitudes Ap(ζ), As(ζ), Ai(ζ) in the parametric DFG-based converter are given by:





  
    
      
        
          
            d
            
              A
              p
            
          
          
            d
            ζ
          
        
        =
        
          
            i
            ⁢
            
              
                2
                ⁢
                
                  ω
                  p
                
              
              
                
                  n
                  p
                
                ⁢
                c
              
            
            ⁢
            
              d
              eff
            
            ⁢
            
              ρ
              ⁡
              
                (
                θ
                )
              
            
            ⁢
            
              A
              i
            
            ⁢
            
              A
              s
            
            ⁢
            
              e
              
                
                  -
                  iΔ
                
                ⁢
                
                  
                
                ⁢
                
                  k
                  linear
                
                ⁢
                ζ
              
            
          
          -
          
            
              Γ
              p
            
            ⁢
            
              A
              p
            
          
        
      
    
    
      
        (
        13
        )
      
    
  
  
    
      
        
          
            d
            
              A
              s
            
          
          
            d
            ζ
          
        
        =
        
          
            i
            ⁢
            
              
                2
                ⁢
                
                  ω
                  s
                
              
              
                
                  n
                  s
                
                ⁢
                c
              
            
            ⁢
            
              d
              eff
            
            ⁢
            
              ρ
              ⁡
              
                (
                θ
                )
              
            
            ⁢
            
              A
              p
            
            ⁢
            
              A
              i
              *
            
            ⁢
            
              e
              
                iΔ
                ⁢
                
                  
                
                ⁢
                
                  k
                  linear
                
                ⁢
                ζ
              
            
          
          -
          
            
              Γ
              s
            
            ⁢
            
              A
              s
            
          
        
      
    
    
      
        (
        14
        )
      
    
  
  
    
      
        
          
            d
            
              A
              i
            
          
          
            d
            ζ
          
        
        =
        
          
            i
            ⁢
            
              
                2
                ⁢
                
                  ω
                  i
                
              
              
                
                  n
                  i
                
                ⁢
                c
              
            
            ⁢
            
              d
              eff
            
            ⁢
            
              ρ
              ⁡
              
                (
                θ
                )
              
            
            ⁢
            
              A
              p
            
            ⁢
            
              A
              s
              *
            
            ⁢
            
              e
              
                iΔ
                ⁢
                
                  
                
                ⁢
                
                  k
                  linear
                
                ⁢
                ζ
              
            
          
          -
          
            
              Γ
              i
            
            ⁢
            
              A
              i
            
          
        
      
    
    
      
        (
        15
        )
      
    
  




where ζ represents the propagation distance along the spiral, deff is the effective second-order nonlinearity, and A{p,s,i} is normalized such that 2ε0n{p,s,i}c|A{p,s,i}|2 corresponds to intensity. The function ρ(θ) defines the variation of the second-order susceptibility along the graphene-covered silicon spiral as experienced by the TE-polarized fields, and is, as specified earlier on, given by ρ(θ)=cos θ with θ defined as the angle between the local field polarization and the direction of the current flow. The terms in Eqs. (13)-(15) containing ejΔklinearζ express the actual DFG interaction. The coefficients Γ{p,s,i} represent the optical losses in the graphene-covered silicon waveguide. These receive contributions from linear propagation losses, two-photon absorption (TPA) and TPA-induced free carrier absorption. In this embodiment one considers for the QPM DFG converter configuration a TE-polarized pump input and a TE-polarized signal input which have the following wavelengths: λp=1.25 μm, λs=2.1 μm. This leads to a generated idler wave with wavelength λi=3.088 μm. The system may have a structure as illustrated in FIG. 1J. When taking an SOI ridge waveguide with a central section of 220 nm height and 1170 nm width, and two side sections of 150 nm height and 180 nm width, and when tuning the chemical potential of the graphene sheet on top to 0.3 eV, the linear phase mismatch between the fundamental pump/signal TE-mode and the 6th-order idler TE-mode is given by Δklinear=kpkski=7×104 m−1 so that QPM can be obtained if the spiral has an average radius equal to





  R
  =
  
    
      -
      
        1
        
          Δ
          ⁢
          
            
          
          ⁢
          
            k
            linear
          
        
      
    
    =
    
      14
      ⁢
      
        
      
      ⁢
      
        μm
        .
      
    
  




When sending a current density of 103 A/m through the graphene sheet, deff ̃100×10−12 m/V. For the remaining device parameters the following values were taken: waveguide modal area Aeff=0.5 μm2, linear loss α=50 dB/cm, effective two-photon absorption coefficient β=25×10−11 m/W, effective free carrier absorption efficiency φ=6×10−12, effective free carrier lifetime τeff=0.5 ns, Ip,in=2×1011 W/m2, Is,in=2×108 W/m2. As in embodiment 3, the relatively small effective free carrier absorption efficiency is due to the fact that at the considered pump wavelength only the graphene sheet contributes to free carrier generation, and only a small fraction of these free carriers effectively diffuse to the silicon waveguide. One then can numerically solve equations (13) to (15) for the QPM DFG converter. FIG. 11 shows the distribution along the spiral for the idler intensity. The output idler intensity emerging from the spiral Ii,out thus equals 4.6×104 W/m2 corresponding to a DFG efficiency Ii,out/Is,in of −36 dB over a propagation distance of only 380 μm. The intensity distribution for the idler in FIG. 11 shows an oscillating build up typical for QPM and similarly to the QPM Raman converter and QPM Kerr converter presented in, respectively, embodiments 1 and 2. As is the case for the QPM Raman converter and QPM Kerr converter in, respectively, embodiments 1 and 2, the QPM concept yields also for the DFG converter considered here performance advantages compared to other phase-matching techniques. It should be noted that whereas the spiral has been designed such that the spiral has an average radius R=14 μm, the actual bend radius along the spiral-shaped waveguide is not constant and varies over a range of values around 14 μm. This means that the QPM scheme used here can also work effectively for values of Δklinear not exactly equal to but around





  
    
      1
      R
    
    =
    
      7
      ×
      
        10
        4
      
      ⁢
      
        m
        
          -
          1
        
      
    
  
  ,
  
    i
    .
    e
    .
  




it can also work effectively for pump, signal and idler wavelengths not exactly equal to but around 1.25 μm, 2.1 μm, and 3.088 μm, respectively.

Whereas the above aspect has been mainly described with reference to system features, as indicated it also relates to a method for obtaining conversion or amplification, using QPM nonlinear optical wave mixing. Such a method comprises receiving a pump radiation beam and a signal radiation beam in a bent structure, a waveguiding portion of the bent structure being made of a uniform nonlinear optical material and the dimensions of the bent structure being selected for obtaining QPM nonlinear optical wave mixing. The method also comprises obtaining an idler radiation beam by interaction of the pump radiation beam and the signal radiation beam using at least one QPM nonlinear optical process such as for example a QPM SFG, a QPM DFG, a QPM Raman-resonant FWM or QPM Kerr-induced FWM process. The method furthermore encloses coupling out an idler radiation beam from the bent structure. Other or more detailed method steps may be present, expressing the functionality of components of the system as described above.
In one aspect, the present invention also relates to a method for designing a converter or amplifier using QPM nonlinear optical wave mixing. The converter or amplifier thereby may be using a pump radiation beam and a signal radiation beam. The method for designing comprises selecting a bent structure suitable for QPM nonlinear optical wave mixing, comprising selecting a uniform material for a radiation propagation portion of the bent structure, e.g. a waveguide, and selecting dimensions of the bent structure taking into account the spatial variation of the nonlinear optical susceptibility along the structure as experienced by radiation travelling along the bent structure. At least one dimension of the bent structure are selected such that QPM nonlinear optical wave mixing is obtained. More particularly, at least one dimension of the radiation propagation portion of the bent structure is selected taking into account the spatial variation of the nonlinear optical susceptibility along the radiation propagation structure as experienced by radiation travelling along the bent structure for obtaining quasi-phase matched nonlinear optical wave mixing in the radiation propagation portion. The dimension may be substantially inverse proportional with the linear phase mismatch for the nonlinear optical wave mixing. The method for designing furthermore may be adapted so that the structure provides cavity enhancement for at least one of the radiation beams that will travel in the system, i.e. for which the system is designed, preferably more or all of the radiation beams are cavity enhanced.
In a further aspect, the above described methods for designing or controlling a system for resonating, converting or amplifying using QPM nonlinear optical wave mixing e.g. the controller may be at least partly implemented in a processing system 500 such as shown in FIG. 12. FIG. 12 shows one configuration of processing system 500 that includes at least one programmable processor 503 coupled to a memory subsystem 505 that includes at least one form of memory, e.g., RAM, ROM, and so forth. It is to be noted that the processor 503 or processors may be a general purpose, or a special purpose processor, and may be for inclusion in a device, e.g., a chip that has other components that perform other functions. Thus, one or more aspects of the present invention can be implemented in digital electronic circuitry, or in computer hardware, firmware, software, or in combinations of them. For example, the determination of test pulse properties may be a computer implemented step. The processing system may include a storage subsystem 507 that has at least one disk drive and/or CD-ROM drive and/or DVD drive. In some implementations, a display system, a keyboard, and a pointing device may be included as part of a user interface subsystem 509 to provide for a user to manually input information. Ports for inputting and outputting data also may be included. More elements such as network connections, interfaces to various devices, and so forth, may be included, but are not illustrated in FIG. 12. The memory of the memory subsystem 505 may at some time hold part or all (in either case shown as 501) of a set of instructions that when executed on the processing system 500 implement the steps of the method embodiments described herein. A bus 513 may be provided for connecting the components. Thus, while a processing system 500 such as shown in FIG. 12 is prior art, a system that includes the instructions to implement aspects of the methods for controlling resonating and/or converting and/or amplifying using a QPM nonlinear optical process is not prior art, and therefore FIG. 12 is not labeled as prior art.
The present invention also includes a computer program product which provides the functionality of any of the methods according to the present invention when executed on a computing device. Such computer program product can be tangibly embodied in a carrier medium carrying machine-readable code for execution by a programmable processor. The present invention thus relates to a carrier medium carrying a computer program product that, when executed on computing means, provides instructions for executing any of the methods as described above. The term “carrier medium” refers to any medium that participates in providing instructions to a processor for execution. Such a medium may take many forms, including but not limited to, non-volatile media, and transmission media. Non-volatile media includes, for example, optical or magnetic disks, such as a storage device which is part of mass storage. Common forms of computer readable media include, a CD-ROM, a DVD, a flexible disk or floppy disk, a tape, a memory chip or cartridge or any other medium from which a computer can read. Various forms of computer readable media may be involved in carrying one or more sequences of one or more instructions to a processor for execution. The computer program product can also be transmitted via a carrier wave in a network, such as a LAN, a WAN or the Internet. Transmission media can take the form of acoustic or light waves, such as those generated during radio wave and infrared data communications. Transmission media include coaxial cables, copper wire and fibre optics, including the wires that comprise a bus within a computer.
It is to be understood that although preferred embodiments, specific constructions and configurations, as well as materials, have been discussed herein for devices according to the present invention, various changes or modifications in form and detail may be made without departing from the scope and spirit of this invention. For example, any formulas given above are merely representative of procedures that may be used. Functionality may be added or deleted from the block diagrams and operations may be interchanged among functional blocks. Steps may be added or deleted to methods described within the scope of the present invention.

## Claims

1. A system for conversion or amplification using a quasi-phase matched nonlinear optical wave mixing, the system comprising
a first radiation source configured for providing a pump radiation beam,
a second radiation source configured for providing a signal radiation beam, and
a bent structure configured for receiving the pump radiation beam and the signal radiation beam, wherein a radiation propagation portion of the bent structure is made of a uniform nonlinear optical material and wherein the radiation propagation portion comprises a dimension taking into account the spatial variation of the nonlinear optical susceptibility along the radiation propagation portion as experienced by radiation travelling along the bent structure for obtaining quasi-phase-matched nonlinear optical wave-mixing in the radiation propagation portion, the dimension being substantially inverse proportional with the linear phase mismatch for the nonlinear optical wave mixing,
an outcoupling radiation propagation portion configured for coupling out an idler radiation beam generated in the bent structure.

2. A system according to claim 1, wherein said nonlinear optical wave mixing is nonlinear optical three wave mixing.

3. A system according to claim 2, wherein the uniform medium is a quadratically nonlinear optical material and wherein the process is a quasi-phase-matched sum-frequency generation or quasi-phase-matched difference-frequency generation.

4. A system according to claim 1, wherein the bent structure is a closed structure.

5. A system according to claim 1, wherein the bent structure is a circular ring, an elliptical ring, a rectangular shaped structure, an octagonally shaped structure, a circular disc or an elliptical shaped disc, a snake-like structure, a sickle-like structure, a spiral-like structure.

6. A system according to claim 5, wherein the structure is a circular ring, and where the radius R of the ring structure is determined substantially inverse proportional with the linear phase mismatch for the nonlinear optical wave mixing.

7. A system according to claim 6, wherein the radius R of the circular ring structure is determined by the relation




  
    R
    =
    
      s
      ⁢
      
        4
        
          Δ
          ⁢
          
            
          
          ⁢
          
            k
            linear
          
        
      
    
  
  ,




with s being a factor equal to +1 or −1 so that R has a positive value and Δklinear being the linear phase mismatch for Raman-resonant four-wave-mixing or being the linear phase mismatch for Kerr-induced four-wave-mixing, and/or, wherein the radius R of the circular ring structure is determined by the relation




  R
  =
  
    s
    ⁢
    
      1
      
        Δ
        ⁢
        
          
        
        ⁢
        
          k
          linear
        
      
    
  




with s being a factor equal to +1 or −1 so that R has a positive value and Δklinear being the linear phase mismatch for sum-frequency generation or being the linear phase mismatch for difference-frequency generation.

8. A system according to claim 1, wherein the bent structure has an inscribed circle and/or circumscribed circle having a radius inversely proportional to the linear phase mismatch for the nonlinear optical wave mixing or wherein the bent structure has an average radius inversely proportional to the linear phase mismatch for the nonlinear optical wave mixing.

9. A system according to claim 1, the system furthermore being arranged for providing a pump radiation beam with wavenumber kp and a signal radiation beam with wavenumber ks and result in an idler radiation beam with wavenumber ki, so that at least one of these beams is at ring resonance and as such at least one of these beams' wavenumbers yields, when multiplying with R, an integer number.

10. A system according to claim 9, wherein the system comprises a heating and/or cooling means and a temperature controller configured for controlling the temperature so that at least one of the pump radiation, the signal radiation and the idler radiation is at ring resonance.

11. A system according to claim 1, wherein the uniform medium is a Raman-active medium, and wherein the process is a quasi-phase-matched Raman-resonant four-wave-mixing process and/or wherein the uniform medium is a Kerr-nonlinear material and wherein the process is a quasi-phase-matched Kerr-induced four-wave-mixing process.

12. A system according to claim 1, wherein the uniform medium is a crystalline material.

13. A system according to claim 1, wherein the uniform medium is (100) grown silicon or (100) grown silicon is a silicon on insulator waveguide.

14. A system according to claim 1, wherein the uniform medium is a silicon on insulator waveguide covered by a graphene layer through which an electric current is flowing.

15. A system according to claim 1, wherein furthermore a controller is provided for tuning the system with respect to an output wavelength, an output power or an obtained bandwidth and/or wherein the system is adapted for selecting a TE or TM output by selecting respectively a TE or TM input.

16. A method for obtaining conversion or amplification, using a quasi-phase-matched nonlinear optical wave mixing process, the method comprising
receiving a pump radiation beam and a signal radiation beam in a bent structure, a radiation propagation portion of the bent structure being made of a uniform nonlinear optical material and comprising a dimension taking into account the spatial variation of the nonlinear optical susceptibility along the radiation propagation portion as experienced by radiation travelling along the bent structure for obtaining the quasi-phase-matched nonlinear optical wave mixing in the radiation propagation portion, the dimension being substantially inverse proportional with the linear phase mismatch for the nonlinear optical wave mixing,
obtaining an idler radiation beam by interaction of the pump radiation beam and the signal radiation beam
coupling out an idler radiation beam from the bent structure.

17. A method according to claim 16, wherein said nonlinear optical wave mixing is nonlinear optical three wave mixing and wherein the uniform medium is a quadratically nonlinear optical material and wherein the process is a quasi-phase-matched sum-frequency generation or quasi-phase-matched difference-frequency generation.

18. A method according to claim 16, wherein the method comprising propagating the radiation beams in the ring structure and obtaining ring resonance for at least one of the different radiation beams and/or wherein the method comprises adjusting the in- and/or outcoupling efficiency for adjusting the cavity-enhancement of the radiation beams inside the ring structure.

19. A method according to claim 16, the method comprising tuning the system with respect to an output wavelength, an output power or an obtained bandwidth.

20. A method for designing a converter or amplifier using a quasi-phase-matched nonlinear optical wave mixing, the converter or amplifier using a pump radiation beam and a signal radiation beam, the method comprising selecting a bent structure suitable for a quasi-phase-matched nonlinear optical wave mixing comprising selecting a uniform nonlinear optical material for a radiation propagation portion of the bent structure and selecting a dimension of the radiation propagation portion taking into account the spatial variation of the nonlinear optical susceptibility along the radiation propagation structure as experienced by radiation travelling along the bent structure for obtaining the quasi-phase-matched nonlinear optical wave mixing in the radiation propagation portion, the dimension being substantially inverse proportional with the linear phase mismatch for the nonlinear optical wave-mixing.