Abstract

A wavelength converter based on counterpropagating quasi-phase matched cascaded sum and difference frequency generation in lossy lithium niobate waveguide is numerically evaluated and compared to a single-pass scheme assuming a large pump wavelength difference of 75 nm. A double-pass device is proposed to improve the conversion efficiency while the response flattening is accomplished by increasing the wavelength tuning of one pump. The criteria for the design of the low-loss waveguide length, and the assignment of power in the pumps to achieve the desired efficiency, ripple and bandwidth are presented.

©2009 Optical Society of America

1. Introduction

There has been increasing interest in broadband wavelength converters based on quasi-phase matched (QPM) lithium niobate waveguides [112] as they are promising for several applications such as future wavelength division multiplexing (WDM) systems because they are signal format independent and can simultaneously convert a group of broadband wavelengths or high-speed signals, with negligible spontaneous emission noise [13]. Recently, interesting wavelength conversion techniques based on single-pass and double-pass cascaded sum and difference frequency generation (SFG + DFG) have been demonstrated theoretically and practically in periodically poled lithium niobate (PPLN) waveguides [1318] and may find applications in broadband wavelength conversion, channel selective wavelength conversion and multiple channel wavelength conversion [13,1820]. Using these techniques, not only can the pumps be out of the conversion bandwidth but also by increasing the difference between pumps wavelengths, the bandwidth can be enhanced, however with decreased efficiency. The double-pass cascaded SFG + DFG scheme has been proposed to overcome the efficiency reduction in the single-pass one, which is also able to cancel out the residual pump wavelengths at the output [18]. Although the double-pass SFG + DFG has been principally investigated, the research on how to choose the length and set the pumps to improve the conversion properties of low-loss waveguides still remains and is of great importance.

Here, we numerically evaluate the properties of the SFG + DFG schemes and show that the increasing detuning of one pump by a small amount to a longer wavelength removes the ripple further and flattens the efficiency response. Moreover, we explain that for the same length, the efficiency enhancement expected due to the use of the double-pass device instead of the single-pass one is slightly reduced for the low-loss waveguide while the conversion efficiency profile has almost the same shape with and without low loss. Finally, the criteria for selection of the waveguide length and total pumps power to obtain the desired efficiency, ripple and bandwidth are presented for the double-pass one with and without pump detuning.

2. Theory and model

In this section, the wavelength converter based on double-pass cascaded SFG + DFG in PPLN shown in Fig. 1 is modeled and theoretically investigated and compared with the single-pass one. With the two pump wavelengths λp 1 and λp 2, and the signal wavelength λs, the wavelengths of the SFG (λSF) and converted signal wave (λc) are equal to λp1λp2/(λp1+λp2) and λsλSF/(λsλSF), respectively. Having a reflective coating at a wavelength λSFλ0/2 and assuming no wavelength-dependent phase shifts upon reflection [21] maximizes the SFG before starting the DFG, where λ0 is almost the mean wavelength of two pumps. For the cascaded SFG + DFG interaction under QPM, the SFG process can be described by the three coupled equations [18]:

ddxAp1(x)=jωp1κSFGAp2(x)ASF(x)exp(jΔkSFGx)12αp1Ap1(x)
ddxAp2(x)=jωp2κSFGAp1(x)ASF(x)exp(jΔkSFGx)12αp2Ap2(x)
ddxASF(x)=jωSFκSFGAp1(x)Ap2(x)exp(jΔkSFGx)12αSFASF(x)
and the DFG process is also expressed as:
ddxASF(x)=jωSFκDFGAs(x)Ac(x)exp(jΔkDFGx)12αSFASF(x)
ddxAs(x)=jωsκDFGASF(x)Ac(x)exp(jΔkDFGx)12αsAs(x)
ddxAc(x)=jωcκDFGASF(x)As(x)exp(jΔkDFGx)12αcAc(x)
where (Ap1,αp1), (Ap2,αp2), (ASF,αSF), (As,αs), (Ac,αc) are the amplitude and propagation loss of the first pump, second pump, sum frequency wave, signal and converted signal (idler) wave, respectively. ΔkSFG=βSFβp1βp22π/Λ and ΔkDFG=βSFβsβc2π/Λ are the SFG and DFG phase-mismatched parameters of the structure where Λ is the poled QPM period. Moreover, κSFG=deff2μ0/cSSFGNSFNp1Np2 and κDFG=deff2μ0/cSDFGNSFNsNc are the coupling coefficients where deff=(2/π)d33 is the effective value of nonlinear coefficient and d33 of lithium niobate is 27  pm/V. Np1, Np2, NSF, Ns, Nc are the effective guided mode indices for the first pump, second pump, sum frequency wave, signal and converted wave, respectively. Also, SSFG and SDFG are the channel waveguide cross sections for SFG and DFG and are calculated to be SDFGSSFG30μm2 using the mode overlap integral of the mode field distributions for a waveguide whose width is 6 µm and depth is 3 µm and only supports first TM mode in 1.55 µm region. The equations describing the double-pass SFG + DFG should be solved numerically with a full depleted model of pumps and sum frequency waves. In the double-pass case, first, Eqs. (1), 2 and 3 (only SFG) for the forward propagation direction and then with Eqs. (4), 5 and 6 (including DFG) for the backward propagation direction are solved. The conversion efficiency is defined as the power ratio of the converted light to the input signal or η=Pc(out)/Ps(in)=|Ac(L)|2/|As(0)|2where L is the waveguide length including QPM gratings. The lithium niobate waveguide loss is assumed to be double for the sum frequency (SF) compared to the pumps, signal and idler and for brevity the SF loss is only mentioned in the text. Throughout this paper, αp1=αp2=αs=αc=0.35  dB/cm and αSF=0.7  dB/cm in the 1550-nm band and 775-nm band, respectively for low-loss waveguides [20] unless otherwise mentioned. Also, for a double-pass device we assume constant 95% and 5% reflectivities at the SF and the pump wavelengths, respectively.

 

Fig. 1 Schematic description of the double-pass SFG + DFG wavelength conversion.

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3. Discussion

For practical applications e.g., in WDM systems, the use of single-pass SFG + DFG wavelength converters to set the pumps wavelengths out of the optical window, which is at least 75 nm, has already been proposed [13]. As a high efficiency is also needed, double-pass converter provides a solution besides filtering out the pump wavelengths. However, in both cases, the efficiency curves show slight spectral efficiency variation or deviation from a spectrally flat efficiency, called ripple from here in the interest of brevity. Figures 2(a) and 2(b) depict the conversion efficiency of single-pass and double-pass SFG + DFG based wavelength converters versus signal wavelength for low-loss waveguide when we set the pumps at wavelengths of λP1=1512.5  nm and λP2=1587.5  +ΔλP2 nm where Δλp2>0 is a slight detuning increase in the wavelength of the second pump. The poled QPM period is calculated to be Λ=14.273  μm when ΔkSFG=0 for the pumps at 1512.5 nm and 1587.5 nm. This is calculated by fitting the refractive indexes with the help of the Sellmeier expression for the crystal at the appropriate temperature and subsequently finding the effective indexes of the waveguide. Also, the total pump powers and signal power are 500 mW and 1 mW, respectively. The problem with these schemes for ΔλP2=0 is that the ripple in the responses (shown with red dotted lines in Figs. 2) even though it is possible to achieve lossless or even amplified responses. In fact, for a signal between two pumps in this case, the SFG, is perfectly phase-matched whilst the DFG is phase-matched only at two points around the wavelengths of the pumps and phase-mismatched between them reaching a maximum at 2λSF. To overcome the non-uniform response, we propose increasing the detuning of pumps for double-pass SFG + DFG to diminish the ripple for a tolerable reduction in the bandwidth and mean efficiency [17]. If one or both of the pump wavelength λp1 or λp2 is increasingly detuned, the conversion response will be changed due to different SFG and DFG phase-matched conditions. In this paper, we consider the increase in the wavelength λp2of the pump although increasing the detuning of both pumps is also possible. With increasing λp2 the new phase-matching conditions are ΔkSFG=βSFβp1βp22π/Λ and ΔkDFG=βSFβsβc2π/Λ for the SFG and DFG, respectively. The phase-mismatch for the SFG and DFG are δkSFG=βSFβSFβp2+βp2 and δkDFG=βSFβSFβc+βc. When the second pump wavelength is detuned such that λp2>λp2, the wavelength of the SF wave increases to λSF. Thus, the reduction of βSF to βSF is more than that of βp2 to βp2 and βc to βc which leads to δkSFGδkDFG<0. For SFG + DFG conversion, the phase-matched conditions for signals between the two pumps are ΔkSFG=0 and ΔkDFG>0. With detuning of the pump wavelength, the phase-mismatch ΔkSFGL and ΔkDFGL are reduced.

 

Fig. 2 Efficiency of (a) single-pass and (b) double-pass SFG + DFG device versus signal wavelength for different pump detuning Δλp 2 when the pumps are set at 1512.5 nm and 1587.5 + Δλp 2 nm; and the length, SF loss and total pump power are 2.5 cm, 0.70 dB/cm and 500 mW.

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However, the conversion efficiencies near the pumps are decreased whilst near 2λSF is increased resulting in a flattening of the response. For the single-pass waveguide with the second pump wavelength slightly detuned by Δλp 2 = 0.450 nm, the phase-matching parameters for both the SFG and DFG decrease and their two new matching points coincide, making the two peaks in efficiency curve move gradually toward 2λSF. In Fig. 2(a), the peak-to-peak ripple in efficiency reduces to around 0.2 dB from 1.25 dB with an efficiency penalty of about 2.5 dB. On the other hand, for the double-pass waveguide, as the SFG and DFG processes are independent, only variation of two DFG phase-matched points contributing to the two peaks in the efficiency curve converge rapidly toward 2λSF as the second pump is detuned to Δλp 2 = 0.225 nm. The peak-to-peak ripple in the efficiency reduces to around 0.2 dB from 1.65 dB with an efficiency penalty of about 2 dB, as seen in Fig. 2(b). Therefore, to achieve the same flatness, the reduction in efficiency is smaller and the mean efficiency is almost 2.5 dB higher for the double-pass scheme in comparison with the single-pass one. The reason for higher mean efficiency in the double-pass device is that the signal and pumps are counter-injected in the waveguide and the available waveguide length is used twice.

Figures 3(a) shows the conversion efficiency of the single- and double-pass device for different losses when the total pump power and the waveguide length is 100 mW and 2.5 cm, respectively. As the loss increases, the efficiency is much reduced for the double-pass device compared with the single-pass one and therefore their efficiencies become the same for a constant loss. For instance, the efficiency enhancement of double-pass scheme compared to single-pass one, drops from almost 5.5 dB to 4 dB showing a 1.5 dB decrease when the SF loss increases from 0 to 0.7 dB/cm in Fig. 3(a). That is because the SF effective path is twofold in the double-pass device compared to the single-pass one. Nonetheless, it is evident that in this case, using a double-pass structure to enhance the efficiency is only feasible when the SF loss is smaller than 2.6 dB/cm. To achieve the efficiency enhancement in double-pass devices with greater SF loss, it is possible to use smaller waveguide lengths with increased pump powers. Figure 3(b) shows the conversion efficiency of the single- and double-pass devices for the different SF loss when the total pump power and the waveguide length is 400 mW and 1.25 cm to achieve almost the same efficiency responses in Fig. 3(a). In this case, the efficiency enhancement based on the double-pass scheme tolerates the same 1.5 dB decrease for an SF loss of 1.4 dB/cm, as shown in Fig. 3(b). Also, the efficiency enhancement is available until the SF loss is smaller than 5.2 dB/cm. Thus, using shorter waveguides with higher input power is more suited to high-loss double-pass devices.

 

Fig. 3 Conversion efficiency of wavelength detuned single-pass (Δλp 2 = 0.450 nm) and double-pass (Δλp 2 = 0.225 nm) SFG + DFG device versus signal wavelength for different loss when the length and total pump power are (a) 2.5 cm and 100 mW and (b) 1.25 cm and 400 mW.

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Figures 4 illustrate the contour map of efficiency, peak-to-peak ripple and bandwidth of the double-pass SFG + DFG device versus waveguide length and total pump power where the pumps are set at wavelengths of 1512.5 nm and 1587.5 + Δλp 2 nm, for a detuning Δλp 2 = 0 and Δλp 2 = 0.225 nm. Figures 4(a) and 4(b) show that almost a constant bandwidth and ripple can be acquired using a constant length and more flattening of the ripples can be achieved in the latter case. In Fig. 4(a), a bandwidth of 115 nm with less than 2-dB ripple is achieved for a 2.5-cm long waveguide and amplification is only possible for the pump powers greater than 344 mW. However, to achieve less than 0.2-dB peak-to-peak ripple with a 2.5-cm waveguide, a pump detuning of 0.225 nm is needed where the 3-dB bandwidth is 98 nm, as shown in Fig. 4(b). Also, as amplification is achieved for the pump powers greater than 433 mW, it demonstrates a need for an 89-mW increase in power in comparison to the similar case in Fig. 4(a). Furthermore, Figs. 4(a) and 4(b) give good information for the design of the lengths of double-pass SFG + DFG wavelength converters, and the assignment of the required total pumps power based on a trade-off between the desired efficiency, ripple and bandwidth. To achieve the bandwidth with the desired efficiency and ripple, one should choose the length and input power on the intersection of the ripple and efficiency curves of the contour map. Hence, the criteria are presented on the contour map and the designer can select the proper length and power. If the ripple is tolerable (r p-p < ~2 dB), Fig. 4(a) (without pump detuning) is chosen, otherwise for a flattop response (r p-p < ~0.2 dB), Fig. 4(b) (with pump detuning) is used. Thus, the designer can select the appropriate waveguide length and input pumps power based on the criteria shown on the contour map.

 

Fig. 4 Contour map of efficiency, peak-to-peak ripple and bandwidth of the cascaded double-pass SFG + DFG device versus length and total power for the SF loss of 0.70 dB/cm when the pumps are set at 1512.5 nm and 1587.5 + Δλp 2 nm for (a) Δλp 2 = 0 and (b) Δλp 2 = 0.225 nm.

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4. Conclusion

Improved-efficiency broadband wavelength converters based on double-pass cascaded sum and difference frequency generation in lossy PPLN waveguides have been analyzed numerically and compared to the single-pass case, with a full depleted model of pumps and sum frequency waves with a large pump wavelength difference. Reasonable pump powers and low-loss waveguide lengths are required to achieve lossless or even amplified flattop responses, suitable for the design of efficient broadband wavelength converters operating in the 1.55-μm spectral window. Moreover, applying pump detuning to the double-pass SFG + DFG configuration, offers a means for reducing the ripples with small bandwidth and efficiency penalties. These reductions can be compensated for easily by decreasing the waveguide length and increasing the pump power, respectively.

Acknowledgements

This research is supported by a Strategic Grant of the National Science and Engineering Research Council of Canada and the Canada Research Chairs Programs.

References and links

1. C. Q. Xu, H. Okayama, K. Shinozaki, K. Watanabe, and M. Kawahara, “Wavelength conversions ~1.5 µm by difference frequency generation in periodically domain-inverted LiNbO3 channel waveguides,” Appl. Phys. Lett. 63(9), 1170–1172 (1993). [CrossRef]  

2. M. H. Chou, J. Hauden, M. A. Arbore, and M. M. Fejer, “1.5-µm-band wavelength conversion based on difference-frequency generation in LiNbO3 waveguides with integrated coupling structures,” Opt. Lett. 23(13), 1004–1006 (1998). [CrossRef]  

3. K. Gallo and G. Assanto, “Analysis of lithium niobate all-optical wavelength shifters for the third spectral window,” J. Opt. Soc. Am. B 16(5), 741–753 (1999). [CrossRef]  

4. I. Brener, M. H. Chou, D. Peale, and M. M. Fejer, “Cascaded χ(2) wavelength converter in LiNbO3 waveguides with counter-propagating beams,” Electron. Lett. 35(14), 1155–1157 (1999). [CrossRef]  

5. M. H. Chou, I. Brener, M. M. Fejer, E. E. Chaban, and S. B. Christman, “1.5-µm-Band Wavelength Conversion Based on Cascaded Second-Order Nonlinearity in LiNbO3 Waveguides,” Photonics Technol. Lett. 11(6), 653–655 (1999). [CrossRef]  

6. X. Liu, H. Zhang, and Y. Guo, “Theoretical analyses and optimizations for wavelength conversion by quasi-phase-matching difference frequency generation,” J. Lightwave Technol. 19(11), 1785–1792 (2001). [CrossRef]  

7. W. Liu, J. Sun, and J. Kurz, “Bandwidth and tunability enhancement of wavelength conversion by quasi-phase-matching difference frequency generation,” Opt. Commun. 216(1-3), 239–246 (2003). [CrossRef]  

8. T. Umeki, M. Asobe, Y. Nishida, O. Tadanaga, K. Magari, T. Yanagawa, and H. Suzuki, “Widely tunable 3.4 µm band difference frequency generation using apodized χ(2) grating,” Opt. Lett. 32(9), 1129–1131 (2007). [CrossRef]   [PubMed]  

9. A. Tehranchi and R. Kashyap, “Design of novel unapodized and apodized step-chirped quasi-phase matched gratings for broadband frequency converters based on second harmonic generation,” IEEE J. Lightwave Technol. 26(3), 343–349 (2008). [CrossRef]  

10. A. Tehranchi, and R. Kashyap, “Engineered gratings for flat broadening of second-harmonic phase-matching bandwidth in MgO-doped lithium niobate waveguides,” Opt. Express 16, 18970–75 (2008). http://www.opticsinfobase.org/abstract.cfm?uri=oe-16-23-18970

11. A. Tehranchi and R. Kashyap, “Novel designs for efficient broadband frequency doublers using singly pump-resonant waveguide and engineered chirped gratings,” IEEE J. Quantum Electron. 45(2), 187–194 (2009). [CrossRef]  

12. S. J. B. Yoo, “Wavelength conversion technologies for WDM network applications,” IEEE J. Lightwave Technol. 14(6), 955–966 (1996). [CrossRef]  

13. B. Chen and C.-Q. Xu, “Analysis of novel cascaded χ(2) (SFG+DFG) wavelength conversions in quasi-phase-matched waveguides,” IEEE J. Quantum Electron. 40, 256–261 (2004). [CrossRef]  

14. Y. Wang, B. Chen, and C.-Q. Xu, “Polarisation-insensitive QPM wavelength converter with out-of-band pump,” Electron. Lett. 40(3), 189–191 (2004). [CrossRef]  

15. Y. L. Lee, B. Yu, C. Jung, Y. Noh, J. Lee, and D. Ko, “All-optical wavelength conversion and tuning by the cascaded sum- and difference frequency generation (cSFG/DFG) in a temperature gradient controlled Ti:PPLN channel waveguide,” Opt. Express 13, 2988–93 (2005). http://www.opticsinfobase.org/abstract.cfm?uri=oe-13-8-2988

16. J. Wang, J. Q. Sun, C. H. Luo, and Q. Z. Sun, “Experimental demonstration of wavelength conversion between ps-pulses based on cascaded sum- and difference frequency generation (SFG+DFG) in LiNbO3 waveguides,” Opt. Express 13, 7405–14 (2005). http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-13-19-7405

17. S. Gao, C. Yang, X. Xiao, Y. Tian, Z. You, and G. Jin, “Bandwidth enhancement and response flattening of cascaded sum- and difference-frequency generation-based wavelength conversion,” Opt. Commun. 266(1), 296–301 (2006). [CrossRef]  

18. S. Yu and W. Gu, “Wavelength conversions in quasi-phase matched LiNbO3 waveguide based on double-pass cascaded χ(2) SFG+DFG interactions,” IEEE J. Quantum Electron. 40(12), 1744 (2004). [CrossRef]  

19. S. Yu and W. Gu, “A tunable wavelength conversion and wavelength add/drop scheme based on cascaded second-order nonlinearity with double-pass configuration,” IEEE J. Quantum Electron. 41(7), 1007–1012 (2005). [CrossRef]  

20. S. Gao, C. Yang, X. Xiao, Y. Tian, Z. You, and G. Jin, “Performance evaluation of tunable channel-selective wavelength shift by cascaded sum- and difference-frequency generation in periodically poled lithium niobate waveguides,” IEEE J. Lightwave Technol. 25(3), 710–718 (2007). [CrossRef]  

21. G. Imeshev, M. Proctor, and M. M. Fejer, “Phase correction in double-pass quasi-phase-matched second-harmonic generation with a wedged crystal,” Opt. Lett. 23(3), 165–167 (1998). [CrossRef]  

References

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  1. C. Q. Xu, H. Okayama, K. Shinozaki, K. Watanabe, and M. Kawahara, “Wavelength conversions ~1.5 µm by difference frequency generation in periodically domain-inverted LiNbO3 channel waveguides,” Appl. Phys. Lett. 63(9), 1170–1172 (1993).
    [Crossref]
  2. M. H. Chou, J. Hauden, M. A. Arbore, and M. M. Fejer, “1.5-µm-band wavelength conversion based on difference-frequency generation in LiNbO3 waveguides with integrated coupling structures,” Opt. Lett. 23(13), 1004–1006 (1998).
    [Crossref]
  3. K. Gallo and G. Assanto, “Analysis of lithium niobate all-optical wavelength shifters for the third spectral window,” J. Opt. Soc. Am. B 16(5), 741–753 (1999).
    [Crossref]
  4. I. Brener, M. H. Chou, D. Peale, and M. M. Fejer, “Cascaded χ(2) wavelength converter in LiNbO3 waveguides with counter-propagating beams,” Electron. Lett. 35(14), 1155–1157 (1999).
    [Crossref]
  5. M. H. Chou, I. Brener, M. M. Fejer, E. E. Chaban, and S. B. Christman, “1.5-µm-Band Wavelength Conversion Based on Cascaded Second-Order Nonlinearity in LiNbO3 Waveguides,” Photonics Technol. Lett. 11(6), 653–655 (1999).
    [Crossref]
  6. X. Liu, H. Zhang, and Y. Guo, “Theoretical analyses and optimizations for wavelength conversion by quasi-phase-matching difference frequency generation,” J. Lightwave Technol. 19(11), 1785–1792 (2001).
    [Crossref]
  7. W. Liu, J. Sun, and J. Kurz, “Bandwidth and tunability enhancement of wavelength conversion by quasi-phase-matching difference frequency generation,” Opt. Commun. 216(1-3), 239–246 (2003).
    [Crossref]
  8. T. Umeki, M. Asobe, Y. Nishida, O. Tadanaga, K. Magari, T. Yanagawa, and H. Suzuki, “Widely tunable 3.4 µm band difference frequency generation using apodized χ(2) grating,” Opt. Lett. 32(9), 1129–1131 (2007).
    [Crossref] [PubMed]
  9. A. Tehranchi and R. Kashyap, “Design of novel unapodized and apodized step-chirped quasi-phase matched gratings for broadband frequency converters based on second harmonic generation,” IEEE J. Lightwave Technol. 26(3), 343–349 (2008).
    [Crossref]
  10. A. Tehranchi, and R. Kashyap, “Engineered gratings for flat broadening of second-harmonic phase-matching bandwidth in MgO-doped lithium niobate waveguides,” Opt. Express 16, 18970–75 (2008). http://www.opticsinfobase.org/abstract.cfm?uri=oe-16-23-18970
  11. A. Tehranchi and R. Kashyap, “Novel designs for efficient broadband frequency doublers using singly pump-resonant waveguide and engineered chirped gratings,” IEEE J. Quantum Electron. 45(2), 187–194 (2009).
    [Crossref]
  12. S. J. B. Yoo, “Wavelength conversion technologies for WDM network applications,” IEEE J. Lightwave Technol. 14(6), 955–966 (1996).
    [Crossref]
  13. B. Chen and C.-Q. Xu, “Analysis of novel cascaded χ(2) (SFG+DFG) wavelength conversions in quasi-phase-matched waveguides,” IEEE J. Quantum Electron. 40, 256–261 (2004).
    [Crossref]
  14. Y. Wang, B. Chen, and C.-Q. Xu, “Polarisation-insensitive QPM wavelength converter with out-of-band pump,” Electron. Lett. 40(3), 189–191 (2004).
    [Crossref]
  15. Y. L. Lee, B. Yu, C. Jung, Y. Noh, J. Lee, and D. Ko, “All-optical wavelength conversion and tuning by the cascaded sum- and difference frequency generation (cSFG/DFG) in a temperature gradient controlled Ti:PPLN channel waveguide,” Opt. Express 13, 2988–93 (2005). http://www.opticsinfobase.org/abstract.cfm?uri=oe-13-8-2988
  16. J. Wang, J. Q. Sun, C. H. Luo, and Q. Z. Sun, “Experimental demonstration of wavelength conversion between ps-pulses based on cascaded sum- and difference frequency generation (SFG+DFG) in LiNbO3 waveguides,” Opt. Express 13, 7405–14 (2005). http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-13-19-7405
  17. S. Gao, C. Yang, X. Xiao, Y. Tian, Z. You, and G. Jin, “Bandwidth enhancement and response flattening of cascaded sum- and difference-frequency generation-based wavelength conversion,” Opt. Commun. 266(1), 296–301 (2006).
    [Crossref]
  18. S. Yu and W. Gu, “Wavelength conversions in quasi-phase matched LiNbO3 waveguide based on double-pass cascaded χ(2) SFG+DFG interactions,” IEEE J. Quantum Electron. 40(12), 1744 (2004).
    [Crossref]
  19. S. Yu and W. Gu, “A tunable wavelength conversion and wavelength add/drop scheme based on cascaded second-order nonlinearity with double-pass configuration,” IEEE J. Quantum Electron. 41(7), 1007–1012 (2005).
    [Crossref]
  20. S. Gao, C. Yang, X. Xiao, Y. Tian, Z. You, and G. Jin, “Performance evaluation of tunable channel-selective wavelength shift by cascaded sum- and difference-frequency generation in periodically poled lithium niobate waveguides,” IEEE J. Lightwave Technol. 25(3), 710–718 (2007).
    [Crossref]
  21. G. Imeshev, M. Proctor, and M. M. Fejer, “Phase correction in double-pass quasi-phase-matched second-harmonic generation with a wedged crystal,” Opt. Lett. 23(3), 165–167 (1998).
    [Crossref]

2009 (1)

A. Tehranchi and R. Kashyap, “Novel designs for efficient broadband frequency doublers using singly pump-resonant waveguide and engineered chirped gratings,” IEEE J. Quantum Electron. 45(2), 187–194 (2009).
[Crossref]

2008 (1)

A. Tehranchi and R. Kashyap, “Design of novel unapodized and apodized step-chirped quasi-phase matched gratings for broadband frequency converters based on second harmonic generation,” IEEE J. Lightwave Technol. 26(3), 343–349 (2008).
[Crossref]

2007 (2)

T. Umeki, M. Asobe, Y. Nishida, O. Tadanaga, K. Magari, T. Yanagawa, and H. Suzuki, “Widely tunable 3.4 µm band difference frequency generation using apodized χ(2) grating,” Opt. Lett. 32(9), 1129–1131 (2007).
[Crossref] [PubMed]

S. Gao, C. Yang, X. Xiao, Y. Tian, Z. You, and G. Jin, “Performance evaluation of tunable channel-selective wavelength shift by cascaded sum- and difference-frequency generation in periodically poled lithium niobate waveguides,” IEEE J. Lightwave Technol. 25(3), 710–718 (2007).
[Crossref]

2006 (1)

S. Gao, C. Yang, X. Xiao, Y. Tian, Z. You, and G. Jin, “Bandwidth enhancement and response flattening of cascaded sum- and difference-frequency generation-based wavelength conversion,” Opt. Commun. 266(1), 296–301 (2006).
[Crossref]

2005 (1)

S. Yu and W. Gu, “A tunable wavelength conversion and wavelength add/drop scheme based on cascaded second-order nonlinearity with double-pass configuration,” IEEE J. Quantum Electron. 41(7), 1007–1012 (2005).
[Crossref]

2004 (3)

S. Yu and W. Gu, “Wavelength conversions in quasi-phase matched LiNbO3 waveguide based on double-pass cascaded χ(2) SFG+DFG interactions,” IEEE J. Quantum Electron. 40(12), 1744 (2004).
[Crossref]

B. Chen and C.-Q. Xu, “Analysis of novel cascaded χ(2) (SFG+DFG) wavelength conversions in quasi-phase-matched waveguides,” IEEE J. Quantum Electron. 40, 256–261 (2004).
[Crossref]

Y. Wang, B. Chen, and C.-Q. Xu, “Polarisation-insensitive QPM wavelength converter with out-of-band pump,” Electron. Lett. 40(3), 189–191 (2004).
[Crossref]

2003 (1)

W. Liu, J. Sun, and J. Kurz, “Bandwidth and tunability enhancement of wavelength conversion by quasi-phase-matching difference frequency generation,” Opt. Commun. 216(1-3), 239–246 (2003).
[Crossref]

2001 (1)

1999 (3)

K. Gallo and G. Assanto, “Analysis of lithium niobate all-optical wavelength shifters for the third spectral window,” J. Opt. Soc. Am. B 16(5), 741–753 (1999).
[Crossref]

I. Brener, M. H. Chou, D. Peale, and M. M. Fejer, “Cascaded χ(2) wavelength converter in LiNbO3 waveguides with counter-propagating beams,” Electron. Lett. 35(14), 1155–1157 (1999).
[Crossref]

M. H. Chou, I. Brener, M. M. Fejer, E. E. Chaban, and S. B. Christman, “1.5-µm-Band Wavelength Conversion Based on Cascaded Second-Order Nonlinearity in LiNbO3 Waveguides,” Photonics Technol. Lett. 11(6), 653–655 (1999).
[Crossref]

1998 (2)

1996 (1)

S. J. B. Yoo, “Wavelength conversion technologies for WDM network applications,” IEEE J. Lightwave Technol. 14(6), 955–966 (1996).
[Crossref]

1993 (1)

C. Q. Xu, H. Okayama, K. Shinozaki, K. Watanabe, and M. Kawahara, “Wavelength conversions ~1.5 µm by difference frequency generation in periodically domain-inverted LiNbO3 channel waveguides,” Appl. Phys. Lett. 63(9), 1170–1172 (1993).
[Crossref]

Arbore, M. A.

Asobe, M.

Assanto, G.

Brener, I.

I. Brener, M. H. Chou, D. Peale, and M. M. Fejer, “Cascaded χ(2) wavelength converter in LiNbO3 waveguides with counter-propagating beams,” Electron. Lett. 35(14), 1155–1157 (1999).
[Crossref]

M. H. Chou, I. Brener, M. M. Fejer, E. E. Chaban, and S. B. Christman, “1.5-µm-Band Wavelength Conversion Based on Cascaded Second-Order Nonlinearity in LiNbO3 Waveguides,” Photonics Technol. Lett. 11(6), 653–655 (1999).
[Crossref]

Chaban, E. E.

M. H. Chou, I. Brener, M. M. Fejer, E. E. Chaban, and S. B. Christman, “1.5-µm-Band Wavelength Conversion Based on Cascaded Second-Order Nonlinearity in LiNbO3 Waveguides,” Photonics Technol. Lett. 11(6), 653–655 (1999).
[Crossref]

Chen, B.

B. Chen and C.-Q. Xu, “Analysis of novel cascaded χ(2) (SFG+DFG) wavelength conversions in quasi-phase-matched waveguides,” IEEE J. Quantum Electron. 40, 256–261 (2004).
[Crossref]

Y. Wang, B. Chen, and C.-Q. Xu, “Polarisation-insensitive QPM wavelength converter with out-of-band pump,” Electron. Lett. 40(3), 189–191 (2004).
[Crossref]

Chou, M. H.

M. H. Chou, I. Brener, M. M. Fejer, E. E. Chaban, and S. B. Christman, “1.5-µm-Band Wavelength Conversion Based on Cascaded Second-Order Nonlinearity in LiNbO3 Waveguides,” Photonics Technol. Lett. 11(6), 653–655 (1999).
[Crossref]

I. Brener, M. H. Chou, D. Peale, and M. M. Fejer, “Cascaded χ(2) wavelength converter in LiNbO3 waveguides with counter-propagating beams,” Electron. Lett. 35(14), 1155–1157 (1999).
[Crossref]

M. H. Chou, J. Hauden, M. A. Arbore, and M. M. Fejer, “1.5-µm-band wavelength conversion based on difference-frequency generation in LiNbO3 waveguides with integrated coupling structures,” Opt. Lett. 23(13), 1004–1006 (1998).
[Crossref]

Christman, S. B.

M. H. Chou, I. Brener, M. M. Fejer, E. E. Chaban, and S. B. Christman, “1.5-µm-Band Wavelength Conversion Based on Cascaded Second-Order Nonlinearity in LiNbO3 Waveguides,” Photonics Technol. Lett. 11(6), 653–655 (1999).
[Crossref]

Fejer, M. M.

I. Brener, M. H. Chou, D. Peale, and M. M. Fejer, “Cascaded χ(2) wavelength converter in LiNbO3 waveguides with counter-propagating beams,” Electron. Lett. 35(14), 1155–1157 (1999).
[Crossref]

M. H. Chou, I. Brener, M. M. Fejer, E. E. Chaban, and S. B. Christman, “1.5-µm-Band Wavelength Conversion Based on Cascaded Second-Order Nonlinearity in LiNbO3 Waveguides,” Photonics Technol. Lett. 11(6), 653–655 (1999).
[Crossref]

M. H. Chou, J. Hauden, M. A. Arbore, and M. M. Fejer, “1.5-µm-band wavelength conversion based on difference-frequency generation in LiNbO3 waveguides with integrated coupling structures,” Opt. Lett. 23(13), 1004–1006 (1998).
[Crossref]

G. Imeshev, M. Proctor, and M. M. Fejer, “Phase correction in double-pass quasi-phase-matched second-harmonic generation with a wedged crystal,” Opt. Lett. 23(3), 165–167 (1998).
[Crossref]

Gallo, K.

Gao, S.

S. Gao, C. Yang, X. Xiao, Y. Tian, Z. You, and G. Jin, “Performance evaluation of tunable channel-selective wavelength shift by cascaded sum- and difference-frequency generation in periodically poled lithium niobate waveguides,” IEEE J. Lightwave Technol. 25(3), 710–718 (2007).
[Crossref]

S. Gao, C. Yang, X. Xiao, Y. Tian, Z. You, and G. Jin, “Bandwidth enhancement and response flattening of cascaded sum- and difference-frequency generation-based wavelength conversion,” Opt. Commun. 266(1), 296–301 (2006).
[Crossref]

Gu, W.

S. Yu and W. Gu, “A tunable wavelength conversion and wavelength add/drop scheme based on cascaded second-order nonlinearity with double-pass configuration,” IEEE J. Quantum Electron. 41(7), 1007–1012 (2005).
[Crossref]

S. Yu and W. Gu, “Wavelength conversions in quasi-phase matched LiNbO3 waveguide based on double-pass cascaded χ(2) SFG+DFG interactions,” IEEE J. Quantum Electron. 40(12), 1744 (2004).
[Crossref]

Guo, Y.

Hauden, J.

Imeshev, G.

Jin, G.

S. Gao, C. Yang, X. Xiao, Y. Tian, Z. You, and G. Jin, “Performance evaluation of tunable channel-selective wavelength shift by cascaded sum- and difference-frequency generation in periodically poled lithium niobate waveguides,” IEEE J. Lightwave Technol. 25(3), 710–718 (2007).
[Crossref]

S. Gao, C. Yang, X. Xiao, Y. Tian, Z. You, and G. Jin, “Bandwidth enhancement and response flattening of cascaded sum- and difference-frequency generation-based wavelength conversion,” Opt. Commun. 266(1), 296–301 (2006).
[Crossref]

Kashyap, R.

A. Tehranchi and R. Kashyap, “Novel designs for efficient broadband frequency doublers using singly pump-resonant waveguide and engineered chirped gratings,” IEEE J. Quantum Electron. 45(2), 187–194 (2009).
[Crossref]

A. Tehranchi and R. Kashyap, “Design of novel unapodized and apodized step-chirped quasi-phase matched gratings for broadband frequency converters based on second harmonic generation,” IEEE J. Lightwave Technol. 26(3), 343–349 (2008).
[Crossref]

Kawahara, M.

C. Q. Xu, H. Okayama, K. Shinozaki, K. Watanabe, and M. Kawahara, “Wavelength conversions ~1.5 µm by difference frequency generation in periodically domain-inverted LiNbO3 channel waveguides,” Appl. Phys. Lett. 63(9), 1170–1172 (1993).
[Crossref]

Kurz, J.

W. Liu, J. Sun, and J. Kurz, “Bandwidth and tunability enhancement of wavelength conversion by quasi-phase-matching difference frequency generation,” Opt. Commun. 216(1-3), 239–246 (2003).
[Crossref]

Liu, W.

W. Liu, J. Sun, and J. Kurz, “Bandwidth and tunability enhancement of wavelength conversion by quasi-phase-matching difference frequency generation,” Opt. Commun. 216(1-3), 239–246 (2003).
[Crossref]

Liu, X.

Magari, K.

Nishida, Y.

Okayama, H.

C. Q. Xu, H. Okayama, K. Shinozaki, K. Watanabe, and M. Kawahara, “Wavelength conversions ~1.5 µm by difference frequency generation in periodically domain-inverted LiNbO3 channel waveguides,” Appl. Phys. Lett. 63(9), 1170–1172 (1993).
[Crossref]

Peale, D.

I. Brener, M. H. Chou, D. Peale, and M. M. Fejer, “Cascaded χ(2) wavelength converter in LiNbO3 waveguides with counter-propagating beams,” Electron. Lett. 35(14), 1155–1157 (1999).
[Crossref]

Proctor, M.

Shinozaki, K.

C. Q. Xu, H. Okayama, K. Shinozaki, K. Watanabe, and M. Kawahara, “Wavelength conversions ~1.5 µm by difference frequency generation in periodically domain-inverted LiNbO3 channel waveguides,” Appl. Phys. Lett. 63(9), 1170–1172 (1993).
[Crossref]

Sun, J.

W. Liu, J. Sun, and J. Kurz, “Bandwidth and tunability enhancement of wavelength conversion by quasi-phase-matching difference frequency generation,” Opt. Commun. 216(1-3), 239–246 (2003).
[Crossref]

Suzuki, H.

Tadanaga, O.

Tehranchi, A.

A. Tehranchi and R. Kashyap, “Novel designs for efficient broadband frequency doublers using singly pump-resonant waveguide and engineered chirped gratings,” IEEE J. Quantum Electron. 45(2), 187–194 (2009).
[Crossref]

A. Tehranchi and R. Kashyap, “Design of novel unapodized and apodized step-chirped quasi-phase matched gratings for broadband frequency converters based on second harmonic generation,” IEEE J. Lightwave Technol. 26(3), 343–349 (2008).
[Crossref]

Tian, Y.

S. Gao, C. Yang, X. Xiao, Y. Tian, Z. You, and G. Jin, “Performance evaluation of tunable channel-selective wavelength shift by cascaded sum- and difference-frequency generation in periodically poled lithium niobate waveguides,” IEEE J. Lightwave Technol. 25(3), 710–718 (2007).
[Crossref]

S. Gao, C. Yang, X. Xiao, Y. Tian, Z. You, and G. Jin, “Bandwidth enhancement and response flattening of cascaded sum- and difference-frequency generation-based wavelength conversion,” Opt. Commun. 266(1), 296–301 (2006).
[Crossref]

Umeki, T.

Wang, Y.

Y. Wang, B. Chen, and C.-Q. Xu, “Polarisation-insensitive QPM wavelength converter with out-of-band pump,” Electron. Lett. 40(3), 189–191 (2004).
[Crossref]

Watanabe, K.

C. Q. Xu, H. Okayama, K. Shinozaki, K. Watanabe, and M. Kawahara, “Wavelength conversions ~1.5 µm by difference frequency generation in periodically domain-inverted LiNbO3 channel waveguides,” Appl. Phys. Lett. 63(9), 1170–1172 (1993).
[Crossref]

Xiao, X.

S. Gao, C. Yang, X. Xiao, Y. Tian, Z. You, and G. Jin, “Performance evaluation of tunable channel-selective wavelength shift by cascaded sum- and difference-frequency generation in periodically poled lithium niobate waveguides,” IEEE J. Lightwave Technol. 25(3), 710–718 (2007).
[Crossref]

S. Gao, C. Yang, X. Xiao, Y. Tian, Z. You, and G. Jin, “Bandwidth enhancement and response flattening of cascaded sum- and difference-frequency generation-based wavelength conversion,” Opt. Commun. 266(1), 296–301 (2006).
[Crossref]

Xu, C. Q.

C. Q. Xu, H. Okayama, K. Shinozaki, K. Watanabe, and M. Kawahara, “Wavelength conversions ~1.5 µm by difference frequency generation in periodically domain-inverted LiNbO3 channel waveguides,” Appl. Phys. Lett. 63(9), 1170–1172 (1993).
[Crossref]

Xu, C.-Q.

B. Chen and C.-Q. Xu, “Analysis of novel cascaded χ(2) (SFG+DFG) wavelength conversions in quasi-phase-matched waveguides,” IEEE J. Quantum Electron. 40, 256–261 (2004).
[Crossref]

Y. Wang, B. Chen, and C.-Q. Xu, “Polarisation-insensitive QPM wavelength converter with out-of-band pump,” Electron. Lett. 40(3), 189–191 (2004).
[Crossref]

Yanagawa, T.

Yang, C.

S. Gao, C. Yang, X. Xiao, Y. Tian, Z. You, and G. Jin, “Performance evaluation of tunable channel-selective wavelength shift by cascaded sum- and difference-frequency generation in periodically poled lithium niobate waveguides,” IEEE J. Lightwave Technol. 25(3), 710–718 (2007).
[Crossref]

S. Gao, C. Yang, X. Xiao, Y. Tian, Z. You, and G. Jin, “Bandwidth enhancement and response flattening of cascaded sum- and difference-frequency generation-based wavelength conversion,” Opt. Commun. 266(1), 296–301 (2006).
[Crossref]

Yoo, S. J. B.

S. J. B. Yoo, “Wavelength conversion technologies for WDM network applications,” IEEE J. Lightwave Technol. 14(6), 955–966 (1996).
[Crossref]

You, Z.

S. Gao, C. Yang, X. Xiao, Y. Tian, Z. You, and G. Jin, “Performance evaluation of tunable channel-selective wavelength shift by cascaded sum- and difference-frequency generation in periodically poled lithium niobate waveguides,” IEEE J. Lightwave Technol. 25(3), 710–718 (2007).
[Crossref]

S. Gao, C. Yang, X. Xiao, Y. Tian, Z. You, and G. Jin, “Bandwidth enhancement and response flattening of cascaded sum- and difference-frequency generation-based wavelength conversion,” Opt. Commun. 266(1), 296–301 (2006).
[Crossref]

Yu, S.

S. Yu and W. Gu, “A tunable wavelength conversion and wavelength add/drop scheme based on cascaded second-order nonlinearity with double-pass configuration,” IEEE J. Quantum Electron. 41(7), 1007–1012 (2005).
[Crossref]

S. Yu and W. Gu, “Wavelength conversions in quasi-phase matched LiNbO3 waveguide based on double-pass cascaded χ(2) SFG+DFG interactions,” IEEE J. Quantum Electron. 40(12), 1744 (2004).
[Crossref]

Zhang, H.

Appl. Phys. Lett. (1)

C. Q. Xu, H. Okayama, K. Shinozaki, K. Watanabe, and M. Kawahara, “Wavelength conversions ~1.5 µm by difference frequency generation in periodically domain-inverted LiNbO3 channel waveguides,” Appl. Phys. Lett. 63(9), 1170–1172 (1993).
[Crossref]

Electron. Lett. (2)

I. Brener, M. H. Chou, D. Peale, and M. M. Fejer, “Cascaded χ(2) wavelength converter in LiNbO3 waveguides with counter-propagating beams,” Electron. Lett. 35(14), 1155–1157 (1999).
[Crossref]

Y. Wang, B. Chen, and C.-Q. Xu, “Polarisation-insensitive QPM wavelength converter with out-of-band pump,” Electron. Lett. 40(3), 189–191 (2004).
[Crossref]

IEEE J. Lightwave Technol. (3)

S. J. B. Yoo, “Wavelength conversion technologies for WDM network applications,” IEEE J. Lightwave Technol. 14(6), 955–966 (1996).
[Crossref]

A. Tehranchi and R. Kashyap, “Design of novel unapodized and apodized step-chirped quasi-phase matched gratings for broadband frequency converters based on second harmonic generation,” IEEE J. Lightwave Technol. 26(3), 343–349 (2008).
[Crossref]

S. Gao, C. Yang, X. Xiao, Y. Tian, Z. You, and G. Jin, “Performance evaluation of tunable channel-selective wavelength shift by cascaded sum- and difference-frequency generation in periodically poled lithium niobate waveguides,” IEEE J. Lightwave Technol. 25(3), 710–718 (2007).
[Crossref]

IEEE J. Quantum Electron. (4)

B. Chen and C.-Q. Xu, “Analysis of novel cascaded χ(2) (SFG+DFG) wavelength conversions in quasi-phase-matched waveguides,” IEEE J. Quantum Electron. 40, 256–261 (2004).
[Crossref]

A. Tehranchi and R. Kashyap, “Novel designs for efficient broadband frequency doublers using singly pump-resonant waveguide and engineered chirped gratings,” IEEE J. Quantum Electron. 45(2), 187–194 (2009).
[Crossref]

S. Yu and W. Gu, “Wavelength conversions in quasi-phase matched LiNbO3 waveguide based on double-pass cascaded χ(2) SFG+DFG interactions,” IEEE J. Quantum Electron. 40(12), 1744 (2004).
[Crossref]

S. Yu and W. Gu, “A tunable wavelength conversion and wavelength add/drop scheme based on cascaded second-order nonlinearity with double-pass configuration,” IEEE J. Quantum Electron. 41(7), 1007–1012 (2005).
[Crossref]

J. Lightwave Technol. (1)

J. Opt. Soc. Am. B (1)

Opt. Commun. (2)

W. Liu, J. Sun, and J. Kurz, “Bandwidth and tunability enhancement of wavelength conversion by quasi-phase-matching difference frequency generation,” Opt. Commun. 216(1-3), 239–246 (2003).
[Crossref]

S. Gao, C. Yang, X. Xiao, Y. Tian, Z. You, and G. Jin, “Bandwidth enhancement and response flattening of cascaded sum- and difference-frequency generation-based wavelength conversion,” Opt. Commun. 266(1), 296–301 (2006).
[Crossref]

Opt. Lett. (3)

Photonics Technol. Lett. (1)

M. H. Chou, I. Brener, M. M. Fejer, E. E. Chaban, and S. B. Christman, “1.5-µm-Band Wavelength Conversion Based on Cascaded Second-Order Nonlinearity in LiNbO3 Waveguides,” Photonics Technol. Lett. 11(6), 653–655 (1999).
[Crossref]

Other (3)

A. Tehranchi, and R. Kashyap, “Engineered gratings for flat broadening of second-harmonic phase-matching bandwidth in MgO-doped lithium niobate waveguides,” Opt. Express 16, 18970–75 (2008). http://www.opticsinfobase.org/abstract.cfm?uri=oe-16-23-18970

Y. L. Lee, B. Yu, C. Jung, Y. Noh, J. Lee, and D. Ko, “All-optical wavelength conversion and tuning by the cascaded sum- and difference frequency generation (cSFG/DFG) in a temperature gradient controlled Ti:PPLN channel waveguide,” Opt. Express 13, 2988–93 (2005). http://www.opticsinfobase.org/abstract.cfm?uri=oe-13-8-2988

J. Wang, J. Q. Sun, C. H. Luo, and Q. Z. Sun, “Experimental demonstration of wavelength conversion between ps-pulses based on cascaded sum- and difference frequency generation (SFG+DFG) in LiNbO3 waveguides,” Opt. Express 13, 7405–14 (2005). http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-13-19-7405

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Figures (4)

Fig. 1
Fig. 1 Schematic description of the double-pass SFG + DFG wavelength conversion.
Fig. 2
Fig. 2 Efficiency of (a) single-pass and (b) double-pass SFG + DFG device versus signal wavelength for different pump detuning Δλp 2 when the pumps are set at 1512.5 nm and 1587.5 + Δλp 2 nm; and the length, SF loss and total pump power are 2.5 cm, 0.70 dB/cm and 500 mW.
Fig. 3
Fig. 3 Conversion efficiency of wavelength detuned single-pass (Δλp 2 = 0.450 nm) and double-pass (Δλp 2 = 0.225 nm) SFG + DFG device versus signal wavelength for different loss when the length and total pump power are (a) 2.5 cm and 100 mW and (b) 1.25 cm and 400 mW.
Fig. 4
Fig. 4 Contour map of efficiency, peak-to-peak ripple and bandwidth of the cascaded double-pass SFG + DFG device versus length and total power for the SF loss of 0.70 dB/cm when the pumps are set at 1512.5 nm and 1587.5 + Δλp 2 nm for (a) Δλp 2 = 0 and (b) Δλp 2 = 0.225 nm.

Equations (6)

Equations on this page are rendered with MathJax. Learn more.

ddxAp1(x)=jωp1κSFGAp2(x)ASF(x)exp(jΔkSFGx)12αp1Ap1(x)
ddxAp2(x)=jωp2κSFGAp1(x)ASF(x)exp(jΔkSFGx)12αp2Ap2(x)
ddxASF(x)=jωSFκSFGAp1(x)Ap2(x)exp(jΔkSFGx)12αSFASF(x)
ddxASF(x)=jωSFκDFGAs(x)Ac(x)exp(jΔkDFGx)12αSFASF(x)
ddxAs(x)=jωsκDFGASF(x)Ac(x)exp(jΔkDFGx)12αsAs(x)
ddxAc(x)=jωcκDFGASF(x)As(x)exp(jΔkDFGx)12αcAc(x)

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