## Design and optimization of fiber optical parametric oscillators for femtosecond pulse generation |

Optics Express, Vol. 18, Issue 16, pp. 17294-17305 (2010)

http://dx.doi.org/10.1364/OE.18.017294

Acrobat PDF (6762 KB)

### Abstract

In this paper, we use a genetic algorithm and pulse-propagation analysis to design and optimize optical parametric oscillators based on soft-glass microstructured optical fibers. The maximum parametric gain, phase-match, walk-off between pump (1560 nm) and signal (880 nm) pulses, signal feedback ratio and signal-pump synchronization of the cavity are optimized. Pulse propagation analysis suggests that one can implement a fiber optical parametric oscillator capable of generating approximately 200-fs pulses at 880 nm with 43% peak-power conversion, high output pulse quality (time-bandwidth product ≈ 0.43) and a wavelength tuning range that is limited only by the glass transmission windows.

© 2010 Optical Society of America

## 1. Introduction

1. J. D. Harvey, R. Leonhardt, S. Coen, G. K. L. Wong, J. Knight, W. J. Wadsworth, and P. S. Russell, “Scalar modulation instability in the normal dispersion regime by use of a photonic crystal fiber,” Opt. Lett. **28**, 2225–2227 (2003). [CrossRef] [PubMed]

1. J. D. Harvey, R. Leonhardt, S. Coen, G. K. L. Wong, J. Knight, W. J. Wadsworth, and P. S. Russell, “Scalar modulation instability in the normal dispersion regime by use of a photonic crystal fiber,” Opt. Lett. **28**, 2225–2227 (2003). [CrossRef] [PubMed]

9. C. Xia, M. Kumar, O. R. Kulkarni, M. N. Islam, F. L. Terry, and M. J. Freeman, “Mid-infrared supercontinuum generation to 4.5µm in ZBLAN fluoride fibers by nanosecond diode pumping,” Opt. Lett. **31**, 2553–2555 (2006). [CrossRef] [PubMed]

15. T. M. Monro and H. Ebendorff-Heidepriem, “Progress in microstructured optical fibers,” Annu. Rev. Mater. Res. **36**, 467–495 (2006). [CrossRef]

15. T. M. Monro and H. Ebendorff-Heidepriem, “Progress in microstructured optical fibers,” Annu. Rev. Mater. Res. **36**, 467–495 (2006). [CrossRef]

17. M. Dantus and V. V. Lozovoy, “Experimental coherent laser control of physicochemical processes,” Chem. Rev. **104**, 1813–1860 (2004). [CrossRef] [PubMed]

5. J. Sharping, “Microstructure fiber based optical parametric oscillators,” J. Lightwave Technol. **26**, 2184–2191 (2008). [CrossRef]

*β*(

*ω*). One can then model the FOPO using a set of coupled wave equations [18]. The adaptive range of this combination of approaches is limited for two reasons. Firstly, truncating the Taylor series fails to follow the true dispersion when the frequency shifts between pump the signal/idler are large. For large frequency shifts, as is the case for state-of-the-art FOPOs, the importance of higher-order terms of

*β*(

*ω*) increases significantly, and each term must be incorporated into any optimization procedure. Secondly, the coupled wave equations, which usually only consider the fields at pump, signal and idler wavelengths, are not suited to simulating complex nonlinear processes with dynamic frequency shifts or large spectral broadening, such as supercontinuum generation (SCG).

*β*(

*ω*), and we use the generalized nonlinear Schrödinger equation [19

19. J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. **78**, 1135–1184 (2006). [CrossRef]

13. Wen Qi Zhang, Sharaam Afshar V., and T. M. Monro, “A genetic algorithm based approach to fiber design for high coherence and large bandwidth supercontinuum generation,” Opt. Express **17**, 19311–19327 (2009). [CrossRef]

21. F. Poletti, V. Finazzi, T.M. Monro, N.G.R. Broderick, V. Tse, and D.J. Richardson, “Inverse design and fabrication tolerances of ultra-flattened dispersion holey fibers,” Opt. Express **13**, 3728–3736 (2005). [CrossRef] [PubMed]

22. M. Gao, C. Jiang, W. Hu, and J. Wang, “Optimized design of two-pump fiber optical parametric amplifier with two-section nonlinear fibers using genetic algorithm”, Opt. Express **12**, 5603–5613 (2004). [CrossRef] [PubMed]

## 2. Fiber design and genetic algorithms

*γ*is the nonlinear coefficient,

*P*

_{0}is the pump power and

*κ*=

*β*+

_{s}*β*− 2

_{i}*β*+ 2

_{p}*γP*

_{0}is the phase mismatch. The quantities

*β*are the mode propagation constants for the signal, idler and pump fields, respectively, within the optical fiber.

_{s,i,p}*γ*and minimize

*κ*. Note that

*κ*is also a function of

*γ*. Besides maximizing the gain, one also must consider the walk-off between the pump and signal pulses as they propagate through the optical fiber. This is especially important for large frequency shift and ultrashort pulses. Thus, the group velocities

*β*

_{1}

^{−1}(where

*β*= ∂

_{i}*/∂*

^{i}β*ω*) must also be included in the optimization. To design a fiber structure optimized for highest output efficiency based on these relations, one has to search the parameter space of

^{i}*β*(

*ω*) and

_{s,i,p}*γ*for an optimal point. However, it is not obvious how best to search for a structure because each structural parameter correlates to all

*β*and

*γ*. To overcome this obstacle, we use a GA to optimize the fiber structure.

### 2.1. GA Background

13. Wen Qi Zhang, Sharaam Afshar V., and T. M. Monro, “A genetic algorithm based approach to fiber design for high coherence and large bandwidth supercontinuum generation,” Opt. Express **17**, 19311–19327 (2009). [CrossRef]

*γ*and minimize both

*κ*and walk-off Δ

*β*

_{1}where Δ

*β*

_{1}=

*β*

_{1,s}−

*β*

_{1,p}, one obvious choice of fitness function

*F*is

*κ*|or |Δ

*β*

_{1}| since either of them will lead to infinite fitness value. To avoid this undesirable behavior in optimization, we set limits to both of them. As an example, if we target the optimized FWM gain to be larger than 97% of the maximum achievable gain (perfect phase matching):

*κ*′ as

*κ*.

*T*

_{0}which corresponds to the width where intensity drops to 1/

*e*

^{2}and the pulses almost stop interacting) and we want the fiber length to be a quarter of this length (25% of total walk-off). For example, for a fiber length of 3 mm (corresponding to approximately

*π*nonlinear phase change in one of our test fibers when pumping with pulses with 5-kW peak power), we have

*β*′

_{1}as follows and use it in the fitness function instead of Δ

*β*

_{1}:

*T*

_{0}= 500 fs) hyperbolic secant pulsed pump source with 5 kW peak power, which gives

*F*

_{target}≈ 1 × 10

^{7}ms

^{−1}W

^{−1}.

*β*(

*ω*), cannot be freely chosen. The bulk material properties are fixed for any given glass type, whereas we have some freedom to modify the waveguide contribution to

*β*(

*ω*) through the design of the waveguide. The goals for phase matching and walk-off may not always be achievable. The idea of setting these targets is to explore how closely they can be achieved via the use of flexible fiber designs and GA modelling techniques.

*β*and

*γ*. A type of tellurite glass (70TeO

_{2}−10Na

_{2}O−20ZnF

_{2}) [23

23. M. D. O’Donnell, K. Richardson, R. Stolen, A. B. Seddon, D. Furniss, V. K. Tikhomirov, C. Rivero, M. Ramme, R. Stegeman, G. Stegeman, M. Couzi, and T. Cardinal, “Tellurite and fluorotellurite glasses for fiberoptic raman amplifiers: Glass characterization, optical properties, raman gain, preliminary fiberization, and fiber characterization,” J. Am. Ceram. Soc. **90**, 1448–1457 (2007). [CrossRef]

*n*

_{2}about 25 times larger than that of silica glass at 1550 nm [24

24. K. S. Kim, R. H. Stolen, W. A. Reed, and K. W. Quoi, “Measurement of the nonlinear index of silica-core and dispersion-shifted fibers,” Opt. Lett. **19**, 257–259 (1994). [CrossRef] [PubMed]

### 2.2. GA results

^{6}after just the second generation (shown in Fig. 2a). There are several possible reasons for this. Firstly, the bulk material properties of the glass limit the range of obtainable fitness parameters. Secondly, the size of the population is large, increasing the likelihood that the GA yields structures with fitness values reach the limit. Finally, there can be many localized maxima of fitness within the parameter space that prevents the population from reaching higher fitness structures. Note also that we observe different saturation behavior with the same fitness function for different optimization wavelengths. The structure with the highest fitness value in the last generation is shown in Fig. 2b. Its corresponding parameter values are:

*R*1 = 1.82 µm,

*R*3 = 2.95 µm,

*R*4 = 6.56 µm,

*r*1 = 0.885 µm,

*r*2 = 0.253 µm and

*L*1 = 1.15 µm.

*σ*R1 = 4.4%,

*σ*R3 = 8.4%,

*σ*R4 = 8.5%,

*σ*r1 = 27.3%,

*σ*r2 = 11.2% and

*σ*L1 = 6.0%. From these results, we conclude that if the distortion during fabrication is kept under 4.4%, the error in the final fiber will be less than 10% in terms of fitness values which is corresponding to 10% in the product of parametric gain and walk-off length.

*β*,

*β*

_{1}and

*γ*) and FWM gain map when the fiber is pumped at 1560 nm with a 5 kWsource. It is interesting that a broad gain bandwidth is obtained automatically even though it is not considered explicitly in the fitness function. While the infrared end of the gain map is not shown in the figure, the gain extends to 8 µm. We discovered that a broad and relatively flat gain bandwidth is likely to be obtained by the GA. When the pump wavelength is not too far from (within the range where structure dispersion can cancel the material dispersion) the ZDW of the material (ZDW ≈ 2.2 µm for our glass) and the pump power is relatively high (≈ 5 kW in this case), the GA tends to converge at the point where the ZDW is close to the pump wavelength. In this situation, phase matching is achieved mainly through high-order dispersion. In our specific case, one observes an s-shape in the gain map (see the boxed region in Fig. 3b), which is due to higher-order

*β*terms beginning with

*β*

_{6}. If

*β*

_{2}is positive, which is the situation here, one cannot achieve phase matching without negative-valued high-order terms. Furthermore, in order for 4

^{th}, 6

^{th}, and higher-order terms to be significant, one must reduce the magnitude of

*β*

_{2}, which, in turn, leads to a large bandwidth [18]. Note that, due to the symmetry of the FWM process, odd orders of

*β*terms do not contribute to phase matching.

## 3. FOPO modeling

*A*(

*z*,

*t*) is the envelope of the electric field of a propagating pulse. The propagation constant

*β*(

*ω*) of fiber modes for specific fiber geometries can be pre-calculated using the finite element method (FEM). The combined material and confinement loss is given by

*α*, and

*γ*is the effective nonlinear coefficient of the fiber. The Raman response function is given by

*R*(

*T*), where we assume a Raman fraction (

*f*) of 0.064 for the tellurite glass [13

_{R}13. Wen Qi Zhang, Sharaam Afshar V., and T. M. Monro, “A genetic algorithm based approach to fiber design for high coherence and large bandwidth supercontinuum generation,” Opt. Express **17**, 19311–19327 (2009). [CrossRef]

*π*. The fiber itself is assumed to be lossless because of its short length.

### 3.1. Optimization of synchronization

### 3.2. The effects of feedback ratios

23. M. D. O’Donnell, K. Richardson, R. Stolen, A. B. Seddon, D. Furniss, V. K. Tikhomirov, C. Rivero, M. Ramme, R. Stegeman, G. Stegeman, M. Couzi, and T. Cardinal, “Tellurite and fluorotellurite glasses for fiberoptic raman amplifiers: Glass characterization, optical properties, raman gain, preliminary fiberization, and fiber characterization,” J. Am. Ceram. Soc. **90**, 1448–1457 (2007). [CrossRef]

23. M. D. O’Donnell, K. Richardson, R. Stolen, A. B. Seddon, D. Furniss, V. K. Tikhomirov, C. Rivero, M. Ramme, R. Stegeman, G. Stegeman, M. Couzi, and T. Cardinal, “Tellurite and fluorotellurite glasses for fiberoptic raman amplifiers: Glass characterization, optical properties, raman gain, preliminary fiberization, and fiber characterization,” J. Am. Ceram. Soc. **90**, 1448–1457 (2007). [CrossRef]

## 4. Conclusion

## References and links

1. | J. D. Harvey, R. Leonhardt, S. Coen, G. K. L. Wong, J. Knight, W. J. Wadsworth, and P. S. Russell, “Scalar modulation instability in the normal dispersion regime by use of a photonic crystal fiber,” Opt. Lett. |

2. | Y. Deng, Q. Lin, F. Lu, G. P. Agrawal, and W. H. Knox, “Broadly tunable femtosecond parametric oscillator using a photonic crystal fiber,” Opt. Lett. |

3. | T. Torounidis and P. Andrekson, “Broadband Single-Pumped Fiber-Optic parametric amplifiers,” IEEE Photon. Technol. Lett. |

4. | K. Garay-Palmett, A. B. U’Ren, R. Rangel-Rojo, R. Evans, and S. Camacho-Lopez, “Ultrabroadband photon pair preparation by spontaneous four-wave mixing in a dispersion-engineered optical fiber,” Phys. Rev. A |

5. | J. Sharping, “Microstructure fiber based optical parametric oscillators,” J. Lightwave Technol. |

6. | H. Chen, H. Wang, M. N. Slipchenko, Y. Jung, Y. Shi, J. Zhu, K. K. Buhman, and J. Cheng, “A multimodal platform for nonlinear optical microscopy and microspectroscopy,” Opt. Express |

7. | K. Kieu, B. G. Saar, G. R. Holtom, X. S. Xie, and F. W. Wise, “High-power picosecond fiber source for coherent raman microscopy,” Opt. Lett. |

8. | A. F. Pegoraro, A. Ridsdale, D. J. Moffatt, J. P. Pezacki, B. K. Thomas, L. Fu, L. Dong, M. E. Fermann, and A. Stolow, “All-fiber CARS microscopy of live cells,” Opt. Express |

9. | C. Xia, M. Kumar, O. R. Kulkarni, M. N. Islam, F. L. Terry, and M. J. Freeman, “Mid-infrared supercontinuum generation to 4.5µm in ZBLAN fluoride fibers by nanosecond diode pumping,” Opt. Lett. |

10. | K. K. Chow, K. Kikuchi, T. Nagashima, T. Hasegawa, S. Ohara, and N. Sugimoto, “Four-wave mixing based widely tunablewavelength conversion using 1-m dispersion-shifted bismuth-oxide photonic crystal fiber,” Opt. Express |

11. | P. Domachuk, N. A. Wolchover, M. Cronin-Golomb, A. Wang, A. K. George, C. M. B. Cordeiro, J. C. Knight, and F. G. Omenetto, “Over 4000 nm bandwidth of mid-IR supercontinuum generation in sub-centimeter segments of highly nonlinear tellurite PCFs,” Opt. Express |

12. | H. Hundertmark, S. Rammler, T. Wilken, R. Holzwarth, T. W. Hänsch, and P. S. Russell, “Octave-spanning supercontinuum generated in SF6-glass PCF by a 1060 nm mode-locked fibre laser delivering 20 pJ per pulse,” Opt. Express |

13. | Wen Qi Zhang, Sharaam Afshar V., and T. M. Monro, “A genetic algorithm based approach to fiber design for high coherence and large bandwidth supercontinuum generation,” Opt. Express |

14. | Xian Feng, Wei H. Loh, Joanne C. Flanagan, Angela Camerlingo, Sonali Dasgupta, Periklis Petropoulos, Peter Horak, Ken E. Frampton, Nicholas M. White, Jonathan H. Price, Harvey N. Rutt, and David J. Richardson, “Single-mode tellurite glass holey fiber with extremely large mode area for infrared nonlinear applications,” Opt. Express |

15. | T. M. Monro and H. Ebendorff-Heidepriem, “Progress in microstructured optical fibers,” Annu. Rev. Mater. Res. |

16. | J. Diels and W. Rudolph, |

17. | M. Dantus and V. V. Lozovoy, “Experimental coherent laser control of physicochemical processes,” Chem. Rev. |

18. | G. Agrawal, |

19. | J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. |

20. | L. Davis, |

21. | F. Poletti, V. Finazzi, T.M. Monro, N.G.R. Broderick, V. Tse, and D.J. Richardson, “Inverse design and fabrication tolerances of ultra-flattened dispersion holey fibers,” Opt. Express |

22. | M. Gao, C. Jiang, W. Hu, and J. Wang, “Optimized design of two-pump fiber optical parametric amplifier with two-section nonlinear fibers using genetic algorithm”, Opt. Express |

23. | M. D. O’Donnell, K. Richardson, R. Stolen, A. B. Seddon, D. Furniss, V. K. Tikhomirov, C. Rivero, M. Ramme, R. Stegeman, G. Stegeman, M. Couzi, and T. Cardinal, “Tellurite and fluorotellurite glasses for fiberoptic raman amplifiers: Glass characterization, optical properties, raman gain, preliminary fiberization, and fiber characterization,” J. Am. Ceram. Soc. |

24. | K. S. Kim, R. H. Stolen, W. A. Reed, and K. W. Quoi, “Measurement of the nonlinear index of silica-core and dispersion-shifted fibers,” Opt. Lett. |

25. | J. Tukey, “An introduction to the calculations of numerical spectrum analysis,” Spectral Analysis of Time Series pp. 25–46 (1967). |

**OCIS Codes**

(060.2280) Fiber optics and optical communications : Fiber design and fabrication

(190.4380) Nonlinear optics : Nonlinear optics, four-wave mixing

(190.4970) Nonlinear optics : Parametric oscillators and amplifiers

(190.5530) Nonlinear optics : Pulse propagation and temporal solitons

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: June 17, 2010

Revised Manuscript: July 17, 2010

Manuscript Accepted: July 17, 2010

Published: July 29, 2010

**Citation**

Wen Qi Zhang, Jay E. Sharping, Richard T. White, Tanya M. Monro, and Shahraam Afshar V., "Design and optimization of fiber optical parametric oscillators for femtosecond pulse generation," Opt. Express **18**, 17294-17305 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-16-17294

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### References

- J. D. Harvey, R. Leonhardt, S. Coen, G. K. L. Wong, J. Knight, W. J. Wadsworth, and P. S. Russell, "Scalar modulation instability in the normal dispersion regime by use of a photonic crystal fiber," Opt. Lett. 28, 2225-2227 (2003). [CrossRef] [PubMed]
- Y. Deng, Q. Lin, F. Lu, G. P. Agrawal, and W. H. Knox, "Broadly tunable femtosecond parametric oscillator using a photonic crystal fiber," Opt. Lett. 30, 1234-1236 (2005). [CrossRef] [PubMed]
- T. Torounidis and P. Andrekson, "Broadband Single-Pumped Fiber-Optic parametric amplifiers," IEEE Photon. Technol. Lett. 19, 650-652 (2007). [CrossRef]
- K. Garay-Palmett, A. B. U’Ren, R. Rangel-Rojo, R. Evans, and S. Camacho-Lopez, "Ultrabroadband photon pair preparation by spontaneous four-wave mixing in a dispersion-engineered optical fiber," Phys. Rev. A 78, 043827 (2008). [CrossRef]
- J. Sharping, "Microstructure fiber based optical parametric oscillators," J. Lightwave Technol. 26, 2184-2191 (2008). [CrossRef]
- H. Chen, H. Wang, M. N. Slipchenko, Y. Jung, Y. Shi, J. Zhu, K. K. Buhman, and J. Cheng, "A multimodal platform for nonlinear optical microscopy and microspectroscopy," Opt. Express 17, 1282-1290 (2009). [CrossRef] [PubMed]
- K. Kieu, B. G. Saar, G. R. Holtom, X. S. Xie, and F. W. Wise, "High-power picosecond fiber source for coherent raman microscopy," Opt. Lett. 34, 2051-2053 (2009). [CrossRef] [PubMed]
- A. F. Pegoraro, A. Ridsdale, D. J. Moffatt, J. P. Pezacki, B. K. Thomas, L. Fu, L. Dong, M. E. Fermann, and A. Stolow, "All-fiber CARS microscopy of live cells," Opt. Express 17, 20700-20706 (2009). [CrossRef] [PubMed]
- C. Xia, M. Kumar, O. R. Kulkarni, M. N. Islam, F. L. Terry, and M. J. Freeman, "Mid-infrared supercontinuum generation to 4.5μm in ZBLAN fluoride fibers by nanosecond diode pumping," Opt. Lett. 31, 2553-2555 (2006). [CrossRef] [PubMed]
- K. K. Chow, K. Kikuchi, T. Nagashima, T. Hasegawa, S. Ohara, and N. Sugimoto, "Four-wave mixing based widely tunablewavelength conversion using 1-m dispersion-shifted bismuth-oxide photonic crystal fiber," Opt. Express 15, 15418-15423 (2007). [CrossRef] [PubMed]
- P. Domachuk, N. A. Wolchover, M. Cronin-Golomb, A. Wang, A. K. George, C. M. B. Cordeiro, J. C. Knight, and F. G. Omenetto, "Over 4000 nm bandwidth of mid-IR supercontinuum generation in sub-centimeter segments of highly nonlinear tellurite PCFs," Opt. Express 16, 7161-7168 (2008). [CrossRef] [PubMed]
- H. Hundertmark, S. Rammler, T. Wilken, R. Holzwarth, T. W. Hansch, and P. S. Russell, "Octave-spanning supercontinuum generated in SF6-glass PCF by a 1060 nm mode-locked fibre laser delivering 20 pJ per pulse," Opt. Express 17, 1919-1924 (2009). [CrossRef] [PubMed]
- W. Q. Zhang, S. Afshar V., and T. M. Monro, "A genetic algorithm based approach to fiber design for high coherence and large bandwidth supercontinuum generation," Opt. Express 17, 19311-19327 (2009). [CrossRef]
- X. Feng, W. H. Loh, J. C. Flanagan, A. Camerlingo, S. Dasgupta, P. Petropoulos, P. Horak, K. E. Frampton, N. M. White, J. H. Price, H. N. Rutt, and D. J. Richardson, "Single-mode tellurite glass holey fiber with extremely large mode area for infrared nonlinear applications," Opt. Express 16, 13651-13656 (2008) [CrossRef] [PubMed]
- T. M. Monro and H. Ebendorff-Heidepriem, "Progress in microstructured optical fibers," Annu. Rev. Mater. Res. 36, 467-495 (2006). [CrossRef]
- J. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena: Fundamentals, Techniques, and Applications on a Femtosecond Time Scale, Optics and photonics (Academic Press, San Diego, 1996).
- M. Dantus and V. V. Lozovoy, "Experimental coherent laser control of physicochemical processes," Chem. Rev. 104, 1813-1860 (2004). [CrossRef] [PubMed]
- G. Agrawal, Nonlinear Fiber Optics (Academic Press, 2001), 3rd ed.
- J. M. Dudley, G. Genty, and S. Coen, "Supercontinuum generation in photonic crystal fiber," Rev. Mod. Phys. 78, 1135-1184 (2006). [CrossRef]
- L. Davis, Handbook Of Genetic Algorithms (Thomson Publishing Group, 1991), 1st ed.
- F. Poletti, V. Finazzi, and T. M. Monro, N.G.R. Broderick, V. Tse, and D.J. Richardson, "Inverse design and fabrication tolerances of ultra-flattened dispersion holey fibers," Opt. Express 13, 3728-3736 (2005). [CrossRef] [PubMed]
- M. Gao, C. Jiang, W. Hu, and J. Wang, "Optimized design of two-pump fiber optical parametric amplifier with twosection nonlinear fibers using genetic algorithm", Opt. Express 12, 5603-5613 (2004). [CrossRef] [PubMed]
- M. D. O’Donnell, K. Richardson, R. Stolen, A. B. Seddon, D. Furniss, V. K. Tikhomirov, C. Rivero, M. Ramme, R. Stegeman, G. Stegeman, M. Couzi, and T. Cardinal, "Tellurite and fluorotellurite glasses for fiberoptic raman amplifiers: Glass characterization, optical properties, raman gain, preliminary fiberization, and fiber characterization," J. Am. Ceram. Soc. 90, 1448-1457 (2007). [CrossRef]
- K. S. Kim, R. H. Stolen, W. A. Reed, and K. W. Quoi, "Measurement of the nonlinear index of silica-core and dispersion-shifted fibers," Opt. Lett. 19, 257-259 (1994). [CrossRef] [PubMed]
- J. Tukey, "An introduction to the calculations of numerical spectrum analysis," Spectral Analysis of Time Series, pp. 25-46 (1967).

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