Abstract

A concept to flexibly adjust the spectral bandwidth of the output pulses of a fiber optical parametric oscillator is presented. By adjusting the chirp of the pump pulses appropriate to the chirp of the resonant pulses, the energy of the output pulses can be transferred into a user-defined spectral bandwidth. For this concept of optical parametric chirped pulse oscillation, we present numerical simulations of a parametric oscillator, which is able to convert pump pulses with a spectral bandwidth of 3.3 nm into output pulses with an adjustable spectral bandwidth between 9 and 0.05 nm. Combined with a wavelength tunability between 1200 and 1300 nm and pulse energies of up to 100 nJ, the concept should allow to adapt a single all-fiber parametric oscillator to a variety of applications, e.g., in multimodal nonlinear microscopy.

© 2017 Optical Society of America

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References

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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
  31. S. Frankinas, A. Michailovas, N. Rusteika, V. Smirnov, R. Vasilieu, and A. Glebov, “Efficient ultrafast fiber laser using chirped fiber Bragg grating and chirped volume Bragg grating stretcher/compressor configuration,” Proc. SPIE 9730, 973017 (2016).
    [Crossref]

2017 (1)

T. Gottschall, T. Meyer, C. Jauregui, F. Just, T. Eidam, M. Schmitt, J. Popp, J. Limpert, and A. Tünnermann, “All-fiber optical parametric oscillator for bio-medical imaging applications,” Proc. SPIE 10083, 100831 (2017).

2016 (2)

S. Frankinas, A. Michailovas, N. Rusteika, V. Smirnov, R. Vasilieu, and A. Glebov, “Efficient ultrafast fiber laser using chirped fiber Bragg grating and chirped volume Bragg grating stretcher/compressor configuration,” Proc. SPIE 9730, 973017 (2016).
[Crossref]

M. Brinkmann, S. Janfrüchte, T. Hellwig, S. Dobner, and C. Fallnich, “Electronically and rapidly tunable fiber-integrable optical parametric oscillator for nonlinear microscopy,” Opt. Lett. 41, 2193 (2016).
[Crossref] [PubMed]

2015 (4)

2014 (2)

2013 (3)

2012 (2)

Y. Takubo and S. Yamashita, “In-vivo OCT imaging using wavelength swept fiber laser based on dispersion tuning,” IEEE Photonic Tech. L. 24, 979 (2012).
[Crossref]

S. Witte and K. S. E. Eikema, “Ultrafast optical parametric chirped-pulse amplification,” IEEE J. Sel. Top. Quantum Electron. 18, 296 (2012).
[Crossref]

2011 (2)

C. Cleff, P. Groß, L. Kleinschmidt, J. Epping, and C. Fallnich, “Optimally chirped CARS using fiber stretchers,” Appl. Phys. B 105, 801 (2011).
[Crossref]

Y. Q. Xu and S. G. Murdoch, “High conversion efficiency fiber optical parametric oscillator,” Opt. Lett. 36, 4266 (2011).
[Crossref] [PubMed]

2010 (1)

2009 (2)

2008 (1)

2007 (1)

2006 (3)

S. Yamashita and M. Asano, “Wide and fast wavelength-tunable mode-locked fiber laser based on dispersion tuning,” Opt. Express 14, 9399 (2006).
[Crossref]

M. Jurna, J. P. Korterik, H. L. Offerhaus, and C. Otto, “Noncritical phase-matched lithium triborate optical parametric oscillator for high resolution coherent anti-Stokes Raman scattering spectroscopy and microscopy,” Appl. Phys. Lett. 89, 2004 (2006).
[Crossref]

J. M. Dudley and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135 (2006).
[Crossref]

2004 (1)

T. Hellerer, A. M. K. Enejder, and A. Zumbusch, “Spectral focusing: High spectral resolution spectroscopy with broad-bandwidth laser pulses,” Appl. Phys. Lett. 85, 25 (2004).
[Crossref]

1999 (1)

K. Osvay and I. N. Ross, “Efficient tuneable bandwidth frequency mixing using chirped pulses,” Opt. Commun. 166, 113 (1999).
[Crossref]

1996 (1)

1987 (1)

O. Martinez, “3000 times grating compressor with positive group velocity dispersion: Application to fiber compensation in 1.3–1.6 micrometre region,” IEEE J. Quantum Electron. 23, 59 (1987).
[Crossref]

1985 (1)

A. P. Piskarskas, A. Stabinis, and A. Yankauskas, “Parametric frequency modulation of picosecond light pulses in quadratically nonlinear crystals,” Sov. J. Quantum Electron. 15, 1179 (1985).
[Crossref]

Agrawal, G.

G. Agrawal, Nonlinear Fiber Optics (Academic, 2013).

Asano, M.

S. Yamashita and M. Asano, “Wide and fast wavelength-tunable mode-locked fiber laser based on dispersion tuning,” Opt. Express 14, 9399 (2006).
[Crossref]

Baumgartl, M.

Bergano, N. S.

Bigourd, D.

Bocklitz, T. W.

N. Vogler, S. Heuke, T. W. Bocklitz, M. Schmitt, and J. Popp, “Multimodal Imaging Spectroscopy of Tissue,” Annu. Rev. Anal. Chem. 8, 359 (2015).
[Crossref]

Brauckmann, N.

Brinkmann, M.

Cheung, K. K. Y.

Chui, P. C.

Cleff, C.

C. Cleff, P. Groß, L. Kleinschmidt, J. Epping, and C. Fallnich, “Optimally chirped CARS using fiber stretchers,” Appl. Phys. B 105, 801 (2011).
[Crossref]

Coen, S.

J. M. Dudley and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135 (2006).
[Crossref]

Davidson, C. R.

Dietzek, B.

Dobner, S.

Dudley, J. M.

J. M. Dudley and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135 (2006).
[Crossref]

Eidam, T.

T. Gottschall, T. Meyer, C. Jauregui, F. Just, T. Eidam, M. Schmitt, J. Popp, J. Limpert, and A. Tünnermann, “All-fiber optical parametric oscillator for bio-medical imaging applications,” Proc. SPIE 10083, 100831 (2017).

Eikema, K. S. E.

S. Witte and K. S. E. Eikema, “Ultrafast optical parametric chirped-pulse amplification,” IEEE J. Sel. Top. Quantum Electron. 18, 296 (2012).
[Crossref]

Enejder, A. M. K.

T. Hellerer, A. M. K. Enejder, and A. Zumbusch, “Spectral focusing: High spectral resolution spectroscopy with broad-bandwidth laser pulses,” Appl. Phys. Lett. 85, 25 (2004).
[Crossref]

Epping, J.

C. Cleff, P. Groß, L. Kleinschmidt, J. Epping, and C. Fallnich, “Optimally chirped CARS using fiber stretchers,” Appl. Phys. B 105, 801 (2011).
[Crossref]

Fallnich, C.

Fermann, M. E.

M. E. Fermann and I. Hartl, “Ultrafast fibre lasers,” Nat. Photonics 7, 868 (2013).
[Crossref]

Fourcade-Dutin, C.

Frankinas, S.

S. Frankinas, A. Michailovas, N. Rusteika, V. Smirnov, R. Vasilieu, and A. Glebov, “Efficient ultrafast fiber laser using chirped fiber Bragg grating and chirped volume Bragg grating stretcher/compressor configuration,” Proc. SPIE 9730, 973017 (2016).
[Crossref]

Glebov, A.

S. Frankinas, A. Michailovas, N. Rusteika, V. Smirnov, R. Vasilieu, and A. Glebov, “Efficient ultrafast fiber laser using chirped fiber Bragg grating and chirped volume Bragg grating stretcher/compressor configuration,” Proc. SPIE 9730, 973017 (2016).
[Crossref]

Gottschall, T.

Groß, P.

C. Cleff, P. Groß, L. Kleinschmidt, J. Epping, and C. Fallnich, “Optimally chirped CARS using fiber stretchers,” Appl. Phys. B 105, 801 (2011).
[Crossref]

M. Kues, N. Brauckmann, T. Walbaum, P. Groß, and C. Fallnich, “Nonlinear dynamics of femtosecond supercontin-uum generation with feedback,” Opt. Express 17, 15827 (2009).
[Crossref] [PubMed]

Hartl, I.

M. E. Fermann and I. Hartl, “Ultrafast fibre lasers,” Nat. Photonics 7, 868 (2013).
[Crossref]

Harvey, J. D.

Hellerer, T.

T. Hellerer, A. M. K. Enejder, and A. Zumbusch, “Spectral focusing: High spectral resolution spectroscopy with broad-bandwidth laser pulses,” Appl. Phys. Lett. 85, 25 (2004).
[Crossref]

Hellwig, T.

Heuke, S.

N. Vogler, S. Heuke, T. W. Bocklitz, M. Schmitt, and J. Popp, “Multimodal Imaging Spectroscopy of Tissue,” Annu. Rev. Anal. Chem. 8, 359 (2015).
[Crossref]

Hugonnot, E.

Janfrüchte, S.

Jauregui, C.

T. Gottschall, T. Meyer, C. Jauregui, F. Just, T. Eidam, M. Schmitt, J. Popp, J. Limpert, and A. Tünnermann, “All-fiber optical parametric oscillator for bio-medical imaging applications,” Proc. SPIE 10083, 100831 (2017).

Ji, M.

Judkins, J. B.

Jurna, M.

M. Jurna, J. P. Korterik, H. L. Offerhaus, and C. Otto, “Noncritical phase-matched lithium triborate optical parametric oscillator for high resolution coherent anti-Stokes Raman scattering spectroscopy and microscopy,” Appl. Phys. Lett. 89, 2004 (2006).
[Crossref]

Just, F.

T. Gottschall, T. Meyer, C. Jauregui, F. Just, T. Eidam, M. Schmitt, J. Popp, J. Limpert, and A. Tünnermann, “All-fiber optical parametric oscillator for bio-medical imaging applications,” Proc. SPIE 10083, 100831 (2017).

Kieu, K.

Kleinschmidt, L.

C. Cleff, P. Groß, L. Kleinschmidt, J. Epping, and C. Fallnich, “Optimally chirped CARS using fiber stretchers,” Appl. Phys. B 105, 801 (2011).
[Crossref]

Korterik, J. P.

M. Jurna, J. P. Korterik, H. L. Offerhaus, and C. Otto, “Noncritical phase-matched lithium triborate optical parametric oscillator for high resolution coherent anti-Stokes Raman scattering spectroscopy and microscopy,” Appl. Phys. Lett. 89, 2004 (2006).
[Crossref]

Kues, M.

M. Brinkmann, M. Kues, and C. Fallnich, “Phase-dependent spectral control of pulsed modulation instability via dichromatic seed fields,” Appl. Phys. B 116, 763 (2014).
[Crossref]

M. Kues, N. Brauckmann, T. Walbaum, P. Groß, and C. Fallnich, “Nonlinear dynamics of femtosecond supercontin-uum generation with feedback,” Opt. Express 17, 15827 (2009).
[Crossref] [PubMed]

Lamb, E. S.

Lefrancois, S.

Lemaire, P. J.

Leonhardt, R.

Li, Q.

Limpert, J.

Marie, V.

Martinez, O.

O. Martinez, “3000 times grating compressor with positive group velocity dispersion: Application to fiber compensation in 1.3–1.6 micrometre region,” IEEE J. Quantum Electron. 23, 59 (1987).
[Crossref]

Maslov, A. V.

Meyer, T.

Michailovas, A.

S. Frankinas, A. Michailovas, N. Rusteika, V. Smirnov, R. Vasilieu, and A. Glebov, “Efficient ultrafast fiber laser using chirped fiber Bragg grating and chirped volume Bragg grating stretcher/compressor configuration,” Proc. SPIE 9730, 973017 (2016).
[Crossref]

Miyawaki, M.

Murdoch, S. G.

Mussot, A.

Nakazaki, Y.

Nguyen, T. N.

Offerhaus, H. L.

M. Jurna, J. P. Korterik, H. L. Offerhaus, and C. Otto, “Noncritical phase-matched lithium triborate optical parametric oscillator for high resolution coherent anti-Stokes Raman scattering spectroscopy and microscopy,” Appl. Phys. Lett. 89, 2004 (2006).
[Crossref]

Osvay, K.

K. Osvay and I. N. Ross, “Efficient tuneable bandwidth frequency mixing using chirped pulses,” Opt. Commun. 166, 113 (1999).
[Crossref]

Otto, C.

M. Jurna, J. P. Korterik, H. L. Offerhaus, and C. Otto, “Noncritical phase-matched lithium triborate optical parametric oscillator for high resolution coherent anti-Stokes Raman scattering spectroscopy and microscopy,” Appl. Phys. Lett. 89, 2004 (2006).
[Crossref]

Pedrazzani, J. R.

Peyghambarian, N.

Piskarskas, A. P.

A. P. Piskarskas, A. Stabinis, and A. Yankauskas, “Parametric frequency modulation of picosecond light pulses in quadratically nonlinear crystals,” Sov. J. Quantum Electron. 15, 1179 (1985).
[Crossref]

Popp, J.

T. Gottschall, T. Meyer, C. Jauregui, F. Just, T. Eidam, M. Schmitt, J. Popp, J. Limpert, and A. Tünnermann, “All-fiber optical parametric oscillator for bio-medical imaging applications,” Proc. SPIE 10083, 100831 (2017).

N. Vogler, S. Heuke, T. W. Bocklitz, M. Schmitt, and J. Popp, “Multimodal Imaging Spectroscopy of Tissue,” Annu. Rev. Anal. Chem. 8, 359 (2015).
[Crossref]

T. Gottschall, T. Meyer, M. Schmitt, J. Popp, J. Limpert, and A. Tünnermann, “Four-wave-mixing-based optical parametric oscillator delivering energetic, tunable, chirped femtosecond pulses for non-linear biomedical applications,” Opt. Express 23, 23968 (2015).
[Crossref] [PubMed]

T. Gottschall, T. Meyer, M. Baumgartl, B. Dietzek, J. Popp, J. Limpert, and A. Tünnermann, “Fiber-based optical parametric oscillator for high resolution coherent anti-Stokes Raman scattering (CARS) microscopy,” Opt. Express 22, 21921 (2014).
[Crossref] [PubMed]

Ross, I. N.

K. Osvay and I. N. Ross, “Efficient tuneable bandwidth frequency mixing using chirped pulses,” Opt. Commun. 166, 113 (1999).
[Crossref]

Rusteika, N.

S. Frankinas, A. Michailovas, N. Rusteika, V. Smirnov, R. Vasilieu, and A. Glebov, “Efficient ultrafast fiber laser using chirped fiber Bragg grating and chirped volume Bragg grating stretcher/compressor configuration,” Proc. SPIE 9730, 973017 (2016).
[Crossref]

Schmitt, M.

T. Gottschall, T. Meyer, C. Jauregui, F. Just, T. Eidam, M. Schmitt, J. Popp, J. Limpert, and A. Tünnermann, “All-fiber optical parametric oscillator for bio-medical imaging applications,” Proc. SPIE 10083, 100831 (2017).

N. Vogler, S. Heuke, T. W. Bocklitz, M. Schmitt, and J. Popp, “Multimodal Imaging Spectroscopy of Tissue,” Annu. Rev. Anal. Chem. 8, 359 (2015).
[Crossref]

T. Gottschall, T. Meyer, M. Schmitt, J. Popp, J. Limpert, and A. Tünnermann, “Four-wave-mixing-based optical parametric oscillator delivering energetic, tunable, chirped femtosecond pulses for non-linear biomedical applications,” Opt. Express 23, 23968 (2015).
[Crossref] [PubMed]

Schnack, M.

Sharping, J. E.

Smirnov, V.

S. Frankinas, A. Michailovas, N. Rusteika, V. Smirnov, R. Vasilieu, and A. Glebov, “Efficient ultrafast fiber laser using chirped fiber Bragg grating and chirped volume Bragg grating stretcher/compressor configuration,” Proc. SPIE 9730, 973017 (2016).
[Crossref]

Stabinis, A.

A. P. Piskarskas, A. Stabinis, and A. Yankauskas, “Parametric frequency modulation of picosecond light pulses in quadratically nonlinear crystals,” Sov. J. Quantum Electron. 15, 1179 (1985).
[Crossref]

Takubo, Y.

Y. Takubo and S. Yamashita, “In-vivo OCT imaging using wavelength swept fiber laser based on dispersion tuning,” IEEE Photonic Tech. L. 24, 979 (2012).
[Crossref]

Trebino, R.

R. Trebino and E. Zeek, Ultrashort Laser Pulses(Springer, 2000).

Tünnermann, A.

Vanvincq, O.

Vasilieu, R.

S. Frankinas, A. Michailovas, N. Rusteika, V. Smirnov, R. Vasilieu, and A. Glebov, “Efficient ultrafast fiber laser using chirped fiber Bragg grating and chirped volume Bragg grating stretcher/compressor configuration,” Proc. SPIE 9730, 973017 (2016).
[Crossref]

Vengsarkar, A. M.

Vogler, N.

N. Vogler, S. Heuke, T. W. Bocklitz, M. Schmitt, and J. Popp, “Multimodal Imaging Spectroscopy of Tissue,” Annu. Rev. Anal. Chem. 8, 359 (2015).
[Crossref]

Wadsworth, W. J.

Walbaum, T.

Wise, F. W.

Witte, S.

S. Witte and K. S. E. Eikema, “Ultrafast optical parametric chirped-pulse amplification,” IEEE J. Sel. Top. Quantum Electron. 18, 296 (2012).
[Crossref]

Wong, G. K. L.

Wong, K. K. Y.

Xie, X. S.

Xu, Y. Q.

Yamashita, S.

Y. Takubo and S. Yamashita, “In-vivo OCT imaging using wavelength swept fiber laser based on dispersion tuning,” IEEE Photonic Tech. L. 24, 979 (2012).
[Crossref]

Y. Nakazaki and S. Yamashita, “Fast and wide tuning range wavelength-swept fiber laser based on dispersion tuning and its application to dynamic FBG sensing,” Opt. Express 17, 8310 (2009).
[Crossref] [PubMed]

S. Yamashita and M. Asano, “Wide and fast wavelength-tunable mode-locked fiber laser based on dispersion tuning,” Opt. Express 14, 9399 (2006).
[Crossref]

Yang, S.

Yankauskas, A.

A. P. Piskarskas, A. Stabinis, and A. Yankauskas, “Parametric frequency modulation of picosecond light pulses in quadratically nonlinear crystals,” Sov. J. Quantum Electron. 15, 1179 (1985).
[Crossref]

Zeek, E.

R. Trebino and E. Zeek, Ultrashort Laser Pulses(Springer, 2000).

Zhou, Y.

Zumbusch, A.

T. Hellerer, A. M. K. Enejder, and A. Zumbusch, “Spectral focusing: High spectral resolution spectroscopy with broad-bandwidth laser pulses,” Appl. Phys. Lett. 85, 25 (2004).
[Crossref]

Annu. Rev. Anal. Chem. (1)

N. Vogler, S. Heuke, T. W. Bocklitz, M. Schmitt, and J. Popp, “Multimodal Imaging Spectroscopy of Tissue,” Annu. Rev. Anal. Chem. 8, 359 (2015).
[Crossref]

Appl. Phys. B (2)

C. Cleff, P. Groß, L. Kleinschmidt, J. Epping, and C. Fallnich, “Optimally chirped CARS using fiber stretchers,” Appl. Phys. B 105, 801 (2011).
[Crossref]

M. Brinkmann, M. Kues, and C. Fallnich, “Phase-dependent spectral control of pulsed modulation instability via dichromatic seed fields,” Appl. Phys. B 116, 763 (2014).
[Crossref]

Appl. Phys. Lett. (2)

M. Jurna, J. P. Korterik, H. L. Offerhaus, and C. Otto, “Noncritical phase-matched lithium triborate optical parametric oscillator for high resolution coherent anti-Stokes Raman scattering spectroscopy and microscopy,” Appl. Phys. Lett. 89, 2004 (2006).
[Crossref]

T. Hellerer, A. M. K. Enejder, and A. Zumbusch, “Spectral focusing: High spectral resolution spectroscopy with broad-bandwidth laser pulses,” Appl. Phys. Lett. 85, 25 (2004).
[Crossref]

IEEE J. Quantum Electron. (1)

O. Martinez, “3000 times grating compressor with positive group velocity dispersion: Application to fiber compensation in 1.3–1.6 micrometre region,” IEEE J. Quantum Electron. 23, 59 (1987).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (1)

S. Witte and K. S. E. Eikema, “Ultrafast optical parametric chirped-pulse amplification,” IEEE J. Sel. Top. Quantum Electron. 18, 296 (2012).
[Crossref]

IEEE Photonic Tech. L. (1)

Y. Takubo and S. Yamashita, “In-vivo OCT imaging using wavelength swept fiber laser based on dispersion tuning,” IEEE Photonic Tech. L. 24, 979 (2012).
[Crossref]

J. Lightwave Technol. (1)

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

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Figures (6)

Fig. 1
Fig. 1

Schematic setup of a dispersion-tuned FOPO. WDM: wavelength-division multiplexer, PCF: photonic crystal fiber, SMF: single-mode fiber.

Fig. 2
Fig. 2

Illustration of the spectral distributions of the resonant signal (blue), pump (green) and output idler (red) pulses, interacting in a FWM process. The distribution of each pulse is normalized to the maximum spectral power of the pulse. The illustrations do not resemble numerical results, but should give an idea of the principal functionality of OPCPO. (a) In case of unchirped pump pulses, the spectral bandwidth ∆νi of the idler pulses is given by the combination of the pump and overlapped signal spectrum. (b) Oppositely chirped signal and pump pulses generate idler pulses with a broad spectral bandwidth. (c) Pump and signal pulses with chirp parameters in the ratio of 1:2 generate idler pulses with a narrow spectral bandwidth.

Fig. 3
Fig. 3

(a) Bandwidth (FWHM) of the output idler pulses as a function of the duration of the pump pulses with a bandwidth of 3.3 nm (FWHM). The simulated data points are connected by the solid lines to guide the eye. The x-axis shows the temporal duration times the sign of the chirp of the pump pulses. Corresponding spectrograms of the signal and pump pulses in front of the PCF and the generated idler pulses after the PCF are shown in (b), (c), (d) for negatively chirped pump pulses with a duration of 150 ps, and in in (e), (f), (g) for positively chirped pump pulses with a duration of 150 ps.

Fig. 4
Fig. 4

(a) Spectral bandwidth (FWHM, blue circles) of the output idler pulses and FWM gain (green line) as a function of the idler center wavelength. The chirp of the pump pulses was optimized and kept constant for the generation of idler pulses with a large bandwidth of 1.72 THz (9.1 nm) at a center wavelength of 1260 nm. The black arrows and red circles exemplify, how the spectral bandwidth can be increased by increasing the pump power as depicted in detail in Fig. 5. (b) Potential bandwidth-limited durations (dark-red circles) of the idler pulses and the minimal durations (blue crosses) reachable by applying only GVD to the idler pulses. The black arrows and red symbols exemplify, how the idler pulse duration can be decreased by increasing the pump power. The green diamonds show the idler pulse energy.

Fig. 5
Fig. 5

Evolution of the idler spectrum as a function of the pump peak power for idler center wavelengths of (a) 1210 nm and (b) 1300 nm; in (a) the pump peak power was incremented from 5.25 to 7 kW in steps of 0.25 kW and in (b) from 5.2 to 6.0 kW in steps of 0.1 kW. The spectra drawn in red exhibited a FWHM bandwidth of 1.73 THz (9.2 nm). The individual spectra are shifted vertically against each other for better visibility; the chirp and the duration of 150 ps of the pump pulses was kept fixed.

Fig. 6
Fig. 6

Spectral bandwidth (FWHM, blue circles) and pulse energy (green crosses) of the idler pulses as a function of the idler center wavelength. The chirp of the pump pulses was optimized for the generation of idler pulses with a narrow bandwidth of 0.01 THz (0.05 nm) at a center wavelength of 1260 nm and kept fixed for the calculated tuning process.

Tables (1)

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Table 1 Fiber parameters. All values are given in SI-units: γ in 1/Wm, βn in sn/m.

Equations (4)

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κ = 2 β ( ν 0 , p ) β ( ν 0 , s ) β ( ν 0 , i ) + 2 γ P 0 0 ,
g ( Ω ) = ( γ P 0 ) 2 ( κ ( Ω ) 2 ) 2 ,
ν 0 , i = 2 ν 0 , p ν 0 , s .
ν i ( t ) = 2 ν 0 , p + 2 b p t ( ν 0 , s + b s t ) = ν 0 , i + ( 2 b p b s ) t .

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