Abstract

We propose a new approach to quasi-phasematching (QPM) design based on convex optimization. We show that with this approach, globally optimum solutions to several important QPM design problems can be determined. The optimization framework is highly versatile, enabling the user to trade-off different objectives and constraints according to the particular application. The convex problems presented consist of simple objective and constraint functions involving a few thousand variables, and can therefore be solved quite straightforwardly. We consider three examples: (1) synthesis of a target pulse profile via difference frequency generation (DFG) from two ultrashort input pulses, (2) the design of a custom DFG transfer function, and (3) a new approach enabling the suppression of spectral gain narrowing in chirped-QPM-based optical parametric chirped pulse amplification (OPCPA). These examples illustrate the power and versatility of convex optimization in the context of QPM devices.

© 2013 OSA

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  32. J. S. Pelc, C. R. Phillips, D. Chang, C. Langrock, and M. M. Fejer, “Efficiency pedestal in quasi-phase-matching devices with random duty-cycle errors,” Opt. Lett.36, 864–866 (2011).
    [CrossRef] [PubMed]
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  38. V. J. Hernandez, C. V. Bennett, B. D. Moran, A. D. Drobshoff, D. Chang, C. Langrock, M. M. Fejer, and M. Ibsen, “104 MHz rate single-shot recording with subpicosecond resolution using temporal imaging,” Opt. Express21, 196–203 (2013).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  44. C. Heese, C. R. Phillips, B. W. Mayer, L. Gallmann, M. M. Fejer, and U. Keller, “75 MW few-cycle mid-infrared pulses from a collinear apodized APPLN-based OPCPA,” Opt. Express20, 26888–26894 (2012).
    [CrossRef] [PubMed]
  45. A. Shirakawa, I. Sakane, M. Takasaka, and T. Kobayashi, “Sub-5-fs visible pulse generation by pulse-front-matched noncollinear optical parametric amplification,” Appl. Phys. Lett.74, 2268–2270 (1999).
    [CrossRef]
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    [CrossRef]
  49. M. Charbonneau-Lefort, B. Afeyan, and M. M. Fejer, “Theory and simulation of gain-guided noncollinear modes in chirped quasi-phase-matched optical parametric amplifiers,” J. Opt. Soc. Am. B27, 824–841 (2010).
  50. C. Heese, C. R. Phillips, L. Gallmann, M. M. Fejer, and U. Keller, “Ultrabroadband, highly flexible amplifier for ultrashort midinfrared laser pulses based on aperiodically poled Mg:LiNbO3,” Opt. Lett.35, 2340–2342 (2010).
    [CrossRef] [PubMed]
  51. C. Heese, C. R. Phillips, L. Gallmann, M. M. Fejer, and U. Keller, “Role of apodization in optical parametric amplifiers based on aperiodic quasi-phasematching gratings,” Opt. Express20, 18066–18071 (2012).
    [CrossRef] [PubMed]
  52. M. N. Rosenbluth, “Parametric instabilities in inhomogeneous media,” Phys. Rev. Lett.29, 565–567 (1972).
    [CrossRef]
  53. C. R. Phillips, C. Langrock, D. Chang, Y. W. Lin, L. Gallmann, and M. M. Fejer, “Apodization of chirped quasi-phsematching devces,” submitted to J. Opt. Soc. Am. B.
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    [CrossRef]
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    [CrossRef]

2013 (1)

2012 (7)

2011 (4)

2010 (5)

2009 (1)

2008 (4)

2007 (3)

2006 (3)

2005 (7)

S. Yang, A. M. Weiner, K. R. Parameswaran, and M. M. Fejer, “Ultrasensitive second-harmonic generation frequency-resolved optical gating by aperiodically poled LiNbO3 waveguides at 1.5 μm,” Opt. Lett.30, 2164–2166 (2005).
[CrossRef] [PubMed]

Z. Jiang, D. S. Seo, S. Yang, D. E. Leaird, R. V. Roussev, C. Langrock, M. M. Fejer, and A. M. Weiner, “Four-User, 2.5-Gb/s, spectrally coded OCDMA system demonstration using Low-Power nonlinear processing,” J. Lightwave Technol.23, 143–158 (2005).
[CrossRef]

M. Charbonneau-Lefort, M. M. Fejer, and B. Afeyan, “Tandem chirped quasi-phase-matching grating optical parametric amplifier design for simultaneous group delay and gain control,” Opt. Lett.30, 634–636 (2005).
[CrossRef] [PubMed]

M. Asobe, O. Tadanaga, H. Miyazawa, Y. Nishida, and H. Suzuki, “Multiple quasi-phase-matched device using continuous phase modulation of χ(2) grating and its application to variable wavelength conversion,” IEEE J. Quant. Electron.41, 1540–1547 (2005).
[CrossRef]

U. Sapaev and D. Reid, “General second-harmonic pulse shaping in grating-engineered quasi-phase-matched nonlinear crystals,” Opt. Express13, 3264–3276 (2005).
[CrossRef] [PubMed]

T. Fuji, J. Rauschenberger, A. Apolonski, V. S. Yakovlev, G. Tempea, T. Udem, C. Gohle, T. W. Haensch, W. Lehnert, M. Scherer, and F. Krausz, “Monolithic carrier-envelope phase-stabilization scheme,” Opt. Lett.30, 332–334 (2005).
[CrossRef] [PubMed]

R. Lifshitz, A. Arie, and A. Bahabad, “Photonic quasicrystals for nonlinear optical frequency conversion,” Phys. Rev. Lett.95, 133901–133904 (2005).
[CrossRef] [PubMed]

2004 (2)

M. Baudrier-Raybaut, R. Haidar, P. Kupecek, P. Lemasson, and E. Rosencher, “Random quasi-phase-matching in bulk polycrystalline isotropic nonlinear materials,” Nature432, 374–376 (2004).
[CrossRef] [PubMed]

M. A. Albota and F. C. Wong, “Efficient single-photon counting at 1.55 μm by means of frequency upconversion,” Opt. Lett.29, 1449–1451 (2004).
[CrossRef] [PubMed]

2001 (3)

2000 (2)

1999 (3)

X. Liu, L. Qian, and F. Wise, “High-energy pulse compression by use of negative phase shifts produced by the cascade χ(2) : χ(2) nonlinearity,” Opt. Lett.24, 1777–1779 (1999).
[CrossRef]

K. Fradkin-Kashi and A. Arie, “Multiple-wavelength quasi-phase-matched nonlinear interactions,” IEEE J. Quant. Electron.35, 1649–1656 (1999).
[CrossRef]

A. Shirakawa, I. Sakane, M. Takasaka, and T. Kobayashi, “Sub-5-fs visible pulse generation by pulse-front-matched noncollinear optical parametric amplification,” Appl. Phys. Lett.74, 2268–2270 (1999).
[CrossRef]

1998 (1)

1997 (1)

S. Zhu, Y. Zhu, and N. Ming, “Quasi-phase-matched third-harmonic generation in a quasi-periodic optical super-lattice,” Science278, 843–846 (1997).
[CrossRef]

1992 (1)

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quant. Electron.28, 2631–2654 (1992).
[CrossRef]

1972 (1)

M. N. Rosenbluth, “Parametric instabilities in inhomogeneous media,” Phys. Rev. Lett.29, 565–567 (1972).
[CrossRef]

1968 (1)

G. D. Boyd, “Parametric interaction of focused gaussian light beams,” Journal of Applied Physics39, 3597–3639 (1968).
[CrossRef]

Afeyan, B.

Albota, M. A.

Angelis, C. D.

Apolonski, A.

Arbore, M. A.

Arie, A.

G. Porat, Y. Silberberg, A. Arie, and H. Suchowski, “Two photon frequency conversion,” Opt. Express20, 3613–3619 (2012).
[CrossRef] [PubMed]

H. Suchowski, V. Prabhudesai, D. Oron, A. Arie, and Y. Silberberg, “Robust adiabatic sum frequencyconversion,” Opt. Express17, 12731–12740 (2009).
[CrossRef] [PubMed]

O. Gayer, Z. Sacks, E. Galun, and A. Arie, “Temperature and wavelength dependent refractive index equations for MgO-doped congruent and stoichiometric LiNbO3,” Appl. Phys. B: Lasers Opt.91, 343–348 (2008).
[CrossRef]

R. Lifshitz, A. Arie, and A. Bahabad, “Photonic quasicrystals for nonlinear optical frequency conversion,” Phys. Rev. Lett.95, 133901–133904 (2005).
[CrossRef] [PubMed]

K. Fradkin-Kashi, A. Arie, P. Urenski, and G. Rosenman, “Multiple nonlinear optical interactions with arbitrary wave vector differences,” Phys. Rev. Lett.88, 023903–023906 (2001).
[CrossRef]

K. Fradkin-Kashi and A. Arie, “Multiple-wavelength quasi-phase-matched nonlinear interactions,” IEEE J. Quant. Electron.35, 1649–1656 (1999).
[CrossRef]

Asobe, M.

M. Asobe, O. Tadanaga, H. Miyazawa, Y. Nishida, and H. Suzuki, “Multiple quasi-phase-matched device using continuous phase modulation of χ(2) grating and its application to variable wavelength conversion,” IEEE J. Quant. Electron.41, 1540–1547 (2005).
[CrossRef]

Bahabad, A.

R. Lifshitz, A. Arie, and A. Bahabad, “Photonic quasicrystals for nonlinear optical frequency conversion,” Phys. Rev. Lett.95, 133901–133904 (2005).
[CrossRef] [PubMed]

Baltuska, A.

Baronio, F.

M. Conforti, F. Baronio, and C. De Angelis, “Nonlinear envelope equation for broadband optical pulses in quadratic media,” Phys. Rev. A81, 053841–053844 (2010).
[CrossRef]

M. Conforti, F. Baronio, and C. D. Angelis, “From femtosecond infrared to picosecond visible pulses: temporal shaping with high-efficiency conversion,” Opt. Lett.32, 1779–1781 (2007).
[CrossRef] [PubMed]

Baudrier-Raybaut, M.

M. Baudrier-Raybaut, R. Haidar, P. Kupecek, P. Lemasson, and E. Rosencher, “Random quasi-phase-matching in bulk polycrystalline isotropic nonlinear materials,” Nature432, 374–376 (2004).
[CrossRef] [PubMed]

Bennett, C. V.

Boyd, G. D.

G. D. Boyd, “Parametric interaction of focused gaussian light beams,” Journal of Applied Physics39, 3597–3639 (1968).
[CrossRef]

Boyd, S.

M. Grant and S. Boyd, CVX: Matlab Software for Disciplined Convex Programming, version 1.21 (2011).

Boyd, S. P.

S. P. Boyd and L. Vandenberghe, Convex Optimization (Cambridge University, 2004).

Byer, R. L.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quant. Electron.28, 2631–2654 (1992).
[CrossRef]

Canalias, C.

C. Canalias and V. Pasiskevicius, “Mirrorless optical parametric oscillator,” Nat. Photon.1, 459–462 (2007).
[CrossRef]

Chang, D.

Charbonneau-Lefort, M.

Conforti, M.

M. Conforti, F. Baronio, and C. De Angelis, “Nonlinear envelope equation for broadband optical pulses in quadratic media,” Phys. Rev. A81, 053841–053844 (2010).
[CrossRef]

M. Conforti, F. Baronio, and C. D. Angelis, “From femtosecond infrared to picosecond visible pulses: temporal shaping with high-efficiency conversion,” Opt. Lett.32, 1779–1781 (2007).
[CrossRef] [PubMed]

De Angelis, C.

M. Conforti, F. Baronio, and C. De Angelis, “Nonlinear envelope equation for broadband optical pulses in quadratic media,” Phys. Rev. A81, 053841–053844 (2010).
[CrossRef]

Deng, Y.

Drobshoff, A. D.

Eikema, K.

S. Witte and K. Eikema, “Ultrafast optical parametric Chirped-Pulse amplification,” Selected Topics in IEEE J. Quant. Electron.18, 296–307 (2012).
[CrossRef]

Fattahi, H.

Fejer, M. M.

V. J. Hernandez, C. V. Bennett, B. D. Moran, A. D. Drobshoff, D. Chang, C. Langrock, M. M. Fejer, and M. Ibsen, “104 MHz rate single-shot recording with subpicosecond resolution using temporal imaging,” Opt. Express21, 196–203 (2013).
[CrossRef] [PubMed]

C. Heese, C. R. Phillips, B. W. Mayer, L. Gallmann, M. M. Fejer, and U. Keller, “75 MW few-cycle mid-infrared pulses from a collinear apodized APPLN-based OPCPA,” Opt. Express20, 26888–26894 (2012).
[CrossRef] [PubMed]

C. Heese, C. R. Phillips, L. Gallmann, M. M. Fejer, and U. Keller, “Role of apodization in optical parametric amplifiers based on aperiodic quasi-phasematching gratings,” Opt. Express20, 18066–18071 (2012).
[CrossRef] [PubMed]

C. R. Phillips and M. M. Fejer, “Adiabatic optical parametric oscillators: steady-state and dynamical behavior,” Opt. Express20, 2466–2482 (2012).
[CrossRef] [PubMed]

J. S. Pelc, Q. Zhang, C. R. Phillips, L. Yu, Y. Yamamoto, and M. M. Fejer, “Cascaded frequency upconversion for high-speed single-photon detection at 1550 nm,” Opt. Lett.37, 476–478 (2012).
[CrossRef] [PubMed]

J. S. Pelc, L. Ma, C. R. Phillips, Q. Zhang, C. Langrock, O. Slattery, X. Tang, and M. M. Fejer, “Long-wavelength-pumped upconversion single-photon detector at 1550 nm: performance and noise analysis,” Opt. Express19, 21445–21456 (2011).
[CrossRef] [PubMed]

C. R. Phillips, C. Langrock, J. S. Pelc, M. M. Fejer, J. Jiang, M. E. Fermann, and I. Hartl, “Supercontinuum generation in quasi-phase-matched LiNbO3 waveguide pumped by a Tm-doped fiber laser system,” Opt. Lett.36, 3912–3914 (2011).
[CrossRef] [PubMed]

J. S. Pelc, C. R. Phillips, D. Chang, C. Langrock, and M. M. Fejer, “Efficiency pedestal in quasi-phase-matching devices with random duty-cycle errors,” Opt. Lett.36, 864–866 (2011).
[CrossRef] [PubMed]

C. R. Phillips, C. Langrock, J. S. Pelc, M. M. Fejer, I. Hartl, and M. E. Fermann, “Supercontinuum generation in quasi-phasematched waveguides,” Opt. Express19, 18754–18773 (2011).
[CrossRef] [PubMed]

C. R. Phillips and M. M. Fejer, “Efficiency and phase of optical parametric amplification in chirped quasi-phase-matched gratings,” Opt. Lett.35, 3093–3095 (2010).
[CrossRef] [PubMed]

C. R. Phillips and M. M. Fejer, “Stability of the singly resonant optical parametric oscillator,” J. Opt. Soc. Am. B27, 2687–2699 (2010).
[CrossRef]

C. Heese, C. R. Phillips, L. Gallmann, M. M. Fejer, and U. Keller, “Ultrabroadband, highly flexible amplifier for ultrashort midinfrared laser pulses based on aperiodically poled Mg:LiNbO3,” Opt. Lett.35, 2340–2342 (2010).
[CrossRef] [PubMed]

M. Charbonneau-Lefort, B. Afeyan, and M. M. Fejer, “Theory and simulation of gain-guided noncollinear modes in chirped quasi-phase-matched optical parametric amplifiers,” J. Opt. Soc. Am. B27, 824–841 (2010).

M. Charbonneau-Lefort, B. Afeyan, and M. M. Fejer, “Competing collinear and noncollinear interactions in chirped quasi-phase-matched optical parametric amplifiers,” J. Opt. Soc. Am. B25, 1402–1413 (2008).
[CrossRef]

M. Charbonneau-Lefort, B. Afeyan, and M. M. Fejer, “Optical parametric amplifiers using chirped quasi-phase-matching gratings I: practical design formulas,” J. Opt. Soc. Am. B25, 463–480 (2008).
[CrossRef]

C. Langrock, M. M. Fejer, I. Hartl, and M. E. Fermann, “Generation of octave-spanning spectra inside reverse-proton-exchanged periodically poled lithium niobate waveguides,” Opt. Lett.32, 2478–2480 (2007).
[CrossRef] [PubMed]

J. Huang, X. P. Xie, C. Langrock, R. V. Roussev, D. S. Hum, and M. M. Fejer, “Amplitude modulation and apodization of quasi-phase-matched interactions,” Opt. Lett.31, 604–606 (2006).
[CrossRef] [PubMed]

M. Charbonneau-Lefort, M. M. Fejer, and B. Afeyan, “Tandem chirped quasi-phase-matching grating optical parametric amplifier design for simultaneous group delay and gain control,” Opt. Lett.30, 634–636 (2005).
[CrossRef] [PubMed]

Z. Jiang, D. S. Seo, S. Yang, D. E. Leaird, R. V. Roussev, C. Langrock, M. M. Fejer, and A. M. Weiner, “Four-User, 2.5-Gb/s, spectrally coded OCDMA system demonstration using Low-Power nonlinear processing,” J. Lightwave Technol.23, 143–158 (2005).
[CrossRef]

S. Yang, A. M. Weiner, K. R. Parameswaran, and M. M. Fejer, “Ultrasensitive second-harmonic generation frequency-resolved optical gating by aperiodically poled LiNbO3 waveguides at 1.5 μm,” Opt. Lett.30, 2164–2166 (2005).
[CrossRef] [PubMed]

L. Gallmann, G. Steinmeyer, U. Keller, G. Imeshev, M. M. Fejer, and J. Meyn, “Generation of sub-6-fs blue pulses by frequency doubling with quasi-phase-matching gratings,” Opt. Lett.26, 614–616 (2001).
[CrossRef]

G. Imeshev, M. M. Fejer, A. Galvanauskas, and D. Harter, “Pulse shaping by difference-frequency mixing with quasi-phase-matching gratings,” J. Opt. Soc. Am. B18, 534–539 (2001).
[CrossRef]

G. Imeshev, M. A. Arbore, S. Kasriel, and M. M. Fejer, “Pulse shaping and compression by second-harmonic generation with quasi-phase-matching gratings in the presence of arbitrary dispersion,” J. Opt. Soc. Am. B17, 1420–1437 (2000).
[CrossRef]

G. Imeshev, M. A. Arbore, M. M. Fejer, A. Galvanauskas, M. Fermann, and D. Harter, “Ultrashort-pulse second-harmonic generation with longitudinally nonuniform quasi-phase-matching gratings: pulse compression and shaping,” J. Opt. Soc. Am. B17, 304–318 (2000).
[CrossRef]

G. Imeshev, A. Galvanauskas, D. Harter, M. A. Arbore, M. Proctor, and M. M. Fejer, “Engineerable femtosecond pulse shaping by second-harmonic generation with fourier synthetic quasi-phase-matching gratings,” Opt. Lett.23, 864–866 (1998).
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M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quant. Electron.28, 2631–2654 (1992).
[CrossRef]

C. R. Phillips, J. S. Pelc, and M. M. Fejer, “Parametric processes in quasi-phasematching gratings with random duty cycle errors,” J. Opt. Soc. Am. B (to be published).

C. R. Phillips, C. Langrock, D. Chang, Y. W. Lin, L. Gallmann, and M. M. Fejer, “Apodization of chirped quasi-phsematching devces,” submitted to J. Opt. Soc. Am. B.

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Forget, N.

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K. Fradkin-Kashi, A. Arie, P. Urenski, and G. Rosenman, “Multiple nonlinear optical interactions with arbitrary wave vector differences,” Phys. Rev. Lett.88, 023903–023906 (2001).
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K. Fradkin-Kashi and A. Arie, “Multiple-wavelength quasi-phase-matched nonlinear interactions,” IEEE J. Quant. Electron.35, 1649–1656 (1999).
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Gallmann, L.

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O. Gayer, Z. Sacks, E. Galun, and A. Arie, “Temperature and wavelength dependent refractive index equations for MgO-doped congruent and stoichiometric LiNbO3,” Appl. Phys. B: Lasers Opt.91, 343–348 (2008).
[CrossRef]

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Gayer, O.

O. Gayer, Z. Sacks, E. Galun, and A. Arie, “Temperature and wavelength dependent refractive index equations for MgO-doped congruent and stoichiometric LiNbO3,” Appl. Phys. B: Lasers Opt.91, 343–348 (2008).
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Ishizuki, H.

Jiang, J.

Jiang, Z.

Jundt, D. H.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quant. Electron.28, 2631–2654 (1992).
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M. Baudrier-Raybaut, R. Haidar, P. Kupecek, P. Lemasson, and E. Rosencher, “Random quasi-phase-matching in bulk polycrystalline isotropic nonlinear materials,” Nature432, 374–376 (2004).
[CrossRef] [PubMed]

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V. J. Hernandez, C. V. Bennett, B. D. Moran, A. D. Drobshoff, D. Chang, C. Langrock, M. M. Fejer, and M. Ibsen, “104 MHz rate single-shot recording with subpicosecond resolution using temporal imaging,” Opt. Express21, 196–203 (2013).
[CrossRef] [PubMed]

C. R. Phillips, C. Langrock, J. S. Pelc, M. M. Fejer, I. Hartl, and M. E. Fermann, “Supercontinuum generation in quasi-phasematched waveguides,” Opt. Express19, 18754–18773 (2011).
[CrossRef] [PubMed]

J. S. Pelc, C. R. Phillips, D. Chang, C. Langrock, and M. M. Fejer, “Efficiency pedestal in quasi-phase-matching devices with random duty-cycle errors,” Opt. Lett.36, 864–866 (2011).
[CrossRef] [PubMed]

C. R. Phillips, C. Langrock, J. S. Pelc, M. M. Fejer, J. Jiang, M. E. Fermann, and I. Hartl, “Supercontinuum generation in quasi-phase-matched LiNbO3 waveguide pumped by a Tm-doped fiber laser system,” Opt. Lett.36, 3912–3914 (2011).
[CrossRef] [PubMed]

J. S. Pelc, L. Ma, C. R. Phillips, Q. Zhang, C. Langrock, O. Slattery, X. Tang, and M. M. Fejer, “Long-wavelength-pumped upconversion single-photon detector at 1550 nm: performance and noise analysis,” Opt. Express19, 21445–21456 (2011).
[CrossRef] [PubMed]

C. Langrock, M. M. Fejer, I. Hartl, and M. E. Fermann, “Generation of octave-spanning spectra inside reverse-proton-exchanged periodically poled lithium niobate waveguides,” Opt. Lett.32, 2478–2480 (2007).
[CrossRef] [PubMed]

J. Huang, X. P. Xie, C. Langrock, R. V. Roussev, D. S. Hum, and M. M. Fejer, “Amplitude modulation and apodization of quasi-phase-matched interactions,” Opt. Lett.31, 604–606 (2006).
[CrossRef] [PubMed]

Z. Jiang, D. S. Seo, S. Yang, D. E. Leaird, R. V. Roussev, C. Langrock, M. M. Fejer, and A. M. Weiner, “Four-User, 2.5-Gb/s, spectrally coded OCDMA system demonstration using Low-Power nonlinear processing,” J. Lightwave Technol.23, 143–158 (2005).
[CrossRef]

C. R. Phillips, C. Langrock, D. Chang, Y. W. Lin, L. Gallmann, and M. M. Fejer, “Apodization of chirped quasi-phsematching devces,” submitted to J. Opt. Soc. Am. B.

Leaird, D. E.

Lehnert, W.

Lemasson, P.

M. Baudrier-Raybaut, R. Haidar, P. Kupecek, P. Lemasson, and E. Rosencher, “Random quasi-phase-matching in bulk polycrystalline isotropic nonlinear materials,” Nature432, 374–376 (2004).
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C. R. Phillips, C. Langrock, D. Chang, Y. W. Lin, L. Gallmann, and M. M. Fejer, “Apodization of chirped quasi-phsematching devces,” submitted to J. Opt. Soc. Am. B.

Liu, X.

Ma, L.

Magel, G. A.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quant. Electron.28, 2631–2654 (1992).
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G. D. Miller, “Periodically poled lithium niobate: modeling, fabrication, and nonlinear-optical performance,” PhD dissertation, Stanford University, Stanford, CA (1998).

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S. Zhu, Y. Zhu, and N. Ming, “Quasi-phase-matched third-harmonic generation in a quasi-periodic optical super-lattice,” Science278, 843–846 (1997).
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M. Asobe, O. Tadanaga, H. Miyazawa, Y. Nishida, and H. Suzuki, “Multiple quasi-phase-matched device using continuous phase modulation of χ(2) grating and its application to variable wavelength conversion,” IEEE J. Quant. Electron.41, 1540–1547 (2005).
[CrossRef]

Moran, B. D.

Moses, J.

Nishida, Y.

M. Asobe, O. Tadanaga, H. Miyazawa, Y. Nishida, and H. Suzuki, “Multiple quasi-phase-matched device using continuous phase modulation of χ(2) grating and its application to variable wavelength conversion,” IEEE J. Quant. Electron.41, 1540–1547 (2005).
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Ossiander, M.

Parameswaran, K. R.

Pasiskevicius, V.

C. Canalias and V. Pasiskevicius, “Mirrorless optical parametric oscillator,” Nat. Photon.1, 459–462 (2007).
[CrossRef]

Pelc, J. S.

Pervak, V.

Phillips, C. R.

C. Heese, C. R. Phillips, B. W. Mayer, L. Gallmann, M. M. Fejer, and U. Keller, “75 MW few-cycle mid-infrared pulses from a collinear apodized APPLN-based OPCPA,” Opt. Express20, 26888–26894 (2012).
[CrossRef] [PubMed]

C. Heese, C. R. Phillips, L. Gallmann, M. M. Fejer, and U. Keller, “Role of apodization in optical parametric amplifiers based on aperiodic quasi-phasematching gratings,” Opt. Express20, 18066–18071 (2012).
[CrossRef] [PubMed]

J. S. Pelc, Q. Zhang, C. R. Phillips, L. Yu, Y. Yamamoto, and M. M. Fejer, “Cascaded frequency upconversion for high-speed single-photon detection at 1550 nm,” Opt. Lett.37, 476–478 (2012).
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C. R. Phillips and M. M. Fejer, “Adiabatic optical parametric oscillators: steady-state and dynamical behavior,” Opt. Express20, 2466–2482 (2012).
[CrossRef] [PubMed]

J. S. Pelc, L. Ma, C. R. Phillips, Q. Zhang, C. Langrock, O. Slattery, X. Tang, and M. M. Fejer, “Long-wavelength-pumped upconversion single-photon detector at 1550 nm: performance and noise analysis,” Opt. Express19, 21445–21456 (2011).
[CrossRef] [PubMed]

C. R. Phillips, C. Langrock, J. S. Pelc, M. M. Fejer, J. Jiang, M. E. Fermann, and I. Hartl, “Supercontinuum generation in quasi-phase-matched LiNbO3 waveguide pumped by a Tm-doped fiber laser system,” Opt. Lett.36, 3912–3914 (2011).
[CrossRef] [PubMed]

J. S. Pelc, C. R. Phillips, D. Chang, C. Langrock, and M. M. Fejer, “Efficiency pedestal in quasi-phase-matching devices with random duty-cycle errors,” Opt. Lett.36, 864–866 (2011).
[CrossRef] [PubMed]

C. R. Phillips, C. Langrock, J. S. Pelc, M. M. Fejer, I. Hartl, and M. E. Fermann, “Supercontinuum generation in quasi-phasematched waveguides,” Opt. Express19, 18754–18773 (2011).
[CrossRef] [PubMed]

C. R. Phillips and M. M. Fejer, “Efficiency and phase of optical parametric amplification in chirped quasi-phase-matched gratings,” Opt. Lett.35, 3093–3095 (2010).
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C. R. Phillips and M. M. Fejer, “Stability of the singly resonant optical parametric oscillator,” J. Opt. Soc. Am. B27, 2687–2699 (2010).
[CrossRef]

C. Heese, C. R. Phillips, L. Gallmann, M. M. Fejer, and U. Keller, “Ultrabroadband, highly flexible amplifier for ultrashort midinfrared laser pulses based on aperiodically poled Mg:LiNbO3,” Opt. Lett.35, 2340–2342 (2010).
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C. R. Phillips, C. Langrock, D. Chang, Y. W. Lin, L. Gallmann, and M. M. Fejer, “Apodization of chirped quasi-phsematching devces,” submitted to J. Opt. Soc. Am. B.

C. R. Phillips, J. S. Pelc, and M. M. Fejer, “Parametric processes in quasi-phasematching gratings with random duty cycle errors,” J. Opt. Soc. Am. B (to be published).

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K. Fradkin-Kashi, A. Arie, P. Urenski, and G. Rosenman, “Multiple nonlinear optical interactions with arbitrary wave vector differences,” Phys. Rev. Lett.88, 023903–023906 (2001).
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Roussev, R. V.

Sacks, Z.

O. Gayer, Z. Sacks, E. Galun, and A. Arie, “Temperature and wavelength dependent refractive index equations for MgO-doped congruent and stoichiometric LiNbO3,” Appl. Phys. B: Lasers Opt.91, 343–348 (2008).
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A. Shirakawa, I. Sakane, M. Takasaka, and T. Kobayashi, “Sub-5-fs visible pulse generation by pulse-front-matched noncollinear optical parametric amplification,” Appl. Phys. Lett.74, 2268–2270 (1999).
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A. Shirakawa, I. Sakane, M. Takasaka, and T. Kobayashi, “Sub-5-fs visible pulse generation by pulse-front-matched noncollinear optical parametric amplification,” Appl. Phys. Lett.74, 2268–2270 (1999).
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M. Asobe, O. Tadanaga, H. Miyazawa, Y. Nishida, and H. Suzuki, “Multiple quasi-phase-matched device using continuous phase modulation of χ(2) grating and its application to variable wavelength conversion,” IEEE J. Quant. Electron.41, 1540–1547 (2005).
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Tadanaga, O.

M. Asobe, O. Tadanaga, H. Miyazawa, Y. Nishida, and H. Suzuki, “Multiple quasi-phase-matched device using continuous phase modulation of χ(2) grating and its application to variable wavelength conversion,” IEEE J. Quant. Electron.41, 1540–1547 (2005).
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Takasaka, M.

A. Shirakawa, I. Sakane, M. Takasaka, and T. Kobayashi, “Sub-5-fs visible pulse generation by pulse-front-matched noncollinear optical parametric amplification,” Appl. Phys. Lett.74, 2268–2270 (1999).
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K. Fradkin-Kashi, A. Arie, P. Urenski, and G. Rosenman, “Multiple nonlinear optical interactions with arbitrary wave vector differences,” Phys. Rev. Lett.88, 023903–023906 (2001).
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S. Witte and K. Eikema, “Ultrafast optical parametric Chirped-Pulse amplification,” Selected Topics in IEEE J. Quant. Electron.18, 296–307 (2012).
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S. Zhu, Y. Zhu, and N. Ming, “Quasi-phase-matched third-harmonic generation in a quasi-periodic optical super-lattice,” Science278, 843–846 (1997).
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S. Zhu, Y. Zhu, and N. Ming, “Quasi-phase-matched third-harmonic generation in a quasi-periodic optical super-lattice,” Science278, 843–846 (1997).
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Appl. Phys. B: Lasers Opt. (1)

O. Gayer, Z. Sacks, E. Galun, and A. Arie, “Temperature and wavelength dependent refractive index equations for MgO-doped congruent and stoichiometric LiNbO3,” Appl. Phys. B: Lasers Opt.91, 343–348 (2008).
[CrossRef]

Appl. Phys. Lett. (1)

A. Shirakawa, I. Sakane, M. Takasaka, and T. Kobayashi, “Sub-5-fs visible pulse generation by pulse-front-matched noncollinear optical parametric amplification,” Appl. Phys. Lett.74, 2268–2270 (1999).
[CrossRef]

IEEE J. Quant. Electron. (3)

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quant. Electron.28, 2631–2654 (1992).
[CrossRef]

K. Fradkin-Kashi and A. Arie, “Multiple-wavelength quasi-phase-matched nonlinear interactions,” IEEE J. Quant. Electron.35, 1649–1656 (1999).
[CrossRef]

M. Asobe, O. Tadanaga, H. Miyazawa, Y. Nishida, and H. Suzuki, “Multiple quasi-phase-matched device using continuous phase modulation of χ(2) grating and its application to variable wavelength conversion,” IEEE J. Quant. Electron.41, 1540–1547 (2005).
[CrossRef]

J. Lightwave Technol. (1)

J. Opt. Soc. Am. B (7)

G. Imeshev, M. A. Arbore, M. M. Fejer, A. Galvanauskas, M. Fermann, and D. Harter, “Ultrashort-pulse second-harmonic generation with longitudinally nonuniform quasi-phase-matching gratings: pulse compression and shaping,” J. Opt. Soc. Am. B17, 304–318 (2000).
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M. Charbonneau-Lefort, B. Afeyan, and M. M. Fejer, “Competing collinear and noncollinear interactions in chirped quasi-phase-matched optical parametric amplifiers,” J. Opt. Soc. Am. B25, 1402–1413 (2008).
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M. Charbonneau-Lefort, B. Afeyan, and M. M. Fejer, “Theory and simulation of gain-guided noncollinear modes in chirped quasi-phase-matched optical parametric amplifiers,” J. Opt. Soc. Am. B27, 824–841 (2010).

C. R. Phillips and M. M. Fejer, “Stability of the singly resonant optical parametric oscillator,” J. Opt. Soc. Am. B27, 2687–2699 (2010).
[CrossRef]

G. Imeshev, M. M. Fejer, A. Galvanauskas, and D. Harter, “Pulse shaping by difference-frequency mixing with quasi-phase-matching gratings,” J. Opt. Soc. Am. B18, 534–539 (2001).
[CrossRef]

M. Charbonneau-Lefort, B. Afeyan, and M. M. Fejer, “Optical parametric amplifiers using chirped quasi-phase-matching gratings I: practical design formulas,” J. Opt. Soc. Am. B25, 463–480 (2008).
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G. Imeshev, M. A. Arbore, S. Kasriel, and M. M. Fejer, “Pulse shaping and compression by second-harmonic generation with quasi-phase-matching gratings in the presence of arbitrary dispersion,” J. Opt. Soc. Am. B17, 1420–1437 (2000).
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Journal of Applied Physics (1)

G. D. Boyd, “Parametric interaction of focused gaussian light beams,” Journal of Applied Physics39, 3597–3639 (1968).
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Nat. Photon. (1)

C. Canalias and V. Pasiskevicius, “Mirrorless optical parametric oscillator,” Nat. Photon.1, 459–462 (2007).
[CrossRef]

Nature (1)

M. Baudrier-Raybaut, R. Haidar, P. Kupecek, P. Lemasson, and E. Rosencher, “Random quasi-phase-matching in bulk polycrystalline isotropic nonlinear materials,” Nature432, 374–376 (2004).
[CrossRef] [PubMed]

Opt. Express (9)

C. R. Phillips, C. Langrock, J. S. Pelc, M. M. Fejer, I. Hartl, and M. E. Fermann, “Supercontinuum generation in quasi-phasematched waveguides,” Opt. Express19, 18754–18773 (2011).
[CrossRef] [PubMed]

J. S. Pelc, L. Ma, C. R. Phillips, Q. Zhang, C. Langrock, O. Slattery, X. Tang, and M. M. Fejer, “Long-wavelength-pumped upconversion single-photon detector at 1550 nm: performance and noise analysis,” Opt. Express19, 21445–21456 (2011).
[CrossRef] [PubMed]

H. Suchowski, V. Prabhudesai, D. Oron, A. Arie, and Y. Silberberg, “Robust adiabatic sum frequencyconversion,” Opt. Express17, 12731–12740 (2009).
[CrossRef] [PubMed]

C. R. Phillips and M. M. Fejer, “Adiabatic optical parametric oscillators: steady-state and dynamical behavior,” Opt. Express20, 2466–2482 (2012).
[CrossRef] [PubMed]

U. Sapaev and D. Reid, “General second-harmonic pulse shaping in grating-engineered quasi-phase-matched nonlinear crystals,” Opt. Express13, 3264–3276 (2005).
[CrossRef] [PubMed]

G. Porat, Y. Silberberg, A. Arie, and H. Suchowski, “Two photon frequency conversion,” Opt. Express20, 3613–3619 (2012).
[CrossRef] [PubMed]

C. Heese, C. R. Phillips, B. W. Mayer, L. Gallmann, M. M. Fejer, and U. Keller, “75 MW few-cycle mid-infrared pulses from a collinear apodized APPLN-based OPCPA,” Opt. Express20, 26888–26894 (2012).
[CrossRef] [PubMed]

C. Heese, C. R. Phillips, L. Gallmann, M. M. Fejer, and U. Keller, “Role of apodization in optical parametric amplifiers based on aperiodic quasi-phasematching gratings,” Opt. Express20, 18066–18071 (2012).
[CrossRef] [PubMed]

V. J. Hernandez, C. V. Bennett, B. D. Moran, A. D. Drobshoff, D. Chang, C. Langrock, M. M. Fejer, and M. Ibsen, “104 MHz rate single-shot recording with subpicosecond resolution using temporal imaging,” Opt. Express21, 196–203 (2013).
[CrossRef] [PubMed]

Opt. Lett. (19)

T. Fuji, N. Ishii, C. Y. Teisset, X. Gu, T. Metzger, A. Baltuska, N. Forget, D. Kaplan, A. Galvanauskas, and F. Krausz, “Parametric amplification of few-cycle carrier-envelope phase-stable pulses at 2.1 μm,” Opt. Lett.31, 1103–1105 (2006).
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Y. Deng, A. Schwarz, H. Fattahi, M. Ueffing, X. Gu, M. Ossiander, T. Metzger, V. Pervak, H. Ishizuki, T. Taira, T. Kobayashi, G. Marcus, F. Krausz, R. Kienberger, and N. Karpowicz, “Carrier-envelope-phase-stable, 1.2 mJ, 1.5 cycle laserpulses at 2.1 μm,” Opt. Lett.37, 4973–4975 (2012).
[CrossRef] [PubMed]

S. Yang, A. M. Weiner, K. R. Parameswaran, and M. M. Fejer, “Ultrasensitive second-harmonic generation frequency-resolved optical gating by aperiodically poled LiNbO3 waveguides at 1.5 μm,” Opt. Lett.30, 2164–2166 (2005).
[CrossRef] [PubMed]

C. Heese, C. R. Phillips, L. Gallmann, M. M. Fejer, and U. Keller, “Ultrabroadband, highly flexible amplifier for ultrashort midinfrared laser pulses based on aperiodically poled Mg:LiNbO3,” Opt. Lett.35, 2340–2342 (2010).
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M. Conforti, F. Baronio, and C. D. Angelis, “From femtosecond infrared to picosecond visible pulses: temporal shaping with high-efficiency conversion,” Opt. Lett.32, 1779–1781 (2007).
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Ł. Kornaszewski, M. Kohler, U. K. Sapaev, and D. T. Reid, “Designer femtosecond pulse shaping using grating-engineered quasi-phase-matching in lithium niobate,” Opt. Lett.33, 378–380 (2008).
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M. A. Albota and F. C. Wong, “Efficient single-photon counting at 1.55 μm by means of frequency upconversion,” Opt. Lett.29, 1449–1451 (2004).
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J. Huang, X. P. Xie, C. Langrock, R. V. Roussev, D. S. Hum, and M. M. Fejer, “Amplitude modulation and apodization of quasi-phase-matched interactions,” Opt. Lett.31, 604–606 (2006).
[CrossRef] [PubMed]

C. R. Phillips and M. M. Fejer, “Efficiency and phase of optical parametric amplification in chirped quasi-phase-matched gratings,” Opt. Lett.35, 3093–3095 (2010).
[CrossRef] [PubMed]

M. Charbonneau-Lefort, M. M. Fejer, and B. Afeyan, “Tandem chirped quasi-phase-matching grating optical parametric amplifier design for simultaneous group delay and gain control,” Opt. Lett.30, 634–636 (2005).
[CrossRef] [PubMed]

L. Gallmann, G. Steinmeyer, U. Keller, G. Imeshev, M. M. Fejer, and J. Meyn, “Generation of sub-6-fs blue pulses by frequency doubling with quasi-phase-matching gratings,” Opt. Lett.26, 614–616 (2001).
[CrossRef]

G. Imeshev, A. Galvanauskas, D. Harter, M. A. Arbore, M. Proctor, and M. M. Fejer, “Engineerable femtosecond pulse shaping by second-harmonic generation with fourier synthetic quasi-phase-matching gratings,” Opt. Lett.23, 864–866 (1998).
[CrossRef]

J. S. Pelc, Q. Zhang, C. R. Phillips, L. Yu, Y. Yamamoto, and M. M. Fejer, “Cascaded frequency upconversion for high-speed single-photon detection at 1550 nm,” Opt. Lett.37, 476–478 (2012).
[CrossRef] [PubMed]

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[CrossRef] [PubMed]

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[CrossRef] [PubMed]

C. R. Phillips, C. Langrock, J. S. Pelc, M. M. Fejer, J. Jiang, M. E. Fermann, and I. Hartl, “Supercontinuum generation in quasi-phase-matched LiNbO3 waveguide pumped by a Tm-doped fiber laser system,” Opt. Lett.36, 3912–3914 (2011).
[CrossRef] [PubMed]

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[CrossRef]

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[CrossRef] [PubMed]

J. S. Pelc, C. R. Phillips, D. Chang, C. Langrock, and M. M. Fejer, “Efficiency pedestal in quasi-phase-matching devices with random duty-cycle errors,” Opt. Lett.36, 864–866 (2011).
[CrossRef] [PubMed]

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[CrossRef]

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[CrossRef]

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[CrossRef]

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[CrossRef]

Other (6)

G. D. Miller, “Periodically poled lithium niobate: modeling, fabrication, and nonlinear-optical performance,” PhD dissertation, Stanford University, Stanford, CA (1998).

C. R. Phillips, C. Langrock, D. Chang, Y. W. Lin, L. Gallmann, and M. M. Fejer, “Apodization of chirped quasi-phsematching devces,” submitted to J. Opt. Soc. Am. B.

J. E. Schaar, “Terahertz sources based on intracavity parametric frequency down-conversion using quasi-phase-matched gallium arsenide,” PhD dissertation, Stanford University, Stanford, CA (2009).

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C. R. Phillips, J. S. Pelc, and M. M. Fejer, “Parametric processes in quasi-phasematching gratings with random duty cycle errors,” J. Opt. Soc. Am. B (to be published).

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Figures (4)

Fig. 1
Fig. 1

Magnitude of the DFG coupling matrix T for several experimental configurations. (a) Narrow-bandwdith pump, nearly group velocity matched signal and idler waves, (b) narrow-bandwidth pump, group velocity mismatched signal and idler, (c) broadband pump, nearly group velocity matched signal and idler, (d) broadband pump and signal, with group velocity mismatch. |T| is plotted in each case, and the parameters are given in the text.

Fig. 3
Fig. 3

Example solution to the optimization problem given in Eq. (18). The parameters are given in the text. (a) Phase mismatch Δk(ω, ωpω) as a function of idler frequency in THz. Vertical dashed black lines indicate the target passband. (b) Grating profile (amplitude and phase of d(z)). The profile returned by the optimization algorithm exhibits a small number of dips in |d(z)| as shown by the black curve. We remove these dips and assume a constant |d(z)| in calculating the transfer function. (c) and (d): Transfer function amplitude and phase.

Fig. 4
Fig. 4

Nonlinear chirp design example using convex optimization. (a) Optimized chirp rate, normalized to the peak gain coefficient of the pump, γ0. The horizontal line indicates the constant chirp rate needed for Eq. (26) to yield the target gain spectrum if γ(Ω) = γ0. (b) Simulated output spectrum. The spectrum with a nominally linear, apodized chirp profile is shown for comparison.

Equations (49)

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d ¯ ( z ) = sgn [ cos ( ϕ ( z ) ) cos ( π D ( z ) ) ]
= ( 2 D ( z ) 1 ) + m = m 0 2 sin ( π m D ( z ) ) π m exp ( i m ϕ ( z ) )
m = d ¯ m ( z ) ,
d ¯ 1 ( z ) = 2 sin ( π D ( z ) ) π m exp ( i ϕ ( z ) ) .
ϕ ( z ) = 0 z K g ( z ) d z .
d A ˜ i ( ω ) d z = i d ¯ ( z ) κ i ( ω ) 0 A ˜ s * ( ω ) A ˜ p ( ω + ω ) exp ( i Δ k ( ω , ω ) z ) d ω 2 π .
Δ k ( ω , ω ) = k p ( ω + ω ) k s ( ω ) k i ( ω ) ,
E ˜ ( z , ω ) u ( ω ) = 1 2 j A ˜ j ( z , ω ) e i k j ( ω ) z
= 1 2 j B ˜ j ( z , ω ) ,
A ˜ i ( L , ω ) = i κ i ( ω ) 0 g ( Δ k ( ω , ω ) ) A ˜ s * ( ω ) A ˜ p ( ω + ω ) d ω 2 π ,
g ( k ) = d ¯ ( z ) e i k z d z .
B ˜ i ( L , ω ) = i κ i ( ω ) e i k i ( ω ) L 0 g ( Δ k ( ω , ω ) ) A ˜ s * ( ω ) A ˜ p ( ω + ω ) d ω 2 π ,
g ( k ) = v ( k ) T g ,
B ˜ i ( L , ω n ) = [ i κ i ( ω n ) e k i ( ω n ) L 0 v ( Δ k ( ω n , ω ) ) T A ˜ s * ( ω ) A ˜ p ( ω n + ω ) d ω 2 π ] g ,
B ω = Tg ,
B i ( L , ω ) A s * ( ω p ω ) = i κ i ( ω ) e i k i ( ω ) L A p 0 g ( Δ k ( ω , ω p ω ) ) , H ( ω )
T n m = i κ i ( ω n ) e i k i ( ω ) L A s * ( ω p ω n ) A p 0 v m ( Δ k ( ω n , ω p ω n ) ) .
g = F k d
B t = F t B ω
d n f d z n D n f
minimize D 2 d 2
subject to : B t B target 2 B target 2 ε
| d | 2 / π .
τ i ( ω ) = τ s ( ω p ω ) + z p m ( ω ) v g , s ( ω p ω ) + L z p m ( ω ) v g , i ( ω )
minimize : max ( | d | )
subject to : H [ P B ] H T [ P B ] 2 H T 2 ε 1
| D 1 d | λ D
D 2 ( H [ P B ] H T [ P B ] max ( H T ) ) 1 ε 2 τ GVM δ ω
d A ˜ i ( ω ) d z = i d ¯ ( z ) κ i ( ω ) 0 A ˜ s * ( ω ) A ˜ p ( ω + ω ) exp ( i Δ k ( ω , ω ) z ) d ω 2 π .
d A ˜ s ( ω ) d z = i d ¯ ( z ) κ s ( ω ) 0 A ˜ i * ( ω ) A ˜ p ( ω + ω ) exp ( i Δ k ( ω , ω ) z ) d ω 2 π .
τ i ( ω p ω ) τ s ( ω )
d A ˜ i ( ω ) d z = i d ¯ ( z ) κ i ( ω ) A p ( τ i ( ω ) ) A ˜ s * ( ω p ω ) e i Δ k ( ω , ω p ω ) z
d A ˜ s ( ω ) d z = i d ¯ ( z ) κ s ( ω ) A p ( τ s ( ω ) ) A ˜ i * ( ω p ω ) e i Δ k ( ω p ω , ω ) z
d a ˜ i ( Ω ) * d z = + i d 1 * ¯ ( z ) κ i ( ω i Ω ) A p * ( τ s ( ω s + Ω ) ) a ˜ s ( Ω ) e i Δ k ( Ω ) z
d a ˜ s ( Ω ) d z = i d ¯ 1 ( z ) κ s ( ω s + Ω ) A p ( τ s ( ω s + Ω ) ) a ˜ i * ( Ω ) e i Δ k ( Ω ) z
Δ k ( Ω ) = k p ( ω p ) k s ( ω s + Ω ) k i ( ω i Ω ) .
G s ( Ω ) exp [ 2 z t p , 1 ( Ω ) z t p , 2 ( Ω ) γ ( Ω ) 2 ( Δ k ( Ω ) K g ( z ) 2 ) 2 d z ] ,
γ ( Ω ) 2 = ( 2 π ) 2 ( ω i Ω ) ( ω s + Ω ) d 0 2 n i ( ω i Ω ) n s ( ω s + Ω ) c 2 2 n p ( ω p ) ε 0 c I p ( τ s ( ω s + Ω ) ) ,
ln ( G s ( Ω ) ) = 2 π Λ ( Ω ) ,
ln ( G s ( Ω ) ) K i K f Γ ( Ω , K ) z K ( K ) d K
Γ ( Ω , K ) = Re [ 2 γ ( Ω ) 2 ( Δ k ( Ω ) K 2 ) 2 ] .
max ( [ K i , K f ] ) = K max + 2 γ 0 min ( [ K i , K f ] ) = K min 2 γ 0
L = Λ 0 [ K max K min γ 0 2 + 4 γ 0 ]
z ( K g ) = K i K g z K ( K ) d K ,
ln ( G s ) = Γ z K .
minimize D 1 z K 2 2
subject to : δ G Γ z K ln ( G T ) δ G
L L max
Λ min γ 0 2 z K Λ max ,

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