Abstract

Chirped quasi-phasematching (QPM) optical devices offer the potential for ultrawide bandwidths, high conversion efficiencies, and high amplification factors across the transparency range of QPM media. In order to properly take advantage of these devices, apodization schemes are required. We study apodization in detail for many regimes of interest, including low-gain difference frequency generation (DFG), high-gain optical parametric amplification (OPA), and high-efficiency adiabatic frequency conversion (AFC). Our analysis is also applicable to second-harmonic generation, sum frequency generation, and optical rectification. In each case, a systematic and optimized approach to grating construction is provided, and different apodization techniques are compared where appropriate. We find that nonlinear chirp apodization, where the poling period is varied smoothly, monotonically, and rapidly at the edges of the device, offers the best performance. We consider the full spatial structure of the QPM gratings in our simulations, but utilize the first order QPM approximation to obtain analytical and semi-analytical results. One application of our results is optical parametric chirped pulse amplification; we show that special care must be taken in this case to obtain high gain factors while maintaining a flat gain spectrum.

© 2013 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  43. 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]
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2013 (3)

2012 (6)

2011 (5)

2010 (4)

2009 (2)

H. Steigerwald, F. Luedtke, and K. Buse, “Ultraviolet light assisted periodic poling of near-stoichiometric, magnesium-doped lithium niobate crystals,” Appl. Phys. Lett. 94, 032906 (2009).
[CrossRef]

H. Suchowski, V. Prabhudesai, D. Oron, A. Arie, and Y. Silberberg, “Robust adiabatic sum frequency conversion,” Opt. Express 17, 12731–12740 (2009).
[CrossRef]

2008 (3)

2007 (3)

2006 (1)

2005 (2)

2003 (1)

2002 (3)

2001 (2)

2000 (2)

1996 (2)

M. Taya, M. C. Bashaw, and M. M. Fejer, “Photorefractive effects in periodically poled ferroelectrics,” Opt. Lett. 21, 857–859 (1996).
[CrossRef]

G. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse compression and solitons,” Opt. Quantum Electron. 28, 1691–1740 (1996).
[CrossRef]

1973 (1)

M. D. Crisp, “Adiabatic-Following approximation,” Phys. Rev. A 8, 2128–2135 (1973).
[CrossRef]

1962 (1)

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Adler, F.

Afeyan, B.

Alber, M. S.

Arbore, M. A.

Arie, A.

Armstrong, J. A.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Artigas, D.

Ashihara, S.

Asobe, M.

Baronio, F.

M. Conforti, F. Baronio, and C. De Angelis, “Nonlinear envelope equation for broadband optical pulses in quadratic media,” Phys. Rev. A 81, 053841 (2010).
[CrossRef]

Bashaw, M. C.

Bender, C. M.

C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory, Vol. 1 (Springer, 1999).

Bennett, C. V.

Bloembergen, N.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Bostani, A.

Boyd, R. W.

R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic, 2008).

Bramati, A.

Buryak, A. V.

A. V. Buryak, P. D. Trapani, D. V. Skryabin, and S. Trillo, “Optical solitons due to quadratic nonlinearities: from basic physics to futuristic applications,” Phys. Rep. 370, 63–235 (2002).
[CrossRef]

Buse, K.

J. R. Schwesyg, M. Falk, C. R. Phillips, D. H. Jundt, K. Buse, and M. M. Fejer, “Pyroelectrically induced photorefractive damage in magnesium-doped lithium niobate crystals,” J. Opt. Soc. Am. B 28, 1973–1987 (2011).
[CrossRef]

H. Steigerwald, F. Luedtke, and K. Buse, “Ultraviolet light assisted periodic poling of near-stoichiometric, magnesium-doped lithium niobate crystals,” Appl. Phys. Lett. 94, 032906 (2009).
[CrossRef]

Chang, D.

Charbonneau-Lefort, M.

Chinaglia, W.

Conforti, M.

M. Conforti, F. Baronio, and C. De Angelis, “Nonlinear envelope equation for broadband optical pulses in quadratic media,” Phys. Rev. A 81, 053841 (2010).
[CrossRef]

Conti, C.

Crisp, M. D.

M. D. Crisp, “Adiabatic-Following approximation,” Phys. Rev. A 8, 2128–2135 (1973).
[CrossRef]

De Angelis, C.

M. Conforti, F. Baronio, and C. De Angelis, “Nonlinear envelope equation for broadband optical pulses in quadratic media,” Phys. Rev. A 81, 053841 (2010).
[CrossRef]

Deng, Y.

Di Trapani, P.

Diddams, S. A.

Drobshoff, A. D.

Ducuing, J.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Falk, M.

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. Express 21, 196–203 (2013).
[CrossRef]

C. R. Phillips, L. Gallmann, and M. M. Fejer, “Design of quasi-phasematching gratings via convex optimization,” Opt. Express 21, 10139–10159 (2013).
[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 30, 982–993 (2013).
[CrossRef]

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. Express 20, 18066–18071 (2012).
[CrossRef]

C. R. Phillips and M. M. Fejer, “Adiabatic optical parametric oscillators: steady-state and dynamical behavior,” Opt. Express 20, 2466–2482 (2012).
[CrossRef]

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. Express 20, 26888–26894 (2012).
[CrossRef]

C. R. Phillips, J. S. Pelc, and M. M. Fejer, “Continuous wave monolithic quasi-phase-matched optical parametric oscillator in periodically poled lithium niobate,” Opt. Lett. 36, 2973–2975 (2011).
[CrossRef]

C. R. Phillips, C. Langrock, J. S. Pelc, M. M. Fejer, I. Hartl, and M. E. Fermann, “Supercontinuum generation in quasi-phasematched waveguides,” Opt. Express 19, 18754–18773 (2011).
[CrossRef]

J. R. Schwesyg, M. Falk, C. R. Phillips, D. H. Jundt, K. Buse, and M. M. Fejer, “Pyroelectrically induced photorefractive damage in magnesium-doped lithium niobate crystals,” J. Opt. Soc. Am. B 28, 1973–1987 (2011).
[CrossRef]

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]

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]

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]

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]

C. R. Phillips and M. M. Fejer, “Stability of the singly resonant optical parametric oscillator,” J. Opt. Soc. Am. B 27, 2687–2699 (2010).
[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. B 25, 463–480 (2008).
[CrossRef]

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. B 25, 1402–1413 (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]

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]

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]

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. B 18, 534–539 (2001).
[CrossRef]

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. 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. B 17, 304–318 (2000).
[CrossRef]

M. Taya, M. C. Bashaw, and M. M. Fejer, “Photorefractive effects in periodically poled ferroelectrics,” Opt. Lett. 21, 857–859 (1996).
[CrossRef]

Fermann, M.

Fermann, M. E.

Gallmann, L.

Galvanauskas, A.

Goldman, R.

R. Goldman, “Curvature formulas for implicit curves and surfaces,” Comput. Aided Geom. Des. 22, 632–658 (2005).
[CrossRef]

Gu, X.

Hagan, D. J.

G. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse compression and solitons,” Opt. Quantum Electron. 28, 1691–1740 (1996).
[CrossRef]

Harter, D.

Hartl, I.

Heese, C.

Hellström, J.

Hernandez, V. J.

Huang, J.

Hum, D. S.

Ibsen, M.

Imeshev, G.

Ishizuki, H.

Jiang, J.

Jundt, D. H.

Karpowicz, N.

Kashyap, R.

Keller, U.

Kienberger, R.

Kilius, J.

Kobayashi, T.

Krausz, F.

Kuroda, K.

Langrock, C.

Laurell, F.

Luedtke, F.

H. Steigerwald, F. Luedtke, and K. Buse, “Ultraviolet light assisted periodic poling of near-stoichiometric, magnesium-doped lithium niobate crystals,” Appl. Phys. Lett. 94, 032906 (2009).
[CrossRef]

Luther, G. G.

Magari, K.

Marcus, G.

Marsden, J. E.

Mayer, B. W.

Metzger, T.

Meyn, J.

Miller, G. D.

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

Minardi, S.

Moran, B. D.

Neely, T. W.

Nishida, Y.

Nishina, J.

Nugent-Glandorf, L.

Oron, D.

Orszag, S. A.

C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory, Vol. 1 (Springer, 1999).

Ossiander, M.

Pasiskevicius, V.

Pelc, J. S.

Pershan, P. S.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Pervak, V.

Phillips, C. R.

C. R. Phillips, L. Gallmann, and M. M. Fejer, “Design of quasi-phasematching gratings via convex optimization,” Opt. Express 21, 10139–10159 (2013).
[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 30, 982–993 (2013).
[CrossRef]

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. Express 20, 18066–18071 (2012).
[CrossRef]

C. R. Phillips and M. M. Fejer, “Adiabatic optical parametric oscillators: steady-state and dynamical behavior,” Opt. Express 20, 2466–2482 (2012).
[CrossRef]

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. Express 20, 26888–26894 (2012).
[CrossRef]

C. R. Phillips, J. S. Pelc, and M. M. Fejer, “Continuous wave monolithic quasi-phase-matched optical parametric oscillator in periodically poled lithium niobate,” Opt. Lett. 36, 2973–2975 (2011).
[CrossRef]

C. R. Phillips, C. Langrock, J. S. Pelc, M. M. Fejer, I. Hartl, and M. E. Fermann, “Supercontinuum generation in quasi-phasematched waveguides,” Opt. Express 19, 18754–18773 (2011).
[CrossRef]

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]

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]

J. R. Schwesyg, M. Falk, C. R. Phillips, D. H. Jundt, K. Buse, and M. M. Fejer, “Pyroelectrically induced photorefractive damage in magnesium-doped lithium niobate crystals,” J. Opt. Soc. Am. B 28, 1973–1987 (2011).
[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]

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]

C. R. Phillips and M. M. Fejer, “Stability of the singly resonant optical parametric oscillator,” J. Opt. Soc. Am. B 27, 2687–2699 (2010).
[CrossRef]

Prabhudesai, V.

Reid, D. T.

Robbins, J. M.

Roussev, R. V.

Schwarz, A.

Schwesyg, J. R.

Shimura, T.

Silberberg, Y.

Skryabin, D. V.

A. V. Buryak, P. D. Trapani, D. V. Skryabin, and S. Trillo, “Optical solitons due to quadratic nonlinearities: from basic physics to futuristic applications,” Phys. Rep. 370, 63–235 (2002).
[CrossRef]

Stegeman, G. I.

G. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse compression and solitons,” Opt. Quantum Electron. 28, 1691–1740 (1996).
[CrossRef]

Steigerwald, H.

H. Steigerwald, F. Luedtke, and K. Buse, “Ultraviolet light assisted periodic poling of near-stoichiometric, magnesium-doped lithium niobate crystals,” Appl. Phys. Lett. 94, 032906 (2009).
[CrossRef]

Steinmeyer, G.

Suchowski, H.

Suzuki, H.

Tadanaga, O.

Taira, T.

Taya, M.

Tehranchi, A.

Tillman, K. A.

Torner, L.

G. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse compression and solitons,” Opt. Quantum Electron. 28, 1691–1740 (1996).
[CrossRef]

Trapani, P. D.

A. V. Buryak, P. D. Trapani, D. V. Skryabin, and S. Trillo, “Optical solitons due to quadratic nonlinearities: from basic physics to futuristic applications,” Phys. Rep. 370, 63–235 (2002).
[CrossRef]

Trillo, S.

C. Conti, S. Trillo, P. Di Trapani, J. Kilius, A. Bramati, S. Minardi, W. Chinaglia, and G. Valiulis, “Effective lensing effects in parametric frequency conversion,” J. Opt. Soc. Am. B 19, 852–857 (2002).
[CrossRef]

A. V. Buryak, P. D. Trapani, D. V. Skryabin, and S. Trillo, “Optical solitons due to quadratic nonlinearities: from basic physics to futuristic applications,” Phys. Rep. 370, 63–235 (2002).
[CrossRef]

Ueffing, M.

Umeki, T.

Valiulis, G.

White, R. B.

R. B. White, Asymptotic Analysis of Differential Equations (World Scientific, 2005).

Xie, X. P.

Yanagawa, T.

Appl. Phys. Lett. (1)

H. Steigerwald, F. Luedtke, and K. Buse, “Ultraviolet light assisted periodic poling of near-stoichiometric, magnesium-doped lithium niobate crystals,” Appl. Phys. Lett. 94, 032906 (2009).
[CrossRef]

Comput. Aided Geom. Des. (1)

R. Goldman, “Curvature formulas for implicit curves and surfaces,” Comput. Aided Geom. Des. 22, 632–658 (2005).
[CrossRef]

J. Lightwave Technol. (1)

J. Opt. Soc. Am. B (11)

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. B 25, 1402–1413 (2008).
[CrossRef]

C. R. Phillips and M. M. Fejer, “Stability of the singly resonant optical parametric oscillator,” J. Opt. Soc. Am. B 27, 2687–2699 (2010).
[CrossRef]

J. R. Schwesyg, M. Falk, C. R. Phillips, D. H. Jundt, K. Buse, and M. M. Fejer, “Pyroelectrically induced photorefractive damage in magnesium-doped lithium niobate crystals,” J. Opt. Soc. Am. B 28, 1973–1987 (2011).
[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. B 17, 304–318 (2000).
[CrossRef]

G. G. Luther, M. S. Alber, J. E. Marsden, and J. M. Robbins, “Geometric analysis of optical frequency conversion and its control in quadratic nonlinear media,” J. Opt. Soc. Am. B 17, 932–941 (2000).
[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. B 18, 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. B 25, 463–480 (2008).
[CrossRef]

C. Conti, S. Trillo, P. Di Trapani, J. Kilius, A. Bramati, S. Minardi, W. Chinaglia, and G. Valiulis, “Effective lensing effects in parametric frequency conversion,” J. Opt. Soc. Am. B 19, 852–857 (2002).
[CrossRef]

S. Ashihara, J. Nishina, T. Shimura, and K. Kuroda, “Soliton compression of femtosecond pulses in quadratic media,” J. Opt. Soc. Am. B 19, 2505–2510 (2002).
[CrossRef]

A. Bostani, A. Tehranchi, and R. Kashyap, “Engineering of effective second-order nonlinearity in uniform and chirped gratings,” J. Opt. Soc. Am. B 29, 2929–2934 (2012).
[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 30, 982–993 (2013).
[CrossRef]

Opt. Express (7)

Opt. Lett. (15)

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]

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]

C. R. Phillips, J. S. Pelc, and M. M. Fejer, “Continuous wave monolithic quasi-phase-matched optical parametric oscillator in periodically poled lithium niobate,” Opt. Lett. 36, 2973–2975 (2011).
[CrossRef]

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]

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]

M. Taya, M. C. Bashaw, and M. M. Fejer, “Photorefractive effects in periodically poled ferroelectrics,” Opt. Lett. 21, 857–859 (1996).
[CrossRef]

K. A. Tillman, D. T. Reid, D. Artigas, J. Hellström, V. Pasiskevicius, and F. Laurell, “Low-threshold femtosecond optical parametric oscillator based on chirped-pulse frequency conversion,” Opt. Lett. 28, 543–545 (2003).
[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]

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]

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, 1129–1131 (2007).
[CrossRef]

K. A. Tillman and D. T. Reid, “Monolithic optical parametric oscillator using chirped quasi-phase matching,” Opt. Lett. 32, 1548–1550 (2007).
[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]

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]

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]

T. W. Neely, L. Nugent-Glandorf, F. Adler, and S. A. Diddams, “Broadband mid-infrared frequency upconversion and spectroscopy with an aperiodically poled LiNbO3 waveguide,” Opt. Lett. 37, 4332–4334 (2012).
[CrossRef]

Opt. Quantum Electron. (1)

G. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse compression and solitons,” Opt. Quantum Electron. 28, 1691–1740 (1996).
[CrossRef]

Phys. Rep. (1)

A. V. Buryak, P. D. Trapani, D. V. Skryabin, and S. Trillo, “Optical solitons due to quadratic nonlinearities: from basic physics to futuristic applications,” Phys. Rep. 370, 63–235 (2002).
[CrossRef]

Phys. Rev. (1)

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Phys. Rev. A (2)

M. D. Crisp, “Adiabatic-Following approximation,” Phys. Rev. A 8, 2128–2135 (1973).
[CrossRef]

M. Conforti, F. Baronio, and C. De Angelis, “Nonlinear envelope equation for broadband optical pulses in quadratic media,” Phys. Rev. A 81, 053841 (2010).
[CrossRef]

Other (4)

R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic, 2008).

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

C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory, Vol. 1 (Springer, 1999).

R. B. White, Asymptotic Analysis of Differential Equations (World Scientific, 2005).

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Figures (8)

Fig. 1.
Fig. 1.

Evolution of the idler a i in a linearly chirped QPM grating, for a DFG interaction. The solid (blue) line shows the numerical solution of Eq. (10a), corresponding to (and indistinguishable from) the analytical solution given by Eq. (15). The dashed (black) line shows the asymptotic solution corresponding to Eq. (18). A normalized propagation coordinate, given by ξ = | Δ k | ( z z pm ) , is used. The idler in each curve is normalized to | a i ( 0 ) | 2 [defined by Eqs. (16) and (18)]. The dashed (black) curve shows [ a i ( eig ) ( z ) a i ( eig ) ( z i ) ] for ξ < 0.5 , and [ a i ( eig ) ( z ) a i ( eig ) ( z i ) + a i ( 0 ) exp ( i φ 1 ( z pm ) ) ] for ξ > 0.5 . The input, phasematching, and output points correspond to ξ = 15 , ξ = 0 , and ξ = 15 , respectively. Because of the way the figure has been normalized, there are no other free parameters in the calculation.

Fig. 2.
Fig. 2.

Apodization examples for DFG, calculated numerically via Eq. (1a), using the full d ¯ = ± 1 grating structure. (a) Nonlinear chirp apodization (NLCA), (b) duty cycle apodization (DCA), (c) deleted domain apodization (DDA) with a domain profile determined from the DCA example in (b). The parameters are given in the text.

Fig. 3.
Fig. 3.

Propagation example for OPA in a linearly chirped QPM grating, showing the signal and idler as a function of normalized position. The gain factor λ R = 2.2 . The position has been normalized to the dephasing length, L deph = 2 γ / | Δ k | . The normalized grating length is given by L / L deph = 8 . The phasematching point z pm is located at the middle of the grating. The dashed lines show the evolution of the signal and idler in a apodized grating with the same parameters; oscillations in a i ( z ) and a s ( z ) near z = z i and z = z f are suppressed in this case. For this example, we apodize via the chirp rate and duty cycle simultaneously to reveal the idler evolution under idealized input conditions; (b) shows the fields on a linear scale to better indicate how the oscillations are suppressed near the output of the grating in the apodized case, but not the unapodized case; the signal and idler magnitudes are indistinguishable on this linear scale due to the high gain involved.

Fig. 4.
Fig. 4.

Chirped-QPM OPA apodization examples. (a) Grating k -vector profile for a NLCA example, (b) normalized grating chirp rate corresponding to (a), and (c) signal gain spectrum. The dashed lines in (b) show min ( ( Δ k 1 / γ 0 ) 2 ) ϵ , with minimization performed with respect to signal wavelength (restricted to the target gain bandwidth that spans the 1450–1650 nm range). This figure indicates that the optimal normalized chirp rate in the apodization region approximately satisfies | Δ k | = min ( Δ k 1 2 ) ϵ . The dashed lines in (c) indicate analytical estimates of the fluctuations in the gain due to the finite value of | γ 0 / Δ k 1 | at the ends of the grating, as described in the text. The gain spectra for DCA and corresponding DDA examples are also shown in (c), for comparison.

Fig. 5.
Fig. 5.

(a) Solution to an example TWM problem, visualized with the geometric description of [28] [and Eq. (46) in particular]. The parameters for this example are K ip = | a p ( z i ) | 2 = 0.9 , g = 1 (50% duty cycle), and Δ k 1 / γ = 1 . The surface shown is φ = 0 , and the curve (blue) represents the trajectory of W . Since Δ k 1 and g are both constant in this example, the curve lies on a plane H = constant . W is initially at the top of the surface ( Z | a p | 2 = K ip = 0.9 ), and the direction of W (with increasing z ) is shown by the blue arrow on the curve. The direction of H is also shown (black arrow). The point where this arrow touches the surface is a nonlinear eigenmode associated with the chosen parameters (i.e., a point where H is in the direction of the surface normal to φ = 0 , φ ). In this unchirped example, the field vector W orbits around the fixed eigenmode W m . In (b), the photon fluxes | a j | 2 are shown for comparison. One period of these fluxes corresponds to a complete traversal of the blue curve in (a); position is normalized to γ , defined in Eq. (11).

Fig. 6.
Fig. 6.

Propagation of field vector W and eigenmode vector W m [the solid and dashed lines, respectively, in (b), (c), (e), and (f)], illustrating the adiabatic following process arising from Eq. (46). The left column is for K ip | a p ( z i ) | 2 = 10 3 (strong input signal), while the right column is for K ip = 0.9 (strong input pump). The first row [(a) and (d)] are cuts through the surface φ = 0 in the plane Y = 0 , corresponding to these two input conditions. The second and third rows show the evolution of the field vectors (versus coordinate ζ = γ z ) under these input conditions, for two different coupling factors: λ R , s = ( 1 , 2 ) for (b) and (c), respectively, and λ R , p = ( 2 , 10 ) for (e) and (f), respectively. In (b), (c), (e) and (f), the x curve is on the bottom, y in the middle, and z on the top.

Fig. 7.
Fig. 7.

q ( ν ) 1 [ q defined in Eq. (56), ν defined in Eq. (33)], shown for several values of ρ , assuming a positively chirped grating. For small ρ , the maximum of q 1 occurs near the first turning point ( | ν | = 1 ) and diverges as ρ 0 .

Fig. 8.
Fig. 8.

AFC apodization example, with ρ = 0.2 and λ R = 5 . The remaining parameters are given in the text. (a) Pump depletion versus signal wavelength for apodized and unapodized cases, (b) grating profile K g ( z ) .

Equations (73)

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d A i d z = i ω i d 0 n i c d ¯ ( z ) A s * A p e i Δ k 0 z ,
d A s d z = i ω s d 0 n s c d ¯ ( z ) A i * A p e i Δ k 0 z ,
d A p d z = i ω p d 0 n p c d ¯ ( z ) A i A s e + i Δ k 0 z ,
E ( z , t ) = 1 2 j A j ( z ) exp [ i ( ω j t k j z ) ] + c.c. ,
d ¯ ( z ) d ( z ) d 0 = sgn [ cos ( ϕ G ( z ) ) cos ( π D ( z ) ) ]
= ( 2 D ( z ) 1 ) + m = m 0 2 sin ( π m D ( z ) ) π m exp ( i m ϕ G ( z ) ) ,
ϕ G ( z ) = ϕ G ( z i ) + z i z K g ( z ) d z ,
d A i d z = i κ i ( z ) A s * A p e i ϕ 1 ( z ) ,
d A s d z = i κ s ( z ) A i * A p e i ϕ 1 ( z ) ,
d A p d z = i κ p ( z ) A i A s e + i ϕ 1 ( z ) .
κ j ( z ) = ω j d 0 n j c 2 π sin ( π D ( z ) ) ,
ϕ 1 ( z ) = ϕ 1 ( z i ) + z i z Δ k 1 ( z ) d z ,
Δ k m ( z ) = k p k s k i m K g ( z ) ,
A j = ω j n j n n n ω n | A n 0 | 2 a j .
d a i d z = i g ( z ) γ a s * a p e i ϕ 1 ( z ) ,
d a s d z = i g ( z ) γ a i * a p e i ϕ 1 ( z ) ,
d a p d z = i g ( z ) γ a i a s e + i ϕ 1 ( z ) ,
γ = ω i ω s ω p n i n s n p j n j ω j | A j 0 | 2 2 d 0 π c .
j | a j ( z i ) | 2 = 1 .
A i ( z f ) = i ω i d 0 n i c A s * A p F [ d ¯ ( z ) ] ( Δ k 0 ) ,
ϕ 1 ( z ) = ϕ 1 ( z i ) + Δ k 2 [ ( z z pm ) 2 ( z i z pm ) 2 ] ,
a i ( z ) = 1 2 a s * a p e i π / 4 e i Δ k ( z p m z i ) 2 / 2 2 π γ 2 g 0 2 Δ k × [ erf ( 2 e i π / 4 Δ k ( z pm z i ) ) + erf ( 2 e i π / 4 Δ k ( z z pm ) ) ] ,
a i ( L ) = { i 2 π γ 2 g 0 2 | Δ k | e i π sgn ( Δ k ) / 4 e [ i Δ k 2 ( z pm z i ) 2 ] + γ g 0 Δ k 1 ( z f ) e [ i Δ k 2 ( ( z pm z i ) 2 ( z f z pm ) 2 ) ] γ g 0 Δ k 1 ( z i ) } a s * a p ,
a i ( eig ) ( z ) = γ g ( z ) Δ k 1 ( z ) a s * a p e i ϕ 1 ( z ) .
a i ( L ) ( z f ) = a i ( 0 ) e i ϕ 1 ( z pm ) + a i ( eig ) ( z f ) a i ( eig ) ( z i ) ,
| ( γ g Δ k 1 ) 1 d d z ( γ g Δ k 1 ) | | d d z ( e i ϕ 1 ) | .
max ω { | 1 g d g d z 1 Δ k 1 d Δ k 1 d z | ϵ | Δ k 1 | } 0 ,
s c d K g d z = ϵ min ω { [ Δ k 0 ( ω ) K g ( z ) ] 2 } ,
s c d K g d z = { ϵ [ K K g ] 2 if Δ k ( z , ω ) > 0 , ϵ [ K + K g ] 2 if Δ k ( z , ω ) < 0 ,
1 K g ( z ) K ± 1 K b , ± K ± = ϵ s c ( z z b , ± ) ,
K nom , z ( z b , ± ) = ϵ s c [ K ± K b , ± ] 2 ,
K b , ± = K nom ( z b , ± ) .
K g ( z ) = { K + [ ϵ s c ( z z b , ) + ( K b , K ) 1 ] 1 for K g ( z ) < K b , . K nom ( z ) for K b , K g ( z ) K b , + . K + + [ ϵ s c ( z z b , + ) + ( K b , + K + ) 1 ] 1 for K g ( z ) > K b , + .
s g 1 g d g d z = min ω { s g 1 Δ k 0 ( ω ) K g ( z ) d K g d z × [ 1 1 + ϵ 2 ( Δ k 0 K g ( z ) ) 4 ( d K g / d z ) 2 ] } ,
d ¯ ( z ) = 1 n 2 g n Π l n / 2 ( z z n ) ,
n g n z i z 2 Π l n / 2 ( z z n ) d z z i z g ( z ) d z .
d 2 b s d z 2 + Q ( z ) b s ,
Q = ( γ g ) 2 + 1 2 d d z ( g g ) i Δ k 2 1 4 ( g g i Δ k ) 2 ,
ln ( G s ( ω ) ) 2 z i z f Re [ g ( z ) γ ( ω ) ] 2 [ Δ k 0 ( ω ) K g ( z ) 2 ] 2 d z ,
{ ω : ( Δ k 0 ( ω ) K g 0 ) [ | Δ k L | 2 + 2 γ ( ω ) , + | Δ k L | 2 2 γ ( ω ) ] } ,
ν ( z ) Δ k 1 ( z ) 2 γ g ( z ) .
| d Q d z 1 Q 3 / 2 | ϵ ,
( d K z d K ) 2 min Δ k 0 { 4 ϵ 2 K z 2 | ( Δ k 2 ) 2 γ 0 2 + i K z 2 | 3 Δ k 2 } min Δ k 0 { f ( Δ k 0 , K , K z ) } ,
min Δ k 0 { f ( Δ k 0 , K apod , j , K z , nom ) } = ( d K z , nom d K ) 2 ,
min Δ k 0 [ | Δ k 0 K apod , j | ] 2 γ 0 ,
dom ( K nom ) = { K : min j ( K apod , j ) K max j ( K apod , j ) } .
K ( z ) = K nom ( z ) , K dom ( K nom ) ,
d K z d K = s K z min Δ k 0 ( f ) , K dom ( K nom ) ,
K z ( K apod , j ) = K z , nom ( K apod , j ) .
z ( K ) = K i K f K z ( K ) 1 d K ,
| K g ( z f ) K g ( z i ) 2 γ | δ ,
X + i Y = a i a s a p * e i ϕ 1 ( z ) Z = | a p | 2 ,
W = [ X , Y , Z ] T .
φ = X 2 + Y 2 Z ( Z K ip ) ( Z K sp ) ,
K j p = | a j | 2 + | a p | 2 ,
d W d ( γ z ) = H × φ ,
H = g X + Δ k 2 γ ( Z ( K ip + K sp ) ) .
φ = 0 ,
φ Z = ν φ X ,
Y = 0 ,
d Z d z d Z m d z .
tan ( θ ) = φ / Z φ / X | W = W m = ν ,
Y = R ( W m ) sin ( θ ) ,
k = 2 | φ | G ¯ ( Z m ) + ν 2 1 + ν 2 ,
k = 2 | φ | ,
G ¯ ( Z ) = 1 + K ip 3 Z .
d Z m d ν = sgn ( d ν d z ) | φ | 2 ( G ¯ ( Z m ) + ν 2 ) ( 1 + ν 2 ) 1 / 2 ,
| d d ( γ z ) ( Δ k 2 γ g ) | 2 g q ( ν ) ,
q ( ν ) = ( 1 + ν 2 ) 1 / 2 ( G ¯ ( Z m ( ν ) ) + ν 2 ) .
| d θ d Ξ | k k .
| d K g d z | = 2 ϵ γ 2 min ω [ q ( ν ( z , ω ) ) ] ,
ν ± ( z ) = Δ k 0 ( ω ± ) K g ( z ) 2 γ g ( z ) ,
| d ν ± d z | = ϵ γ q ( ν ± ) ,

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