Abstract

The influence of parasitic processes on the performance of ultra-broadband noncollinear optical parametric amplifiers (NOPA’s) is investigated for walk-off and non-walk-off compensating configurations. Experimental results with a white-light–seeded NOPA agree well with numerical simulations. The same model shows that 10% of the output energy of an amplified signal can be transferred into a parasitic second harmonic of the signal. These findings are supported by quantitative measurements on a few-cycle NOPA, where a few percent of the signal energy is converted to its second harmonic in the walk-off compensating case. This effect is reduced by an order of magnitude in the non-walk-off compensating configuration. A detailed study of the phase-matching conditions of the most common nonlinear crystals provides guidelines for designing NOPA systems.

© 2011 OSA

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References

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  1. C. Schriever, S. Lochbrunner, P. Krok, and E. Riedle, “Tunable pulses from below 300 to 970 nm with durations down to 14 fs based on a 2 MHz ytterbium-doped fiber system,” Opt. Lett. 33(2), 192–194 (2008).
    [CrossRef] [PubMed]
  2. J. Piel, E. Riedle, L. Gundlach, R. Ernstorfer, and R. Eichberger, “Sub-20 fs visible pulses with 750 nJ energy from a 100 kHz noncollinear optical parametric amplifier,” Opt. Lett. 31(9), 1289–1291 (2006).
    [CrossRef] [PubMed]
  3. G. Cerullo and S. De Silvestri, “Ultrafast optical parametric amplifiers,” Rev. Sci. Instrum. 74(1), 1–18 (2003).
    [CrossRef]
  4. A. Steinmann, A. Killi, G. Palmer, T. Binhammer, and U. Morgner, “Generation of few-cycle pulses directly from a MHz-NOPA,” Opt. Express 14(22), 10627–10630 (2006).
    [CrossRef] [PubMed]
  5. S. Hädrich, S. Demmler, J. Rothhardt, C. Jocher, J. Limpert, and A. Tünnermann, “High-repetition-rate sub-5-fs pulses with 12 GW peak power from fiber-amplifier-pumped optical parametric chirped-pulse amplification,” Opt. Lett. 36(3), 313–315 (2011).
    [CrossRef] [PubMed]
  6. J. Rothhardt, S. Hädrich, D. N. Schimpf, J. Limpert, and A. Tünnermann, “High repetition rate fiber amplifier pumped sub-20 fs optical parametric amplifier,” Opt. Express 15(25), 16729–16736 (2007).
    [CrossRef] [PubMed]
  7. J. Rothhardt, S. Hädrich, F. Röser, J. Limpert, and A. Tünnermann, “500 MW peak power degenerated optical parametric amplifier delivering 52 fs pulses at 97 kHz repetition rate,” Opt. Express 16(12), 8981–8988 (2008).
    [CrossRef] [PubMed]
  8. J. Rothhardt, S. Hädrich, E. Seise, M. Krebs, F. Tavella, A. Willner, S. Düsterer, H. Schlarb, J. Feldhaus, J. Limpert, J. Rossbach, and A. Tünnermann, “High average and peak power few-cycle laser pulses delivered by fiber pumped OPCPA system,” Opt. Express 18(12), 12719–12726 (2010).
    [CrossRef] [PubMed]
  9. A. Dubietis, R. Butkus, and A. P. Piskarskas, “Trends in chirped pulse optical parametric amplification,” IEEE J. Sel. Top. Quantum Electron. 12(2), 163–172 (2006).
    [CrossRef]
  10. F. Tavella, A. Marcinkevičius, and F. Krausz, “90 mJ parametric chirped pulse amplification of 10 fs pulses,” Opt. Express 14(26), 12822–12827 (2006).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  14. G. Arisholm, J. Biegert, P. Schlup, C. P. Hauri, and U. Keller, “Ultra-broadband chirped-pulse optical parametric amplifier with angularly dispersed beams,” Opt. Express 12(3), 518–530 (2004).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  17. M. Bradler, P. Baum, and E. Riedle, “Femtosecond continuum generation in bulk laser host materials with sub-μJ pump pulses,” Appl. Phys. B 97(3), 561–574 (2009).
    [CrossRef]
  18. G. M. Gale, M. Cavallari, T. J. Driscoll, and F. Hache, “Sub-20-fs tunable pulses in the visible from an 82-MHz optical parametric oscillator,” Opt. Lett. 20(14), 1562–1564 (1995).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  21. For more system details, seeJ. Bromage, J. M. Fini, C. Dorrer, and J. D. Zuegel, “Characterization and optimization of Yb-doped photonic-crystal fiber rod amplifiers using spatially resolved spectral interferometry,” Appl. Opt. 50(14), 2001–2007 (2011).
    [CrossRef] [PubMed]
  22. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical recipes in FORTRAN: The art of scientific computing, 2nd ed. (Cambridge University Press, 1992), pp. 710–722.
  23. G. G. Gurzadian, V. G. Dmitriev, and D. N. Nikogosian, Handbook of Nonlinear Optical Crystals, 3rd rev. ed., Springer Series in Optical Sciences, vol. 64 (Springer-Verlag, 1999).
  24. D. Eimerl, L. Davis, S. Velsko, E. K. Graham, and A. Zalkin, “Optical, mechanical, and thermal properties of barium borate,” J. Appl. Phys. 62(5), 1968–1983 (1987).
    [CrossRef]
  25. G. Rustad, O. Farsund, and G. Arisholm, “Optical parametric oscillators with idler absorption,” in Proceedings of Advances in Optical Materials (AIOM), Istanbul, Turkey (2011), paper AWA25.

2011 (2)

2010 (1)

2009 (1)

M. Bradler, P. Baum, and E. Riedle, “Femtosecond continuum generation in bulk laser host materials with sub-μJ pump pulses,” Appl. Phys. B 97(3), 561–574 (2009).
[CrossRef]

2008 (3)

2007 (2)

2006 (5)

2005 (1)

2004 (1)

2003 (1)

G. Cerullo and S. De Silvestri, “Ultrafast optical parametric amplifiers,” Rev. Sci. Instrum. 74(1), 1–18 (2003).
[CrossRef]

2001 (1)

2000 (1)

E. Riedle, M. Beutter, S. Lochbrunner, J. Piel, S. Schenkl, S. Spörlein, and W. Zinth, “Generation of 10 to 50 fs pulses tunable through all of the visible and the NIR,” Appl. Phys. B 71, 457–465 (2000).
[CrossRef]

1995 (1)

1994 (1)

G. C. Bhar, P. K. Datta, A. M. Rudra, and U. Chatterjee, “Tangentially phase-matched efficient difference frequency generation in beta barium borate crystal,” Opt. Commun. 105(1-2), 95–98 (1994).
[CrossRef]

1987 (1)

D. Eimerl, L. Davis, S. Velsko, E. K. Graham, and A. Zalkin, “Optical, mechanical, and thermal properties of barium borate,” J. Appl. Phys. 62(5), 1968–1983 (1987).
[CrossRef]

Arisholm, G.

Baum, P.

M. Bradler, P. Baum, and E. Riedle, “Femtosecond continuum generation in bulk laser host materials with sub-μJ pump pulses,” Appl. Phys. B 97(3), 561–574 (2009).
[CrossRef]

Beutter, M.

E. Riedle, M. Beutter, S. Lochbrunner, J. Piel, S. Schenkl, S. Spörlein, and W. Zinth, “Generation of 10 to 50 fs pulses tunable through all of the visible and the NIR,” Appl. Phys. B 71, 457–465 (2000).
[CrossRef]

Bhar, G. C.

G. C. Bhar, P. K. Datta, A. M. Rudra, and U. Chatterjee, “Tangentially phase-matched efficient difference frequency generation in beta barium borate crystal,” Opt. Commun. 105(1-2), 95–98 (1994).
[CrossRef]

Biegert, J.

Binhammer, T.

Bradler, M.

M. Bradler, P. Baum, and E. Riedle, “Femtosecond continuum generation in bulk laser host materials with sub-μJ pump pulses,” Appl. Phys. B 97(3), 561–574 (2009).
[CrossRef]

Bromage, J.

Butkus, R.

A. Dubietis, R. Butkus, and A. P. Piskarskas, “Trends in chirped pulse optical parametric amplification,” IEEE J. Sel. Top. Quantum Electron. 12(2), 163–172 (2006).
[CrossRef]

Cavallari, M.

Cerullo, G.

G. Cerullo and S. De Silvestri, “Ultrafast optical parametric amplifiers,” Rev. Sci. Instrum. 74(1), 1–18 (2003).
[CrossRef]

Charukhchev, A. V.

Chatterjee, U.

G. C. Bhar, P. K. Datta, A. M. Rudra, and U. Chatterjee, “Tangentially phase-matched efficient difference frequency generation in beta barium borate crystal,” Opt. Commun. 105(1-2), 95–98 (1994).
[CrossRef]

Chekhlov, O. V.

Collier, J. L.

Datta, P. K.

G. C. Bhar, P. K. Datta, A. M. Rudra, and U. Chatterjee, “Tangentially phase-matched efficient difference frequency generation in beta barium borate crystal,” Opt. Commun. 105(1-2), 95–98 (1994).
[CrossRef]

Davis, L.

D. Eimerl, L. Davis, S. Velsko, E. K. Graham, and A. Zalkin, “Optical, mechanical, and thermal properties of barium borate,” J. Appl. Phys. 62(5), 1968–1983 (1987).
[CrossRef]

De Silvestri, S.

G. Cerullo and S. De Silvestri, “Ultrafast optical parametric amplifiers,” Rev. Sci. Instrum. 74(1), 1–18 (2003).
[CrossRef]

Demmler, S.

Dorrer, C.

Driscoll, T. J.

Dubietis, A.

A. Dubietis, R. Butkus, and A. P. Piskarskas, “Trends in chirped pulse optical parametric amplification,” IEEE J. Sel. Top. Quantum Electron. 12(2), 163–172 (2006).
[CrossRef]

Dunn, M. H.

Düsterer, S.

Eichberger, R.

Eimerl, D.

D. Eimerl, L. Davis, S. Velsko, E. K. Graham, and A. Zalkin, “Optical, mechanical, and thermal properties of barium borate,” J. Appl. Phys. 62(5), 1968–1983 (1987).
[CrossRef]

Ernstorfer, R.

Feldhaus, J.

Fini, J. M.

Freidman, G. I.

Gale, G. M.

Garanin, S. G.

Gerke, R. R.

Ginzburg, V. N.

Graham, E. K.

D. Eimerl, L. Davis, S. Velsko, E. K. Graham, and A. Zalkin, “Optical, mechanical, and thermal properties of barium borate,” J. Appl. Phys. 62(5), 1968–1983 (1987).
[CrossRef]

Gundlach, L.

Hache, F.

Hädrich, S.

Hanna, D. C.

Hauri, C. P.

Hellstrã M, J.

Hernandez-Gomez, C.

Jocher, C.

Karlsson, G.

Katin, E. V.

Keller, U.

Khazanov, E. A.

Killi, A.

Kirsanov, A. V.

Krausz, F.

Krebs, M.

Krok, P.

Laurell, F.

Limpert, J.

Lochbrunner, S.

C. Schriever, S. Lochbrunner, P. Krok, and E. Riedle, “Tunable pulses from below 300 to 970 nm with durations down to 14 fs based on a 2 MHz ytterbium-doped fiber system,” Opt. Lett. 33(2), 192–194 (2008).
[CrossRef] [PubMed]

E. Riedle, M. Beutter, S. Lochbrunner, J. Piel, S. Schenkl, S. Spörlein, and W. Zinth, “Generation of 10 to 50 fs pulses tunable through all of the visible and the NIR,” Appl. Phys. B 71, 457–465 (2000).
[CrossRef]

Lozhkarev, V. V.

Luchinin, G. A.

Mal’shakov, A. N.

Marcinkevicius, A.

Martyanov, M. A.

Matousek, P.

Morgner, U.

Musgrave, I.

New, G. H. C.

Palashov, O. V.

Palmer, G.

Pasiskevicius, V.

Piel, J.

J. Piel, E. Riedle, L. Gundlach, R. Ernstorfer, and R. Eichberger, “Sub-20 fs visible pulses with 750 nJ energy from a 100 kHz noncollinear optical parametric amplifier,” Opt. Lett. 31(9), 1289–1291 (2006).
[CrossRef] [PubMed]

E. Riedle, M. Beutter, S. Lochbrunner, J. Piel, S. Schenkl, S. Spörlein, and W. Zinth, “Generation of 10 to 50 fs pulses tunable through all of the visible and the NIR,” Appl. Phys. B 71, 457–465 (2000).
[CrossRef]

Piskarskas, A. P.

A. Dubietis, R. Butkus, and A. P. Piskarskas, “Trends in chirped pulse optical parametric amplification,” IEEE J. Sel. Top. Quantum Electron. 12(2), 163–172 (2006).
[CrossRef]

Poteomkin, A. K.

Rae, C. F.

Riedle, E.

M. Bradler, P. Baum, and E. Riedle, “Femtosecond continuum generation in bulk laser host materials with sub-μJ pump pulses,” Appl. Phys. B 97(3), 561–574 (2009).
[CrossRef]

C. Schriever, S. Lochbrunner, P. Krok, and E. Riedle, “Tunable pulses from below 300 to 970 nm with durations down to 14 fs based on a 2 MHz ytterbium-doped fiber system,” Opt. Lett. 33(2), 192–194 (2008).
[CrossRef] [PubMed]

J. Piel, E. Riedle, L. Gundlach, R. Ernstorfer, and R. Eichberger, “Sub-20 fs visible pulses with 750 nJ energy from a 100 kHz noncollinear optical parametric amplifier,” Opt. Lett. 31(9), 1289–1291 (2006).
[CrossRef] [PubMed]

E. Riedle, M. Beutter, S. Lochbrunner, J. Piel, S. Schenkl, S. Spörlein, and W. Zinth, “Generation of 10 to 50 fs pulses tunable through all of the visible and the NIR,” Appl. Phys. B 71, 457–465 (2000).
[CrossRef]

Röser, F.

Ross, I. N.

Rossbach, J.

Rothhardt, J.

Rudra, A. M.

G. C. Bhar, P. K. Datta, A. M. Rudra, and U. Chatterjee, “Tangentially phase-matched efficient difference frequency generation in beta barium borate crystal,” Opt. Commun. 105(1-2), 95–98 (1994).
[CrossRef]

Rukavishnikov, N. N.

Schenkl, S.

E. Riedle, M. Beutter, S. Lochbrunner, J. Piel, S. Schenkl, S. Spörlein, and W. Zinth, “Generation of 10 to 50 fs pulses tunable through all of the visible and the NIR,” Appl. Phys. B 71, 457–465 (2000).
[CrossRef]

Schimpf, D. N.

Schlarb, H.

Schlup, P.

Schriever, C.

Seise, E.

Sergeev, A. M.

Shaykin, A. A.

Spörlein, S.

E. Riedle, M. Beutter, S. Lochbrunner, J. Piel, S. Schenkl, S. Spörlein, and W. Zinth, “Generation of 10 to 50 fs pulses tunable through all of the visible and the NIR,” Appl. Phys. B 71, 457–465 (2000).
[CrossRef]

Steinmann, A.

Sukharev, S. A.

Tang, Y.

Tavella, F.

Terry, J. A. C.

Tünnermann, A.

Velsko, S.

D. Eimerl, L. Davis, S. Velsko, E. K. Graham, and A. Zalkin, “Optical, mechanical, and thermal properties of barium borate,” J. Appl. Phys. 62(5), 1968–1983 (1987).
[CrossRef]

Willner, A.

Yakovlev, I. V.

Yashin, V. E.

Zalkin, A.

D. Eimerl, L. Davis, S. Velsko, E. K. Graham, and A. Zalkin, “Optical, mechanical, and thermal properties of barium borate,” J. Appl. Phys. 62(5), 1968–1983 (1987).
[CrossRef]

Zinth, W.

E. Riedle, M. Beutter, S. Lochbrunner, J. Piel, S. Schenkl, S. Spörlein, and W. Zinth, “Generation of 10 to 50 fs pulses tunable through all of the visible and the NIR,” Appl. Phys. B 71, 457–465 (2000).
[CrossRef]

Zuegel, J. D.

Appl. Opt. (1)

Appl. Phys. B (2)

E. Riedle, M. Beutter, S. Lochbrunner, J. Piel, S. Schenkl, S. Spörlein, and W. Zinth, “Generation of 10 to 50 fs pulses tunable through all of the visible and the NIR,” Appl. Phys. B 71, 457–465 (2000).
[CrossRef]

M. Bradler, P. Baum, and E. Riedle, “Femtosecond continuum generation in bulk laser host materials with sub-μJ pump pulses,” Appl. Phys. B 97(3), 561–574 (2009).
[CrossRef]

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

A. Dubietis, R. Butkus, and A. P. Piskarskas, “Trends in chirped pulse optical parametric amplification,” IEEE J. Sel. Top. Quantum Electron. 12(2), 163–172 (2006).
[CrossRef]

J. Appl. Phys. (1)

D. Eimerl, L. Davis, S. Velsko, E. K. Graham, and A. Zalkin, “Optical, mechanical, and thermal properties of barium borate,” J. Appl. Phys. 62(5), 1968–1983 (1987).
[CrossRef]

J. Opt. Soc. Am. B (2)

Opt. Commun. (1)

G. C. Bhar, P. K. Datta, A. M. Rudra, and U. Chatterjee, “Tangentially phase-matched efficient difference frequency generation in beta barium borate crystal,” Opt. Commun. 105(1-2), 95–98 (1994).
[CrossRef]

Opt. Express (7)

V. V. Lozhkarev, G. I. Freidman, V. N. Ginzburg, E. V. Katin, E. A. Khazanov, A. V. Kirsanov, G. A. Luchinin, A. N. Mal’shakov, M. A. Martyanov, O. V. Palashov, A. K. Poteomkin, A. M. Sergeev, A. A. Shaykin, I. V. Yakovlev, S. G. Garanin, S. A. Sukharev, N. N. Rukavishnikov, A. V. Charukhchev, R. R. Gerke, and V. E. Yashin, “200 TW 45 fs laser based on optical parametric chirped pulse amplification,” Opt. Express 14(1), 446–454 (2006).
[CrossRef] [PubMed]

G. Arisholm, J. Biegert, P. Schlup, C. P. Hauri, and U. Keller, “Ultra-broadband chirped-pulse optical parametric amplifier with angularly dispersed beams,” Opt. Express 12(3), 518–530 (2004).
[CrossRef] [PubMed]

F. Tavella, A. Marcinkevičius, and F. Krausz, “90 mJ parametric chirped pulse amplification of 10 fs pulses,” Opt. Express 14(26), 12822–12827 (2006).
[CrossRef] [PubMed]

J. Rothhardt, S. Hädrich, D. N. Schimpf, J. Limpert, and A. Tünnermann, “High repetition rate fiber amplifier pumped sub-20 fs optical parametric amplifier,” Opt. Express 15(25), 16729–16736 (2007).
[CrossRef] [PubMed]

J. Rothhardt, S. Hädrich, F. Röser, J. Limpert, and A. Tünnermann, “500 MW peak power degenerated optical parametric amplifier delivering 52 fs pulses at 97 kHz repetition rate,” Opt. Express 16(12), 8981–8988 (2008).
[CrossRef] [PubMed]

J. Rothhardt, S. Hädrich, E. Seise, M. Krebs, F. Tavella, A. Willner, S. Düsterer, H. Schlarb, J. Feldhaus, J. Limpert, J. Rossbach, and A. Tünnermann, “High average and peak power few-cycle laser pulses delivered by fiber pumped OPCPA system,” Opt. Express 18(12), 12719–12726 (2010).
[CrossRef] [PubMed]

A. Steinmann, A. Killi, G. Palmer, T. Binhammer, and U. Morgner, “Generation of few-cycle pulses directly from a MHz-NOPA,” Opt. Express 14(22), 10627–10630 (2006).
[CrossRef] [PubMed]

Opt. Lett. (6)

Rev. Sci. Instrum. (1)

G. Cerullo and S. De Silvestri, “Ultrafast optical parametric amplifiers,” Rev. Sci. Instrum. 74(1), 1–18 (2003).
[CrossRef]

Other (3)

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical recipes in FORTRAN: The art of scientific computing, 2nd ed. (Cambridge University Press, 1992), pp. 710–722.

G. G. Gurzadian, V. G. Dmitriev, and D. N. Nikogosian, Handbook of Nonlinear Optical Crystals, 3rd rev. ed., Springer Series in Optical Sciences, vol. 64 (Springer-Verlag, 1999).

G. Rustad, O. Farsund, and G. Arisholm, “Optical parametric oscillators with idler absorption,” in Proceedings of Advances in Optical Materials (AIOM), Istanbul, Turkey (2011), paper AWA25.

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Figures (12)

Fig. 1
Fig. 1

Relative position of the wave vectors for the pump (green), signal (red), and idler (black) for critical phase matching in a negative uniaxial crystal. The noncollinear angle between the pump and signal wave vectors is α. The walk-off angle between the pump Poynting vector [S P (blue)] and signal wave vector is ρ. (a) Definition of the walk-off compensating (WC) configuration for negative uniaxial crystals. (b) Non-walk-off compensating (NWC) configuration. The vertical dimension has been expanded 3 × for clarity.

Fig. 2
Fig. 2

NOPA phase-matching curves (red) for BBO pumped at 523 nm for several noncollinear angles α: (a) the walk-off compensating (WC) and (b) the non-walk-off compensating (NWC) configurations. The phase-matching curves for signal SHG are shown in blue.

Fig. 3
Fig. 3

The idler wavelengths and internal angles for NOPA phase matching in BBO pumped at 523 nm for two values of α (in red): (a) the WC and (b) the NWC configurations. The phase-matching curves for idler SHG are shown in blue.

Fig. 4
Fig. 4

(a) Experimental setup. Yb: ytterbium-doped fiber amplifiers; AOM: acousto-optic modulator; WLC: white-light continuum; SHG: second-harmonic generation; NOPA: noncollinear optical parametric amplifier. Near-field profiles of the amplified beams for (b) walk-off compensating (WC) and (c) non-walk-off compensating (NWC) configurations.

Fig. 5
Fig. 5

Signal spectra for (a) the WC and (b) the NWC configurations. The pump angle θ P and noncollinear angle α were similar in both cases. In the WC configuration, the signal angle θ S of 28.2° made it possible for parasitic second-harmonic generation to produce a narrow spectral line at 410 nm, leaving notches in the signal spectrum in the 780- to 870-nm range.

Fig. 6
Fig. 6

Spectra measured from idler SHG for the WC configuration. The spectrum spans much of the visible spectrum, as shown by the color bar above.

Fig. 7
Fig. 7

Comparison of experimental and numerical simulations for a range of signal angles in the WC configuration.

Fig. 8
Fig. 8

Spectra for simulation case study. Signal SHG (a) turned on in the model and (b) turned off.

Fig. 9
Fig. 9

Simulated signal energy along the crystal, (a) with and (b) without signal SHG. In both cases signal reconversion means that the length for optimum efficiency is 2.3 mm. When SHG is included in the simulations, 10% of the signal energy at this length is lost to the parasitic process.

Fig. 10
Fig. 10

A noncollinear OPCPA setup pumped by the second harmonic of a high-energy FCPA system.

Fig. 11
Fig. 11

Measured spectra for (a) the WC and (b) the NWC configurations containing the signal and its second harmonic. Note that both the fundamental and the second-harmonic spectra have been normalized to their peak value, independently.

Fig. 12
Fig. 12

Phase-matching curves for signal and idler SHG for other common NOPA crystals: [(a),(b)] DKDP; [(c),(d)] LBO; and [(e),(f)] BiBO.

Tables (2)

Tables Icon

Table 1 Input Parameters for Signal SHG Case Study

Tables Icon

Table 2 SHG Phase-matched Signal and Idler Wavelengths for BBO, LBO, BiBO, and DKDP in both WC and NWC Configurations for NOPA’s Pumped by the Second Harmonic of Yb-based Systems ( P = 510 to 525 nm)

Equations (3)

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

d A S d z = i 2 ω S d eff n S c ( A I A P A S A SSH ) , d A I d z = i 2 ω I d eff n I c ( A S A P A I A ISH ) , d A P d z = i 2 ω P d eff n P c A S A I , d A SSH d z = i ω SSH d eff n SSH c A S 2 , d A ISH d z = i ω ISH d eff n ISH c A I 2 ,
a j ( z , Ω ) = FFT 1 [ A j ( z , t ) , t Ω ] .
a j ( z + h , Ω ) = a j ( z , Ω ) e i n j h ( ω j + Ω ) / c i h Ω / v S ,

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