Abstract

Finding an optimal phase pattern in a multidimensional solution landscape becomes easier and faster if local optima are suppressed and contour lines are tailored towards closed convex patterns. Using wideband second harmonic generation as a coherent control test case, we show that a linear combination of spectral phase basis functions can result in such improvements and also in separable phase terms, each of which can be found independently. The improved shapes are attributed to a suppressed nonlinear shear, changing the relative orientation of contour lines. The first order approximation of the process shows a simple relation between input and output phase profiles, useful for pulse shaping at ultraviolet wavelengths.

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    [CrossRef]

2009 (6)

A. C. W. van Rhijn, S. Postma, J. P. Korterik, J. L. Herek, and H. L. Offerhaus, “Chemically selective imaging by spectral phase shaping for broadband CARS around 3000 cm−1,” J. Opt. Soc. Am. B 26(3), 559–563 (2009).
[CrossRef]

P. van der Walle, M. T. W. Milder, L. Kuipers, and J. L. Herek, “Quantum control experiment reveals solvation-induced decoherence,” Proc. Natl. Acad. Sci. U.S.A. 106(19), 7714–7717 (2009).
[CrossRef] [PubMed]

P. van der Walle, J. Savolainen, L. Kuipers, and J. L. Herek, “Learning from evolutionary optimization by retracing search paths,” Chem. Phys. Lett. 483(1-3), 164–167 (2009), doi:.
[CrossRef]

J. Roslund and H. Rabitz, “Gradient algorithm applied to laboratory quantum control,” Phys. Rev. A 79(5), 053417 (2009).
[CrossRef]

A. Jafarpour, J. Savolainen, R. de Jong, J. Middag, D. P. Sprünken, P. van der Walle, D. Yang, and J. L. Herek, “Robust orthogonal parameterization of evolution strategy for adaptive laser pulse shaping,” Opt. Express 17(14), 11986–12000 (2009).
[CrossRef] [PubMed]

I. A. Walmsley and C. Dorrer, “Characterization of ultrashort electromagnetic pulses,” Adv. Opt. Photon. 1(2), 308–437 (2009).
[CrossRef]

2008 (3)

R. Fanciulli, L. Willmes, J. Savolainen, P. Walle, T. Bäck, and J. L. Herek, “Evolution strategies for laser pulse compression,” Lect. Notes Comput. Sci. 4926, 219–230 (2008).
[CrossRef]

J. Savolainen, R. Fanciulli, N. Dijkhuizen, A. L. Moore, J. Hauer, T. Buckup, M. Motzkus, and J. L. Herek, “Controlling the efficiency of an artificial light-harvesting complex,” Proc. Natl. Acad. Sci. U.S.A. 105(22), 7641–7646 (2008).
[CrossRef] [PubMed]

S. Postma, A. C. W. van Rhijn, J. P. Korterik, P. Gross, J. L. Herek, and H. L. Offerhaus, “Application of spectral phase shaping to high resolution CARS spectroscopy,” Opt. Express 16(11), 7985–7996 (2008).
[CrossRef] [PubMed]

2007 (1)

J. Werschnik and E. K. U. Gross, “Quantum optimal control theory,” J. Phys. At. Mol. Opt. Phys. 40(18), R175–R211 (2007).
[CrossRef]

2006 (1)

J. Roslund, M. Roth, and H. Rabitz, “Laboratory observation of quantum control level sets,” Phys. Rev. A 74(4), 043414 (2006).
[CrossRef]

2005 (1)

V. V. Lozovoy and M. Dantus, “Systematic control of nonlinear optical processes using optimally shaped femtosecond pulses,” ChemPhysChem 6(10), 1970–2000 (2005).
[CrossRef] [PubMed]

2003 (1)

2002 (1)

J. L. Herek, W. Wohlleben, R. J. Cogdell, D. Zeidler, and M. Motzkus, “Quantum control of energy flow in light harvesting,” Nature 417(6888), 533–535 (2002).
[CrossRef] [PubMed]

2001 (4)

Z. Zheng and A. M. Weiner, “Coherent control of second harmonic generation using spectrally phase coded femtosecond waveforms,” Chem. Phys. 267(1–3), 161–171 (2001).
[CrossRef]

B. J. Pearson, J. L. White, T. C. Weinacht, and P. H. Bucksbaum, “Coherent control using adaptive learning algorithms,” Phys. Rev. A 63(6), 063412 (2001).
[CrossRef]

D. Zeidler, S. Frey, K.-L. Kompa, and M. Motzkus, “Evolutionary algorithms and their application to optimal control studies,” Phys. Rev. A 64(2), 023420 (2001).
[CrossRef]

T. Brixner, B. Kiefer, and G. Gerber, “Problem complexity in femtosecond quantum control,” Chem. Phys. 267(1–3), 241–246 (2001).
[CrossRef]

1999 (1)

D. Meshulach and Y. Silberberg, “Coherent Quantum Control of Multiphoton Transitions by Shaped Ultrashort Optical Pulses,” Phys. Rev. A 60(2), 1287–1292 (1999).
[CrossRef]

1995 (2)

B. Kohler, V. V. Yakovlev, J. Che, J. L. Krause, M. Messina, K. R. Wilson, N. Schwentner, R. M. Whitnell, and Y. J. Yan, “Quantum control of wave packet evolution with tailored femtosecond pulses,” Phys. Rev. Lett. 74(17), 3360–3363 (1995).
[CrossRef] [PubMed]

C. J. Bardeen, Q. Wang, and C. V. Shank, “Selective excitation of vibrational wave packet motion using chirped pulses,” Phys. Rev. Lett. 75(19), 3410–3413 (1995).
[CrossRef] [PubMed]

1992 (1)

R. S. Judson and H. Rabitz, “Teaching lasers to control molecules,” Phys. Rev. Lett. 68(10), 1500–1503 (1992).
[CrossRef] [PubMed]

1988 (1)

S. Shi, A. Woody, and H. Rabitz, “Optimal control of selective vibrational excitation in harmonic linear chain molecules,” J. Chem. Phys. 88(11), 6870–6883 (1988).
[CrossRef]

1986 (2)

P. Brumer and M. Shapiro, “Control of unimolecular reactions using coherent light,” Chem. Phys. Lett. 126(6), 541–546 (1986).
[CrossRef]

A. Paeth, “A fast algorithm for general raster rotation,” Graphics Interface 86, 77–81 (1986).

Bäck, T.

R. Fanciulli, L. Willmes, J. Savolainen, P. Walle, T. Bäck, and J. L. Herek, “Evolution strategies for laser pulse compression,” Lect. Notes Comput. Sci. 4926, 219–230 (2008).
[CrossRef]

Bardeen, C. J.

C. J. Bardeen, Q. Wang, and C. V. Shank, “Selective excitation of vibrational wave packet motion using chirped pulses,” Phys. Rev. Lett. 75(19), 3410–3413 (1995).
[CrossRef] [PubMed]

Brixner, T.

T. Brixner, B. Kiefer, and G. Gerber, “Problem complexity in femtosecond quantum control,” Chem. Phys. 267(1–3), 241–246 (2001).
[CrossRef]

Brumer, P.

P. Brumer and M. Shapiro, “Control of unimolecular reactions using coherent light,” Chem. Phys. Lett. 126(6), 541–546 (1986).
[CrossRef]

Bucksbaum, P. H.

B. J. Pearson, J. L. White, T. C. Weinacht, and P. H. Bucksbaum, “Coherent control using adaptive learning algorithms,” Phys. Rev. A 63(6), 063412 (2001).
[CrossRef]

Buckup, T.

J. Savolainen, R. Fanciulli, N. Dijkhuizen, A. L. Moore, J. Hauer, T. Buckup, M. Motzkus, and J. L. Herek, “Controlling the efficiency of an artificial light-harvesting complex,” Proc. Natl. Acad. Sci. U.S.A. 105(22), 7641–7646 (2008).
[CrossRef] [PubMed]

Che, J.

B. Kohler, V. V. Yakovlev, J. Che, J. L. Krause, M. Messina, K. R. Wilson, N. Schwentner, R. M. Whitnell, and Y. J. Yan, “Quantum control of wave packet evolution with tailored femtosecond pulses,” Phys. Rev. Lett. 74(17), 3360–3363 (1995).
[CrossRef] [PubMed]

Cogdell, R. J.

J. L. Herek, W. Wohlleben, R. J. Cogdell, D. Zeidler, and M. Motzkus, “Quantum control of energy flow in light harvesting,” Nature 417(6888), 533–535 (2002).
[CrossRef] [PubMed]

Dantus, M.

V. V. Lozovoy and M. Dantus, “Systematic control of nonlinear optical processes using optimally shaped femtosecond pulses,” ChemPhysChem 6(10), 1970–2000 (2005).
[CrossRef] [PubMed]

de Jong, R.

Dijkhuizen, N.

J. Savolainen, R. Fanciulli, N. Dijkhuizen, A. L. Moore, J. Hauer, T. Buckup, M. Motzkus, and J. L. Herek, “Controlling the efficiency of an artificial light-harvesting complex,” Proc. Natl. Acad. Sci. U.S.A. 105(22), 7641–7646 (2008).
[CrossRef] [PubMed]

Dorrer, C.

Fanciulli, R.

R. Fanciulli, L. Willmes, J. Savolainen, P. Walle, T. Bäck, and J. L. Herek, “Evolution strategies for laser pulse compression,” Lect. Notes Comput. Sci. 4926, 219–230 (2008).
[CrossRef]

J. Savolainen, R. Fanciulli, N. Dijkhuizen, A. L. Moore, J. Hauer, T. Buckup, M. Motzkus, and J. L. Herek, “Controlling the efficiency of an artificial light-harvesting complex,” Proc. Natl. Acad. Sci. U.S.A. 105(22), 7641–7646 (2008).
[CrossRef] [PubMed]

Frey, S.

D. Zeidler, S. Frey, K.-L. Kompa, and M. Motzkus, “Evolutionary algorithms and their application to optimal control studies,” Phys. Rev. A 64(2), 023420 (2001).
[CrossRef]

Gerber, G.

T. Brixner, B. Kiefer, and G. Gerber, “Problem complexity in femtosecond quantum control,” Chem. Phys. 267(1–3), 241–246 (2001).
[CrossRef]

Gross, E. K. U.

J. Werschnik and E. K. U. Gross, “Quantum optimal control theory,” J. Phys. At. Mol. Opt. Phys. 40(18), R175–R211 (2007).
[CrossRef]

Gross, P.

Hauer, J.

J. Savolainen, R. Fanciulli, N. Dijkhuizen, A. L. Moore, J. Hauer, T. Buckup, M. Motzkus, and J. L. Herek, “Controlling the efficiency of an artificial light-harvesting complex,” Proc. Natl. Acad. Sci. U.S.A. 105(22), 7641–7646 (2008).
[CrossRef] [PubMed]

Herek, J. L.

P. van der Walle, M. T. W. Milder, L. Kuipers, and J. L. Herek, “Quantum control experiment reveals solvation-induced decoherence,” Proc. Natl. Acad. Sci. U.S.A. 106(19), 7714–7717 (2009).
[CrossRef] [PubMed]

P. van der Walle, J. Savolainen, L. Kuipers, and J. L. Herek, “Learning from evolutionary optimization by retracing search paths,” Chem. Phys. Lett. 483(1-3), 164–167 (2009), doi:.
[CrossRef]

A. C. W. van Rhijn, S. Postma, J. P. Korterik, J. L. Herek, and H. L. Offerhaus, “Chemically selective imaging by spectral phase shaping for broadband CARS around 3000 cm−1,” J. Opt. Soc. Am. B 26(3), 559–563 (2009).
[CrossRef]

A. Jafarpour, J. Savolainen, R. de Jong, J. Middag, D. P. Sprünken, P. van der Walle, D. Yang, and J. L. Herek, “Robust orthogonal parameterization of evolution strategy for adaptive laser pulse shaping,” Opt. Express 17(14), 11986–12000 (2009).
[CrossRef] [PubMed]

R. Fanciulli, L. Willmes, J. Savolainen, P. Walle, T. Bäck, and J. L. Herek, “Evolution strategies for laser pulse compression,” Lect. Notes Comput. Sci. 4926, 219–230 (2008).
[CrossRef]

S. Postma, A. C. W. van Rhijn, J. P. Korterik, P. Gross, J. L. Herek, and H. L. Offerhaus, “Application of spectral phase shaping to high resolution CARS spectroscopy,” Opt. Express 16(11), 7985–7996 (2008).
[CrossRef] [PubMed]

J. Savolainen, R. Fanciulli, N. Dijkhuizen, A. L. Moore, J. Hauer, T. Buckup, M. Motzkus, and J. L. Herek, “Controlling the efficiency of an artificial light-harvesting complex,” Proc. Natl. Acad. Sci. U.S.A. 105(22), 7641–7646 (2008).
[CrossRef] [PubMed]

J. L. Herek, W. Wohlleben, R. J. Cogdell, D. Zeidler, and M. Motzkus, “Quantum control of energy flow in light harvesting,” Nature 417(6888), 533–535 (2002).
[CrossRef] [PubMed]

Jafarpour, A.

Judson, R. S.

R. S. Judson and H. Rabitz, “Teaching lasers to control molecules,” Phys. Rev. Lett. 68(10), 1500–1503 (1992).
[CrossRef] [PubMed]

Keusters, D.

Kiefer, B.

T. Brixner, B. Kiefer, and G. Gerber, “Problem complexity in femtosecond quantum control,” Chem. Phys. 267(1–3), 241–246 (2001).
[CrossRef]

Kohler, B.

B. Kohler, V. V. Yakovlev, J. Che, J. L. Krause, M. Messina, K. R. Wilson, N. Schwentner, R. M. Whitnell, and Y. J. Yan, “Quantum control of wave packet evolution with tailored femtosecond pulses,” Phys. Rev. Lett. 74(17), 3360–3363 (1995).
[CrossRef] [PubMed]

Kompa, K.-L.

D. Zeidler, S. Frey, K.-L. Kompa, and M. Motzkus, “Evolutionary algorithms and their application to optimal control studies,” Phys. Rev. A 64(2), 023420 (2001).
[CrossRef]

Korterik, J. P.

Krause, J. L.

B. Kohler, V. V. Yakovlev, J. Che, J. L. Krause, M. Messina, K. R. Wilson, N. Schwentner, R. M. Whitnell, and Y. J. Yan, “Quantum control of wave packet evolution with tailored femtosecond pulses,” Phys. Rev. Lett. 74(17), 3360–3363 (1995).
[CrossRef] [PubMed]

Kuipers, L.

P. van der Walle, J. Savolainen, L. Kuipers, and J. L. Herek, “Learning from evolutionary optimization by retracing search paths,” Chem. Phys. Lett. 483(1-3), 164–167 (2009), doi:.
[CrossRef]

P. van der Walle, M. T. W. Milder, L. Kuipers, and J. L. Herek, “Quantum control experiment reveals solvation-induced decoherence,” Proc. Natl. Acad. Sci. U.S.A. 106(19), 7714–7717 (2009).
[CrossRef] [PubMed]

Lozovoy, V. V.

V. V. Lozovoy and M. Dantus, “Systematic control of nonlinear optical processes using optimally shaped femtosecond pulses,” ChemPhysChem 6(10), 1970–2000 (2005).
[CrossRef] [PubMed]

Meshulach, D.

D. Meshulach and Y. Silberberg, “Coherent Quantum Control of Multiphoton Transitions by Shaped Ultrashort Optical Pulses,” Phys. Rev. A 60(2), 1287–1292 (1999).
[CrossRef]

Messina, M.

B. Kohler, V. V. Yakovlev, J. Che, J. L. Krause, M. Messina, K. R. Wilson, N. Schwentner, R. M. Whitnell, and Y. J. Yan, “Quantum control of wave packet evolution with tailored femtosecond pulses,” Phys. Rev. Lett. 74(17), 3360–3363 (1995).
[CrossRef] [PubMed]

Middag, J.

Milder, M. T. W.

P. van der Walle, M. T. W. Milder, L. Kuipers, and J. L. Herek, “Quantum control experiment reveals solvation-induced decoherence,” Proc. Natl. Acad. Sci. U.S.A. 106(19), 7714–7717 (2009).
[CrossRef] [PubMed]

Moore, A. L.

J. Savolainen, R. Fanciulli, N. Dijkhuizen, A. L. Moore, J. Hauer, T. Buckup, M. Motzkus, and J. L. Herek, “Controlling the efficiency of an artificial light-harvesting complex,” Proc. Natl. Acad. Sci. U.S.A. 105(22), 7641–7646 (2008).
[CrossRef] [PubMed]

Motzkus, M.

J. Savolainen, R. Fanciulli, N. Dijkhuizen, A. L. Moore, J. Hauer, T. Buckup, M. Motzkus, and J. L. Herek, “Controlling the efficiency of an artificial light-harvesting complex,” Proc. Natl. Acad. Sci. U.S.A. 105(22), 7641–7646 (2008).
[CrossRef] [PubMed]

J. L. Herek, W. Wohlleben, R. J. Cogdell, D. Zeidler, and M. Motzkus, “Quantum control of energy flow in light harvesting,” Nature 417(6888), 533–535 (2002).
[CrossRef] [PubMed]

D. Zeidler, S. Frey, K.-L. Kompa, and M. Motzkus, “Evolutionary algorithms and their application to optimal control studies,” Phys. Rev. A 64(2), 023420 (2001).
[CrossRef]

Offerhaus, H. L.

O'Shea, P.

Paeth, A.

A. Paeth, “A fast algorithm for general raster rotation,” Graphics Interface 86, 77–81 (1986).

Pearson, B. J.

B. J. Pearson, J. L. White, T. C. Weinacht, and P. H. Bucksbaum, “Coherent control using adaptive learning algorithms,” Phys. Rev. A 63(6), 063412 (2001).
[CrossRef]

Postma, S.

Rabitz, H.

J. Roslund and H. Rabitz, “Gradient algorithm applied to laboratory quantum control,” Phys. Rev. A 79(5), 053417 (2009).
[CrossRef]

J. Roslund, M. Roth, and H. Rabitz, “Laboratory observation of quantum control level sets,” Phys. Rev. A 74(4), 043414 (2006).
[CrossRef]

R. S. Judson and H. Rabitz, “Teaching lasers to control molecules,” Phys. Rev. Lett. 68(10), 1500–1503 (1992).
[CrossRef] [PubMed]

S. Shi, A. Woody, and H. Rabitz, “Optimal control of selective vibrational excitation in harmonic linear chain molecules,” J. Chem. Phys. 88(11), 6870–6883 (1988).
[CrossRef]

Roslund, J.

J. Roslund and H. Rabitz, “Gradient algorithm applied to laboratory quantum control,” Phys. Rev. A 79(5), 053417 (2009).
[CrossRef]

J. Roslund, M. Roth, and H. Rabitz, “Laboratory observation of quantum control level sets,” Phys. Rev. A 74(4), 043414 (2006).
[CrossRef]

Roth, M.

J. Roslund, M. Roth, and H. Rabitz, “Laboratory observation of quantum control level sets,” Phys. Rev. A 74(4), 043414 (2006).
[CrossRef]

Savolainen, J.

A. Jafarpour, J. Savolainen, R. de Jong, J. Middag, D. P. Sprünken, P. van der Walle, D. Yang, and J. L. Herek, “Robust orthogonal parameterization of evolution strategy for adaptive laser pulse shaping,” Opt. Express 17(14), 11986–12000 (2009).
[CrossRef] [PubMed]

P. van der Walle, J. Savolainen, L. Kuipers, and J. L. Herek, “Learning from evolutionary optimization by retracing search paths,” Chem. Phys. Lett. 483(1-3), 164–167 (2009), doi:.
[CrossRef]

J. Savolainen, R. Fanciulli, N. Dijkhuizen, A. L. Moore, J. Hauer, T. Buckup, M. Motzkus, and J. L. Herek, “Controlling the efficiency of an artificial light-harvesting complex,” Proc. Natl. Acad. Sci. U.S.A. 105(22), 7641–7646 (2008).
[CrossRef] [PubMed]

R. Fanciulli, L. Willmes, J. Savolainen, P. Walle, T. Bäck, and J. L. Herek, “Evolution strategies for laser pulse compression,” Lect. Notes Comput. Sci. 4926, 219–230 (2008).
[CrossRef]

Schwentner, N.

B. Kohler, V. V. Yakovlev, J. Che, J. L. Krause, M. Messina, K. R. Wilson, N. Schwentner, R. M. Whitnell, and Y. J. Yan, “Quantum control of wave packet evolution with tailored femtosecond pulses,” Phys. Rev. Lett. 74(17), 3360–3363 (1995).
[CrossRef] [PubMed]

Shank, C. V.

C. J. Bardeen, Q. Wang, and C. V. Shank, “Selective excitation of vibrational wave packet motion using chirped pulses,” Phys. Rev. Lett. 75(19), 3410–3413 (1995).
[CrossRef] [PubMed]

Shapiro, M.

P. Brumer and M. Shapiro, “Control of unimolecular reactions using coherent light,” Chem. Phys. Lett. 126(6), 541–546 (1986).
[CrossRef]

Shi, S.

S. Shi, A. Woody, and H. Rabitz, “Optimal control of selective vibrational excitation in harmonic linear chain molecules,” J. Chem. Phys. 88(11), 6870–6883 (1988).
[CrossRef]

Silberberg, Y.

D. Meshulach and Y. Silberberg, “Coherent Quantum Control of Multiphoton Transitions by Shaped Ultrashort Optical Pulses,” Phys. Rev. A 60(2), 1287–1292 (1999).
[CrossRef]

Sprünken, D. P.

Tan, H. S.

Trebino, R.

van der Walle, P.

A. Jafarpour, J. Savolainen, R. de Jong, J. Middag, D. P. Sprünken, P. van der Walle, D. Yang, and J. L. Herek, “Robust orthogonal parameterization of evolution strategy for adaptive laser pulse shaping,” Opt. Express 17(14), 11986–12000 (2009).
[CrossRef] [PubMed]

P. van der Walle, J. Savolainen, L. Kuipers, and J. L. Herek, “Learning from evolutionary optimization by retracing search paths,” Chem. Phys. Lett. 483(1-3), 164–167 (2009), doi:.
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Figures (12)

Fig. 1
Fig. 1

(left) 1D landscapes of the SHG fitness for different phase orders φ(ω) = φnωn/n!, where ω is the radian frequency of the pulse envelope function and 2≤n≤6, and (right) 1D landscapes of the cubic phase in two cases with the second- and the fifth-order background phase.

Fig. 2
Fig. 2

2D landscapes of the SHG energy (phase orders with opposite parities).

Fig. 3
Fig. 3

2D landscapes of the SHG energy (phase orders with similar parities). Contour plots use the same color map as in Fig. 2.

Fig. 4
Fig. 4

The evolution of the φ24 SHG landscape by changing the polynomial basis of the phase function φ(ω) = (φ2/2!)ω2 + (φ4/4!)ω4 = (φ2/2!)(Δω)2ωn 2 + (φ4/4!)(Δω)4ωn 4, where ω is the radian frequency of the pulse envelope function, Δω is the FWHM bandwidth, and ωn = ω/Δω. The first five plots on the left show the effect of modification of the fourth order basis function, as it is changed to ωn 4-ηωn 2, and the parameter η is varied. The two rightmost plots correspond to an additional change in the second order basis function by modifying it to ωn 2-η'ωn 4, and also scaling the fourth order basis function as η”(ωn 4-ηωn 2). The general expression for the phase function in all cases is φ(ω) = (φ2/2!)(Δω)2n 2-η'ωn 4) + (φ4/4!)(Δω)4η”(ωn 4-ηωn 2), The numerical values of the modification parameters in the right-most plot are η0 = 2.22, η'0 = 1.24, and η”0 = 0.6. The effect of these transforms on shear is discussed in Section 5.1.

Fig. 5
Fig. 5

Original (top) and modified (bottom) 2D landscapes. All contour plots use the same color map.

Fig. 6
Fig. 6

The level sets of the 3D SHG landscape of the original (top) and modified (bottom) polynomials, corresponding to fitness values of 0.95 (left), 0.90 (middle), and 0.85 (right). All 3D level sets are viewed in the same Cartesian coordinate and from the same angle (azimuth = −37.5°, elevation = 30°). See the attached animated GIF for the evolution of the level sets.

Fig. 7
Fig. 7

Contour plots of the elliptic paraboloid z = 1-(x2/22 + y2) in the original, rotated, and sheared forms.

Fig. 8
Fig. 8

The five plots of the left panel show the effect of linear shear ratio (LSR) on the contour plots of an elliptic paraboloid distorted with nonlinear shear. The rightmost figure shows the effect of LSR on the range/amplitude of a third order function.

Fig. 9
Fig. 9

Phasor diagram showing the contributions of different orders of fundamental spectral phase to the SHG electric field

Fig. 10
Fig. 10

The original (left) and the modified (right) surfaces of the total SHG energy with the second order and the fourth order polynomial functions. The middle plot shows the original (red) and modulated (blue) input spectra used for the simulations.

Fig. 11
Fig. 11

The impacts of different levels of phase noise on the original (top) and modified (bottom) φ24 landscapes.

Fig. 12
Fig. 12

Four different realizations of the original (top) and modified (bottom) φ24 landscapes for a given level of phase noise.

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