Abstract

Many spectroscopic applications of femtosecond laser pulses require properly-shaped spectral phase profiles. The optimal phase profile can be programmed on the pulse by adaptive pulse shaping. A promising optimization algorithm for such adaptive experiments is evolution strategy (ES). Here, we report a four fold increase in the rate of convergence and ten percent increase in the final yield of the optimization, compared to the direct parameterization approach, by using a new version of ES in combination with Legendre polynomials and frequency-resolved detection. Such a fast learning rate is of paramount importance in spectroscopy for reducing the artifacts of laser drift, optical degradation, and precipitation.

© 2009 OSA

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2008

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]

J. W. Wilson, P. Schlup, M. Lunacek, D. Whitley, and R. A. Bartels, “Calibration of liquid crystal ultrafast pulse shaper with common-path spectral interferometry and application to coherent control with a covariance matrix adaptation evolutionary strategy,” Rev. Sci. Instrum. 79(3), 033103 (2008).
[CrossRef]

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

2007

C. Riziotis and A. V. Vasilakos, “Computational intelligence in photonics technology and optical networks: A survey and future perspectives,” Inf. Sci. 177(23), 5292–5315 (2007).
[CrossRef]

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

2006

Ch. Siedschlaga, O. M. Shirb, Th. Bäckb, and M. J. J. Vrakkinga, “Evolutionary algorithms in the optimization of dynamic molecular alignment,” Opt. Commun. 264(2), 511–518 (2006).
[CrossRef]

J. Roslund, M. Roth, and H. Rabitz, “Laboratory observation of quantum control level sets,” Phys. Rev. A 74(4), 043414–11 (2006).
[CrossRef]

2004

D. Cardoza, F. Langhojer, C. Trallero-Herrero, O. L. A. Monti, and T. Weinacht, “Changing pulse-shape basis for molecular learning control,” Phys. Rev. A 70(5), 053406 (2004).
[CrossRef]

2002

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]

2001

B. J. Pearson, J. L. White, T. C. Weinacht, and P. H. Bucksbaum, “Coherent control using adaptive learning algorithms,” Phys. Rev. A 63(6), 063412–12 (2001).

D. Zeidler, S. Frey, K.-L. Kompa, and M. Motzkus, “Evolutionary algorithms and their application to optimal control studies,” Phys. Rev. A 64(2), 023420–13 (2001).
[CrossRef]

N. Hansen and A. Ostermeier, “Completely derandomized self-adaptation in evolution strategies,” Evol. Comput. 9(2), 159–195 (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]

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]

2000

A. M. Weiner, “Femtosecond pulse shaping using spatial light modulators,” Rev. Sci. Instrum. 71(5), 1929–1960 (2000).
[CrossRef]

H.-G. Beyer, “Evolutionary algorithms in noisy environments: theoretical issues and guidelines for practice,” Comput. Methods Appl. Mech. Eng. 186(2-4), 239–267 (2000).
[CrossRef]

1998

A. Assion, T. Baumert, M. Bergt, T. Brixner, B. Kiefer, V. Seyfried, M. Strehle, and G. Gerber, “Control of chemical reactions by feedback-optimized phase-shaped femtosecond laser pulses,” Science 282(5390), 919–922 (1998).
[CrossRef]

1997

D. Meshulach, D. Yelin, and Y. Silberberg, “Adaptive ultrashort pulse compression and shaping,” Opt. Commun. 138(4-6), 345–348 (1997).
[CrossRef]

1992

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

1988

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

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

Assion, A.

A. Assion, T. Baumert, M. Bergt, T. Brixner, B. Kiefer, V. Seyfried, M. Strehle, and G. Gerber, “Control of chemical reactions by feedback-optimized phase-shaped femtosecond laser pulses,” Science 282(5390), 919–922 (1998).
[CrossRef]

Bäck, T.

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

Bäckb, Th.

Ch. Siedschlaga, O. M. Shirb, Th. Bäckb, and M. J. J. Vrakkinga, “Evolutionary algorithms in the optimization of dynamic molecular alignment,” Opt. Commun. 264(2), 511–518 (2006).
[CrossRef]

Bartels, R. A.

J. W. Wilson, P. Schlup, M. Lunacek, D. Whitley, and R. A. Bartels, “Calibration of liquid crystal ultrafast pulse shaper with common-path spectral interferometry and application to coherent control with a covariance matrix adaptation evolutionary strategy,” Rev. Sci. Instrum. 79(3), 033103 (2008).
[CrossRef]

Baumert, T.

A. Assion, T. Baumert, M. Bergt, T. Brixner, B. Kiefer, V. Seyfried, M. Strehle, and G. Gerber, “Control of chemical reactions by feedback-optimized phase-shaped femtosecond laser pulses,” Science 282(5390), 919–922 (1998).
[CrossRef]

Bergt, M.

A. Assion, T. Baumert, M. Bergt, T. Brixner, B. Kiefer, V. Seyfried, M. Strehle, and G. Gerber, “Control of chemical reactions by feedback-optimized phase-shaped femtosecond laser pulses,” Science 282(5390), 919–922 (1998).
[CrossRef]

Beyer, H.-G.

H.-G. Beyer, “Evolutionary algorithms in noisy environments: theoretical issues and guidelines for practice,” Comput. Methods Appl. Mech. Eng. 186(2-4), 239–267 (2000).
[CrossRef]

Brixner, T.

A. Assion, T. Baumert, M. Bergt, T. Brixner, B. Kiefer, V. Seyfried, M. Strehle, and G. Gerber, “Control of chemical reactions by feedback-optimized phase-shaped femtosecond laser pulses,” Science 282(5390), 919–922 (1998).
[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), 063412–12 (2001).

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]

Cardoza, D.

D. Cardoza, F. Langhojer, C. Trallero-Herrero, O. L. A. Monti, and T. Weinacht, “Changing pulse-shape basis for molecular learning control,” Phys. Rev. A 70(5), 053406 (2004).
[CrossRef]

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]

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]

Fanciulli, R.

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]

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

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), 023420–13 (2001).
[CrossRef]

Gerber, G.

A. Assion, T. Baumert, M. Bergt, T. Brixner, B. Kiefer, V. Seyfried, M. Strehle, and G. Gerber, “Control of chemical reactions by feedback-optimized phase-shaped femtosecond laser pulses,” Science 282(5390), 919–922 (1998).
[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]

Hansen, N.

N. Hansen and A. Ostermeier, “Completely derandomized self-adaptation in evolution strategies,” Evol. Comput. 9(2), 159–195 (2001).
[CrossRef]

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]

Herek, J. 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]

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

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]

Judson, R. S.

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

Kiefer, B.

A. Assion, T. Baumert, M. Bergt, T. Brixner, B. Kiefer, V. Seyfried, M. Strehle, and G. Gerber, “Control of chemical reactions by feedback-optimized phase-shaped femtosecond laser pulses,” Science 282(5390), 919–922 (1998).
[CrossRef]

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), 023420–13 (2001).
[CrossRef]

Langhojer, F.

D. Cardoza, F. Langhojer, C. Trallero-Herrero, O. L. A. Monti, and T. Weinacht, “Changing pulse-shape basis for molecular learning control,” Phys. Rev. A 70(5), 053406 (2004).
[CrossRef]

Lunacek, M.

J. W. Wilson, P. Schlup, M. Lunacek, D. Whitley, and R. A. Bartels, “Calibration of liquid crystal ultrafast pulse shaper with common-path spectral interferometry and application to coherent control with a covariance matrix adaptation evolutionary strategy,” Rev. Sci. Instrum. 79(3), 033103 (2008).
[CrossRef]

Meshulach, D.

D. Meshulach, D. Yelin, and Y. Silberberg, “Adaptive ultrashort pulse compression and shaping,” Opt. Commun. 138(4-6), 345–348 (1997).
[CrossRef]

Monti, O. L. A.

D. Cardoza, F. Langhojer, C. Trallero-Herrero, O. L. A. Monti, and T. Weinacht, “Changing pulse-shape basis for molecular learning control,” Phys. Rev. A 70(5), 053406 (2004).
[CrossRef]

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]

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]

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]

D. Zeidler, S. Frey, K.-L. Kompa, and M. Motzkus, “Evolutionary algorithms and their application to optimal control studies,” Phys. Rev. A 64(2), 023420–13 (2001).
[CrossRef]

Ostermeier, A.

N. Hansen and A. Ostermeier, “Completely derandomized self-adaptation in evolution strategies,” Evol. Comput. 9(2), 159–195 (2001).
[CrossRef]

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), 063412–12 (2001).

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]

Rabitz, H.

J. Roslund, M. Roth, and H. Rabitz, “Laboratory observation of quantum control level sets,” Phys. Rev. A 74(4), 043414–11 (2006).
[CrossRef]

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

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]

Riziotis, C.

C. Riziotis and A. V. Vasilakos, “Computational intelligence in photonics technology and optical networks: A survey and future perspectives,” Inf. Sci. 177(23), 5292–5315 (2007).
[CrossRef]

Roslund, J.

J. Roslund, M. Roth, and H. Rabitz, “Laboratory observation of quantum control level sets,” Phys. Rev. A 74(4), 043414–11 (2006).
[CrossRef]

Roth, M.

J. Roslund, M. Roth, and H. Rabitz, “Laboratory observation of quantum control level sets,” Phys. Rev. A 74(4), 043414–11 (2006).
[CrossRef]

Savolainen, J.

R. Fanciulli, L. Willmes, J. Savolainen, P. van der 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]

Schlup, P.

J. W. Wilson, P. Schlup, M. Lunacek, D. Whitley, and R. A. Bartels, “Calibration of liquid crystal ultrafast pulse shaper with common-path spectral interferometry and application to coherent control with a covariance matrix adaptation evolutionary strategy,” Rev. Sci. Instrum. 79(3), 033103 (2008).
[CrossRef]

Seyfried, V.

A. Assion, T. Baumert, M. Bergt, T. Brixner, B. Kiefer, V. Seyfried, M. Strehle, and G. Gerber, “Control of chemical reactions by feedback-optimized phase-shaped femtosecond laser pulses,” Science 282(5390), 919–922 (1998).
[CrossRef]

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]

Shirb, O. M.

Ch. Siedschlaga, O. M. Shirb, Th. Bäckb, and M. J. J. Vrakkinga, “Evolutionary algorithms in the optimization of dynamic molecular alignment,” Opt. Commun. 264(2), 511–518 (2006).
[CrossRef]

Siedschlaga, Ch.

Ch. Siedschlaga, O. M. Shirb, Th. Bäckb, and M. J. J. Vrakkinga, “Evolutionary algorithms in the optimization of dynamic molecular alignment,” Opt. Commun. 264(2), 511–518 (2006).
[CrossRef]

Silberberg, Y.

D. Meshulach, D. Yelin, and Y. Silberberg, “Adaptive ultrashort pulse compression and shaping,” Opt. Commun. 138(4-6), 345–348 (1997).
[CrossRef]

Strehle, M.

A. Assion, T. Baumert, M. Bergt, T. Brixner, B. Kiefer, V. Seyfried, M. Strehle, and G. Gerber, “Control of chemical reactions by feedback-optimized phase-shaped femtosecond laser pulses,” Science 282(5390), 919–922 (1998).
[CrossRef]

Trallero-Herrero, C.

D. Cardoza, F. Langhojer, C. Trallero-Herrero, O. L. A. Monti, and T. Weinacht, “Changing pulse-shape basis for molecular learning control,” Phys. Rev. A 70(5), 053406 (2004).
[CrossRef]

van der Walle, P.

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

Vasilakos, A. V.

C. Riziotis and A. V. Vasilakos, “Computational intelligence in photonics technology and optical networks: A survey and future perspectives,” Inf. Sci. 177(23), 5292–5315 (2007).
[CrossRef]

Vrakkinga, M. J. J.

Ch. Siedschlaga, O. M. Shirb, Th. Bäckb, and M. J. J. Vrakkinga, “Evolutionary algorithms in the optimization of dynamic molecular alignment,” Opt. Commun. 264(2), 511–518 (2006).
[CrossRef]

Weinacht, T.

D. Cardoza, F. Langhojer, C. Trallero-Herrero, O. L. A. Monti, and T. Weinacht, “Changing pulse-shape basis for molecular learning control,” Phys. Rev. A 70(5), 053406 (2004).
[CrossRef]

Weinacht, T. C.

B. J. Pearson, J. L. White, T. C. Weinacht, and P. H. Bucksbaum, “Coherent control using adaptive learning algorithms,” Phys. Rev. A 63(6), 063412–12 (2001).

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]

Weiner, A. M.

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]

A. M. Weiner, “Femtosecond pulse shaping using spatial light modulators,” Rev. Sci. Instrum. 71(5), 1929–1960 (2000).
[CrossRef]

Werschnik, J.

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Figures (10)

Fig. 1.
Fig. 1.

A model block diagram for an adaptive laser pulse shaper.

Fig. 2.
Fig. 2.

(Left) Second harmonic spectrum of a transform-limited Gaussian pulse centered at 800nm with a bandwidth of 35nm (dot), along with second harmonic spectra of the same pulse with second order (dash-dot), third order (dash), and fourth order (solid) phase terms. (Right) Variations of the total second harmonic energy as the second order (dash-dot), third order (dash), and fourth order (solid) phase terms are scanned one at a time.

Fig. 3.
Fig. 3.

Variations of the total second harmonic energy of a laser pulse (a) when two phase orders exist simultaneously in the input phase profile. (b) The level set of the total second harmonic energy generated by the second, the third, and the fourth order phase terms for a normalized energy of 0.8 and (c) 0.65.

Fig. 4.
Fig. 4.

A typical phase profile found using our previous optimization experiments and used here as the phase of the input laser pulse in our simulations.

Fig. 5.
Fig. 5.

The first few orders of the four basis functions used in our simulations: (top left) Legendre, (top right), Polynomial, (bottom left) Chebyshev, and (bottom right) Fourier basis functions

Fig. 6.
Fig. 6.

Variations of the mean fitness found after 150 generations using (left) Legendre and (right) polynomial basis functions. In each case, five sets of simulations have been done with the maximum order of each basis function being four (star), five (solid), six (dash-dot), seven (dash), and eight (dot).

Fig. 7.
Fig. 7.

The effect of different parameterizations on the learning dynamics of the algorithm in the absence (left) and presence (right) of noise. The employed parameterizations are direct (grey), Fourier series (black), Chebyshev (green), polynomial (blue), and Legendre (red) functions.

Fig. 8.
Fig. 8.

Experimental results of using different parameterizations for adaptive second harmonic generation: (Left) Fitness vs. generation curves for Legendre (cross) and direct (circle) parameterizations, and (Right) the retrieved spectral phase profiles using Legendre (dot) and direct (dash-dot) parameterizations. The phase profile retrieved using Legendre parameterization is also shown in the wrapped form (solid) to be comparable with the profile obtained by direct parameterization.

Fig. 9.
Fig. 9.

Experimental results of using different parameterizations for adaptive second harmonic generation: (Left) Fitness vs. generation curves for Legendre (cross) and polynomial (circle) parameterizations, and (Right) the retrieved spectral phase profiles using Legendre (solid) and polynomial (dash-dot) parameterizations.

Fig. 10.
Fig. 10.

The modified contour plots of the total SHG energy parameterized with the second order and the fourth order modified polynomial functions. The parameters η and η’ are the coefficients of the second/fourth order polynomial added to a fourth/second order polynomial function of the normalized frequency. The horizontal/vertical axes represent the fourth/second order phase terms, respectively. The ranges of coefficients and the color maps are the same in all contour plots. Dark blue in the color map represents zero, and dark red represents one.

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