## Interaction of a single-cycle laser pulse with a bound electron without ionization |

Optics Express, Vol. 18, Issue 14, pp. 15155-15168 (2010)

http://dx.doi.org/10.1364/OE.18.015155

Acrobat PDF (1354 KB)

### Abstract

In this paper, interaction of an ultrashort single-cycle pulse (USCP) with a bound electron without ionization is reported for the first time. For a more realistic mathematical description of USCPs, Hermitian polynomials and combination of Laguerre functions are used for two different single cycle excitation cases. These single cycle pulse models are used as driving functions for the classical approach to model the interaction of a bound electron with an applied electric field. A new novel time-domain technique was developed for modifying the classical Lorentz damped oscillator model in order to make it compatible with USCP excitation. This modification turned the Lorentz oscillator model equation into a Hill-like function with non-periodic time varying damping and spring coefficients. Numerical results are presented for two different excitation models and for varying spring and damping constants. Our two driving model excitations provide quite different time response of the bound electron. Different polarization response will subsequently result in relative differences in the time dependent index of refraction.

© 2010 OSA

## 1. Introduction

2. M. A. Porras, “Nonsinusoidal few-cycle pulsed light beams in free space,” J. Opt. Soc. Am. B **16**(9), 1468 (1999). [CrossRef]

3. P. B. Corkum and F. Krausz, “Attosecond science,” Nat. Phys. **3**(6), 381–387 (2007). [CrossRef]

12. R. M. Joseph, S. C. Hagness, and A. Taflove, “Direct time integration of Maxwell’s equations in linear dispersive media with absorption for scattering and propagation of femtosecond electromagnetic pulses,” Opt. Lett. **16**(18), 1412–1414 (1991). [CrossRef] [PubMed]

24. D. Hovhannisyan, “Propagation of a femtosecond laser pulse of a few optical oscillations in a uniaxial crystal,” Microw. Opt. Technol. Lett. **36**(4), 280–285 (2003). [CrossRef]

## 2. Mathematical model

- i) Arbitrary transient steepness: The rising and the falling times of the signal can be essentially unequal.
- ii) Varying zero spacing: The distances between zero-crossing points may be essentially unequal.
- iii) Both the waveform envelope and its first spatial and temporal derivatives are continuous.
- iv) Arbitrary envelope asymmetry: USCP waveforms can be classified conventionally for two groups.

*c*is the velocity of light in vacuum,

*z*is the propagation direction and

^{nd}and 4

^{th}order Laguerre functions are used to define a single USCP:where the phase term is defined as

*ϕ*is the initial phase [Fig. 1(a) ].

22. H. Xiao and K. E. Oughstun, “Failure of the group velocity description for ultrawideband pulse propagation in a casually dispersive, absorptive dielectric,” J. Opt. Soc. Am. B **16**(10), 1773 (1999). [CrossRef]

29. J. E. Rothenberg, “Space-time focusing: Breakdown of the slowly varying envelope approximation in the self-focusing of femtosecond pulses,” Opt. Lett. **17**(19), 1340 (1992). [CrossRef] [PubMed]

31. M. D. Crisp, “Propagation of small-area pulses of coherent light through a resonant medium,” Phys. Rev. A **1**(6), 1604–1611 (1970). [CrossRef]

37. K. S. Cole and R. H. Cole, “Dispersion and absorption in dielectrics,” J. Chem. Phys. **9**(4), 341–351 (1941). [CrossRef]

38. A. B. Djurišic and E. H. Li, “Modeling the index of refraction of insulating solids with a modified Lorentz oscillator model,” Appl. Opt. **37**(22), 5291 (1998). [CrossRef]

8. A. Couairon, M. Franco, A. Mysyrowicz, J. Biegert, and U. Keller, “Pulse self-compression to the single-cycle limit by filamentation in a gas with a pressure gradient,” Opt. Lett. **30**(19), 2657–2659 (2005). [CrossRef] [PubMed]

10. G. Steinmeyer, D. H. Sutter, L. Gallmann, N. Matuschek, and U. Keller, “Frontiers in Ultrashort Pulse Generation: Pushing the Limits in Linear and Nonlinear Optics,” Science **286**(5444), 1507–1512 (1999). [CrossRef] [PubMed]

42. K. E. Oughstun and G. C. Sherman, “Uniform asymptotic description of electromagnetic pulse propagation in a linear dispersive medium with absorption (the Lorentz medium),” J. Opt. Soc. Am. A **6**(9), 1394–1420 (1989). [CrossRef]

^{2}/volt which is equivalent to Coulomb*meter/Newton. So physically, modifier function defines dipole moment per unit force. Plugging Eq. (7) into Eq. (6), we obtainPerforming the necessary calculations in Eq. (8), we obtain Eq. (10):

## 3. Numerical results and discussions

^{−20}m – 10

^{−21}m which is in the scale of electron radius length. Finally, as the spring constant is increased to relatively higher values, the Laguerre interaction settles down into the inverted phase time profile of the excitation pulse, too (inverted Laguerre pulse) [see Fig. 4(i)]. Figure 4 shows a very clear distinction between the interaction characteristics of Laguerre and Hermitian USCPs until the spring constant is 2500 N/m (after this value, we obtain only the inverted phase time profile of the excitation source for the oscillation). The oscillation characteristics of bound electron under different single USCP sources originates from modifier function approach. The Hill-like equation, which is the result of the modification on the classic Lorentz damped oscillator model with the modifier function approach, causes the time varying physical parameters to come into play during the interaction process. Since these physical parameters (time varying damping and spring coefficients) are absolutely source dependent, they behave differently in the pulse duration of each different USCP source. As a result of this, we see different oscillation profiles for a bound electron under a single Laguerre and Hermitian USCP excitations.

^{16}) than the previous case (Fig. 4). An interesting feature here in Fig. 5(a) and Fig. 4(g) is that although they are at the same spring constant value, they show different oscillation characteristics. Due to a higher dampimg coefficient in Fig. 5(a), while the oscillation attenuates quicker at the second half cycle of the Laguerre USCP than in Fig. 4(g), it hits to a higher peak at the first half cycle of the excitation pulse than in Fig. 4(g). So, for a reasonable value of spring constant, while relatively higher damping coefficient makes the first half cycle of the Laguerre USCP more efficient in the means of interaction, it makes the second half cycle less efficient. In order to compare oscillation results more detailly between Figs. 5(a) and 4(g), it is necessary to look at their physical parameter solutions such as time varying damping and time varying spring coefficients. As it is explained above, these time varying parameters come into play due to the nature of “Modifier Function Approach”. In Fig. 6 , time varying damping coefficient, time varying spring coefficient and the modifier function solutions ofFigures 4(g) and 5(a) are shown respectively for two different damping constant values with a fixed spring constant at 525 N/m. In Figs. 6(a) and 6(c), a sudden jump is seen in the time varying damping coefficient profiles at the time point where the excitation pulse changes itspolarization direction. Although they look identical, the magnified views [see Figs. 7(a) , 7(b), 7(c), 7(d)] of the left and right wings of the damping coefficient show the difference between two different damping constant cases. Here, the left wing corresponds to the first half cycle, right wing corresponds to the second half cycle of the Laguerre excitation pulse. Comparing the amount of the change on the y-axis with the time duration on the x-axis between Figs. 7(a) - 7(b), and 7(c) - 7(d), it is easy to see the reasonable amount of difference to affect the solution of modifier function [see Figs. 6(i), 6(j)]. For time varying spring coefficients [see Figs. 6(e), 6(g)], a significant difference is seen in the time profile although the spring constant values are the same for both cases. The jump in Fig. 6(g) hits a higher peak than the jump in Fig. 6(e). This can be a reasonable explanation for a relatively low oscillation tendency in the second half cycle of Fig. 5(a) than the Fig. 4(g). It can be said that, due to the dissipation of higher energy, this jump causes a lower oscillation profile for the bound electron during its interaction with the second half cycle of the Laguerre pulse in Fig. 5(a) than in Fig. 4(g). In Fig. 5(c), as the spring constant is increased to a relatively higher values, same as in Fig. 4(i), the oscillation profile settles down into the inverted time phase profile of the excitation pulse. Different from Fig. 4(i), the oscillation settles down at a relativley lower spring constant value. So, it can be said that, for a higher damping constant, a lower spring constant is enough to stabilize the oscillation profile in time domain.

^{17}(Fig. 8), very different oscillation behaviors are seen than the previuos cases (Fig. 4) of Hermitian pulse excitation. The most prominent feature in Figs. 8(a), 8(b) and 8(c) is the high frequency oscillation profile with a phase delay wrt. excitation pulse. In Fig. 8, spring constant is increased gradually from 8(a) to 8(c) while keeping the damping value constant. For a relatively low value of spring constant in Fig. 8(a), the main lobe and the trailing tail of the excitation pulse have almost no effect on the oscillation of the electron. The bound electron starts sensing the leading tail of the Hermitian excitation after a phase delay of 5 fs. In Fig. 9 , the modifier function solutions for the Hermitian pulse excitation for Fig. 8 is shown. As it is clearly seen in Fig. 9(a), modifier function suppresses the interaction effect of main lobe and the trailing tail of Hermitian function. As a result of this, the bound electron starts sensing the excitation pulse with a phase delay [Fig. 8(a)] associated with the modifier function. Same behaviour of the modifier function is seen in Figs. 9(b) and 9(c), too. As a result of this, approximately 2fs phase delay occurs in Figs. 8(b) and 8(c). In Fig. 9(d), the type of modifier function is seen that gives a completely phase inverted time profile of the excitation pulse for the oscillation of the bound electron. In Fig. 8(d), the stabilized oscillation profile is seen as a result of this modifier function.

## 4. Conclusion

## References and links

1. | A. Couairon, J. Biegert, C. P. Hauri, W. Kornelis, F. W. Helbing, U. Keller, and A. Mysyrowicz, “Self-compression of ultrashort laser pulses down to one optical cycle by filamentation,” J. Mod. Opt. |

2. | M. A. Porras, “Nonsinusoidal few-cycle pulsed light beams in free space,” J. Opt. Soc. Am. B |

3. | P. B. Corkum and F. Krausz, “Attosecond science,” Nat. Phys. |

4. | A. Zewail, “Femtochemistry: atomic-scale dynamics of the chemical bond,” J. Phys. Chem. A |

5. | H. Niikura, F. Légaré, R. Hasbani, A. D. Bandrauk, M. Y. Ivanov, D. M. Villeneuve, and P. B. Corkum, “Sub-laser-cycle electron pulses for probing molecular dynamics,” Nature |

6. | J. Itatani, J. Levesque, D. Zeidler, H. Niikura, H. Pépin, J. C. Kieffer, P. B. Corkum, and D. M. Villeneuve, “Tomographic imaging of molecular orbitals,” Nature |

7. | G. Krauss, S. Lohss, T. Hanke, A. Sell, S. Eggert, R. Huber, and A. Leitenstorfer, “Synthesis of a single cycle of light with compact erbium-doped fibre technology,” Nat. Photonics |

8. | A. Couairon, M. Franco, A. Mysyrowicz, J. Biegert, and U. Keller, “Pulse self-compression to the single-cycle limit by filamentation in a gas with a pressure gradient,” Opt. Lett. |

9. | Y. Yan, E. B. Gamble Jr, and K. A. Nelson, “Impulsive stimulated scattering: General importance in femtosecond laser pulse interactions with matter, and spectroscopic applications,” J. Chem. Phys. |

10. | G. Steinmeyer, D. H. Sutter, L. Gallmann, N. Matuschek, and U. Keller, “Frontiers in Ultrashort Pulse Generation: Pushing the Limits in Linear and Nonlinear Optics,” Science |

11. | K. Akimoto, Properties and Applications of ultrashort electromagnetic mono- and sub- cycle waves. |

12. | R. M. Joseph, S. C. Hagness, and A. Taflove, “Direct time integration of Maxwell’s equations in linear dispersive media with absorption for scattering and propagation of femtosecond electromagnetic pulses,” Opt. Lett. |

13. | S. L. Dvorak, D. G. Dudley. Propagation of ultrawideband electromagnetic pulses through dispersive media. |

14. | S. A. Kozlov and S. V. Sazanov, “Nonlinear propagation of optical pulses of a few oscillations duration in dielectric media,” Sov. Phys. JETP |

15. | H. Wilkelmsson, J. H. Trombert, and J. F. Eloy, “Dispersive and dissipative medium response to an ultrashort pulse: A green’s function approach,” Phys. Scr. |

16. | P. Kinsler and G. H. C. New, “Few-cycle pulse propagation,” Phys. Rev. A |

17. | J. F. Eloy, and F. Moriamez, Spectral analysis of EM ultrashort pulses at coherence limit. Modelling. |

18. | J. F. Eloy and H. Wilhelmsson, “Response of a bounded plasma to ultrashort pulse excitation,” Phys. Scr. |

19. | M. Pietrzyk, I. Kanattsikov, and U. Bandelow, “On the propagation of vector ultrashort pulses,” J. Nonlinear Math. Phys. |

20. | B. Macke and B. Segard, “Propagation of light pulses at a negative group velocity,” Eur. Phys. J. D |

21. | Q. Zou and B. Lu, “Propagation properties of ultrashort pulsed beams with constant waist width in free space,” Opt. Laser Technol. |

22. | H. Xiao and K. E. Oughstun, “Failure of the group velocity description for ultrawideband pulse propagation in a casually dispersive, absorptive dielectric,” J. Opt. Soc. Am. B |

23. | A. L. Gutman, “Electrodynamics of short pulses for pulse durations comparable to relaxation times of a medium,” Dokl. Phys. |

24. | D. Hovhannisyan, “Propagation of a femtosecond laser pulse of a few optical oscillations in a uniaxial crystal,” Microw. Opt. Technol. Lett. |

25. | A. B. Shvartsburg, Single-cycle waveforms and non-periodic waves in dispersive media (exactly solvable models). |

26. | Z. Wang, Z. Zhang, Z. Xu, and Q. Lin, “Space-time profiles of an ultrashort pulsed Gaussian beam,” IEEE J. Quantum Electron. |

27. | A. B. Shvartsburg, Time-Domain Optics of Ultrashort Waveforms. |

28. | A. B. Shvartsburg, Impulse Time-Domain Electromagnetic of Continuos Media. |

29. | J. E. Rothenberg, “Space-time focusing: Breakdown of the slowly varying envelope approximation in the self-focusing of femtosecond pulses,” Opt. Lett. |

30. | H. Kumagai, S. H. Cho, K. Ishikawa, K. Midorikawa, M. Fujimoto, S. Aoshima, and Y. Tsuchiya, “Observation of the comples propagation of a femtosecond laser pulse in a dispersive transparent bulk material,” J. Opt. Soc. Am. |

31. | M. D. Crisp, “Propagation of small-area pulses of coherent light through a resonant medium,” Phys. Rev. A |

32. | B. K. P. Scaife, Principles of Dielectrics, Oxford University Press, Oxford, 1989. |

33. | A. L. Gutman, Passage of short pulse throughout oscillating circuit with dielectric in condenser. |

34. | V. V. Daniel, Dielectric Relaxation. |

35. | A. B. Shvartsburg, Optics of nonstationary media, |

36. | A. B. Shvartsburg, G. Petite. Progress in Optics, Vol. 44 (Ed. E Wolf), p. 143, Elsevier Sci, 2002. |

37. | K. S. Cole and R. H. Cole, “Dispersion and absorption in dielectrics,” J. Chem. Phys. |

38. | A. B. Djurišic and E. H. Li, “Modeling the index of refraction of insulating solids with a modified Lorentz oscillator model,” Appl. Opt. |

39. | S. P. Blanc, R. Sauerbrey, S. C. Rae, and K. Burnett, “Spectral blue shifting of a femtosecond laser pulse propagating through a high-pressure gas,” J. Opt. Soc. Am. |

40. | G. P. Agrawal and N. A. Olsson, “Self-phase modulation and spectral broadening of optical pulses in semiconductor laser amplifiers,” IEEE J. Quantum Electron. |

41. | C. B. Schaffer, Interaction of femtosecond laser pulses with transparent materials, Ph.D. Thesis. Harvard University, May 2001. |

42. | K. E. Oughstun and G. C. Sherman, “Uniform asymptotic description of electromagnetic pulse propagation in a linear dispersive medium with absorption (the Lorentz medium),” J. Opt. Soc. Am. A |

43. | L. N. Hand, and J. D. Finch, Analytical Mechanics. |

**OCIS Codes**

(000.3860) General : Mathematical methods in physics

(260.5430) Physical optics : Polarization

(320.2250) Ultrafast optics : Femtosecond phenomena

(320.5550) Ultrafast optics : Pulses

(320.7090) Ultrafast optics : Ultrafast lasers

(320.7120) Ultrafast optics : Ultrafast phenomena

**ToC Category:**

Ultrafast Optics

**History**

Original Manuscript: May 6, 2010

Revised Manuscript: June 17, 2010

Manuscript Accepted: June 18, 2010

Published: June 30, 2010

**Citation**

Ufuk Parali and Dennis R. Alexander, "Interaction of a single-cycle laser pulse with a bound electron without ionization," Opt. Express **18**, 15155-15168 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-14-15155

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### References

- A. Couairon, J. Biegert, C. P. Hauri, W. Kornelis, F. W. Helbing, U. Keller, and A. Mysyrowicz, “Self-compression of ultrashort laser pulses down to one optical cycle by filamentation,” J. Mod. Opt. 53(1–2), (2006).
- M. A. Porras, “Nonsinusoidal few-cycle pulsed light beams in free space,” J. Opt. Soc. Am. B 16(9), 1468 (1999). [CrossRef]
- P. B. Corkum and F. Krausz, “Attosecond science,” Nat. Phys. 3(6), 381–387 (2007). [CrossRef]
- A. Zewail, “Femtochemistry: atomic-scale dynamics of the chemical bond,” J. Phys. Chem. A 104(24), 5660–5694 (2000). [CrossRef]
- H. Niikura, F. Légaré, R. Hasbani, A. D. Bandrauk, M. Y. Ivanov, D. M. Villeneuve, and P. B. Corkum, “Sub-laser-cycle electron pulses for probing molecular dynamics,” Nature 417(6892), 917–922 (2002). [CrossRef] [PubMed]
- J. Itatani, J. Levesque, D. Zeidler, H. Niikura, H. Pépin, J. C. Kieffer, P. B. Corkum, and D. M. Villeneuve, “Tomographic imaging of molecular orbitals,” Nature 432(7019), 867–871 (2004). [CrossRef] [PubMed]
- G. Krauss, S. Lohss, T. Hanke, A. Sell, S. Eggert, R. Huber, and A. Leitenstorfer, “Synthesis of a single cycle of light with compact erbium-doped fibre technology,” Nat. Photonics 4(1), 33–36 (2010). [CrossRef]
- A. Couairon, M. Franco, A. Mysyrowicz, J. Biegert, and U. Keller, “Pulse self-compression to the single-cycle limit by filamentation in a gas with a pressure gradient,” Opt. Lett. 30(19), 2657–2659 (2005). [CrossRef] [PubMed]
- Y. Yan, E. B. Gamble, and K. A. Nelson, “Impulsive stimulated scattering: General importance in femtosecond laser pulse interactions with matter, and spectroscopic applications,” J. Chem. Phys. 83(11), 5391 (1985). [CrossRef]
- G. Steinmeyer, D. H. Sutter, L. Gallmann, N. Matuschek, and U. Keller, “Frontiers in Ultrashort Pulse Generation: Pushing the Limits in Linear and Nonlinear Optics,” Science 286(5444), 1507–1512 (1999). [CrossRef] [PubMed]
- K. Akimoto, Properties and Applications of ultrashort electromagnetic mono- and sub- cycle waves. Journal ofthe Physical Society of Japan, Vol. 65, No. 7, 2020–2032, 1996.
- R. M. Joseph, S. C. Hagness, and A. Taflove, “Direct time integration of Maxwell’s equations in linear dispersive media with absorption for scattering and propagation of femtosecond electromagnetic pulses,” Opt. Lett. 16(18), 1412–1414 (1991). [CrossRef] [PubMed]
- S. L. Dvorak, D. G. Dudley. Propagation of ultrawideband electromagnetic pulses through dispersive media. IEEE Transaction of Electromagnetic Compatibility, Vol. 37. No. 2, May 1995.
- S. A. Kozlov and S. V. Sazanov, “Nonlinear propagation of optical pulses of a few oscillations duration in dielectric media,” Sov. Phys. JETP 84(2), 221–228 (1997). [CrossRef]
- H. Wilkelmsson, J. H. Trombert, and J. F. Eloy, “Dispersive and dissipative medium response to an ultrashort pulse: A green’s function approach,” Phys. Scr. 52(1), 102–107 (1995). [CrossRef]
- P. Kinsler and G. H. C. New, “Few-cycle pulse propagation,” Phys. Rev. A 67(2), 023813 (2003). [CrossRef]
- J. F. Eloy, and F. Moriamez, Spectral analysis of EM ultrashort pulses at coherence limit. Modelling. SPIE Intense Microwave and Particle Beams III, Vol. 1629, 1992.
- J. F. Eloy and H. Wilhelmsson, “Response of a bounded plasma to ultrashort pulse excitation,” Phys. Scr. 55(4), 475–477 (1997). [CrossRef]
- M. Pietrzyk, I. Kanattsikov, and U. Bandelow, “On the propagation of vector ultrashort pulses,” J. Nonlinear Math. Phys. 15(2), 162–170 (2008). [CrossRef]
- B. Macke and B. Segard, “Propagation of light pulses at a negative group velocity,” Eur. Phys. J. D 23, 125–141 (2003). [CrossRef]
- Q. Zou and B. Lu, “Propagation properties of ultrashort pulsed beams with constant waist width in free space,” Opt. Laser Technol. 39(3), 619–625 (2007). [CrossRef]
- H. Xiao and K. E. Oughstun, “Failure of the group velocity description for ultrawideband pulse propagation in a casually dispersive, absorptive dielectric,” J. Opt. Soc. Am. B 16(10), 1773 (1999). [CrossRef]
- A. L. Gutman, “Electrodynamics of short pulses for pulse durations comparable to relaxation times of a medium,” Dokl. Phys. 43(6), 343–345 (1998).
- D. Hovhannisyan, “Propagation of a femtosecond laser pulse of a few optical oscillations in a uniaxial crystal,” Microw. Opt. Technol. Lett. 36(4), 280–285 (2003). [CrossRef]
- A. B. Shvartsburg, Single-cycle waveforms and non-periodic waves in dispersive media (exactly solvable models). Physics – Uspekhi, Vol. 41, No. 1, 77–94, 1998.
- Z. Wang, Z. Zhang, Z. Xu, and Q. Lin, “Space-time profiles of an ultrashort pulsed Gaussian beam,” IEEE J. Quantum Electron. 33(4), (1997).
- A. B. Shvartsburg, Time-Domain Optics of Ultrashort Waveforms. Clarendon Press, Oxford, 1996.
- A. B. Shvartsburg, Impulse Time-Domain Electromagnetic of Continuos Media. Birkhauser Verlag, Boston, 1999.
- J. E. Rothenberg, “Space-time focusing: Breakdown of the slowly varying envelope approximation in the self-focusing of femtosecond pulses,” Opt. Lett. 17(19), 1340 (1992). [CrossRef] [PubMed]
- H. Kumagai, S. H. Cho, K. Ishikawa, K. Midorikawa, M. Fujimoto, S. Aoshima, and Y. Tsuchiya, “Observation of the comples propagation of a femtosecond laser pulse in a dispersive transparent bulk material,” J. Opt. Soc. Am. 20(3), (2003).
- M. D. Crisp, “Propagation of small-area pulses of coherent light through a resonant medium,” Phys. Rev. A 1(6), 1604–1611 (1970). [CrossRef]
- B. K. P. Scaife, Principles of Dielectrics, Oxford University Press, Oxford, 1989.
- A. L. Gutman, Passage of short pulse throughout oscillating circuit with dielectric in condenser. Ultra-Wideband, Short-Pulse Electromagnetics 4, Kluwer Academic / Plenum Publishers, New York, 1999.
- V. V. Daniel, Dielectric Relaxation. Academic Press, New York, 1967.
- A. B. Shvartsburg, Optics of nonstationary media, Physics – Uspekhi, Vol. 48, No. 8, 797–823, 2005.
- A. B. Shvartsburg, G. Petite. Progress in Optics, Vol. 44 (Ed. E Wolf), p. 143, Elsevier Sci, 2002.
- K. S. Cole and R. H. Cole, “Dispersion and absorption in dielectrics,” J. Chem. Phys. 9(4), 341–351 (1941). [CrossRef]
- A. B. Djurišic and E. H. Li, “Modeling the index of refraction of insulating solids with a modified Lorentz oscillator model,” Appl. Opt. 37(22), 5291 (1998). [CrossRef]
- S. P. Blanc, R. Sauerbrey, S. C. Rae, and K. Burnett, “Spectral blue shifting of a femtosecond laser pulse propagating through a high-pressure gas,” J. Opt. Soc. Am. 10(10), (1993).
- G. P. Agrawal and N. A. Olsson, “Self-phase modulation and spectral broadening of optical pulses in semiconductor laser amplifiers,” IEEE J. Quantum Electron. 25(11), 2297–2306 (1989). [CrossRef]
- C. B. Schaffer, Interaction of femtosecond laser pulses with transparent materials, Ph.D. Thesis. Harvard University, May 2001.
- K. E. Oughstun and G. C. Sherman, “Uniform asymptotic description of electromagnetic pulse propagation in a linear dispersive medium with absorption (the Lorentz medium),” J. Opt. Soc. Am. A 6(9), 1394–1420 (1989). [CrossRef]
- L. N. Hand, and J. D. Finch, Analytical Mechanics. Cambridge University Press, 7th edition, Cambridge, 2008.

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