## Influence of precursor fields on ultrashort pulse autocorrelation measurements and pulse width evolution

Optics Express, Vol. 8, Issue 8, pp. 481-491 (2001)

http://dx.doi.org/10.1364/OE.8.000481

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### Abstract

The influence of the precursor fields of a double resonance Lorentz model dielectric on ultrashort pulse autocorrelation measurements and the resultant dynamical pulse width evolution is presented.

© Optical Society of America

## 1. Introduction

1. D. J. Bradley and G. H. C. New, “Ultrashort pulse measurements,” Proc. IEEE **62**, 313–345 (1974). [CrossRef]

3. F. Hache, T. J. Driscoll, M. Cavallari, and G. M. Gale, “Measurement of ultrashort pulse durations by interferometric autocorrelation: influence of various parameters,” Appl. Opt. **35**, 3230–3236 (1996). [CrossRef] [PubMed]

4. A. Baltuska, Z. Wei, M. S. Pshenichnikov, and D. A. Wiersma, “Optical pulse compression to 5 fs at a 1-MHz repetition rate,” Opt. Lett. **22**, 102–104 (1997). [CrossRef] [PubMed]

6. P. N. Butcher and D. Cotter, *The elements of nonlinear optics* (Cambridge U. Press, Cambridge, 1990) *Chap. 2.* [CrossRef]

7. K. E. Oughstun and C. M. Balictsis, “Gaussian pulse propagation in a dispersive, absorbing dielectric,” Phys. Rev. Lett. **77**, 2210–2213 (1996). [CrossRef] [PubMed]

10. H. Xiao and K. E. Oughstun, “Failure of the group velocity description for ultrawideband pulse propagation in a double resonance Lorentz model dielectric,” J. Opt. Soc. Am. B **16**, 1773–1785 (1999). [CrossRef]

9. K. E. Oughstun and H. Xiao, “Failure of the quasimonochromatic approximation for ultrashort pulse propagation in a dispersive, attenuative medium,” Phys. Rev. Lett. **78**, 642–645 (1997). [CrossRef]

10. H. Xiao and K. E. Oughstun, “Failure of the group velocity description for ultrawideband pulse propagation in a double resonance Lorentz model dielectric,” J. Opt. Soc. Am. B **16**, 1773–1785 (1999). [CrossRef]

7. K. E. Oughstun and C. M. Balictsis, “Gaussian pulse propagation in a dispersive, absorbing dielectric,” Phys. Rev. Lett. **77**, 2210–2213 (1996). [CrossRef] [PubMed]

11. K. E. Oughstun and G. C. Sherman, *Electromagnetic pulse propagation in causal dielectrics* (Springer-Verlag, Berlin, 1994) *Chapters 3 & 9.* [CrossRef]

## 2. Analytical Descriptions of Dispersive Pulse Propagation

*A*(0,

*t*)=

*u*(

*t*)sin(

*ω*) that is initiated at the plane

_{c}t*z*=0 with fixed carrier frequency

*ω*>0 and is propagating in the positive

_{c}*z*-direction through a double resonance Lorentz model dielectric [10

10. H. Xiao and K. E. Oughstun, “Failure of the group velocity description for ultrawideband pulse propagation in a double resonance Lorentz model dielectric,” J. Opt. Soc. Am. B **16**, 1773–1785 (1999). [CrossRef]

*ω*is the undamped resonance frequency,

_{j}*b*the plasma frequency, and

_{j}*δ*the phenomenological damping constant of the

_{j}*j*resonance line (

^{th}*j*=

*0*,

*2*). This causal model provides an accurate description of both normal and anomalous dispersion in homogeneous, isotropic, locally linear optical materials when the input pulse carrier frequency is situated within the medium passband between the two absorption bands [

*ω*

_{0},

*ω*

_{1}] and [

*ω*

_{2},

*ω*

_{3}], where

*ω*

_{0}=1.74×10

^{14}

*r*/

*s*,

*b*

_{0}=1.21×10

^{14}

*r*/

*s*,

*δ*

_{0}=4.96×10

^{13}

*r*/

*s*) and visible (

*ω*

_{2}=9.14×10

^{15}

*r*/

*s*,

*b*

_{2}=6.72×10

^{15}

*r*/

*s*,

*δ*

_{2}=1.43×10

^{15}

*r*/

*s*) resonance lines, with associated relaxation times

*τ*

_{r0}~2

*π*/

*δ*

_{0}=126.8

*f*sec and

*τ*

_{r2}~2

*π*/

*δ*

_{2}=4.4

*f*sec, respectively. The frequency dispersion of the real and imaginary parts of the complex index of refraction (1) is illustrated in Fig. 1 along the positive real angular frequency axis. The indicated angular frequency

*ω*=

_{c}*ω*

_{min}=1.615×10

^{15}

*r*/

*s*denotes the frequency value at which

*n*(

_{r}*ω*) has an inflection point in the passband (

*ω*

_{1},

*ω*

_{2}) and the material dispersion is a minimum.

*A*(

*z*,

*t*) is given by the Fourier-Laplace integral [10

**16**, 1773–1785 (1999). [CrossRef]

*z*≥0, where

*u*̃(

*ω*) is the temporal frequency spectrum of the initial pulse envelope function

*u*(

*t*), and where

*a*is a constant that is greater than the abscissa of absolute convergence [11

11. K. E. Oughstun and G. C. Sherman, *Electromagnetic pulse propagation in causal dielectrics* (Springer-Verlag, Berlin, 1994) *Chapters 3 & 9.* [CrossRef]

*u*(

*t*). Here

*ℜ*{∗} denotes the real part of the quantity appearing in brackets. The frequency spectrum

*A*̃(

*z*,

*ω*) of

*A*(

*z*,

*t*) satisfies the Helmholtz equation (∇

^{2}+

*k*̃

^{2}(

*ω*))

*A*̃(

*z*,

*ω*)=0, where

*k*̃(

*ω*)=

*β*(

*ω*)+

*iα*(

*ω*)=

*ωn*(

*ω*)/

*c*is the complex wavenumber of the plane wave field with propagation factor

*β*(

*ω*) given by the real part and attenuation factor

*α*(

*ω*) given by the imaginary part of the complex wavenumber.

*k*̃(

*ω*) about the carrier frequency

*ω*

_{c}*k*̃

^{(j)}(

*ω*)≡

*∂*̃(

^{j}k*ω*)/

*∂ω*, is truncated after only a few terms with some undefined error. Typically, the quartic and higher-order terms in Eq. (3) are neglected [5,6

^{j}6. P. N. Butcher and D. Cotter, *The elements of nonlinear optics* (Cambridge U. Press, Cambridge, 1990) *Chap. 2.* [CrossRef]

*cubic dispersion approximation*, the pulse is found to propagate at the classical group velocity

*v*(

_{g}*ω*)=1/(

*∂β*(

*ω*)/

*∂ω*), while the quantity

*k*̃

^{(2)}(

*ω*) results in the so-called group velocity dispersion and the cubic term yields asymmetric pulse distortion [5].

_{c}9. K. E. Oughstun and H. Xiao, “Failure of the quasimonochromatic approximation for ultrashort pulse propagation in a dispersive, attenuative medium,” Phys. Rev. Lett. **78**, 642–645 (1997). [CrossRef]

**16**, 1773–1785 (1999). [CrossRef]

11. K. E. Oughstun and G. C. Sherman, *Electromagnetic pulse propagation in causal dielectrics* (Springer-Verlag, Berlin, 1994) *Chapters 3 & 9.* [CrossRef]

*ω*=

*ω*. A correct description of dispersive pulse propagation phenomena is provided by the modern asymptotic theory [9

_{c}9. K. E. Oughstun and H. Xiao, “Failure of the quasimonochromatic approximation for ultrashort pulse propagation in a dispersive, attenuative medium,” Phys. Rev. Lett. **78**, 642–645 (1997). [CrossRef]

*Electromagnetic pulse propagation in causal dielectrics* (Springer-Verlag, Berlin, 1994) *Chapters 3 & 9.* [CrossRef]

*ϕ*(

*ω*,

*θ*)=(

*c*/

*z*)

*i*(

*k*̃(

*ω*)

*z*-

*ωt*)=

*iω*(

*n*(

*ω*)-

*θ*). Here

*θ*=

*ct*/

*z*denotes a nondimensional space-time parameter which, for any fixed propagation distance

*z*>0, describes the temporal evolution. According to the mathematically well-defined asymptotic theory [11

*Electromagnetic pulse propagation in causal dielectrics* (Springer-Verlag, Berlin, 1994) *Chapters 3 & 9.* [CrossRef]

*ϕ*(

*ω*,

*θ*) and their interaction (if any) with the pole singularities of

*u*̃(

*ω*-

*ω*).

_{c}*u*(

*t*)=exp{1+[

*T*

^{2}(4

*t*(

*t*-

*T*))]}, 0≤

*t*≤

*T*, and is zero elsewhere, which has compact temporal support. Since the Fourier spectrum

*u*̃(

*ω*) of this input pulse envelope is an entire function of complex

*ω*, the asymptotic description of the propagated field is then due to just the saddle points of

*ϕ*(

*ω*,

*θ*). The propagated field may then be expressed in the form [10

**16**, 1773–1785 (1999). [CrossRef]

*Electromagnetic pulse propagation in causal dielectrics* (Springer-Verlag, Berlin, 1994) *Chapters 3 & 9.* [CrossRef]

*z*→∞, where

*j*=

*S*,

*m*,

*B*. Here

*A*is the Sommerfeld precursor due to the positive distant saddle point

_{S}*θ*in the lower right-half of the complex

*ω*-plane above the upper absorption band of the medium,

*A*is the middle precursor due to the positive middle saddle point

_{m}*θ*in the region of the complex

*ω*-plane below the upper absorption band, and

*A*is the Brillouin precursor due to the positive near saddle point

_{B}*θ*in the region of the complex

*ω*-plane below the lower absorption band of the medium. Each saddle point is a solution of the saddle point equation

*dϕ*/

*dω*=0. Details of the dynamical evolution of these saddle points may be found in Refs. [9

**78**, 642–645 (1997). [CrossRef]

*Electromagnetic pulse propagation in causal dielectrics* (Springer-Verlag, Berlin, 1994) *Chapters 3 & 9.* [CrossRef]

## 3. Interferometric Autocorrelation Calculations

*normalized interferometric autocorrelation function*(also referred to as the

*second-order interferometric autocorrelation*) of any particular pulse is given by the expression [1

1. D. J. Bradley and G. H. C. New, “Ultrashort pulse measurements,” Proc. IEEE **62**, 313–345 (1974). [CrossRef]

2. J. C. Diels, J. J. Fontaine, I. C. McMichael, and F. Simoni, “Control and measurement of ultrashort pulse shapes (in amplitude and phase) with femtosecond accuracy,” Appl. Opt. **24**, 1270–1282 (1985). [CrossRef] [PubMed]

2. J. C. Diels, J. J. Fontaine, I. C. McMichael, and F. Simoni, “Control and measurement of ultrashort pulse shapes (in amplitude and phase) with femtosecond accuracy,” Appl. Opt. **24**, 1270–1282 (1985). [CrossRef] [PubMed]

*e*points and comparing that measure to the same measurement for the initial pulse to determine the appropriate deconvolution factor for that particular pulse shape [3

3. F. Hache, T. J. Driscoll, M. Cavallari, and G. M. Gale, “Measurement of ultrashort pulse durations by interferometric autocorrelation: influence of various parameters,” Appl. Opt. **35**, 3230–3236 (1996). [CrossRef] [PubMed]

*ω*=

_{c}*ω*

_{min}=1.615×10

^{15}

*r*/

*s*, the pulse width is

*w*

_{0}=

*T*=10

*π*/

*ω*=19.45

_{c}*fs*. The pulse duration, measured at the 1/

*e*points of the computed second-order interferometric autocorrelation function for this pulse, is (

*w*

_{0})

*=16.3*

_{measured}*fs*, resulting in a deconvolution factor of 1.19. Furthermore, in this particular case, Δ

*ω*/

*ω*≈0.62 and 99.98% of the input pulse spectral energy is contained in the medium passband, as evident in Fig. 1. The dynamical field evolution and associated interferometric autocorrelation functions due to this input pulse are depicted in Figures 2 through 5, each at a fixed propagation distance

_{c}*z*relative to the

*e*

^{-1}absorption depth

*z*=

_{d}*α*

^{-1}(

*ω*) in the dispersive medium at the input pulse carrier frequency. The time

_{c}*t*=(

_{SM}*z*/

*c*)

*θ*indicated in the upper-half of each figure indicates the instant at which the middle precursor becomes dominant over the Sommerfeld precursor (notice that the Sommerfeld precursor is entirely negligible in this below resonance case), the time

_{SM}*t*=(

_{MB}*z*/

*c*)

*θ*indicates the instant when the Brillouin precursor becomes dominant over the middle precursor, and the time

_{MB}*t*

_{0}=(

*z*/

*c*)

*θ*

_{0}indicates the instant at which the Brillouin precursor experiences zero exponential attenuation [10

**16**, 1773–1785 (1999). [CrossRef]

*Electromagnetic pulse propagation in causal dielectrics* (Springer-Verlag, Berlin, 1994) *Chapters 3 & 9.* [CrossRef]

*θ*

_{0}=

*n*(0).

*z*exceeds the absorption depth

*z*=

_{d}*α*

^{-1}(

*ω*), as described in Eq. (6). The rapidly oscillating field structure at the leading and trailing edges of the propagated pulse is due to the middle precursor field

_{c}*A*(

_{m}*z*,

*t*) when the carrier frequency

*ω*satisfies the inequality

_{c}*ω*

_{1}≤

*ω*

_{c}≤

*ω*

_{2}so that it is in the passband between the two absorption bands of the dielectric material [10

**16**, 1773–1785 (1999). [CrossRef]

*ω*, most closely corresponds to this asymptotic contribution when

_{c}*ω*

_{1}≤

*ω*≤

_{c}*ω*

_{2}since it is due to the middle saddle point

*θ*in the region of the complex

*ω*-plane below the upper absorption band. Although the group velocity description yields a propagated pulse structure that is qualitatively similar to that provided by the middle precursor field

*A*(

_{m}*z*,

*t*) alone, its quantitative accuracy decreases as the propagation distance increases because the expansion given in Eq. (4) is taken about a point that does not provide the dominant contribution to the integral representation (2); that point is given by the dominant saddle point [11

*Electromagnetic pulse propagation in causal dielectrics* (Springer-Verlag, Berlin, 1994) *Chapters 3 & 9.* [CrossRef]

*A*(

_{B}*z*,

*t*). Because this contribution is due to the positive near saddle point

*θ*in the region of the complex

*ω*plane below the lower absorption band of the medium, the group velocity description is incapable of describing it.

*z*/

*z*≥5, while for a single cycle pulse, this typically occurs when

_{d}*z*/

*z*≥1. Similar results are obtained for other pulse shapes.

_{d}## 4. Pulse Width Evolution

**78**, 642–645 (1997). [CrossRef]

*ω*. For longer input pulses, the pulse width evolution is complicated by the fact that the pulse spectrum

_{j}*u*̃(

*ω*-

*ω*) becomes sharply peaked about the carrier frequency

_{c}*ω*. The interplay between these two frequency components, which changes as the pulse evolves, then produces the complicated pulse width evolution observed in Fig. 6 for the ten and twenty cycle input pulses. In understanding this evolution, notice that each precursor field component of Eq. (6) decays with propagation distance at a different rate [7

_{c}7. K. E. Oughstun and C. M. Balictsis, “Gaussian pulse propagation in a dispersive, absorbing dielectric,” Phys. Rev. Lett. **77**, 2210–2213 (1996). [CrossRef] [PubMed]

*Electromagnetic pulse propagation in causal dielectrics* (Springer-Verlag, Berlin, 1994) *Chapters 3 & 9.* [CrossRef]

*t*

_{0}=(

*z*/

*c*)

*θ*

_{0}decays only as

*z*

^{-½}, begins to completely overshadow the middle precursor contribution. For all larger propagation distances, the dynamical pulse evolution is dominated by the Brillouin precursor and this is reflected in the linear increase in the logarithmic pulse width observed in Fig. 6.

**78**, 642–645 (1997). [CrossRef]

**16**, 1773–1785 (1999). [CrossRef]

*ω*=

*ω*and the minimum in exponentail attenuation at the upper middle saddle point

_{c}## 5. Conclusions

## Acknowledgements

## References and links

1. | D. J. Bradley and G. H. C. New, “Ultrashort pulse measurements,” Proc. IEEE |

2. | J. C. Diels, J. J. Fontaine, I. C. McMichael, and F. Simoni, “Control and measurement of ultrashort pulse shapes (in amplitude and phase) with femtosecond accuracy,” Appl. Opt. |

3. | F. Hache, T. J. Driscoll, M. Cavallari, and G. M. Gale, “Measurement of ultrashort pulse durations by interferometric autocorrelation: influence of various parameters,” Appl. Opt. |

4. | A. Baltuska, Z. Wei, M. S. Pshenichnikov, and D. A. Wiersma, “Optical pulse compression to 5 fs at a 1-MHz repetition rate,” Opt. Lett. |

5. | G. P. Agrawal, |

6. | P. N. Butcher and D. Cotter, |

7. | K. E. Oughstun and C. M. Balictsis, “Gaussian pulse propagation in a dispersive, absorbing dielectric,” Phys. Rev. Lett. |

8. | C. M. Balictsis and K. E. Oughstun, “Generalized asymptotic description of the propagated field dynamics in Gaussian pulse propagation in a linear, causally dispersive medium,” Phys. Rev. E |

9. | K. E. Oughstun and H. Xiao, “Failure of the quasimonochromatic approximation for ultrashort pulse propagation in a dispersive, attenuative medium,” Phys. Rev. Lett. |

10. | H. Xiao and K. E. Oughstun, “Failure of the group velocity description for ultrawideband pulse propagation in a double resonance Lorentz model dielectric,” J. Opt. Soc. Am. B |

11. | K. E. Oughstun and G. C. Sherman, |

12. | J. Van Bladel, |

**OCIS Codes**

(260.2030) Physical optics : Dispersion

(320.7100) Ultrafast optics : Ultrafast measurements

**ToC Category:**

Research Papers

**History**

Original Manuscript: February 22, 2001

Published: April 9, 2001

**Citation**

Kurt Oughstun and Hong Xiao, "Influence of precursor fields on ultrashort pulse autocorrelation measurements and pulse width evolution," Opt. Express **8**, 481-491 (2001)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-8-8-481

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### References

- J.Bradley and G.H.C.New,�Ultrashort pulse measurements,� Proc..IEEE 62 ,313-345 (1974). [CrossRef]
- J.C.Diels,J.J.Fontaine,I.C.McMichael,and F.Simoni,�Control and measurement of ultrashort pulse shapes (in amplitude and phase)with femtosecond accuracy,� Appl..Opt.24 ,1270-1282 (1985). [CrossRef] [PubMed]
- F.Hache,T.J.Driscoll,M.Cavallari,and G.M.Gale,�Measurement of ultrashort pulse durations by interferometric autocorrelation:influence of various parameters,� Appl..Opt.35 ,3230-3236 (1996). [CrossRef] [PubMed]
- A.Baltuska,Z.Wei,M.S.Pshenichnikov,and D.A.Wiersma,�Optical pulse compression to 5 fs at a 1- MHz repetition rate,� Opt..Lett.22 ,102-104 (1997). [CrossRef] [PubMed]
- G.P.Agrawal,Nonlinear fiber optics (Academic,Boston,1989)Chapters 2-5.
- P.N.Butcher and D.Cotter,The elements of nonlinear optics (Cambridge U.Press,Cambridge,1990) Chap.2. [CrossRef]
- K.E.Oughstun and C.M.Balictsis,�Gaussian pulse propagation in a dispersive,absorbing dielectric,� Phys.Rev.Lett.77 ,2210-2213 (1996). [CrossRef] [PubMed]
- C.M.Balictsis and K.E.Oughstun,�Generalized asymptotic description of the propagated field dynamics in Gaussian pulse propagation in a linear,causally dispersive medium,� Phys..Rev.E 55 ,1910-1921 (1997). [CrossRef]
- K.E.Oughstun and H.Xiao,�Failure of the quasimonochromatic approximation for ultrashort pulse propagation in a dispersive,attenuative medium,� Phys..Rev.Lett.78 ,642-645 (1997). [CrossRef]
- H.Xiao and K.E.Oughstun,�Failure of the group velocity description for ultrawideband pulse propagation in a double resonance Lorentz model dielectric,� J..Opt.Soc.Am.B 16 ,1773-1785 (1999). [CrossRef]
- K.E.Oughstun and G.C.Sherman,Electromagnetic pulse propagation in causal dielectrics (Springer- Verlag,Berlin,1994)Chapters 3 &9. [CrossRef]
- J.Van Bladel,Singular Electromagnetic Fields and Sources (Oxford U.Press,Oxford,1991)Chap.1.

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