## 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

A practical experimental measure of the fundamental characteristics of ultrashort pulse evolution is provided by interferometric autocorrelation techniques [1–3]. Optical pulse widths as short as 5 femtoseconds have been measured [4] with this technique. The interpretation of these measurements has been almost exclusively based upon the popular group velocity description [5,6] of dispersive pulse propagation phenomena. However, it has recently been shown [7–10] that this approximate description breaks down in either the ultrashort pulse or ultrawideband signal limits when the propagation distance exceeds a single absorption depth in the dispersive, attenuative medium. As a consequence, it cannot be used to accurately interpret the experimental results obtained in either the ultrawideband or ultrashort regimes. In particular, the group velocity description is incapable [9,10] of providing an accurate description of the precursor fields that are a characteristic of the dispersive medium and are primarily responsible for ultrashort pulse distortion [7–11]. In this paper, we present the manner in which the precursor fields that are characteristic of a double resonance Lorentz model dielectric influence the interferometric autocorrelation function of an ultrashort optical pulse as it propagates through that linear dispersive medium. The resultant pulse width measurements are also presented as a function of the propagation distance into the dispersive, attenuative medium.

## 2. Analytical Descriptions of Dispersive Pulse Propagation

Consider a plane wave pulse *A*(0,*t*)=*u*(*t*)sin(*ω _{c}t*) that is initiated at the plane

*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] with complex index of refraction

Here *ω _{j}* is the undamped resonance frequency,

*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 ${\omega}_{1}=\sqrt{{\omega}_{0}^{2}+{b}_{0}^{2}}$ and ${\omega}_{3}=\sqrt{{\omega}_{2}^{2}+{b}_{2}^{2}}$. The medium parameters considered here are representative of a fluoride glass with both infrared (

*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.

The propagated optical field *A*(*z*,*t*) is given by the Fourier-Laplace integral [10]

for all *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] for *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.

The analytical description of the dynamical pulse evolution is usually determined under the slowly-varying envelope approximation in which the Taylor series approximation of the complex wavenumber *k*̃(*ω*) about the carrier frequency *ω _{c}*

where *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], so that

^{j}Within this *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}It has recently been established [9,10] that the group velocity description of ultrashort pulse propagation breaks down as the propagation distance exceeds a few absorption depths into a dispersive, attenuative medium. This failure is due to its inability to properly describe the precursor field structures that are a characteristic of the dispersive medium [11] and is a direct consequence of the unfounded assumption that the dominant contribution to the integral appearing in Eq. (2) arises from the region about *ω*=*ω _{c}*. A correct description of dispersive pulse propagation phenomena is provided by the modern asymptotic theory [9–11]. In this description the integral representation appearing in Eq. (2) is rewritten in the form

where *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], the dominant contribution to the integral appearing in Eq. (5) arises from the saddle points of the complex phase function *ϕ*(*ω*,*θ*) and their interaction (if any) with the pole singularities of *u*̃(*ω*-*ω _{c}*).

A canonical problem of central importance in the asymptotic theory is provided by the infinitely smooth, unit amplitude Van Bladel envelope function [12], defined by *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 [10,11]

as *z*→∞, where

for *j*=*S*,*m*,*B*. Here *A _{S}* is the Sommerfeld precursor due to the positive distant saddle point ${\omega}_{\mathit{S}}={\omega}_{{\mathit{SP}}_{\mathit{d}}}^{+}(\theta )$ that evolves with

*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 ${\omega}_{m}={\omega}_{{\mathit{SP}}_{m}}^{+}(\theta )$ that evolves with

_{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 ${\omega}_{B}={\omega}_{{\mathit{SP}}_{n}}^{+}(\theta )$ that evolves with

_{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. [9–11].

## 3. Interferometric Autocorrelation Calculations

The *normalized interferometric autocorrelation function* (also referred to as the *second-order interferometric autocorrelation*) of any particular pulse is given by the expression [1,2]

which contains both intensity autocorrelation and third-order correlation information. A typical second-order autocorrelation measurement employs a 50/50 beam splitter and an optical delay line to split the original optical pulse into a sequential pair of equal pulses with some relative time delay. A nonlinear crystal such as KDP is then used to generate the second harmonic of the interference pattern produced by that pulse sequence. The relative pulse width may then be directly obtained [2] from the resultant intensity autocorrelation trace by measuring the full width at either the half-maximum points or the 1/*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].

For a five cycle Van Bladel envelope pulse with *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 [10,11], where

*θ*

_{0}=

*n*(0).

At three absorption depths the only distinctive differences between the exact and approximate pulse dynamics occur at the leading and trailing edges of the pulse (a 30% error in peak amplitude at the leading edge and a complete misrepresentation of the trailing edge). The only discernible differences between the corresponding interferometric autocorrelation functions occur in the outermost peaks, as seen in Fig. 2. In this case, the approximate pulse width measurement yields a value that is 4% larger than the exact result.

At five absorption depths, the group velocity description of the propagated field structure has broken down (an 85% error in peak amplitude at the leading edge and an increased misrepresentation of the trailing edge of the pulse), as clearly seen in Fig. 3. Easily discernible differences are now evident in the interferometric autocorrelation, particularly in the outer peaks. In this case, the approximate pulse width measurement yields a value that is 16% smaller than the exact result.

The observed differences between the exact and approximate results are due to the inherent inability of the group velocity description to properly describe the precursor field structures that comprise the pulse when the propagation distance *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]. The group velocity approximation of the propagated pulse evolution, which relies upon the local behavior of the material dispersion along the real frequency axis about

*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 ${\omega}_{m}={\omega}_{{\mathit{SP}}_{m}}^{+}(\theta )$ that evolves with

*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]. Finally, the slowly oscillating, field structure that appears at the trailing edge of the propagated pulse is due to the Brillouin precursor field

*A*(

_{B}*z*,

*t*). Because this contribution is due to the positive near saddle point ${\omega}_{B}={\omega}_{{\mathit{SP}}_{n}}^{+}(\theta )$ that evolves with

*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.

At larger propagation distances, the middle precursor field is found to dominate the field structure as the group velocity description rapidly decreases in accuracy, as is clearly evident in Figs. 4 and 5. The structure of this middle precursor is reflected in the outer peaks of the interferometric autocorrelation function. At seven absorption depths (Fig. 5), the peak field amplitude described by the group velocity description is in error by 120% and the approximate pulse width measurement yields a value that is 26.5% smaller than the exact result. At ten absorption depths (Fig. 5), the peak field amplitude described by the group velocity description is in error by 183% and the approximate pulse width measurement yields a value that is 36% smaller than the exact result.

These results clearly show that interferometric autocorrelation measurements can reveal the presence of the precursor field structures that are a characteristic of the dispersive medium for ultrashort pulse propagation when the propagation distance is sufficiently large that the group velocity description has broken down. For five and ten-cycle Van Bladel envelope pulses, this typically occurs when *z*/*z _{d}*≥5, while for a single cycle pulse, this typically occurs when

*z*/

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

_{d}## 4. Pulse Width Evolution

The numerically determined relative pulse width evolution with propagation distance is depicted in Fig. 6 for several different initial pulse widths. For initial widths at or below 5 cycles, the pulse rapidly evolves into a single precursor field dominated structure [9] and this is reflected in the observed monotonic increase in measured pulse width with propagation distance. By a single precursor dominated structure it is meant that the asymptotic description of the pulse given in Eq. (6) is dominated by the contribution from a single saddle point *ω _{j}*. For longer input pulses, the pulse width evolution is complicated by the fact that the pulse spectrum

*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 [7–11].

_{c}For the single-cycle pulse the increase in pulse width is initially very rapid as the pulse quickly evolves into a form that is dominated by the middle precursor field. At approximately twelve absorption depths into the medium, this middle precursor component has decayed to a sufficiently small value that the Brillouin precursor contribution has now become significant by comparison. As the propagation distance increases beyond this point, a slow transition is made from a pulse field dominated by the middle precursor to a pulse field dominated by the Brillouin precursor. This transition is reflected in the plateau in pulse width evolution between twelve and seventeen absorption depths. For larger propagation distances, the field evolution is dominated by the Brillouin precursor with a steadily diminishing relative contribution from the middle precursor. This continues until approximately twenty-five absorption depths when the Brillouin precursor, whose peak amplitude at *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.

For the five-cycle pulse the initial spectrum is almost sufficiently localized about the carrier frequency for the appearance of a component in the middle precursor oscillating at (or very near to) the carrier frequency, as seen in Figs. 2 and 3. Because this component has larger attenuation than that at the peak of the middle precursor, it is nearly absent at seven absorption depths (Fig. 4) and has become entirely negligible at ten absorption depths (Fig. 5). The pulse width evolution is then still similar to that for shorter pulses but with only a brief plateau occurring between fifteen and sixteen absorption depths marking the transition from a pulse field dominated by the middle precursor to a pulse field dominated by the Brillouin precursor. For larger propagation distances, the field evolution is dominated by the Brillouin precursor with a steadily diminishing relative contribution from the middle precursor. This is reflected in Fig. 6 as the logarithmic pulse width approaches a linearly increasing behavior with increasing propagation distance above twenty absorption depths.

For longer input pulses the initial pulse evolution is found to be reasonably well described by the group velocity approximation until the middle precursor begins to dominate the propagated field structure [9,10]. This occurs at roughly five absorption depths for the ten-cycle pulse and at roughly seven absorption depths for the twenty-cycle pulse. Since the initial pulse spectrum is now sharply peaked about the carrier frequency (Fig. 1), a somewhat more complicated evolution in pulse width is obtained in the mature dispersion limit where the asymptotic description (6) applies, as is clearly evident in Fig. 6 for both the ten and twenty-cycle pulse cases. The transition from the immature dispersion regime containing the initial pulse evolution where the group velocity approximation applies to the mature dispersion regime where the asymptotic description allies is marked by the rapid rise in pulse width evolution preceding a sloped plateau region. In this sloped plateau region, the pulse evolution is dominated by the interplay between the peak in spectral amplitude at *ω*=*ω _{c}* and the minimum in exponentail attenuation at the upper middle saddle point ${\omega}_{m}={\omega}_{{\mathit{SP}}_{m}}^{+}(\theta )$. A rapid decrease in pulse width then occurs as the Brillouin precursor begins to dominate the dynamical pulse evolution. This transition occurs between approximately sixteen and twenty absorption depths for the ten cycle pulse while for the twenty cycle pulse it occurs between eighteen and twenty absorption depths, as seen in Fig. 6. For larger propagation distances, the pulse field evolution is dominated by the Brillouin precursor with a steadily diminishing relative contribution from the middle precursor. This is reflected in Fig. 6 as the logarithmic pulse width approaches a linearly increasing behavior with increasing propagation distance above twenty absorption depths.

## 5. Conclusions

The analysis presented here shows that the influence of the middle and Brillouin precursor fields on the dynamical pulse evolution can be observed in a carefully designed interferometric autocorrelation experiment. In addition, it has been shown that the measured pulse width (from a numerical experiment) exhibits a complicated evolutionary structure due to these precursor fields that is dependent upon the initial temporal pulse width. When the initial pulse width is sufficiently short, the asymptotic behavior of the dynamical pulse evolution is primarily determined by the saddle point contributions throughout the mature dispersion regime, resulting in a monotonic increase in pulse width with increasing propagation distance (monotonic in the sense that it never decreases). Its rate of increase, however, changes several times during the dynamical pulse evolution, each time indicating a change in dominance of the precursor fields that comprise the propagated pulse field. As the initial pulse width is increased, the asymptotic behavior becomes complicated by the increased localization of the pulse spectrum about the carrier frequency, resulting in a complicated dependence of the measured pulse width on propagation distance.

Although the results presented here were determined for a specific ultrashort pulse shape, similar results should be obtained for other pulse envelope shapes such as gaussian, rectangular and double exponential functions. It is hoped that these results will help to stimulate much needed experimental research into this phenomena.

## Acknowledgements

The research presented in this paper has been supported, in part, by AFOSR Grant #F49620-98-1-0067. Helpful critiques of an earlier version of this paper by both peer reviewers are gratefully acknowledged for their help in producing a more accessible report.

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