## Abstract

The impact of high-frequency spectral phase modulation on the temporal intensity of optical pulses is derived analytically and simulated in two different regimes. The temporal contrast of an optical pulse close to the Fourier-transform limit is degraded by a pedestal related to the power spectral density of the spectral phase modulation. When the optical pulse is highly chirped, its intensity modulation is directly related to the spectral phase variations with a transfer function depending on the second-order dispersion of the chirped pulse. The metrology of the spectral phase of an optical pulse using temporal-intensity measurements performed after chirping the pulse is studied. The effect of spatial averaging is also discussed.

© 2008 Optical Society of America

## 1. Introduction

Short optical pulses are used in a large variety of applications including plasma physics, chemistry, coherent control, and optical telecommunications. The generation, amplification, and propagation of a pulse can lead to modulation of the pulse shape. It is important to understand how this modulation arises for system optimization. An optical pulse is fundamentally described by its electric field [1]. The analytic signal of the electric field can be represented equivalently in the time or frequency domain. The Fourier transform relation between these two representations generally links high-frequency modulation of the spectral representation of the field to features appearing at delays significantly larger than the coherence time of the pulse. Simulations have shown that high-frequency spectral phase modulation leads to a degradation of the temporal contrast of an optical pulse, i.e., generate a high-intensity pedestal that extends over a temporal range much longer than the pulse duration [2]. This can limit the temporal contrast of chirped-pulse amplification and optical parametric chirped-pulse–amplification systems [3,4]. In these systems, stretching and compression devices typically rely on angular spectral dispersion [5], and reflection of the spectrally dispersed optical pulse on a scattering surface can induce spectral phase modulation. Scattering is inherent to finished optical surfaces [6]. Evidence of coupling between the wavefront of optical elements composing an Öffner triplet used in a stretcher [7] and the spectral phase of an optical pulse propagating in this stretcher has been demonstrated [8]. There might not be a direct coupling between high-frequency variations of a surface and the spectral phase of a spatially dispersed optical pulse because of the nonzero beam size of a monochromatic component of the pulse. However, proper quantification of the effects of high-frequency spectral phase modulation is useful to understand the limitations of current systems delivering optical pulses and to develop new systems.

This paper analytically links high-frequency spectral phase modulation to the temporal contrast of short optical pulses. Simulations confirm that replicas of the pulse are induced in the case of a sinusoidal spectral phase (i.e., a discrete Fourier component), while a pedestal related to the power spectral density (PSD) of the phase degrades the temporal contrast of the pulse for a continuous range of Fourier components. The effect of high-frequency spectral phase modulation on the temporal profile of a chirped optical pulse is derived. These modulations lead to temporal intensity modulation with a direct relation between the Fourier components of the spectral phase and the temporal intensity modulations. Such derivation is important since the amplification and propagation of chirped optical pulses might be impacted by these modulations. The characteristics of the temporal intensity make it possible to identify and quantity the spectral phase modulation. This would allow the direct measurement of high-frequency spectral phase modulation using time-domain intensity-only measurements. Finally, a general discussion of the effect of spatial averaging is presented.

## 2. Effect of high-frequency spectral phase modulation on the temporal intensity of an optical pulse

#### 2.1 Analytic derivation

The optical pulse is represented by its analytic signal *E*(*t*) corresponding to the spectral representation

where *Ĩ* and *φ* are the spectral intensity and phase of the field, respectively. The spectral phase of the pulse is decomposed as *φ*(*ω*)=*φ*
_{0}(*ω*)+*δφ*(*ω*), where *φ*
_{0} represents a slowly varying spectral phase (describing, for example, the chirp) and *δφ* represents a high-frequency spectral phase modulation of small amplitude (in a sense precisely defined below). The spectral representation of the field is

Assuming that the amplitude of δφ is small relatively to unity, one can write exp[*iδφ*(*ω*)]≈1+*iδφ*(*ω*). Equation (2) can be written

The Fourier transform of Eq. (3) can be written as a function of the Fourier transform of the spectral phase *δ$\tilde{\phi}$*(*t*):

where ⊗ is a convolution. The intensity of the pulse is

We assume that the product *E*
^{*}
_{0}(*t*)·[*E*
_{0}⊗*δ$\tilde{\phi}$*(*t*)] is zero over the temporal range of interest. This assumption derives from the properties of the spectral modulation, i.e., the mo dulation represented by *δφ* has a high-frequency content relative to the pulse *Ẽ*
_{0} Equation (5) simplifies to

If the spectral phase has a single discrete Fourier component, i.e., *δφ*(*ω*)=*δφ*
_{0}cos(*ω*τ), one has *δ$\tilde{\phi}$*(*t*)=*δφ*
_{0}[*δ*(*t*-τ)+*δ*(*t*+τ)/2, and Eq. (6) leads to

Equation (7) shows that the sinusoidal spectral phase modulation with period 2*π*/τ leads to two replicas of the optical pulse located at *t*=τ and *t*=-τ. The relative intensity of these replicas is proportional to the square of the phase modulation amplitude. Equation (7) is easily extended to a discrete sum of sinusoidal phases with respective period τ* _{n}* and amplitude

*δφ*

_{0,n}, leading to a discrete set of replicas of the optical pulse located at times τ

*and -τ*

_{n}*, with relative intensity*

_{n}*δφ*

^{2}

_{0,n}/4,

This extension is valid only if there is no temporal overlap between the different replicas of the electric field composing the sum ${E}_{0}\left(t\right)+\frac{\sum _{n}\delta {\phi}_{0,n}\left[{E}_{0}\left(t-{\tau}_{n}\right)+{E}_{0}\left(t+{\tau}_{n}\right)\right]}{2}$ , so that the resulting intensity is given by the sum of the intensities of each individual replica.

When the spectral phase is described by a continuous power spectral density, Eqs. (7) and (8) are not applicable. The convolution in the right-hand side of Eq. (6) can be written as ∫ *E*
_{0}(*t*-*t*′) *E*
^{*}
_{0}(*t*-*t*″)*δ$\tilde{\theta}$*(*t*′)*δ$\tilde{\theta}$*
^{*}(*t*″)*dt*′*dt*″ *E*
_{0}(*t*-*t*″)*E*
^{*}
_{0}(*t*-*t*″) can be replaced by *I*
_{0}(*t*-*t*′)*δ* (*t*′-*t*?) when the variations of *δ$\tilde{\phi}$*(*t*′)*δ$\tilde{\phi}$**(*t*″) are small in a time interval given approximately by the duration of the pulse. This allows one to simplify

which itself can be simplified as

where *ε*
_{0}=∫*I*
_{0}(*t*)*dt* is the energy of the pulse.

The intensity of the pulse with high-frequency spectral phase modulation *I*(*t*) is the sum of the intensity calculated without phase modulation *I*
_{0}(*t*) and a temporal pedestal given by the square of the Fourier transform of the modulation |*δ$\tilde{\phi}$*(*t*)|^{2}; i.e., the PSD of the phase modulation. The intensity of the pedestal increases with the square of the amplitude of the phase modulation. The temporal extent of the pedestal is proportional to the bandwidth of the phase modulation, i.e., higher frequencies lead to temporal features farther away from the peak of the pulse, as is generally expected from the Fourier relation between the spectral and temporal representations of the field. Since *φ* is a real function, its Fourier transform is symmetric, and the pedestal induced by a high-frequency, low-amplitude spectral phase modulation is symmetric in time. This is analogous to the far-field description of an optical field after scattering on a surface. This analogy is due to the Fourier transform relation between the spectral and temporal representations of the field, which is similar to the Fourier transform relation between near- and far-field representations of the field in the Fraunhofer approximation [6].

#### 2.2 Simulations

The effect of sinusoidal spectral phase modulation is illustrated in Fig. 1, which displays the temporal intensity of the optical pulse on a logarithmic scale in two different cases. The spectrum of the pulse is Gaussian with full-width at half-maximum equal to 6 nm and the spectrum centered at 1053 nm. A spectral phase with period 2*π*/τ, where τ=30 ps, has been considered. The amplitude *δφ*
_{0} is chosen equal to 0.01 rad [Fig. 1(a)] or 0.1 rad [Fig. 1(b)]. As predicted by Eq. (7), there is a set of replicas at -30 ps and 30 ps. The intensity of these replicas is properly predicted as *δφ*
^{2}
_{0}/4, i.e., -46 dB and -26 dB, respectively. Other replicas are visible at times corresponding to multiples of τ, which are due to harmonics of the spectral phase modulation generated by the nonlinearity of the exponential function. Linearity around 0 has been explicitly assumed when simplifying Eq. (2) into Eq. (3).

The effect of various continuous PSD’s of the spectral phase of an optical pulse has been simulated for illustration purposes. The normalized power spectral density PSD1 corresponds to |*δ$\tilde{\phi}$*(*t*)|^{2}=exp(-|*t*|/*T*
_{1}), with *T*
_{1}=5 ps, and the normalized power spectral density PSD2 corresponds to |*δ$\tilde{\phi}$*(*t*)|^{2}=exp(-*t*
^{2}/*T*
^{2}
_{2}) with *T*
_{2}=20 ps. These normalized PSD’s are plotted in Fig. 2(a). Spectral phases with such PSD’s and standard deviation of 0.1 rad are plotted in Fig. 2(b). For each PSD, two different phase standard deviations, 0.1 rad and 0.01 rad, have been considered. A Gaussian spectral intensity with full width at half maximum equal to 6 nm centered at 1053 nm was assumed for the optical pulse. According to Eq. (10), the choice of the Gaussian spectral intensity impacts only the shape of the intensity of the pulse around *t*=0 via *I*
_{0}(*t*), but the induced temporal pedestal depends on only the spectral phase modulation. The temporal intensity of the corresponding pulses is plotted in Fig. 3 on a logarithmic scale, and is compared to the intensity calculated with Eq. (10). The comparison is excellent. This confirms that the intensity of the pedestal scales like the square of the amplitude of the spectral phase modulation. Even relatively small spectral phase variations of the order of 0.01 rad lead to a pedestal that can be measured easily with state-of-the-art temporal diagnostics such as a third-order high-dynamic range temporal cross-correlator.

## 3. Effect of high-frequency spectral phase modulation on the temporal intensity of a chirped optical pulse

#### 3.1 Analytic derivation

The spectral representation of the field is

The spectral phase modulation can be decomposed as a Fourier sum
$\delta \phi \left(\omega \right)=\sum _{n}{\alpha}_{n}\mathrm{cos}\left({\tau}_{n}\omega +{\beta}_{n}\right)$
, where *α _{n}*, τ

*, and*

_{n}*β*are real constants. The assumption that the added phase is relatively small compared to 1 leads to

_{n}Equation 12 can be developed as

If every *α _{n}* is small compared to 1, the intensity of the stretched pulse is

$$\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.6em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}+{E}_{0}^{*}\left(t+{\tau}_{n}\right)\mathrm{exp}\left(i{\beta}_{n}\right)]+\frac{i}{2}{E}_{0}^{*}\left(t\right)\times $$

$$\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\sum _{n}{\alpha}_{n}\left[{E}_{0}\left(t-{\tau}_{n}\right)\mathrm{exp}\left(i{\beta}_{n}\right)+{E}_{0}\left(t+{\tau}_{n}\right)\mathrm{exp}\left(-i{\beta}_{n}\right)\right].$$

For a highly chirped pulse *E*
_{0} with second-order dispersion *φ*? (i.e., *φ*
_{0}(*ω*)=*φ*?*ω*
^{2}/2),
${E}_{0}\left(t\right)=\left(\frac{1}{\sqrt{{\phi}^{\u2033}}}\right)\stackrel{~}{{E}_{0}}\left(\frac{t}{{\phi}^{\u2033}}\right)\mathrm{exp}\left(\frac{-{\mathrm{it}}^{2}}{2{\phi}^{\u2033}}\right)$
. The interference of *E*
_{0}(*t*) with *E*
_{0}(t-τ* _{n}*) leads to

where it has been assumed that *Ẽ*
_{0} is slowly varying, so that *Ẽ*
_{0}(*t*/*φ*″)*Ẽ*
^{*}
_{0}[(*t*-τ* _{n}*)/

*φ*″] is replaced by

*Ĩ*

_{0}(

*t*/

*φ*″). The interference of two time -delayed replicas of a chirped pulse leads to intensity modulation depending on the dispersion of the chirped pulse that have been previously used for chromatic dispersion measurements [9]. Regrouping the terms of Eq. (14) with the simplification of Eq. (15) leads to

Equation (16) shows that the relative intensity modulation of the chirped optical pulse is given by a scaled representation of the spectral phase modulation of the optical pulse before stretching. A spectral phase component with amplitude *α _{n}* and period 2

*π*/τ

*leads to a relative temporal-intensity modulation with amplitude 2*

_{n}*α*sin(τ

_{n}^{2}

_{n}/2

*φ*″) and period 2

*πφ*″/τ

*. Similar to the time-to-frequency correspondence between the temporal intensity of the stretched pulse and the spectral intensity of the pulse, there is a time-to-frequency correspondence between the relative intensity modulations of the stretched pulse and the spectral phase modulation. The ratio of the period of the temporal modulation induced by a specific sinusoidal spectral phase modulation to the duration of the chirped pulse does not depend on the second-order dispersion. Equivalently, the period of the induced temporal modulation increases linearly with*

_{n}*φ*?. For a given spectral modulation of period 2

*π*/τ

*, the temporal intensity modulation is maximal for τ*

_{n}^{2}

_{n}/2

*φ*″=

*π*/2+

*mπ*, where

*m*is an integer. The amplitude of all components of the temporal intensity modulation converges to zero when

*φ*? tends towards infinity, confirming that, for a large frequency chirp, the temporal intensity of the stretched optical pulse is a scaled representation of the spectral density of the pulse, with no dependence on its spectral phase.

#### 3.2 Simulations

A pulse with a 20th-order super-Gaussian spectrum with FWHM equal to 6 nm and a sinusoidal spectral phase illustrates the induced temporal modulation. The spectral phase has a period 2*π*/τ with τ=10 ps and amplitude of 0.1 rad (i.e., 0.2 rad peak-to-valley). Second-order dispersions *φ*?=τ^{2}/3*π*, τ^{2}/2*π*, and τ^{2}/*π*, have been added. These dispersions respectively, correspond to sin(τ^{2}/2*φ*?)=-1, 0, and 1; i.e., to an intensity modulation of maximal amplitude with sign inverted relatively to the spectral phase modulation, no intensity modulation, and intensity modulation of maximal amplitude with sign identical to that of the spectral phase modulation. The intensities of the resulting pulses are plotted in Figs. 4(a)–4(c). Close-ups of the simulated modulation and the modulation calculated analytically using Eq. (16) are plotted in Fig. 4(d). On this subplot, the intensities have been plotted as a function of the variable *t*/2*πω*? (each intensity having its own *ω*?), in units of frequency in Hz, to provide proper scaling. An excellent match between the simulated and derived modulation is obtained. A spectral phase variation with peak-to-valley amplitude of 0.2 rad can, for some values of the second-order dispersion, lead to a relative intensity peak-to-valley modulation of 40%.

A pulse with 20th-order super-Gaussian spectrum and spectral phase described by the normalized PSD |*δ$\tilde{\phi}$*(*t*)|^{2}=exp(-*t*
^{2}/*T*
^{2}
_{2} with *T*
_{2}=20 ps and a standard deviation of 0.1 rad was simulated [Fig. 5(a)]. Second-order dispersion *φ*?=τ^{2}/3*π*, *φ*?=τ^{2}/2*π*, and *φ*?=τ^{2}/*π*, where τ=10 ps was added to the pulse. The corresponding temporal intensities are plotted in Figs. 5(b), 5(c), and 5(d), respectively. In contrast to the results of Fig. 4, the temporal intensity is modulated for all three values of the second-order dispersion because the temporal modulation induced by the different Fourier components of the spectral phase do not cancel evenly for a given value of the dispersion. This implies that, in the general case, for a continuous PSD, spectral phase variations always lead to modulation on the temporal intensity of the chirped pulse [apart from the regime when the second-order dispersion is large enough that sin(τ^{2}/2*φ*?) tends toward zero for all values of τ corresponding to a significant Fourier component of the spectral phase]. A standard deviation of 0.1 rad leads to significant intensity modulation of the chirped pulse for the tested range of second-order dispersions. The correlation between the modulation of the spectral phase and the intensity modulation in the case of a continuous PSD is not obvious, as can be concluded by visually comparing the phase plotted on Fig. 5(a) with the intensity modulations plotted on Figs. 5(b)–5(d).

#### 3.3 Application to the measurement of high-frequency spectral phase modulation of an optical pulse

The measurement of the temporal characteristics of optical pulses is important in many applications. Measuring high-frequency spectral phase variations on a short optical pulse is typically difficult, because such a measurement requires good spectral resolution over a large spectral bandwidth. The analytic relationship between the Fourier components of the spectral phase and the components of the intensity expressed by Eq. (16) indicates the possibility of reconstructing the high-frequency spectral phase modulation present on an optical pulse using intensity-only measurements after chromatic dispersion. The intensity of the pulse under test after second-order dispersion *φ*? is measured with a photodetector capable of resolving all the components of the signal, as expressed by Eq. (16). The measured intensity can be scaled by the temporal intensity calculated from the measured optical spectrum and second-order dispersion *Ĩ*
_{0}(*t*/*φ*′)/*φ*′. The temporal axis can be scaled to an optical-frequency axis by dividing by *φ*? to obtain the modulation
$M\left(\omega \right)=2\sum _{n}{\alpha}_{n}\mathrm{cos}\left(\omega {\tau}_{n}+{\beta}_{n}\right)\mathrm{sin}\left(\frac{{\tau}_{n}^{2}}{2{\phi}^{\u2033}}\right)$
. *M* is Fourier-transformed and scaling is performed to remove the attenuation effect of sin(τ^{2}
* _{n}*/2

*φ*′); i.e., by dividing the amplitude of the Fourier component at

*t*=τ

*by sin(τ*

_{n}^{2}

*/2*

_{n}*φ*′). Finally, a Fourier transform back to the optical-frequency domain yields $\sum _{n}{\alpha}_{n}\mathrm{cos}\left(\omega {\tau}_{n}+{\beta}_{n}\right)$ , which is a representation of the spectral phase of the optical pulse under test.

Direct reconstruction of the spectral phase from the temporal intensity of the chirped pulse is possible only if sin(τ^{2}/2*φ*′) is non-zero over the range of values of τ, corresponding to the frequencies present on the spectral phase. Assuming that such a range is included in the continuous interval [τ_{min}, τ_{max}], there exists at least one value of the second-order dispersion *φ*? verifying *α*<τ^{2}/2*φ*?<*π*-*α* for all τ in this interval only if
$\frac{{\tau}_{max}}{{\tau}_{min}}<\sqrt{\frac{\left(\pi -\alpha \right)}{\alpha}}$
The arbitrary choice *α*=0.1 leads to |sin(τ^{2}/2*φ*′)|>0.1; i.e., to a relative modulation of the Fourier components of the spectral phase approximately between 0.1 and 1. This implies the condition τ_{max}/τ_{min}<5.5, which appears to be a strong limitation of the range of components of the spectral phase that can be reconstructed in this manner. The reconstruction of a discrete or small range of Fourier components of the spectral phase is possible with the temporal intensity measured with a single second-order dispersion. The reconstruction of a larger range of Fourier components can be performed by dividing the range into smaller intervals and performing a temporal intensity for each range with an appropriate second-order dispersion. Temporal intensities measured for different second-order dispersions would be required to characterize the phase described in Fig. 5(a). While such a procedure appears theoretically appealing, it may be difficult to implement in practice.

## 4. Spatial-averaging effects for spatially dependent electric fields

In this section, a formalism for spatially extended electric fields with high-frequency spectral phase modulation is outlined to extend the predictions of Eqs. (8), (10), and (16). It is beyond the scope of this paper to precisely study mechanisms and formalisms for space-time coupling, such as those found when spatial modulation occurs at the Fourier plane of a zero-dispersion line. The spatially extended field *Ẽ*(*x*,*y*,*ω*) depends on the spatial variables *x* and *y*, and the spectral variable *ω*.

The temporal representation of this field is *E*(*x*,*y*,*t*) and the corresponding intensity is given, by analogy, to Eq. (6) as

where *δ$\tilde{\phi}$* is the Fourier transform of the function *δφ*(*x*,*y*,*ω*) with respect to the optical frequency and ⊗* _{t}* is a convolution with respect to the temporal variable. This can be decomposed in expressions of the intensity

*I*(

*x*,

*y*,

*t*) analogous to Eqs. (8) and (10), which predict the temporal-contrast degradation at a given point in the beam. The temporal intensity of the pulse

*I*(

*t*) is defined as the sum of the function

*I*(

*x*,

*y*,

*t*) over the spatial variables. Since the contrast degradation at all points in the beam is a positive function, it does not cancel during the averaging process. The contras t degradation due to the high-frequency spectral phase modulation at different points in the beam adds up in the spatial domain.

In the presence of a large second-order dispersion *φ*?, the intensity at a given point in the beam is given by an equivalent of Eq. (16) as

$$\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\times \mathrm{cos}\left[\frac{t{\tau}_{n}(x,y)}{{\phi}^{\u2033}}+{\beta}_{n}(x,y)\right]\mathrm{sin}\left[\frac{{\tau}_{n}^{2}(x,y)}{2{\phi}^{\u2033}}\right]\}.$$

In Eq. (18), the real functions *α _{n}*, τ

*, and*

_{n}*β*are spatially varying. In contrast to Eq. (17), the temporal modulation induced by the high-frequency spectral phase modulations is not strictly positive. It is easy to exhibit sets of these functions that will cancel out the temporal modulations when spatial averaging is performed. On the other hand, sets of real functions

_{n}*α*, τ

_{n}*, and*

_{n}*β*that are not spatially varying lead to a spatially averaged modulation identical to the modulation at each point in the beam. In the general case, nothing can be said of the temporal intensity of a spatially extended chirped pulse. Spatial averaging of the temporal intensity of a chirped optical pulse with spatially varying high-frequency spectral phase modulation might effectively lead to the attenuation or cancellation of the temporal modulation.

_{n}## 5. Conclusion

The impact of high-frequency spectral phase modulation on the temporal intensity of optical pulses has been studied. For a short pulse, high-frequency spectral phase modulation leads to a reduction of the temporal contrast of the pulse, and the induced pedestal is directly linked to the power spectral density of the spectral phase. The intensity of the pedestal scales quadratically with the amplitude of the spectral phase. For a highly chirped pulse, the high-frequency spectral phase modulation leads to intensity modulations, and the respective Fourier components of these two modulations are analytically linked. The amplitude of the temporal modulation depends on the second-order dispersion of the chirped pulse, and tends toward zero for large dispersions. A direct technique to reconstruct the high-frequency spectral phase modulation of an optical pulse based on this formalism has been presented. An extension of these results to spatially varying fields show that contrast degradation due to spectral phase modulation in some portion of the beam leads to contrast degradation when the full temporal intensity of the beam is considered. However, the temporal modulation present at different locations in the beam of a highly chirped pulse can cancel out when spatial averaging is performed.

## Acknowledgments

This work was supported by the U.S. Department of Energy Office of Inertial Confinement Fusion under Cooperative Agreement No. DE-FC52-92SF19460, the University of Rochester, and the New York State Energy Research and Development Authority. The support of DOE does not constitute an endorsement by DOE of the views expressed in this article.

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