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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 5 — Mar. 3, 2008
  • pp: 3058–3068
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Impact of high-frequency spectral phase modulation on the temporal profile of short optical pulses

Christophe Dorrer and Jake Bromage  »View Author Affiliations


Optics Express, Vol. 16, Issue 5, pp. 3058-3068 (2008)
http://dx.doi.org/10.1364/OE.16.003058


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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

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

E~(ω)=I~(ω)exp[iφ(ω)],
(1)

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

E~(ω)=I~(ω)exp[iφ0(ω)]exp[iδφ(ω)]=E~0(ω)exp[iδφ(ω)].
(2)

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

E~(ω)=E~0(ω)[1+iδφ(ω)].
(3)

The Fourier transform of Eq. (3) can be written as a function of the Fourier transform of the spectral phase δφ˜(t):

E(t)=E0(t)+iE0δφ~(t),
(4)

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

I(t)=I0(t)+E0δφ˜(t)22Im{E0*(t)[E0δφ˜(t)]}.
(5)

We assume that the product E * 0(t)·[E 0δφ˜(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 0 Equation (5) simplifies to

I(t)=I0(t)+Eδφ~(t)2.
(6)

If the spectral phase has a single discrete Fourier component, i.e., δφ(ω)=δφ 0cos(ωτ), one has δφ˜(t)=δφ 0[δ(t-τ)+δ(t+τ)/2, and Eq. (6) leads to

I(t)=I0(t)+δφ02[I0(tτ)+I0(t+τ)]4.
(7)

Equation (7) shows that the sinusoidal spectral phase modulation with period 2π/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 τn and amplitude δφ 0,n, leading to a discrete set of replicas of the optical pulse located at times τn and -τn, with relative intensity δφ 2 0,n/4,

I(t)=I0(t)+nδφ0,n2[I0(tτn)+I0(t+τn)]4.
(8)

This extension is valid only if there is no temporal overlap between the different replicas of the electric field composing the sum E0(t)+nδφ0,n[E0(tτn)+E0(t+τn)]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″)δθ˜(t′)δθ˜ *(t″)dtdtE 0(t-t″)E * 0(t-t″) can be replaced by I 0(t-t′)δ (t′-t?) when the variations of δφ˜(t′)δφ˜*(t″) are small in a time interval given approximately by the duration of the pulse. This allows one to simplify

I(t)=I0(t)+I0δφ~2(t),
(9)

which itself can be simplified as

I(t)=I0(t)+ε0δφ~(t)2,
(10)

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 |δφ˜(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

6 . J. C. Stover, Optical Scattering: Measurement and Analysis, 2nd ed., Vol. PM24 (SPIE, Bellingham, WA, 1995). [CrossRef]

].

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 2π/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 |δφ˜(t)|2=exp(-|t|/T 1), with T 1=5 ps, and the normalized power spectral density PSD2 corresponds to |δφ˜(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.

Fig. 1. Temporal intensity of a pulse with sinusoidal spectral phase modulation of period 2π/τ with τ=30 ps on a logarithmic scale in the case of (a) an amplitude δφ 0=0.01 rad and (b) an amplitude δφ 0=0.1 rad.
Fig. 2. (a) Normalized power spectral density for PSD1 (blue line) and PSD2 (red line). (b) Spectral intensity of the pulse (black continuous line) and realizations of the spectral phase with a standard deviation of 0.1 rad and normalized power spectral densities PSD1 (blue continuous line) and PSD2 (red continuous line). The realizations of the spectral phase have been vertically separated to ease comparison.
Fig. 3. Temporal intensity of the pulse on a logarithmic scale for spectral phase modulation with normalized power spectral density PSD1 and standard deviation (a) 0.01 rad and (b) 0.1 rad, and with normalized power spectral density PSD2 and standard deviation (c) 0.01 rad and (d) 0.1 rad.

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

E~(ω)=I~(ω)exp[iφ0(ω)]exp[iδφ(ω)].
(11)

The spectral phase modulation can be decomposed as a Fourier sum δφ(ω)=nαncos(τnω+βn) , where αn, τn, and βn are real constants. The assumption that the added phase is relatively small compared to 1 leads to

E~(ω)=E0~(ω)[1+inαncos(τnω+βn)].
(12)

Equation 12 can be developed as

E(t)=E0(t)+i2nαn[E0(tτn)exp(iβn)+E0(t+τn)exp(iβn)].
(13)

If every αn is small compared to 1, the intensity of the stretched pulse is

I(t)=I0(t)i2E0(t)nαn[E0*(tτn)exp(iβn)
+E0*(t+τn)exp(iβn)]+i2E0*(t)×
nαn[E0(tτn)exp(iβn)+E0(t+τn)exp(iβn)].
(14)

For a highly chirped pulse E 0 with second-order dispersion φ? (i.e., φ 0(ω)=φ?ω 2/2), E0(t)=(1φ)E0~(tφ)exp(it22φ) . The interference of E 0(t) with E 0(t-τn) leads to

E0(t)E0*(tτn)=1φI˜0(tφ)exp(itτnφ)exp(iτn22φ),
(15)

where it has been assumed that 0 is slowly varying, so that 0(t/φ″) * 0[(tn)/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

9. C. Dorrer, “Chromatic dispersion characterization by direct instantaneous frequency measurement,” Opt. Lett. 29, 204–206 (2004). [CrossRef] [PubMed]

]. Regrouping the terms of Eq. (14) with the simplification of Eq. (15) leads to

I(t)=1φI˜0(tφ)[1+2nαncos(tτnφ+βn)sin(τn22φ)].
(16)

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 αn and period 2πn leads to a relative temporal-intensity modulation with amplitude 2αnsin(τ2 n/2φ″) and period 2πφ″/τn. 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 φ?. For a given spectral modulation of period 2πn, the temporal intensity modulation is maximal for τ2 n/2φ″=π/2+, 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 2π/τ with τ=10 ps and amplitude of 0.1 rad (i.e., 0.2 rad peak-to-valley). Second-order dispersions φ?=τ2/3π, τ2/2π, and τ2/π, have been added. These dispersions respectively, correspond to sin(τ2/2φ?)=-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/2πω? (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 |δφ˜(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?=τ2/3π, φ?=τ2/2π, and φ?=τ2/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(τ2/2φ?) 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).

Fig. 4. Temporal intensity of a pulse with sinusoidal spectral phase modulation having a period 2π/τ with τ=10 ps and amplitude 0.1 rad after dispersion (a)φ?=τ2/3π, (b) φ?=τ2/2π, and (c) φ?=τ2/π. (d) Displays a comparison between the simulated temporal intensity and the intensity calculated analytically for φ?=τ2/3π and φ?=τ2/π.
Fig. 5. (a) Spectral intensity (black line) and phase (red line) of a pulse corresponding to a FWHM of 6 nm centered at 1053 nm and a continuous PSD with |δφ˜(t)|2=exp(-t 2/T 2 2 with T 2=20 ps and a standard deviation of 0.1 rad. (b)–(d) represent close-ups of the temporal intensity of the pulse after second-order dispersion φ?=τ2/3π, φ?=τ2/2π, and φ?=τ2/π, where τ=10 ps, respectively.

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(ω)=2nαncos(ωτn+βn)sin(τn22φ) . M is Fourier-transformed and scaling is performed to remove the attenuation effect of sin(τ2 n/2φ′); i.e., by dividing the amplitude of the Fourier component at tn by sin(τ2 n/2φ′). Finally, a Fourier transform back to the optical-frequency domain yields nαncos(ωτn+βn) , 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(τ2/2φ′) 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 [τmin, τmax], there exists at least one value of the second-order dispersion φ? verifying α2/2φ?<π-α for all τ in this interval only if τmaxτmin<(πα)α The arbitrary choice α=0.1 leads to |sin(τ2/2φ′)|>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 τmaxmin<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 (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

I(x,y,t)=I0(x,y,t)+Etδφ~(x,y,t)2,
(17)

where δφ˜ 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

I(x,y,t)=1φI˜0(x,y,tφ){1+2nαn(x,y)
×cos[tτn(x,y)φ+βn(x,y)]sin[τn2(x,y)2φ]}.
(18)

In Eq. (18), the real functions αn, τn, and βn 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 αn, τn, and βn 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.

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.

References and links

1.

C. Dorrer and I. A. Walmsley, “Concepts for the temporal characterization of short optical pulses,” Eurasip J. Appl. Signal Process. 2005, 1541–1553 (2005). [CrossRef]

2.

K.-H. Hong, B. Hou, J. A. Nees, E. Power, and G. A. Mourou, “Generation and measurement of>108 intensity contrast ratio in a relativistic khz chirped-pulse amplified laser,” Appl. Phys. B 81, 447–457 (2005). [CrossRef]

3.

S. Backus, C. G. Durfee III, M. M. Murnane, and H. C. Kapteyn, “High power ultrafast lasers,” Rev. Sci. Instrum. 69, 1207–1223 (1998). [CrossRef]

4.

A. Dubietis, R. Butkus, and A. P. Piskarskas, “Trends in chirped pulse optical parametric amplification,” IEEE J. Sel. Top. Quantum Electron. 12, 163–172 (2006). [CrossRef]

5.

I. Walmsley, L. Waxer, and C. Dorrer, “The role of dispersion in ultrafast optics,” Rev. Sci. Instrum. 72, 1–29 (2001). [CrossRef]

6 .

J. C. Stover, Optical Scattering: Measurement and Analysis, 2nd ed., Vol. PM24 (SPIE, Bellingham, WA, 1995). [CrossRef]

7.

G. Chériaux, P. Rousseau, F. Salin, J. P. Chambaret, B. Walker, and L. F. Dimauro, “Aberration-free stretcher design for ultrashort -pulse amplification,” Opt. Lett. 21, 414–416 (1996). [CrossRef] [PubMed]

8.

V. Bagnoud and F. Salin, “Influence of optical quality on chirped-pulse amplification: Characterization of a 150-nm-bandwidth stretcher,” J. Opt. Soc. Am. B 16, 188–193 (1999). [CrossRef]

9.

C. Dorrer, “Chromatic dispersion characterization by direct instantaneous frequency measurement,” Opt. Lett. 29, 204–206 (2004). [CrossRef] [PubMed]

OCIS Codes
(320.5550) Ultrafast optics : Pulses
(320.7160) Ultrafast optics : Ultrafast technology

ToC Category:
Ultrafast Optics

History
Original Manuscript: October 31, 2007
Revised Manuscript: January 17, 2008
Manuscript Accepted: January 17, 2008
Published: February 20, 2008

Citation
Christophe Dorrer and Jake Bromage, "Impact of high-frequency spectral phase modulation on the temporal profile of short optical pulses," Opt. Express 16, 3058-3068 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-5-3058


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References

  1. C. Dorrer and I. A. Walmsley, "Concepts for the temporal characterization of short optical pulses," Eurasip J. Appl. Signal Process. 2005, 1541?1553 (2005). [CrossRef]
  2. K.-H. Hong, B. Hou, J. A. Nees, E. Power, and G. A. Mourou, "Generation and measurement of >108 intensity contrast ratio in a relativistic khz chirped-pulse amplified laser," Appl. Phys. B 81, 447?457 (2005). [CrossRef]
  3. S. Backus, C. G. DurfeeIII, M. M. Murnane, and H. C. Kapteyn, "High power ultrafast lasers," Rev. Sci. Instrum. 69, 1207?1223 (1998). [CrossRef]
  4. A. Dubietis, R. Butkus, and A. P. Piskarskas, "Trends in chirped pulse optical parametric amplification," IEEE J. Sel. Top. Quantum Electron. 12, 163?172 (2006). [CrossRef]
  5. I. Walmsley, L. Waxer, and C. Dorrer, "The role of dispersion in ultrafast optics," Rev. Sci. Instrum. 72, 1?29 (2001). [CrossRef]
  6. J. C. Stover, Optical Scattering: Measurement and Analysis, 2nd ed., Vol. PM24 (SPIE, Bellingham, WA, 1995). [CrossRef]
  7. G. Chériaux, P. Rousseau, F. Salin, J. P. Chambaret, B. Walker, and L. F. Dimauro, "Aberration-free stretcher design for ultrashort-pulse amplification," Opt. Lett. 21, 414?416 (1996). [CrossRef] [PubMed]
  8. V. Bagnoud and F. Salin, "Influence of optical quality on chirped-pulse amplification: Characterization of a 150-nm-bandwidth stretcher," J. Opt. Soc. Am. B 16, 188?193 (1999). [CrossRef]
  9. C. Dorrer, "Chromatic dispersion characterization by direct instantaneous frequency measurement," Opt. Lett. 29, 204?206 (2004). [CrossRef] [PubMed]

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