## Impact of high-frequency spectral phase modulation on the temporal profile of short optical pulses

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

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>10^{8} 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]

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

### 2.1 Analytic derivation

*E*(

*t*) corresponding to the spectral representation

*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

*iδφ*(

*ω*)]≈1+

*iδφ*(

*ω*). Equation (2) can be written

*δ*φ ˜ (

*t*):

*E*

^{*}

_{0}(

*t*)·[

*E*

_{0}⊗

*δ*φ ˜ (

*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

*δφ*(

*ω*)=

*δφ*

_{0}cos(

*ω*τ), one has

*δ*φ ˜ (

*t*)=

*δφ*

_{0}[

*δ*(

*t*-τ)+

*δ*(

*t*+τ)/2, and Eq. (6) leads to

*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

*and amplitude*

_{n}*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,

*E*

_{0}(

*t*-

*t*′)

*E*

^{*}

_{0}(

*t*-

*t*″)

*δ*θ ˜ (

*t*′)

*δ*θ ˜

^{*}(

*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*φ ˜ (

*t*′)

*δ*φ ˜ *(

*t*″) are small in a time interval given approximately by the duration of the pulse. This allows one to simplify

*ε*

_{0}=∫

*I*

_{0}(

*t*)

*dt*is the energy of the pulse.

*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*φ ˜ (

*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

*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).

*δ*φ ˜ (

*t*)|

^{2}=exp(-|

*t*|/

*T*

_{1}), with

*T*

_{1}=5 ps, and the normalized power spectral density PSD2 corresponds to |

*δ*φ ˜ (

*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

*α*, τ

_{n}*, and*

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

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

_{n}*E*

_{0}with second-order dispersion

*φ*? (i.e.,

*φ*

_{0}(

*ω*)=

*φ*?

*ω*

^{2}/2),

*E*

_{0}(

*t*) with

*E*

_{0}(t-τ

*) leads to*

_{n}*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

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

*α*and period 2

_{n}*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

*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%.

*δ*φ ˜ (

*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

*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*is Fourier-transformed and scaling is performed to remove the attenuation effect of sin(τ

^{2}

*/2*

_{n}*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

^{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

*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

*Ẽ*(

*x*,

*y*,

*ω*) depends on the spatial variables

*x*and

*y*, and the spectral variable

*ω*.

*E*(

*x*,

*y*,

*t*) and the corresponding intensity is given, by analogy, to Eq. (6) as

*δ*φ ˜ is the Fourier transform of the function

*δφ*(

*x*,

*y*,

*ω*) with respect to the optical frequency and ⊗

*is a convolution with respect to the temporal variable. This can be decomposed in expressions of the intensity*

_{t}*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.

*φ*?, the intensity at a given point in the beam is given by an equivalent of Eq. (16) as

*α*, τ

_{n}*, 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

## Acknowledgments

## References and links

1. | C. Dorrer and I. A. Walmsley, “Concepts for the temporal characterization of short optical pulses,” Eurasip J. Appl. Signal Process. |

2. | K.-H. Hong, B. Hou, J. A. Nees, E. Power, and G. A. Mourou, “Generation and measurement of>10 |

3. | S. Backus, C. G. Durfee III, M. M. Murnane, and H. C. Kapteyn, “High power ultrafast lasers,” Rev. Sci. Instrum. |

4. | A. Dubietis, R. Butkus, and A. P. Piskarskas, “Trends in chirped pulse optical parametric amplification,” IEEE J. Sel. Top. Quantum Electron. |

5. | I. Walmsley, L. Waxer, and C. Dorrer, “The role of dispersion in ultrafast optics,” Rev. Sci. Instrum. |

6 . | J. C. Stover, |

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. |

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 |

9. | C. Dorrer, “Chromatic dispersion characterization by direct instantaneous frequency measurement,” Opt. Lett. |

**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

- 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]
- 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]
- S. Backus, C. G. DurfeeIII, M. M. Murnane, and H. C. Kapteyn, "High power ultrafast lasers," Rev. Sci. Instrum. 69, 1207?1223 (1998). [CrossRef]
- 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]
- I. Walmsley, L. Waxer, and C. Dorrer, "The role of dispersion in ultrafast optics," Rev. Sci. Instrum. 72, 1?29 (2001). [CrossRef]
- J. C. Stover, Optical Scattering: Measurement and Analysis, 2nd ed., Vol. PM24 (SPIE, Bellingham, WA, 1995). [CrossRef]
- 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]
- 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]
- C. Dorrer, "Chromatic dispersion characterization by direct instantaneous frequency measurement," Opt. Lett. 29, 204?206 (2004). [CrossRef] [PubMed]

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