# Recent Developments in Experimental Techniques for Measuring Two Pulses Simultaneously

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Blind FROG and CRAB

_{sig}(t) represents the signal field from the nonlinear interaction happening inside the nonlinear medium. The expression of E

_{sig}(t) varies depending on the gating geometry in the experimental apparatus. For example, in PG geometry:

_{1}(t) and E

_{2}(t) are the electric field of the two unknown pulses to be measured. The retrieval algorithm begins with guesses of E

_{1}(t) and E

_{2}(t) and generate E

_{sig}(ω,τ) by Fourier transforming (2) with respect to t. The magnitude of E

_{sig}(ω,τ) is replaced by the square root of the measured Blind FROG trace with the phase unchanged to generate a modified signal field, . is generated by an inverse Fourier transform. The method of generalized projections is used to generate new guesses for the fields. An error function, Z, is defined as [18]:

**Figure 1.**The schematic of single-shot polarization-gate (PG) cross-correlation frequency-resolved optical gating (XFROG) setup. The orange beam and red beam represent the known reference pulse and unknown pulse, respectively. The experimental apparatus is essentially a PG FROG setup with the known reference pulse replacing one of the replicas. The apparatus becomes a Blind PG FROG setup when the known reference is replaced by an unknown one.

**Figure 2.**Electric fields of two pulses with different amount of chirp retrieved by second harmonic generation (SHG) Blind FROG (circles) compared with results retrieved by SHG FROG (lines) [18].

_{2}, and a new guess for E

_{1}(t) is generated by minimizing with Z respect to E

_{1}(t). On even iterations, the operations on pulse 1 and pulse 2 reverse. The algorithm continues until the root-mean-square (rms) error between the measured and retrieved Blind FROG traces is minimized. In the original work, the Blind FROG apparatus was implemented with second harmonic generation (SHG) geometry. Figure 2 shows the two pulses with different amounts of chirp retrieved by SHG Blind FROG. The Blind FROG retrieval results were compared with measurement independently made by SHG FROG showing good agreement. However, independently measured spectra of the two pulses were also required in order to achieve convergence.

_{R}(ω,τ), is:

_{X}(t), and the vector potential of the probe pulse, A(t), by:

_{e}is the electron energy and v

_{o}is a constant obtained by the reconstruction algorithm that minimizes the error between S(ω,τ) and S

_{R}(ω,τ). The algorithm starts with setting X

^{(i=0)}(t) and G

^{(i=0)}(t) to unity, where i represents the iteration number. is generated from the X

^{(i)}(t) and G

^{(i)}(t). The intensity of is replaced by the experimentally measured S(ω,τ). New guesses for and G

^{(i+1)}(t) are generated from by a generalized projections approach. The algorithm continues until the rms error between measured and reconstructed photoelectron spectra has stabilized.

^{14}W/cm

^{2}and 3.2 × 10

^{14}W/cm

^{2}. Figure 3 shows the measured and reconstructed photoelectron spectra for a 33 fs probe pulse and attosecond-pulse trains with rms error of 0.014 and 0.012 with low- and high-intensity cases, respectively. The reconstructed temporal profiles are shown in Figure 4 which are consistent with the prediction made by numerical simulations.

**Figure 3.**Photoelectron spectra of He ionized by attosecond harmonic pulses with a probe laser pulse. High harmonics were obtained with a driving laser at an intensity of (

**a**) 1.6 × 10

^{14}W/cm

^{2}and (

**b**) 3.2 × 10

^{14}W/cm

^{2}. The reconstructed spectra are shown in (

**c**) and (

**d**) for cases (

**a**) and (

**b**), respectively [22].

**Figure 4.**Reconstructed temporal profiles of attosecond-pulse train corresponding to the cases of low (Figure 3a) and high intensities (Figure 3b), as indicated by the blue and red solid lines, respectively. The temporal profile of the reconstructed probe pulse is shown by the red dashed line. The duration of each pulse in the train is shown by squares and circles for low and high intensities, respectively [22].

## 3. VAMPIRE

**Figure 5.**Optical schematic of the very advanced method for phase and intensity retrieval of e-fields (VAMPIRE) technique [24].

_{1}and ϕ

_{2}are the spectral phase of the pulses. The spectra of the pulses are measured independently and used as a constraint to the reconstruction problem, thus ϕ

_{1}and ϕ

_{2}are the unknown functions to be determined. The reconstruction algorithm starts with solving the 1D phase-retrieval problem for an arbitrary row, i.e., the arbitrary value of Ω, in the trace by using a Gerchberg-Saxton algorithm:

_{Ω}(τ)| is given by the square root of I(Ω,τ) and G

_{Ω}(ω) is given by the measured spectrum of the pulse. The solution of the 1D phase-retrieval problem is used as the initial guess for the neighboring row until all the rows are covered. The phase information is contained in a function, h(ω,Ω) = P(ω,Ω) + F(Ω), with P defined in Equation (6) and F being an arbitrary function. The error of each row between the measured data and the computed result is calculated. A modified function G

_{mod}(ω,Ω) is generated by omitting rows with high error, which reduces the computational complexity in the next step. The spectral phase ϕ

_{1}and ϕ

_{2}are reconstructed by using a singular value decomposition (SVD) algorithm with G

_{mod}and h as the input. The arbitrary function F is eliminated in the last step.

**Figure 6.**(

**A**) Simulated VAMPIRE spectrogram generated with the signal in (

**B**) and a filtered signal Gaussian. (

**C**) Rate of convergence plots. The plots on the right side show the convergence behavior of principle component generalized projection algorithm (PCGPA) with random initial guesses, and the plot on the lower left corner shows convergence of the VAMPIRE algorithm [24].

## 4. Double Blind FROG

^{(3)}nonlinear medium, such as a fused silica window. The polarizations of pulse 1 and pulse 2 are set to be 45° relative to each other, for example, pulse 1 at 0° and pulse 2 at 45°, as shown in Figure 9. When the two pulses with 45° relative polarization interact in a χ

^{(3)}nonlinear medium, polarization rotation occurs due to birefringence induced via the third-order nonlinear polarization. This interaction is automatically phase-matched for any crossing-angles, wavelengths and bandwidths. Both pulses experience polarization rotation and will pass through the crossed-polarizer pair in each arm. A spectrometer and camera are placed in each beam path to record a spectrogram (FROG trace), one from each pulse, so two FROG traces are generated in one measurement.

**Figure 9.**The schematic of single-shot Double Blind (DB) PG FROG for measuring an unknown pulse pair [25].

_{1}(ω,τ) represents trace 1, with pulse 1 and pulse 2 acting as the unknown and the known reference pulses, respectively. In trace 2, the roles of pulse 1 and 2 are reversed. Each of the DB PG FROG traces is essentially a PG XFROG trace, but with the known reference pulse replaced by an unknown pulse. In the standard generalized-projections XFROG phase-retrieval algorithm, the known reference pulse is the gate pulse, and the unknown is retrieved by using this gate pulse. DB FROG has two unknown pulses, instead of one known and one unknown pulse. The DB FROG retrieval problem consists of two linked XFROG retrieval problems: in order to retrieve one pulse correctly, the other pulse must be known, and vice versa.

_{1}(t) and E

_{2}(t). In the first half of the cycle, the DB FROG retrieval algorithm assumes E

_{1}(t) is the unknown pulse and E

_{2}(t) is the gate pulse (even though a random guess is usually not close to the correct gate, it is a sufficient starting point). The algorithm retrieves pulse 1 from trace 1 as a PG XFROG problem. After the first half cycle, the retrieved pulse 1 is not the correct pulse, because pulse 2 is not the correct gate pulse to begin with, but it is a better estimate of it than the initial guess, because this estimate more closely satisfies the trace I

_{1}(ω,τ). In the second half of the cycle, the roles of 1 and 2 are switched. Trace 2 and the improved version of E

_{1}(t) (acting as the gate pulse) are used to retrieve E

_{2}(t). After one complete cycle, both improved versions of E

_{1}(t) and E

_{2}(t) are used as the inputs for the next cycle. This process is repeated, until the two DB PG FROG traces generated from E

_{1}(t) and E

_{2}(t) match the experimentally measured traces, that is, the difference between the measured and retrieved traces is minimized. Standard noise-reduction and background subtraction are performed on the two measured traces before running the retrieval algorithm. The convergence of DB FROG is defined, similarly to other standard FROG techniques, by the G-error (the root-mean-square difference of the measured and retrieved traces), one for each DB PG FROG traces.

**Figure 10.**Simulations of DB PG FROG for two complex pulses. The two “measured” traces (

**a**) and (

**c**) are shown in the traces labeled “Simulation”, and their retrieved traces (

**b**) and (

**d**) are shown to their immediate right. The retrieved intensities and phases are shown in (

**e**) to (

**h**) by the solid-color lines. The actual intensity and phase of the simulated pulses are shown as dashed black lines. Both of these simulated complex pulses have time-bandwidth products of about seven, and 1% additive Poisson noise was added to the simulated traces to simulate noisy measurements [25].

**Figure 11.**The measured trace for (

**a**) a chirped pulse train and (

**c**) a chirped double pulse. The retrieved trace (

**b**) and (

**d**) with FROG error of 0.81% and 0.74% for (

**a**) and (

**c**), respectively. (

**e**) and (

**g**), retrieved pulse intensity and phase in temporal domain showing structures from chirped pulse beating. (

**f**) and (

**h**), the measured spectrum and the spectral phase compared with measurement made by a spectrometer [26].

**Figure 12.**The schematic of single-shot DB PG FROG for measuring an unknown pulse pair at different wavelengths (400 nm and 800 nm in the actual experimental demonstration) [26].

**Figure 13.**The measured DB PG FROG traces for (

**a**) a simple pulse at 400nm and (

**c**) a well separated double pulse at 800nm. The retrieved traces (

**b**) and (

**d**) with a FROG error of 0.83% and 0.52% for (a) and (c), respectively. (

**e**) and (

**g**), retrieved pulse intensity and phase in temporal domain. (

**f**) and (

**h**), the measured spectrum and the spectral phase compared with a measurement made by a spectrometer [26].

## 5. Summary

## Acknowledgments

## Conflict of Interest

## References

- Trebino, R.; Kane, D.J. Using phase retrieval to measure the intensity and phase of ultrashort pulses: Frequency-resolved optical gating. J. Opt. Soc. Am. A
**1993**, 10, 1101–1111. [Google Scholar] [CrossRef] - Richman, B.A.; DeLong, K.W.; Trebino, R. Temporal characterization of the stanford mid-IR FEL micropulses by “FROG”. Nuclear Instrum. Methods Phys. Res.
**1995**, A358, 268–271. [Google Scholar] [CrossRef] - Trebino, R. Frequency-Resolved Optical Gating: The Measurement of Ultrashort Laser Pulses; Springer: Manhattan, NY, USA, 2002. [Google Scholar]
- O’Shea, P.; Kimmel, M.; Gu, X.; Trebino, R. Highly simplified device for ultrashort-pulse measurement. Opt. Lett.
**2001**, 26, 932–934. [Google Scholar] [CrossRef] - Akturk, S.; Kimmel, M.; O’Shea, P.; Trebino, R. Extremely simple device for measuring 20-fs pulses. Opt. Lett.
**2004**, 29, 1025–1027. [Google Scholar] [CrossRef] - Bowlan, P.; Trebino, R. Complete single-shot measurement of arbitrary nanosecond laser pulses in time. Opt. Express
**2011**, 19, 1367–1377. [Google Scholar] [CrossRef] - Lozovoy, V.V.; Pastirk, I.; Dantus, M. Multiphoton intrapulse interference. IV. Ultrashort laserpulse spectral phase characterization and compensation. Opt. Lett.
**2004**, 29, 775–777. [Google Scholar] [CrossRef] - Xu, B.; Gunn, J.M.; Cruz, J.M.D.; Lozovoy, V.V.; Dantus, M. Quantitative investigation of the multiphoton intrapulse interference phase scan method for simultaneous phase measurement and compensation of femtosecond laser pulses. J. Opt. Soc. Am. B
**2006**, 23, 750–759. [Google Scholar] [CrossRef] - Iaconis, C.; Walmsley, I.A. Spectral phase interferometry for direct electric-field reconstruction of ultrashort optical pulses. Opt. Lett.
**1998**, 23, 792–794. [Google Scholar] [CrossRef] - Baum, P.; Lochbrunner, S.; Riedle, E. Zero-additional-phase SPIDER: Full characterization of visible and sub-20-fs ultraviolet pulses. Opt. Lett.
**2004**, 29, 210–212. [Google Scholar] [CrossRef] - Polli, D.; Brida, D.; Mukamel, S.; Lanzani, G.; Cerullo, G. Effective temporal resolution in pump-probe spectroscopy with strongly chirped pulses. Phys. Rev. A
**2010**, 82, 053809. [Google Scholar] [CrossRef] - Zewail, A.H. Femtochemistry: Atomic-scale dynamics of the chemical bond. J. Phys. Chem. A
**2000**, 104, 5660–5694. [Google Scholar] [CrossRef] - Fushitani, M. Applications of pump-probe spectroscopy. Annu. Rep. Sec. C
**2008**, 104, 272–297. [Google Scholar] [CrossRef] - Kohler, B.; Yakovlev, V.V.; Che, J.; Krause, J.L.; Messina, M.; Wilson, K.R.; Schwentner, N.; Whitnell, R.M.; Yan, Y. Quantum control of wave packet evolution with tailored femtosecond pulses. Phys. Rev. Lett.
**1995**, 74, 3360–3363. [Google Scholar] [CrossRef] - Gu, X.; Xu, L.; Kimmel, M.; Zeek, E.; O’Shea, P.; Shreenath, A.P.; Trebino, R.; Windeler, R.S. Frequency-resolved optical gating and single-shot spectral measurements reveal fine structure in microstructure-fiber continuum. Opt. Lett.
**2002**, 27, 1174–1176. [Google Scholar] [CrossRef] - Cao, Q.; Gu, X.; Zeek, E.; Kimmel, M.; Trebino, R.; Dudley, J.; Windeler, R.S. Measurement of the intensity and phase of supercontinuum from an 8-mm-long microstructure fiber. Appl. Phys. B
**2003**, 77, 239–244. [Google Scholar] [CrossRef] - Zhang, J.-Y.; Lee, C.-K.; Huang, J.Y.; Pan, C.-Y. Sub-femto-joule sensitive single-shot OPA-XFROG and its application in study of white-light supercontinuum generation. Opt. Express
**2004**, 12, 574–581. [Google Scholar] [CrossRef] - DeLong, K.W.; Trebino, R.; White, W.E. Simultaneous recovery of two ultrashort laser pulses from a single spectrogram. J. Opt. Soc. Am. B
**1995**, 12, 2463–2466. [Google Scholar] [CrossRef] - Kane, D.J.; Rodriguez, G.; Taylor, A.J.; Clement, T.S. Simultaneous measurement of two ultrashort laser pulses from a single spectrogram in a single shot. J. Opt. Soc. Am. B
**1997**, 14, 935–943. [Google Scholar] [CrossRef] - Field, J.J.; Durfee, C.G.; Squier, J.A. Blind frequency-resolved optical-gating pulse characterization for quantitative differential multiphoton microscopy. Opt. Lett.
**2010**, 35, 3369–3371. [Google Scholar] [CrossRef] - Mairesse, Y.; Quéré, F. Frequency-resolved optical gating for complete reconstruction of attosecond bursts. Phys. Rev. A
**2005**, 71, 011401. [Google Scholar] [CrossRef] - Kim, K.T.; Ko, D.H.; Park, J.; Tosa, V.; Nam, C.H. Complete temporal reconstruction of attosecond high-harmonic pulse trains. New J. Phys.
**2010**, 12, 083019. [Google Scholar] [CrossRef] - Hause, A.; Hartwig, H.; Seifert, B.; Stolz, H.; Böhm, M.; Mitschke, F. Phase structure of soliton molecules. Phys. Rev. A
**2007**, 75, 063836. [Google Scholar] [CrossRef] - Birger, S.; Heinrich, S. A method for unique phase retrieval of ultrafast optical fields. Meas. Sci. Technol.
**2009**, 20, 015303. [Google Scholar] [CrossRef] - Wong, T.C.; Ratner, J.; Chauhan, V.; Cohen, J.; Vaughan, P.M.; Xu, L.; Consoli, A.; Trebino, R. Simultaneously measuring two ultrashort laser pulses on a single-shot using double-blind frequency-resolved optical gating. J. Opt. Soc. Am. B
**2012**, 29, 1237–1244. [Google Scholar] - Wong, T.C.; Ratner, J.; Trebino, R. Simultaneous measurement of two different-color ultrashort pulses on a single shot. J. Opt. Soc. Am. B
**2012**, 29, 1889–1893. [Google Scholar] [CrossRef] - Gu, X.; Akturk, S.; Shreenath, A.; Cao, Q.; Trebino, R. The measurement of ultrashort light pulses—Simple devices, complex pulses. Opt. Rev.
**2004**, 11, 141–152. [Google Scholar] - Anderson, M.E.; Monmayrant, A.; Gorza, S.P.; Wasylczyk, P.; Walmsley, I.A. SPIDER: A decade of measuring ultrashort pulses. Laser Phys. Lett.
**2008**, 5, 259–266. [Google Scholar] [CrossRef] - Ratner, J.; Steinmeyer, G.; Wong, T.C.; Bartels, R.; Trebino, R. Coherent artifact in modern pulse measurements. Opt. Lett.
**2012**, 37, 2874–2876. [Google Scholar] [CrossRef] - Diels, J.C.P.; Diels, J.C.; Rudolph, W. Ultrashort Laser Pulse Phenomena; Elsevier Science: Waltham, MA, USA, 2006. [Google Scholar]
- Linden, S.; Giessen, H.; Kuhl, J. XFROG-a new method for amplitude and phase characterization of weak ultrashort pulses. Phys. Stat. Solidi B
**1998**, 206, 119–124. [Google Scholar] [CrossRef] - Seifert, B.; Stolz, H.; Tasche, M. Nontrivial ambiguities for blind frequency-resolved optical gating and the problem of uniqueness. J. Opt. Soc. Am. B
**2004**, 21, 1089–1097. [Google Scholar] [CrossRef]

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Wong, T.C.; Trebino, R. Recent Developments in Experimental Techniques for Measuring Two Pulses Simultaneously. *Appl. Sci.* **2013**, *3*, 299-313.
https://doi.org/10.3390/app3010299

**AMA Style**

Wong TC, Trebino R. Recent Developments in Experimental Techniques for Measuring Two Pulses Simultaneously. *Applied Sciences*. 2013; 3(1):299-313.
https://doi.org/10.3390/app3010299

**Chicago/Turabian Style**

Wong, Tsz Chun, and Rick Trebino. 2013. "Recent Developments in Experimental Techniques for Measuring Two Pulses Simultaneously" *Applied Sciences* 3, no. 1: 299-313.
https://doi.org/10.3390/app3010299