A Simple Method for the Evaluation of the Pulse Width of an Ultraviolet Femtosecond Laser Used in Two-Photon Ionization Mass Spectrometry

Tomoko Imasaka 1,*, Akifumi Hamachi 2, Tomoya Okuno 2 and Totaro Imasaka 2,3 1 Laboratory of Chemistry, Graduate School of Design, Kyushu University, Fukuoka 815-8540, Japan 2 Department of Applied Chemistry, Graduate School of Engineering, Kyushu University, Fukuoka 819-0395, Japan; hamachi.akifumi.967@s.kyushu-u.ac.jp (A.H.); tomoyawashere@yahoo.co.jp (T.O.); imasaka@cstf.kyushu-u.ac.jp (T.I.) 3 Division of Optoelectronics and Photonics, Center for Future Chemistry, Kyushu University, Fukuoka 819-0395, Japan * Correspondence: imasaka@design.kyushu-u.ac.jp; Tel.: +81-92-553-4462


Introduction
Ultrashort laser pulses have been used successfully in a variety of applications including trace analysis of organic compounds [1].Several techniques have been developed to measure the pulse width of the femtosecond laser, including autocorrelation (AC), spectral phase interferometry for direct electric field reconstruction (SPIDER), and frequency-resolved optical gating (FROG) [2].Among these methods, the FROG technique is widely used and has several variations in its implementation.For example, second harmonic generation (SHG), polarization gating (PG), or self-diffraction (SD) can be used for the nonlinear optical effect.To expand the spectral domain to measure shorter pulse widths, third harmonic generation (THG), cross-correlation (X), and four-wave mixing (FWM) have all been used to date [3][4][5][6].On the other hand, the pulse width of the UV femtosecond laser has been measured based on autocorrelation using a sold material, e.g., a CaF 2 plate, as a nonlinear optical device [7][8][9][10].Moreover, the vacuum-ultraviolet (VUV) laser pulse has been measured by focusing it with a near-infrared (NIR) probe pulse into xenon to observe the cross-correlation signal of ionization [11][12][13][14][15].However, the pulse duration would be changed by traveling even in ambient air from the location of the device used to perform the pulse width measurement to the point of application.Complex techniques such as attosecond streaking (AS) or FROG for complete reconstruction of attosecond bursts (FROG-CRAB) can be used to measure the pulse width in a vacuum chamber used for extreme ultraviolet (EUV) pulse generation [16,17].However, these techniques are rather complicated for use in practical applications.
For simple on-site measurement of pulse widths, a device based on fringe-resolved autocorrelation (FRAC) was developed, which consisted of an interferometer and a time-of-flight mass spectrometer that was used as a two-photon-response detector to measure a non-resonant two-photon ionization signal [18][19][20][21].However, the laser beam must be split into two parts and recombined after the interferometer, which reduces the laser pulse energy and then the sensitivity of the mass spectrometer.Also, the low-dispersion aluminum mirrors used in the interferometric system have substantial reflection losses in the deep-ultraviolet (DUV) region.In addition, the mass spectrometer must be operated for long periods, e.g., for periods of days, without maintenance to compare the data from repeated measurements obtained on the same day.Thus, it requires sufficient optical system stability for the pulse width measurements to be performed.In fact, we tried to measure an autocorrelation trace for a DUV pulse generated through several nonlinear optical processes combined in series, but the attempt was unsuccessful due to low pulse energy, poor beam quality, and instability of the laser pulse energy although performance of the laser was sufficient for application to mass spectrometry.Therefore, a simple and rugged instrument that does not contain an interferometer is highly desirable for evaluation of the widths of pulses from DUV femtosecond lasers.On the other hand, DUV femtosecond lasers have been used successfully for two-photon ionization applications in mass spectrometry [1].Because the dispersion caused by the optical components or even by the ambient air cannot be negligible in the DUV region, a device such as a prism pair must be used for pulse compression in the laser system to generate a nearly-transform-limited pulse in the mass spectrometer [22].It should be noted that a technique referred to as multiphoton intrapulse interference phase scan (MIIPS) or dispersion scan (d-can) has been reported, in which the SHG spectrum is measured by changing the dispersion in the beam path, e.g., by translating a wedge or prism, or changing the angle of a grating [23][24][25][26][27][28][29].This method is useful for observing a two-dimensional display of the delay (dispersion) and the spectrum, allowing the full characterization of the chirp of the pulse.However, the use of a crystal for SHG limits the spectral range to the NIR-visible region and also makes the on-site measurement of the pulse width difficult.
In this study, we propose a simple method to evaluate the widths of pulses from a DUV femtosecond laser without use of an interferometer.This technique is based on measurement of a non-resonant two-photon ionization signal in mass spectrometry by translating the prism in the laser's pulse compressor.This method was applied to femtosecond optical pulses emitted at 241 and 219 nm, in addition to pulses emitted at 267 nm, which were generated by four-wave Raman mixing in hydrogen gas.The results obtained herein were compared with the transform-limited pulse widths that were calculated from the spectral bandwidths of the laser used in the experiments.

Theoretical Calculations
The electric field and intensity characteristics of an optical pulse with respect to time (t) can be assumed to have Gaussian profiles [2].

"
Eptq " E 0 e ´at 2 e ipωt`bt 2 q (1) where "a" is related to the pulse duration, ∆t, as shown below, and "b" is a chirp parameter.
The second-order FRAC signal can be expressed as where τ is the time delay between the two pulses in the interferometer [21].This parameter can be set to zero because the beam is not separated in this study (i.e., no interferometer is used), and this leads to Equation (5).
where the parameter, A, is defined as a pulse energy by the following equation.
When the signal intensity becomes one half of this value via a chirp of the pulse, where I 2 FTL and I 2 CP are the signal intensities of the transform-limited and chirped pulses, respectively.The following equation is then obtained.

?
a FTL " 2 ? a CP (8) where a FTL and a CP are the parameters that were calculated using Equation (3).Then, where ∆t FTL and ∆t CP are the pulse durations of the transform-limited and chirped pulses, respectively.The relationship between ∆t CP and ∆t FTL can be expressed as Equation (10) [21].
where ϕ 2 is the group delay dispersion (GDD), which can be expressed as a product of the group velocity dispersion (GVD) determined using the Sellmeier equation [30,31] and the effective length of the optical material in the beam path, ´ 0 (see Figure 1a).Then, where = (2 tan 34 ˝) ˆL for a Brewster prism used at approximately 250 nm.The factor of 2 in Equation ( 11) is multiplied due to a double pass of the beam in the prism.When the dispersion is canceled by a prism pair, GDD = 0 and 0 = (2 tan 34 ˝) ˆL0 where 0 and L 0 are the parameters of and L at which the dispersion is canceled (the two-photon ionization signal is maximal).A parameter of X can be defined as a displacement of the prism position from the optimum location that cancels the dispersion (i.e., GDD = 0), then X " L ´L0 .When the intensity of the two-photon ionization signal decreases to one half of the maximum value, L = L 1/2 and then X 1/2 = L 1/2 ´L0 .From Equations ( 9) and ( 10), ∆t FTL can then be rewritten as where ϕ 2 (X 1/2 ) is the value of GDD at X = X 1/2 .This equation suggests that the transform-limited pulse width can be calculated by measuring the parameter of X 1/2 in the experiment.The actual pulse width is, however, broadened by several reasons: (1) an initial laser pulse would be neither Gaussian-shaped nor transform-limited; (2) the laser pulse would be chirped even at the optimal prism position because of the third-order dispersion (TOD) of the fused silica of the prisms used for pulse compression; and (3) the pulse front would be deformed by mode-change and selffocusing during beam transmission in a hollow capillary filled with hydrogen gas (many hot spots were observed in the beam pattern).In order to take account of the deterioration of the laser beam, a parameter, α, can be introduced into Equation (12).
where ∆tREAL is the pulse width observed in the experiment and α is the parameter showing the deviation from the transform-limited pulse.When TOD is only a factor responsible for the distortion of the pulse, α can be written as 1 + TOD/GVD where Equations ( 11)-( 13) lead to the following equations.
As shown in Equation ( 16), the value of X1/2 to be obtained for a transform-limited pulse is actually expanded to a value of αX1/2 in the experiment by some undesirable effects arising from additional chirps and deterioration of the laser pulse.The graph showing the relationship between ΔtREAL and αX1/2 is the same as that of ΔtFTL vs. X1/2, as shown in Equations ( 15) and ( 16).As can be recognized from Equations (15) and ( 16), ΔtREAL is expanded by a factor of α from ΔtFTL that can be calculated from the spectral bandwidth of the laser beam.The parameter, α, can be calculated from The actual pulse width is, however, broadened by several reasons: (1) an initial laser pulse would be neither Gaussian-shaped nor transform-limited; (2) the laser pulse would be chirped even at the optimal prism position because of the third-order dispersion (TOD) of the fused silica of the prisms used for pulse compression; and (3) the pulse front would be deformed by mode-change and self-focusing during beam transmission in a hollow capillary filled with hydrogen gas (many hot spots were observed in the beam pattern).In order to take account of the deterioration of the laser beam, a parameter, α, can be introduced into Equation (12).
where ∆t REAL is the pulse width observed in the experiment and α is the parameter showing the deviation from the transform-limited pulse.When TOD is only a factor responsible for the distortion of the pulse, α can be written as 1 + TOD/GVD where Equations ( 11)-( 13) lead to the following equations.
When α " 1, ∆t FTL " ?16ln2tan34 When α ą 1, ∆t REAL " ?16ln2tan34 As shown in Equation ( 16), the value of X 1/2 to be obtained for a transform-limited pulse is actually expanded to a value of αX 1/2 in the experiment by some undesirable effects arising from additional chirps and deterioration of the laser pulse.The graph showing the relationship between ∆t REAL and αX 1/2 is the same as that of ∆t FTL vs. X 1/2 , as shown in Equations ( 15) and ( 16).As can be recognized from Equations ( 15) and ( 16), ∆t REAL is expanded by a factor of ?α from ∆t FTL that can be calculated from the spectral bandwidth of the laser beam.The parameter, α, can be calculated from the ratio of ∆t REAL and ∆t FTL , suggesting a degree of the deviation from the ideal transform-limited Gaussian pulse.A train of ultrashort optical pulses is generated by superposition of the laser emissions, which are phase-locked to each other.In this study, three beams, i.e., 9ω (267 nm), 10ω (241 nm), and 11ω (219 nm), where ω = 4155 cm ´1, were superimposed and were then used as one of the test beams.The spectral width, ∆ω n , and the parameter, a n , where n = 1, 2, 3, for each beam, were assumed to be ∆ω 1 = ∆ω 2 = ∆ω 3 = ∆ω and a 1 = a 2 = a 3 = a for the purposes of this study.A train of ultrashort pulses is formed under these conditions if the three emission lines are phase locked.Thus, the phase locking expected to occur during the process of four-wave Raman mixing, can be confirmed by comparing the experimental data with the simulation results.The GDD can then be expressed as follows [2,21].
When the laser emissions are phase-locked together, I 2 FRAC (τ = 0) can be written as By assuming that E 0,1 = E 0,2 = E 0,3 = 1 for simplicity, ´8 ˇˇˇp e ´a1 t 2 e ipω 1 t`b 1 t 2 q `e´a 2 t 2 e ipω 2 t`b 2 t 2 q `e´a 3 t 2 e ipω 3 t`b 3 t 2 q q 2 ˇˇˇ2 dt (20) It should be noted that the three transform-limited pulses are assumed to be superimposed in-phase without any chirp in Equation (20).In contrast, when the laser emissions are not phase-locked to each other (i.e., they are randomly phased), I 2 FRAC (τ = 0) can then be expressed as The spectral width can be assumed to be ∆ω = 2ω, although the actual spectral shape is far beyond a Gaussian profile.The experimental data can then be compared with the data calculated using Equations ( 20) and (21).

Experimental
An optical parametric amplifier (OPA, OPerA-Solo, <50 fs, Coherent, Inc., Santa Clara, CA, USA) was pumped by a Ti:sapphire laser (800 nm, 35 fs, 4 mJ, 1 kHz, Elite, Coherent, Santa Clara, CA, USA).The beam of the Ti:sapphire laser that remained for mixing with the OPA output was used for third harmonic generation (267 nm).The remaining fundamental beam (800 nm) and the signal beam of the OPA (1200 nm) were spatially and temporally superimposed on each other and were focused into a hollow capillary filled with hydrogen gas for vibrational molecular modulation.The third harmonic emission (267 nm) was then focused into the hollow capillary to provide frequency modulation and generate vibrational Raman emissions at 241 and 219 nm.The three-color beam (267, 241, 219 nm) was separated into three individual beams using three dielectric mirrors (Sigma Koki, Tokyo, Japan) placed in series, as shown in Figure 1b.The laser beams passed through three pairs of prisms for pulse compression.The beams were then reflected by a pair of roof mirrors and were recombined into a single beam using the dielectric mirrors.The three-color beam was then focused by a concave mirror into a molecular beam in a mass spectrometer (Hikari Gijyutsu Corp., Fukuoka, Japan).The energy of each pulse measured in front of the mass spectrometer was ca. 1 µJ.In the experiment, 1,4-dioxane was introduced into the mass spectrometer for recording of a non-resonant two-photon ionization signal [20].The signal intensity was measured by translating the second prism to alter the positive dispersion in the compressor.The parameter X 1/2 was determined to be the half width at half maximum of the observed data.The spectral bandwidth of the laser was measured using a spectrometer (Maya2000pro, spectral resolution 1.5 nm, Ocean Optics, Dunedin, FL, USA), the resolution of which was calibrated using a mercury lamp (Ocean Optics, Dunedin, FL, USA) at 254 nm.

Calculations
Figure 2 shows the dependence of ∆t REAL on αX 1/2 (or ∆t FTL on X 1/2 ), which was calculated in the range from 2ω to 16ω (ω = 4155 cm ´1) using Equation ( 16).Because the dispersion increases at higher frequencies, the pulse width increases even at the same value of αX 1/2 .For example, when αX 1/2 = 1 mm, the pulse width becomes 5.0 fs at 2ω and 95 fs at 16ω.Another example would be αX 1/2 = 10 mm, providing pulse widths of 16 fs at 2ω and 300 fs at 16ω.Therefore, the pulse width can be evaluated based on the observed data to show the dependence of the two-photon ionization signal on the displacement (X) of the second prism in the pulse compressor.
Appl.Sci.2016, 6, x 6 of 10 positive dispersion in the compressor.The parameter X1/2 was determined to be the half width at half maximum of the observed data.The spectral bandwidth of the laser was measured using a spectrometer (Maya2000pro, spectral resolution 1.5 nm, Ocean Optics, Dunedin, FL, USA), the resolution of which was calibrated using a mercury lamp (Ocean Optics, Dunedin, FL, USA) at 254 nm.

Calculations
Figure 2 shows the dependence of ΔtREAL on αX1/2 (or ΔtFTL on X1/2), which was calculated in the range from 2ω to 16ω (ω = 4155 cm −1 ) using Equation ( 16).Because the dispersion increases at higher frequencies, the pulse width increases even at the same value of αX1/2.For example, when αX1/2 = 1 mm, the pulse width becomes 5.0 fs at 2ω and 95 fs at 16ω.Another example would be αX1/2 = 10 mm, providing pulse widths of 16 fs at 2ω and 300 fs at 16ω.Therefore, the pulse width can be evaluated based on the observed data to show the dependence of the two-photon ionization signal on the displacement (X) of the second prism in the pulse compressor.

Puse Width Evaluation
Figure 3 shows the dependence of the signal intensities measured as a function of the prism displacement, X, for laser beams emitted at 9ω (=267 nm), 10ω (=241 nm), and 11ω (=219 nm).Because αX1/2 = 4.9 mm at 9ω, the pulse width, ∆tREAL, can be calculated to be 64 fs from the data shown in Figure 2.For the values of αX1/2 = 5.6 mm at 10ω and 1.7 mm at 11ω, the corresponding values of ∆tREAL can be calculated to be 78 and 49 fs, respectively.The spectral bandwidth was measured using a spectrometer, and the estimated values were ca.3.3, 2.9, and 2.5 nm after calibration of the resolution (1.5 nm) of the spectrometer at 9ω, 10ω, and 11ω, respectively.From the relationship of ∆t × ∆ν ≥ k (where k = 0.441 for a Gaussian pulse), ∆tFTL can be calculated to be 36, 34, and 35 fs at 9ω, 10ω, and 11ω, respectively.The parameter, α, can then be calculated to ca. 3.2, 5.3, and 2.0 for 9ω, 10ω, and 11ω, respectively.These results suggest that the anti-Stokes beam has the shortest pulse width among them and has a slightly (1.4 times) larger pulse width than the width expected for a transform-limited pulse.
2. Calculated dependence of the parameter, ∆t REAL , on the parameter, αX 1/2 .The laser frequency, nω, is shown in the figure, where n and ω are the order of Raman sidebands and the Raman shift frequency of molecular hydrogen (ω = 4155 cm ´1), respectively.The definitions of the parameters, ∆t REAL and αX 1/2 , are given in the text.

Puse Width Evaluation
Figure 3 shows the dependence of the signal intensities measured as a function of the prism displacement, X, for laser beams emitted at 9ω (=267 nm), 10ω (=241 nm), and 11ω (=219 nm).Because αX 1/2 = 4.9 mm at 9ω, the pulse width, ∆t REAL , can be calculated to be 64 fs from the data shown in Figure 2.For the values of αX 1/2 = 5.6 mm at 10ω and 1.7 mm at 11ω, the corresponding values of ∆t REAL can be calculated to be 78 and 49 fs, respectively.The spectral bandwidth was measured using a spectrometer, and the estimated values were ca.3.3, 2.9, and 2.5 nm after calibration of the resolution (1.5 nm) of the spectrometer at 9ω, 10ω, and 11ω, respectively.From the relationship of ∆t ˆ∆ν ě k (where k = 0.441 for a Gaussian pulse), ∆t FTL can be calculated to be 36, 34, and 35 fs at 9ω, 10ω, and 11ω, respectively.The parameter, α, can then be calculated to ca. 3.2, 5.3, and 2.0 for 9ω, 10ω, and 11ω, respectively.These results suggest that the anti-Stokes beam has the shortest pulse width among them and has a slightly (1.4 times) larger pulse width than the width expected for a transform-limited pulse.While the spectral profile of the three-color beam composed from the three spectral lines (267, 241, 219 nm) was far beyond the Gaussian shape, the presented technique was applied to this type of beam on a trial basis.The predicted results were calculated for two cases: (1) where the phases of the emissions are random; and (2) where the emission lines are phase-locked.In the former case, a broad band profile was obtained, as shown in Figure 4a, in which αX1/2 = 2.8 mm.In the latter case, a very sharp peak, i.e., αX1/2 = 0.015 mm, was obtained, as shown in Figure 4b.The experimental data shown in Figure 4c consisted of two components, where one is a sharp peak (αX1/2 = 0.12 mm) at the center, and the other is a broad band observed as a pedestal.It should be noted that the signal intensity was highly sensitive to the positions of the second prisms in the vicinity of the maximum value, and that this result was obtained by carefully translating these prisms simultaneously after critical optimization of their positions.
It is possible to consider that a sharp peak would arise from the phase-locked components while another would arise from the phase-random components.The former peak width (0.12 mm), corresponding to a pulse width of 11 fs, was apparently broader than the theoretically predicted value (0.015 mm), corresponding to a pulse width of 4.0 fs.This discrepancy is likely to arise from insufficient precision in phase control of the three emissions, during which the prisms were moved simultaneously using translation stages equipped with manually driven differential micrometers (see Figure 1b).Another explanation for the discrepancy, and the appearance of the pedestal, could be the spectral phase and amplitude fluctuations between the three pulses used in this study [32].While the spectral profile of the three-color beam composed from the three spectral lines (267, 241, 219 nm) was far beyond the Gaussian shape, the presented technique was applied to this type of beam on a trial basis.The predicted results were calculated for two cases: (1) where the phases of the emissions are random; and (2) where the emission lines are phase-locked.In the former case, a broad band profile was obtained, as shown in Figure 4a, in which αX 1/2 = 2.8 mm.In the latter case, a very sharp peak, i.e., αX 1/2 = 0.015 mm, was obtained, as shown in Figure 4b.The experimental data shown in Figure 4c consisted of two components, where one is a sharp peak (αX 1/2 = 0.12 mm) at the center, and the other is a broad band observed as a pedestal.It should be noted that the signal intensity was highly sensitive to the positions of the second prisms in the vicinity of the maximum value, and that this result was obtained by carefully translating these prisms simultaneously after critical optimization of their positions.
It is possible to consider that a sharp peak would arise from the phase-locked components while another would arise from the phase-random components.The former peak width (0.12 mm), corresponding to a pulse width of 11 fs, was apparently broader than the theoretically predicted value (0.015 mm), corresponding to a pulse width of 4.0 fs.This discrepancy is likely to arise from insufficient precision in phase control of the three emissions, during which the prisms were moved simultaneously using translation stages equipped with manually driven differential micrometers (see Figure 1b).Another explanation for the discrepancy, and the appearance of the pedestal, could be the spectral phase and amplitude fluctuations between the three pulses used in this study [32].

Conclusions
A simple method for evaluation of the widths of pulses from a UV femtosecond laser was proposed that was based on measurement of a two-photon ionization signal by translation of the second prism in the laser's pulse compressor.The pulse widths that were observed experimentally showed reasonably good agreement with the values that were calculated from the spectral bandwidths, which were measured using a spectrometer.The method presented here could be applied to lasers with shorter pulse widths by using a prism with lower TOD, e.g., a prism made of CaF2 or MgF2.The main advantage of this method is the minimal loss of pulse energy during measurement of the pulse width.Therefore, this technique can be used to evaluate the widths of pulses from the UV femtosecond laser that is used as an ionization source in mass spectrometry for practical trace analysis.FRAC (τ = 0), on the parameter, X.The parameter of X 1/2 can be obtained by calculating the half width at the half maximum of the signal peak.In this case, a three-color beam emitting at 267, 241, and 219 nm was introduced into the molecular beam in the mass spectrometer.

Conclusions
A simple method for evaluation of the widths of pulses from a UV femtosecond laser was proposed that was based on measurement of a two-photon ionization signal by translation of the second prism in the laser's pulse compressor.The pulse widths that were observed experimentally showed reasonably good agreement with the values that were calculated from the spectral bandwidths, which were measured using a spectrometer.The method presented here could be applied to lasers with shorter pulse widths by using a prism with lower TOD, e.g., a prism made of CaF 2 or MgF 2 .The main advantage of this method is the minimal loss of pulse energy during measurement of the pulse width.Therefore, this technique can be used to evaluate the widths of pulses from the UV femtosecond laser that is used as an ionization source in mass spectrometry for practical trace analysis.

Figure 1 .
Figure 1.(a) Prism parameters; (b) configuration of the prisms in a pulse compressor.Dotted arrows show the direction of displacement of the prisms.The parameters, ℓ and L, are the path length of the laser beam in the prism and the distance from the top of the prism to the laser beam, respectively.The angle of the prism is specified in the figure.Three prisms in (b) can be manually translated by rotating the nobs of the differential micrometers independently.

Figure 1 .
Figure 1.(a) Prism parameters; (b) configuration of the prisms in a pulse compressor.Dotted arrows show the direction of displacement of the prisms.The parameters, and L, are the path length of the laser beam in the prism and the distance from the top of the prism to the laser beam, respectively.The angle of the prism is specified in the figure.Three prisms in (b) can be manually translated by rotating the nobs of the differential micrometers independently.

Figure 2 .
Figure 2. Calculated dependence of the parameter, ΔtREAL, on the parameter, αX1/2.The laser frequency, nω, is shown in the figure, where n and ω are the order of Raman sidebands and the Raman shift frequency of molecular hydrogen (ω = 4155 cm −1 ), respectively.The definitions of the parameters, ΔtREAL and αX1/2, are given in the text.

Figure 3 .
Figure 3. Observed dependence of the signal intensity, I 2 FRAC (τ = 0), on the parameter, X.The dots in the figure are the observed data, and a solid curve is a guide to the eye.The parameter of X1/2 can be obtained by calculating the half width at the half maximum of the signal peak.Laser operating wavelengths: (a) 267; (b) 241; and (c) 219 nm.

Figure 3 .
Figure 3. Observed dependence of the signal intensity, I 2 FRAC (τ = 0), on the parameter, X.The dots in the figure are the observed data, and a solid curve is a guide to the eye.The parameter of X 1/2 can be obtained by calculating the half width at the half maximum of the signal peak.Laser operating wavelengths: (a) 267; (b) 241; and (c) 219 nm.

Figure 4 .
Figure 4. Calculated dependence of the parameter, I 2 FRAC (τ = 0), on the parameter, X: (a) random phase; and (b) phase-locked; (c) Observed dependence of the signal intensity, I 2 FRAC (τ = 0), on the parameter, X.The parameter of X1/2 can be obtained by calculating the half width at the half maximum of the signal peak.In this case, a three-color beam emitting at 267, 241, and 219 nm was introduced into the molecular beam in the mass spectrometer.

Figure 4 .
Figure 4. Calculated dependence of the parameter, I 2 FRAC (τ = 0), on the parameter, X: (a) random phase; and (b) phase-locked; (c) Observed dependence of the signal intensity, I 2FRAC (τ = 0), on the parameter, X.The parameter of X 1/2 can be obtained by calculating the half width at the half maximum of the signal peak.In this case, a three-color beam emitting at 267, 241, and 219 nm was introduced into the molecular beam in the mass spectrometer.