Simplified Chirp Characterization in Single-shot Supercontinuum Spectral Interferometry
SSimplified Chirp Characterization in Single-shotSupercontinuum Spectral Interferometry
DinhDuy Tran Vu , Dogeun Jang , and Ki-Yong Kim ∗ Institute for Research in Electronics and Applied Physics,University of Maryland, College Park, MD, 20742
Single-shot supercontinuum spectral interferometry (SSSI) is an optical technique that can measureultrafast transients in the complex index of refraction. This method uses chirped supercontinuumreference/probe pulses that need to be pre-characterized prior to use. Conventionally, the spectralphase (or chirp) of those pulses can be determined from a series of phase or spectral measurementstaken at various time delays with respect to a pump-induced modulation. Here we propose a novelmethod to simplify this process and characterize reference/probe pulses up to the third order disper-sion from a minimum of 2 snapshots taken at different pump-probe delays. Alternatively, withoutany pre-characterization, our method can retrieve both unperturbed and perturbed reference/probephases, including the pump-induced modulation, from 2 time-delayed snapshots. From numericalsimulations, we show that our retrieval algorithm is robust and can achieve high accuracy evenwith 2 snapshots. Without any apparatus modification, our method can be easily applied to anyexperiment that uses SSSI.
I. INTRODUCTION
Single-shot spectral interferometry (SSI) is an ultrafastoptical method that can measure ultra-rapid refractiveindex transients induced by ultrashort laser pulses [1, 2].This measurement can provide a direct view of how alaser-induced perturbation evolves in time and space ina single-shot. In this technique, a pump pulse inducesa refractive index transient in a medium, and a chirpedreference and a time-delayed replica (probe) pulse, uponwhich the pump-pulse-induced phase shift has been im-posed, interfere in an imaging spectrometer, producing aspectral interferogram. The corresponding spatiotempo-ral (time and 1-dimensional space) evolution of refractiveindex transient is then reconstructed with frequency-to-time mapping or full Fourier transform methods [2]. Inparticular, supercontinuum (SC) light has been used forthe reference and probe pulses to provide a temporal res-olution better than ∼
10 fs [2]. This single-shot super-continuum spectral interferometry (SSSI) and SSI tech-niques have been successfully applied to capture laser-induced double step ionization of helium [3], laser-heatedcluster explosion dynamics [4], laser wakefields [5], op-tical nonlinearity near the ionization threshold [6], andelectronic and inertial nonlinear responses in moleculargases [7–10]. SSI has been also used to capture tera-hertz waveforms in snapshots without pump-probe scan-ning [11, 12].Unlike self-referencing nonlinear diagnostics such asFROG [13, 14] and SPIDER [15, 16], linear spectral in-terferometery including SSSI requires pre-characterizedreference/probe pulses prior to its use. One methodto pre-characterize a chirped probe in SSSI is to scanthe delay between the pump-probe pulses while track-ing a characteristic central extremum in the modulated ∗ [email protected] probe phase or spectrum [2]. This method can determinethe spectral phase of SC probe light to arbitrary order[17]. However it relies on stationary-phase and small-perturbation approximations, which can be problematicwhen the phase modulation is too large or too asymmet-ric. In addition, this requires repetitive measurementsover the entire chirp window for an accurate extractionof higher order dispersion coefficients. In particular, for aSC probe that has an extremely large bandwidth, a largenumber of pump-probe scans are necessary. This scan-ning method is impractical for use with low-repetition-rate laser sources.For these reasons, we aim to develop a method thatcan simplify or avoid the pre-characterization process ifpossible and potentially to characterize both SC refer-ence/probe pulses and pump-induced transients with aminimal number of repetitions. II. BACKGROUND THEORY
In SSSI, the probe SC pulse, E ( t ), is perturbed by apump-induced modulation, ∆Φ( t − τ ), applied at a timedelay τ with respect to the probe pulse (see Fig. 1). Thereference SC pulse, E r ( t ), precedes the pump in time andthus remains unaffected.The perturbed probe pulse E ( t ) can be written as E ( t ) = E ( t ) e i ∆Φ( t − τ ) , (1)where E ( t ) is the unperturbed probe pulse. Then ∆Φ( t ),the same pump-induced modulation but applied at τ =0, can be extracted from the interference between thereference and probe pulses in the frequency domain as∆Φ( t ) = − i ln (cid:34) F (cid:8)(cid:12)(cid:12) E ( ω ) (cid:12)(cid:12) e i (∆ ϕ τ ( ω )+ ϕ ( ω )) e − iωτ (cid:9) F (cid:8) | E ( ω ) | e iϕ ( ω ) e − iωτ (cid:9) (cid:35) , (2)where F denotes the Fourier transform, | E ( ω ) | and | E ( ω ) | is the spectral amplitude of the perturbed and un- a r X i v : . [ phy s i c s . i n s - d e t ] J un FIG. 1. Schematic of SSSI consisting of a chirped super-continuum (SC) pulse (reference, E r ( t )) and a time-delayedreplica pulse (probe, E ( t )) upon which a pump-induced ul-trafast modulation ∆Φ( t − τ ) has been imparted at a timedelay τ with respect to the probe. perturbed probe pulses, respectively, that can be directlymeasured by a spectrometer; ∆ ϕ τ ( ω ) is the phase differ-ence between the modulated probe and reference pulsesat a given τ that can be obtained from an interferometer;and ϕ ( ω ) is the spectral phase of the unperturbed probe(or reference) pulse. Here to retrieve ∆Φ( t ), the spectralphase ϕ ( ω ) needs to be characterized. In general, thespectral phase of a chirped pulse can be expressed in aTaylor expansion around the central frequency ω c as ϕ ( ω ) = ϕ + b ( ω − ω c )+ b ( ω − ω c ) + b ( ω − ω c ) + ..., (3)where ϕ is the absolute spectral phase; b is the first or-der dispersion coefficient related to a pulse shift in time; b and b are the second and third order dispersion co-efficients, respectively. Here the first two terms are notrequired in retrieving ∆Φ( t ), but b and b need to becharacterized for SSSI operation.In Eq. (2), it is important to note that the modula-tion ∆Φ( t ) remains unchanged even if the time delay τ changes. This is because the term e − iωτ shifts the modu-lation occurring at t − τ back to t . In other words, ∆Φ( t )must be uniquely retrieved from many different τ delayedshots if the spectral phase ϕ is correctly characterized.For illustration, we consider a Gaussian-type phasemodulation given by∆Φ( t − τ ) = Ae − ( t − τ ) / (2 σ ) , (4)where we choose A = 0 . σ = 50 fs. Here the probepulse is also a Gaussian pulse centered at 800 nm witha full-width-at-half-maximum (FWHM) bandwidth of170 nm and chirped with b = 1000 fs and b = 400 fs .Figure 2(a) shows a series of differential probe powerspectrum [17], ∆ I ( ω ), as a function of the pump-probedelay τ . Figure 2(b) shows two spectral line-outs at τ = 0 fs and 400 fs. One prominent feature is that theposition of the central minimum ω shifts with respectto the pump-probe delay τ . Here the central minimumis defined as the point where ∆ I ( ω ) oscillates slowest;mathematically, it is given by the condition ϕ (cid:48) ( ω ) = 2 b ( ω − ω c ) + 3 b ( ω − ω c ) = τ. (5) Therefore, by tracing ω at each τ , one can determine b and b with a polynomial fit [17]. This method, however,is limited by the validity of the stationary phase approx-imation. Moreover, it is inefficient as only the centralminimum/ maximum point or at most some adjacent ex-trema are used in each shot for characterization. FIG. 2. (a) Simulated differential probe power spectrum,∆ I ( ω, τ ), modulated by a phase transient given by Eq. (4)as a function of pump-probe delay. The probe is chirped with b = 1000 fs and b = 400 fs . (b) Differential probe spectralline-outs at τ = 0 fs and τ = 400 fs It is obvious that one needs the correct values of thesecond and third dispersion coefficients to properly char-acterize the phase modulation, but what is the conse-quence if the known values differ by ∆ b and ∆ b fromthe true ones? Our simulation shows that nonzero ∆ b or ∆ b lead to ambiguity in the retrieved modulation∆Φ( t ). For illustration, we show the retrieved ∆Φ( t )’sfrom two different time delays τ = 0 fs and τ = 400 fswith ∆ b = 0 fs , ∆ b = −
40 fs in Fig. 3(a) and∆ b = −
60 fs , ∆ b = 40 fs in Fig. 3(b). Those tworetrieved ∆Φ( t )’s are different, and furthermore neitheris identical to the true modulation. For a wider rangeof ∆ b and ∆ b , we quantify the difference in shape ofthe retrieved modulations obtained from multiple time-delayed shots by∆ S = (cid:90) ∞−∞ (cid:88) τ (cid:2) ∆Φ τ ( t ) − ∆Φ( t ) (cid:3) dt, (6)where ∆Φ( t ) is the average of ∆Φ τ retrieved from alldifferent time delays τ . This ∆ S strongly depends onhow well the probe phase is characterized. For example,the dependence of ∆ S on both ∆ b and ∆ b is com-puted and shown in Fig. 3(c). It clearly shows a deepglobal minimum located at (0 , b = 1000 fs and b = 400 fs ). This shows that the modulations obtainedfrom all different time delays converge only when thepulse’s phase used for retrieval matches the true form.At the same time, the converged function represents thereal form of modulation. A mathematical proof of thisobservation is provided in Appendix. FIG. 3. (a), (b) Extracted modulations ∆Φ( t ) obtained fromtwo different time-delayed shots at τ = 0 fs (green dashed line)and τ = 400 fs (red dotted line) using intentionally incorrectspectral phase coefficients, along with the correct modulation(black line). The modulations ∆Φ( t ) obtained from two shotsof different time delay are nonidentical when ∆ b or ∆ b isnon-zero. (c) The dependence of ∆ S on ∆ b and ∆ b showsa deep global minimum located at (0, 0). The square and thetriangle correspond to (∆ b , ∆ b ) = (0 , −
40) and ( − ,
40) asillustrated in (a), (b) respectively.
III. ALGORITHM DETAILS
Experimentally, it is possible that modulations from dif-ferent shots are similar in shape but slightly different inmagnitude due to pump pulse power fluctuation. There-fore, the modulation extracted from each shot is normal-ized prior to comparison. We also emphasize that theretrieved modulation often exhibits smooth variations inthe vicinity of the central extremum, but it is very noisyin the far away region. Therefore, in practice, only a re-gion of interest is used for an input. This should cover asmuch meaningful features of modulation as possible but be narrow enough to avoid too much noise.One feature needs to be discussed is how to choosedifferent time delays τ to optimize the operation of ouralgorithm. In a stationary phase approximation, the per-turbed probe pulse can be expressed as [17] E ( ω ) = E ( ω ) − C ∆Φ( ω − ω ) (cid:112) b (cid:48) | E ( ω ) |× exp (cid:104) ib (cid:48) ( ω − ω ) + ib ( ω − ω ) (cid:105) , (7)where C is a constant, ∆Φ( ω ) is the Fourier transform of∆Φ( t ) in the frequency domain, b (cid:48) = b + 3 b ( ω − ω c ),and ω is given by ϕ (cid:48) ( ω ) = τ . In the case of a small τ , ω becomes close to ω c and b (cid:48) ≈ b , and the dominant partcontaining the third-order dispersion 3 b ( ω c − ω )( ω − ω ) becomes insignificant. In that case, the third orderdispersion is hard to be determined. Therefore, for aneffective operation, we want the change caused by thethird-order dispersion to be greater than its measurementerror ε b ( ω − ω c ) b > ε, (8)where ω − ω c ≈ τ / (2 b ). Therefore, the time delay sep-aration between two shots should be∆ τ > εb b . (9)Equation (9) establishes the relation between the timedelay and experimental conditions. Furthermore, the up-per limit of the time delay is fundamentally set by theprobe pulse duration.As illustrated in Fig. 3, the probe spectral phase canbe found by minimizing ∆ S . This process can be per-formed by a genetic algorithm (GA). Figure 4 shows adiagram of our algorithm routine to characterize bothprobe and modulation simultaneously. First, the spectralmodulations of probe (∆ ϕ s , ∆ ϕ s , .., ∆ ϕ sN ) are mea-sured at multiple pump-probe time delays τ i . Along withan initial population of b ’s and b ’s, the correspondingtemporal modulation functions (∆Φ , ∆Φ , .., ∆Φ N ) areconstructed within Eq. (2). Then the GA is used to min-imize ∆ S defined by Eq. (6). Finally, those b , b thatprovides the global minimum of ∆ S will be selected forthe best fitting parameters. Simultaneously, the ∆Φ( t )calculated from these optimized values is deemed to bethe correct form of the applied modulation. IV. PERFORMANCE TEST
In this section, we test the reliability of our algorithmwith numerical simulations. Here we simulate two typesof modulations. The first one mimics a Kerr-inducedrefractive index modulation, where the modulation isproportional to the intensity of a co-propagating pump
FIG. 4. Algorithm routine for simultaneous characterizationof both probe chirp ( b and b ) and modulation (∆Φ( t )). Ituses a genetic algorithm (GA) to minimize ∆ S such that themodulation functions obtained from all pump-probe delayscan converge to equality. pulse. The second one simulates a femtosecond photo-ionization process, where the modulation asymptoticallyapproaches zero at t → −∞ and a non-zero value at t → ∞ . In both cases, the probe pulse is set to be thesame as in the previous sections with b = 1000 fs and b = 400 fs . IV.1. Kerr-like Modulation
An intense laser pulse can induce a transient in the indexof refraction of a medium it propagates through, leadingto a phase modulation on the co-propagating probe pulse.We assume the modulation has a form of∆Φ( t ) = A e − at [1 − b ( t − t )] + iA e − ct , (10)with A = 0 . , a = 1 . × − fs − , b = 5 × − fs − , t = 30 fs, A = 0 .
1, and c = 2 . × − fs − . The imag-inary part (second term) represents nonzero absorptionin the medium. Here we assume the modulation is notsymmetric in time. We also introduce a random errorof ≤
5% to the simulated spectrogram to test the stabil-ity of our algorithm and a fluctuation of ≤
10% to themagnitude of ∆Φ for each time-delayed shot.We first test the convergence speed of our GA. In thissimulation, each generation comprises 80 sets of b , b with the search range of 600 − for b and 200 −
800 fs for b . We perform the simulation in three caseswith different numbers of time-delayed shots, namely twoshots at τ = 0 and 400 fs, three shots at τ = 0, 400 and600 fs, and four shots at τ = 0, 400, 600 and -200 fs.In each case, we use the same initial population that isintentionally chosen to be far away from the convergedvalues. The optimized ∆ S at each generation is shownin Fig. 3a. Despite the unfavorable condition we set, ∆ S converges fast in all three cases after 15 generations.In Figs. 5(b)-5(d), we show the optimized set of b ∗ , b ∗ ,and the average modulation ∆Φ( t ) obtained from two, FIG. 5. (a) Minimal ∆ S after each generation when us-ing data from two shots at τ = 0 and 400 fs (black solidline), three shots at τ = 0, 400 and 600 fs (red dashed line),and four shots at τ = 0, 400, 600 and -200 fs (green dottedline). ∆ S is normalized to the final converged value. (b)-(d) The real part of reconstructed average modulation (greensolid line) compared to the exact function (red dashed line)given by Eq. (10), with its best fitting parameters ( b ∗ and b ∗ )obtained from 2, 3, and 4 shots, respectively, as defined in(a). (e)-(g) Distribution of retrieved ( b , b ) after 300 trials,corresponding to (b-d) respectively. three, and four delayed shots. In this example, the sec-ond order dispersion coefficient b can be characterizedwithin a 1% error, and the shape of modulation can bereconstructed fairly well even with 2 shots. However, thethird order dispersion coefficient b , less significant com-pared to b , suffers from a 5% error when only two shotsare used only. It is noticeable that when 3 and 4 shotsare used, the retrieval errors of b and b reduce to lessthan 1% and 2%, respectively.We emphasize that the GA is so effective that it con-verges quickly to the almost exact global minimum of∆ S regardless of the number of shots used. Note thatwe also introduce ≤
10% fluctuations to the modulationamplitude, but it is neutralized by the normalization stepin our algorithm. Therefore, the retrieval error as shownpreviously is solely due to the random error introducedto the spectrogram. To examine how this error affectsthe retrieved values, we repeat the same simulation for300 times and plot the extracted b and b in Figs. 5(e)-5(g). Firstly, compared to the typical uncertainty in b (2%), b spans much wider with the standard deviationof ∼
10% in the case of 2 shots. This is understandableas the effect of third order dispersion on the spectra isquite small and can be overwhelmed by the random noise.Secondly, the overall certainty is diminished by increasingthe number of snapshots. The error margins of retrieved b and b shrink significantly when the number of shotsincrease from 2 to 4, specially from ∼
10% to ∼
2% for b . This is not surprising as the effect of random noisecan be lessened by repetition. We note that our GA canalways retrieve the exact b and b , and ∆Φ( t ) when norandom fluctuation is included in the simulations. IV.2. Femtosecond Stepwise Modulation
As a second example, we consider an ultrafast transientcommonly observed in optical field ionization. In stronglaser electric fields, atoms and molecules can be tunnelionized, producing free electrons in continuum states.Macroscopically, the free electron density rises in timeuntil the intense pulse passes by. The density modula-tion induced by the pump pulse can be picked up by aco-propagating probe pulse. For simplicity, we considerthe following phase modulation caused by tunneling ion-ization, | ∆Φ( t ) | = t ≤ −
20 fs0 . t + 20) −
20 fs < t ≤
20 fs0 . t >
20 fs . (11)Similar to the previous section, we simulate the spec-trograms at different time delays and use data from 2,3 or 4 shots to reconstruct the modulation. The spec-trograms are also subject to ≤
5% random fluctuations.Our simulation results are presented in Figs. 6(a)-6(c).In the 2-shot case, the optimized b ∗ and b ∗ exhibit 1%and 10% errors, respectively. With the FWHM band-width of 170 nm, the fastest resolvable phase transientis ∼ / | ω | fora step function in time). This leads to an even worsetemporal resolution. Thus a relatively large uncertaintyis expected in the extraction process. However, when us-ing three or four shots, highly accurate characterizationis possible.In conclusion, our algorithm works well for two ex-amples of ultrafast modulations even with ≤
5% randomnoise applied in the spectrograms. It will work equallywell, we believe, for any reasonably shaped modulations.However, depending on the modulation shape, more than2 shots are needed to obtain very high accuracy, espe-cially when non-negligible random errors are present.
FIG. 6. Same as Figs. 5(b)-5(d) but the modulation functionis given by Eq. (11).
IV.3. Extending the Number of Fitting parameters
In this section, we test the flexibility of the algorithmwhen more fitting parameters are introduced. In onepossible scenario, the probe and reference pulses can benon-identical with different b and/or b . This dispar-ity can occur when the reference and probe pulses passthrough a beam splitter different numbers of times, thusleading to unequal dispersion. In that case, Eq. (2) needsto be modified as∆Φ( t ) = − i ln (cid:34) F (cid:8) | E ( ω ) | e i (∆ ϕ τ + ϕ − ωτ ) (cid:9) F (cid:8) | E ( ω ) | e i ( ϕ + δϕ − ωτ ) (cid:9) (cid:35) , (12)where δϕ is the phase difference between the probe andreference pulses. We estimate δϕ ≈ B ( ω − ω c ) , (13)where B has an order of 10 fs . In this example, thereare three parameters to be optimized ( b , b and B ).Here we choose b = 1000 fs , b = 400 fs , and B =30 fs , with the same modulation and noise ( ≤ b , b and B retrieved from 2, 3 and 4 time-delayed shotsare presented in Tab. I.As shown in Tab. I, b and B can be determinedwithin 2% regardless of the number of shots. Noticeably,when 3 or 4 shots are used, all three parameters can beobtained within 1% error. Note that b is retrieved witha 3% difference, which is comparable to the 5% error ob-tained when the reference and probe pulses are set tobe identical ( B = 0) in Section IV.1. Therefore, theaddition of more chirp parameters does not significantlyaffect the performance of our algorithm. TABLE I. Best-fit parameters ( b ∗ , b ∗ , and B ∗ ) retrieved with2, 3, and 4 time-delayed shots when the probe and referencepulses are allowed to have second order dispersion coefficientsdifferent by B . b ∗ (fs ) b ∗ (fs ) B ∗ (fs )2 shots 1000 389 30.53 shots 997 398 29.84 shots 998 398 29.8 V. CONCLUSION
In summary, we have presented a simple method to de-termine both probe spectral phases and pump-inducedmodulations in a conventional SSSI setup. Our GA-basedroutine is shown to work for typical ultrafast modulationsand capable of characterizing the probe phase with highaccuracy. Also our algorithm can be easily modified toinclude more chirp parameters if necessary. With numer-ical simulations, we show that our algorithm is robustagainst random errors ( ≤ APPENDIX: THE UNIQUENESS OF ∆S MINIMUM
This section attempts to prove mathematically that thestandard deviation ∆ S exhibits a zero value only whenthe phase function used in extraction has the correctform. Suppose that ϕ is slightly deviated as ϕ → ϕ + δϕ ,then we have | E ( ω ) | e i ( ϕ +∆ ϕ ) e iδϕ ( ω ) e − iωτ (14)= e iδϕ ( ω ) e − iωτ F − (cid:104) E ( t ) e i ∆Φ( t − τ ) (cid:105) = e iδϕ ( ω ) (cid:90) M ( ω − ω (cid:48) ) e iω (cid:48) τ | E ( ω (cid:48) ) | e iϕ ( ω (cid:48) ) dω (cid:48) , where M ( ω ) = F − [ e i ∆Φ( t ) ]. The Fourier transform of this term is given by F (cid:110) E ( ω ) e i ( ϕ +∆ ϕ ) e iδϕ ( ω ) e − iωτ (cid:111) (15) ∝ (cid:90) F (cid:20)(cid:90) M ( ω − ω (cid:48) ) E ( ω (cid:48) ) e i [ ϕ ( ω (cid:48) ) − ω (cid:48) τ ] dω (cid:48) (cid:21) C ( t (cid:48) ) e iδ ˜ ϕ ( t (cid:48) ) dt (cid:48) ∝ (cid:90) e i ∆Φ( t − t (cid:48) ) | E ( t − t (cid:48) − τ ) | e i ˜ ϕ ( t − t (cid:48) − τ ) C ( t (cid:48) ) e iδ ˜ ϕ ( t (cid:48) ) dt (cid:48) , where F (cid:8) e iδϕ ( ω ) (cid:9) = C ( t ) e iδ ˜ ϕ ( t ) and E ( t ) = | E ( t ) | e i ˜ ϕ ( t ) .Generally, the modulation varies in a shorter time scalethan the probe pule, so ˜ ϕ and δ ˜ ϕ vary much faster than∆Φ. Using the stationary phase approximation, Eq. (15)can be approximated as F (cid:110) E ( ω ) e i ( ϕ +∆ ϕ ) e iδϕ ( ω ) e − iωτ (cid:111) (16) ∝ e i ∆Φ( t − g ) (cid:90) | E ( t − t (cid:48) − τ ) | e i ˜ ϕ ( t − t (cid:48) − τ ) C ( t (cid:48) ) e iδ ˜ ϕ ( t (cid:48) ) dt (cid:48) , where g is the point that contributes the most to the in-tegral. Within the stationary phase approximation, thispoint can be given by δ ˜ ϕ (cid:48) ( g ) − ˜ ϕ (cid:48) ( t − g − τ ) = 0 . (17)As a result, Eq. (2) now gives F { E ( ω ) exp [ i (∆ ϕ + ϕ + δϕ )] exp( − iωτ ) } F { E ( ω ) exp( iϕ + iδϕ ) exp( − iωτ ) } (18) ≈ exp [ i ∆Φ( t − g )] . Because g depends on τ according to Eq. (17), the ex-traction now produces different results at various timedelays. Only when δϕ = 0, making C ( t ) e iδ ˜ φ → δ ( t ) and g →
0, Eq. (18) yields the same function form regardlessof τ . FUNDING INFORMATION
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