Accurate characterization of complex pulses by delay-controlled fringe-free interferometry of spectral high-resolution technique

Optik 125 (2014) 5524–5528

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Accurate characterization of complex pulses by delay-controlled fringe-free interferometry of spectral high-resolution technique夽 Liang Lei a,∗ , Ziqiang Liu a , Xiaobo Xing b , Qu Wang a , Kunhua Wen a a b

School of Physics and Optoelectronic Engineering, Guangdong University of Technology, Guangzhou 510006, China MOE Key Laboratory of Laser Life Science and Institute of Laser Life Science, South China Normal University, Guangzhou 510631, China

a r t i c l e

i n f o

Article history: Received 29 September 2013 Accepted 1 May 2014 Keywords: Femtosecond pulse Ultrafast measurements Spectral phase interferometry

a b s t r a c t We discuss the disadvantage in conventional Spectral Phase Interferometry for Direct Electric-field Reconstruction (SPIDER) technology in complex femtosecond pulse measurement. An improved version of conventional technology named DC-FISH is presented, where single replica of the unknown pulse upconverts synchronously with two frequency-shifted narrow-banded long pulses. The spectral phase of the unknown pulse can be directly calculated from the fringe-free spectra with the introduction of a suitable small delay between the upconverted pulses. The numerical simulation results are achieved to identify a higher efficiency and lower requirements on measurement in novel approach. © 2014 Elsevier GmbH. All rights reserved.

1. Introduction During the past decade, continuous progress in the field of ultrashort pulse generation has lead to pulse durations below 6 fs in the visible and near-infrared spectral ranges [1,2]. Meanwhile, the use of ultrashort pulses for both fundamental studies and applications is spreading rapidly [3]. In addition, interest in the synthesis of ultrashort pulses with arbitrarily controllable pulse shapes has also arisen. Numerous applications have emerged for shaped ultrashort pulses, such as all-optical switching, pulse coding for communications, control of THz radiation, and coherent control of chemical reactions [4,5]. All these make characterization of ultrashort pulses a demanding task. Fortunately, self-referencing measurement techniques for characterizing the temporal electric field of a short optical pulse such as Frequency-Resolved Optical Gating (FROG) [6] and Spectral Phase Interferometry for Direct Electric-field Reconstruction (SPIDER) [7] are successfully used in many areas of ultrafast sciences. The most common technique, FROG can reconstruct pulses with various properties, simple or complex, from a few femtoseconds to many picoseconds. Yet it does not perform in real time, as it requires taking a lot of data. The experimental trace must be processed by use of an iterative algorithm, which takes

夽 Supported by the National Natural Science Foundation of China Grant Nos. 61107029 and 61177077. ∗ Corresponding author. Tel.: +86 20 39322266; fax: +86 20 39322265. E-mail address: [email protected] (L. Lei). http://dx.doi.org/10.1016/j.ijleo.2014.06.081 0030-4026/© 2014 Elsevier GmbH. All rights reserved.

a long period for complex pulses. Contrarily, SPIDER is intrinsic a fast technique. It requires acquisition of only two spectra for the pulse reconstruction and naturally operates single-shot. Continuous efforts have been made to extend the applicability of this technique such as ZAP-SPIDER [8], SEA-SPIDER [9], CAR-SPIDER [10], SEA-CAR-SPIDER [11], 2DSI [12], Filter-SPIDER [13], Wide-time-range-SPIDER [14]. Our team also experimentally reported a modified technology based on ZAP-SPIDER two years ago [15]. However, these improved SPIDER technology still have some limitations on their application areas. They are robust at the characterization of simple pulses with smooth spectra and phases, while the accuracy drops significantly for complex pulses with abrupt variations in phase and spectrum. They are also hard to measure pulses longer than 1 ps and limited as a powerful tool for pulse shaping. Recently we step forward to achieve some improvement in ultrashort pulse measurement. To eliminate SPIDER’s limitations in complex pulse’s characterization, in this paper we theoretically propose a novel spectral phase interferometry, called DelayControlled Fringe-free Interferometry of Spectral High-resolution (DC-FISH). We designed an experimental system where single replica of the unknown pulse upconverts synchronously with two frequency-shifted narrow-banded long pulses. With the introduction of a suitable small delay between the upconverted pulses, the spectral phase of the unknown pulse can be directly calculated from the fringe-free spectra. By means of numerical simulation, the accuracy of phase retrieval of complex pulses is compared between the DC-FISH and conventional SPIDER. Finally some concluding remarks are presented.

L. Lei et al. / Optik 125 (2014) 5524–5528

2. The origin of the limitations of conventional SPIDER

denser more, likely irresolvable by the spectrometer. As a result, conventional SPIDER is hard to measure pulses longer than 1 ps.

In SPIDER, two time-separated replicas of the unknown pulse upconvert with a strongly chirped long pulse, the resulting spectral interference pattern of the upconverted pulses is of the form [7]



S(ω) = S1 (ω) + S2 (ω) + 2

S1 (ω)S2 (ω) × cos[(ω) + ω]

5525

(1)

where S1 (ω), S2 (ω) and  are, respectively, the individual spectra of the upconverted pulses and the delay between them; (ω) = (ω + ıω) − (ω) is the difference of the spectral phase, with ω, ıω and (ω) denoting the circular frequency, the spectral shear, and the spectral phase, respectively. In Eq. (1), the terms S1 (ω), S2 (ω), S(ω) and  are measurable in the experiment, only (ω) is unknown. For most pulses to be measured, −/2 < (ω) < /2 is usually true at each ω when ıω is not selected too large. Therefore, if cos[(ω) + ω] is univalent, the (ω) trace can be deduced directly from Eq. (1), while it is not the case in conventional SPIDER. Because the delay  is as large as 1–5 ps, so that ω·   (ω denoting the bandwidth of the unknown pulse), the values of ω spread around a large number of univalent regions of the cosine function, resulting in obvious fringes in the spectrum. At each spectral point ω with ω = n + ˇ (n is a large integer, |ˇ |  /2), there is a relationship of cos[(ω) + ω] = cos[−2ˇ − (ω) + ω], due to the periodicity of cosine function. Therefore, unique (ω) cannot be deduced from related cos[(ω) + ω] value. Contrarily, at any spectral point with ω ∼ n + /2, there is no ambiguity to calculate (ω) from cos[(ω) + ω]. Such phenomena should repeat many times within the spectral range of the pulse. As a result, it is impossible to calculate the (ω) trace with Eq. (1) directly. Fortunately, when simple pulses with smooth phases and spectra are characterized, the ambiguities can be significantly reduced by virtue of continuous variations of (ω) at adjacent frequencies. With the presumption that (ω), S1 (ω) and S2 (ω) should vary much slowly than ω, the terms in Eq. (1) can be divided into “dc” and “ac” parts. By performing Fourier transforming, filtering and then inverse Fourier transforming at the interferogram, the trace of (ω) + ω can be extracted. Then the (ω) trace is obtained by removing the term ω. Finally, (ω) retrieval can be accomplished by the phase concatenating procedure. Unfortunately, the presumption above is unlikely fulfilled for complex pulses. If near some frequencies of a complex pulse, the variations of (ω) and/or S1 (ω) are as fast as (or even faster than) ω, there will be no clear division between the so-called “ac” and “dc” terms in Eq. (1). This leads to the intermingling of the Fourier transform peaks. For example, if the derivative |ı(ω)/ıω| somewhere is comparable to , they will beat and yield new Fourier transform components at t =  − ı(ω)/ıω and t =  + ı(ω)/ıω. So the Fourier transform peak at t = + will spread in both directions. It will spread down and partially overlap with the peak at t = 0, and it will spread up to a larger delay time. As a result, the finite window of the filtering function is hard to gather all the information of the (ω) trace. Similarly, if |ıS1 (ω)/ıω| is comparable to , the Fourier transform peak at t = 0 will also expand upwards, resulting in partial mixture with the peak at t = +. Thus, some unwanted information will join in the filtering window. As one can imagine, either of these situations can introduce errors to the reconstruction of the (ω) trace. As the (ω) trace is deduced by concatenation, it can be observably distorted with deviations of (ω) values at a few spectral points. In addition, the denseness of the interference fringes in the upconverted spectrum is related to the delay . When long pulses are characterized, even larger delay should be adjusted to ensure that the pair of pulse replicas upconvert, respectively, with separate spectral components of the chirped pulse. This makes the fringes

3. The principle of DC-FISH technique As discussed above, the limitations of the conventional SPIDER mainly come from its configuration, where two spectral-sheared pulses are prepared at the cost of a considerable delay added on them. This makes cos[(ω) + ω] a periodic function, from which the (ω) trace cannot be deduced directly with Eq. (1). To make cos[(ω) + ω] a univalent function, the following conditions should be fulfilled: (1)  should be very small, so that |ω ·|  . With such a small delay, the upconverted spectrum is fringe-free. All the values of ω are approximately equal, likely within the same univalent region {m, (m + 1)}, where m is a small integer. (2)  should be precisely controlled, so that each ω is set near the middle point of the univalent region (i.e., ω ∼ m + /2). This gives sufficient area for each (ω) to be positive or negative, roughly from −/2 to /2. Let ωa denote an arbitrary frequency of the upconverted pulses. If  is precisely adjusted so that ωa  = m + /2, the cosine term in Eq. (1) cos[(ω) + ω] = cos[(ω) + (ω − ωa ) + m + /2]



=

sin[(ω) + a(ω)]

(m = ±1, ±3, ±5, ...)

− sin[(ω) + a(ω)]

(m = 0, ±2, ±4, ...) (2)

where a(ω) = (ω − ωa ) = (m + /2)(ω − ωa )/ωa . It reflects at certain frequency ω, the offset of ω from the middle point of {m, (m + 1)}, and hence the asymmetry of positive and negative univalent areas for (ω). |ω/ωa |  1 is true for most pulses to be measured, so |a(ω)|  /2 can be fulfilled. With Eqs. (1) and (2), one obtains



(ω) = ± sin

−1

[S(ω) − S1 (ω) − S2 (ω)]



2

S1 (ω)S2 (ω)



− a(ω)

(3)

This is the principle of DC-FISH. The phase retrieval can be accomplished simply and directly, as the (ω) trace can be calculated point-by-point with Eq. (3). No Fourier-transform filtering is needed. The effective range for each (ω) is {−/2 − a(ω), /2 − a(ω)}, which is sufficient for most measured pulses. In DC-FISH, the (ω) values at adjacent frequencies are calculated independently, regardless of their differences in sign and amplitude. Furthermore, abrupt changes in the spectrum should not affect the accuracy of reconstruction, as S1 (ω) and S2 (ω) are involved in the calculation of Eq. (3). These properties are highly advantageous to the measurement of complex pulses. No matter how complicated the pulse can be, few distortion should exist at the phase retrieval, as long as |(ω) + a(ω)| < /2 is always true at each ω. In order to achieve a controllable small delay, the configuration of SPIDER should be significantly changed. The sketch of DC-FISH system is shown in Fig. 1. The pulse to be measured is split into two beams. One beam is stretched by a grating pair and then passes through a double slit, yielding two narrow-banded long pulses with different frequencies ω1 and ω2 = ω1 + ıω. A glass piece is used to roughly compensate the delay between them. The other beam, as the single pulse replica, is polarization-rotated and then reflected parallel to the long pulses. Then, all three pulses are focused at the same position of the type II nonlinear crystal, yielding two upconverted pulses synchronously. Before the upconverted pulses are focused into the spectrometer, a small delay is precisely introduced between them by a piezo delay stage, or by a suitable step mirror.

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PR G2

4

PZ

GP

1#

SP

G1

BS

0

4

Fig. 1. DC-FISH setup: BS: beam splitter; G1, G2: grating pair; DS: double-slit; GP: glass piece for delay compensation; PR: periscope for polarization rotation in type II upconversion; SFG: type II upconversion crystal. PZ: piezo delay stage; SP: spectrometer. The dot-lines denote the optical paths of the upconverted pulses.

0 720

730

1 0

0 -1

-1

730

380

390

400

750

760

770

780

-2

revealing that abrupt changes in S1 (ω) and S2 (ω) do not affect the results of phase retrieval. In this case, cos[(ω) + ω] is a univalent function, as all the values of (ω) + ω are within the univalent region {0, }. Thus, unique (ω) at each ω can be directly deduced with Eq. (3). Ideally, there should be no error in the reconstruction of these pulses. On the other hand, when the delay  becomes considerably large, obvious fringes will present in the upconverted spectra. Fig. 5 shows the spectra with almost the same parameters except  = 1300.25 fs. The interferograms are mainly determined by ω, while the abrupt changes in (ω), S1 (ω) and S2 (ω) are not apparent. As cos[(ω) + ω] becomes a periodic function, the (ω) trace cannot be reconstructed point-by-point. We follow the standard procedure for conventional SPIDER. The array of the interferogram is increased to 4096 elements before going to the time 4

1#

S( ω) (a.u.)

2

φ(ω) (rad)

Intensity (a.u.)

0

370

740

Fig. 4. The actual (ω) curve of pulses 1# and 2#, and the cos[(ω) + ω] trace retrieved by DC-FISH with  = 0.33 fs.

1

360

780

2

2

350

770

ω/2π (THz)

To elucidate the advantages of DC-FISH on the characterization of complex pulses, we make a comparison with conventional SPIDER by numerical simulation. Suppose that two pulses are generated by means of pulse shaping. As shown in Fig. 2, pulses-1# and 2# have identical phase structures, with a jump somewhere. Their spectra are almost the same, except that there is an abrupt change in pulse-2#. Their durations are about 50 fs. The spectra and phases are defined on a 512-element array. Fig. 3 shows the upconverted spectra of the measured pulses expected to be recorded by DC-FISH system with ω1 = 2 × 375 THz, ıω = 2 × 1.0 THz, and  = 0.33 fs (i.e., ωa  = /2 for ωa = 2 × 750 THz). It is seen that with extremely small , the upconverted spectra S(ω) of both pulses are fringe-free. Fine structures of (ω), S1 (ω) and S2 (ω) are fully present in the spectra, while ω only contributes a slow linear modulation. Fig. 4 draws the calculated cos[(ω) + ω] trace (actually the −sin[(ω) + a(ω)] curve), which fully exhibits the profile of the (ω) trace. Note that the cos[(ω) + ω] traces of pulses-1# and 2# are identical,

340

760

1

4. Comparison on the accuracy of complex pulse reconstruction

0

750

Fig. 3. The upconverted spectra recorded by DC-FISH with  = 0.33 fs.

720

1

740

ω/2π (THz)

cos[Δφ(ω)+ωτ]

After the delay is adjusted to a suitable value, three spectra S1 (ω), S2 (ω) and S(ω) can be recorded with a two-dimensional CCD camera. The (ω) trace is then be calculated point-by-point with Eq. (3). The spectra S1 (ω) and S2 (ω) can be readily obtained by two means: (1) Block alternately either of the upconverted pulses and then record the other. (2) Insert a beam-splitter to pick up two more beams from the upconverted pulses, and then focus them at different positions of the slit of the spectrometer. With the latter arrangement, one can gather all the spectra at the same time, so DC-FISH may conserve the advantages of SPIDER as a fast and single-shot technique. In addition, because the introduction of the delay  is independent of the duration of the unknown pulse, few limitations exist to long pulses measurement.

2#

Δφ(ω) (rad)

W2 W1

S( ω) (a.u.)

SFG

DS

0 4

-1

2

-2

0

410

ω/2π ( TH z) Fig. 2. Pulses are supposed to be measured: pulses 1# and 2# are with different spectra, but with the same phases.

2#

720

730

740

750

760

770

780

ω/2π (THz) Fig. 5. The upconverted spectra of the measured pulses obtained by SPIDER with  = 1300.25 fs.

L. Lei et al. / Optik 125 (2014) 5524–5528 2

1#

0.1

window

5527 1.0

(a)

1#

0.5

0.0

δφ(ω) (rad)

φ(ω) (rad)

Re[S(t)]

x20

0

1300.60 fs -1

(b) 1300.60 fs

0.0

1

1300.25 fs

-2 340

1300.43 fs

0.0 -0.5 0.5

1300.25 fs

0.0

1300.43 fs 350

-0.5 0.5

360

370

-0.5 380

390

400

340 350 360 370 380 390 400 410

410

ω/2π (THz)

-0.1

-4

-3

-2

-1 0 1 Time (ps)

2

3

Fig. 8. (a) The (ω) traces of pulse 1# with slightly different delays retrieved by SPIDER. The bold-dash-line denotes the actual curve. (b) The consequently errors of phase retrieval.

4

Fig. 6. The Fourier transform of the interferogram of pulse-1# in Fig. 5. The width of the filtering window is equal to .

ε=

1 2



1/2

+∞

2 ˜ 2 + |ıϕ|2 |E(ω)| ˜ dω(|ıE| )

(4)

−∞

0.5

1#

0.0

2

2# 1

Δφ(ω) (rad)

domain. The Fourier transform of the interferogram of pulse-1# is shown in Fig. 6, where the peak of t = + appears spread in both directions. Although the width of the filtering window is set as large as the delay , it is still unable to gather all the information of the (ω) trace. With very small amplitude, the lost information is likely covered up by the random noises that are uniform in the time domain. So it is hard to recognize that errors have occurred during the process of phase retrieval. Fig. 7 draws the distorted sections of calculated (ω) traces of the measured pulses. It shows that the deviations always occur near the spectral positions with abrupt variations in phase or spectrum. The reconstructed (ω) traces of pulse-1# exhibit apparent distortions compared to the actual (ω) curve, as shown in Fig. 8(a) and (b). The sensitivity of the calculation results to the delay is also presented in these figures. It seems that sometimes the errors of (ω) may cancel out during the concatenation procedure, resulting in an approximate (ω) trace ( = 1300.43 fs); while in other cases, the errors of (ω) will accumulate, leading to obvious deviations from the actual curves. With one more abrupt change in its spectrum, pluse-2# experiences more distortions than pluse-1#, as shown in Fig. 9. In order to evaluate the accuracy of conventional SPIDER for the measurement of complex pulses, we adopt the RMS field error that is defined as [16]



ω/2π (THz)

0

-1

-2 340

350

360

370

380

390

400

410

ω/2π (THz) Fig. 9. The (ω) trace of pulse 2# retrieved by SPIDER with ␶ = 1300.43 fs. The boldline denotes the actual (ω) curve. 2 is the energy at frequency ω, while ı|E(ω)| ˜ ˜ where |E(ω)| and ı(ω) are small variations of the spectral amplitude and phase compared to the actual values. ˜ = 0 in condition that the accuracy of phase retrieval Let ı|E(ω)| is considered. Thus, ε = 0.079, 0.049, 0.068 can, respectively, be obtained for pulse-1# at  = 1300.25, 1300.43, 1300.60 fs. They all correspond to average reconstructions. Furthermore, ε = 0.150 is deduced for pulse-2# at  = 1300.25 fs, corresponding to poor reconstruction. The temporal intensities and phases of the measured pulses reconstructed at  = 1300.25 fs are shown in Fig. 10, where apparent distortions can be seen compared to the actual curves. Obviously, the accuracy of conventional SPIDER for complex pulses is relevant to three parameters: the delay , the spectral shear ıω, and the width of the filtering window. For example, Fig. 11 draws the RMS field error of pulse-1# as a function of the delay . The larger the delay, the better the reconstruction could achieve.

-0.5

1#

1

2 1

2#

0.0 -0.5

-1 -2

0 1

2#

2 1

Phase (rad)

0

-1.5 0.5

Intensity (a.u.)

Δφ(ω) (rad)

-1.0

0

-1.0

-1

-1.5 735

-2

0

740

745

750

755

760

ω/2π (THz) Fig. 7. Distorted sections of (ω) traces of pulses 1# and 2# calculated by SPIDER with  = 1300.25 fs, compared to the actual (ω) traces (bold-line).

-150

-100

-50

0

50

100

Time (fs) Fig. 10. The temporal intensities and phases for actual (solid-line) and reconstructed (dashed-line) of the measured pulses.

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5. Conclusion

0.15

RMS error ε

1# 0.10

0.05

0.00 0.5

1.0

1.5

2.0

2.5

Delay τ (ps) Fig. 11. The RMS field error of pulse-1# as a function of the delay .

In this paper we present an improved version of SPIDER, in order to overcome the difficulties of SPIDER on the characterization of complex pulses and long pulses. DC-FISH conserves the conventional advantages as a fast technique. More significantly, by controlling the delay to a suitable small value, the (ω) trace can be deduced point-by-point from the fringe-free upconverted spectrum. Two complex pulses are selected as examples to show that when there are abrupt changes in the spectra and phases, errors would occur in SPIDER, while DC-FISH can remain high accuracy. Recently we are trying to achieve actual complex pulse, the experiment result to verify the capability of DC-FISH would be presented soon. References

In order to achieve high accuracy in conventional SPIDER, the preconditions   |ı(ω)/ıω| and   |ıS1 (ω)/ıω| should be fulfilled. The steeper the variations in (ω) and S1 (ω), the larger the delay should be selected. Unfortunately, when a pulse is to be measured, its structures of spectrum and phase are unknown. So it is hard to adjust the suitable delay. Although the safety may rise to adjust a sufficiently large delay, limitations from the resolution of the spectrometer may draw it back. Wider filtering window can gather more (ω) information, meanwhile it gathers more noises either and the risk of the interfusion with unwanted S1 (ω) information will also increase. In addition, the shape of the (ω) trace is relevant to ıω. If ıω is changed, the distribution of the Fourier transform of the interferogram should also change. Certainly it will affect the accuracy of SPIDER. Actually, pulses-1# and 2# are not very complex, as there are only one and two abrupt variations in their phases and spectra. When more complicated pulses are measured, the accuracy of conventional SPIDER should be quite limited. Contrarily, with simple and point-by-point algorithm, DC-FISH can reconstruct various pulses with high accuracy. In DC-FISH, as the (ω) trace is calculated point-by-point with Eq. (3), the noise will directly add on it. Fortunately, as the noise is random and uncorrelated, it will not significantly distort the profile of the (ω) trace. Therefore, by means of data smoothing and filtering, the effects of the noise can be greatly reduced. In addition, as three spectra S1 (ω), S2 (ω) and S(ω) should be recorded to extract the (ω) trace, fluctuations in the spectrum of the unknown pulse may also affect the accuracy of reconstruction. If these spectra are not recorded at the same time, the recorded traces of S1 (ω) and S2 (ω) may not be the same as those contributing to the interferogram S(ω). This will introduce some errors to the retrieved (ω) trace. Therefore, it is favorable to record the spectra S1 (ω), S2 (ω) and S(ω) synchronously.

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