## Abstract

We show through experiments and simulations that parallel phase modulation, a technique developed in the field of adaptive optics, can be employed to quickly determine the spectral phase profile of ultrafast laser pulses and to perform phase compensation as well as pulse shaping. Different from many existing ultrafast pulse measurement methods, the technique reported here requires no spectrum measurements of nonlinear signals. Instead, the power of nonlinear signals is used directly to quickly measure the spectral phase, a convenient feature for applications such as two-photon fluorescence microscopy. The method is found to work with both smooth and even completely random distortions. The experimental results are verified with MIIPS measurements.

© 2011 OSA

## 1. Introduction

Ultrafast laser pulse measurement and compression are crucial to many research fields of ultrafast sciences and technologies [1–4]. Over the past two decades, various techniques have been developed, such as FROG [5–8], SPIDER [9,10] and MIIPS [11–13]. A research field analogous to ultrafast pulse measurements and compressions is coherent optical adaptive techniques (COAT) [14,15], in which the goal is often to determine the phase profile of an aberration distorted light beam in the spatial frequency domain and to form a diffraction limited focus in the spatial domain. In ultrafast optics, the spectral phase of laser pulses can be controlled at will by using a conventional phase-only 4*f* pulse shaper. After determining the phase profile of a distorted laser pulse in the spectral domain, we can directly compensate for its phase distortion and compress the pulse to its transform limited duration in the time domain. In this work, we show through both experiments and simulations that by measuring only the power of the generated nonlinear signals, we can use a well established adaptive optics technique [14–16] to quickly measure the spectral phase of ultrafast laser pulses and to provide compensation for its phase distortion. Different from genetic algorithm and other optimization algorithms based techniques [17–19], the technique reported here allows a direct phase measurement through nonlinear spectral interferometry and therefore operates at higher speed.

The idea is based on multidither COAT [14,15], a technique developed at the Hughes Research Laboratory in the 1970s to focus a laser beam through air turbulence. The idea is to use a phase only spatial light modulator to control the wavefront of light, which is incident on a target. Each phase element is dithered at a unique frequency. The scattered light from the target is demodulated for each modulation frequency and the phase values are extracted and used as the feedback signals to drive the phase elements such that they all interfere constructively at the target position. The method works with a single point target (size smaller than the diffraction limit) and multiple isolated point targets. In the case of multiple isolated point targets, the iterative feedback nature of this technique can force the light to focus onto the strongest target through atmospheric aberration. However, this method will fail to form a diffraction limited focus if the target is uniform and extended over a large area. In such cases, the phase-only wavefront modulation cannot generate any variation on the total scattered signal. The scenario is similar in ultrafast optics. We cannot directly use the detector to sample one point in the time domain (time interval shorter than the transform limited pulse duration). Instead, the detector is only fast enough to capture the total power integrated over a time interval much longer than the transform limited pulse duration, which seemingly prevents the application of COAT to ultrafast pulse measurements.

We show in this work that with the help of nonlinearity the multidither COAT can actually form transform limited pulses. Suppose the laser pulse is transform limited and therefore its spectral phase can only be decomposed to 0th order and 1st order terms. For simplicity, let’s also assume the 0th order and 1st order terms are 0 such that the spectral phase is flat and equal to 0. With a conventional 4*f* pulse shaper, the phase of different wavelength can be controlled. If one of the phase elements of the pulse shaper is changed from 0 to π, the pulse will become longer due to the nonconstructive interference between the out of phase wavelength and all the rest of the wavelengths at the temporal peak position. If the pulse interacts with a nonlinear medium and generates nonlinear signals, the signal strength will be lower than if all the wavelengths are in phase. If the phase of one of the wavelengths was originally φ out of phase with respect to all the rest of wavelength, adding –φ to the out of phase wavelength with the pulse shaper can increase the nonlinear signal to the maximum value. By applying a continuous phase modulation and monitoring the nonlinear signal variation, we can determine the phase value of the out of phase element, which is the basic principle of the phase resolved interferometric spectral modulation (PRISM) reported here. Although the physical picture becomes more complicated if the pulse is initially severely distorted, we found through both experiments and simulations that such a scheme can indeed quickly determine the spectral phase profile of ultrafast laser pulses regardless of the initial phase profile (smooth or even completely random). Since only the power of nonlinear signals is measured, PRISM can be used in multi-photon fluorescence microscopy to directly use fluorescence signals for pulse measurement and compression.

## 2. Method

In this work, we adapted the method reported in Ref. [16] for ultrafast pulse measurement and compression. The spectral phase of the laser pulse is controlled by a 4*f* pulse shaper with N independent pixels. The task of pulse measurement is to determine the phase values of the N pixels and the pulse compression can be simply achieved by applying the phase values with opposite sign to the N pixels.

To determine the phase values, we first divide the N pixels into a few groups (group number ≥ 2). During the measurement, only one group is modulated and all the other groups are kept stationary such that the unmodulated pixels provide a stable reference field for the nonlinear spectral interferometry measurement. The phase values of the pixels in the modulated group are modulated in parallel, each at a unique rate or phase advance per step. As an example, we show the modulation phase profile for 32 pixels in 128 modulation steps in Fig. 1(*a*)
, in which the phase advance per step is uniformly distributed between π/2 and π. In each step, the 32 phase values are applied to the 32 pixels and the power of the generated nonlinear signal is recorded. At the end of the 128 modulation steps, 128 nonlinear power data are recorded. The nonlinear power data are Fourier transformed and the 32 phase values at the corresponding frequencies are extracted and sign reversed before being applied to the 32 pixels. The same procedure is applied to every group, which concludes one round of PRISM measurement. To yield transform limited pulses, the PRISM measurement is often repeated once or twice. The reason is that at the start of PRISM operation, the reference field (controlled by stationary pixels) is not transform limited. As more pixels are measured and phase compensated, the reference field approaches the transform limited form. Therefore, more than one round of PRISM is required. The entire PRISM operation is summarized in Fig. 1(*b*). Typically, group number is 2-4 and the iteration number is 2-3.

## 3. Simulation

We used numerical simulation to investigate the performance of PRISM with both smooth and completely random phase distortions. We defined a transform limited pulse in time domain. The pulse duration is 40 femtosecond (fs) and the pulse center wavelength is 800 nanometer (nm). The temporal pulse intensity profile is shown in Fig. 2(*a*)
. The pulse was then Fourier transformed to the spectral domain, where 7000 fs^{2} group delay dispersion was added. The pulse was chirped to ~500 fs in duration, which is shown in Fig. 2(*b*) (note the scale is different). At this point, we started PRISM to measure its spectral phase and to compensate for the measured phase distortion. In the spectral domain, 136 phase elements evenly distributed between 766 nm and 838 nm were used to control the pulse. The 136 phase elements were randomly divided to four modulation groups (34 elements per group). During the phase measurement, one of the four groups was modulated and the other three groups were stationary. Similar to [16], each element was modulated at a unique frequency. In principle, PRISM can utilize a variety of nonlinear signals. In this simulation, we assumed that the laser pulse generated second order nonlinear signals (for example, second harmonic generation (SHG) or two-photon excited fluorescence) and the power of the second order signal was recorded during the phase modulation. The recorded nonlinear signal was Fourier transformed at the end of the modulation and the phase value at each modulation frequency was the determined spectral phase of that modulated element. The sign of the measured spectral phase was reversed before the phase profile was applied to this group for compensation. At this point, we found that the nonlinear signal increased and we then used the same procedure to determine the phase profile and to apply compensation for the other three groups one by one, which concluded one round of PRISM. It is interesting to see at the end of the first round of PRISM, the peak intensity of the pulse reached ~50% of its transform limited value, as shown in Fig. 2(*c*). The entire process was then repeated and the temporal profiles of the pulse after the second and the third round of PRISM are shown in Figs. 2(*d*) and 2(*e*) respectively. The spectral phase residual after three rounds of PRISM and the power spectrum are shown in Fig. 2(*f*).

To study the performance of PRISM with completely random phase distortions, we assigned a random number between 0 and 2π to the spectral phase controlled by each phase element. The goal was to test if PRISM can correctly generate the opposite phase profile to completely cancel the spectral phase distortion and yield a transform limited pulse. The added random phase distortion and the distorted pulse are shown in Figs. 2(*g*) and 2(*h*) respectively. The phase compensated pulses after one, two and three rounds of PRISM are shown in Figs. 2(*i*), 2(*j*), and 2(*k*) respectively and the phase residual after three rounds of PRISM is shown in Fig. 2(*l*). These numerical simulation results suggest that PRISM can quickly and accurately measure spectral phase distortions and provide phase compensation for both smooth and completely random distortions.

## 4. Experiments

We experimentally tested the performance of PRISM using the setup shown in Fig. 3
. A Coherent Chameleon Ultra II Femtosecond oscillator was employed as the light source. The output of the laser propagated through a prism pair compressor (Coherent Chameleon precomp) before entering a reflective 4*f* pulse shaper. A 128 elements spatial light modulator (SLM) located at the Fourier plane (between 784.5 nm and 798.3 nm) was configured to provide phase-only modulation. The output of the pulse shaper was split to two beams with one beam directly monitored by a photodiode and the other beam used for nonlinear signal generation. In this test, the SHG signal generated by a BBO crystal was used as the nonlinear signal. A short pass filter was employed to block the red fundamental beam and let the blue SHG signal recorded by a photodiode. The signals from both photodiodes were low-pass filtered at 785 kHz and digitized at 2 MHz simultaneously by a NI PCI 6110 data acquisition card. The SHG power signal was normalized by the square of the fundamental power signal.

During the PRISM measurement, 100 milliseconds (ms) were used for each step, in which the first 50 ms were used to update the SLM phase profile and the second 50 ms were used for the nonlinear signal measurement. The minimum number of phase modulation required for determining the phase value of 128 SLM pixels is 256 (Nyquist–Shannon theorem). In the experiments, 512 measurements were used. As in the simulation, the 128 SLM pixels were randomly divided to four groups (32 pixels per group). One group was modulated while the other three groups were stationary. At the end of the modulation, the recorded nonlinear power signal was Fourier transformed and the phase profile was sign reversed and applied to the 32 pixels of the modulated group. The process was repeated for the other three groups one by one and the phase for the entire 128 pixels were all determined, which concluded one round of PRISM measurement and compensation.

To study the performance of PRISM with smooth spectral phase distortion, the prism pair compressor was configured to provide minimum negative dispersion. The spectral compensation phase profile determined by three rounds of PRISM is shown in Fig. 4(*a*)
(dark cross) and the power spectrum of the laser source is shown in Fig. 4(*b*). As a comparison, we used commercially available MIIPS software (Biophotonics Solutions Inc) to control the pulse shaper system and used a calibrated spectrometer to provide the SHG spectrum to the MIIPS software. The spectral compensation phase profile determined from the MIIPS measurement is also shown in Fig. 4(*a*) (red circle) for a direct comparison.

To study the performance of PRISM with completely random spectral phase distortions, we used the same prism pair compressor configuration as in the previous measurement and added a random number between 0 and 2π to every pixel of the SLM. This random phase distortion was included by all the PRISM modulation phase patterns during the measurements such that the single SLM phase array behaved as two independent SLM phase arrays, one phase array providing the distortion and the other phase array implementing the PRISM measurement. The added random phase distortion profile and the PRISM determined compensation profile are shown in Figs. 4(*c*) and 4(*d*) respectively. The two phase profiles were summed and unwrapped, as shown in Fig. 4(*e*). The PRISM determined phase profile should compensate for not only the added random distortion but also the existing smooth distortion. To verify this point, we removed the 0th order and 1st order phase difference between the phase in Fig. 4(*e*) and the MIIPS phase in Fig. 4(*a*) and show them in Fig. 4(*f*) for a clear comparison.

With the same input laser power at the BBO crystal, the generated SHG power was 2.22 ± 0.01 a.u. with the MIIPS compensation profile, 2.20 ± 0.01 a.u. with the PRISM compensation profile shown in Fig. 4(*a*), and 2.22 ± 0.01 a.u. with the phase profile shown in Fig. 4(*e*). Without the compensation profile (flat phase on SLM), the SHG signal was 0.90 ± 0.01 a.u. With the additional random phase distortion, the SHG was further reduced to 0.062 ± 0.001 a.u.

## 5. Conclusion and discussion

In summary, we propose and demonstrate a new and fast ultrafast pulse measurement and compensation technique, which determines the spectral phase profile of ultrafast laser pulses through interferometric spectral domain modulation and is named phase resolved interferometric spectral modulation (PRISM). We investigated its performance through both numerical simulations and experimental comparisons with MIIPS, and found PRISM can indeed quickly determine the spectral phase distortion and provide compensation. In the demonstration, each measurement step took 100 ms and 512 measurements were used in each round of PRISM. We found that three rounds of PRISM were sufficient for both smooth and even completely random phase distortions. The total measurement time was ~150 seconds. With a commercially available high speed SLM or a linear MEMS segmented deformable mirror array, the total measurement time can be potentially reduced by ~2-3 orders of magnitude. In addition, the number of measurements can be further reduced if the phase distortion is known to be smooth and the phase difference between adjacent pixels is much smaller than π. In such cases, we can sparsely measure a few pixels and use interpolation to determine the phase values of the pixels located between the measured pixels.

The key difference between PRISM and many other ultrafast pulse measurement techniques is that PRISM requires no measurements of coherent nonlinear spectra. Even incoherent signals such as two-photon excited fluorescence emission can be used for PRISM. Such a property allows PRISM to be combined with multiphoton microscopy to directly use the nonlinear signal (two-photon fluorescence, SHG, etc) measured by a PMT to determine and compensate for spectral phase distortions inside samples.

Conventional genetic algorithm and other optimization based pulse compression methods also only utilize the power of nonlinear signals. Compared to these optimization algorithms based methods, PRISM allows a direct phase measurement through nonlinear spectral interferometry and is therefore fast and robust. For measuring the phase values of N pixels, PRISM requires at least 2N modulations in each round of measurement. Typically only 2-3 rounds of PRISM are sufficient to yield near transform limited pulses. Therefore, the total modulations required are 4N-6N, which is valid even for completely random phase distortions. If the phase distortion is known to be smooth and the phase variation between adjacent pixels is much less than π, the total number of modulation can be greatly reduced. In addition, the algorithm of PRISM is very simple and robust since only Fourier transform is involved. In comparison, conventional optimization algorithms based methods often converge at a slower rate [17–19] and do not always yield transform limited pulses [18–21]. To our best knowledge, no optimization based pulse compression techniques have experimentally shown the capability of restoring pulses distorted by a completely random phase aberration. PRISM makes no assumptions of phase distortion profiles and is proved to work with both smooth and even completely random phase distortions, which is beyond the capability of many well established techniques.

## Acknowledgment

The authors thank Na Ji for helpful discussions, Eric Betzig for providing the prism pair compressor, and Charles Shank for providing the MIIPS system and the ultrafast laser source. This work is supported by Howard Hughes Medical Institute.

## References and links

**1. **V. V. Lozovoy and M. Dantus, “Systematic control of nonlinear optical processes using optimally shaped femtosecond pulses,” ChemPhysChem **6**(10), 1970–2000 (2005). [CrossRef] [PubMed]

**2. **J. P. Ogilvie, D. Débarre, X. Solinas, J. L. Martin, E. Beaurepaire, and M. Joffre, “Use of coherent control for selective two-photon fluorescence microscopy in live organisms,” Opt. Express **14**(2), 759–766 (2006). [CrossRef] [PubMed]

**3. **T. C. Weinacht, R. Bartels, S. Backus, P. H. Bucksbaum, B. Pearson, J. M. Geremia, H. Rabitz, H. C. Kapteyn, and M. M. Murnane, “Coherent learning control of vibrational motion in room temperature molecular gases,” Chem. Phys. Lett. **344**(3-4), 333–338 (2001). [CrossRef]

**4. **H. C. Kapteyn, R. Bartels, S. Backus, E. Zeek, L. Misoguti, G. Vdovin, I. P. Christov, and M. M. Murnane, “Shaped-pulse optimization of coherent emission of high-harmonic soft X-rays,” Nature **406**(6792), 164–166 (2000). [CrossRef] [PubMed]

**5. **D. J. Kane and R. Trebino, “Characterization of arbitrary femtosecond pulses using frequency-resolved optical gating,” IEEE J. Quantum Electron. **29**(2), 571–579 (1993). [CrossRef]

**6. **D. J. Kane and R. Trebino, “Single-shot measurement of the intensity and phase of an arbitrary ultrashort pulse by using frequency-resolved optical gating,” Opt. Lett. **18**(10), 823–825 (1993). [CrossRef] [PubMed]

**7. **R. Trebino and D. J. Kane, “Using phase retrieval to measure the intensity and phase of ultrashort pulses - frequency-resolved optical gating,” J. Opt. Soc. Am. A **10**(5), 1101–1111 (1993). [CrossRef]

**8. **R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbugel, B. A. Richman, and D. J. Kane, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. **68**(9), 3277–3295 (1997). [CrossRef]

**9. **C. Iaconis and I. A. Walmsley, “Spectral phase interferometry for direct electric-field reconstruction of ultrashort optical pulses,” Opt. Lett. **23**(10), 792–794 (1998). [CrossRef] [PubMed]

**10. **C. Iaconis and I. A. Walmsley, “Self-referencing spectral interferometry for measuring ultrashort optical pulses,” IEEE J. Quantum Electron. **35**(4), 501–509 (1999). [CrossRef]

**11. **V. V. Lozovoy, I. Pastirk, K. A. Walowicz, and M. Dantus, “Multiphoton intrapulse interference. II. Control of two- and three-photon laser induced fluorescence with shaped pulses,” J. Chem. Phys. **118**(7), 3187–3196 (2003). [CrossRef]

**12. **V. V. Lozovoy, I. Pastirk, and M. Dantus, “Multiphoton intrapulse interference. IV. Ultrashort laser pulse spectral phase characterization and compensation,” Opt. Lett. **29**(7), 775–777 (2004). [CrossRef] [PubMed]

**13. **B. Xu, J. M. Gunn, J. M. D. Cruz, V. V. Lozovoy, and M. Dantus, “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 **23**(4), 750–759 (2006). [CrossRef]

**14. **W. B. Bridges, P. T. Brunner, S. P. Lazzara, T. A. Nussmeier, T. R. O’Meara, J. A. Sanguinet, and W. P. Brown Jr., “Coherent optical adaptive techniques,” Appl. Opt. **13**(2), 291–300 (1974). [CrossRef] [PubMed]

**15. **J. E. Pearson, W. B. Bridges, S. Hansen, T. A. Nussmeier, and M. E. Pedinoff, “Coherent optical adaptive techniques: design and performance of an 18-element visible multidither COAT system,” Appl. Opt. **15**(3), 611–621 (1976). [CrossRef] [PubMed]

**16. **M. Cui, “Parallel wavefront optimization method for focusing light through random scattering media,” Opt. Lett. **36**(6), 870–872 (2011). [CrossRef] [PubMed]

**17. **E. Zeek, R. Bartels, M. M. Murnane, H. C. Kapteyn, S. Backus, and G. Vdovin, “Adaptive pulse compression for transform-limited 15-fs high-energy pulse generation,” Opt. Lett. **25**(8), 587–589 (2000). [CrossRef] [PubMed]

**18. **D. Yelin, D. Meshulach, and Y. Silberberg, “Adaptive femtosecond pulse compression,” Opt. Lett. **22**(23), 1793–1795 (1997). [CrossRef] [PubMed]

**19. **D. Meshulach, D. Yelin, and Y. Silberberg, “Adaptive ultrashort pulse compression and shaping,” Opt. Commun. **138**(4-6), 345–348 (1997). [CrossRef]

**20. **E. Zeek, K. Maginnis, S. Backus, U. Russek, M. Murnane, G. Mourou, H. Kapteyn, and G. Vdovin, “Pulse compression by use of deformable mirrors,” Opt. Lett. **24**(7), 493–495 (1999). [CrossRef] [PubMed]

**21. **A. Efimov, M. D. Moores, N. M. Beach, J. L. Krause, and D. H. Reitze, “Adaptive control of pulse phase in a chirped-pulse amplifier,” Opt. Lett. **23**(24), 1915–1917 (1998). [CrossRef] [PubMed]