All-optical self-referencing measurement of vectorial optical arbitrary waveform

We propose the Vectorial E-field Characterization Through allOptical and self-Referenced (VECTOR) method to characterize vectorial optical arbitrary waveform with up to 100% duty cycle, which is free of ambiguity, iteration, radio-frequency or external optical reference, restriction on repetition rate, and requirement of external interferometric stabilization. The feasibility of VECTOR is experimentally verified by different waveforms created by a phase-modulated CW comb source and a built-in polarization line-by-line pulse shaper. ©2014 Optical Society of America OCIS codes: (320.7100) Ultrafast measurements; (320.7110) Ultrafast nonlinear optics; (120.2130) Ellipsometry and polarimetry. References and links 1. C.-C. Chen, I.-C. Hsieh, S.-D. Yang, and C.-B. Huang, “Polarization line-by-line pulse shaping for the implementation of vectorial temporal Talbot effect,” Opt. Express 20(24), 27062–27070 (2012). 2. S. T. Cundiff and A. M. Weiner, “Optical arbitrary waveform generation,” Nat. 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Introduction
The marriage between optical frequency combs and polarization line-by-line pulse shaping opens access to vectorial optical arbitrary waveform (V-OAW) with ultrafast evolution of amplitude, phase, state of polarization (SOP) that can span up to the entire repetition period (100% duty cycle) [1].Scalar OAW (with static SOP) and vectorial ultrashort pulse (with small duty cycle) have been applied to radio-frequency (RF) photonics [2], intensity repetition rate multiplication of a pulse train [3], isolated attosecond burst generation [4], and selective spatiotemporal excitations with nanometer and femtosecond resolutions [5], respectively.The increased degree of freedom of V-OAW is expected to enable unique applications in ultrafast plasmonics.
A limited number of methods have been developed to measure either scalar OAW or vectorial ultrashort pulse, while none of them is readily applicable to V-OAW characterizations.The 100% duty cycle of scalar OAW prohibits the employment of conventional femtosecond pulse measurement methods requiring the generation of isolated pulse replicas [6][7][8][9].As a result, a synchronized and well-characterized optical reference pulse [10] or a coherent RF reference wave [11][12][13] is often used in generating a set of interferograms, from which the relative phases of individual comb lines can be retrieved.However, the required optical or RF reference could be unavailable especially when the OAW comes from a Kerr frequency comb of large (>100 GHz) mode spacing [14].This restriction was recently overcome by orthogonally probed dual quadrature spectral shearing interferometry (DQ-SSI) [15], where the reference probe field was produced by shaping the signal OAW itself.
To resolve V-OAW, not only the individual temporal shapes of x-and y-polarizations [i.e.
the power spectra |A x,y (ω)| 2 and the nonlinear components of spectral phases φ x,y (ω)] but their relative delay τ xy and constant phase θ have to be retrieved.This can be achieved by dualchannel spectral interferometry [16,17] if a synchronized reference optical frequency comb with known spectral phase and a bandwidth broader than or equal to that of the signal is available.It is a linear technique with high sensitivity, however, restricted by the requirement of interferometric stability (the transient SOP is vulnerable to the environmental perturbations) unless special polarization demultiplexing geometry and fast data acquisition are used [10].Tomographic ultrafast retrieval of transverse light E-fields (TURTLE) is a nonlinear, self-referenced technique that can recover τ xy and θ by using the spectrogram due to a different polarization (typically linear polarization at 45°) as the constraint for optimization [18,19].TURTLE is robust against interferometric perturbations but fails to measure V-OAW for a well-behaved spectrogram (surrounded by zeros) does not exist when the signal waveform is of 100% duty cycle.It was recently proposed that the data trace of multiphoton intrapulse interference phase scan (MIIPS) [20]  measuring vectorial ultrashort pulse or V-OAW.In our earlier demonstration of vectorial temporal Talbot effect [1], φ x (ω), φ y (ω), τ xy and θ were individually measured by optimizing the second-harmonic yields with a pulse shaper, dual quadrature spectral interferometry (DQ-SI), and monochromatic phase-scanning interferometry, respectively (please refer to Section 3.1).The 4-step approach may not be practically useful because of its high complexity, slow data acquisition, and the requirement of a well-characterized optical reference with interferometric stability.
In this paper, we report the Vectorial E-field Characterization Through all-Optical and self-Referenced (VECTOR) method that can synthesize and analytically characterize V-OAW without any external reference or ambiguity for the first time (to the best of our knowledge).It consists of one scalar OAW measurement by orthogonally probed DQ-SSI [15] and one SOP spectrum measurement by a wavelength-parallel polarimeter (WPP) [22].The selfreferenced, nearly common-path, all-optical configuration makes it robust against interferometric perturbations and compatible with Kerr frequency combs of large mode spacing.The requirement of only one nonlinear measurement (compared with three or four in TURTLE) and free of iteration or optimization greatly improve the complexity, update rate, and sensitivity of the system.The feasibility of VECTOR was experimentally verified by measuring three types of 20 GHz V-OAWs with ~2.8 ps temporal structures, but is readily applicable to >100 GHz waveforms with femtosecond local features [14].

Experiment
The experimental setup is shown in Fig. 1.A phase-modulated continuous-wave (PMCW) comb of ~17 spectral lines and 20 GHz mode spacing (50 ps repetition period) was generated by injecting a 1 kHz-linewidth CW laser (NKT Adjustik) centered at 1545 nm (ω 0 = 2π × 194.17 THz) into an optical phase modulator driven by an RF tone of amplitude 1.89V π .A polarization line-by-line pulse shaper independently controlled the amplitudes and phases of the x-and y-polarized comb lines [23].The single-source and nearly common-path configuration ensured stable τ xy and θ though the carrier-envelope phase of the PMCW comb was unlocked.In the measurement of φ x (ω) by DQ-SSI, two probe lines spaced by 20 GHz were generated in the y-polarization [15].The shaper output was sent to a 2-mm-thick Type II BBO for SFG, recorded by a high-resolution spectrograph and an intensified CCD camera (Fig. 1, Path-1A).In performing WPP measurement, the shaper output was connected to a polarization manipulator (consisting of two quarter-wave plates and a polarization beamsplitter) for SOP component sampling [22].The four power spectra I x (ω), I y (ω), I 45 (ω), I RHC (ω) were recorded by the same spectrograph and an InGaAs detector array (Fig. 1, Path-2A).electric field e(t) is depicted in Fig. 2(d), where the instantaneous frequencies (defined by taking the inverse of the piecewise time required for the field in passing through the maximum twice along the polarization ellipse) are displayed by different color tones.The xand y-polarized field components, e x,y (t) = Re[a x,y (t) × exp(jω 0 t)], are independently depicted using the black projection traces onto the corresponding axes.The reconstructed e(t) clearly exhibits the signatures of V-OAW, including the time-varying SOP, strong chirp, and 100% duty cycle.The polarization line-by-line pulse shaper in our setup enables independent measurement of e(t) by a 4-step method similar to that used in [1].In Steps 1 and 2, φ x (ω) and φ y (ω) [Figs.

Measurement of V-OAW with 100% duty cycle
3(a) and 3(b)] were determined by maximizing the SHG yields of the x-and y-polarized combs, respectively (Fig. 1, Path-2B).The accuracy was verified by performing shaperassisted intensity autocorrelation (IA) [24] for the phase-compensated transform-limited (TL) pulses at x-and y-polarizations (Fig. 1, Path-2B).The insets of Figs.3(a) and 3(b) show that the experimentally measured IA functions (solid) are in good agreement with the simulated ones assuming TL pulses (dashed).In Step 3, the x-and y-polarized TL pulses were sent into the Type II BBO for shaper-assisted intensity cross-correlation to characterize τ xy (Fig. 1, Path-1B).Figure 3(c) shows that the SFG yield was maximized when an extra delay of −6.227 ps was added to e y (t).The value of τ xy is actually −6.277 ps by considering the 50 fs group velocity mismatch walk-off between the fundamental e-wave (y-polarization) and owave (x-polarization) in the Type II BBO [25].In Step 4, all but the central comb lines of the x-and y-polarizations were blocked by the pulse shaper and the interference signal I 45 (ω 0 ) [Fig.3(d those retrieved by VECTOR.The values of (τ xy ,θ) are (−6.277ps, 0.317π) (4-step method) and (−6.306 ps, 0.316π) (VEVTOR), respectively.This shows that VECTOR can accurately characterize V-OAW with greatly simplified configuration.

Measurement of vectorial temporal Talbot effect
In the second case, we generated an intensity-rate doubled vectorial pulse train by applying

Conclusions
We demonstrated an integrated system that can simultaneously synthesize and characterize V-OAW (conceptually the optical field of extreme complexity in the time/frequency domain if a large number of comb lines are accessed) without RF or external optical reference, restriction on the comb spacing, and requirement of external interferometric stabilization.The field retrieval is non-iterative and unambiguous, only utilizing the data sets from one nonlinear measurement (DQ-SSI) and one linear measurement (WPP).

Figure 2
Figure2shows the measurement results for a raw PMCW comb, where φ x , φ y change abruptly and Δφ mainly results from the imbalanced paths of the system.The circles in Figs.2(a-c) illustrate φ x , SOP, and φ y,tot of the 17 comb lines measured by VECTOR.The reconstructed

Fig. 3 .
Fig. 3.The raw V-OAW measured by the four-step method.(a-b) Spectral intensities (shaded) and phases (circles) of (a) A x (ω) and (b) rA y (ω), respectively.The insets show the intensityautocorrelation traces of x-and y-polarized TL pulses obtained by experiment (solid) and simulation (dashed), respectively.(c) Intensity cross-correlation trace between x-and ypolarized TL pulses for τ xy measurement.(d) Interference signal I 45 (ω 0 ) versus the extra phase Φ applied to the y-polarization used for θ measurement.