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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 24 — Nov. 19, 2012
  • pp: 27062–27070
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Polarization line-by-line pulse shaping for the implementation of vectorial temporal Talbot effect

Chi-Cheng Chen, I-Chun Hsieh, Shang-Da Yang, and Chen-Bin Huang  »View Author Affiliations


Optics Express, Vol. 20, Issue 24, pp. 27062-27070 (2012)
http://dx.doi.org/10.1364/OE.20.027062


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Abstract

Vectorial optical arbitrary waveform generation is experimentally demonstrated by applying polarization line-by-line pulse shaping on a phase-modulated continuous laser frequency comb. Polarization shaped optical waveforms extending a 50-ps time window are successfully synthesized. Temporal Talbot effect is extended into the vectorial regime, where the distinct periodic temporal phases of the two orthogonally polarized pulse trains are exploited. In one example, we generate repetition-rate doubled circularly polarized pulses with alternating pulse-by-pulse handedness. In another example, complex instantaneous field polarizations are synthesized through the combination of line-by-line amplitude and temporal Talbot phase shaping. Our experimental results are measured through a dual-quadrature spectral interferometry system and are found in excellent agreements to the applied shaping controls.

© 2012 OSA

1. Introduction

It is well-known that optical fields are vectorial in nature. Various quantum, physiochemical, and biophysical systems are known to react adaptively according to the input optical waveforms [11

11. Y. Silberberg, “Quantum coherent control for nonlinear spectroscopy and microscopy,” Annu. Rev. Phys. Chem. 60(1), 277–292 (2009). [CrossRef] [PubMed]

]. The ability to control the two orthogonally polarized fields for the synthesis of optical waveforms with desired instantaneous field polarizations is therefore vital. Polarization optical pulse shapers with various geometries [12

12. T. Brixner and G. Gerber, “Femtosecond polarization pulse shaping,” Opt. Lett. 26(8), 557–559 (2001). [CrossRef] [PubMed]

16

16. M. Ninck, A. Galler, T. Feurer, and T. Brixner, “Programmable common-path vector field synthesizer for femtosecond pulses,” Opt. Lett. 32(23), 3379–3381 (2007). [CrossRef] [PubMed]

] and different targeted wavelength regimes [17

17. M. T. Seidel, Z. Zhang, S. Yan, and H.-S. Tan, “Ultraviolet polarization pulse shaping using sum-frequency generation,” J. Opt. Soc. Am. B 28(5), 1146–1151 (2011). [CrossRef]

20

20. K. Lee, M. Yi, J. D. Song, and J. Ahn, “Polarization shaping of few-cycle terahertz waves,” Opt. Express 20(11), 12463–12472 (2012). [CrossRef] [PubMed]

] have been developed in this context. However, in these seminal works, optical frequency combs were not used and the pulse shapers were not designed to operate in the line-by-line regime. It is noteworthy that high resolution polarization pulse shapers have been implemented, however focusing on the compensation of polarization mode dispersions [21

21. L. Xu, H. Miao, and A. M. Weiner, “All-order polarization-mode-dispersion compensation at 40 Gb/s via hyperfine resolution optical pulse shaping,” IEEE Photon. Technol. Lett. 22(15), 1078–1080 (2010). [CrossRef]

] rather than on vectorial waveform synthesis.

Temporal Talbot effect, in which the pulse train repetition-rate may be multiplied through the temporal self-imaging, has been investigated extensively by Azaña and associates [22

22. J. Azaña and M. A. Muriel, “Temporal self-imaging effects: theory and application for multiplying pulse repetition rates,” IEEE J. Sel. Top. Quantum Electron. 7(4), 728–744 (2001). [CrossRef]

,23

23. J. Azaña and S. Gupta, “Complete family of periodic Talbot filters for pulse repetition rate multiplication,” Opt. Express 14(10), 4270–4279 (2006). [CrossRef] [PubMed]

]. Temporal Talbot repetition-rate multiplication is achieved as the pulses propagate through a first-order dispersive medium, or equivalently when the spectral lines are applied with periodic spectral phase values [24

24. C.-B. Huang and Y. Lai, “Loss-less pulse intensity repetition-rate multiplication using optical all-pass filtering,” IEEE Photon. Technol. Lett. 12(2), 167–169 (2000). [CrossRef]

]. We note here that temporal Talbot effect enables repetition-rate multiplication for the pulse intensity. The resulting temporal phase repetition-rate remains the same as the input pulse train, and develops a periodicity which can be deterministically derived [22

22. J. Azaña and M. A. Muriel, “Temporal self-imaging effects: theory and application for multiplying pulse repetition rates,” IEEE J. Sel. Top. Quantum Electron. 7(4), 728–744 (2001). [CrossRef]

]. Various theoretical and experimental works on repetition-rate multiplication have been carried out using fiber Bragg gratings [25

25. D. Pudo and L. R. Chen, “Tunable passive all-optical pulse repetition rate multiplier using fiber Bragg gratings,” J. Lightwave Technol. 23(4), 1729–1733 (2005). [CrossRef]

27

27. M. A. Preciado and M. A. Muriel, “Ultrafast all-optical Nth-order differentiator and simultaneous repetition-rate multiplier of periodic pulse train,” Opt. Express 15(19), 12102–12107 (2007). [CrossRef] [PubMed]

], planar lightwave circuits [28

28. D. E. Leaird, S. Shen, A. M. Weiner, A. Sugita, S. Kamei, M. Ishii, and K. Okamoto, “Generation of high repetition rate WDM pulse trains from an arrayed-waveguide grating,” IEEE Photon. Technol. Lett. 13(3), 221–223 (2001). [CrossRef]

, 29

29. P. Samadi, L. R. Chen, I. A. Kostko, P. Dumais, C. L. Callender, S. Jacob, and B. Shia, “Generating 4x20 and 4x40 GHz pulse trains from a single 10-GHz mode-locked laser using a tunable planar lightwave circuit,” IEEE Photon. Technol. Lett. 22(5), 281–282 (2010). [CrossRef]

], and spectral line-by-line shaping [9

9. H.-P. Chuang and C.-B. Huang, “Generation and delivery of 1-ps optical pulses with ultrahigh repetition-rates over 25-km single mode fiber by a spectral line-by-line pulse shaper,” Opt. Express 18(23), 24003–24011 (2010). [CrossRef] [PubMed]

]. However in these reports, temporal Talbot effect has not been implemented in the vectorial regime.

In this Letter, vectorial optical arbitrary waveform generations are experimentally demonstrated. Polarization spectral line-by-line pulse shaping is applied on a phase-modulated continuous-wave (PMCW) laser frequency comb with 20 GHz comb spacing. The procedures in characterizing the spectral phase difference between the two orthogonal polarizations are described in details. Vectorially synthesized optical waveforms with a 50-ps time window, corresponding to a 100% duty cycle of the PMCW comb, are experimentally achieved. Temporal Talbot effect is extended into the vectorial regime for the first time to our best knowledge, where we exploit the periodic temporal phases for interesting field polarization synthesis. In one example, we generate intensity repetition-rate doubled circularly polarized pulses with alternating pulse-by-pulse handedness. In another example, deterministic syntheses of complex vectorially shaped electric fields are also performed through the combination of Talbot phase shaping and line-by-line amplitude pulse shaping. Our experimental results are measured through a dual-quadrature spectral interferometry (DQSI) system [30

30. V. R. Supradeepa, D. E. Leaird, and A. M. Weiner, “Optical arbitrary waveform characterization via dual-quadrature spectral interferometry,” Opt. Express 17(1), 25–33 (2009). [CrossRef] [PubMed]

] and are found in excellent agreements to the applied shaping controls. Our results manifest the capacity of vectorial optical arbitrary waveform generation, and may find interesting applications such as the optimization of quantum ionization yield [31

31. T. Brixner, G. Krampert, T. Pfeifer, R. Selle, G. Gerber, M. Wollenhaupt, O. Graefe, C. Horn, D. Liese, and T. Baumert, “Quantum control by ultrafast polarization shaping,” Phys. Rev. Lett. 92(20), 208301 (2004). [CrossRef] [PubMed]

] and the control of optical near-fields [32

32. M. Aeschlimann, M. Bauer, D. Bayer, T. Brixner, S. Cunovic, F. Dimler, A. Fischer, W. Pfeiffer, M. Rohmer, C. Schneider, F. Steeb, C. Strüber, and D. V. Voronine, “Spatiotemporal control of nanooptical excitations,” Proc. Natl. Acad. Sci. U.S.A. 107(12), 5329–5333 (2010). [CrossRef] [PubMed]

].

2. Experimental setup

Figure 1
Fig. 1 Experimental setup. PC: polarization controller; WP: Wollaston prism; LCM: liquid crystal modulator; DQSI: dual-quadrature spectral interferometry; LP: linear polarizer; PD: photodetector.
shows the schematics of our experimental setup. A PMCW laser frequency comb with 20 GHz spacing (50 ps repetition period) is generated by injecting a 1 kHz-linewidth CW laser (NKT Adjustik) centered at λ0 = 1545 nm into a low-Vπ optical phase modulator [2

2. S. T. Cundiff and A. M. Weiner, “Optical arbitrary waveform generation,” Nat. Photonics 4(11), 760–766 (2010). [CrossRef]

,6

6. C.-B. Huang, Z. Jiang, D. E. Leaird, J. Caraquitena, and A. M. Weiner, “Spectral line-by-line shaping for optical and microwave arbitrary waveform generations,” Laser Photonics Rev. 2(4), 227–248 (2008). [CrossRef]

]. The frequency comb is sent to a homemade reflective polarization line-by-line pulse shaper. The x- and y-polarized frequency comb lines are spatially separated by a Wollaston prism and the incident angles upon a 52.67 grooves/mm gold-coated Echelle grating working in 22nd diffraction order are adjustable through the lateral positioning of the telescope [16

16. M. Ninck, A. Galler, T. Feurer, and T. Brixner, “Programmable common-path vector field synthesizer for femtosecond pulses,” Opt. Lett. 32(23), 3379–3381 (2007). [CrossRef] [PubMed]

]. The diffraction efficiency is about 70% for both the x- and y-polarizations. Discrete comb lines are diffracted by the grating and focused by a lens with 500 mm focal length. A fiberized polarization controller is used to ensure balanced amplitudes between the two polarizations. A computer controllable 2 × 640 pixel liquid crystal modulator array (LCM, CRi SLM-640-D-NM) with 100 μm pitch and 2 μm inter-pixel gap is positioned before the mirror (the Fourier plane) to provide independent control to the amplitude and phase of each comb line for both polarizations. Our pulse shaper is designed so that a LCM pixel width of 100 μm corresponds to a 10 GHz frequency separation. On the Fourier plane, each comb line is focused to 160 μm diameter (1/e2) and controlled using two LCM pixels. The Wollaston prism provides gray-level intensity control with a nominal extinction ratio of 27 dB. The total insertion loss, including the circulator, is 8 dB.

Prior to performing polarization pulse shaping, the phase difference between the two orthogonal polarizations due to dispersion and the slight path difference must be carefully characterized. The spectral phase difference can be mathematically formulated as
ΔΨ(ω˜)=θ+τxyω˜+[Ψx,NL(ω˜)Ψy,NL(ω˜)].
(1)
Here, ω˜denotes the frequency detuning from λ0, θ and τxy denote the absolute phase difference and the relative temporal delay between the two polarizations, respectively. Ψx,NL and Ψy,NL stand for the nonlinear spectral phase terms for the x- and y-polarizations, respectively.

3. Experimental result and discussion

3.1 Characterization of spectral phase difference

Our procedures in retrieving/compensating the three phase difference terms in Eq. (1) are described as follows. First, nonlinear spectral phases ΨNL of both the x- and y-polarized comb lines are compensated so that transform-limited (TL) pulse trains are generated. This is achieved by sending in a single polarized comb at a time while maximizing the second-harmonic yield using an intensity autocorrelator positioned at zero time delay through an automated shaper LCM phase control process [1

1. Z. Jiang, C.-B. Huang, D. E. Leaird, and A. M. Weiner, “Optical arbitrary waveform processing of more than 100 spectral comb lines,” Nat. Photonics 1(8), 463–467 (2007). [CrossRef]

,9

9. H.-P. Chuang and C.-B. Huang, “Generation and delivery of 1-ps optical pulses with ultrahigh repetition-rates over 25-km single mode fiber by a spectral line-by-line pulse shaper,” Opt. Express 18(23), 24003–24011 (2010). [CrossRef] [PubMed]

]. After this process, only θ and τxy in Eq. (1) remain undetermined.

Secondly, the relative temporal delay τxy between the two orthogonally polarized TL pulse trains is determined through a dual-quadrature spectral interferometry (DQSI) system [30

30. V. R. Supradeepa, D. E. Leaird, and A. M. Weiner, “Optical arbitrary waveform characterization via dual-quadrature spectral interferometry,” Opt. Express 17(1), 25–33 (2009). [CrossRef] [PubMed]

] as depicted in Fig. 1. The temporal delays τx and τy for the x- and y-polarized pulse train relative to a TL reference pulse train (derived by using another homemade reflective line-by-line pulse shaper) are individually measured. The measured spectral phases are depicted in Fig. 2(a)
Fig. 2 (a) Linear spectral phases measured by DQSI (open circles) for x- and y-polarizations. (b) Characterization of the absolute phase difference between x- and y-polarizations.
, where the circles denote the average values after 10 DQSI measurements. The average retrieved spectral phase standard deviations (for each comb line over 10 separate measurements) are 0.05 and 0.04 radian for x- and y-polarizations, respectively. The slopes of the two fit traces give τx = 11.88 ps and τy = −11.97 ps. The relative temporal delay τxy = 23.85 ps between the two polarizations is thus experimentally determined. A linear spectral phase of τxyω˜ is applied to the y-polarized comb lines to ensure the two orthogonally polarized pulse trains have zero temporal delay.

Lastly, the absolute phase difference θ is determined for complete knowledge over the instantaneous vectorial field. Our pulse shaper is programmed to block all the comb lines except the central one at λ0 for both polarizations. The shaper amplitude control is applied to ensure equal amplitudes between the two polarizations. The combined x- and y-polarized single comb line passes through a linear polarizer positioned at 45 degree. The resulting interferometric signal strength versus the extra phase added to the y-polarized comb line by the pulse shaper is shown in Fig. 2(b). The perfect agreement between the measurement (circles, averaged over 50 interferometric data) against the cosine-squared fit (line) is a manifestation of our excellent LCM calibration. The absolute phase difference of θ = 1.375π is determined when the applied LCM phase gives the in-phase condition between the two polarizations, leading to a maximum interferometric signal. After this procedure, all of the y-polarized comb lines are applied with θ = 1.375π to completely eliminate the spectral phase difference between the two orthogonally polarized pulse trains.

3.2 Polarization shaped waveform filling 100% duty cycle

We now demonstrate full-vectorial optical arbitrary waveform generation. In the first example, the y-polarized frequency comb is unaltered after the phase calibration process, corresponding to a TL pulse train. The x-polarized frequency comb lines are modulated by a quadratic spectral phase (coefficient of −236.87 ps2) via the LCM to generate a linearly positively-chirped waveform filling the entire 50-ps window (100% duty cycle). The shaped waveform is then carefully characterized by the DQSI system. Figure 3(a)
Fig. 3 (a) Characterization of the strongly chirped x-polarized field. (b) Quasi-three-dimensional electric field representation for the vectorially shaped waveform. The color tone represents the instantaneous frequency of the waveform.
shows that the measured (circles) and the LCM-applied spectral phases (line) for the shaped x-polarized comb are in excellent agreement. The average phase error (defined by the modulus of the difference between the phases measured by DQSI and applied by LCM, respectively) over all frequency comb lines is 0.07 radian, and the averaged standard deviation of the retrieved phases over 10 measurements is 0.05 radian.

Based on the experimentally measured power spectra and spectral phases shown in Fig. 3(a), the vectorially-synthesized OAWG is displayed in Fig. 3(b) using the colored quasi-three-dimensional electric field representation [13

13. T. Brixner, G. Krampert, P. Niklaus, and G. Gerber, “Generation and characterization of polarization-shaped femtosecond laser pulses,” Appl. Phys. B 74(9), S133–S144 (2002). [CrossRef]

]. The x- and y-polarized pulses are independently depicted using the black projection traces onto the corresponding axes. The instantaneous field near 0 ps time location is magnified within the inset, being left-hand elliptically polarized. The instantaneous frequency of the waveform, defined by taking the inverse of the piecewise time required for the field amplitude in passing through the maximum twice along the polarization ellipse [13

13. T. Brixner, G. Krampert, P. Niklaus, and G. Gerber, “Generation and characterization of polarization-shaped femtosecond laser pulses,” Appl. Phys. B 74(9), S133–S144 (2002). [CrossRef]

], is represented by the color tone (blue and red for higher and lower frequencies). In this example, the instantaneous frequency is dominated by the x-polarized up-chirped pulse, increasing linearly with time across the entire 50 ps period (the y-polarized TL pulse has constant instantaneous frequency).

We make two notes here over the unique characteristics of the temporal waveforms derived from a PMCW comb: (1) The inherent highly structured power spectrum of the PMCW comb is determined by Bessel function of the first kind [33

33. T. Kobayashi, H. Yao, K. Amano, Y. Fukushima, A. Morimoto, and T. Sueta, “Optical pulse compression using high-frequency electrooptic phase modulation,” IEEE J. Quantum Electron. 24(2), 382–387 (1988). [CrossRef]

]. This in term produces sinc-like temporal pulse TL envelope, and is evident in the projection in the y-polarization. (2) The linearly chirped x-polarized waveform does not have the maximum amplitude at time 0. With the strong linear chirp applied here, the temporal waveform resembles the PMCW power spectrum. This in turn can be understood as a manifestation of direct frequency-to-time mapping [34

34. I. S. Lin, J. D. McKinney, and A. M. Weiner, “Photonic synthesis of broadband microwave arbitrary waveforms applicable to ultra-wideband communication,” IEEE Microw. Wirel. Compon. Lett. 15(4), 226–228 (2005). [CrossRef]

]. These are in contrast to a typical laser with Gaussian-like spectral envelope, which still gives a Gaussian temporal envelope under frequency-to-time mapping.

3.3 Vectorial temporal Talbot effect

Temporal Talbot effect is now implemented in the vectorial regime. It is well-known that although the pulse intensity rate can be multiplied through temporal Talbot effect, the temporal phase rate remains fixed and develops a deterministic periodical variation [22

22. J. Azaña and M. A. Muriel, “Temporal self-imaging effects: theory and application for multiplying pulse repetition rates,” IEEE J. Sel. Top. Quantum Electron. 7(4), 728–744 (2001). [CrossRef]

]. We exploit this unique periodic temporal phase behavior in generating circularly polarized pulses with alternating pulse-by-pulse handedness. We generate intensity-rate doubled pulse trains by applying periodic spectral phases of {0, π/2} and {π/2, π} [24

24. C.-B. Huang and Y. Lai, “Loss-less pulse intensity repetition-rate multiplication using optical all-pass filtering,” IEEE Photon. Technol. Lett. 12(2), 167–169 (2000). [CrossRef]

] to the x- and y-polarized comb lines, respectively. Application of such periodic spectral phase is only possible through clean line-by-line control. Figures 4(a,b)
Fig. 4 (a,b) Characterization of repetition-rate doubled fields for the x- and y-polarizations, respectively. (c) Quasi-three-dimensional electric field representation.
show the measured power spectra and spectral phases of the shaped x- and y-polarized combs, respectively. The average phase error is 0.08 radian, and the average standard deviation of the retrieved phases for all spectral lines is 0.04 radian.

The resulting intensity-rate doubled x-polarized pulses have periodic pulse-by-pulse temporal phases of [0, 3π/2]. On the other hand, the y-polarized pulses have periodic temporal phase of [π/2, −π]. Through this design, adjacent pulses will have alternating handedness since the pulse-by-pulse phase difference is [-π/2, π/2]. Figure 4(c) shows the resulting instantaneous electrical field. The intensity-rate doubling is evident: the temporal period is reduced to 25 ps. To confirm our design principle, the pulses at time locations of −25, 0, and 25 ps are magnified within the insets. It is evident that the handedness of the - 25 and + 25 ps pulses is right-handed, while the 0 ps pulse being left-handed. The instantaneous frequency in this example remains constant.

3.4 Vectorial amplitude and Talbot phase shaping

We now generate intensity-rate multiplied pulse trains through a mixture of temporal Talbot shaping (implemented through line-by-line phase control) to the x-polarized comb lines and line-by-line amplitude shaping to the y-polarized comb lines. The x-polarized comb lines are applied with periodic spectral phases of {0, 2π/3, 2π/3} for the generation of intensity-rate tripled (60 GHz) pulse train [24

24. C.-B. Huang and Y. Lai, “Loss-less pulse intensity repetition-rate multiplication using optical all-pass filtering,” IEEE Photon. Technol. Lett. 12(2), 167–169 (2000). [CrossRef]

]. Figure 5(a)
Fig. 5 (a) Characterization of intensity-rate tripled fields through line-by-line phase control for the x-polarization. (b) Characterization of repetition-rate doubled fields using line-by-line amplitude control for the y-polarization. (c) Resulting quasi-three-dimensional shaped electric fields generating circularly and elliptically polarized pulses.
shows the retrieved optical power spectrum and the spectral phase of the shaped x-polarized comb. The average phase error and the average standard deviation of the retrieved phases for all spectral lines are 0.04 and 0.02 radian, respectively. The y-polarized comb lines are applied with line-by-line amplitude control with every other line being suppressed to yield a 40 GHz comb. Such comb generates intensity-rate doubled (compared to the initial 20 GHz comb) optical pulse train. Figure 5(b) shows the retrieved optical power spectrum and the spectral phase of the amplitude shaped y-polarized comb. In our experiment, a nominal 27 dB extinction ratio is achieved for the line suppression. The average phase error and the average standard deviation of the retrieved phases for all spectral lines are 0.03 and 0.02 radian, respectively.

The resulting vectorial arbitrarily shaped electric field within a 50 ps original comb period is depicted in Fig. 5(c) and discussed as follows: Four x-polarized main pulses at times [0, 16.7, 33.3 and 50] ps are generated through the application of temporal Talbot intensity-rate tripling. The corresponding pulse temporal phases are [0.5π, −0.77π, 0.45π, and 0.5π]. The observed increased fields between two adjacent main pulses are attributed to two reasons: (1) Finite number of comb lines and (2) the unequal amplitudes of the comb lines. On the other hand, three y-polarized pulses at times [0, 25, and 50] ps are generated through the amplitude intensity-rate doubling. As a result of pure amplitude filtering, these pulses have a common temporal phase. The quasi-three dimensional electric field representation reveals the instantaneous polarization of the five main pulses at time [0, 16.7, 25, 33.3, and 50] ps. The corresponding temporal phase-to-pulse phase difference between the two polarizations are [0.5π, −0.77π, 0.6π, 0.45π, and 0.5π], respectively. In making the pulse at times 0 and 50 ps right-hand circularly polarized, an amplitude suppression of 1.8 dB is applied to all the remaining y-polarized comb lines (shown in Fig. 5(b)) to equalize the field amplitudes of the two polarizations. The pulses generated at 16.7, 25 and 33.3 ps are slightly left-, right-, and right-hand elliptically polarized, respectively. Magnified views for closer inspection over the pulse handedness are provided within the insets of Fig. 5(c). The slight ellipticity observed here is due to the inherent PMCW spectral envelope, in accord to the discussion made in section 3.2. If a broad laser frequency comb with smooth spectral envelope (e.g. Gaussian with twice the optical bandwidth) is used to perform the vectorial OAWG, the 16.7 ps and 33.3 ps pulses could then be x-linearly polarized while the 25 ps pulse y-linearly polarized.

4. Conclusion

In summary, vectorial optical arbitrary waveforms are experimentally demonstrated by applying polarization line-by-line pulse shaping to a 20 GHz phase-modulated CW laser frequency comb. Polarization shaped waveforms spanning a 100% duty cycle (50 ps) is successfully synthesized and characterized. Vectorial temporal Talbot effect is also experimentally realized. In one demonstration, repetition-rate doubled circularly polarized pulses with alternating pulse-by-pulse handedness are generated through the unique pulse-by-pulse periodic temporal phase relation. In another example, complicated instantaneous field polarizations are synthesized through the combination of line-by-line amplitude and temporal Talbot phase shaping. Our experiments are measured using a dual-quadrature spectral interferometry system. The retrieved spectral phase and optical spectra are found in excellent agreements to the experimentally applied shaping controls. We note here our vectorial pulse shaping results may find interesting correspondence to the Pancharatnam-Berry phase, in which the adiabatic polarization variations of a light traversing along a closed loop along the Poincaré sphere acquires an extra geometric phase [35

35. M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proc. R. Soc. Lond. A Math. Phys. Sci. 392(1802), 45–57 (1984). [CrossRef]

]. Typical experiment in extracting such geometric phase would require measuring the interference between an unaltered field with the altered field in the time- or spatial-domains [36

36. P. G. Kwiat and R. Y. Chiao, “Observation of a nonclassical Berry’s phase for the photon,” Phys. Rev. Lett. 66(5), 588–591 (1991). [CrossRef] [PubMed]

]. Here our phase retrievals are performed in the spectral domain. Recently, closed-form expression for the Pancharatnam-Berry phase generated by an arbitrary polarization control device has been formulated [37

37. J. C. Gutiérrez-Vega, “Pancharatnam-Berry phase of optical systems,” Opt. Lett. 36(7), 1143–1145 (2011). [CrossRef] [PubMed]

], which may facilitate in finding the direct linkage between vectorial optical waveform generations and the associated geometric phase.

Acknowledgment

This work was supported by the National Science Council in Taiwan under grants NSC 100-2112-M-007-007-MY3, 101-2120-M-007-002, 100-2221-E-007-093-MY3 and by the National Tsing Hua University under grant 101N2081E1.

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C. T. Middleton, D. B. Strasfeld, and M. T. Zanni, “Polarization shaping in the mid-IR and polarization-based balanced heterodyne detection with application to 2D IR spectroscopy,” Opt. Express 17(17), 14526–14533 (2009). [CrossRef] [PubMed]

19.

M. T. Seidel, S. Yan, and H.-S. Tan, “Mid-infrared polarization pulse shaping by parametric transfer,” Opt. Lett. 35(4), 478–480 (2010). [CrossRef] [PubMed]

20.

K. Lee, M. Yi, J. D. Song, and J. Ahn, “Polarization shaping of few-cycle terahertz waves,” Opt. Express 20(11), 12463–12472 (2012). [CrossRef] [PubMed]

21.

L. Xu, H. Miao, and A. M. Weiner, “All-order polarization-mode-dispersion compensation at 40 Gb/s via hyperfine resolution optical pulse shaping,” IEEE Photon. Technol. Lett. 22(15), 1078–1080 (2010). [CrossRef]

22.

J. Azaña and M. A. Muriel, “Temporal self-imaging effects: theory and application for multiplying pulse repetition rates,” IEEE J. Sel. Top. Quantum Electron. 7(4), 728–744 (2001). [CrossRef]

23.

J. Azaña and S. Gupta, “Complete family of periodic Talbot filters for pulse repetition rate multiplication,” Opt. Express 14(10), 4270–4279 (2006). [CrossRef] [PubMed]

24.

C.-B. Huang and Y. Lai, “Loss-less pulse intensity repetition-rate multiplication using optical all-pass filtering,” IEEE Photon. Technol. Lett. 12(2), 167–169 (2000). [CrossRef]

25.

D. Pudo and L. R. Chen, “Tunable passive all-optical pulse repetition rate multiplier using fiber Bragg gratings,” J. Lightwave Technol. 23(4), 1729–1733 (2005). [CrossRef]

26.

J. Magné, J. Bolger, M. Rochette, S. LaRochelle, L. R. Chen, B. J. Eggleton, and J. Azaña, “Generation of a 4x100 GHz pulse train generation from a single-wavelength 10-GHz mode-locked laser using superimposed fiber gratings and nonlinear conversion,” J. Lightwave Technol. 24, 2091–2099 (2006). [CrossRef]

27.

M. A. Preciado and M. A. Muriel, “Ultrafast all-optical Nth-order differentiator and simultaneous repetition-rate multiplier of periodic pulse train,” Opt. Express 15(19), 12102–12107 (2007). [CrossRef] [PubMed]

28.

D. E. Leaird, S. Shen, A. M. Weiner, A. Sugita, S. Kamei, M. Ishii, and K. Okamoto, “Generation of high repetition rate WDM pulse trains from an arrayed-waveguide grating,” IEEE Photon. Technol. Lett. 13(3), 221–223 (2001). [CrossRef]

29.

P. Samadi, L. R. Chen, I. A. Kostko, P. Dumais, C. L. Callender, S. Jacob, and B. Shia, “Generating 4x20 and 4x40 GHz pulse trains from a single 10-GHz mode-locked laser using a tunable planar lightwave circuit,” IEEE Photon. Technol. Lett. 22(5), 281–282 (2010). [CrossRef]

30.

V. R. Supradeepa, D. E. Leaird, and A. M. Weiner, “Optical arbitrary waveform characterization via dual-quadrature spectral interferometry,” Opt. Express 17(1), 25–33 (2009). [CrossRef] [PubMed]

31.

T. Brixner, G. Krampert, T. Pfeifer, R. Selle, G. Gerber, M. Wollenhaupt, O. Graefe, C. Horn, D. Liese, and T. Baumert, “Quantum control by ultrafast polarization shaping,” Phys. Rev. Lett. 92(20), 208301 (2004). [CrossRef] [PubMed]

32.

M. Aeschlimann, M. Bauer, D. Bayer, T. Brixner, S. Cunovic, F. Dimler, A. Fischer, W. Pfeiffer, M. Rohmer, C. Schneider, F. Steeb, C. Strüber, and D. V. Voronine, “Spatiotemporal control of nanooptical excitations,” Proc. Natl. Acad. Sci. U.S.A. 107(12), 5329–5333 (2010). [CrossRef] [PubMed]

33.

T. Kobayashi, H. Yao, K. Amano, Y. Fukushima, A. Morimoto, and T. Sueta, “Optical pulse compression using high-frequency electrooptic phase modulation,” IEEE J. Quantum Electron. 24(2), 382–387 (1988). [CrossRef]

34.

I. S. Lin, J. D. McKinney, and A. M. Weiner, “Photonic synthesis of broadband microwave arbitrary waveforms applicable to ultra-wideband communication,” IEEE Microw. Wirel. Compon. Lett. 15(4), 226–228 (2005). [CrossRef]

35.

M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proc. R. Soc. Lond. A Math. Phys. Sci. 392(1802), 45–57 (1984). [CrossRef]

36.

P. G. Kwiat and R. Y. Chiao, “Observation of a nonclassical Berry’s phase for the photon,” Phys. Rev. Lett. 66(5), 588–591 (1991). [CrossRef] [PubMed]

37.

J. C. Gutiérrez-Vega, “Pancharatnam-Berry phase of optical systems,” Opt. Lett. 36(7), 1143–1145 (2011). [CrossRef] [PubMed]

OCIS Codes
(070.6760) Fourier optics and signal processing : Talbot and self-imaging effects
(260.5430) Physical optics : Polarization
(320.5540) Ultrafast optics : Pulse shaping

ToC Category:
Ultrafast Optics

History
Original Manuscript: September 17, 2012
Revised Manuscript: November 9, 2012
Manuscript Accepted: November 11, 2012
Published: November 16, 2012

Citation
Chi-Cheng Chen, I-Chun Hsieh, Shang-Da Yang, and Chen-Bin Huang, "Polarization line-by-line pulse shaping for the implementation of vectorial temporal Talbot effect," Opt. Express 20, 27062-27070 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-24-27062


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References

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  17. M. T. Seidel, Z. Zhang, S. Yan, and H.-S. Tan, “Ultraviolet polarization pulse shaping using sum-frequency generation,” J. Opt. Soc. Am. B28(5), 1146–1151 (2011). [CrossRef]
  18. C. T. Middleton, D. B. Strasfeld, and M. T. Zanni, “Polarization shaping in the mid-IR and polarization-based balanced heterodyne detection with application to 2D IR spectroscopy,” Opt. Express17(17), 14526–14533 (2009). [CrossRef] [PubMed]
  19. M. T. Seidel, S. Yan, and H.-S. Tan, “Mid-infrared polarization pulse shaping by parametric transfer,” Opt. Lett.35(4), 478–480 (2010). [CrossRef] [PubMed]
  20. K. Lee, M. Yi, J. D. Song, and J. Ahn, “Polarization shaping of few-cycle terahertz waves,” Opt. Express20(11), 12463–12472 (2012). [CrossRef] [PubMed]
  21. L. Xu, H. Miao, and A. M. Weiner, “All-order polarization-mode-dispersion compensation at 40 Gb/s via hyperfine resolution optical pulse shaping,” IEEE Photon. Technol. Lett.22(15), 1078–1080 (2010). [CrossRef]
  22. J. Azaña and M. A. Muriel, “Temporal self-imaging effects: theory and application for multiplying pulse repetition rates,” IEEE J. Sel. Top. Quantum Electron.7(4), 728–744 (2001). [CrossRef]
  23. J. Azaña and S. Gupta, “Complete family of periodic Talbot filters for pulse repetition rate multiplication,” Opt. Express14(10), 4270–4279 (2006). [CrossRef] [PubMed]
  24. C.-B. Huang and Y. Lai, “Loss-less pulse intensity repetition-rate multiplication using optical all-pass filtering,” IEEE Photon. Technol. Lett.12(2), 167–169 (2000). [CrossRef]
  25. D. Pudo and L. R. Chen, “Tunable passive all-optical pulse repetition rate multiplier using fiber Bragg gratings,” J. Lightwave Technol.23(4), 1729–1733 (2005). [CrossRef]
  26. J. Magné, J. Bolger, M. Rochette, S. LaRochelle, L. R. Chen, B. J. Eggleton, and J. Azaña, “Generation of a 4x100 GHz pulse train generation from a single-wavelength 10-GHz mode-locked laser using superimposed fiber gratings and nonlinear conversion,” J. Lightwave Technol.24, 2091–2099 (2006). [CrossRef]
  27. M. A. Preciado and M. A. Muriel, “Ultrafast all-optical Nth-order differentiator and simultaneous repetition-rate multiplier of periodic pulse train,” Opt. Express15(19), 12102–12107 (2007). [CrossRef] [PubMed]
  28. D. E. Leaird, S. Shen, A. M. Weiner, A. Sugita, S. Kamei, M. Ishii, and K. Okamoto, “Generation of high repetition rate WDM pulse trains from an arrayed-waveguide grating,” IEEE Photon. Technol. Lett.13(3), 221–223 (2001). [CrossRef]
  29. P. Samadi, L. R. Chen, I. A. Kostko, P. Dumais, C. L. Callender, S. Jacob, and B. Shia, “Generating 4x20 and 4x40 GHz pulse trains from a single 10-GHz mode-locked laser using a tunable planar lightwave circuit,” IEEE Photon. Technol. Lett.22(5), 281–282 (2010). [CrossRef]
  30. V. R. Supradeepa, D. E. Leaird, and A. M. Weiner, “Optical arbitrary waveform characterization via dual-quadrature spectral interferometry,” Opt. Express17(1), 25–33 (2009). [CrossRef] [PubMed]
  31. T. Brixner, G. Krampert, T. Pfeifer, R. Selle, G. Gerber, M. Wollenhaupt, O. Graefe, C. Horn, D. Liese, and T. Baumert, “Quantum control by ultrafast polarization shaping,” Phys. Rev. Lett.92(20), 208301 (2004). [CrossRef] [PubMed]
  32. M. Aeschlimann, M. Bauer, D. Bayer, T. Brixner, S. Cunovic, F. Dimler, A. Fischer, W. Pfeiffer, M. Rohmer, C. Schneider, F. Steeb, C. Strüber, and D. V. Voronine, “Spatiotemporal control of nanooptical excitations,” Proc. Natl. Acad. Sci. U.S.A.107(12), 5329–5333 (2010). [CrossRef] [PubMed]
  33. T. Kobayashi, H. Yao, K. Amano, Y. Fukushima, A. Morimoto, and T. Sueta, “Optical pulse compression using high-frequency electrooptic phase modulation,” IEEE J. Quantum Electron.24(2), 382–387 (1988). [CrossRef]
  34. I. S. Lin, J. D. McKinney, and A. M. Weiner, “Photonic synthesis of broadband microwave arbitrary waveforms applicable to ultra-wideband communication,” IEEE Microw. Wirel. Compon. Lett.15(4), 226–228 (2005). [CrossRef]
  35. M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proc. R. Soc. Lond. A Math. Phys. Sci.392(1802), 45–57 (1984). [CrossRef]
  36. P. G. Kwiat and R. Y. Chiao, “Observation of a nonclassical Berry’s phase for the photon,” Phys. Rev. Lett.66(5), 588–591 (1991). [CrossRef] [PubMed]
  37. J. C. Gutiérrez-Vega, “Pancharatnam-Berry phase of optical systems,” Opt. Lett.36(7), 1143–1145 (2011). [CrossRef] [PubMed]

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