Proposal for CEP measurement based on terahertz air photonics

Abstract

Single-shot carrier envelope phase (CEP) measurement is a challenge in the research field of ultrafast optics. We theoretically investigate how an intense terahertz pulse modulates second harmonic emission (SH) from a gas plasma induced by a few-cycle laser pulse (FCL). Results show that the modulation quantity of SH intensity has a cosinoidal dependence on the CEP of FCL pulses, based on which we propose a low energy, all-optical method for single-shot CEP measurements via using a known intense terahertz pulse. Moreover, we propose an experimental realization.

Keywords

ultrafast measurements / far-infrared or terahertz / ultrafast nonlinear optics / harmonic generation and mixing

Introduction

Terahertz (THz) air photonics is one of the most active research areas in THz science during the past two decades, including of generation and detection of ultra-broadband intense THz waves by using laser-induced gas-plasma [¹–¹⁷]. Recent advances of laser technology have made it possible to generate high-power laser pulses with the duration approaching a single optical cycle, i.e., few-cycle laser (FCL) pulses [¹⁸–²⁰]. Thus the emergence of such ultra-short pulse, to our knowledge, leads to the generation of THz waves whose peak electric field strengths exceed a few of MV/cm [¹⁶]. Meanwhile, the bandwidth of detected THz pulses reaches from 0.1 to 200 THz via measuring the second harmonic (SH) radiation from the gas-plasma [¹³]. In 2013, Liu et al. propose an approach for THz coherent detection via using FCL pulse with very low laser energies [¹⁵]. Specifically, the SH intensity emitted from the gas plasma can well replicate THz waveform if the laser pulse has a fixed carrier envelope phase (CEP). Here an inverse question emerges: if the detailed information of THz waveform is given, can this fixed CEP of FCL pulse be identified?

In the research field of ultrafast optics, improving the ability to effectively measure and stabilize the CEP of FCL pulse is a key subject [²¹–²⁴]. Current widely used method is the in-loop f-to-2f interferometer [²¹]. Alternatively, quantum interference in photocurrents and linear optical interferometry are also invented to derive the CEP’s evolution [²²,²³]. However, the CEP measured by these systems is not a real value but its relative change. In addition, all of these methods are not single-shot measurements. In other words, a large number of laser shots are needed to obtain a single measurement value of the CEP. In 2009, Wittmann et al. performed a single-shot CEP measurement called stereo ATI. By detecting photoelectrons in opposing directions parallel to the polarization of the FCL and comparing their left–right asymmetries at different energies, they can retrieve the real value of CEP without any phase ambiguity [²⁴]. Enlightened by the thought in this work, we propose an all-optical approach for single-shot CEP measurements based on THz air photonics, i.e., retrieving the real values of CEP of consecutive non-phase-stabilized pulses. An experimental realization about our approach will be also proposed.

Simulation results and discussion

First, we need introduce three kinds of symmetries between the FCL pulses with different CEP. The electric field of a linearly polarized FCL readsE_{{\omega }_{\mathrm{0}}}(t)=E_{\mathrm{0}}(t)cos({\omega }_{\mathrm{0}}t+{\varphi }_{CEP}), where{\omega }_{\mathrm{0}}is the angular frequency of carrier wave, E_{\mathrm{0}}(t) is supposed to be a Gaussian envelope function, and {\phi }_{CEP}is CEP. Figures 1(a)−1(e) plot the electric field of FCL pulses with different CEP. Pulses with{\phi }_{CEP}^{}=\mathrm{0}and π have maximum electric field strength at t=\mathrm{0} and are asymmetric along the positive and negative directions, respectively, while pulses with {\phi }_{CEP}^{}=\pm \pi /\mathrm{2} have two smaller field peaks with opposite sign and retain the field orientation symmetry, as shown in Figs. 1(a) and 1(b). For{\phi }_{CEP}=\pi /\mathrm{4}and-\pi /\mathrm{4} (see Fig. 1(c)), the electric fieldE_{{\omega }_{\mathrm{0}}}(t)=E_{{\omega }_{\mathrm{0}}}(-t), implying a time reversal symmetry (TRS). Figure 1(d) gives the polarity reversal symmetry (PRS), i.e., such illustrated two pulses have the same electric field strength distribution but opposite sign. Center symmetry (CS), seeing Fig. 1(e), can be interpreted as a symmetry after a successive time and polarity reversal, i.e., CS= TRS+ PRS. Previous research shows that two pulses with one of these three symmetries will usually yield the same electron and harmonic emission, making it difficult to determine the values of CEP [¹⁸–²⁰].

Using well-developed photocurrent (PC) model [⁵,⁷], we numerically investigate how the FCLs with different CEP influence the SH intensity, see Fig. 2. The simulation parameters can be found in Ref.[¹⁵]. Comparing two curves in Fig. 2(a), one can clearly see that THz field effectively modulate the SH intensity. When a strong THz pulse is simultaneously incident on the plasma with a fixed time interval with respect to the FCL pulse, the dependence of SH emission intensity on CEP will be changed (see the blue curve in Fig. 2(a)). Note that this strong THz field can be regarded as static field with a particular polarity since the THz wave pulse width is much longer than that of the FCL pulses. To further manifest the modulation of THz field, we introduce a quantity\Delta I_{\mathrm{2}{\omega }_{\mathrm{0}}}^{}=I_{\mathrm{2}{\omega }_{\mathrm{0}}}^{{\omega }_{\mathrm{0}}+T}-I_{\mathrm{2}{\omega }_{\mathrm{0}}}^{{\omega }_{\mathrm{0}}}, and plot it versus CEP in Fig. 2(b). When the absolute values of CEP are smaller than\pi /\mathrm{2}, THz waves enhance the SH intensity. For\left|{\phi }_{CEP}^{}\right|>\pi /\mathrm{2}, \Delta I_{\mathrm{2}{\omega }_{\mathrm{0}}}^{}is decreased. Fitting calculation shows that\Delta I_{\mathrm{2}{\omega }_{\mathrm{0}}}^{}has a cosinoidal dependence on the CEP, implying that the absolute value (not the real value) of CEP can be retrieved if\Delta I_{\mathrm{2}{\omega }_{\mathrm{0}}}^{}is measured.

To distinguish the contributions from the positive or negative electric field of a FCL pulse, a polarized ionization rate r\left(t\right)is defined, i.e., r\left(t\right)=\mathrm{d}{N}_{\mathrm{e}}/\mathrm{d}t\times E_{{\omega }_{\mathrm{0}}}/ \left|E_{{\omega }_{\mathrm{0}}}\right|, N_e is the electron density that can be calculated by static tunneling (ST) model [⁷]. Figures 3(a) and 3(b) show the time dependence of r\left(t\right)with and without a positive 1 MV/cm THz field E_T applied on the gas target for {\varphi }_{\mathrm{C}\mathrm{E}\mathrm{P}}^{}=\mathrm{0}, \mathrm{\pi } and \pm \mathrm{\pi }/\mathrm{2}, respectively. For FCL pulses with {\varphi }_{\mathrm{C}\mathrm{E}\mathrm{P}}^{}=\mathrm{0}, as shown in the insets of Fig. 3(a), the positive polarized electric field contributes more to gas ionization processes than the negative field, while {\varphi }_{\mathrm{C}\mathrm{E}\mathrm{P}}^{}=\mathrm{\pi } is on the contrary. So the total increment of SH intensities has a positive extremum for {\varphi }_{\mathrm{C}\mathrm{E}\mathrm{P}}^{}=\mathrm{0} and negative extremum for {\varphi }_{\mathrm{C}\mathrm{E}\mathrm{P}}^{}=\mathrm{\pi } (see Fig. 2(b)), showing that the THz field breaks PRS of FCL pulses. As a result, {\varphi }_{\mathrm{C}\mathrm{E}\mathrm{P}}^{}=\mathrm{0}and {\varphi }_{\mathrm{C}\mathrm{E}\mathrm{P}}^{}=\mathrm{\pi } can be retrieved. However, for FCL pulses with {\varphi }_{\mathrm{C}\mathrm{E}\mathrm{P}}^{}=\pm \mathrm{\pi }/\mathrm{2}, i.e., TRS pulses, r\left(t\right)has same incremental quantity but with opposite sign for positive and negative field of FCL due to the presence of THz fields, as shown in Fig. 3(b). So the TRS of FCL is not broken since they have the same asymmetry and equal THz increment, leading to that {\varphi }_{\mathrm{C}\mathrm{E}\mathrm{P}}^{}=\pm \mathrm{\pi }/\mathrm{2} cannot be resolved (see Fig. 2(b)). Since CS is a successive action of PRS and TRS, the real values of CEP with CS cannot be also retrieved.

Actually, the situation similar to our results, i.e., two pulses with TRS cannot be resolved, is also appeared in Ref. [²⁴]. To solve this problem, another FCL pulse with a fixed phase shift {\phi }_S^{} to the former one is introduced. Referring to this idea, we also introduce another FCL pulse which has shift {\phi }_S^{} with respect to that of E_{{\omega }_{\mathrm{0}}}(t), i.e., {E^{\prime }}_{{\omega }_{\mathrm{0}}}(t)=E_{\mathrm{0}}(t)cos({\omega }_{\mathrm{0}}t+{\phi }_{CEP}+{\phi }_s). To elucidate how to break the TRS, a value of {\phi }_S^{}=\pi /\mathrm{4}is chosen to be an example to distinguish pulses with{\phi }_{CEP}\mathrm{=\; +}\pi /\mathrm{2}and-\pi /\mathrm{2}. Note that the CEP of {E^{\prime }}_{{\omega }_{\mathrm{0}}}(t)corresponding to former E_{{\omega }_{\mathrm{0}}}(t)with+\pi /\mathrm{2} (-\pi /\mathrm{2}) is changed to be\mathrm{+3}\pi /\mathrm{4} (-\pi /\mathrm{4}) now. It is cleared that the former symmetric r(t) with{\phi }_{CEP}^{}=\pm \pi /\mathrm{2} become negative and positive asymmetric, respectively, as illustrated by the arrows in Fig. 4(a). Further calculation shows that \Delta I_{\mathrm{2}{\omega }_{\mathrm{0}}}^Tis not symmetric about the axis{\phi }_{CEP}^{}=\mathrm{0}, showing that one value of \Delta I_{\mathrm{2}{\omega }_{\mathrm{0}}}^Tcorrespond to two CEP with different absolute values (green line in Fig. 4(b)), instead of former same absolute values (black line in Fig. 4(b)). Therefore, the three asymmetries have been broken, enabling a determination of CEP in the full \mathrm{2}\pi range without ambiguity.

A proposed experimental realization

Here we propose an experimental realization, illustrated in Fig. 5(a), to elucidate our approach to measure the real values of CEP. Four detectors (D1−D4, e.g., photomultiplier tube) are used to measure intensities of SH signals emitted from four gas plasma. Narrow band filter (BNF) is used to block residual FCL and THz waves. Note that four FCLs can be obtained from one laser pulses via using beam splitters. Two fully known intense THz waves are impinged on two gas plasma to modulate the SH signals. The FCLs in C1 and C2 correspond toE_{{\omega }_{\mathrm{0}}}(t) and{E^{\prime }}_{{\omega }_{\mathrm{0}}}(t), respectively. The shift{\phi }_S^{}betweenE_{{\omega }_{\mathrm{0}}}(t)and{E^{\prime }}_{{\omega }_{\mathrm{0}}}(t)can be controlled by using a pair of glass wedges [²⁵,²⁶]. To ensure the identical spatial distributions of four plasmas, the lens in front of plasmas must be placed precisely by observing plasmas through CCD camera or micro-bolometer camera.

We assume that the FCL pulses are generated from a non-phase-stabilized FCL. As also performed in Ref. [²⁴], we can map the CEP of consecutive laser shots on a parametric plot in which the axes are\Delta I_{\mathrm{2}{\omega }_{\mathrm{0}}}^{}and\Delta {I^{\prime }}_{\mathrm{2}{\omega }_{\mathrm{0}}}^{}, respectively. In this representation, the CEP of each shot can be mapped to one point on the plot. Let’s take{\phi }_S^{}=\pi /\mathrm{4}again as an example, calculation results show that a large number of shots will form an oblique elliptic curve, as depicted in Fig. 5(b). Since the horizontal axis is the parameter derived from Cl, i.e.,\Delta I_{\mathrm{2}{\omega }_{\mathrm{0}}}^{}, the rightmost point represents{\phi }_{CEP}=\mathrm{0}.The leftmost point corresponds to{\phi }_{CEP}^{}=\pi. Considering that\Delta {I^{\prime }}_{\mathrm{2}{\omega }_{\mathrm{0}}}^{}is obtained from C2 corresponding to {\phi }_S^{}=\pi /\mathrm{4}shift, the points{\phi }_{CEP}^{}=\pm \pi /\mathrm{2},\pm \pi /\mathrm{4} and \pm \mathrm{3}\pi /\mathrm{4}are easily identified by referring to the black curve in Fig. 4(b). Other CEP points fall among these special points. For a single shot pulse, once the two signals from the two channels are obtained, i.e., \Delta I_{\mathrm{2}{\omega }_{\mathrm{0}}}^{}and\Delta {I^{\prime }}_{\mathrm{2}{\omega }_{\mathrm{0}}}^{}, the absolute value of CEP can be calculated according to the cosine relation,\left|{\phi }_{CEP}^{}\right|=\arccos [\Delta I_{\mathrm{1}}/\Delta I_{\mathrm{1}}({\phi }_{CEP}^{}=\mathrm{0})] according to Fig. 2(b). If and, as shown in Fig. 5(b), we can derive{\phi }_{CEP}^{}=\left|{\phi }_{CEP}^{}\right|, else{\phi }_{CEP}^{}=-\left|{\phi }_{CEP}^{}\right|. Similarly, the sign of CEP for can be recognized by comparing \Delta {I^{\prime }}_{\mathrm{2}{\omega }_{\mathrm{0}}}^{}and-\Delta {I^{\prime }}_{\mathrm{2}{\omega }_{\mathrm{0}}}^{}({\phi }_{CEP}^{}=\mathrm{0}). In this way, the CEP of a single shot can be retrieved without ambiguity. Figure 5(c) shows parametric plots for other values of{\phi }_S^{}. Since a circular curve is more convenient for the determination of the CEP, we suggest that{\phi }_S^{}should be set around\pi /\mathrm{2}.

Conclusion

In conclusion, a method for single-shot CEP measurements is proposed. By detecting the intense THz pulse-induced SH emission increment and adding a similar signal channel, all symmetries between FCL pulses can be broken. Thus the CEP of a single-shot FCL pulse can be retrieved without ambiguity. Our proposed method is all-optical and easily implemented at ambient conditions with only a small fraction of the FCL energy. Although the key physical processes associated with the proposed method have been discussed, other problems such as nonlinear diffraction and propagation phenomena still need to be discussed further, which should use a higher dimensional model. In addition, some technical issues such as signal-to-noise ratio and detection precision and accuracy also need to be addressed in the future.

References

[1] Hamster H, Sullivan A, Gordon S, White W, Falcone R W. Subpicosecond, electromagnetic pulses from intense laser-plasma interaction. Physical Review Letters, 1993, 71(17): 2725–2728

[2] Cook D J, Hochstrasser R M. Intense terahertz pulses by four-wave rectification in air. Optics Letters, 2000, 25(16): 1210–1212

[3] Dai J, Xie X, Zhang X C. Detection of broadband terahertz waves with a laser-induced plasma in gases. Physical Review Letters, 2006, 97(10): 103903

[4] Roskos H G, Thomson M D, Kreß M, Löffler T. Broadband THz emission from gas plasmas induced by femtosecond optical pulses: from fundamentals to applications. Laser & Photonics Reviews, 2007, 1(4): 349–368

[5] Kim K Y, Glownia J H, Taylor A J, Rodriguez G. Terahertz emission from ultrafast ionizing air in symmetry-broken laser fields. Optics Express, 2007, 15(8): 4577–4584

[6] Karpowicz N, Dai J, Lu X, Chen Y, Yamaguchi M, Zhao H, Zhang X C, Zhang L, Zhang C, Price-Gallagher M, Fletcher C, Mamer O, Lesimple A, Johnson K. Coherent heterodyne time-domain spectrometry covering the entire “terahertz gap”. Applied Physics Letters, 2008, 92(1): 011131

[7] Kim K Y. Generation of coherent terahertz radiation in ultrafast laser-gas interactions. Physics of Plasmas, 2009, 16(5): 056706

[8] Dai J, Karpowicz N, Zhang X C. Coherent polarization control of terahertz waves generated from two-color laser-induced gas plasma. Physical Review Letters, 2009, 103(2): 023001

[9] Wang T J, Marceau C, Chen Y, Yuan S, Théberge F, Châteauneuf M, Dubois J, Chin S L. Terahertz emission from a dc-biased two-color femtosecond laser-induced filament in air. Applied Physics Letters, 2010, 96(21): 211113

[10] Babushkin I, Kuehn W, Köhler C, Skupin S, Bergé L, Reimann K, Woerner M, Herrmann J, Elsaesser T. Ultrafast spatiotemporal dynamics of terahertz generation by ionizing two-color femtosecond pulses in gases. Physical Review Letters, 2010, 105(5): 053903

[11] Dai J, Liu J, Zhang X C. Terahertz wave air photonics: terahertz wave generation and detection with laser-induced gas plasma. IEEE Journal of Selected Topics in Quantum Electronics, 2011, 17(1): 183–190

[12] Clough B, Dai J, Zhang X C. Laser air photonics: beyond the terahertz gap. Materials Today, 2012, 15(1–2): 50–58

[13] Matsubara E, Nagai M, Ashida M. Ultrabroadband coherent electric field from far infrared to 200 THz using air plasma induced by 10 fs pulses. Applied Physics Letters, 2012, 101(1): 011105

[14] Wang H, Wang K, Liu J, Dai H, Yang Z. Theoretical research on terahertz air-breakdown coherent detection with the transient photocurrent model. Optics Express, 2012, 20(17): 19264–19270

[15] Liu J, Wang H, Wang K, Yang Z, Wang S. Coherent detection of terahertz pulses via gas plasma induced by few-cycle laser pulses with fixed carrier envelope phase. Optics Letters, 2013, 38(7): 1104–1106

[16] Oh T I, Yoo Y J, You Y S, Kim K Y. Generation of strong terahertz fields exceeding 8 MV/cm at 1 kHz and real-time beam profiling. Applied Physics Letters, 2014, 105(4): 041103

[17] Hafez H A, Chai X, Ibrahim A, Mondal S, Férachou D, Ropagnol X, Ozaki T. Intense terahertz radiation and their applications. Journal of Optics, 2016, 18(9): 093004

[18] Goulielmakis E, Schultze M, Hofstetter M, Yakovlev V S, Gagnon J, Uiberacker M, Aquila A L, Gullikson E M, Attwood D T, Kienberger R, Krausz F, Kleineberg U. Single-cycle nonlinear optics. Science, 2008, 320(5883): 1614–1617

[19] Krausz F, Ivanov M. Attosecond physics. Reviews of Modern Physics, 2009, 81(1): 163–234

[20] Takahashi E J, Lan P, Mücke O D, Nabekawa Y, Midorikawa K. Nonlinear attosecond metrology by intense isolated attosecond pulses. IEEE Journal of Selected Topics in Quantum Electronics, 2015, 21(5): 8800112

[21] Yu T J, Nam C H. Carrier-envelope phase stabilization of femtosecond lasers by the direct locking method. Progress in Quantum Electronics, 2012, 36(4−6): 541–565

[22] Roos P A, Li X, Smith R P, Pipis J A, Fortier T M, Cundiff S T. Solid-state carrier-envelope phase stabilization via quantum interference control of injected photocurrents. Optics Letters, 2005, 30(7): 735–737

[23] Osvay K, Görbe M, Grebing C, Steinmeyer G. Bandwidth-independent linear method for detection of the carrier-envelope offset phase. Optics Letters, 2007, 32(21): 3095–3097

[24] Wittmann T, Horvath B, Helml W, Schätzel M G, Gu X, Cavalieri A L, Paulus G G, Kienberger R. Single-shot carrier–envelope phase measurement of few-cycle laser pulses. Nature Physics, 2009, 5(5): 357–362

[25] Vernaleken A, Schmidt B, Wolferstetter M, Hänsch T W, Holzwarth R, Hommelhoff P. Carrier-envelope frequency stabilization of a Ti:sapphire oscillator using different pump lasers. Optics Express, 2012, 20(16): 18387–18396

[26] Piglosiewicz B, Schmidt S, Park D J, Vogelsang J, Groß P, Manzoni C, Farinello P, Cerullo G, Lienau C. Carrier-envelope phase effects on the strong-field photoemission of electrons from metallic nanostructures. Nature Photonics, 2014, 8(1): 37–42

RIGHTS & PERMISSIONS

Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature