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Phase-retrieval of femtosecond pulses utilizing ω/2ω quantum interference control of electrical currents Elmar Sternemann,1 Markus Betz,1,* and Claudia Ruppert1,2 1
2
Experimentelle Physik 2, TU Dortmund, 44227 Dortmund, Germany Department of Physics, Columbia University, New York, New York 10027 USA *Corresponding author: markus.betz@tu‑dortmund.de Received April 29, 2014; revised May 9, 2014; accepted May 10, 2014; posted May 13, 2014 (Doc. ID 210733); published June 12, 2014
We propose and implement a versatile scheme to analyze the phase structure of femtosecond pulses. It relies on second harmonic generation in combination with phase-sensitive χ 3 -current injection driven by two time-delayed portions of the emerging ω∕2ω pulse pair. Most strikingly, the group velocity dispersions of both the ω and 2ω components can be unambiguously determined from a simple Fourier transformation of the resulting current interferogram. We test the concept for 45 fs pulses at 1.45 μm and directly compare it to second harmonic frequency resolved optical gating. By choosing appropriate frequency doublers and semiconductor detectors, the scheme is applicable to many wavelength regimes and to pulses as short as a few optical cycles. © 2014 Optical Society of America OCIS codes: (030.1670) Coherent optical effects; (190.7110) Ultrafast nonlinear optics. http://dx.doi.org/10.1364/OL.39.003654
The characterization of ultrashort laser pulses typically relies on self-referencing approaches. The rapid development of increasingly sophisticated femtosecond sources has constantly triggered the search for new pulse characterization techniques that—in addition to a pulse’s spectrum and temporal envelope—also resolve its spectral phase. The most advanced concepts for this ultrafast pulse metrology are provided by frequency resolved optical gating (FROG) [1,2], spectral phase interferometry for direct electric-field reconstruction (SPIDER) [3], and related sophisticated methods, examples of which are reported in Refs. [4–7]. The joint aspect of those concepts is the analysis of an optical spectrum emerging from the nonlinear optical interaction of two portions of the laser pulse train. A numerical algorithm then extracts the spectral amplitude and phase with no or little ambiguity. However, the underlying optical nonlinearity—in many cases sum-frequency generation— depends on the momentary light intensity rather than on optical phases per se. In contrast, certain optical nonlinearities such as optical rectification or, more generally, harmonic mixing phenomena, are directly sensitive to phase differences of interacting optical beams. In particular, a superposition of an ultrashort fundamental pulse (ω) and its second harmonic (2ω) incident on a semiconductor induces strong lateral electrical currents with a vector direction governed by the pulses’ polarizations and the relative phases [8,9]. This third-order optical nonlinearity arises from a quantum interference of one- and two-photon absorption pathways. While its phase dependence is well known, its potential for phase retrieval has barely been appreciated to date. In this Letter, we conceive a versatile concept to characterize femtosecond optical pulses based on a ω∕2ω coherent control approach. After standard second harmonic generation, the emerging ω∕2ω beam is split up and superimposed again to result in two time-delayed collinear beams. While one part consists of the entire ω∕2ω spectrum, the other one is filtered to contain the ω part only. This pulse sequence induces a coherently controlled current in a semiconductor detector. Governed by the 0146-9592/14/123654-04$15.00/0
scaling of the injection process, interferograms obtained by varying the relative temporal delay are modulated by both ω and 2ω carrier frequencies. In essence, a Fourier transformation then extracts important information about, e.g., the group velocity dispersion (GVD) of both ω and 2ω spectral components without a time-reversal ambiguity. This concept is demonstrated by characterizing the 45 fs output of an optical parametric amplifier (OPA) operating at 1.45 μm. A direct comparison to FROG indicates a similar sensitivity to GVDs. However, our scheme neither requires spectral resolution nor a phase-retrieval algorithm. Instead, we acquire a one-dimensional data set and use fast Fourier transformation (FFT) to obtain detailed amplitude and phase information. The scheme for pulse characterization is exemplarily realized for the output of a coherent OPA 9850. It operates at a 1.45 μm central wavelength and delivers 20 mW of tp;ω 45 fs pulses (FWHM) at a repetition rate of 250 kHz. The pulses are frequency-doubled in a 0.5 mm thick BBO crystal. As shown in Fig. 1, the ω∕2ω pulse pair is passed through a Michelson interferometer. In one arm, a low-dispersive, 140 μm thick Si-wafer blocks the second harmonic while a λ∕2 plate tilts the remaining fundamental E 2 ω to horizontal (x) polarization. The second arm contains an unaltered pulse pair consisting of the y-polarized fundamental E 1 ω and the x-polarized second harmonic E 2ω . A motorized stage precisely controls the relative delay τ of the two arms. Note that we use reflective optics and a pellicle beamsplitter to minimize dispersion. The pulse combination emerging from the interferometer is focused onto a contacted, low temperature grown GaAs specimen at room temperature using a f 25 mm parabolic mirror. Its bandgap E G satisfies ℏω < E G < 2ℏω such that these ω∕2ω pulses induce substantial currents. In a contacted device, they lead to a conveniently measurable electrical signal reflecting the optically induced charge separation [9]. More specifically, we use two gold contacts spaced ∼10 μm apart and tilted by ∼45° in the x–y plane (cf. Fig. 1). Figure 2(a) depicts an example for the τ-dependent photoresponse. It shows an interferogram with a Gaussian envelope that © 2014 Optical Society of America
June 15, 2014 / Vol. 39, No. 12 / OPTICS LETTERS
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↔
Fig. 1. Experimental scheme: A 0.5 mm thick BBO generates the second harmonic (blue). An interferometer controls the delay τ of the fundamental portion E 2 ω (red) relative to a pulse pair composed of the remaining fundamental (E 1 ω ) and the second harmonic E 2ω . The emergent beam is focused onto a LT-GaAs sample with gold electrodes spaced 10 μm apart. A lock-in amplifier measures the photoresponse. Inset: geometry of the contacts and the optical polarizations.
arises from the temporal convolution of the interacting pulses. As also evident from the FFT in Fig. 2(b), it is strongly modulated with two carrier waves ω0 and 2ω0 , i.e., the pulses’ center frequencies. We will illustrate below that those two modulation frequencies arise from different beam combinations for current injection and their spectral phases ultimately reveal the phase structure of both ω and 2ω pulses. Note that the Si-filter in the interferometer effectively suppresses any disturbing interference of two 2ω pulses. To elucidate the theoretical background of phase retrieval, we consider a superposition of ω and 2ω pulses incident on a semiconductor satisfying ℏω < E G < 2ℏω. It induces a current with an injection rate of [8]: _ ↔ ⃗ ⃗ ⃗ J⃗ η E ω Eω E2ω ;
current I (nA)
0.3
(1)
(a)
0.0 -0.3 -0.6
norm. FFT amp.
-100 -80 -60 -40 -20 0 20 40 pulse delay (fs) 0.80 1.0
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energy 0.92
(eV) 1.68
60
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80 100
1.76
(b)
0.5 0
2
0
0.0 1.25 1.30 1.35 1.40 2.55 2.60 2.65 2.70 frequency (1015 Hz)
Fig. 2. (a) Current interferogram Iτ recorded for the 45 fs pulse at 1.45 μm. (b) FFT amplitude of Iτ. ω0 and 2ω0 are the center frequencies of the ω and 2ω pulses.
where η is a fourth-rank tensor proportional to the imaginary part of the third-order susceptibility χ 3 0; ω; ω; −2ω. Note that we work with complex fields ⃗ ω . Physically and denote the complex conjugate as, e.g., E relevant quantities are given by the real part. As a result of Eq. (1), current injection driven by two ω and 2ω monochromatic waves with phases φω;2ω scales as J_ ∝ E 2ω E 2ω sin2φω − φ2ω . For the present pulsed configuration currents are induced with one y-polarized 2 field E 1 ω t τ and two x-polarized fields E ω t and E 2ω t τ with relative timing τ. For a zinc blende crystal, such as GaAs, the largest elements of the current injection tensor are ηxxxx ≈ 2η↔yyxx [8]. We implicitly assume a frequency independent η and crystallographic axes aligned to the lab frame. While this geometry is not taken care of, deviations do not affect the underlying principles here. We now restrict the consideration to the two current contributions that depend on the relative timing τ. They read: 2 J_ x ηxxxx E 2 ω tE ω t E 2ω t τ;
(2)
2 J_ y ηyyxx E 1 ω t τE ω t E 2ω t τ.
(3)
Intuitively, the term in Eq. (2) reflects a situation where the two ω driving fields in Eq. (1) are derived from the same interferometer arm. In contrast, the two ω fields originate from different interferometer arms to yield the current in Eq. (3). Current detection is performed in a time-integrating fashion, i.e., by analyzing charge accumulation at contacts on the semiconductor specimen [9]. In the experiment, we interferometrically scan the delay τ between the two interferometer arms and detect signatures Iτ of the coherently controlled current. Therefore, we exR R pect signals I x τ ∝ J_ x dt and I y τ ∝ J_ y dt. As the detector is tilted in the x–y plane, we do not separately measure I x τ and I x τ but a superposition Iτ. Due to the different τ-dependence, the interferograms I x τ and I y τ feature carrier frequencies 2ω0 and ω0 , respectively, as seen in Fig. 2. We now analyze how phase distortions of the ω and 2ω pulses manifest in the interference pattern Iτ and its ˆ FFT Iω. The above expressions I x τ and I y τ represent different field convolutions of the driving ω∕2ω pulse combinations. Utilizing the convolution theorem their spectrum can be expressed by the Fourier transformations of the ω∕2ω fields (⊗ denotes a convolution): ˆ ˆ 2 Iˆ x ω ∝ Eˆ 2 ω ω ⊗ E ω ω · E 2ω ω;
(4)
ˆ 1 ˆ Iˆ y ω ∝ Eˆ 2 ω ω · E ω −ω ⊗ E 2ω ω.
(5)
Such convolutions can be evaluated analytically for transform-limited and linearly chirped Gaussian pulses. Higher order dispersions are incorporated numerically. The result of such a computation is a relation between the spectral phases of the incoming ω∕2ω pulses and ˆ the phase properties of Iω. Most importantly, we find GVDs ϕ00 ω0 and ϕ00 2ω0 to manifest as corresponding FFT phase curvatures Dω0 and D2ω0 at the respective
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center frequencies ω0 and 2ω0 . It is instructive to first analyze the situation of moderate chirp jϕ00 ω0 j ≪ tp;ω 2 2tp;2ω 2 where tp;ω∕2ω denotes the durations of the unchirped, transform-limited pulses. In this limit, the phase curvatures Dω0 and D2ω0 are linked to the GVDs by Dω0
10 00 4 ϕ ω0 − ϕ00 2ω0 ; 9 9
(6)
1 D2ω0 ϕ00 ω0 − ϕ00 2ω0 . 2
(7)
0
curvature D (fs2)
A further simplification is achieved by assuming ϕ00 2ω0 1∕2ϕ00 ω0 typical of second harmonic generation in a thin frequency doubler [10]. In this situation, Eqs. (6) and (7) imply Dω0 8∕9ϕ00 ω0 and D2ω0 0. The more general setting of arbitrary (linear) chirp can still be treated analytically. However, the results for Dω0 and D2ω0 constitute lengthy expressions that explicitly depend on the pulse durations tp;ω and tp;2ω . The full result for Dω0 as a function of the GVD ϕ00 ω0 is shown in exemplary pulse durations tp;ω Fig. 3(a) forptwo and tp;2ω 1∕ 2tp;ω . We also include the expectation Dω0 8∕9ϕ00 ω0 as a solid line for comparison. Most importantly, we see a clear one-to-one correspondence between ϕ00 ω0 and the phase curvature Dω0 , i.e., a quanˆ tity directly accessible from the FFT Iω of the interferogram Iτ. It is also possible to reveal departures from ϕ00 2ω0 1∕2ϕ00 ω0 by determining both Dω0 and D2ω0 and evaluating the outcome with Eqs. (6) and (7). Sufficient knowledge about tp;ω∕2ω can be extracted from conventional optical spectra under the assumption of transform-limited pulses. In principle, similar relations 3 exist between the coefficients D3 ω0 , D2ω0 of the cubic 1000
(a)
500 0
tp, → ∞ tp, = 45 fs tp, = 30 fs
-500
-1000
phase of fourier spectrum (rad)
-1500
-1000
(b)
-500 0 500 1000 theoretical GVD ''( 0) (fs2) 0
2
1500
0
0.1
0.0 no material SF11 5.3 mm Infrasil 10 mm
-0.1 1280
1300 1320 2560 2580 frequency (THz)
2600
2620
Fig. 3. (a) Theoretical phase curvature Dω0 as a function of the GVD ϕ00 ω0 . Solid line: expectation Dω0 8∕9ϕ00 ω0 valid for long pulses (tp → ∞). Dashed lines: full result for two exemplary tp;ω assuming ϕ00 2ω0 1∕2ϕ00 ω0 and tp;2ω p 1∕ 2tp;ω . (b) Spectral phases of the FFT of Iτ-traces for pulses of different chirp. Positive (negative) chirp is introduced by 5.3 mm SF11 (10 mm Infrasil). The lines are polynomial fits.
phase terms and the p third-order dispersion. For weak chirp and tp;2ω 1∕ 2tp;ω they read D3 ω0 26∕27 000 000 ϕ000 ω0 − 8∕27ϕ000 2ω0 , D3 1∕4ϕ ω 0 − ϕ 2ω0 . 2ω0 Overall our approach has conceptual similarities to our recent concept targeting phase resolution within the 2ω spectrum [11,12]. In contrast to the present technique, however, it is not capable of retrieving the spectral phase within the ω pulse. We now apply the above interferometric technique for the phase-sensitive characterization of chirped OPA pulses. To this end, we insert SF11 (Infrasil) glass slabs of different thicknesses and impose a positive (negative) GVD. Values of ϕ00 ω0 620 fs2 ∕cm (−160 fs2 ∕cm) are expected from their Sellmeier equations. Figure 3(b) depicts the spectral phase of FFTs of exemplary current interferograms Iτ for unaltered, positively and negatively chirped pulses. The linear contributions to the spectral phase have been subtracted as they only reflect temporal delays owing to the refractive indices. The phase within the fundamental peak around ω0 depends pronouncedly on the material introduced into the beam path. In comparison to the reference measurement without additional material, the SF11 (Infrasil) adds a substantial positive (negative) phase curvature. In contrast, the spectral phase of the peak around 2ω0 depends little on the introduced material. This finding is in line with the picture put forward in Eq. (7) since frequency doubling of a chirped pulse implies ϕ00 2ω0 1∕2ϕ00 ω0 [10]. For a more quantitative analysis, we apply third-order polynomial fits to the spectral phases that are shown in Fig. 3(b). The extracted phase curvatures together with Eq. (6) and (7) finally determine the GVDs ϕ00 ω0 and ϕ00 2ω0 . We note that we do not further analyze the third-order terms here because they are small for the present configurations. To provide a benchmark comparison to an established phase-retrieval scheme, we also perform phase retrieval with second harmonic FROG [2]. The setup uses a 0.5 mm BBO and a fiber-coupled spectrometer. We use the freely available FROG code [13] to reconstruct the spectral phase of the pulse and to extract the GVD from its curvature. Because of the time-reversal ambiguity, the sign of the GVD is unknown a priori. Figure 4 depicts the final result for the measured GVD ϕ00 ω0 for various chirp configurations as imposed by SF11 and Infrasil glass slabs of different thicknesses. The error bars are related to the scatter of three independent sets of measurements. The abscissa is calibrated using the corresponding Sellmeier equations. The FROG data nicely reveals the theoretically expected GVD even though our FROG setup is not particularly sensitive in unveiling very small GVDs. The GVDs extracted with the coherent control technique also agree very well with the theoretical expectation and show a similar small scatter around the dashed line. We note that the analysis of the spectral phase of the FFT of Iτ utilizes Eqs. (6) and (7). For the most strongly chirped configurations presented in Fig. 4 the limit jϕ00 ω0 j ≪ tp;ω 2 starts to break down. Here the simplified analysis of the coherent control data suggests a somewhat smaller absolute value of the GVD compared to the theoretical expectation. This finding is in line with the more general analysis seen in
June 15, 2014 / Vol. 39, No. 12 / OPTICS LETTERS
spectrally flat nonlinear response [8,16], our scheme is useful for pulses as short as a few optical cycles as long as ℏω < E G < 2ℏω is satisfied for the entire spectrum. Also weak (sub-nJ) pulses of ∼100 fs duration still yield detailed interferograms useful for sensitive phase retrieval [11].
coherent control setup FROG theoretical expectation with -50 fs2 offset
measured GVD ''(
0
) (fs2)
1000
3657
500
This work is supported by the DFG within the priority program SPP1391 “Ultrafast Nanooptics.” We thank T. Jostmeier and J. Lohrenz for useful discussions. M. B. and C. R. acknowledge financial support from the Alexander von Humboldt Foundation.
0
-500
-500
0 500 theoretical GVD ''( 0) (fs2)
1000
Fig. 4. Results for the GVD ϕ00 ω0 measured by the coherent control technique (red symbols) and by FROG (blue symbols) for different chirp configurations. The abscissa is calibrated by the Sellmeier equations for SF11 and Infrasil. The dashed line shows the theoretical expectation assuming that the unaltered OPA output has a residual GVD of ϕ00 ω0 −50 fs2 .
Fig. 3(a). In fact, an evaluation of the observed Dω0 with the red dashed line in Fig. 3(a) improves the agreement with the expected GVD. In conclusion, we have developed a new phaseretrieval scheme for ultrashort laser pulses. It relies on second harmonic generation in combination with phasedependent current injection driven by two time-delayed portions of the emerging ω∕2ω pulse pair. The phase information can be extracted from a simple Fourier transformation of a one-dimensional current interferogram. These ideas are tested for different chirp configurations of 45 fs pulses centered at 1.45 μm wavelength. A direct comparison to FROG indicates a similar sensitivity to the pulse’s GVD. Our scheme is applicable to a wide range of optical wavelengths and pulse durations. As an example, CdSe is an efficient detector crystal [14] and has a bandgap convenient for characterizing Ti:sapphire sources. The small-bandgap materials InAs and InSb are appropriate to analyze mid-IR pulses and have been predicted to exhibit a huge current injection nonlinearity [15]. Since current injection offers an ultrabroadband and
References 1. D. J. Kane and R. Trebino, Opt. Lett. 18, 823 (1993). 2. R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, B. A. Richman, and D. J. Kane, Rev. Sci. Instrum. 68, 3277 (1997). 3. C. Iaconis and I. A. Walmsley, Opt. Lett. 23, 792 (1998). 4. P. O’Shea, M. Kimmel, X. Gu, and R. Trebino, Opt. Lett. 26, 932 (2001). 5. P. Baum, S. Lochbrunner, and E. Riedle, Opt. Lett. 29, 210 (2004). 6. A. Pasquazi, M. Peccianti, Y. Park, B. E. Little, S. T. Chu, R. Morandotti, J. Azana, and D. J. Moss, Nat. Photonics 5, 618 (2011). 7. Y. Nomura, H. Shirai, and T. Fuji, Nat. Commun. 4, 2820 (2013). 8. R. Atanasov, A. Haché, J. L. P. Hughes, H. M. van Driel, and J. E. Sipe, Phys. Rev. Lett. 76, 1703 (1996). 9. A. Haché, Y. Kostoulas, R. Atanasov, J. L. P. Hughes, J. E. Sipe, and H. M. van Driel, Phys. Rev. Lett. 78, 306 (1997). 10. E. Sidick, A. Dienes, and A. Knoesen, J. Opt. Soc. Am. B 12, 1713 (1995). 11. S. Thunich, C. Ruppert, A. W. Holleitner, and M. Betz, Opt. Lett. 36, 1791 (2011). 12. C. Ruppert, J. Lohrenz, S. Thunich, and M. Betz, Opt. Lett. 37, 3879 (2012). 13. Code for Retrieving a Pulse Intensity and Phase from Its FROG Trace, available from the Trebino group at http:// frog.gatech.edu/code.html. 14. N. Laman, A. I. Shkrebtii, J. E. Sipe, and H. M. van Driel, Appl. Phys. Lett. 75, 2581 (1999). 15. M. Sheik-Bahae, Phys. Rev. B 60, 11257 (1999). 16. L. Costa, M. Betz, M. Spasenović, A. D. Bristow, and H. M. van Driel, Nat. Phys. 3, 632 (2007).
OPTICS LETTERS / Vol. 39, No. 12 / June 15, 2014
Phase-retrieval of femtosecond pulses utilizing ω/2ω quantum interference control of electrical currents Elmar Sternemann,1 Markus Betz,1,* and Claudia Ruppert1,2 1
2
Experimentelle Physik 2, TU Dortmund, 44227 Dortmund, Germany Department of Physics, Columbia University, New York, New York 10027 USA *Corresponding author: markus.betz@tu‑dortmund.de Received April 29, 2014; revised May 9, 2014; accepted May 10, 2014; posted May 13, 2014 (Doc. ID 210733); published June 12, 2014
We propose and implement a versatile scheme to analyze the phase structure of femtosecond pulses. It relies on second harmonic generation in combination with phase-sensitive χ 3 -current injection driven by two time-delayed portions of the emerging ω∕2ω pulse pair. Most strikingly, the group velocity dispersions of both the ω and 2ω components can be unambiguously determined from a simple Fourier transformation of the resulting current interferogram. We test the concept for 45 fs pulses at 1.45 μm and directly compare it to second harmonic frequency resolved optical gating. By choosing appropriate frequency doublers and semiconductor detectors, the scheme is applicable to many wavelength regimes and to pulses as short as a few optical cycles. © 2014 Optical Society of America OCIS codes: (030.1670) Coherent optical effects; (190.7110) Ultrafast nonlinear optics. http://dx.doi.org/10.1364/OL.39.003654
The characterization of ultrashort laser pulses typically relies on self-referencing approaches. The rapid development of increasingly sophisticated femtosecond sources has constantly triggered the search for new pulse characterization techniques that—in addition to a pulse’s spectrum and temporal envelope—also resolve its spectral phase. The most advanced concepts for this ultrafast pulse metrology are provided by frequency resolved optical gating (FROG) [1,2], spectral phase interferometry for direct electric-field reconstruction (SPIDER) [3], and related sophisticated methods, examples of which are reported in Refs. [4–7]. The joint aspect of those concepts is the analysis of an optical spectrum emerging from the nonlinear optical interaction of two portions of the laser pulse train. A numerical algorithm then extracts the spectral amplitude and phase with no or little ambiguity. However, the underlying optical nonlinearity—in many cases sum-frequency generation— depends on the momentary light intensity rather than on optical phases per se. In contrast, certain optical nonlinearities such as optical rectification or, more generally, harmonic mixing phenomena, are directly sensitive to phase differences of interacting optical beams. In particular, a superposition of an ultrashort fundamental pulse (ω) and its second harmonic (2ω) incident on a semiconductor induces strong lateral electrical currents with a vector direction governed by the pulses’ polarizations and the relative phases [8,9]. This third-order optical nonlinearity arises from a quantum interference of one- and two-photon absorption pathways. While its phase dependence is well known, its potential for phase retrieval has barely been appreciated to date. In this Letter, we conceive a versatile concept to characterize femtosecond optical pulses based on a ω∕2ω coherent control approach. After standard second harmonic generation, the emerging ω∕2ω beam is split up and superimposed again to result in two time-delayed collinear beams. While one part consists of the entire ω∕2ω spectrum, the other one is filtered to contain the ω part only. This pulse sequence induces a coherently controlled current in a semiconductor detector. Governed by the 0146-9592/14/123654-04$15.00/0
scaling of the injection process, interferograms obtained by varying the relative temporal delay are modulated by both ω and 2ω carrier frequencies. In essence, a Fourier transformation then extracts important information about, e.g., the group velocity dispersion (GVD) of both ω and 2ω spectral components without a time-reversal ambiguity. This concept is demonstrated by characterizing the 45 fs output of an optical parametric amplifier (OPA) operating at 1.45 μm. A direct comparison to FROG indicates a similar sensitivity to GVDs. However, our scheme neither requires spectral resolution nor a phase-retrieval algorithm. Instead, we acquire a one-dimensional data set and use fast Fourier transformation (FFT) to obtain detailed amplitude and phase information. The scheme for pulse characterization is exemplarily realized for the output of a coherent OPA 9850. It operates at a 1.45 μm central wavelength and delivers 20 mW of tp;ω 45 fs pulses (FWHM) at a repetition rate of 250 kHz. The pulses are frequency-doubled in a 0.5 mm thick BBO crystal. As shown in Fig. 1, the ω∕2ω pulse pair is passed through a Michelson interferometer. In one arm, a low-dispersive, 140 μm thick Si-wafer blocks the second harmonic while a λ∕2 plate tilts the remaining fundamental E 2 ω to horizontal (x) polarization. The second arm contains an unaltered pulse pair consisting of the y-polarized fundamental E 1 ω and the x-polarized second harmonic E 2ω . A motorized stage precisely controls the relative delay τ of the two arms. Note that we use reflective optics and a pellicle beamsplitter to minimize dispersion. The pulse combination emerging from the interferometer is focused onto a contacted, low temperature grown GaAs specimen at room temperature using a f 25 mm parabolic mirror. Its bandgap E G satisfies ℏω < E G < 2ℏω such that these ω∕2ω pulses induce substantial currents. In a contacted device, they lead to a conveniently measurable electrical signal reflecting the optically induced charge separation [9]. More specifically, we use two gold contacts spaced ∼10 μm apart and tilted by ∼45° in the x–y plane (cf. Fig. 1). Figure 2(a) depicts an example for the τ-dependent photoresponse. It shows an interferogram with a Gaussian envelope that © 2014 Optical Society of America
June 15, 2014 / Vol. 39, No. 12 / OPTICS LETTERS
3655
↔
Fig. 1. Experimental scheme: A 0.5 mm thick BBO generates the second harmonic (blue). An interferometer controls the delay τ of the fundamental portion E 2 ω (red) relative to a pulse pair composed of the remaining fundamental (E 1 ω ) and the second harmonic E 2ω . The emergent beam is focused onto a LT-GaAs sample with gold electrodes spaced 10 μm apart. A lock-in amplifier measures the photoresponse. Inset: geometry of the contacts and the optical polarizations.
arises from the temporal convolution of the interacting pulses. As also evident from the FFT in Fig. 2(b), it is strongly modulated with two carrier waves ω0 and 2ω0 , i.e., the pulses’ center frequencies. We will illustrate below that those two modulation frequencies arise from different beam combinations for current injection and their spectral phases ultimately reveal the phase structure of both ω and 2ω pulses. Note that the Si-filter in the interferometer effectively suppresses any disturbing interference of two 2ω pulses. To elucidate the theoretical background of phase retrieval, we consider a superposition of ω and 2ω pulses incident on a semiconductor satisfying ℏω < E G < 2ℏω. It induces a current with an injection rate of [8]: _ ↔ ⃗ ⃗ ⃗ J⃗ η E ω Eω E2ω ;
current I (nA)
0.3
(1)
(a)
0.0 -0.3 -0.6
norm. FFT amp.
-100 -80 -60 -40 -20 0 20 40 pulse delay (fs) 0.80 1.0
0.84
0.88
energy 0.92
(eV) 1.68
60
1.72
80 100
1.76
(b)
0.5 0
2
0
0.0 1.25 1.30 1.35 1.40 2.55 2.60 2.65 2.70 frequency (1015 Hz)
Fig. 2. (a) Current interferogram Iτ recorded for the 45 fs pulse at 1.45 μm. (b) FFT amplitude of Iτ. ω0 and 2ω0 are the center frequencies of the ω and 2ω pulses.
where η is a fourth-rank tensor proportional to the imaginary part of the third-order susceptibility χ 3 0; ω; ω; −2ω. Note that we work with complex fields ⃗ ω . Physically and denote the complex conjugate as, e.g., E relevant quantities are given by the real part. As a result of Eq. (1), current injection driven by two ω and 2ω monochromatic waves with phases φω;2ω scales as J_ ∝ E 2ω E 2ω sin2φω − φ2ω . For the present pulsed configuration currents are induced with one y-polarized 2 field E 1 ω t τ and two x-polarized fields E ω t and E 2ω t τ with relative timing τ. For a zinc blende crystal, such as GaAs, the largest elements of the current injection tensor are ηxxxx ≈ 2η↔yyxx [8]. We implicitly assume a frequency independent η and crystallographic axes aligned to the lab frame. While this geometry is not taken care of, deviations do not affect the underlying principles here. We now restrict the consideration to the two current contributions that depend on the relative timing τ. They read: 2 J_ x ηxxxx E 2 ω tE ω t E 2ω t τ;
(2)
2 J_ y ηyyxx E 1 ω t τE ω t E 2ω t τ.
(3)
Intuitively, the term in Eq. (2) reflects a situation where the two ω driving fields in Eq. (1) are derived from the same interferometer arm. In contrast, the two ω fields originate from different interferometer arms to yield the current in Eq. (3). Current detection is performed in a time-integrating fashion, i.e., by analyzing charge accumulation at contacts on the semiconductor specimen [9]. In the experiment, we interferometrically scan the delay τ between the two interferometer arms and detect signatures Iτ of the coherently controlled current. Therefore, we exR R pect signals I x τ ∝ J_ x dt and I y τ ∝ J_ y dt. As the detector is tilted in the x–y plane, we do not separately measure I x τ and I x τ but a superposition Iτ. Due to the different τ-dependence, the interferograms I x τ and I y τ feature carrier frequencies 2ω0 and ω0 , respectively, as seen in Fig. 2. We now analyze how phase distortions of the ω and 2ω pulses manifest in the interference pattern Iτ and its ˆ FFT Iω. The above expressions I x τ and I y τ represent different field convolutions of the driving ω∕2ω pulse combinations. Utilizing the convolution theorem their spectrum can be expressed by the Fourier transformations of the ω∕2ω fields (⊗ denotes a convolution): ˆ ˆ 2 Iˆ x ω ∝ Eˆ 2 ω ω ⊗ E ω ω · E 2ω ω;
(4)
ˆ 1 ˆ Iˆ y ω ∝ Eˆ 2 ω ω · E ω −ω ⊗ E 2ω ω.
(5)
Such convolutions can be evaluated analytically for transform-limited and linearly chirped Gaussian pulses. Higher order dispersions are incorporated numerically. The result of such a computation is a relation between the spectral phases of the incoming ω∕2ω pulses and ˆ the phase properties of Iω. Most importantly, we find GVDs ϕ00 ω0 and ϕ00 2ω0 to manifest as corresponding FFT phase curvatures Dω0 and D2ω0 at the respective
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OPTICS LETTERS / Vol. 39, No. 12 / June 15, 2014
center frequencies ω0 and 2ω0 . It is instructive to first analyze the situation of moderate chirp jϕ00 ω0 j ≪ tp;ω 2 2tp;2ω 2 where tp;ω∕2ω denotes the durations of the unchirped, transform-limited pulses. In this limit, the phase curvatures Dω0 and D2ω0 are linked to the GVDs by Dω0
10 00 4 ϕ ω0 − ϕ00 2ω0 ; 9 9
(6)
1 D2ω0 ϕ00 ω0 − ϕ00 2ω0 . 2
(7)
0
curvature D (fs2)
A further simplification is achieved by assuming ϕ00 2ω0 1∕2ϕ00 ω0 typical of second harmonic generation in a thin frequency doubler [10]. In this situation, Eqs. (6) and (7) imply Dω0 8∕9ϕ00 ω0 and D2ω0 0. The more general setting of arbitrary (linear) chirp can still be treated analytically. However, the results for Dω0 and D2ω0 constitute lengthy expressions that explicitly depend on the pulse durations tp;ω and tp;2ω . The full result for Dω0 as a function of the GVD ϕ00 ω0 is shown in exemplary pulse durations tp;ω Fig. 3(a) forptwo and tp;2ω 1∕ 2tp;ω . We also include the expectation Dω0 8∕9ϕ00 ω0 as a solid line for comparison. Most importantly, we see a clear one-to-one correspondence between ϕ00 ω0 and the phase curvature Dω0 , i.e., a quanˆ tity directly accessible from the FFT Iω of the interferogram Iτ. It is also possible to reveal departures from ϕ00 2ω0 1∕2ϕ00 ω0 by determining both Dω0 and D2ω0 and evaluating the outcome with Eqs. (6) and (7). Sufficient knowledge about tp;ω∕2ω can be extracted from conventional optical spectra under the assumption of transform-limited pulses. In principle, similar relations 3 exist between the coefficients D3 ω0 , D2ω0 of the cubic 1000
(a)
500 0
tp, → ∞ tp, = 45 fs tp, = 30 fs
-500
-1000
phase of fourier spectrum (rad)
-1500
-1000
(b)
-500 0 500 1000 theoretical GVD ''( 0) (fs2) 0
2
1500
0
0.1
0.0 no material SF11 5.3 mm Infrasil 10 mm
-0.1 1280
1300 1320 2560 2580 frequency (THz)
2600
2620
Fig. 3. (a) Theoretical phase curvature Dω0 as a function of the GVD ϕ00 ω0 . Solid line: expectation Dω0 8∕9ϕ00 ω0 valid for long pulses (tp → ∞). Dashed lines: full result for two exemplary tp;ω assuming ϕ00 2ω0 1∕2ϕ00 ω0 and tp;2ω p 1∕ 2tp;ω . (b) Spectral phases of the FFT of Iτ-traces for pulses of different chirp. Positive (negative) chirp is introduced by 5.3 mm SF11 (10 mm Infrasil). The lines are polynomial fits.
phase terms and the p third-order dispersion. For weak chirp and tp;2ω 1∕ 2tp;ω they read D3 ω0 26∕27 000 000 ϕ000 ω0 − 8∕27ϕ000 2ω0 , D3 1∕4ϕ ω 0 − ϕ 2ω0 . 2ω0 Overall our approach has conceptual similarities to our recent concept targeting phase resolution within the 2ω spectrum [11,12]. In contrast to the present technique, however, it is not capable of retrieving the spectral phase within the ω pulse. We now apply the above interferometric technique for the phase-sensitive characterization of chirped OPA pulses. To this end, we insert SF11 (Infrasil) glass slabs of different thicknesses and impose a positive (negative) GVD. Values of ϕ00 ω0 620 fs2 ∕cm (−160 fs2 ∕cm) are expected from their Sellmeier equations. Figure 3(b) depicts the spectral phase of FFTs of exemplary current interferograms Iτ for unaltered, positively and negatively chirped pulses. The linear contributions to the spectral phase have been subtracted as they only reflect temporal delays owing to the refractive indices. The phase within the fundamental peak around ω0 depends pronouncedly on the material introduced into the beam path. In comparison to the reference measurement without additional material, the SF11 (Infrasil) adds a substantial positive (negative) phase curvature. In contrast, the spectral phase of the peak around 2ω0 depends little on the introduced material. This finding is in line with the picture put forward in Eq. (7) since frequency doubling of a chirped pulse implies ϕ00 2ω0 1∕2ϕ00 ω0 [10]. For a more quantitative analysis, we apply third-order polynomial fits to the spectral phases that are shown in Fig. 3(b). The extracted phase curvatures together with Eq. (6) and (7) finally determine the GVDs ϕ00 ω0 and ϕ00 2ω0 . We note that we do not further analyze the third-order terms here because they are small for the present configurations. To provide a benchmark comparison to an established phase-retrieval scheme, we also perform phase retrieval with second harmonic FROG [2]. The setup uses a 0.5 mm BBO and a fiber-coupled spectrometer. We use the freely available FROG code [13] to reconstruct the spectral phase of the pulse and to extract the GVD from its curvature. Because of the time-reversal ambiguity, the sign of the GVD is unknown a priori. Figure 4 depicts the final result for the measured GVD ϕ00 ω0 for various chirp configurations as imposed by SF11 and Infrasil glass slabs of different thicknesses. The error bars are related to the scatter of three independent sets of measurements. The abscissa is calibrated using the corresponding Sellmeier equations. The FROG data nicely reveals the theoretically expected GVD even though our FROG setup is not particularly sensitive in unveiling very small GVDs. The GVDs extracted with the coherent control technique also agree very well with the theoretical expectation and show a similar small scatter around the dashed line. We note that the analysis of the spectral phase of the FFT of Iτ utilizes Eqs. (6) and (7). For the most strongly chirped configurations presented in Fig. 4 the limit jϕ00 ω0 j ≪ tp;ω 2 starts to break down. Here the simplified analysis of the coherent control data suggests a somewhat smaller absolute value of the GVD compared to the theoretical expectation. This finding is in line with the more general analysis seen in
June 15, 2014 / Vol. 39, No. 12 / OPTICS LETTERS
spectrally flat nonlinear response [8,16], our scheme is useful for pulses as short as a few optical cycles as long as ℏω < E G < 2ℏω is satisfied for the entire spectrum. Also weak (sub-nJ) pulses of ∼100 fs duration still yield detailed interferograms useful for sensitive phase retrieval [11].
coherent control setup FROG theoretical expectation with -50 fs2 offset
measured GVD ''(
0
) (fs2)
1000
3657
500
This work is supported by the DFG within the priority program SPP1391 “Ultrafast Nanooptics.” We thank T. Jostmeier and J. Lohrenz for useful discussions. M. B. and C. R. acknowledge financial support from the Alexander von Humboldt Foundation.
0
-500
-500
0 500 theoretical GVD ''( 0) (fs2)
1000
Fig. 4. Results for the GVD ϕ00 ω0 measured by the coherent control technique (red symbols) and by FROG (blue symbols) for different chirp configurations. The abscissa is calibrated by the Sellmeier equations for SF11 and Infrasil. The dashed line shows the theoretical expectation assuming that the unaltered OPA output has a residual GVD of ϕ00 ω0 −50 fs2 .
Fig. 3(a). In fact, an evaluation of the observed Dω0 with the red dashed line in Fig. 3(a) improves the agreement with the expected GVD. In conclusion, we have developed a new phaseretrieval scheme for ultrashort laser pulses. It relies on second harmonic generation in combination with phasedependent current injection driven by two time-delayed portions of the emerging ω∕2ω pulse pair. The phase information can be extracted from a simple Fourier transformation of a one-dimensional current interferogram. These ideas are tested for different chirp configurations of 45 fs pulses centered at 1.45 μm wavelength. A direct comparison to FROG indicates a similar sensitivity to the pulse’s GVD. Our scheme is applicable to a wide range of optical wavelengths and pulse durations. As an example, CdSe is an efficient detector crystal [14] and has a bandgap convenient for characterizing Ti:sapphire sources. The small-bandgap materials InAs and InSb are appropriate to analyze mid-IR pulses and have been predicted to exhibit a huge current injection nonlinearity [15]. Since current injection offers an ultrabroadband and
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