Revealing carrier-envelope phase through frequency mixing and interference in frequency resolved optical gating

E. W. Snedden; D. A. Walsh; S. P. Jamison

doi:10.1364/OE.23.008507

1. Introduction

Few-cycle electromagnetic pulses offer a means of generating and controlling physical processes on sub-femtosecond timescales relevant in atomic and condensed matter physics. The characterisation of such short pulses remains a crucial aspect of the experimental process and has a rich history [1–3]. Of the many techniques available today, Frequency Resolved Optical Gating (FROG) has found widespread application due to its robustness and relatively simple experimental implementation. An expansive array of related FROG techniques have been developed in the last two decades [4–8]; each offers specific advantages in varying experimental conditions, yet share a common mathematical and physical basis: the measurement of an intensity spectrogram from a non-linear combination of pulses. This is widely held to make determination of the carrier-envelope phase (CEP, ϕ^CE) beyond FROG techniques [2,9–12].

The value of ϕ^CE is equal to the zero-order term in a Taylor expansion of spectral phase at positive frequencies [6]. For electromagnetic pulses with many cycles, ϕ^CE can be identified with a simple time shift of the carrier with respect to the temporal envelope. For the shortest of pulses, such as single-cycle THz pulses, the separation of carrier and envelope is no longer appropriate and the intuitive understanding of ϕ^CE as a time-shift breaks down. In such cases ϕ^CE instead plays a more fundamental role in defining the absolute electric field temporal profile.

For repetitive pulse trains possessing an underlying frequency comb, the beating between fundamental and second harmonic waves in a f-2f interferometer reveals changes in ϕ^CE with time (∂ϕ^CE/∂t) [13–16]. While widely and successfully used in oscillator CEP stabilisation schemes, such an approach does not reveal the absolute phase of a stabilised system [3, 16]. A distinct approach for CEP control is employed in low repetition rate amplified systems [17,18]; by spectrally interfering the fundamental and second harmonic (or higher harmonics [19]), the spectral intensity in the region of overlap is dependent on the CEP. Quantitative characterisation of the CEP can be performed by applying a chirp to the interfering pulses, through which the CEP is revealed as spectral fringes. This method enables single-shot detection capability and current state-of-the-art techniques are able to stabilise CEP to better than 200 mrad [20]. The requirement for chirp in the measured pulses – as opposed to a general desire for near transformlimited pulses for experimental exploitation – together with challenges in the calibration of dispersion associated with the measurement process [17] restricts the wide application of f-2f interferometry to one of CEP stabilisation and control.

In current ultrafast metrology, full temporal characterisation of the electric field typically requires a separate and often complex measurement of CEP in addition to information such as that provided by FROG. Stereo-ATI has been used to measure CEP by measuring the direction of photoelectrons emitted following ionisation in a time-of-flight spectrometer [9,16,21,22]. CEP measurement in also directly achieved in so-called attosecond streaking [23,24], which incorporates the spectral analysis of photoelectrons excited by sub-femtosecond high harmonic pulses. In this latter case the principles of FROG have been applied to the emitted photoelectrons (as opposed to the input electromagnetic pulse as per usual) in the FROG-CRAB algorithm [25].

Following the definition of ϕ^CE as the zero-order term in an expansion of spectral phase, if ϕ^CE is determined at one frequency it is therefore known uniquely. Recently, Nomura [10] and Shira [11] have utilised this, determining ϕ^CE at low frequencies through THz electrooptic sampling (EOS) techniques, obtaining complete pulse information when combined with traditional FROG techniques.

Here we extend FROG retrieval to encompass conditions of spectral overlap between harmonic components, including those used in f-2f interferometry. Through this we demonstrate that self and cross-referenced FROG techniques can be extended to be directly capable of full characterisation of the electric field temporal profile of few-cycle electromagnetic pulses including CEP. By directly resolving the interference between harmonic pulse replicas in a FROG spectrogram, we theoretically and experimentally demonstrate absolute electric field retrieval, even in cross-referenced cases when the higher-frequency probe is of similar duration and unknown carrier-envelope phase. There is no requirement for the to-be-determined pulse to contain low-frequency content accessible to δ-function sampling, or for the interfering pulses to be chirped.

2. ReD-FROG: theory and retrieval algorithm

FROG for electromagnetic pulse characterisation consists of a spectrally resolved auto- or cross-correlation of pulses mediated by a non-linear mixing process. Herein the detailed discussion assumes second-order mixing through χ⁽²⁾, although the approach and results can be readily modified to higher order mixing. For observable, real, electric fields in the time domain, E₁(t), E₂(t) the complex spectra ${\tilde{E}}_{1} (ω)$ , ${\tilde{E}}_{2} (ω)$ are defined from the Fourier transform, introducing the mathematical construct of negative frequencies. For the χ⁽²⁾ interaction between pulses the spectrogram, the spectrum of the χ⁽²⁾ generated field as a function of relative delay τ between input pulses can be given as [26]:

I (ω; τ) = {| \tilde{R} (ω) \int_{- \infty}^{\infty} d Ω {\tilde{E}}_{1} (ω - Ω) {\tilde{E}}_{2} (Ω) \exp (i Ω τ) |}^{2},

in which we take the common approximation that the linear and non-linear material response functions can be collected outside the integral in the response function R(ω) [12, 26]. Techniques such as surface harmonic generation [27, 28] or phase-match angle-dithering [29] are capable of very high bandwidth response, as would be necessary for few cycle pulses. As the specific form of R(ω) does not affect either analysis or conclusion it is omitted from the following discussion.

The carrier envelope phase ϕ^CE is the phase contribution that is widely accepted to be unmeasurable through FROG phase retrieval; to highlight conditions for observation and retrieval, ϕ^CE is explicitly separated out from the total spectral phase. All electric fields are explicitly constrained to be real in the time domain, which gives:

{\tilde{E}}_{total} (ω) = {\begin{array}{l} \tilde{E} (ω) \exp (i ϕ^{C E}) & ; ω > 0 \\ \tilde{E} * (| ω |) \exp (- i ϕ^{C E}) & ; ω < 0 \end{array}

The imposition of the physical constraint for all fields to be purely real in the time domain and in particular the Heaviside functional form of ϕ^CE(ω) gives rise to the observability of CEP. Introducing the functional form of Eq. (2) for the electric fields, the spectrogram can be expressed as:

\begin{array}{l} I (ω; τ) = | S F G (ω; τ) |^{2} + | D F G_{+} (ω; τ) |^{2} + | D F G_{-} (ω; τ) |^{2} \\ + 2 ℜ {S F G (ω; τ) D F G_{+}^{*} (ω; τ) e^{i 2 ϕ_{2}^{C E}}} \\ + 2 ℜ {S F G (ω; τ) D F G_{-}^{*} (ω; τ) e^{i 2 ϕ_{1}^{C E}}} \\ + 2 ℜ {D F G_{+} (ω; τ) D F G_{-}^{*} (ω; τ) e^{i 2 ϕ_{1}^{C E} - i 2 ϕ_{2}^{C E}}} \end{array}

where

\begin{array}{l} S F G (ω; τ) \equiv \int_{0}^{ω} d Ω {\tilde{E}}_{1} (ω - Ω) {\tilde{E}}_{2} (Ω; τ), \\ D F G_{-} (ω; τ) \equiv \int_{0}^{\infty} d Ω {\tilde{E}}_{1}^{*} (Ω - ω) {\tilde{E}}_{2} (Ω; τ) \\ D F G_{+} (ω; τ) \equiv \int_{0}^{+ \infty} d Ω {\tilde{E}}_{1} (ω + Ω) {\tilde{E}}_{2}^{*} (Ω; τ) . \end{array}

The contributions arising from sum-frequency and difference frequency generation are represented by SFG and (DFG₊,DFG₋) respectively. When both sum and difference frequency contributions spectrally overlap the absolute carrier phase becomes observable.

In this work we address two specific cases; that of a single broad-band pulse cross-referenced with itself, and, as part of a proof-of-concept demonstration, the cross-correlation of two spectrally distinguishable pulses. The first case applies to the measurement of few-cycle optical pulses; the second case describes the unambiguous determination of the electric field of a pulse through sampling with another (unknown) optical field, where the sampling pulse may be of similar or longer duration than the lower frequency pulse. Representative field amplitude spectra of these two cases and the relative extent of the sum and difference frequency terms are shown schematically in Fig. 1. The principles discussed with these example can be readily extended to address other specific spectral considerations, such as spectrally overlapping pulses.

Fig. 1 Schematic demonstrating bandwidth conditions for carrier-envelope phase observation, for the two examples discussed in the text. (a) Self referenced mixing of a few cycle pulse: (top) spectrum for input pulse; (bottom) the sum and difference spectra. DFG₊ = DFG₋ for this configuration. (b) Cross-correlation of spectrally distinct pulses: (top) input spectra; (bottom) frequency mixed output, for which DFG₋ ≡ 0.

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To demonstrate the complete electric field retrieval for the self-referenced measurement of a single pulse, test pulses with complex spectra differing only in carrier-envelope phase have been constructed, as shown in Figs. 2(c) and 2(f). Simulated ‘experimental’ spectrograms were constructed directly as the product of the time domain fields. Modified FROG retrieval algorithms, accounting for the constraint of a strictly real time domain field, are then applied to the experimental spectrograms. Retrieval has been examined in presence of both measurement noise and spectral truncation.

Fig. 2 Numerical demonstration of the self-referencing retrieval of a transform-limited single-cycle optical pulse (110 THz bandwidth Gaussian with peak centred at 375 THz). (a) Simulated experimental spectrogram with ϕ^CE = 0; (b) retrieved spectrogram, in which the retrieval has been performed over a truncated spectral region (frequencies above 375 THz) and (c) comparison of input and retrieved electric field profile. (d–f) as above, with ϕ^CE = π/2 rad. All spectrograms are shown with logarithmic intensity scale.

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Conventional FROG algorithms follow a two stage iterative procedure, the first being the comparison of the field amplitudes of the product spectrogram and the data. The second ‘numerical-constraint’ draws on the physics of the relevant frequency mixing process to extract the individual time domain fields from the field-product spectrogram. This constraint is conventionally undertaken from a functional minimisation between the intensity spectrogram and the expected spectrogram, inferred from iterated test fields and the physical process giving rise to frequency mixing.

We have modified a PCGPA FROG retrieval algorithm [30] to include the physical constraint of a real time-domain field, which together with the frequency-mixing spectral overlap, we label as ReD-FROG (Real Domain FROG) retrieval for distinction from conventional algorithms. The use of the PCGPA in providing the necessary deconvolution step decreases both the processing time and complexity of ReD-FROG, but is generally limited to measurements obeying Eq. (1). For more complex mixing schemes, for example instances in which dispersion of the mixing medium cannot be neglected, a more complex version of the deconvolution algorithm may be required [31]. The spectral constraint of real time-domain fields can be achieved through extending the measured spectrogram by mirror reflection about zero-frequency and applying the so-called numerical constraint minimisation procedure on the extended data set. Through effective inclusion of negative frequencies, and the pinning of the spectrogram to have mirror symmetry, the input field retrieved is constrained to be real in the time domain. In this case the mirror symmetry constrains the retrieved fields to have the Hermitian property necessary for the time domain fields to be purely real.

The retrieval algorithm was applied on a spectral truncation of the measurement data; such a spectral limited data set is to be expected in any practicable implementation. As shown in Figs. 2(c) and 2(f), the correct and full (including CEP) electric field temporal profile is recovered from the modified retrieval algorithm. All ReD-FROG retrievals were based on a 512×512 grid encompassing mirror reflection. For Figs. 2(c) and 2(f), FROG errors of 5×10⁻³ and 6 × 10⁻³ were respectively obtained. In the given case the to-be-determined pulse was centred at 375 THz, and only the spectrogram data for frequencies > 375 THz was included in the retrieval; to fully investigate the effect of truncation, retrieval was repeated and found to be effective for a range of cut-off frequencies between 325 and 375 THz. Similar agreement in the retrieved profile was obtained across the full range of cut-off frequencies investigated. For an unambiguous electric field determination it is sufficient to include the sum-frequency region of the spectrogram in the presence of some overlapping difference-frequency mixing spectral content. Stable convergence was recorded with and without use of a spectral constraint corresponding to a measurement of spectral intensity [6] applied over the truncated frequency range, although the rate of convergence was improved with its implementation.

Up to this point in our analysis we have taken the conventional form of the FROG spectrogram (Eq. (1)), in which the only the non-linear generated fields are included in the measurement and retrieval. In this arrangement the bandwidth (Δ ≡ ω^U −ω^L) necessary for carrier envelope observability is Δ ≥ 2ω^L. By inclusion of the input field in the measurement process, additional ϕ^CE-dependent interference terms arise which allow the bandwidth requirements to be relaxed. As a simplification we assume that only one of the input fields spectrally overlaps with the mixing fields; under these conditions the spectrogram can be expressed as:

I (ω; τ) = {| {\tilde{E}}_{1} (ω) + \int_{- \infty}^{\infty} d Ω {\tilde{E}}_{1} (ω - Ω) {\tilde{E}}_{2} (Ω) \exp (i Ω τ) |}^{2} .

Explicit introduction of the CEP and expansion of the integral (as performed for Eq. (3)) yields

\begin{array}{l} I (ω; τ) = {| {\tilde{E}}_{1} (ω) |}^{2} + {| S F G (ω; τ) |}^{2} + {| D F G_{+} (ω; τ) |}^{2} + {| D F G_{-} (ω; τ) |}^{2} \\ + 2 ℜ {{\tilde{E}}_{1} (ω) S F G^{*} (ω; τ) e^{- i ϕ_{2}^{C E}}} \\ + 2 ℜ {{\tilde{E}}_{1} (ω) D F G_{+}^{*} (ω; τ) e^{i ϕ_{2}^{C E}}} \\ + 2 ℜ {{\tilde{E}}_{1} (ω) D F G_{-}^{*} (ω; τ) e^{i 2 ϕ_{1}^{C E} - i ϕ_{2}^{C E}}} \\ + 2 ℜ {S F G (ω; τ) D F G_{+}^{*} (ω; τ) e^{i 2 ϕ_{2}^{C E}}} \\ + 2 ℜ {S F G (ω; τ) D F G_{-}^{*} (ω; τ) e^{i 2 ϕ_{1}^{C E}}} \\ + 2 ℜ {D F G_{+} (ω; τ) D F G_{-}^{*} (ω; τ) e^{i 2 ϕ_{1}^{C E} - i 2 ϕ_{2}^{C E}}} \end{array}

The additional interference terms in the above expression allow for carrier envelope phase observation even when there is no spectral overlap between sum and difference frequency mixing. This arrangement and the observed interference is closely related to the f-2f interferometry of amplified laser systems for CEP stabilisation. In the FROG configuration however it is no longer necessary for the pulses to have chirp; the CEP measurement is incorporated within a simultaneous measurement of higher-order spectral phase. The first two interference terms in Eq. (6) yield measurement of the phase $ϕ_{2}^{C E}$ through interference between either the fundamental field ${\tilde{E}}_{1}$ and the sum-frequency field, or between the fundamental and ${\tilde{E}}_{1}$ and difference frequency fields. The relative bandwidth requirements are shown schematically in Fig. 3(a); the carrier envelope phase is observable for a bandwidth satisfying Δ ≥ ω^L.

Fig. 3 (a) Schematic of bandwidth requirements for carrier phase observation through interference between fundamental and sum (or difference) frequency fields. (b,c) Spectrograms for a self-referencing measurement of a Gaussian transform-limited femtosecond pulse (375 THz carrier frequency, bandwidth 75 THz), ϕ^CE = 0 and ϕ^CE = π/2 rad respectively; (d,e) electric field temporal profile for for (b) and (c) respectively.

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Figure 3(b) presents an example of the spectrogram arising from a self-referencing measurement of a transform-limited Gaussian pulse (75 THz bandwidth) in which overlap with the input field occurs. The overlap between sum and difference frequency generated fields is absent in this case and the interference features arise from the first three terms in Eq. (6). In this example spectrogram of a transform-limited pulse the CEP is revealed through the oscillations along the delay time axis at frequencies centred on 375 THz. No oscillations can be observed along the frequency axis for any given time delay. If however a linear (or higher order) chirp is added to the pulses, the interference will be observed as a function of both frequency and time delay; with sufficient chirp a f-2f interference will be observed at a fixed delay, as described by existing f-2f interferometry schemes for amplified laser systems [17,18].

3. Experiment: characterisation of single-cycle THz radiation

We now consider the ReD-FROG process applied to the case of spectrally distinct pulses, as in Fig. 1(b). For this configuration DFG₋(ω;τ) ≡ 0, and therefore $ϕ_{1}^{C E}$ and $ϕ_{1}^{C E} - ϕ_{2}^{C E}$ are not observable. Conversely $ϕ_{2}^{C E}$ is observable and independent of the (unknown) CEP phase of the upper band pulse. To observe $ϕ_{2}^{C E}$ we require $Δ_{1} \equiv ω_{1}^{U} - ω_{1}^{L} > ω_{2}^{L}$ .

In this case a benchmarked proof-of-concept experiment is feasible; a quasi-single cycle THz pulse can serve as the lower frequency pulse to be determined, while an optical pulse of similar bandwidth acts as the higher frequency probe. For the benchmarking, a significantly broader bandwidth δ-function like optical pulse can be employed for electro-optic sampling (EOS) [32,33], obtaining an independent measure of the true electric field profile.

Such an experimental demonstration has been performed, with the experimental arrangement shown in Fig. 4. An amplified femtosecond Ti:Sapphire laser system (Coherent Micra/Legend, amplifier output: 45 fs, 1 mJ at 800 nm) was used to drive a large-area semi-insulating GaAs photo-conductive antenna, producing quasi-single cycle sub-ps THz pulses. The pulses were detected within the near-field region of the antenna, allowing the electric field oscillation to have a unipolar appearance. The optical probe pulse was selected prior to the PCA using a beam-splitter and directed through a zero-dispersion 4-f filter, allowing the pulse bandwidth to be selected as required. The benchmarking measurement of the temporal profile of the THz pulse was performed using a standard balanced detection EOS arrangement using a 45 fs optical probe [34]. For the spectrogram retrieval experiments the optical bandwidth was restricted to 1 THz, less than the full bandwidth of the THz pulse, with corresponding increase in duration to 500 fs. Optical probe temporal envelopes were confirmed with a custom-built SHG-FROG setup with 100 μm thick BBO.

Fig. 4 (a) Schematic of the experimental arrangement for carrier phase retrieval and benchmarking. (b) Measured spectrogram with a 1 THz bandwidth, 500 fs duration optical probe; (c) ReD-FROG retrieved spectrogram. (d) Simulated ReD-FROG spectrogram for THz pulse with $ϕ^{C E} = ϕ^{C E_{r e t r i e v e r}} + π / 2$ . (e) Comparison of the true electric field (determined by EOS with a 45 fs probe) and that from ReD-FROG retrieval. The EOS measurement for the spectrally narrowed 500 fs probe is also shown for comparison. (f) THz pulse with $ϕ^{C E} = ϕ^{C E_{r e t r i e v e r}} + π / 2$ corresponding to spectrogram (d). Spectrograms are shown with logarithmic intensity scale.

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Both optical and THz beams were focused into a $[\bar{1} 10]$ orientated ZnTe crystal. The frequency-mixing signal was isolated from residual input-probe light using a Glan-Thompson calcite polariser and spectrally analysed as a function of the relative delay between the THz and optical probe beams using an imaging spectrometer (Jobin Yvon, iHR550) and CCD camera (PCO DiCamPRO). A ReD-FROG algorithm as described earlier was applied to retrieve the field, using only the measured spectrogram as input.

The measured and retrieved spectrograms are shown in Figs. 4(b) and 4(c), together with the ReD-FROG retrieved field time profile in Fig. 4(e). The FROG error in the retrieved spectrogram (512×512 grid) was 0.01. Interference fringes arising from the overlap between SFG and DFG are observed in the measured spectrogram at frequencies close to the optical carrier frequency (375 THz) and are accurately replicated in the retrieved spectrogram. This fringe pattern is sensitive to the value of CEP; a simulated ReD-FROG spectrogram corresponding to the retrieved electric field but with an additional π/2 in CEP is presented in Fig. 4(d) for comparison.

Despite the complex form of the THz pulse, both the main peak and oscillations over a period of several picoseconds (due to THz absorption by ambient water vapour) are recovered, with the obtained value of $ϕ_{m e a s u r e d}^{C E} ≃ 2 °$ consistent with that of the input single-cycle pulse. For comparison the field profile inferred by direct EOS with the 1 THz bandwidth probe, without spectrally resolved phase retrieval, is also shown; as expected from an EOS measurement in which the optical bandwidth is not sufficiently larger than the THz, the method overestimates the pulse duration.

Whilst this experiment characterises low-frequency THz, for which alternative ultrafast sampling techniques are available, it explicitly confirms the validity of CEP retrieval using ReD-FROG. The approach may immediately be extended to the CEP characterisation of mid-IR pulses for which sampling may be infeasible. In the THz regime, ReD-FROG may also find immediate application in surpassing probe-duration limitations present in high time-resolution EOS. EOS conventionally requires an optical probe duration smaller than the shortest temporal feature in the THz pulse to be analysed; continuing the notation of section 2, these limitations can be expressed in the frequency domain as Δ≫ω^H. For example, EOS of THz pulse features 30 fs in duration to an accuracy better than 5 % requires use of a 10 fs optical sampling pulse; with ReD-FROG (Δ ≥ ω^L) the same THz features could be resolved with a 50 fs sampling pulse, which is significantly less challenging to generate and manipulate. We note that such THz sampling limitations are also overcome in a recently reported FROG-based scheme in which the use of cascaded non-linear interactions overcomes the need for direct FROG CEP retrieval [35].

4. Conclusions

We have demonstrated that a FROG measurement which resolves the superposition of nonlinear frequency mixing processes is capable of unambiguous electric field temporal characterisation, including the measurement of carrier envelope phase. Numerical simulations have been used to demonstrate the capability for few-cycle optical pulses. A proof-of-concept benchmarked experiment using single-cycle THz radiation was performed and confirmed the validity of the formalism. This method opens up new avenues of possibility in the measurement of few-cycle electric fields and demonstrates that, contrary to previous expectations, FROG is directly capable of measuring absolute phase. Potential applications for ReD-FROG in high time-resolution THz detection are also envisaged as a means to surpass the probe duration limitations of electro-optic sampling techniques.

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Revealing carrier-envelope phase through frequency mixing and interference in frequency resolved optical gating

Abstract

1. Introduction

2. ReD-FROG: theory and retrieval algorithm

3. Experiment: characterisation of single-cycle THz radiation

4. Conclusions

References and links

Cited By

Figures (4)

Equations (6)

Optics Express