1. Introduction

Since the original theoretical basis for optical parametric amplification in a three-wave mixing process was established over forty years ago [1 – 4], the conservation of photon energy in a parametric interaction has been accepted as true [5]. Experimental devices rely on the fact that the pump, signal and idler frequencies are related by ω_p = ω_s + ω_i , and laboratory measurements routinely authenticate this to the resolution of typical laboratory spectrometers (eg 0.01 - 0.1nm). The experimental verification of the energy conservation equation in an optical parametric oscillator (OPO), i.e. the accuracy of an OPO as an optical frequency divider, has been explored to a precision of 5×10^-18 [6], where a doubly-resonant OPO was operated in the frequency degenerate region and pumped by the second harmonic of a cw Nd:YAG laser with bandwidth of 5kHz. In that paper, the frequency differences between the signal, idler and the Nd:YAG laser were in the radio-frequency (RF) range, offering a heterodyne beat measurement scheme. In any OPO, both linear and nonlinear processes can introduce slight frequency shifts which may violate the energy conservation equation [7]. In particular, ultrashort pulses are liable to experience parasitic nonlinear processes that occur simultaneously with the main parametric three-wave interaction. These processes operate at the photon level in the nonlinear media, for example Raman scattering, Brillouin scattering and four-wave mixing, and may imbalance the energy conservation equation, while leaving the number of input and output photons unchanged. In this paper we describe an experimental verification of energy conservation in a free-running, singly-resonant femtosecond parametric oscillator with an optical frequency precision of approximately 200 kHz (< 10^-6 nm) which is made possible by comparing the frequency offsets of the frequency combs from the OPO with the broadband supercontinuum of the pump laser. In contrast to earlier work, this approach allows measurements to be recorded at any combination of signal and idler wavelengths available from the OPO and, in principle, by phase controlling both the pump laser and the OPO, a frequency precision at the sub-Hz level could be achieved.

2. Carrier-envelope phase-slip in optical parametric interaction

It is well known that pulses from a mode-locked laser have a spectrum comprising tens of thousands of individual modes which are uniformly separated at a frequency interval equal to the longitudinal mode spacing of the laser resonator, Δω. The offset of this comb of modes with respect to 0 Hz is determined by the difference between the intracavity phase and group velocities which results in the carrier wave slipping relative to the pulse envelope during each cavity roundtrip, as illustrated by Fig. 1(a). The rate of change of the carrier-envelope phase is the comb offset frequency and this can be precisely measured using a self-referencing nonlinear interferometer [8], and controlled using electronic feedback. By using a modelocked laser to synchronously pump a OPO [9] it is possible to create frequency combs in the near-infrared (signal output) and the mid-infrared (idler output) which have an intermode spacing identical to that of the pump laser and offset frequencies related by the energy conservation law [see Fig. 1(b)].

 

Fig.1. (a). Illustration of the change in the carrier-envelope phase between successive pulses (pump, signal or idler), and the relationship between the roundtrip carrier envelope phase-slip, Δϕ, and the carrier-envelope phase-slip frequency, Ω^CEP . (b). The carrier-envelope phase-slip frequencies of the pump, signal and idler pulses equal their comb-offset frequencies. The modes of these pulses (black lines) are illustrated relative to an idealized frequency scale with zero DC-offset and a scale spacing equal to the laser inter-mode frequency separation, Δω (grey lines). The pump offset frequency equals the sum of the offset frequencies of the signal and idler pulses. (c). Measurement of the pump, signal and idler carrier-envelope phase-slip frequencies is made using the pump super-continuum (grey lines) which itself has a DC-offset equal to Ω^CEP _p Beating the pump super-continuum against its own second-harmonic, the signal second-harmonic and the pump-idler sum-frequency light results in detectable beat frequencies at Ω^CEP _p , 2Ω^CEP _s −Ω^CEP _p and Ω^CEP _i respectively.

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The coupled-wave equations that are used to describe any second-order parametric interaction imply a fixed relationship between the phases of the pump (p), signal (s) and idler (i) waves according to [10]:

The time derivative of Eq. (1) can be applied to relate the rate of change of the carrier-envelope phase of the pump laser to those of the signal and idler pulses,

and so Eq. (2) can equally be interpreted as relating the comb offset frequencies of the pump, signal and idler pulses. The absolute mode frequencies of the pump, signal and idler pulses are therefore,

where j, k and l are integers used to identify an individual mode, and Δω is the intermode frequency interval. The parametric energy conservation law, ω_p = ω_s + ω_i, can therefore be verified by confirming experimentally that the carrier-envelope phase-slip frequencies of the pump, signal and idler pulses are related by Eq. (2). Equation (2) has been invoked in schemes for locking the carrier-envelope phase among the outputs of an OPO with a pump : signal : idler frequency ratio of 3:2:1 [11] and of an optical parametric amplifier seeded with white light [12]. Such demonstrations clearly support the relation, but to our knowledge there has been no direct and general measurement in which the pump, signal and idler carrier-envelope phase-slip frequencies have been simultaneously recorded using a common reference. The experimental tools for making such a measurement are readily available because techniques for the detection and control of the rate of phase-slip in the pulses emitted by a mode-locked laser [8] or femtosecond OPO [9, 13, 14] are well developed. In this work we have measured the pump, signal and idler comb offset frequencies using a common broadband super-continuum reference, which itself has a comb offset frequency equal to that of the pump pulses from which it is generated. Figure 1(c) illustrates the principle of this measurement.

3. Experimental configuration

The experimental system is illustrated in Fig. 2 and was based on a self-mode-locked Ti:sapphire laser that generated 50fs pulses with a repetition frequency of 200 MHz and an average power of 1.3W. The laser spectrum was centred at 800nm with a FWHM width of 20nm. The rate of carrier-envelope phase-slip in the pump laser was detected by using a self-referencing nonlinear interferometer [8] (Fig. 2, bottom left box). Using a beamsplitter, 250mW of the laser output was diverted into a 30cm section of photonic crystal fibre [15] (PCF; Crystal Fibre A/S, NL-2.0-740) and its spectrum broadened by super-continuum generation so that it spanned a spectral range from 500 − 1100nm. This super-continuum preserves the comb offset of the pump pulses, and by frequency doubling a portion of the super-continuum around 1040nm and beating this with components at 520nm we detected the comb offset frequency of the pump laser. Control of the phase-slip frequency was made by moving a fused-silica wedge of 1° apex angle into and out of the cavity beam [16].

The Ti: sapphire laser was used to synchronously pump a femtosecond OPO consisting of a 1mm long crystal (HCP Photonics) of magnesium-doped periodically-poled lithium niobate (MgO:PPLN) in a 5-mirror X-cavity (Fig. 2, top left box). The crystal had a grating period of 20.6μm and was operated at room temperature. A Gires-Tournois Interferometer (GTI) mirror coating with a group-delay dispersion of ∂ ² ϕ/∂ω ² = −200 fs² compensated for positive dispersion in the OPO cavity; all other cavity mirrors had high-reflectivity dielectric coatings centered at 1300nm. Cavity length tuning allowed the OPO to resonate at signal centre wavelengths from 1.25 − 1.34 μm. The average signal output power after the 2.5 % output coupler (OC) was 80mW and the output pulse durations were approximately 110fs. In addition to the signal and idler pulses, the OPO generated visible outputs by second-harmonic generation (SHG) and sum-frequency mixing (SFM) in the PPLN crystal. The strongest of these, the signal SHG (red) and the pump + idler SFM (yellow), had good spectral overlap with the PCF super-continuum spectrum and were used to measure the signal and idler comb offset frequencies as illustrated in Fig. 1(c). In order to do this a second interferometer was constructed (Fig. 2, right box). Because the red and yellow pulses left the OPO slightly separated in time, they were individually delayed in a prism-pair arrangement (P1, P2) to achieve temporal overlap with the super-continuum pulses at beam-splitter PBS2. The red and yellow components of the combined OPO and super-continuum beams were detected without filtering by a single avalanche photo-diode (APD 2; Becker & Hickl GmbH, APM-400N).

 

Fig. 2. Schematic of the experimental system. GTI, Gires-Tournois interferometer mirror; OC, output coupler; PBS, polarising beam-splitter; IF, interference filter; APD, avalanche photodiode; CM ,cold mirror (highly reflecting at 520nm and highly transmitting at 1040nm); KTP, potassium titanyl phosphate nonlinear crystal; P, BK7 glass prism.

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The pump laser, self-referencing interferometer and OPO were each housed in perspex boxes to minimise fluctuations in the pulse phase-slips caused by air currents, and the whole apparatus was mounted on a vibration-isolated bench. Despite these precautions, the beat frequencies from the OPO outputs typically varied over a range of ∼12 MHz (red) and ∼4 MHz (yellow) in 5 seconds; in the same period the beat frequency from the pump (green) jittered over a range of only 0.2 MHz. In order to improve the accuracy of our measurements we connected detectors APD1 and APD2 to the same RF spectrum analyser (Agilent E4411B) and captured all three beat frequencies in a single screen shot (Fig. 3) over a sweep time of 5ms, during which the jitter of the red beat was only ∼2MHz. The beat frequencies from the red and yellow OPO outputs were controlled by translating a light end-mirror (M3) in the OPO cavity with a piezo-electric transducer (PZT). This action simultaneously tuned the peak output wavelengths of the OPO [17] but this gross tuning effect only reflects a change in the relative intensities of the signal and idler modes.

The beat frequencies of interest were observed on a RF spectrum analyzer as high and low frequency side-bands of the laser repetition frequency, F (200MHz), at values mF ± (Ω_beat /2_π) where m is an integer and F = Δω/2π . All the beats were therefore observable (modulo F) on a linear frequency range from 0 - F/2 and we used this fact to record a broad range of beat frequencies (spanning ∼ 400MHz for the red beat) while making measurements within a span of 100MHz and at a resolution bandwidth of 250kHz. To accomplish this we kept the pump beat constant at ∼50MHz while tuning the OPO monotonically, and recorded the frequencies of the red and yellow side-bands as they appeared within the observable frequency range. Recorded values were then replaced by equivalent frequencies to obtain monotonically increasing (yellow) and decreasing (red) data series. Negative beat frequencies are acceptable because they describe phase slips which are ambiguously negative or positive.

 

Fig. 3. RF spectrum analyser screen showing frequency sidebands generated by interference between the super-continuum reference spectrum and: (1) red frequency-doubled signal pulses (33.55MHz), (2) yellow pump + idler sum-frequency mixing light (41.4MHz) and, (3) frequency-doubled pump super-continuum light (49.15MHz). The resolution bandwidth of the measurements was 100kHz. When the negative sideband of the red beat is used (-33.55MHz) the data illustrate that Ω^red _beat −Ω^pump _beat = −2Ω^yellow _beat.

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4. Results and analysis

Energy conservation in the parametric process can be tested in the following way. If energy is not conserved then Eq. (2) must be modified to take account of an energy deficit, ħΔΩ, so that:

Correspondingly, a graph of Ω^red _beat −Ω^pump _beat against Ω^yellow _beatshould have a slope of -2 and an intercept of −2ΔΩ . Figure 4 shows such a plot of our experimental data and a linear least-squares fit to the data (solid line) has slope -1.995 ± 0.004 and intercept -0.1 ± 0.4 MHz. The measured slope is around one standard deviation away from the predicted value but in the context of the experimental uncertainties this is probably not significant. The predicted intercept is within the measured error bound and is also within one resolution bandwidth of the origin. For each point on the graph it is possible to calculate Ω^red _beat −Ω^pump _beat + 2Ω^yellow _beat which is zero in the case of perfect data. Analyzing the data in this way showed that the mean value of Ω^red _beat −Ω^pump _beat + 2Ω^yellow _beat was -0.4MHz and its standard deviation was 2.2MHz, so the data can be considered to imply energy conservation to within the limits of experimental precision. In conclusion we believe our results verify that energy is conserved in the parametric process to a frequency uncertainty of around 200kHz.

The experimental accuracy of our test of the energy conservation equation can be improved in a number of ways. Neither the Ti:sapphire nor the optical parametric oscillator were carrier-envelope phase stabilized for these experiments, but techniques exist to allow such stabilization to be applied to both sources. In a mode-locked Ti:sapphire laser, the high nonlinearity of the gain medium and the intense intra-cavity field means that the carrier-envelope phase-slip frequency is sensitive to the pump power and can be stabilized to an absolute value using a servo-loop to drive an electro-optic or acousto-optic modulator inserted in the laser pump beam [8]. No equivalent performance has yet been reported for a femtosecond OPO, but in work yet to be published we have stabilized the carrier-envelope phase-slip frequencies of the pump, signal and idler pulses of a femtosecond OPO to a bandwidth of less than 3kHz. By using an atomic clock as a frequency reference it is possible to apply precision counting techniques to accurately compare two beat frequencies [18], and such a method could allow sub-Hz testing (∼10-33 J) of the energy conservation law.

 

Fig. 4. Difference between the pump beat linear frequency (Ω^pump _beat/2π)and the signal SHG beat linear frequency (Ω^red _beat/2π) shown as a function of the yellow beat linear frequency(Ω^yellow _beat/2π). The slope is -1.995 +;/− 0.004 and the intercept is −0.1 +/− 0.4 MHz.

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5. Conclusion

In summary, we have made simultaneous measurements of the pump, signal and idler phase-slip frequencies in a femtosecond OPO and have compared the relation between them with that predicted by coupled-wave theory and obtained excellent agreement. Direct verification of the phase relation (1) would be far more challenging, even though it is now possible to detect the absolute phase of a train of identical pulses. Our experiment demonstrates how measurements of the carrier-envelope phase could potentially reveal the presence of subtle loss processes operating a photon level. In particular it should be possible to apply such techniques to investigate the role of acoustic phonons which, through processes such as stimulated Brillouin scattering, can make very small changes to the optical carrier frequency, shifting it by only a few GHz and causing line-broadening effects with typical bandwidths of ∼30 MHz [19, 20]. Although our results have shown, to the limits of the experimental error, that energy is conserved in a parametric interaction, more precise tests may yet reveal small discrepancies with significant consequences. For example, such an observation would have important implications for quantum information experiments based on using parametric down-conversion to create entangled photon pairs [21]. If the energy conservation law were broken then the generated photons would no longer be perfectly anti-correlated in energy, meaning that measuring the state of one entangled photon would no longer yield complete information about the other. Beyond testing the energy conservation law, the direct verification of the phase relation (1) presents a considerably greater challenge which would require the measurement of the absolute optical phases of the pump, signal and idler pulses. Although strong-field processes like high-harmonic generation are sensitive to the absolute phase [22], there are no equivalent processes that can be driven by the low energy pulses from mode-locked oscillators of the kind described in this work. In principle, an optical parametric amplifier measurement using phase detection enabled by a strong-field process would be possible, albeit technically demanding.