1. Introduction

As a result of a complex interplay of a number of nonlinear effects, phase-coherent supercontinua (SC) are generated, when ultrashort pulses obtained from a modelocked laser are launched into a microstructure (MSF) or photonic crystal fiber [1–7]. Comprising the entire visible wavelength range and extending into the near infrared, thus generated phase-coherent comb spectra have found wide-spread use in various fields, e.g., absolute frequency metrology or pulse synthesis [8–13]. A future application might employ a phase-coherent comb spectrum in heterodyne CARS. Here the SC would serve as an ensemble of phase-coherent oscillators covering the entire difference frequency interval needed for CARS (<120 THz). Simultaneously, the phase-coherent local oscillator needed for the heterodyne detection of the anti-Stokes signal would also be contained in the SC. The heterodyne detection of the CARS signal would improve the signal to noise ratio, possibly by an order of magnitude.

The mentioned applications will eventually be limited by phase and/or intensity noise generated in the MSF used to generate the SC. (In frequency metrology phase noise in the SC limits the achievable short term stability. Both, phase and power noise, will have detrimental effects on stable pulse synthesis, and in applications using a SC in heterodyne pump-probe-experiments phase noise will limit detectability, while power noise adversely affects measurement accuracy.)

In system/network analysis, the (complex) transfer function (η(λ)) is introduced to describe, how input intensity modulations are transferred to the output as intensity and/or phase modulations. Separating η(λ) into a phase modulation coefficient Δϕ(λ) and a power modulation gain g(λ), we investigated the carrier wavelength dependence of the transfer function for SC generation in MSF.

2. Measurements of intensity-to-phase transfer

 

Fig. 1. Setup used to measure the phase modulation in the output spectrum of microstructure fiber.

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The setup used for the measurements is shown in Fig. 1. A prism compensated Kerr-lens modelocked laser (not shown) is used to generate pulses of 30 fs duration at a repetition rate of 100 MHz. The pulses are intensity modulated using a thin fused silica acousto-optical modulator (AOM). The modulation frequency of 50 MHz is derived (divide-by-two) from the optically detected repetition frequency of the modelocked laser. Thus, the intensity of the pulses alternates for consecutive pulses, which therefore undergo alternating phase shifts in the (highly nonlinear) microstructure fiber (MSF). In our experiment, we used 120 mm of Lucent microstructure fiber [14]. (This fiber’s dispersion characteristics enhance (in combination with a small core diameter) the nonlinear spectral broadening during pulse propagation in the fiber.)

The alternating phase shifts (as result of the alternating intensity) of the pulses are detected using a matched Mach-Zehnder interferometer (MZI) as a “slope discriminator” for phase modulations in the spectral windows defined by the interference filter (IF) (see following subsection for a detailed explanation).

 

Fig. 2. Phase modulation coefficient Δϕ in rad per percent of pulse energy change as function of carrier wavelength, measured for spectral windows of 10 nm FWHM (solid squares) and by beating the supercontinuum with single frequency lasers (open triangle); pulse energy 200pJ.

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Figure 2 shows the thus determined intensity-to-phase modulation coefficient during super continuum generation in rad per percent of pulse energy change as function of carrier wavelength (solid squares). Three points (open triangles) were validated in independent control experiments: With beat measurements of the super continuum against (i) a Nd:YAG laser (at 1064 nm) and (ii) a He-Ne-laser (at 633 nm) the reversal of the sign of the phase excursion of the lines in the near infrared as compared to those in the visible was verified. Additionally, the value of the intensity-to-phase transfer for lines at 1064 nm was used to calculate the phase excursion of lines around 532 nm from (iii) measured phase excursions of the carrier-envelope-offset beat. (This beat is generated by superposition of frequency doubled lines from the low frequency wing of the SC with lines of the upper frequency wing. Therefore it can be used to infer the phase excursion of one of the superimposed beat constituents, when the other constituent’s phase modulation is already known).

In all spectral areas of the SC the intensity-to-phase transfer was found not to exceed 40 rad per percent of pulse energy change (similar to values reported in [15]), and the transfer function does not exhibit any fine spectral details; adjacent comb lines essentially sustain equal phase modulations.

2.1 Mach-Zehnder interferometer as slope discriminator for phase modulation

In Fig. 3 the transmission of the Mach-Zehnder interferometer (for spectral windows of 10 nm FWHM) is shown on the lower part as function of the delay between the two arms. In the upper part, the power of the 50 MHz intensity modulation behind the interferometer is shown in dB. Figure 3 additionally contains numbers in circles denoting interferometer-delay positions. Position 1 is a bright-fringe-delay, i.e. the delay between interferometer arms is adjusted such, that pulses from both arms interfere constructively on the recombination beam splitter BS2. Adjusted to the dark fringe, position 3, pulses from both arms interfere destructively. Between these two delay positions, the transmission curve of the interferometer exhibits the common sinusoidal discrimination characteristic of a two-beam-interferometer.

 

Fig. 3. Transmission of MZI as function of delay (left y-axis) and power of the modulation signal as result of phase modulation (right y-axis) (both for spectral windows of 10 nm FWHM).

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The right axis shows the power of the 50 MHz modulation of the throughput of the MZI. Our Mach-Zehnder interferometer is inherently AM balanced, as the beam splitters BS1 and BS2 are identical. Therefore the 50 MHz intensity modulation of the MZI-throughput does not arise from the alternating intensity of the pulses coupled into the MZI, because the sum of the energies of the interfering pulses detected behind the MZI is always identical. Instead, the 50 MHz signal results from the phase modulation of the pulses.

 

Table 1. Phasor description for interfering fields at the recombination beamsplitter BS2, shown for two scenarios: left without phase modulation (PM) of the pulses, right with phase modulation in arm a or arm b (for further details, see text)

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The phase modulation, however, can only be detected in positions where the symmetry of the phase discriminator is broken with respect to phase modulation of the interfering pulses. This is not the case for positions 1 and 3, therefore the 50 MHz signal vanishes, which is explained in table 1 above. The table gives a phasor description of the fields interfering on the recombination beam splitter BS2. The numbers denote the MZI delay setting (as in Fig. 3). In the left part of the table the description is given for interference without phase modulation. In the right part, the phase modulation of the interfering pulses is shown as an angular deviation of one of the phasors (solid) from the original position (dashed). The row letters a^(‘) and b^(‘) discern between the two alternating interference scenarios: The brighter pulse can come from the long arm (a’) or the short arm (b’) of the interferometer. The bottom line shows the difference in field strength between the two scenarios for the three MZI delay settings. Only for position 2 this difference does not vanish, and hence an intensity modulation can be detected behind the MZI. Phase-synchronous detection allows the sign of the phase modulation to be determined. In the way thus described, the data of Fig. 2 was obtained.

3. Intensity-to-intensity transfer

 

Fig. 4. Power spectral density and power modulation gain factors g(λ) as function of wave length measured with different input pulse energies of a) 1 pJ, b) 25 pJ, c) 200 pJ (supercontinuum).

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The intensity-to-intensity transfer, on the other hand, is expected to have sharper spectral features due to possible interference of different SC generation scenarios. Therefore the MZI was replaced by a grating spectrometer. It offers one-nanometer-resolution for the measurement of the power spectral density at a given wavelength and the relative intensity modulation, which is calculated by dividing the demodulated ‘ac’ component’s intensity by the respective ‘dc’ power spectral density. (To avoid confusion: The above described experiment to measure the modulation of the optical phase was carried out in the optical frequency domain. The experiments of this chapter in order to measure the power modulation gain factors, in contrast, represent measurements in the base band domain, i.e. the ‘phases’ in both experiments are physically different.) Typical results obtained with different pulse energies are shown in Fig. 4. The plots show the power spectral density and the intensity-to-intensity modulation transfer (power modulation gain factor, g) as function of wavelength. For plot a) pulses of 1 pJ are coupled into the MSF. At these energies the MSF essentially behaves as a linear medium. Spectral broadening is negligible and intensity-to-intensity-transfer is spectrally uniform and equals unity. Increasing the pulse energy -see plot b)- shows the “onset of nonlinear behavior” with spectral broadening and a varying intensity-to-intensity transfer. Especially, the shifting of spectral entities, presumably constituting solitons, to larger wavelengths with increasing pulse energy is visible: On the upper frequency wing of these entities negative intensity transfer is found, while unity is measured at the maximum, where the effect of wavelength shifting vanishes.

Note: If the phase index dispersion of the MSF is assumed to equal the value of fused silica, -0.012/µm [16], the observed shifting of spectral entities in the outermost low frequency wing of the spectrum leads to phase delays of -15 rad -i.e. phase advances of 15 rad- per percent of pulse energy change. This is similar to the values shown in Fig. 3 for the extreme low frequency wing of the spectrum around one micron. For SC generation (plot c), the intensity-to-intensity-transfer coefficient deviates from unity in both directions with values representing power modulation amplifications of up to the order of 100 in some spectral areas. Note: Similar values were reported in [17, 18] for the amplification of technical input noise. However, in the cited reference, the values were obtained by comparing measured relative intensity noise to input noise. Those noise power –as opposed to our amplitude- measurements out ruled the possibility to study the spectral evolution dynamics of SC generation presented here.

4. Conclusion

Pulses obtained from a modelocked Ti:Sa-laser were intensity-modulated before being coupled into a microstructure fiber (MSF) for supercontinuum (SC) generation. We phase-synchronously detected the intensity and phase modulations in the spectrally broadened output spectrum behind the fiber. We found intensity modulation amplifications of up to the order of 100, and we determined intensity-to-phase transfer in all spectral areas of the SC not to exceed 40 rad per percent of pulse energy change. (Both values apply for SC generation by pulses with energies of some 200 pJ coupled into 120 mm of Lucent MSF [14]).

For our laser with an rms intensity noise of 0.1% the following estimates for SC applications become possible:

a) In applications sensitive to intensity fluctuations in the SC, such as pump-probe-experiments, intensity fluctuations as large as ten percent in certain spectral areas have to be included in an estimate of the expected measurement accuracy.

b) For applications relying on the phase-coherence of the SC modes, phase excursions of a few rad have to be taken into account. For example: If a self-referenced SC frequency comb is to be used as an optical clockwork, the phase deviations of a few rad correspond to an achievable stability better than 10^-15 in one second, as the circular frequencies in the optical are of the order of a few Prad/s. The achievable stability lies within reach of the quantum projection limit of ultrastable optical frequency standards, e.g. a Ca atomic clock [19]. These optical standards’ high short term stability can therefore be transferred to the radio frequency domain using femtosecond comb generators based on self-referenced supercontinua generated in microstructure fibers.