Unveiling the dynamics of optical frequency combs from phase-amplitude correlations

The noise dynamics of an optical frequency comb based on a mode-locked Ti-sapphire laser is analyzed in terms of noise modes. A spectrally resolved multipixel homodyne detection enables the simultaneous measurement of the amplitude and phase noises of several optical frequency channels, from which the covariance matrices of the amplitude and phase quadratures of the laser field are calculated. The decomposition of these matrices into the four most significant time/frequency modes of the field enables the tracking of the origin of the noises and the correlations between the noise modes. In particular, the correlations between the amplitude and phase noises are measured. These measurements are well reproduced by a model taking into account the correlations between the carrier envelope offset phase noise and the amplitude noise induced by the group velocity dispersion of the laser cavity.

Figure 1

Experimental setup: The experiment is composed of three parts. (a) Reference filtering, to generate the reference beam via a high finesse cavity on which the repetition rate of the laser is locked. The CEO is also locked thanks to an f-2f interferometer to ensure maximum transmission through the cavity. The laser is free running above the locking bandwidth of a few kHz. (b) Spectral phase compensation with a pulse shaper to ensure a flat phase for all spectral bands. That way, the homodyne is locked on the phase quadrature for all the spectral bands. (c) Multipixel homodyne detection to measure the amplitude and phase noise in eight different spectral bands. A demodulation stage is used to acquire noise at a given offset frequency selected by the radio-frequency local oscillator (RF LO) whose frequency can be swept. All data are acquired by an acquisition card (DAQ) and postprocessed on a computer to recover the covariance matrices.

Figure 2

Experimental covariance matrices. (a) Amplitude and (b) phase covariance matrices. The matrices are measured at 500 kHz. The noise is expressed as noise relative to shot noise (NRSN) on a linear scale. Correlations can be seen between the different spectral bands. In inset, matrices for noises at 4 MHz. At that frequency, the laser is only affected by the shot noise. (c) Noise on the physical parameters relative to the shot noise. Projection of the covariance matrices on the modes corresponding to the noise on the CEO, central, and repetition rate frequencies as well as the mean power. The noises are expressed in dB relative to the shot noise. The shot noise is measured by blocking the reference beam and taking the difference of the photocurrents.

Figure 3

Noise in physical units (a) CEO frequency noise, (b) timing jitter noise, (c) mean power noise, and (d) spectrum center frequency noise. The spectral mode corresponding to each quantity is reproduced in each plot. The shot noise is plotted as a dashed black line.

Figure 4

Phase and amplitude correlations: (a) Correlation matrix between amplitude and phase at an offset frequency of 500 kHz. In inset the same but at 4 MHz where the correlations vanish. (b) Schmidt decomposition of the correlation matrices. The Schmidt number as a function of the offset frequency is plotted together with the amplitude and phase singular modes for the frequencies where it is equal to one. (c) Projection of the amplitude and phase singular modes on the detection modes.

Figure 5

Experimental versus model for the CEO noise: In dashed blue, the model from Eq. (22), the shaded zone corresponds to the uncertainty on the GVD determined in the laser. Plain line: Measured CEO frequency fluctuation given by $\delta f_{\text{CEO}}=\sqrt{S_{f_{\text{CEO}}}}$ from Eq. (16).