Deterministic Nonlinear Transformations of Phase Noise in Quantum-Limited Frequency Combs

Optical phase noise of femtosecond lasers is analyzed over various steps of broadband nonlinear frequency conversion. The intrinsic phase jitter of our system originates from quantum statistics in the mode-locked oscillator. Supercontinuum generation by four-wave-mixing processes preserves a noise minimum at the optical carrier frequency. From there, a quadratic increase of the comb linewidth results with mutually anticorrelated phase fluctuations of both spectral wings. Passive phase locking by difference frequency generation strongly enhances the optical phase noise to a level equaling the carrier-envelope phase jitter of the fundamental comb. The same value results from quadratic extrapolation of the optical phase noise to radio frequencies. Our findings are consistent with a fully deterministic transformation of phase noise according to the elastic tape model.

Figure 1

(a) Scheme of the mode-locked Er:fiber laser system and setup for broadband investigation of phase noise and linewidth characteristics. Single-line references and their subharmonics are depicted in magenta. (b) Spectral scheme of generating a radio-frequency beat $f_{\text{beat}}$ between a narrow-line reference and the closest line in a frequency comb with a spacing given by the laser repetition rate $f_{\text{rep}}$. (c) Spectral scheme when temporally interfering the fundamental comb where $f_{\mathrm{CEO}}\ne 0$ with the phase-stable comb generated via DFG ($f_{\mathrm{CEO}}=0$), thus providing access to the carrier-envelope offset frequency $f_{\mathrm{CEO}}$.

Figure 2

Evolution of the optical linewidth of two DFG combs (blue and red) at 193 THz when the spectral width $\mathrm{\Delta }\nu$ of the soliton within the all-fiber oscillator is varied. Both fit well to the theoretical prediction for quantum-limited phase noise characteristics (black dashed) up to the point where intracavity soliton fission crucially affects the pulse characteristics.

Figure 3

(a) Spectral width of several individual comb teeth. $f_{\mathrm{CEO}}$ (gray) is measured by interfering the fundamental and DFG comb. The linewidth of the latter at 193 THz is shown in black. (b) Frequency noise spectral density of a comb line at 282 THz (pink), $f_{\mathrm{CEO}}$ (gray) and the DFG comb at 193 THz (black). (c) and (d) Sketch of the elastic tape behavior of the comb modes for a fix point at the carrier and zero frequency, respectively.

Figure 4

(a) and (b) Narrowed spectra of the dispersive wave and the soliton, respectively, used for DFG at varying frequencies. (c) CEO linewidth when the DFG is centered at 193 THz (gray marker) and if mixed with spectra of same color shown in (a) and (b). Black line and shading depict the optical linewidth of the DFG comb at 193 THz and its statistical uncertainty, respectively. (d) Frequency noise spectral density of $f_{\mathrm{CEO}}$ if the DFG is centered at two different spectral regions.