Quiet broadband light

The interference of waves evenly separated in frequency generates a periodic signal, whose intensity fluctuations depend on the phases of the individual waves. A fundamental question in telecommunications and acoustics is the minimization of the peak-to-average-power ratio (PAR): for a given spectrum, how to arrange the phases of the individual frequency components to ensure minimal intensity fluctuations of the resulting signal? For a flat spectrum a near-optimal solution is brought by the so-called Newman phases, commonly used in acoustic and radio waves. Here we transpose this property into the optical domain and prove the possibility to suppress the intensity fluctuations resulting from intermodal beating of a broadband comb of optical frequencies, whose phases are set according to a generalization of the Newman phases. We demonstrate experimentally a broadband laser, whose intensity contrast (defined as the standard deviation-to-mean ratio) is reduced down to 0.54 as compared to 1 when the phases are set randomly.

Figure 1

Reduction of the intensity fluctuations of a flat comb of $N$ optical frequencies arranged according to the Newman phases. The evolution of $I\left(\theta \right)=|{\sum }_{n=1}^N{e}^{-in\theta +i{\phi }_n}{|}^2$ with the dimensionless time $\theta$ is computed for both Newman phases (i.e., ${\phi }_n=-\pi n^2/N$, in blue) and random phases (in green) for different values of $N$ (10, 50, 250, and 1250). The degree of second-order coherence at zero delay $g^2\left(0\right)$ (see text) is equal to $2.0\pm 0.1$ in all four cases of random phases, while it is equal respectively to 1.21, 1.09, 1.04, and 1.02 in the cases of the Newman phases.

Figure 2

Top: comparison of the degree of second-order coherence $g^{\left(2\right)}\left(\tau \right)$ in the case of a flat spectrum of 100 equidistant optical frequencies with phases set as random (green, left) and according to the Newman phases ${\phi }_n=-\pi n^2/N$ (blue, right). Bottom: corresponding normalized IAC $|{[E\left(\theta \right)+E(\theta +{\theta }_{\tau })]}^2{|}^2$. The red trace is the average over the fringes. The peak-to-baseline ratio of the fringes are close to $4:1$ and $8:3$ in the case of the random and Newman phases respectively while the peak-to-baseline ratio of the average traces are $3:2$ and $1:1$ respectively.

Figure 3

Top: Plot of $g_{\phi }^{\left(2\right)}\left(0\right)$ as a function of $\phi$ in the case of $N=100$ modes with equal amplitude. Bottom left: zoom of $g_{\phi }^{\left(2\right)}\left(0\right)$ in the vicinity of $\phi /2\pi =p/q$ with $p/q=0,1/2$ and $3/5$ from top to bottom. The horizontal span is $6\pi /100$. The dips correspond to the generalized Newman phases $\widehat{\phi }=2\pi (p/q\pm 1/qN)$. The corresponding intensities $I_{\widehat{\phi }}\left(\theta \right)$ are plotted in the right column and the values of $g_{\widehat{\phi }}^{\left(2\right)}\left(0\right)$ are equal to 1.06, 1.09, and 1.15 respectively (top to bottom).

Figure 4

Sketch of the experimental setup (see text). The output coupler (OC) of the injection-seeded FSF Ti:Sa cavity is mounted on a translation stage (TS1). A Michelson interferometer (MI) and a BBO crystal tuned for SHG at 780 nm enable us to record the IAC. BS, PD, and PMT stand respectively for beamsplitter, photodiode, and photomultiplier tube.

Figure 5

Experimental comparison of the FSF laser operating without (in green) and with external seeding (in blue) when the cavity length is adjusted according to the generalized Newman phase $\widehat{\phi }=2\pi (p/q\pm 1/qN)$ where $p=4,q=5$, and $N\approx 2500$ so as to minimize the intensity fluctuations. Top left: optical spectra of the laser without (i.e., modeless laser) and with external seeding. The sharp peak corresponds to the injection laser. Top right and bottom left: comparison of the RF spectra in the 0–1-GHz range and the IAC of the laser without and with external seeding. Bottom right: corresponding SHG signals in the same experimental conditions. The solid lines are quadratic fits.

Figure 6

Experimental IAC traces of the injection seeded laser in the vicinity of the minimum of fluctuations. Left: the vertical scale [$g^{\left(2\right)}(\tau =0)$] has been adjusted by comparison to the unseeded laser.