Analogy of harmonic modelocked pulses to trapped Brownian particles improves laser performance

Abstract

High-repetition-rate ultrafast lasers are needed for diverse applications. Harmonic modelocking, where multiple identical, equidistant pulses circulate in the cavity, reaches beyond the practical limitations of reducing the cavity length. However, it suffers from stochastic deviations that manifest as supermodes in the radio-frequency spectrum and difficulties in maintaining the same harmonic state, often coupled with trade-offs in pulse energy, duration, or noise performance. Here, we first show that deviations in the temporal positions of the pulses contribute disproportionately more to the supermodes than deviations in their amplitudes. Then, we argue that these fluctuations are analogous to those of trapped Brownian particles. This analogy reveals that supermodes are suppressed by stronger spectral filtering, which corresponds to fluid viscosity, and higher pulse energy reduces the noise, akin to lower temperature. Guided by this intuitive picture, we construct a Yb-fibre laser incorporating strong filtering and high intracavity energies by limiting nonlinear polarisation evolution to a short section of ordinary fibre. The rest of the all-fibre cavity comprises polarisation-maintaining fibre, which additionally improves environmental robustness. We report record-high supermode suppression ratios, reaching 80 dB, excellent long-term and environmental stability, and pulse energy, duration, and noise characteristics that are similar to fundamentally modelocked lasers.

Contributions of energy and position deviations to the supermode suppression ratio

The radio-frequency (RF) signal is the Fourier transform of the time signal. The RF trace, at frequency f in logarithmic scale, is

$$\begin{aligned} \text {RF} (f)&= 10 \log _{10} \left( C \times \vert {\tilde{P}} (f) \vert ^2 \right) \\\left|{\tilde{P}} (f)\right|^2&= \left|\frac{1}{T_\text {c}} \int _{0}^{T_\text {c}} \exp \left( - 2 \pi \text{i} \frac{n}{T_\text {c}} \tau \right) \sum _{j = 1}^{N} E_j u \left( \tau - \tau _j \right) \text {d} \tau \right|^2 . \end{aligned}$$

(A1)

Here, ${\tilde{P}} (f)$ is the Fourier transform of the photocurrent from the photodiode, C is a proportionality constant adding a vertical shift to the RF trace, $T_{\textrm{c}}$ is the cavity round-trip time, n is a positive integer enumerating the maxima in the RF trace such that $f_n = n / T_{\textrm{c}}$ represents all frequencies allowed by the periodicity of the cavity. Furthermore, $\tau$ is the delay coordinate, N is the number of pulses in the cavity, $E_j$ and $\tau _j$ are the energy and temporal position of the $j$th pulse, respectively, and u describes the temporal profile of the electrical current pulse generated by the photodetector in response to an optical pulse as measured by the RF spectrum analyser. Approximating the electrical pulse shape u as a delta function leads to

$$\begin{aligned} \left| {\tilde{P}} (f) \right| ^2 = \left| \frac{1}{T_{\textrm{c}}} \sum _{j = 1}^{N} E_j \exp \left( - 2 \pi \text {i} \frac{n}{T_{\textrm{c}}} \tau _j \right) \right| ^2 \, . \end{aligned}$$

(A2)

With N identical pulses in the cavity ($N$th harmonic state), each with an energy E, and positioned equidistantly ($\tau _j = jT_{\textrm{c}} /N$), the Fourier transform becomes

$$\begin{aligned} \left| {\tilde{P}} (f) \right| ^2 = \left( \frac{NE}{T_{\textrm{c}}} \right) ^2, \end{aligned}$$

(A3)

if n/N is an integer, otherwise zero. The maxima in the RF trace at frequencies $n/T_{\textrm{c}}$ other than integer multiples of $N/T_{\textrm{c}}$ result from supermodes. The supermode-maxima only appear in case of deviations from the ideal pulse train. A relative (fractional) energy deviation of $\delta _E$ in one pulse (e.g., the first pulse, $j = 1$) leads to

$$\begin{aligned} \left| {\tilde{P}} (f) \right| ^2&= \left( \frac{E}{T_{\textrm{c}}} \delta _\textrm{E} \right) ^2, \nonumber \\&\quad \text {if }n/N\text { is not an integer (at supermodes)}, \end{aligned}$$

(A4)

$$\begin{aligned} \left| {\tilde{P}} (f) \right| ^2&= \left( \frac{E}{T_{\textrm{c}}} \left( N + \delta _E \right) \right) ^2, \nonumber \\&\quad \text {if }n/N\text { is an integer (at the harmonic repetition rate).} \end{aligned}$$

(A5)

The supermode suppression ratio (SSR) (in logarithmic scale) due to the energy deviation is the ratio between Eqs. A4 and A5:

$$\begin{aligned} \text {SSR}&= 10 \log _{10} \left( \frac{ \vert {\tilde{P}} (N/T_{\textrm{c}}) \vert ^2}{\vert {\tilde{P}} (n/T_{\textrm{c}})\vert ^2} \right) = 10 \log _{10} \left( \left( \frac{N}{\delta _E} +1 \right) ^2 \right) \nonumber \\&\approx 20 \log _{10} \left( \left| \frac{N}{\delta _E} \right| \right), \end{aligned}$$

(A6)

$$\begin{aligned} \delta _E&= \frac{1}{N} \times 10^{- \frac{\text {SSR}}{20}} , \end{aligned}$$

(A7)

where n corresponds to the supermode for which the SSR is to be measured. In the case of a position deviation $\delta _\tau$ in the first pulse, by linearizing with respect to $\delta _\tau$, $\vert {\tilde{P}} (f) \vert ^2$ becomes

$$\begin{aligned} \left| {\tilde{P}} (f) \right| ^2&\approx \left( 2\pi \frac{n}{T_\textrm{c}^2} \delta _\tau T_{\rm R} E\right) ^2=\left(\frac{E}{T_\textrm{c}}\right)^2\left( 2\pi \frac{n}{N} \delta _\tau \right) ^2,\nonumber \\&\quad \text {if }n/N\text { is not an integer (at supermodes)} \end{aligned}$$

(A8)

$$\begin{aligned} \left| {\tilde{P}} (f) \right| ^2&\approx \left( \frac{E}{T_{\textrm{c}}} \right) ^2 \left( \left( 2\pi \frac{n}{N} \delta _\tau \right) ^2 + N^2 \right) , \nonumber \\&\quad \text {if }n/N\text { is an integer.} \end{aligned}$$

(A9)

The resulting SSR then is again the ratio of Eqs. A8 and A9:

$$\begin{aligned} \text {SSR} &= 10 \log _{10} \left( \frac{ \vert {\tilde{P}} (N/T_{\textrm{c}}) \vert ^2}{\vert {\tilde{P}} (n/T_{\textrm{c}})\vert ^2} \right) = 10 \log _{10} \left( \frac{\left( \frac{E}{T_{\textrm{c}}} \right) ^2 \left( \left( 2\pi \frac{n}{N} \delta _\tau \right) ^2 + N^2 \right)}{\left(\frac{E}{T_\textrm{c}}\right)^2\left( 2\pi \frac{n}{N} \delta _\tau \right) ^2} \right) \nonumber \\&\approx 20 \log _{10} \left( \left| \frac{N^2}{2 \pi n} \frac{1}{\delta _\tau } \right| \right) , \end{aligned}$$

(A10)

$$\begin{aligned} \delta _\tau&= \frac{2\pi n}{N^2} \times 10^{-\frac{\text {SSR}}{20}} . \end{aligned}$$

(A11)