# Mode locking based on the temporal Talbot effect

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## Abstract

We propose a new laser mode locking state in which the pulse disperses quickly and then revives after a certain time. This mechanism is based on the temporal Talbot effect and requires a large amount of intra-cavity group velocity dispersion. Similar to the usual mode locking it should be possible to employ the Kerr effect to force the laser into this mode, even when the cold cavity dispersion is not exactly matched. We show that the mode spectrum of such a laser is not equidistant but increases linearly with very high precision. This Talbot frequency comb can be self referenced. The beating with the adjacent modes uniquely defines the optical mode frequency, which means that the optical spectrum is directly mapped into the radio frequency domain. This is similar to the dual frequency comb technique, albeit without the limiting relative jitter between two combs.

## Keywords

Fiber Bragg Grating Mode Spacing Group Velocity Dispersion Frequency Comb Beat Note## 1 Introduction

The Talbot effect has been described for the first time in 1836 as a peculiar phenomenon observed in the near field of an optical grating. Summing over contributions of the individual rulings to the total field in the Fresnel approximation, a term of the form \(\exp(-ikl^2 a^2/2z)\) appears with the wave number *k*, the rulings numbered by *l* and spaced by *a* and the distance from the grating *z*. Summing over *l* generally yields a rather chaotic intensity distribution. Talbot noted though that all these terms reduce to \(\exp(-i2\pi l^2)=1\) at a distance of \(z=k a^2/4 \pi\). The remaining terms add up to the intensity at \(z=0\) provided that this intensity is periodic with *a* [1]. The same phenomenon can be observed in the time domain with a periodic pulse train signal that is subject to group velocity dispersion \(\phi ''\) that provides the quadratic phase evolution. The pulses first spread out in time, an then reassemble after propagation the distance \(t_r^2/2\pi \vert \phi '' \vert\), where \(t_r\) is the pulse repetition time [2, 3]. The same behaviour should be observable with a single pulse that is on a repetitive path in an optical cavity. In contrast to a free space pulse train, higher order dispersion is required as we will show below.

## 2 Talbot Comb

*n*which leads to an equally spaced comb of radio frequency (RF) components:

These RF components are the result of beating between adjacent optical modes and can be seen in the power spectrum of the laser output. Higher order mode beatings like \(\omega _{n+2}-\omega _n\) etc. are separated in the power spectrum by about \(\omega _r\). This is illustrated in Fig. 1 and similar to the harmonics of the repetition rate in a regular frequency comb. The optical mode spacing at the carrier frequency \(\omega _0\) is given approximately by \(\omega _r\) for large *m*. It becomes the spacing for all modes for \(m \rightarrow \infty\) for which (1) turns into a regular frequency comb [4]. Note however that \(\omega _r\) is not the usual repetition frequency. Nevertheless \(\omega _r\) and \(\omega _0\) can be measured in very much the same way as for a regular frequency comb (see Sect. 3). Besides representing an all new mode locking regime, the interesting aspect of (1) is that each mode beating uniquely belongs to on particular pair of modes. For example the RF signal at \(\Delta _0=\omega _r(1+1/m)\) belongs to the beating between \(\omega _0\) and \(\omega _1\) and so on. Hence a RF spectrum recorded with a photo detector and a radio frequency spectrum analyzer directly displays a scaled down version of the optical spectrum of the laser. By placing a sample between the laser and the photo detector and recording the change of the RF spectrum one gets the absorption spectrum of the sample. This is similar to a dual frequency comb setting with a linearly increasing spacing between the modes of the two frequency combs [5], albeit with a single laser avoiding problems due to the relative jitter of the combs.

*m*is an exact integer however, the pulse will revive up to a carrier–envelope phase of \(\varphi _{ce}=2 \pi m \omega _0/\omega _r\) after the time \(T=2 \pi m/\omega _r\):

*m*multiple of the cavity round trip phase delay of the \(n=0\) mode at \(\omega _0\). Figure 2 shows an example of the resulting laser power for a special case as a function of time. Like in a regular mode locked laser, the peak power enhancement over the time averaged power is roughly given by the number of active modes. In contrast to the latter, the large peak power occurs not every cavity round trip but only every

*m*-th cavity round trip. The usual Kerr lens mode locking mechanism might be used to enforce an integer

*m*by reduced loss of the high peak intensity pulse. In the case of the Talbot comb, that large peak intensity occurs only every

*m*-th round trip. We therefore expect that the self amplitude modulation of the Kerr effect has to be stronger for successful mode locking. The mode spacing nominally becomes negative for \(n<-m/2\) (see Fig. 3). Physically this means that the corresponding spectral region possesses a negative group velocity. While this is possible in principle we exclude this for a reasonable laser design and assume that the active modes of the laser are limited to \(n>-m/2\).

*n*:

*L*, the round trip phase \(\phi (\omega )\) at frequency \(\omega\) has to fulfill the boundary condition:

*m*, i.e. a long pulse revival time (for a given \(\omega _r\)). However we expect that this would lead to a reduced Kerr effect and hence weakens the mode locking mechanism. We expect that once the laser is set up for a particular value of

*m*, it will be reproduced every time it is put in the mode locked state.

## 3 Self-referencing

For self-referencing the two parameters of the Talbot comb, \(\omega _r\) and \(\omega _0\) need to be measured and ideally stabilized. We assume that the recurrence index *m* is known. One might get an estimate of it and then fix it to be an integer, by measuring the recurrence time and compare it to the cavity length. A more reliable method would be to measure a known optical frequency with a self-referenced Talbot comb and then identify the proper *m* compatible with that measurement. This is the same method often applied with regular combs to determine the correct mode number.

*n*as expressed by (2), i.e. by the second order mode differences:

## 4 Non-linear interactions

*n*, the \(\chi ^{(2)}\) process is expected to be less efficient than for a regular frequency comb. Of course this can also be understood in the time domain where a short pulse is formed only after

*m*cavity round trips. The mode combinations in (14) with \(n' \ne n\) do not belong to the doubled Talbot comb. This probably means that the frequency doubled Talbot comb are not good for the type of spectroscopy described above. However, the usual \(f-2f\) self referencing is possible using these processes if one finds ways for a more efficient non-linear interaction. Since the doubling process is expected to be weak, it might be advisable to work with an auxiliary frequency doubled cw laser beating the fundamental and second harmonic with the Talbot comb and the doubled Talbot comb respectively. It should be mentioned that even without self-referencing the Talbot comb could be a useful tool. One might instead reference one of its modes to a wavemeter or to a atomic or molecular line. In fact for the intended broad band spectroscopy referencing to an atomic clock is usually not necessary.

*m*: \((n+n'-n'')^2=n^2+n'^2-n''^2\). Solving this equation yields two solution \(n''=n\) and \(n''=n'\). In neither case additional modes are added to the initial Talbot comb.

## 5 Example design

*m*times smaller than the conventional lasers. Therefore either strong nonlinear effects or matching the overall dispersion to (6) with very good accuracy is required to enforce the Talbot mode-locking. Intracavity dispersion can be introduced through a fiber Bragg grating (FBG) that can be designed with very large values for the group velocity dispersion and precise values for the higher order dispersions. To manufacture a FBG a grating is written into a photo sensitive fiber with a UV laser. Up to the 6th order dispersion is commercially available.

One or even several FBGs can be conveniently implemented into the fiber-laser cavity using an optical circulator. The cavity should include a gain fiber, a pump beam combiner, an optical isolator and an output coupler as shown in Fig. 5. An example for design parameters might be \(m=10^6\), \(\omega _r=2 \pi \times 100\) MHz assuming a spectral width of \(\delta \omega = 2 \pi \times 10\) THz and an optical carrier frequency of \(\omega _0=2 \pi \times 300\) THz (\(\lambda = 1~\upmu\)m) which is close to the Ytterbium gain maximum. With this we obtain from (11) \(\phi ''_{\omega _0}=-3.2\times 10^7\) fs\(^2\), \(\phi '''_{\omega _0}=3.0\times 10^8\) fs\(^3\) and \(\phi ^{(4)}_{\omega _0}=-4.8\times 10^9\) fs\(^4\) etc. Rather than this expansion one might use the first term in (7) to directly compute the required dispersion function.

To estimate the order of magnitude of the required length of the FBG \(\Delta z\) we calculate the difference of the round trip phase delay for the two ends of the spectrum (full width \(\Delta \omega)\) using (7): \(\Delta \phi =\phi (\omega _0+\Delta \omega /2)-\phi (\omega _0-\Delta \omega /2)\). This phase difference has to be divided by the wavenumber \(2\pi /\lambda\) to obtain the required path length difference between the two extreme colors. With the parameters above, dividing by the refractive index of 1.5 and taking into account that the light travels twice through the FBG we obtain \(\Delta z=3.3\) cm. The real length might then also depend on the requirements for the reflectivity. Fiber lasers generally come with a large optical gain so that it may be possible to compromise on that parameter. The first design may not be the rather optimistic one of this proposal but could be a trade off between large *m* (=low dispersion) and small *m* (=stronger mode locking). To start operation at a very large value of *m* it may also conceivable to include an intracavity modulator that mimics all or parts of the temporal envelope shown in Fig. 2.

## 6 Conclusions

This article is dedicated to Theodor Hänsch on the occasion of his 75th birthday. Among the many other laser tricks that he has invented is the optical frequency comb. Originally intended as a tool to measure laser frequencies, it has found several other applications for example in attosecond science and astronomy. Further it is used for large bandwidth direct comb spectroscopy. The current work tries to extends these possibilities, even though we have to admit that more ideas are required to turn it into a useful tool. Nevertheless we hope that this idea provides a new playground and entertainment for those who like curiosity driven research. In this spirit Theodor Hänsch has been our guide for many decades and we hope that there will be many more to follow. Happy Birthday!

## Notes

### Acknowledgements

Open access funding provided by Max Planck Society.

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