# Frequency Comb characterization with High-Resolution Fourier Transform Spectroscopy (FTS)

**– Explanation and applications of Frequency Combs –**

Frequency combs are very precise light sources which can be used for the measurement of absolute optical frequencies, so they can serve as an **optical clockwork** for precision spectroscopy, optical sensing, distance measurements, laser noise characterization, telecommunications, and in fundamental physics.

The spectrum of an optical frequency comb consists of a **train of equidistant lines**, i.e. it has equidistant optical frequency components, while the intensity of the comb lines can vary substantially (see header image of this article). Usually, such kind of optical spectrum is associated with a** regular train of ultrashort pulses** (see Figure 1). This train is generally generated using a mode-locked laser oscillator, having a **fixed pulse repetition rate **which determines the inverse line spacing in the spectrum.

*Figure 1. An ultrashort pulse of light in the time domain: the electric field is a sinusoid with a Gaussian envelope; the pulse length is on the order of a few 10 ps.*

The frequency domain representation of a perfect laser frequency comb is **a series of delta functions** spaced according to: *f(n) = f _{0 }+ n f_{r , }*where

*n*is an integer,

*f*is the comb tooth spacing (equal to the mode-locked laser’s repetition rate or, alternatively, the AM frequency), and

_{r}*f*is the carrier offset frequency, which is less than

_{0}*f*.

_{r}The development and the application of **laser-based precision spectroscopy, including the optical frequency comb technique**, was rewarded ^{[1]} by the Nobel Prize in Physics 2005 (Roy J. Glauber, John L. Hall and Theodor W. Hänsch). Current developments based on Frequency Combs look now for **broadband spectroscopy with sub-Hz precision** ^{[2]}.

**Broadband spectroscopy with sub-Hz precision based on Frequency Comb and Fourier Transform Spectrometer**

The analysis of laser Frequency Combs is particularly interesting with **High-Resolution Fourier Transform Spectrometer (FTS)**^{ [3]}, since the repetition of the laser pulses can be directly detected. Fourier spectrometry can be physically seen as the **most direct spectral analysis technique**, requiring no frequency-deviation conversion introduced by the dispersion spectrometers.

The time-bandwidth product (TBWP) of a pulse is the product of its temporal duration Δτ and spectral width Δν (in frequency space): Δτ.Δν = TBWP. In ultrafast laser physics, it is common to specify the full width at half-maximum (FWHM) in both time and frequency domains*.* The minimum possible time-bandwidth product is obtained for bandwidth-limited pulses. For example, it is ~ 0.315 for bandwidth-limited sech²-shaped pulses and ~ 0.441 for Gaussian-shaped pulses. This means that **for a given spectral width, there is a lower limit for the pulse duration**. This limitation is essentially a property of the Fourier transform.

With FTS technology, the signal to noise ratio of the spectrum is linked to the **spectral bandwidth of analysis**: the larger it is, the noisier the spectrum, since it requires a mathematical inversion operation with sometimes small defects. But all the necessary information for the characterization of a frequency comb can be **directly deduced from the spatial signal of the interference pattern of a static FTS** (see Figure 2) in function of its optical path difference (OPD).

**– Frequency comb analysis with FTS –**

The coherence length *L _{c} *of each wavepacket is inversely proportional to the

**bandwidth**

**Δλ**

**of the frequency comb**centered at the wavelength λ:

*L*= (4.ln2/π) * (λ²/Δλ)

_{c}The spatial optical distance between each wavepacket δOPD gives a **direct access to the frequency rate δν of the comb**: δOPD = *c */ δν , with *c* the speed of light.

The maximal spatial optical distance OPD_{max} of the FTS determines its equivalent **spectral resolution** dν_{max} = *c* / OPD_{max} (e.g. 5 GHz for 6 cm maximal OPD).

*Figure 2. Representation of 10 ps laser pulses repetitions from 15 GHz repetition rate frequency comb, limited to 1.5 nm spectral width (bottom), detected by a SWIFTS spectrometer (static high-resolution FTS with ~ 5 GHz resolution), with the detection of about 7 wavepackets on ± 6 cm Optical Path Difference (OPD) on the interference pattern (top).*

With high-resolution FTS, the **magnitude** *m *= √(*Re*²+*Im*²) and the **phase** ϕ = arctan(*Im*/*Re*) **information of each peak of the frequency comb are available**, with real *Re* and imaginary *Im* parts of the complex Fourier transform of the interference pattern.

Impacts of jitter in amplitudes and/or in frequencies of the frequency comb affects all the OPD samples of the FTS (see Figure 3): with this analysis, scanning a frequency comb over an absorption line (e.g. Rubidium) can result in a **very-high resolution spectrometer with sub-Hz tuning range and accuracy**.

*Figure 3. Jitters on lines amplitudes (top graph) or on frequency repetition (bottom graph) of a frequency comb translate into spatial information distributed on each optical path difference (OPD) of a Fourier spectrometer. *

**– References –**

[1] “The Nobel Prize in Physics 2005”. Nobelprize.org. Nobel Media AB 2014.

[2] Rutkowski, L. et all, “Optical frequency comb Fourier transform spectroscopy with sub-nominal resolution and precision beyond the Voigt profile”, in Journal of Quantitative Spectroscopy and Radiative Transfer, Volume 204, 2018, Pages 63-73, ISSN 0022-4073.

[3] Mandon J, Guelachvili G, Picque N. “Fourier transform spectroscopy with a laser frequency comb.” Nat Photon 2009;3(2):99-102.