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

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– 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

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) = f0 + n fr  , where n is an integer, fr is the comb tooth spacing (equal to the mode-locked laser’s repetition rate or, alternatively, the AM frequency), and f0 is the carrier offset frequency, which is less than fr.

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 Lc of each wavepacket is inversely proportional to the bandwidth Δλ of the frequency comb centered at the wavelength λ: Lc = (4.ln2/π) * (λ²/Δλ)

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 OPDmax of the FTS determines its equivalent spectral resolutionmax = c / OPDmax (e.g. 5 GHz for 6 cm maximal OPD).

Figure 2-v2

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 = √(Re²+Im²) 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 3a-v2

Figure 3b-v2

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.


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