# Noise, Coherence and Dynamics Physical Study

This study requires the investigation of all the axes on which the coherence state can be projected (**|k>**, **|z>** and **|E>**); thus, the polarization state, the beam propagation, the frequency stability and the intensity noise have to be studied.

### Study of the Spatial Coherence

Studying the spatial coherence is similar to study the transverse phase repartition/fluctuations in a plane perpendicular to the propagation. Indeed, in the case of gaussian shaped beams, an ideal, theoretical, TEM00 gaussian beam which phase repartition is perfectly determined in any transverse plane. Unfortunately, a real beam exhibits some additional fluctuations, which strongly impact the beam propagation (leaded by the diffraction laws), thus the M^{2} value.

In VeCSELs, this transverse phase fluctuation is very low, even in thermal-lens stabilized lasers, which exhibit phase fluctuations lower than 2% of wavelength. This was numerically simulated and measured by means of a wavefront sensor based on lateral shearing interferometry technology (SID4-PHASICS), and lead to M^{2} factors lower than 1.2 as shown below :

### Study of the Time Domain Properties

The time domain properties requires the study of two fundamental informations :

- the frequency noise, i.e. the study of the laser frequency fluctuations as a function of time, which allows to define the laser linewidth and lineshape,
- the intensity noise, i.e. the study of the intensity fluctuations as a function of time.

Physically, the fundamental quantum limit of these two parameters are linked to the spontaneous emission which is superimposed on the stimulated emission field. They are thus intimately linked to the photon lifetime in the cavity, thus to both the cavity finesse and cavity length. Because the VeCSEL is based on a very high finesse cavity compared to all other laser technologies, the light emitted by these lasers will experience very low quantum limits, similar to meter-length cavities in more traditional DPSSL lasers.

### Frequency Noise and Linewidth

Thanks to high finesse free-space cavity design, the fundamental frequency noise / linewidth limit, induced by spontaneous emission phase fluctuations and given by the Schawlow-Townes formula, is far below 1 Hz for cm-length VeCSEL cavities. This value is similar to 1m long DPSSL and far below the fundamental limit exhibited by conventional laser diodes or ECDLs (Extended Cavity Laser Diodes).

The frequency noise of VeCSELs is usually measured by injecting the fundamental mode of a Fabry-Perot cavity : each longitudinal mode of the analysis Fabry-Perot cavity can be used, in the neighborhood of a linear section of its wings, as a slope that converts the laser’s frequency fluctuations into intensity fluctuations, which can be measured by means of a simple photodiode. The resulting time-domain signal represents the frequency variations of the laser line as a function of time, and it can be analyzed by calculating the power spectral density, like any noise signal. The typical resulting spectrum is given in the following picture :

Below 1 kHz, the frequency noise is limited by thermal and mechanical contributions (characteristic peaks around 200 Hz) and 1/f technical noise. The frequency noise spectral density is measured up to the cutoff frequency of the analysis Fabry-Perot interferometer, where the frequency noise comes close to the fundamental quantum white noise limit (? 0.1Hz2/Hz up to 200MHz as calculated in our case). However it is experimentally limited here by the VECSEL RIN level above 2 MHz. Above this frequency, the signal reaches the background level, with preeminent shot noise contribution. We believe that the main contribution of the frequency noise above f=200 Hz is the index fluctuations induced by the pump intensity noise through thermal fluctuations. Reducing the frequency noise thus requires enhanced thermal management and low noise pumping. From the experiment, we obtain a rms laser frequency noise of 32 kHz (over 1 ms) for L=7.5mm without any active frequency stabilization. Finally, the linewidth was obtained thanks to integral computations based on the frequency noise spectral density. This leads to a Gaussian–like shape and a linewidth of ??L - 37 kHz (FWHM) over 1 ms, with a slight enlargement of wings due to relatively strong noise peaks in the frequency noise spectrum around f=100 kHz. This linewidth value is similar to what would be measured using a standard heterodyne technique.

### Intensity Noise

Owing to long photon lifetime in the VeCSEL cavity, the laser exhibits a class-A dynamics with a cut-off frequency at low frequency (typically <40 MHz). This allows to obtain a shot-noise limited operation over a wide bandwidth, which is a very interesting property for many tricky applications (such as telecommunications). At higher frequencies, the noise spectrum can experience only beatings between the fundamental mode and the side modes : the VeCSEL thus exhibits a shot-noise limited operation over a broadband frequency range, typically 40 MHz-20 GHz. The RMS noise calculated in the laser bandwith is <0.1%. We show below a Relative Intensity Noise spectrum.