## Frequency References

The concept of phase and frequency stabilization relies on the possibility to detect changes of the quantity that is wished to be stabilized. To do so this quantity has to be measured continuously, i.e. has to be compared to a reference. In the case of a laser´s phase or frequency, among others, this reference can be an atomic or molecular transition, an optical cavity with fixed mirror distance, another light source that is believed to be more stable, e.g. A second, Stable Laser or a frequency comb, or a wavelength meter. In this application note, different references are presented and their advantages and disadvantages in terms of stability and convenience are discussed.

#### a) Atomic or Molecular Transitions – Spectroscopy

For many applications, a good and simple way to stabilize a laser´s frequency is to compare it to an atomic or molecular transition. One advantage of this method is that optical transitions are rather narrow (typically several MHz). Furthermore, for experiments working with atoms, this method suggests itself because lasers can be locked directly to the atomic transition of interest. Therefore it is a proven method e.g. for laser cooling of atoms.

The most straightforward realization of a spectroscopy lock is to send part of the laser light through a cell which contains an atomic gas and measure the transmission with a photo diode. Light will be absorbed if the laser´s frequency is close to the resonance frequency of the atom, the closer it is the lower the transmission will be. The width of these spectral features are typically on the order of several MHz. However, the Doppler effect will lead to a broadening of these spectral features. At room temperature, e.g. for rubidium 87, this leads to a spectral width of approximately 0.3 GHz. How an error signal, which eliminates the effect of the Doppler broadening, can be generated, will be presented in the application note “Error Signal Generation”.

### TOPTICA solution

With the CoSy, TOPTICA offers a plug-and-play spectroscopy package which is based on Doppler free saturation spectroscopy. It includes the gas cell as a reference together with the optical setup and a fast photo diode for error signal generation. As the CoSy is fibre coupled no alignment is necessary.

#### b) Optical Cavities

An alternative approach of measuring a laser´s frequency is to link it to the geometrical properties of an optical cavity. The most common type for this purpose is the Fabry-Pérot interferometer, i.e. two parallel mirrors facing each other. The approach is based on the idea that light can only resonate, and thus be transmitted, if twice the distance between the two mirrors, i.e. the optical path length of a round-trip, is an integer multiple of the wavelength. Deviations of the laser frequency from this condition will decrease the transmission. Close to resonance the relation between transmission and frequency deviation is given by a Lorentzian function with a full-width-at-half-maximum (FWHM) linewidth

$$\Delta\nu_c=\frac{\Delta\nu_{FSR}}{\mathcal{F}}$$

where $$\Delta\nu_{FSR}=\frac{c}{2l}$$ is the free spectral range of the cavity, i.e. the frequency difference between adjacent resonances, c is the speed of light, l is the cavity length and $$\mathcal{F}=\frac{\pi\sqrt{R}}{1-R}$$ is the Finesse of the cavity. R is the intensity reflectivity of the mirrors. Therefore, to obtain small cavity linewidths, the reflectivity of the mirrors has to be large. For a cavity with a length of 10 cm and a Finesse of 100.000 this leads to a cavity linewidth of only approximately 15 kHz.

Typically, to achieve a long-term stability of the resonator length, the mirrors are mounted in a material called ultra-low expansion (ULE) glass. It has the property that its heat expansion coefficient can be brought close to zero by fine-tuning its temperature.

If it is the goal to reduce the linewidth of the laser as much as possible, a high-finesse ULE cavity certainly is the right choice. However, engineering such a cavity which fulfils the stringent requirements concerning long-term stability and finesse is a rather sophisticated undertaking. Beyond that, despite of the ULE material, on a long time scale small frequency drifts occur.

#### c) Frequency Comb

Another way to stabilize a laser´s phase or frequency is to compare it to a stable light source. A tool which has been developed for exactly this purpose is the frequency comb. Its spectrum exhibits peaks at fixed equidistant frequencies which is realized by mode-locking a spectrally broad laser. In the time domain this corresponds to a regular train of laser pulses with repetition rate $$f_{rep}$$, which equals the separation between the peaks of the comb spectrum. The relative stability between individual peaks is obtained by locking $$f_{rep}$$ to a radio frequency (RF) reference. To obtain absolute stability, in addition, either the offset frequency $$f_{CEO}$$, called carrier-envelope offset, of the periodic pattern from zero has to be stabilized to a reference or a method called difference frequency generation, which passively sets $$f_{CEO}=0$$, has to be implemented Therefore the long-term stability of the individual peaks is inherited from the underlying RF reference. It is therefore possible to transfer the long-term stability of e.g. an atomic clock to the regime of optical frequencies. Such a degree of stability cannot be offered by any other optical reference.

The frequency comb is the perfect tool if many lasers are to be stabilized to a common reference. Locking different lasers to a common reference is a particularly good idea if relative fluctuations between different lasers are of importance. Using a frequency comb, relative stability can be achieved between lasers separated in wavelength by several hundreds of nm. Additionally stabilizing a frequency comb to a short-term stable optical reference results in the best laser reference system available.

### TOPTICA Solution

The TOPTICA DFC CORE + is a frequency comb which is based on Difference Frequency Generation (DFG), i.e. the carrier-envelop offset is passively fixed to $$f_{CEO}=0$$. It is therefore inherently stable and combines high robustness and high-end performance. Several wavelength extensions are available to bridge the wavelength range between 420 and 2200 nm. On top of that the DFC CORE + can be extended by further hardware for beat detection and processing. Together with fast locking electronics and a convenient control software this forms a complete reference system for controlling and locking several lasers.

#### d) A Second, Stable Laser

A laser locked to one of the above mentioned references can serve as a reference itself. Difference frequencies are limited by the bandwidth of the employed photodetector, typically up to several GHz, unless a frequency comb is employed. With this method it is possible to achieve near-perfect relative phase stability between the two lasers, which is advantageous for many applications. A typical application example is a phase stable Raman pair addressing two hyperfine levels.

#### e) Wavelength Meter

A very convenient way to determine the frequency of a laser is to use a wavelength meter. With typical measurement ranges covering more than an octave, they offer the possibility to lock the laser to almost any frequency. However, the accuracy of a wavelength meter is typically limited to several MHz. Also, the time needed for measuring the wavelength and generating a control signal is much larger than for other schemes. Wavelength meters are therefore mostly used for stabilizing the laser against drifts occurring on time scales of seconds and more but not for reducing the laser´s linewidth.