# Mode coupling

Mode coupling (English: mode locking) is the synchronization of the laser oscillating eigenstates ( modes ) for the generation of extremely short pulses of light down to the femtosecond range. The term mode coupling refers to the fact that a constant phase relationship to one another is sought for the largest possible number of modes . The individual modes are then phase-locked.

## Bandwidth and number of modes

The bandwidth of a laser is primarily determined by the laser medium used. The gain bandwidth of a helium-neon laser (wavelength λ = 632.8 nm) is approx. 0.002 nm. However, there are also extremely broadband laser media such as. B. the titanium-sapphire laser , which has a bandwidth of approx. 400 nm (670-1070 nm).

In a simple 2-mirror resonator, the distance between the mirrors is usually very large compared to the wavelength. The number of possible longitudinal modes in this resonator corresponds to the natural frequencies of the resonator and is also very large.

The frequency between two neighboring modes can also be calculated, where is the speed of light and the resonator length. In a resonator 1 meter long, this frequency is 0.15 GHz. With a helium-neon laser (0.002 nm bandwidth at a central wavelength λ = 632.8 nm corresponds to 1.5 GHz bandwidth) this would mean that 10 modes could exist at the same time. With a titanium-sapphire laser with a bandwidth of approx. 128 THz, approx. 850,000 modes would be possible simultaneously. ${\ displaystyle f = {\ tfrac {c} {2L}}}$${\ displaystyle c}$${\ displaystyle L}$

## principle

The intensity of 75 oscillators with a random phase reference is shown in red. The oscillators have a center frequency of  Hz and a bandwidth of 1 Hz. In green, the same oscillators vibrate in phase and their amplitude is modulated with 10 GHz.${\ displaystyle 10 ^ {14}}$

If several modes start to oscillate in a laser, they have no fixed phase relationship without further measures. There is continuous multi-mode operation (or continuous wave = cw).

By modulating the light power in the resonator with a frequency that corresponds to the length of the round trip time of the light in the resonator , sidebands are formed that correspond to the resonator modes and oscillate in phase with one another (see amplitude modulation ). The modulation generates additional sidebands that are phase-locked and propagate over the entire mode spectrum. The constructive interference of the individual modes creates short pulses. The pulse interval is . ${\ displaystyle f = {\ tfrac {c} {2L}}}$${\ displaystyle L}$${\ displaystyle T = {\ tfrac {2L} {c}}}$

## Number of modes, coherence and pulse duration

Mode-locked pulse train: as the number of modes increases, the pulses become increasingly narrow

In principle, the more eigenstates (modes) of the resonator oscillate coherently with one another in the resonator, the shorter the laser pulses become. Two things have to be guaranteed:

1. The individual modes must remain coherent with one another despite their different frequency, that is, the dispersion of the resonator must be compensated. Every pulse is a soliton .
2. The normally more stable cw radiation must be suppressed compared to the unstable pulses (see "Generation").

For the time course of the intensity of a mode-locked pulse train, the following applies:

${\ displaystyle I (t) \ propto {\ frac {\ sin ^ {2} \ left ({\ frac {N \ omega t} {2}} \ right)} {\ sin ^ {2} \ left ({ \ frac {\ omega t} {2}} \ right)}}}$

( : Repetition rate,: number of modes,: time). The maximum intensity of this function increases quadratically with the number of modes. The width of the peaks decreases to the same extent (see illustration). ${\ displaystyle \ omega}$${\ displaystyle N}$${\ displaystyle t}$

The minimum achievable pulse duration depends on the achievable bandwidth of the laser radiation and the constancy of the phase relationship between the individual modes (the smallest possible residual dispersion / chirp ). The uncertainty principle therefore applies to ultrashort laser pulses

${\ displaystyle \ Delta \ nu \ Delta t \ geq K}$.

${\ displaystyle \ Delta \ nu}$is the frequency bandwidth of the pulse, the pulse duration. is a number that depends on the pulse shape. The shape of the pulse is determined by a number of factors, e.g. B. the design of the resonator. If the pulse has a Gaussian profile, e.g. B. . ${\ displaystyle \ Delta t}$${\ displaystyle K}$${\ displaystyle K = {\ tfrac {2 \ ln 2} {\ pi}} \ approx 0 {,} 441}$

In the inequality, the equal sign applies when there is no longer any chirp (no delay dispersion or frequency modulation) on the pulse. The pulse is then called bandwidth-limited.

## generation

A distinction is made between active mode coupling through acousto-optical modulators (AOM) or electro-optical modulators (EOM) such as the Pockels cell and passive mode coupling through the Kerr lens effect and saturable absorbers . The shortest pulses can be achieved with the passive method. The most widespread ultrashort pulse laser is the titanium: sapphire laser , in which the mode coupling is brought about with the help of the Kerr lens effect.

The short-term pulses generated are a prerequisite for building a frequency comb .