2.1 Ultrafast solid-state lasers - ETH - the Keller Group

74 2.1.3 Overview of ultrafast solid-state lasers [Ref. p. 134
by diffraction, and ultimately pump-induced damage to the crystal. However, diode laser arrays
or bars do not generate diffraction-limited pump beams, which makes the situation a bit more
complicated and is therefore explained next.
The propagation of diffraction-limited Gaussian laser beams is extensively described in many
text books, as for example in [91Sal2] and [86Sie]. A beam quality factor M 2 was introduced to
describe the propagation of non-diffraction-limited beams [89Sas]. The objective was to provide
propagation equations for non-diffraction-limited beams that retain the simplicity of the fundamental
Gaussian mode beam equations. The M 2 -factor is given by
M 2 ≡ θ
λ
with θ G ≡ , (2.1.3)
θ G π W 0,G
where θ is the actual far-field divergence angle of any beam with any mixtures of modes, θ G the
far-field Gaussian beam divergence angle, W 0,G the beam waist of a Gaussian beam which is set
equal to the beam waist of the non-diffraction-limited beam. The quantity M 2 is then a numerical
expression of (inverse) beam quality with 1 being a perfect Gaussian beam and higher values
indicating “poorer” quality. This is entirely equivalent to the “number of times diffraction limit”
quantity introduced by Siegman [86Sie]. The beam quality does not give any information about the
details of higher-order mode content in the beam. The propagation of laser beams with M 2 larger
than 1, can be reduced to standard Gaussian beam propagation after substituting the wavelength
λ n with a new “effective wavelength” λ eff given by
λ eff = M 2 · λ n , (2.1.4)
where λ n is the wavelength in the dispersive medium (i.e. λ n = λ/n) in which the beam is propagating.
Physically, this means that non-diffraction-limited beams propagate like an ideal diffractionlimited
Gaussian beam but with the new, longer “effective wavelength”. Beams with larger M 2
have larger “effective wavelengths”, and therefore a smaller depth of focus for a given beam waist.
The output beam of a laser diode array or broad-stripe diode suffers from poor beam quality.
In the so-called “fast” axis, perpendicular to the pn-junction of the diode laser, the light diverges
with a large angle (25 ◦ to 40 ◦ typically) from a narrow aperture of ≈ 1 μm. However, in this
direction the light is nearly diffraction-limited with Mfast 2 ≈ 1. Thus, even though the light in the
fast axis is highly divergent, it can be efficiently collected with a “fast” high-numerical aperture
lens and tightly focused due to its diffraction-limited nature. In the “slow” axis, parallel to the pnjunction
of the diode laser, the beam typically has a divergence of ≈ 10 ◦ . For single-stripe diodes,
the emitting aperture is ≈ 3 μm, resulting in a beam close to diffraction-limited. For higher-power
“arrays” of such apertures, the divergence is also ≈ 10 ◦ , but the total aperture has increased to
typically 50 μm to more than 200 μm, or in case of “arrays of arrays” (i.e. bars) to a width of
approximately 1 cm. The diode laser light in the slow axis is therefore many times worse than
diffraction-limited. High-brightness diode arrays with ≈ 1 W output power and ≈ 100 μm stripe
width typically have Mslow 2 ≈ 10, whereas low brightness bars with ≈ 20 W and ≈ 1 cm stripe
width have Mslow 2 > 1000. The slow axis ultimately limits the spot size of focused pump due to
the requirements of mode matching to the laser mode.
With such pump lasers, the lowest pump threshold can be achieved with the following
Optimized-Mode-Matching (OMM) design guidelines applied to both fast and slow axis of the
diode pump laser [90Fan, 97Kop3]:
1. Determine M 2 for the pump beam (2.1.3) where 2W 0,G is set equal to the width of the pump
source D p . The width of the pump source is approximately given by the stripe width of a diode
array or bar or more accurately by the second-order intensity moment.
2. Determine the “effective wavelength” λ eff (2.1.4).
3. Set the depth of focus or confocal parameter b of the pump beam approximately equal to the
absorption length L a of the pump beam in the laser medium, i.e. b ≈ L a .
Landolt-Börnstein
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