2.1 Ultrafast solid-state lasers - ETH - the Keller Group
2.1 Ultrafast solid-state lasers - ETH - the Keller Group
2.1 Ultrafast solid-state lasers - ETH - the Keller Group
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74 <strong>2.1</strong>.3 Overview of ultrafast <strong>solid</strong>-<strong>state</strong> <strong>lasers</strong> [Ref. p. 134<br />
by diffraction, and ultimately pump-induced damage to <strong>the</strong> crystal. However, diode laser arrays<br />
or bars do not generate diffraction-limited pump beams, which makes <strong>the</strong> situation a bit more<br />
complicated and is <strong>the</strong>refore explained next.<br />
The propagation of diffraction-limited Gaussian laser beams is extensively described in many<br />
text books, as for example in [91Sal2] and [86Sie]. A beam quality factor M 2 was introduced to<br />
describe <strong>the</strong> propagation of non-diffraction-limited beams [89Sas]. The objective was to provide<br />
propagation equations for non-diffraction-limited beams that retain <strong>the</strong> simplicity of <strong>the</strong> fundamental<br />
Gaussian mode beam equations. The M 2 -factor is given by<br />
M 2 ≡ θ<br />
λ<br />
with θ G ≡ , (<strong>2.1</strong>.3)<br />
θ G π W 0,G<br />
where θ is <strong>the</strong> actual far-field divergence angle of any beam with any mixtures of modes, θ G <strong>the</strong><br />
far-field Gaussian beam divergence angle, W 0,G <strong>the</strong> beam waist of a Gaussian beam which is set<br />
equal to <strong>the</strong> beam waist of <strong>the</strong> non-diffraction-limited beam. The quantity M 2 is <strong>the</strong>n a numerical<br />
expression of (inverse) beam quality with 1 being a perfect Gaussian beam and higher values<br />
indicating “poorer” quality. This is entirely equivalent to <strong>the</strong> “number of times diffraction limit”<br />
quantity introduced by Siegman [86Sie]. The beam quality does not give any information about <strong>the</strong><br />
details of higher-order mode content in <strong>the</strong> beam. The propagation of laser beams with M 2 larger<br />
than 1, can be reduced to standard Gaussian beam propagation after substituting <strong>the</strong> wavelength<br />
λ n with a new “effective wavelength” λ eff given by<br />
λ eff = M 2 · λ n , (<strong>2.1</strong>.4)<br />
where λ n is <strong>the</strong> wavelength in <strong>the</strong> dispersive medium (i.e. λ n = λ/n) in which <strong>the</strong> beam is propagating.<br />
Physically, this means that non-diffraction-limited beams propagate like an ideal diffractionlimited<br />
Gaussian beam but with <strong>the</strong> new, longer “effective wavelength”. Beams with larger M 2<br />
have larger “effective wavelengths”, and <strong>the</strong>refore a smaller depth of focus for a given beam waist.<br />
The output beam of a laser diode array or broad-stripe diode suffers from poor beam quality.<br />
In <strong>the</strong> so-called “fast” axis, perpendicular to <strong>the</strong> pn-junction of <strong>the</strong> diode laser, <strong>the</strong> light diverges<br />
with a large angle (25 ◦ to 40 ◦ typically) from a narrow aperture of ≈ 1 μm. However, in this<br />
direction <strong>the</strong> light is nearly diffraction-limited with Mfast 2 ≈ 1. Thus, even though <strong>the</strong> light in <strong>the</strong><br />
fast axis is highly divergent, it can be efficiently collected with a “fast” high-numerical aperture<br />
lens and tightly focused due to its diffraction-limited nature. In <strong>the</strong> “slow” axis, parallel to <strong>the</strong> pnjunction<br />
of <strong>the</strong> diode laser, <strong>the</strong> beam typically has a divergence of ≈ 10 ◦ . For single-stripe diodes,<br />
<strong>the</strong> emitting aperture is ≈ 3 μm, resulting in a beam close to diffraction-limited. For higher-power<br />
“arrays” of such apertures, <strong>the</strong> divergence is also ≈ 10 ◦ , but <strong>the</strong> total aperture has increased to<br />
typically 50 μm to more than 200 μm, or in case of “arrays of arrays” (i.e. bars) to a width of<br />
approximately 1 cm. The diode laser light in <strong>the</strong> slow axis is <strong>the</strong>refore many times worse than<br />
diffraction-limited. High-brightness diode arrays with ≈ 1 W output power and ≈ 100 μm stripe<br />
width typically have Mslow 2 ≈ 10, whereas low brightness bars with ≈ 20 W and ≈ 1 cm stripe<br />
width have Mslow 2 > 1000. The slow axis ultimately limits <strong>the</strong> spot size of focused pump due to<br />
<strong>the</strong> requirements of mode matching to <strong>the</strong> laser mode.<br />
With such pump <strong>lasers</strong>, <strong>the</strong> lowest pump threshold can be achieved with <strong>the</strong> following<br />
Optimized-Mode-Matching (OMM) design guidelines applied to both fast and slow axis of <strong>the</strong><br />
diode pump laser [90Fan, 97Kop3]:<br />
1. Determine M 2 for <strong>the</strong> pump beam (<strong>2.1</strong>.3) where 2W 0,G is set equal to <strong>the</strong> width of <strong>the</strong> pump<br />
source D p . The width of <strong>the</strong> pump source is approximately given by <strong>the</strong> stripe width of a diode<br />
array or bar or more accurately by <strong>the</strong> second-order intensity moment.<br />
2. Determine <strong>the</strong> “effective wavelength” λ eff (<strong>2.1</strong>.4).<br />
3. Set <strong>the</strong> depth of focus or confocal parameter b of <strong>the</strong> pump beam approximately equal to <strong>the</strong><br />
absorption length L a of <strong>the</strong> pump beam in <strong>the</strong> laser medium, i.e. b ≈ L a .<br />
Landolt-Börnstein<br />
New Series VIII/1B1