Technical Note » Custom Resonators I

Custom Resonators I

Part 1: Spatial mode selection

Introduction

Historically, optical resonantors have comprised an open cavity formed with otpical mirrors. Except for a few particlular cases, the mirrors used are either spherical or flat. In stable type resonantors, the eigenmodes are determined by those modes with wavefronts that exactly match the curvature of the resonator mirrors. Because all the Hermite-Gauss modes have spherical wavefronts, using flat or spherical mirrors in a resonator implies that all the Hermite-Gauss modes will be stable spatial eigenmodes of the system. These modes are referred to as the TEM_nm modes, with higher order modes having higher values of n and m. The above density plots show the mode field distribution for the TEM_oo (top left), TEM₁₀ (top right) and TEM₁₁ (bottom left) modes, as well as an arbitrary sum of several low order modes (bottom right). It is not possible to induce losses for higher order modes by phase alone, since their wavefronts are identical except for a constant phase offset. In most resonators, the fact that the higher order modes are larger in size than the lower order modes is exploited, and mode selection is achieved using amplitude filtering, i.e., by introducing amplitude losses rather than phase matching losses.

Graded reflectivity mirrors and intra-cavity apertures are examples of components used to achieve mode selection by amplitude filtering.

Gaussian Beams

For many applications, where “good” beam quality and low divergence is needed, an intra-cavity aperture will be used to suppress the higher order spatial modes. There is however a balance between keeping the divergence low, and maximising the energy extraction from the laser. All coherent radiation obeys the Fourier relation.

where ω_o and θ_o are the beam waist and far field divergence , defined as second intensity moments, and the M² parameter is called the beam quality factor. An M² of 1 indicates a perfect Gaussian beam. This is usually referred to as a TEM_oo output.

Higher values for the M² parameter imply that the beam deviates from Gaussian, and can be made up of higher order modes (often many of them). Such outputs from resonators are commonly called multimode. Because the selection between the two is done by amplitude filtering, the result is always that multimode beams have higher energy than TEM_oo beams, but TEM_oo beams will have a lower divergence for any given beam size.

In conventional resonator design, it is not possible to specify an arbitrary output intensity distribution and phase. This is because although the TEM modes form an orthogonal set, the amplitudes of each cannot be controlled easily within the resonator. However, it is possible to build a resonator that will output arbitrary beams, by using aspheric reflective elements and/or diffractive optical elements.

The idea is based on the fact that if the phase of a mirror is not spherical, then the reflected wave front will depend not only on the incoming beam’s wave front, but also on it’s (electric) field distribution. Because the mirror acts differently for different field amplitudes, it can be designed to act as a phase only mode selector. It can be shown that a resonator with such a mirror will have an eigenmode with a wave front that exactly matches the phase of the mirror. An alternative view is that the resonator acts as a phase-conjugate resonator, but with the difference that the mirror can phase conjugate only a particular field distribution.

Design example

The intensity and phase of the desired beam is specified at the output coupler. In this example, a top hat beam is desired for material ablation (see our Application Note titled Optical beams for material processing). This field is then propagated to the back mirror, and the phase of the beam immediately in front of the mirror is extracted. Say this incoming beam has a phase given by Φ₁ and after reflection from the mirror, the phase of the beam is Φ₂. If the
mirror has a phase profile given by Φ₁, then clearly ...

Moreover, for this resonator to have the desired output as a stable mode, it must act as a phase conjugate mirror for this particular field distribution. Thus and therefore the phase profile of the mirror is given by ...

This last equation also states that the wave front of the incoming beam will match the shape of the mirror; a requirement in all resonators for the fundamental mode under consideration to also be an eigenmode.

The top hat above is specified as a super-Gaussian beam of order 10, with a 5mm beam radius. The phase at the output coupler is specified to be planar. The resonator consists of a flat output coupler, and a back reflector with a shape that is to be determined.

Using the Kirchoff-Fresnel diffraction integral, the field is propagated back to the mirror from the output coupler, and the phase extracted. The shape of the mirror for this particular beam can then be calculated, and is shown below.

This can either be fabricated by direct cutting of a spherical surface, or by using a diffractive optical element. This particular mirror surface is close to spherical in the central region. In calculating this profile, the piston phase term has been left out (recall that the wave fronts must match within a constant).

There are a few constraints placed on the designer in such systems. Firstly, using the mirror alone as the phase selecting element means that the physical length of the resonator becomes important. If the length is far greater than the Rayleigh length for the specified field, then the wave front for all fields will approach that of a spherical wave and the mode selectivity will be affected adversely.

If on the other hand the length of the resonator is far shorter than the Rayleigh length, then it can be that the fabrication of the element becomes difficult.

In most cases, a combination of phase and amplitude filtering techniques are used to design a resonator that will have good energy extraction, and the desired beam shape. At SDILasers we can offer resonator solutions for optimal energy, divergence, and beam size, as well as custom designs to deliver an user defined output intensity (within the laws of physics of course).

Applications

One very prominent application for shaped beams is materials processing. In processes where ablation is the primary mechanism, such as in paint stripping or pollutant removal from buildings, the peak intensity of the beam needs to be close to the threshold intensity of the process.

If the beam fluence is higher than the threshold, this energy is effectively lost to momentum of the substrate. A Gaussian beam used in this application would be inefficient: the peak has too much energy, whereas the edges have too little. This is because the peak fluence in Gaussian beams is double the average fluence, with the fluence falling off towards the edges of the beam.

Therefore the effective area of the beam that can be used would be small. Top hat beams have the useful quality that their peak fluence is the same as their average fluence, therefore not only can all of the beam area be used, but in fact the size of the top hat can be made somewhat larger than that of a Gaussian beam. The efficiency of ablation with such a beam would be significantly better…

Further Information

For further information on this topic, or on our custom laser solutions, contact us.