Unstable confocal optical resonators with high magnification (about 100x) are commonly used with Metal Vapor Lasers (MVL) with high-aperture active media. The main problem in modeling these devices is the large single-pass loss,  which requires an accurate resampling process. Other problems include modelling hard apertures when using Fast-Fourier Transform (FFT) calculations, as sharp edges produce a high-frequency drift of the resulting field. 

This article will demonstrate how to handle both these problems. Here is the specification of the laser to be modelled:
      
Specifications:
Laser:

• Copper vapor (CVL)
• Wavelengths: 510.6 nm and 578.2 nm
• Active medium length: 600 mm
• Active medium diameter: 14 mm

Resonator:

• Confocal with output coupler represented by afocal meniscus with reflective coating of 3 mm in diameter
• Primary mirror: R2032mm
• Secondary (output) mirror: R20.04mm
• Distance: 1005.98mm

Experimental and previous theoretical investigations have shown the output radiation to be independent of  starting beam characteristics after at least 2 passes through the resonator. Only diffraction effects and manufacturing and adjustment errors are significant . The following model may be used for system sensitivity analysis of these factors.

The file resonator.zmx is included in the zip archive that may be downloaded from the last page of this article. Note that it has the following system parameters:
System unit to mm (System > General > Units)
Wavelength to 0.5106um and 0.5782um (System > Wavelengths)
Set one field with values X=0 and Y=0 (System > Fields)
Set system aperture as Entrance Pupil Diameter of 0.14 mm (System > General > Aperture)

It is set to the geometry equaling to 2 full passes in sequential mode:

On the surfaces 2, 3, 7, 8, 10, 11, 15, 16 are set circular apertures with semi-diameters 7mm representing the apertures of the active medium.

Surface 17 is a user-defined surface which is supplied with Zemax. It represents a soft edged aperture on the central mirror at output lens. The optical density function is given by D =  0.5D_max (1.0 + cos(πr )), where  r is the radial coordinate normalized to the semi-diamater of the surface.

Surfaces 18 and 19 represent the output coupler and may be omitted. The distance 5e4mm at the19^th surface is used for analysis at far field (actually it is not real far field, but the distance is good for analysis).

Coordinate breaks are inserted to model the primary mirror misalignment. Note that if this mirror is decentered, then the chief ray will miss the center of the output surface (this is usually compensated by adjustment and there is no case when only offset of tilt only is present). So the needed tilt may be found by simple optimization. For modelling of significant offset is must be changed iteratively or also by optimization. Operands of merit function are the following:

      
      Here REAY controls the needed tilt and PMVA must be modified to desired target offset (shown +10mm).

 If one starts now Analysis>Physical optics>Physical Optics Propagation with the following settings, than will get an odd result (note the small point in the center of the POp output drawing):
    

 

The reason is in the high losses seen by each pass: if the magnification is about 100x then only 1/(100^2) of the overall number of rays are kept in the resonator for the next pass. This problem may be overcome by following actions:

The most crucial place is reflection by the output mirror of the beam after each pass. Here we have only one such reflection. The radiation to this surface must come with the largest sampling that the PC can provide (I use 1024x1024 at 1Gb RAM PC). To propagate further after this reflection the radiation must be resampled to make effective use of the data. Let’s study propagation from the entrance pupil:

•  The initial beam is defined a top hat with sampling 512x512 and diameter of 0.4 mm because after first reflection at secondary mirror will be kept only
  
  d_in = d_max/b = 14/101 ~ 0.14 mm
  
  and the rest (0.4-0.14=0.26mm) diameter is used to provide a good FFT working guard band to prevent aliassing
• At the primary mirror we have only 180x180 (because (0.14/.4)*512 ~ 180) effective sampling. We keep in mind that next reflection will be with huge spatial losses, so here we may resample the beam again. Apertures between primary and secondary mirrors will not influence the final field distribution, so the sampling to width ratio must be as large as possible considering computer RAM size and stability of output beam profile: if the sampling per mm is not enough, then non-symmetrical modes will appear as shown below (sampling 1024 and width 20 used):
  
  
  
• Physical optics settings at surface 5 will be as follows for satisfactory results (if sampling of 1024 is not acceptable due to insufficient RAM, a smaller width can be used):
  
  

After 2 apertures, radiation is incident on the secondary mirror and only 7x7 effective beam sampling is kept ((0.14/10.2)*1024 ~ 7!). We need to resample to get back to the original sampling:



After next reflection from primary mirror at 2-passes system the final beam will be produced, so the width of the beam array must be larger than the beam width for good FFT functionality (a good guard band).



The next 2 apertures are accounted without any additional settings.
Then light hits surface 17, which has a variable transmission filter surface representing the output coupler. Because of this surface transmission function, the pilot beam parameters must be recalculated for correct propagation after this surface:

With these changes, the final field distribution will be as follows (at the surface 19 with settings as for first analysis):

This is in excellent agreement with experimental measurements.

A comprehensive treatment of a high magnification output coupler has been presented. It is important to resample the POP beam at the primary mirror to account for losses due to the magnification of the cavity. 

The model can be easily extended:

• Further beam propagation may be analyzed by setting appropriate surface and distance of propagation
• Propagation of both wavelengths in the CVL can be studied
• The influence of output meniscus lens can be studied
• The influence of the reflection coefficients

The Physical Optics model allows detaild modelling of the cavity.