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The strict definition of an afocal system is a system in which both object and image conjugates are at infinity. Such systems would include, for example, a laser beam expander in which both input and output beams are collimated. Another example is a system like binoculars where light is brought to focus by the eye, and the binocular design itself relays light from an infinite object conjugate to an infinite image conjugate, with some angular magnification.

The term "afocal" is also used sometimes to mean any system in which the image conjugate is at infinity. ZEMAX uses the term "afocal image space" to describe any system in this category or in the full afocal category. The major consequence is that the units we use to describe optical performance in the image space change from spatial units to angular units. Different units are used in different applications, and the choice of units is made on the Units tab of the System > General Dialog box:

As a result, the various ZEMAX analysis features will report in different units:

Analysis Type	Focal Image Space	Afocal Image Space
Transverse Aberrations	Micrometers (µm)	As chosen above
Modulation Transfer Function	cycles per millimeter	cycles per angular unit
Field Curvature	length units	diopters (inverse meters)

Whether ZEMAX uses focal or afocal units is set by a control on the General > System> Aperture tab, "Afocal Image Space":

Other than the change of units, most ZEMAX features work exactly the same with focal or afocal image spaces. Some features are specific to focal systems: relative illumination, for example, has no physical meaning in an afocal system. In addition there are default merit functions for either mode: spot radius, spot in x only, spot in y only for focal system and angular radius, angular x only angular y only for afocal. Wavefront error can be used in either mode.

In this article we will design two simple systems: a laser beam expander which is a true afocal system, and a cylindrical lens which is focal in one direction and afocal in the other.

The zip archive which accompanies this article (which can be downloaded from the final page of the article) contains a starting point design beam_expander.zmx. This is intended to be a 5x beam expander, working at the red He-Ne line, and to have minimum RMS wavefront error. In the starting design there is no power in the optics and therefore no beam expansion:

Click on System > General  and choose "Afocal Image Space" so that ZEMAX computes all parameters in afocal units.



Then open the merit function (Editors > Merit Function) and select Tools > Default Merit Function:



Note that we can build a default merit function to minimize wavefront error, spot radius (and x, y individually) or angular error as a radius or as x and y separately. In this case, we will choose Wavefront, and use 5 rings in the Gaussian quadrature algorithm because we want a well-corrected system. Press OK to build the default merit function.

The only extra information ZEMAX needs is the size of the output beam. The input beam is 5 mm, and the magnification is x5, so the output beam should have a diameter of 25 mm. Insert a new operand before the DMFS statement in the merit function, and enter the REAY operand as follows:



This requires the real ray y-coordinate on surface 6 (the image surface) to have a height of 12.5 mm. Then click Tools >  Optimization > Optimization and press the Automatic button!



ZEMAX quickly optimizes the afocal system:

So how good is our afocal system? Look in the merit function, at the value of the REAY operand. It should show a value of exactly 12.5. So, we are getting the beam expansion we asked for. Then open the OPD, Ray-Fan, Point Spread Function and Modulation Transfer Function windows.

The OPD should appear like so:

this shows focus, spherical and higher-order spherical all balancing, and the system's PTV wavefront error to be less than l/1000.  The ray-fan plot is also interesting:



Note the focus, spherical and higher order spherical are also clearly shown, but also note that the units used are µR: microradians. This plot is showing the deviation from perfect collimation directly. The spot diagram also shows this:

The RMS angular deviation is 0.265 µR. Of course, diffraction effects are much larger: on the settings of the spot diagram, click the button "Show Airy Disc". Diffraction effects limit the resolution to about 30 µR.

To see this, look at Analysis > PSF > FFT PSF Cross-section:

This shows that diffraction effects produce an Airy disc of around 30 µR. Analysis > MTF > FFT MTF shows the contrast ratio of the system in units of cycles per milliradian:

Cylindrical systems are only a little more complex, because these systems are focal in one plane and afocal in the other plane. From the zip archive, open the file cylindrical_lens.zmx:



This shows a cylindrical lens, which has a flat rear surface, and a toroidal front surface. This lens is intended to produce a line focus, with the smallest spatial extent in y and the smallest angular divergence in x. This is easy to do.

Again, open the merit function editor and the default merit function tool. Set this as follows:



This will build a merit function that will minimize the y-spot size. Scroll to the end of the merit function, and note that ZEMAX has entered 42 lines of operands. Then use the default merit function tool again:



This build the operands to control the angular spread of the beam in x. The reason we start at line 43 is that we want to keep the spot-in-y operands: so, this merit function will require the smallest spatial extent in y, and the smallest angular extent in x: a line focus. The optimization variables are the y-radius, x-radius and back focal distance. Optimize, and ZEMAX again quickly produces the best system.

Note that this technique can be extended using the IMSF operand. IMSF allows the image surface to be re-defined on the fly in the merit function. Therefore, if a system is focal on surface 10, but afocal on surface 6, it can be easily modelled by building an Angular Radius merit function, with IMSF=6 immediately before it in the merit function, and then adding an RMS Spot merit function with IMSF=10 immediately before that.

Note also that the multi-configuration operand AFOC allows the afocal mode to be zoomed between configurations. The ZPL keywords GETSYSTEMDATA and SETSYSTEMPROPERTY allow control of the Afocal Image Space switch from within a ZPL macro.

This article has discussed how to design afocal systems in ZEMAX. The main points are:

• The major difference between focal and afocal systems is that focal systems are measured in spatial units and afocal systems are measured in angular units
• The Afocal Image Space switch switches ZEMAX from using spatial units to angular units where appropriate
• The default merit function tool allows you to easily target both spatial and angular properties of the beam
• The IMSF operand allows you to target spatial properties at one surface, and angular properties at another (for example) to control both focal and afocal properties of the optical system at any surface
• The AFOC multi-configuration operand lets you define focal and afocal configurations of a design.
• The ZPL keywords GETSYSTEMDATA and SETSYSTEMPROPERTY allow control of the Afocal Image Space switch from within a ZPL macro

References

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