This article covers afocal system design and optimization in OpticStudio. In particular, we discuss what afocal systems are, how to analyze afocal systems in terms of angular units, how to handle cylindrical systems, and how to work with systems with multiple focal and afocal spaces.

**Authored By Mark Nicholson**

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## Introduction

The strict definition of an afocal system is one where both object and image conjugates are at infinity. Examples include a laser beam expander where the input and output beams are collimated, or, even, a set of binoculars since the design itself relays light from an infinite object conjugate to an infinite image conjugate (with some angular magnification). The term "afocal" is also sometimes used to mean any system in which the image conjugate is at infinity.

"Afocal image space" in OpticStudio covers both definitions. During analysis of these systems, a different reference or different units are usually required. OpticStudio handles this behind the scenes when the user enables Afocal Image Space.

Using Afocal Image Space

To enable Afocal Image Space, navigate to the **System Explorer...Units** and check the setting.

The major consequence of doing this 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 at **System Explorer...Units**.

As a result, the various OpticStudio analysis features will report in different units.

Other than the change of units, the other primary difference between focal and afocal mode is the reference wavefront (the “perfect” wavefront against which the actual wavefront of the system is compared). In focal mode, the reference wavefront is spherical, whereas in afocal mode, the reference wavefront is planar. This affects the results of all wavefront-based analyses, as well as wavefront optimization.

Most OpticStudio 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 can be used for focal systems, and Angular for afocal systems. Wavefront error can be used in either mode; the reference wavefront is either spherical or planar, depending on if afocal mode is being used.

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.

Optimizing afocal systems

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 Explorer...Aperture** and choose **Afocal Image Space** so that OpticStudio computes all parameters in afocal units.

Then open the Merit Function via **Optimize...Merit Function Editor** and select **Optimization Wizard** from the settings.

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 OpticStudio 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 **Optimize...Optimize!** and press the **Start** button.

OpticStudio quickly optimizes the afocal system.

Analyzing data in angular units

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 as follows.

This shows focus, spherical and higher-order spherical all balancing, and the system's PTV wavefront error to be less than 5.0*10-4 waves. The Ray Fan plot is also interesting, as the spherical and higher order spherical aberrations are also clearly shown, but with units of arc-min. That is, this plot is showing the deviation from perfect collimation directly.

The spot diagram also shows that the RMS angular deviation is less than 0.001 arc-min. However, diffraction effects are much larger than this, which can be displayed by enabling **Show Airy Disc** in the Spot Diagram settings.

Upon further analysis, the diffraction effects limit the resolution to about 0.107 arc-min, which can also be visualized using the FFT PSF Cross Section, available at **Analyze...PSF...FFT Cross-Section**.

Finally, we can view the system's MTF by navigating to **Analyze...MTF...FFT MTF**. This shows the contrast ratio of the system in units of cycles per arc-min.

Modeling cylindrical systems

Cylindrical systems are only a little more complex, because these systems are focal in one plane and afocal in the other plane. From the Article Attachments, download and 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. To do this, open the **Optimization Wizard** and define the following settings.

This will build a Merit Function that will minimize the Y-spot size. Scroll to the end of the merit function and note that OpticStudio has entered 41 lines of operands. We can then use the Optimization Wizard again to create a Merit Function to minimize the divergence in X.

This builds 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 OpticStudio 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.

KA-01529

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