Petzval Lens Geometric Modulation Transfer Function

Petzval Lens Geometric Modulation
Transfer Function

In this tutorial we demonstrate how to use an Application Method to compute the modulation transfer function (MTF) of a lens. The optical transfer function (OTF) is a measure of an optical systems ability to resolve an object at a given spatial frequency. The OTF is defined as

(1)

where the contrast is defined in terms of the intensity as

(2)

The OTF is a vector quantity including a phase term. When referring to the modulation transfer function, only the amplitude is considered.

The modulation transfer function at a given spatial frequency ν can be computed from the line spread function (LSF). In the context of an image of a point source (or, a collimated source at infinity), the LSF is an integration in a single direction of the point spread function (PSF). The MTF is then given by (Ref. 1)

(3)

in which the expressions for Lc and Ls are

(4)

and

(5)

The LSF is given in terms of δ, the spatial location on the detector plane. The MTF, which is computed in the sagittal (x) or tangential (y) directions, is the response of the LSF to a signal that is spatially periodic at the frequency ν in either of these directions.

Geometric Modulation Transfer Function

The MTF can be calculated using the results of a Geometrical Optics ray trace. In this method for computing the geometric MTF, the LSF is generated using the number density of ray intersections on the focal plane (that is, the spot diagram; see Figure 1). That is, for each of the sagittal (x) or tangential (y) directions, the LSF is given by

(6)

and

(7)

where δ is the distance relative to the centroid of the spot along either the x or y directions. The LSF of the spot diagram shown in Figure 1 can be seen in Figure 2.

Figure 1: On-axis spot diagram. The line spread function (LSF) of this spot can be seen below.

Figure 2: Line spread function (LSF) computed using the spot from the previous figure.

Model Definition

The model used for this example is the Petzval Lens, which can be found in the Ray Optics Module Application Libraries. The lens has a 100 mm focal length and includes a field flattening lens to provide good image quality over a ±10° field of view (Ref. 2). An overview can be seen in Figure 3.

The computeMTF is used to compute and plot the LSF and MTF curves and it can be added to the model using the . The method computeMTF also uses the mtfutil and plotutil. The Java code snippets for all of these methods can be found in the Appendix.

Detailed step-by-step instructions for creating and running the method can be found in the section Modeling Instructions. First the sagittal and tangential LSFs are captured in functions with piecewise cubic interpolation. Next, a set of functions are created to be used in the integration of the expressions for Lc and Ls. Computing the MTF involves iterating over spatial frequency with a of these expressions. The MTF results are also placed in functions for plotting.

Figure 3: The Petzval lens with an on-axis ray trace.

The LSF and MTF plots each use a with pointing to datasets to create the curves. Each of the grid datasets uses the corresponding LSF or MTF functions as a source. The method creates and updates these features as required. The settings interface (see Figure 4) provide the user with control over several options.

Figure 4: The Settings window for the MTF method call.

Results and Discussion

The default settings for this method will compute the MTF for the first active release feature. The spot diagram for this release was seen in Figure 1 together with the resulting LSF (Figure 2). These were used to compute the MTF seen in Figure 5. Note that the calculations presented here are all monochromatic with λ = 550 nm.

Figure 5: The modulation transfer function (MTF) of the Petzval lens for an on-axis field.

After enabling all three release features, the method can be run again. The spot diagram shown in Figure 6 now includes two off-axis fields. The line spread function for these fields are shown in Figure 7 and the resulting MTF is seen in Figure 8.

These results show how the spatial resolution of the Petzval lens can be analyzed as a function of field of view. The use of an MTF evaluation also allows the impact of the off-axis aberrations on the MTF in sagittal and tangential directions to be displayed.

Figure 6: The Petzval lens spot diagram with three fields.

Figure 7: The line spread function (LSF) of the spots shown above.

Figure 8: The modulation transfer function (MTF) of all three fields for the Petzval lens.

References

1. W.J. Smith, Modern lens design, vol. 2. New York, NY, USA: McGraw-Hill, 2005.

2. M.J. Kidger, Fundamental Optical Design, SPIE Press, 2001.

Ray_Optics_Module/Lenses_Cameras_and_Telescopes/petzval_lens_geometric_modulation_transfer_function

Modeling Instructions

Application Libraries

From the menu, choose .

In the window, select in the tree.

Click

Global Definitions

To increase the accuracy of the MTF calculation we increase the number of rays to be traced.

Parameters 2: General

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
N_ring	30	30	Number of hexapolar rings

Application Builder

The geometric MTF can be computed using an Application Method. These may be added to an existing model using the . Note that the is only available in the Windows® version of the COMSOL Desktop.

In the toolbar, click

Methods

The code for the computing and plotting the MTF can be simplified by using utility classes.

util1

In the toolbar, click

and choose .

In the window, click .

In the window for , type mtfutil in the text field.

mtfutil

Right-click and choose .

Copy the code for the MTF utilities getReleaseList, createGroups, getSpotRadius, and evaluateLSFRays and paste it into the editor for mtfutil. This code may be found in the Appendix to this document.

util1

In the toolbar, click

and choose .

In the window for , type plotutil in the text field.

plotutil

Right-click and choose .

Copy the code for the plot utilities plotLSF, plotMTF, getPlotFeature, and updateDatasets and paste it into the editor for plotutil This code may be found in an appendix to this document.

Global Method

Next, create the Application Method computeMTF.

In the toolbar, click and choose .

In the dialog box, type computeMTF in the text field.

Click .

computeMTF

In the window, under click .

In the window for , locate the section.

Find the subsection. Click

. Repeat this action to add a total of 9 .

In the table, enter the following settings:


Name	Type	Default	Description	Unit
runstudy	Boolean	true	Always run the study?
ismono	Boolean	true	Compute monochromatic MTF?
lambda_mono	String	550[nm]	Monochromatic wavelength
useall	Boolean	false	Compute MTF for all enabled releases? (Default is to use the first enabled release)
nustep_str	String	21	Number of frequency steps
numax_str	String	125	Maximum spatial frequency (1/mm)
orient	String	xy	Focal plane orientation (xy, yz, or zx)
lsfplot	Boolean	true	Create LSF plot?
newplot	Boolean	false	Create new plots? (Default is to update existing plots)