Description

Although FRED does not have an automated calculation for generating the Modulation Transfer Function (MTF) of an optical system, the calculation can be performed within the software using the procedure described in this example.  The MTF of an optical system is the normalized Fourier Transform of the systems Point Spread Function (PSF) and is an image quality metric that indicates how faithfully the optical system can reproduce features of an object of different spatial frequencies.  Assuming an optical system is prepared in the software, the procedure for calculating the MTF takes the following steps:

1.Perform a coherent raytrace of a point object (or plane wave) through the optical system

2.Calculate the Irradiance Spread Function, which gives the system PSF

3.Fourier Transform the PSF

4.Normalize the distribution to get the MTF

Geometry and Source Construction

The optical system to be analyzed consists of an idealized lens element, a detector plane, and an analysis surface.  First, create a new FRED document by going to File > New > FRED Document, or by clicking on the New Document button on the FRED Dashboard.

To create the ideal lens, right mouse click on the Geometry folder and choose, "Create Element Primitive > Ideal Lens".  In the resulting dialog, set the parameter #5 (the lens type) to be "Perfect Infinite Object".  Set the X and Y semi-apertures to be 10.1 mm and set the focal length (parameter #6) to be 100.  Hit OK on the dialog to create the lens element.

To create the detector plane, right mouse click on the Geometry folder and chose, "Create Element Primitive > Plane" from the context menu.  Provide the name as "Detector" and then hit OK to accept the default settings.

Now, lets place the detector at the focus of the ideal lens.  Right mouse click on the Detector node on your object tree and select, "Position/Orientation" from the context menu.  Left mouse click in the mint-green box for operation #0 to activate the picker control and then use the black arrow on the right hand side to select the ideal lens element as the reference object.  Right mouse click in the position/orientation list and select "Append" from the context menu.  For the new operation #1, set the z-shift value to be 100.  Hit OK to accept the changes.  The position/orientation dialog for the detector element is shown below.

In order to calculate the PSF of the system, we will input a coherent plane wave source into the lens so that it is brought to focus on the detector plane.  The source MUST be coherent so that the effects of diffraction are captured during the propagation.  Right mouse click on the Optical Sources folder and choose, "Create New Source Primitive > Plane Wave (coherent)" from the context menu.

Configure the source as shown in the dialog below so that it has a 10 mm semi-aperture, 41x41 samples in the x and y axes, uses the 0.5875618 micron wavelength and is positioned z = -0.1 mm off the origin (this breaks any coincidence with the ideal lens at the global origin).

Analysis Surface Setup

The first step in the MTF calculation is performing an Irradiance Spread Function analysis, which requires that an Analysis Surface be defined in the model so that the spatial distribution of power/area can be evaluated over the focal plane.  It is desirable to calculate the MTF out to a maximum spatial frequency at or beyond the cutoff frequency of the optical system, which requires that the size and number of pixels in the analysis grid be properly configured prior to calculating the Irradiance Spread Function.  The equation below can be used to determine the analysis surface semi-width, w_semi, given the wavelength (system units), F/# of the optical system, number of pixels in the analysis grid, N_grid, and number of pixels in the Fourier Transform, N_FFT.  The value of N_FFT is the nearest 2ⁿ value that is greater than N_grid.  If, for example, N_grid = 200, the value of N_FFT would be 256.

In this example, the wavelength of the analysis is 0.0005875618 mm, the system F/# is 5.0, and the number of pixels in the analysis grid will be 201 for both the X and Y axes.  Using the equation above, N_grid = 201, N_FFT = 256 and the semi-width of the analysis surface is calculated as w_semi = 0.11591 mm.

Right mouse click on the Analysis Surface(s) folder and choose, "New Analysis Surface" from the context menu.  Configure the Analysis Surface as shown in the image below and then hit OK on the dialog.

Assign the analysis surface to Geometry.Detector.Surface by using the drag and drop technique on the object tree.  Confirm that the analysis surface has shown up at the detector position in the 3D view.

Performing the MTF Analysis

To perform the MTF analysis, take the following steps on the completed model:

1.Raytrace > Trace and Render

2.Analyses > Irradiance Spread Function, hit OK on the dialog

3.In the main chart window, right mouse click and choose, "Fourier Transform (FFT)" from the context menu.  Hit OK on the dialog when prompted.

4.Right mouse click in the main chart window and choose, "Scale Data" from the context menu and then choose the "Normalize max to value" option and hit OK.

Note that the last step in the above sequence is not required for the ideal system in this example but would is generally required when performing the MTF analysis.

The chart axes are now transformed into spatial frequencies and the X and Y profile plots show the characteristic MTF curves for the ideal lens used in this example.  There is no requirement that the source be monochromatic - a polychromatic MTF can calculated using the exact same procedure where the source is configured to have a list of sampled wavelengths over the spectral band of interest.

MTF from an Analysis Result Node (ARN)

The previous section described how to retrieve the MTF by calling the FFT operation on an Irradiance Spread Function result being displayed in the chart window.  It is possible to achieve the same result by operating directly on an irradiance spread function result being stored in an analysis result node (ARN) using the following steps:

1.Right mouse click on the ARN containing the irradiance spread function result and choose, "Modify Data > Perform 2D FFT (...)" from the context menu

2.Double click on the new ARN, which contains the FFT result, to display it in the chart

3.Right mouse click in the main chart window and choose, "Show Field Amplitude" from the context menu