How to include detector resolution in MTF calculations

MTF is commonly used to describe the performance of an imaging system, but the finite resolution of the detection system is often ignored. This article describes how to account for detector pixel sizes and position shifts to give a full-system MTF measurement.

Authored By Mark Nicholson

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Introduction

Modulation Transfer Function (MTF) is an important method of describing the performance of an optical system. MTF describes the imaging of the system, but an important system parameter is usually neglected: the resolution of the detector. If the detector's pixels are significantly bigger than the resolvable spot size, the optical system is said to be detector limited, and the overall system MTF is reduced compared to the MTF the optical system itself can achieve.

Within OpticStudio, we use a method very similar to the typical experiment: the partially coherent image analysis feature is used to image a small bar chart through the system onto a pixelated detector, and the MTF is computed directly from this. This article will provide an example of this method.

An example

The example file used here can be downloaded via the link on the last page of this article. It is a derivative of the Cooke triplet sample file:

Shaded_model

We image a bar-chart through the system. The bar chart must be small because the optical transfer function of the lens should not vary significantly over the target pattern.

Partially_Coherent_Image_Analysis

The full width of the image is 0.5 mm, and the optical performance of the lens does not vary significantly over this field of view:

Ray_Fan

We will now look at the cross section of the Partially Coherent Image Analysis, configured like so:

Partially_Coherent_Image_Analysis_configuration

so we use 500 1 mm wide pixels to view our image. The resulting image is like so:

Partially_Coherent_Image_Analysis_2

The cross section of the false color map is shown above. Note we have ten cycles of the bar pattern over about a 200 mm region: this corresponds to 50 cycles/mm. The MTF can then be estimated by determining the maximum and minimum relative intensity across the cross section.

To reduce the effects of edges, the analysis parameters should be set to provide at least 5 well defined peaks across the cross section. The MTF is computed by looking for the minimum and maximum intensity at all points between the second and second-to-last local peaks in the intensity data. By considering only data within these two peaks, the effects of the edges is somewhat reduced.

The estimated MTF is then given by the usual computation of (Imax-Imin)/(Imax+Imin). Finally, note that if a bar target is used, the resulting MTF is the square-wave, not sine-wave modulation:

FFT_MTF

Note there is excellent agreement between the two analysis features, with an estimated MTF of 0.66 from both the Partially Coherent Image Analysis and the FFT MTF plot at 50 cycles/mm (note that we are only approximately at 50 cycles/mm in the Partially Coherent Image Analysis). This is to be expected, as the 1 mm detector pixel size is smaller than the 5 mm RMS spot size and 3.5 mm Airy disk radius. This combination of optical system and detector is optics limited, not detector limited.

Instead, repeat the Partially Coherent Image Analysis, but use an array of 100x100 pixels each of 5 mm width. Now the MTF is 0.42:

Partially_Coherent_Image_Analysis_3

The MTF is clearly degraded by the coarser detector resolution. Equally important, consider what happens if the detector is shifted in the image plane. Because the pixel size is close to the resolution limit, the measured MTF will be sensitive to shifts of the detector array of the order of one pixel. If we decenter the detector by a half-pixel in x... 

Decenter_the_detector_by_a_half-pixel

...the MTF improves to 0.58 because there is less cross-talk between light and dark regions of the image as they are integrated by the detector array:

Partially_Coherent_Image_Analysis_4

KA-01404

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