How to Include Detector Resolution in MTF Calculations
http://www.zemax.com/kb/articles/233/1/How-to-Include-Detector-Resolution-in-MTF-Calculations/Page1.html
By Mark Nicholson
Published on 2 May 2008
MTF is a 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.
Introduction
Modulation Transfer Function (MTF) is an important method of describing the performance of an optical system. A consequence of applying Fourier theory to image forming optical systems, MTF describes the contrast in the image of a spatial frequency presented in the scene being viewed. See this article for more information on what MTF is.
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 is capable of achieving.
Experimentally MTF can be measured by imaging a small bar (or single-frequency sine) chart through the lens and onto the detector. The bar chart must be small because the optical transfer function of the lens should not vary significantly over the target pattern. Within ZEMAX we can use the same method: the diffraction image analysis feature is used to image a small bar chart through the system onto a pixellated detector, and the MTF is computed directly from this.
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 is capable of achieving.
Experimentally MTF can be measured by imaging a small bar (or single-frequency sine) chart through the lens and onto the detector. The bar chart must be small because the optical transfer function of the lens should not vary significantly over the target pattern. Within ZEMAX we can use the same method: the diffraction image analysis feature is used to image a small bar chart through the system onto a pixellated detector, and the MTF is computed directly from this.
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:
We image a bar-chart through the system:
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:
We will now look at the cross-section of the Diffraction Image Analysis, configured like so:
so we use 500 1m-wide pixels to view our image. The resulting image is like so:
This is the cross-section of the false color map shown above. Note we have ten cycles of the bar patterm over about a 200m 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 effect 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:
Note there is excellent agreement between the two analysis features, with an estimated MTF of 0.68 from both the Diffraction Image Analysis and the FFT MTF plot at 50 cycles/mm (note that we are only approximately at 50 cycles/mm in the Diffraction Image Analysis). This is to be expected, as the 1m detector pixel size is smaller than the 5m RMS spot size and 3.5m Airy disk radius. This combination of optical system and detector is optics-limited, not detector limited.
Instead, repeat the Diffraction Image Analysis, but use an array of 100x100 pixels each of 5m width. Now the MTF is 0.43:
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
The MTF improves to 0.59 because there is less cross-talk between light and dark regions of the image as they are integrated by the detector array:
We image a bar-chart through the system:
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:
We will now look at the cross-section of the Diffraction Image Analysis, configured like so:
so we use 500 1m-wide pixels to view our image. The resulting image is like so:
This is the cross-section of the false color map shown above. Note we have ten cycles of the bar patterm over about a 200m 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 effect 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:
Note there is excellent agreement between the two analysis features, with an estimated MTF of 0.68 from both the Diffraction Image Analysis and the FFT MTF plot at 50 cycles/mm (note that we are only approximately at 50 cycles/mm in the Diffraction Image Analysis). This is to be expected, as the 1m detector pixel size is smaller than the 5m RMS spot size and 3.5m Airy disk radius. This combination of optical system and detector is optics-limited, not detector limited.
Instead, repeat the Diffraction Image Analysis, but use an array of 100x100 pixels each of 5m width. Now the MTF is 0.43:
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
The MTF improves to 0.59 because there is less cross-talk between light and dark regions of the image as they are integrated by the detector array:
Summary
If an optical system's minimum resolvable spot is comparable to or less than the detector size, it is important to consider the effects of integrating the spatial signal on the detector array when computing MTF. The Diffraction Image Analysis feature provides this capability.