Modulation Transfer Function (MTF)
Abu Bucker Siddik
New Mexico State University
Presentation Objectives �
Modulation Transfer Function (MTF)
From Introduction to Lens Design with Practical ZEMAX Examples, Geary and Willmann-Bell
What is Modulation?�
From Introduction to Lens Design with Practical ZEMAX Examples, Geary and Willmann-Bell
Figure: Sinusoidal object pattern
Figure: Profile of sinusoidal object pattern
The mathematical description of the object profile:
Modulation (or contrast) of the profile:
Inherent modulation (contrast) of the object:
Perfect imaging system: profile across the image of object will mimic that of object
Imperfect imaging system: reduced modulation, image shift
Convolution and Fourier Transform�
From Introduction to Lens Design with Practical ZEMAX Examples, Geary and Willmann-Bell
Figure: Star image due to aberrated optical system
Figure: Convolution of aberrated PSF with perfect geometric image
For finite-sized object: convolution of aberrated PSF with the perfect image.
For one dimensional analog, we can express the convolution as
Fourier transform of f(x) can be defined as -
Optical Transfer Function�
From Introduction to Lens Design with Practical ZEMAX Examples, Geary and Willmann-Bell
Figure: Transferring the object through the optical imaging system.
Figure: Fourier Transform of the object.
Fourier Transform of object function f(x) is given by:
Figure: Fourier Transform of the image.
Fourier transform of the image function g(x) is given by:
Image modulation at the spatial frequency v0:
Optical Transfer Function�
From Introduction to Lens Design with Practical ZEMAX Examples, Geary and Willmann-Bell
H(v) is complex and can be expressed by Euler formula:
OTF is also complex and can be written as:
where:
Modulation in image at the spatial frequency v0:
Consider same procedure for all sinusoidal object
spatial frequencies, we can write:
where:
Optical Transfer Function�
From Introduction to Lens Design with Practical ZEMAX Examples, Geary and Willmann-Bell
Figure: Effect of MTF and PTF on image for one spatial frequency
Figure: Typical MTF plot
OTF can be expressed as:
MTF is the modulus of the OTF which is responsible for the reduction in image modulation (contrast).
Phase transfer function (PTF) is responsible for the literal shifting of the pattern.
where:
MTF value range: 0 – 1
Cut-off frequency: spatial frequency where the MTF
becomes zero
Importance of MTF
MTF in ZEMAX
Fraunhofer (FFT) MTF:
(a) large enough f/# to apply scalar diffraction theory properly
(b) large enough sampling to accurately model PSF
(c) uniform distribution of rays is present for both the entrance pupil and the exit pupil
(d) the significant energy region of diffraction PSF is small compared to the distance from the exit pupil of the optical system to the image surface
From https://support.zemax.com/hc/en-us/articles/1500005575102-Methods-for-analyzing-MTF-in-OpticStudio
MTF in ZEMAX
Geometric MTF:
Huygens MTF:
From https://support.zemax.com/hc/en-us/articles/1500005575102-Methods-for-analyzing-MTF-in-OpticStudio
MTF in ZEMAX
From Introduction to Lens Design with Practical ZEMAX Examples, Geary and Willmann-Bell
MTF for an Ideal Focusing Surface
MTF for an Ideal Focusing Surface
Figure: Cross-section
Figure: FFT MTF
From Fourier optics, MTF cutoff spatial frequency for an ideal aperture is 1/(wavelength * f/#)
For f/5 and wavelength 0.587 um, MTF cutoff spatial frequency = 340.7 cycles per mm
MTF for an Ideal Focusing Surface
Figure: Huygens MTF
Figure: Geometric MTF
MTF for a Real Lens
MTF for a Real Lens
Figure: FFT MTF before optimization
Figure: FFT MTF after optimization
MTF for a Real Lens with Different Field Angles
Figure: FFT MTF with different field angles
Conclusions �