MTFs¶
prysm models often include analysis of Modulation Transfer Function (MTF) data. The MTF is formally defined as:
the normalized magnitude of the Fourier transform of the Point Spread Function
It is nothing more and nothing less. It may not be negative, complex-valued, or equal to any value other than unity at the origin.
Initializing an MTF model should feel similar to a PSF,
[1]:
import numpy as np
from prysm import MTF
x = y = 1/np.linspace(-1,1,128)
z = np.random.random((128,128))
mt = MTF(data=z, unit_x=x, unit_y=y)
MTFs are usually created from a PSF instance
[2]:
from prysm import Pupil, PSF
pu = Pupil(dia=10, wavelength=0.5)
ps = PSF.from_pupil(pu, efl=20)
mt = MTF.from_psf(ps)
If modeling the MTF directly from a pupil plane, the intermediate PSF plane may be skipped;
[3]:
mt = MTF.from_pupil(pu, Q=2, efl=20) # Q, efl same as PSF.from_pupil
Much like a PSF or other Convolvable, MTFs have quick-access slices
[4]:
mt.tan, mt.sag
[4]:
((array([ 0. , 7.87401575, 15.7480315 , 23.62204724,
31.49606299, 39.37007874, 47.24409449, 55.11811024,
62.99212598, 70.86614173, 78.74015748, 86.61417323,
94.48818898, 102.36220472, 110.23622047, 118.11023622,
125.98425197, 133.85826772, 141.73228346, 149.60629921,
157.48031496, 165.35433071, 173.22834646, 181.1023622 ,
188.97637795, 196.8503937 , 204.72440945, 212.5984252 ,
220.47244094, 228.34645669, 236.22047244, 244.09448819,
251.96850394, 259.84251969, 267.71653543, 275.59055118,
283.46456693, 291.33858268, 299.21259843, 307.08661417,
314.96062992, 322.83464567, 330.70866142, 338.58267717,
346.45669291, 354.33070866, 362.20472441, 370.07874016,
377.95275591, 385.82677165, 393.7007874 , 401.57480315,
409.4488189 , 417.32283465, 425.19685039, 433.07086614,
440.94488189, 448.81889764, 456.69291339, 464.56692913,
472.44094488, 480.31496063, 488.18897638, 496.06299213,
503.93700787, 511.81102362, 519.68503937, 527.55905512,
535.43307087, 543.30708661, 551.18110236, 559.05511811,
566.92913386, 574.80314961, 582.67716535, 590.5511811 ,
598.42519685, 606.2992126 , 614.17322835, 622.04724409,
629.92125984, 637.79527559, 645.66929134, 653.54330709,
661.41732283, 669.29133858, 677.16535433, 685.03937008,
692.91338583, 700.78740157, 708.66141732, 716.53543307,
724.40944882, 732.28346457, 740.15748031, 748.03149606,
755.90551181, 763.77952756, 771.65354331, 779.52755906,
787.4015748 , 795.27559055, 803.1496063 , 811.02362205,
818.8976378 , 826.77165354, 834.64566929, 842.51968504,
850.39370079, 858.26771654, 866.14173228, 874.01574803,
881.88976378, 889.76377953, 897.63779528, 905.51181102,
913.38582677, 921.25984252, 929.13385827, 937.00787402,
944.88188976, 952.75590551, 960.62992126, 968.50393701,
976.37795276, 984.2519685 , 992.12598425, 1000. ]),
array([1.00000000e+00, 9.90034799e-01, 9.80069598e-01, 9.70104397e-01,
9.60139196e-01, 9.50173996e-01, 9.40208795e-01, 9.30243594e-01,
9.20278393e-01, 9.10313192e-01, 9.00347991e-01, 8.90382790e-01,
8.80417589e-01, 8.70452388e-01, 8.60487188e-01, 8.50521987e-01,
8.40556786e-01, 8.30591585e-01, 8.20626384e-01, 8.10661183e-01,
8.00695982e-01, 7.90730781e-01, 7.80765581e-01, 7.70958557e-01,
7.61151534e-01, 7.51344511e-01, 7.41537488e-01, 7.31730465e-01,
7.21923442e-01, 7.12116419e-01, 7.02309396e-01, 6.92502373e-01,
6.82695350e-01, 6.73046504e-01, 6.63397659e-01, 6.53748814e-01,
6.44099968e-01, 6.34451123e-01, 6.24802278e-01, 6.15311610e-01,
6.05820943e-01, 5.96330275e-01, 5.86839608e-01, 5.77348940e-01,
5.67858273e-01, 5.58525783e-01, 5.49193293e-01, 5.39860804e-01,
5.30528314e-01, 5.21195824e-01, 5.11863334e-01, 5.02689022e-01,
4.93514711e-01, 4.84340399e-01, 4.75166087e-01, 4.66149953e-01,
4.57133818e-01, 4.48117684e-01, 4.39101550e-01, 4.30243594e-01,
4.21385637e-01, 4.12527681e-01, 4.03669725e-01, 3.94969946e-01,
3.86270168e-01, 3.77570389e-01, 3.68870611e-01, 3.60329010e-01,
3.51787409e-01, 3.43403986e-01, 3.35020563e-01, 3.26637140e-01,
3.18253717e-01, 3.10028472e-01, 3.01803227e-01, 2.93736159e-01,
2.85669092e-01, 2.77760202e-01, 2.69851313e-01, 2.61942423e-01,
2.54033534e-01, 2.46282822e-01, 2.38532110e-01, 2.30939576e-01,
2.23347042e-01, 2.15912686e-01, 2.08478330e-01, 2.01202151e-01,
1.93925973e-01, 1.86807972e-01, 1.79689972e-01, 1.72730149e-01,
1.65770326e-01, 1.58968681e-01, 1.52167036e-01, 1.45523568e-01,
1.38880101e-01, 1.32394812e-01, 1.25909522e-01, 1.19582411e-01,
1.13255299e-01, 1.07244543e-01, 1.01233787e-01, 9.53812085e-02,
8.95286302e-02, 8.38342297e-02, 7.81398292e-02, 7.27617842e-02,
6.73837393e-02, 6.21638722e-02, 5.69440051e-02, 5.20404935e-02,
4.71369820e-02, 4.25498260e-02, 3.79626700e-02, 3.36918697e-02,
2.94210693e-02, 2.54666245e-02, 2.15121797e-02, 1.80322683e-02,
1.45523568e-02, 1.15469788e-02, 8.54160076e-03, 6.01075609e-03,
3.47991142e-03, 1.73995571e-03, 5.95103509e-17, 4.58025254e-17])),
(array([ 0. , 7.87401575, 15.7480315 , 23.62204724,
31.49606299, 39.37007874, 47.24409449, 55.11811024,
62.99212598, 70.86614173, 78.74015748, 86.61417323,
94.48818898, 102.36220472, 110.23622047, 118.11023622,
125.98425197, 133.85826772, 141.73228346, 149.60629921,
157.48031496, 165.35433071, 173.22834646, 181.1023622 ,
188.97637795, 196.8503937 , 204.72440945, 212.5984252 ,
220.47244094, 228.34645669, 236.22047244, 244.09448819,
251.96850394, 259.84251969, 267.71653543, 275.59055118,
283.46456693, 291.33858268, 299.21259843, 307.08661417,
314.96062992, 322.83464567, 330.70866142, 338.58267717,
346.45669291, 354.33070866, 362.20472441, 370.07874016,
377.95275591, 385.82677165, 393.7007874 , 401.57480315,
409.4488189 , 417.32283465, 425.19685039, 433.07086614,
440.94488189, 448.81889764, 456.69291339, 464.56692913,
472.44094488, 480.31496063, 488.18897638, 496.06299213,
503.93700787, 511.81102362, 519.68503937, 527.55905512,
535.43307087, 543.30708661, 551.18110236, 559.05511811,
566.92913386, 574.80314961, 582.67716535, 590.5511811 ,
598.42519685, 606.2992126 , 614.17322835, 622.04724409,
629.92125984, 637.79527559, 645.66929134, 653.54330709,
661.41732283, 669.29133858, 677.16535433, 685.03937008,
692.91338583, 700.78740157, 708.66141732, 716.53543307,
724.40944882, 732.28346457, 740.15748031, 748.03149606,
755.90551181, 763.77952756, 771.65354331, 779.52755906,
787.4015748 , 795.27559055, 803.1496063 , 811.02362205,
818.8976378 , 826.77165354, 834.64566929, 842.51968504,
850.39370079, 858.26771654, 866.14173228, 874.01574803,
881.88976378, 889.76377953, 897.63779528, 905.51181102,
913.38582677, 921.25984252, 929.13385827, 937.00787402,
944.88188976, 952.75590551, 960.62992126, 968.50393701,
976.37795276, 984.2519685 , 992.12598425, 1000. ]),
array([1.00000000e+00, 9.90034799e-01, 9.80069598e-01, 9.70104397e-01,
9.60139196e-01, 9.50173996e-01, 9.40208795e-01, 9.30243594e-01,
9.20278393e-01, 9.10313192e-01, 9.00347991e-01, 8.90382790e-01,
8.80417589e-01, 8.70452388e-01, 8.60487188e-01, 8.50521987e-01,
8.40556786e-01, 8.30591585e-01, 8.20626384e-01, 8.10661183e-01,
8.00695982e-01, 7.90730781e-01, 7.80765581e-01, 7.70958557e-01,
7.61151534e-01, 7.51344511e-01, 7.41537488e-01, 7.31730465e-01,
7.21923442e-01, 7.12116419e-01, 7.02309396e-01, 6.92502373e-01,
6.82695350e-01, 6.73046504e-01, 6.63397659e-01, 6.53748814e-01,
6.44099968e-01, 6.34451123e-01, 6.24802278e-01, 6.15311610e-01,
6.05820943e-01, 5.96330275e-01, 5.86839608e-01, 5.77348940e-01,
5.67858273e-01, 5.58525783e-01, 5.49193293e-01, 5.39860804e-01,
5.30528314e-01, 5.21195824e-01, 5.11863334e-01, 5.02689022e-01,
4.93514711e-01, 4.84340399e-01, 4.75166087e-01, 4.66149953e-01,
4.57133818e-01, 4.48117684e-01, 4.39101550e-01, 4.30243594e-01,
4.21385637e-01, 4.12527681e-01, 4.03669725e-01, 3.94969946e-01,
3.86270168e-01, 3.77570389e-01, 3.68870611e-01, 3.60329010e-01,
3.51787409e-01, 3.43403986e-01, 3.35020563e-01, 3.26637140e-01,
3.18253717e-01, 3.10028472e-01, 3.01803227e-01, 2.93736159e-01,
2.85669092e-01, 2.77760202e-01, 2.69851313e-01, 2.61942423e-01,
2.54033534e-01, 2.46282822e-01, 2.38532110e-01, 2.30939576e-01,
2.23347042e-01, 2.15912686e-01, 2.08478330e-01, 2.01202151e-01,
1.93925973e-01, 1.86807972e-01, 1.79689972e-01, 1.72730149e-01,
1.65770326e-01, 1.58968681e-01, 1.52167036e-01, 1.45523568e-01,
1.38880101e-01, 1.32394812e-01, 1.25909522e-01, 1.19582411e-01,
1.13255299e-01, 1.07244543e-01, 1.01233787e-01, 9.53812085e-02,
8.95286302e-02, 8.38342297e-02, 7.81398292e-02, 7.27617842e-02,
6.73837393e-02, 6.21638722e-02, 5.69440051e-02, 5.20404935e-02,
4.71369820e-02, 4.25498260e-02, 3.79626700e-02, 3.36918697e-02,
2.94210693e-02, 2.54666245e-02, 2.15121797e-02, 1.80322683e-02,
1.45523568e-02, 1.15469788e-02, 8.54160076e-03, 6.01075609e-03,
3.47991142e-03, 1.73995571e-03, 1.09720259e-17, 6.67479670e-17])))
The tangential MTF is a slice through the x=0 axis, and assumes the usual optics sign convention of an object extended in y. The sagittal MTF is a slice through the y=0 axis.
The MTF at exact frequencies may be queried through any of the following methods: exact_polar, takes arguments of freqs and azimuths. If there is a single frequency and multiple azimuths, the MTF at each azimuth and and the specified radial spatial frequency will be returned. The reverse is true for a single azimuth and multiple frequencies. exact_xy follows the same semantics, but with Cartesian coordinates instead of polar. exact_tan and exact_sag both take a single argument of
freq, which may be an int, float, or ndarray.
Finally, MTFs may be plotted:
[5]:
mt.plot_tan_sag(max_freq=1000, fig=None, ax=None, labels=('Tangential', 'Sagittal'))
mt.plot2d(max_freq=1000, power=2, fig=None, ax=None)
[5]:
(<Figure size 640x480 with 2 Axes>,
<matplotlib.axes._subplots.AxesSubplot at 0x7f004ce12c88>)
all arguments have these default values. The axes of plot2d will span (-max_freq, max_freq) on both x and y.
This example should be familiar as the diffraction limited MTF of a circular aperture.
[ ]:
