340:
onto a sensor or photographic film, by converting the planar wave front to a spherical wave front, centered onto the image plane. The pupil function of such an ideal system is equal to one at every point within the pupil, and zero out with it. In case of a circular pupil, this can be written mathematically as:
339:
In a homogeneous medium, a point source emits light with spherical wave fronts. A lens that is focused onto the point source will have optics that change the spherical wave front into a planar wave before it passes through the pupil or aperture stop. Often, additional lens element refocus the light
325:
Since the (incoherent) point spread function is also related to the optical transfer function via a
Fourier transform, a direct relationship exists between the pupil function and the optical transfer function. In the case of an incoherent optical imaging system, the optical transfer function is the
36:
pupil function, in which case pupil function only indicates whether light is transmitted or not. Imperfections in the optics typically have a direct effect on the pupil function, it is therefore an important tool to study optical imaging systems and their performance.
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describes how a light wave is affected upon transmission through an optical imaging system such as a camera, microscope, or the human eye. More specifically, it is a
849:
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When the point source is out of focus, the spherical wave will not be completely made planar by the optics, but will have an approximately parabolic wave front:
556:
238:
that occur between the image plane and the focal plane in the scene or sample. The light may also be attenuated differently at different positions
441:
346:
1179:
322:. As such, the effect of aberrations on the point spread function can be described mathematically using the concept of the pupil function.
813:
It is thus possible to deduce the point-spread function of the out of focus point source as the
Fourier transform of the pupil function.
1163:
1138:
1110:
1030:
Such a variation in optical path length will create an image that is blurred only in one dimension as is typical of systems with
32:) that indicates the relative change in amplitude and phase of the light wave. Sometimes this function is referred to as the
615:. Such a variation in optical path length corresponds to a radial variation in the complex argument of the pupil function:
1031:
1053:
200:
25:
755:{\displaystyle \mathrm {P} (u,v)=\mathrm {exp} (i\,k(u^{2}+v^{2})),\forall u,v:{\sqrt {u^{2}+v^{2}}}\leq R}
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The spherical wave could also be deformed by imperfect optics to an approximately cylindrical wave front:
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48:
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is the phase change (in radians) introduced by the optics, or the surrounding medium. It captures all
569:
972:{\displaystyle \mathrm {P} (u,v)=\mathrm {exp} (i\,ku^{2}),\forall u,v:{\sqrt {u^{2}+v^{2}}}\leq R}
235:
186:{\displaystyle \mathrm {P} (u,v)=\mathrm {A} (u,v)\cdot \mathrm {exp} (i\,\mathrm {\Theta } (u,v))}
29:
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83:
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1180:"Optics Course Notes on the calculation of the OTF from the Pupil function"
528:{\displaystyle \mathrm {P} (u,v)=0,\forall u,v:{\sqrt {u^{2}+v^{2}}}>R,}
430:{\displaystyle \mathrm {P} (u,v)=1,\forall u,v:{\sqrt {u^{2}+v^{2}}}\leq R}
274:. Such change in amplitude of the light wave is described by the factor
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in the pupil, sometimes deliberately for the purpose of
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314:The pupil function is also directly related to the
28:of the position in the pupil or aperture (often an
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1133:(2nd ed.). The McGraw-Hill Companies, Inc.
1105:(3rd ed.). Roberts & Co Publishers.
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41:Relationship with other functions in optics
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326:auto correlation of the pupil function.
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227:{\displaystyle \mathrm {\Theta } (u,v)}
7:
1019:{\displaystyle \mathrm {P} (u,v)=0,}
802:{\displaystyle \mathrm {P} (u,v)=0,}
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304:{\displaystyle \mathrm {A} (u,v)}
75:{\displaystyle \mathrm {P} (u,v)}
1156:Handbook of confocal microscopy
1103:Introduction to Fourier Optics
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608:{\displaystyle k(u^{2}+v^{2})}
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1079:. SPIE Press, Bellingham, WA
1073:Kidger, Michael J. (2001).
45:The complex pupil function
1226:
1158:(3rd ed.). Springer.
1076:Fundamental Optical Design
86:using two real functions:
1154:Pawley, James B. (2006).
1054:Optical transfer function
1101:Goodman, Joseph (2005).
1129:Fisher, Robert (2008).
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844:{\displaystyle ku^{2}}
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1131:Optical System Design
1049:Point spread function
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263:{\displaystyle (u,v)}
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236:optical aberrations
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82:can be written in
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551:{\displaystyle R}
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320:Fourier transform
84:polar coordinates
22:aperture function
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817:Aberrated Optics
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26:complex function
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1044:Fourier optics
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18:pupil function
13:
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1165:0-387-25921-X
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1140:9780071472487
1136:
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1112:0-9747077-2-4
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40:
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27:
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19:
1186:. Retrieved
1174:
1155:
1149:
1130:
1102:
1081:. Retrieved
1075:
1068:
1029:
820:
812:
565:
562:Out of focus
537:
338:
324:
313:
196:
44:
33:
21:
17:
15:
1083:10 November
1032:astigmatism
272:apodization
34:generalized
1188:2 February
1060:References
1026:otherwise.
809:otherwise.
964:≤
922:∀
747:≤
705:∀
475:∀
422:≤
380:∀
206:Θ
162:Θ
140:⋅
1204:Category
1038:See also
335:In focus
330:Examples
318:by its
1210:Optics
1162:
1137:
1109:
538:where
197:where
1183:(PDF)
1190:2022
1160:ISBN
1135:ISBN
1107:ISBN
1085:2013
517:>
30:iris
16:The
20:or
1206::
1121:^
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311:.
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1014:,
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1008:=
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