Pupil function
The pupil function or aperture function 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 complex function of the position in the pupil[1] or aperture (often an iris) that indicates the relative change in amplitude and phase of the light wave. Sometimes this function is referred to as the generalized pupil function, in which case pupil function only indicates whether light is transmitted or not.[2] 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.[3]
Contents
1 Relationship with other functions in optics
2 Examples
2.1 In focus
2.2 Out of focus
2.3 Aberrated Optics
3 See also
4 References
Relationship with other functions in optics
The complex pupil function P(u,v){displaystyle mathrm {P} (u,v)}
P(u,v)=A(u,v)⋅exp(iΘ(u,v)){displaystyle mathrm {P} (u,v)=mathrm {A} (u,v)cdot mathrm {exp} (i,mathrm {Theta } (u,v))},
where Θ(u,v){displaystyle mathrm {Theta } (u,v)}
The pupil function is also directly related to the point spread function by its Fourier transform. As such, the effect of aberrations on the point spread function can be described mathematically using the concept of the pupil function.
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 auto correlation of the pupil function.[2][5]
Examples
In focus
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 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:
- P(u,v)=1,∀u,v:u2+v2≤R{displaystyle mathrm {P} (u,v)=1,forall u,v:{sqrt {u^{2}+v^{2}}}leq R}
- P(u,v)=0,∀u,v:u2+v2>R,{displaystyle mathrm {P} (u,v)=0,forall u,v:{sqrt {u^{2}+v^{2}}}>R,}
where R{displaystyle R}
Out of focus
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: k(u2+v2){displaystyle k(u^{2}+v^{2})}
- P(u,v)=exp(ik(u2+v2)),∀u,v:u2+v2≤R{displaystyle mathrm {P} (u,v)=mathrm {exp} (i,k(u^{2}+v^{2})),forall u,v:{sqrt {u^{2}+v^{2}}}leq R}
P(u,v)=0,{displaystyle mathrm {P} (u,v)=0,}otherwise.
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.
Aberrated Optics
The spherical wave could also be deformed by imperfect optics to an approximately cylindrical wave front: ku2{displaystyle ku^{2}}
- P(u,v)=exp(iku2),∀u,v:u2+v2≤R{displaystyle mathrm {P} (u,v)=mathrm {exp} (i,ku^{2}),forall u,v:{sqrt {u^{2}+v^{2}}}leq R}
P(u,v)=0,{displaystyle mathrm {P} (u,v)=0,}
Such a variation in optical path length will create an image that is blurred only in one dimension as is typical of systems with astigmatism.
See also
- Fourier optics
- Point spread function
- Optical transfer function
References
^ Kidger, Michael J. (2001). Fundamental Optical Design. SPIE Press, Bellingham, WA. Retrieved 10 November 2013..mw-parser-output cite.citation{font-style:inherit}.mw-parser-output q{quotes:"""""""'""'"}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-limited a,.mw-parser-output .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}
^ ab Goodman, Joseph (2005). Introduction to Fourier Optics (3rd ed, ed.). Roberts & Co Publishers. ISBN 0-9747077-2-4.
^ ab Fisher, Robert (2008). Optical System Design (2nd ed, ed.). The McGraw-Hill Companies, Inc. ISBN 9780071472487.
^ Pawley, James B. (2006). Handbook of confocal microscopy (3rd ed, ed.). Springer. ISBN 0-387-25921-X.
^ "Optics Course Notes on the calculation of the OTF from the Pupil function" (PDF). Retrieved 7 November 2013.
