In classical optics or Fourier optics, we use the Fresnel law or diffraction theory to explain the imaging via a thin lens. The interference phenomenons like the Young's Double-slit interference can be explained by the diffraction theory.
In quantum theory, interference derives from indistinguish. The quantum imaging (entanglement ghost imaging) can be explained by the two photons interference.
In the classical or quantum optics, what is the relationship between interference and imaging?
There is very close relationship between imaging and interference, but probably this fact is not very commonly known as it is much easier to think about imaging in the geometric optics framework.
Typically, in classical optics or photography we image objects scattering incoherent light. Incoherent here means that the coherent length is very short - comparable to the wavelength. However such light also demonstrate the ability to interfere, but at a very short length scale. In particular it perfectly interferes constructively with its copy while the delay between the each other are zero.
Imaging means that one point of the object is mapped onto another point - in the image plane. All paths from one point to the other have the same optical length. It means they correspond to the same phase shift. If you add many copies of even incoherent light but with the same phase shift you simply got the perfect constructive interference.
Therefore an image is just a constructive interference of many independent sources.
As you see the relationship is very close.
In quantum imaging methods you additionally utilize a strong, quantum correlation between two photons. But the principle of forming an image is the same as I described above.
Image is formed in the approximation of geometrical optics where interference is not important. If the size of the object is comparable with the wavelength and the diffraction start to play role, a good image is not possible. One gets an interference pattern instead.
Entanglement is not interference. If two photons are in entangled state then change of the state of one photon (particularly, due to interaction with a body) leads to the change of the state of the second photon. It allows to reconstruct the image of the body by using of the first photon which does not interact with the body.It is quantum imaging and it is not interference There are a few interesting experiments in this field (see, e.g., papers attached or Phys Rev. A 52, p.R3429).
There is very close relationship between imaging and interference, but probably this fact is not very commonly known as it is much easier to think about imaging in the geometric optics framework.
Typically, in classical optics or photography we image objects scattering incoherent light. Incoherent here means that the coherent length is very short - comparable to the wavelength. However such light also demonstrate the ability to interfere, but at a very short length scale. In particular it perfectly interferes constructively with its copy while the delay between the each other are zero.
Imaging means that one point of the object is mapped onto another point - in the image plane. All paths from one point to the other have the same optical length. It means they correspond to the same phase shift. If you add many copies of even incoherent light but with the same phase shift you simply got the perfect constructive interference.
Therefore an image is just a constructive interference of many independent sources.
As you see the relationship is very close.
In quantum imaging methods you additionally utilize a strong, quantum correlation between two photons. But the principle of forming an image is the same as I described above.
imaging means the formation of amplitude point-to-point field by using focusing lenses. interference is a method to create faze field, that is a hologram, which may be released without focusing optics. The information about each object point is contained in all space.
Dear Yaokun Xu,
in the old geometrical optics images are constructed by having recourse to ray diagrams. In the same geometrical framework the Huygens principle of wave front propagation allows for both, "near field" diffraction and "far field" interference.
Later on, Joseph Lagrange gave an algebraic reformulation to mechanics, which is called analytic mechanics. Following this direction, William Rowan Hamilton derived the laws of reflection and refraction of light rays from the principle of least action. In that algebraic framework a kind of ray-wave duality can be exhibited when starting from Fermat's principle of stationary action, in stead of using the principle of least action. The first geometrical representation of that theory in the configuration space is possibly due to Lipschitz, who also led Hamilton's optics back to mechanics. The cited ray-wave duality is purely theoretical, and has nothing to do with the celebrated double-slit experiment of quantum mechanics.
Hamilton's theory was taken up again by M. Born in his “Optik” (in the later English reprints the title was changed to “Principles of Optics” by Born & Wolf,). He used it to explain optical aberrations in a perturbative manner, taking diffraction into account. In my opinion the book “Fundamentals of quantum mechanics” by Sidney Borowitz gives a fair account of how quantum mechanics is connected to Hamilton's theory.
Although the principles of Fourier optics can be likewise traced back to E. Abbe's studies on the resolution limits of microscopes, it is not a perturbation theory. In fact, when speaking about spectral data people often mean that their time domain representation can be recovered "exactly" by the mathematical process of Fourier transform. Fourier optics can be related to the double-slit experiment. But the same framework is also akin to the practices and theories of telecommunications.
I hope that my reply addresses the main point in your question.
> Entanglement is not interference.
This is true.
> If two photons are in entangled state then change of the state of one photon (particularly, due to interaction with a body) leads to the change of the state of the second photon.
This is not true, see
arxiv.org/abs/quant-ph/9906007
and
http://arxiv.org/abs/1109.6223.
In entanglement, observables of a quantum system contain some information that can't be revealed by measurements on that system alone because the expectation values of the observables of that system alone don't depend on that information - locally inaccessible information. That information can only be revealed by comparing results of measurements on the two entangled systems. The systems that carry those measurement results carry the locally inaccessible information too, even if the carriers are decoherent.
This quantum information is revealed when the results of measurements on the entangled systems are compared. That is when the correlations between the results are established.
> It allows to reconstruct the image of the body by using of the first photon which does not interact with the body.It is quantum imaging and it is not interference There are a few interesting experiments in this field (see, e.g., papers attached or Phys Rev. A 52, p.R3429).
Those experiments are interesting, and can be explained entirely in terms of local transfers of information.
Radislaw Chrapkiewicz's explanation is good, and probably adequate for what you seek.
What Radislaw describes is imaging in an aberration-free system. You might note that, even in such a system geometrical optics alone cannot describe the precise form of image as it ignores the effects of diffraction (which is then followed by interference to form a finite spread of the image of any part of the object field). For a rotationally symmetric imaging system with a circular aperture and no axial aberration the image of an on-axis, point-source object is not a point (as geometric optics alone would suggest) but a disc with surrounding annuli (the Airy function, amongst several names). This arises due to interference consequent on diffraction at the aperture. The interference is fully constructive on axis (as described by Radislaw), but as you move to any specific point off-axis in the image plane not all paths for light that reaches that point has travelled the same optical distance and so there is partially destructive interference, resulting in an overall reduction in the time-averaged flux. If the optical system is not aberration free then non-equal optical path distances (and hence imperfectly matched phases of arriving wavefronts or photons) will occur anywhere and hence images are not exact geometric replicas of the objects being imaged. I hope this is of some value, rather than just confusing!
It's the same thing ;) and it's well-known in astronomy: the image of a point-source through a telescope results from the interferences of all points in the pupil (or aperture) of the telescope. And the optical transfer function of a diffraction-limited telescope is a mesasure of the redundancy of its pupil.
This allows one to make the connection between optical long-baseline interferometers and (monolithic) telescopes. This connection is well-explained by Roddier in his 1981 contribution to Progress in Optics entitled "the effects of atmospheric turbulence in optical astronomy" (see Fig. 7 p. 316) and later by Mariotti in his chapter entitled "Introduction to Fourier Optics and Coherence" of the book "Diffraction-limited imaging with very large telescopes".
BibTeX entries given below. Hope this helps.
@incollection{Roddier-81,
crossref = {PO-XIX},
author = {F. Roddier},
title = {The effects of atmospherical turbulence in optical astronomy},
pages = {281--376}
}
@book{PO-XIX,
editor = {E. Wolf},
title = {Progress in Optics},
booktitle = {Progress in Optics},
year = {1981},
publisher = {North Holland},
address = {Amsterdam},
volume = {XIX},
}
@INCOLLECTION{Mariotti-89,
author = {J.-M. Mariotti},
title = {Introduction to {F}ourier Optics and Coherence},
pages = {3--31},
crossref = {Alloin-l-89},
}
@BOOK{Alloin-l-89,
title = {Diffraction-limited imaging with very large telescopes},
publisher = {Kluwer Academic Publishers},
year = {1989},
editor = {D. M. Alloin and J.-M. Mariotti},
volume = {C-274},
series = {NATO ASI Series},
address = {Dordrecht},
booktitle = {Diffraction-limited imaging with very large telescopes},
organization = {NATO},
volume_title = {Proceedings of the NATO Advanced Study Institute, held in Cargese,
Corsica, Sep 13--23, 1988},
}
Wow, many complicated wordings for such a simple question:
1. Interference is wave phenomena
In classical (in same way in quantum, but with somewhat more abstract mathematics) optics, image formation is a result of interference of all waves scattered be an object.
Now: it is very convenient to look at a diffraction grating as an object. Once illuminated by a plain wave, it produces plain waves propagating to very particular directions those are called diffraction orders.
Put a lens at distance of 2f from this grating (for the very simplicity think of the lens to be infinite in its extent) and see how the wave-fronts of the diffraction orders are interfering to reproduce the image of your grating.
Next huge conceptual jump: any object can be represented as superposition of diffraction gratings. It is called Fourier Transformation, and corresponding branch of optics, is , naturally called Fourier Optics.
Particularly, to describe a single point object you need really lots of diffraction gratings, having lots of different periods.
Or, you can simply draw couple of lines and build a high-school image diagrams. It will be good, unless you need to exactly calculate EM fields close to focus of your lens. There you will need the full formalism.
So much of imaging and interference.
2. Entanglement is not interference, and change of state of one photon does not change the state of the other.
OK, lets go to the main question: what is photon. Please show me a physicist who still thinks that photon is a "ball of light". In such a view, there are many miracles happening, and some phenomena are hard to explain.
Photon is a mathematical concept. There is no such a physical thing. Those are indeed very useful to describe statistical nature of measurement of very weak electromagnetic fields. Now, you can indeed create two electromagnetic waves that do not interfere if you perform your measurement in one way, but do, if you perform it in another. Say two counter-rotating circularly polarized waves measured with a polarizer will either output double intensity on either P or S output or single intensity on both outputs simultaneously(depending on initial phase between those). If you will put a quarter wave plate prior to polarizer, and orient it appropriately, you can discard such interference, so you will have always equal intensity on both outputs. But now rotate your polarizer by 45 degrees, and you are back to situation where you can have all light on one port.
Here you start to say clever quantum wording of distinguishability: like well, projecting of the free space hamiltonian eigen states onto eigen states modified by the polarizer can mix , since the two Hamiltonians do not commute (in other words - orthogonally polarized plain waves have common component when measured with polarizer at 45 degrees)
WOW is it a quantum phenomena? what if I use 100W laser? is it still quantum? Well, yeh, if, you do have such a cool coherent state (very far displaced vacuum) :)
Good luck.
I would like to give another example which relates directly interference to imaging. This is the case in holography when we talk about holographic imaging. One may encode the 3D and color information about an object in a hologram - analog or digital. The hologram is formed as a result of interference of the light beam scattered from the object with a mutually coherent reference beam. In a computer generated hologram (CGH) the information about the object is encoded as a 2D array of integer numbers which represent digitally an interference or fringe pattern. The CGH can be fed to a phase or amplitude spatial-light modulator. When the modulator is illuminated with a coherent light, the light diffracted from the modulator forms in space a 3D image of the object reconstructed from the hologram. The CGH can be also printed with a holographic printer on a holographic emulsion.
Imaging requires a lens. The lens MTF is considered "a measurement of the optical performance potential of a lens." (http://www.nikonusa.com/en/Learn-And-Explore/Article/h3c4udzf/what-is-a-lens-mtf-chart-how-do-i-read-it.html)
The MTF may, in theory, be derived from the interference of all point-pairs within the lens aperture. The more point-pairs within the aperture of a given distance apart, the better that ideal lens can produce the corresponding spatial frequency in the image plane.
MTF is the key to understanding that a lens, like all optical components, is an interferometer. Goodman's "Introduction to Fourier Optics" is the classic reference. Gaskill's "Linear Systems, Fourier Transforms, and Optics" adds some more detailed examples.
Real lenses have aberrations. The lens, designed and assembled, that produces the MTF closest to the theoretical limit for that aperture will produce the best images. This is because aberrations cause imperfect interference between the point-pairs within the aperture.
Neither does camera obscura have much of an MTF. KLA-Tencor, for example, does not decide to employ camera obscura for imaging. But yes, pinhole cameras form images with only a pinhole, so yes, my first statement is false. It should have stated "Typical imaging applications require a lens."
John, well indeed optical systems almost never use lens-less cameras. There are systems operating in x-ray and gamma ranges, though that do (Mirrors are rather inefficient, and lenses in x-ray or gamma aren't best choice as well), simply because far-field regime in shorter wavelength is much closer. In fact, you can define MTF for camera obscura, and its physical meaning will be rather similar to a lens system. As always, for any optical system, you define field and pupil planes, effective magnification etc. (Just did the exercise, purely of curiosity), so I take my comment back, MTF can be defined throughout any imaging system. The thing is, that for a lens-less system point spread function is much better definition, but you can always Fourier it, to get MTF.
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