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Since the diffraction pattern only depends on the width of the slit and the wavelength of light, could we see a diffraction pattern if we use an extremely small (to the order of micrometers) light source? Why or why not?

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    $\begingroup$ Yes. At the most basic level: how would the light know the difference between the size of a source and the size of an aperture? $\endgroup$ Commented Feb 1 at 11:01
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    $\begingroup$ A laser diode is an extremely small crystal/source and there is huge diffraction when it is radiating .... the only reason your laser pointer works is because of a small lens in front. The lens diameter/aperature is ~3mm, the crystal aperture is ~~10um. $\endgroup$ Commented Feb 1 at 11:43

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In case you have in mind a point-source, then - no. By definition of diffraction (you can check it in wiki) it's light bending around obstacles corners or aperture. By definition, point-source of light has no any obstacles or apertures, hence, diffraction is not possible, because light has no object to diffract upon to.

On the other hand, if what you mean is not a point-like source, but some finite dimensions "micro-source", which has some miniature parts inside like micro-lenses, micro-filters, or anything on which light can diffract, then yes,- in such micro-source diffraction will happen, as long as it have diffractable parts.

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    $\begingroup$ For diffraction to happen, light needs to encounter some obstacles. Mathematically points are defined such objects which have zero dimensions and so can't include any diffractable parts. Hence light point-source itself can't produce diffraction pattern, unless you put some diffractable parts near point source,- slits, micro apertures, any "micro-thing". So technically not the light source is producing diffraction, but some obstacles to the light which can scatter light off. Hope that makes sense. Light surely doesn't have to know anything and it doesn't. $\endgroup$ Commented Feb 1 at 13:36
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    $\begingroup$ This is merely semantics. The diffraction pattern from a small aperture is spherical, like a point source.. $\endgroup$
    – John Doty
    Commented Feb 1 at 14:09
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    $\begingroup$ @JohnDoty Not exactly. You talk about second point sources, while OP asks what it would happen if you would continuously squeeze primary light source (bulb, light diode, etc) down to the point. When dimensions of source reach comparable levels to EM wavelengths, then you will probably get double-diffraction,- one from source apertures and second from say slits after source. Then again when you squeeze it finally to the point (hypothetically) , this again will result just in a single diffraction because point-source can't have any apertures by itself. $\endgroup$ Commented Feb 1 at 14:27
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    $\begingroup$ @JohnDoty, In addition OP hasn't specified squeezed light source dimensions, so depending on them, source apertures affected diffraction wavefront may not be spherical and change along the way of source compactification. $\endgroup$ Commented Feb 1 at 14:42
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    $\begingroup$ Is an antenna a "primary source"? We analyze antenna patterns using diffraction theory. $\endgroup$
    – John Doty
    Commented Feb 1 at 15:36
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All objects diffract and diffraction is light's interaction with obstacles, specifically with their edges. The reason we see the effect more pronounced with smaller slits/openings is because the amount of diffracted light relative to the non-diffracted is much larger. Think of a slit opening in an otherwise opaque (reflecting or absorbing) very thin screen. The edge that interacts with light is more or less 1D, while the opening is 2D. As you increase the width of the slit the light in the middle grows with the area, while the diffracted light grows only linearly and will be swamped by the former. So if you want to see the diffraction clearly then you have to reduce the undiffracted amount.

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  • $\begingroup$ To this enlightening answer one could add that a fringe pattern also appears on the observation screen behind a single corner. $\endgroup$ Commented Feb 2 at 3:59
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Let us consider some practical example. There exist microscopic lasers (https://doi.org/10.1364/OPTICA.476758), they have mirrors, so there is diffraction on a mirror.

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