# How light from a thermal source gets coherent behind a small hole?

To this question Why are there interference patterns from black body radiation? was given a statement:

Incoherent light going through a small hole creates coherence, and that is how Young got coherence for the double slit experiment.

I'm curious how from a thermic and - let it be a monochromatic - source the light gets coherent inside a small hole.

Light has an oscillating electric field component (and of course a magnetic too). In my understanding incoherent light means that there have to exist constituents of the light, called photons and the phases of their electric fields are randomly distributed (which has to due with the randomly distributed emission process from the involved electrons of the radiating body). So how a small hole makes this phases equal at any point behind the hole?

• How light gets coherent behind a sharpe edge? In my elaboration about Deflection of electrons behind edges on page 8 there is a picture from the 1900th showing interference of electrons behind a single edge. – HolgerFiedler Feb 11 '17 at 7:18
• While coherence behind a small hole is coherent wouldn't be this the better laser? :-) – HolgerFiedler Feb 11 '17 at 8:44

Spatial coherence can be seen as a Classic effect of EM wavefronts, with no necessity to model light as photons.[*] We can use a wave-tank to produce incoherent water waves, or a stretched rubber sheet to produce incoherent rubber-waves, or an array of loudspeakers to produce an incoherent sound field.

First, a simplified explanation: to be spatially incoherent, waves must issue from multiple sources having random frequency/phase relationship, and producing a radiation field with spatial interference patterns with changing node positions as relative phase is changing randomly. An extended source with diameter>>wavelength is able to act as multiple uncorrelated sources. However, a point-source with diameter<< wavelength acts as a single source with a single well-defined phase ...and so an ideal point-source with infinitesimal diameter behaves as a source of spatially-coherent waves.

Second, when light passes through a pinhole with diameter<< wavelength, diffraction converts the light into sphere-waves, and all the original multiple uncorrelated spatial wavefronts of the light are erased. They're converted to sphere-waves, and sphere-waves are spatially coherent; lacking spatial interference patterns where nodes move to random positions.

Thought experiment: imagine two wave-sources having slightly different frequencies, therefore randomized phase relationship. Place them many wavelengths apart, and observe their interference patterns. The spatial patterns are varying, with nodes sweeping at right angles to the average direction of the traveling waves. This is our model of incoherent light. Next, bring the two sources slowly together. As they approach, the nodes of the varying interference patterns become wider, and the average shape of the combined wavefronts become more like sphere-waves. Finally, place the two sources together. The nodes in the interference pattern spread out and vanish, and the radiated waves become sphere-waves, but with temporal 'beat notes' caused by the frequency-difference of the two sources. Perfect sphere-waves represent spatially-coherent light, with all points on the spherical wave-front having equal phase.

In practical terms, slits and pinholes only provide light with partial spatial-coherence. The smaller the diameter of a pinhole aperture, the more spatially coherent is the light passing through. D. Gabor used this phenomenon to invent holography well before the invention of lasers. Young's experiment uses it to convert sunlight into a spatially-coherent point source (slot-source.)

Also, as light travels away from a spatially-incoherent extended source, the light becomes more and more spatially coherent, eventually approaching ideal sphere-waves in character. This phenomenon leads to the entire field of Stellar Interferometry, where stars behave as point-sources having transverse coherence-length in thousands of KM, even though the stars themselves are incoherent extended sources perhaps millions of KM in diameter.

Note that temporal coherence length is measured longitudinally along the direction of wave propagation, while spatial coherence length is measured transverse to propagation. To increase the spatial coherence length, pass light through a pinhole. To increase the temporal coherence length, pass light through a bandpass filter having extremely narrow linewidth (a "pinhole in frequency" rather than a pinhole in space.)

Rule of thumb: clear glass incandescent light bulbs with very short filaments are far more spatially coherent than frosted light bulbs. Starlight is more coherent than laser light.

[*]Also, a classic misconception is that laser "coherence" occurs when photons are in phase. No, wrong. Instead, when photons are in phase, amplification has occurred, as with the stimulated gain-medium in a laser. Stimulated emission increases the amplitude of light, but does not alter the coherence of the amplified light. Because of in-phase emission, the coherence of the stimulating light is preserved, and so a laser medium is "transparent-but-amplifying." The in-phase photon emission is not the source of a laser's coherence. Instead, spatial coherence is produced by the cavity modes: by the mirrors. See laser-cavity diagrams, and note that any light not passing through these "virtual pinholes" cannot remain in the cavity to be amplified by the gain medium: WP: optical cavities

• Wbeaty Thanks for a so detailed answer. Now apply what you wrote to thermal sources, please. How such light gets converted not only coherent but with not moving phase out of a small hole? Or would it be all with same phase Outgoing from the hole? – HolgerFiedler Feb 11 '17 at 9:26
• @HolgerFiedler atoms emitting at different frequencies, this models thermal broadband light. An ideal, zero-diameter pinhole converts all waves into spatially-coherent waves. Sphere-waves and plane-waves are spatially coherent, lacking all spatial interference patterns. Temporal coherence and monochromatic light, that's a partially-separate issue. Also, for Young's double-slit, thermal radiation from the first slit must travel the same distance to each of the two slits, else EM waves at each slit lack correlation, and no stripes are observed. "Same" means few-wavelengths accuracy. – wbeaty Feb 11 '17 at 9:46
• wbeaty, and temporal incoherent wouldn't produce patterns that "are varying, sweeping at right angles to the average direction of the traveling waves." Or more bad a blurry area big und slits. Prove it, the equations for the intensity distribution behind slits are using temporal coherence, didn't it? – HolgerFiedler Feb 11 '17 at 9:50
• @HolgerFiedler spatial coherence length is measured transverse to wave propagation, temporal coherence measured longitudinally. With Young's double-slit, if 1st slit is centered and two paths are equal-length, then temporal incoherence has no effect. Fringe spacing measured via color filter, only quasi-monochromatic. Equations: find homework problems on Michelson's measurement of stellar diameter of Betelgeuse, where fringes vanish if "double slit" spaced 306cm apart (where the star diameter serves as the first slit in Young's experiment, producing light only partial-spatially-coherent.) – wbeaty Feb 11 '17 at 10:59
• @HolgerFiedler : check out wbeaty's most excellent web site here and here. – John Duffield Feb 12 '17 at 14:19