# Emergence - what constitutes a minimally coherent source that "just begins" to produce stationary interference patterns?

• A superposition of two signals with different frequencies will never produce visible (i.e. stationary) interference patterns. Such waveforms will produce spatiotemporal beat patterns, but they rapidly propagate through space such that a stationary detector (e.g. screen or antenna array) never sees any interference fringes whatsoever. One can easily verify this e.g. by simulating two point sources emitting waves of slightly differing frequencies.

• AFAIK, simply increasing the number of signals with discrete frequencies (e.g. to one billion) will also never produce static interference.

• However, real life experience shows that it's quite straightforward to generate visible interference fringes from virtually any source of almost arbitrary spectral properties and coherence, such as black body radiators (burning candle, lightbulbs, glowing metal surfaces, sunlight), broadband LEDs etc.

QUESTION: Where exactly between these extreme cases do stable patterns start emerging out of chaos, i.e. what is mathematically the minimal condition for the emergence of stationary interference fringes?

Exactly how many frequencies and exactly what kind of phase correlations are required? What mathematically constitutes a minimally coherent source that will "just begin" to produce static/visible interference patterns?

What does a burning candle or white LED provide that a superposition of a billion or trillion sine waves cannot?

To better visualize the problem, have a look at the following 2D wave fields generated with William Leu's WaveToy:

Overview

Five point emitters and two square barriers rotated 45° to emulate a single slit. In the subsequent snapshots, spatial coherence was reduced by increasing the slit width, and temporal coherence was reduced by adding a frequency step $$\Delta f$$ from one emitter to the next. Case I: Full spatiotemporal coherence

Fully coherent wave field, both in front and behind the single slit.

On a screen placed anywhere in front or behind the slit, a stationary intensity pattern would be observed (a more complex pattern in front and a single, broad lobe behind the slit), independent of the time scale of observation. Case II: Spatial incoherence, temporal coherence ($$\Delta f = 0$$)

Same setup as above, but larger slit width. The wave field gets broken up in smaller areas of spatial coherence.

Important: These coherent areas are stationary and would produce visible/static interference fringes on a screen. Case III: Spatial coherence, temporal incoherence ($$\Delta f = 0.1$$)

Spatially coherent wave fronts, but now the wave field is broken up into "wave packets" propagating with the speed of light (image intensity rescaled to increase visibility of diffracted fields).

Important: The wave fronts between subsequent "packets" are no longer in phase (not easy to see here)! For a larger number of emitters, the phase relation would be essentially random. On a screen, one would observe either random flashes or a homogeneous illumination, depending on the speed of the detector (camera sensor, eyes). Case IV: Spatiotemporal incoherence ($$\Delta f = 0.1$$)

Wave field is both spatially and temporally incoherent. Random$$^*$$ patches of (transiently?) coherent areas rapidly propagate through space.

On a screen, one would observe either essentially random$$^*$$ fluctuations (speckle noise) or, over more realistic integration times, a homogenous illumination. No visible interference fringes. As far as I understand, it is no longer possible to generate stationary intensity patterns from such a source by any (passive) means.

...yet, nothing is easier than generating stationary interference patterns from virtually any real light source, such as sunlight or a piece of red-hot metal! $$^*$$Here: Only quasi-random, see 2nd remark below.

Remark 1: Note that the coherence characteristics of the wave field behind the slit are a reflection of the "source field" characteristics before it. In those cases, where visible interference patterns can (or potentially could) be observed - namely I & II, the source field itself has a stationary intensity distribution. Whereas in cases III & IV, the source field itself does not have a stationary intensity distribution. To my current understanding, passive elements such as a diffraction slit therefore only serves to select a subset of field constituents with have a higher correlation (coherence) than the underlying ensemble.

Remark 2: The wave fields in cases II-IV are, in fact, not 100% incoherent when measured in the respective quantities (coherence area for spatial coherence, coherence time/length for temporal coherence). Rather, they are still partially coherent due to the small number of emitters and frequencies used in the simulation, so the term "incoherent" here is used as a relative one to compare different scenarios.

The latter point also results in a periodicity of the wave fields. Without actually (continuously) randomizing the emitters, the generated fields will repeat as can be seen in the following updated example using 10 sources with 10 different frequencies and phases (A, B, C representing repeating "wave packets"): In this sense, the whole problem/question could also be formulated backwards:

• On one hand, we have to add more and more constituent waves in order to further and further randomize the wave field and thereby reduce correlation.
• On the other hand, upon superposing sufficiently (infinitely?) many components, we "suddenly" get precisely what we've tried to eliminate, namely a strong correlation in the form of stationary intensity distributions.

UPDATE: Two more simulations, this time using only 2 emitters with different frequencies (image intensities rescaled to increase visibility of diffracted fields).

Case IV.b: Spatiotemporal incoherence, single slit ($$\Delta f = 0.1$$)

The wave field looks similar as in case IV above, but with larger coherent patches. Again, these patches rapidly propagate through space. As one increases the number of emitters, the field is broken up into smaller and smaller patches. Case IV.c: Spatiotemporal incoherence, double slit ($$\Delta f = 0.1$$)

Important: The distinct "fringe patterns" rapidly propagate through space and do not represent stationary interference fringes.

As the number of emitters is increased, the resulting wave fields will transition more and more into chaos. UPDATE 2: Proof for the 1st observation in the question, namely that a superposition of two waves with different frequencies never produces a stationary interference pattern.

Starting with the modulus squared of a complex number

$$|z|^2 = \sqrt{zz^*}^2$$

we insert the superposition $$z = z_1 + z_2$$ with

$$z_1 = \tfrac{1}{2} e^{i k_1 x}e^{-i \omega_1 t} \\ z_2 = \tfrac{1}{2} e^{i k_2 x}e^{-i \omega_2 t}$$

using half-amplitudes for convenience (just to normalize the resulting intensity) and get

\begin{align} |z|^2 &= \sqrt{ (z_1 + z_2)(z_1 + z_2)^* }^2 \\ &= \sqrt{ z_1 z_1^* + z_2 z_2^* + z_1 z_2^* + z_1^* z_2 }^2 \\ &= \sqrt{ 1 + z_1 z_2^* + z_1^* z_2 }^2 \\ &= \sqrt{ 1 + z_1 z_2^* + (z_1 z_2^*)^* }^2 \\ &= \sqrt{ 1 + 2 \ \Re (z_1 z_2^*) }^2 \\ &= \sqrt{ 1 + 2 \ \Re (\tfrac{1}{4} e^{i k_1 x} e^{-i \omega_1 t} e^{-i k_2 x} e^{i \omega_2 t}) }^2 \\ &= \sqrt{ 1 + \tfrac{1}{2} \ \cos (k_1 x) \ \cos (-k_2 x) \ \cos (-\omega_1 t) \ \cos (\omega_2 t) }^2 \\ &= \sqrt{ 1 + \tfrac{1}{8} \ [\cos ((k_1 - k_2)x) + \cos ((k_1 + k_2)x)] \times [ \cos ((-\omega_1 + \omega_2)t) + \cos ((-\omega_1 - \omega_2)t)] }^2 \\ &= \sqrt{ 1 + \tfrac{1}{8} \ [\cos (k_- x) + \cos (k_+ x)] \times [ \cos (\omega_- t) + \cos (\omega_+ t)] }^2 \\ &= \sqrt{ 1 + \tfrac{1}{8} \ \Psi_{\text{spat. beat}}(x) \ \Psi_{\text{temp. beat}}(t) }^2 \\ \end{align}

with $$k_1 \pm k_2 =: k_\pm$$, $$\omega_1 \pm \omega_2 =: \omega_\pm$$, and using the cosine's symmetry and a prosthaphaeresis relation in the final steps.

The result is not simply a stationary sum of the intensity distributions from individual sources, but a time-dependent/oscillatory sum of "crossed" products of waves, i.e. a spatiotemporal beat pattern that rapidly propagates through space. This is also visualized in the simulations above (scenarios IV, IV.b, and IV.c).

Also, both spatial and temporal beat components are multiplicative (they're in a product term), so averaging over either one returns zero spatiotemporal modulation and, therefore, zero fringe contrast:

\begin{align} \langle |z|^2 \rangle_T \ &= \ \langle \sqrt{ 1 + \tfrac{1}{8} \ [\cos (k_- x) + \cos (k_+ x)] \times [ \cos (\omega_- t) + \cos (\omega_+ t)] }^2 \rangle_T \\ \ &= \ \sqrt{ 1 + \tfrac{1}{8} \ [\cos (k_- x) + \cos (k_+ x)] \ \times \langle [ \cos (\omega_- t) + \cos (\omega_+ t)] \rangle_T }^2 \\ \ &= \ \sqrt{ 1 + \tfrac{1}{8} \ [\cos (k_- x) + \cos (k_+ x)] \cdot 0 }^2 \\ \ &= \ 1 \end{align}

For the plane waves considered here, the result is simply the sum of source intensities ($$\tfrac{1}{2} + \tfrac{1}{2} = 1$$), as one would expect due to energy conservation. In the case of e.g. point sources, one would get the sum of $$\tfrac{1}{r^2}$$ laws for the source intensities. Experimentally, temporal averaging corresponds to the signal integration (limited time resolution/finite exposure time) inherent to any realistic sensor or detector, and spatial averaging corresponds to a finite detection volume (limited spatial resolution).

...to round things off (and prove point 3 in the introduction), here is an actual Michelson interferogram taken from the flame of a lighter, so basically combusting butane gas with black body radiation from glowing soot particles: 