Effects of coherence length on optical interference filter

[shiweiliu]

I have an optical interference filter that is used to block (reflect) solar light heat between 800nm and 1100nm. That filter is made by using 200 layers of quarter wave high and low index coatings on plastic film. The total thickness of the coating is 25 micrometers or even 50 micrometers.

I was told that interference effect can only happen when the thickness of coating is less than coherence length of light. How can my filter work when the total thickness of interference coating is much larger than the coherence length of solar light that is only about one micrometer?

I can understand locally the adjacent coating layers can still have interference effect but how can the light beam reflected by the top coating layer interfere with light beam reflected by bottom coating layer since the optical path difference between these two beams is far larger than coherence length of solar light?

 

[renormalize]

I was told that interference effect can only happen when the thickness of coating is less than coherence length of light. How can my filter work when the total thickness of interference coating is much larger than the coherence length of solar light that is only about one micrometer?

The coherence length of sunlight is significantly larger than $1 \mu\text{m}$.
From this reference: Spatial coherence of sunlight... I quote an excerpt (from pg. 98):

So for a solar wavelength of $0.5 \mu\text{m}$ the coherence length $L_\bot$ amounts to $40 \mu\text{m}$.

 

[shiweiliu]

The coherence length of sunlight is significantly larger than $1 \mu\text{m}$.
From this reference: Spatial coherence of sunlight... I quote an excerpt (from pg. 98):
View attachment 355358
So for a solar wavelength of $0.5 \mu\text{m}$ the coherence length $L_\bot$ amounts to $40 \mu\text{m}$.

Oh. I estimated it according to formula: wavelength^2/Delta wavelength. Wavelength is about 500nm and Delta wavelength is about 300nm. It is about one micrometer. Something is wrong in my method?

 

[Gleb1964]

defining the coherence length L⊥ as the slit separation for which theamplitude of η⊥ decays to 1∕2, we find that L⊥ 80λ fordirect sunshine

It looks like they are defining something that they call for "a coherence length", but I am not sure that is the same as coherence length defined as "the maximum optical path difference to still observe interference".
For unfiltered sunlight the interference would disappear withing 1um or less path difference. Using a narrow bandpass filter (or spectral device) it is possible to get interference with much longer path difference.
However, there is a limit for thermal sources that I have seen in a physics books, which is connected to the maximum coherence time not exceeding 10^-8s or 2..3m of path difference.

 

[Cthugha]

The coherence length of sunlight is significantly larger than $1 \mu\text{m}$.
From this reference: Spatial coherence of sunlight... I quote an excerpt (from pg. 98):
View attachment 355358
So for a solar wavelength of $0.5 \mu\text{m}$ the coherence length $L_\bot$ amounts to $40 \mu\text{m}$.

That is spatial coherence or the transverse coherence length. It tells you how point-like the light source is or what the maximal separation of slits may be to observe interference in a double slit experiment.

For interference filters, the longitudinal coherence length (coherence time times the speed of light) is the relevant quantity. The coherence time is essentially given by the Fourier transform of the power spectral density of your light field. Simply speaking: a broad spectrum results in a short coherence time. Therefore, one may improve coherence time by spectral filtering. Depending on what is going on in the atmosphere, sunlight filtered by the atmosphere is usually considered to have a coherence time on the order of one to few femtoseconds:

https://www.nature.com/articles/s41598-022-08693-0

A coherence time of 1 femtosecond corresponds to a longitudinal coherence length of about 300 nm.

 

[Gleb1964]

Intersting article in Nature, but it doesn't give the answer to the topic starter question, asking why a thick multilayer coating is working while sunlight has a very short coherence length.

 

[Charles Link]

The Fabry-Perot effect (present in interference filters) is often taught as multiple reflections that occur that then might need fairly long coherence lengths to be completely effective. You might find this Insights article that I authored a few years ago of interest, where the interference effects can occur at a single dielectric interface, and where multiple reflections are not the fundamental reason why the interference occurs. The multiple layers with the multiple interfaces do play a role, but as this article explains, interference occurs from two mutually coherent sinusoidal sources incident onto a single dielectric interface from opposite directions.

See https://www.physicsforums.com/insights/fabry-perot-michelson-interferometry-fundamental-approach/

 

[Charles Link]

I need to add one more item to the above=at each dielectric interface in the multi-layer system there is a right-going wave and also a left-going wave (amplitude) on the left side of the interface, and also a left-going wave and a right-going wave on the right side of the interface. There is a phase relation depending on the path distance of each layer that also comes into play, (between the right-going waves at adjacent interfaces, etc. and between the left-going waves at adjacent interfaces). This is how the multi-layer problem is solved rather than trying to compute all possible multiple reflections.