Can n-slit interference produce subwavelength stripes?

[genxium]

TL;DR Summary

If n(number of slits) is around 1000, can we produce an intensity pattern whose adjacent local minimas are separated by a distance less than the wavelength of incident light?

According to the intensity formula
$$I = I_0 \cdot \frac{sin^2(n\phi/2)}{sin^2(\phi/2)}$$
(quoted from Feynman Lectures Vol I)

Therefore the 2 local minimas adjacent to each global maxima are angularly $\Delta \phi_{localmin} = \frac{4\pi}{n}$ apart. When trying to think of a use case to exploit this formula I got something like this

Given an imaging medium whose intensity response is only non-zero when the intensity is above certain threshold, can I cast subwavelength stripes of monochromatic light on it by an n-slit apparatus?

(the orange dash line marks the threshold intensity below which the imaging medium wouldn't respond)

It seems possible to me that if we use a monochromatic red laser with wavelength $\lambda=650nm$, a reasonably large $n = 1500$, and a reasonably small "slit-to-imaging-plate distance" $r = 20um$, by far-field approximation it produces a stripe width $w = r \cdot \Delta \phi_{localmin} = 167.55nm < \lambda$ on the presumed imaging medium.

Is there something I overlooked here that'll prohibit such subwavelength stripe?

There're two possible caveats of the derivation that I'm aware of but not sure whether they're critical enough to break the argument (still checking them theoretically).

• The formula above doesn't take into account the impact from intensity distribution of single slit diffraction, by regarding each single slit just as a perfect point source.
• When having $n = 1500$, the "slit-to-imaging-plate distance" $r = 20um$ might not be an eligible far-field compared to the total length of all slits, e.g. $n \cdot (SlitWidth + SlitPitch)$ which easily exceeds $r$ for slits of a few microns.

Any advice is appreciated :)

 

[sophiecentaur]

TL;DR Summary: If n(number of slits) is around 1000, can we produce an intensity pattern whose adjacent local minimas are separated by a distance less than the wavelength of incident light?

Is there something I overlooked here that'll prohibit such subwavelength stripe?

Your "sub wavelength" term seems to have some special significance for you. The sin(Nx)/sin(x) type of pattern simply shows the distribution of angles at which minima can be found. Those are directions of the features of just a pattern.

If you were a radio antenna designer you wouldn't be at all surprised that there can be very tight patterns of side lobes from a directional antenna. There is no spacial significance in this.

You need to think where you could actually measure this pattern. The pattern you have calculated is essentially a far field (Fraunhofer diffraction) pattern. The width of your multiple slit array would be much greater than a wavelength and you'd need to be a long way away to see your minima - many wavelengths away - and the separation of the minima would be larger than the wavelength of your light / radio / sea waves / (and probably gravity waves, if you could make the equipment).. In close, there would be a different (Fresnel) pattern.

 

[BvU]

reasonably small "slit-to-imaging-plate distance" r=20 $\mu$m

How wide is the beam spot at the position of grating if it illuminates 1500 lines ?

I think 'reasonably small' is the understatement of the month !

ah, beaten by sophie

$\$

 

[sophiecentaur]

In close, there would be a different (Fresnel) pattern.

Furthermore, iirc, when you're close into the array of slits, the pattern will be complicated but not have nulls in it. I think this is what was nudging the back of the mind of @genxium : there's still no 'violation' of anything because there will be few or no nulls near the array.

 

[genxium]

Furthermore, iirc, when you're close into the array of slits, the pattern will be complicated but not have nulls in it. I think this is what was nudging the back of the mind of @genxium : there's still no 'violation' of anything because there will be few or no nulls near the array.

Thanks sophie that was a thorough answer. As both you and @BvU pointed out the "reasonably small $r = 20um$" part is what looks unreasonable here, I'd look more into it :)

I used to put some efforts into searching for a near-field intensity formula of the n-slit apparatus, and come across something useful like this paper (which does contain examples like an aperture or a convex lens, but not the n-slit) and later this other paper (which contains exactly the n-slit).

From the second paper, it seems like there're still some nulls in the pattern predicted by the near-field model

, and even if there were no null I'm happy to see that the peeks are easily distinguishable by drawing a certain threshold.

The term "sub-wavelength" does have some special significance for me. The processing capability I can access would only produce slit width & pitch as well as positioning resolution at a few microns -- the feasibility of "sub wavelength stripes" just intrigues me out of no reason.

There're already many existing ways to achieve "sub-wavelength" interference

• with already sub-wavelength slit and surface plasmon polariton
• with electromagnetically induced transparency (don't really understand this)
• whatever tricks used by the photolithography industry, e.g. large NA and wet-litho

, yet I'm still quite interested in how "minimum" the setup for these "sub-wavelength stripes" can be.

 

[sophiecentaur]

From the second paper, it seems like there're still some nulls in the pattern predicted by the near-field model

The total width of the array is many wavelengths so is the result a surprise? The relevant condition for 'stripes' is not the absence or presence of nulls but a variation of resultant amplitude. What you can see in "Fig 7" is a variation in amplitude of the resultant. I would say that having no variation in the Fresnel patterns would actually be very unexpected.

So the sums look right and the result is not unexpected.

 

[sophiecentaur]

So the sums look right and the result is not unexpected.

You have to be careful when looking at the result image. There is zero amplitude in between the slits because of the individual slit pattern (2 wavelengths wide), which will launch no energy parallel to the surface. I couldn't see any numbers for the resultant amplitudes of the pattern close in and the density of the blacks and greys is not specified. The troughs near-in are very low level and are spaced by less than a wavelength but they do not represent, in themselves, a travelling wave; they are the result of a number of waves, adding vectorally.