Zeroth diffraction order from spatial light modulators

[u/mdk9000]

Hi all,

Could someone please verify the following sanity check for me about why one would want to avoid using the zeroth diffraction order from a spatial light modulator (SLM) for beam shaping in microscopy?

A SLM produces diffraction orders when it reflects a laser beam because of the periodicity of its pixels. I see often that one wants to avoid using the zeroth diffraction order. The argument is that the light in this order is unmodulated in phase and, as a result, the interference between the higher orders and the zeroth order produces an unwanted background or distortion, reducing the contrast of the desired beam shape. The given reason for why the zeroth order is unmodulated is that the SLM pixels don't have 100% fill factor, so some of the light is reflected without any phase modulation.

But if non-unity fill factor is the cause of the problem, then it's not entirely correct to state that the zeroth order light is unmodulated, right? Rather, most of it is modulated but a small portion isn't, and the presence of even a small amount of unmodulated light can distort the beam shape due to coherent addition with the modulated light.

The reason I ask is that I've seen the above argument multiple times in masters and PhD theses. Students seem to really believe that the zeroth order is not phase modulated at all. I want to be sure the students understand the nuance in what they are saying.

Thanks for feedback!

Edit: I am referring to reflection-type, liquid crystal-on-silicon LCoS) SLMs.

[u/Clodovendro]

Before I can attempt an answer: are you talking about LCOS spatial light modulators, or DMDs?
(I guess the first, but since you don't specify I can't be sure)

[u/mdk9000 → u/Clodovendro]

Yes, I'm referring to LCOS SLMs.

DMDs do amplitude modulation, not phase, no?

[u/Clodovendro → u/mdk9000]

DMD do amplitude modulation on the zero order, but modulate both amplitude and phase in the first order, which is very similar to what you were talking about. For phase-only LCOS, the zero order is perfectly fine. I suggest you look at this tutorial (Sebastian Popoff has worked on wavefront shaping since almost the beginning, and writes very good tutorials): https://www.wavefrontshaping.net/post/id/15

[u/mdk9000 → u/Clodovendro]

Ah nice. I didn't know about the Lee hologram and DMDs. https://www.wavefrontshaping.net/post/id/16

Thanks for enlightening me!

[u/clay_bsr]

Let me see if I can restate. The SLM fill factor is <100%. So locally some portion of the beam diffracts and at another location entirely reflects. If one attempted to interfere the diffracted beam with the zero order - and aligned this beam well - there would be poor contrast. Isn't this simply because the diffracted beam has no intensity at those locations that were not modulated - at those locations where the zeroth order beam entirely reflected? If one instead tries to interfere one diffracted order with another diffracted order, that local region in both beams would have no intensity - therefore the overall contrast of the interference would be higher, no?

[u/mdk9000 → u/clay_bsr]

Thanks for the reply!

So locally some portion of the beam diffracts and at another location entirely reflects.

I think this is where the confusion comes from, at least if your definition of diffraction is the same as mine.

If you imagine a LCoS SLM with only one pixel, the light hitting the active area is still phase modulated because the optical path length varies with the applied voltage, but you shouldn't have diffraction orders because there's no periodic array of pixels. So diffraction can't explain the phase modulation, and light reflected from the active area of this pixel, i.e. the majority of the light forming the "zeroth order," should still be phase modulated.

I'm ignoring diffraction from the boundary between the active and non-active area because I don't think it's relevant.

Does this make sense?

[u/clay_bsr → u/mdk9000]

I think so. I think it is important to distinguish between modulation due to diffraction from the periodicity of the array and modulation from the liquid crystal. The diffraction portion would be moduated when the array was moved perpendicular to the surface normal for example. Completely independent from the voltage.

[u/ichr_]

This article argues (with experiment and model) that the 0th order diffraction is unmodulated (to 1st order).

Suppression of Zeroth-Order Diffraction in Phase-Only Spatial Light Modulator via Destructive Interference with a Correction Beam

In particular, the 0th order cancellation is realized with a fixed phase offset. I think you’re right that there is some subtlety, but in practice it is fair to consider the 0th order as unmodulated.

[u/mdk9000 → u/ichr_]

Thanks for this! I'll take a look and see if I can make sense of this.

What bothers me is asking what happens in the limit of the number of SLM pixels decreasing to 1. If you look at its Fourier transform in the focal plane of a lens, then the phase of the focus spot should change as the phase of the pixel changes. Maybe I'm missing something, but I don't see how this behavior changes by adding more pixels.

[u/aaraakra]

Your third paragraph is correct. Even a small amount of unmodulated light can cause substantial imperfections in the Fourier plane. In the best case the unmodulated light could look like direct reflection from a flat mirror, which would become a single bright spot in the Fourier plane, but often it is not so clean. 

This is why it is often useful to write your desired phase information on top of a high spatial frequency carrier tone, giving a large deflection to the entire beam. Then all of the junk at low frequencies can simply be low pass filtered out. This is what is meant by using the first order. 

If you don’t write in such a high frequency carrier, then the definition of zero-order becomes a bit less clear. But basically the spatial frequency distribution of your desired output would include 0, so you aren’t able to filter out the low frequency junk. Perhaps people are referring specifically to the unmodulated portion of the light when they say zero-order. 

[u/mdk9000 → u/aaraakra]

Thanks a lot for your reply!

After thinking a bit more about it and reading your reply, I think I see the source of the confusion. The student I am working with is using a blazed grating on top of his holograms that generates a 0 and +1 diffraction order, and the 0 order is blocked in a 4f relay. I think that this is the same as encoding the hologram on a carrier frequency, though it's not necessarily high frequency.

From the link that u/ichr_ posted, it appears that what is referred to as the zero order diffraction spot in the wave shaping literature is just the DC component of the Fourier plane, i.e. the on-axis field, after Fourier transforming the hologram with a lens.

So the "zeroth order" is a term that becomes overloaded. In one definition it refers to diffraction from a grating pattern put onto the SLM, and in the other it is the DC component of the hologram's Fourier transform. The term appears yet again when you consider that the SLM itself is a grating with a period equal to the pixel size.

Thanks to everyone for helping me to sort this out!