Mr Ceffa is correct on the first question. The extra factor of 1/2 is there to make the max[t(x,y)] = 1. If the max[t(x,y)] > 1, then the grating would be amplifying the transiting field which is nonphysical for the situation described in the attached paper.

I've attached a quick derivation for the answer to the second question. The key is to evaluate the spatial Fourier transform of t(x,y). This is actually what the first half of the 4f system in Fig. 1 does in the attached paper--the field in the spatial filter plane is the spatial Fourier transform of the grating on the SLM. Once you evaluate the Fourier transform of t(x,y), the diffraction order phases are quite clear.

Thanks for the explanation but I still have two queries. First why do we take the two phases and as constant? They have a dependence on x and y co-ordinates.

Second, the paper says that the selected first diffraction order beam coming out will carry a phase of exp{i*δ1(x,y)}.what about the term 2πfx ? In the derivation provided by Mr. Hyde, the diffraction orders have the 2πfx term too, but in the paper the beam is expressed to carry only the exp{i*δ1(x,y)} term.

If we see the full paper (screen shot below) we see that a Ronchi grating is placed after the 4f system. What could be the function of the Ronchi grating which has its frequency matched with the spatial frequency of the HG? Does it somehow help in removing the 2πfx term?

I'm going to answer your questions out of order, but I'll cover them all. First, let me explain the optical system in Fig. 1. The spatial light modulator (SLM) applies the grating [complex transmittance function t(x,y) given in the paper] to the transiting optical field. The light then propagates a distance f to a lens, passes through a lens of focal length f, and propagates an additional f to a spatial filter. This optical set-up/system spatially Fourier transforms the input field. Thus, the field in the spatial filter plane is the spatial Fourier transform of the grating function t(x,y).

If you accept this, then we can think of t(x,y) as a signal and the optical system as a device that takes the analog Fourier transform of that signal. The signal t(x,y) has a carrier frequency f0, which has units of 1/m, orientated in the x and y directions and is frequency or phase modulated with signals delta1 and delta2 in the x and y directions, respectively. If the max spatial frequencies of delta1 and delta2 are much less than the carrier f0 [This is why I assumed they were constant in the original derivation. If you don't, the math gets nasty and you lose the physical picture. I've attached a quick derivation of the Fourier transform of a phase modulated cosine function to show you what I mean.], then the Fourier transform of t(x,y) will have 5 spots. One of those spots will be on axis (i.e., the DC or 0 1/m component) and in grating terminology, is referred to as the zeroth diffraction order. The other four will be symmetrically spaced away from the center spot, two in the horizontal direction (due to carrier frequency f0 in the x direction) and two in the vertical direction (due to carrier frequency f0 in the y direction). In grating terminology, these are referred to as the first diffraction orders. The spots in the positive x and y directions will possess the phase modulations delta1 and delta2, respectively; the spots in the negative x and y directions possess the conjugates of delta1 and delta2. Note that unless delta1 and delta2 are constant, the phases of the x and y first diffraction order beams will not be delta1 and delta2, respectively. In general, the x and y spots appear as "fuzzy blobs" centered on the grating's carrier frequency f0. The spatial filter, which is literally nothing more than an opaque screen with holes cut into it, allows only the positive x and y spots to pass--it blocks all others.

Before continuing, I would like to point out the similarity of all this with FM radio. In FM radio, the desired signal is the phase modulation placed on a much higher frequency carrier wave. The Fourier transform of an FM signal has two spikes, one at the positive carrier frequency and one at the negative carrier frequency. The positive frequency component contains the desired signal which shows up as a "fuzzy blob" centered on the carrier frequency; the negative frequency component contains the conjugate of that signal. To isolate the desired signal, we simply apply a filter centered on the positive carrier frequency. Of course, FM radio is modulation in time; whereas, the above description is modulation in space. Nevertheless, the analogy is valid and applicable.

After passing through the spatial filter, where only the positive x and y spots were allowed to pass, the light propagates a distance f, passes through another lens of focal length f, and travels an additional f to a Ronchi grating. Note that this is the same optical system described in the first paragraph. Thus, the field in the plane of the Ronchi grating is the spatial Fourier transform of the spatially filtered field. The Fourier transform of a Fourier transform is the original signal reflected through (rotated 180-degrees about) the origin. The field in the Ronchi grating plane is not exactly the original SLM grating field (rotated 180 degrees) due to the spatial filter (I've attached a derivation of the field in the Ronchi grating plane), but you get the idea. The purpose of the Ronchi grating is to make the x and y beams overlap and stay overlapped thereafter.

Hi Milo Hyde , thanks for your detailed explanation. Here I have another question realted to this problem.

Assume that I just output a phase distribution function \delta(x,y) in SLM plane. Then after Fourier lens, the multiple diffration orders with the same \delta(x,y) are projected onto the back focal plane (Fourier domain).

My question is, multiple diffration orders are formed by the diffration grating property of the SLM plane. Hence in principle, the \delta(x,y) should be the same in each diffraction order, only with distance lambda f/d? But according to observation, it seems that the \delta(x,y) looks different in 0th order and 1st order. I am curious why this happens.

Just so I'm clear, you're just commanding the function \delta(x,y) to the SLM? No grating?

Milo Hyde Yes, for example, I just command the random Gaussian patterns to the SLM. But SLM is not an ideal device (not 100% fill factor) which would lead to diffraction grating

Yes, SLMs are not 100% fill factor, although Meadowlark Optics advertises one that is in the high 90s. I've attached a simple MATLAB simulation of an SLM with an 87.5% fill factor. The diffraction pattern changes depending on whether the "dead space" is transmissive/reflective (depending on whether the SLM is a transmissive or reflective SLM) or absorptive. In the code, I assumed an amplitude-only SLM, like the one in the paper above [note that t(x,y) models an amplitude grating] , and made \delta(x,y) an elliptically-shaped Gaussian function. When the dead space is absorptive, a nice Gaussian ellipse is formed in the zeroth diffraction order. On the other hand, when the dead space is transmissive/reflective, the pattern is distorted and looks "sinc-like" from the rectangular SLM grid. I'm unfamiliar with amplitude SLMs, but my experience with phase-only SLMs leads me to believe that the dead space is probably transmissive/reflective meaning that the zeroth order is unusable.

If the SLM is a phase-only SLM, which are much more common than amplitude SLMs, the results will change. In short, for phase-only SLMs, the zeroth diffraction order is practically unusable. Standard practice is to command the SLM to form a stepped, blazed grating and use the first diffraction order.

Milo Hyde Thanks for your simulation code and your explanation!

I still have two problems:

1. The current SLM/ DMD are all reflective/ tramsmissive. You mention absorptive for the dead area. Is there any practical absorptive device for dead zone?

2. I can see the central spot of the 0th diffraction order of the Gaussian function is distorted when dead zone is on. But I am not sure I have the same concept of diffraction order with you. It seems that you mean the central spot with distance \lambda f/d when you mention the 0th, 1st diffraction order and so on. When I mention diffraction order, I mean the whole area (square area) with heigh and width of \lambda f/d.

1. I'm unaware of any SLM where the dead space is absorptive. I think this would be a bad thing, i.e., the laser would heat up the device. For phase-only reflective SLMs (I use these all the time), the dead space manifests as reflections from the silicon backplane.

2. I've attached a picture of what I mean by diffraction orders. For this simulation, I "turned off" the SLM (active area set to 0). The pattern in the focal plane is precisely what you'd see in an actual experiment (I've seen it many times). The pattern forms from the dead space between active pixels.

Hi Milo Hyde , thanks for your answer! My understanding of diffraction order is actually different from yours. Now I see what you mean. Thanks!