Abstract

When a hologram is desired from an object which does not exist physically but is known in mathematical terms, one can compute the hologram. An automatic plotter will make a drawing at a large scale which is then reduced photographically. Since the drawing can contain only black and white areas, we have developed a theory for binary holograms. They are equivalent in terms of image reconstruction with ordinary holograms. This has been proven theoretically and verified experimentally.

© 1967 Optical Society of America

Introduction

An ordinary hologram is produced by recording the interference pattern which is caused by an object wave and a reference wave. Here we want to show how a computer-guided plotter can draw holograms. This new way for making holograms is sometimes advantageous, mainly because no object has to exist. It is enough if the object is known in mathematical terms.

Our computer-generated holograms are binary, i.e., they consist of many transparent dots on an opaque background. The binary character of our holograms facilitates their production. First the holograms are plotted in black and white on a large scale, then they are optically reduced in size and recorded on film. Although the transmittance of our holograms can assume only the values zero or one, the reconstructed images are as good as those from ordinary gray holograms of comparable dimensions.

Our holograms, as well as ordinary holograms, have a real nonnegative amplitude transmittance; nevertheless both types are able to influence the phase of a light wave. The trick which achieves a phase shift, is called detour phase and has been explained earlier.[1] In brief, one considers a Fraunhofer hologram as a grating with irregularities. Rowland and Lord Rayleigh have shown in their explanation of grating ghosts, that a lateral displacement of some grating grooves causes a deformation of the diffracted wavefront, i.e., a phase shift. Ordinarily these phase shifts are undesirable in spectroscopic gratings, but sometimes, as in a hologram, these phase shifts are essential.[2][4] One can get an intuitive introduction into holography, based on the detour phase effect,[1] but here we prefer a deductive approach because our intuitive explanation left many readers unconvinced. Furthermore, the intuitive approach was afflicted by some approximations; their severity had not been estimated. Yet another approach has been presented recently by Waters.[5]

As has been said before, our binary holograms and the ordinary gray holograms are alike in their manner of producing phase shifts, but they are different in the way they influence the amplitude of the light wave. In our holograms, the gray scale is simulated by the sizes of many fine dots, as it is also done in halftone printing. A study by Marquet and Tsujiuchi[6] demonstrates very nicely the diffraction aspects of halftone print patterns.

It is our plan to explain the principle of computer generated Fraunhofer holograms, using a deductive approach. First we describe the selection of a proper format of the hologram. Then follows the diffraction theory with and without approximations. Next we comment on the computational and plotting procedures, and finally we show some experimental results. The formulation of the theory is chosen such that it will be easy to generalize it toward spatial filters[1] and binary image holograms.[7]

Format Selection

We want to produce a complex amplitude distribution u(x,y) in the image plane IMG (see Fig. 1). To do this we illuminate a properly designed binary hologram H(νx,νy) with a tilted plane wave exp(2πix0νx), abbreviated as E(x0,νy). [The actual coordinates (xH,yH) in the plane of the hologram are related to the reduced coordinates (νx,νy) by xH = λx;yH = λy.] The lens between HOLO and IMG performs a two-dimensional Fourier transform of the complex amplitude; hence, the Fourier transform ũ(νx,νy) of the desired image amplitude u(x,y) is required as complex amplitude in the plane HOLO, whereby u is defined as

u˜(νx,νy)=u(x,y)E(-xνx-yνy)dxdy.

We attempt to produce the complex amplitude ũ by illuminating the binary hologram H by means of a tilted plane wave. In mathemetical terms we hope to find a binary function H which satisfies the following equation:

constu˜(νx,νy)=H(νx,νy)E(x0νx).

This will not be possible in the most general case, since ũ could be any complex function, whereas the amplitude transmission H(νx,νy) of the binary hologram can assume only the values one or zero. Hence HE is severely limited in its generality.

In practice, however, u(x,y) will not be unrestricted in generality, thus making our problem solvable. The two restrictions for u(x,y), which we assume from now on are a finite extension Δx = Δy in area and a limited resolution δx = δy = 1/Δν:

u(x,y)=0outsideofxΔx/2;yΔy/2;u˜(νx,νy)0outsideofνxΔν/2;vyΔνν2.

The resolution condition tells us that the size of the hologram can be limited to |νx| ≤ Δν/2; |νy| ≤ Δν/2. The condition for the finite size of the image means that the light which is diffracted from the hologram shall build up a complex amplitude u(x,y) within the area |x|,|y| ≤ Δx/2 of the image plane, but we do not care whether there is any light outside of this area. If there is any light we can eliminate it by means of a square shaped window frame of size (Δx)2. Our ambiguity about the light outside of the window is the reason why the actual amplitude HE in the plane HOLO does not have to be equal to ũ.

Another way to state our problem is by defining in Eq. (4) a function h(x,y), which is the diffracted amplitude from the hologram H(νx,νy) in the region of the image u(x,y). Then we require h(x,y) to be proportional to u(x,y) [Eq. (6)].

h(x,y)=rect(x/Δx)rect(y/Δx)H(νx,νy)E[(x+x0)νx+yνy]dνxdν;.
rect(z)={1;inz120;otherwise
h(x,y)=constu(x,y).

The finite size of the image allows us to draw some conclusions about the size of the resolution element (δν)2 of the hologram. The number of resolvable points in the image is:

N2=(ΔxΔy)/(δxδy)=(Δx/δx)2=(ΔxΔν)2=SW.

This can also be considered as the space–bandwidth product SW or as the number of degrees of freedom in the image. Obviously there must be at least as many degrees of freedom in the hologram as in the image, which shall be generated from this hologram. In other words, the number of resolution elements in the hologram has to be larger (or equal) than the number of resolution elements in the image:

(Δν/δν)2(Δx/δx)2=(ΔxΔν)2;   δν1/Δx.

The sampling theorem gives us a good reason to believe that the largest permissible δν is good enough, i.e., δν = 1/Δx. Since u(x,y) is different from zero only in a square of size (Δx)2, its Fourier transform ũ can be written as:

u˜(νx,νy)=ΣΣ(n,m)u˜(n/Δx,m/Δx)×sinc(νxΔx-n)sinc(νyΔx-m).

The sinc function is defined here as sinc(z) = sin(πz)/(πz). In other words, ũ(νx,yy) is completely specified by a set of complex parameters ũ(nx,mx), which represent ũ at a mesh of points: νx = nx; νy = mx. A quadratic area of size (1/Δx)2 = (δν)2, surrounding each of these sampling points, will be called a cell.

Since the density of free complex parameters is one per cell for ũ(νx,νy), we need at least the same density of free parameters in the binary hologram H(νx,νy). Considering that H can contain only real parameters, we require two real parameters per cell for H, as compared to one complex parameter per cell for ũ. The total number of real parameters in H is then equal to twice the number of cells: 2(Δν/δν) = 2N2.

There are infinitely many different ways in which one can assign two parameters to the size, shape, and position of one or several spots within each cell of the hologram. In most of our recent experiments we choose one rectangular spot per cell with variable height Wδν, with fixed width cδν, and with variable position Pδν (Fig. 2):

νx-(n+Pnm)δνcδν/2;νy-mδνWnmδν/2.

In other words, W is the relative height, c the relative width of the rectangular dot, and P its shift off center, measured with the cell size δν as unit length. Two other shapes were described earlier.[1] Our choice was dictated by the abilities of the plotter we used. Another consideration is that some spot shapes cause less mathematical complications than others. For example, the following symmetry relation, which singles out a specific class of spot shapes, helps to keep the mathematics simple:

(m-12)δν(m+12)δνH(νx,νy)dνy=Hm(νx);Hm[(n+Pnm)δν+μ]=Hm[(n+Pnm)δν-μ],

within each cell (n,m).

Furthermore, the choice of the cell shape has some influence on the brightness of the image, as we shall see in the next section.

Diffraction Theory of Binary Fraunhofer Holograms

As stated before, a tilted plane wave E(x0νx) falls onto the binary hologram. The binary amplitude transmission H of the hologram can be described by means of the rect function:

H(νx,νy)=ΣΣ(n,m)rect[νx-(n+Pnm)δνcδν]rect[νy-mδνWnmδν].

Each product of two rect functions accounts for one of the rectangular openings in the cells (Fig. 2). The complex amplitude behind the hologram is: H(νx,νy) E(x0νx). Its Fourier transform appears as complex amplitude in the image plane IMG (Fig. 1):

H(νx,νy)E[(x+x0)νx+yνy]dνxdνy=c(δν)2sinc[cδν(x+x0)]ΣΣ(n,m)Wnmsinc(yWnmδν)E{δν[(x+x0)(n+Pnm)+ym]}.

We want to choose the 2N2 free parameters Wnm, Pnm and the two constants x0,c such that the complex amplitude in IMG, which comes from the hologram, equals the desired image u(x,y), at least within the region |x| ≤ Δx/2; |y| ≤ Δx/2:

rect(x/Δx)rect(y/Δx)HEdνxdνy=h(x,y);h(x,y)=constu(x,y).

For this purpose it is practical to rewrite u(x,y) in a different form. We insert the sampling representation of ũ(νx,νy) [see Eq. (9)] into the Fourier representation u(x,y) and get:

u(x,y)=u˜(νx,νy)E(xνx+yνy)dνxdνy=rect(x/Δx)rect(y/Δx)(n,m)u˜(nδν),mδν)E[δν(xn+ym)].

As we can see from Eq. (15) u(x,y) is represented as a Fourier series within the (Δx, Δy) area, and h(x,y), which is defined in Eq. (14) and represented in detail by Eq. (13), is almost a Fourier series except for the two sinc functions with the arguments cδν(x + x0) and yWδν, and for the phase factor E(xPnmδν). Let us assume for the moment that these three factors are all close to unity so that h is represented with sufficient accuracy by the Fourier series, then we could achieve the equality of the two series simply by equalizing the Fourier coefficients one by one:

c(δν)2WnmE[x0δν(n+Pnm)]constu˜(nδν,mδν);constu˜(nδν,mδν)=c(δν)2AnmE(φnm/2π);WnmAnm;Pnm+nφnm/2πx0δν.

This means that the height W and the position P of the dot in the cell (n,m) are directly responsible for generating the amplitude A and the phase φ of the complex amplitude ũ at the cell. The equalization of the phase factors E[x0δν(n + Pnm)] and E(φnm/2π) leads to a simpler equation if x0 is chosen such that x0δν is an integer M:

Pnmφnm/2πM.

This simplification is also justified for the following practical reason: it is desirable to place the dots close to the center of the cells, at least in average: P¯=0. If the phase φ has a zero average, φ¯=0, then P¯=0 is achieved by x0δν = M:

E[x0δν(n+Pnm)]=E(MPnm);φnm=2πMPnm;φ¯=2πMP¯.

These Eqs. (16) for the positions Pnm and heights Wnm of the rectangular dots in the hologram have been derived earlier[1]; however, it was not stated then that their validity is contingent upon these three approximations:

(a)sinc[cδν(x+x0)]const,inxΔx/2;(b)sinc(yWnmδν)1inyΔx/2;(c)E(xPnmδν)1inxΔx/2.

As we have just seen those approximations did allow us to consider h(x,y) as a Fourier series which leads immediately to simple formulas for the shape of the dots in the hologram. In the following two sections, we discuss under what circumstances these approximations are justified, and as an alternative, how one can solve the fundamental equation h = constu without relying on the assumptions (17 a,b,c).

Discussion of the Approximations

Before discussing the approximations (17 a,b,c) in detail, we need a better understanding of the two parameters M = x0x and c. Physically M = x0δν = x0x is related to the tilting angle of the illuminating plane wave, coming from point x0 (Fig. 1). We get an interrelation between M and the relative dot width c, when considering that the phase φ has to be able to assume every value between −π and +π:

φπ;φ=2πMP;P12M.

Herein P describes the position of the center (Fig. 2). If we want to prevent a dot from extending beyond the edge of its cell, we must require P+c/212; c ≤ 1 − 1/M. Actually we disobeyed this rule in many of our holograms by setting c=12 and M = 1. It may happen that the dot from the cell (n,m) extends into the cell (n + 1,m) and overlaps with the adjacent dot. For example, P(n,m)=12;P(n+1,m)=-12 would cause an overlap. If treated properly, these overlap area should be implemented as doubled transmittance; however, overlapping occurred so rarely, that we did not pay any attention to this problem in most of our holograms. The experimental results justified this attitude.

The reason for disobeying the rule c ≤ 1 − 1/M was that we wanted to get an image as bright as possible. Looking at the formulas (14) and (13) for h(x,y), we find the image amplitude will be proportional to c sinc [cδν(x+ x0)]. The square of this term describes the image brightness. At the center (x = 0) this factor becomes (with x0δν = M) sin(πcM)/(πM). As far as the relative dot width c is concerned, the brightness is optimized for cM=12,32,, yielding a brightness factor (1/πM)2; hence, |M| = 1 is desirable. The only choice of c is 12, since c=32>1 would mean a dot width larger than the width of the cell. Actually c=12 is the largest meaningful dot width, because c and 1 − c are equivalent in terms of Babinet’s principle.

Now let us discuss the implication of the first approximation (17a), which, among the three approximations, is most noticeable in the experiments, but also most easily avoided by mathematical precautions.

sinc[cδν(x+x0)]const,inxΔx/2.

Neglecting this sine function means that the diffracted intensity |h(x,y)|2 might deviate from the wanted image intensity |u(x,y)|2 by the square of this sine function. Remembering x0δν = xx = M we get as the two extreme values at x = ±Δx/2: sinc(cM ± c/2). Obviously a small dot width c is desirable, which however would reduce the brightness of the image. For cM=12 the ratio of the two extreme values becomes (1 + c)/(1 − c) which corresponds in the worst case of c=12 to a brightness range of 9:1. A dot width of c=13 and M = 1 means only a reduction in brightness of about in the center of the image, and a ratio of about 2:1 of the two extreme values of the sine function squared. The observation of this effect will be reported in the chapter on experiments; however, sometimes this effect is of no importance. For one thing, the sine function varies smoothly. Secondly, if the object u(x,y) is binary (0,1), the nonlinear response of the photographic plate or of the eye can be adjusted to suppress the inhomogeneity of the image brightness, and even for continuous tone images one can tolerate quite a bit of brightness inhomogeneity. For example, most photographic lenses produce only 30% brightness in the corners of the image field, obviously without offending the spectator.

The second approximation (17b) which we want to discuss next is

sinc(yWδν)1,

where y varies within ±Δx/2 and W within zero and one. Because of Δxδν = 1, the argument of the sine function stays within +12 which means that the sine function may drop down to 0.64. In some of our experiments, we simply calculated the holograms, as if sinc(yWδν) = 1. No decrease in image quality was recognizable, not even close to the upper and lower edges |y| = Δx/2. This is qualitatively understandable, because the sine function acts as if the aperture (the hologram), when looked at from the image point y, has a slightly decreased amplitude transmission at (νx = nδν; νy = mδν) by a factor of sinc(yWnmδν). It is well known that amplitude errors in the aperture of coherent image forming systems do little to the image because they do not deviate the rays as phase errors would do. We realize of course that this qualitative statement cannot be taken as a reliable protection against deterioration of any image, though up to now we have had no experience to the contrary. If one wants to be on the safe side however, one can make every W smaller by a factor 2/π for example. Then the sine function never drops below 0.84. The price one has to pay for this precaution is that the image brightness will decrease by a factor (2/π) = 0.4.

Now let us discuss the third approximation:

E(xPnmδν)1xinxΔx/2.

If we neglect this phase factor, the image will be deteriorated as if there were a phase error in the aperture (= hologram) at (νx = nδν; νy = mδν). The amount of the phase error varies with the location x in the image plane. The range of this phase error 2πxPδν depends on the range of x and of Pnm. The range of |x| is Δx/2=12δν. The range of |P| has to be large enough so that the phase 2πMP of the Fourier coefficients of h(x,y) can vary within ±π. Hence |P| varies between zero and 12M. Therefore the range of the phase error 2πxPδν lies within zero and π/2M which corresponds to a range in optical path (to be multiplied by λ/2π) of λ/4M. In other words the phase error introduced by approximation (17c) is within the limits of Rayleigh’s λ/4 criterion for wave aberrations, even for the lowest integer M = 1. We did not pay any attention to approximation (17c) in most of our holograms, without seeing any determental effect, but this might have been due to a lucky coincidence. In fact, it is not only the amount of the phase error which counts, for even very small phase errors can be harmful when distributed in certain particular ways over the aperture. Rayleigh did not claim that his criterion would guarantee good image quality for the most general distribution of wave aberrations but he considered only smoothly varying functions, represented by low order polynomials as function of the aperture coordinates. Therefore we present a more rigorous solution of the fundamental Eq. (14) h(x,y) = constu(x,y) which is not based on any of the three approximations (17 a, bc).

Before going on to the more rigorous treatment we want to mention another situation in which our approximations are better than usual. This situation, which may come up naturally when Fourier holograms are used as spatial filters, is a further restriction in the size of the image u(x,y), which heretofore has extended over a square of size (Δx)2. Now we assume that:

u(x,y)0onlyinx,yξΔx/2;ξ1.

The approximation (17a) now leaves as extreme values for the sine function sinc(cM ± ξc/2), which yields as the ratio of the extreme intensities (for cM=12): (1 + )2/(1 − )2. This is about 2:1 for c=12 and ξ=13.

The sinc(yWδν) in approximation (17b) will not drop below sinc(ξWmax/2), which is 0.95 for ξ=13and Wmax = 1.

Also the phase error |2πxPδν| in (17c) becomes very limited in range πξ/2M. This corresponds to a range in optical path length of λξ/4M. For M = 1 and ξ=13, this is only λ/12.

A More Rigorous Solution of h = constu [Eq. (14)]

The first approximation (17a) can be dealt with in a very simple way because the function sine [cδν(x + x0) ] acts simply as an amplitude reducing mask in the image plane as we have seen earlier. The influence of the sine function can be compensated experimentally by inserting an amplitude transmittance of 1/sinc into the image plane. This eliminates the need to consider the sine function in the rigorous solution of the equation h(x,y) = constu(x,y).

Instead of compensating the sine function experimentally, one can do the same thing analytically. To this end we define:

u(x,y)=v(x,y)sinc[cδν(x+x0)].

The functions v and u are similar because sinc is smooth and never zero within our range |x| ≤ Δx/2. The solution of h = constu now reduces to

h(x,y)/sinc[cδν(x+x0)]=constv(x,y)

wherein the sine function is canceled. The following is a complete description of the functions h and v:

v(x,y)=u(x,y)/sinc[cδν(x+x0)]=rect(x/Δx)rect(y/Δx)(n,m)v˜nmE[δν(xn+ym)];
v˜nm=(1/Δx)2u/sincE[-δν(xn+ym)]dxdy;
h(x,y)/sinc[cδν(x+x0)]=rect(x/Δx)rect(y/Δx)c(δν)2ΣΣ(n,m)Wnmsinc(yWnmδν)E[MPnm+(xn+ym)δν+xPnmδν].

Instead of equating the two functions, h/sinc and v, one can as well equalize their Fourier transforms. Because both functions are unequal only within |x|,|y| ≤ Δx/2, their Fourier transforms can be written as sampling series, but we are not explicitly interested in these sampling series. We mention them only for reasons of mathematical economy, i.e., we need to know the Fourier transforms only at the sampling points: νx = jx = jδν; νy = kδν; |j|,|k| = 0,1,2,…,N/2.

We need only a finite number N2 of sampling values, since it was assumed in the beginning that ũ(νx,νy) and hence v˜(νx,νy) would be essentially zero outside of |νx|,|νy| ≤ Nδν/2. Also the Fourier transform of h(x,y)/sinc will be essentially zero outside, because one can interpret it as an image of the hologram, which has a square size of (Nδν)2. This hypothetical image is in plane FREQ (Fig. 1), formed by the lens between HOLO and IMG (Fig. 1), through the (Δx)2 window, in plane IMG, which is covered by an amplitude mask and then by means of the dotted lens which is responsible for the last Fourier transform.

On the basis of these considerations the equalization (21) of h/sinc = constv is now replaced by the following system of equations for the Fourier coefficients:

constv˜jk=c(δν/Δx)2(n,m)Wnm×(-Δx/2)(+Δx/2)sinc(yWnmδν)E[]dxdy;

wherein the stands for:

=MPnm+(n-j)xδν+(m-k)yδν+xPnmδν.

The result of the integrations (22) assumes the form:

constv˜jk=c(δν)2(m,n)WnmE(MPnm)σ(n,j;m,k);
σ(n,j;m,k)=σx(n,j;m)σy(m,k;n);
σx=(1/Δx)-Δx/2+Δx/2E[xδν(n-j+Pnm)]dx=sinc(n-j+Pnm);
σy=(1/Δx)-Δx/2+Δx/2sinc(yWnmδν)E[yδν(m-k)]dy=[Si(m-k+Wnm/2)-Si(m-k-Wnm/2)/Wnm.

Herein the Si function is defined as:

Si(z)=0zsinc(z)dz.

The result of the σy integral (23c) is easily obtained, if one introduces the Fourier representation of the sine function:

sinc(yWδν)=(1/Wδν)-+rect(μ/Wδν)E(μy)dμ.

Ultimately, we want to solve the system of equations (23) and find the parameters Wnm and Pnm of the dots in the hologram. We do this by means of an iterative approach, which can be carried as far as one wants to approximate the exact solution. It is convenient to rewrite the system of Eqs. (23) by introducing the following abbreviations:

constv˜jk/c(δν)2=AjkE(φjk/2π);
(n,m)WnmE(MPnm)σ(n,j;m,k)=WjkE(MPjk)σ(j,j;k,k)+Rjk(Wnm,Pnm).

The terms Rjk contain all but the (n = j,m = k) term of the ∑∑(n,m) in Eq. (24b). The system of equations now has the form

AjkE(φjk/2π)=WjkE(MPjk)σ(j,j;k,k)+Rjk;
σ(j,j;k,k)=sinc(Pjk)Si(Wjk/2)/(Wjk/2).

In zero order approximation we set:

Rjk=0;σ(j,j;k,k)=1.

This gives us the solution, which we obtained earlier[1]:

AjkE(φjk/2π)=Wjk(0)E(MPjk(0));Wjk(0)=Ajk;Pjk(0)=φjk/2πM.

The superscript (0) indicates that this is the solution in zeroth order approximation. The first order solution of Eq. (25) is based upon the fact that σ(j,j;k,k) varies only slowly as a function of W and P. Hence we insert W(0) and P(0) into σ [Eq. (25a)] but still assume Rjk = 0:

AjkE(φjk/2π)=Wjk(1)E(MPjk(1))σ(Wjk(0),Pjk(0));
Pjk(1)=φjk/2πM=Pjk(0);
Wjk(1)=Ajk/σ(Wjk(0),Pjk(0))=Wjk(0)/σ(Wjk(0),Pjk(0)).

In the second order solution Rjk is also taken into account. Since Rjk varies only slowly as a function of all Wnm, Pnm, and since Rjk is anyway smaller than the two main terms of the system of equations, it is good enough to insert Pnm(0), and Wnm(0) or Wnm(1) into Eq. (25), giving:

AjkE(φjk/2π)=Wjk(2)E(MPjk(2))σjk(Wjk(1),Pjk(1))+Rjk(Wnm(1),Pnm(1));
AjkE(φjk/2π)-Rjk(Wnm(1),Pnm(1))=BjkE(βjk/2π);
Wjk(2)=Bjk/σjk(Wjk(1),Pjk(1));Pjk(2)=βjk/2πM.

The third order solution can be obtained by inserting Wjk(2) and Pjk(2) into σ and R of Eq. (25), and so on. The convergence of the iteration follows from |R| ≪A, and from R and σ being smooth functions of W and P.

While calculating the terms Rjk for the iterative solution of Eq. (25) one can introduce the following simplifications:

σ(n,j;m,k)=σx(n,j;m)σy(m,k;n);σx(nj;mm)(-1)n-jsin(πPnm)/π(n-j);
σy(mk,n)(-1)m-k-1[Wnm/2(m-k)]2/3.

This indicates that the ∑∑(n,m), which builds up Rjk in Eq. (24b), has to be taken only over a few terms. Both the denominators (nj) and (mk)2, as well as the alternating factors (−1) provide for a rapid convergence of ∑∑(n,m). Since both sin(πP) and W cover the range (0,1), the σy(mk) terms will be in average 4|nj| times smaller than σx(nj) for |ni| = |mk|. This suggests an intermediate level of approximation, in which one sets σy(mk) = 0 but not σx. Then, the double sum ∑∑(n,m) in Rjk reduces to a single sum ∑(n), which abbreviates the computation. This means neglect of the influence of approximation (17b).

Computational and Plotting Procedures

In order to shorten the treatment, we assume that both the image and the hologram have a quadratic shape. If we want to make a hologram H(νx,νy), from which a prescribed image u(x,y) can be reconstructed, we have to determine first the resolution length δx, which is needed for sufficient image quality. In the appendix we discuss the risk of choosing δx too large. Since δx is related to the bandwidth Δν = 1/δx of the image spectrum ũ(νx,νy), we can find ΔxH, the size of the hologram (after reduction):

u˜(νx,νy)0forνx,νyΔν/2=1/2δx;ΔxH=ΔyH=λfΔν.

The input for the computer is the following matrix of complex image amplitudes:

u(nδx,mδx)=unm.

It consists of N2 complex numbers, if N = Δx/δx describes the number of resolution elements in one direction across the image field of size (Δx)2. Next we divide by the sine function, if we want to avoid the brightness inhomogeneity due to approximation (17a);

vnm=unm/sinc[cδν(x0+nδx)]=unm/sinc[c(M+n/N)];δν=1/Δx;x0δν=M.

Now we have to chose the parameters M and c, where M = x0x is related to the tilt angle of the illuminating plane wave, and λfcδν describes the horizontal width of the dots in the hologram cells of size (λfδν)2. As explained before, M and c influence the brightness of the image and also the severity of the approximations (a,b,c). If one wants to keep the calculations simple, i.e., be satisfied with the solutions {W(0), P(0)}, {W(1), P(1)} in zero order or first order approximation one has to sacrifice either brightness for image quality or vice versa. The maximal brightness at the image center is achieved for M = 1, c=12, and Wmax = 1. But M = 2, c=14, Wmax = 2/π would improve the validity of the approximations (b,c) considerably, if one is willing to sacrifice a loss in brightness by a factor of twenty. Another situation in which the approximations are very well satisfied without reduction of brightness (with M = 1, c=12, Wmax = 1) was mentioned before [Eq. (19)]. It appears if the image u(x,y) is zero in the outer portions of the image square of size (Δx)2.

For performing the Fourier transform

vnmv˜(jδν,kδν)=v˜jkE(φjk/2π)

on the digital computer it is very convenient to use the Cooley-Tukey algorithm.[7] It reduces the computer time by a factor of the order of 4 log2N/N2, as compared to more conventional Fourier transform programs. In our case, where the number of image points N2 was mostly 642 (sometimes 1282), this meant a factor 1:170.

Next we search for the maximum value of all | v˜jk|. In case of an amplitude object [u(x,y) real and nonnegative], this is v˜jkmax=v˜oo.. We need this maximum value in order to fix the open constant in h = constu [Eq. (14)]:

constv˜max=c(δν)2Wmax.

After inserting this constant we define the normalized amplitude Ajk as in Eq. (24a).

constνjk/c(δν)2=Ajk;AjkWmax.

Now we are ready to enter the interaction procedure contained in Eqs. (25, 26, 27, 23a, 23b′, 23c′) to calculate {W(0), P(0)}jk {W(1), P(1)}jk {W(2), P(2)}jk from the known {A,φ)jk. In order to keep the time for the computation of the correction terms Rjk reasonably short one can set rules such that the summation goes only over a few terms: |nj|≤nx; |mk|≤ny. These numbers nx and even more so ny can be considerably smaller than N. Inspecting the terms in Rjk [Eqs. (23b′, 23c′)] one sees that a cutoff rule of the form

n-j(m-j)2n03

would lead to even fewer terms in Rjk.

We used a plotter which is common in computer installations. The pertinent features of the plotter are: stepwise movements and finite stroke width of the pen. When dividing the width of the paper stripe (about 25 cm) into sixty-four cells, the plotter makes fifteen discrete steps across a cell. This means a quantization of the amplitude with 7% intervals, and a phase quantization of 24° intervals. We convinced ourselves that this amount of quantization is harmless by occasionally using a better plotter (artwork generator). The width of the stroke was normally two or three steps; therefore, one has to introduce a clipping procedure for low amplitude values. For example, all calculated dots heights W, which are smaller than 0.05, are treated by the plotter program as if they were zero. Another influence of the finite stroke width is that the height W of each dot would be too large by one stroke width if the pen was moved a distance W when drawing the outline of the dot. This can be easily anticipated if the stroke width is known and is consistent.

Sometimes the finite stroke width was detrimental to the image quality because almost half of the cells in the hologram remained empty due to the 5% clipping procedure. To alleviate this occurrence we suggest incorporation of a second clipping procedure, which acts upon the highest amplitudes. The reason for this procedure is that for some images u(x,y) (e.g., many letters) only 5% of all amplitudes Ajk of u fall within the range of 0.3 to 1 whereas more than 50% are below 0.05. In such a case, we multiply all amplitudes Ajk by a suitable factor, say three. Whenever 3Ajk is larger than one, the plotter will implement a height of Wjk = 1. This type of clipping acts similar to the saturation portion of the H & D curve, whereas the low level clipping due to the finite stroke width corresponds to the threshold portion of the H & D curve. Choosing a proper factor, say three, as in the previous example, corresponds to choosing the proper working point on the H & D curve. We did this when necessary after inspection of the first plotting attempt. But it would be easy to include an additional feature into the plotter program, such that clipping at high and low amplitude values appears automatically in a well balanced way. An alternative is to multiply the input u(x,y) by a random phase factor; by this the distribution of heights W between zero and one becomes more uniform. This phase factor is unimportant for the image, since only the intensity |u(x,y)|2 can be observed. The idea of introducing a random phase is of course induced by Leith and Upatnieks,[8] who used a ground glass for better adapting the amplitude distribution of the light to the dynamic range of the holographic plate.

Experimental Results

In a typical drawing of a hologram the cell size was usually 4 mm. The drawing was reduced photographically by a factor of 1:100 up to 1:400. A good photo objective lens (f = 50 mm, F/2, stopped down to F/4) achieves a hologram which appears fairly sharp under the microscope. But even if the sharpness of the edges is not very good anymore, the quality of the reconstructed image is not influenced markedly. Only the brightness of the uninteresting higher order diffraction spots besides the image decreases.

Normally we recorded the hologram on 649 F film or plate. Immersion helps somewhat to reduce the background light, but it is not necessary. Some holograms were bleached in order to improve the brightness of the image. Theoretically the gain in brightness should be up to a factor four, if the phase shift is 180°. In one case, we found an even higher increase in brightness: six. This was probably due to smoothening of the film surface as a side effect of the bleaching procedure; hence less light was lost due to scattering.

In most reconstructions, we used light from a He–Ne laser. But filtered Hg light is suitable as well. The size of the source pinhole at x0 has to be smaller than λf/2Δx, and the spectral purity λ/Δλ has to be better than 2NM.

The first hologram shown here [Fig. 3(a)] contains the letters ICO. In this case it did not matter whether we based the construction of the hologram on the lowest order approximation (W(0), P(0)) or on (W(2), P(2)). This is understandable based on the considerations around Eq. (19) because the letters ICO filled only the innermost portion of the mage area of size (Δx)2 which is obvious from Fig. 3(b), since M = 1 = x0x. Obviously, the image can be considered as the first order diffraction from the gratinglike hologram.

When looking at the hologram in Fig. 3(a), one sees that most of the dot heights Wn are fairly small. This accounts for a fairly dim image. (The zero order diffraction spot was covered by a neutral density filter.) The brightness can be improved by multiplying the image u(x,y), which was so far binary, by a random phase factor. The corresponding hologram has a much more irregular structure [Fig. 4(a)]. The reconstructed image is considerably brighter [Fig. 4(b)]. It contains a wormlike microstructure similar to ordinary holographic images, made from objects, which were superposed by a ground glass.

In the next holographic image (Fig. 5) one sees the detrimental effects owing to the approximations (17 a,b,c). For one thing, the brightness is inhomogeneous, furthermore, the quality decreases toward the outer parts of the image as might be expected from our discussion of the approximations.

Finally we show in Fig. 6 two holographic images, constructed from the same hypothetical object, but with different parameters M which describes the diffraction order of the holographic grating: Fig. 6(a), M = 1; Fig. 6(b), M = 3. Since the position P of the dots suffers from the quantization error of 11   5, this means a quantization error for the phase φ = 2πPM of 24° (M = 1) and 72°(M = 3). The amplitude quantization is independent of M, at least for the dot shape assumed here (Fig. 2).

Conclusions

We have shown, that one can reconstruct images from binary Fraunhofer holograms. The necessary number of free parameters in the hologram does not have to exceed the number of free parameters in the image we want to see. Since this is fundamentally the best one could hope to achieve in any hologram, we conclude that our binary holograms are as efficient as ordinary thin-emulsion holograms possibly can be. The only difference is that our holograms are similar to square wave gratings, whereas ordinary holograms are similar to sinusoidal gratings. This is the reason that we always see higher order spots besides the images. Similar spots sometimes appear in the reconstruction from an ordinary hologram when the ordinarily sinusoidal interference fringes in the hologram are distorted due to the nonlinear recording process. These higher orders are as far away from the image as is the always present zero order spot. Hence the avoidance of overlap with the zero order spot also guarantees the avoidance of overlap with the higher order spots. In other words, the image field limitations Δx of ordinary and binary holograms are both determined in the same way by the grating constant d(= cell width), focal length f, and wavelength λ: Δx ≤ λf/d.

Sometimes binary holograms might be even better, in terms of signal-to-noise, than ordinary holograms of comparable size. A photographic emulsion exhibits more grain noise at medium gray transmittance than at zero and one transmittance; hence the amplitude distribution of binary holograms is better suited from this point of view.

Binary holograms are also advantageous in terms of light economy. A square wave grating representing a binary hologram will send (4/π)2 times more light into the first diffraction order as compared with a fully modulated sine wave grating, representing the brightest possible thin emulsion hologram. Furthermore, for binary holograms it is conceivable to synthesize mathematically the most efficient random phase distribution, whereas in ordinary holography one has to rely on available ground glasses, which might not be optimal in terms of image brightness. Yet another advantage of binary holograms is that they are easier to copy photographically. On the other hand, it takes about 10 min on the IBM 7094 to synthesize a hologram for an image consisting of 64 × 64 resolvable points.

Since the feasibility study presented in this paper provides a full understanding of the theoretical and experimental problems, one can now plan to make computer generated holograms for all kinds of purposes. This is meaningful if the object does not exist physically, but only in mathematical terms. Some situations where this might occur are: spatial filtering with Fraunhofer holograms,[10],[11] 3D display from Fresnel holograms,[12] prototype wavefronts from image holograms.[7] Some of these applications have been presented orally. They will be reported in subsequent articles.

Appendix: The Risk of Undersampling

If the desired image u(x) varies only slowly as a function of x, we can define u(x) sufficiently well by specifying only its sampling values at intervals δx: u(x) → u(nδx). We do this for reasons of economy in the computations. However we must be careful to choose δx fine enough. Otherwise we risk producing a hologram that creates an image with some defects. In this appendix we want to study the nature of these defects. The formalism used is the same as in electrical communication theory where this effect is called under-sampling or aliasing.

Sampling means to replace the original image u(x) by its product with the Dirac comb:

u(x)u(x)Σ(n)δ(x-nδx)=v(x).

This facilitates the calculation of the Fourier transform, which reduces from an integral to a sum:

u˜(ν)=u(x)E(-vx)dx;
v˜(ν)=Σ(n)u(x)δ(x-nδx)E(-νx)dx=Σ(n)u(nδx)E(-νnδx).

If we had replaced in the Fourier integral leading to v˜(ν) the Dirac comb by its Fourier series we would have obtained:

Σ(n)δ(x-nδx)=Σ(m)E(mx/δx);
v˜(ν)=Σ(m)u(x)E[x(-ν+m/δx)]dx=Σ(m)u˜(ν-m/δx).

Our problem is now to find conditions under which v˜(ν) can replace ũ(ν), at least within the finite width ΔνH of the hologram. We have to make sure that only the (m = 0) term falls within ΔνH, not the adjacent terms (|m| = 1). This is accomplished if 1/δx − Δν0/2 ≤ ΔνH/2; or δx ≤ 2/(Δν0 + ΔνH). Here Δν0 means the bandwidth of the image spectrum ũ(ν). If we want the reconstructed image to contain the whole bandwidth Δν0 of the image, upon which our calculations are based, the width ΔνH of the hologram has to be at least ΔνH = Δν0. A broader hologram would be a waste, and furthermore a risk of intrusion of the (|m| = 1) terms. So we get as condition for the sampling step:

δx1/Δν0.

We have treated the problem of undersampling in one dimension only. Extension toward two dimensions is trivial. It shows that the intrusion of (n ≠ 0, m ≠ 0) terms can become even more severe, because there are now eight instead of two next neighbors:

v˜(νx,νy)=ΣΣ(n,m)u˜(νx-n/δx,νy-m/δx).

We would like to thank H. Werlich for his skillful performance of the experimental work, and J. H. Eaton for discussions about the sampling theorem. B. R. Brown helped us to prepare the manuscript.

Figures

 figure: Fig. 1

Fig. 1 Setup for the construction of an image IMG from a binary hologram HOLO, illuminated by a point source x0 in SOU.

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 figure: Fig. 2

Fig. 2 Structure of the (n,m) cell in the binary hologram. The rectangular opening has a width cδν, a height Wnmδν, and it is displaced from the center of the cell at (nδν,mδν) by Pnmδν.

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 figure: Fig. 3

Fig. 3 Binary Fraunhofer holography: (a) the hologram; (b) the image.

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 figure: Fig. 4

Fig. 4 Simulation of a groundglass, i.e., object with random phase: (a) the hologram; (b) the image.

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 figure: Fig. 5

Fig. 5 A holographic image with defects owing to zero-order approximation in the calculation.

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 figure: Fig. 6

Fig. 6 Images from binary holograms of different orders: (a) M = 1; (b) M = 3.

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References

1. B. R. Brown and A. W. Lohmann, Appl. Opt. 5, 967 (1966) [CrossRef]   [PubMed]  .

2. D. Hauk and A. Lohmann, Optik 15, 275 (1958).

3. C. A. Taylor and H. Lipson, Optical Transforms (Bell, London, 1964); G. Harburn, K. Walkley, and C. A. Taylor, Nature 205, 1096 (1965) [CrossRef]  .

4. A. Kozma and D. L. Kelly, Appl. Opt. 4, 37 (1965) [CrossRef]  .

5. J. P. Waters, Appl. Phys. Letters 9, 405 (1957) [CrossRef]  .

6. M. Marquet and J. Tsujiuchi, Opt. Acta 8, 267 (1961) [CrossRef]  .

7. A. W. Lohmann and D. P. Paris, J. Opt. Soc. Am. 56, 537A (1966).

8. J. W. Cooley and J. W. Tukey, Math. Comp. 19, 297 (1965); [CrossRef]  IBM Share Libr. HARM 3425.

9. E. N. Leith and J. Upatnieks, J. Opt. Soc. Am. 54, 1295 (1964) [CrossRef]  .

10. B. R. Brown, A. W. Lohmann, and D. P. Paris, Opt. Acta 13, 377 (1966).

11. A. W. Lohmann and D. P. Paris, J. Opt. Soc. Am. 56, 1413 A (1966).

12. A. W. Lohmann and D. P. Paris, IEEE J. Quantum Electron. QE-2, LXV (1966).

References

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  1. B. R. Brown, A. W. Lohmann, Appl. Opt. 5, 967 (1966).
    [Crossref] [PubMed]
  2. D. Hauk, A. Lohmann, Optik 15, 275 (1958).
  3. C. A. Taylor, H. Lipson, Optical Transforms (Bell, London, 1964); G. Harburn, K. Walkley, C. A. Taylor, Nature 205, 1096 (1965).
    [Crossref]
  4. A. Kozma, D. L. Kelly, Appl. Opt. 4, 37 (1965).
    [Crossref]
  5. J. P. Waters, Appl. Phys. Letters 9, 405 (1957).
    [Crossref]
  6. M. Marquet, J. Tsujiuchi, Opt. Acta 8, 267 (1961).
    [Crossref]
  7. A. W. Lohmann, D. P. Paris, J. Opt. Soc. Am. 56, 537A (1966).
  8. J. W. Cooley, J. W. Tukey, Math. Comp. 19, 297 (1965); IBM Share Libr. HARM 3425.
    [Crossref]
  9. E. N. Leith, J. Upatnieks, J. Opt. Soc. Am. 54, 1295 (1964).
    [Crossref]
  10. B. R. Brown, A. W. Lohmann, D. P. Paris, Opt. Acta 13, 377 (1966).
  11. A. W. Lohmann, D. P. Paris, J. Opt. Soc. Am. 56, 1413 A (1966).
  12. A. W. Lohmann, D. P. Paris, IEEE J. Quantum Electron. QE-2, LXV (1966).

1966 (5)

B. R. Brown, A. W. Lohmann, Appl. Opt. 5, 967 (1966).
[Crossref] [PubMed]

A. W. Lohmann, D. P. Paris, J. Opt. Soc. Am. 56, 537A (1966).

B. R. Brown, A. W. Lohmann, D. P. Paris, Opt. Acta 13, 377 (1966).

A. W. Lohmann, D. P. Paris, J. Opt. Soc. Am. 56, 1413 A (1966).

A. W. Lohmann, D. P. Paris, IEEE J. Quantum Electron. QE-2, LXV (1966).

1965 (2)

J. W. Cooley, J. W. Tukey, Math. Comp. 19, 297 (1965); IBM Share Libr. HARM 3425.
[Crossref]

A. Kozma, D. L. Kelly, Appl. Opt. 4, 37 (1965).
[Crossref]

1964 (1)

1961 (1)

M. Marquet, J. Tsujiuchi, Opt. Acta 8, 267 (1961).
[Crossref]

1958 (1)

D. Hauk, A. Lohmann, Optik 15, 275 (1958).

1957 (1)

J. P. Waters, Appl. Phys. Letters 9, 405 (1957).
[Crossref]

Brown, B. R.

B. R. Brown, A. W. Lohmann, Appl. Opt. 5, 967 (1966).
[Crossref] [PubMed]

B. R. Brown, A. W. Lohmann, D. P. Paris, Opt. Acta 13, 377 (1966).

Cooley, J. W.

J. W. Cooley, J. W. Tukey, Math. Comp. 19, 297 (1965); IBM Share Libr. HARM 3425.
[Crossref]

Hauk, D.

D. Hauk, A. Lohmann, Optik 15, 275 (1958).

Kelly, D. L.

A. Kozma, D. L. Kelly, Appl. Opt. 4, 37 (1965).
[Crossref]

Kozma, A.

A. Kozma, D. L. Kelly, Appl. Opt. 4, 37 (1965).
[Crossref]

Leith, E. N.

Lipson, H.

C. A. Taylor, H. Lipson, Optical Transforms (Bell, London, 1964); G. Harburn, K. Walkley, C. A. Taylor, Nature 205, 1096 (1965).
[Crossref]

Lohmann, A.

D. Hauk, A. Lohmann, Optik 15, 275 (1958).

Lohmann, A. W.

B. R. Brown, A. W. Lohmann, Appl. Opt. 5, 967 (1966).
[Crossref] [PubMed]

A. W. Lohmann, D. P. Paris, J. Opt. Soc. Am. 56, 537A (1966).

B. R. Brown, A. W. Lohmann, D. P. Paris, Opt. Acta 13, 377 (1966).

A. W. Lohmann, D. P. Paris, J. Opt. Soc. Am. 56, 1413 A (1966).

A. W. Lohmann, D. P. Paris, IEEE J. Quantum Electron. QE-2, LXV (1966).

Marquet, M.

M. Marquet, J. Tsujiuchi, Opt. Acta 8, 267 (1961).
[Crossref]

Paris, D. P.

A. W. Lohmann, D. P. Paris, J. Opt. Soc. Am. 56, 537A (1966).

A. W. Lohmann, D. P. Paris, IEEE J. Quantum Electron. QE-2, LXV (1966).

A. W. Lohmann, D. P. Paris, J. Opt. Soc. Am. 56, 1413 A (1966).

B. R. Brown, A. W. Lohmann, D. P. Paris, Opt. Acta 13, 377 (1966).

Taylor, C. A.

C. A. Taylor, H. Lipson, Optical Transforms (Bell, London, 1964); G. Harburn, K. Walkley, C. A. Taylor, Nature 205, 1096 (1965).
[Crossref]

Tsujiuchi, J.

M. Marquet, J. Tsujiuchi, Opt. Acta 8, 267 (1961).
[Crossref]

Tukey, J. W.

J. W. Cooley, J. W. Tukey, Math. Comp. 19, 297 (1965); IBM Share Libr. HARM 3425.
[Crossref]

Upatnieks, J.

Waters, J. P.

J. P. Waters, Appl. Phys. Letters 9, 405 (1957).
[Crossref]

Appl. Opt. (2)

Appl. Phys. Letters (1)

J. P. Waters, Appl. Phys. Letters 9, 405 (1957).
[Crossref]

IEEE J. Quantum Electron. (1)

A. W. Lohmann, D. P. Paris, IEEE J. Quantum Electron. QE-2, LXV (1966).

J. Opt. Soc. Am. (3)

A. W. Lohmann, D. P. Paris, J. Opt. Soc. Am. 56, 1413 A (1966).

A. W. Lohmann, D. P. Paris, J. Opt. Soc. Am. 56, 537A (1966).

E. N. Leith, J. Upatnieks, J. Opt. Soc. Am. 54, 1295 (1964).
[Crossref]

Math. Comp. (1)

J. W. Cooley, J. W. Tukey, Math. Comp. 19, 297 (1965); IBM Share Libr. HARM 3425.
[Crossref]

Opt. Acta (2)

B. R. Brown, A. W. Lohmann, D. P. Paris, Opt. Acta 13, 377 (1966).

M. Marquet, J. Tsujiuchi, Opt. Acta 8, 267 (1961).
[Crossref]

Optik (1)

D. Hauk, A. Lohmann, Optik 15, 275 (1958).

Other (1)

C. A. Taylor, H. Lipson, Optical Transforms (Bell, London, 1964); G. Harburn, K. Walkley, C. A. Taylor, Nature 205, 1096 (1965).
[Crossref]

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Figures (6)

Fig. 1
Fig. 1 Setup for the construction of an image IMG from a binary hologram HOLO, illuminated by a point source x0 in SOU.
Fig. 2
Fig. 2 Structure of the (n,m) cell in the binary hologram. The rectangular opening has a width cδν, a height Wnmδν, and it is displaced from the center of the cell at (nδν,mδν) by Pnmδν.
Fig. 3
Fig. 3 Binary Fraunhofer holography: (a) the hologram; (b) the image.
Fig. 4
Fig. 4 Simulation of a groundglass, i.e., object with random phase: (a) the hologram; (b) the image.
Fig. 5
Fig. 5 A holographic image with defects owing to zero-order approximation in the calculation.
Fig. 6
Fig. 6 Images from binary holograms of different orders: (a) M = 1; (b) M = 3.

Equations (65)

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u ˜ ( ν x , ν y ) = u ( x , y ) E ( - x ν x - y ν y ) d x d y .
const u ˜ ( ν x , ν y ) = H ( ν x , ν y ) E ( x 0 ν x ) .
u ( x , y ) = 0 outside of x Δ x / 2 ; y Δ y / 2 ; u ˜ ( ν x , ν y ) 0 outside of ν x Δ ν / 2 ; v y Δ ν ν 2.
h ( x , y ) = rect ( x / Δ x ) rect ( y / Δ x ) H ( ν x , ν y ) E [ ( x + x 0 ) ν x + y ν y ] d ν x d ν ; .
rect ( z ) = { 1 ; in z 1 2 0 ; otherwise
h ( x , y ) = const u ( x , y ) .
N 2 = ( Δ x Δ y ) / ( δ x δ y ) = ( Δ x / δ x ) 2 = ( Δ x Δ ν ) 2 = S W .
( Δ ν / δ ν ) 2 ( Δ x / δ x ) 2 = ( Δ x Δ ν ) 2 ;     δ ν 1 / Δ x .
u ˜ ( ν x , ν y ) = Σ Σ ( n , m ) u ˜ ( n / Δ x , m / Δ x ) × sinc ( ν x Δ x - n ) sinc ( ν y Δ x - m ) .
ν x - ( n + P n m ) δ ν c δ ν / 2 ; ν y - m δ ν W n m δ ν / 2.
( m - 1 2 ) δ ν ( m + 1 2 ) δ ν H ( ν x , ν y ) d ν y = H m ( ν x ) ; H m [ ( n + P n m ) δ ν + μ ] = H m [ ( n + P n m ) δ ν - μ ] ,
H ( ν x , ν y ) = Σ Σ ( n , m ) rect [ ν x - ( n + P n m ) δ ν c δ ν ] rect [ ν y - m δ ν W n m δ ν ] .
H ( ν x , ν y ) E [ ( x + x 0 ) ν x + y ν y ] d ν x d ν y = c ( δ ν ) 2 sinc [ c δ ν ( x + x 0 ) ] Σ Σ ( n , m ) W n m sinc ( y W n m δ ν ) E { δ ν [ ( x + x 0 ) ( n + P n m ) + y m ] } .
rect ( x / Δ x ) rect ( y / Δ x ) H E d ν x d ν y = h ( x , y ) ; h ( x , y ) = const u ( x , y ) .
u ( x , y ) = u ˜ ( ν x , ν y ) E ( x ν x + y ν y ) d ν x d ν y = rect ( x / Δ x ) rect ( y / Δ x ) ( n , m ) u ˜ ( n δ ν ) , m δ ν ) E [ δ ν ( x n + y m ) ] .
c ( δ ν ) 2 W n m E [ x 0 δ ν ( n + P n m ) ] const u ˜ ( n δ ν , m δ ν ) ; const u ˜ ( n δ ν , m δ ν ) = c ( δ ν ) 2 A n m E ( φ n m / 2 π ) ; W n m A n m ; P n m + n φ n m / 2 π x 0 δ ν .
P n m φ n m / 2 π M .
E [ x 0 δ ν ( n + P n m ) ] = E ( M P n m ) ; φ n m = 2 π M P n m ; φ ¯ = 2 π M P ¯ .
( a ) sinc [ c δ ν ( x + x 0 ) ] const , in x Δ x / 2 ; ( b ) sinc ( y W n m δ ν ) 1 in y Δ x / 2 ; ( c ) E ( x P n m δ ν ) 1 in x Δ x / 2.
φ π ; φ = 2 π M P ; P 1 2 M .
sinc [ c δ ν ( x + x 0 ) ] const , in x Δ x / 2.
sinc ( y W δ ν ) 1 ,
E ( x P n m δ ν ) 1 x in x Δ x / 2.
u ( x , y ) 0 only in x , y ξ Δ x / 2 ; ξ 1.
u ( x , y ) = v ( x , y ) sinc [ c δ ν ( x + x 0 ) ] .
h ( x , y ) / sinc [ c δ ν ( x + x 0 ) ] = const v ( x , y )
v ( x , y ) = u ( x , y ) / sinc [ c δ ν ( x + x 0 ) ] = rect ( x / Δ x ) rect ( y / Δ x ) ( n , m ) v ˜ n m E [ δ ν ( x n + y m ) ] ;
v ˜ n m = ( 1 / Δ x ) 2 u / sinc E [ - δ ν ( x n + y m ) ] d x d y ;
h ( x , y ) / sinc [ c δ ν ( x + x 0 ) ] = rect ( x / Δ x ) rect ( y / Δ x ) c ( δ ν ) 2 Σ Σ ( n , m ) W n m sinc ( y W n m δ ν ) E [ M P n m + ( x n + y m ) δ ν + x P n m δ ν ] .
const v ˜ j k = c ( δ ν / Δ x ) 2 ( n , m ) W n m × ( - Δ x / 2 ) ( + Δ x / 2 ) sinc ( y W n m δ ν ) E [ ] d x d y ;
= M P n m + ( n - j ) x δ ν + ( m - k ) y δ ν + x P n m δ ν .
const v ˜ j k = c ( δ ν ) 2 ( m , n ) W n m E ( M P n m ) σ ( n , j ; m , k ) ;
σ ( n , j ; m , k ) = σ x ( n , j ; m ) σ y ( m , k ; n ) ;
σ x = ( 1 / Δ x ) - Δ x / 2 + Δ x / 2 E [ x δ ν ( n - j + P n m ) ] d x = sinc ( n - j + P n m ) ;
σ y = ( 1 / Δ x ) - Δ x / 2 + Δ x / 2 sinc ( y W n m δ ν ) E [ y δ ν ( m - k ) ] d y = [ Si ( m - k + W n m / 2 ) - Si ( m - k - W n m / 2 ) / W n m .
Si ( z ) = 0 z sinc ( z ) d z .
sinc ( y W δ ν ) = ( 1 / W δ ν ) - + rect ( μ / W δ ν ) E ( μ y ) d μ .
const v ˜ j k / c ( δ ν ) 2 = A j k E ( φ j k / 2 π ) ;
( n , m ) W n m E ( M P n m ) σ ( n , j ; m , k ) = W j k E ( M P j k ) σ ( j , j ; k , k ) + R j k ( W n m , P n m ) .
A j k E ( φ j k / 2 π ) = W j k E ( M P j k ) σ ( j , j ; k , k ) + R j k ;
σ ( j , j ; k , k ) = sinc ( P j k ) Si ( W j k / 2 ) / ( W j k / 2 ) .
R j k = 0 ; σ ( j , j ; k , k ) = 1.
A j k E ( φ j k / 2 π ) = W j k ( 0 ) E ( M P j k ( 0 ) ) ; W j k ( 0 ) = A j k ; P j k ( 0 ) = φ j k / 2 π M .
A j k E ( φ j k / 2 π ) = W j k ( 1 ) E ( M P j k ( 1 ) ) σ ( W j k ( 0 ) , P j k ( 0 ) ) ;
P j k ( 1 ) = φ j k / 2 π M = P j k ( 0 ) ;
W j k ( 1 ) = A j k / σ ( W j k ( 0 ) , P j k ( 0 ) ) = W j k ( 0 ) / σ ( W j k ( 0 ) , P j k ( 0 ) ) .
A j k E ( φ j k / 2 π ) = W j k ( 2 ) E ( M P j k ( 2 ) ) σ j k ( W j k ( 1 ) , P j k ( 1 ) ) + R j k ( W n m ( 1 ) , P n m ( 1 ) ) ;
A j k E ( φ j k / 2 π ) - R j k ( W n m ( 1 ) , P n m ( 1 ) ) = B j k E ( β j k / 2 π ) ;
W j k ( 2 ) = B j k / σ j k ( W j k ( 1 ) , P j k ( 1 ) ) ; P j k ( 2 ) = β j k / 2 π M .
σ ( n , j ; m , k ) = σ x ( n , j ; m ) σ y ( m , k ; n ) ; σ x ( n j ; m m ) ( - 1 ) n - j sin ( π P n m ) / π ( n - j ) ;
σ y ( m k , n ) ( - 1 ) m - k - 1 [ W n m / 2 ( m - k ) ] 2 / 3.
u ˜ ( ν x , ν y ) 0 for ν x , ν y Δ ν / 2 = 1 / 2 δ x ; Δ x H = Δ y H = λ f Δ ν .
u ( n δ x , m δ x ) = u n m .
v n m = u n m / sinc [ c δ ν ( x 0 + n δ x ) ] = u n m / sinc [ c ( M + n / N ) ] ; δ ν = 1 / Δ x ; x 0 δ ν = M .
v n m v ˜ ( j δ ν , k δ ν ) = v ˜ j k E ( φ j k / 2 π )
const v ˜ max = c ( δ ν ) 2 W max .
const ν j k / c ( δ ν ) 2 = A j k ; A j k W max .
n - j ( m - j ) 2 n 0 3
u ( x ) u ( x ) Σ ( n ) δ ( x - n δ x ) = v ( x ) .
u ˜ ( ν ) = u ( x ) E ( - v x ) d x ;
v ˜ ( ν ) = Σ ( n ) u ( x ) δ ( x - n δ x ) E ( - ν x ) d x = Σ ( n ) u ( n δ x ) E ( - ν n δ x ) .
Σ ( n ) δ ( x - n δ x ) = Σ ( m ) E ( m x / δ x ) ;
v ˜ ( ν ) = Σ ( m ) u ( x ) E [ x ( - ν + m / δ x ) ] d x = Σ ( m ) u ˜ ( ν - m / δ x ) .
δ x 1 / Δ ν 0 .
v ˜ ( ν x , ν y ) = Σ Σ ( n , m ) u ˜ ( ν x - n / δ x , ν y - m / δ x ) .

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