This is a sketch of a paper published in J.O.S.A. (1997) with John Sadi, D. Vukicevic, and M. Torzynski.

Smoothing speckle by three-colour recording

Imaging by coherent light leads to diffractions patterns such as Airy spots or speckle. Often such patterns are prejudicial to the quality of imaging. Incoherent light cancels these patterns by averaging. Incoherent light can indeed be considered as a superposition of a very large number of coherent components, whose phase factors are distributed at random. This averaging effect by the law of large numbers can be represented mathematically by a mean value of complex numbers

where the a_j's are independent and uniformly distributed random variables. It is then a well-known result that the probability density of |S| will be approximately gaussian, zero-centred, with standard deviation of order 1/ÖN. In other words, S will rarely deviate from zero of more than a few 1/ÖN.

Now the question arises whether such an averaging effect will occur if we superpose a rather small number of coherent light components; this question is important in colour holography, where three different coherent interference patterns will be recorded successively and overlap incoherently.

In the following figures, we have represented some one-dimensional patterns, which look like typical diffraction or speckle patterns. These patterns are on the first lines of each figure. Then we have blown up (or shrunk down) seven times the abscissa with arbitrary (but close to 1) scale factors, and represented the successive mean values: the first lines represent the mean values of the original signal and its first rescaled version, the second lines the mean values of the original signal and its first two rescaled versions, and so on. The law of large numbers ensures that the limit (far beyond the eighth line) will be flat: this would be the perfect state of averaging. But here we wanted to test whether or not the third line can be considered as significantly better than the first one.

figure 1

The first line represents a one-dimensional oscillating function, which can be considered as a model for a typical diffraction or speckle pattern. Then we have blown up (or shrunk down) seven times the abscissa with arbitrary (but close to 1) scale factors, and represented the successive mean values: the second line represents the mean values of the original signal and its first rescaled version, the third line the mean values of the original signal and its first two rescaled versions, and so on.

In fact, the first two scale factors, which correspond to lines 2 and 3, are not arbitrary, but equal respectively to 540/460 and 633/460, the wave-lengthes ratios of the three recording colours.

Of course, this should not be considered as a true experiment: the curve represented on the first line is not a section of a true diffraction pattern. Moreover, even if it were the case, the mean values represented on the lower lines could only be a model for the pattern recorded in the hologram (the modulation). But there is no simple and direct link between the fidelity in reproducing the modulation, and the fidelity in image restitution. To explain the quality of the holographic image will require a complete theoretical analysis of the light diffracted by the recorded modulation.

figure 2

figure 3

Here and in the two next figures, in addition to the rescaling of abscissas, we have shifted each rescaled copy of the original signal by an arbitrary additive constant. This additional operation randomizes the beginning of the lines.

figure 4

figure 5

The values of the function represented in the first lines of the previous pictures at given abscissas can be considered as distributed at random, if these abscissas are chosen such that their mutual distances are larger than a few wavelengthes. This is true for any kind of speckle, and remains true in two or three dimensions. If we superpose three of them the statistical frequencies of the deviations will be described by the probability density of a certain random variable S, which is defined by

S = (1/3)[cos q₁ + cos q₂ + cos q₃]

where q₁,q₂,q₃ are uniformly distributed in the angle interval [ 0, 2p].

This probability density is represented on figure 6. We see that the quasi totality of deviations are less than the half of the maximum possible deviation. This is in accordance with the numerical simulations of figures 1 - 5: we can easily see that the fluctuations on the third line are half as large as on the first one.

Annex: computation of the probability density

q₁, q₂, and q₃ are three independent random variables with uniform distributions over the angle interval [ 0, 2p]. and S = [1/ 3][ cosq₁ + cosq₂ + cosq₃ ]. S will take values between -1 and +1, but its probability density will not be uniform on this interval. To compute this density, we consider its primitive F(x) (the repartition function). We have:

This integral can be computed by changing the variables; we put u₁ = cosq₁, u₂ = cosq₂, and u₃ = cosq₃. Then

and

The integration domain in this three-dimensional integral is a cube, truncated at its corner by the plane u₁ + u₂ + u₃ = x. The three-dimensional integral can be reduced to three successive one-dimensional integrations; but only the first can be performed by primitive: the primitive of 1/[Ö(1-u²)] is indeed easy, but then we knock against elliptical integrals. So we will have to compute numerically for the two remaining dimensions. We then obtain the density as the derivative F¢(x) (in practice stepwise finite differences). This has been done for figure 6.

File translated from T_EX by T_TH, version 1.1. I have replaced the formulas by JPEG images, because the translation was awkward. (J.H.)