1. Introduction

This paper is basically concerned with the problem of imaging through or within a thick light scattering medium.

Holography has been used or proposed since its early times in order to image objects hidden by thin and thick random screens or media [1–8].

On the other side, time integrated (or time average) holography is currently used for such purposes as vibration analysis [9] but it does not seem to have been used or proposed in the field of optical tomography, i.e. as a mean to visualize objects placed beyond or inside a thick highly scattering medium.

The basic principle, the generalization and the limitations of the technique are exposed in Sections 2 to 4, while Section 5 is devoted to a preliminary discussion of possible applications, in particular to medical imaging.

2. Basic technique

The proposed application of TIH can be conveniently presented starting from the particular case of fig 1, which shows a holographic arrangement where a light wave with complex amplitude a(x,y) impinges on a light scattering medium supposed bounded by planes xy and x’y’ perpendicular to the axis of propagation z, while the light emerging from plane x’y’ is recorded on the holographic plate H, with reference wave R.

 

Fig. 1. Basic setup.

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Fig.2. One-dimensional transparency hidden by an expanded polyethylene slice.

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An object in the form of a transparency with real and one-dimensional transmission function M(x) is placed on plane xy. The following relationship is assumed to hold between the complex amplitudes of the impinging and emerging wave-fronts:

In the (1),the random function θ(x,y;x’,y’) indicates the phase change undergone by a photon travelling from point x,y to point x’,y’ and the assumption is implicit that on summing up the contributions of the surface elements dxdy to a(x’,y’),only phase changes may be considered , while the intensity changes,supposed uniform, are expressed by a constant factor, here omitted as non significant. Such factors will be systematically omitted also in all the following equations.

Let us suppose now that time varying phase changes are added to θ(x,y;x’,y’). One simple way of doing so is to rigidly rotate the body of the medium around a fixed axis ,for example the axis,parallel to the y direction and crossing the z axis,, whose trace on plane xz is O. Supposing that the rotation speed ω is constant and considering the positive directions of x and ω, the (1) becomes:

where $k=\frac{2\pi }{\lambda },\lambda$ is the light wavelenght, t the time.

The (2) is exact only if the impinging wave is a plane one and if the thickness of the medium is vanishingly small, however the error introduced is negligible if the thickness and the extension of the medium in the xy direction are small in comparison to the distance between the x’y’ plane and the holographic plate and to the distance between the xy plane and the origin of a spherical wave.

By the well known properties of holography, which we do not extensively recall here, the holographically significant part of a(x’,y’) which is actually recorded is:

where the asterisk means conjugate complex.

Since R is supposted constant with time, the complex amplitude of the virtual image reconstructed by R is:

Starting with time the arbitrary order of integration in (4),integrating from 0 to T we obtain:

The function $\frac{\sin \left[\mathit{k}\left(x-x\text{'}\right)\omega \frac{T}{2}\right]}{\mathit{k}\left(x-x\text{'}\right)\omega \frac{T}{2}}$ is a good approximation of the Dirac delta function δ(x-x’) for values of the wavelenght in the visible or near infrared and for values of the angle $\omega \frac{T}{2}$ around 10^-2rad or more. For example, setting the latter value for this angle and 0.633 μm for λ, the denominator of the above function takes the value 99.2(x-x’); (x,x’ in mm). This means,pursuing the example and anticipating the integration according to x in the (5), that the amplitude of the light coming to points x’ from points of the xy plane for which (x-x’)= 10^-1 mm is one tenth or less of the amplitude of the light coming from points for which x=x’.Of course the resolution can be increased by increasing the angle of rotation.

Substituting δ(x-x’) for the above function in the (5) and integrating according to x, we have:

According to the (6),the virtual image reconstructed by R contains a random structure or speckle (the intensity of the integral function) modulated by the intensity of the function M(x).From the point of view of image visibility this is not different from the condition of a classical hologram, where speckle is also present. Fig 2 shows the photograph of the virtual image of a one dimensional transparency composed of straight lines of different thickness (0.2,0.2,0.5,1,1.5,2 mm respectively) hidden by a 5 mm thick slice of expanded polyethylene of the type used for heat insulation, fairly homogeneous in appearance . This is so diffusive a material that nothing at all can be discerned by direct observation, while the visibility and resolution in the holographic image is excellent.

 

Fig. 3. Illustration of the condition x=x’.

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Fig.4. Two-dimensional transparency.

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The process formally described by eq. (1)–(6) can be summarized as follows. By virtue of the phase changes introduced by the rotation and the possibility offered by holography of integrating these changes over time,the hologram selectively records only that fraction of the light that a line of coordinate x’ receives from the line of plane xy having the same coordinate x=x’. Fig 3 is intended to make this concept clearer. Of course, also the light coming to the line x’ from lines x ≠ x’reaches the holographic plate and is recorded as noise, thus degrading the efficiency of the hologram.The latter can be however still sufficiently good, as Fig 2 and following prove,if the usual rules of holographic technique are followed.

The main operating features and steps for this as well for the following Figures 4, 5a and 5b were: He-Ne laser of 30 mw, exposure ratio reference/object 8-10, Slavich PFG-01 plates, SM-6 developer recommended by Slavich, no bleaching (actually bleaching seems to reduce the visibility). Since,for various technical reasons, the direct recording of holographic images by a digital camera gives unsatisfactory results, the images were first recorded on Ilford FP3 film by a conventional camera and a digital camera was eventually used to store the positive in the computer memory. No special successive image processing step was used, save the “image automatic balancing” of the Kodak Photo Editor software. The total rotation of the medium was about 10^-2 rad. and, since the small power of the laser required exposure times of a few seconds, the rotation speed ω was correspondingly small and it was obtained by means of a simple hydraulic apparatus made in the laboratory.

It must by emphasized that the image obtained by the technique just exposed retains the three dimensional properties (concerning the dimension z) of holographic images, as a correct interpretation of the (6) suggests and as some tests performed on the virtual and real images of the same object have proved. Such tests were performed by placing a transparency with a symbol (appearing as a rotated H in fig 2) in the x’y’ plane. This symbol is imaged as in a ordinary hologram , as the previous analysis suggests. A clear parallax effect between the symbol and the bars can be seen in the virtual image reconstruction, while in the real image recontruction the planes of maximum resolution of the symbol and of the bars are well separated.

It must be also pointed out that a random oscillating motion of the medium can be an alternative to constant speed rotation. In this case the time dependent factor in Eq.(4) is to be replaced by:

where φ(t) is a random adimensional function of time expressing the variation of the angle around axis O.

The time integral of the cosine and sine components of the exponential (7) tends to 0 if (x-x’) ≠ 0 , if φ(t) oscillates with sufficient amplitude around an average value and if the integral is carried out for a sufficient time interval T, while if (x-x’) = 0 the sine component vanishes and the cosine component is equal to T. Thus, if the above conditions are satisfied, the exponential (7) reduces to the delta function δ(x-x’).

If φ(t) is a harmonic function of time of the type:

setting k(x-x’)φ_o = w, if $T\gg \frac{2\pi }{\omega }$ the integral of the sine component of (7) is always 0, while the integral of the cosine component is:

where J_o(w) is the Bessel function of 0 order of w.

Harmonic oscillation appears less favourable than random oscillation, since J_o(w) tends rather slowly to 0 (in a oscillating fashion) with increasing φ_o and since it is not dependent on exposure time.

3. The two-dimensional case

By the technique just described, only the features of the object depending on one coordinate,say x, can be obtained, while if the object is two-dimensional the features depending on y are lost. Fig 4 is an example of what can be obtained for a two-dimensional transparency containing the first six capital letters of the alphabet. The letters are recognizable,although the dispersion of the information in the y direction is evident.

The technique could be in principle extended to the two-dimensional case through an arrangement where the medium rotates around two perpendicular axes O_x and O_y with independent random angular motions φ_x(t) and φ_y(t). This can be practically realized by placing the medium on a cardan suspension. In this case the time dependent factor in the (4) becomes:

and since the two random functions φ_x(t) and φ_y(t) are not correlated, the time integral of (10) over the interval 0-T is different from 0 and equal to T only if the condition: x =x’ ; y =y’ is satisfied. This condition implies that the hologram records only the light coming to any point x’,y’ of plane x’y’ from the correspondent point x=x’, y=y’ of plane xy. Since the holographic plate records also,as noise,the light coming to point x’,y’ from all other points of plane xy, it is evident that a much stronger degradation of hologram efficiency occurs than in the one-dimensional case. As experimental tests have confirmed, this degradation causes the just exposed development of the technique to be practically useless.

From the practical viewpoint it seems more adequate to treat the two-dimensional case by making two holograms of the same subject with rotations around two perpendicular axes O_x and O_y respectively, with subsequent combination of the one-dimensional features by a suitable (computerized) procedure, which could be the subject of further study.

4. Embedded object

If the object to be imaged is embedded into the scattering medium, the consideration already made over Fig 3 suggests that only details of the object (again supposed to be one-dimensional) very close to one of the surfaces xy and x’y’ can be imaged to some extent.

Supposing that a real transparency M(ξ) is placed on the internal plane ξ,η with the same assumptions that led to equations (1)–(6) and applying the same technique, in place of the (6) we would obtain:

were to avoid formal complexities the treatment has been limited to one dimension and the case of a plane impinging wave has been assumed.

Equation (11) describes more formally what can be intuitively drawn from Fig 3. The object cannot be in general imaged because the relationship between the ξη and the x’y’ planes is analogous to the relationship between the xy and x’y’ planes before the application of TIH, expressed by the (1). Yet some remarks can be made on this result.

By comparison of equations (1) and (11) it appears that the process is a sort of scanning (in one dimension)of the plane xy, while the complex amplitude at points x’ of the x’y’ plane is the resultant of the radiation coming from points x = x’. This somewhat reproduces the technique of Continuous Wave Optical Tomography [10] where the planes xy and x’y’ are contemporarily scanned by a point source and by a point receiver (the entrance of an optical fiber) respectively, while the radiation picked up by the latter interferes with a fixed reference and the resultant intensity is stored in a computer memory. The difference in negative is that by CWOT the two dimensions x and y can be explored, although point-wise,and that, for a fixed point x,y, all x’,y’ points may be scanned. The difference in positive is that by FIH the complex amplitude (intensity and phase)of the emerging wave is stored in a single whole field record. The latter can be the source of subsequent numerical treatment by computer if suitable algorithms in the domain of tomography are used. This is however a subject for a future development.

 

Fig.5a. TI Hologram of a small fish.

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Fig.5b. Same object without TI.

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A possibility of direct imaging is left for details close to the surfaces xy and x’y’. The obtainable resolution strongly depends on the nature of the scattering medium. Fig 5a presents an application of the same technique employed for obtaining Fig 2. The object is a small fish (10 cm long,5 cm wide,7 mm thick) whose skin has been stripped out. For comparison the same object transilluminated by the same laser light but without application of TIH is presented in Fig 5b. This is clearly a case where TIH, even with the limitations pointed out, furnishes interesting results.

5. Discussion and conclusions

The concept of TIH applied to the optical tomographic (or transillumination) problem, its implementation in form of holographic techniques accompanied by some experimental applications and some discussion about their advantages and limits have been presented in the preceding Sections. A few remarks can be here devoted to a preliminary evaluation of possible developments, particularly in the important field of medical imaging. A detailed discussion of this topic appears premature, since it should have a much broader theoretical and experimental basis.

As already pointed out, the two main drawbacks of TIH are its one-dimensional way of operation and its ability to detect only details close to the limiting surfaces of the medium or outside the illuminated surface. While the first limitation can be overcome, the second appears to restrict the usefulness of the technique to the particular case where thin tissues are examined.

However a more rigorous theoretical analysis of the problem (based, in particular, on a less simplified model of the scattering medium) and more adequate experimental instruments could prove that this conclusion is too pessimistic. An experimental development to be pursued is the use of a laser emitting the radiation least absorbed by biological tissues, in the range of 800 nm. This radiation can be obtained, with sufficient coherence lenght, also from relatively inexpensive laser diodes. The real problem is to dispose of a hologram recording medium of sufficient size and efficiency at this wavelenght. Although, as far as the author knows, nothing of this kind is commercially available, at least two types of such media have been tested in the laboratory, namely MQW (multiple quantum well) photorefractives [10] and polymer photorefractives [11]. Only further investigation can say whether TIH can advance on these lines.