## Abstract

In a previous paper [Opt. Express **10**, 663 (2002)], the current author presented a principle and a technique allowing the transformation of a random relationship between the complex amplitudes at two wave fronts into a selective one. As the current paper shows, the same principle can be readily implemented to transmit one-dimensional images through a single multimode optical fiber.

© 2003 Optical Society of America

## 1. Introduction

The transmission of images by purely optical means (mainly in the field of endoscopy) is currently carried out by coherent bundles of optical fibers. The attractive idea of replacing the bundle by a single fiber, either monomode or multimode, has been actively investigated, and several principles and techniques have been proposed. Wavelength encoding [1,2,3], spatial coherence encoding [4], optical phase conjugation [5], and space-time conversion [6] are among the principles successfully tested in the laboratory.

This paper presents another technique, based on a principle already published by the author [7], which can be stated on general grounds as follows. Let two wave fronts of an optical system be randomly correlated (such as at the entrance and exit surfaces of a diffusive medium or at the entrance and exit cross sections of a multimode fiber). Let phase changes be introduced on both surfaces,varying linearly with time and (with opposite signs) according to one coordinate *x*.

If a time-integrated hologram is made of the complex amplitude at the exit surface, in the reconstructed image the intensity varies according to *x* proportionally to the corresponding variations at the entrance surface.

The following section shows how this principle can be implemented to transmit one-dimensional (1D) images through a fiber of 1 mm diameter.

## 2. Experimental demonstration

Figure 1 presents the basic features of the experimental arrangement. A collimated laser beam of sufficient cross section is focused by a lens L on the entrance section of an optical fiber after illuminating a transparency with real and 1D transmission function *M*(*x*) placed at the plane *xy*. A device producing a linear phase change according to x is placed at the same plane. In our case the device (Fig. 2) was composed of two glass plates separated by a deformable rubber gasket, one of them fixed, the other rotating around a fixed pivot with constant rotation speed ω_{1}. The space between the plates was filled with a liquid of refractive index n.

A similar device is placed on plane *x*’*y*’at a suitable distance from the exit section of the fiber, the angular speed of the rotating plate being ω_{2}.A hologram of the complex amplitude at plane *x*’*y*’ is made by means of the holographic plate H with reference beam *R* and exposure time *T*. Let the complex amplitude of the light wave impinging on plane *xy* be *a*(*x,y*). The complex amplitude at plane *x’y’*, just after the second device, can be expressed as

where *k*=2*π*/*λ*, *t* is time, and the random function θ(*x,y;x’,y’*) indicates the phase change undergone by a photon traveling from a point *x,y* on plane *xy* to a point *x’,y’* on plane *x’y’*. Intensity changes are supposed to be uniform and should be expressed by a constant factor, here omitted as nonsignificant. Such factors will be systematically omitted also in the following equations.

The holographically significant part of *a*(*x*’,*y*’) recorded in the hologram is

where the asterisk means conjugate complex. Since *R* is assumed constant with time and since in multiple integrals the order of integration is arbitrary, the complex amplitude of the virtual image reconstructed by *R* is obtained by integrating *a*(*x*’,*y*’,*t*) as expressed by Eq. (1), according to time and for the time interval 0-*T*.

The result is

where

The above function has value 1 for *ω*1*x*+*ω*2*x*'=0, i.e., for

For a fixed *x’* and *x* differing from *x _{m}* the modulus of

*φ*(

*x*',

*y*') drops quickly to fractions of unity, depending on the values of ω

_{1}and T.

Setting, for instance, *x*-*x _{m}*=0.1 mm, λ=0.633 µm,

*n*=1.33 (water as the phase changing medium), the above modulus has a maximum value of 0.1 for ω

_{1}

*T*=0.06 rad. In other words this slight rotation of device 1 (accompanied of course by a rotation of device 2 according to the fixed ratio of angular speeds) suffices to attenuate at least by a factor of ten the light coming to the line

*x*’ from a line

*x*displaced from

*x*by only 0.1 mm.

_{m}The above numerical discussion justifies the approximation consisting in treating the function φ(*x*’,*y*’) in Eq. (4) as a Dirac delta function, so that, upon integration according to *x* we obtain

The virtual image reconstructed by R contains a random structure [the integral function in (6)], modulated by the real function *M*(*x*). The latter is transmitted from the object *xy* plane to the image *x*’*y*’ plane with a magnification (or reduction) factor that includes the sign. The images are identical if the rotation speeds of the transmitting and receiving devices have the same values but opposite signs. For equal signs the image is inverted, as in the case of a lens.

Figure 3 shows an object consisting of straight lines of different thicknesses varying from 0.3 to 1.2 mm as transmitted by an optical fiber of 1 mm diameter and 1 m length. The transmitted image can be recorded by photographing either the virtual or the real holographic displays, visualized by a diffusive screen. Recording the real display is preferable if a digital camera is used, in order to avoid conflicts with the focusing automatisms.

Other operating features were as follows: ω_{1}=-ω_{2} ; ω_{1}
*T*=0.03 rad ; He-Ne laser of 30 mw; reference/object exposure ratio 8; Slavich PFG-01 holoplates ; SM-6 developer recommended by Slavich, no bleaching; transmitted image recorded by a digital camera on the real hologram.

No special image-processing step was used, save the “image automatic balancing” of the Kodak Photo Editor software.

A small lens (not drawn in Fig. 1) was placed at the entrance section of the fiber, with the task of reducing the lateral displacement of the focal point of L caused by the first linear phase variation.

It is worthwhile to point out that such a lens has no imaging purposes, since the object and the image stay in the *xy* and *x*’*y*’ planes, respectively, and the image is not modified by any change in the optical system between them.

## 3. Advantages and drawbacks of the technique

Since the aim of this paper is limited to contributing (supposedly) new concepts in the field, the discussion on these points will be very concise.

The main advantages of the technique seem to be its simplicity and the good resolution obtainable.

As for the main drawbacks, namely, the delayed time and the 1D limit, here are a few comments:

Delayed time: This disadvantage could in principle be overcome by the use of a photorefractive crystal in place of a conventional holographic plate.

Two-dimensional capability seems in practice unattainable (see Ref. [7] for a discussion on this point) unless an hybrid and delayed-time technique is used. A two-dimensional subject can be studied for instance by making a series of holograms for a series of directions of the rotation axes *y* and *y*’ and by combining them by means of a tomographic algorithm.

## References and links

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**2. **A. Friesem and U. Levy, “Parallel image transmission by a single optical fiber,” Opt. Lett. **2**, 133 (1978). [CrossRef] [PubMed]

**3. **T. C. Yang and C. Frohely, “Optical image transmission through a single fiber by chromatic coding,” in *Optical Waveguide Science*, H. C. Huang and A. W. Snaider, eds. (Martinus Nijhoff, LaHague, 1983), pp. 333–334.

**4. **P. Naulleau and E. Leith, “Imaging through optical fibers by spacial coherence encoding methods,” J. Opt. Soc. Am. **13**, 2096–2101 (1996). [CrossRef]

**5. **G. J. Dunning and R. C. Lind, “Demonstration of image transmission through fibers by optical phase conjugation,” Opt. Lett. **7**, 558–560 (1982). [CrossRef] [PubMed]

**6. **P. C. Sun, Y. T. Mazurenko, and Y. Fainman, “Real time one-dimensional coherent imaging through single mode fibers by space-time conversion processors,” Opt. Lett. **22**, 1861–1863 (1997). [CrossRef]

**7. **L. Pirodda, “Time—integrated holography and optical tomography,” Opt. Express **10**, 663–669 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-15-663. [CrossRef] [PubMed]