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The generation of hollow beams by a multimode light guiding device is analyzed. The light propagation through the light guide is simulated by ray tracing. It is shown that hollow beams are generated by light rays that propagate along a spiral path through the light guide. The properties of the hollow beam depend on the tilt angle and location of the input beam on the front surface of the light guide. The properties of the output beam are investigated experimentally.
©2010 Optical Society of America
1. Introduction
Recently, there has been a substantial interest in dark hollow beams (DHB). Such beams can be used for guiding cold atoms [1–10]or trapping of particles with low index of refraction [11–13]. Various methods have been developed to generate DHB. Phase plates [14–16], axicons [2, 17–20] lenses, holograms [11, 15, 21–24], and hollow optical wave guides [8, 25–31] have been used. Recently it was shown that a DHB can also be generated by coupling a focussed laser beam off-axis into a multimode fiber [32, 33]. In the following we will give an explanation of the formation of a dark hollow beam in a multimode fiber by off-axis coupling using geometrical optics. In addition to the theoretical considerations, we have investigated the near field of such hollow beams experimentally.
2. Theoretical model
In the theoretical model presented below, we assume that the dimensions of the wave guide are large compared to the wavelength and that the simplifications of geometrical optics are applicable. We are primarily interested on the geometrical shape of the light field generated by the wave guide and ignore reflection losses. For simplicity, we consider the light guide as a straight cylinder. In this case, the angle of incidence of a light ray entering the light guiding cylinder is the same as the exit angle. No ray, that is not directed toward the axis of the light guide will ever cross the axis. All these rays will propagate on a spiral shaped path through the light guide, as sketched in Fig. 1 .
Fig. 1 Light path of a single ray (left) within a cylindrical light guide for an off-axis obliquely incident ray and beam broadening by multiple reflections on the side wall (right)
Also shown in the insert of Fig. 1 on the left is the projection of the light ray to the plain z = 0, where z coincides with the cylinder axis. The ray enters the wave guide at x0, y0 and its projection on the x,y-plane is a polygon, that generates a circular caustic. The caustic radius, rc, corresponds to the shortest possible distance of the projection and is given by the relation: rc = x0⋅sinα − y0⋅cosα, where α is the angle between the direction of the ray in the x,y-plane and the x-axis, see Fig. 1. In this figure the elliptical intersection of the beam with the front surface of the wave guide is shown. In contrast to a ray a light beam has a final width. Beam broadening is shown for coupling on the cylinder axis and off-axis after a couple of reflections on the wall of the wave guide. After sufficient reflections on the curved sidewall the beam fills complete the annulus between the caustic
circle and the wall. In the case of on-axis incidence, the caustic radius is zero and the wave guide is completely filled with light. Obviously, a shift of the location of incidence in the direction of propagation, or more precisely its projection in the z = 0 plane, has nearly no effect on the propagation properties of the beam. For example, shifting the point of incidence, x0,y0 along the y – axis moves the reflection points along the z – axis but changes nothing in the z = 0 plane except that the oval intersection of the beam with the front surface is shifted along the y – axis. The refraction on the front and end face of the light guide does not change the x,y-component of the direction vector. As a consequence, no ray can come closer to the axis of the wave guide than given by its specific caustic radius.
This holds also for rays leaving the wave guide. All possible exit directions for rays with the same caustic and the same angle of incidence are located on the surface of a rotational hyperboloid, such as shown in Fig. 2(a) .
Fig. 2 a) Rotational hyperboloid formed by rays leaving the end face of the light guide having the same caustic, b) construction of the beam boundaries, c) beam profile for ϑ = 10°, rc = 0, d) beam profile for ϑ = 20°, rc/D = 0.2.
The waist radius of the hyperboloid is equal to the caustic radius. The inclination of the rays relative to the end face, is the same as that of the incident beam. The surface of the hyperboloid can be described by the following relation:
where rc is the waist radius, which is identical with the caustic radius, ϑ is the incidence angle, and ze is the z-coordinate of the waist plane, that contains the location of the vertices in an axial cut through the rotational hyperboloid of Fig. 2(a). The origin of the z-coordinate is the end face. Obviously, the half aperture angle of the hyperboloid is equal to the angle of incidence ϑ. The ray family shown in Fig. 2b is one of a group forming the output beam. This group comprises all ray families with the same caustic whose waist radius is located between and including:We can now construct the shape of the output beam. We assume the beam enters the wave guide off-axis. We only consider two rays of the incident beam. The incident ray closest to the axis generates the hyperboloid marked by the red lines in Fig. 2(b)). The hyperboloid drown in black is generated by the ray of the incident beam with the largest distance from the axis. These rays generate families of output rays that form the inner and outer boundary of the output beam. The two rays hitting the front surface at different distances from the axis have obviously different caustic radii. As we see in Fig. 2 the ray with the smaller caustic radius determines the boundaries (red lines). The inner limit is given by the rays that are reflected on the rim of the wave guide and generate the rotational hyperboloid with the largest downstream waist distance, marked by the letter A. The outer limit is given by the rays that just pass the rim without reflection and form the rotational hyperboloid with the largest upstream waist distance marked by the letter B. Also shown in Fig. 2(b) are rays that have their caustic in the exit plane. They form the central part of the output beam. With this procedure the profiles of the two output-beams shown in Fig. 2(c) and 2(d) where constructed.
All rays whose vertex has a positive z- coordinate converge toward the axis, have there closest distance from the axis at z = ze, and diverge again. All other rays diverge from the end face. The thickness, d, of the ring can easily be calculated by aid of Fig. 2 and is given by d = D [1-(2rc/D)2]½. Obviously, if the caustic radius is zero the ring thickness is equal to the thickness of the light guide and remains constant independently of the distance from the exit. For an off-axis incident ray a hollow beam is formed inside the light guide. The diameter of the dark region is twice the caustic radius. This hollow region continues in the exit beam with constant diameter to the maximum downstream distance of the waist, z = zcmax and transits into a ring beam with an aperture angle equal to twice the angle of incidence of the input beam.
From the considerations presented above, we conclude that the properties of the light field depend only on the angle of incidence of the beam coupled into the wave guide, its lowest caustic radius, and the diameter of the wave guide. The first defines the aperture angle, the two others the thickness of the light ring. In the near field a hollow dark region with constant diameter is formed. Its radius is equal to the smallest caustic radius. We have ignored so far finite aperture angles of the incident beam and the quantitative distribution of radiation in the output beam. The qualitative effect of a finite aperture angle will be, that the ring thickness will not remain constant with increasing distance but will become broader. For a prediction of the quantitative light distribution numerical simulations will be necessary.
3. Experimental set-up
To verify the theoretical model the light field generated by a straight light guiding PMMA rod was investigated with the experimental set-up sketched in Fig. 3 . The rod was mounted on a motorized goniometer. The position could be controlled in all three coordinate directions by precision translation stages. The beam of a HeNe- laser was coupled into the front surface of the rod via two mirrors.
A straight rod with 10 mm diameter and a length of 620 mm was chosen as multimode light guiding device. A straight rod was chosen to avoid effects of bending on the light transport. The light guide had no cladding. The large refractive index jump on the walls of the rod allowed relative big input angles (inclination of the input beam). This was important to have sufficient reflections within the relatively short rod to fill sufficiently the end face with radiation. Incidence angles of 10° and 15° were used.
Before starting the experiment, the axis of the laser beam was adjusted to the axis of the light guiding rod by the following procedure: The rod was removed and replaced by two pinholes separated approximately 30 cm from each other. The goniometer was set into its zero position. By properly adjusting the mirrors the laser beam was fed through the pinholes. After this adjustment the pinholes were replaced by the PMMA rod and fixed in a position, where the front surface coincided with the axis of the goniometer. In the next step the PMMA rod was turned by the goniometer to the selected input angle. A transparent screen was mounted downstream from the end face of the rod at different distances. The pictures generated by the output beam on the screen were recorded by a digital camera. After the measurements the content of the camera memory was transferred to a computer for evaluation.
4. Results
A selection of the pictures as they were recorded are shown in Fig. 4 . For better picture quality some of the pictures were γ-adjusted. The two photo series on the left were recorded with a laser beam diameter of w ~4 mm. The two photo series on the right were taken with a laser beam diameter of w ~2 mm. Obviously, the filling of the cross section with the smaller beam is worse than with the larger beam. For this reason the angle of incidence was increased to 15°. It is interesting to notice that the incomplete filling of the ring after only a few reflections as illustrated in Fig. 2 is qualitatively well reproduced in the right columns in Fig. 6 .
Fig. 4 Screen shots of the spots generated by the output beam at different downstream distances. The position and direction of the input beam is indicated on the bottom. The angle of incidence for the left two column is ϑ = 10°, for the right two column ϑ = 15°.
Fig. 6 Pictures of the hollow beam 150 mm downstream from the end of the wave guide at various tilt angles ϑ = 14°,ϑ = 12°,ϑ = 10° (from left to right) of the incident beam.
All main features predicted by the theoretical model are clearly visible in the pictures. In the case of the two picture series on the left, the theory predicts the formation of the dark core at a downstream distance of z (ϑ = 10°) ~28 mm. In the case of off axis coupling, the two columns on the right show that the dark region generated already in the light guiding rod continues with the radius of the caustic until a downstream distance of z (ϑ = 15°) ~18 mm. After this distance the aperture angle of the ring is the same as the tilt angle of the incident beam. We also see that the thickness of the ring remains constant and has the same size as the wave guide, in the case of on-axis coupling and is slightly smaller in the case of off-axis coupling. As already mentioned, only a selection of the recorded images is reproduced in Fig. 4. The complete data set for the measurements with on-axis coupling was evaluated numerically and the colour coded pictures given in Fig. 5 were calculated. The image quality for the off-axis results could not be evaluated by this procedure.
Fig. 5 Colour coded light field emerging from a cylindrical wave guiding rod, left figure the input beam is coupled on axis to the rod, right picture the input beam is shifted in the input direct from the axis. Angle of incidence ϑ = 10°.
To verify the prediction from the theoretical analyses that the aperture angle is the same as the tilt angle of the incident beam the transparent screen was mounted 150 mm downstream of the exit plane of the light guiding rod and images were taken for different tilt angles of the incident beam. Results are shown in Fig. 6. The ratio of the three rings is 29:26:21. This corresponds very well to the ratio tg14:tg12:tg10 = 0.249:0.212:0.176. Due to the increase in size the intensity of the largest ring is low and the uncertainty in the measurement of the size has increased. One can also see that the width of the ring is constant.
5. Discussion and conclusion
We have shown that geometrical optics is well suited to explain all major features of light fields generated by coupling a tilted beam into a multimode fiber. The aperture angle of the emerging field is identical with the tilt angle of the incident beam. This is certainly only valid as long as the tilt angle is smaller than the aperture of the light guide. In the fare field – at distances large compared to the light guide diameter – a dark hollow beam is formed with nearly constant width of the ring. The width is identical to the core diameter of the light guide or slightly less in the case of off axis coupling. If the incident beam is smaller than the diameter of the light guide and its propagation direction has no radial component, the core region of the light guide is free of radiation and a dark hollow core is generated in the near as well as in the far field. The diameter of the radius of the dark region in the near field is identical to the shortest distance of the propagation direction of the incident beam from the axis of the light guide. The diameter of the dark region remains nearly constant from the end of the light guide to a distance given by the light guide radius times the cotangents of the tilt angle. At larger distances the core increases linearly with distance. We have not investigated the effect of the length of the light guide explicitly. However, from the description of the model in section 2 it becomes clear that the main effect of increasing the length of the light guide is the homogenizing of the intensity distribution in the light ring due to the increasing number of reflections.
Although the model takes not into account the final aperture of the incident beam, the main features should remain unaffected by the final beam aperture. One can assume, that the results are not only valid for a straight cylindrical wave guide but also for multimode bended fibers as long as diffraction effects will not be dominant. It is interesting to notice, that very much the same results would be achieved by replacing the massive light guiding rod by a hollow rod with reflecting walls. In this case, by off-axis coupling a light tube can be formed within the hollow rod that continues on the end into the surroundings. In such a light tube reflecting objects may be transported free of friction. While in a hollow fiber only a narrow evanescent light field close to the wall can be generated, with a light guiding tube a high intensity light field can be generated whose thickness can be adjusted from a narrow region close to the wall until complete filling of the tube.
Acknowledgement
The authors express their gratitude to the Deutsche Forschungsgemeinschaft for support of the work.
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