The Orbital Geometric Phase: A New Geometric Phase of Optical Beams Carrying Orbital Angular Momentum

Geometric phase arises when a physical system undergoes a closed path in state space or parameter space. The formalism for this phase was first developed by Michael Berry for quantum systems [1]. Since then  this phase has been known as Berry's phase. The initial formalism of Berry was limited to quantum systems. Since then it has been generalized and shown to have a classical counterpart, sometimes referred to as the Hannay angle [2]. This concept applies to a large range of quantum and classical systems, from the Aharonov-Bohm effect to the Foucault pendulum. It also loosely referred to as geometric phase.

In optics there are two well studied geometric phases, spin-redirection or coiled-light phase and Pancharatnam phase. Both show the interesting contrasts of geometric phase. In spin redirection phase the light acquires a phase after traveling a three dimensional path in real space. For example is we send a light beam with circular polarization through a single mode optical fiber, the final phase of the wave will depend on how the fiber was wound. The first experiments on this topic were performed by Chiao and co-workers [3]. Consider the figure below. In (a) we have a fiber wound once.

If we map the propagation vectors of a beam as it travels through a coiled fiber onto a single point, we see that the tip of the vector follows a closed path in the space of directions, as shown in (b). The geometric phase is given by the solid angle described by the propagation vector. Our group has done much work on the phase acquired by reflecting off mirrors, which affect both the polarization of the light and the orientation of images in image-bearing beams. For example, the image inversion effected by the Porro prisms in binoculars is due to this geometric phase [4].

A second geometric phase in optics is due to a path in polarization state of the light. This phase is called Pancharatnam phase, after S. Pancharatnam, who discovered this phase in the 1950's. The phase arises when the state of polarization of the light is transformed following a closed path in the space of states of polarization. The latter space is the Poincare sphere, shown below.

Each point on the surface of the sphere corresponds to a unique state of polarization: the north and south poles are the states of circular polarization, and points along the equator correspond to linear polarization of different orientations. Points on the northern and southern hemispheres correspond to elliptically polarized light of different handedness. By transforming the polarization of an input beam following a closed path on the Poincare sphere the wave acquires a geometric phase. This phase has been studied extensively and has led to many applications [5]. For more information on this phase see a review article by Bhandari [6].

It is important to note that in both cases we can vary the phase by changing the either the shape of the path of the light or the state of polarization without changing the optical length of the path. Changing the optical path length introduces dynamical phases. This aspect makes geometric phase boundless. That is, we can keep increasing it without limit.

In this work we introduced a new geometric phase by changing the mode of the light beam. Our measurements were restricted to "first-order" modes. We can also map these modes onto the "Orbital" Poincare sphere, shown below.

The north and the south pole correspond to the LG₀⁺¹ and LG₀^-1 modes, respectively. Points along the equator correspond to the HG modes in different orientations. In our experiments we started with the mode LG₀^-1 (point A) and followed the path ABCA. Points B and C form an angle 2q with the center of the sphere. In our experiments, by changing q we changed  the path and thus changed the geometric phase that was introduced.

We measured the phase interferometrically. The LG₀^-1 mode was generated by a forked grating with its "handle" pointing up. If we send the output of the grating to a Mach-Zehnder interferometer we can select to have two different modes pass through each arm, and have them interfere at the output, as shown below. The first-order diffraction to the left of the zero order was mode LG₀^-1, which we selected.

In the top path we select LG₀⁰, and in the bottom path we select LG₀^-1. Contour profiles of the intensity of each mode are shown in the color pictures: a symmetric peak for LG₀⁰ and a doughnut for LG₀^-1. At the output of the interferometer we will have the interference between the two. Since the phase of LG₀^-1 varies with angle and the phase of LG₀⁰ does not, then the interference pattern will have a region of constructive interference on one side of the rim of the doughnut, and destructive interference at the opposite side, as shown in the simulation below.

 If we change the phase of the two beams  the point where the constructive interference is located moves around the rim of the doughnut. Click on the figure above to see a simulation.

To transform the modes on paths AB and CA we used well known p/2 converters. These consist of pairs of cylindrical lenses in between spherical lenses. The arrangement rephases the wavefront of the beam via the Gouy phase. 

For more information refer to the article by Beijensberger [6]. The transformation from B to C was done by rotating the HG mode using a pair of Dove prisms.

The experiment that we did had many challenges. It took us almost two years to overcome an important one: elliminating dynamical phases. Earlier experiments had all of the converters in one arm of the Mach-Zehnder interferometer. Chaging q meant rotating the converter for path CA. Since this is an astigmatic device, the tolerances were extremely high; slight misalignments of the light beam with the centers of the cylindrical lenses and their axis of rotation would deflect the beam, introducing dynamical phases that masked the phase we were trying to measure. Our breakthrough came this past summer when we realized that LG₀⁰ does not gain any phase by going through the converters; LG₀⁰ is the only mode in its space of order 0. The solution was then to send LG₀⁰ together with LG₀^-1. Any dynamical phases would be introduced to both beams, and thus would cancel out. Thus our interferometer is strikingly different than the usual one: the interfering beams are combined before passing through the phase shifting devices. A schematic of the apparatus is shown below.

Our measurements consisted of images taken by a CCD camera. In order to make sure we were doing the correct measurements we introduced a dynamical phase between the two beams. This was done by either shortening or lengthening the path traveled by the LG₀⁰ beam by means of a piezoelectric attached to the top mirror labeled P. When wwe increased the voltage on the piezo this advanced the phase of LG₀⁰ relative to LG₀^-1. Since the phase of LG₀^-1 forms a left-handed corkscrew, delaying it meant that the interference pattern should rotate counter-clockwise. A mirror put before the camera (not shown in the diagram) changed this to clockwise. A series of interferograms as a function of the dynamical phase is shown below.

 The two beams that generated these interferogram are shown below:

If we turned the piezo voltage to zero and changed the optical components so as to increase q, we increased the phase of LG₀^-1 relative to LG₀⁰, and thus changed the orientation of the interference maximum so that it rotated counter-clockwise. A selection of interferograms is shown below, was taken for increasing values of q.

Extracting the phases from these data yields excellent agreement with theory. More information can be obtained from our upcoming publication in Physical Review Letters [7]. An article describing our first attempts at these measurements will also appear in Coherence and Quantum Optics VIII  [8].

The picture below shows Patrick Crawford '04, who worked on this project last summer.

Current research is focusing on the geometric phase of higher-order modes.

This work has received support by an award from Research Corporation: "Geometric Phase of Optical Beams Possessing Angular Momentum" (2000-2002) and from NSF grant RUI-9988004.

Latest: Is geometric phase linked to the exchange of orbital angular momentum  between the light and the optical system? Our work on first-order modes (Refs. 7 and 8) is consistent with this conjecture. More startling, our more recent work on the geometric phase of second-order modes gives NO GEOMETRIC PHASE for paths that DO NOT involve exchange of orbital angular momentum [9].

[1] M.V. Berry, "Quantal Phase Factors Accompanying Adiabatic Changes," Proceedings of the Royal Society of London A 392, 54-57 (1984).
[2] J.H. Hannay, "Angle Variable Holonomy in Adiabatic Excursion if an Integrable Hamiltonian," Journal of Physics A 18, 221-230 (1985).
[3] A. Tomita and R. Chiao, "Observation of Berry's Topological Phase by Use of an Optical Fiber," Physical Review Letters 57, 937-940 (1986).
[4] E.J. Galvez and C.D. Holmes, "Geometric Phase of Optical Rotators," Journal of the Optical Society of America A 16, 1981-5 (1999).
[5] E.J. Galvez, "Applications of Geometric Phase in Optics,"in Recent Research Developments in Optics 2, 165-182 (Research Signpost, Kerala, 2002).
[5] R. Bhandari, "Polarization of Light and Topological Phases," Physics Reports 281, 1-64 (1997).
[6] M.W. Beijersbergen, L. Allen, H.E.L.O. van der Veen, and J.P. Woerdman, "Astigmatic Laser Mode Converters and Transfer of Orbital Angular Momentum," Optics Communications 96, 123-132 (1993).
[7] E.J. Galvez, P.R. Crawford, H.I. Sztul, P.J. Pysher, P.J. Haglin and R.E. Williams, "Geometric Phase Associated to Mode Transformations of Optical Beams Bearing Orbital Angular Momentum," Physical Review Letters, 90, 203901 (2003). Link to PRL issue. PDF of reprint. See also AIP Physics News Update article.
[8] E.J. Galvez, H.I. Sztul and P.J. Haglin, "Measurements of the Geometric Phase of First-Order Gaussian Beams," Proceedings of the Eight Rochester Conference on Coherence and Quantum Optics (preprint available).
[9] E.J. Galvez and M. O’Connell, “Existence and Absence of Geometric Phases Due to Mode Transformations of High-Order Modes,”  Proceedings of SPIE 5736, 166-172 (2005). Reprint.
 

[Background image is a spiral interference pattern produced by a beam in a LG01 mode.]

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