Optical Physics

Light Beams in High-Order Modes

A recent project that we have been working on involves the geometric phase that results from cyclic changes in the modes of a higher-order Gaussian beam. A set of the beam modes of a Gaussian beam carry orbital angular momentum. Patrick Crawford (Summer 2002), Matt Pysher (Summer 2002), Henry Sztul (Spring 2000 and 2001) and PJ Haglin (Summer 2000) have been working on this project. We have done the first measurements of this phase for optical beams. An article on this work appeared recently in Physical Review Letters [PRL 90, 203901 (2003)].

When you have a collimated beam of light the wave equation in three dimensions becomes the paraxial wave equation. This equation can be solved in either Cartesian or cylindrical coordinates. The solution in Cartesian coordinates gives a family of solutions expressed in terms of Hermite polynomials multiplied by a Gaussian envelope. These Hermite-Gauss solutions (or modes) are categorized by their indices (n,m) and "order" N=n+m. The Hermite-Gauss (HG_nm) of order zero, HG₀₀ is the usual output of an ordinary laser (TEM₀₀): All the points on a transverse plane have the same phase:

That is, if we show the surfaces containing points that have the same phase (e.g. crests, like in the figure), these surfaces are planes separated by a wavelength. Higher order solutions have more complex profiles, like HG₁₀: HG₂₀: and HG₃₀: . Hermite-Gauss modes do not carry orbital angular momentum.

The solutions of the paraxial wave equation in cylindrical coordinates give rise to a family of modes, expressed in terms of the product of a Laguerre polynomial of radial index p and axial index l, a Gaussian envelope and a phase term that depends on the product of the transverse angular coordinate and the radial index l. Light beams in Laguerre-Gauss (LG_p^l) beams carry orbital angular momentum ( l h/2p per photon) . The lowest order mode LG₀⁰ is the same as the HG₀₀. The first order modes LG₀⁺¹ and LG₀^-1 have intensity profiles that are radially symmetric , with zero intensity in the center: . When the light beam in this mode is projected on a screen it looks like a "doughnut:" . But more interesting is the phase structure of this wave. The phase winds as a function of the angle: points at opposite sides of the doughnut are 180 degrees out of phase, and points 90 degrees from each other with respect to the center are 90 degrees out of phase. In a transverse plane, the phase smoothly advances with angle, counter-clockwise for LG₀⁺¹ and clockwise for LG₀^-1. One way to represent this phase structure is the following figure: . Points of equal color have the same phase. Another view is shown below, where the phase of the wave on a transverse cut is shown. The red dots mark the intersection of the wave with the transverse plane: see how the phase advances in a circle around the center of the plane.

Since the center has "all phases" it is a phase singularity. This explains why the intensity of the beam is zero at that point. If we were to show the points of equal phase as the wave propagates, from a right-handed corkscrew spiral for LG₀⁺¹ we would get: , and for a left-handed corkscrew spiral for LG₀^-1 : . It is important to understand that the light is not following a helical path. The phase of the light is changing in such a way that it describes a helix, as shown in the figure below. The contour plot gives the intensity distribution of the doughnut. Points of the beam that have the same phase (e.g., crests in the figure) describe a helix. The pitch of the helix is one wavelength.

The orbital angular momentum arises from this phase structure. The local linear momentum density of the wave (p=e₀ExB) has an axial component. For an expanding Gaussian beam in the lowest order (TEM₀₀ or LG₀⁰) the linear momentum has radial and longitudinal components, as shown below.

For a beam with orbital angular momentum, the spiral phase front gives rise to an axial component of the linear momentum.

Thus, an integration of the angular momentum (L=rxp) over the entire profile gives rise to a component of angular momentum along the propagation direction [For more on this see L. Allen and M. Padgett, Optics Communications 184, 67-71 (2000) and G.A. Turnbull et al. Optics Communications 127, 183-188 (1996).].

We can make these beams in the laboratory in different ways. The easiest way involves using a computer-generated hologram. If we calculate the interference pattern generated by combining LG₀⁰ and LG₀⁺¹,

and print out an fringe pattern in binary form, we get a "forked" pattern, shown below. The fork reveals the phase singularity.

If we photo-reduce this image, we can use it as a diffraction grating. If we then shine a laser beam onto it we get

We can verify the first order is indeed the LG₀⁺¹ mode by interfering it with the zero order LG₀⁰:

[If you want to generate your own doughnuts click here.]

Our interference experiments can get fancier but also much more fun. If we set up a Mach Zehnder interferometer, like the one shown below

and put the forked grating before the first beam splitter, we will have copies of the interference pattern shown above traveling both arms of the interferometer. We can align the mirrors so that after the interferometer we have the modes LG₀⁰ and LG₀⁺¹ superimposed. Then... we expand the LG₀⁰ mode by placing a lens in the corresponding arm of the interferometer (you can do this also in a Michelson interferometer). If both interfering beams were LG₀⁰ the pattern would be concentric circles: , but if the unexpanded beam is in the LG₀⁺¹ mode then the interference pattern looks like a spiral: . If we change the length of one of the arms get the ...Mesmerizer! (Short mpg video made by Henry Sztul.).

So far we have discussed first order modes. Second order modes such as LG₀⁺² also have a doughnut profile but with a phase that winds twice as fast as the LG₀⁺¹ mode. The points of equal phase wind about the axis forming a double helix: . An interference pattern with an expanding reference beam yields a double spiral: .

HG modes can be transformed into LG modes and vice versa using "mode transformers." These are made of cylindrical lenses. They exploit the Gouy phase to insert the proper phases that convert one set of modes into the other. The possibility of making these conversions was first reported by Allen and co-workers [L.Allen, M.W. Beijersberger, R.J.C. Spreew and J.P. Woerdman, Phys. Rev. A 45, 8185-9 (1992)]. For first order modes we have the following identities:

One "equation" can quickly become the other by inserting a 90-degree phase between HG₁₀ and HG₀₁. The transformation is made via the Gouy phase shift. This phase is a little known "secret" of wave propagation. It arises when a light wave goes through a waist after being focused by a lens, as shown below.

We have measured this phase using the Mach-Zehnder interferometer shown above. We interfere an LG₀⁰ beam with a copy of itself that is being focused by a positive lens. We then move the camera before and after the focal plane of the lens. One can see that the phase between the two waves changes as we go through the waist of the beam being focused. You can see it for yourself in this mpg video made by Henry Sztul.