“Nondiffracting” Light Beam
Sergio Johnson
College of Optical Sciences, University of Arizona, Tucson, Arizona 85721
Solutions will be presented to the wave equation for beams that are non-diffracting. Bessel
functions will be discussed as these solutions. An experimental method of generating a finite
energy Bessel, which avoids an infinite energy problem, will be shown. The experiment
creates a nearly non-diffracting beam. The propagation distance of the central lobe of this
beam will be compared with the Rayleigh range of a Gaussian beam of the same radius as this
lobe.
Introduction
Diffraction is a feature of the wave nature of light. It can occur anytime a beam of
light passes through an aperture that is large with respect to its wavelength1. According to the
Huygens-Fresnel Principle2, every point of the wavefront that is unobstructed by the
hindrance serves as a source of spherical wavelets that constructively and destructively
interfere with each other depending on their optical path length. This creates what is called a
diffraction pattern. Diffraction causes the intensity profile of a laser to spread out as it
propagates through free space. The Rayleigh range ZR gives a measurement of this spreading
of a monochromatic beam. It gives the range over which a Gaussian beam increases its beam
waist size by a factor of 2, where the beam waist is the radius that has the minimum cross
section of the beam. The Rayleigh range is give by [3]:
(1)
where w0 is the beam waist size, and  is the wavelength.
Durnin4 showed that Bessel functions could be used to create exact solutions to the
free space wave equation given by [4]:
(2)
These solutions are non-diffracting in their propagation. The ideal Bessel beam solution is
given when the electric field is proportional to the zeroth-order Bessel function. This
discovery and explanation of this diffraction-free propagation has helped in gaining further
understanding of the nature of the electromagnetic field as well as have had many applications
in the world of optics.
Bessel Beam Theory
Bessel functions of the first kind Jn(x) are defined as solutions to the Bessel
differential equation [5]:
(3)
The zeroth-order Bessel function Jo(x), which will later be shown of interest, can be
represented in the integral form of [5]:
(4)
These solutions are what comprise the Bessel beam theory of non-diffracting beams.
Figure 1 Graph of zeroth-order Bessel function.
The Bessel beam solution results in a beam profile that has a narrow central area with
concentric rings surrounding it. This ideal solution has an electric field of the form [3]:
(5)
where k|| = (2/)cos(), k = (2/)sin(), r = (x2+y2)1/2, and J0 is a Bessel function of the
zeroth-order. The wave vector k indicates the angular propagation of the beam. The
associated intensity is proportional to the square of the electric field, I(x,y,z)  |E(x,y,z)|2. By
multiplying the electric field by the complex conjugate of itself (assuming k is real), the
resulting intensity distribution is proportional to J02(kr). This result is completely
independent from the location of the solution in the propagation direction, z. This means that
the intensity profile of an ideal Bessel beam does not change under free-space propagation.
These solutions cannot be physically possible since they would require a beam of infinite
energy6. However, it is possible to approximate a Bessel beam over a finite extent in many
ways.
Figure 2 Intensity distribution of a finite energy Bessel beam.
Bessel Beam Experiments
One of these experiments is explained by Durnin et al6 by using the experimental set
up in Figure 3. It involves a narrow circular slit, or annulus, on the order of 10m, which has
a diameter of a few millimeters. This slit is placed at the back focal plane of a lens. When it
is illuminated by a plane wave, such as a He-Ne laser, every point in the aperture acts like a
point source. These point sources create spherical wavelets. These wavelets then go through
the lens and are transformed into plane waves. The wave vectors of these plane waves lie on
a cone, which is a property of Bessel beams. This is shown in Figure 4. Since this is only
experimental, it does not retain its characteristics forever.
Figure 3 Experimental set up used by Durnin et al.
Figure 4 Spherical wavelets transformed into plane waves
There exists a theoretical Zmax over which this experimental Bessel beam will
propagate without the central max exhibiting diffractive spreading3. The plane waves form a
cone with a half angle . According to Figure 3 and through similar triangles, the diffractionfree distance is:
Zmax = 2Rf/D
(6)
where D is the diameter of the annulus, and f and R are focal length and radius of the lens
respectively. This should be a greater distance than the Rayleigh range of a Gaussian beam.
Durnin et al6 used an annulus with a diameter of 2.5mm and a slit width of 10m. The lens
had a radius 3.5mm, and a focal length of 305mm. The source had a wavelength of 633nm.
The central spot radius was approximated to be r0  1/k7, therefore, r0 was 25m. By using
Eq. (6) the maximum beam propagation distance is 85.4cm. In comparison, by using Eq. (1)
the Rayleigh range of a 25m waist beam is only 3.1mm. These nearly non-diffracting beams
clearly have a central peak that propagates much further that the Rayleigh distance for
Gaussian beams. Finite energy Bessel beams can give great insight into the wave nature of
light and electromagnetic fields.
Citations
1. E. B. Brown, Modern Optics (Reinhold Publishing Corporation, New York, 1965), pp. 5-8.
2. E. Hecht, Optics (Addison Wesley, San Francisco, 2002), pp. 444-445, 474-476.
3. C. A. McQueen, J. Arlt, and K. Dholakia, “An experiment to study a ‘nondiffracting’ light
beam,” Am. J. Phys. 67, 912-915 (1999).
4. J. Durnin, “Exact solution for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am.
A 4, 651-654 (1987).
5. E. W. Weisstein. MathWorld: Bessel Function of the First Kind. Wolfram Research. 29
Oct 2005 <http://mathworld.wolfram.com/BesselFunctionoftheFirstKind.html>.
6. J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, “Diffraction-free Beams,” Phys. Rev. Lett. 58,
1499-1501 (1987).
7. M. R. Lapointe, “Review of non-diffracting Bessel beam experiments,” Opt. Laser
Technol. 24, 315-321 (1992).