“Nondiffracting” Light Beam
Sergio Johnson
College of Optical Sciences, University of Arizona, Tucson, Arizona 85721
Solutions will be presented to the wave equation for beam that are non-diffracting.
Bessel functions will be discussed as these solutions. These solutions are not physically
possible. An experimental method of generating a finite energy Bessel, which avoids
these problems 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 an interesting wonder that is a feature of the wave nature of light. It
has the possibility of happening anytime when a beam of light passes through an opening
or aperture that is large with respect to its wavelength1. According to the HuygensFresnel 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.
This singularity affects Gaussian light beams. Diffraction causes the intensity
profile of the laser to spread out as it propagates through free space. This is studied under
Gaussian beam theory. The Raleigh range ZR gives a measurement of this spreading of a
monochromatic beam. This beam property gives one the range over which the Gaussian
beam increases its beam waist size by a factor of 2, where the beam waist is defined as
the radius that gives 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. However, these diffracting
solutions are not the only ones that exist for the wave equation.
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 surprisingly 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 Functions
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 by the infinite power series [5]:
(4)
or in the integral form of:
(5)
Solutions of this kind are what comprise the Bessel beam theory of non-diffracting
beams.
Figure 1 Graph of zeroth-order Bessel function.
Bessel Beam Theory
The zeroth-order Bessel beam solution results in a beam profile that has a very
narrow central area. Concentric rings surround this central area. This ideal Bessel beam
has an electric field of the form [3]:
(6)
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 with the Bessel beam 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 its self (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 cross-sectional intensity
profile of an ideal Bessel beam does not change under free space propagation. This is in
stark contrast to many other beam solutions to the wave equation such as the Gaussian.
The amount of energy contained in each lobe of the Bessel distribution is almost equal to
the amount of energy contained in the central peak6. This cannot be physically possible
since it would require a beam of infinite energy. 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 McQueen et al3 by using the
experimental set up in Figure 3. This set up is the same as was used by Durnin et al6. 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 Bessel beam 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 where the central max will not exhibit diffractive spreading3. Say the halfangle of the cone that the plane waves form is . Then tan = (D/(2f)), where D is the
diameter of the annulus and f is the focal length of the lens. According to Figure 2, tan
= R/Zmax where R is the radius of the lens. Through similar triangles, the Bessel beam
diffraction-free distance is:
Zmax = 2Rf/D
(7)
This should be a much greater distance than the Rayleigh range of a Gaussian beam.
Durnin et al6 used an annulus of a diameter of a = 2.5mm, a slit width of 10m, a lens of
radius R = 3.5mm and a focal length of f = 305mm. The plane wave source had a
wavelength  = 633nm. By approximating the central spot radius of the Bessel beam to
be r0  1/k7, the central radius is 25m. Using Eq. (7), the theoretical 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. This is an astonishing difference.
Conclusion
Bessel function solutions to Eq. 2 require an infinite amount of energy in order for
them to propagate diffraction-free. There have been many experiments that have created
a Bessel beam of finite energy. The simplest of which can be done by placing an annulus
at the rear focal point of a lens and illuminating it with plane waves. However, this
aperture naturally introduces the problem of diffraction in a beam that ideally should
propagate without spreading. These nearly non-diffracting beams that are created with
the circular aperture have a central peak that propagates much further that the Rayleigh
distance for Gaussian beams with a waist size equal to the radii of the central spot of the
Bessel beams. Finite energy Bessel beams can give great insight into the wave nature of
light and electromagnetic fields despite the fact that they are not ideal.
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 ‘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).
Figure 5
Figure 6