“Nondiffracting” Light Beam

 

 

Sergio Johnson

sergioj@email.arizona.edu

 

College of Optical Sciences, University of Arizona, Tucson, Arizona 85721

 

Abstract

 

The wave equation supports non-diffracting Bessel beams.  An experimental method of generating a finite energy Bessel beam, which avoids an infinite energy problem, is described.  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.  However, when taking into account the entire beam radius, the results are not as conclusive.

 

Introduction

 

            Diffraction is a feature of the wave nature of light.  It occurs when 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 21/2, where the beam waist is the radius that has the minimum cross section of the beam.  The Rayleigh range is given by [3]:         

ZR=pw01/2/l                                                                                      (1)

where w­0 is the beam waist size, and l is the wavelength.

  Durnin4 showed that Bessel functions could be used to create exact solutions to the free space wave equation.  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.  The zeroth-order Bessel function Jo(x), which will later be shown of interest,  can be represented in the integral form of [5]:                      

J0(z)=(1/p)òexp(izcosq)dq                                                         (2)

where the integral goes from 0 to p.  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]:

E(r,f,z,t)=E0exp(i(-wt+k||z))J0(k^r)                                             (3)

where k|| = (2p/l)cos(q), k^ = (2p/l)sin(q), 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) a |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, which is proportional to the square of the zeroth-order Bessel function.

 

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 10mm, 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.  In practice the beam does not retain its characteristics forever.

Figure 3  Experimental set up used by Durnin et al.  D is the diameter of the annulus, and f an R are the focal length and radius of the lens respectively.

Figure 4  Spherical wavelets transformed into plane waves of the same wavelength.  Wavevectors of plane waves lie on a cone.

           

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 q.  According to Figure 3 and through similar triangles, the diffraction-free distance is:

Zmax = 2Rf/D                                                                  (4)

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 10mm.  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 25mm.  By using Eq. (4) the maximum beam propagation distance is 85.4cm.  In comparison, by using Eq. (1) the Rayleigh range of a 25mm waist beam is only 3.1mm. 

 

Conclusion

These nearly non-diffracting beams clearly have a central peak that propagates much further that the Rayleigh distance for Gaussian beams.  However, a proper comparison of the propagation distances requires the use of the full Bessel beam radius.  These reported propagation distances have been found to be smaller than the Rayleigh distances when comparing beams with equivalent radii7.  Despite the fact that nearly non-diffracting beam propagation has not been convincingly illustrated, finite energy Bessel beams can give great insight into the wave nature of light and electromagnetic fields.

 

References

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).