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Propagation invariant light fields such as Bessel light beams are of interest in a variety of current areas such as micromanipulation of atoms and mesoscopic particles, laser plasmas, and the study of optical angular momentum. Considering the optical fields as a superposition of conical waves, we discuss how the coherence properties of light play a key role in their formation. As an example, we show that Bessel beams can be created from temporally incoherent broadband light sources including a halogen bulb. By using a supercontinuum source we elucidate how the beam behaves as a function of bandwidth of the incident light field.
©2005 Optical Society of America
1. Introduction
It is well known that diffraction is a characteristic of the wave nature of light that occurs when any wavefront is spatially modulated in amplitude and/or phase. Sections of the wavefront that propagate beyond the modulation interfere and a diffraction pattern arises. From a quantum mechanical viewpoint, diffraction is central to the understanding of the Heisenberg uncertainty principle and is directly linked with de Broglie’s notion of particles possessing a wavelength that is inversely proportional to particle momentum. In this respect solutions to the Helmholtz equation that are propagation invariant or “pseudo-non diffracting” have gained substantial interest and application in recent years. Durnin et. al. [1] were the first to point out that a set of solutions could be obtained for the free-space scalar wave equation that were “non-diffracting.” The zeroth-order Bessel beam is one such solution and results in a beam with a narrow central region surrounded by a series of concentric rings.
The phase of a plane wave propagating a distance Δz along the z-axis undergoes a shift of kzΔz, kz being the component of the wave vector that points along the propagation direction. This results in a changing interference pattern when a superposition of plane waves propagates. There exist, however, optical elements which impose the same phase shift on all the superimposed plane waves, if the light propagates on a cone for example. In that case, the interference pattern does not change with distance, and these beams are therefore propagation invariant [2]. Such a cone can be formed readily, for example, by illuminating an Axicon which may create an approximation to a Bessel beam in the laboratory. This point also implies that light forming the central maximum of this beam has traveled equidistant paths which in turn mean we may consider the use of temporally incoherent light for their formation. Indeed, prior to the identification of the Bessel beam, MacLeod created a white light focal line using the Axicon as a focusing element [3].
Bessel beams have been studied from the ultrashort pulse regime through to the continuous wave regime. They have found important application in atom optics [2], optical micromanipulation [4], laser plasmas [5], non-linear optics [6] and interferometry [7]. Studies with ultrashort-pulse lasers have indicated the important role of pulse duration for the creation of Bessel beams [8], and in fact these are linked with so-called X-waves which maintain their spatio-temporal localisation upon propagation [9]. In this paper we discuss in detail how the coherence properties of the incident light field influence the Bessel beam formation in the absence of dispersion for both the CW and femtosecond regimes: Our studies show that true white light sources can indeed still generate Bessel beams and that we can use the beam formation to interpret information on the temporal and spatial coherence properties of the incident light field. Our insights show that the key criterion for propagation length of a Bessel or non–diffracting light field is the need for spatial coherence of the light source, and we realise a pseudo non-diffracting light field using spatially coherent but temporally incoherent light sources. We compare the properties of non-diffracting light fields created using a variety of light sources. The work presented may offer new applications of these beams particularly in imaging and micromanipulation.
2. Theory
Firstly, we develop a simple theory of broadband Bessel beams to elucidate our observations. In particular, we consider propagation of a pulsed lowest-order Bessel beam along the z-axis through air for which chromatic dispersion will be negligible. The electric field is described in the scalar and slowly-varying approximations by the field envelope E(r, z, t), where the carrier exp(iω0(z-ct)/c) centred around frequency ω0 has been removed, and we have assumed radial symmetry around the z-axis. Then the fluence profile, time-integrated intensity, of the pulse may be written as
where Parsevals identity has been used, and the spectrally resolved field envelope E(r,z,Ω) is obtained from E(r,z,t) by Fourier transformation, Ω representing the frequency detuning from the carrier Ω=ω-ω0. For the situation described here the pulse at the input z = 0 has a collimated Gaussian profile, and we write the spectrally resolved field envelope as
with w0 the input Gaussian spot size, S(Ω) the normalized spectrum for the pulse, Φ(Ω) a frequency dependent phase, and E0 the amplitude of the field. The input field is first passed through an axicon of refractive-index n and angle γ, and then propagated along the z-axis. It is well known [8, 10] that the initial Gaussian is transformed into a Bessel beam for propagation distances in the vicinity of z = zmax/2 with zmax = w0/θ , and θ = (n-1)γ. (For this paper we neglect chromatic dispersion in the Axicon material.) Then, the spectrally resolved field at the observation plane may be approximated using the results of Ref. [8] as
where f(z, Ω) ≈ f(z, 0) is a slowly varying function of frequency and z in the vicinity of zmax/2, and kr (Ω) = (ω0 + Ω)(n - 1)γ/c is the radial wave vector component. Finally, by inserting the approximate Bessel beam solution (3) in Eq. (1) we obtain the normalized transverse fluence profile
where F 0(zmax / 2) = |E 0 f(z max / 2,0)|2 .
The normalized fluence profile F(r) in Eq. (4) is to be compared against the experimental fluence profiles for pulsed Bessel beams with different spectra S(Ω). As a simple model we here consider a Gaussian spectrum
with Δω the pulse bandwidth. Note that in Eq. (4) for the fluence profile both the spectrum S(Ω) and the argument of the Bessel function J0(kr(Ω)r) depend on Ω, and this is how the pulse spectrum influences the fluence profile. Basically, the fluence profile is a weighted sum over the bandwidth of the pulse of squared Bessel beams terms (kr(Ω)r) which vary as cos2(kr(ω)r -π/4) = [1+cos(2kr(Ω)r -π/2)]/2 for kr(Ω)r ≫ 1. When the spectral average over the pulse bandwidth Δω is performed the oscillatory terms cos(2kr(Ω)r -π/2) from the Bessel beams will be largely cancelled for radii such that a π phase difference or greater exists between the spectrum edge Ω = Δω and center Ω = 0, 2(kr (Δω)-kr (0))r ≥π, which defines the critical radius for a π phase difference
with the FWHM frequency bandwidth given by ΔωFWHM = Thus, what is predicted is that for a bandwidth Δω the Bessel beam fringes will be extinguished beyond the critical radius rcr, where the field will display a 1/r2 fall-off [8]. Alternatively, the number of fringes in the white-light Bessel beam can be viewed as a measure of the pulse bandwidth. The number of fringes may be approximated as
An example of this is shown in Fig. 1 where λ= 790 nm = 2πc/ω0, ΔωFWHM/ω0=0.11, n = 1.5, and γ = 1°, giving rcr = 331 μm and Nfr ≈ 7. The dashed line shows the CW Bessel result (ΔωFWHM/ω0 → 0) with multiple rings, whereas the solid line shows the pulsed Bessel Beam result with ΔωFWHM/ω0 = 0.114 and there are around six rings which are indeed extinguished beyond rcr = 330 μm. We note for an initial Gaussian beam of spot size w0 the collapse of the rings will only be observed if w0 > rcr, since w0 sets the spatial scale of the finite realisation of the Bessel beam generated by the axicon.
Fig. 1. Calculated Bessel beam profiles with λ,=0.79 μm, Δω/ω=0.19, n=1.5 and γ=1°, giving rcr=330 μm and Nfr≈ 7.
3. Experimental results
We performed experiments to generate Bessel light beams from a wide variety of sources to explore the role of coherence for their formation. Pseudo Bessel beams were generated in all instances illuminating a standard glass (BK7) Axicon (n=1.5, opening angle 1 ° or 5°). We note that there are several methods by which a Bessel beam may be created including placing an annular slit in the back focal plane of a convex lens [2], using holographic techniques [11] or diffractive elements [12]. Each of these techniques results in a pseudo-Bessel beam profile but the Axicon is by far the most efficient method for beam generation. In our experiment, the propagation and beam characteristics recorded using a CCD camera and spectrum analyser. Data was taken for a variety of input sources that are listed in Table 1. Light source characterisation was performed using both optical spectrometry as well as a white light interferometer. In all instances, we observed a Gaussian like variation of the on-axis intensity (and no spreading) as reported in [2] and expected for any Axicon generated Bessel beam, coherent or otherwise.
Table 1. Overview and characterisation of the light sources used, where λ is the wavelength, Δλ. the bandwidth, ΔωFWHM/ω0 the relative bandwidth, and lc the coherence length.
3.1 Continuous wave and modelocked Ti:Sapphire laser
As our first example we consider a Ti:Sapphire laser in both continuous wave (CW) and modelocked (ML) operation, which should display high levels of both spatial and temporal coherence. The CW bandwidth of the Ti:Sapphire laser is <3nm centred at 780nm, which increases to 90 nm centered at 790 nm for ML operation. The ML pulses have duration 10 fs FWHM (measured using a Femtochrome FR-103MN Autocorrelator) which increases to 13 fs after propagation through the Axicon (apex angle 1°).
Fig. 2. Beam profiles modelocked (Bandwidth 90 nm (FWHM) centred around 790nm ΔωFWHM/ω0 = 0.114) and continuous wave (Bandwidth 3 nm (FWHM) centred around 780nm, ΔωFWHM/ω0 = 0.0038.
Figure 2 shows the measured fluence profiles for both CW (blue) and ML (red) operation and the corresponding theoretical results are shown in Fig. 1 (CW dashed line, ML solid line). Here we see that the ML laser shows considerably fewer fringes, around 5, than the CW laser, thereby validating the theoretical prediction that increasing bandwidth leads to a reduction in the number of fringes. (The modulation on the fringes for the CW case is attributed to reflections within the Axicon.) Referring to Table 2 which summarises the measured beam characteristics, we see that there is also good agreement between the predicted and measured range for the CW and ML Bessel beam ranges zmax meaning that the increased bandwidth of the ML laser has not degraded the range of the Bessel beam.
3.2 Super luminescent diode
As an example of a non-laser source we considered an AlGaAs superluminescent diode (SLD) emitting at a wavelength of 830nm. This diode had a spectral width of 14nm (FWHM) resulting in a temporal coherence length of 46μm at an optical output power of 5mW and was used to generate a Bessel beam.
Fig. 3. (a) Measured fluence profiles of generated Bessel beams for propagation distance between 13–21 mm, and (b) the calculated beam profile from the model.
Figure 3(a) shows the propagation of the Bessel beam formed after an Axicon with an apex angle of 5° for a range of propagation distances past the Axicon. The calculated fluence profile is displayed in Fig. 3(b) and matches the observed beam pattern very well for a distance 21 mm. Even though the SLD source is not a laser, the spatial coherence seems large enough to have a propagation distance of about 2 cm. Figure 4 displays the Bessel beam generated from the SLD as captured by a camera. The summary in Table 2 for the SLD confirms that the theoretical and experimental Bessel beam ranges are in excellent agreement.
3.3 Halogen Bulb
We next turn our attention to the creation of a Bessel beam using a standard halogen bulb that has poor spatial and temporal coherence. In order to investigate the influence of the spatial coherence on the beam pattern, different sizes of pinholes were placed in the light path between the light source and the Axicon. As the radius of the central spot of a Bessel beam at frequency detuning Ω is given by r0(Ω)=2.405/kr(Ω), we expect to see a central white spot, to which all frequencies contribute, followed by a series of coloured rings due to the spatially varying combination of the different coloured Bessel beam components. Each of the rings should show a similar spectrum with the red part at the outer radius of each ring. Figure 5(a) displays a picture of a Bessel beam generated from such a halogen bulb with the expected coloured ring structure. For the generation of this beam, the radiation was coupled into a single mode fibre with a core diameter of less than 10 microns to increase the spatial coherence.
In order to investigate on the influence of the spatial coherence on the Beam patterns produced, different sized pinholes were used to spatially filter the light source, the larger the pinhole the larger the effect of spatial incoherence is expected to be. With the smallest pinhole (53 μm) a central maximum and 5 rings are visible, but the first ring is subsumed into the central maximum. Already with a pinhole of 65 μm the central spot and the first two rings are superimposed. With pinholes bigger than 80 μm, the ring structure is washed out and no longer visible, but still a central spot propagating over about 11 mm could be observed (see Table 2). This establishes spatial coherence as a key criterion for the formation and propagation of broadband Bessel beams.
3.4 Supercontinuum source
Finally, we consider a broadband supercontinuum (SC) source as an example of a spatially coherent source but with limited temporal coherence. The SC was generated using a length of photonic crystal fibre, and Fig. 6(a) shows the visible part of the spectrum after being split up by a prism.
Fig. 6. (a) Visible part of the spectrum split by a prism. (b) Spectrum measured with a spectrometer (sensitive in the range 500 – 1100 nm).
The SC beam was used to generate a Bessel beam by propagating the initial Gaussian of spot size 1 mm through an Axicon with an apex angle of 1°. Figure 7(a) shows the resulting fluence profile which exhibits a bright central spot surrounded by coloured rings separated by black rings. We remark that the detector used for this example is most sensitive in the visible range 500 to 800 μm, and Fig. 6(b) shows that in this region the spectrum is dominated by a peak at 580 nm with a width of around 20 nm. The detector bandwidth to the optical frequency is therefore small ≈0.03, and that is why fringes are observed. Using interference filters, each with 40nm bandwidth, allows inspection of the Bessel beam intensity profiles for individual spectral components.
Fig. 7. (a) Picture of the visible part of the Bessel beam generated from a supercontinuum. (b) Model predictions for the spectrum in fig 9a where a dominant peak around λ=580nm with a width of 20 nm is present of the fluence profile (ΔωFWHM/ω0 = 0.0344).
Figure 8 is a display of the spectrally resolved Bessel beam profiles for (a) 500 nm, (b) 600 nm (c) 700 nm, and (d) 850 nm. From Fig. 8 we clearly see the distinct spatial frequencies kr(Ω) for each components, shorter wavelengths having higher spatial frequencies. This leads to the expected natural splitting of the colours as shown in figure 7(a), which is discussed in more detail elsewhere. Furthermore, we point out that the diameter of the central spot of the Bessel beams displayed in figure 8 increases with wavelength, according to z0 = 2.405/kr. However, in the superposition of all the beams displayed in figure 8, we do have a chromatic aberration free white light line in the centre. This verifies the basic theoretical idea employed here that the fluence profiles of broadband Bessel beams can be viewed as spectrally weighted sums of Bessel beam intensity patterns with distinct spatial frequencies.
Fig. 8. Intensity profiles taken with interference filters at (a) 500nm (b) 600nm (c) 700nm (d) 850nm, each with a bandwidth of 40nm. [Media 1]
4. Discussion
Our detailed experimental results show the role of temporal and spatial coherence for the formation of the Bessel light beam. In terms of temporal coherence, the Ti:Sapphire laser at 780nm provides a source that without beam misalignment permits us to switch directly between high temporal coherence (CW) and low temporal coherence (ML) regimes with both having good spatial coherence. Although the results of Fig. 2 show that the ML source displays fewer rings, by virtue of its large bandwidth, the propagation range zmax=115 mm was found to be identical for operating the laser in CW and ML modes, within experimental error. This verifies that temporal coherence is not a strong prerequisite for Bessel beam propagation. Furthermore, inspection of Table 2 shows good agreement between the experimental and theoretical values for zmax for a wide range of sources, reinforcing this point that temporal coherence is not a key limitation to the propagation range of broadband Bessel beams.
The illustration of the relevance of spatial coherence is given by the use of a white light emitting halogen bulb as our input source. This has a measured temporal coherence length of 2.3 microns. This source was directed to the Axicon and the output recorded. When the white light is made spatially coherent by coupling into a single mode fibre, the coloured ring structure becomes very visible as displayed in figure 5. In order to investigate the influence of the degree of spatial coherence on the beam structure, the fibre was removed and the source was apertured using a pinhole of various sizes to control the spatial coherence. This showed a marked difference in the output light field observed from the Axicon. We observed that as we decrease the size of the pinhole (increasing the spatial coherence), the contrast on the outer rings of the Bessel light field increases dramatically: This is an indication that the formation of the central maximum and the appearance of rings within the Bessel beam is a measure of the spatial coherence of the light field.
Our data clearly shows that spatial coherence is a key criterion for Bessel beam propagation. The influence we observe on the number of Bessel beam rings is in accord with the very recent paper of Basano and Ottonello [13], who give a paedagogical discussion of how the incident field coherence impacts the BB profile. Our detailed observations that even spatially incoherent sources, such as the halogen lamp, can produce pseudo-nondiffracting fields is given theoretical support by the work of Kowarz and Agarwal [14] and Shchegrov and Wolf [15] for Gaussian-Schell sources. In particular, Kowarz and Agarwal introduce a Bessel-beam representation for partially coherent sources, and discuss the generation of nondiffracting fields starting from a set of concentric spatial ring-like sources that are mutually incoherent, and Shchegrov and Wolf introduced a class of partially coherent conical waves, and showed that they can propagate over large distances before diffracting. Although these works do not address how the partially coherent incident light is modified by the action of an Axicon, they nonetheless show that partial coherence does not rule out the production of nondiffracting fields, as indeed our observations support.
5. Summary
We have demonstrated that the generation of interference patterns such as Bessel beams with a various range of incoherent light sources is possible, and we have discussed the influence of both spatial and temporal coherence on the pattern of the created beams and compare the experimental results with an analytical model.
Table 2. Overview of the Bessel beams generated from the different light sources, where γ is the apex angle of the Axicon, w0 the incoming beam spot size, exp. zmax and theor. zmax the experimental and theoretical ranges for the Bessel beam, r0 the radius of the central spot, rcritical the critical radius (Eq. 6), and Nfr the number of fringes (Eq. 7).
We find that fringe visibility decreases as the spatial coherence of the source diminishes, and a key requirement for the propagation range of non–diffracting light fields is the need for spatial coherence. Temporally incoherent Bessel beams may find application in high resolution incoherent interferometry [16] and large depth range [17] resolution in optical coherence tomography.
Acknowledgments
P. Fischer acknowledges the Swiss National Funds. This work, as part of the European Science Foundation EUROCORES programme 02-PE-SONS-063-NOMSAN which was supported by funds from the UK Engineering and Physical Sciences Research Council and the EC Sixth Framework programme. We thank V. Garcés-Chávez, C.G. Leburn and D. McGloin from University of St Andrews and Prof. M. Mansuripur of the College of Optical Sciences, University of Arizona for useful discussions. The work of E. M. Wright is sponsored by the U. S. Air Force Office of Scientific Research (AFOSR) under Grant No. F49620-00-1-0312. C. Lopez-Mariscal acknowledges support from Tecnologico de Monterrey Research Chair in Optics and Conacyt Mexico grant 42808.
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