In this paper it is shown how one can use Bessel beams to obtain a stationary localized wave field with high transverse localization, and whose longitudinal intensity pattern can assume any desired shape within a chosen interval 0≤z≤L of the propagation axis. This intensity envelope remains static, i.e., with velocity υ=0; and because of this we call “Frozen Waves” the new solutions to the wave equations (and, in particular, to the Maxwell equations). These solutions can be used in many different and interesting applications, such as optical tweezers, atom guides, optical or acoustic bistouries, various important medical purposes, etc.
©2004 Optical Society of America
Over many years the theory of localized waves (LW), or nondiffracting waves, has been developed, generalized, and experimentally verified in many fields, such as optics, microwaves and acoustics. These waves have the surprising characteristic of resisting the diffraction effects for long distances, i.e., of possessing a large depth of field.
These waves can be divided into two classes: the localized beams and the localized pulses. With regard to the beams, the most popular is the Bessel beam.
Much work have been made about the properties and applications of single Bessel beams. By contrast, only a few papers have been addressed to the properties and applications of superpositions of Bessel beams with the same frequency, but with different longitudinal wave numbers. The few works on this subject have shown some surprising possibilities related with this type of superpositions, mainly the possibility of controlling the transverse shape of the resulting beam[2, 3]. The other important point, i.e., that of controlling the longitudinal shape, has been very rarely analyzed, and the relevant papers have been confined to numerical optimization processes[4, 5], to find out one appropriate computer-generated hologram.
In this work we develop a very simple method** that makes possible the control of the beam intensity longitudinal shape within a chosen interval 0≤z≤L, where z is the propagation axis and L can be much greater than the wavelength λ of the monochromatic light which is being used. Inside such a space interval, we can construct a stationary envelope with many different shapes, including one or more high-intensity peaks (with distances between them much larger than λ). This intensity envelope remains static, i.e., with velocity υ=0; and because of this we call “Frozen Waves” such new solutions to the wave equations (and, in particular, to the Maxwell equations).
We also suggest a simple apparatus capable of generating these stationary fields.
Static wave solutions like these can have many different and interesting applications, as optical tweezers, atom guides, optical or acoustic bistouries, electromagnetic or ultrasound high-intensity fields for various important medical purposes, etc. **
2. The mathematical methodology
We start from the well known axis-symmetric Bessel beam
where ω, kρ and β are the angular frequency, the transverse and the longitudinal wave numbers, respectively. We also impose the conditions
(which imply ω/β≥c) to ensure forward propagation only, as well as a physical behavior of the Bessel function.
Now, let us make a superposition of 2N+1 Bessel beams with the same frequency ω 0, but with different (and still unknown) longitudinal wave numbers βn:
where An are constant coefficients. For each n, the parameters ω 0, kρn and βn must satisfy Eq. (2), and, because of conditions (3), when considering ω 0>0, we must have
Now our goal is to find out the values of the longitudinal wave numbers βn and of the coefficients An in order to reproduce approximately, inside the interval 0≤z≤L (on the axis ρ=0), a chosen longitudinal intensity pattern that we call |F(z)|2. In other words, we want to have
Following Eq. (6), one might be tempted to take βn=2πn/L, thus obtaining a truncated Fourier series, expected to represent the desired pattern F(z). Superpositions of Bessel beams with βn=2πn/L have been actually used in some works to obtain a large set of transverse amplitude profiles[2, 3]. However, for our purposes, this choice is not appropriate due to two principal reasons: 1) It yields negative values for βn (when n<0), which implies backwards propagating components (since ω 0>0); 2) In the cases when L≫λ0, which are of our interest here, the main terms of the series would correspond to very small values of βn, which results in a very short field depth of the corresponding Bessel beams(when generated by finite apertures), impeding the creation of the desired envelopes far form the source.
Therefore, we need to make a better choice for the values of βn, which allows forward propagation components only, and a good depth of field. This problem can be solved by putting
where Q>0 is a value to be chosen (as we shall see) according to the given experimental situation, and the desired degree of transverse field localization. Due to Eq. (5), we get
Inequality (8) determines the maximum value of n, that we call N, once Q, L and ω 0 have been chosen.
As a consequence, for getting a longitudinal intensity pattern approximately equal to the desired one, F(z), in the interval 0≤z≤L, Eq. (4) should be rewritten as:
Obviously, one obtains only an approximation to the desired longitudinal pattern, because the trigonometric series given by Eq. (9) is necessarily truncated. Its total number of terms, let us repeat, will be fixed once the values of Q, L and ω 0 are chosen.
When ρ≠0, the wave field Ψ(ρ, z, t) becomes
The coefficients An will yield the amplitudes and the relative phases of each Bessel beam in the superposition.
Because we are adding together zero order Bessel functions, we can expect a high field concentration around ρ=0.
3. Some examples
In this section we shall present two examples of our methodology.
Let us suppose that we want an optical wave field with λ0=0.632 µm, that is, with ω 0=2.98 1015 Hz), whose longitudinal pattern (along its z-axis) in the range 0≤z≤L is given by the function
where l 1=L/10, l 2=3L/10, l 3=4L/10, l 4=6L/10, l 5=7L/10 and l 6=9L/10. In other words, the desired longitudinal shape, in the range 0≤z≤L, is a parabolic function for l 1≤z≤ l 2, a unitary step function for l 3≤z≤l 4, and again a parabola in the interval l 5≤z≤ l 6, it being zero elsewhere (in the interval 0≤z≤L). In this example, let us put L=0.5m.
We can then calculate the coefficients An, which appear in the superposition Eq. (11), by inserting Eq. (13) into Eq. (10). Let us choose, for instance, Q=0.9998 ω 0/c: This choice allows the maximum value N=158 of n, as one can infer from Eq. (8). Let us specify that, in such a case, one is not obliged to use just N=158, but one can adopt for N any values smaller than it; more generally, any value smaller than that calculated via Eq. (8). Of course, using the maximum value allowed for N, one will get a better result.
In the present case, let us adopt the value N=20. In Fig.1(a) we compare the intensity of the desired longitudinal function F(z) with that of the Frozen Wave (FW), Ψ(ρ=0, z, t), obtained from Eq. (9) by using the mentioned value N=20.
One can verify that a good agreement between the desired longitudinal behavior and our approximate Frozen Wave is already obtained with N=20. Obviously, the use of higher values for N will improve the approximation.
We can expect that, for a desired longitudinal pattern of the field intensity, by choosing smaller values of the parameter Q one will get FWs with higher transverse width (for the same number of terms in the series in Eq. (11)), because of the fact that the Bessel beams in Eq. (11) will possess a larger transverse wave number, and consequently higher transverse concentrations. We can verify this expectation by considering, for instance, a desired longitudinal pattern, in the range 0≤z≤L, given by the function
with l 1=L/2-ΔL and l 2=L/2+ΔL. Such a function has a parabolic shape, with the peak centered at L/2 and a width of 2ΔL. By adopting λ0=0.632 µm (that is, ω 0=2.98 1015 Hz), let us use the superposition Eq. (11) with two different values of Q: we shall obtain two different FWs that, in spite of having the same longitudinal intensity pattern, will have different transverse localizations. Namely, let us consider L=0.5m and ΔL=L/50, and the two values Q=0.99996 ω 0/c and Q=0.99980 ω 0/c. In both cases the coefficients An will be the same, calculated from Eq. (10), on using this time the value N=30 in the superposition Eq. (11). The results are shown in Figs. (2a) and (2b). One can observe that both FWs have the (same) longitudinal intensity pattern, but the one with the smaller Q is endowed with the higher transverse localization.
4. Generation of Frozen Waves
Concerning the generation of Frozen Waves, we have to recall that the superpositions given by Eq. (11), which define them, consists of sums of Bessel beams. Let us also recall that a Bessel beam, when generated by finite apertures (as it must be, in any real situations), maintains its nondiffracting properties till a certain distance only (its field depth), given by
where R is the aperture radius and θ is the so-called axicon angle, related with the longitudinal wave number by the known expression cos θ=cβ/ω.
So, given an apparatus whatsoever capable of generating a single (truncated) Bessel beam, we can use an array of such apparatuses to generate a sum of them, with the appropriate longitudinal wave numbers and amplitudes/phases (as required by Eq. (11)), thus producing the desired FW. Here, it is worthwhile to notice that we shall be able to generate the desired FW in the the range 0≤z≤L if all Bessel beams entering the superposition Eq. (11) are able to reach the distance L resisting the diffraction effects. We can guarantee this if L≤Z min, where Z min is the field depth of the Bessel beam with the smallest longitudinal wave number βn =-N=Q-2πN/L, that is, with the shortest depth of field. In such a way, once we have the values of L, ω 0, Q, N, from Eq. (15) and the above considerations it results that the radius R of the finite aperture has to be
The simplest apparatus capable of generating a Bessel beam is that adopted by Durnin et al., which consists in an annular slit located at the focus of a convergent lens and illuminated by a cw laser. Then, an array of such annular rings, with the appropriate radii and transfer functions able to yield both the correct longitudinal wave numbers* and the coefficients An of the fundamental superposition Eq. (11), can generate the desired FW. These questions will be analyzed in more detail elsewhere.
Obviously, other powerful tools, like the computer generated holograms (ROACH’s approach, for instance), may be used to generated our FWs.
In this work we have shown how Bessel beams can be used to obtain stationary localized wave fields, with high transverse localization, whose longitudinal intensity pattern can assume any desired shape within a chosen space interval 0≤z≤L. The produced envelope remains static, i.e., with velocity v=0, and because of this we have called Frozen Waves such news solutions.
The present results can find applications in many fields: For instance, in the optical tweezers modelling, since we can construct stationary optical fields with a great variety of shapes, capable, e.g., of trapping particles or tiny objects at different locations. This topic is being studied and will be reported elsewhere.
The author is very grateful to Erasmo Recami, Hugo E. Hernández-Figueroa, C. A. Dartora, Marco Mattiuzi and V. Abate for continuous discussions and collaboration. This work was supported by FAPESP (Brazil).
|*||Once a value for Q has been chosen.|
References and links
1. For a review, see: E. Recami, M. Zamboni-Rached, K.Z. Nóbrega, C.A. Dartora, and H.E. Hernández-Figueroa, “On the localized superluminal solutions to the Maxwell equations,” IEEE Journal of Selected Topics in Quantum Electronics 9, 59–73 (2003); and references therein. [CrossRef]
2. Z. Bouchal and J. Wagner, “Self-reconstruction effect in free propagation wavefield,” Opt. Commun. 176, 299–307 (2000). [CrossRef]
3. Z. Bouchal, “Controlled spatial shaping of nondiffracting patterns and arrays,” Opt. Lett. 27, 1376–1378 (2002). [CrossRef]
5. R. Piestun, B. Spektor, and J. Shamir, “Unconventional light distributions in three-dimensional domains,” J. Mod. Opt. 43, 1495–1507 (1996). [CrossRef]