The Bessel beam can be considered in terms of an angular spectrum of plane waves. Because of ... annular mask as the pupil of a lens, or using a phase mask.
Sheppard CJR (2013) Localized beams and localized pulses: Generation using the angular spectrum, in Non-Diffracting Waves, E. Hernandez-Figueroa, E Recami, M Zamboni-Rached, eds. Wiley-VCH, Berlin, pp. 365-380. ISBN-10: 352741195X, ISBN-13: 978-3527411955
Localized beams and localized pulses: Generation using the angular spectrum
CJR Sheppard Abstract The Bessel beam can be considered in terms of an angular spectrum of plane waves. Because of linearity and superposition, as each plane wave component is a solution of the Helmholtz equation (or of Maxwell’s equations for the electromagnetic case), Bessel beams are also rigorous solutions. Pulsed beams can be generated by coherent superposition of Bessel beams over a spectral range with a specified spectral distribution. Assumption of different relationships between the propagation angle and wavelength leads to different types of pulse. These can be compared using the concept of the three-‐ dimensional (generalized) pupil. Other beams, such as Gaussian beams and their nonparaxial generalizations, can be generated by superposition of Bessel beams. Applications in microscopy and tomography are discussed. 1. Bessel beams The Bessel beam has a long history, dating back to Airy [1] and Rayleigh [2]. Airy calculated numerically the field in a focal cross section of a point object for a circular pupil, and describes how for a narrow annular pupil the dark rings are of smaller radius and the side-‐lobes stronger (Fig. 1(a)). Rayleigh proposed a narrow annulus as a way to decrease the intensity in the image of a bright object (the sun) while maintaining good spatial resolution. He explicitly gives the image amplitude as the Bessel function J 0 , and mentions the reduction in the effects of spherical aberration. Neither Airy or Rayleigh mentioned defocus effects, but Steward [3] and later Steel [4], gave detailed discussions of the effects of defocus and aberrations, and showed that depth of focus increased as the width of the annulus reduced. Stratton showed that a propagation invariant Bessel solution of the Helmholtz equation or Maxwell’s equations holds for systems of cylindrical symmetry, such as waveguides [5]. Linfoot and Wolf presented the intensity in the focal region for an annular lens [6]. Welford noted that the side-‐lobes were too strong to image extended objects [7], and this was later demonstrated experimentally [8] .
Fig. 1. Meanwhile, McLeod described the axicon, a conical prism that also produces a beam that is approximately propagation invariant [9]. Fujiwara showed that the axicon produces an approximation to a Bessel beam [10]. Kelly pointed out the similarity between axicons and annular masks (Fig.1(b)) [11]. Dyson proposed a diffractive axicon, similar to a zone plate but with equal width zones [12] (Fig.1(c)). He also showed that spiral zone plates can be used to generate higher order Bessel beams. Bessel beams, or approximations to them, can thus be formed using an annular mask as the pupil of a lens, or using a phase mask. Steel showed that a particular example of a phase mask is a lens with spherical aberration, which produces approximately an axicon behaviour over a zone of the lens [13]. In the mid-‐1970s we started development of a confocal microscope at Oxford University. We investigated theoretically and experimentally the use of Bessel beams in the confocal microscope. In a confocal microscope, the effective point spread function of the system is equal to the product of the point spread functions for the illuminating and detection lenses [14]. Rayleigh had pointed out that the maxima in the side-‐lobes for a thin annular pupil coincide with the zeros of a plain circular aperture [2], so we realized that a confocal microscope with one annular pupil and one circular pupil should give a good, improved resolution, image of an extended object [8, 14-‐17]. In order to investigate how the cross-‐section of a Bessel beam transforms with defocus from a Bessel beam to an annulus, we considered the case of a pupil that is an annulus convolved with a Gaussian beam [18], which is now known as a Bessel-‐Gauss(ian) beam [19]. The field was expanded in Laguerre-‐Gaussian (LG) beams, and summed to give an analytic expression for the field at any point in space. A narrow annulus was termed a δ ring. The paper states: “The radial distribution for a δ ring is given by a zero-order Bessel function in any plane … perpendicular to the optic axis. That this is so is not surprising because such a wave is the circularly symmetric mode of free space. We are acquainted with modes of this form in circular waveguides, and we can consider free space as the
limiting case of a waveguide of very large diameter. Such an overmoded waveguide has an infinity of circularly symmetric modes, that is the scale of the Bessel functions may be chosen at will. A wave with zero-order Bessel-function radial distribution propagates without change.” The propagation invariance of Bessel beams is thus clearly stated in this paper. We also investigated the properties of electromagnetic Bessel beams, generated by illumination of a narrow annulus by a plane-‐polarized wave [20]. It was shown that as the numerical aperture of the system increases, the relative strength of a longitudinal field component increases, so that by an angular semi-‐ aperture α = 60 the central spot has split into two. For the limiting case α → 180 (which could be achieved in practice using a mirror rather than a lens), the intensity varies as J 22 , and the beam exhibits a central dark core. In 1987, the nondiffracting (or diffraction-‐free) beam was proposed [21, 22]. Actually, these names are not strictly accurate and are misleading, as of course diffraction always occurs, except for an infinite plane wave. What actually happens is that dynamical equilibrium is maintained, so that the diffraction outwards from the central lobe is exactly cancelled by the inward diffraction from the strong side-‐lobes. According to Durnin [21]: “Only 5% of the total energy of the J 0 beam is initially contained within the central maximum, yet this is sufficient to create a sharply defined central spot with an unchanging 200 µm diameter over a distance of approximately 1 m.” This statement suggests an incorrect interpretation of the propagation of a Bessel beam. It implies that the central lobe travels along the axis, whereas in fact the central spot is continuously refreshed from the side-‐lobes. The so-‐called self-‐healing, or self-‐reconstructing, property of Bessel beams stems directly from this mechanism, as does also the superluminality of Bessel pulses described later. We mentioned that Bessel-‐like beams can be formed using a phase mask. Binary phase masks consisting of an array of rings used as pupil filters can be designed to produce a flat axial intensity over a limited range [23-‐29]. These have been termed maximally-‐flat filters. In [29], for a filter with 5 elements the axial intensity is very flat over a range 7.8 times the depth of focus of a circular pupil. The Strehl ratio is 0.133, as compared with 0.016 for an annular pupil of the same axial uniformity. So phase masks are an efficient approach for generating Bessel beams over a limited range. 3. The Bessel-Gauss beam In [18], the amplitude at any point of a paraxial zero-‐order Bessel-‐Gauss (BG) beam was written in the form, with the exponentials separated into modulus and phase components, ⎛ v 2 a 2 cos 2 ζ ⎞ U(v,ζ ) = cosζ J 0 (v cosζ e−iζ )exp ⎜ − ⎟⎠ 2 ⎝ (1) ⎛ sin 2 ζ ⎞ i(kz − ζ ) ⎡ (1 − a 4 v 2 )sin 2ζ ⎤ × exp ⎢ −i , ⎥ exp ⎜ − 2a 2 ⎟ e 4a 2 ⎝ ⎠ ⎦ ⎣
where the normalized radial coordinate v, the Gaussian beam waist w0 , and ζ are defined by 2ρ 2 λz (2) v = k ρ sin α = , w0 = , tanζ = = ua 2 . 2 w0 a ak sin α π w0 Here, k = 2π / λ , α is the semi-‐angular aperture, ρ and z are cylindrical coordinates, and a is a small parameter specifying the relative widths of the Gaussian and Bessel function. v and u are the transverse and axial optical coordinates defined by Born and Wolf [30], and w0 is the waist of the Gaussian. In this Chapter, we restrict our attention mainly to the case of rotationaly symmetric beams. The amplitude can also be written in a form with complex argument for the exponential: ⎛ ⎞ ⎡ v 2 a 4 + i tan ζ ⎤ ikz 1 v (3) U(v,ζ ) = J0 ⎜ exp ⎥e . ⎢− 2 (1 + i tanζ ) ⎝ 1 + i tanζ ⎟⎠ ⎣ 2a (1 + i tanζ ) ⎦ In terms of the axial optical coordinate u, we then have ⎡ v 2 a 2 − iu ⎤ ikz 1 v ⎞ ⎛ (4) e . U(v,u) = J exp ⎜ ⎟ 0 ⎢− 2 ⎥ (1 + iua 2 ) ⎝ 1 + iua 2 ⎠ ⎣ 2(1 + iua ) ⎦ In the near-‐field, tanζ 1 , and ⎛ v2a2 ⎞ u⎞⎤ ⎡ ⎛ (5) U(v,u) ≈ J 0 ( v ) exp ⎜ − exp ⎢ i ⎜ kz − ⎟ ⎥ , ⎟ 2 ⎠ ⎝ 2⎠ ⎦ ⎣ ⎝ i.e. it behaves as a Bessel beam multiplied by a Gaussian. Using an asymptotic expression given by Porras et al. [31], this can be written for small a, u⎞⎤ ⎡ ⎛ (6) U(v,u) ≈ L1/ 4 a ( v 2 a 2 ) exp ( −v 2 a 2 ) exp ⎢ i ⎜ kz − ⎟ ⎥ , 2⎠ ⎦ ⎣ ⎝ which establishes the equivalence with the circularly-‐symmetric elegant LG beam of order 1 / 4a 2 [31, 32]. 2
In the far-‐field, tanζ 1 , and ⎡ 1 ⎛v ⎞ ⎤ ⎛ iv 2 ⎞ ikz i ⎧ ⎛ v ⎞ ⎛ v ⎞⎫ (7) U(v,u) = − 2 ⎨ I 0 ⎜ 2 ⎟ exp ⎜ − 2 ⎟ ⎬ exp ⎢ − 2 ⎜ − 1⎟ ⎥ exp ⎜ ⎟ e . ⎝ ua ⎠ ⎭ ua ⎩ ⎝ ua ⎠ ⎝ 2u ⎠ ⎢⎣ 2a ⎝ u ⎠ ⎥⎦ The Gaussian is only appreciable when v ≈ u , and then the expression in curly brackets is approximately unity. So the amplitude has the form of a ring with approximately Gaussian cross-‐section. The phase variation is the same as for a conventional Gaussian beam. In fact the amplitude in the far-‐field is the convolution of a Gaussian beam with an annulus. In [18] it was shown that the annular nature of the BG beam is well-‐ developed by a distance from the waist tanζ = 1 , i.e. in the mid-‐field region. If 2
v cosζ 1 , i.e. if v 2 (1 + u 2 a 4 ) , but also the more restrictive condition vua 2 (1 + u 2 a 4 ) is satisfied, we can use the asymptotic expression for the Bessel function J 0 , i(1 + iua 2 ) ⎛ v ⎞ ⎛ iv ⎞ J0 ⎜ exp ⎜ ≈ − , 2⎟ ⎝ 1 + iua ⎠ ⎝ 1 + iua 2 ⎟⎠ 2π v
to give
(8)
2 ⎧ 2 2 ⎡⎛ v ⎞ ⎤2 ⎫ ⎡ 4⎛ 1 ⎞ ⎤ v + iua ⎪ u a ⎢ ⎜ ⎟ − 1⎥ ⎪ ⎢ ⎜⎝ 4⎟ ⎥ ⎠ −i i ⎞ ua ⎛ ⎣ ⎝ u ⎠ ⎦ ⎪ eikz . ⎥ exp ⎪⎨− U(v,u) ≈ exp ⎜ − exp ⎢ ⎬ 2 4⎟ 2 4 ⎝ 2ua ⎠ ⎢ 2(1 + u a ) ⎥ 2π v(1 + iua ) 2(1 + u 2 a 4 ) ⎪ ⎪ ⎢ ⎥ ⎪⎩ ⎪⎭ ⎣ ⎦ (9) From this expression we can see that in the region where the asymptotic approximation to the Bessel function is valid, the modulus exhibits a peak value around v ≈ u , the breadth of the peak increasing as we move into the mid-‐field region from the far-‐field region. The phase is parabolic, but centred about v = −1 / ua 4 , i.e. it is approximately constant on a parabolic toroidal surface. The transverse phase gradient is
∂Φ(v,ζ ) 1 + vua 4 = . 1 + u 2 a4 ∂v If v 2 / ua 4 , the parabolic term can be neglected, and
(10)
⎧ 2 2 ⎡⎛ v ⎞ ⎤ 2 ⎫ ⎪ u a ⎢⎜ ⎟ − 1⎥ ⎪ i ⎡ i(v − u / 2) ⎤ ⎪ ⎣⎝ u ⎠ ⎦ ⎪ eikz . (11) exp ⎨− U(v,u) ≈ − exp ⎢ ⎬ 2 2 4 ⎥ 2 4 2π v(1 + iua ) 2(1 + u a ) ⎣ (1 + u a ) ⎦ ⎪ ⎪ ⎪⎩ ⎪⎭ The combined conditions 2 / ua 4 v (1 + u 2 a 4 ) / ua 2 show that the phase is
linear as long as u 2 / a 2 , or ζ π / 3 . So the phase is linear in the mid-‐field region, which explains the similarity between a BG beam and an axicon. The near field has been found numerically to set in at about u ≈ 1.67a −1.13 , in the range 0.01 < a < 0.2 . The Bessel-‐Gauss beam can be generated using a lens illuminated by the dual of a BG beam, that we have called dBG for short [33]. A dBG beam has a waist that is the convolution of a Gaussian with an annulus. Then if a dBG is placed with its waist in the front-‐focal plane of a lens, a BG is formed with its waist in the back focal plane. The parameter a is conserved upon focusing. Then if the value u f of u corresponding to the focal length f of the lens satisfies the condition u f a 2 1 , a BG beam is formed with a depth of focus large compared with f. At the lens
(
)
I(v) ≈ J 02 ( v ) exp −2v 2 a 2 ,
(12)
so the radius of the lens must be larger than about v ≈ 2 / a in order not to truncate the beam appreciably. 4. Pulsed Bessel beams A pulsed Bessel beam can be generated by integrating Bessel beams of different frequency over an assumed spectral distribution. As each component Bessel beam is an exact solution of the Helmholtz equation (or Maxwell’s equations for the electromagnetic case), the pulsed beam is also an exact solution. A monochromatic Bessel beam can be considered as an in-‐phase sum of plane wave components incident in different directions on the surface of a cone: ∞
U(r, z,t) =
∫ f (k) J 0
0
(kr sinθ )exp(ikz cosθ )exp(−ikct)dk .
(13)
The angle θ of the cone is in general a function of the frequency. Different choices for this functional relationship give different types of pulsed beam [34]. Note that the wave number k is positive semi-‐definite. Four main types appear in the literature. Type 1. The Bessel beams all have the same width, so k sinθ = kρ is a constant [35, 36]. This type of pulse is produced by a diffractive axicon. Then ∞
U( ρ, z,t) = J 0 (kρ ρ ) ∫ f (k) exp(ikz cosθ )exp(−ikct)dk .
(14)
0
or ∞
U( ρ, z,t) = J 0 (kρ ρ ) ∫ f (k) exp ⎡⎢ i ⎣ 0
(
)
k 2 − kρ2 z − kct ⎤⎥ dk . ⎦
(15)
The Bessel structure is seen to be independent of time. Gaussian beams of this type have also been investigated [37]. Type 2. The Bessel beams have constant θ = α , so that cosθ = cos α = 1 / β . This corresponds to the X-‐wave [38]. The group and phase velocities of X-‐waves are axicon, if material both β c . This type of pulse is produced by a refractive dispersion can be neglected. Then ∞ ⎛ kρ ⎞ (16) U[ ρ, t ′ = t − z / β c] = ∫ f (k) J 0 ⎜ exp(−ikct ′ )dk , ⎝ Γβ ⎟⎠ 0 where Γ = 1 / β 2 − 1 . For the extreme case when f (k) = 1 [38], β . U( ρ,t) = 2 2 ρ / Γ − (z − β ct)2
(17)
The observed field is the real part of this expression. Generalizing to the case when f (k) = exp[−(k − k0 )z0 ] , k > k0 ≥ 0 , we have β exp[ik0 (z − β ct) / β ]exp(−k0 z0 ) . (18) U( ρ,t) = ρ 2 / Γ 2 − [(z + iβ z0 ) − β ct]2 Gaussian beams of this type have also been investigated [39]. Type 3. The Bessel beams are related to each other, so that they travel in step at the speed of light, a property sometimes called isodiffracting [40]. This type of pulse is a generalization of the original focus wave mode (FWM) [41-‐43]. Eq.12 can be written U( ρ, z, t ′) =
∞
∫ f (k) J
0
(k ρ sinθ )exp ⎡⎣ −ikz (1 − cosθ ) ⎤⎦ exp(−ikct ′)dk , (19)
0
where the local time t ′ = t − z / c has been introduced. The Type 3 pulse requires that k(1 − cosθ ) = kc , a constant [44]. Then exp(−ikc z) can be taken outside of the integral to give
U( ρ, z, t ′) = exp ( −ikc z )
∞
∫
kc /2
f (k) J 0 (k ρ sin θ )exp(−ikct ′ )dk . (20)
We have k sinθ = [kc (2k − kc )]1/2 , k > kc / 2 [45], which provides the lower limit of integration. In the paraxial approximation, k sinθ = (2kc k)1/2 [44], i.e. 2k kc . On the axis, ∞
U(0, z, t ′) = exp ( −ikc z ) ∫ f (k) exp(−ikct ′ )dk ,
(21)
0
so that the temporal and spatial variations have been separated. Gaussian beams of this type have also been investigated [34, 46]. This type of pulse can be produced using a combination of diffractive and refractive axicons [47]. It is interesting to note that Durnin mentions [21] that he was investigating pulsed beams with spectral components “all having the same α”, i.e. X-‐waves rather than FWM. Type 4. The Bessel beams travel in step at a speed different from the speed of light [48, 49]. These have been termed generalized focus wave modes (GFWM). Then k(1 / β − cos α ) = kc , where β is the ratio of the speed to the speed of light, and Eqs. 17,18 still hold, where now t ′ = t − z / β c , and
⎛ k2 ⎞ 2k k k sinθ = ⎜ − 2 2 + c − kc2 ⎟ β ⎝ γ β ⎠
1/ 2
,
(22)
where γ = 1 / 1 − β 2 . For the paraxial case for β > 1 , k sinθ ≈ k / Γβ , i.e. it reduces to the X-‐wave solution (Type 2) with cos α = 1 / β . Durnin et al. [22] mentioned that their Bessel beam solution is not the packet-‐like solution of Brittingham [41]. But actually the FWM is a pulsed Bessel beam, although this is perhaps not obvious from the Gaussian form of the equations in Brittingham’s paper. The connection comes from an integral relationship of the form ∞ 1 (23) Ln ( ρ 2 )exp(− ρ 2 ) = ∫ e− t t n J 0 (2 ρ t )dt , n! 0 where Ln is a Laguerre polynomial, which shows that integrating over Bessel beams with an appropriate spectral distribution can results in a LG, or Gaussian for the case when n = 0 . An important observation is that the strong side-‐lobes of the Bessel beam can tend to cancel out upon integration. Fig. 2. shows the intensity at the centre of a pulse for four different cases, FWM and X-‐waves, and for similar incoherent integration over the corresponding spectral distributions. For the pulsed cases, because the Bessel function can go negative, cancellation can occur, while for the incoherent summation only averaging of the side-‐lobes occurs. The (un-‐normalized) spectral distribution is taken as (24) f (k) = exp(−kz0 )(k − k0 )s , k > k0 , with s = 1 , 2k0 = kc and k0 z0 = 0.1 , which are chosen to avoid significant backward propagating components, and also negative values of frequency. For s large, the spectral distribution tends to a Gaussian. For the X-‐waves, α is taken as 20 .
Fig. 2. Putting the spectral distribution of Eq. 24 into Eq. 21, we obtain for 2k0 ≥ kc , exp(−ikc z) , (25) U(0, z, t ′) ∝ [(z + iz0 ) − ct]s which represents an envelope moving in the positive z direction at the speed of light, with fringes traveling in the negative z direction. Putting the spectral distribution into Eq. 20, and using the integral in Eq. 23, the amplitude at any point for the case when 2k0 = kc is ⎛ kc ρ 2 ⎞ Γ(s + 1)exp(−k0 z0 )exp[−i(kc z − k0 ct ′ )] ⎛ kc ρ 2 ⎞ , exp ⎜ − U( ρ, z, t ′) = Ln ⎜ (z0 + ict ′ )s ⎝ z0 + ict ′ ⎟⎠ ⎝ z0 + ict ′ ⎟⎠
(26) i.e. it is equivalent to an elegant (complex argument) LG pulse [42, 44, 50, 51]. As it is known that an elegant LG beam is generated by sums of complex source-‐sink multipole pairs [51-‐54], so the FWM solutions can be generated by such source-‐ sink pairs traveling along the axis [48]. Note that a source-‐sink pairs are necessary, rather than sources alone, to avoid nonphysical singularities [54, 55]. The scalar form of the original FWM results if we take the parameter s as zero. Then the field can be shown to be equivalent to that of a simple complex source-‐ sink pair traveling along the axis at the speed of light. When looking into the history of the FWM, many papers have been concerned with their physical realizability [56-‐58]. The original FWM consisted of forward and backward propagating components, which although they are physically realizable (e.g. as in 4Pi microscopy [59, 60]), can be a problem for many applications. However, choice of an appropriate spectral distribution can eliminate the backward propagating components completely [48, 61]. An instructive approach to appreciate the different types of pulsed beam is using the three-‐dimensional (3D) spatial frequency spectrum. Monochromatic waves can be represented in k -‐space, each point on a sphere radius
2π / λ representing a plane wave traveling in a different direction. This is called the generalized pupil [62]. The amplitude at any point in space is then given by a 3D Fourier transform of the 3D spatial frequency spectrum. A monochromatic Bessel beam is represented by a circle on the k-‐space sphere. This approach can be generalized to the different cases of pulsed beam. A Type 1 pulsed Bessel beam has a constant transverse k component, and so must lie on the surface of a cylinder, radius kρ in k-‐space [63], as shown in Fig. 3. A Type 2 pulsed Bessel
beam (X-‐wave) has constant θ = α , and hence lies on the surface of a cone with its vertex at the origin. The k-‐vector of a Type 3 pulsed Bessel beam (FWM) lies on the surface of a paraboloid of revolution with its focus at the origin. In general both forward and backward components are thus allowed, but as the spectral components are independent, the spectral distribution can be chosen at will to avoid completely backward propagating waves. Alternatively, forward propagating waves can be eliminated [61]. Bélanger [42] showed that the FWM is a special case of a more general solution (our Type 3 beam) that can be written as the product of any solution of the paraxial wave equation with a backward propagating plane wave. This can be shown very simply from Eq.20, which can be written in the form ∞ ⎛ ikρ2 ct ′ ⎞ ⎡ ik ⎤ U( ρ, z, t ′) = exp ⎢ − c (z + ct) ⎥ ∫ g(kρ ) J 0 (kρ ρ )exp ⎜ − ⎟ kρ dk ρ , (27) ⎣ 2 ⎦0 ⎝ 2kc ⎠ where 2 2 1 ⎛ k ρ + kc ⎞ , (28) g(kρ ) = f kc ⎜⎝ 2kc ⎟⎠ and the integral in Eq.27 can be recognized as the usual form for the Debye theory for paraxial focusing [64]. Different solutions of the paraxial wave equation can be used to generate pulses with different spectral distributions. The spectral distribution of Eq. 23 results in elegant LG (eLG) pulses. Similarly we can generate the dual of the elegant LG (deLG) [33] pulses using a spectral distribution Ln [(k − kc / 2)z0 ]exp(−kz0 ) . Conventional LG pulses require a spectral distribution Ln [(2k − kc )z0 ]exp(−kz0 ) . BG pulses [65, 66] need a spectral distribution I 0 [(k − kc / 2)z0 / 2a]exp(−kz0 ) . Lommel pulses arise from f (k) = exp[−(k − k1 )z0 ] , k2 > k > k1 , where k1 , k2 are constants [61]. It is known that solutions of the paraxial wave equation can be based on any confluent hypergeometric function [33, 67, 68]. In k-‐space, solutions of the paraxial wave equation lie on the surface of a paraboloid, and the plane wave gives a shift in the negative kz direction. Different solutions can be generated by choice of different spectral distributions, equivalent to different angular spectra. We find that [44] k 4k dk θ θ (29) = − c cot csc 2 = −k −1 . kc dθ 4 2 2 The spectral distribution in Eq. 23 for 2k0 = kc is then 1⎛k ⎞ f (k)dk = − ⎜ c ⎟ 4⎝ 2 ⎠
s +1
cot 2 s
θ θ θ⎞ ⎛ k csc 4 exp ⎜ − c csc2 ⎟ sin θ dθ . ⎝ 2 2 2 2⎠
(28)
For a Type 4 pulsed beam, kz = k / β − kc / 2 , and the k-‐vector lies on the surface of a surface of revolution of a conic section, a prolate spheroid for β < 1 (subluminal), and one branch of a hyperboloid of two sheets for β > 1 (superluminal) [48]. The prolate spheroid solution requires that there is a maximum as well as a minimum value of k. For pulses of Type 4 and β < 1 , from Eq.20, completing the square in the argument of the Bessel function, and performing the integral over k 1 (31) p= − , 2 2 kcγ β β we have ⎡ ⎛ ct ⎞ ⎤ U( ρ, z, t ′) = kcγ 2 β 2 exp ⎢ikc Γ 2 β 2 ⎜ z − ⎟ ⎥ β ⎠⎦ ⎝ ⎣ (32) 1
(
)
(
)
× ∫ f (k) J 0 kcγ β ρ 1 − p 2 exp −ikcγ 2 β 2 ct ′p dp. −1
As for Type 3 pulses, the amplitude is the product of two terms, one of which is a plane wave, this time traveling at a speed c / β . Putting p = cos χ , U( ρ, z, t ′) = −kcγ 2 β 2 exp ⎡⎣ ikcγ 2 β ( β z − ct ) ⎤⎦ π
(
(33)
)
× ∫ f (k) J 0 ( kcγ β ρ sin χ ) exp −ikcγ 2 β 2 ct ′ cos χ sin χ dχ . 0
We recognize that Eq.34 is of the same form as the scalar version of the Richards & Wolf diffraction integral for nonparaxial focusing [69]. For some spectral distributions it can be evaluated analytically using an integral given by Ref.([70], p. 1467) for spherical harmonics: ⎛ ⎞ z 2 2 (cos χ ) J m ( k ρ sin χ ) exp ( ikz cos χ ) sin χ dχ = 2i n − m Pnm ⎜ ⎟ jn k ρ + z , 2 2 ⎝ ρ +z ⎠ 0 (34) m where Pn is an associated Legendre polynomial, and jn is a spherical Bessel
(
π
∫P
m n
)
function. The function Pn0 (cos χ ) corresponds to a spectral distribution Pn (k / kcγ 2 β 2 − 1 / β ) , and we obtain
⎡ ⎛ ct ⎞ ⎤ ⎛ z′ ⎞ Un ( ρ, z′ ) = −2i n kcγ 2 β 2 exp −kcγ 2 β z0 exp ⎢ ikcγ 2 β 2 ⎜ z − ⎟ ⎥ Pn ⎜ ⎟ jn kcγ 2 β R′ , ⎝ β ⎠ ⎦ ⎝ R′ ⎠ ⎣ (35) where
(
)
(
)
(36) z′ = (z + iβ z0 ) − β ct , R′ = ρ 2 / γ 2 + z′ 2 . Here we have introduced a complex displacement z0 to include traveling complex source-‐sink multipole solutions [48]. Similarly for β > 1 , introducing the variable k 1 (37) p′ = + , 2 2 kc Γ β β we have
U( ρ, z, t ′) = exp ⎡⎣ −ikc Γ 2 β ( β z − ct ) ⎤⎦
(
∞
)
(
(38)
)
× ∫ k1′ f (k) J 0 kc Γβ ρ p′ 2 − 1 exp −ikc Γ 2 β 2 ct ′p′ dp′, 1
and putting p′ = cosh χ ′ ,
U( ρ, z, t ′) = kc Γ 2 β 2 exp ⎡⎣ −ikc Γ 2 β ( β z − ct ) ⎤⎦ ∞
(
)
× ∫ f (k) J 0 ( kc Γβ ρ sinh χ ′ ) exp −ikc Γ 2 β 2 ct ′ cosh χ ′ sinh χ ′dχ ′.
(39)
0
A particular solution can be obtained by taking f (k) = exp{−[k − β kc / (β + 1)]z0 } , k > β kc / (β + 1) , which gives from Eq.40 U( ρ, z,t) =
(
{
β exp −Γ 2 β kc i[ β (z + iz0 ) − ct] + ρ 2 / Γ 2 − [(z + iβ z0 ) − β ct] ρ 2 / Γ 2 − [(z + iβ z0 ) − β ct]
}) .
(41) It can be seen that if kc = 0 , the solution becomes identical to that for the Type 2 (X wave) in Eq.18 with k0 = 0 .
Fig. 3. It is straightforward to calculate the dispersion diagram, a plot of k = ω / c versus kz , from the 3D spatial frequency spectrum (Fig. 4.). It is seen that the group velocity, given by c(∂k / ∂k z ) , is independent of frequency for pulses of Types 2, 3 or 4. Type 2 pulses (X-‐waves) are superluminal, and also have a phase velocity that is independent of frequency. For Types 3, 4, the phase velocity is different from the group velocity, so that the pulse exhibits fringes that move through the envelope. Components that exhibit negative phase velocity, but positive group velocity, are allowed. For Type 4, β < 1 , the magnitude of the phase velocity is greater than c. For Type 1 pulses, the group and phase velocities have the same sign.
Fig. 4. 5. Applications in biomedical imaging Localized waves have numerous potential applications, and of these the applications in biological and medical imaging are important examples. We mentioned that Bessel beams can in principle result in improved spatial resolution compared using a circular pupil of the same dimension [8, 14]. However, cross-‐polarization effects rule out use of plane-‐polarized illumination for high numerical aperture systems [20, 71]. This problem can be overcome by illumination with other polarization distributions such as radially-‐polarized light [72]. Radially-‐polarized Bessel beams were, in fact, proposed many years ago for particle acceleration [73]. They produce a longitudinal, on-‐axis electric field. Another option is TE polarized illumination [74], which produces a concentrated transverse electric field. The original FWM was TE polarized [41]. Bessel beams have been used successfully for illumination in optical coherence tomography (OCT) [75, 76]. Usually, OCT employs lenses of low numerical aperture, so that polarization effects are unimportant. Further, the axial resolution in OCT results from the finite coherence length of the source, so that good 3D imaging results even with low numerical aperture, unlike in confocal microscopy where high numerical aperture is necessary. Using a Bessel beam allows a higher numerical aperture to be used to improve spatial resolution, while maintaining focusing depth. Binary phase masks have also been used for similar applications [77, 78]. Bessel beams also have the advantage of the ‘self-‐healing’ effect in scattering media, allowing deep penetration into biological tissue [79]. References 1. G. B. Airy, "The diffraction of an annular aperture," Phil. Mag. Ser. 3 18, 1-‐ 10 (1841). 2. J. W. S. Rayleigh, "On the diffraction of object glasses," M. Notes of the R. Astron. Soc. 33, 59-‐63 (1872). 3. G. C. Steward, "IV Aberration diffraction effects," Phil. Trans. Royal Soc. London, Ser. A 225, 131-‐198 (1926). 4. W. H. Steel, "Etude des effets combinés des aberrations et d'une obturation centrale de la pupille sur le contraste des images optiques," Revue d'Optique 32, 4, 143, 269 (1953). 5. J. Stratton, Electromagnetic Theory (McGraw-‐Hill, New York, 1941). 6. E. H. Linfoot and E. Wolf, "Diffraction images in systems with an annular aperture," Proc. Phys. Soc. B 66, 145-‐149 (1953). 7. W. T. Welford, "Use of annular apertures to increase focal depth," J. Opt. Soc. Am. 50, 749-‐753 (1960).
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Figure Captions Fig. 1. Bessel-‐like beams can be produced using (a) a narrow annular pupil, (b) an axicon, or (c) a diffractive axicon. In all cases the rays in the image space all travel in the same direction relative to the axis. Fig. 2. The intensity in the central plane of a Bessel pulse for four different cases; (a) FWM, (b) incoherent superposition of FWMs, and (c) X-‐waves, and (d) incoherent superposition of X-‐waves. Parameters are s = 1 , 2k0 = kc ,
k0 z0 = 0.1 , α = 20 . The cross-‐section of a monochromatic Bessel beam is shown as a dashed line. Fig. 3. Different types of pulsed Bessel beam shown in a 3D Fourier representation. Type 1 corresponds to identical Bessel function for all spectral components. Type 2 is X-‐wave. Type 4 with β = 1 reduces to Type 3 (FWM). Type 4 with β = 2 is superluminal. Type 4 with β = 1 / 2 is subluminal. Fig. 4. The dispersion curves k versus kz for different types of pulsed beam. Type 1 corresponds to identical Bessel function for all spectral components. Type 2 is X-‐wave. Type 4 with β = 1 reduces to Type 3 (FWM). Type 4 with β = 2 is superluminal. Type 4 with β = 1 / 2 is subluminal.
