Abstract

The Airy beams are analyzed in order to provide a cogent physical explanation to their intriguing features which include weak diffraction, curved propagation trajectories in free-space, and self healing. The asymptotically exact analysis utilizes the method of uniform geometrical optics (UGO), and it is also verified via a uniform asymptotic evaluation of the Kirchhoff-Huygens integral. Both formulations are shown to fully agree with the exact Airy beam solution in the paraxial zone where the latter is valid, but they are also valid outside this zone. Specifically it is shown that the beam along the curved propagation trajectory is not generated by contributions from the main lobe in the aperture, i.e., it is not described by a local wave-dynamics along this trajectory. Actually, this beam is identified as a caustic of rays that emerge sideways from points in the initial aperture that are located far away from the main lobe. The field of these focusing rays, described here by the UGO, fully agrees with the Airy beam solution. These observations explain that the “weak-diffraction” and the “self healing” properties are generated, in fact, by a continuum of sideways contributions to the field, and not by local self-curving dynamics. The uniform ray representation provides a systematic framework to synthesize aperture sources for other beam solutions with similar properties in uniform or in non-uniform media.

© 2010 Optical Society of America

1. Introduction

Recently, a class of “curving beam” solutions of the paraxial wave equation has been introduced [1–7]. These solutions were also termed “accelerating beams,” borrowing from the original work of Berry and Balazs [8] that has dealt with accelerating wavepacket solutions of the time-dependent potential-free Schrödinger equation, and the similarity between that equation and the paraxial (parabolic) wave equation. Originally, these beams were formulated in a 2D coordinate space, say (x,z), and were generated by setting an Airy function as the initial field distribution in the aperture plane z = 0. Later on, finite energy beams were obtained by multiplying the Airy function distribution by an exponential window or by a Gaussian window. Closed form solutions for the radiated field in these cases were derived in [2] and [3], respectively. Extensions to 3D-space have been presented in [2,6,7]. Henceforth we shell refer mainly to the “Airy beams” as representatives of this class of waves.

The Airy beams attracted a lot of attention because of their intriguing features. Their most distinctive characteristic is the propagation along curved trajectories in free-space [2,9]. These beams are also weakly diffractive along their trajectories, i.e., they retain their structure and remain essentially diffraction-free for distances that are much longer than the conventional diffraction length of Gaussian beams with the same beamwidth [2]. Another interesting feature, pointed out in [10], is the ability of the Airy beam to “heal” itself, i.e., regenerate itself when part of the aperture is obstructed.

Our aim here is to clarify these features viz a viz what we consider to be the basic definition of a “beam.” We regard a beam as a local wave function whose evolution along its propagation trajectory is described by a local self-dynamics (for a broader perspective see [11] and references therein). Thus we would like to identify the wave mechanisms that give rise to the special properties of the Airy beams.

While previous studies of the Airy beams were based on their closed form solutions, here we shall explore their properties using basic wave theory. We present an asymptotically exact wave analysis based on the method of uniform geometrical optics (UGO) [12,13], and also on the Kirchhoff-Huygens integral. Both formulations are shown to fully agree with the exact Airy beam solution in the paraxial zone where the latter is valid, but they are also valid outside this zone. The results provide a full wave explanation to the various phenomena associated with the Airy beams. For simplicity, we shall present here only the 2D case and focus on the Airy beam as a representative of this class of waves, although these concepts can easily be extended to the 3D case or to the finite energy Airy beams.

The presentation starts in Sec. 2 with a short review of the Airy beams. In Sec. 3 we consider the field generated by the main lobe of the Airy function initial distribution. It is demonstrated that this contribution decays faster than a Gaussian beam with the same width, thus confirming that the Airy beam is not generated by a self-curving dynamical evolution of this main lobe. The field radiated by the other part of the initial field distribution is calculated next in Sec. 4 using uniform geometrical optics (UGO), and the results are also verified by a uniform asymptotic evaluation of the Kirchhoff-Huygens integral. The curving Airy beam is shown to be a caustic of rays emerging sideways from points in the aperture which are located far away from the main lobe. The total field is calculated in Sec. 5, where it is shown that the UGO formulation agrees extremely well with the closed form solution of the parabolic wave equation in the range where the latter is valid. Beyond this range, the closed form solution fails whereas the UGO formulation remains valid. This ray interpretation verifies that the curving Airy beams are not “beam fields” in the sense discussed above. This interpretation also explains the weak diffraction and the self healing properties. It also provides a framework for synthesizing other aperture distributions that give rise to beam fields that propagate along any desired trajectory, and may also be applied to non-uniform media. These and other conclusions are discussed in Sec. 6.

2. Preliminaries

We begin by considering the 2D paraxial wave equation in free-space, that governs the propagation of the electric field amplitude ϕ associated with planar optical beams

x2ϕ+2ikzϕ=0,

where z is the coordinates along the propagation axis, x is the transverse coordinate, and k = 2π/λ is the wavenumber. A time dependence exp(-iωt) is assumed and suppressed throughout. The Airy beam is generated by an initial field distribution in the aperture plane z = 0, given by

ϕ|z=0ϕ0(x)=Ai(x/x0),

where Ai is the Airy function and x0 is an arbitrary scaling factor. Since the L2 norm of the Airy function is unbounded, one may multiply the initial distribution in Eq. (2) by a window function F(x) in order to generate a finite energy beam. Possible choices for F(x) are the exponential and Gaussian functions, resulting in the exponential Airy beam [2] or the Airy-Gauss beam [3], respectively. For simplicity, we shell study only the basic Airy beam resulting from the initial distribution in Eq. (2) but the same techniques may be used for the finite energy Airy beams as discussed in the conclusions.

The field U at r = (x,z) associated with the initial distribution in Eq. (2) is given by

U(r)=ϕ(r)eikz=Ai(x̃z̃2/4)exp(ikz+ix̃z̃/2iz̃3/12),

where we used the normalized coordinates x̃ = x/x0 and z̃ = z/z0 with z0 = kx20. One readily verifies that the beam envelope shifts transversely without change along a parabolic trajectory x̃ = z̃2/4, as may also be seen in Fig. 1.

In this work we shall employ scalar diffraction theory in order to calculate the field generated by the aperture field at z = 0. Using the Kirchhoff-Huygens integral with the Rayleigh-Sommerfeld Green function, the radiated field is given by

U(r)=2dxϕ0(x)nG(r,r),

where r′ = (x′, z′) with z′ = 0 denotes points in the aperture, ∂n is the derivative with respect to the normal at r′, G = (i/4)H(1)0() is the Green function of the 2D free-space wave equation, H0(1) is the zeroth order first kind Hankel function, and ρ = [(x-x′)2+z2]1/2 = ∣rr′∣. For observation points where ≫ 1, G~exp(ikρ+/4)/(8πkρ)1/2. In this case n′G~ −ikcosθG where θ = cos−1z/ρ is the angle between the vector from r′ to r and the z axis.

 

Fig. 1. Ray description of the Airy beam plotted on a background of the intensity of the Airy beam in Eq. (3) (in Colors). As discussed after Eq. (7), the z and x axes are normalized with respect to zF and W, respectively. Here and in all other figures, λ = 0.5μm, x0 = 200λ, giving W = 139.4λ and zF = 2kW2 = 24.4104λ.

The following ray species are depicted (see discussion after Eq. (11)): the ϕ0+ species (black solid lines) that converge to a caustic (thick dashed line); the ϕ0 species (dashed-dotted lines); the artificial shadow boundaries SB± of the ϕ0± ray species, introduced by the aperture truncation at xep [see Eq. (5)]. Regions I = shadow of the caustic; Regions II = lit side of the caustic and also lit side of SB+; Regions III = shadow side of SB+ and of SB; Regions IV = lit side of SB. At a typical observation point in region II (shown in the figure), there are two ray contributions: “ray 1”, which has yet to touch the caustic, and “ray 2”, which has already touched it. Note that additional rays of the ϕ0+ species emerging from points at the aperture which are located beyond the figure frame (on the left hand side) have been removed for clarity.

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In the sequel we consider separately the fields U1 and U2 representing, respectively, the contributions of the main lobe in the aperture and of the rest of the aperture. They are defined as

U1(r)=2xepdxϕ0(x)nG(r,r),U2(r)=2xepdxϕ0(x)nG(r,r),

where xep = α1x0 with α1 = -2.338 being the first zero of Ai(x). These two contributions will be calculated, separately, in Secs. 3 and 4, respectively.

3. The field U1 generated by the main lobe

The field U1 due to the main lobe of the Airy function initial distribution is calculated numerically via Eq. (5), and compared with the field of a Gaussian beam (GB) having the same effective half-width, defined as

W=[1Ixepdxϕ0(x)2(xx)2]1/2.

Here 〈x〉 and I are, respectively, the centroid and the intensity of the aperture distribution of U1, defined via

x=1Ixepdxxϕ0(x)2,I=xepdxϕ0(x)2.

As a reference, we compare U1 to the field due to a Gaussian initial distribution UGBz=0 = I01/2 exp [ − (x − 〈x〉)2/2W20], centered about x = 〈x〉, where W0 is the e−1 half width. This distribution has the same effective width W and intensity I as in Eqs. (6)–(7) if W0 = √2W and I0=I/(W2π).. The GB field radiating by this distribution is UGB(r)=I01/2[(−izF)/(zizF)]1/2 exp[ik(z+(x−〈x〉)2/2(zizF))] where zF = kW20 is the Fresnel (or Rayleigh) length.

Figure 2 compares the intensity of U1 calculated via Eq. (5) to the intensity of the field UGB mentioned above and also to the Airy beam of Eq. (3), at different ranges. Here and throughout λ = 0.5μm and x0 = 200λ, giving W = 139.4λ and zF = 2kW2 = 24.4104λ. In order to present the results using physically meaningful scales, we normalize the z and x axes with respect to zF and W, respectively. One readily observes that U1 does not generate the weakly diffractive Airy beam but it rather behaves like the matched UGB above. In fact, it diffracts a little faster: At z = √3zF, the effective width of UGB and of U1 increase by the factors of 2 and 2.3, respectively, while at z = √8zF, the factors are 3 and 3.6.

To conclude this section we note that the main curved lobe of the Airy beam is not generated by the first main lobe of the aperture field, but by some other part, which is identified in the next section. In other words, the curved lobe of the Airy beam is not described by a local self-curving wave dynamics, as was explain in the Introduction.

4. The field U2 generated by the aperture beyond the main lobe

We now turn to calculate the field U2 in Eq. (5). The contribution of this part will be described in terms of rays that emerge from the aperture in directions specified by the local phase gradient. Special care should be given to the field near the curving main beam, where the rays focus to a caustic. To this end, the ray field is expressed in Sec. 4.1 using uniform geometrical optics (UGO). This provides a complete and accurate representation for the field of the Airy beam.

The artificial separation of the aperture field distribution into two parts at xxep gives rise to a spurious end-point diffraction at xep which should also be added to the asymptotic representation of U2. This contribution is derived in Sec. 4.2 using the uniform asymptotic theory (UAT), which is the uniform variant of the geometrical theory of diffraction (GTD). Clearly, when the total field U1 + U2 is calculated, this end-point contribution is canceled out by the truncation effect which is included implicitly in the numerical calculation of U1, leaving the total field described only by the UGO contribution, as discussed in Sec. 5.

The field representation above is obtained by following the rules of uniform ray theory. For completeness, they are verified rigorously in Sec. 4.3 using uniform asymptotic evaluation of the kirchhoff integral in Eq. (5).

Readers who are not interested in the analytic details may skip Secs. 4.2–4.3 and proceed directly to Sec. 5 which includes a summary of the main results and of the numerical results.

4.1. The uniform geometrical optics (UGO) field

In order to facilitate ray analysis, we use the asymptotic expression of Ai(x) for x≪−1 and decompose the aperture field at xxep into a sum of two local plane wave constituents, viz,

ϕ0(x)(π2x̃)1/4sin[2/3(x̃)3/2+π/4]
=A0+exp(ikψ0+)+A0exp(ikψ0)
ϕ0+(x)+ϕ0(x),
 

Fig. 2. The intensity ∣U12 of the field radiated by the main lobe of the Airy beam initial distribution (solid blue line), compared with the intensity ∣U2 of the total field calculated from the paraxial closed-form solution in Eq. (3) (solid magenta line), and the intensity ∣UGB2 of the Gaussian beam that has the same effective width as the main lobe of the Airy beam initial distribution (see discussion after Eq. (7); dashed green line). All intensities are normalized with respect to the maximal intensity of the close-form solution. The fields are calculated at several ranges from the aperture: (a) z = 0; (b) z = √3zF; and z = √8zF. At z = √3zF the effective width of the GB doubles whereas the effective width of the U1 field increases by a factor of 2.3. At z = √8zF the effective width of the GB and the U1 field increase by a factor of 3 and 3.6, respectively.

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where ψ±0 and A±0, the initial phase and amplitude functions, are given by

kψ0±(x)=[2/3(x̃)3/2+π/4],A0±(x)=±(i/2)(π2x̃)1/4.

The initial fields ϕ0+ and ϕ0 in Eq. (8) give rise to rays that emerge from points x′ ≤ xep in the aperture at local angles (measured with respect to the z-axis)

θ±(x)=sin1[xψ0±(x)]=±sin1[(x/x0)1/2/(kx0)].

The rays follow the trajectories

(x,z)=(x+ρsinθ(x),ρcosθ(x)),

where x′ is the exit point and ρ measures the distance along the ray (see Figs. 1 and 3). The ray species ϕ+0 emerges sideways, converging to a caustic along the curving main beam trajectory x̃ = z̃2/4 (this expression is valid within the parabolic approximation; a more accurate expression is discussed below). Each observation point on the lit side of the caustic (region II in Fig. 1) is illuminated by two rays of species ϕ+0, one which has touched the caustic before reaching the observer, and another which has not. They are denoted, respectively as rays 2 and 1, referring to the fact that the phase of ray 1 is smaller than that of ray 2 (implying that in the time domain, ray 1 reaches the observer first).

Due to the artificial truncation of the aperture at xep, the rays leaving this point form artificial shadow boundaries for the ϕ+0 and ϕ0 species denoted, respectively, as SB+ and SB (see Fig. 1). Referring first to species ϕ+0, this implies that the caustic is truncated at the point where it touches SB+ (see Fig. 3). The shadow side of SB+ (region III) is illuminated only by ray 1 of this species.

Finally, we note that the lit side of SB corresponding to species ϕ0 is far from the main beam region and therefore this contribution will not be considered below.

The geometrical optics (GO) field at points r along a ray of the ϕ+0 species emerging from a given point x′ [see Eq. (11)] is given by

UGO+(r)=A+(r)exp[ikψ+(r)].

The phase along this ray is

ψ+(r)=ψ0+(x)+ρ,

where the initial phase ψ0+ is given in Eq. (9) and ρ is the distance along this ray. The ray amplitude is

A+(r)=A0+(J0/J)1/2=A0+[1+ρ(x2ψ0+/cos2θ)]1/2=A0+[1ρ/ρc]1/2,

where J = ∂(x,z)/∂(ρ,x′) is the Jacobian for the transformation (ρ,x′) → (x,z) evaluated at the observation point r, and J0 is the Jacobian at the exit point. We also identify ρc = − cos2 θ/∂2x ψ+0 as the distance from x′ to the caustic (note that ∂2x ψ+0 < 0 so that ρc > 0). Eq. (14) fails near the caustic at ρρc. For ρ > ρc, the square root in Eq. (14) becomes −i∣1 − ρ/ρc∣−1/2. The caustic coordinates are obtained by setting ρρc in Eq. (11) for each exit point x′.

 

Fig. 3. Definition of the ray coordinates (ρ1,2,x1,2,θ1,2) for rays 1 and 2 (green lines) that are shown in Fig. 1. Also shown (in magenta) are the coordinates of the diffracting ray (ρd,xep,θd) and (in black) the shadow boundaries angles θSB± defined in Sec. 4.2. ρ are the distances traversed by the rays, θ are the ray exit angles and x′ are the ray exit points.

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The exit points x′ of the rays that are associated with a given observation point r are found via

xψ0+(x)=sinθ(x)=(xx)/ρ.

For observation points in the lit region of the caustic (region II in Fig. 1), this equation has two solutions, denoted as x1,2 in accord with rays 1 and 2 discussed above (see Fig. 3). On the shadow side of SB+ (region III) only the x1 solution exists, while the other solution x2 is beyond the end point (i.e., x2 > xep). In the paraxial region one may replace in Eq. (15) ρ by z, leading to a simple algebraic expression for x1,2.

As an illustrative example we depict in Fig. 4 the intensity of the initial field distribution at z = 0 and the exit points corresponding to certain observation points at the observation plane z = 10zF. The observation points are tagged by their location in the 1st, 2nd, or 3rd lobes of the field at this observation plane. One observes, for example, that the exit points x1 and x2 corresponding to an observation point in the first lobe at z = 10zF are located at the 44th and the 13th lobes of the initial field distribution, respectively.

The field on the lit side of the caustic is the sum of the two ray fields UGO1+ + UGO2+ [recall that ray 2 is beyond the caustic hence its amplitude function A+2 contains a factor of -i as discussed after Eq. (14)]. The representation fails near the caustic at ρρc where A+1,2 in Eq. (14)explode. A uniform expression for the field, valid through the caustic, is obtained by replacing the GO fields, derived above, by the uniform geometrical optics (UGO) field [12,13]

UUGO+=exp(ikφ0)[gAi(σk2/3)+ihk1/3Ai(σk2/3)],

where Ai′ is the derivative of the Ai function with respect to its argument. The parameters in Eq. (16) are functions of r and defined in terms of the GO parameters via

φ0=12(ψ1++ψ2+),σ=[34(ψ1++ψ2+)]2/3
g=12σ1/4(A1++A2+),h=12σ1/4(A2+A1+).
 

Fig. 4. Exit points of the rays corresponding to certain observation points at the observation plane z = 10zF. The observation points are tagged by their location in the 1st, 2nd, or 3rd lobes of the field at this observation plane. As a reference, we also show the intensity of the initial field distribution on the z = 0 plane.

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They are denoted, respectively, as the average and differential phases and the average and differential amplitudes. In the lit region, σ is real and positive and σ1/4 is taken to be the real and positive root. In the shadow region near the caustic, σ is real and negative and σ1/4 is taken such that its argument is π/4.

Equation (16) describes the field on both sides of the caustic. Far from the caustic on the lit side it reduces uniformly to UGO1+ + UGO2+ while on the shadow side it describes the decaying field there (see numerical results in Sec. 5). Note also that the Ai′ term can be neglected on the caustic.

4.2. The uniform diffracted field

As mentioned earlier, the artificial end point x = xep gives rise to edge diffraction fields. They have the GTD form

Ud±=A0d±exp(ikψ0d±)D±G(ρd),

where the ± sign corresponds to ϕ0±, ρd(r) = [(xxep)2+z2]1/2 is the distance from the end point at x = xep to the observation point r (see Fig. 3), ψ0d± and A0d± are the initial phase and amplitude functions at the end point, obtained by substituting x′ = xep in Eq. (9). G(ρd) ~ exp(ikρd+/4)/(8πkρd)1/2 is the asymptotic expression for the Green function, and D± are the diffraction coefficients

D±=2icosθd/(sinθSB±sinθd),

where θd(r) = sin−1[(xxep)/ρd] is the exit angle of the ray from xep to r and θSB± = θ±(x′) are the exit angles of rays SB± from xep (see Eq. (10) and Fig. 3; note that θSB = − θSB+). The expression in Eq. (20) is the GTD diffraction coefficient under the physical optics approximation for a Neumann boundary condition at the edge (the Neumann boundary condition ∂nU = 0 is implied by the Rayleigh-Sommerfeld formulation used in the the Kirchhoff integral in Eq. 4).

The edge diffraction fields in Eq. (19) explode near the shadow boundaries where θdθSB±. A uniform expression for the fields, valid near the shadow boundary, is obtained via the uniform asymptotic theory (UAT) [14,15]. Henceforth we consider only the field near SB+ since, as noted earlier, SB is far from the region of interest (see Figs. 1 and 3). Recalling the discussion following Eq. (11), the ray that is shadowed by SB+ is ray 2 in species ϕ0+, hence the UAT replaces the fields Ud++UGO2+. It is given by [14,15]

UUAT+=Ud++[F(η)F̂(η)]UGO2+,

where UGO2+ is given in Eqs. (12)-(14), Ud+ is given in Eqs. (19)-(20), and

F(η)=exp(iπ/4)π1/2ηdtexp(it)2,F̂(η)=(2ηπ)1exp(iη2/4),
η=sgn(θSB+θ2)[k(ψ2+ψd+)]1/2.

F is the Fresnel integral, F̂ is the asymptotic end point contribution in the Fresnel integral, sgn(x) = ±1 for x ≷ 0, θSB+ is defined after Eq. (20), θ2 is the exit angle of ray 2 as defined in Eq. (15), ψd+ = ρd + ψ0d+ is the phase of the diffracted ray [see Eq. (19)]. Note that ψ2+ > ψd+ hence the square root in Eq. (23) is taken to be positive.

One may show that the singularity of Ud+ and of F̂ near SB+ (i.e., η ≈ 0) are canceled, leaving the field there described by the F term. For observation points far enough from SB+, the large η asymptotics F(η) ~H(-η)+F̂(η), where H(x) = 0 or 1 for x ≶ 0, respectively, reduces Eq. (21) to UUAT+~UGO2++Ud+ or to Ud+ in the lit or shadow sides of SB+, respectively.

4.3. Asymptotic evaluation of the Kirchhoff integral

The results presented in the previous sections have been derived directly via the rules of the UGO and the UAT. Here they will be re-derived by asymptotic evaluation of the kirchhoff integrals in Eq. (5). We present only an outline of the analysis since the final expressions have been given above. For further analytical details the readers are referred to [16, Ch. 4].

Substituting the decomposition ϕ0 = ϕ0++ϕ0 of Eq. (8) into the kirchhoff integral for U2 in Eq. (5), it is expressed as a sum of two generalized ray integrals U2 = U2++U2, with

U2±=xepdx'A±(x')exp(ikψ±(x'))
A±=A0±cosθexp(iπ/4)(k/2πρ)1/2,ψ±=ρ+ψ0±,

where we have used the asymptotic expression for ∂n G presented after Eq. (4). A±0 and ψ0± are defined in Eq. (9).

We start with the evaluation of U2+ which, as discussed above, describes the main contribution near the caustic. We utilize the complex steepest descent path (SDP) method. The staddle points are found by solving the equation ∂x ψ+ (x′) = 0. In view of Eq. (25), this equation is identical to Eq. (15). There are, therefore, two real saddle points xs1,2 which are the same as the exit points xs1,2 found in Sec. 4.1. The saddle point contributions are given by

Us1,2+=A±(x)[2π/ikx2ψ+(x)]1/2exp(ikψ+(x))|x=x1,2.

Noting that ∂x2ψ+(x′) ≷ 0 at x1,2, respectively, it follows that the sign of the square root in Eq. (26) is exp(±/4), so that in the total amplitude in Eq. (26), A0+ is multiplied by a real positive factor for Us1+, and by a negative imaginary factor for Us2+.

The expressions in Eq. (26) are identical with UGO1,2+ discussed in Eq. (12)–(15). They fail near the caustic where the two saddle points coincide to a 2nd order saddle point with ∂x2ψ+ = 0. A uniform expression, valid near the caustic, is derived via the Chester-Friedman-Ursell method [16, Sec. 4.5], [17]. The method utilizes a mapping of U+ to a canonical integral whose phase is a 3rd order polynomial that can be matched to both saddle points xs1,2, simultaneously. The canonical integral can be expressed in terms of the Airy function and its derivative and it depends on the phases and amplitudes at the stationary points. Applying the procedure of [16, Sec. 4.5] to the integral in Eq. (24), using the the phases and amplitudes from Eq. (26), one obtains the UGO result in Eq. (16).

The asymptotic evaluation of U± in Eq. (24) also yields the end-point contributions

Uep±=A±(x)[ikxψ±(x)]1exp(ikψ±(x))|x=xep.

These expressions are identical to the GTD fields Ud± in Eq. (19). U+ep fails near the shadow boundary SB+ where ∂x ψ+→0. A uniform asymptotic expression valid through this region is obtained by following the procedure in [16, Sec. 4.6] for a stationary point near an end point. The result can be expressed in the UAT form of Eq. (21).

5. Numerical results

Summarizing the results of the previous sections, the total radiated field is given by

U~U1+U2=U1+UGO1++UGO2++Ud++Ud,

where it is assumed here that the observation point is in region II of Fig. 1 (lit side of the caustic but not too close to the caustic or to SB+). Near the caustic UGO1++UGO2 are replaced by UUGO+ as in Eq. (16). Near SB+, UGO2++Ud+ are replaced by UUAT+ as in Eq. (21).

The separate contribution of U1 and U2 as well as their sum are depicted in Fig. (5), together with the paraxial closed form solution. The sum U1 + U2 agrees well with the closed form solution except for a small 1% error around the peak of the main lobe. This error is due to a small error in Ud± arising from the approximation of the initial field in Eq. (8) near the end point xep. Note that the first lobes are generated mainly by U2 and that U1 is negligible there.

As noted in Sec. 4, the artificial separation of the aperture field distribution into two parts at xxep gives rise to a spurious end-point effect in the numerical calculation of U1. This contribution is canceled by the asymptotic end point diffractions Ud++Ud in Eq. (28). Since the contribution of the main lobe in U1 is otherwise negligible, we obtain from Eq. (28) only the contribution of UGO1+ and of UGO2+, which, as mentioned above, may be replaced by the UGO field, i.e.

U~UUGO+.

Figure 6 compares the paraxial closed form solution of Eq. (3) to the UGO solution. An excellent agreement between the two is seen.

 

Fig. 5. The intensities of the fields radiated by the main lobe of the initial distribution (∣U12 – blue line) and by the rest of the initial distribution (∣U22 – red line), calculated at the observation plane z = 10zF. The intensity of their sum ∣U1+U22 (black line) agrees extremely well with the intensity of the total field ∣U2 calculated from the paraxial closed-form solution in Eq. (3) (magenta line); a difference is seen only near the peak of the main lobe. The locations of the caustic and of the artificial shadow boundary are indicated by tags on the x-axis.

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6. Discussion and conclusions

The Airy beam belongs to a new class of curving beam solutions with intriguing features such as curved propagation trajectories in free-space, weak diffraction, and self healing. Our goal in this paper has been to provide a cogent physical description to the Airy beam solutions viz a viz what we consider to be the basic definition of a “beam,” namely a local wave function, whose evolution along its propagation trajectory is described by a local wave-dynamics.

 

Fig. 6. The intensity ∣UUGO2 of the UGO field (green line) compared to the intensity ∣U2 of the total field calculated from the paraxial closed-form solution in Eq. (3) (magenta line), at the observation plane z = 10zF. The two results are indistinguishable within the scale of the figure. The location of the caustic is indicated by a tag on the x-axis.

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In order to establish the wave mechanism of the Airy beam we explored its properties within the rigorous framework of the Kirchhoff-Huygens integral, and also provided an asymptotically exact analysis utilizing the method of uniform geometrical optics (UGO). In Sec. 3 we calculated the field U1 radiated only by the main lobe of the Airy function initial field distribution, and demonstrated that it diffracts faster than a Gaussian beam with the same effective width (Fig. 2). The field U2 radiated by the other part of the initial field distribution has been studied in Sec. 4 using a uniform asymptotic evaluation of the corresponding Kirchhoff-Huygens integral. It was found that the Airy beam is described by sideways radiating rays that focus onto a caustic along the curved beam trajectory, and thereafter diverge away from it. It has been shown in Fig. 6 that the field of these focusing rays, expressed via the UGO, fully agrees with the Airy beam solution, thus providing a simple physical interpretation for its properties. It has thus been established that the evolution of the main lobe of the Airy beam along the curved trajectory is not described by a local wave dynamics, and it cannot be regarded as a “beam field” in the sense discussed above.

In addition to these focusing rays, U2 also consists of spurious contributions of end point diffraction, due to the “artificial” aperture truncation after it has been separated into two parts: the main lobe part and the rest of the aperture. These spurious contributions which are calculated asymptotically via a uniform version of the GTD, are canceled out by the numerical truncation effect in U1, as has been discussed in Sec. 5, leaving the field described only by the UGO field.

The ray representation readily explains the “self-healing” property of the Airy beams reported in [10], i.e., its regeneration when the main lobe is obstructed. Noting that the main lobe field is not generated by a local field evolution but rather by sideways ray contributions, it follows that obstructing the main lobe at some point will have no effect on the forming of the main lobe at some range further on.

The ray interpretation can also be used to synthesize curving beam solutions that follow any prescribed (convex) trajectory, by synthesizing the aperture field distribution that gives rise to a caustic along this trajectory (for a recent physical realization see [18]). This is done by backtracking rays that emerge tangentially from that caustic to the aperture plane z = 0. The angle of these rays with respect to the aperture plane define the local gradient of the phase distribution there (see Eq. (10)). This concept can readily be generalized to the synthesis of curving beams in non-uniform media.

The concepts above may also be used to synthesize finite-energy Airy beam solutions. As discussed after Eq. (3), such beam fields are obtained by multiplying the Airy function aperture field distribution in Eq. (2) by a proper window function F(x). In view of the ray interpretation above, F(x) must be wide enough on the scale of x0 to incorporate many lobes of the Airy function aperture field distribution, and hence it can be regarded as a slowly varying amplitude function in the asymptotic evaluation of the Kirchhoff-Huygens integral. The resulting field is the same as the UGO result in Eq (16) except that A+1,2 in Eq. (14) and in Eq. (25) are now multiplied by F(x1,2), the value of F at the ray exit points x1,2.

Finally, it should be noted that the closed form Airy function solution in Eq. (3) is valid only within the paraxial zone where the parabolic wave equation in Eq. (1) is valid. The UGO solution, on the other hand, is not restricted by the paraxial approximation and hence it is valid beyond the paraxial zone. One may show that the error in the paraxial solution in Eq. (3) is mainly due to the phase, causing a transversal shift of the solution. An estimate of the region of validity of the paraxial approximation can be obtained, therefore, by calculating the transversal shift of the paraxial caustic x̃ = z̃2/4 with respect to the exact caustic, obtained via the ray procedure outlined in Sec. 4.1.

Acknowledgement

This work is supported in part by the Israeli Science Foundation under Grant No. 674/07.

References and links

1. G. A. Siviloglou, J. Brokly, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99 (2007). [CrossRef]  

2. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32, 979–981 (2007). [CrossRef]   [PubMed]  

3. M. A. Bandres and J. C. Gutierrez-Vega, “Airy-Gauss beams and their transformation by paraxial optical systems,” Opt. Express 15, 16719–16728 (2007). [CrossRef]   [PubMed]  

4. I. M. Besieris and A. M. Shaarawi, “A note on an accelerating finite energy Airy beam,” Opt. Lett. 32, 2447–2449 (2007). [CrossRef]   [PubMed]  

5. M. A. Bandres, “Observation of accelerating parabolic beams,” Opt. Express 16, 12866–12871 (2008). [CrossRef]   [PubMed]  

6. M. A. Bandres, “Accelarating parabolic beams,” Opt. Lett. 33, 1678–1680 (2008). [CrossRef]   [PubMed]  

7. M. A. Bandres, “Accelerating beams,” Opt. Lett. 34, 3791–3793 (2009). [CrossRef]   [PubMed]  

8. M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47, 264–267 (1979). [CrossRef]  

9. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Ballistic dynamics of Airy beams,” Opt. Lett. 33, 207–209 (2008). [CrossRef]   [PubMed]  

10. J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express 16, 12880–12891 (2008). [CrossRef]   [PubMed]  

11. E. Heyman and L. B. Felsen, “Gaussian beam and pulsed beam dynamics: complex-source and complex-spectrum formulations within and beyond paraxial asymptotics,” J. Opt. Soc. Am. 181588–1611 (2001). [CrossRef]  

12. D. Ludwig, “Wave propagation near a smooth caustic,” Bull. Amer. Math. Soc. 71, 776–779 (1965). [CrossRef]  

13. Yu. A. Kravtsov, “A modification of the geometrical optics method,” Izv. VUZ Radiofiz. Engl. transl., Radiophys. Quantum Electron. 7, 664–673 (1964).

14. R. M. Lewis and J. Boersma, “Uniform asymptotic theory of edge diffraction,” Math. Phys. 10, 2291–2305 (1969). [CrossRef]  

15. S. W. Lee and G.A. Deschamps, “A uniform asymptotic theory of electromagnetic diffraction by a curved wedge,” IEEE Trans. Antennas Propag. AP-24, 25–34 (1976).

16. L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, New Jersey, 1973).

17. C. Chester, B. Friedman, and F. Ursell, “An extension of the method of steepest descenets,” Proc. of Cambridge Phi.Soc. 53, 599–611 (1957). [CrossRef]  

18. T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie “Nonlinear generation and manipulation of Airy beams,” Nature Photonics 3, 395–398 (2009). [CrossRef]  

References

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  1. G. A. Siviloglou, J. Brokly, A. Dogariu, and D. N. Christodoulides, "Observation of accelerating Airy beams," Phys. Rev. Lett. 99 (2007).
    [CrossRef]
  2. G. A. Siviloglou, and D. N. Christodoulides, "Accelerating finite energy Airy beams," Opt. Lett. 32, 979-981 (2007).
    [CrossRef] [PubMed]
  3. M. A. Bandres and J. C. Gutierrez-Vega, "Airy-Gauss beams and their transformation by paraxial optical systems," Opt. Express 15, 16719-16728 (2007).
    [CrossRef] [PubMed]
  4. I. M. Besieris and A. M. Shaarawi, "A note on an accelerating finite energy Airy beam," Opt. Lett. 32, 2447-2449 (2007).
    [CrossRef] [PubMed]
  5. M. A. Bandres, "Observation of accelerating parabolic beams," Opt. Express 16, 12866-12871 (2008).
    [CrossRef] [PubMed]
  6. M. A. Bandres, "Accelarating parabolic beams," Opt. Lett. 33, 1678-1680 (2008).
    [CrossRef] [PubMed]
  7. M. A. Bandres, "Accelerating beams," Opt. Lett. 34, 3791-3793 (2009).
    [CrossRef] [PubMed]
  8. M. V. Berry, and N. L. Balazs, "Nonspreading wave packets," Am. J. Phys. 47, 264-267 (1979).
    [CrossRef]
  9. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, "Ballistic dynamics of Airy beams," Opt. Lett. 33, 207-209 (2008).
    [CrossRef] [PubMed]
  10. J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, "Self-healing properties of optical Airy beams," Opt. Express 16, 12880-12891 (2008).
    [CrossRef] [PubMed]
  11. E. Heyman and L. B. Felsen, "Gaussian beam and pulsed beam dynamics: complex-source and complexspectrum formulations within and beyond paraxial asymptotics," J. Opt. Soc. Am. 181588-1611 (2001).
    [CrossRef]
  12. D. Ludwig, "Wave propagation near a smooth caustic," Bull. Amer. Math. Soc. 71, 776-779 (1965).
    [CrossRef]
  13. Yu. A. Kravtsov, "A modification of the geometrical optics method," Izv. VUZ Radiofiz. Engl. transl., Radiophys. Quantum Electron. 7, 664-673 (1964).
  14. R. M. Lewis and J. Boersma, "Uniform asymptotic theory of edge diffraction," Math. Phys. 10, 2291-2305 (1969).
    [CrossRef]
  15. S. W. Lee, and G. A. Deschamps, "A uniform asymptotic theory of electromagnetic diffraction by a curved wedge," IEEE Trans. Antennas Propag. AP-24, 25-34 (1976).
  16. L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, New Jersey, 1973).
  17. C. Chester, B. Friedman, and F. Ursell, "An extension of the method of steepest descenets," Proc. of Cambridge Phi.Soc. 53, 599-611 (1957).
    [CrossRef]
  18. T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie "Nonlinear generation and manipulation of Airy beams," Nature Photonics 3, 395-398 (2009).
    [CrossRef]

2009

M. A. Bandres, "Accelerating beams," Opt. Lett. 34, 3791-3793 (2009).
[CrossRef] [PubMed]

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie "Nonlinear generation and manipulation of Airy beams," Nature Photonics 3, 395-398 (2009).
[CrossRef]

2008

2007

2001

E. Heyman and L. B. Felsen, "Gaussian beam and pulsed beam dynamics: complex-source and complexspectrum formulations within and beyond paraxial asymptotics," J. Opt. Soc. Am. 181588-1611 (2001).
[CrossRef]

1979

M. V. Berry, and N. L. Balazs, "Nonspreading wave packets," Am. J. Phys. 47, 264-267 (1979).
[CrossRef]

1976

S. W. Lee, and G. A. Deschamps, "A uniform asymptotic theory of electromagnetic diffraction by a curved wedge," IEEE Trans. Antennas Propag. AP-24, 25-34 (1976).

1969

R. M. Lewis and J. Boersma, "Uniform asymptotic theory of edge diffraction," Math. Phys. 10, 2291-2305 (1969).
[CrossRef]

1965

D. Ludwig, "Wave propagation near a smooth caustic," Bull. Amer. Math. Soc. 71, 776-779 (1965).
[CrossRef]

1964

Yu. A. Kravtsov, "A modification of the geometrical optics method," Izv. VUZ Radiofiz. Engl. transl., Radiophys. Quantum Electron. 7, 664-673 (1964).

1957

C. Chester, B. Friedman, and F. Ursell, "An extension of the method of steepest descenets," Proc. of Cambridge Phi.Soc. 53, 599-611 (1957).
[CrossRef]

Arie, A.

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie "Nonlinear generation and manipulation of Airy beams," Nature Photonics 3, 395-398 (2009).
[CrossRef]

Balazs, N. L.

M. V. Berry, and N. L. Balazs, "Nonspreading wave packets," Am. J. Phys. 47, 264-267 (1979).
[CrossRef]

Bandres, M. A.

Berry, M. V.

M. V. Berry, and N. L. Balazs, "Nonspreading wave packets," Am. J. Phys. 47, 264-267 (1979).
[CrossRef]

Besieris, I. M.

Boersma, J.

R. M. Lewis and J. Boersma, "Uniform asymptotic theory of edge diffraction," Math. Phys. 10, 2291-2305 (1969).
[CrossRef]

Brokly, J.

G. A. Siviloglou, J. Brokly, A. Dogariu, and D. N. Christodoulides, "Observation of accelerating Airy beams," Phys. Rev. Lett. 99 (2007).
[CrossRef]

Broky, J.

Chester, C.

C. Chester, B. Friedman, and F. Ursell, "An extension of the method of steepest descenets," Proc. of Cambridge Phi.Soc. 53, 599-611 (1957).
[CrossRef]

Christodoulides, D. N.

Deschamps, G. A.

S. W. Lee, and G. A. Deschamps, "A uniform asymptotic theory of electromagnetic diffraction by a curved wedge," IEEE Trans. Antennas Propag. AP-24, 25-34 (1976).

Dogariu, A.

Ellenbogen, T.

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie "Nonlinear generation and manipulation of Airy beams," Nature Photonics 3, 395-398 (2009).
[CrossRef]

Felsen, L. B.

E. Heyman and L. B. Felsen, "Gaussian beam and pulsed beam dynamics: complex-source and complexspectrum formulations within and beyond paraxial asymptotics," J. Opt. Soc. Am. 181588-1611 (2001).
[CrossRef]

Friedman, B.

C. Chester, B. Friedman, and F. Ursell, "An extension of the method of steepest descenets," Proc. of Cambridge Phi.Soc. 53, 599-611 (1957).
[CrossRef]

Ganany-Padowicz, A.

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie "Nonlinear generation and manipulation of Airy beams," Nature Photonics 3, 395-398 (2009).
[CrossRef]

Gutierrez-Vega, J. C.

Heyman, E.

E. Heyman and L. B. Felsen, "Gaussian beam and pulsed beam dynamics: complex-source and complexspectrum formulations within and beyond paraxial asymptotics," J. Opt. Soc. Am. 181588-1611 (2001).
[CrossRef]

Kravtsov, Yu. A.

Yu. A. Kravtsov, "A modification of the geometrical optics method," Izv. VUZ Radiofiz. Engl. transl., Radiophys. Quantum Electron. 7, 664-673 (1964).

Lee, S. W.

S. W. Lee, and G. A. Deschamps, "A uniform asymptotic theory of electromagnetic diffraction by a curved wedge," IEEE Trans. Antennas Propag. AP-24, 25-34 (1976).

Lewis, R. M.

R. M. Lewis and J. Boersma, "Uniform asymptotic theory of edge diffraction," Math. Phys. 10, 2291-2305 (1969).
[CrossRef]

Ludwig, D.

D. Ludwig, "Wave propagation near a smooth caustic," Bull. Amer. Math. Soc. 71, 776-779 (1965).
[CrossRef]

Shaarawi, A. M.

Siviloglou, G. A.

Ursell, F.

C. Chester, B. Friedman, and F. Ursell, "An extension of the method of steepest descenets," Proc. of Cambridge Phi.Soc. 53, 599-611 (1957).
[CrossRef]

Voloch-Bloch, N.

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie "Nonlinear generation and manipulation of Airy beams," Nature Photonics 3, 395-398 (2009).
[CrossRef]

Am. J. Phys.

M. V. Berry, and N. L. Balazs, "Nonspreading wave packets," Am. J. Phys. 47, 264-267 (1979).
[CrossRef]

Bull. Amer. Math. Soc.

D. Ludwig, "Wave propagation near a smooth caustic," Bull. Amer. Math. Soc. 71, 776-779 (1965).
[CrossRef]

IEEE Trans. Antennas Propag.

S. W. Lee, and G. A. Deschamps, "A uniform asymptotic theory of electromagnetic diffraction by a curved wedge," IEEE Trans. Antennas Propag. AP-24, 25-34 (1976).

J. Opt. Soc. Am.

E. Heyman and L. B. Felsen, "Gaussian beam and pulsed beam dynamics: complex-source and complexspectrum formulations within and beyond paraxial asymptotics," J. Opt. Soc. Am. 181588-1611 (2001).
[CrossRef]

Math. Phys.

R. M. Lewis and J. Boersma, "Uniform asymptotic theory of edge diffraction," Math. Phys. 10, 2291-2305 (1969).
[CrossRef]

Nature Photonics

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie "Nonlinear generation and manipulation of Airy beams," Nature Photonics 3, 395-398 (2009).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. Lett.

G. A. Siviloglou, J. Brokly, A. Dogariu, and D. N. Christodoulides, "Observation of accelerating Airy beams," Phys. Rev. Lett. 99 (2007).
[CrossRef]

Proc. of Cambridge Phi.Soc.

C. Chester, B. Friedman, and F. Ursell, "An extension of the method of steepest descenets," Proc. of Cambridge Phi.Soc. 53, 599-611 (1957).
[CrossRef]

Radiophys. Quantum Electron.

Yu. A. Kravtsov, "A modification of the geometrical optics method," Izv. VUZ Radiofiz. Engl. transl., Radiophys. Quantum Electron. 7, 664-673 (1964).

Other

L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, New Jersey, 1973).

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Figures (6)

Fig. 1.
Fig. 1.

Ray description of the Airy beam plotted on a background of the intensity of the Airy beam in Eq. (3) (in Colors). As discussed after Eq. (7), the z and x axes are normalized with respect to zF and W, respectively. Here and in all other figures, λ = 0.5μm, x0 = 200λ, giving W = 139.4λ and zF = 2kW2 = 24.4104λ.

Fig. 2.
Fig. 2.

The intensity ∣U12 of the field radiated by the main lobe of the Airy beam initial distribution (solid blue line), compared with the intensity ∣U2 of the total field calculated from the paraxial closed-form solution in Eq. (3) (solid magenta line), and the intensity ∣UGB2 of the Gaussian beam that has the same effective width as the main lobe of the Airy beam initial distribution (see discussion after Eq. (7); dashed green line). All intensities are normalized with respect to the maximal intensity of the close-form solution. The fields are calculated at several ranges from the aperture: (a) z = 0; (b) z = √3zF; and z = √8zF. At z = √3zF the effective width of the GB doubles whereas the effective width of the U1 field increases by a factor of 2.3. At z = √8zF the effective width of the GB and the U1 field increase by a factor of 3 and 3.6, respectively.

Fig. 3.
Fig. 3.

Definition of the ray coordinates (ρ1,2,x1,2,θ1,2) for rays 1 and 2 (green lines) that are shown in Fig. 1. Also shown (in magenta) are the coordinates of the diffracting ray (ρd,xep,θd) and (in black) the shadow boundaries angles θSB± defined in Sec. 4.2. ρ are the distances traversed by the rays, θ are the ray exit angles and x′ are the ray exit points.

Fig. 4.
Fig. 4.

Exit points of the rays corresponding to certain observation points at the observation plane z = 10zF. The observation points are tagged by their location in the 1st, 2nd, or 3rd lobes of the field at this observation plane. As a reference, we also show the intensity of the initial field distribution on the z = 0 plane.

Fig. 5.
Fig. 5.

The intensities of the fields radiated by the main lobe of the initial distribution (∣U12 – blue line) and by the rest of the initial distribution (∣U22 – red line), calculated at the observation plane z = 10zF. The intensity of their sum ∣U1+U22 (black line) agrees extremely well with the intensity of the total field ∣U2 calculated from the paraxial closed-form solution in Eq. (3) (magenta line); a difference is seen only near the peak of the main lobe. The locations of the caustic and of the artificial shadow boundary are indicated by tags on the x-axis.

Fig. 6.
Fig. 6.

The intensity ∣UUGO2 of the UGO field (green line) compared to the intensity ∣U2 of the total field calculated from the paraxial closed-form solution in Eq. (3) (magenta line), at the observation plane z = 10zF. The two results are indistinguishable within the scale of the figure. The location of the caustic is indicated by a tag on the x-axis.

Equations (31)

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x 2 ϕ + 2 i k z ϕ = 0 ,
ϕ | z = 0 ϕ 0 ( x ) = Ai ( x / x 0 ) ,
U ( r ) = ϕ ( r ) e ikz = Ai ( x ̃ z ̃ 2 / 4 ) exp ( ikz + i x ̃ z ̃ / 2 i z ̃ 3 / 12 ) ,
U ( r ) = 2 d x ϕ 0 ( x ) n G ( r , r ) ,
U 1 ( r ) = 2 x ep d x ϕ 0 ( x ) n G ( r , r ) , U 2 ( r ) = 2 x ep d x ϕ 0 ( x ) n G ( r , r ) ,
W = [ 1 I x ep d x ϕ 0 ( x ) 2 ( x x ) 2 ] 1 / 2 .
x = 1 I x ep d x x ϕ 0 ( x ) 2 , I = x ep d x ϕ 0 ( x ) 2 .
ϕ 0 ( x ) ( π 2 x ̃ ) 1 / 4 sin [ 2 / 3 ( x ̃ ) 3 / 2 + π / 4 ]
= A 0 + exp ( i k ψ 0 + ) + A 0 exp ( i k ψ 0 )
ϕ 0 + ( x ) + ϕ 0 ( x ) ,
k ψ 0 ± ( x ) = [ 2 / 3 ( x ̃ ) 3 / 2 + π / 4 ] , A 0 ± ( x ) = ± ( i / 2 ) ( π 2 x ̃ ) 1 / 4 .
θ ± ( x ) = sin 1 [ x ψ 0 ± ( x ) ] = ± sin 1 [ ( x / x 0 ) 1 / 2 / ( k x 0 ) ] .
( x , z ) = ( x + ρ sin θ ( x ) , ρ cos θ ( x ) ) ,
U GO + ( r ) = A + ( r ) exp [ i k ψ + ( r ) ] .
ψ + ( r ) = ψ 0 + ( x ) + ρ ,
A + ( r ) = A 0 + ( J 0 / J ) 1 / 2 = A 0 + [ 1 + ρ ( x 2 ψ 0 + / cos 2 θ ) ] 1 / 2 = A 0 + [ 1 ρ / ρ c ] 1 / 2 ,
x ψ 0 + ( x ) = sin θ ( x ) = ( x x ) / ρ .
U UGO + = exp ( ik φ 0 ) [ g Ai ( σ k 2 / 3 ) + i h k 1 / 3 Ai ( σ k 2 / 3 ) ] ,
φ 0 = 1 2 ( ψ 1 + + ψ 2 + ) , σ = [ 3 4 ( ψ 1 + + ψ 2 + ) ] 2 / 3
g = 1 2 σ 1 / 4 ( A 1 + + A 2 + ) , h = 1 2 σ 1 / 4 ( A 2 + A 1 + ) .
U d ± = A 0 d ± exp ( ik ψ 0 d ± ) D ± G ( ρ d ) ,
D ± = 2 i cos θ d / ( sin θ SB ± sin θ d ) ,
U UAT + = U d + + [ F ( η ) F ̂ ( η ) ] U GO 2 + ,
F ( η ) = exp ( i π / 4 ) π 1 / 2 η d t exp ( it ) 2 , F ̂ ( η ) = ( 2 η π ) 1 exp ( i η 2 / 4 ) ,
η = sgn ( θ SB + θ 2 ) [ k ( ψ 2 + ψ d + ) ] 1 / 2 .
U 2 ± = x ep d x ' A ± ( x ' ) exp ( ik ψ ± ( x ' ) )
A ± = A 0 ± cos θ exp ( i π / 4 ) ( k / 2 π ρ ) 1 / 2 , ψ ± = ρ + ψ 0 ± ,
U s 1,2 + = A ± ( x ) [ 2 π / i k x 2 ψ + ( x ) ] 1 / 2 exp ( ik ψ + ( x ) ) | x = x 1,2 .
U ep ± = A ± ( x ) [ ik x ψ ± ( x ) ] 1 exp ( ik ψ ± ( x ) ) | x = x ep .
U ~ U 1 + U 2 = U 1 + U GO 1 + + U GO 2 + + U d + + U d ,
U ~ U UGO + .

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