1. Introduction

The broadening of wave packets, or diffraction, is such a general effect in wave dynamics that the situations were it is absent attract a relevant research interest, especially in connection with possible applications where signal spreading is detrimental. In optics, perhaps the most striking example is that of solitons, where complete diffraction compensation mediated by nonlinear self-action leads to localized waves that do not suffer spatial or temporal spreading. It is possible to consider undistorted fields also in the linear domain. For example, it is well known that Bessel beams in a homogeneous medium are monochromatic nondiffracting fields, as predicted by Stratton [1] and experimentally observed by Durnin et al. [2]. Recently, Bessel beams have been exploited both for optical manipulation of micron-sized particles [3] and for optical trapping of atoms [4]. A variety of diffraction-free beams has also been investigated in the form of Matthieu [5], parabolic [6] and rotating beams [7]. In the polychromatic realm, the family of propagation invariant fields also encompasses X waves [8, 9], which are rigidly travelling wave packets, and focus waves modes [10] whose intensity profiles do not suffer any change during propagation.

Bessel beams in homogeneous and isotropic media do not suffer diffraction since they are the result of the superposition of those plane waves that have the same propagation constant, and this leads to a factorization of the field into a plane wave carrier and a transverse profile. This specific spectral superposition technique can in fact be exploited also whenever a plane wave dispersion relation f(k,ω) = 0 is available, this proving useful for example in the construction of nondiffracting beams also in homogeneous anisotropic media [11]. In turn, this generalization to inhomogeneous media is generally hampered by the fact that monochromatic modes are no longer plane waves. However, if the refractive index has a two-dimensional profile n(x,y) (i.e. the medium is translation invariant in the direction of z) the modes of the field factorizes into a plane wave carrier exp(iβz) (where β propagation constant) and a transverse mode profile U(x,y) and, if degeneracy occurs, different modes characterized by the same propagation constant can be superimposed to yield a nondiffracting beam [12].

In this Letter we investigate the propagation of diffraction-free beams through a reflection grating, i.e. a refractive index exhibiting a periodic modulation along the propagation direction of the optical field. In the paraxial regime, we represent the field as the superposition of two counterpropagating beams whose interaction is mediated by a shallow grating. Exploiting the quasi-Bragg matching between the optical plane wave carriers and the grating vector we consider the coupled mode equations for the slowly-varying envelopes of the counterpropagating beams. Within this description the counterpropagating beams have eigenmodes which are coupled plane waves, the resulting dispersion relation exhibiting three different regimes tunable by means of the relative magnitude of the Bragg-matching detuning parameter and the grating depth. Therefore, the design of nondiffracting beams is attained by suitably superimposing those plane waves with a prescribed component of their wave vectors along the propagation direction. We analytically investigate the resulting nondiffracting beams by relating them to their appropriate input profiles launched at the opposite facets of the medium, thus describing their reflection and transmission properties. It is worth noting that the nontrivial structure of the plane wave dispersion relation yields a variety of structurally different nondiffracting beams in the three regimes. We are able to identify the conditions in which each one of the counterpropagating beam is longitudinally invariant or exhibiting a periodic or exponential longitudinal profile. In addition, the degeneracy of the propagation constant resulting from the structure of the dispersion relation in one of the mentioned regimes allows us to consider nondiffracting beams characterized by a double ring spectral structure which result, in the physical space, into the superposition of two Bessel beams of different width.

 

Fig. 1. Plot of the plane wave dispersion relation β ² = (κ ² + χ)² - 1 in the three different regimes: (a) χ > 1, (b) |χ| < 1 and (c) χ < -1. We have used black and red lines to report pure real (homogeneous plane waves) and pure imaginary (inhomogeneous plane waves) β, respectively. Note in (c) the longitudinal degeneracy occurring for β(κ ₁) = β(κ ₂). The labels reported along the κ and β axes are relevant points characterizing the dispersion curves.

Download Full Size | PDF

2. Transmission and reflection of paraxial beams

The complex amplitude E(r) of the monochromatic electric field Re[E(r)exp( -iωt)] propagating through a medium hosting a linear grating is described by the Maxwell’s equations ∇ × ∇ × E = k ²(1 + δn/n ₀)E, where n ₀ is a uniform refractive index background, k = ωn ₀/c and δn(z) = n ₀∑^n=+∞ _n=-∞ c_nexp(i2πnz/L_g) is a periodic refractive index profile (of spatial period L_g) describing a shallow reflection grating (δ_n ≪ n ₀). Here we assume c ₀ = 0 since a uniform contribution to δn can always be added to n ₀ and we focus on a real modulation for which c _-n = c_n ^*. In order to describe a physical situation where two optical beams counter-propagate through the grating (along the z- axis) mediating their interaction, we consider a paraxial scheme where the two beams plane wave carriers are exp(ikz) and exp(-ikz) and are quasi-Bragg-matched with the N-th Fourier grating component c_N, that is to say the relation k(1 + δ) = πN/L_g holds for a small detuning parameter |δ| ≪ 1. Hence, we consider the electric field

where A ₊ and A _- are slowly varying envelopes associated to the counterpropagating beams, ϕ is the phase of the N-th Fourier grating coefficient c_N = |c_N|exp(iϕ) and we have chosen the linear polarization along the x-axis. Substituting the field of Eq.(1) into Maxwell’s equations and exploiting both paraxial and Bragg resonance approximation (which amounts to gathering together the resultant synchronous terms and, for shallow grating, discarding higher order terms) we obtain the coupled mode equations [13, 14]

where we have introduced the dimensionless coordinates $\xi =\sqrt{2\mid c_N\mid }\mathrm{kx}$, $\eta =\sqrt{2\mid c_N\mid }\mathrm{ky}$ and ζ = |c_N|kz along with the relative detuning parameter χ = δ/|c_N|. The grating mediated coupling between the two beams is described by the terms A _- and A ₊ in the first and second of Eqs.(2), respectively, and far from the Bragg-matching condition they do not appear.

Plane wave solution A _±(ξ,η, ζ) = exp[i(κ_ξξ + κ_ηη) + βζ)]U _±, once inserted into the coupled mode Equations (2), yield an algebraic homogeneous system for the amplitudes U ± which has nontrivial solution if the dispersion relation

is satisfied, where $\kappa =\sqrt{{\kappa }_{\xi }^2+{\kappa }_{\eta }^2}$. Note that, since β ² can be either positive or negative, both homogeneous (β real) and inhomogeneous (β imaginary) plane waves occur. Besides, the dependence of the dispersion relation on the detuning parameter χ yields three different regimes for the optical phenomenology (see Fig. 1). Once β is known from the dispersion relation for a given (κ_ξ, κ_η), the same algebraic system yields the plane wave amplitudes $U_{+}=U\sqrt{{\kappa }^2+\chi -\beta }$ and $U_{-}=U\sqrt{{\kappa }^2+\chi +\beta }$ where U is an arbitrary complex constant.

Note that paraxial approximation and quasi-Bragg matching condition are guaranteed by the conditions |β| ≪ 1/|c_N| and χ ≪ 1/|c_N|, respectively, which in turn implies that grating shallowness favors both.

The general solution of Eqs.(2) can be obtained by suitably superimposing the considered plane waves and, focusing on the situation where the two counterpropagating beams are launched through the opposite facets ζ = 0 and ζ = Λ of the medium (of longitudinal length Λ), the fields read

where ρ = ξ ê_x + η ê_y, κ = κ_ξ ê_x + κ_η e _y and

are the well-known coefficient describing grating reflection and transmission of plane-waves. Here β is obtained from the dispersion relation and it has, hereafter, positive real or imaginary part. Besides we have defined $V_{+}\left(\kappa \right)=\int \frac{d^2\rho }{{\left(2\pi \right)}^2}\exp \left(-\mathrm{i\kappa }\bullet \rho \right)A_{+}(\rho ,0)$ and $V_{-}\left(\kappa \right)=\int \frac{d^2\rho }{{\left(2\pi \right)}^2}\exp (-\mathrm{i\kappa }\bullet \rho )A_{-}(\rho ,\Lambda )$ to be the Fourier spectra of the input field profiles A ₊(ρ, 0) and A _-(ρ, Λ) compatibly with the relations T(0,κ) = 1, R(0,κ) = 0.

3. Nondiffracting fields

From Eqs.(4) it is evident that, even in the presence of a grating causing reflection, the counterpropagating beams generally broaden during propagation as a consequence of the continuous superposition of plane-wave modes with different β. This implies that, generalizing the procedure adopted in longitudinally translation invariant media, we can consider here nondiffracting beams by simply selecting the modes corresponding to a degenerate β. This can be done by launching at ζ = 0 and ζ = Λ two beams whose Fourier spectra are not vanishing only at a ring of radius κ ₀, or V _±(κ) = [δ(κ-κ ₀)/(2πκ ₀)]∑^n=+∞ _n=-∞ i ^-n v _± ⁽ⁿ⁾ exp(inθ) (where κ = κ(cos θ ê_x + sin θ ê_y)), with arbitrary angular Fourier coefficients v _± ⁽ⁿ⁾. Substituting these boundary spectra into Eqs.(4) and performing the radial integrals we obtain

where the boundary field distributions are A ₊(ρ,0)= ∑^n=+∞ _n=-∞ v ₊ ⁽ⁿ⁾ exp(inθ)J_n(κ ₀ ρ), A _-(ρ,Λ) = ∑^n=+∞ _n=-∞ v _- ⁽ⁿ⁾ exp(inθ)J_n(κ ₀ ρ), ρ = ρ(cos φ ê_x + sin φê_y) and J_n(ξ) is the Bessel function of the first kind of order n. The nondiffracting structure of the fields in Eqs.(6), which are generally not propagation invariant, reveals in the fact that each contribution is the product of a function of ζ (characterizing transmission and reflection) and a transverse profile (the externally launched beams). Considering the case $v_{+}^{\left(n\right)}=v\sqrt{{\kappa }_0^2+\chi -{\beta }_0}{\delta }_{n,0}$ and $v_{-}^{\left(n\right)}=v\sqrt{{\kappa }_0^2+\chi +{\beta }_0}\exp \left(i{\beta }_0\Lambda \right){\delta }_{n,0}$, Eqs.(6) yield $A_{+}(\rho ,\zeta )=v\sqrt{{\kappa }_0^2+\chi -{\beta }_0}\exp \left(i{\beta }_0\zeta \right)J_0\left({\kappa }_0\rho \right)$ and $A_{-}(\rho ,\zeta )=v\sqrt{{\kappa }_0^2+\chi +{\beta }_0}\exp \left(i{\beta }_0\zeta \right)J_0\left({\kappa }_0\rho \right)$ which are, for real and imaginary β ₀, genuinely propagation invariant and exponentially decaying Bessel beams, respectively, for the two counterpropagating fields and they are excited by launching at ζ = 0 and ζ = Λ two Bessel beams whose peak intensity ratio is |A ₊ (0,0)/A _- (0, Λ)|² = |β ₀ ² + χ - β ₀|² (fixed by κ ₀). Their propagation invariance or exponential decay stem from the longitudinal compensation between a transmitted beam (say T(ζ)A ₊(ρ,0) in A ₊) and its co-propagating field arising from the reflection of the counterpropagating beam (say R(ζ)A _-(ρ,Λ) in A ₊). From an equivalent point of view, the considered boundary field profiles select only those modal plane wave propagating through the grating characterized by the single β ₀ propagation constant instead of the pair β ₀ and -β ₀.

In a reflection scheme where v ₊ ⁽ⁿ⁾ = v ₊ ⁽⁰⁾ and v _- ⁽ⁿ⁾ = 0, Eqs.(6) yield A ₊(ρ,ζ) = T(ζ, κ ₀)v ₊ ⁽⁰⁾ J ₀(κ ₀ ρ) and A _-(ρ, ζ) = R(Λ - ζ, κ ₀)v ₊ ⁽⁰⁾ J ₀(κ ₀ ρ) which implies that if a single nondiffracting beam impinges onto a facet of the medium, it gives rise to a transmitted and reflected nondiffracting beam showing the same transverse profile of the input field and whose longitudinal dynamics is given by T and R.

As far as the longitudinal dynamics, there are two main situations according to whether the propagation constant β ₀ = β(κ ₀) is real or imaginary where the functions T and R of Eqs.(5) show periodic or exponential behavior in ζ, respectively. It is worth noting that the detuning parameter χ plays a major role on the structure of the counterpropagating beams because of the three regimes characterizing the dispersion relation.

(a) For χ > 1 (see Fig. 1(a)) plane waves are homogeneous at any κ and β, always real, and the values $\mid \beta \mid <\sqrt{{\chi }^2-1}$ are forbidden (forbidden gap) and, as a consequence, nondiffracting beams can have arbitrary width and solely exhibit periodic (or propagation invariant) longitudinal profile with a period $P=\frac{2\pi }{\beta }<P_{max}=\frac{2\pi }{\sqrt{{\chi }^2-1}}$. In a typical experimental setup where |c_N| ≈ 10^-4, δ ≈ 10^-3, λ ≈ 0.5 μm and n ₀ ≈ 1 the dimensional maximum period is P_max/(k|c_N|) ≈ 0.5 mm which is comparable with a typical medium length.

(b) For |χ| < 1 (see Fig. 1(b)) the dispersion relation has a single homogeneous branch for $\kappa >\sqrt{1-\chi }$ and a inhomogeneous branch for $0<\kappa <\sqrt{1-\chi }$. Correspondingly there are longitudinally periodic nondiffracting beams for $\kappa >\sqrt{1-\chi }$ whose period can attain any value (since there is no forbidden band for β) and whose width $w=\left(\frac{1}{\kappa }\right)<\tilde{w}=\frac{1}{\sqrt{1-\chi }}$. In addition, for a beam width greater than $\tilde{w}\left(\kappa <\sqrt{1-\chi }\right)$ longitudinally exponential nondiffracting beams occur and are characterized by longitudinal length $\frac{1}{\mid \beta \mid }>\frac{1}{\sqrt{1-{\chi }^2}}$. In the same conditions as in the numerical examples of (a) and at exact Bragg-matching δ = 0, the threshold transverse width between exponential and periodic nondiffracting beam is $\frac{\tilde{w}}{\left(k\sqrt{2\mid c_N\mid }\right)}\approx 0.5$ mm whereas for evanescent nondiffracting beams the minimum longitudinal length is 1/(k|c_N|) ≈ 1mm.

(c) For χ < -1 (see Fig. 1(c)) the dispersion relation has two homogeneous branches located at $0<\kappa <\sqrt{-\chi -1}$ and $\kappa >\sqrt{-\chi +1}$, respectively, together with a single inhomogeneous branch for $\sqrt{-\chi -1}<\kappa <\sqrt{-\chi +1}$. Therefore, longitudinally periodic nondiffracting beams occur if their width is either greater than $\frac{1}{\sqrt{-\chi -1}}$ (with longitudinal period $\frac{2\pi }{\beta }>\frac{1}{\sqrt{{\chi }^2-1}}$ or smaller than $\frac{1}{\sqrt{-\chi +1}}$ (with longitudinal period which can attain any value) whereas exponential nondiffracting beams (with longitudinal length 1/|β| < 1) can be excited if their width is between these two thresholds. Also in this case, the dimensional threshold lengths are of the order of a millimeter for the considered numerical example.

 

Fig. 2. Example of counterpropagating nondiffracting beams (movie: counterprop.avi, 3154 KB) of Eqs.(1) for v ₊ ⁽ⁿ⁾ = δ _n,1 v ₊ ⁽¹⁾ and v _- ⁽ⁿ⁾ = δ _n,-1 v _- ^(-1) (with v _- ^(-1) = 2v ₊ ⁽⁺¹⁾) where the periodic regime has been selected with a longitudinal period 2π/β = 3Λ/4. [Media 1]

Download Full Size | PDF

In general, each counterpropagating beam is nondiffracting and exhibiting a slow (as compared to the wavelength) longitudinal dynamic. In Fig. 2 we report an example where Eqs.(6) are specialized to the case where two Bessel beams A ₊(ρ,0) = v ₊ ⁽¹⁾ exp(iφ)J ₁(κ ₀ ρ) and A _-(ρ,Λ) = v _- exp(-iφ)J ₁(κ ₀ ρ) with opposite topological charge and in the periodic regime are launched into the crystal.

4. Double spectral radius nondiffracting beams

Nondiffracting beams described in the above section result from the superposition of plane waves whose transverse wave vectors lie on a circle (single spectral radius nondiffracting beams) as a consequence of the rotational degeneracy of the dispersion relation β = β(κ). In homogeneous media where β(κ) is generally a monotonic function, there is no other degeneracy which can be exploited to construct different nondiffracting beams. The situation is different in periodic media since the coupling of plane waves is the physical ingredient yielding a dispersion relation characterized by nontrivial topological features. In the case of the reflection grating we are considering, the dispersion relation is not monotonic within the third considered regime χ < -1 (see Fig. 1(c)) and both its homogeneous and inhomogeneous branches, in addition to the rotational degeneracy, show a “longitudinal” degeneracy if $\beta <\sqrt{{\chi }^2-1}$ and |β| < 1, respectively. The physical origin of such a non-monotonic behavior of the dispersion relation is easily grasped by noting that a plane wave is exactly Bragg-matched with the grating whenever the relation k_z = πN/L_g holds, where k_z is the component of its wavevector along the grating axis. The quasi-Bragg-matching condition k(1 + δ) = πN/L_g (see Section 1), which defines the detuning δ, physically implies that a plane wave travelling along the z-axis (k_z = k) is not exactly matched with the grating. On other hand, a paraxial plane wave not travelling along the z-axis with $k_z=\sqrt{k^2-k_{\perp }^2}\simeq k\left(\frac{1-k_{\perp }^2}{2k^2}\right)$ (k _⊥ being the component of the wavevector orthogonal to the z-axis) is exactly Bragg-matched if k(1 -k _⊥ ²/2k ²) = πN/L_g or, exploiting the quasi-Bragg matching condition, if the relation k _⊥ ²/2k ² = -δ holds. In the dimensionless coordinates we have chosen this relation reads $\kappa =\sqrt{-\chi }$ from which it is obvious that in the regime χ < -1 an angled plane wave exists which is exactly Bragg-matched and reflected by the grating (see Fig. 1(c) where it is shown that the imaginary part of β attains its maximum at the point $\kappa =\sqrt{-\chi }$). As a direct consequence, both on the left and the right of the point $\kappa =\sqrt{-\chi }$ the dispersion relation has to exhibit homogeneous branches stemming from the departure from Bragg-matching and this explains the non-monotonic behavior of the dispersion relation.

The non-monotonic behavior of the dispersion relation for χ < -1 can be profitably exploited to investigate a further novel class of nondiffracting beams. The forementioned “longitudinal degeneracy” implies that the two values ${\kappa }_1=\sqrt{-\chi -\sqrt{{\beta }^2+1}}$ and ${\kappa }_2=\sqrt{-\chi -\sqrt{{\beta }^2+1}}$ correspond to the same real value of β ₀ = β(κ ₁) = β(κ ₂) (see Fig. 1(c)) so that nondiffracting beams can be obtained with a double ring spectral shape. For example, for the spectral field distributions $V_{+}\left(\kappa \right)={\sum }_{j=1}^{j=2}\left[\frac{\sqrt{{\kappa }_j^2+\chi -{\beta }_0}}{\left(2\pi {\kappa }_j\right)}\right]\delta \left(\kappa -{\kappa }_j\right)v_j$ and $V_{-}\left(\kappa \right)=\exp \left(i{\beta }_0\Lambda \right){\sum }_{j=1}^{j=2}\left[\frac{\sqrt{{\kappa }_j^2+\chi +{\beta }_0}}{\left(2\pi {\kappa }_j\right)}\right]\delta \left(\kappa -{\kappa }_j\right)v_j$, Eqs.(4) yield

so that each counterpropagating beams is strictly propagation invariant and it results as the superposition of two Bessel beams of different radii. Double spectral radius nondiffracting beams of Eqs.(7) can be regarded as a family of fields parameterized by β spanning the range $0\le \beta \le \sqrt{{\chi }^2-1}$. Note that, the expressions of κ ₁(β ₀) and κ ₂(β ₀) implies that κ ² ₁ + κ ² ₂ = -2χ which result into the relation 1/w ² ₁ + 1/w ² ₂ = -2χ joining the widths w ₁ = 1/κ ₁ and the w ₂ = 1/κ ₂ of the two Bessel components appearing in each of the two nondiffracting beams of Eqs.(7). In the limiting β = 0 case the two Bessel components attain their minimum $w_1=\frac{1}{\sqrt{-\chi -1}}$ and maximum $w_2=\frac{1}{\sqrt{-\chi +1}}$ allowed widths, respectively and the counterpropagating beams are identical A ₊(ρ, ζ) = A _-(ρ, ζ), in agreement with the observation that β (as a common correction to the wavevectors k(1 + δ) and -k(1 + δ) of the carriers) marks a structural difference between the two counterpropagating beams. Of some interest is the opposite limiting case $\beta =\sqrt{{\chi }^2-1}$ at which the fields of Eqs.(7) turn into

where each beam is the superposition of a plane wave and a Bessel component sharing the same propagation constant so that they strictly do not interfere. This field configuration is allowed by the property of the dispersion relation ${\kappa }_1\left({\beta }_0=\sqrt{{\chi }^2-1}\right)=0$, so that the infinitely broad first Bessel component coincides with a plane wave. Note that in Eqs.(8) the ratios between the amplitudes of the counterpropagating plane waves and the counterpropagating Bessel components are fixed by the grating (through the parameter χ) as usual in the reflection geometry we are considering. On the other hand, the parameters v ₁ and v ₂ are arbitrary and can be chosen to adjust two of the four amplitudes appearing in the fields of Eqs.(8).

5. Conclusions

Counterpropagating nondiffracting beams can be excited through a reflection grating and their interaction (mediated by grating reflectivity) is generally responsible for a slow longitudinal dynamic not accompanied by a beam broadening. The structure of the plane-wave dispersion relation essentially determines the possible kinds of nondiffracting beams (longitudinally invariant, with periodic or exponential intensity profiles) and, in this, the detuning parameter from Bragg-matching plays a dominant role. In addition, the non monotonic behavior of the dispersion relation for certain values of the detuning parameters allows the investigation of peculiar family of nondiffracting beams characterized by a double ring spectral structure.