## Counterpropagating nondiffracting beams through reflection gratings

Optics Express, Vol. 15, Issue 21, pp. 14163-14170 (2007)

http://dx.doi.org/10.1364/OE.15.014163

Acrobat PDF (645 KB)

### Abstract

We investigate the counterpropagation of paraxial nondiffracting optical beams through a medium hosting a bulk reflection grating in the quasi-Bragg matching condition. The impact of the relative magnitude of the Bragg detuning parameter and the grating depth on the plane wave dispersion relation allows us to identify three distinct regimes where counterpropagation and interaction of nondiffracting beams show qualitatively different features, encompassing longitudinally invariant, periodic or exponential intensity profiles. In one of the identified regimes the dispersion relation is not monotonic and the consequent “longitudinal degeneracy” allows the investigation of new class of nondiffracting beams characterized by a double spectral ring profile.

© 2007 Optical Society of America

## 1. Introduction

*et al*. [2

2. J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. **58**, 1499–1501 (1987). [CrossRef] [PubMed]

3. D. McGloin, V. Garcs-Chvez, and K. Dholakia, “Interfering Bessel beams for optical micromanipulation,” Opt. Lett. **28**, 657–659 (2003). [CrossRef] [PubMed]

4. J. Arlt, K. Dholakia, J. Soneson, and E. M. Wright, “Optical dipole traps and atomic waveguides based on Bessel light beams,” Phys. Rev. A **63**063602 (2001). [CrossRef]

6. M. A. Bandres, J. C. Gutirrez-Vega, and S. Chvez-Cerda, “Parabolic nondiffracting optical wave fields,” Opt. Lett. **29**, 44–46 (2004). [CrossRef] [PubMed]

7. R. Piestun and J. Shamir, “Generalized propagation-invariant wave fields,” J. Opt. Soc. Am. A **15**, 3039–3044 (1998). [CrossRef]

8. J. Lu and J. F. Greenleaf, “Ultrasonic nondiffracting transducer for medical imaging,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control **37**, 438–447 (1990). [CrossRef] [PubMed]

9. A. Ciattoni, C. Conti, and P. Di Porto, “Vector electomagnetic X waves,” Phys. Rev. E **69**, 036608 (2004). [CrossRef]

10. J. N. Brittingham, “Focus waves modes in homogeneous Maxwell’s equations: Transverse electric mode,” J. Appl. Phys. **54**, 1179–1189 (1983). [CrossRef]

*f*(

**k**,

*ω*) = 0 is available, this proving useful for example in the construction of nondiffracting beams also in homogeneous anisotropic media [11

11. A. Ciattoni and C. Palma, “Nondiffracting beams in uniaxial media propagating orthogonally to the optical axis,” Opt. Commun. **224**, 175–183 (2003). [CrossRef]

*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

12. O. Manela, M. Segev, and D. N. Christodoulides, “Nondiffracting beams in periodic media,” Opt. Lett. **30**, 2611–2613 (2005). [CrossRef] [PubMed]

## 2. Transmission and reflection of paraxial beams

**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*exp(

_{n}*i*2

*πnz*/

*L*) is a periodic refractive index profile (of spatial period

_{g}*L*) describing a shallow reflection grating (

_{g}*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*, that is to say the relation

_{N}*k*(1 +

*δ*) =

*πN*/

*L*holds for a small detuning parameter |

_{g}*δ*| ≪ 1. Hence, we consider the electric field

*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*|exp(

_{N}*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

14. A. Ciattoni, C. Rizza, E. DelRe, and E. Palange, “Counterpropagating spatial Kerr soliton in reflection gratings,” Opt. Lett. **31**, 1507–1509 (2006) [CrossRef] [PubMed]

*ζ*= |

*c*|

_{N}*kz*along with the relative detuning parameter

*χ*=

*δ*/|

*c*|. The grating mediated coupling between the two beams is described by the terms

_{N}*A*

_{-}and

*A*

_{+}in the first and second of Eqs.(2), respectively, and far from the Bragg-matching condition they do not appear.

*A*

_{±}(

*ξ*,

*η*,

*ζ*) = exp[

*i*(

*κ*+

_{ξ}ξ*κ*) +

_{η}η*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

*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*), the same algebraic system yields the plane wave amplitudes

_{η}*U*is an arbitrary complex constant.

*β*| ≪ 1/|

*c*| and

_{N}*χ*≪ 1/|

*c*|, respectively, which in turn implies that grating shallowness favors both.

_{N}*ζ*= 0 and

*ζ*= Λ of the medium (of longitudinal length Λ), the fields read

*ρ*=

*ξ*

**e**̂

_{x}+

*η*

**e**̂

_{y},

*κ*=

*κ*

_{ξ}**e**̂

_{x}+

*κ*

_{η}**e**

_{y}and

*β*is obtained from the dispersion relation and it has, hereafter, positive real or imaginary part. Besides we have defined

*A*

_{+}(

*ρ*, 0) and

*A*

_{-}(

*ρ*, Λ) compatibly with the relations

*T*(0,

*κ*) = 1,

*R*(0,

*κ*) = 0.

## 3. Nondiffracting fields

*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})/(2

*πκ*

_{0})]∑

^{n=+∞}

_{n=-∞}

*i*

^{-n}

*v*

_{±}

^{(n)}exp(

*inθ*) (where

*κ*=

*κ*(cos

*θ*

**e**̂

_{x}+ sin

*θ*

**e**̂

_{y})), with arbitrary angular Fourier coefficients

*v*

_{±}

^{(n)}. Substituting these boundary spectra into Eqs.(4) and performing the radial integrals we obtain

*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*

**e**̂

_{x}+ sin φ

**e**̂

_{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

*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}.

*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*.

*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.

*χ*> 1 (see Fig. 1(a)) plane waves are homogeneous at any

*κ*and

*β*, always real, and the values

*c*| ≈ 10

_{N}^{-4},

*δ*≈ 10

^{-3},

*λ*≈ 0.5

*μ*m and

*n*

_{0}≈ 1 the dimensional maximum period is

*P*/(

_{max}*k*|

*c*|) ≈ 0.5 mm which is comparable with a typical medium length.

_{N}*χ*| < 1 (see Fig. 1(b)) the dispersion relation has a single homogeneous branch for

*β*) and whose width

*δ*= 0, the threshold transverse width between exponential and periodic nondiffracting beam is

*k*|

*c*|) ≈ 1mm.

_{N}*χ*< -1 (see Fig. 1(c)) the dispersion relation has two homogeneous branches located at

*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.

*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

*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

*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*holds, where

_{g}*k*is the component of its wavevector along the grating axis. The quasi-Bragg-matching condition

_{z}*k*(1 +

*δ*) =

*πN*/

*L*(see Section 1), which defines the detuning

_{g}*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**

_{⊥}being the component of the wavevector orthogonal to the

*z*-axis) is exactly Bragg-matched if

*k*(1 -

*k*

_{⊥}

^{2}/2

*k*

^{2}) =

*πN*/

*L*or, exploiting the quasi-Bragg matching condition, if the relation

_{g}*k*

_{⊥}

^{2}/2

*k*

^{2}= -

*δ*holds. In the dimensionless coordinates we have chosen this relation reads

*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

*χ*< -1 can be profitably exploited to investigate a further novel class of nondiffracting beams. The forementioned “longitudinal degeneracy” implies that the two values

*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

*β*spanning the range

*κ*

_{1}(

*β*

_{0}) and

*κ*

_{2}(

*β*

_{0}) implies that

*κ*

^{2}

_{1}+

*κ*

^{2}

_{2}= -2

*χ*which result into the relation 1/

*w*

^{2}

_{1}+ 1/

*w*

^{2}

_{2}= -2

*χ*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

*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

*same*propagation constant so that they strictly do not interfere. This field configuration is allowed by the property of the dispersion relation

*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

## References and links

1. | J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941) |

2. | J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. |

3. | D. McGloin, V. Garcs-Chvez, and K. Dholakia, “Interfering Bessel beams for optical micromanipulation,” Opt. Lett. |

4. | J. Arlt, K. Dholakia, J. Soneson, and E. M. Wright, “Optical dipole traps and atomic waveguides based on Bessel light beams,” Phys. Rev. A |

5. | J. C. Gutirrez-Vega, M. D. Iturbe-Castillo, and S. Chvez-Cerda, “Alternative formulation for invariant optical fields: Mathieu beams,” Opt. Lett. |

6. | M. A. Bandres, J. C. Gutirrez-Vega, and S. Chvez-Cerda, “Parabolic nondiffracting optical wave fields,” Opt. Lett. |

7. | R. Piestun and J. Shamir, “Generalized propagation-invariant wave fields,” J. Opt. Soc. Am. A |

8. | J. Lu and J. F. Greenleaf, “Ultrasonic nondiffracting transducer for medical imaging,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control |

9. | A. Ciattoni, C. Conti, and P. Di Porto, “Vector electomagnetic X waves,” Phys. Rev. E |

10. | J. N. Brittingham, “Focus waves modes in homogeneous Maxwell’s equations: Transverse electric mode,” J. Appl. Phys. |

11. | A. Ciattoni and C. Palma, “Nondiffracting beams in uniaxial media propagating orthogonally to the optical axis,” Opt. Commun. |

12. | O. Manela, M. Segev, and D. N. Christodoulides, “Nondiffracting beams in periodic media,” Opt. Lett. |

13. | A. Yariv and P. Yeh, Optical Waves in crystals (Wiley, New York, 1984). |

14. | A. Ciattoni, C. Rizza, E. DelRe, and E. Palange, “Counterpropagating spatial Kerr soliton in reflection gratings,” Opt. Lett. |

**OCIS Codes**

(050.1950) Diffraction and gratings : Diffraction gratings

(350.2770) Other areas of optics : Gratings

(350.7420) Other areas of optics : Waves

**ToC Category:**

Diffraction and Gratings

**History**

Original Manuscript: July 17, 2007

Revised Manuscript: August 20, 2007

Manuscript Accepted: August 20, 2007

Published: October 12, 2007

**Citation**

A. Ciattoni, C. Rizza, E. DelRe, and E. Palange, "Counterpropagating nondiffracting beams through reflection gratings," Opt. Express **15**, 14163-14170 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-21-14163

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### References

- J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941)
- J. Durnin, J. J. Miceli, and J. H. Eberly, "Diffraction-free beams," Phys. Rev. Lett. 58, 1499-1501 (1987). [CrossRef] [PubMed]
- D. McGloin, V. Garcs-Chvez, and K. Dholakia, "Interfering Bessel beams for optical micromanipulation," Opt. Lett. 28, 657-659 (2003). [CrossRef] [PubMed]
- J. Arlt, K. Dholakia, J. Soneson, and E. M. Wright, "Optical dipole traps and atomic waveguides based on Bessel light beams," Phys. Rev. A 63, 063602 (2001). [CrossRef]
- J. C. Gutirrez-Vega, M. D. Iturbe-Castillo, and S. Chvez-Cerda, "Alternative formulation for invariant optical fields: Mathieu beams," Opt. Lett. 25, 1492-1494 (2000).
- M. A. Bandres, J. C. Gutirrez-Vega, and S. Chvez-Cerda, "Parabolic nondiffracting optical wave fields," Opt. Lett. 29, 44-46 (2004). [CrossRef] [PubMed]
- R. Piestun and J. Shamir, "Generalized propagation-invariant wave fields," J. Opt. Soc. Am. A 15, 3039-3044 (1998). [CrossRef]
- J. Lu and J. F. Greenleaf, "Ultrasonic nondiffracting transducer for medical imaging," IEEE Trans. Ultrason. Ferroelectr. Freq. Control 37, 438-447 (1990). [CrossRef] [PubMed]
- A. Ciattoni, C. Conti, and P. Di Porto, "Vector electomagnetic X waves," Phys. Rev. E 69, 036608 (2004). [CrossRef]
- J. N. Brittingham, "Focus waves modes in homogeneous Maxwell’s equations: Transverse electric mode," J. Appl. Phys. 54, 1179-1189 (1983). [CrossRef]
- A. Ciattoni, and C. Palma, "Nondiffracting beams in uniaxial media propagating orthogonally to the optical axis," Opt. Commun. 224, 175-183 (2003). [CrossRef]
- O. Manela, M. Segev, and D. N. Christodoulides, "Nondiffracting beams in periodic media," Opt. Lett. 30, 2611- 2613 (2005). [CrossRef] [PubMed]
- A. Yariv, and P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).
- A. Ciattoni, C. Rizza, E. DelRe and E. Palange, "Counterpropagating spatial Kerr soliton in reflection gratings," Opt. Lett. 31, 1507-1509 (2006) [CrossRef] [PubMed]

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