## Creation and annihilation of phase singularities near a sub-wavelength slit

Optics Express, Vol. 11, Issue 4, pp. 371-380 (2003)

http://dx.doi.org/10.1364/OE.11.000371

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### Abstract

The anomalously-high transmission of light through sub-wavelength apertures is a phenomenon which has been observed in numerous experiments, but whose theoretical explanation is incomplete. In this article we present a numerical analysis of the power flow (characterized by the Poynting vector)of the electromagnetic field near a sub-wavelength sized slit in a thin metal plate, and demonstrate that the enhanced transmission is accompanied by the annihilation of phase singularities in the power flow near the slit.

© 2003 Optical Society of America

## 1. Introduction

1. J.F. Nye and M.V. Berry, “Dislocations in wave trains,” Proc. Roy. Soc. Lond. A **336**, 165–190 (1974). [CrossRef]

## 2. Singular optics of electromagnetic fields

*t*]).

_{S}of the Poynting vector is given by the pair of relations

**S**| is the modulus of

**S**. It follows from these equations that Φ

_{S}(

*x, z*)is singular whenever

**S**= 0. Electromagnetic systems which exhibit singularities of power flow have been known of for some time [6

6. A. Boivin, J. Dow, and E. Wolf, “Energy flow in the neighborhood of the focusof a coherent beam,” J. Opt. Soc. Am. **57**, 1171–1175 (1967). [CrossRef]

8. G.P. Karman, M.W. Beijersbergen, A. van Duijl, and J.P. Woerdman, “Creation and annihilation of phase singularities in a focal field,” Opt. Lett. **22**, 1503–1505 (1997). [CrossRef]

_{y}, it can be readily shown using Eq. (1) and Maxwell’s equations that

_{y}as an amplitude and phase,

_{y}correspond to vortices (also referred to as centers)of the power flow

**S**, around which the power flow circulates (see Fig. 1(b)). A center is referred to as right-handed (left-handed) if it is counterclockwise with respect to the positive (negative)

*y*-axis. It is to be noted that in all the figures in this article the y-axis points into the page; a left-handed or right-handed center therefore corresponds to a positive or negative phase vortex, respectively.

_{y}correspond to saddle points of the power flow, as illustrated in Fig. 1(d). A phase maximum (Fig. 1(e)) of Ê

_{y}corresponds to a sink of power flow (Fig. 1(f)), and a phase minimum corresponds to a source of power flow; it is to be noted that sinks and sources do not occur in free space.

_{y}, and several conserved quantities can be associated with each topological feature. The first of these is the so-called

*topological charge s*

_{E}of the field, defined as the integral of ∇Φ

_{E}around a closed loop enclosing the feature such that

*C*is a closed counterclockwise path of winding number 1. It can be shown that the topological charge of a given phase singularity takes on a unique positive or negative integer value, independent of the choice of the enclosing path

*C*. Likewise, the topological charge of a phase saddle, maximum or minimum is always zero.

*topological index t*

_{E}, which is defined as the topological charge of the phase singularities of the vector field ∇Φ

_{E}. It can be shown that for a positive or negative vortex

*t*

_{E}= +1, while for a phase saddle

*t*

_{E}= -1. The topological index of a phase maximum or minimum is

*t*

_{E}= +1.

*s*

_{S}and index

*t*

_{S}can also be associated with the phase Φ

_{S}of the power flow. It follows directly from Eq. (5)t hat the topological

*charge*of

**S**for a given feature is equal to the topological

*index*of Ê

_{y}. The topological charge of a vortex of power flow is therefore

*s*

_{S}= +1 regardless of whether it is a positive or negative vortex of Ê

_{y}. Similarly, the topological charge of a saddle point of power flow is

*s*

_{S}= -1, and the topological charge of a source or sink is

*s*

_{S}= +1. A topological index may be defined for the singularities of power flow, but is not necessary for our interests and will not be considered here.

*s*

_{E}= +1,

*t*

_{E}= +1), a negative vortex (

*s*

_{E}= -1,

*t*

_{E}= +1), and two phase saddles (

*s*

_{E}= 0,

*t*

_{E}= -1 for each). This event may also be described in terms of the field of power flow as the creation (annihiliation)o f two centers of opposite direction (

*s*

_{S}= +1 for each) with two saddle points (

*s*

_{S}= -1). Other, more complex, events are possible, but are not typical.

## 3. Integral equation solution for the electromagnetic field near a slit

*d*and permittivity ε

_{plate}from the negative

*z*-direction. A single slit of width

*w*, infinitely long in the

*y*-direction, is present in the plate. Because the system is invariant with respect to

*y*-translations, we may treat the problem as two-dimensional, with relevant coordinates

*x*and

*z*.

^{inc}, and the scattered field, Ê

^{scatt}. The incident field is here taken to be the field that would occur in the absence of the slit in the plate; it can readily be calculated analytically by use of the electromagnetic boundary conditions.

9. T.D. Visser, H. Blok, and D. Lenstra, “Theory of polarization-dependent amplification in a slab waveguide with anisotropic gain and losses,” IEEE J. Quant. Elect. **35**, 240–249 (1999). [CrossRef]

*i*th component of the total field, denoted by

*Ê*, satisfies an integral equation of the form

_{i}(*x*,*z*)_{plate}is the difference between the vacuum permittivity and the permittivity of the metal plate,

*D*. For points which lie within the slit, Eq. (7) is a Fredholm equation of the second kind for Ê, which can be solved numerically by the collocation method with piecewise-constant basis functions. The electric field outside the domain of the slit may then be calculated by substituting this solution back into Eq. (7). With the electric field determined everywhere, the magnetic field everywhere follows directly from Maxwell’s equations. The Poynting vector may then be calculated using Eq. (1).

**S**over the slit, and the second is the difference of the normal components of the actual time-averaged Poynting vector and that of the Poynting vector in the absence of the slit,

**S**

^{inc}, integrated over the dark side of the plate (not the region of the slit). The result is normalized by the normal component of

**S**

^{(0)}, the Poynting vector of the field emitted by the source and impinging on the slit, i.e.

## 4. Phase singularities near sub-wavelength slits

*T*= 1.11. When the slit width is increased in a continuous manner, the four singularities below the slit (in the region indicated by the box)mo ve together and finally annihilate each other. After the annihilation takes place, a smoother field of power flow results, corresponding to a greater transmission coefficient. This process is shown in Fig. 4. It can be seen that the transmission takes on a maximum value of

*T*= 1.33 when the slit width is

*w*= 0.5λ.

*w*and the corresponding power transmission coefficient

*T*are also shown. Three groups of singularities obstruct the power flow directly in front of the slit; these groups each annihilate at

*w*= 0.42λ,

*w*= 0.43λ, and

*w*= 0.46λ. Only when they have disappeared does the transmission show anomalously high behavior.

## Acknowledgements

## References and links

1. | J.F. Nye and M.V. Berry, “Dislocations in wave trains,” Proc. Roy. Soc. Lond. A |

2. | J.F. Nye, |

3. | M.S. Soskin and M.V. Vasnetsov, “Singular Optics”, in |

4. | T.W. Ebbesen, H.J. Lezec, H.F. Ghaemi, T. Thio, and P.A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature |

5. | T. Thio, K.M. Pellerin, R.A. Linke, H.J. Lezec, and T.W. Ebbesen, “Enhanced light transmission through a single subwavelength aperture,” Opt. Lett. |

6. | A. Boivin, J. Dow, and E. Wolf, “Energy flow in the neighborhood of the focusof a coherent beam,” J. Opt. Soc. Am. |

7. | F. Landstorfer, H. Meinke, and G. Niedermair, “Ringförmiger Energiewirbel im Nahfeld einer Richtantenne,” Nachrichtentechn. Z. |

8. | G.P. Karman, M.W. Beijersbergen, A. van Duijl, and J.P. Woerdman, “Creation and annihilation of phase singularities in a focal field,” Opt. Lett. |

9. | T.D. Visser, H. Blok, and D. Lenstra, “Theory of polarization-dependent amplification in a slab waveguide with anisotropic gain and losses,” IEEE J. Quant. Elect. |

10. | H.F. Schouten, T.D. Visser, D. Lenstra, and H. Blok, “Light transmission through a sub-wavelength slit: waveguiding and optical vortices,” Phys. Rev. E, in press. |

11. | D.E. Gray, ed., |

**OCIS Codes**

(050.1220) Diffraction and gratings : Apertures

(050.1960) Diffraction and gratings : Diffraction theory

(260.2110) Physical optics : Electromagnetic optics

**ToC Category:**

Research Papers

**History**

Original Manuscript: January 27, 2003

Revised Manuscript: February 17, 2003

Published: February 24, 2003

**Citation**

Hugo Schouten, Taco Visser, Greg Gbur, Daan Lenstra, and Hans Blok, "Creation and annihilation of phase singularities near a sub-wavelength slit," Opt. Express **11**, 371-380 (2003)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-4-371

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### References

- J.F. Nye and M.V. Berry, �??Dislocations in wave trains,�?? Proc. Roy. Soc. Lond. A 336, 165-190 (1974). [CrossRef]
- J.F. Nye, Natural Focusing and the Fine Structure of Light (Institute of Physics, Bristol, 1999).
- M.S. Soskin and M.V. Vasnetsov, �??Singular Optics,�?? in Progress in Optics 42, ed. E. Wolf (Elsevier, Amsterdam, 2001), 219-276.
- T.W. Ebbesen, H.J. Lezec, H.F. Ghaemi, T. Thio and P.A. Wol., �??Extraordinary optical transmission through sub-wavelength hole arrays,�?? Nature 391, 667-669 (1998). [CrossRef]
- T. Thio, K.M. Pellerin, R.A. Linke, H.J. Lezec and T.W. Ebbesen, �??Enhanced light transmission through a single subwavelength aperture,�?? Opt. Lett. 26, 1972-1974 (2001). [CrossRef]
- A. Boivin, J. Dow and E. Wolf, �??Energy flow in the neighborhood of the focusof a coherent beam,�?? J. Opt. Soc. Am. 57, 1171-1175 (1967). [CrossRef]
- F. Landstorfer, H. Meinke and G. Niedermair, �??Ringformiger Energiewirbel im Nahfeld einer Richtantenne,�?? Nachrichtentechn. Z. 25, 537-576 (1972).
- G.P. Karman, M.W. Beijersbergen, A. van Duijl, and J.P. Woerdman, �??Creation and annihilation of phase singularities in a focal fleld,�?? Opt. Lett. 22, 1503-1505 (1997). [CrossRef]
- T.D. Visser, H. Blok and D. Lenstra, �??Theory of polarization-dependent amplification in a slab waveguide with anisotropic gain and losses,�?? IEEE J. Quantum Electron. 35, 240-249 (1999). [CrossRef]
- H.F. Schouten, T.D. Visser, D. Lenstra and H. Blok, �??Light transmission through a sub-wavelength slit: waveguiding and optical vortices,�?? Phys. Rev. E, in press.
- D.E. Gray, ed., American Institute of Physics Handbook (McGraw-Hill, New York, 1972, 3rd edition).

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