## Highly nonparaxial (1+1)-D subwavelength optical fields

Optics Express, Vol. 18, Issue 8, pp. 7617-7624 (2010)

http://dx.doi.org/10.1364/OE.18.007617

Acrobat PDF (311 KB)

### Abstract

A general approach for describing (1 + 1)-D subwavelength optical field whose waist is much smaller than the wavelength is presented. Exploiting the vectorial Rayleigh-Sommerfeld diffraction theory, a suitable expansion in the ratio between the beam waist and the wavelength allows us to prove the a (1+1)D highly nonparaxial field is generally the product of a cylindrical wave carrier and an envelope which is angularly slowly varying. We apply our general approach to the case of highly nonparaxial Hermite-Gaussian beams whose description is fully analytical.

© 2010 Optical Society of America

## 1. Introduction

1. J. B. Pendry, “Negative Refraction Makes a Perfect Lens,” Phys. Rev. Lett. **85**, 3966 (2000). [CrossRef] [PubMed]

2. J. Takahara, S. Yamagishi, H. Taki, A. Morimoto, and T. Kobayashi, “Guiding of a one-dimensional optical beam with nanometer diameter,” Opt. Lett. **22**, 475 (1997). [CrossRef] [PubMed]

3. H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming Light from a Subwavelength Aperture,” Science **297**, 820 (2002). [CrossRef] [PubMed]

4. L. Verslegers, P.B. Catrysse, Z. Yu, J. S. White, E. S. Barnard, M. L. Brongersma, and S. Fan, “Planar Lenses Based on Nanoscale Slit Arrays in a Metallic Film,” Nano Lett. **9**, 235 (2009). [CrossRef]

5. A. Ciattoni, B. Crosignani, and P. Di Porto, “Vectorial free-space optical propagation: a simple approach for generating all-order nonparaxial corrections,” Opt. Commun. **177**9 (2000). [CrossRef]

6. A. Yu Savchencko and B. Ya Zel’dovich, “Wave propagation in a guiding structure: one step beyond the paraxial approximation,” J. Opt. Soc. Am. B **13**, 273 (1996). [CrossRef]

7. A. Ciattoni, B. Crosignani, and P. Di Porto, “Vectorial analytical description of propagation of a highly nonparaxial beam,” Opt. Commun. **202**, 17 (2002). [CrossRef]

11. E. Moreno, F. J. García-Vidal, and L. Martín-Moreno, “Enhanced transmission and beaming of light via photonic crystal surface modes,” Phys. Rev. B **69**, 121402 (2004). [CrossRef]

*w*is much smaller than the wavelength

*λ*and it is based on the expansion of the exact vectorial Rayleigh-Sommerfeld diffraction formulas in the parameter

*w*/

*λ*≪ 1. By its general derivation, our scheme is such that the smaller the waist, the higher the accuracy of the description so that it proves to be suitable for describing field configurations occurring in nano-optical applications. We show that (1 + 1)-D highly nonparaxial field is the product between a cylindrical wave carrier and a slowly varying envelope as opposed to the (1 + 2)-D situation [7

7. A. Ciattoni, B. Crosignani, and P. Di Porto, “Vectorial analytical description of propagation of a highly nonparaxial beam,” Opt. Commun. **202**, 17 (2002). [CrossRef]

7. A. Ciattoni, B. Crosignani, and P. Di Porto, “Vectorial analytical description of propagation of a highly nonparaxial beam,” Opt. Commun. **202**, 17 (2002). [CrossRef]

## 2. One-dimensional highly nonparaxial fields

*Re*[

**E**(

*x*,

*z*)

*e*

^{-iωt}] (of frequency

*ω*) through a homogenous medium whose refractive index is

*n*, is described by the Rayleigh-Sommerfeld diffraction formulas

**E**

_{⊥}=

*E*

_{x}**e**̂

_{x}+

*E*

_{y}**e**̂

_{y}[14]. Consider the field [7

**202**, 17 (2002). [CrossRef]

*y*-component of the magnetic field is

*B*= (

_{y}*i*/

*ω*)(

*∂*

^{2}

_{z}+

*∂*

^{2}

_{x})

*F*so that, exploiting the fact that

_{x}**F**

_{⊥}satisfies the 2D Helmholtz equation (

*∂*

^{2}

_{x}+

*∂*

^{2}

_{z}+

*k*

^{2})

**F**

_{⊥}= 0, we have

*F*.

_{x}**202**, 17 (2002). [CrossRef]

**E**

_{⊥}(

*x*, 0) has a width

*w*much smaller than the wavelength

*λ*. It is convenient to replace in the field of Eq. (2) the Hankel function with its asymptotic expansion thus getting

*kR*> 1. Since the integrand of Eq. (5) is not negligible only for ∣

*x*′∣/

*λ*<

*w*/

*λ*≪ 1, we expand

*R*in the smallness parameter

*x*′/

*λ*, thus obtaining

*z*≫

*w*. On the other hand, since

*w*< 1/

*k*for the fields we are considering, we conclude that our approach is valid for

*z*> 1/

*k*[condition allowing both the series truncation and the asymptotic expansion of the Hankel function in Eq. (2)]. Note that the cylindrical wave

**202**, 17 (2002). [CrossRef]

8. Z. Mei and D. Zhao, “Nonparaxial analysis of vectorial Laguerre-Bessel-Gaussian beams,” Opt. Express **15**, 11942 (2007). [CrossRef] [PubMed]

**F**

_{⊥}of Eq. (6) is the product of the cylindrical wave carrier and an envelope

**G**

_{⊥}which is a slowly varying function of variables

*x*/

*r*and (1/

*kr*)(

*z*/

*r*)

^{2}, i.e. it is angularly slowly varying. In addition, it is evident from Eq. (7) that the field

**G**

_{⊥}asymptotically (i.e. for

*r*→ ∞) approaches a function of

*x*/

*r*so that the field

**F**

_{⊥}asymptotically reproduces an angular modulated cylindrical wave.

**E**

_{⊥}(

*x*, 0) =

*E*

_{0}

*wδ*(

*x*)

**e**̂

_{x}where

*δ*(

*x*) is the Dirac delta function,

*E*

_{0}is a complex constant and

*w*is an arbitrary length. Substituting this boundary profile into Eq. (6), we obtain

**G**

_{⊥}=

*E*

_{0}

*w*

**e**̂

_{x}is constant as expected for an infinitely narrow source.

## 3. Highly nonparaxial Hermite-Gaussian fields

*z*= 0 is

*x*

_{0}is a real constant,

*H*is the Hermite polynomial of order

_{m}*m*defined by the Rodrigues’ formula

**G**

_{⊥}into Eq. (6) and exploiting Eqs. (3) it is straightforward to deduce the highly nonparaxial Hermite-Gaussian electric field. For the fundamental Hermite-Gaussian field (

*m*= 0) we obtain (for

*x*

_{0}= 0)

*x*=

*r*sin

*θ*and

*z*=

*r*cos

*θ*have been introduced and, from the second of Eqs. (11),

*P*= 1 - (

*ikw*

^{2}/

*r*) cos

^{2}

*θ*. From the first of Eqs. (12), we note that, in agreement with the general discussion of Section 2, the envelope

**G**

_{⊥}is a slowly varying function of the variables sin

*θ*=

*x*/

*r*and (1/

*kr*)cos

^{2}

*θ*= (1/

*kr*)(

*z*/

*r*)

^{2}(since

*kw*≪ 1). In addition

**G**

_{⊥}asymptotically (i.e. for

*r*→ ∞) contributes to the total optical field as pure angular distribution (since

*P*→ 1). From the second and the third of Eqs. (12) we note that

*E*and

_{x}*E*have definite spatial parity (even and odd respectively) and that their orders of magnitude are comparable, a distinct feature of the nonparaxial (in this case highly nonpariaxial) regime.

_{z}*F*for the

_{x}*m*= 0 [Fig. 1(a)] and

*m*= 1 [Fig. 1(b)] Hermite-Gaussian fields and the plots show the main features of (1 + 1)D highly nonparaxial fields. Note that

*F*[and hence the magnetic field, see Eq. (4)], vanishes for

_{x}*x*→ ∞ as

*x*

^{-1/2}in agreement with the general behavior of these fields. In Fig. 2 and Fig. 3 we plot the moduli of the

*x*- and

*z*-component of the electric field in the case of the fundamental Gaussian field and the first order Hermite-Gaussian field, respectively. It is worth noting that the

*x*and

*z*- components of the electric field, for ∣

*x*∣ → ∞, vanish as

*x*

^{-3/2}and

*x*

^{-1/2}, as opposed to the (1 + 2)-D situation where

*E*and

_{x}*E*vanish as

_{z}*x*

^{-2}and as

*x*

^{-1}, respectively.

*y*-axis (transverse magnetic case), the average Poynting vector is

*m*= 0 and

*m*= 1 Hermite-Gaussian fields, respectively. Note that presence of a large

*x*- component of the Poynting vector (which is spatially odd along the

*x*- axis) corresponds to an energy flow diverging from the origin. This is also consistent with the fact that the field asymptotically coincides with an angularly modulated cylindrical wave so that the field lines of the Poynting vector are asymptotically radial straight lines.

## 4. Conclusions

## References and links

1. | J. B. Pendry, “Negative Refraction Makes a Perfect Lens,” Phys. Rev. Lett. |

2. | J. Takahara, S. Yamagishi, H. Taki, A. Morimoto, and T. Kobayashi, “Guiding of a one-dimensional optical beam with nanometer diameter,” Opt. Lett. |

3. | H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming Light from a Subwavelength Aperture,” Science |

4. | L. Verslegers, P.B. Catrysse, Z. Yu, J. S. White, E. S. Barnard, M. L. Brongersma, and S. Fan, “Planar Lenses Based on Nanoscale Slit Arrays in a Metallic Film,” Nano Lett. |

5. | A. Ciattoni, B. Crosignani, and P. Di Porto, “Vectorial free-space optical propagation: a simple approach for generating all-order nonparaxial corrections,” Opt. Commun. |

6. | A. Yu Savchencko and B. Ya Zel’dovich, “Wave propagation in a guiding structure: one step beyond the paraxial approximation,” J. Opt. Soc. Am. B |

7. | A. Ciattoni, B. Crosignani, and P. Di Porto, “Vectorial analytical description of propagation of a highly nonparaxial beam,” Opt. Commun. |

8. | Z. Mei and D. Zhao, “Nonparaxial analysis of vectorial Laguerre-Bessel-Gaussian beams,” Opt. Express |

9. | R. Martnez-Herrero, P. M. Mejas, and A. Carnicer, “Evanescent field of vectorial highly non-paraxial beams,” Opt. Express , |

10. | P. Liu, B. Lü, and K. Duan, “Propagation of vectorial nonparaxial Gaussian beams through an annular aperture,” Opt. Laser Technol. |

11. | E. Moreno, F. J. García-Vidal, and L. Martín-Moreno, “Enhanced transmission and beaming of light via photonic crystal surface modes,” Phys. Rev. B |

12. | S. K. Morrison and Y. S. Kivshar, “Engineering of directional emission from photonic-crystal waveguides,” Appl. Phys. Lett. |

13. | C. Liu, N. Chen, and C. Sheppard, “Nanoillumination based on self-focus and field enhancement inside a sub-wavelength metallic structure,” Appl. Phys. Lett. |

14. | V. I. Smirnov, A Course of Higher Mathematics, Vol. 4 (Pergamon, Oxford, 1975). |

**OCIS Codes**

(050.1960) Diffraction and gratings : Diffraction theory

(050.6624) Diffraction and gratings : Subwavelength structures

**ToC Category:**

Physical Optics

**History**

Original Manuscript: October 14, 2009

Revised Manuscript: December 10, 2009

Manuscript Accepted: January 10, 2010

Published: March 29, 2010

**Citation**

C. Rizza, A. Ciattoni, and E. Palange, "Highly nonparaxial (1+1)-D subwavelength optical fields," Opt. Express **18**, 7617-7624 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-8-7617

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### References

- J. B. Pendry, "Negative Refraction Makes a Perfect Lens," Phys. Rev. Lett. 85, 3966 (2000). [CrossRef] [PubMed]
- J. Takahara, S. Yamagishi, H. Taki, A. Morimoto, and T. Kobayashi, "Guiding of a one-dimensional optical beam with nanometer diameter," Opt. Lett. 22, 475 (1997). [CrossRef] [PubMed]
- H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, "Beaming Light from a Subwavelength Aperture," Science 297, 820 (2002). [CrossRef] [PubMed]
- L. Verslegers, P.B. Catrysse, Z. Yu, J. S. White, E. S. Barnard, M. L. Brongersma, and S. Fan, "Planar Lenses Based on Nanoscale Slit Arrays in a Metallic Film," Nano Lett. 9, 235 (2009). [CrossRef]
- A. Ciattoni, B. Crosignani, and P. Di Porto, "Vectorial free-space optical propagation: a simple approach for generating all-order nonparaxial corrections," Opt. Commun. 1779 (2000). [CrossRef]
- A. Yu. Savchencko and B. Ya. Zel’dovich, "Wave propagation in a guiding structure: one step beyond the paraxial approximation," J. Opt. Soc. Am. B 13, 273 (1996). [CrossRef]
- A. Ciattoni, B. Crosignani, and P. Di Porto, "Vectorial analytical description of propagation of a highly nonparaxial beam," Opt. Commun. 202, 17 (2002). [CrossRef]
- Z. Mei and D. Zhao, "Nonparaxial analysis of vectorial Laguerre-Bessel-Gaussian beams," Opt. Express 15, 11942 (2007). [CrossRef] [PubMed]
- R. Martnez-Herrero, P. M. Mejas, and A. Carnicer, "Evanescent field of vectorial highly non-paraxial beams," Opt. Express 16, 2845 (2008). [CrossRef]
- P. Liu, B. Lü, and K. Duan, "Propagation of vectorial nonparaxial Gaussian beams through an annular aperture," Opt. Laser Technol. 38, 133 (2006). [CrossRef]
- E. Moreno, F. J. García-Vidal, and L. Martín-Moreno, "Enhanced transmission and beaming of light via photonic crystal surface modes," Phys. Rev. B 69, 121402 (2004). [CrossRef]
- S. K. Morrison and Y. S. Kivshar, "Engineering of directional emission from photonic-crystal waveguides," Appl. Phys. Lett. 86, 081110 (2005). [CrossRef]
- C. Liu, N. Chen, and C. Sheppard, "Nanoillumination based on self-focus and field enhancement inside a subwavelength metallic structure," Appl. Phys. Lett. 90, 011501 (2007). [CrossRef]
- V. I. Smirnov, A Course of Higher Mathematics, Vol. 4 (Pergamon, Oxford, 1975).

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